id sid tid token lemma pos ejpam-5896 1 1 european european PROPN ejpam-5896 1 2 journal journal PROPN ejpam-5896 1 3 of of ADP ejpam-5896 1 4 pure pure ADJ ejpam-5896 1 5 and and CCONJ ejpam-5896 1 6 applied applied ADJ ejpam-5896 1 7 mathematics mathematic NOUN ejpam-5896 1 8 2025 2025 NUM ejpam-5896 1 9 , , PUNCT ejpam-5896 1 10 vol vol NOUN ejpam-5896 1 11 . . PROPN ejpam-5896 1 12 18 18 NUM ejpam-5896 1 13 , , PUNCT ejpam-5896 1 14 issue issue NOUN ejpam-5896 1 15 2 2 NUM ejpam-5896 1 16 , , PUNCT ejpam-5896 1 17 article article NOUN ejpam-5896 1 18 number number NOUN ejpam-5896 1 19 5896 5896 NUM ejpam-5896 1 20 issn issn PROPN ejpam-5896 1 21 1307 1307 NUM ejpam-5896 1 22 - - SYM ejpam-5896 1 23 5543 5543 NUM ejpam-5896 1 24 – – PUNCT ejpam-5896 1 25 ejpam.com ejpam.com X ejpam-5896 1 26 published publish VERB ejpam-5896 1 27 by by ADP ejpam-5896 1 28 new new PROPN ejpam-5896 1 29 york york PROPN ejpam-5896 1 30 business business PROPN ejpam-5896 1 31 global global ADJ ejpam-5896 1 32 applications application NOUN ejpam-5896 1 33 of of ADP ejpam-5896 1 34 the the DET ejpam-5896 1 35 supra supra PROPN ejpam-5896 1 36 soft soft ADJ ejpam-5896 1 37 sd sd NOUN ejpam-5896 1 38 - - PUNCT ejpam-5896 1 39 closure closure NOUN ejpam-5896 1 40 operator operator NOUN ejpam-5896 1 41 to to ADP ejpam-5896 1 42 soft soft ADJ ejpam-5896 1 43 connectedness connectedness NOUN ejpam-5896 1 44 and and CCONJ ejpam-5896 1 45 compactness compactness NOUN ejpam-5896 2 1 alaa alaa PROPN ejpam-5896 2 2 m. m. PROPN ejpam-5896 2 3 abd abd PROPN ejpam-5896 2 4 el el PROPN ejpam-5896 2 5 - - PROPN ejpam-5896 2 6 latif1 latif1 PROPN ejpam-5896 2 7 , , PUNCT ejpam-5896 2 8 radwan radwan VERB ejpam-5896 2 9 abu abu PROPN ejpam-5896 2 10 - - PUNCT ejpam-5896 2 11 gdairi2 gdairi2 PROPN ejpam-5896 2 12 , , PUNCT ejpam-5896 2 13 a. a. NOUN ejpam-5896 2 14 a. a. PROPN ejpam-5896 2 15 azzam3,4,∗ azzam3,4,∗ PROPN ejpam-5896 2 16 , , PUNCT ejpam-5896 2 17 khaled khaled PROPN ejpam-5896 2 18 a. a. PROPN ejpam-5896 2 19 aldwoah5 aldwoah5 PROPN ejpam-5896 2 20 , , PUNCT ejpam-5896 2 21 m. m. PROPN ejpam-5896 2 22 aldawood3 aldawood3 PROPN ejpam-5896 2 23 , , PUNCT ejpam-5896 2 24 shaaban shaaban ADJ ejpam-5896 2 25 m. m. NOUN ejpam-5896 2 26 shaaban6 shaaban6 PROPN ejpam-5896 3 1 1 1 NUM ejpam-5896 3 2 department department NOUN ejpam-5896 3 3 of of ADP ejpam-5896 3 4 mathematics mathematic NOUN ejpam-5896 3 5 , , PUNCT ejpam-5896 3 6 college college NOUN ejpam-5896 3 7 of of ADP ejpam-5896 3 8 science science NOUN ejpam-5896 3 9 , , PUNCT ejpam-5896 3 10 northern northern ADJ ejpam-5896 3 11 border border NOUN ejpam-5896 3 12 university university NOUN ejpam-5896 3 13 , , PUNCT ejpam-5896 3 14 arar arar NOUN ejpam-5896 3 15 91431 91431 NUM ejpam-5896 3 16 , , PUNCT ejpam-5896 3 17 saudi saudi PROPN ejpam-5896 3 18 arabia arabia PROPN ejpam-5896 3 19 2 2 NUM ejpam-5896 3 20 mathematics mathematics PROPN ejpam-5896 3 21 department department NOUN ejpam-5896 3 22 , , PUNCT ejpam-5896 3 23 faculty faculty NOUN ejpam-5896 3 24 of of ADP ejpam-5896 3 25 science science NOUN ejpam-5896 3 26 , , PUNCT ejpam-5896 3 27 zarqa zarqa PROPN ejpam-5896 3 28 university university PROPN ejpam-5896 3 29 , , PUNCT ejpam-5896 3 30 zarqa zarqa NOUN ejpam-5896 3 31 13132 13132 NUM ejpam-5896 3 32 , , PUNCT ejpam-5896 3 33 jordan jordan PROPN ejpam-5896 3 34 3 3 NUM ejpam-5896 3 35 department department PROPN ejpam-5896 3 36 of of ADP ejpam-5896 3 37 mathematics mathematic NOUN ejpam-5896 3 38 , , PUNCT ejpam-5896 3 39 faculty faculty NOUN ejpam-5896 3 40 of of ADP ejpam-5896 3 41 science science NOUN ejpam-5896 3 42 and and CCONJ ejpam-5896 3 43 humanities humanity NOUN ejpam-5896 3 44 , , PUNCT ejpam-5896 3 45 prince prince PROPN ejpam-5896 3 46 sattam sattam PROPN ejpam-5896 3 47 bin bin PROPN ejpam-5896 3 48 abdulaziz abdulaziz PROPN ejpam-5896 3 49 university university PROPN ejpam-5896 3 50 , , PUNCT ejpam-5896 3 51 alkharj alkharj VERB ejpam-5896 3 52 11942 11942 NUM ejpam-5896 3 53 , , PUNCT ejpam-5896 3 54 saudi saudi PROPN ejpam-5896 3 55 arabia arabia PROPN ejpam-5896 3 56 4 4 NUM ejpam-5896 3 57 department department NOUN ejpam-5896 3 58 of of ADP ejpam-5896 3 59 mathematics mathematic NOUN ejpam-5896 3 60 , , PUNCT ejpam-5896 3 61 faculty faculty NOUN ejpam-5896 3 62 of of ADP ejpam-5896 3 63 science science NOUN ejpam-5896 3 64 , , PUNCT ejpam-5896 3 65 new new ADJ ejpam-5896 3 66 valley valley NOUN ejpam-5896 3 67 university university NOUN ejpam-5896 3 68 , , PUNCT ejpam-5896 3 69 elkharga elkharga NOUN ejpam-5896 3 70 72511 72511 NUM ejpam-5896 3 71 , , PUNCT ejpam-5896 3 72 egypt egypt PROPN ejpam-5896 3 73 5 5 NUM ejpam-5896 3 74 department department NOUN ejpam-5896 3 75 of of ADP ejpam-5896 3 76 mathematics mathematic NOUN ejpam-5896 3 77 , , PUNCT ejpam-5896 3 78 faculty faculty NOUN ejpam-5896 3 79 of of ADP ejpam-5896 3 80 science science NOUN ejpam-5896 3 81 , , PUNCT ejpam-5896 3 82 islamic islamic PROPN ejpam-5896 3 83 university university PROPN ejpam-5896 3 84 of of ADP ejpam-5896 3 85 madinah madinah PROPN ejpam-5896 3 86 , , PUNCT ejpam-5896 3 87 medinah medinah PROPN ejpam-5896 3 88 , , PUNCT ejpam-5896 3 89 saudi saudi PROPN ejpam-5896 3 90 arabia arabia PROPN ejpam-5896 3 91 6 6 NUM ejpam-5896 3 92 center center NOUN ejpam-5896 3 93 for for ADP ejpam-5896 3 94 scientific scientific ADJ ejpam-5896 3 95 research research NOUN ejpam-5896 3 96 and and CCONJ ejpam-5896 3 97 entrepreneurship entrepreneurship NOUN ejpam-5896 3 98 , , PUNCT ejpam-5896 3 99 northern northern ADJ ejpam-5896 3 100 border border NOUN ejpam-5896 3 101 university university NOUN ejpam-5896 3 102 , , PUNCT ejpam-5896 3 103 arar arar PROPN ejpam-5896 3 104 73213 73213 NUM ejpam-5896 3 105 , , PUNCT ejpam-5896 3 106 saudi saudi PROPN ejpam-5896 3 107 arabia arabia PROPN ejpam-5896 3 108 abstract abstract NOUN ejpam-5896 3 109 . . PUNCT ejpam-5896 4 1 the the DET ejpam-5896 4 2 concepts concept NOUN ejpam-5896 4 3 of of ADP ejpam-5896 4 4 supra supra PROPN ejpam-5896 4 5 soft soft ADJ ejpam-5896 4 6 somewhere somewhere ADV ejpam-5896 4 7 dense dense ADJ ejpam-5896 4 8 closure closure NOUN ejpam-5896 4 9 , , PUNCT ejpam-5896 4 10 also also ADV ejpam-5896 4 11 known know VERB ejpam-5896 4 12 as as ADP ejpam-5896 4 13 ss ss NOUN ejpam-5896 4 14 - - PUNCT ejpam-5896 4 15 sd sd NOUN ejpam-5896 4 16 - - PUNCT ejpam-5896 4 17 closure closure NOUN ejpam-5896 4 18 , , PUNCT ejpam-5896 4 19 are be AUX ejpam-5896 4 20 first first ADV ejpam-5896 4 21 applied apply VERB ejpam-5896 4 22 in in ADP ejpam-5896 4 23 this this DET ejpam-5896 4 24 study study NOUN ejpam-5896 4 25 to to ADP ejpam-5896 4 26 the the DET ejpam-5896 4 27 problem problem NOUN ejpam-5896 4 28 of of ADP ejpam-5896 4 29 connectedness connectedness NOUN ejpam-5896 4 30 in in ADP ejpam-5896 4 31 supra supra PROPN ejpam-5896 4 32 soft soft ADJ ejpam-5896 4 33 topological topological ADJ ejpam-5896 4 34 spaces space NOUN ejpam-5896 4 35 , , PUNCT ejpam-5896 4 36 or or CCONJ ejpam-5896 4 37 sstss sstss NOUN ejpam-5896 4 38 . . PUNCT ejpam-5896 5 1 to to PART ejpam-5896 5 2 be be AUX ejpam-5896 5 3 more more ADV ejpam-5896 5 4 specific specific ADJ ejpam-5896 5 5 , , PUNCT ejpam-5896 5 6 we we PRON ejpam-5896 5 7 use use VERB ejpam-5896 5 8 the the DET ejpam-5896 5 9 concepts concept NOUN ejpam-5896 5 10 of of ADP ejpam-5896 5 11 ss ss NOUN ejpam-5896 5 12 - - PUNCT ejpam-5896 5 13 sd sd NOUN ejpam-5896 5 14 - - PUNCT ejpam-5896 5 15 components component NOUN ejpam-5896 5 16 to to PART ejpam-5896 5 17 describe describe VERB ejpam-5896 5 18 a a DET ejpam-5896 5 19 novel novel ADJ ejpam-5896 5 20 method method NOUN ejpam-5896 5 21 of of ADP ejpam-5896 5 22 connectedness connectedness NOUN ejpam-5896 5 23 via via ADP ejpam-5896 5 24 ss ss NOUN ejpam-5896 5 25 - - PUNCT ejpam-5896 5 26 sd sd NOUN ejpam-5896 5 27 - - PUNCT ejpam-5896 5 28 sets set NOUN ejpam-5896 5 29 , , PUNCT ejpam-5896 5 30 which which PRON ejpam-5896 5 31 we we PRON ejpam-5896 5 32 refer refer VERB ejpam-5896 5 33 to to ADP ejpam-5896 5 34 as as ADP ejpam-5896 5 35 ssl ssl ADJ ejpam-5896 5 36 - - PUNCT ejpam-5896 5 37 sd sd NOUN ejpam-5896 5 38 - - PUNCT ejpam-5896 5 39 connectedness connectedness NOUN ejpam-5896 5 40 , , PUNCT ejpam-5896 5 41 or or CCONJ ejpam-5896 5 42 supra supra NOUN ejpam-5896 5 43 soft soft ADJ ejpam-5896 5 44 locally locally ADV ejpam-5896 5 45 sd sd NOUN ejpam-5896 5 46 - - PUNCT ejpam-5896 5 47 connectedness connectedness NOUN ejpam-5896 5 48 . . PUNCT ejpam-5896 6 1 sssd sssd NOUN ejpam-5896 6 2 - - PUNCT ejpam-5896 6 3 hyperconnectedness hyperconnectedness PROPN ejpam-5896 6 4 , , PUNCT ejpam-5896 6 5 a a DET ejpam-5896 6 6 different different ADJ ejpam-5896 6 7 kind kind NOUN ejpam-5896 6 8 of of ADP ejpam-5896 6 9 connection connection NOUN ejpam-5896 6 10 via via ADP ejpam-5896 6 11 ss ss NOUN ejpam-5896 6 12 - - PUNCT ejpam-5896 6 13 sd sd NOUN ejpam-5896 6 14 - - PUNCT ejpam-5896 6 15 sets set NOUN ejpam-5896 6 16 in in ADP ejpam-5896 6 17 sstss sstss NOUN ejpam-5896 6 18 , , PUNCT ejpam-5896 6 19 is be AUX ejpam-5896 6 20 shown show VERB ejpam-5896 6 21 . . PUNCT ejpam-5896 7 1 we we PRON ejpam-5896 7 2 investigate investigate VERB ejpam-5896 7 3 the the DET ejpam-5896 7 4 master master NOUN ejpam-5896 7 5 characteristics characteristic NOUN ejpam-5896 7 6 of of ADP ejpam-5896 7 7 different different ADJ ejpam-5896 7 8 kinds kind NOUN ejpam-5896 7 9 of of ADP ejpam-5896 7 10 connectedness connectedness NOUN ejpam-5896 7 11 . . PUNCT ejpam-5896 8 1 we we PRON ejpam-5896 8 2 found find VERB ejpam-5896 8 3 out out ADP ejpam-5896 8 4 that that SCONJ ejpam-5896 8 5 , , PUNCT ejpam-5896 8 6 ss ss NOUN ejpam-5896 8 7 - - PUNCT ejpam-5896 8 8 sd sd NOUN ejpam-5896 8 9 - - PUNCT ejpam-5896 8 10 hyperconnected hyperconnecte VERB ejpam-5896 8 11 spaces space NOUN ejpam-5896 8 12 are be AUX ejpam-5896 8 13 equivalent equivalent ADJ ejpam-5896 8 14 to to ADP ejpam-5896 8 15 ss ss NOUN ejpam-5896 8 16 - - PUNCT ejpam-5896 8 17 sd sd NOUN ejpam-5896 8 18 - - PUNCT ejpam-5896 8 19 connected connect VERB ejpam-5896 8 20 spaces space NOUN ejpam-5896 8 21 , , PUNCT ejpam-5896 8 22 which which PRON ejpam-5896 8 23 distinguishes distinguish VERB ejpam-5896 8 24 our our PRON ejpam-5896 8 25 concepts concept NOUN ejpam-5896 8 26 from from ADP ejpam-5896 8 27 their their PRON ejpam-5896 8 28 counterparts counterpart NOUN ejpam-5896 8 29 . . PUNCT ejpam-5896 9 1 furthermore furthermore ADV ejpam-5896 9 2 , , PUNCT ejpam-5896 9 3 we we PRON ejpam-5896 9 4 present present VERB ejpam-5896 9 5 two two NUM ejpam-5896 9 6 new new ADJ ejpam-5896 9 7 forms form NOUN ejpam-5896 9 8 of of ADP ejpam-5896 9 9 compactness compactness NOUN ejpam-5896 9 10 , , PUNCT ejpam-5896 9 11 called call VERB ejpam-5896 9 12 ss ss NOUN ejpam-5896 9 13 - - PUNCT ejpam-5896 9 14 sd sd NOUN ejpam-5896 9 15 - - PUNCT ejpam-5896 9 16 compactness compactness NOUN ejpam-5896 9 17 and and CCONJ ejpam-5896 9 18 ss ss NOUN ejpam-5896 9 19 - - PUNCT ejpam-5896 9 20 sd sd NOUN ejpam-5896 9 21 - - PUNCT ejpam-5896 9 22 lindelöfness lindelöfness NOUN ejpam-5896 9 23 , , PUNCT ejpam-5896 9 24 which which PRON ejpam-5896 9 25 are be AUX ejpam-5896 9 26 based base VERB ejpam-5896 9 27 on on ADP ejpam-5896 9 28 the the DET ejpam-5896 9 29 concepts concept NOUN ejpam-5896 9 30 of of ADP ejpam-5896 9 31 super super ADJ ejpam-5896 9 32 soft soft ADJ ejpam-5896 9 33 sd sd NOUN ejpam-5896 9 34 - - PUNCT ejpam-5896 9 35 sets set NOUN ejpam-5896 9 36 in in ADP ejpam-5896 9 37 the the DET ejpam-5896 9 38 context context NOUN ejpam-5896 9 39 of of ADP ejpam-5896 9 40 sstss sstss NOUN ejpam-5896 9 41 . . PUNCT ejpam-5896 10 1 we we PRON ejpam-5896 10 2 get get VERB ejpam-5896 10 3 into into ADP ejpam-5896 10 4 a a DET ejpam-5896 10 5 lot lot NOUN ejpam-5896 10 6 of of ADP ejpam-5896 10 7 detail detail NOUN ejpam-5896 10 8 about about ADP ejpam-5896 10 9 their their PRON ejpam-5896 10 10 primary primary ADJ ejpam-5896 10 11 features feature NOUN ejpam-5896 10 12 . . PUNCT ejpam-5896 11 1 in in ADP ejpam-5896 11 2 particular particular ADJ ejpam-5896 11 3 , , PUNCT ejpam-5896 11 4 we we PRON ejpam-5896 11 5 demonstrate demonstrate VERB ejpam-5896 11 6 that that SCONJ ejpam-5896 11 7 an an DET ejpam-5896 11 8 ss ss NOUN ejpam-5896 11 9 - - PUNCT ejpam-5896 11 10 sd sd NOUN ejpam-5896 11 11 - - PUNCT ejpam-5896 11 12 compact compact ADJ ejpam-5896 11 13 ( ( PUNCT ejpam-5896 11 14 lindelöf lindelöf NOUN ejpam-5896 11 15 ) ) PUNCT ejpam-5896 11 16 soft soft ADJ ejpam-5896 11 17 set set NOUN ejpam-5896 11 18 is be AUX ejpam-5896 11 19 the the DET ejpam-5896 11 20 soft soft ADJ ejpam-5896 11 21 intersection intersection NOUN ejpam-5896 11 22 of of ADP ejpam-5896 11 23 an an DET ejpam-5896 11 24 ss ss ADJ ejpam-5896 11 25 - - PUNCT ejpam-5896 11 26 sc sc NOUN ejpam-5896 11 27 - - PUNCT ejpam-5896 11 28 set set NOUN ejpam-5896 11 29 and and CCONJ ejpam-5896 11 30 an an DET ejpam-5896 11 31 ss ss NOUN ejpam-5896 11 32 - - PUNCT ejpam-5896 11 33 sd sd NOUN ejpam-5896 11 34 - - PUNCT ejpam-5896 11 35 compact compact ADJ ejpam-5896 11 36 ( ( PUNCT ejpam-5896 11 37 lindelöf lindelöf PROPN ejpam-5896 11 38 ) ) PUNCT ejpam-5896 11 39 soft soft ADJ ejpam-5896 11 40 set set NOUN ejpam-5896 11 41 . . PUNCT ejpam-5896 12 1 furthermore furthermore ADV ejpam-5896 12 2 , , PUNCT ejpam-5896 12 3 an an DET ejpam-5896 12 4 sssd sssd NOUN ejpam-5896 12 5 - - PUNCT ejpam-5896 12 6 compact compact ADJ ejpam-5896 12 7 ( ( PUNCT ejpam-5896 12 8 lindelöf lindelöf NOUN ejpam-5896 12 9 ) ) PUNCT ejpam-5896 12 10 ssts sst NOUN ejpam-5896 12 11 with with ADP ejpam-5896 12 12 the the DET ejpam-5896 12 13 soft soft ADJ ejpam-5896 12 14 finite finite NOUN ejpam-5896 12 15 ( ( PUNCT ejpam-5896 12 16 countable countable ADJ ejpam-5896 12 17 ) ) PUNCT ejpam-5896 12 18 intersection intersection NOUN ejpam-5896 12 19 property property NOUN ejpam-5896 12 20 also also ADV ejpam-5896 12 21 known know VERB ejpam-5896 12 22 as as ADP ejpam-5896 12 23 sfip sfip NOUN ejpam-5896 12 24 ( ( PUNCT ejpam-5896 12 25 scip scip PROPN ejpam-5896 12 26 ) ) PUNCT ejpam-5896 12 27 has have AUX ejpam-5896 12 28 been be AUX ejpam-5896 12 29 demonstrated demonstrate VERB ejpam-5896 12 30 in in ADP ejpam-5896 12 31 terms term NOUN ejpam-5896 12 32 of of ADP ejpam-5896 12 33 its its PRON ejpam-5896 12 34 behaviour behaviour NOUN ejpam-5896 12 35 . . PUNCT ejpam-5896 13 1 finally finally ADV ejpam-5896 13 2 , , PUNCT ejpam-5896 13 3 we we PRON ejpam-5896 13 4 compare compare VERB ejpam-5896 13 5 our our PRON ejpam-5896 13 6 novel novel ADJ ejpam-5896 13 7 soft soft ADJ ejpam-5896 13 8 versions version NOUN ejpam-5896 13 9 of of ADP ejpam-5896 13 10 connectedness connectedness NOUN ejpam-5896 13 11 and and CCONJ ejpam-5896 13 12 compactness compactness NOUN ejpam-5896 13 13 using use VERB ejpam-5896 13 14 ss ss NOUN ejpam-5896 13 15 - - PUNCT ejpam-5896 13 16 sd sd NOUN ejpam-5896 13 17 - - PUNCT ejpam-5896 13 18 sets set NOUN ejpam-5896 13 19 with with ADP ejpam-5896 13 20 earlier early ADJ ejpam-5896 13 21 research research NOUN ejpam-5896 13 22 and and CCONJ ejpam-5896 13 23 add add VERB ejpam-5896 13 24 two two NUM ejpam-5896 13 25 topological topological ADJ ejpam-5896 13 26 charts chart NOUN ejpam-5896 13 27 to to PART ejpam-5896 13 28 figures figure NOUN ejpam-5896 13 29 1 1 NUM ejpam-5896 13 30 and and CCONJ ejpam-5896 13 31 2 2 NUM ejpam-5896 13 32 to to PART ejpam-5896 13 33 illustrate illustrate VERB ejpam-5896 13 34 the the DET ejpam-5896 13 35 main main ADJ ejpam-5896 13 36 ideas idea NOUN ejpam-5896 13 37 of of ADP ejpam-5896 13 38 this this DET ejpam-5896 13 39 study study NOUN ejpam-5896 13 40 . . PUNCT ejpam-5896 14 1 concrete concrete ADJ ejpam-5896 14 2 examples example NOUN ejpam-5896 14 3 and and CCONJ ejpam-5896 14 4 counterexamples counterexample NOUN ejpam-5896 14 5 have have AUX ejpam-5896 14 6 verified verify VERB ejpam-5896 14 7 that that SCONJ ejpam-5896 14 8 the the DET ejpam-5896 14 9 arrows arrow NOUN ejpam-5896 14 10 in in ADP ejpam-5896 14 11 these these DET ejpam-5896 14 12 charts chart NOUN ejpam-5896 14 13 are be AUX ejpam-5896 14 14 non non ADJ ejpam-5896 14 15 - - ADJ ejpam-5896 14 16 reversible reversible ADJ ejpam-5896 14 17 . . PUNCT ejpam-5896 15 1 2020 2020 NUM ejpam-5896 15 2 mathematics mathematic NOUN ejpam-5896 15 3 subject subject NOUN ejpam-5896 15 4 classifications classification NOUN ejpam-5896 15 5 : : PUNCT ejpam-5896 15 6 54a05 54a05 NUM ejpam-5896 15 7 , , PUNCT ejpam-5896 15 8 54c10 54c10 NUM ejpam-5896 15 9 , , PUNCT ejpam-5896 15 10 03e72 03e72 X ejpam-5896 15 11 key key ADJ ejpam-5896 15 12 words word NOUN ejpam-5896 15 13 and and CCONJ ejpam-5896 15 14 phrases phrase NOUN ejpam-5896 15 15 : : PUNCT ejpam-5896 15 16 supra supra PROPN ejpam-5896 15 17 soft soft ADJ ejpam-5896 15 18 sd sd NOUN ejpam-5896 15 19 - - PUNCT ejpam-5896 15 20 operators operator NOUN ejpam-5896 15 21 , , PUNCT ejpam-5896 15 22 ssl ssl PROPN ejpam-5896 15 23 - - PUNCT ejpam-5896 15 24 sd sd NOUN ejpam-5896 15 25 - - PUNCT ejpam-5896 15 26 connectedness connectedness NOUN ejpam-5896 15 27 , , PUNCT ejpam-5896 15 28 ss ss NOUN ejpam-5896 15 29 - - PUNCT ejpam-5896 15 30 sd sd NOUN ejpam-5896 15 31 - - PUNCT ejpam-5896 15 32 hyperconnectedness hyperconnectedness NOUN ejpam-5896 15 33 , , PUNCT ejpam-5896 15 34 ss ss NOUN ejpam-5896 15 35 - - PUNCT ejpam-5896 15 36 almost almost ADV ejpam-5896 15 37 compactness compactness NOUN ejpam-5896 15 38 , , PUNCT ejpam-5896 15 39 ss ss NOUN ejpam-5896 15 40 - - PUNCT ejpam-5896 15 41 sd sd NOUN ejpam-5896 15 42 - - PUNCT ejpam-5896 15 43 compactness compactness NOUN ejpam-5896 15 44 , , PUNCT ejpam-5896 15 45 ss ss NOUN ejpam-5896 15 46 - - PUNCT ejpam-5896 15 47 sd sd NOUN ejpam-5896 15 48 - - PUNCT ejpam-5896 15 49 lindelöfness lindelöfness NOUN ejpam-5896 15 50 1 1 NUM ejpam-5896 15 51 . . X ejpam-5896 15 52 introduction introduction NOUN ejpam-5896 15 53 the the DET ejpam-5896 15 54 approach approach NOUN ejpam-5896 15 55 of of ADP ejpam-5896 15 56 supra supra PROPN ejpam-5896 15 57 topological topological ADJ ejpam-5896 15 58 spaces space NOUN ejpam-5896 15 59 [ [ X ejpam-5896 15 60 1 1 NUM ejpam-5896 15 61 ] ] PUNCT ejpam-5896 15 62 was be AUX ejpam-5896 15 63 defined define VERB ejpam-5896 15 64 by by ADP ejpam-5896 15 65 mashhour mashhour PROPN ejpam-5896 15 66 et et PROPN ejpam-5896 15 67 al al PROPN ejpam-5896 15 68 . . PROPN ejpam-5896 15 69 , , PUNCT ejpam-5896 15 70 in in ADP ejpam-5896 15 71 1983 1983 NUM ejpam-5896 15 72 . . PUNCT ejpam-5896 16 1 numerous numerous ADJ ejpam-5896 16 2 applications application NOUN ejpam-5896 16 3 of of ADP ejpam-5896 16 4 this this DET ejpam-5896 16 5 study study NOUN ejpam-5896 16 6 have have AUX ejpam-5896 16 7 been be AUX ejpam-5896 16 8 proposed propose VERB ejpam-5896 16 9 . . PUNCT ejpam-5896 17 1 in in ADP ejpam-5896 17 2 [ [ X ejpam-5896 17 3 2–4 2–4 NUM ejpam-5896 17 4 ] ] X ejpam-5896 17 5 . . PUNCT ejpam-5896 18 1 novel novel ADJ ejpam-5896 18 2 rough rough ADJ ejpam-5896 18 3 sets set NOUN ejpam-5896 18 4 models model NOUN ejpam-5896 18 5 inspired inspire VERB ejpam-5896 18 6 by by ADP ejpam-5896 18 7 these these DET ejpam-5896 18 8 spaces space NOUN ejpam-5896 18 9 have have AUX ejpam-5896 18 10 been be AUX ejpam-5896 18 11 explored explore VERB ejpam-5896 18 12 in in ADP ejpam-5896 18 13 [ [ X ejpam-5896 18 14 5 5 NUM ejpam-5896 18 15 ] ] PUNCT ejpam-5896 18 16 . . PUNCT ejpam-5896 19 1 ∗corresponding ∗corresponde VERB ejpam-5896 19 2 author author NOUN ejpam-5896 19 3 . . PUNCT ejpam-5896 20 1 doi doi NOUN ejpam-5896 20 2 : : PUNCT ejpam-5896 20 3 https://doi.org/10.29020/nybg.ejpam.v18i2.5896 https://doi.org/10.29020/nybg.ejpam.v18i2.5896 PRON ejpam-5896 20 4 email email NOUN ejpam-5896 20 5 addresses address NOUN ejpam-5896 20 6 : : PUNCT ejpam-5896 20 7 alaa.ali@nbu.edu.sa alaa.ali@nbu.edu.sa PROPN ejpam-5896 20 8 ; ; PUNCT ejpam-5896 20 9 alaa alaa PROPN ejpam-5896 20 10 8560@yahoo.com 8560@yahoo.com PROPN ejpam-5896 21 1 ( ( PUNCT ejpam-5896 21 2 a. a. PROPN ejpam-5896 21 3 m. m. PROPN ejpam-5896 21 4 abd abd PROPN ejpam-5896 21 5 el el PROPN ejpam-5896 21 6 - - PROPN ejpam-5896 21 7 latif latif PROPN ejpam-5896 21 8 ) ) PUNCT ejpam-5896 21 9 , , PUNCT ejpam-5896 21 10 rgdairi@zu.edu.jo rgdairi@zu.edu.jo PROPN ejpam-5896 21 11 ( ( PUNCT ejpam-5896 21 12 r. r. PROPN ejpam-5896 21 13 abu abu PROPN ejpam-5896 21 14 - - PUNCT ejpam-5896 21 15 gdairi gdairi PROPN ejpam-5896 21 16 ) ) PUNCT ejpam-5896 21 17 , , PUNCT ejpam-5896 21 18 azzam0911@yahoo.com azzam0911@yahoo.com X ejpam-5896 21 19 ( ( PUNCT ejpam-5896 21 20 a. a. NOUN ejpam-5896 21 21 a. a. PROPN ejpam-5896 21 22 azzam azzam PROPN ejpam-5896 21 23 ) ) PUNCT ejpam-5896 21 24 , , PUNCT ejpam-5896 21 25 aldwoah@yahoo.com aldwoah@yahoo.com X ejpam-5896 21 26 ( ( PUNCT ejpam-5896 21 27 k. k. PROPN ejpam-5896 21 28 a. a. PROPN ejpam-5896 21 29 aldwoah aldwoah PROPN ejpam-5896 21 30 ) ) PUNCT ejpam-5896 21 31 , , PUNCT ejpam-5896 21 32 m.aldawood@psau.edu.sa m.aldawood@psau.edu.sa PROPN ejpam-5896 21 33 ( ( PUNCT ejpam-5896 21 34 m. m. NOUN ejpam-5896 21 35 aldawood aldawood PROPN ejpam-5896 21 36 ) ) PUNCT ejpam-5896 21 37 , , PUNCT ejpam-5896 21 38 shabaan27@gmail.com shabaan27@gmail.com PROPN ejpam-5896 21 39 ( ( PUNCT ejpam-5896 21 40 s. s. PROPN ejpam-5896 21 41 m. m. PROPN ejpam-5896 21 42 shaaban shaaban PROPN ejpam-5896 21 43 ) ) PUNCT ejpam-5896 21 44 https://www.ejpam.com https://www.ejpam.com NOUN ejpam-5896 21 45 1 1 NUM ejpam-5896 21 46 copyright copyright NOUN ejpam-5896 21 47 : : PUNCT ejpam-5896 21 48 © © PROPN ejpam-5896 21 49 2025 2025 NUM ejpam-5896 21 50 the the DET ejpam-5896 21 51 author(s author(s NOUN ejpam-5896 21 52 ) ) PUNCT ejpam-5896 21 53 . . PUNCT ejpam-5896 22 1 ( ( PUNCT ejpam-5896 22 2 cc cc NOUN ejpam-5896 22 3 by by ADP ejpam-5896 22 4 - - PUNCT ejpam-5896 22 5 nc nc PROPN ejpam-5896 22 6 4.0 4.0 NUM ejpam-5896 22 7 ) ) PUNCT ejpam-5896 22 8 abd abd PROPN ejpam-5896 22 9 el el PROPN ejpam-5896 22 10 - - PROPN ejpam-5896 22 11 latif latif PROPN ejpam-5896 22 12 et et PROPN ejpam-5896 22 13 al al PROPN ejpam-5896 22 14 . . PUNCT ejpam-5896 22 15 / / SYM ejpam-5896 22 16 eur eur PROPN ejpam-5896 22 17 . . PUNCT ejpam-5896 23 1 j. j. PROPN ejpam-5896 23 2 pure pure PROPN ejpam-5896 23 3 appl appl PROPN ejpam-5896 23 4 . . PROPN ejpam-5896 23 5 math math PROPN ejpam-5896 23 6 , , PUNCT ejpam-5896 23 7 18 18 NUM ejpam-5896 23 8 ( ( PUNCT ejpam-5896 23 9 2 2 NUM ejpam-5896 23 10 ) ) PUNCT ejpam-5896 23 11 ( ( PUNCT ejpam-5896 23 12 2025 2025 NUM ejpam-5896 23 13 ) ) PUNCT ejpam-5896 23 14 , , PUNCT ejpam-5896 23 15 5896 5896 NUM ejpam-5896 23 16 2 2 NUM ejpam-5896 23 17 of of ADP ejpam-5896 23 18 20 20 NUM ejpam-5896 23 19 a a DET ejpam-5896 23 20 crucial crucial ADJ ejpam-5896 23 21 alternative alternative ADJ ejpam-5896 23 22 tool tool NOUN ejpam-5896 23 23 to to PART ejpam-5896 23 24 fuzzy fuzzy ADJ ejpam-5896 23 25 , , PUNCT ejpam-5896 23 26 crisp crisp ADJ ejpam-5896 23 27 , , PUNCT ejpam-5896 23 28 and and CCONJ ejpam-5896 23 29 rough rough ADJ ejpam-5896 23 30 set set NOUN ejpam-5896 23 31 theories theory NOUN ejpam-5896 23 32 all all PRON ejpam-5896 23 33 of of ADP ejpam-5896 23 34 which which PRON ejpam-5896 23 35 struggle struggle VERB ejpam-5896 23 36 with with ADP ejpam-5896 23 37 handling handling NOUN ejpam-5896 23 38 uncertainties uncertainty NOUN ejpam-5896 23 39 the the DET ejpam-5896 23 40 soft soft ADJ ejpam-5896 23 41 set set NOUN ejpam-5896 23 42 theory theory NOUN ejpam-5896 23 43 [ [ X ejpam-5896 23 44 6 6 NUM ejpam-5896 23 45 ] ] PUNCT ejpam-5896 23 46 . . PUNCT ejpam-5896 24 1 maji maji PROPN ejpam-5896 24 2 et et PROPN ejpam-5896 24 3 al al PROPN ejpam-5896 24 4 . . PUNCT ejpam-5896 25 1 [ [ X ejpam-5896 25 2 7 7 X ejpam-5896 25 3 ] ] PUNCT ejpam-5896 25 4 improved improve VERB ejpam-5896 25 5 the the DET ejpam-5896 25 6 soft soft ADJ ejpam-5896 25 7 set set NOUN ejpam-5896 25 8 theory theory NOUN ejpam-5896 25 9 by by ADP ejpam-5896 25 10 defining define VERB ejpam-5896 25 11 new new ADJ ejpam-5896 25 12 operations operation NOUN ejpam-5896 25 13 . . PUNCT ejpam-5896 26 1 several several ADJ ejpam-5896 26 2 concrete concrete ADJ ejpam-5896 26 3 applications application NOUN ejpam-5896 26 4 on on ADP ejpam-5896 26 5 soft soft ADJ ejpam-5896 26 6 sets set NOUN ejpam-5896 26 7 have have AUX ejpam-5896 26 8 been be AUX ejpam-5896 26 9 introduced introduce VERB ejpam-5896 26 10 in in ADP ejpam-5896 26 11 rough rough ADJ ejpam-5896 26 12 set set NOUN ejpam-5896 26 13 models model NOUN ejpam-5896 26 14 [ [ X ejpam-5896 26 15 8 8 NUM ejpam-5896 26 16 ] ] PUNCT ejpam-5896 26 17 , , PUNCT ejpam-5896 26 18 medical medical ADJ ejpam-5896 26 19 sciences science NOUN ejpam-5896 27 1 [ [ X ejpam-5896 27 2 9 9 NUM ejpam-5896 27 3 ] ] PUNCT ejpam-5896 27 4 , , PUNCT ejpam-5896 27 5 and and CCONJ ejpam-5896 27 6 decision decision NOUN ejpam-5896 27 7 making make VERB ejpam-5896 27 8 problems problem NOUN ejpam-5896 27 9 [ [ X ejpam-5896 27 10 10 10 NUM ejpam-5896 27 11 ] ] PUNCT ejpam-5896 27 12 . . PUNCT ejpam-5896 28 1 in in ADP ejpam-5896 28 2 2011 2011 NUM ejpam-5896 28 3 , , PUNCT ejpam-5896 28 4 the the DET ejpam-5896 28 5 soft soft ADJ ejpam-5896 28 6 topological topological ADJ ejpam-5896 28 7 spaces space NOUN ejpam-5896 28 8 ( ( PUNCT ejpam-5896 28 9 or or CCONJ ejpam-5896 28 10 stss stss NOUN ejpam-5896 28 11 ) ) PUNCT ejpam-5896 29 1 [ [ X ejpam-5896 29 2 11–13 11–13 NUM ejpam-5896 29 3 ] ] PUNCT ejpam-5896 29 4 were be AUX ejpam-5896 29 5 introduced introduce VERB ejpam-5896 29 6 as as ADP ejpam-5896 29 7 a a DET ejpam-5896 29 8 parameterized parameterized ADJ ejpam-5896 29 9 collections collection NOUN ejpam-5896 29 10 of of ADP ejpam-5896 29 11 crisp crisp ADJ ejpam-5896 29 12 topological topological ADJ ejpam-5896 29 13 spaces space NOUN ejpam-5896 29 14 . . PUNCT ejpam-5896 30 1 in in ADP ejpam-5896 30 2 the the DET ejpam-5896 30 3 same same ADJ ejpam-5896 30 4 year year NOUN ejpam-5896 30 5 , , PUNCT ejpam-5896 30 6 the the DET ejpam-5896 30 7 notions notion NOUN ejpam-5896 30 8 of of ADP ejpam-5896 30 9 soft soft ADJ ejpam-5896 30 10 continuity continuity NOUN ejpam-5896 30 11 were be AUX ejpam-5896 30 12 introduced introduce VERB ejpam-5896 30 13 by by ADP ejpam-5896 30 14 ahmad ahmad NOUN ejpam-5896 30 15 and and CCONJ ejpam-5896 30 16 kharal kharal ADJ ejpam-5896 30 17 [ [ X ejpam-5896 30 18 14 14 NUM ejpam-5896 30 19 ] ] PUNCT ejpam-5896 30 20 . . PUNCT ejpam-5896 31 1 after after ADP ejpam-5896 31 2 that that PRON ejpam-5896 31 3 , , PUNCT ejpam-5896 31 4 some some DET ejpam-5896 31 5 classes class NOUN ejpam-5896 31 6 of of ADP ejpam-5896 31 7 soft soft ADJ ejpam-5896 31 8 functions function NOUN ejpam-5896 31 9 [ [ X ejpam-5896 31 10 15 15 NUM ejpam-5896 31 11 , , PUNCT ejpam-5896 31 12 16 16 NUM ejpam-5896 31 13 ] ] PUNCT ejpam-5896 31 14 have have AUX ejpam-5896 31 15 been be AUX ejpam-5896 31 16 presented present VERB ejpam-5896 31 17 . . PUNCT ejpam-5896 32 1 later later ADV ejpam-5896 32 2 , , PUNCT ejpam-5896 32 3 more more ADJ ejpam-5896 32 4 investigations investigation NOUN ejpam-5896 32 5 of of ADP ejpam-5896 32 6 soft soft ADJ ejpam-5896 32 7 continuity continuity NOUN ejpam-5896 32 8 were be AUX ejpam-5896 32 9 disused disuse VERB ejpam-5896 32 10 [ [ PUNCT ejpam-5896 32 11 17 17 NUM ejpam-5896 32 12 , , PUNCT ejpam-5896 32 13 18 18 NUM ejpam-5896 32 14 ] ] PUNCT ejpam-5896 32 15 . . PUNCT ejpam-5896 33 1 several several ADJ ejpam-5896 33 2 studies study NOUN ejpam-5896 33 3 related relate VERB ejpam-5896 33 4 to to ADP ejpam-5896 33 5 generalized generalize VERB ejpam-5896 33 6 soft soft ADJ ejpam-5896 33 7 open open ADJ ejpam-5896 33 8 sets set NOUN ejpam-5896 33 9 and and CCONJ ejpam-5896 33 10 generalized generalized ADJ ejpam-5896 33 11 soft soft ADJ ejpam-5896 33 12 continuity continuity NOUN ejpam-5896 33 13 , , PUNCT ejpam-5896 33 14 named name VERB ejpam-5896 33 15 soft soft ADJ ejpam-5896 33 16 b b NOUN ejpam-5896 33 17 - - PUNCT ejpam-5896 33 18 open open ADJ ejpam-5896 33 19 sets set NOUN ejpam-5896 33 20 and and CCONJ ejpam-5896 33 21 soft soft ADJ ejpam-5896 33 22 b b NOUN ejpam-5896 33 23 - - PUNCT ejpam-5896 33 24 continuous continuous ADJ ejpam-5896 33 25 functions function NOUN ejpam-5896 33 26 [ [ X ejpam-5896 33 27 19 19 NUM ejpam-5896 33 28 ] ] PUNCT ejpam-5896 33 29 , , PUNCT ejpam-5896 33 30 soft soft ADJ ejpam-5896 33 31 semi semi ADJ ejpam-5896 33 32 - - ADJ ejpam-5896 33 33 open open ADJ ejpam-5896 33 34 sets set NOUN ejpam-5896 33 35 and and CCONJ ejpam-5896 33 36 soft soft ADJ ejpam-5896 33 37 semi semi ADJ ejpam-5896 33 38 irresolute irresolute ADJ ejpam-5896 33 39 soft soft ADJ ejpam-5896 33 40 functions function NOUN ejpam-5896 33 41 [ [ X ejpam-5896 33 42 20 20 NUM ejpam-5896 33 43 , , PUNCT ejpam-5896 33 44 21 21 NUM ejpam-5896 33 45 ] ] PUNCT ejpam-5896 33 46 , , PUNCT ejpam-5896 33 47 several several ADJ ejpam-5896 33 48 kinds kind NOUN ejpam-5896 33 49 of of ADP ejpam-5896 33 50 soft soft ADJ ejpam-5896 33 51 continuity[22 continuity[22 NOUN ejpam-5896 33 52 ] ] X ejpam-5896 33 53 , , PUNCT ejpam-5896 33 54 soft soft ADJ ejpam-5896 33 55 sd sd NOUN ejpam-5896 33 56 - - PUNCT ejpam-5896 33 57 sets set NOUN ejpam-5896 33 58 [ [ X ejpam-5896 33 59 23 23 NUM ejpam-5896 33 60 ] ] PUNCT ejpam-5896 33 61 , , PUNCT ejpam-5896 33 62 and and CCONJ ejpam-5896 33 63 nearly nearly ADV ejpam-5896 33 64 soft soft ADJ ejpam-5896 33 65 βopen βopen ADJ ejpam-5896 33 66 sets set NOUN ejpam-5896 33 67 [ [ X ejpam-5896 33 68 24 24 NUM ejpam-5896 33 69 ] ] PUNCT ejpam-5896 33 70 have have AUX ejpam-5896 33 71 been be AUX ejpam-5896 33 72 presented present VERB ejpam-5896 33 73 . . PUNCT ejpam-5896 34 1 al al PROPN ejpam-5896 34 2 - - PUNCT ejpam-5896 34 3 shami shami PROPN ejpam-5896 34 4 et et PROPN ejpam-5896 34 5 al al PROPN ejpam-5896 34 6 . . PUNCT ejpam-5896 35 1 [ [ X ejpam-5896 35 2 25 25 NUM ejpam-5896 35 3 ] ] PUNCT ejpam-5896 35 4 applied apply VERB ejpam-5896 35 5 these these DET ejpam-5896 35 6 notions notion NOUN ejpam-5896 35 7 in in ADP ejpam-5896 35 8 compactness compactness NOUN ejpam-5896 35 9 and and CCONJ ejpam-5896 35 10 connectedness connectedness NOUN ejpam-5896 35 11 in in ADP ejpam-5896 35 12 2014 2014 NUM ejpam-5896 35 13 , , PUNCT ejpam-5896 35 14 the the DET ejpam-5896 35 15 soft soft ADJ ejpam-5896 35 16 ideal ideal ADJ ejpam-5896 35 17 notion notion NOUN ejpam-5896 35 18 was be AUX ejpam-5896 35 19 introduced introduce VERB ejpam-5896 35 20 by by ADP ejpam-5896 35 21 kandil kandil PROPN ejpam-5896 35 22 et et PROPN ejpam-5896 35 23 al al PROPN ejpam-5896 35 24 . . PUNCT ejpam-5896 36 1 [ [ X ejpam-5896 36 2 26 26 NUM ejpam-5896 36 3 ] ] PUNCT ejpam-5896 36 4 . . PUNCT ejpam-5896 37 1 they they PRON ejpam-5896 37 2 used use VERB ejpam-5896 37 3 the the DET ejpam-5896 37 4 soft soft ADJ ejpam-5896 37 5 local local ADJ ejpam-5896 37 6 functions function NOUN ejpam-5896 37 7 to to PART ejpam-5896 37 8 define define VERB ejpam-5896 37 9 the the DET ejpam-5896 37 10 class class NOUN ejpam-5896 37 11 of of ADP ejpam-5896 37 12 soft soft ADJ ejpam-5896 37 13 ideal ideal ADJ ejpam-5896 37 14 topological topological ADJ ejpam-5896 37 15 spaces space NOUN ejpam-5896 37 16 . . PUNCT ejpam-5896 38 1 these these DET ejpam-5896 38 2 notions notion NOUN ejpam-5896 38 3 have have AUX ejpam-5896 38 4 been be AUX ejpam-5896 38 5 generalized generalize VERB ejpam-5896 38 6 by by ADP ejpam-5896 38 7 using use VERB ejpam-5896 38 8 the the DET ejpam-5896 38 9 approach approach NOUN ejpam-5896 38 10 of of ADP ejpam-5896 38 11 soft soft ADJ ejpam-5896 38 12 semi semi ADJ ejpam-5896 38 13 - - ADJ ejpam-5896 38 14 open open ADJ ejpam-5896 38 15 sets set NOUN ejpam-5896 38 16 [ [ X ejpam-5896 38 17 27 27 NUM ejpam-5896 38 18 , , PUNCT ejpam-5896 38 19 28 28 NUM ejpam-5896 38 20 ] ] PUNCT ejpam-5896 38 21 . . PUNCT ejpam-5896 39 1 novel novel ADJ ejpam-5896 39 2 soft soft ADJ ejpam-5896 39 3 ideal ideal NOUN ejpam-5896 39 4 rough rough ADJ ejpam-5896 39 5 topological topological ADJ ejpam-5896 39 6 spaces space NOUN ejpam-5896 39 7 and and CCONJ ejpam-5896 39 8 application application NOUN ejpam-5896 39 9 in in ADP ejpam-5896 39 10 diabetes diabetes NOUN ejpam-5896 39 11 mellitus mellitus NOUN ejpam-5896 40 1 [ [ X ejpam-5896 40 2 29 29 NUM ejpam-5896 40 3 ] ] PUNCT ejpam-5896 40 4 have have AUX ejpam-5896 40 5 been be AUX ejpam-5896 40 6 explored explore VERB ejpam-5896 40 7 by by ADP ejpam-5896 40 8 abd abd PROPN ejpam-5896 40 9 el el PROPN ejpam-5896 40 10 - - PROPN ejpam-5896 40 11 latif latif PROPN ejpam-5896 40 12 in in ADP ejpam-5896 40 13 2018 2018 NUM ejpam-5896 40 14 . . PUNCT ejpam-5896 41 1 after after ADP ejpam-5896 41 2 that that PRON ejpam-5896 41 3 , , PUNCT ejpam-5896 41 4 many many ADJ ejpam-5896 41 5 researchers researcher NOUN ejpam-5896 41 6 used use VERB ejpam-5896 41 7 the the DET ejpam-5896 41 8 soft soft ADJ ejpam-5896 41 9 ideal ideal ADJ ejpam-5896 41 10 notions notion NOUN ejpam-5896 41 11 to to PART ejpam-5896 41 12 generalize generalize VERB ejpam-5896 41 13 several several ADJ ejpam-5896 41 14 weaker weak ADJ ejpam-5896 41 15 classes class NOUN ejpam-5896 41 16 of of ADP ejpam-5896 41 17 soft soft ADJ ejpam-5896 41 18 open open ADJ ejpam-5896 41 19 sets set NOUN ejpam-5896 41 20 [ [ X ejpam-5896 41 21 30–32 30–32 NUM ejpam-5896 41 22 ] ] PUNCT ejpam-5896 41 23 and and CCONJ ejpam-5896 41 24 soft soft ADJ ejpam-5896 41 25 separation separation NOUN ejpam-5896 41 26 axioms axiom NOUN ejpam-5896 41 27 [ [ X ejpam-5896 41 28 33 33 NUM ejpam-5896 41 29 , , PUNCT ejpam-5896 41 30 34 34 NUM ejpam-5896 41 31 ] ] PUNCT ejpam-5896 41 32 . . PUNCT ejpam-5896 42 1 since since SCONJ ejpam-5896 42 2 connectedness connectedness NOUN ejpam-5896 42 3 has have VERB ejpam-5896 42 4 a a DET ejpam-5896 42 5 vital vital ADJ ejpam-5896 42 6 role role NOUN ejpam-5896 42 7 in in ADP ejpam-5896 42 8 discriminating discriminate VERB ejpam-5896 42 9 among among ADP ejpam-5896 42 10 different different ADJ ejpam-5896 42 11 stss stss NOUN ejpam-5896 42 12 , , PUNCT ejpam-5896 42 13 weijian weijian NOUN ejpam-5896 42 14 and and CCONJ ejpam-5896 42 15 lin lin PROPN ejpam-5896 42 16 [ [ X ejpam-5896 42 17 35 35 NUM ejpam-5896 42 18 ] ] PUNCT ejpam-5896 42 19 presented present VERB ejpam-5896 42 20 the the DET ejpam-5896 42 21 notions notion NOUN ejpam-5896 42 22 of of ADP ejpam-5896 42 23 connectedness connectedness NOUN ejpam-5896 42 24 in in ADP ejpam-5896 42 25 stss stss NOUN ejpam-5896 42 26 in in ADP ejpam-5896 42 27 2013 2013 NUM ejpam-5896 42 28 . . PUNCT ejpam-5896 43 1 more more ADJ ejpam-5896 43 2 investigations investigation NOUN ejpam-5896 43 3 for for ADP ejpam-5896 43 4 this this DET ejpam-5896 43 5 new new ADJ ejpam-5896 43 6 approach approach NOUN ejpam-5896 43 7 have have AUX ejpam-5896 43 8 been be AUX ejpam-5896 43 9 discussed discuss VERB ejpam-5896 43 10 later later ADV ejpam-5896 43 11 in in ADP ejpam-5896 43 12 [ [ X ejpam-5896 43 13 36 36 NUM ejpam-5896 43 14 , , PUNCT ejpam-5896 43 15 37 37 NUM ejpam-5896 43 16 ] ] PUNCT ejpam-5896 43 17 . . PUNCT ejpam-5896 44 1 an an DET ejpam-5896 44 2 application application NOUN ejpam-5896 44 3 of of ADP ejpam-5896 44 4 soft soft ADJ ejpam-5896 44 5 connectedness connectedness NOUN ejpam-5896 44 6 in in ADP ejpam-5896 44 7 decision decision NOUN ejpam-5896 44 8 making make VERB ejpam-5896 44 9 problems problem NOUN ejpam-5896 44 10 has have AUX ejpam-5896 44 11 been be AUX ejpam-5896 44 12 introduced introduce VERB ejpam-5896 44 13 in in ADP ejpam-5896 44 14 [ [ X ejpam-5896 44 15 38 38 NUM ejpam-5896 44 16 ] ] PUNCT ejpam-5896 44 17 . . PUNCT ejpam-5896 45 1 kandil kandil PROPN ejpam-5896 45 2 et et PROPN ejpam-5896 45 3 al al PROPN ejpam-5896 45 4 . . PUNCT ejpam-5896 46 1 [ [ X ejpam-5896 46 2 39 39 NUM ejpam-5896 46 3 ] ] PUNCT ejpam-5896 46 4 used use VERB ejpam-5896 46 5 the the DET ejpam-5896 46 6 soft soft ADJ ejpam-5896 46 7 ideal ideal ADJ ejpam-5896 46 8 notions notion NOUN ejpam-5896 46 9 to to PART ejpam-5896 46 10 define define VERB ejpam-5896 46 11 a a DET ejpam-5896 46 12 new new ADJ ejpam-5896 46 13 class class NOUN ejpam-5896 46 14 of of ADP ejpam-5896 46 15 soft soft ADJ ejpam-5896 46 16 connectedness connectedness NOUN ejpam-5896 46 17 , , PUNCT ejpam-5896 46 18 named name VERB ejpam-5896 46 19 soft soft ADJ ejpam-5896 46 20 i i NOUN ejpam-5896 46 21 - - PUNCT ejpam-5896 46 22 connectedness connectedness NOUN ejpam-5896 46 23 . . PUNCT ejpam-5896 47 1 abd abd PROPN ejpam-5896 47 2 el el PROPN ejpam-5896 47 3 - - PROPN ejpam-5896 47 4 latif latif PROPN ejpam-5896 47 5 [ [ X ejpam-5896 47 6 40 40 NUM ejpam-5896 47 7 ] ] PUNCT ejpam-5896 47 8 used use VERB ejpam-5896 47 9 the the DET ejpam-5896 47 10 notions notion NOUN ejpam-5896 47 11 of of ADP ejpam-5896 47 12 soft soft ADJ ejpam-5896 47 13 β β NOUN ejpam-5896 47 14 - - ADJ ejpam-5896 47 15 open open ADJ ejpam-5896 47 16 sets set NOUN ejpam-5896 47 17 to to PART ejpam-5896 47 18 present present VERB ejpam-5896 47 19 a a DET ejpam-5896 47 20 new new ADJ ejpam-5896 47 21 generalization generalization NOUN ejpam-5896 47 22 to to ADP ejpam-5896 47 23 this this DET ejpam-5896 47 24 notion notion NOUN ejpam-5896 47 25 . . PUNCT ejpam-5896 48 1 in in ADP ejpam-5896 48 2 2023 2023 NUM ejpam-5896 48 3 , , PUNCT ejpam-5896 48 4 al al PROPN ejpam-5896 48 5 - - PUNCT ejpam-5896 48 6 ghour ghour PROPN ejpam-5896 48 7 and and CCONJ ejpam-5896 48 8 al al PROPN ejpam-5896 48 9 - - PUNCT ejpam-5896 48 10 saadi saadi NOUN ejpam-5896 48 11 [ [ X ejpam-5896 48 12 41 41 NUM ejpam-5896 48 13 ] ] PUNCT ejpam-5896 48 14 defined define VERB ejpam-5896 48 15 a a DET ejpam-5896 48 16 new new ADJ ejpam-5896 48 17 approach approach NOUN ejpam-5896 48 18 of of ADP ejpam-5896 48 19 connectedness connectedness NOUN ejpam-5896 48 20 in in ADP ejpam-5896 48 21 sts st NOUN ejpam-5896 48 22 , , PUNCT ejpam-5896 48 23 named name VERB ejpam-5896 48 24 soft soft ADJ ejpam-5896 48 25 weakly weakly ADJ ejpam-5896 48 26 connected connected ADJ ejpam-5896 48 27 sets set NOUN ejpam-5896 48 28 . . PUNCT ejpam-5896 49 1 aygünoglu aygünoglu NUM ejpam-5896 50 1 and and CCONJ ejpam-5896 50 2 aygün aygün NOUN ejpam-5896 50 3 [ [ X ejpam-5896 50 4 42 42 NUM ejpam-5896 50 5 ] ] PUNCT ejpam-5896 50 6 in in ADP ejpam-5896 50 7 2013 2013 NUM ejpam-5896 50 8 , , PUNCT ejpam-5896 50 9 defined define VERB ejpam-5896 50 10 the the DET ejpam-5896 50 11 notions notion NOUN ejpam-5896 50 12 of of ADP ejpam-5896 50 13 soft soft ADJ ejpam-5896 50 14 compactness compactness NOUN ejpam-5896 50 15 . . PUNCT ejpam-5896 51 1 a a DET ejpam-5896 51 2 stronger strong ADJ ejpam-5896 51 3 notions notion NOUN ejpam-5896 51 4 of of ADP ejpam-5896 51 5 soft soft ADJ ejpam-5896 51 6 compactness compactness NOUN ejpam-5896 51 7 [ [ X ejpam-5896 51 8 43 43 NUM ejpam-5896 51 9 ] ] PUNCT ejpam-5896 51 10 were be AUX ejpam-5896 51 11 introduced introduce VERB ejpam-5896 51 12 , , PUNCT ejpam-5896 51 13 in in ADP ejpam-5896 51 14 2014 2014 NUM ejpam-5896 51 15 , , PUNCT ejpam-5896 51 16 which which PRON ejpam-5896 51 17 generalized generalize VERB ejpam-5896 51 18 by by ADP ejpam-5896 51 19 using use VERB ejpam-5896 51 20 using use VERB ejpam-5896 51 21 the the DET ejpam-5896 51 22 soft soft ADJ ejpam-5896 51 23 ideal ideal ADJ ejpam-5896 51 24 notion notion NOUN ejpam-5896 51 25 in in ADP ejpam-5896 51 26 [ [ X ejpam-5896 51 27 44 44 NUM ejpam-5896 51 28 ] ] PUNCT ejpam-5896 51 29 . . PUNCT ejpam-5896 52 1 recently recently ADV ejpam-5896 52 2 , , PUNCT ejpam-5896 52 3 al al PROPN ejpam-5896 52 4 - - PUNCT ejpam-5896 52 5 shami shami PROPN ejpam-5896 52 6 et et PROPN ejpam-5896 52 7 al al PROPN ejpam-5896 52 8 . . PUNCT ejpam-5896 53 1 [ [ X ejpam-5896 53 2 45 45 NUM ejpam-5896 53 3 ] ] PUNCT ejpam-5896 53 4 used use VERB ejpam-5896 53 5 the the DET ejpam-5896 53 6 notion notion NOUN ejpam-5896 53 7 of of ADP ejpam-5896 53 8 soft soft ADJ ejpam-5896 53 9 sd sd NOUN ejpam-5896 53 10 - - PUNCT ejpam-5896 53 11 sets set NOUN ejpam-5896 53 12 to to PART ejpam-5896 53 13 define define VERB ejpam-5896 53 14 six six NUM ejpam-5896 53 15 types type NOUN ejpam-5896 53 16 of of ADP ejpam-5896 53 17 compactness compactness NOUN ejpam-5896 53 18 in in ADP ejpam-5896 53 19 sts st NOUN ejpam-5896 53 20 in in ADP ejpam-5896 53 21 2021 2021 NUM ejpam-5896 53 22 . . PUNCT ejpam-5896 54 1 the the DET ejpam-5896 54 2 ss ss NOUN ejpam-5896 54 3 - - PUNCT ejpam-5896 54 4 operators operator NOUN ejpam-5896 54 5 are be AUX ejpam-5896 54 6 not not PART ejpam-5896 54 7 only only ADV ejpam-5896 54 8 important important ADJ ejpam-5896 54 9 in in ADP ejpam-5896 54 10 accuracy accuracy NOUN ejpam-5896 54 11 measures measure NOUN ejpam-5896 54 12 [ [ X ejpam-5896 54 13 46 46 NUM ejpam-5896 54 14 ] ] PUNCT ejpam-5896 54 15 , , PUNCT ejpam-5896 54 16 but but CCONJ ejpam-5896 54 17 also also ADV ejpam-5896 54 18 tools tool NOUN ejpam-5896 54 19 to to PART ejpam-5896 54 20 introduce introduce VERB ejpam-5896 54 21 many many ADJ ejpam-5896 54 22 topological topological ADJ ejpam-5896 54 23 properties property NOUN ejpam-5896 54 24 sstss sstss VERB ejpam-5896 54 25 like like ADP ejpam-5896 54 26 , , PUNCT ejpam-5896 54 27 generating generate VERB ejpam-5896 54 28 supra supra ADJ ejpam-5896 54 29 soft soft ADJ ejpam-5896 54 30 topologies topology NOUN ejpam-5896 54 31 via via ADP ejpam-5896 54 32 soft soft ADJ ejpam-5896 54 33 set set VERB ejpam-5896 54 34 operators operator NOUN ejpam-5896 54 35 [ [ X ejpam-5896 54 36 47 47 NUM ejpam-5896 54 37 ] ] PUNCT ejpam-5896 54 38 , , PUNCT ejpam-5896 54 39 continuity continuity NOUN ejpam-5896 54 40 [ [ X ejpam-5896 54 41 48 48 NUM ejpam-5896 54 42 ] ] PUNCT ejpam-5896 54 43 , , PUNCT ejpam-5896 54 44 connectedness connectedness NOUN ejpam-5896 54 45 [ [ X ejpam-5896 54 46 49 49 NUM ejpam-5896 54 47 ] ] PUNCT ejpam-5896 54 48 , , PUNCT ejpam-5896 54 49 compactness compactness NOUN ejpam-5896 54 50 [ [ X ejpam-5896 54 51 50], 50], NUM ejpam-5896 54 52 ...... ...... SYM ejpam-5896 54 53 etc etc X ejpam-5896 54 54 . . X ejpam-5896 55 1 this this PRON ejpam-5896 55 2 encouraged encourage VERB ejpam-5896 55 3 el el PROPN ejpam-5896 55 4 - - PUNCT ejpam-5896 55 5 sheikh sheikh PROPN ejpam-5896 55 6 and and CCONJ ejpam-5896 55 7 abd abd PROPN ejpam-5896 55 8 el el PROPN ejpam-5896 55 9 - - PROPN ejpam-5896 55 10 latif latif PROPN ejpam-5896 55 11 [ [ X ejpam-5896 55 12 51 51 NUM ejpam-5896 55 13 ] ] PUNCT ejpam-5896 55 14 , , PUNCT ejpam-5896 55 15 in in ADP ejpam-5896 55 16 2014 2014 NUM ejpam-5896 55 17 , , PUNCT ejpam-5896 55 18 to to PART ejpam-5896 55 19 define define VERB ejpam-5896 55 20 the the DET ejpam-5896 55 21 notions notion NOUN ejpam-5896 55 22 of of ADP ejpam-5896 55 23 sstss sstss NOUN ejpam-5896 55 24 as as ADP ejpam-5896 55 25 a a DET ejpam-5896 55 26 parameterized parameterized ADJ ejpam-5896 55 27 collections collection NOUN ejpam-5896 55 28 of of ADP ejpam-5896 55 29 crisp crisp ADJ ejpam-5896 55 30 supra supra ADJ ejpam-5896 55 31 topological topological ADJ ejpam-5896 55 32 spaces space NOUN ejpam-5896 55 33 . . PUNCT ejpam-5896 56 1 they they PRON ejpam-5896 56 2 also also ADV ejpam-5896 56 3 defined define VERB ejpam-5896 56 4 several several ADJ ejpam-5896 56 5 types type NOUN ejpam-5896 56 6 of of ADP ejpam-5896 56 7 ss ss NOUN ejpam-5896 56 8 - - PUNCT ejpam-5896 56 9 operators operator NOUN ejpam-5896 56 10 . . PUNCT ejpam-5896 57 1 in in ADP ejpam-5896 57 2 addition addition NOUN ejpam-5896 57 3 , , PUNCT ejpam-5896 57 4 they they PRON ejpam-5896 57 5 defined define VERB ejpam-5896 57 6 different different ADJ ejpam-5896 57 7 types type NOUN ejpam-5896 57 8 of of ADP ejpam-5896 57 9 ss ss NOUN ejpam-5896 57 10 - - NOUN ejpam-5896 57 11 continuity continuity NOUN ejpam-5896 57 12 . . PUNCT ejpam-5896 58 1 later later ADV ejpam-5896 58 2 , , PUNCT ejpam-5896 58 3 many many ADJ ejpam-5896 58 4 generalized generalized ADJ ejpam-5896 58 5 ss ss NOUN ejpam-5896 58 6 - - PUNCT ejpam-5896 58 7 operators operator NOUN ejpam-5896 58 8 via via ADP ejpam-5896 58 9 ss ss NOUN ejpam-5896 58 10 - - PUNCT ejpam-5896 58 11 b b NOUN ejpam-5896 58 12 - - PUNCT ejpam-5896 58 13 open open ADJ ejpam-5896 58 14 sets set NOUN ejpam-5896 58 15 [ [ X ejpam-5896 58 16 52 52 NUM ejpam-5896 58 17 ] ] PUNCT ejpam-5896 58 18 , , PUNCT ejpam-5896 58 19 ss ss ADJ ejpam-5896 58 20 - - PUNCT ejpam-5896 58 21 δi δi ADV ejpam-5896 58 22 - - PUNCT ejpam-5896 58 23 open open ADJ ejpam-5896 58 24 sets set NOUN ejpam-5896 58 25 [ [ X ejpam-5896 58 26 53 53 NUM ejpam-5896 58 27 , , PUNCT ejpam-5896 58 28 54 54 NUM ejpam-5896 58 29 ] ] PUNCT ejpam-5896 58 30 , , PUNCT ejpam-5896 58 31 ss ss ADJ ejpam-5896 58 32 - - PUNCT ejpam-5896 58 33 swopen swopen NOUN ejpam-5896 58 34 sets set NOUN ejpam-5896 58 35 [ [ X ejpam-5896 58 36 55 55 NUM ejpam-5896 58 37 ] ] PUNCT ejpam-5896 58 38 , , PUNCT ejpam-5896 58 39 and and CCONJ ejpam-5896 58 40 ss ss NOUN ejpam-5896 58 41 - - PUNCT ejpam-5896 58 42 sd sd NOUN ejpam-5896 58 43 - - PUNCT ejpam-5896 58 44 sets set NOUN ejpam-5896 58 45 [ [ X ejpam-5896 58 46 56 56 NUM ejpam-5896 58 47 ] ] PUNCT ejpam-5896 58 48 have have AUX ejpam-5896 58 49 been be AUX ejpam-5896 58 50 studied study VERB ejpam-5896 58 51 . . PUNCT ejpam-5896 59 1 abd abd PROPN ejpam-5896 59 2 el el PROPN ejpam-5896 59 3 - - PROPN ejpam-5896 59 4 latif latif PROPN ejpam-5896 59 5 and and CCONJ ejpam-5896 59 6 alqahtani alqahtani ADJ ejpam-5896 60 1 [ [ X ejpam-5896 60 2 57 57 NUM ejpam-5896 60 3 ] ] PUNCT ejpam-5896 60 4 introduced introduce VERB ejpam-5896 60 5 different different ADJ ejpam-5896 60 6 types type NOUN ejpam-5896 60 7 of of ADP ejpam-5896 60 8 ss ss ADJ ejpam-5896 60 9 - - ADJ ejpam-5896 60 10 continuous continuous ADJ ejpam-5896 60 11 functions function NOUN ejpam-5896 60 12 inspired inspire VERB ejpam-5896 60 13 by by ADP ejpam-5896 60 14 ss ss NOUN ejpam-5896 60 15 - - PUNCT ejpam-5896 60 16 sd sd NOUN ejpam-5896 60 17 - - PUNCT ejpam-5896 60 18 sets set NOUN ejpam-5896 60 19 and and CCONJ ejpam-5896 60 20 ss ss NOUN ejpam-5896 60 21 - - PUNCT ejpam-5896 60 22 sd sd NOUN ejpam-5896 60 23 - - PUNCT ejpam-5896 60 24 operators operator NOUN ejpam-5896 60 25 . . PUNCT ejpam-5896 61 1 abd abd PROPN ejpam-5896 61 2 ellatif ellatif PROPN ejpam-5896 62 1 [ [ X ejpam-5896 62 2 58 58 NUM ejpam-5896 62 3 ] ] PUNCT ejpam-5896 62 4 presented present VERB ejpam-5896 62 5 different different ADJ ejpam-5896 62 6 kinds kind NOUN ejpam-5896 62 7 of of ADP ejpam-5896 62 8 connectedness connectedness NOUN ejpam-5896 62 9 in in ADP ejpam-5896 62 10 sstss sstss NOUN ejpam-5896 62 11 based base VERB ejpam-5896 62 12 on on ADP ejpam-5896 62 13 ss ss PROPN ejpam-5896 62 14 - - PUNCT ejpam-5896 62 15 b b NOUN ejpam-5896 62 16 - - PUNCT ejpam-5896 62 17 open open ADJ ejpam-5896 62 18 sets set NOUN ejpam-5896 62 19 . . PUNCT ejpam-5896 63 1 he he PRON ejpam-5896 63 2 also also ADV ejpam-5896 63 3 defined define VERB ejpam-5896 63 4 different different ADJ ejpam-5896 63 5 kinds kind NOUN ejpam-5896 63 6 of of ADP ejpam-5896 63 7 compactness compactness NOUN ejpam-5896 63 8 [ [ X ejpam-5896 63 9 59 59 NUM ejpam-5896 63 10 ] ] PUNCT ejpam-5896 63 11 in in ADP ejpam-5896 63 12 sstss sstss NOUN ejpam-5896 63 13 . . PUNCT ejpam-5896 64 1 al al PROPN ejpam-5896 64 2 - - PUNCT ejpam-5896 64 3 shami shami PROPN ejpam-5896 64 4 and and CCONJ ejpam-5896 64 5 el el PROPN ejpam-5896 64 6 - - PROPN ejpam-5896 64 7 shafei shafei NOUN ejpam-5896 64 8 [ [ X ejpam-5896 64 9 60 60 NUM ejpam-5896 64 10 ] ] PUNCT ejpam-5896 64 11 extended extend VERB ejpam-5896 64 12 these these DET ejpam-5896 64 13 ideas idea NOUN ejpam-5896 64 14 by by ADP ejpam-5896 64 15 using use VERB ejpam-5896 64 16 the the DET ejpam-5896 64 17 notions notion NOUN ejpam-5896 64 18 of of ADP ejpam-5896 64 19 ss ss NOUN ejpam-5896 64 20 - - PUNCT ejpam-5896 64 21 pre pre ADJ ejpam-5896 64 22 - - ADJ ejpam-5896 64 23 open open ADJ ejpam-5896 64 24 set set NOUN ejpam-5896 64 25 . . PUNCT ejpam-5896 65 1 el el PROPN ejpam-5896 65 2 - - PUNCT ejpam-5896 65 3 shafei shafei PROPN ejpam-5896 65 4 and and CCONJ ejpam-5896 65 5 al al PROPN ejpam-5896 65 6 - - PUNCT ejpam-5896 65 7 shami shami PROPN ejpam-5896 66 1 [ [ X ejpam-5896 66 2 61 61 NUM ejpam-5896 66 3 ] ] PUNCT ejpam-5896 66 4 defined define VERB ejpam-5896 66 5 the the DET ejpam-5896 66 6 ss ss NOUN ejpam-5896 66 7 - - NOUN ejpam-5896 66 8 sdconnectedness sdconnectedness NOUN ejpam-5896 66 9 , , PUNCT ejpam-5896 66 10 as as ADP ejpam-5896 66 11 a a DET ejpam-5896 66 12 generalization generalization NOUN ejpam-5896 66 13 to to ADP ejpam-5896 66 14 many many ADJ ejpam-5896 66 15 famous famous ADJ ejpam-5896 66 16 studies study NOUN ejpam-5896 66 17 . . PUNCT ejpam-5896 67 1 abd abd PROPN ejpam-5896 67 2 el el PROPN ejpam-5896 67 3 - - PROPN ejpam-5896 67 4 latif latif PROPN ejpam-5896 67 5 et et PROPN ejpam-5896 67 6 al al PROPN ejpam-5896 67 7 . . PUNCT ejpam-5896 68 1 [ [ X ejpam-5896 68 2 62 62 NUM ejpam-5896 68 3 ] ] PUNCT ejpam-5896 68 4 defined define VERB ejpam-5896 68 5 the the DET ejpam-5896 68 6 notions notion NOUN ejpam-5896 68 7 of of ADP ejpam-5896 68 8 ss ss NOUN ejpam-5896 68 9 - - PUNCT ejpam-5896 68 10 sd sd NOUN ejpam-5896 68 11 - - PUNCT ejpam-5896 68 12 connectedness connectedness NOUN ejpam-5896 68 13 in in ADP ejpam-5896 68 14 the the DET ejpam-5896 68 15 frame frame NOUN ejpam-5896 68 16 of of ADP ejpam-5896 68 17 sstss sstss NOUN ejpam-5896 68 18 . . PUNCT ejpam-5896 69 1 this this DET ejpam-5896 69 2 paper paper NOUN ejpam-5896 69 3 is be AUX ejpam-5896 69 4 arranged arrange VERB ejpam-5896 69 5 as as SCONJ ejpam-5896 69 6 follows follow VERB ejpam-5896 69 7 : : PUNCT ejpam-5896 69 8 in in ADP ejpam-5896 69 9 preliminaries preliminary NOUN ejpam-5896 69 10 , , PUNCT ejpam-5896 69 11 we we PRON ejpam-5896 69 12 provide provide VERB ejpam-5896 69 13 the the DET ejpam-5896 69 14 approaches approach NOUN ejpam-5896 69 15 and and CCONJ ejpam-5896 69 16 terminologies terminology NOUN ejpam-5896 69 17 which which PRON ejpam-5896 69 18 shall shall AUX ejpam-5896 69 19 needed need VERB ejpam-5896 69 20 in in ADP ejpam-5896 69 21 the the DET ejpam-5896 69 22 sequels sequel NOUN ejpam-5896 69 23 . . PUNCT ejpam-5896 70 1 in in ADP ejpam-5896 70 2 section section NOUN ejpam-5896 70 3 3 3 NUM ejpam-5896 70 4 : : PUNCT ejpam-5896 70 5 we we PRON ejpam-5896 70 6 investigate investigate VERB ejpam-5896 70 7 more more ADJ ejpam-5896 70 8 characterizations characterization NOUN ejpam-5896 70 9 of of ADP ejpam-5896 70 10 the the DET ejpam-5896 70 11 notions notion NOUN ejpam-5896 70 12 of of ADP ejpam-5896 70 13 ss ss NOUN ejpam-5896 70 14 - - PUNCT ejpam-5896 70 15 sd sd NOUN ejpam-5896 70 16 - - PUNCT ejpam-5896 70 17 connectedness connectedness NOUN ejpam-5896 70 18 . . PUNCT ejpam-5896 71 1 in in ADP ejpam-5896 71 2 special special ADJ ejpam-5896 71 3 , , PUNCT ejpam-5896 71 4 we we PRON ejpam-5896 71 5 define define VERB ejpam-5896 71 6 the the DET ejpam-5896 71 7 concept concept NOUN ejpam-5896 71 8 of of ADP ejpam-5896 71 9 ss ss NOUN ejpam-5896 71 10 - - PUNCT ejpam-5896 71 11 sd sd NOUN ejpam-5896 71 12 - - PUNCT ejpam-5896 71 13 component component NOUN ejpam-5896 71 14 , , PUNCT ejpam-5896 71 15 and and CCONJ ejpam-5896 71 16 use use VERB ejpam-5896 71 17 it it PRON ejpam-5896 71 18 to to PART ejpam-5896 71 19 define define VERB ejpam-5896 71 20 a a DET ejpam-5896 71 21 new new ADJ ejpam-5896 71 22 type type NOUN ejpam-5896 71 23 of of ADP ejpam-5896 71 24 connectedness connectedness NOUN ejpam-5896 71 25 in in ADP ejpam-5896 71 26 ssts sst NOUN ejpam-5896 71 27 , , PUNCT ejpam-5896 71 28 named name VERB ejpam-5896 71 29 ssl ssl PROPN ejpam-5896 71 30 - - PUNCT ejpam-5896 71 31 sd sd NOUN ejpam-5896 71 32 - - PUNCT ejpam-5896 71 33 connectedness connectedness NOUN ejpam-5896 71 34 . . PUNCT ejpam-5896 72 1 another another DET ejpam-5896 72 2 type type NOUN ejpam-5896 72 3 of of ADP ejpam-5896 72 4 connectedness connectedness NOUN ejpam-5896 72 5 in in ADP ejpam-5896 72 6 sstss sstss PROPN ejpam-5896 72 7 , , PUNCT ejpam-5896 72 8 named name VERB ejpam-5896 72 9 ss ss NOUN ejpam-5896 72 10 - - PUNCT ejpam-5896 72 11 sd sd NOUN ejpam-5896 72 12 - - PUNCT ejpam-5896 72 13 hyperconnectedness hyperconnectedness NOUN ejpam-5896 72 14 has have AUX ejpam-5896 72 15 been be AUX ejpam-5896 72 16 defined define VERB ejpam-5896 72 17 . . PUNCT ejpam-5896 73 1 we we PRON ejpam-5896 73 2 show show VERB ejpam-5896 73 3 that that SCONJ ejpam-5896 73 4 it it PRON ejpam-5896 73 5 is be AUX ejpam-5896 73 6 equivalent equivalent ADJ ejpam-5896 73 7 to to ADP ejpam-5896 73 8 the the DET ejpam-5896 73 9 notions notion NOUN ejpam-5896 73 10 of of ADP ejpam-5896 73 11 ss ss NOUN ejpam-5896 73 12 - - PUNCT ejpam-5896 73 13 sd sd NOUN ejpam-5896 73 14 - - PUNCT ejpam-5896 73 15 connectedness connectedness NOUN ejpam-5896 73 16 [ [ X ejpam-5896 73 17 62 62 NUM ejpam-5896 73 18 ] ] PUNCT ejpam-5896 73 19 . . PUNCT ejpam-5896 74 1 moreover moreover ADV ejpam-5896 74 2 , , PUNCT ejpam-5896 74 3 the the DET ejpam-5896 74 4 relationships relationship NOUN ejpam-5896 74 5 among among ADP ejpam-5896 74 6 them they PRON ejpam-5896 74 7 , , PUNCT ejpam-5896 74 8 in in ADP ejpam-5896 74 9 addition addition NOUN ejpam-5896 74 10 to to ADP ejpam-5896 74 11 the the DET ejpam-5896 74 12 abd abd PROPN ejpam-5896 74 13 el el PROPN ejpam-5896 74 14 - - PROPN ejpam-5896 74 15 latif latif PROPN ejpam-5896 74 16 et et PROPN ejpam-5896 74 17 al al PROPN ejpam-5896 74 18 . . PUNCT ejpam-5896 74 19 / / SYM ejpam-5896 74 20 eur eur PROPN ejpam-5896 74 21 . . PUNCT ejpam-5896 75 1 j. j. PROPN ejpam-5896 75 2 pure pure PROPN ejpam-5896 75 3 appl appl PROPN ejpam-5896 75 4 . . PROPN ejpam-5896 75 5 math math PROPN ejpam-5896 75 6 , , PUNCT ejpam-5896 75 7 18 18 NUM ejpam-5896 75 8 ( ( PUNCT ejpam-5896 75 9 2 2 NUM ejpam-5896 75 10 ) ) PUNCT ejpam-5896 75 11 ( ( PUNCT ejpam-5896 75 12 2025 2025 NUM ejpam-5896 75 13 ) ) PUNCT ejpam-5896 75 14 , , PUNCT ejpam-5896 75 15 5896 5896 NUM ejpam-5896 75 16 3 3 NUM ejpam-5896 75 17 of of ADP ejpam-5896 75 18 20 20 NUM ejpam-5896 75 19 relationships relationship NOUN ejpam-5896 75 20 with with ADP ejpam-5896 75 21 previous previous ADJ ejpam-5896 75 22 studies study NOUN ejpam-5896 75 23 , , PUNCT ejpam-5896 75 24 have have AUX ejpam-5896 75 25 been be AUX ejpam-5896 75 26 provided provide VERB ejpam-5896 75 27 . . PUNCT ejpam-5896 76 1 furthermore furthermore ADV ejpam-5896 76 2 , , PUNCT ejpam-5896 76 3 we we PRON ejpam-5896 76 4 present present VERB ejpam-5896 76 5 a a DET ejpam-5896 76 6 topological topological ADJ ejpam-5896 76 7 chart chart NOUN ejpam-5896 76 8 to to PART ejpam-5896 76 9 declare declare VERB ejpam-5896 76 10 the the DET ejpam-5896 76 11 key key ADJ ejpam-5896 76 12 concepts concept NOUN ejpam-5896 76 13 presented present VERB ejpam-5896 76 14 in in ADP ejpam-5896 76 15 figure figure NOUN ejpam-5896 76 16 1 1 NUM ejpam-5896 76 17 . . PUNCT ejpam-5896 77 1 also also ADV ejpam-5896 77 2 , , PUNCT ejpam-5896 77 3 we we PRON ejpam-5896 77 4 provide provide VERB ejpam-5896 77 5 counterexamples counterexample NOUN ejpam-5896 77 6 to to PART ejpam-5896 77 7 elucidate elucidate VERB ejpam-5896 77 8 that that SCONJ ejpam-5896 77 9 the the DET ejpam-5896 77 10 arrows arrow NOUN ejpam-5896 77 11 in in ADP ejpam-5896 77 12 this this DET ejpam-5896 77 13 chart chart NOUN ejpam-5896 77 14 are be AUX ejpam-5896 77 15 non non ADJ ejpam-5896 77 16 - - ADJ ejpam-5896 77 17 reversible reversible ADJ ejpam-5896 77 18 . . PUNCT ejpam-5896 78 1 in in ADP ejpam-5896 78 2 section section NOUN ejpam-5896 78 3 4 4 NUM ejpam-5896 78 4 : : PUNCT ejpam-5896 78 5 we we PRON ejpam-5896 78 6 introduce introduce VERB ejpam-5896 78 7 two two NUM ejpam-5896 78 8 different different ADJ ejpam-5896 78 9 types type NOUN ejpam-5896 78 10 of of ADP ejpam-5896 78 11 compactness compactness NOUN ejpam-5896 78 12 in in ADP ejpam-5896 78 13 sstss sstss NOUN ejpam-5896 78 14 , , PUNCT ejpam-5896 78 15 named name VERB ejpam-5896 78 16 ss ss NOUN ejpam-5896 78 17 - - ADJ ejpam-5896 78 18 sdcompactness sdcompactness ADJ ejpam-5896 78 19 and and CCONJ ejpam-5896 78 20 ss ss NOUN ejpam-5896 78 21 - - PUNCT ejpam-5896 78 22 sd sd NOUN ejpam-5896 78 23 - - PUNCT ejpam-5896 78 24 lindelöfness lindelöfness NOUN ejpam-5896 78 25 . . PUNCT ejpam-5896 79 1 we we PRON ejpam-5896 79 2 study study VERB ejpam-5896 79 3 several several ADJ ejpam-5896 79 4 of of ADP ejpam-5896 79 5 their their PRON ejpam-5896 79 6 essential essential ADJ ejpam-5896 79 7 properties property NOUN ejpam-5896 79 8 . . PUNCT ejpam-5896 80 1 moreover moreover ADV ejpam-5896 80 2 , , PUNCT ejpam-5896 80 3 we we PRON ejpam-5896 80 4 discuss discuss VERB ejpam-5896 80 5 the the DET ejpam-5896 80 6 relationships relationship NOUN ejpam-5896 80 7 between between ADP ejpam-5896 80 8 supra supra PROPN ejpam-5896 80 9 soft soft ADJ ejpam-5896 80 10 - - PUNCT ejpam-5896 80 11 sd sd NOUN ejpam-5896 80 12 - - PUNCT ejpam-5896 80 13 compact compact ADJ ejpam-5896 80 14 ( ( PUNCT ejpam-5896 80 15 lindelöf lindelöf NOUN ejpam-5896 80 16 ) ) PUNCT ejpam-5896 80 17 ssts sst NOUN ejpam-5896 80 18 and and CCONJ ejpam-5896 80 19 their their PRON ejpam-5896 80 20 parametric parametric ADJ ejpam-5896 80 21 supra supra PROPN ejpam-5896 80 22 topological topological PROPN ejpam-5896 80 23 spaces space NOUN ejpam-5896 80 24 , , PUNCT ejpam-5896 80 25 which which PRON ejpam-5896 80 26 are be AUX ejpam-5896 80 27 defined define VERB ejpam-5896 80 28 on on ADP ejpam-5896 80 29 any any DET ejpam-5896 80 30 universal universal ADJ ejpam-5896 80 31 set set NOUN ejpam-5896 80 32 and and CCONJ ejpam-5896 80 33 finite finite ADJ ejpam-5896 80 34 set set NOUN ejpam-5896 80 35 of of ADP ejpam-5896 80 36 parameters parameter NOUN ejpam-5896 80 37 . . PUNCT ejpam-5896 81 1 finally finally ADV ejpam-5896 81 2 , , PUNCT ejpam-5896 81 3 a a DET ejpam-5896 81 4 topological topological ADJ ejpam-5896 81 5 chart chart NOUN ejpam-5896 81 6 to to PART ejpam-5896 81 7 illustrate illustrate VERB ejpam-5896 81 8 the the DET ejpam-5896 81 9 key key ADJ ejpam-5896 81 10 concepts concept NOUN ejpam-5896 81 11 are be AUX ejpam-5896 81 12 presented present VERB ejpam-5896 81 13 in in ADP ejpam-5896 81 14 figure figure NOUN ejpam-5896 81 15 2 2 NUM ejpam-5896 81 16 , , PUNCT ejpam-5896 81 17 and and CCONJ ejpam-5896 81 18 confirmed confirm VERB ejpam-5896 81 19 by by ADP ejpam-5896 81 20 concrete concrete ADJ ejpam-5896 81 21 counterexamples counterexample NOUN ejpam-5896 81 22 . . PUNCT ejpam-5896 82 1 2 2 X ejpam-5896 82 2 . . X ejpam-5896 82 3 preliminaries preliminary NOUN ejpam-5896 82 4 the the DET ejpam-5896 82 5 concepts concept NOUN ejpam-5896 82 6 and and CCONJ ejpam-5896 82 7 terms term NOUN ejpam-5896 82 8 that that PRON ejpam-5896 82 9 will will AUX ejpam-5896 82 10 be be AUX ejpam-5896 82 11 used use VERB ejpam-5896 82 12 in in ADP ejpam-5896 82 13 this this DET ejpam-5896 82 14 manuscript manuscript NOUN ejpam-5896 82 15 are be AUX ejpam-5896 82 16 introduced introduce VERB ejpam-5896 82 17 in in ADP ejpam-5896 82 18 this this DET ejpam-5896 82 19 section section NOUN ejpam-5896 82 20 ; ; PUNCT ejpam-5896 82 21 for for ADP ejpam-5896 82 22 further further ADJ ejpam-5896 82 23 information information NOUN ejpam-5896 82 24 , , PUNCT ejpam-5896 82 25 see see VERB ejpam-5896 82 26 [ [ X ejpam-5896 82 27 11 11 NUM ejpam-5896 82 28 , , PUNCT ejpam-5896 82 29 51 51 NUM ejpam-5896 82 30 , , PUNCT ejpam-5896 82 31 56 56 NUM ejpam-5896 82 32 , , PUNCT ejpam-5896 82 33 57 57 NUM ejpam-5896 82 34 ] ] PUNCT ejpam-5896 82 35 . . PUNCT ejpam-5896 83 1 definition definition NOUN ejpam-5896 83 2 1 1 NUM ejpam-5896 83 3 . . PUNCT ejpam-5896 84 1 [ [ X ejpam-5896 84 2 6 6 NUM ejpam-5896 84 3 ] ] PUNCT ejpam-5896 84 4 a a DET ejpam-5896 84 5 soft soft ADJ ejpam-5896 84 6 set set NOUN ejpam-5896 84 7 is be AUX ejpam-5896 84 8 a a DET ejpam-5896 84 9 pair pair NOUN ejpam-5896 84 10 ( ( PUNCT ejpam-5896 84 11 k,∆ k,∆ PROPN ejpam-5896 84 12 ) ) PUNCT ejpam-5896 84 13 ; ; PUNCT ejpam-5896 84 14 represented represent VERB ejpam-5896 84 15 by by ADP ejpam-5896 84 16 k∆ k∆ PROPN ejpam-5896 84 17 , , PUNCT ejpam-5896 84 18 over over ADP ejpam-5896 84 19 the the DET ejpam-5896 84 20 initial initial ADJ ejpam-5896 84 21 universe universe NOUN ejpam-5896 84 22 u u NOUN ejpam-5896 84 23 and and CCONJ ejpam-5896 84 24 the the DET ejpam-5896 84 25 set set NOUN ejpam-5896 84 26 of of ADP ejpam-5896 84 27 parameters parameter NOUN ejpam-5896 84 28 ∆. ∆. X ejpam-5896 84 29 it it PRON ejpam-5896 84 30 is be AUX ejpam-5896 84 31 characterized characterize VERB ejpam-5896 84 32 by by ADP ejpam-5896 84 33 k∆ k∆ PROPN ejpam-5896 84 34 = = SYM ejpam-5896 84 35 { { PUNCT ejpam-5896 84 36 k(γ k(γ PROPN ejpam-5896 84 37 ) ) PUNCT ejpam-5896 84 38 : : PUNCT ejpam-5896 84 39 γ γ X ejpam-5896 84 40 ∈ ∈ PROPN ejpam-5896 84 41 ∆ ∆ PROPN ejpam-5896 84 42 , , PUNCT ejpam-5896 84 43 k k X ejpam-5896 84 44 : : PUNCT ejpam-5896 84 45 ∆ ∆ PROPN ejpam-5896 85 1 → → X ejpam-5896 85 2 p p X ejpam-5896 85 3 ( ( PUNCT ejpam-5896 85 4 u u NOUN ejpam-5896 85 5 ) ) PUNCT ejpam-5896 85 6 } } PUNCT ejpam-5896 85 7 . . PUNCT ejpam-5896 86 1 if if SCONJ ejpam-5896 86 2 for for ADP ejpam-5896 86 3 all all DET ejpam-5896 86 4 γ γ PROPN ejpam-5896 86 5 ∈ ∈ PROPN ejpam-5896 86 6 ∆ ∆ PROPN ejpam-5896 86 7 , , PUNCT ejpam-5896 86 8 k(γ k(γ PROPN ejpam-5896 86 9 ) ) PUNCT ejpam-5896 86 10 = = SYM ejpam-5896 86 11 φ φ PROPN ejpam-5896 86 12 ( ( PUNCT ejpam-5896 86 13 respectively respectively ADV ejpam-5896 86 14 , , PUNCT ejpam-5896 86 15 k(γ k(γ PROPN ejpam-5896 86 16 ) ) PUNCT ejpam-5896 87 1 = = SYM ejpam-5896 87 2 u u NOUN ejpam-5896 87 3 ) ) PUNCT ejpam-5896 87 4 , , PUNCT ejpam-5896 87 5 then then ADV ejpam-5896 87 6 ( ( PUNCT ejpam-5896 87 7 k,∆ k,∆ PROPN ejpam-5896 87 8 ) ) PUNCT ejpam-5896 87 9 is be AUX ejpam-5896 87 10 referred refer VERB ejpam-5896 87 11 to to ADP ejpam-5896 87 12 as as ADP ejpam-5896 87 13 a a DET ejpam-5896 87 14 null null NOUN ejpam-5896 87 15 ( ( PUNCT ejpam-5896 87 16 or or CCONJ ejpam-5896 87 17 absolute absolute ADJ ejpam-5896 87 18 ) ) PUNCT ejpam-5896 87 19 soft soft ADJ ejpam-5896 87 20 set set NOUN ejpam-5896 87 21 , , PUNCT ejpam-5896 87 22 and and CCONJ ejpam-5896 87 23 it it PRON ejpam-5896 87 24 is be AUX ejpam-5896 87 25 represented represent VERB ejpam-5896 87 26 by by ADP ejpam-5896 87 27 φ̃ φ̃ PROPN ejpam-5896 87 28 ( ( PUNCT ejpam-5896 87 29 or or CCONJ ejpam-5896 87 30 ũ ũ PROPN ejpam-5896 87 31 , , PUNCT ejpam-5896 87 32 respectively respectively ADV ejpam-5896 87 33 ) ) PUNCT ejpam-5896 87 34 . . PUNCT ejpam-5896 88 1 from from ADP ejpam-5896 88 2 now now ADV ejpam-5896 88 3 on on ADV ejpam-5896 88 4 , , PUNCT ejpam-5896 88 5 we we PRON ejpam-5896 88 6 will will AUX ejpam-5896 88 7 use use VERB ejpam-5896 88 8 s(u)∆ s(u)∆ NOUN ejpam-5896 88 9 to to PART ejpam-5896 88 10 represent represent VERB ejpam-5896 88 11 the the DET ejpam-5896 88 12 class class NOUN ejpam-5896 88 13 of of ADP ejpam-5896 88 14 all all DET ejpam-5896 88 15 soft soft ADJ ejpam-5896 88 16 sets set NOUN ejpam-5896 88 17 . . PUNCT ejpam-5896 89 1 definition definition NOUN ejpam-5896 89 2 2 2 NUM ejpam-5896 89 3 . . PUNCT ejpam-5896 90 1 [ [ X ejpam-5896 90 2 11 11 NUM ejpam-5896 90 3 ] ] PUNCT ejpam-5896 90 4 if if SCONJ ejpam-5896 90 5 the the DET ejpam-5896 90 6 collection collection NOUN ejpam-5896 90 7 τ τ PROPN ejpam-5896 90 8 ⊆ ⊆ NUM ejpam-5896 90 9 s(u)∆ s(u)∆ NOUN ejpam-5896 90 10 comprises comprise VERB ejpam-5896 90 11 ũ ũ PROPN ejpam-5896 90 12 , , PUNCT ejpam-5896 90 13 φ̃ φ̃ PROPN ejpam-5896 90 14 , , PUNCT ejpam-5896 90 15 and and CCONJ ejpam-5896 90 16 is be AUX ejpam-5896 90 17 closed close VERB ejpam-5896 90 18 under under ADP ejpam-5896 90 19 arbitrary arbitrary ADJ ejpam-5896 90 20 soft soft ADJ ejpam-5896 90 21 union union NOUN ejpam-5896 90 22 and and CCONJ ejpam-5896 90 23 finite finite VERB ejpam-5896 90 24 soft soft ADJ ejpam-5896 90 25 intersection intersection NOUN ejpam-5896 90 26 , , PUNCT ejpam-5896 90 27 then then ADV ejpam-5896 90 28 it it PRON ejpam-5896 90 29 is be AUX ejpam-5896 90 30 referred refer VERB ejpam-5896 90 31 to to ADP ejpam-5896 90 32 as as ADP ejpam-5896 90 33 a a DET ejpam-5896 90 34 soft soft ADJ ejpam-5896 90 35 topology topology NOUN ejpam-5896 90 36 on on ADP ejpam-5896 90 37 u u PROPN ejpam-5896 90 38 . . PUNCT ejpam-5896 91 1 over over ADP ejpam-5896 91 2 u u PROPN ejpam-5896 91 3 , , PUNCT ejpam-5896 91 4 the the DET ejpam-5896 91 5 triplet triplet NOUN ejpam-5896 91 6 ( ( PUNCT ejpam-5896 91 7 u u NOUN ejpam-5896 91 8 , , PUNCT ejpam-5896 91 9 τ,∆ τ,∆ NOUN ejpam-5896 91 10 ) ) PUNCT ejpam-5896 91 11 is be AUX ejpam-5896 91 12 referred refer VERB ejpam-5896 91 13 to to ADP ejpam-5896 91 14 as as ADP ejpam-5896 91 15 an an DET ejpam-5896 91 16 sts st NOUN ejpam-5896 91 17 . . PUNCT ejpam-5896 92 1 definition definition NOUN ejpam-5896 92 2 3 3 NUM ejpam-5896 92 3 . . PUNCT ejpam-5896 93 1 [ [ X ejpam-5896 93 2 11 11 NUM ejpam-5896 93 3 ] ] PUNCT ejpam-5896 93 4 assume assume VERB ejpam-5896 93 5 that that SCONJ ejpam-5896 93 6 ( ( PUNCT ejpam-5896 93 7 k,∆ k,∆ PROPN ejpam-5896 93 8 ) ) PUNCT ejpam-5896 93 9 ∈ ∈ PROPN ejpam-5896 93 10 s(u)∆ s(u)∆ NOUN ejpam-5896 93 11 and and CCONJ ejpam-5896 93 12 ( ( PUNCT ejpam-5896 93 13 u u NOUN ejpam-5896 93 14 , , PUNCT ejpam-5896 93 15 τ,∆ τ,∆ NOUN ejpam-5896 93 16 ) ) PUNCT ejpam-5896 93 17 are be AUX ejpam-5896 93 18 sts st NOUN ejpam-5896 93 19 . . PUNCT ejpam-5896 94 1 the the DET ejpam-5896 94 2 intersection intersection NOUN ejpam-5896 94 3 of of ADP ejpam-5896 94 4 all all DET ejpam-5896 94 5 soft soft ADJ ejpam-5896 94 6 closed closed ADJ ejpam-5896 94 7 supersets superset NOUN ejpam-5896 94 8 of of ADP ejpam-5896 94 9 ( ( PUNCT ejpam-5896 94 10 k,∆ k,∆ PROPN ejpam-5896 94 11 ) ) PUNCT ejpam-5896 94 12 is be AUX ejpam-5896 94 13 the the DET ejpam-5896 94 14 soft soft ADJ ejpam-5896 94 15 closure closure NOUN ejpam-5896 94 16 of of ADP ejpam-5896 94 17 ( ( PUNCT ejpam-5896 94 18 k,∆ k,∆ PROPN ejpam-5896 94 19 ) ) PUNCT ejpam-5896 94 20 ; ; PUNCT ejpam-5896 94 21 it it PRON ejpam-5896 94 22 is be AUX ejpam-5896 94 23 represented represent VERB ejpam-5896 94 24 by by ADP ejpam-5896 94 25 cl(k,∆ cl(k,∆ NOUN ejpam-5896 94 26 ) ) PUNCT ejpam-5896 94 27 . . PUNCT ejpam-5896 95 1 additionally additionally ADV ejpam-5896 95 2 , , PUNCT ejpam-5896 95 3 the the DET ejpam-5896 95 4 union union NOUN ejpam-5896 95 5 of of ADP ejpam-5896 95 6 all all DET ejpam-5896 95 7 soft soft ADJ ejpam-5896 95 8 open open ADJ ejpam-5896 95 9 subsets subset NOUN ejpam-5896 95 10 of of ADP ejpam-5896 95 11 ( ( PUNCT ejpam-5896 95 12 g,∆ g,∆ PROPN ejpam-5896 95 13 ) ) PUNCT ejpam-5896 95 14 is be AUX ejpam-5896 95 15 the the DET ejpam-5896 95 16 soft soft ADJ ejpam-5896 95 17 interior interior NOUN ejpam-5896 95 18 of of ADP ejpam-5896 95 19 ( ( PUNCT ejpam-5896 95 20 g,∆ g,∆ PROPN ejpam-5896 95 21 ) ) PUNCT ejpam-5896 95 22 ; ; PUNCT ejpam-5896 95 23 this this PRON ejpam-5896 95 24 is be AUX ejpam-5896 95 25 represented represent VERB ejpam-5896 95 26 as as ADP ejpam-5896 95 27 int(g,∆ int(g,∆ NOUN ejpam-5896 95 28 ) ) PUNCT ejpam-5896 95 29 . . PUNCT ejpam-5896 96 1 definition definition NOUN ejpam-5896 96 2 4 4 NUM ejpam-5896 96 3 . . PUNCT ejpam-5896 97 1 [ [ X ejpam-5896 97 2 11 11 NUM ejpam-5896 97 3 , , PUNCT ejpam-5896 97 4 20 20 NUM ejpam-5896 97 5 ] ] PUNCT ejpam-5896 97 6 the the DET ejpam-5896 97 7 soft soft ADJ ejpam-5896 97 8 set set NOUN ejpam-5896 97 9 ( ( PUNCT ejpam-5896 97 10 g,∆ g,∆ PROPN ejpam-5896 97 11 ) ) PUNCT ejpam-5896 97 12 ∈ ∈ PROPN ejpam-5896 97 13 s(u)∆ s(u)∆ NOUN ejpam-5896 97 14 is be AUX ejpam-5896 97 15 referred refer VERB ejpam-5896 97 16 to to ADP ejpam-5896 97 17 as as ADP ejpam-5896 97 18 a a DET ejpam-5896 97 19 soft soft ADJ ejpam-5896 97 20 point point NOUN ejpam-5896 97 21 in in ADP ejpam-5896 97 22 ũ ũ PROPN ejpam-5896 97 23 ; ; PUNCT ejpam-5896 97 24 it it PRON ejpam-5896 97 25 is be AUX ejpam-5896 97 26 represented represent VERB ejpam-5896 97 27 as as ADP ejpam-5896 97 28 rγ rγ NOUN ejpam-5896 97 29 , , PUNCT ejpam-5896 97 30 provided provide VERB ejpam-5896 97 31 that that SCONJ ejpam-5896 97 32 r r NOUN ejpam-5896 97 33 ∈ ∈ PROPN ejpam-5896 97 34 u u NOUN ejpam-5896 97 35 and and CCONJ ejpam-5896 97 36 γ γ PROPN ejpam-5896 97 37 ∈ ∈ PROPN ejpam-5896 97 38 ∆ ∆ PROPN ejpam-5896 97 39 exist exist VERB ejpam-5896 97 40 such such ADJ ejpam-5896 97 41 that that SCONJ ejpam-5896 97 42 g(γ g(γ PROPN ejpam-5896 97 43 ) ) PUNCT ejpam-5896 97 44 = = PRON ejpam-5896 98 1 { { PUNCT ejpam-5896 98 2 r r NOUN ejpam-5896 98 3 } } PUNCT ejpam-5896 98 4 and and CCONJ ejpam-5896 98 5 g(γ′ g(γ′ NOUN ejpam-5896 98 6 ) ) PUNCT ejpam-5896 98 7 = = PUNCT ejpam-5896 99 1 φ φ PROPN ejpam-5896 99 2 for for ADP ejpam-5896 99 3 every every DET ejpam-5896 99 4 γ′ γ′ PROPN ejpam-5896 99 5 ∈ ∈ PROPN ejpam-5896 99 6 ∆− ∆− NOUN ejpam-5896 99 7 { { PUNCT ejpam-5896 99 8 γ γ X ejpam-5896 99 9 } } PUNCT ejpam-5896 99 10 . . PUNCT ejpam-5896 100 1 moreover moreover ADV ejpam-5896 100 2 , , PUNCT ejpam-5896 100 3 sγ∈̃(f,∆ sγ∈̃(f,∆ PROPN ejpam-5896 100 4 ) ) PUNCT ejpam-5896 100 5 , , PUNCT ejpam-5896 100 6 if if SCONJ ejpam-5896 100 7 g(γ g(γ PROPN ejpam-5896 100 8 ) ) PUNCT ejpam-5896 100 9 ⊆ ⊆ NUM ejpam-5896 100 10 f f X ejpam-5896 100 11 ( ( PUNCT ejpam-5896 100 12 γ γ PROPN ejpam-5896 100 13 ) ) PUNCT ejpam-5896 100 14 for for ADP ejpam-5896 100 15 the the DET ejpam-5896 100 16 element element NOUN ejpam-5896 100 17 γ γ PROPN ejpam-5896 100 18 ∈ ∈ PROPN ejpam-5896 100 19 ∆. ∆. PROPN ejpam-5896 100 20 theorem theorem NOUN ejpam-5896 100 21 1 1 NUM ejpam-5896 100 22 . . PUNCT ejpam-5896 101 1 [ [ X ejpam-5896 101 2 14 14 NUM ejpam-5896 101 3 ] ] PUNCT ejpam-5896 101 4 for for ADP ejpam-5896 101 5 the the DET ejpam-5896 101 6 soft soft ADJ ejpam-5896 101 7 map map NOUN ejpam-5896 101 8 ψsd ψsd VERB ejpam-5896 101 9 : : PUNCT ejpam-5896 101 10 ( ( PUNCT ejpam-5896 101 11 u u NOUN ejpam-5896 101 12 , , PUNCT ejpam-5896 101 13 τ,∆ τ,∆ NOUN ejpam-5896 101 14 ) ) PUNCT ejpam-5896 101 15 → → SYM ejpam-5896 101 16 ( ( PUNCT ejpam-5896 101 17 v v NOUN ejpam-5896 101 18 , , PUNCT ejpam-5896 101 19 σ σ PROPN ejpam-5896 101 20 , , PUNCT ejpam-5896 101 21 λ λ PROPN ejpam-5896 101 22 ) ) PUNCT ejpam-5896 101 23 , , PUNCT ejpam-5896 101 24 the the DET ejpam-5896 101 25 following follow VERB ejpam-5896 101 26 statements statement NOUN ejpam-5896 101 27 hold hold VERB ejpam-5896 101 28 . . PUNCT ejpam-5896 102 1 ( ( PUNCT ejpam-5896 102 2 1 1 X ejpam-5896 102 3 ) ) PUNCT ejpam-5896 102 4 ψ−1 ψ−1 PROPN ejpam-5896 102 5 sd sd ADP ejpam-5896 102 6 ( ( PUNCT ejpam-5896 102 7 t t PROPN ejpam-5896 102 8 c̃,λ c̃,λ PROPN ejpam-5896 102 9 ) ) PUNCT ejpam-5896 103 1 = = PRON ejpam-5896 104 1 ( ( PUNCT ejpam-5896 104 2 ψ−1 ψ−1 PROPN ejpam-5896 104 3 sd sd PROPN ejpam-5896 104 4 ( ( PUNCT ejpam-5896 104 5 t t PROPN ejpam-5896 104 6 , , PUNCT ejpam-5896 104 7 λ λ NOUN ejpam-5896 104 8 ) ) PUNCT ejpam-5896 104 9 ) ) PUNCT ejpam-5896 105 1 c̃ c̃ NOUN ejpam-5896 105 2 ∀ ∀ X ejpam-5896 105 3 ( ( PUNCT ejpam-5896 105 4 t t PROPN ejpam-5896 105 5 , , PUNCT ejpam-5896 105 6 λ λ NOUN ejpam-5896 105 7 ) ) PUNCT ejpam-5896 105 8 ∈ ∈ PROPN ejpam-5896 105 9 s(v s(v PROPN ejpam-5896 105 10 ) ) PUNCT ejpam-5896 106 1 λ λ PROPN ejpam-5896 106 2 . . PUNCT ejpam-5896 107 1 ( ( PUNCT ejpam-5896 107 2 2 2 X ejpam-5896 107 3 ) ) PUNCT ejpam-5896 107 4 ψsd(ψ ψsd(ψ NOUN ejpam-5896 107 5 −1 −1 NOUN ejpam-5896 107 6 sd sd ADP ejpam-5896 107 7 ( ( PUNCT ejpam-5896 107 8 t t PROPN ejpam-5896 107 9 , , PUNCT ejpam-5896 107 10 λ))⊆̃(t λ))⊆̃(t ADV ejpam-5896 107 11 , , PUNCT ejpam-5896 107 12 λ λ NOUN ejpam-5896 107 13 ) ) PUNCT ejpam-5896 107 14 ∀ ∀ X ejpam-5896 107 15 ( ( PUNCT ejpam-5896 107 16 t t PROPN ejpam-5896 107 17 , , PUNCT ejpam-5896 107 18 λ λ NOUN ejpam-5896 107 19 ) ) PUNCT ejpam-5896 107 20 ∈ ∈ PROPN ejpam-5896 107 21 s(v s(v PROPN ejpam-5896 107 22 ) ) PUNCT ejpam-5896 108 1 λ λ PROPN ejpam-5896 108 2 . . PROPN ejpam-5896 109 1 in in ADP ejpam-5896 109 2 the the DET ejpam-5896 109 3 event event NOUN ejpam-5896 109 4 that that PRON ejpam-5896 109 5 ψsd ψsd VERB ejpam-5896 109 6 is be AUX ejpam-5896 109 7 surjective surjective ADJ ejpam-5896 109 8 , , PUNCT ejpam-5896 109 9 we we PRON ejpam-5896 109 10 obtain obtain VERB ejpam-5896 109 11 equality equality NOUN ejpam-5896 109 12 . . PUNCT ejpam-5896 110 1 ( ( PUNCT ejpam-5896 110 2 3 3 X ejpam-5896 110 3 ) ) PUNCT ejpam-5896 110 4 ( ( PUNCT ejpam-5896 110 5 m,∆)⊆̃ψ−1 m,∆)⊆̃ψ−1 NOUN ejpam-5896 110 6 sd sd PROPN ejpam-5896 110 7 ( ( PUNCT ejpam-5896 110 8 ψsd(m,∆ ψsd(m,∆ NUM ejpam-5896 110 9 ) ) PUNCT ejpam-5896 110 10 ) ) PUNCT ejpam-5896 110 11 ∀ ∀ X ejpam-5896 110 12 ( ( PUNCT ejpam-5896 110 13 m,∆ m,∆ X ejpam-5896 110 14 ) ) PUNCT ejpam-5896 110 15 ∈ ∈ NOUN ejpam-5896 110 16 s(u)∆. s(u)∆. NOUN ejpam-5896 110 17 in in ADP ejpam-5896 110 18 the the DET ejpam-5896 110 19 event event NOUN ejpam-5896 110 20 that that PRON ejpam-5896 110 21 ψsd ψsd VERB ejpam-5896 110 22 is be AUX ejpam-5896 110 23 injective injective ADJ ejpam-5896 110 24 , , PUNCT ejpam-5896 110 25 we we PRON ejpam-5896 110 26 obtain obtain VERB ejpam-5896 110 27 equality equality NOUN ejpam-5896 110 28 . . PUNCT ejpam-5896 111 1 ( ( PUNCT ejpam-5896 111 2 4 4 X ejpam-5896 111 3 ) ) PUNCT ejpam-5896 111 4 ψsd(ũ)⊆̃ṽ ψsd(ũ)⊆̃ṽ NOUN ejpam-5896 111 5 . . PUNCT ejpam-5896 112 1 in in ADP ejpam-5896 112 2 the the DET ejpam-5896 112 3 event event NOUN ejpam-5896 112 4 that that PRON ejpam-5896 112 5 ψsd ψsd VERB ejpam-5896 112 6 is be AUX ejpam-5896 112 7 surjective surjective ADJ ejpam-5896 112 8 , , PUNCT ejpam-5896 112 9 we we PRON ejpam-5896 112 10 obtain obtain VERB ejpam-5896 112 11 equality equality NOUN ejpam-5896 112 12 . . PUNCT ejpam-5896 113 1 definition definition NOUN ejpam-5896 113 2 5 5 NUM ejpam-5896 113 3 . . PUNCT ejpam-5896 114 1 [ [ X ejpam-5896 114 2 51 51 NUM ejpam-5896 114 3 ] ] PUNCT ejpam-5896 114 4 an an DET ejpam-5896 114 5 ssts sst NOUN ejpam-5896 114 6 on on ADP ejpam-5896 114 7 u u NOUN ejpam-5896 114 8 is be AUX ejpam-5896 114 9 defined define VERB ejpam-5896 114 10 as as ADP ejpam-5896 114 11 the the DET ejpam-5896 114 12 collection collection NOUN ejpam-5896 114 13 µ µ PROPN ejpam-5896 114 14 ⊆ ⊆ NUM ejpam-5896 114 15 s(u)∆ s(u)∆ NOUN ejpam-5896 114 16 if if SCONJ ejpam-5896 114 17 µ µ NOUN ejpam-5896 114 18 contains contain VERB ejpam-5896 114 19 ũ ũ PROPN ejpam-5896 114 20 , , PUNCT ejpam-5896 114 21 φ̃ φ̃ PROPN ejpam-5896 114 22 , , PUNCT ejpam-5896 114 23 and and CCONJ ejpam-5896 114 24 is be AUX ejpam-5896 114 25 closed close VERB ejpam-5896 114 26 under under ADP ejpam-5896 114 27 an an DET ejpam-5896 114 28 arbitrary arbitrary ADJ ejpam-5896 114 29 soft soft ADJ ejpam-5896 114 30 union union NOUN ejpam-5896 114 31 . . PUNCT ejpam-5896 115 1 also also ADV ejpam-5896 115 2 , , PUNCT ejpam-5896 115 3 the the DET ejpam-5896 115 4 ss ss NOUN ejpam-5896 115 5 - - NOUN ejpam-5896 115 6 interior interior ADJ ejpam-5896 115 7 of of ADP ejpam-5896 115 8 a a DET ejpam-5896 115 9 soft soft ADJ ejpam-5896 115 10 subset subset NOUN ejpam-5896 115 11 ( ( PUNCT ejpam-5896 115 12 t,∆ t,∆ NUM ejpam-5896 115 13 ) ) PUNCT ejpam-5896 115 14 , , PUNCT ejpam-5896 115 15 is be AUX ejpam-5896 115 16 indicated indicate VERB ejpam-5896 115 17 by by ADP ejpam-5896 115 18 ints(t,∆ ints(t,∆ NOUN ejpam-5896 115 19 ) ) PUNCT ejpam-5896 115 20 , , PUNCT ejpam-5896 115 21 which which PRON ejpam-5896 115 22 is be AUX ejpam-5896 115 23 the the DET ejpam-5896 115 24 soft soft ADJ ejpam-5896 115 25 union union NOUN ejpam-5896 115 26 of of ADP ejpam-5896 115 27 all all DET ejpam-5896 115 28 ss ss ADJ ejpam-5896 115 29 - - ADJ ejpam-5896 115 30 open open ADJ ejpam-5896 115 31 subsets subset NOUN ejpam-5896 115 32 of of ADP ejpam-5896 115 33 ( ( PUNCT ejpam-5896 115 34 t,∆ t,∆ NUM ejpam-5896 115 35 ) ) PUNCT ejpam-5896 115 36 . . PUNCT ejpam-5896 116 1 additionally additionally ADV ejpam-5896 116 2 , , PUNCT ejpam-5896 116 3 ss ss NOUN ejpam-5896 116 4 - - NOUN ejpam-5896 116 5 closure closure NOUN ejpam-5896 116 6 of of ADP ejpam-5896 116 7 ( ( PUNCT ejpam-5896 116 8 t,∆ t,∆ X ejpam-5896 116 9 ) ) PUNCT ejpam-5896 116 10 is be AUX ejpam-5896 116 11 represented represent VERB ejpam-5896 116 12 as as ADP ejpam-5896 116 13 cls(t,∆ cls(t,∆ PROPN ejpam-5896 116 14 ) ) PUNCT ejpam-5896 116 15 , , PUNCT ejpam-5896 116 16 which which PRON ejpam-5896 116 17 is be AUX ejpam-5896 116 18 the the DET ejpam-5896 116 19 soft soft ADJ ejpam-5896 116 20 intersection intersection NOUN ejpam-5896 116 21 of of ADP ejpam-5896 116 22 all all DET ejpam-5896 116 23 supra supra PROPN ejpam-5896 116 24 closed close VERB ejpam-5896 116 25 soft soft ADJ ejpam-5896 116 26 supersets superset NOUN ejpam-5896 116 27 of of ADP ejpam-5896 116 28 ( ( PUNCT ejpam-5896 116 29 t,∆ t,∆ NUM ejpam-5896 116 30 ) ) PUNCT ejpam-5896 116 31 . . PUNCT ejpam-5896 117 1 moreover moreover ADV ejpam-5896 117 2 , , PUNCT ejpam-5896 117 3 ( ( PUNCT ejpam-5896 117 4 t,∆ t,∆ X ejpam-5896 117 5 ) ) PUNCT ejpam-5896 117 6 is be AUX ejpam-5896 117 7 referred refer VERB ejpam-5896 117 8 to to ADP ejpam-5896 117 9 as as ADP ejpam-5896 117 10 an an DET ejpam-5896 117 11 ss ss NOUN ejpam-5896 117 12 - - PUNCT ejpam-5896 117 13 open open ADJ ejpam-5896 117 14 set set NOUN ejpam-5896 117 15 if if SCONJ ejpam-5896 117 16 ( ( PUNCT ejpam-5896 117 17 t,∆ t,∆ X ejpam-5896 117 18 ) ) PUNCT ejpam-5896 117 19 ∈ ∈ PROPN ejpam-5896 117 20 µ µ NOUN ejpam-5896 117 21 , , PUNCT ejpam-5896 117 22 and and CCONJ ejpam-5896 117 23 its its PRON ejpam-5896 117 24 soft soft ADJ ejpam-5896 117 25 complement complement NOUN ejpam-5896 117 26 ( ( PUNCT ejpam-5896 117 27 t t PROPN ejpam-5896 117 28 c̃,∆ c̃,∆ PROPN ejpam-5896 117 29 ) ) PUNCT ejpam-5896 117 30 is be AUX ejpam-5896 117 31 referred refer VERB ejpam-5896 117 32 to to ADP ejpam-5896 117 33 as as ADP ejpam-5896 117 34 an an DET ejpam-5896 117 35 ss ss NOUN ejpam-5896 117 36 - - PUNCT ejpam-5896 117 37 closed closed ADJ ejpam-5896 117 38 set set NOUN ejpam-5896 117 39 . . PUNCT ejpam-5896 118 1 definition definition NOUN ejpam-5896 118 2 6 6 NUM ejpam-5896 118 3 . . PUNCT ejpam-5896 119 1 [ [ X ejpam-5896 119 2 51 51 NUM ejpam-5896 119 3 ] ] PUNCT ejpam-5896 119 4 consider consider VERB ejpam-5896 119 5 the the DET ejpam-5896 119 6 sts st NOUN ejpam-5896 119 7 ( ( PUNCT ejpam-5896 119 8 u u NOUN ejpam-5896 119 9 , , PUNCT ejpam-5896 119 10 τ,∆ τ,∆ NOUN ejpam-5896 119 11 ) ) PUNCT ejpam-5896 119 12 and and CCONJ ejpam-5896 119 13 the the DET ejpam-5896 119 14 ssts sst NOUN ejpam-5896 119 15 ( ( PUNCT ejpam-5896 119 16 u u NOUN ejpam-5896 119 17 , , PUNCT ejpam-5896 119 18 µ,∆ µ,∆ NUM ejpam-5896 119 19 ) ) PUNCT ejpam-5896 119 20 . . PUNCT ejpam-5896 120 1 if if SCONJ ejpam-5896 120 2 τ τ PROPN ejpam-5896 120 3 ⊂ ⊂ PROPN ejpam-5896 120 4 µ µ PROPN ejpam-5896 120 5 , , PUNCT ejpam-5896 120 6 then then ADV ejpam-5896 120 7 µ µ NOUN ejpam-5896 120 8 is be AUX ejpam-5896 120 9 an an DET ejpam-5896 120 10 ssts sst NOUN ejpam-5896 120 11 associated associate VERB ejpam-5896 120 12 with with ADP ejpam-5896 120 13 τ τ PROPN ejpam-5896 120 14 . . PUNCT ejpam-5896 121 1 abd abd PROPN ejpam-5896 121 2 el el PROPN ejpam-5896 121 3 - - PROPN ejpam-5896 121 4 latif latif PROPN ejpam-5896 121 5 et et PROPN ejpam-5896 121 6 al al PROPN ejpam-5896 121 7 . . PUNCT ejpam-5896 121 8 / / SYM ejpam-5896 121 9 eur eur PROPN ejpam-5896 121 10 . . PUNCT ejpam-5896 122 1 j. j. PROPN ejpam-5896 122 2 pure pure PROPN ejpam-5896 122 3 appl appl PROPN ejpam-5896 122 4 . . PROPN ejpam-5896 122 5 math math PROPN ejpam-5896 122 6 , , PUNCT ejpam-5896 122 7 18 18 NUM ejpam-5896 122 8 ( ( PUNCT ejpam-5896 122 9 2 2 NUM ejpam-5896 122 10 ) ) PUNCT ejpam-5896 122 11 ( ( PUNCT ejpam-5896 122 12 2025 2025 NUM ejpam-5896 122 13 ) ) PUNCT ejpam-5896 122 14 , , PUNCT ejpam-5896 122 15 5896 5896 NUM ejpam-5896 122 16 4 4 NUM ejpam-5896 122 17 of of ADP ejpam-5896 122 18 20 20 NUM ejpam-5896 122 19 definition definition NOUN ejpam-5896 122 20 7 7 NUM ejpam-5896 122 21 . . PUNCT ejpam-5896 123 1 [ [ X ejpam-5896 123 2 51 51 NUM ejpam-5896 123 3 ] ] PUNCT ejpam-5896 123 4 a a DET ejpam-5896 123 5 soft soft ADJ ejpam-5896 123 6 map map NOUN ejpam-5896 123 7 ψsd ψsd NOUN ejpam-5896 123 8 : : PUNCT ejpam-5896 123 9 ( ( PUNCT ejpam-5896 123 10 u u NOUN ejpam-5896 123 11 , , PUNCT ejpam-5896 123 12 τ,∆ τ,∆ NOUN ejpam-5896 123 13 ) ) PUNCT ejpam-5896 123 14 → → SYM ejpam-5896 123 15 ( ( PUNCT ejpam-5896 123 16 v v NOUN ejpam-5896 123 17 , , PUNCT ejpam-5896 123 18 σ σ PROPN ejpam-5896 123 19 , , PUNCT ejpam-5896 123 20 λ λ NOUN ejpam-5896 123 21 ) ) PUNCT ejpam-5896 123 22 with with ADP ejpam-5896 123 23 µ µ PRON ejpam-5896 123 24 an an DET ejpam-5896 123 25 associated associate VERB ejpam-5896 123 26 ssts sst NOUN ejpam-5896 123 27 with with ADP ejpam-5896 123 28 τ τ PROPN ejpam-5896 123 29 is be AUX ejpam-5896 123 30 claimed claim VERB ejpam-5896 123 31 to to PART ejpam-5896 123 32 be be AUX ejpam-5896 123 33 ss ss NOUN ejpam-5896 123 34 - - ADJ ejpam-5896 123 35 continuous continuous ADJ ejpam-5896 123 36 if if SCONJ ejpam-5896 123 37 ψ−1 ψ−1 PROPN ejpam-5896 123 38 sd sd ADP ejpam-5896 123 39 ( ( PUNCT ejpam-5896 123 40 g g NOUN ejpam-5896 123 41 , , PUNCT ejpam-5896 123 42 λ λ NOUN ejpam-5896 123 43 ) ) PUNCT ejpam-5896 123 44 ∈ ∈ PROPN ejpam-5896 123 45 µ µ X ejpam-5896 123 46 ∀ ∀ X ejpam-5896 123 47 ( ( PUNCT ejpam-5896 123 48 g g NOUN ejpam-5896 123 49 , , PUNCT ejpam-5896 123 50 λ λ NOUN ejpam-5896 123 51 ) ) PUNCT ejpam-5896 123 52 ∈ ∈ PROPN ejpam-5896 123 53 σ σ PROPN ejpam-5896 123 54 . . PUNCT ejpam-5896 123 55 definition definition NOUN ejpam-5896 123 56 8 8 NUM ejpam-5896 123 57 . . PUNCT ejpam-5896 124 1 [ [ X ejpam-5896 124 2 56 56 NUM ejpam-5896 124 3 ] ] PUNCT ejpam-5896 124 4 given give VERB ejpam-5896 124 5 a a DET ejpam-5896 124 6 soft soft ADJ ejpam-5896 124 7 set set NOUN ejpam-5896 124 8 ( ( PUNCT ejpam-5896 124 9 w,∆ w,∆ NOUN ejpam-5896 124 10 ) ) PUNCT ejpam-5896 124 11 ∈ ∈ PROPN ejpam-5896 124 12 s(u)∆. s(u)∆. INTJ ejpam-5896 124 13 if if SCONJ ejpam-5896 124 14 ints(cls(w,∆))eqφ̃ ints(cls(w,∆))eqφ̃ PROPN ejpam-5896 124 15 , , PUNCT ejpam-5896 124 16 then then ADV ejpam-5896 124 17 ( ( PUNCT ejpam-5896 124 18 w,∆ w,∆ NOUN ejpam-5896 124 19 ) ) PUNCT ejpam-5896 124 20 is be AUX ejpam-5896 124 21 called call VERB ejpam-5896 124 22 an an DET ejpam-5896 124 23 ss ss NOUN ejpam-5896 124 24 - - PUNCT ejpam-5896 124 25 sd sd NOUN ejpam-5896 124 26 - - PUNCT ejpam-5896 124 27 set set NOUN ejpam-5896 124 28 . . PUNCT ejpam-5896 125 1 additionally additionally ADV ejpam-5896 125 2 , , PUNCT ejpam-5896 125 3 ( ( PUNCT ejpam-5896 125 4 w w NOUN ejpam-5896 125 5 c̃,∆ c̃,∆ NOUN ejpam-5896 125 6 ) ) PUNCT ejpam-5896 125 7 is be AUX ejpam-5896 125 8 also also ADV ejpam-5896 125 9 known know VERB ejpam-5896 125 10 as as ADP ejpam-5896 125 11 ss ss PROPN ejpam-5896 125 12 - - PUNCT ejpam-5896 125 13 sc sc NOUN ejpam-5896 125 14 - - PUNCT ejpam-5896 125 15 set set NOUN ejpam-5896 125 16 . . PUNCT ejpam-5896 126 1 furthermore furthermore ADV ejpam-5896 126 2 , , PUNCT ejpam-5896 126 3 sd(u)∆ sd(u)∆ ADJ ejpam-5896 126 4 ( ( PUNCT ejpam-5896 126 5 or or CCONJ ejpam-5896 126 6 sc(u)∆ sc(u)∆ PROPN ejpam-5896 126 7 , , PUNCT ejpam-5896 126 8 respectively respectively ADV ejpam-5896 126 9 ) ) PUNCT ejpam-5896 126 10 will will AUX ejpam-5896 126 11 be be AUX ejpam-5896 126 12 used use VERB ejpam-5896 126 13 to to PART ejpam-5896 126 14 represent represent VERB ejpam-5896 126 15 the the DET ejpam-5896 126 16 category category NOUN ejpam-5896 126 17 of of ADP ejpam-5896 126 18 all all DET ejpam-5896 126 19 ss ss NOUN ejpam-5896 126 20 - - PUNCT ejpam-5896 126 21 sd sd NOUN ejpam-5896 126 22 - - PUNCT ejpam-5896 126 23 sets set NOUN ejpam-5896 126 24 ( ( PUNCT ejpam-5896 126 25 or or CCONJ ejpam-5896 126 26 ss ss PROPN ejpam-5896 126 27 - - PUNCT ejpam-5896 126 28 sc sc NOUN ejpam-5896 126 29 - - PUNCT ejpam-5896 126 30 sets set NOUN ejpam-5896 126 31 ) ) PUNCT ejpam-5896 126 32 . . PUNCT ejpam-5896 127 1 moreover moreover ADV ejpam-5896 127 2 , , PUNCT ejpam-5896 127 3 ( ( PUNCT ejpam-5896 127 4 w,∆ w,∆ NOUN ejpam-5896 127 5 ) ) PUNCT ejpam-5896 127 6 is be AUX ejpam-5896 127 7 referred refer VERB ejpam-5896 127 8 to to ADP ejpam-5896 127 9 as as ADP ejpam-5896 127 10 ss ss NOUN ejpam-5896 127 11 - - PUNCT ejpam-5896 127 12 nowhere nowhere ADV ejpam-5896 127 13 dense dense ADJ ejpam-5896 127 14 if if SCONJ ejpam-5896 127 15 it it PRON ejpam-5896 127 16 is be AUX ejpam-5896 127 17 not not PART ejpam-5896 127 18 ss ss NOUN ejpam-5896 127 19 - - PUNCT ejpam-5896 127 20 sd sd NOUN ejpam-5896 127 21 - - PUNCT ejpam-5896 127 22 set set NOUN ejpam-5896 127 23 . . PUNCT ejpam-5896 128 1 finally finally ADV ejpam-5896 128 2 , , PUNCT ejpam-5896 128 3 if if SCONJ ejpam-5896 128 4 ( ( PUNCT ejpam-5896 128 5 w,∆ w,∆ NOUN ejpam-5896 128 6 ) ) PUNCT ejpam-5896 128 7 is be AUX ejpam-5896 128 8 both both DET ejpam-5896 128 9 ss ss NOUN ejpam-5896 128 10 - - PUNCT ejpam-5896 128 11 sd sd NOUN ejpam-5896 128 12 - - PUNCT ejpam-5896 128 13 set set VERB ejpam-5896 128 14 and and CCONJ ejpam-5896 128 15 ss ss NOUN ejpam-5896 128 16 - - ADJ ejpam-5896 128 17 sc sc NOUN ejpam-5896 128 18 - - PUNCT ejpam-5896 128 19 set set NOUN ejpam-5896 128 20 , , PUNCT ejpam-5896 128 21 then then ADV ejpam-5896 128 22 it it PRON ejpam-5896 128 23 is be AUX ejpam-5896 128 24 called call VERB ejpam-5896 128 25 ss ss NOUN ejpam-5896 128 26 - - PUNCT ejpam-5896 128 27 sdc sdc NOUN ejpam-5896 128 28 - - PUNCT ejpam-5896 128 29 set set NOUN ejpam-5896 128 30 . . PUNCT ejpam-5896 129 1 corollary corollary ADJ ejpam-5896 129 2 1 1 NUM ejpam-5896 129 3 . . PUNCT ejpam-5896 130 1 [ [ X ejpam-5896 130 2 56 56 NUM ejpam-5896 130 3 ] ] PUNCT ejpam-5896 130 4 any any DET ejpam-5896 130 5 ss ss PROPN ejpam-5896 130 6 - - ADJ ejpam-5896 130 7 sc sc NOUN ejpam-5896 130 8 - - PUNCT ejpam-5896 130 9 set set VERB ejpam-5896 130 10 ( ( PUNCT ejpam-5896 130 11 ss ss NOUN ejpam-5896 130 12 - - PUNCT ejpam-5896 130 13 sd sd NOUN ejpam-5896 130 14 - - PUNCT ejpam-5896 130 15 set set NOUN ejpam-5896 130 16 ) ) PUNCT ejpam-5896 130 17 has have VERB ejpam-5896 130 18 a a DET ejpam-5896 130 19 soft soft ADJ ejpam-5896 130 20 subset subset NOUN ejpam-5896 130 21 ( ( PUNCT ejpam-5896 130 22 superset superset NOUN ejpam-5896 130 23 ) ) PUNCT ejpam-5896 130 24 that that PRON ejpam-5896 130 25 is be AUX ejpam-5896 130 26 also also ADV ejpam-5896 130 27 ss ss PROPN ejpam-5896 130 28 - - ADJ ejpam-5896 130 29 sc sc NOUN ejpam-5896 130 30 - - PUNCT ejpam-5896 130 31 set set VERB ejpam-5896 130 32 ( ( PUNCT ejpam-5896 130 33 ss ss NOUN ejpam-5896 130 34 - - PUNCT ejpam-5896 130 35 sd sd NOUN ejpam-5896 130 36 - - PUNCT ejpam-5896 130 37 set set NOUN ejpam-5896 130 38 ) ) PUNCT ejpam-5896 130 39 . . PUNCT ejpam-5896 131 1 definition definition NOUN ejpam-5896 131 2 9 9 NUM ejpam-5896 131 3 . . PUNCT ejpam-5896 132 1 [ [ X ejpam-5896 132 2 56 56 NUM ejpam-5896 132 3 ] ] PUNCT ejpam-5896 132 4 the the DET ejpam-5896 132 5 greatest great ADJ ejpam-5896 132 6 ss ss NOUN ejpam-5896 132 7 - - PUNCT ejpam-5896 132 8 sd sd NOUN ejpam-5896 132 9 - - PUNCT ejpam-5896 132 10 subsets subset NOUN ejpam-5896 132 11 of of ADP ejpam-5896 132 12 ( ( PUNCT ejpam-5896 132 13 t,∆ t,∆ X ejpam-5896 132 14 ) ) PUNCT ejpam-5896 132 15 are be AUX ejpam-5896 132 16 represented represent VERB ejpam-5896 132 17 by by ADP ejpam-5896 132 18 intssd(t,∆ intssd(t,∆ NOUN ejpam-5896 132 19 ) ) PUNCT ejpam-5896 132 20 , , PUNCT ejpam-5896 132 21 which which PRON ejpam-5896 132 22 is be AUX ejpam-5896 132 23 the the DET ejpam-5896 132 24 ss ss NOUN ejpam-5896 132 25 - - PUNCT ejpam-5896 132 26 sd sd NOUN ejpam-5896 132 27 - - PUNCT ejpam-5896 132 28 interior interior NOUN ejpam-5896 132 29 of of ADP ejpam-5896 132 30 a a DET ejpam-5896 132 31 non non ADJ ejpam-5896 132 32 - - ADJ ejpam-5896 132 33 null null ADJ ejpam-5896 132 34 soft soft ADJ ejpam-5896 132 35 subset subset NOUN ejpam-5896 132 36 ( ( PUNCT ejpam-5896 132 37 t,∆ t,∆ NUM ejpam-5896 132 38 ) ) PUNCT ejpam-5896 132 39 of of ADP ejpam-5896 132 40 an an DET ejpam-5896 132 41 ssts sst NOUN ejpam-5896 132 42 ( ( PUNCT ejpam-5896 132 43 u u NOUN ejpam-5896 132 44 , , PUNCT ejpam-5896 132 45 µ,∆ µ,∆ NUM ejpam-5896 132 46 ) ) PUNCT ejpam-5896 132 47 . . PUNCT ejpam-5896 133 1 additionally additionally ADV ejpam-5896 133 2 , , PUNCT ejpam-5896 133 3 clssd(t,∆ clssd(t,∆ ADJ ejpam-5896 133 4 ) ) PUNCT ejpam-5896 133 5 , , PUNCT ejpam-5896 133 6 the the DET ejpam-5896 133 7 smallest small ADJ ejpam-5896 133 8 ss ss ADJ ejpam-5896 133 9 - - ADJ ejpam-5896 133 10 sc sc NOUN ejpam-5896 133 11 - - PUNCT ejpam-5896 133 12 superset superset NOUN ejpam-5896 133 13 of of ADP ejpam-5896 133 14 ( ( PUNCT ejpam-5896 133 15 t,∆ t,∆ NUM ejpam-5896 133 16 ) ) PUNCT ejpam-5896 133 17 , , PUNCT ejpam-5896 133 18 represents represent VERB ejpam-5896 133 19 the the DET ejpam-5896 133 20 ss ss NOUN ejpam-5896 133 21 - - PUNCT ejpam-5896 133 22 sd sd NOUN ejpam-5896 133 23 - - PUNCT ejpam-5896 133 24 closure closure NOUN ejpam-5896 133 25 of of ADP ejpam-5896 133 26 ( ( PUNCT ejpam-5896 133 27 t,∆ t,∆ NUM ejpam-5896 133 28 ) ) PUNCT ejpam-5896 133 29 . . PUNCT ejpam-5896 134 1 theorem theorem NOUN ejpam-5896 134 2 2 2 NUM ejpam-5896 134 3 . . PUNCT ejpam-5896 135 1 [ [ X ejpam-5896 135 2 56 56 NUM ejpam-5896 135 3 ] ] PUNCT ejpam-5896 135 4 given give VERB ejpam-5896 135 5 a a DET ejpam-5896 135 6 soft soft ADJ ejpam-5896 135 7 set set NOUN ejpam-5896 135 8 ( ( PUNCT ejpam-5896 135 9 t,∆ t,∆ NUM ejpam-5896 135 10 ) ) PUNCT ejpam-5896 135 11 ∈ ∈ PROPN ejpam-5896 135 12 s(u)∆ s(u)∆ NOUN ejpam-5896 135 13 , , PUNCT ejpam-5896 135 14 we we PRON ejpam-5896 135 15 have have VERB ejpam-5896 135 16 that that PRON ejpam-5896 135 17 ( ( PUNCT ejpam-5896 135 18 1 1 X ejpam-5896 135 19 ) ) PUNCT ejpam-5896 135 20 clssd(t clssd(t NOUN ejpam-5896 135 21 c̃,∆ c̃,∆ NOUN ejpam-5896 135 22 ) ) PUNCT ejpam-5896 135 23 = = PUNCT ejpam-5896 136 1 [ [ X ejpam-5896 136 2 intssd(t,∆)]c̃ intssd(t,∆)]c̃ NOUN ejpam-5896 136 3 and and CCONJ ejpam-5896 136 4 intssd(t intssd(t NOUN ejpam-5896 136 5 c̃,∆ c̃,∆ NOUN ejpam-5896 136 6 ) ) PUNCT ejpam-5896 136 7 = = PUNCT ejpam-5896 137 1 [ [ X ejpam-5896 137 2 clssd(t,∆)]c̃. clssd(t,∆)]c̃. PROPN ejpam-5896 137 3 ( ( PUNCT ejpam-5896 137 4 2 2 NUM ejpam-5896 137 5 ) ) PUNCT ejpam-5896 137 6 clssd(t,∆)⊆̃cls(t,∆ clssd(t,∆)⊆̃cls(t,∆ NOUN ejpam-5896 137 7 ) ) PUNCT ejpam-5896 137 8 . . PUNCT ejpam-5896 138 1 ( ( PUNCT ejpam-5896 138 2 3 3 X ejpam-5896 138 3 ) ) PUNCT ejpam-5896 138 4 ints(t,∆)⊆̃intssd(t,∆ ints(t,∆)⊆̃intssd(t,∆ ADJ ejpam-5896 138 5 ) ) PUNCT ejpam-5896 138 6 . . PUNCT ejpam-5896 139 1 definition definition NOUN ejpam-5896 139 2 10 10 NUM ejpam-5896 139 3 . . PUNCT ejpam-5896 140 1 [ [ X ejpam-5896 140 2 62]given 62]given NUM ejpam-5896 140 3 ( ( PUNCT ejpam-5896 140 4 g,∆ g,∆ PROPN ejpam-5896 140 5 ) ) PUNCT ejpam-5896 140 6 ̸= ̸= PROPN ejpam-5896 140 7 φ̃ φ̃ PROPN ejpam-5896 140 8 and and CCONJ ejpam-5896 140 9 ( ( PUNCT ejpam-5896 140 10 h,∆ h,∆ ADJ ejpam-5896 140 11 ) ) PUNCT ejpam-5896 140 12 ̸= ̸= PROPN ejpam-5896 140 13 φ̃. φ̃. PROPN ejpam-5896 140 14 if if SCONJ ejpam-5896 140 15 ( ( PUNCT ejpam-5896 140 16 g,∆)∩̃clssd(h,∆ g,∆)∩̃clssd(h,∆ PROPN ejpam-5896 140 17 ) ) PUNCT ejpam-5896 140 18 = = SYM ejpam-5896 140 19 φ̃ φ̃ PROPN ejpam-5896 140 20 and and CCONJ ejpam-5896 140 21 clssd(g,∆)∩̃((h,∆ clssd(g,∆)∩̃((h,∆ NOUN ejpam-5896 140 22 ) ) PUNCT ejpam-5896 140 23 ) ) PUNCT ejpam-5896 141 1 = = PUNCT ejpam-5896 141 2 φ̃ φ̃ PROPN ejpam-5896 141 3 , , PUNCT ejpam-5896 141 4 then then ADV ejpam-5896 141 5 ( ( PUNCT ejpam-5896 141 6 g,∆ g,∆ PROPN ejpam-5896 141 7 ) ) PUNCT ejpam-5896 141 8 , , PUNCT ejpam-5896 141 9 ( ( PUNCT ejpam-5896 141 10 h,∆ h,∆ X ejpam-5896 141 11 ) ) PUNCT ejpam-5896 141 12 are be AUX ejpam-5896 141 13 called call VERB ejpam-5896 141 14 ss ss NOUN ejpam-5896 141 15 - - PUNCT ejpam-5896 141 16 sd sd NOUN ejpam-5896 141 17 - - PUNCT ejpam-5896 141 18 separated separate VERB ejpam-5896 141 19 . . PUNCT ejpam-5896 142 1 definition definition NOUN ejpam-5896 142 2 11 11 NUM ejpam-5896 142 3 . . PUNCT ejpam-5896 143 1 [ [ X ejpam-5896 143 2 62]given 62]given NUM ejpam-5896 143 3 ( ( PUNCT ejpam-5896 143 4 g,∆ g,∆ PROPN ejpam-5896 143 5 ) ) PUNCT ejpam-5896 143 6 ̸= ̸= PROPN ejpam-5896 143 7 φ̃ φ̃ PROPN ejpam-5896 143 8 and and CCONJ ejpam-5896 143 9 ( ( PUNCT ejpam-5896 143 10 h,∆ h,∆ ADJ ejpam-5896 143 11 ) ) PUNCT ejpam-5896 143 12 ̸= ̸= PROPN ejpam-5896 143 13 φ̃ φ̃ PROPN ejpam-5896 143 14 are be AUX ejpam-5896 143 15 ss ss NOUN ejpam-5896 143 16 - - PUNCT ejpam-5896 143 17 sd sd NOUN ejpam-5896 143 18 - - PUNCT ejpam-5896 143 19 separated separate VERB ejpam-5896 143 20 sets set NOUN ejpam-5896 143 21 . . PUNCT ejpam-5896 144 1 if if SCONJ ejpam-5896 144 2 ( ( PUNCT ejpam-5896 144 3 g,∆ g,∆ X ejpam-5896 144 4 ) ) PUNCT ejpam-5896 144 5 and and CCONJ ejpam-5896 144 6 ( ( PUNCT ejpam-5896 144 7 h,∆ h,∆ X ejpam-5896 144 8 ) ) PUNCT ejpam-5896 144 9 can can AUX ejpam-5896 144 10 not not PART ejpam-5896 144 11 be be AUX ejpam-5896 144 12 expressed express VERB ejpam-5896 144 13 as as ADP ejpam-5896 144 14 a a DET ejpam-5896 144 15 soft soft ADJ ejpam-5896 144 16 union union NOUN ejpam-5896 144 17 of of ADP ejpam-5896 144 18 ũ ũ PROPN ejpam-5896 144 19 , , PUNCT ejpam-5896 144 20 then then ADV ejpam-5896 144 21 the the DET ejpam-5896 144 22 ssts sst NOUN ejpam-5896 144 23 ( ( PUNCT ejpam-5896 144 24 u u NOUN ejpam-5896 144 25 , , PUNCT ejpam-5896 144 26 µ,∆ µ,∆ NUM ejpam-5896 144 27 ) ) PUNCT ejpam-5896 144 28 is be AUX ejpam-5896 144 29 considered consider VERB ejpam-5896 144 30 as as ADP ejpam-5896 144 31 an an DET ejpam-5896 144 32 ss ss NOUN ejpam-5896 144 33 - - PUNCT ejpam-5896 144 34 sd sd NOUN ejpam-5896 144 35 - - PUNCT ejpam-5896 144 36 connected connect VERB ejpam-5896 144 37 . . PUNCT ejpam-5896 145 1 if if SCONJ ejpam-5896 145 2 not not PART ejpam-5896 145 3 , , PUNCT ejpam-5896 145 4 ( ( PUNCT ejpam-5896 145 5 u u NOUN ejpam-5896 145 6 , , PUNCT ejpam-5896 145 7 µ,∆ µ,∆ NUM ejpam-5896 145 8 ) ) PUNCT ejpam-5896 145 9 is be AUX ejpam-5896 145 10 considered consider VERB ejpam-5896 145 11 as as ADP ejpam-5896 145 12 an an DET ejpam-5896 145 13 ss ss NOUN ejpam-5896 145 14 - - PUNCT ejpam-5896 145 15 sd sd NOUN ejpam-5896 145 16 - - PUNCT ejpam-5896 145 17 disconnected disconnected ADJ ejpam-5896 145 18 . . PUNCT ejpam-5896 146 1 additionally additionally ADV ejpam-5896 146 2 , , PUNCT ejpam-5896 146 3 if if SCONJ ejpam-5896 146 4 ỹ ỹ PROPN ejpam-5896 146 5 is be AUX ejpam-5896 146 6 ss ss NOUN ejpam-5896 146 7 - - PUNCT ejpam-5896 146 8 sd sd NOUN ejpam-5896 146 9 - - PUNCT ejpam-5896 146 10 connected connect VERB ejpam-5896 146 11 subspace subspace NOUN ejpam-5896 146 12 of of ADP ejpam-5896 146 13 ũ ũ PROPN ejpam-5896 146 14 , , PUNCT ejpam-5896 146 15 then then ADV ejpam-5896 146 16 it it PRON ejpam-5896 146 17 is be AUX ejpam-5896 146 18 ss ss NOUN ejpam-5896 146 19 - - PUNCT ejpam-5896 146 20 sd sd NOUN ejpam-5896 146 21 - - PUNCT ejpam-5896 146 22 connected connect VERB ejpam-5896 146 23 . . PUNCT ejpam-5896 147 1 definition definition NOUN ejpam-5896 147 2 12 12 NUM ejpam-5896 147 3 . . PUNCT ejpam-5896 148 1 [ [ X ejpam-5896 148 2 57 57 NUM ejpam-5896 148 3 ] ] PUNCT ejpam-5896 148 4 given give VERB ejpam-5896 148 5 µ µ NOUN ejpam-5896 148 6 , , PUNCT ejpam-5896 148 7 µ∗ µ∗ VERB ejpam-5896 148 8 as as SCONJ ejpam-5896 148 9 associated associated ADJ ejpam-5896 148 10 sstss sstss NOUN ejpam-5896 148 11 with with ADP ejpam-5896 148 12 τ τ PROPN ejpam-5896 148 13 , , PUNCT ejpam-5896 148 14 σ σ PROPN ejpam-5896 148 15 , , PUNCT ejpam-5896 148 16 respectively respectively ADV ejpam-5896 148 17 , , PUNCT ejpam-5896 148 18 a a DET ejpam-5896 148 19 soft soft ADJ ejpam-5896 148 20 map map NOUN ejpam-5896 148 21 ψsd ψsd NOUN ejpam-5896 148 22 : : PUNCT ejpam-5896 148 23 ( ( PUNCT ejpam-5896 148 24 u u NOUN ejpam-5896 148 25 , , PUNCT ejpam-5896 148 26 τ,∆ τ,∆ NOUN ejpam-5896 148 27 ) ) PUNCT ejpam-5896 148 28 → → SYM ejpam-5896 148 29 ( ( PUNCT ejpam-5896 148 30 v v NOUN ejpam-5896 148 31 , , PUNCT ejpam-5896 148 32 σ σ PROPN ejpam-5896 148 33 , , PUNCT ejpam-5896 148 34 λ λ PROPN ejpam-5896 148 35 ) ) PUNCT ejpam-5896 148 36 is be AUX ejpam-5896 148 37 stated state VERB ejpam-5896 148 38 to to PART ejpam-5896 148 39 be be AUX ejpam-5896 148 40 : : PUNCT ejpam-5896 148 41 ( ( PUNCT ejpam-5896 148 42 1 1 X ejpam-5896 148 43 ) ) PUNCT ejpam-5896 148 44 ss ss NOUN ejpam-5896 148 45 - - PUNCT ejpam-5896 148 46 sd sd NOUN ejpam-5896 148 47 - - PUNCT ejpam-5896 148 48 continuous continuous ADJ ejpam-5896 148 49 if if SCONJ ejpam-5896 148 50 and and CCONJ ejpam-5896 148 51 only only ADV ejpam-5896 148 52 if if SCONJ ejpam-5896 148 53 either either CCONJ ejpam-5896 148 54 ψ−1 ψ−1 PROPN ejpam-5896 148 55 sd sd PROPN ejpam-5896 148 56 ( ( PUNCT ejpam-5896 148 57 t t PROPN ejpam-5896 148 58 , , PUNCT ejpam-5896 148 59 λ λ NOUN ejpam-5896 148 60 ) ) PUNCT ejpam-5896 148 61 = = SYM ejpam-5896 148 62 φ̃ φ̃ PROPN ejpam-5896 148 63 or or CCONJ ejpam-5896 148 64 ψ−1 ψ−1 PROPN ejpam-5896 148 65 sd sd PROPN ejpam-5896 148 66 ( ( PUNCT ejpam-5896 148 67 t t PROPN ejpam-5896 148 68 , , PUNCT ejpam-5896 148 69 λ λ NOUN ejpam-5896 148 70 ) ) PUNCT ejpam-5896 148 71 ∈ ∈ PROPN ejpam-5896 148 72 sd(u)∆ sd(u)∆ PROPN ejpam-5896 148 73 , , PUNCT ejpam-5896 148 74 ∀(t ∀(t NUM ejpam-5896 148 75 , , PUNCT ejpam-5896 148 76 λ λ NOUN ejpam-5896 148 77 ) ) PUNCT ejpam-5896 148 78 ∈ ∈ PROPN ejpam-5896 148 79 σ σ PROPN ejpam-5896 148 80 . . PUNCT ejpam-5896 149 1 ( ( PUNCT ejpam-5896 149 2 2 2 X ejpam-5896 149 3 ) ) PUNCT ejpam-5896 149 4 ss ss NOUN ejpam-5896 149 5 - - PUNCT ejpam-5896 149 6 sd sd NOUN ejpam-5896 149 7 - - PUNCT ejpam-5896 149 8 irresolute irresolute ADJ ejpam-5896 150 1 if if SCONJ ejpam-5896 150 2 either either CCONJ ejpam-5896 150 3 ψ−1 ψ−1 PROPN ejpam-5896 150 4 sd sd PROPN ejpam-5896 150 5 ( ( PUNCT ejpam-5896 150 6 t t PROPN ejpam-5896 150 7 , , PUNCT ejpam-5896 150 8 λ λ NOUN ejpam-5896 150 9 ) ) PUNCT ejpam-5896 150 10 = = SYM ejpam-5896 151 1 φ̃ φ̃ PROPN ejpam-5896 151 2 or or CCONJ ejpam-5896 151 3 ψ−1 ψ−1 PROPN ejpam-5896 151 4 sd sd PROPN ejpam-5896 151 5 ( ( PUNCT ejpam-5896 151 6 t t PROPN ejpam-5896 151 7 , , PUNCT ejpam-5896 151 8 λ λ NOUN ejpam-5896 151 9 ) ) PUNCT ejpam-5896 151 10 ∈ ∈ PROPN ejpam-5896 151 11 sd(u)∆ sd(u)∆ PROPN ejpam-5896 151 12 , , PUNCT ejpam-5896 151 13 ∀(t ∀(t NUM ejpam-5896 151 14 , , PUNCT ejpam-5896 151 15 λ λ NOUN ejpam-5896 151 16 ) ) PUNCT ejpam-5896 151 17 ∈ ∈ PROPN ejpam-5896 151 18 sd(v sd(v PUNCT ejpam-5896 151 19 ) ) PUNCT ejpam-5896 151 20 λ λ NOUN ejpam-5896 151 21 . . PUNCT ejpam-5896 151 22 definition definition NOUN ejpam-5896 151 23 13 13 NUM ejpam-5896 151 24 . . PUNCT ejpam-5896 152 1 [ [ X ejpam-5896 152 2 62]given 62]given NUM ejpam-5896 152 3 ( ( PUNCT ejpam-5896 152 4 g,∆ g,∆ PROPN ejpam-5896 152 5 ) ) PUNCT ejpam-5896 152 6 ̸= ̸= PROPN ejpam-5896 152 7 φ̃ φ̃ PROPN ejpam-5896 152 8 and and CCONJ ejpam-5896 152 9 ( ( PUNCT ejpam-5896 152 10 h,∆ h,∆ ADJ ejpam-5896 152 11 ) ) PUNCT ejpam-5896 152 12 ̸= ̸= PROPN ejpam-5896 152 13 φ̃ φ̃ PROPN ejpam-5896 152 14 are be AUX ejpam-5896 152 15 ss ss NOUN ejpam-5896 152 16 - - PUNCT ejpam-5896 152 17 sd sd NOUN ejpam-5896 152 18 - - PUNCT ejpam-5896 152 19 separated separate VERB ejpam-5896 152 20 sets set NOUN ejpam-5896 152 21 . . PUNCT ejpam-5896 153 1 if if SCONJ ejpam-5896 153 2 ( ( PUNCT ejpam-5896 153 3 g,∆ g,∆ X ejpam-5896 153 4 ) ) PUNCT ejpam-5896 153 5 and and CCONJ ejpam-5896 153 6 ( ( PUNCT ejpam-5896 153 7 h,∆ h,∆ X ejpam-5896 153 8 ) ) PUNCT ejpam-5896 153 9 can can AUX ejpam-5896 153 10 not not PART ejpam-5896 153 11 be be AUX ejpam-5896 153 12 expressed express VERB ejpam-5896 153 13 as as ADP ejpam-5896 153 14 a a DET ejpam-5896 153 15 soft soft ADJ ejpam-5896 153 16 union union NOUN ejpam-5896 153 17 of of ADP ejpam-5896 153 18 ũ ũ PROPN ejpam-5896 153 19 , , PUNCT ejpam-5896 153 20 then then ADV ejpam-5896 153 21 the the DET ejpam-5896 153 22 ssts sst NOUN ejpam-5896 153 23 ( ( PUNCT ejpam-5896 153 24 u u NOUN ejpam-5896 153 25 , , PUNCT ejpam-5896 153 26 µ,∆ µ,∆ NUM ejpam-5896 153 27 ) ) PUNCT ejpam-5896 153 28 is be AUX ejpam-5896 153 29 considered consider VERB ejpam-5896 153 30 as as ADP ejpam-5896 153 31 an an DET ejpam-5896 153 32 ss ss NOUN ejpam-5896 153 33 - - PUNCT ejpam-5896 153 34 sd sd NOUN ejpam-5896 153 35 - - PUNCT ejpam-5896 153 36 connected connect VERB ejpam-5896 153 37 . . PUNCT ejpam-5896 154 1 if if SCONJ ejpam-5896 154 2 not not PART ejpam-5896 154 3 , , PUNCT ejpam-5896 154 4 ( ( PUNCT ejpam-5896 154 5 u u NOUN ejpam-5896 154 6 , , PUNCT ejpam-5896 154 7 µ,∆ µ,∆ NUM ejpam-5896 154 8 ) ) PUNCT ejpam-5896 154 9 is be AUX ejpam-5896 154 10 considered consider VERB ejpam-5896 154 11 as as ADP ejpam-5896 154 12 an an DET ejpam-5896 154 13 ss ss NOUN ejpam-5896 154 14 - - PUNCT ejpam-5896 154 15 sd sd NOUN ejpam-5896 154 16 - - PUNCT ejpam-5896 154 17 disconnected disconnected ADJ ejpam-5896 154 18 . . PUNCT ejpam-5896 155 1 additionally additionally ADV ejpam-5896 155 2 , , PUNCT ejpam-5896 155 3 if if SCONJ ejpam-5896 155 4 ỹ ỹ PROPN ejpam-5896 155 5 is be AUX ejpam-5896 155 6 ss ss NOUN ejpam-5896 155 7 - - PUNCT ejpam-5896 155 8 sd sd NOUN ejpam-5896 155 9 - - PUNCT ejpam-5896 155 10 connected connect VERB ejpam-5896 155 11 subspace subspace NOUN ejpam-5896 155 12 of of ADP ejpam-5896 155 13 ũ ũ PROPN ejpam-5896 155 14 , , PUNCT ejpam-5896 155 15 then then ADV ejpam-5896 155 16 it it PRON ejpam-5896 155 17 is be AUX ejpam-5896 155 18 ss ss NOUN ejpam-5896 155 19 - - PUNCT ejpam-5896 155 20 sd sd NOUN ejpam-5896 155 21 - - PUNCT ejpam-5896 155 22 connected connect VERB ejpam-5896 155 23 . . PUNCT ejpam-5896 156 1 theorem theorem VERB ejpam-5896 156 2 3 3 NUM ejpam-5896 156 3 . . PUNCT ejpam-5896 157 1 [ [ X ejpam-5896 157 2 62 62 NUM ejpam-5896 157 3 ] ] PUNCT ejpam-5896 157 4 the the DET ejpam-5896 157 5 following follow VERB ejpam-5896 157 6 characteristics characteristic NOUN ejpam-5896 157 7 are be AUX ejpam-5896 157 8 equivalent equivalent ADJ ejpam-5896 157 9 for for ADP ejpam-5896 157 10 any any DET ejpam-5896 157 11 ssts sst NOUN ejpam-5896 157 12 ( ( PUNCT ejpam-5896 157 13 u u NOUN ejpam-5896 157 14 , , PUNCT ejpam-5896 157 15 µ,∆ µ,∆ NUM ejpam-5896 157 16 ): ): PUNCT ejpam-5896 157 17 ( ( PUNCT ejpam-5896 157 18 1 1 X ejpam-5896 157 19 ) ) PUNCT ejpam-5896 157 20 ũ ũ PROPN ejpam-5896 157 21 is be AUX ejpam-5896 157 22 ss ss NOUN ejpam-5896 157 23 - - PUNCT ejpam-5896 157 24 sd sd NOUN ejpam-5896 157 25 - - PUNCT ejpam-5896 157 26 connected connect VERB ejpam-5896 157 27 . . PUNCT ejpam-5896 158 1 ( ( PUNCT ejpam-5896 158 2 2 2 X ejpam-5896 158 3 ) ) PUNCT ejpam-5896 158 4 the the DET ejpam-5896 158 5 soft soft ADJ ejpam-5896 158 6 union union NOUN ejpam-5896 158 7 of of ADP ejpam-5896 158 8 any any DET ejpam-5896 158 9 two two NUM ejpam-5896 158 10 disjoint disjoint ADJ ejpam-5896 158 11 ss ss NOUN ejpam-5896 158 12 - - PUNCT ejpam-5896 158 13 sd sd NOUN ejpam-5896 158 14 - - PUNCT ejpam-5896 158 15 sets set NOUN ejpam-5896 158 16 can can AUX ejpam-5896 158 17 not not PART ejpam-5896 158 18 be be AUX ejpam-5896 158 19 expressed express VERB ejpam-5896 158 20 as as ADP ejpam-5896 158 21 ũ ũ PROPN ejpam-5896 158 22 . . PUNCT ejpam-5896 159 1 ( ( PUNCT ejpam-5896 159 2 3 3 X ejpam-5896 159 3 ) ) PUNCT ejpam-5896 159 4 the the DET ejpam-5896 159 5 soft soft ADJ ejpam-5896 159 6 union union NOUN ejpam-5896 159 7 of of ADP ejpam-5896 159 8 any any DET ejpam-5896 159 9 two two NUM ejpam-5896 159 10 disjoint disjoint NOUN ejpam-5896 159 11 ss ss PROPN ejpam-5896 159 12 - - ADJ ejpam-5896 159 13 sc sc NOUN ejpam-5896 159 14 - - PUNCT ejpam-5896 159 15 sets set NOUN ejpam-5896 159 16 can can AUX ejpam-5896 159 17 not not PART ejpam-5896 159 18 be be AUX ejpam-5896 159 19 expressed express VERB ejpam-5896 159 20 as as ADP ejpam-5896 159 21 ũ ũ PROPN ejpam-5896 159 22 . . PUNCT ejpam-5896 160 1 ( ( PUNCT ejpam-5896 160 2 4 4 X ejpam-5896 160 3 ) ) PUNCT ejpam-5896 160 4 there there PRON ejpam-5896 160 5 is be VERB ejpam-5896 160 6 n’t not PART ejpam-5896 160 7 a a DET ejpam-5896 160 8 proper proper ADJ ejpam-5896 160 9 ss ss NOUN ejpam-5896 160 10 - - PUNCT ejpam-5896 160 11 sdc sdc NOUN ejpam-5896 160 12 - - PUNCT ejpam-5896 160 13 subset subset NOUN ejpam-5896 160 14 of of ADP ejpam-5896 160 15 ũ ũ PROPN ejpam-5896 160 16 . . PUNCT ejpam-5896 161 1 ( ( PUNCT ejpam-5896 161 2 5 5 X ejpam-5896 161 3 ) ) PUNCT ejpam-5896 161 4 the the DET ejpam-5896 161 5 soft soft ADJ ejpam-5896 161 6 union union NOUN ejpam-5896 161 7 of of ADP ejpam-5896 161 8 any any DET ejpam-5896 161 9 two two NUM ejpam-5896 161 10 non non ADJ ejpam-5896 161 11 - - ADJ ejpam-5896 161 12 null null ADJ ejpam-5896 161 13 ss ss PROPN ejpam-5896 161 14 - - PUNCT ejpam-5896 161 15 sd sd NOUN ejpam-5896 161 16 - - PUNCT ejpam-5896 161 17 separated separate VERB ejpam-5896 161 18 sets set NOUN ejpam-5896 161 19 can can AUX ejpam-5896 161 20 not not PART ejpam-5896 161 21 be be AUX ejpam-5896 161 22 expressed express VERB ejpam-5896 161 23 as as ADP ejpam-5896 161 24 ũ ũ PROPN ejpam-5896 161 25 . . PUNCT ejpam-5896 162 1 abd abd PROPN ejpam-5896 162 2 el el PROPN ejpam-5896 162 3 - - PROPN ejpam-5896 162 4 latif latif PROPN ejpam-5896 162 5 et et PROPN ejpam-5896 162 6 al al PROPN ejpam-5896 162 7 . . PUNCT ejpam-5896 162 8 / / SYM ejpam-5896 162 9 eur eur PROPN ejpam-5896 162 10 . . PUNCT ejpam-5896 163 1 j. j. PROPN ejpam-5896 163 2 pure pure PROPN ejpam-5896 163 3 appl appl PROPN ejpam-5896 163 4 . . PROPN ejpam-5896 163 5 math math PROPN ejpam-5896 163 6 , , PUNCT ejpam-5896 163 7 18 18 NUM ejpam-5896 163 8 ( ( PUNCT ejpam-5896 163 9 2 2 NUM ejpam-5896 163 10 ) ) PUNCT ejpam-5896 163 11 ( ( PUNCT ejpam-5896 163 12 2025 2025 NUM ejpam-5896 163 13 ) ) PUNCT ejpam-5896 163 14 , , PUNCT ejpam-5896 163 15 5896 5896 NUM ejpam-5896 163 16 5 5 NUM ejpam-5896 163 17 of of ADP ejpam-5896 163 18 20 20 NUM ejpam-5896 163 19 3 3 NUM ejpam-5896 163 20 . . PUNCT ejpam-5896 163 21 novel novel ADJ ejpam-5896 163 22 generalized generalize VERB ejpam-5896 163 23 connectedness connectedness NOUN ejpam-5896 163 24 types type NOUN ejpam-5896 163 25 derived derive VERB ejpam-5896 163 26 from from ADP ejpam-5896 163 27 the the DET ejpam-5896 163 28 supra supra PROPN ejpam-5896 163 29 soft soft ADJ ejpam-5896 163 30 sd sd NOUN ejpam-5896 163 31 - - PUNCT ejpam-5896 163 32 closure closure NOUN ejpam-5896 163 33 operator operator NOUN ejpam-5896 163 34 here here ADV ejpam-5896 163 35 , , PUNCT ejpam-5896 163 36 we we PRON ejpam-5896 163 37 continue continue VERB ejpam-5896 163 38 studying study VERB ejpam-5896 163 39 the the DET ejpam-5896 163 40 properties property NOUN ejpam-5896 163 41 of of ADP ejpam-5896 163 42 connectedness connectedness NOUN ejpam-5896 163 43 via via ADP ejpam-5896 163 44 ss ss NOUN ejpam-5896 163 45 - - PUNCT ejpam-5896 163 46 sd sd NOUN ejpam-5896 163 47 - - PUNCT ejpam-5896 163 48 sets set NOUN ejpam-5896 163 49 in in ADP ejpam-5896 163 50 ssts sst NOUN ejpam-5896 163 51 [ [ X ejpam-5896 163 52 62 62 NUM ejpam-5896 163 53 ] ] PUNCT ejpam-5896 163 54 . . PUNCT ejpam-5896 164 1 we we PRON ejpam-5896 164 2 define define VERB ejpam-5896 164 3 the the DET ejpam-5896 164 4 concept concept NOUN ejpam-5896 164 5 of of ADP ejpam-5896 164 6 ss ss NOUN ejpam-5896 164 7 - - PUNCT ejpam-5896 164 8 sd sd NOUN ejpam-5896 164 9 - - PUNCT ejpam-5896 164 10 component component NOUN ejpam-5896 164 11 . . PUNCT ejpam-5896 165 1 we we PRON ejpam-5896 165 2 demonstrate demonstrate VERB ejpam-5896 165 3 that that SCONJ ejpam-5896 165 4 an an DET ejpam-5896 165 5 ss ss NOUN ejpam-5896 165 6 - - PUNCT ejpam-5896 165 7 sd sd NOUN ejpam-5896 165 8 - - PUNCT ejpam-5896 165 9 component component NOUN ejpam-5896 165 10 ’s ’s PART ejpam-5896 165 11 image image NOUN ejpam-5896 165 12 under under ADP ejpam-5896 165 13 a a DET ejpam-5896 165 14 bijective bijective ADJ ejpam-5896 165 15 ss*-sd ss*-sd NOUN ejpam-5896 165 16 - - PUNCT ejpam-5896 165 17 open open ADJ ejpam-5896 165 18 map map NOUN ejpam-5896 165 19 is be AUX ejpam-5896 165 20 an an DET ejpam-5896 165 21 ss ss VERB ejpam-5896 165 22 - - PUNCT ejpam-5896 165 23 sd sd NOUN ejpam-5896 165 24 - - PUNCT ejpam-5896 165 25 component component NOUN ejpam-5896 165 26 . . PUNCT ejpam-5896 166 1 additionally additionally ADV ejpam-5896 166 2 , , PUNCT ejpam-5896 166 3 we we PRON ejpam-5896 166 4 prove prove VERB ejpam-5896 166 5 that that SCONJ ejpam-5896 166 6 the the DET ejpam-5896 166 7 category category NOUN ejpam-5896 166 8 of of ADP ejpam-5896 166 9 all all DET ejpam-5896 166 10 ss ss NOUN ejpam-5896 166 11 - - PUNCT ejpam-5896 166 12 sd sd NOUN ejpam-5896 166 13 - - PUNCT ejpam-5896 166 14 components component NOUN ejpam-5896 166 15 of of ADP ejpam-5896 166 16 an an DET ejpam-5896 166 17 ssts sst NOUN ejpam-5896 166 18 ( ( PUNCT ejpam-5896 166 19 u u NOUN ejpam-5896 166 20 , , PUNCT ejpam-5896 166 21 µ,∆ µ,∆ NUM ejpam-5896 166 22 ) ) PUNCT ejpam-5896 166 23 forms form VERB ejpam-5896 166 24 a a DET ejpam-5896 166 25 partition partition NOUN ejpam-5896 166 26 to to ADP ejpam-5896 166 27 it it PRON ejpam-5896 166 28 . . PUNCT ejpam-5896 167 1 moreover moreover ADV ejpam-5896 167 2 , , PUNCT ejpam-5896 167 3 we we PRON ejpam-5896 167 4 apply apply VERB ejpam-5896 167 5 it it PRON ejpam-5896 167 6 to to PART ejpam-5896 167 7 define define VERB ejpam-5896 167 8 a a DET ejpam-5896 167 9 new new ADJ ejpam-5896 167 10 approach approach NOUN ejpam-5896 167 11 of of ADP ejpam-5896 167 12 connectedness connectedness NOUN ejpam-5896 167 13 via via ADP ejpam-5896 167 14 ss ss NOUN ejpam-5896 167 15 - - PUNCT ejpam-5896 167 16 sd sd NOUN ejpam-5896 167 17 - - PUNCT ejpam-5896 167 18 sets set NOUN ejpam-5896 167 19 , , PUNCT ejpam-5896 167 20 named name VERB ejpam-5896 167 21 ssl ssl PROPN ejpam-5896 167 22 - - PUNCT ejpam-5896 167 23 sd sd NOUN ejpam-5896 167 24 - - PUNCT ejpam-5896 167 25 connectedness connectedness NOUN ejpam-5896 167 26 . . PUNCT ejpam-5896 168 1 we we PRON ejpam-5896 168 2 prove prove VERB ejpam-5896 168 3 that that SCONJ ejpam-5896 168 4 the the DET ejpam-5896 168 5 property property NOUN ejpam-5896 168 6 of of ADP ejpam-5896 168 7 ssl ssl ADJ ejpam-5896 168 8 - - PUNCT ejpam-5896 168 9 sd sd NOUN ejpam-5896 168 10 - - PUNCT ejpam-5896 168 11 connectedness connectedness NOUN ejpam-5896 168 12 is be AUX ejpam-5896 168 13 hereditary hereditary ADJ ejpam-5896 168 14 w.r.t w.r.t NOUN ejpam-5896 168 15 ss ss PROPN ejpam-5896 168 16 - - PUNCT ejpam-5896 168 17 sd sd NOUN ejpam-5896 168 18 - - PUNCT ejpam-5896 168 19 subspaces subspace NOUN ejpam-5896 168 20 . . PUNCT ejpam-5896 169 1 also also ADV ejpam-5896 169 2 , , PUNCT ejpam-5896 169 3 we we PRON ejpam-5896 169 4 define define VERB ejpam-5896 169 5 the the DET ejpam-5896 169 6 concept concept NOUN ejpam-5896 169 7 of of ADP ejpam-5896 169 8 ss ss NOUN ejpam-5896 169 9 - - PUNCT ejpam-5896 169 10 sd sd NOUN ejpam-5896 169 11 - - PUNCT ejpam-5896 169 12 hyperconnectedness hyperconnectedness NOUN ejpam-5896 169 13 . . PUNCT ejpam-5896 170 1 we we PRON ejpam-5896 170 2 discovered discover VERB ejpam-5896 170 3 that that SCONJ ejpam-5896 170 4 , , PUNCT ejpam-5896 170 5 ss ss NOUN ejpam-5896 170 6 - - PUNCT ejpam-5896 170 7 sd sd NOUN ejpam-5896 170 8 - - PUNCT ejpam-5896 170 9 hyperconnected hyperconnecte VERB ejpam-5896 170 10 spaces space NOUN ejpam-5896 170 11 are be AUX ejpam-5896 170 12 identical identical ADJ ejpam-5896 170 13 to to ADP ejpam-5896 170 14 ss ss VERB ejpam-5896 170 15 - - PUNCT ejpam-5896 170 16 sd sd NOUN ejpam-5896 170 17 - - PUNCT ejpam-5896 170 18 connected connect VERB ejpam-5896 170 19 spaces space NOUN ejpam-5896 170 20 , , PUNCT ejpam-5896 170 21 which which PRON ejpam-5896 170 22 distinguishes distinguish VERB ejpam-5896 170 23 our our PRON ejpam-5896 170 24 notions notion NOUN ejpam-5896 170 25 from from ADP ejpam-5896 170 26 their their PRON ejpam-5896 170 27 counterparts counterpart NOUN ejpam-5896 170 28 . . PUNCT ejpam-5896 171 1 finally finally ADV ejpam-5896 171 2 , , PUNCT ejpam-5896 171 3 we we PRON ejpam-5896 171 4 provide provide VERB ejpam-5896 171 5 a a DET ejpam-5896 171 6 topological topological ADJ ejpam-5896 171 7 chart chart NOUN ejpam-5896 171 8 to to PART ejpam-5896 171 9 declare declare VERB ejpam-5896 171 10 the the DET ejpam-5896 171 11 key key ADJ ejpam-5896 171 12 concepts concept NOUN ejpam-5896 171 13 presented present VERB ejpam-5896 171 14 in in ADP ejpam-5896 171 15 this this DET ejpam-5896 171 16 section section NOUN ejpam-5896 171 17 in in ADP ejpam-5896 171 18 figure figure NOUN ejpam-5896 171 19 1 1 NUM ejpam-5896 171 20 . . PUNCT ejpam-5896 172 1 the the DET ejpam-5896 172 2 arrows arrow NOUN ejpam-5896 172 3 in in ADP ejpam-5896 172 4 this this DET ejpam-5896 172 5 chart chart NOUN ejpam-5896 172 6 are be AUX ejpam-5896 172 7 non non ADJ ejpam-5896 172 8 - - ADJ ejpam-5896 172 9 reversible reversible ADJ ejpam-5896 172 10 , , PUNCT ejpam-5896 172 11 as as SCONJ ejpam-5896 172 12 has have AUX ejpam-5896 172 13 been be AUX ejpam-5896 172 14 confirmed confirm VERB ejpam-5896 172 15 by by ADP ejpam-5896 172 16 concrete concrete ADJ ejpam-5896 172 17 counterexamples counterexample NOUN ejpam-5896 172 18 . . PUNCT ejpam-5896 173 1 definition definition NOUN ejpam-5896 173 2 14 14 NUM ejpam-5896 173 3 . . PUNCT ejpam-5896 174 1 let let VERB ejpam-5896 174 2 ( ( PUNCT ejpam-5896 174 3 u u NOUN ejpam-5896 174 4 , , PUNCT ejpam-5896 174 5 µ,∆ µ,∆ NUM ejpam-5896 174 6 ) ) PUNCT ejpam-5896 174 7 be be VERB ejpam-5896 174 8 an an DET ejpam-5896 174 9 ssts sst NOUN ejpam-5896 174 10 and and CCONJ ejpam-5896 174 11 ( ( PUNCT ejpam-5896 174 12 a,∆)⊆̃ũ a,∆)⊆̃ũ NOUN ejpam-5896 174 13 with with ADP ejpam-5896 174 14 sγ∈̃ũ sγ∈̃ũ NOUN ejpam-5896 174 15 , , PUNCT ejpam-5896 174 16 then then ADV ejpam-5896 174 17 the the DET ejpam-5896 174 18 ss ss PROPN ejpam-5896 174 19 - - PUNCT ejpam-5896 174 20 sd sd NOUN ejpam-5896 174 21 - - PUNCT ejpam-5896 174 22 component component NOUN ejpam-5896 174 23 of of ADP ejpam-5896 174 24 ( ( PUNCT ejpam-5896 174 25 a,∆ a,∆ VERB ejpam-5896 174 26 ) ) PUNCT ejpam-5896 174 27 related relate VERB ejpam-5896 174 28 to to ADP ejpam-5896 174 29 sγ sγ PROPN ejpam-5896 174 30 is be AUX ejpam-5896 174 31 the the DET ejpam-5896 174 32 finer fine ADJ ejpam-5896 174 33 ss ss NOUN ejpam-5896 174 34 - - PUNCT ejpam-5896 174 35 sd sd NOUN ejpam-5896 174 36 - - PUNCT ejpam-5896 174 37 connected connect VERB ejpam-5896 174 38 subset subset NOUN ejpam-5896 174 39 of of ADP ejpam-5896 174 40 ( ( PUNCT ejpam-5896 174 41 a,∆ a,∆ VERB ejpam-5896 174 42 ) ) PUNCT ejpam-5896 174 43 containing contain VERB ejpam-5896 174 44 sγ sγ NOUN ejpam-5896 174 45 and and CCONJ ejpam-5896 174 46 will will AUX ejpam-5896 174 47 denoted denote VERB ejpam-5896 174 48 by by ADP ejpam-5896 174 49 c̃s c̃s NOUN ejpam-5896 174 50 sd(asγ sd(asγ NOUN ejpam-5896 174 51 , , PUNCT ejpam-5896 174 52 ∆ ∆ PROPN ejpam-5896 174 53 ) ) PUNCT ejpam-5896 174 54 . . PUNCT ejpam-5896 175 1 example example NOUN ejpam-5896 176 1 1 1 NUM ejpam-5896 176 2 . . PUNCT ejpam-5896 176 3 let let VERB ejpam-5896 176 4 u u PRON ejpam-5896 176 5 = = PUNCT ejpam-5896 176 6 { { PUNCT ejpam-5896 176 7 x x PROPN ejpam-5896 176 8 , , PUNCT ejpam-5896 176 9 y y PROPN ejpam-5896 176 10 , , PUNCT ejpam-5896 176 11 z z NOUN ejpam-5896 176 12 } } PUNCT ejpam-5896 176 13 , , PUNCT ejpam-5896 176 14 ∆ ∆ X ejpam-5896 176 15 = = SYM ejpam-5896 176 16 { { PUNCT ejpam-5896 176 17 γ1 γ1 PROPN ejpam-5896 176 18 , , PUNCT ejpam-5896 176 19 γ2 γ2 NOUN ejpam-5896 176 20 } } PUNCT ejpam-5896 176 21 and and CCONJ ejpam-5896 176 22 µ µ X ejpam-5896 176 23 = = SYM ejpam-5896 176 24 { { PUNCT ejpam-5896 176 25 ũ ũ PROPN ejpam-5896 176 26 , , PUNCT ejpam-5896 176 27 φ̃ φ̃ PROPN ejpam-5896 176 28 , , PUNCT ejpam-5896 176 29 ( ( PUNCT ejpam-5896 176 30 oi,∆ oi,∆ PROPN ejpam-5896 176 31 ) ) PUNCT ejpam-5896 176 32 , , PUNCT ejpam-5896 176 33 i i PRON ejpam-5896 176 34 = = NOUN ejpam-5896 176 35 1 1 NUM ejpam-5896 176 36 , , PUNCT ejpam-5896 176 37 2 2 NUM ejpam-5896 176 38 , , PUNCT ejpam-5896 176 39 , , PUNCT ejpam-5896 176 40 ... ... PUNCT ejpam-5896 176 41 , , PUNCT ejpam-5896 176 42 5 5 X ejpam-5896 176 43 } } PUNCT ejpam-5896 176 44 be be AUX ejpam-5896 176 45 an an DET ejpam-5896 176 46 ssts sst NOUN ejpam-5896 176 47 over over ADP ejpam-5896 176 48 n n CCONJ ejpam-5896 176 49 , , PUNCT ejpam-5896 176 50 where where SCONJ ejpam-5896 176 51 : : PUNCT ejpam-5896 176 52 o1(γ1 o1(γ1 NUM ejpam-5896 176 53 ) ) PUNCT ejpam-5896 176 54 = = SYM ejpam-5896 176 55 u u NOUN ejpam-5896 176 56 , , PUNCT ejpam-5896 176 57 o1(γ2 o1(γ2 NOUN ejpam-5896 176 58 ) ) PUNCT ejpam-5896 176 59 = = SYM ejpam-5896 177 1 φ φ PROPN ejpam-5896 177 2 . . PUNCT ejpam-5896 178 1 o2(γ1 o2(γ1 NOUN ejpam-5896 178 2 ) ) PUNCT ejpam-5896 179 1 = = SYM ejpam-5896 179 2 φ φ NUM ejpam-5896 179 3 , , PUNCT ejpam-5896 179 4 o2(γ2 o2(γ2 NOUN ejpam-5896 179 5 ) ) PUNCT ejpam-5896 179 6 = = SYM ejpam-5896 180 1 u. u. NOUN ejpam-5896 180 2 o3(γ1 o3(γ1 VERB ejpam-5896 180 3 ) ) PUNCT ejpam-5896 180 4 = = SYM ejpam-5896 180 5 u u NOUN ejpam-5896 180 6 , , PUNCT ejpam-5896 180 7 o3(γ2 o3(γ2 NOUN ejpam-5896 180 8 ) ) PUNCT ejpam-5896 180 9 = = SYM ejpam-5896 180 10 { { PUNCT ejpam-5896 180 11 y y PROPN ejpam-5896 180 12 , , PUNCT ejpam-5896 180 13 z z NOUN ejpam-5896 180 14 } } PUNCT ejpam-5896 180 15 . . PUNCT ejpam-5896 181 1 o4(γ1 o4(γ1 NOUN ejpam-5896 181 2 ) ) PUNCT ejpam-5896 181 3 = = PRON ejpam-5896 181 4 { { PUNCT ejpam-5896 181 5 z z NOUN ejpam-5896 181 6 } } PUNCT ejpam-5896 181 7 , , PUNCT ejpam-5896 181 8 o4(γ2 o4(γ2 NOUN ejpam-5896 181 9 ) ) PUNCT ejpam-5896 181 10 = = VERB ejpam-5896 181 11 u. u. NOUN ejpam-5896 181 12 o5(γ1 o5(γ1 VERB ejpam-5896 181 13 ) ) PUNCT ejpam-5896 181 14 = = SYM ejpam-5896 181 15 { { PUNCT ejpam-5896 181 16 x x NOUN ejpam-5896 181 17 , , PUNCT ejpam-5896 181 18 y y NOUN ejpam-5896 181 19 } } PUNCT ejpam-5896 181 20 , , PUNCT ejpam-5896 181 21 o5(γ2 o5(γ2 NOUN ejpam-5896 181 22 ) ) PUNCT ejpam-5896 181 23 = = VERB ejpam-5896 182 1 u. u. PROPN ejpam-5896 182 2 then then ADV ejpam-5896 182 3 , , PUNCT ejpam-5896 182 4 ũ ũ PROPN ejpam-5896 182 5 = = SYM ejpam-5896 182 6 ( ( PUNCT ejpam-5896 182 7 o1,∆)∪̃(o2,∆ o1,∆)∪̃(o2,∆ PROPN ejpam-5896 182 8 ) ) PUNCT ejpam-5896 182 9 whereas whereas SCONJ ejpam-5896 182 10 ( ( PUNCT ejpam-5896 182 11 o1,∆ o1,∆ ADJ ejpam-5896 182 12 ) ) PUNCT ejpam-5896 182 13 , , PUNCT ejpam-5896 182 14 ( ( PUNCT ejpam-5896 182 15 o2,∆ o2,∆ PROPN ejpam-5896 182 16 ) ) PUNCT ejpam-5896 182 17 ∈ ∈ PROPN ejpam-5896 182 18 sd(u)∆ sd(u)∆ NOUN ejpam-5896 182 19 which which PRON ejpam-5896 182 20 are be AUX ejpam-5896 182 21 disjoint disjoint ADJ ejpam-5896 182 22 . . PUNCT ejpam-5896 183 1 it it PRON ejpam-5896 183 2 follows follow VERB ejpam-5896 183 3 that that SCONJ ejpam-5896 183 4 , , PUNCT ejpam-5896 183 5 ũ ũ PROPN ejpam-5896 183 6 is be AUX ejpam-5896 183 7 an an DET ejpam-5896 183 8 ss ss NOUN ejpam-5896 183 9 - - PUNCT ejpam-5896 183 10 sd sd NOUN ejpam-5896 183 11 - - PUNCT ejpam-5896 183 12 disconnected disconnected ADJ ejpam-5896 183 13 . . PUNCT ejpam-5896 184 1 now now ADV ejpam-5896 184 2 , , PUNCT ejpam-5896 184 3 c̃s c̃s ADJ ejpam-5896 184 4 sd(xγ1 sd(xγ1 PUNCT ejpam-5896 184 5 ) ) PUNCT ejpam-5896 185 1 = = SYM ejpam-5896 185 2 c̃s c̃s NOUN ejpam-5896 185 3 sd(yγ1 sd(yγ1 NUM ejpam-5896 185 4 ) ) PUNCT ejpam-5896 185 5 = = VERB ejpam-5896 185 6 c̃s c̃s NOUN ejpam-5896 185 7 sd(zγ1 sd(zγ1 NOUN ejpam-5896 185 8 ) ) PUNCT ejpam-5896 186 1 = = PUNCT ejpam-5896 186 2 ( ( PUNCT ejpam-5896 186 3 o1,∆ o1,∆ ADJ ejpam-5896 186 4 ) ) PUNCT ejpam-5896 186 5 and and CCONJ ejpam-5896 186 6 c̃s c̃s NOUN ejpam-5896 186 7 sd(xγ2 sd(xγ2 NOUN ejpam-5896 186 8 ) ) PUNCT ejpam-5896 186 9 = = PUNCT ejpam-5896 186 10 c̃s c̃s NOUN ejpam-5896 186 11 sd(yγ2 sd(yγ2 NOUN ejpam-5896 186 12 ) ) PUNCT ejpam-5896 186 13 = = SYM ejpam-5896 186 14 c̃s c̃s NOUN ejpam-5896 186 15 sd(zγ2 sd(zγ2 NOUN ejpam-5896 186 16 ) ) PUNCT ejpam-5896 186 17 = = SYM ejpam-5896 186 18 ( ( PUNCT ejpam-5896 186 19 o2,∆ o2,∆ PROPN ejpam-5896 186 20 ) ) PUNCT ejpam-5896 186 21 . . PUNCT ejpam-5896 187 1 lemma lemma PROPN ejpam-5896 187 2 1 1 X ejpam-5896 187 3 . . PUNCT ejpam-5896 188 1 let let VERB ejpam-5896 188 2 ( ( PUNCT ejpam-5896 188 3 u u NOUN ejpam-5896 188 4 , , PUNCT ejpam-5896 188 5 µ,∆ µ,∆ NUM ejpam-5896 188 6 ) ) PUNCT ejpam-5896 188 7 be be VERB ejpam-5896 188 8 an an DET ejpam-5896 188 9 ssts sst NOUN ejpam-5896 188 10 , , PUNCT ejpam-5896 188 11 then then ADV ejpam-5896 188 12 ( ( PUNCT ejpam-5896 188 13 1 1 X ejpam-5896 188 14 ) ) PUNCT ejpam-5896 188 15 ũ ũ PROPN ejpam-5896 188 16 is be AUX ejpam-5896 188 17 ss ss NOUN ejpam-5896 188 18 - - PUNCT ejpam-5896 188 19 sd sd NOUN ejpam-5896 188 20 - - PUNCT ejpam-5896 188 21 connected connect VERB ejpam-5896 188 22 if if SCONJ ejpam-5896 188 23 and and CCONJ ejpam-5896 188 24 only only ADV ejpam-5896 188 25 if if SCONJ ejpam-5896 188 26 it it PRON ejpam-5896 188 27 is be AUX ejpam-5896 188 28 ss ss NOUN ejpam-5896 188 29 - - PUNCT ejpam-5896 188 30 sd sd NOUN ejpam-5896 188 31 - - PUNCT ejpam-5896 188 32 component component NOUN ejpam-5896 188 33 . . PUNCT ejpam-5896 189 1 ( ( PUNCT ejpam-5896 189 2 2 2 X ejpam-5896 189 3 ) ) PUNCT ejpam-5896 189 4 if if SCONJ ejpam-5896 189 5 ũ ũ PROPN ejpam-5896 189 6 is be AUX ejpam-5896 189 7 ss ss NOUN ejpam-5896 189 8 - - PUNCT ejpam-5896 189 9 sd sd NOUN ejpam-5896 189 10 - - PUNCT ejpam-5896 189 11 connected connect VERB ejpam-5896 189 12 , , PUNCT ejpam-5896 189 13 then then ADV ejpam-5896 189 14 ũ ũ PROPN ejpam-5896 189 15 is be AUX ejpam-5896 189 16 the the DET ejpam-5896 189 17 only only ADJ ejpam-5896 189 18 ss ss NOUN ejpam-5896 189 19 - - PUNCT ejpam-5896 189 20 sd sd NOUN ejpam-5896 189 21 - - PUNCT ejpam-5896 189 22 component component NOUN ejpam-5896 189 23 of of ADP ejpam-5896 189 24 each each PRON ejpam-5896 189 25 of of ADP ejpam-5896 189 26 its its PRON ejpam-5896 189 27 soft soft ADJ ejpam-5896 189 28 points point NOUN ejpam-5896 189 29 . . PUNCT ejpam-5896 190 1 proof proof NOUN ejpam-5896 190 2 . . PUNCT ejpam-5896 191 1 obvious obvious ADJ ejpam-5896 191 2 from from ADP ejpam-5896 191 3 definition definition NOUN ejpam-5896 191 4 14 14 NUM ejpam-5896 191 5 . . PUNCT ejpam-5896 192 1 proposition proposition NOUN ejpam-5896 192 2 1 1 NUM ejpam-5896 192 3 . . PUNCT ejpam-5896 193 1 [ [ X ejpam-5896 193 2 62 62 NUM ejpam-5896 193 3 ] ] PUNCT ejpam-5896 193 4 if if SCONJ ejpam-5896 193 5 an an DET ejpam-5896 193 6 ssts sst NOUN ejpam-5896 193 7 ( ( PUNCT ejpam-5896 193 8 u u NOUN ejpam-5896 193 9 , , PUNCT ejpam-5896 193 10 µ,∆ µ,∆ NUM ejpam-5896 193 11 ) ) PUNCT ejpam-5896 193 12 has have VERB ejpam-5896 193 13 a a DET ejpam-5896 193 14 soft soft ADJ ejpam-5896 193 15 subset subset NOUN ejpam-5896 193 16 ( ( PUNCT ejpam-5896 193 17 g,∆ g,∆ PROPN ejpam-5896 193 18 ) ) PUNCT ejpam-5896 193 19 which which PRON ejpam-5896 193 20 is be AUX ejpam-5896 193 21 ss ss NOUN ejpam-5896 193 22 - - PUNCT ejpam-5896 193 23 sd sd NOUN ejpam-5896 193 24 - - PUNCT ejpam-5896 193 25 connected connect VERB ejpam-5896 193 26 , , PUNCT ejpam-5896 193 27 then then ADV ejpam-5896 193 28 clssd(g,∆ clssd(g,∆ ADJ ejpam-5896 193 29 ) ) PUNCT ejpam-5896 193 30 is be AUX ejpam-5896 193 31 also also ADV ejpam-5896 193 32 . . PUNCT ejpam-5896 194 1 lemma lemma PROPN ejpam-5896 194 2 2 2 X ejpam-5896 194 3 . . PUNCT ejpam-5896 195 1 let let VERB ejpam-5896 195 2 ( ( PUNCT ejpam-5896 195 3 u u NOUN ejpam-5896 195 4 , , PUNCT ejpam-5896 195 5 µ,∆ µ,∆ NUM ejpam-5896 195 6 ) ) PUNCT ejpam-5896 195 7 be be VERB ejpam-5896 195 8 an an DET ejpam-5896 195 9 ssts sst NOUN ejpam-5896 195 10 , , PUNCT ejpam-5896 195 11 then then ADV ejpam-5896 195 12 ( ( PUNCT ejpam-5896 195 13 1 1 X ejpam-5896 195 14 ) ) PUNCT ejpam-5896 195 15 every every DET ejpam-5896 195 16 ss ss NOUN ejpam-5896 195 17 - - PUNCT ejpam-5896 195 18 sd sd NOUN ejpam-5896 195 19 - - PUNCT ejpam-5896 195 20 component component NOUN ejpam-5896 195 21 subset subset NOUN ejpam-5896 195 22 of of ADP ejpam-5896 195 23 ũ ũ PROPN ejpam-5896 195 24 is be AUX ejpam-5896 195 25 the the DET ejpam-5896 195 26 finer fine ADJ ejpam-5896 195 27 ss ss NOUN ejpam-5896 195 28 - - PUNCT ejpam-5896 195 29 sd sd NOUN ejpam-5896 195 30 - - PUNCT ejpam-5896 195 31 connected connect VERB ejpam-5896 195 32 subset subset NOUN ejpam-5896 195 33 of of ADP ejpam-5896 195 34 ũ ũ PROPN ejpam-5896 195 35 . . PUNCT ejpam-5896 196 1 ( ( PUNCT ejpam-5896 196 2 2 2 X ejpam-5896 196 3 ) ) PUNCT ejpam-5896 196 4 every every DET ejpam-5896 196 5 ss ss NOUN ejpam-5896 196 6 - - PUNCT ejpam-5896 196 7 sd sd NOUN ejpam-5896 196 8 - - PUNCT ejpam-5896 196 9 component component NOUN ejpam-5896 196 10 subset subset NOUN ejpam-5896 196 11 of of ADP ejpam-5896 196 12 ũ ũ PROPN ejpam-5896 196 13 is be AUX ejpam-5896 196 14 an an DET ejpam-5896 196 15 ss ss ADJ ejpam-5896 196 16 - - PUNCT ejpam-5896 196 17 sc sc NOUN ejpam-5896 196 18 - - PUNCT ejpam-5896 196 19 set set NOUN ejpam-5896 196 20 . . PUNCT ejpam-5896 197 1 proof proof NOUN ejpam-5896 197 2 . . PUNCT ejpam-5896 198 1 obvious obvious ADJ ejpam-5896 198 2 from from ADP ejpam-5896 198 3 definition definition NOUN ejpam-5896 198 4 14 14 NUM ejpam-5896 198 5 and and CCONJ ejpam-5896 198 6 proposition proposition NOUN ejpam-5896 198 7 1 1 NUM ejpam-5896 198 8 . . PUNCT ejpam-5896 198 9 theorem theorem VERB ejpam-5896 198 10 4 4 NUM ejpam-5896 198 11 . . X ejpam-5896 199 1 for for ADP ejpam-5896 199 2 each each DET ejpam-5896 199 3 soft soft ADJ ejpam-5896 199 4 point point NOUN ejpam-5896 199 5 sγ sγ NOUN ejpam-5896 199 6 in in ADP ejpam-5896 199 7 an an DET ejpam-5896 199 8 ssts sst NOUN ejpam-5896 199 9 ( ( PUNCT ejpam-5896 199 10 u u NOUN ejpam-5896 199 11 , , PUNCT ejpam-5896 199 12 µ,∆ µ,∆ NUM ejpam-5896 199 13 ) ) PUNCT ejpam-5896 199 14 , , PUNCT ejpam-5896 199 15 there there PRON ejpam-5896 199 16 is be VERB ejpam-5896 199 17 only only ADV ejpam-5896 199 18 one one NUM ejpam-5896 199 19 ss ss NOUN ejpam-5896 199 20 - - PUNCT ejpam-5896 199 21 sd sd NOUN ejpam-5896 199 22 - - PUNCT ejpam-5896 199 23 component component NOUN ejpam-5896 199 24 subset subset NOUN ejpam-5896 199 25 of of ADP ejpam-5896 199 26 ũ ũ PROPN ejpam-5896 199 27 which which PRON ejpam-5896 199 28 contains contain VERB ejpam-5896 199 29 sγ sγ PROPN ejpam-5896 199 30 . . PUNCT ejpam-5896 200 1 proof proof NOUN ejpam-5896 200 2 . . PUNCT ejpam-5896 201 1 consider consider VERB ejpam-5896 201 2 the the DET ejpam-5896 201 3 class class NOUN ejpam-5896 201 4 ω ω PROPN ejpam-5896 201 5 = = SYM ejpam-5896 201 6 { { PUNCT ejpam-5896 201 7 ( ( PUNCT ejpam-5896 201 8 k,∆)⊆̃ũ k,∆)⊆̃ũ PROPN ejpam-5896 201 9 : : PUNCT ejpam-5896 201 10 sγ∈̃(k,∆ sγ∈̃(k,∆ PROPN ejpam-5896 201 11 ) ) PUNCT ejpam-5896 201 12 , , PUNCT ejpam-5896 201 13 ( ( PUNCT ejpam-5896 201 14 k,∆ k,∆ PROPN ejpam-5896 201 15 ) ) PUNCT ejpam-5896 201 16 is be AUX ejpam-5896 201 17 an an DET ejpam-5896 201 18 ss−sd−connected ss−sd−connecte VERB ejpam-5896 201 19 } } PUNCT ejpam-5896 201 20 . . PUNCT ejpam-5896 202 1 since since SCONJ ejpam-5896 202 2 sγ∈̃(k,∆ sγ∈̃(k,∆ PROPN ejpam-5896 202 3 ) ) PUNCT ejpam-5896 202 4 for for ADP ejpam-5896 202 5 each each PRON ejpam-5896 202 6 ( ( PUNCT ejpam-5896 202 7 k,∆ k,∆ PROPN ejpam-5896 202 8 ) ) PUNCT ejpam-5896 202 9 ∈ ∈ PROPN ejpam-5896 202 10 ω ω PROPN ejpam-5896 202 11 , , PUNCT ejpam-5896 202 12 ω ω PROPN ejpam-5896 202 13 ̸= ̸= PROPN ejpam-5896 202 14 φ̃. φ̃. PROPN ejpam-5896 202 15 it it PRON ejpam-5896 202 16 follows follow VERB ejpam-5896 202 17 that that SCONJ ejpam-5896 202 18 , , PUNCT ejpam-5896 202 19 the the DET ejpam-5896 202 20 soft soft ADJ ejpam-5896 202 21 set set VERB ejpam-5896 202 22 abd abd PROPN ejpam-5896 202 23 el el PROPN ejpam-5896 202 24 - - PROPN ejpam-5896 202 25 latif latif PROPN ejpam-5896 202 26 et et PROPN ejpam-5896 202 27 al al PROPN ejpam-5896 202 28 . . PUNCT ejpam-5896 202 29 / / SYM ejpam-5896 202 30 eur eur PROPN ejpam-5896 202 31 . . PUNCT ejpam-5896 203 1 j. j. PROPN ejpam-5896 203 2 pure pure PROPN ejpam-5896 203 3 appl appl PROPN ejpam-5896 203 4 . . PROPN ejpam-5896 203 5 math math PROPN ejpam-5896 203 6 , , PUNCT ejpam-5896 203 7 18 18 NUM ejpam-5896 203 8 ( ( PUNCT ejpam-5896 203 9 2 2 NUM ejpam-5896 203 10 ) ) PUNCT ejpam-5896 203 11 ( ( PUNCT ejpam-5896 203 12 2025 2025 NUM ejpam-5896 203 13 ) ) PUNCT ejpam-5896 203 14 , , PUNCT ejpam-5896 203 15 5896 5896 NUM ejpam-5896 203 16 6 6 NUM ejpam-5896 203 17 of of ADP ejpam-5896 203 18 20 20 NUM ejpam-5896 203 19 ( ( PUNCT ejpam-5896 203 20 g,∆ g,∆ X ejpam-5896 203 21 ) ) PUNCT ejpam-5896 203 22 = = SYM ejpam-5896 203 23 ⋃̃ ⋃̃ PROPN ejpam-5896 203 24 { { PUNCT ejpam-5896 203 25 ( ( PUNCT ejpam-5896 203 26 k,∆)⊆̃ũ k,∆)⊆̃ũ PROPN ejpam-5896 203 27 : : PUNCT ejpam-5896 203 28 sγ∈̃(k,∆ sγ∈̃(k,∆ PROPN ejpam-5896 203 29 ) ) PUNCT ejpam-5896 203 30 , , PUNCT ejpam-5896 203 31 ( ( PUNCT ejpam-5896 203 32 k,∆ k,∆ PROPN ejpam-5896 203 33 ) ) PUNCT ejpam-5896 203 34 is be AUX ejpam-5896 203 35 an an DET ejpam-5896 203 36 ss ss NOUN ejpam-5896 203 37 − − NOUN ejpam-5896 203 38 sd− sd− NUM ejpam-5896 203 39 connected connect VERB ejpam-5896 203 40 } } PUNCT ejpam-5896 203 41 has have AUX ejpam-5896 203 42 a a DET ejpam-5896 203 43 non non ADJ ejpam-5896 203 44 - - ADJ ejpam-5896 203 45 null null ADJ ejpam-5896 203 46 soft soft ADJ ejpam-5896 203 47 intersection intersection NOUN ejpam-5896 203 48 and and CCONJ ejpam-5896 203 49 ss ss NOUN ejpam-5896 203 50 - - PUNCT ejpam-5896 203 51 sd sd NOUN ejpam-5896 203 52 - - PUNCT ejpam-5896 203 53 connected connect VERB ejpam-5896 203 54 subset subset NOUN ejpam-5896 203 55 of of ADP ejpam-5896 203 56 ũ ũ PROPN ejpam-5896 203 57 containing contain VERB ejpam-5896 203 58 sγ sγ PRON ejpam-5896 203 59 . . PUNCT ejpam-5896 204 1 hence hence ADV ejpam-5896 204 2 , , PUNCT ejpam-5896 204 3 ( ( PUNCT ejpam-5896 204 4 g,∆ g,∆ X ejpam-5896 204 5 ) ) PUNCT ejpam-5896 204 6 is be AUX ejpam-5896 204 7 the the DET ejpam-5896 204 8 ss ss NOUN ejpam-5896 204 9 - - PUNCT ejpam-5896 204 10 sd sd NOUN ejpam-5896 204 11 - - PUNCT ejpam-5896 204 12 component component NOUN ejpam-5896 204 13 c̃s c̃s NOUN ejpam-5896 204 14 sd(usγ sd(usγ NOUN ejpam-5896 204 15 , , PUNCT ejpam-5896 204 16 ∆ ∆ PROPN ejpam-5896 204 17 ) ) PUNCT ejpam-5896 204 18 of of ADP ejpam-5896 204 19 ũ ũ PROPN ejpam-5896 204 20 w.r.t w.r.t NOUN ejpam-5896 204 21 sγ sγ NOUN ejpam-5896 204 22 . . PUNCT ejpam-5896 205 1 now now ADV ejpam-5896 205 2 , , PUNCT ejpam-5896 205 3 if if SCONJ ejpam-5896 205 4 c̃s∗ c̃s∗ PROPN ejpam-5896 205 5 sd(usγ sd(usγ PROPN ejpam-5896 205 6 , , PUNCT ejpam-5896 205 7 ∆ ∆ PROPN ejpam-5896 205 8 ) ) PUNCT ejpam-5896 205 9 is be AUX ejpam-5896 205 10 another another DET ejpam-5896 205 11 ss ss NOUN ejpam-5896 205 12 - - PUNCT ejpam-5896 205 13 sd sd NOUN ejpam-5896 205 14 - - PUNCT ejpam-5896 205 15 component component NOUN ejpam-5896 205 16 containing contain VERB ejpam-5896 205 17 sγ sγ NOUN ejpam-5896 205 18 , , PUNCT ejpam-5896 205 19 then then ADV ejpam-5896 205 20 c̃s∗ c̃s∗ VERB ejpam-5896 205 21 sd(usγ sd(usγ PROPN ejpam-5896 205 22 , , PUNCT ejpam-5896 205 23 ∆ ∆ PROPN ejpam-5896 205 24 ) ) PUNCT ejpam-5896 205 25 is be AUX ejpam-5896 205 26 ss ss ADV ejpam-5896 205 27 - - PUNCT ejpam-5896 205 28 sdconnected sdconnecte VERB ejpam-5896 205 29 subset subset NOUN ejpam-5896 205 30 of of ADP ejpam-5896 205 31 ũ ũ PROPN ejpam-5896 205 32 containing contain VERB ejpam-5896 205 33 sγ sγ PRON ejpam-5896 205 34 . . PUNCT ejpam-5896 206 1 however however ADV ejpam-5896 206 2 , , PUNCT ejpam-5896 206 3 c̃s c̃s NOUN ejpam-5896 206 4 sd(usγ sd(usγ NOUN ejpam-5896 206 5 , , PUNCT ejpam-5896 206 6 ∆ ∆ PROPN ejpam-5896 206 7 ) ) PUNCT ejpam-5896 206 8 is be AUX ejpam-5896 206 9 an an DET ejpam-5896 206 10 ss ss VERB ejpam-5896 206 11 - - PUNCT ejpam-5896 206 12 sd sd NOUN ejpam-5896 206 13 - - PUNCT ejpam-5896 206 14 component component NOUN ejpam-5896 206 15 , , PUNCT ejpam-5896 206 16 and and CCONJ ejpam-5896 206 17 so so ADV ejpam-5896 206 18 c̃s∗ c̃s∗ PROPN ejpam-5896 206 19 sd(usγ sd(usγ PROPN ejpam-5896 206 20 , , PUNCT ejpam-5896 206 21 ∆)⊆̃c̃s ∆)⊆̃c̃s PROPN ejpam-5896 206 22 sd(usγ sd(usγ NOUN ejpam-5896 206 23 , , PUNCT ejpam-5896 206 24 ∆ ∆ PROPN ejpam-5896 206 25 ) ) PUNCT ejpam-5896 206 26 . . PUNCT ejpam-5896 207 1 by by ADP ejpam-5896 207 2 a a DET ejpam-5896 207 3 similar similar ADJ ejpam-5896 207 4 argument argument NOUN ejpam-5896 207 5 we we PRON ejpam-5896 207 6 can can AUX ejpam-5896 207 7 get get VERB ejpam-5896 207 8 c̃s c̃s ADJ ejpam-5896 207 9 sd(usγ sd(usγ NOUN ejpam-5896 207 10 , , PUNCT ejpam-5896 207 11 ∆)⊆̃c̃s∗ ∆)⊆̃c̃s∗ PROPN ejpam-5896 207 12 sd(usγ sd(usγ PROPN ejpam-5896 207 13 , , PUNCT ejpam-5896 207 14 ∆ ∆ PROPN ejpam-5896 207 15 ) ) PUNCT ejpam-5896 207 16 . . PUNCT ejpam-5896 208 1 thus thus ADV ejpam-5896 208 2 , , PUNCT ejpam-5896 208 3 sγ sγ PROPN ejpam-5896 208 4 is be AUX ejpam-5896 208 5 contained contain VERB ejpam-5896 208 6 in in ADP ejpam-5896 208 7 one one NUM ejpam-5896 208 8 ss ss NOUN ejpam-5896 208 9 - - PUNCT ejpam-5896 208 10 sd sd NOUN ejpam-5896 208 11 - - PUNCT ejpam-5896 208 12 component component NOUN ejpam-5896 208 13 subset subset NOUN ejpam-5896 208 14 of of ADP ejpam-5896 208 15 ũ ũ PROPN ejpam-5896 208 16 only only ADV ejpam-5896 208 17 . . PUNCT ejpam-5896 209 1 proposition proposition NOUN ejpam-5896 209 2 2 2 NUM ejpam-5896 209 3 . . PUNCT ejpam-5896 210 1 ( ( PUNCT ejpam-5896 210 2 1 1 X ejpam-5896 210 3 ) ) PUNCT ejpam-5896 210 4 any any DET ejpam-5896 210 5 two two NUM ejpam-5896 210 6 ss ss NOUN ejpam-5896 210 7 - - PUNCT ejpam-5896 210 8 sd sd NOUN ejpam-5896 210 9 - - PUNCT ejpam-5896 210 10 components component NOUN ejpam-5896 210 11 related relate VERB ejpam-5896 210 12 to to ADP ejpam-5896 210 13 two two NUM ejpam-5896 210 14 distinct distinct ADJ ejpam-5896 210 15 soft soft ADJ ejpam-5896 210 16 points point NOUN ejpam-5896 210 17 are be AUX ejpam-5896 210 18 either either CCONJ ejpam-5896 210 19 identical identical ADJ ejpam-5896 210 20 or or CCONJ ejpam-5896 210 21 disjoint disjoint NOUN ejpam-5896 210 22 . . PUNCT ejpam-5896 211 1 ( ( PUNCT ejpam-5896 211 2 2 2 X ejpam-5896 211 3 ) ) PUNCT ejpam-5896 211 4 the the DET ejpam-5896 211 5 class class NOUN ejpam-5896 211 6 of of ADP ejpam-5896 211 7 all all DET ejpam-5896 211 8 ss ss NOUN ejpam-5896 211 9 - - PUNCT ejpam-5896 211 10 sd sd NOUN ejpam-5896 211 11 - - PUNCT ejpam-5896 211 12 components component NOUN ejpam-5896 211 13 of of ADP ejpam-5896 211 14 an an DET ejpam-5896 211 15 ssts sst NOUN ejpam-5896 211 16 ( ( PUNCT ejpam-5896 211 17 u u NOUN ejpam-5896 211 18 , , PUNCT ejpam-5896 211 19 µ,∆ µ,∆ NUM ejpam-5896 211 20 ) ) PUNCT ejpam-5896 211 21 forms form VERB ejpam-5896 211 22 a a DET ejpam-5896 211 23 partition partition NOUN ejpam-5896 211 24 to to ADP ejpam-5896 211 25 ũ ũ PROPN ejpam-5896 211 26 . . PUNCT ejpam-5896 212 1 proof proof NOUN ejpam-5896 212 2 . . PUNCT ejpam-5896 213 1 ( ( PUNCT ejpam-5896 213 2 1 1 X ejpam-5896 213 3 ) ) PUNCT ejpam-5896 213 4 clear clear ADJ ejpam-5896 213 5 from from ADP ejpam-5896 213 6 definition definition NOUN ejpam-5896 213 7 14 14 NUM ejpam-5896 213 8 . . PUNCT ejpam-5896 214 1 ( ( PUNCT ejpam-5896 214 2 2 2 X ejpam-5896 214 3 ) ) PUNCT ejpam-5896 214 4 consider consider VERB ejpam-5896 214 5 the the DET ejpam-5896 214 6 class class NOUN ejpam-5896 214 7 { { PUNCT ejpam-5896 214 8 c̃s c̃s NOUN ejpam-5896 214 9 sd(usγ sd(usγ NOUN ejpam-5896 214 10 , , PUNCT ejpam-5896 214 11 ∆ ∆ PROPN ejpam-5896 214 12 ) ) PUNCT ejpam-5896 214 13 : : PUNCT ejpam-5896 214 14 sγ∈̃ũ sγ∈̃ũ NOUN ejpam-5896 214 15 } } PUNCT ejpam-5896 214 16 of of ADP ejpam-5896 214 17 all all DET ejpam-5896 214 18 ss ss NOUN ejpam-5896 214 19 - - PUNCT ejpam-5896 214 20 sd sd NOUN ejpam-5896 214 21 - - PUNCT ejpam-5896 214 22 components component NOUN ejpam-5896 214 23 of of ADP ejpam-5896 214 24 an an DET ejpam-5896 214 25 ssts sst NOUN ejpam-5896 214 26 ( ( PUNCT ejpam-5896 214 27 u u NOUN ejpam-5896 214 28 , , PUNCT ejpam-5896 214 29 µ,∆ µ,∆ NUM ejpam-5896 214 30 ) ) PUNCT ejpam-5896 214 31 . . PUNCT ejpam-5896 215 1 then then ADV ejpam-5896 215 2 , , PUNCT ejpam-5896 215 3 ⋃̃ ⋃̃ PROPN ejpam-5896 215 4 { { PUNCT ejpam-5896 215 5 c̃s c̃s NOUN ejpam-5896 215 6 sd(usγ sd(usγ NOUN ejpam-5896 215 7 , , PUNCT ejpam-5896 215 8 ∆ ∆ PROPN ejpam-5896 215 9 ) ) PUNCT ejpam-5896 215 10 : : PUNCT ejpam-5896 215 11 sγ∈̃ũ sγ∈̃ũ VERB ejpam-5896 215 12 } } PUNCT ejpam-5896 215 13 = = SYM ejpam-5896 215 14 ũ ũ PROPN ejpam-5896 215 15 . . PUNCT ejpam-5896 216 1 assume assume VERB ejpam-5896 216 2 that that SCONJ ejpam-5896 216 3 sγ sγ INTJ ejpam-5896 216 4 , , PUNCT ejpam-5896 216 5 rϑ rϑ X ejpam-5896 216 6 are be AUX ejpam-5896 216 7 any any DET ejpam-5896 216 8 two two NUM ejpam-5896 216 9 distinct distinct ADJ ejpam-5896 216 10 soft soft ADJ ejpam-5896 216 11 points point NOUN ejpam-5896 216 12 in in ADP ejpam-5896 216 13 ũ ũ PROPN ejpam-5896 216 14 . . PUNCT ejpam-5896 217 1 from from ADP ejpam-5896 217 2 ( ( PUNCT ejpam-5896 217 3 1 1 NUM ejpam-5896 217 4 ) ) PUNCT ejpam-5896 217 5 , , PUNCT ejpam-5896 217 6 c̃s c̃s NOUN ejpam-5896 217 7 sd(usγ sd(usγ NOUN ejpam-5896 217 8 , , PUNCT ejpam-5896 217 9 ∆ ∆ PROPN ejpam-5896 217 10 ) ) PUNCT ejpam-5896 217 11 , , PUNCT ejpam-5896 217 12 c̃s c̃s NOUN ejpam-5896 217 13 sd(urϑ sd(urϑ NOUN ejpam-5896 217 14 , , PUNCT ejpam-5896 217 15 ∆ ∆ X ejpam-5896 217 16 ) ) PUNCT ejpam-5896 217 17 are be AUX ejpam-5896 217 18 either either CCONJ ejpam-5896 217 19 identical identical ADJ ejpam-5896 217 20 or or CCONJ ejpam-5896 217 21 disjoint disjoint NOUN ejpam-5896 217 22 . . PUNCT ejpam-5896 218 1 if if SCONJ ejpam-5896 218 2 c̃s c̃s NOUN ejpam-5896 218 3 sd(usγ sd(usγ NOUN ejpam-5896 218 4 , , PUNCT ejpam-5896 218 5 ∆ ∆ PROPN ejpam-5896 218 6 ) ) PUNCT ejpam-5896 218 7 , , PUNCT ejpam-5896 218 8 c̃s c̃s NOUN ejpam-5896 218 9 sd(urϑ sd(urϑ NOUN ejpam-5896 218 10 , , PUNCT ejpam-5896 218 11 ∆ ∆ X ejpam-5896 218 12 ) ) PUNCT ejpam-5896 218 13 are be AUX ejpam-5896 218 14 identical identical ADJ ejpam-5896 218 15 , , PUNCT ejpam-5896 218 16 then then ADV ejpam-5896 218 17 it it PRON ejpam-5896 218 18 contradicts contradict VERB ejpam-5896 218 19 the the DET ejpam-5896 218 20 ũ ũ PROPN ejpam-5896 218 21 is be AUX ejpam-5896 218 22 the the DET ejpam-5896 218 23 finer fine ADJ ejpam-5896 218 24 ss ss NOUN ejpam-5896 218 25 - - PUNCT ejpam-5896 218 26 sd sd NOUN ejpam-5896 218 27 - - PUNCT ejpam-5896 218 28 connected connect VERB ejpam-5896 218 29 set set NOUN ejpam-5896 218 30 containing contain VERB ejpam-5896 218 31 sγ sγ NOUN ejpam-5896 218 32 , , PUNCT ejpam-5896 218 33 rϑ rϑ PROPN ejpam-5896 218 34 from from ADP ejpam-5896 218 35 lemma lemma PROPN ejpam-5896 218 36 2 2 NUM ejpam-5896 218 37 . . PUNCT ejpam-5896 219 1 hence hence ADV ejpam-5896 219 2 , , PUNCT ejpam-5896 219 3 c̃s c̃s NOUN ejpam-5896 219 4 sd(usγ sd(usγ NOUN ejpam-5896 219 5 , , PUNCT ejpam-5896 219 6 ∆ ∆ PROPN ejpam-5896 219 7 ) ) PUNCT ejpam-5896 219 8 , , PUNCT ejpam-5896 219 9 c̃s c̃s NOUN ejpam-5896 219 10 sd(urϑ sd(urϑ NOUN ejpam-5896 219 11 , , PUNCT ejpam-5896 219 12 ∆ ∆ X ejpam-5896 219 13 ) ) PUNCT ejpam-5896 219 14 are be AUX ejpam-5896 219 15 disjoint disjoint ADJ ejpam-5896 219 16 , , PUNCT ejpam-5896 219 17 and and CCONJ ejpam-5896 219 18 so so ADV ejpam-5896 219 19 we we PRON ejpam-5896 219 20 get get VERB ejpam-5896 219 21 the the DET ejpam-5896 219 22 proof proof NOUN ejpam-5896 219 23 . . PUNCT ejpam-5896 220 1 theorem theorem ADJ ejpam-5896 220 2 5 5 NUM ejpam-5896 220 3 . . PUNCT ejpam-5896 221 1 every every DET ejpam-5896 221 2 non non ADJ ejpam-5896 221 3 - - ADJ ejpam-5896 221 4 null null ADJ ejpam-5896 221 5 ss ss PROPN ejpam-5896 221 6 - - PUNCT ejpam-5896 221 7 sd sd NOUN ejpam-5896 221 8 - - PUNCT ejpam-5896 221 9 connected connect VERB ejpam-5896 221 10 subset subset NOUN ejpam-5896 221 11 of of ADP ejpam-5896 221 12 an an DET ejpam-5896 221 13 ssts sst NOUN ejpam-5896 221 14 ( ( PUNCT ejpam-5896 221 15 u u NOUN ejpam-5896 221 16 , , PUNCT ejpam-5896 221 17 µ,∆ µ,∆ NUM ejpam-5896 221 18 ) ) PUNCT ejpam-5896 221 19 in in ADP ejpam-5896 221 20 which which PRON ejpam-5896 221 21 is be AUX ejpam-5896 221 22 ss ss NOUN ejpam-5896 221 23 - - PUNCT ejpam-5896 221 24 sdc sdc NOUN ejpam-5896 221 25 - - PUNCT ejpam-5896 221 26 set set NOUN ejpam-5896 221 27 is be AUX ejpam-5896 221 28 ss ss NOUN ejpam-5896 221 29 - - PUNCT ejpam-5896 221 30 sd sd NOUN ejpam-5896 221 31 - - PUNCT ejpam-5896 221 32 component component NOUN ejpam-5896 221 33 . . PUNCT ejpam-5896 222 1 proof proof NOUN ejpam-5896 222 2 . . PUNCT ejpam-5896 223 1 let let VERB ejpam-5896 223 2 ( ( PUNCT ejpam-5896 223 3 t,∆ t,∆ X ejpam-5896 223 4 ) ) PUNCT ejpam-5896 223 5 is be AUX ejpam-5896 223 6 ss ss NOUN ejpam-5896 223 7 - - PUNCT ejpam-5896 223 8 sd sd NOUN ejpam-5896 223 9 - - PUNCT ejpam-5896 223 10 connected connect VERB ejpam-5896 223 11 and and CCONJ ejpam-5896 223 12 ss ss NOUN ejpam-5896 223 13 - - PUNCT ejpam-5896 223 14 sdc sdc NOUN ejpam-5896 223 15 - - PUNCT ejpam-5896 223 16 set set NOUN ejpam-5896 223 17 . . PUNCT ejpam-5896 224 1 assume assume VERB ejpam-5896 224 2 that that SCONJ ejpam-5896 224 3 there there PRON ejpam-5896 224 4 is be VERB ejpam-5896 224 5 an an DET ejpam-5896 224 6 arbitrary arbitrary ADJ ejpam-5896 224 7 ss ss NOUN ejpam-5896 224 8 - - PUNCT ejpam-5896 224 9 sd sd NOUN ejpam-5896 224 10 - - PUNCT ejpam-5896 224 11 connected connect VERB ejpam-5896 224 12 superset superset NOUN ejpam-5896 224 13 ( ( PUNCT ejpam-5896 224 14 s,∆ s,∆ NUM ejpam-5896 224 15 ) ) PUNCT ejpam-5896 224 16 of of ADP ejpam-5896 224 17 ( ( PUNCT ejpam-5896 224 18 t,∆ t,∆ NUM ejpam-5896 224 19 ) ) PUNCT ejpam-5896 224 20 . . PUNCT ejpam-5896 225 1 since since SCONJ ejpam-5896 225 2 clssd(t,∆)∩̃[(t clssd(t,∆)∩̃[(t PROPN ejpam-5896 225 3 c̃,∆)∩̃(s,∆ c̃,∆)∩̃(s,∆ PROPN ejpam-5896 225 4 ) ) PUNCT ejpam-5896 225 5 ] ] PUNCT ejpam-5896 226 1 = = PUNCT ejpam-5896 226 2 φ̃ φ̃ PROPN ejpam-5896 226 3 , , PUNCT ejpam-5896 226 4 ( ( PUNCT ejpam-5896 226 5 t,∆)∩̃clssd[(t t,∆)∩̃clssd[(t PROPN ejpam-5896 226 6 c̃,∆)∩̃(s,∆)]⊆̃(t,∆)∩̃clssd(t c̃,∆)∩̃(s,∆)]⊆̃(t,∆)∩̃clssd(t NOUN ejpam-5896 226 7 c̃,∆ c̃,∆ NOUN ejpam-5896 226 8 ) ) PUNCT ejpam-5896 226 9 = = SYM ejpam-5896 226 10 φ̃ φ̃ PROPN ejpam-5896 226 11 and and CCONJ ejpam-5896 226 12 ( ( PUNCT ejpam-5896 226 13 s,∆ s,∆ NUM ejpam-5896 226 14 ) ) PUNCT ejpam-5896 226 15 = = SYM ejpam-5896 226 16 ( ( PUNCT ejpam-5896 226 17 t,∆)∪̃[(t t,∆)∪̃[(t PROPN ejpam-5896 226 18 c̃,∆)∩̃(s,∆ c̃,∆)∩̃(s,∆ PROPN ejpam-5896 226 19 ) ) PUNCT ejpam-5896 226 20 ] ] PUNCT ejpam-5896 226 21 . . PUNCT ejpam-5896 227 1 therefore therefore ADV ejpam-5896 227 2 , , PUNCT ejpam-5896 227 3 ( ( PUNCT ejpam-5896 227 4 t,∆ t,∆ X ejpam-5896 227 5 ) ) PUNCT ejpam-5896 227 6 and and CCONJ ejpam-5896 227 7 [ [ X ejpam-5896 227 8 ( ( PUNCT ejpam-5896 227 9 t t PROPN ejpam-5896 227 10 c̃,∆)∩̃(s,∆ c̃,∆)∩̃(s,∆ PROPN ejpam-5896 227 11 ) ) PUNCT ejpam-5896 227 12 ] ] PUNCT ejpam-5896 227 13 form form VERB ejpam-5896 227 14 an an DET ejpam-5896 227 15 ss ss NOUN ejpam-5896 227 16 - - PUNCT ejpam-5896 227 17 sd sd NOUN ejpam-5896 227 18 - - PUNCT ejpam-5896 227 19 disconnection disconnection NOUN ejpam-5896 227 20 of of ADP ejpam-5896 227 21 ( ( PUNCT ejpam-5896 227 22 s,∆ s,∆ NUM ejpam-5896 227 23 ) ) PUNCT ejpam-5896 227 24 , , PUNCT ejpam-5896 227 25 which which PRON ejpam-5896 227 26 runs run VERB ejpam-5896 227 27 counter counter ADV ejpam-5896 227 28 to to ADP ejpam-5896 227 29 our our PRON ejpam-5896 227 30 hypothesis hypothesis NOUN ejpam-5896 227 31 . . PUNCT ejpam-5896 228 1 thus thus ADV ejpam-5896 228 2 , , PUNCT ejpam-5896 228 3 ( ( PUNCT ejpam-5896 228 4 t,∆ t,∆ X ejpam-5896 228 5 ) ) PUNCT ejpam-5896 228 6 is be AUX ejpam-5896 228 7 an an DET ejpam-5896 228 8 ss ss VERB ejpam-5896 228 9 - - PUNCT ejpam-5896 228 10 sd sd NOUN ejpam-5896 228 11 - - PUNCT ejpam-5896 228 12 component component NOUN ejpam-5896 228 13 . . PUNCT ejpam-5896 229 1 definition definition NOUN ejpam-5896 229 2 15 15 NUM ejpam-5896 229 3 . . PUNCT ejpam-5896 230 1 let let VERB ejpam-5896 230 2 ψsd ψsd VERB ejpam-5896 230 3 : : PUNCT ejpam-5896 230 4 ( ( PUNCT ejpam-5896 230 5 u u NOUN ejpam-5896 230 6 , , PUNCT ejpam-5896 230 7 τ,∆ τ,∆ NOUN ejpam-5896 230 8 ) ) PUNCT ejpam-5896 230 9 → → SYM ejpam-5896 230 10 ( ( PUNCT ejpam-5896 230 11 v v NOUN ejpam-5896 230 12 , , PUNCT ejpam-5896 230 13 σ σ PROPN ejpam-5896 230 14 , , PUNCT ejpam-5896 230 15 λ λ PROPN ejpam-5896 230 16 ) ) PUNCT ejpam-5896 230 17 be be VERB ejpam-5896 230 18 a a DET ejpam-5896 230 19 soft soft ADJ ejpam-5896 230 20 mapping mapping NOUN ejpam-5896 230 21 with with ADP ejpam-5896 230 22 µ µ NUM ejpam-5896 230 23 , , PUNCT ejpam-5896 230 24 µ∗ µ∗ VERB ejpam-5896 230 25 as as ADP ejpam-5896 230 26 associated associated ADJ ejpam-5896 230 27 sstss sstss NOUN ejpam-5896 230 28 with with ADP ejpam-5896 230 29 τ τ PROPN ejpam-5896 230 30 , , PUNCT ejpam-5896 230 31 σ σ PROPN ejpam-5896 230 32 , , PUNCT ejpam-5896 230 33 respectively respectively ADV ejpam-5896 230 34 . . PUNCT ejpam-5896 231 1 then then ADV ejpam-5896 231 2 , , PUNCT ejpam-5896 231 3 ψsd ψsd PROPN ejpam-5896 231 4 is be AUX ejpam-5896 231 5 called call VERB ejpam-5896 231 6 ss∗-sd ss∗-sd NOUN ejpam-5896 231 7 - - ADJ ejpam-5896 231 8 open open ADJ ejpam-5896 231 9 if if SCONJ ejpam-5896 231 10 ψsd(g,∆ ψsd(g,∆ ADJ ejpam-5896 231 11 ) ) PUNCT ejpam-5896 231 12 ∈ ∈ NOUN ejpam-5896 231 13 sd(v sd(v PUNCT ejpam-5896 231 14 ) ) PUNCT ejpam-5896 231 15 λ λ NOUN ejpam-5896 231 16 for for ADP ejpam-5896 231 17 each each DET ejpam-5896 231 18 φ̃ φ̃ PROPN ejpam-5896 231 19 ̸= ̸= PROPN ejpam-5896 231 20 ( ( PUNCT ejpam-5896 231 21 g,∆ g,∆ PROPN ejpam-5896 231 22 ) ) PUNCT ejpam-5896 231 23 ∈ ∈ PROPN ejpam-5896 231 24 sd(u)∆. sd(u)∆. NOUN ejpam-5896 231 25 theorem theorem VERB ejpam-5896 231 26 6 6 NUM ejpam-5896 231 27 . . PUNCT ejpam-5896 232 1 under under ADP ejpam-5896 232 2 an an DET ejpam-5896 232 3 injective injective ADJ ejpam-5896 232 4 ss∗-sd ss∗-sd ADJ ejpam-5896 232 5 - - PUNCT ejpam-5896 232 6 open open ADJ ejpam-5896 232 7 map map NOUN ejpam-5896 232 8 , , PUNCT ejpam-5896 232 9 the the DET ejpam-5896 232 10 pre pre NOUN ejpam-5896 232 11 - - NOUN ejpam-5896 232 12 image image NOUN ejpam-5896 232 13 of of ADP ejpam-5896 232 14 an an DET ejpam-5896 232 15 ss ss NOUN ejpam-5896 232 16 - - PUNCT ejpam-5896 232 17 sd sd NOUN ejpam-5896 232 18 - - PUNCT ejpam-5896 232 19 connected connect VERB ejpam-5896 232 20 set set NOUN ejpam-5896 232 21 is be AUX ejpam-5896 232 22 ss ss NOUN ejpam-5896 232 23 - - PUNCT ejpam-5896 232 24 sd sd NOUN ejpam-5896 232 25 - - PUNCT ejpam-5896 232 26 connected connect VERB ejpam-5896 232 27 . . PUNCT ejpam-5896 233 1 abd abd PROPN ejpam-5896 233 2 el el PROPN ejpam-5896 233 3 - - PROPN ejpam-5896 233 4 latif latif PROPN ejpam-5896 233 5 et et PROPN ejpam-5896 233 6 al al PROPN ejpam-5896 233 7 . . PUNCT ejpam-5896 233 8 / / SYM ejpam-5896 233 9 eur eur PROPN ejpam-5896 233 10 . . PUNCT ejpam-5896 234 1 j. j. PROPN ejpam-5896 234 2 pure pure PROPN ejpam-5896 234 3 appl appl PROPN ejpam-5896 234 4 . . PROPN ejpam-5896 234 5 math math PROPN ejpam-5896 234 6 , , PUNCT ejpam-5896 234 7 18 18 NUM ejpam-5896 234 8 ( ( PUNCT ejpam-5896 234 9 2 2 NUM ejpam-5896 234 10 ) ) PUNCT ejpam-5896 234 11 ( ( PUNCT ejpam-5896 234 12 2025 2025 NUM ejpam-5896 234 13 ) ) PUNCT ejpam-5896 234 14 , , PUNCT ejpam-5896 234 15 5896 5896 NUM ejpam-5896 234 16 7 7 NUM ejpam-5896 234 17 of of ADP ejpam-5896 234 18 20 20 NUM ejpam-5896 234 19 proof proof NOUN ejpam-5896 234 20 . . PUNCT ejpam-5896 235 1 let let VERB ejpam-5896 235 2 ψsd ψsd VERB ejpam-5896 235 3 : : PUNCT ejpam-5896 235 4 ( ( PUNCT ejpam-5896 235 5 u u NOUN ejpam-5896 235 6 , , PUNCT ejpam-5896 235 7 τ,∆ τ,∆ NOUN ejpam-5896 235 8 ) ) PUNCT ejpam-5896 235 9 → → SYM ejpam-5896 235 10 ( ( PUNCT ejpam-5896 235 11 v v NOUN ejpam-5896 235 12 , , PUNCT ejpam-5896 235 13 σ σ PROPN ejpam-5896 235 14 , , PUNCT ejpam-5896 235 15 λ λ PROPN ejpam-5896 235 16 ) ) PUNCT ejpam-5896 235 17 is be AUX ejpam-5896 235 18 an an DET ejpam-5896 235 19 injective injective ADJ ejpam-5896 235 20 ss∗-sd ss∗-sd ADJ ejpam-5896 235 21 - - ADJ ejpam-5896 235 22 open open ADJ ejpam-5896 235 23 map map NOUN ejpam-5896 235 24 with with ADP ejpam-5896 235 25 µ µ NOUN ejpam-5896 235 26 , , PUNCT ejpam-5896 235 27 µ∗ µ∗ VERB ejpam-5896 235 28 as as SCONJ ejpam-5896 235 29 associated associated ADJ ejpam-5896 235 30 sstss sstss NOUN ejpam-5896 235 31 with with ADP ejpam-5896 235 32 τ τ PROPN ejpam-5896 235 33 , , PUNCT ejpam-5896 235 34 σ σ PROPN ejpam-5896 235 35 , , PUNCT ejpam-5896 235 36 respectively respectively ADV ejpam-5896 235 37 , , PUNCT ejpam-5896 235 38 and and CCONJ ejpam-5896 235 39 ( ( PUNCT ejpam-5896 235 40 w w PROPN ejpam-5896 235 41 , , PUNCT ejpam-5896 235 42 λ λ NOUN ejpam-5896 235 43 ) ) PUNCT ejpam-5896 235 44 is be AUX ejpam-5896 235 45 ss ss NOUN ejpam-5896 235 46 - - PUNCT ejpam-5896 235 47 sd sd NOUN ejpam-5896 235 48 - - PUNCT ejpam-5896 235 49 connected connect VERB ejpam-5896 235 50 subset subset NOUN ejpam-5896 235 51 of of ADP ejpam-5896 235 52 ṽ ṽ PROPN ejpam-5896 235 53 . . PUNCT ejpam-5896 236 1 assume assume VERB ejpam-5896 236 2 conversely conversely ADV ejpam-5896 236 3 ψ−1 ψ−1 PROPN ejpam-5896 236 4 sd sd ADP ejpam-5896 236 5 ( ( PUNCT ejpam-5896 236 6 ( ( PUNCT ejpam-5896 236 7 w w PROPN ejpam-5896 236 8 , , PUNCT ejpam-5896 236 9 λ λ NOUN ejpam-5896 236 10 ) ) PUNCT ejpam-5896 236 11 be be VERB ejpam-5896 236 12 an an DET ejpam-5896 236 13 ss ss NOUN ejpam-5896 236 14 - - PUNCT ejpam-5896 236 15 sd sd NOUN ejpam-5896 236 16 - - PUNCT ejpam-5896 236 17 disconnected disconnected ADJ ejpam-5896 236 18 subset subset NOUN ejpam-5896 236 19 of of ADP ejpam-5896 236 20 ũ ũ PROPN ejpam-5896 236 21 , , PUNCT ejpam-5896 236 22 then then ADV ejpam-5896 236 23 there there PRON ejpam-5896 236 24 are be VERB ejpam-5896 236 25 two two NUM ejpam-5896 236 26 disjoint disjoint NOUN ejpam-5896 236 27 ss ss NOUN ejpam-5896 236 28 - - NOUN ejpam-5896 236 29 sdsubsets sdsubset NOUN ejpam-5896 236 30 ( ( PUNCT ejpam-5896 236 31 a,∆ a,∆ NOUN ejpam-5896 236 32 ) ) PUNCT ejpam-5896 236 33 , , PUNCT ejpam-5896 236 34 ( ( PUNCT ejpam-5896 236 35 b,∆ b,∆ NOUN ejpam-5896 236 36 ) ) PUNCT ejpam-5896 236 37 of of ADP ejpam-5896 236 38 ũ ũ PROPN ejpam-5896 236 39 such such ADJ ejpam-5896 236 40 that that SCONJ ejpam-5896 236 41 ψ−1 ψ−1 PROPN ejpam-5896 236 42 sd sd ADP ejpam-5896 236 43 ( ( PUNCT ejpam-5896 236 44 w w PROPN ejpam-5896 236 45 , , PUNCT ejpam-5896 236 46 λ λ NOUN ejpam-5896 236 47 ) ) PUNCT ejpam-5896 236 48 = = SYM ejpam-5896 236 49 ( ( PUNCT ejpam-5896 236 50 a,∆)∪̃(b,∆ a,∆)∪̃(b,∆ PROPN ejpam-5896 236 51 ) ) PUNCT ejpam-5896 236 52 . . PUNCT ejpam-5896 237 1 since since SCONJ ejpam-5896 237 2 ψsd ψsd PROPN ejpam-5896 237 3 is be AUX ejpam-5896 237 4 ss∗-sd ss∗-sd ADJ ejpam-5896 237 5 - - ADJ ejpam-5896 237 6 open open ADJ ejpam-5896 237 7 , , PUNCT ejpam-5896 237 8 ψsd(a,∆ ψsd(a,∆ NOUN ejpam-5896 237 9 ) ) PUNCT ejpam-5896 237 10 and and CCONJ ejpam-5896 237 11 ψsd(b,∆ ψsd(b,∆ X ejpam-5896 237 12 ) ) PUNCT ejpam-5896 237 13 are be AUX ejpam-5896 237 14 disjoint disjoint ADJ ejpam-5896 237 15 ss ss NOUN ejpam-5896 237 16 - - PUNCT ejpam-5896 237 17 sd sd NOUN ejpam-5896 237 18 - - PUNCT ejpam-5896 237 19 subsets subset NOUN ejpam-5896 237 20 of of ADP ejpam-5896 237 21 ṽ ṽ PROPN ejpam-5896 237 22 . . PUNCT ejpam-5896 238 1 hence hence ADV ejpam-5896 238 2 , , PUNCT ejpam-5896 238 3 ( ( PUNCT ejpam-5896 238 4 w w PROPN ejpam-5896 238 5 , , PUNCT ejpam-5896 238 6 λ λ NOUN ejpam-5896 238 7 ) ) PUNCT ejpam-5896 238 8 = = SYM ejpam-5896 239 1 ψsd(ψ ψsd(ψ NOUN ejpam-5896 239 2 −1 −1 NOUN ejpam-5896 239 3 sd sd INTJ ejpam-5896 239 4 ( ( PUNCT ejpam-5896 239 5 ( ( PUNCT ejpam-5896 239 6 w w PROPN ejpam-5896 239 7 , , PUNCT ejpam-5896 239 8 λ λ NOUN ejpam-5896 239 9 ) ) PUNCT ejpam-5896 239 10 ) ) PUNCT ejpam-5896 240 1 = = SYM ejpam-5896 240 2 ψsd(a,∆)∪̃ψsd(b,∆ ψsd(a,∆)∪̃ψsd(b,∆ PROPN ejpam-5896 240 3 ) ) PUNCT ejpam-5896 240 4 , , PUNCT ejpam-5896 240 5 ψsd ψsd PROPN ejpam-5896 240 6 is be AUX ejpam-5896 240 7 surjective surjective ADJ ejpam-5896 240 8 , , PUNCT ejpam-5896 240 9 which which PRON ejpam-5896 240 10 is be AUX ejpam-5896 240 11 a a DET ejpam-5896 240 12 contradiction contradiction NOUN ejpam-5896 240 13 to to ADP ejpam-5896 240 14 our our PRON ejpam-5896 240 15 hypothesis hypothesis NOUN ejpam-5896 240 16 . . PUNCT ejpam-5896 241 1 therefore therefore ADV ejpam-5896 241 2 , , PUNCT ejpam-5896 241 3 ψ−1 ψ−1 PROPN ejpam-5896 241 4 sd sd ADP ejpam-5896 241 5 ( ( PUNCT ejpam-5896 241 6 ( ( PUNCT ejpam-5896 241 7 w w PROPN ejpam-5896 241 8 , , PUNCT ejpam-5896 241 9 λ λ NOUN ejpam-5896 241 10 ) ) PUNCT ejpam-5896 241 11 is be AUX ejpam-5896 241 12 an an DET ejpam-5896 241 13 ss ss NOUN ejpam-5896 241 14 - - PUNCT ejpam-5896 241 15 sd sd NOUN ejpam-5896 241 16 - - PUNCT ejpam-5896 241 17 connected connect VERB ejpam-5896 241 18 subset subset NOUN ejpam-5896 241 19 of of ADP ejpam-5896 241 20 ũ ũ PROPN ejpam-5896 241 21 . . PUNCT ejpam-5896 242 1 theorem theorem VERB ejpam-5896 242 2 7 7 NUM ejpam-5896 242 3 . . PUNCT ejpam-5896 243 1 under under ADP ejpam-5896 243 2 a a DET ejpam-5896 243 3 bijective bijective ADJ ejpam-5896 243 4 ss∗-sd ss∗-sd ADJ ejpam-5896 243 5 - - PUNCT ejpam-5896 243 6 open open ADJ ejpam-5896 243 7 map map NOUN ejpam-5896 243 8 , , PUNCT ejpam-5896 243 9 the the DET ejpam-5896 243 10 image image NOUN ejpam-5896 243 11 of of ADP ejpam-5896 243 12 an an DET ejpam-5896 243 13 ss ss NOUN ejpam-5896 243 14 - - PUNCT ejpam-5896 243 15 sd sd NOUN ejpam-5896 243 16 - - PUNCT ejpam-5896 243 17 component component NOUN ejpam-5896 243 18 is be AUX ejpam-5896 243 19 ss ss NOUN ejpam-5896 243 20 - - PUNCT ejpam-5896 243 21 sdcomponent sdcomponent NOUN ejpam-5896 243 22 . . PUNCT ejpam-5896 244 1 proof proof NOUN ejpam-5896 244 2 . . PUNCT ejpam-5896 245 1 let let VERB ejpam-5896 245 2 ψsd ψsd VERB ejpam-5896 245 3 : : PUNCT ejpam-5896 245 4 ( ( PUNCT ejpam-5896 245 5 u u NOUN ejpam-5896 245 6 , , PUNCT ejpam-5896 245 7 τ,∆ τ,∆ NOUN ejpam-5896 245 8 ) ) PUNCT ejpam-5896 245 9 → → SYM ejpam-5896 245 10 ( ( PUNCT ejpam-5896 245 11 v v NOUN ejpam-5896 245 12 , , PUNCT ejpam-5896 245 13 σ σ PROPN ejpam-5896 245 14 , , PUNCT ejpam-5896 245 15 λ λ PROPN ejpam-5896 245 16 ) ) PUNCT ejpam-5896 245 17 is be AUX ejpam-5896 245 18 a a DET ejpam-5896 245 19 bijective bijective ADJ ejpam-5896 245 20 ss∗-sd ss∗-sd ADJ ejpam-5896 245 21 - - ADJ ejpam-5896 245 22 open open ADJ ejpam-5896 245 23 map map NOUN ejpam-5896 245 24 with with ADP ejpam-5896 245 25 µ µ NOUN ejpam-5896 245 26 , , PUNCT ejpam-5896 245 27 µ∗ µ∗ VERB ejpam-5896 245 28 as as SCONJ ejpam-5896 245 29 associated associated ADJ ejpam-5896 245 30 sstss sstss NOUN ejpam-5896 245 31 with with ADP ejpam-5896 245 32 τ τ PROPN ejpam-5896 245 33 , , PUNCT ejpam-5896 245 34 σ σ PROPN ejpam-5896 245 35 , , PUNCT ejpam-5896 245 36 respectively respectively ADV ejpam-5896 245 37 , , PUNCT ejpam-5896 245 38 such such ADJ ejpam-5896 245 39 that that SCONJ ejpam-5896 245 40 ( ( PUNCT ejpam-5896 245 41 g,∆ g,∆ X ejpam-5896 245 42 ) ) PUNCT ejpam-5896 245 43 is be AUX ejpam-5896 245 44 ss ss NOUN ejpam-5896 245 45 - - PUNCT ejpam-5896 245 46 sd sd NOUN ejpam-5896 245 47 - - PUNCT ejpam-5896 245 48 component component NOUN ejpam-5896 245 49 subsets subset NOUN ejpam-5896 245 50 of of ADP ejpam-5896 245 51 ũ ũ PROPN ejpam-5896 245 52 . . PUNCT ejpam-5896 246 1 assume assume VERB ejpam-5896 246 2 instead instead ADV ejpam-5896 246 3 that that PRON ejpam-5896 246 4 , , PUNCT ejpam-5896 246 5 ψsd(g,∆ ψsd(g,∆ PUNCT ejpam-5896 246 6 ) ) PUNCT ejpam-5896 246 7 is be AUX ejpam-5896 246 8 not not PART ejpam-5896 246 9 ss ss NOUN ejpam-5896 246 10 - - PUNCT ejpam-5896 246 11 sd sd NOUN ejpam-5896 246 12 - - PUNCT ejpam-5896 246 13 component component NOUN ejpam-5896 246 14 subsets subset NOUN ejpam-5896 246 15 of of ADP ejpam-5896 246 16 ṽ ṽ PROPN ejpam-5896 246 17 . . PUNCT ejpam-5896 247 1 hence hence ADV ejpam-5896 247 2 , , PUNCT ejpam-5896 247 3 there there PRON ejpam-5896 247 4 is be VERB ejpam-5896 247 5 an an DET ejpam-5896 247 6 ss ss NOUN ejpam-5896 247 7 - - PUNCT ejpam-5896 247 8 sdconnected sdconnecte VERB ejpam-5896 247 9 subset subset NOUN ejpam-5896 247 10 ( ( PUNCT ejpam-5896 247 11 h h NOUN ejpam-5896 247 12 , , PUNCT ejpam-5896 247 13 λ λ NOUN ejpam-5896 247 14 ) ) PUNCT ejpam-5896 247 15 of of ADP ejpam-5896 247 16 ṽ ṽ PROPN ejpam-5896 247 17 such such ADJ ejpam-5896 247 18 that that SCONJ ejpam-5896 247 19 ψsd(g,∆)⊆̃(h ψsd(g,∆)⊆̃(h NOUN ejpam-5896 247 20 , , PUNCT ejpam-5896 247 21 λ λ NOUN ejpam-5896 247 22 ) ) PUNCT ejpam-5896 247 23 which which PRON ejpam-5896 247 24 follows follow VERB ejpam-5896 247 25 ( ( PUNCT ejpam-5896 247 26 g,∆)⊆̃ψ−1 g,∆)⊆̃ψ−1 NOUN ejpam-5896 247 27 sd sd ADP ejpam-5896 247 28 ( ( PUNCT ejpam-5896 247 29 ψsd(g,∆))⊆̃ψ−1 ψsd(g,∆))⊆̃ψ−1 INTJ ejpam-5896 247 30 sd sd PROPN ejpam-5896 247 31 ( ( PUNCT ejpam-5896 247 32 h h NOUN ejpam-5896 247 33 , , PUNCT ejpam-5896 247 34 λ λ NOUN ejpam-5896 247 35 ) ) PUNCT ejpam-5896 247 36 , , PUNCT ejpam-5896 247 37 ψsd ψsd PROPN ejpam-5896 247 38 is be AUX ejpam-5896 247 39 surjective surjective ADJ ejpam-5896 247 40 . . PUNCT ejpam-5896 248 1 since since SCONJ ejpam-5896 248 2 ψsd ψsd PROPN ejpam-5896 248 3 is be AUX ejpam-5896 248 4 injective injective ADJ ejpam-5896 248 5 ss∗-sd ss∗-sd NOUN ejpam-5896 248 6 - - ADJ ejpam-5896 248 7 open open ADJ ejpam-5896 248 8 , , PUNCT ejpam-5896 248 9 ψ−1 ψ−1 PROPN ejpam-5896 248 10 sd sd ADP ejpam-5896 248 11 ( ( PUNCT ejpam-5896 248 12 h h NOUN ejpam-5896 248 13 , , PUNCT ejpam-5896 248 14 λ λ NOUN ejpam-5896 248 15 ) ) PUNCT ejpam-5896 248 16 is be AUX ejpam-5896 248 17 an an DET ejpam-5896 248 18 ss ss NOUN ejpam-5896 248 19 - - PUNCT ejpam-5896 248 20 sd sd NOUN ejpam-5896 248 21 - - PUNCT ejpam-5896 248 22 connected connect VERB ejpam-5896 248 23 subset subset NOUN ejpam-5896 248 24 of of ADP ejpam-5896 248 25 ũ ũ PROPN ejpam-5896 248 26 , , PUNCT ejpam-5896 248 27 from from ADP ejpam-5896 248 28 theorem theorem NOUN ejpam-5896 248 29 6 6 NUM ejpam-5896 248 30 , , PUNCT ejpam-5896 248 31 it it PRON ejpam-5896 248 32 conflicts conflict VERB ejpam-5896 248 33 with with ADP ejpam-5896 248 34 that that PRON ejpam-5896 248 35 ( ( PUNCT ejpam-5896 248 36 g,∆ g,∆ PROPN ejpam-5896 248 37 ) ) PUNCT ejpam-5896 248 38 is be AUX ejpam-5896 248 39 an an DET ejpam-5896 248 40 ss ss VERB ejpam-5896 248 41 - - PUNCT ejpam-5896 248 42 sd sd NOUN ejpam-5896 248 43 - - PUNCT ejpam-5896 248 44 component component NOUN ejpam-5896 248 45 . . PUNCT ejpam-5896 249 1 therefore therefore ADV ejpam-5896 249 2 , , PUNCT ejpam-5896 249 3 ψsd(g,∆ ψsd(g,∆ PUNCT ejpam-5896 249 4 ) ) PUNCT ejpam-5896 249 5 is be AUX ejpam-5896 249 6 an an DET ejpam-5896 249 7 ss ss NOUN ejpam-5896 249 8 - - PUNCT ejpam-5896 249 9 sdcomponent sdcomponent NOUN ejpam-5896 249 10 subset subset NOUN ejpam-5896 249 11 of of ADP ejpam-5896 249 12 ṽ ṽ PROPN ejpam-5896 249 13 . . PUNCT ejpam-5896 250 1 definition definition NOUN ejpam-5896 250 2 16 16 NUM ejpam-5896 250 3 . . PUNCT ejpam-5896 251 1 [ [ X ejpam-5896 251 2 58 58 NUM ejpam-5896 251 3 ] ] PUNCT ejpam-5896 251 4 given give VERB ejpam-5896 251 5 the the DET ejpam-5896 251 6 soft soft ADJ ejpam-5896 251 7 point point NOUN ejpam-5896 251 8 sγ sγ NOUN ejpam-5896 251 9 in in ADP ejpam-5896 251 10 an an DET ejpam-5896 251 11 ssts sst NOUN ejpam-5896 251 12 ( ( PUNCT ejpam-5896 251 13 u u NOUN ejpam-5896 251 14 , , PUNCT ejpam-5896 251 15 µ,∆ µ,∆ NUM ejpam-5896 251 16 ) ) PUNCT ejpam-5896 251 17 . . PUNCT ejpam-5896 252 1 then then ADV ejpam-5896 252 2 ũ ũ PROPN ejpam-5896 252 3 is be AUX ejpam-5896 252 4 stated state VERB ejpam-5896 252 5 to to PART ejpam-5896 252 6 be be AUX ejpam-5896 252 7 an an DET ejpam-5896 252 8 ssl ssl ADJ ejpam-5896 252 9 - - VERB ejpam-5896 252 10 connected connected ADJ ejpam-5896 252 11 at at ADP ejpam-5896 252 12 sγ sγ PRON ejpam-5896 252 13 if if SCONJ ejpam-5896 252 14 each each DET ejpam-5896 252 15 ss ss NOUN ejpam-5896 252 16 - - PUNCT ejpam-5896 252 17 neighbourhood neighbourhood NOUN ejpam-5896 252 18 ( ( PUNCT ejpam-5896 252 19 a,∆ a,∆ NOUN ejpam-5896 252 20 ) ) PUNCT ejpam-5896 252 21 of of ADP ejpam-5896 252 22 sγ sγ PRON ejpam-5896 252 23 there there PRON ejpam-5896 252 24 is be VERB ejpam-5896 252 25 ( ( PUNCT ejpam-5896 252 26 g,∆)⊆̃(a,∆ g,∆)⊆̃(a,∆ ADJ ejpam-5896 252 27 ) ) PUNCT ejpam-5896 252 28 , , PUNCT ejpam-5896 252 29 which which PRON ejpam-5896 252 30 is be AUX ejpam-5896 252 31 ss ss NOUN ejpam-5896 252 32 - - PUNCT ejpam-5896 252 33 connected connect VERB ejpam-5896 252 34 and and CCONJ ejpam-5896 252 35 containing contain VERB ejpam-5896 252 36 sγ sγ NOUN ejpam-5896 252 37 . . PUNCT ejpam-5896 253 1 if if SCONJ ejpam-5896 253 2 ũ ũ PROPN ejpam-5896 253 3 is be AUX ejpam-5896 253 4 ssl ssl ADJ ejpam-5896 253 5 - - ADJ ejpam-5896 253 6 connected connect VERB ejpam-5896 253 7 at at ADP ejpam-5896 253 8 all all PRON ejpam-5896 253 9 of of ADP ejpam-5896 253 10 its its PRON ejpam-5896 253 11 soft soft ADJ ejpam-5896 253 12 points point NOUN ejpam-5896 253 13 , , PUNCT ejpam-5896 253 14 then then ADV ejpam-5896 253 15 it it PRON ejpam-5896 253 16 is be AUX ejpam-5896 253 17 stated state VERB ejpam-5896 253 18 to to PART ejpam-5896 253 19 be be AUX ejpam-5896 253 20 an an DET ejpam-5896 253 21 ssl ssl ADJ ejpam-5896 253 22 - - VERB ejpam-5896 253 23 connected connect VERB ejpam-5896 253 24 . . PUNCT ejpam-5896 254 1 definition definition NOUN ejpam-5896 254 2 17 17 NUM ejpam-5896 254 3 . . PUNCT ejpam-5896 255 1 an an DET ejpam-5896 255 2 ssts sst NOUN ejpam-5896 255 3 ( ( PUNCT ejpam-5896 255 4 u u NOUN ejpam-5896 255 5 , , PUNCT ejpam-5896 255 6 µ,∆ µ,∆ NUM ejpam-5896 255 7 ) ) PUNCT ejpam-5896 255 8 is be AUX ejpam-5896 255 9 said say VERB ejpam-5896 255 10 to to PART ejpam-5896 255 11 be be AUX ejpam-5896 255 12 ssl ssl ADJ ejpam-5896 255 13 - - PUNCT ejpam-5896 255 14 sd sd NOUN ejpam-5896 255 15 - - PUNCT ejpam-5896 255 16 connected connect VERB ejpam-5896 255 17 at at ADP ejpam-5896 255 18 a a DET ejpam-5896 255 19 soft soft ADJ ejpam-5896 255 20 point point NOUN ejpam-5896 255 21 sγ sγ INTJ ejpam-5896 255 22 if if SCONJ ejpam-5896 255 23 every every DET ejpam-5896 255 24 ss ss NOUN ejpam-5896 255 25 - - PUNCT ejpam-5896 255 26 sd sd NOUN ejpam-5896 255 27 - - PUNCT ejpam-5896 255 28 neighbourhood neighbourhood NOUN ejpam-5896 255 29 ( ( PUNCT ejpam-5896 255 30 a,∆ a,∆ NOUN ejpam-5896 255 31 ) ) PUNCT ejpam-5896 255 32 of of ADP ejpam-5896 255 33 sγ sγ NOUN ejpam-5896 255 34 there there PRON ejpam-5896 255 35 is be VERB ejpam-5896 255 36 an an DET ejpam-5896 255 37 ss ss NOUN ejpam-5896 255 38 - - PUNCT ejpam-5896 255 39 sd sd NOUN ejpam-5896 255 40 - - PUNCT ejpam-5896 255 41 connected connect VERB ejpam-5896 255 42 subset subset NOUN ejpam-5896 255 43 ( ( PUNCT ejpam-5896 255 44 g,∆ g,∆ PROPN ejpam-5896 255 45 ) ) PUNCT ejpam-5896 255 46 of of ADP ejpam-5896 255 47 ( ( PUNCT ejpam-5896 255 48 a,∆ a,∆ NOUN ejpam-5896 255 49 ) ) PUNCT ejpam-5896 255 50 containing contain VERB ejpam-5896 255 51 sγ sγ NOUN ejpam-5896 255 52 . . PUNCT ejpam-5896 256 1 if if SCONJ ejpam-5896 256 2 ũ ũ PROPN ejpam-5896 256 3 is be AUX ejpam-5896 256 4 ssl ssl ADJ ejpam-5896 256 5 - - PUNCT ejpam-5896 256 6 sd sd NOUN ejpam-5896 256 7 - - PUNCT ejpam-5896 256 8 connected connect VERB ejpam-5896 256 9 at at ADP ejpam-5896 256 10 all all PRON ejpam-5896 256 11 of of ADP ejpam-5896 256 12 its its PRON ejpam-5896 256 13 soft soft ADJ ejpam-5896 256 14 points point NOUN ejpam-5896 256 15 , , PUNCT ejpam-5896 256 16 then then ADV ejpam-5896 256 17 it it PRON ejpam-5896 256 18 is be AUX ejpam-5896 256 19 stated state VERB ejpam-5896 256 20 to to PART ejpam-5896 256 21 be be AUX ejpam-5896 256 22 an an DET ejpam-5896 256 23 ssl ssl ADJ ejpam-5896 256 24 - - PUNCT ejpam-5896 256 25 sd sd NOUN ejpam-5896 256 26 - - PUNCT ejpam-5896 256 27 connected connect VERB ejpam-5896 256 28 . . PUNCT ejpam-5896 257 1 proposition proposition NOUN ejpam-5896 257 2 3 3 NUM ejpam-5896 257 3 . . PUNCT ejpam-5896 258 1 ( ( PUNCT ejpam-5896 258 2 1 1 X ejpam-5896 258 3 ) ) PUNCT ejpam-5896 258 4 every every DET ejpam-5896 258 5 ss ss NOUN ejpam-5896 258 6 - - PUNCT ejpam-5896 258 7 sd sd NOUN ejpam-5896 258 8 - - PUNCT ejpam-5896 258 9 connected connect VERB ejpam-5896 258 10 is be AUX ejpam-5896 258 11 ssl ssl ADJ ejpam-5896 258 12 - - PUNCT ejpam-5896 258 13 sd sd NOUN ejpam-5896 258 14 - - PUNCT ejpam-5896 258 15 connected connect VERB ejpam-5896 258 16 . . PUNCT ejpam-5896 259 1 ( ( PUNCT ejpam-5896 259 2 2 2 X ejpam-5896 259 3 ) ) PUNCT ejpam-5896 259 4 every every DET ejpam-5896 259 5 ssl ssl PROPN ejpam-5896 259 6 - - PUNCT ejpam-5896 259 7 sd sd NOUN ejpam-5896 259 8 - - PUNCT ejpam-5896 259 9 connected connect VERB ejpam-5896 259 10 is be AUX ejpam-5896 259 11 ssl ssl ADJ ejpam-5896 259 12 - - VERB ejpam-5896 259 13 connected connect VERB ejpam-5896 259 14 . . PUNCT ejpam-5896 260 1 proof proof NOUN ejpam-5896 260 2 . . PUNCT ejpam-5896 261 1 ( ( PUNCT ejpam-5896 261 2 1 1 X ejpam-5896 261 3 ) ) PUNCT ejpam-5896 261 4 let let AUX ejpam-5896 261 5 sγ∈̃ũ sγ∈̃ũ VERB ejpam-5896 261 6 such such ADJ ejpam-5896 261 7 that that SCONJ ejpam-5896 261 8 ũ ũ PROPN ejpam-5896 261 9 is be AUX ejpam-5896 261 10 ss ss NOUN ejpam-5896 261 11 - - PUNCT ejpam-5896 261 12 sd sd NOUN ejpam-5896 261 13 - - PUNCT ejpam-5896 261 14 connected connect VERB ejpam-5896 261 15 . . PUNCT ejpam-5896 262 1 consequently consequently ADV ejpam-5896 262 2 , , PUNCT ejpam-5896 262 3 ũ ũ PROPN ejpam-5896 262 4 does do AUX ejpam-5896 262 5 not not PART ejpam-5896 262 6 have have VERB ejpam-5896 262 7 a a DET ejpam-5896 262 8 proper proper ADJ ejpam-5896 262 9 ss ss NOUN ejpam-5896 262 10 - - NOUN ejpam-5896 262 11 sdcset sdcset NOUN ejpam-5896 262 12 . . PUNCT ejpam-5896 263 1 hence hence ADV ejpam-5896 263 2 , , PUNCT ejpam-5896 263 3 sγ∈̃ũ⊆̃ũ sγ∈̃ũ⊆̃ũ PROPN ejpam-5896 263 4 . . PUNCT ejpam-5896 264 1 therefore therefore ADV ejpam-5896 264 2 , , PUNCT ejpam-5896 264 3 ũ ũ PROPN ejpam-5896 264 4 is be AUX ejpam-5896 264 5 an an DET ejpam-5896 264 6 ssl ssl ADJ ejpam-5896 264 7 - - PUNCT ejpam-5896 264 8 sd sd NOUN ejpam-5896 264 9 - - PUNCT ejpam-5896 264 10 connected connect VERB ejpam-5896 264 11 at at ADP ejpam-5896 264 12 the the DET ejpam-5896 264 13 arbitrary arbitrary ADJ ejpam-5896 264 14 soft soft ADJ ejpam-5896 264 15 point point NOUN ejpam-5896 264 16 sγ sγ NOUN ejpam-5896 265 1 and and CCONJ ejpam-5896 265 2 hence hence ADV ejpam-5896 265 3 it it PRON ejpam-5896 265 4 is be AUX ejpam-5896 265 5 an an DET ejpam-5896 265 6 ssl ssl ADJ ejpam-5896 265 7 - - PUNCT ejpam-5896 265 8 sd sd NOUN ejpam-5896 265 9 - - PUNCT ejpam-5896 265 10 connected connect VERB ejpam-5896 265 11 . . PUNCT ejpam-5896 266 1 ( ( PUNCT ejpam-5896 266 2 2 2 X ejpam-5896 266 3 ) ) PUNCT ejpam-5896 266 4 follows follow VERB ejpam-5896 266 5 from from ADP ejpam-5896 266 6 ( ( PUNCT ejpam-5896 266 7 1 1 NUM ejpam-5896 266 8 ) ) PUNCT ejpam-5896 266 9 . . PUNCT ejpam-5896 267 1 remark remark PROPN ejpam-5896 267 2 1 1 NUM ejpam-5896 267 3 . . PUNCT ejpam-5896 268 1 in in ADP ejpam-5896 268 2 general general ADJ ejpam-5896 268 3 , , PUNCT ejpam-5896 268 4 the the DET ejpam-5896 268 5 opposite opposite NOUN ejpam-5896 268 6 of of ADP ejpam-5896 268 7 proposition proposition NOUN ejpam-5896 268 8 3 3 NUM ejpam-5896 268 9 is be AUX ejpam-5896 268 10 not not PART ejpam-5896 268 11 satisfied satisfied ADJ ejpam-5896 268 12 , , PUNCT ejpam-5896 268 13 as as SCONJ ejpam-5896 268 14 demonstrated demonstrate VERB ejpam-5896 268 15 by by ADP ejpam-5896 268 16 the the DET ejpam-5896 268 17 example example NOUN ejpam-5896 268 18 that that PRON ejpam-5896 268 19 follows follow VERB ejpam-5896 268 20 . . PUNCT ejpam-5896 269 1 abd abd PROPN ejpam-5896 269 2 el el PROPN ejpam-5896 269 3 - - PROPN ejpam-5896 269 4 latif latif PROPN ejpam-5896 269 5 et et PROPN ejpam-5896 269 6 al al PROPN ejpam-5896 269 7 . . PUNCT ejpam-5896 269 8 / / SYM ejpam-5896 269 9 eur eur PROPN ejpam-5896 269 10 . . PUNCT ejpam-5896 270 1 j. j. PROPN ejpam-5896 270 2 pure pure PROPN ejpam-5896 270 3 appl appl PROPN ejpam-5896 270 4 . . PROPN ejpam-5896 270 5 math math PROPN ejpam-5896 270 6 , , PUNCT ejpam-5896 270 7 18 18 NUM ejpam-5896 270 8 ( ( PUNCT ejpam-5896 270 9 2 2 NUM ejpam-5896 270 10 ) ) PUNCT ejpam-5896 270 11 ( ( PUNCT ejpam-5896 270 12 2025 2025 NUM ejpam-5896 270 13 ) ) PUNCT ejpam-5896 270 14 , , PUNCT ejpam-5896 270 15 5896 5896 NUM ejpam-5896 270 16 8 8 NUM ejpam-5896 270 17 of of ADP ejpam-5896 270 18 20 20 NUM ejpam-5896 270 19 example example NOUN ejpam-5896 270 20 2 2 NUM ejpam-5896 270 21 . . PUNCT ejpam-5896 271 1 let let VERB ejpam-5896 271 2 u u PRON ejpam-5896 271 3 = = X ejpam-5896 271 4 { { PUNCT ejpam-5896 271 5 f1 f1 NOUN ejpam-5896 271 6 , , PUNCT ejpam-5896 271 7 f2 f2 PROPN ejpam-5896 271 8 , , PUNCT ejpam-5896 271 9 f3 f3 PROPN ejpam-5896 271 10 } } PUNCT ejpam-5896 271 11 , , PUNCT ejpam-5896 271 12 ∆ ∆ X ejpam-5896 271 13 = = SYM ejpam-5896 271 14 { { PUNCT ejpam-5896 271 15 γ1 γ1 PROPN ejpam-5896 271 16 , , PUNCT ejpam-5896 271 17 γ2 γ2 NOUN ejpam-5896 271 18 } } PUNCT ejpam-5896 271 19 and and CCONJ ejpam-5896 271 20 µ µ X ejpam-5896 271 21 = = SYM ejpam-5896 271 22 { { PUNCT ejpam-5896 271 23 ũ ũ PROPN ejpam-5896 271 24 , , PUNCT ejpam-5896 271 25 φ̃ φ̃ PROPN ejpam-5896 271 26 , , PUNCT ejpam-5896 271 27 ( ( PUNCT ejpam-5896 271 28 xi,∆ xi,∆ PROPN ejpam-5896 271 29 ) ) PUNCT ejpam-5896 271 30 , , PUNCT ejpam-5896 271 31 i i PRON ejpam-5896 271 32 = = NOUN ejpam-5896 271 33 1 1 NUM ejpam-5896 271 34 , , PUNCT ejpam-5896 271 35 2 2 NUM ejpam-5896 271 36 , , PUNCT ejpam-5896 271 37 3 3 NUM ejpam-5896 271 38 , , PUNCT ejpam-5896 271 39 4 4 NUM ejpam-5896 271 40 } } PUNCT ejpam-5896 271 41 be be AUX ejpam-5896 271 42 an an DET ejpam-5896 271 43 ssts sst NOUN ejpam-5896 271 44 on on ADP ejpam-5896 271 45 u u NOUN ejpam-5896 271 46 , , PUNCT ejpam-5896 271 47 where where SCONJ ejpam-5896 271 48 : : PUNCT ejpam-5896 271 49 x1(γ1 x1(γ1 X ejpam-5896 271 50 ) ) PUNCT ejpam-5896 271 51 = = SYM ejpam-5896 271 52 { { PUNCT ejpam-5896 271 53 f1 f1 NOUN ejpam-5896 271 54 } } PUNCT ejpam-5896 271 55 , , PUNCT ejpam-5896 271 56 x1(γ2 x1(γ2 NOUN ejpam-5896 271 57 ) ) PUNCT ejpam-5896 271 58 = = PUNCT ejpam-5896 272 1 φ φ PROPN ejpam-5896 272 2 . . PUNCT ejpam-5896 273 1 x2(γ1 x2(γ1 NOUN ejpam-5896 273 2 ) ) PUNCT ejpam-5896 273 3 = = SYM ejpam-5896 273 4 { { PUNCT ejpam-5896 273 5 f1 f1 NOUN ejpam-5896 273 6 } } PUNCT ejpam-5896 273 7 , , PUNCT ejpam-5896 273 8 x2(γ2 x2(γ2 PROPN ejpam-5896 273 9 ) ) PUNCT ejpam-5896 273 10 = = SYM ejpam-5896 273 11 { { PUNCT ejpam-5896 273 12 f1 f1 NOUN ejpam-5896 273 13 } } PUNCT ejpam-5896 273 14 . . PUNCT ejpam-5896 274 1 x3(γ1 x3(γ1 NOUN ejpam-5896 274 2 ) ) PUNCT ejpam-5896 274 3 = = SYM ejpam-5896 274 4 { { PUNCT ejpam-5896 274 5 f1 f1 NOUN ejpam-5896 274 6 , , PUNCT ejpam-5896 274 7 f2 f2 PROPN ejpam-5896 274 8 } } PUNCT ejpam-5896 274 9 , , PUNCT ejpam-5896 274 10 x3(γ2 x3(γ2 NOUN ejpam-5896 274 11 ) ) PUNCT ejpam-5896 274 12 = = SYM ejpam-5896 274 13 { { PUNCT ejpam-5896 274 14 f1 f1 NOUN ejpam-5896 274 15 , , PUNCT ejpam-5896 274 16 f3 f3 PROPN ejpam-5896 274 17 } } PUNCT ejpam-5896 274 18 . . PUNCT ejpam-5896 275 1 x4(γ1 x4(γ1 NOUN ejpam-5896 275 2 ) ) PUNCT ejpam-5896 275 3 = = SYM ejpam-5896 275 4 { { PUNCT ejpam-5896 275 5 f2 f2 PROPN ejpam-5896 275 6 } } PUNCT ejpam-5896 275 7 , , PUNCT ejpam-5896 275 8 x4(γ2 x4(γ2 NOUN ejpam-5896 275 9 ) ) PUNCT ejpam-5896 275 10 = = SYM ejpam-5896 275 11 { { PUNCT ejpam-5896 275 12 f1 f1 NOUN ejpam-5896 275 13 , , PUNCT ejpam-5896 275 14 f3 f3 PROPN ejpam-5896 275 15 } } PUNCT ejpam-5896 275 16 . . PUNCT ejpam-5896 276 1 for for ADP ejpam-5896 276 2 the the DET ejpam-5896 276 3 soft soft ADJ ejpam-5896 276 4 sets set NOUN ejpam-5896 276 5 ( ( PUNCT ejpam-5896 276 6 g,∆ g,∆ PROPN ejpam-5896 276 7 ) ) PUNCT ejpam-5896 276 8 , , PUNCT ejpam-5896 276 9 ( ( PUNCT ejpam-5896 276 10 h,∆ h,∆ NUM ejpam-5896 276 11 ) ) PUNCT ejpam-5896 276 12 , , PUNCT ejpam-5896 276 13 where where SCONJ ejpam-5896 276 14 : : PUNCT ejpam-5896 276 15 g(γ1 g(γ1 NOUN ejpam-5896 276 16 ) ) PUNCT ejpam-5896 277 1 = = PRON ejpam-5896 277 2 { { PUNCT ejpam-5896 277 3 f2 f2 PROPN ejpam-5896 277 4 , , PUNCT ejpam-5896 277 5 f3 f3 ADJ ejpam-5896 277 6 } } PUNCT ejpam-5896 277 7 , , PUNCT ejpam-5896 277 8 g(γ2 g(γ2 NOUN ejpam-5896 277 9 ) ) PUNCT ejpam-5896 277 10 = = SYM ejpam-5896 277 11 { { PUNCT ejpam-5896 277 12 f1 f1 NOUN ejpam-5896 277 13 , , PUNCT ejpam-5896 277 14 f3 f3 PROPN ejpam-5896 277 15 } } PUNCT ejpam-5896 277 16 . . PUNCT ejpam-5896 278 1 h(γ1 h(γ1 NOUN ejpam-5896 278 2 ) ) PUNCT ejpam-5896 279 1 = = PRON ejpam-5896 279 2 { { PUNCT ejpam-5896 279 3 f1 f1 NOUN ejpam-5896 279 4 } } PUNCT ejpam-5896 279 5 , , PUNCT ejpam-5896 279 6 h(γ2 h(γ2 NOUN ejpam-5896 279 7 ) ) PUNCT ejpam-5896 279 8 = = SYM ejpam-5896 279 9 { { PUNCT ejpam-5896 279 10 f2 f2 PROPN ejpam-5896 279 11 } } PUNCT ejpam-5896 279 12 . . PUNCT ejpam-5896 280 1 it it PRON ejpam-5896 280 2 ’s ’ VERB ejpam-5896 280 3 simple simple ADJ ejpam-5896 280 4 to to PART ejpam-5896 280 5 verify verify VERB ejpam-5896 280 6 that that SCONJ ejpam-5896 280 7 ũ ũ PROPN ejpam-5896 280 8 is be AUX ejpam-5896 280 9 an an DET ejpam-5896 280 10 ssl ssl ADJ ejpam-5896 280 11 - - PUNCT ejpam-5896 280 12 sd sd NOUN ejpam-5896 280 13 - - PUNCT ejpam-5896 280 14 connected connect VERB ejpam-5896 280 15 . . PUNCT ejpam-5896 281 1 however however ADV ejpam-5896 281 2 , , PUNCT ejpam-5896 281 3 we we PRON ejpam-5896 281 4 have have VERB ejpam-5896 281 5 ũ ũ PROPN ejpam-5896 281 6 = = SYM ejpam-5896 281 7 ( ( PUNCT ejpam-5896 281 8 g,∆)∪̃(h,∆ g,∆)∪̃(h,∆ PROPN ejpam-5896 281 9 ) ) PUNCT ejpam-5896 281 10 , , PUNCT ejpam-5896 281 11 whereas whereas SCONJ ejpam-5896 281 12 ( ( PUNCT ejpam-5896 281 13 g,∆ g,∆ NUM ejpam-5896 281 14 ) ) PUNCT ejpam-5896 281 15 , , PUNCT ejpam-5896 281 16 ( ( PUNCT ejpam-5896 281 17 h,∆ h,∆ X ejpam-5896 281 18 ) ) PUNCT ejpam-5896 281 19 are be AUX ejpam-5896 281 20 ss ss NOUN ejpam-5896 281 21 - - PUNCT ejpam-5896 281 22 sd sd NOUN ejpam-5896 281 23 - - PUNCT ejpam-5896 281 24 separated separate VERB ejpam-5896 281 25 sets set NOUN ejpam-5896 281 26 . . PUNCT ejpam-5896 282 1 therefore therefore ADV ejpam-5896 282 2 , , PUNCT ejpam-5896 282 3 ( ( PUNCT ejpam-5896 282 4 u u NOUN ejpam-5896 282 5 , , PUNCT ejpam-5896 282 6 µ,∆ µ,∆ NUM ejpam-5896 282 7 ) ) PUNCT ejpam-5896 282 8 is be AUX ejpam-5896 282 9 an an DET ejpam-5896 282 10 ss ss NOUN ejpam-5896 282 11 - - PUNCT ejpam-5896 282 12 sd sd NOUN ejpam-5896 282 13 - - PUNCT ejpam-5896 282 14 disconnected disconnected ADJ ejpam-5896 282 15 . . PUNCT ejpam-5896 283 1 theorem theorem ADJ ejpam-5896 283 2 8 8 NUM ejpam-5896 283 3 . . PUNCT ejpam-5896 284 1 the the DET ejpam-5896 284 2 ss ss NOUN ejpam-5896 284 3 - - PUNCT ejpam-5896 284 4 sd sd NOUN ejpam-5896 284 5 - - PUNCT ejpam-5896 284 6 component component NOUN ejpam-5896 284 7 of of ADP ejpam-5896 284 8 an an DET ejpam-5896 284 9 ssl ssl ADJ ejpam-5896 284 10 - - PUNCT ejpam-5896 284 11 sd sd NOUN ejpam-5896 284 12 - - PUNCT ejpam-5896 284 13 connected connect VERB ejpam-5896 284 14 ssts sst NOUN ejpam-5896 284 15 is be AUX ejpam-5896 284 16 a a DET ejpam-5896 284 17 ss ss VERB ejpam-5896 284 18 - - PUNCT ejpam-5896 284 19 sd sd NOUN ejpam-5896 284 20 - - PUNCT ejpam-5896 284 21 set set NOUN ejpam-5896 284 22 . . PUNCT ejpam-5896 285 1 proof proof NOUN ejpam-5896 285 2 . . PUNCT ejpam-5896 286 1 let let VERB ejpam-5896 286 2 ( ( PUNCT ejpam-5896 286 3 u u NOUN ejpam-5896 286 4 , , PUNCT ejpam-5896 286 5 µ,∆ µ,∆ NUM ejpam-5896 286 6 ) ) PUNCT ejpam-5896 286 7 be be VERB ejpam-5896 286 8 an an DET ejpam-5896 286 9 ssl ssl ADJ ejpam-5896 286 10 - - ADJ ejpam-5896 286 11 connected connected ADJ ejpam-5896 286 12 ssts sst NOUN ejpam-5896 286 13 and and CCONJ ejpam-5896 286 14 c̃s c̃s NOUN ejpam-5896 286 15 sd(usγ sd(usγ NOUN ejpam-5896 286 16 , , PUNCT ejpam-5896 286 17 ∆ ∆ PROPN ejpam-5896 286 18 ) ) PUNCT ejpam-5896 286 19 is be AUX ejpam-5896 286 20 ss ss NOUN ejpam-5896 286 21 - - PUNCT ejpam-5896 286 22 sd sd NOUN ejpam-5896 286 23 - - PUNCT ejpam-5896 286 24 component component NOUN ejpam-5896 286 25 related relate VERB ejpam-5896 286 26 to to AUX ejpam-5896 286 27 sγ∈̃ũ sγ∈̃ũ VERB ejpam-5896 286 28 . . PUNCT ejpam-5896 287 1 since since SCONJ ejpam-5896 287 2 ũ ũ PROPN ejpam-5896 287 3 is be AUX ejpam-5896 287 4 ssl ssl ADJ ejpam-5896 287 5 - - PUNCT ejpam-5896 287 6 sd sd NOUN ejpam-5896 287 7 - - PUNCT ejpam-5896 287 8 connected connect VERB ejpam-5896 287 9 , , PUNCT ejpam-5896 287 10 every every DET ejpam-5896 287 11 sssd sssd NOUN ejpam-5896 287 12 - - PUNCT ejpam-5896 287 13 neighbourhood neighbourhood NOUN ejpam-5896 287 14 of of ADP ejpam-5896 287 15 sγ sγ PROPN ejpam-5896 287 16 contains contain VERB ejpam-5896 287 17 an an DET ejpam-5896 287 18 ss ss NOUN ejpam-5896 287 19 - - PUNCT ejpam-5896 287 20 sdconnected sdconnecte VERB ejpam-5896 287 21 neighbourhood neighbourhood NOUN ejpam-5896 287 22 ( ( PUNCT ejpam-5896 287 23 g,∆ g,∆ PROPN ejpam-5896 287 24 ) ) PUNCT ejpam-5896 287 25 of of ADP ejpam-5896 287 26 sγ sγ PROPN ejpam-5896 287 27 . . PUNCT ejpam-5896 288 1 however however ADV ejpam-5896 288 2 , , PUNCT ejpam-5896 288 3 c̃s c̃s NOUN ejpam-5896 288 4 sd(usγ sd(usγ NOUN ejpam-5896 288 5 , , PUNCT ejpam-5896 288 6 ∆ ∆ PROPN ejpam-5896 288 7 ) ) PUNCT ejpam-5896 288 8 is be AUX ejpam-5896 288 9 the the DET ejpam-5896 288 10 largest large ADJ ejpam-5896 288 11 ss ss NOUN ejpam-5896 288 12 - - PUNCT ejpam-5896 288 13 sd sd NOUN ejpam-5896 288 14 - - PUNCT ejpam-5896 288 15 connected connect VERB ejpam-5896 288 16 neighbourhood neighbourhood NOUN ejpam-5896 288 17 of of ADP ejpam-5896 288 18 sγ sγ PROPN ejpam-5896 288 19 . . PUNCT ejpam-5896 289 1 therefore therefore ADV ejpam-5896 289 2 , , PUNCT ejpam-5896 289 3 sγ∈̃(g,∆)⊆̃c̃s sγ∈̃(g,∆)⊆̃c̃s ADJ ejpam-5896 289 4 sd(usγ sd(usγ NOUN ejpam-5896 289 5 , , PUNCT ejpam-5896 289 6 ∆ ∆ PROPN ejpam-5896 289 7 ) ) PUNCT ejpam-5896 289 8 which which PRON ejpam-5896 289 9 follows follow VERB ejpam-5896 289 10 c̃s c̃s NOUN ejpam-5896 289 11 sd(usγ sd(usγ NOUN ejpam-5896 289 12 , , PUNCT ejpam-5896 289 13 ∆ ∆ PROPN ejpam-5896 289 14 ) ) PUNCT ejpam-5896 289 15 is be AUX ejpam-5896 289 16 an an DET ejpam-5896 289 17 sssd sssd NOUN ejpam-5896 289 18 - - PUNCT ejpam-5896 289 19 neighbourhood neighbourhood NOUN ejpam-5896 289 20 of of ADP ejpam-5896 289 21 sγ sγ PROPN ejpam-5896 289 22 . . PUNCT ejpam-5896 290 1 hence hence ADV ejpam-5896 290 2 , , PUNCT ejpam-5896 290 3 c̃s c̃s NOUN ejpam-5896 290 4 sd(usγ sd(usγ NOUN ejpam-5896 290 5 , , PUNCT ejpam-5896 290 6 ∆ ∆ PROPN ejpam-5896 290 7 ) ) PUNCT ejpam-5896 290 8 is be AUX ejpam-5896 290 9 an an DET ejpam-5896 290 10 ss ss VERB ejpam-5896 290 11 - - PUNCT ejpam-5896 290 12 sd sd NOUN ejpam-5896 290 13 - - PUNCT ejpam-5896 290 14 neighbourhood neighbourhood NOUN ejpam-5896 290 15 of of ADP ejpam-5896 290 16 each each DET ejpam-5896 290 17 its its PRON ejpam-5896 290 18 soft soft ADJ ejpam-5896 290 19 points point NOUN ejpam-5896 290 20 . . PUNCT ejpam-5896 291 1 thus thus ADV ejpam-5896 291 2 , , PUNCT ejpam-5896 291 3 c̃s c̃s NOUN ejpam-5896 291 4 sd(usγ sd(usγ NOUN ejpam-5896 291 5 , , PUNCT ejpam-5896 291 6 ∆ ∆ PROPN ejpam-5896 291 7 ) ) PUNCT ejpam-5896 291 8 is be AUX ejpam-5896 291 9 ss ss NOUN ejpam-5896 291 10 - - PUNCT ejpam-5896 291 11 sd sd NOUN ejpam-5896 291 12 - - PUNCT ejpam-5896 291 13 set set NOUN ejpam-5896 291 14 . . PUNCT ejpam-5896 292 1 theorem theorem VERB ejpam-5896 292 2 9 9 NUM ejpam-5896 292 3 . . PUNCT ejpam-5896 293 1 the the DET ejpam-5896 293 2 property property NOUN ejpam-5896 293 3 of of ADP ejpam-5896 293 4 ssl ssl ADJ ejpam-5896 293 5 - - PUNCT ejpam-5896 293 6 sd sd NOUN ejpam-5896 293 7 - - PUNCT ejpam-5896 293 8 connectedness connectedness NOUN ejpam-5896 293 9 is be AUX ejpam-5896 293 10 hereditary hereditary ADJ ejpam-5896 293 11 w.r.t w.r.t NOUN ejpam-5896 293 12 ss ss PROPN ejpam-5896 293 13 - - PUNCT ejpam-5896 293 14 sd sd NOUN ejpam-5896 293 15 - - PUNCT ejpam-5896 293 16 subspaces subspace NOUN ejpam-5896 293 17 . . PUNCT ejpam-5896 294 1 proof proof NOUN ejpam-5896 294 2 . . PUNCT ejpam-5896 295 1 assume assume VERB ejpam-5896 295 2 that that SCONJ ejpam-5896 295 3 ( ( PUNCT ejpam-5896 295 4 u u NOUN ejpam-5896 295 5 , , PUNCT ejpam-5896 295 6 µu µu NUM ejpam-5896 295 7 , , PUNCT ejpam-5896 295 8 ∆ ∆ X ejpam-5896 295 9 ) ) PUNCT ejpam-5896 295 10 be be VERB ejpam-5896 295 11 an an DET ejpam-5896 295 12 ss ss NOUN ejpam-5896 295 13 - - PUNCT ejpam-5896 295 14 sd sd NOUN ejpam-5896 295 15 - - PUNCT ejpam-5896 295 16 subspace subspace NOUN ejpam-5896 295 17 of of ADP ejpam-5896 295 18 an an DET ejpam-5896 295 19 ssl ssl ADJ ejpam-5896 295 20 - - PUNCT ejpam-5896 295 21 sd sd NOUN ejpam-5896 295 22 - - PUNCT ejpam-5896 295 23 connected connect VERB ejpam-5896 295 24 ssts sst NOUN ejpam-5896 295 25 ( ( PUNCT ejpam-5896 295 26 u u NOUN ejpam-5896 295 27 , , PUNCT ejpam-5896 295 28 µ,∆ µ,∆ NUM ejpam-5896 295 29 ) ) PUNCT ejpam-5896 295 30 and and CCONJ ejpam-5896 295 31 sγ∈̃ũ sγ∈̃ũ VERB ejpam-5896 295 32 . . PUNCT ejpam-5896 296 1 since since SCONJ ejpam-5896 296 2 ũ ũ PROPN ejpam-5896 296 3 is be AUX ejpam-5896 296 4 ssl ssl ADJ ejpam-5896 296 5 - - PUNCT ejpam-5896 296 6 sd sd NOUN ejpam-5896 296 7 - - PUNCT ejpam-5896 296 8 connected connect VERB ejpam-5896 296 9 , , PUNCT ejpam-5896 296 10 there there PRON ejpam-5896 296 11 is be VERB ejpam-5896 296 12 an an DET ejpam-5896 296 13 ss ss NOUN ejpam-5896 296 14 - - PUNCT ejpam-5896 296 15 sd sd NOUN ejpam-5896 296 16 - - PUNCT ejpam-5896 296 17 connected connect VERB ejpam-5896 296 18 neighbourhood neighbourhood NOUN ejpam-5896 296 19 ( ( PUNCT ejpam-5896 296 20 g,∆ g,∆ PROPN ejpam-5896 296 21 ) ) PUNCT ejpam-5896 296 22 of of ADP ejpam-5896 296 23 sγ sγ PRON ejpam-5896 296 24 such such ADJ ejpam-5896 296 25 that that SCONJ ejpam-5896 296 26 sγ sγ PROPN ejpam-5896 296 27 ∈ ∈ PROPN ejpam-5896 296 28 ( ( PUNCT ejpam-5896 296 29 g,∆)⊆̃ũ g,∆)⊆̃ũ NOUN ejpam-5896 296 30 . . PUNCT ejpam-5896 297 1 since since SCONJ ejpam-5896 297 2 ( ( PUNCT ejpam-5896 297 3 g,∆ g,∆ PROPN ejpam-5896 297 4 ) ) PUNCT ejpam-5896 297 5 ∈ ∈ PROPN ejpam-5896 297 6 sd(u)∆ sd(u)∆ PROPN ejpam-5896 297 7 , , PUNCT ejpam-5896 297 8 ( ( PUNCT ejpam-5896 297 9 g,∆)∩̃ g,∆)∩̃ VERB ejpam-5896 297 10 ˜̃u ˜̃u X ejpam-5896 297 11 ∈ ∈ PROPN ejpam-5896 297 12 sd(u)∆ sd(u)∆ PROPN ejpam-5896 297 13 and and CCONJ ejpam-5896 297 14 ( ( PUNCT ejpam-5896 297 15 g,∆ g,∆ PROPN ejpam-5896 297 16 ) ) PUNCT ejpam-5896 297 17 ∩ ∩ PROPN ejpam-5896 297 18 ũ ũ PROPN ejpam-5896 297 19 is be AUX ejpam-5896 297 20 ss ss NOUN ejpam-5896 297 21 - - PUNCT ejpam-5896 297 22 sd sd NOUN ejpam-5896 297 23 - - PUNCT ejpam-5896 297 24 connected connect VERB ejpam-5896 297 25 subset subset NOUN ejpam-5896 297 26 of of ADP ejpam-5896 297 27 ũ ũ PROPN ejpam-5896 297 28 . . PUNCT ejpam-5896 298 1 hence hence ADV ejpam-5896 298 2 , , PUNCT ejpam-5896 298 3 ũ ũ PROPN ejpam-5896 298 4 is be AUX ejpam-5896 298 5 an an DET ejpam-5896 298 6 ssl ssl ADJ ejpam-5896 298 7 - - PUNCT ejpam-5896 298 8 sd sd NOUN ejpam-5896 298 9 - - PUNCT ejpam-5896 298 10 connected connect VERB ejpam-5896 298 11 for for ADP ejpam-5896 298 12 each each DET ejpam-5896 298 13 sγ∈̃ũ sγ∈̃ũ NOUN ejpam-5896 298 14 . . PUNCT ejpam-5896 299 1 therefore therefore ADV ejpam-5896 299 2 , , PUNCT ejpam-5896 299 3 ũ ũ PROPN ejpam-5896 299 4 is be AUX ejpam-5896 299 5 an an DET ejpam-5896 299 6 ssl ssl ADJ ejpam-5896 299 7 - - PUNCT ejpam-5896 299 8 sd sd NOUN ejpam-5896 299 9 - - PUNCT ejpam-5896 299 10 connected connect VERB ejpam-5896 299 11 . . PUNCT ejpam-5896 300 1 proposition proposition NOUN ejpam-5896 300 2 4 4 NUM ejpam-5896 300 3 . . PUNCT ejpam-5896 301 1 the the DET ejpam-5896 301 2 ss ss NOUN ejpam-5896 301 3 - - PUNCT ejpam-5896 301 4 sd sd NOUN ejpam-5896 301 5 - - PUNCT ejpam-5896 301 6 components component NOUN ejpam-5896 301 7 of of ADP ejpam-5896 301 8 every every DET ejpam-5896 301 9 ss ss NOUN ejpam-5896 301 10 - - PUNCT ejpam-5896 301 11 sd sd NOUN ejpam-5896 301 12 - - PUNCT ejpam-5896 301 13 subspace subspace NOUN ejpam-5896 301 14 of of ADP ejpam-5896 301 15 an an DET ejpam-5896 301 16 ssl ssl ADJ ejpam-5896 301 17 - - PUNCT ejpam-5896 301 18 sd sd NOUN ejpam-5896 301 19 - - PUNCT ejpam-5896 301 20 connected connect VERB ejpam-5896 301 21 ssts sst NOUN ejpam-5896 301 22 are be AUX ejpam-5896 301 23 ss ss NOUN ejpam-5896 301 24 - - PUNCT ejpam-5896 301 25 sd sd NOUN ejpam-5896 301 26 - - PUNCT ejpam-5896 301 27 set set NOUN ejpam-5896 301 28 . . PUNCT ejpam-5896 302 1 proof proof NOUN ejpam-5896 302 2 . . PUNCT ejpam-5896 303 1 direct direct ADJ ejpam-5896 303 2 result result NOUN ejpam-5896 303 3 of of ADP ejpam-5896 303 4 theorems theorem NOUN ejpam-5896 303 5 8 8 NUM ejpam-5896 303 6 and and CCONJ ejpam-5896 303 7 9 9 NUM ejpam-5896 303 8 . . X ejpam-5896 303 9 theorem theorem VERB ejpam-5896 303 10 10 10 NUM ejpam-5896 303 11 . . PUNCT ejpam-5896 304 1 [ [ X ejpam-5896 304 2 62 62 NUM ejpam-5896 304 3 ] ] PUNCT ejpam-5896 304 4 an an DET ejpam-5896 304 5 ss ss NOUN ejpam-5896 304 6 - - PUNCT ejpam-5896 304 7 sd sd NOUN ejpam-5896 304 8 - - PUNCT ejpam-5896 304 9 connected connect VERB ejpam-5896 304 10 set set NOUN ejpam-5896 304 11 is be AUX ejpam-5896 304 12 represented represent VERB ejpam-5896 304 13 by by ADP ejpam-5896 304 14 its its PRON ejpam-5896 304 15 image image NOUN ejpam-5896 304 16 under under ADP ejpam-5896 304 17 an an DET ejpam-5896 304 18 ss ss NOUN ejpam-5896 304 19 - - PUNCT ejpam-5896 304 20 sd sd NOUN ejpam-5896 304 21 - - PUNCT ejpam-5896 304 22 irresolute irresolute NOUN ejpam-5896 304 23 map map NOUN ejpam-5896 304 24 . . PUNCT ejpam-5896 305 1 theorem theorem VERB ejpam-5896 305 2 11 11 NUM ejpam-5896 305 3 . . PUNCT ejpam-5896 306 1 if if SCONJ ejpam-5896 306 2 ψsd ψsd ADV ejpam-5896 306 3 : : PUNCT ejpam-5896 306 4 ( ( PUNCT ejpam-5896 306 5 u u NOUN ejpam-5896 306 6 , , PUNCT ejpam-5896 306 7 τ,∆ τ,∆ NOUN ejpam-5896 306 8 ) ) PUNCT ejpam-5896 306 9 → → SYM ejpam-5896 306 10 ( ( PUNCT ejpam-5896 306 11 v v NOUN ejpam-5896 306 12 , , PUNCT ejpam-5896 306 13 σ σ PROPN ejpam-5896 306 14 , , PUNCT ejpam-5896 306 15 λ λ PROPN ejpam-5896 306 16 ) ) PUNCT ejpam-5896 306 17 is be AUX ejpam-5896 306 18 a a DET ejpam-5896 306 19 surjective surjective ADJ ejpam-5896 306 20 ss ss NOUN ejpam-5896 306 21 - - PUNCT ejpam-5896 306 22 sd sd NOUN ejpam-5896 306 23 - - PUNCT ejpam-5896 306 24 irresolute irresolute ADJ ejpam-5896 306 25 map map NOUN ejpam-5896 306 26 with with ADP ejpam-5896 306 27 µ µ NOUN ejpam-5896 306 28 , , PUNCT ejpam-5896 306 29 µ∗ µ∗ VERB ejpam-5896 306 30 as as ADP ejpam-5896 306 31 associated associated ADJ ejpam-5896 306 32 sstss sstss NOUN ejpam-5896 306 33 with with ADP ejpam-5896 306 34 τ τ PROPN ejpam-5896 306 35 , , PUNCT ejpam-5896 306 36 σ σ PROPN ejpam-5896 306 37 , , PUNCT ejpam-5896 306 38 respectively respectively ADV ejpam-5896 306 39 , , PUNCT ejpam-5896 306 40 and and CCONJ ejpam-5896 306 41 ũ ũ PROPN ejpam-5896 306 42 is be AUX ejpam-5896 306 43 ssl ssl ADJ ejpam-5896 306 44 - - PUNCT ejpam-5896 306 45 sd sd NOUN ejpam-5896 306 46 - - PUNCT ejpam-5896 306 47 connected connect VERB ejpam-5896 306 48 , , PUNCT ejpam-5896 306 49 then then ADV ejpam-5896 306 50 also also ADV ejpam-5896 306 51 ṽ ṽ PROPN ejpam-5896 306 52 . . PUNCT ejpam-5896 307 1 proof proof NOUN ejpam-5896 307 2 . . PUNCT ejpam-5896 308 1 let let VERB ejpam-5896 308 2 ψsd ψsd VERB ejpam-5896 308 3 : : PUNCT ejpam-5896 308 4 ( ( PUNCT ejpam-5896 308 5 u u NOUN ejpam-5896 308 6 , , PUNCT ejpam-5896 308 7 τ,∆ τ,∆ NOUN ejpam-5896 308 8 ) ) PUNCT ejpam-5896 308 9 → → SYM ejpam-5896 308 10 ( ( PUNCT ejpam-5896 308 11 v v NOUN ejpam-5896 308 12 , , PUNCT ejpam-5896 308 13 σ σ PROPN ejpam-5896 308 14 , , PUNCT ejpam-5896 308 15 λ λ PROPN ejpam-5896 308 16 ) ) PUNCT ejpam-5896 308 17 is be AUX ejpam-5896 308 18 a a DET ejpam-5896 308 19 surjective surjective ADJ ejpam-5896 308 20 ssl ssl ADJ ejpam-5896 308 21 - - PUNCT ejpam-5896 308 22 sd sd NOUN ejpam-5896 308 23 - - PUNCT ejpam-5896 308 24 irresolute irresolute ADJ ejpam-5896 308 25 map map NOUN ejpam-5896 308 26 with with ADP ejpam-5896 308 27 µ µ NOUN ejpam-5896 308 28 , , PUNCT ejpam-5896 308 29 µ∗ µ∗ VERB ejpam-5896 308 30 as as ADP ejpam-5896 308 31 associated associated ADJ ejpam-5896 308 32 sstss sstss NOUN ejpam-5896 308 33 with with ADP ejpam-5896 308 34 τ τ PROPN ejpam-5896 308 35 , , PUNCT ejpam-5896 308 36 σ σ PROPN ejpam-5896 308 37 , , PUNCT ejpam-5896 308 38 respectively respectively ADV ejpam-5896 308 39 , , PUNCT ejpam-5896 308 40 and and CCONJ ejpam-5896 308 41 ũ ũ PROPN ejpam-5896 308 42 is be AUX ejpam-5896 308 43 an an DET ejpam-5896 308 44 ssl ssl ADJ ejpam-5896 308 45 - - PUNCT ejpam-5896 308 46 sd sd NOUN ejpam-5896 308 47 - - PUNCT ejpam-5896 308 48 connected connect VERB ejpam-5896 308 49 . . PUNCT ejpam-5896 309 1 let let VERB ejpam-5896 309 2 ( ( PUNCT ejpam-5896 309 3 h h NOUN ejpam-5896 309 4 , , PUNCT ejpam-5896 309 5 λ λ NOUN ejpam-5896 309 6 ) ) PUNCT ejpam-5896 309 7 is be AUX ejpam-5896 309 8 an an DET ejpam-5896 309 9 ss ss VERB ejpam-5896 309 10 - - PUNCT ejpam-5896 309 11 sd sd NOUN ejpam-5896 309 12 - - PUNCT ejpam-5896 309 13 neighbourhood neighbourhood NOUN ejpam-5896 309 14 of of ADP ejpam-5896 309 15 sλ∈̃ṽ sλ∈̃ṽ NOUN ejpam-5896 309 16 . . PUNCT ejpam-5896 310 1 it it PRON ejpam-5896 310 2 follows follow VERB ejpam-5896 310 3 that that SCONJ ejpam-5896 310 4 , , PUNCT ejpam-5896 310 5 ψ−1 ψ−1 PROPN ejpam-5896 310 6 sd sd ADP ejpam-5896 310 7 ( ( PUNCT ejpam-5896 310 8 h h NOUN ejpam-5896 310 9 , , PUNCT ejpam-5896 310 10 λ λ NOUN ejpam-5896 310 11 ) ) PUNCT ejpam-5896 310 12 is be AUX ejpam-5896 310 13 ss ss NOUN ejpam-5896 310 14 - - PUNCT ejpam-5896 310 15 sd sd NOUN ejpam-5896 310 16 - - PUNCT ejpam-5896 310 17 neighbourhood neighbourhood NOUN ejpam-5896 310 18 of of ADP ejpam-5896 310 19 ψ−1 ψ−1 PROPN ejpam-5896 310 20 sd sd ADP ejpam-5896 310 21 ( ( PUNCT ejpam-5896 310 22 sλ)∈̃ũ sλ)∈̃ũ NOUN ejpam-5896 310 23 . . PUNCT ejpam-5896 311 1 since since SCONJ ejpam-5896 311 2 ũ ũ PROPN ejpam-5896 311 3 is be AUX ejpam-5896 311 4 ssl ssl ADJ ejpam-5896 311 5 - - PUNCT ejpam-5896 311 6 sd sd NOUN ejpam-5896 311 7 - - PUNCT ejpam-5896 311 8 connected connect VERB ejpam-5896 311 9 , , PUNCT ejpam-5896 311 10 there there PRON ejpam-5896 311 11 is be VERB ejpam-5896 311 12 an an DET ejpam-5896 311 13 ss ss NOUN ejpam-5896 311 14 - - PUNCT ejpam-5896 311 15 sd sd NOUN ejpam-5896 311 16 - - PUNCT ejpam-5896 311 17 connected connect VERB ejpam-5896 311 18 subset subset NOUN ejpam-5896 311 19 ( ( PUNCT ejpam-5896 311 20 g,∆ g,∆ PROPN ejpam-5896 311 21 ) ) PUNCT ejpam-5896 311 22 of of ADP ejpam-5896 311 23 ψ−1 ψ−1 PROPN ejpam-5896 311 24 sd sd ADP ejpam-5896 311 25 ( ( PUNCT ejpam-5896 311 26 h h NOUN ejpam-5896 311 27 , , PUNCT ejpam-5896 311 28 λ λ NOUN ejpam-5896 311 29 ) ) PUNCT ejpam-5896 311 30 containing contain VERB ejpam-5896 311 31 ψ−1 ψ−1 PROPN ejpam-5896 311 32 sd sd ADP ejpam-5896 311 33 ( ( PUNCT ejpam-5896 311 34 sλ sλ NOUN ejpam-5896 311 35 ) ) PUNCT ejpam-5896 311 36 which which PRON ejpam-5896 311 37 follows follow VERB ejpam-5896 311 38 ψsd(g,∆)⊆̃ψsd(ψ ψsd(g,∆)⊆̃ψsd(ψ X ejpam-5896 311 39 −1 −1 ADV ejpam-5896 311 40 sd sd ADP ejpam-5896 311 41 ( ( PUNCT ejpam-5896 311 42 h h NOUN ejpam-5896 311 43 , , PUNCT ejpam-5896 311 44 λ λ NOUN ejpam-5896 311 45 ) ) PUNCT ejpam-5896 311 46 ) ) PUNCT ejpam-5896 312 1 = = PRON ejpam-5896 312 2 ( ( PUNCT ejpam-5896 312 3 h h NOUN ejpam-5896 312 4 , , PUNCT ejpam-5896 312 5 λ λ NOUN ejpam-5896 312 6 ) ) PUNCT ejpam-5896 312 7 , , PUNCT ejpam-5896 312 8 ψsd ψsd PROPN ejpam-5896 312 9 is be AUX ejpam-5896 312 10 surjective surjective ADJ ejpam-5896 312 11 . . PUNCT ejpam-5896 313 1 since since SCONJ ejpam-5896 313 2 ψsd ψsd PROPN ejpam-5896 313 3 is be AUX ejpam-5896 313 4 ss ss NOUN ejpam-5896 313 5 - - PUNCT ejpam-5896 313 6 sd sd NOUN ejpam-5896 313 7 - - PUNCT ejpam-5896 313 8 irresolute irresolute ADJ ejpam-5896 313 9 , , PUNCT ejpam-5896 313 10 ψsd(g,∆ ψsd(g,∆ PUNCT ejpam-5896 313 11 ) ) PUNCT ejpam-5896 313 12 is be AUX ejpam-5896 313 13 ss ss NOUN ejpam-5896 313 14 - - PUNCT ejpam-5896 313 15 sd sd NOUN ejpam-5896 313 16 - - PUNCT ejpam-5896 313 17 connected connect VERB ejpam-5896 313 18 neighbourhood neighbourhood NOUN ejpam-5896 313 19 of of ADP ejpam-5896 313 20 ψ−1 ψ−1 PROPN ejpam-5896 313 21 sd sd ADP ejpam-5896 313 22 ( ( PUNCT ejpam-5896 313 23 sλ sλ NOUN ejpam-5896 313 24 ) ) PUNCT ejpam-5896 313 25 from from ADP ejpam-5896 313 26 theorem theorem ADJ ejpam-5896 313 27 10 10 NUM ejpam-5896 313 28 . . PUNCT ejpam-5896 314 1 therefore therefore ADV ejpam-5896 314 2 , , PUNCT ejpam-5896 314 3 ṽ ṽ PROPN ejpam-5896 314 4 is be AUX ejpam-5896 314 5 an an DET ejpam-5896 314 6 ssl ssl ADJ ejpam-5896 314 7 - - PUNCT ejpam-5896 314 8 sd sd NOUN ejpam-5896 314 9 - - PUNCT ejpam-5896 314 10 connected connect VERB ejpam-5896 314 11 . . PUNCT ejpam-5896 315 1 abd abd PROPN ejpam-5896 315 2 el el PROPN ejpam-5896 315 3 - - PROPN ejpam-5896 315 4 latif latif PROPN ejpam-5896 315 5 et et PROPN ejpam-5896 315 6 al al PROPN ejpam-5896 315 7 . . PUNCT ejpam-5896 315 8 / / SYM ejpam-5896 315 9 eur eur PROPN ejpam-5896 315 10 . . PUNCT ejpam-5896 316 1 j. j. PROPN ejpam-5896 316 2 pure pure PROPN ejpam-5896 316 3 appl appl PROPN ejpam-5896 316 4 . . PROPN ejpam-5896 316 5 math math PROPN ejpam-5896 316 6 , , PUNCT ejpam-5896 316 7 18 18 NUM ejpam-5896 316 8 ( ( PUNCT ejpam-5896 316 9 2 2 NUM ejpam-5896 316 10 ) ) PUNCT ejpam-5896 316 11 ( ( PUNCT ejpam-5896 316 12 2025 2025 NUM ejpam-5896 316 13 ) ) PUNCT ejpam-5896 316 14 , , PUNCT ejpam-5896 316 15 5896 5896 NUM ejpam-5896 316 16 9 9 NUM ejpam-5896 316 17 of of ADP ejpam-5896 316 18 20 20 NUM ejpam-5896 316 19 definition definition NOUN ejpam-5896 316 20 18 18 NUM ejpam-5896 316 21 . . PUNCT ejpam-5896 317 1 if if SCONJ ejpam-5896 317 2 for for ADP ejpam-5896 317 3 all all DET ejpam-5896 317 4 ( ( PUNCT ejpam-5896 317 5 g,∆ g,∆ PROPN ejpam-5896 317 6 ) ) PUNCT ejpam-5896 317 7 , , PUNCT ejpam-5896 317 8 ( ( PUNCT ejpam-5896 317 9 h,∆ h,∆ X ejpam-5896 317 10 ) ) PUNCT ejpam-5896 317 11 ∈ ∈ NOUN ejpam-5896 317 12 sd(u)∆ sd(u)∆ NOUN ejpam-5896 317 13 such such ADJ ejpam-5896 317 14 that that SCONJ ejpam-5896 317 15 ( ( PUNCT ejpam-5896 317 16 g,∆)∩̃(h,∆ g,∆)∩̃(h,∆ X ejpam-5896 317 17 ) ) PUNCT ejpam-5896 317 18 ̸= ̸= PROPN ejpam-5896 317 19 φ̃ φ̃ PROPN ejpam-5896 317 20 , , PUNCT ejpam-5896 317 21 then then ADV ejpam-5896 317 22 ũ ũ PROPN ejpam-5896 317 23 is be AUX ejpam-5896 317 24 called call VERB ejpam-5896 317 25 an an DET ejpam-5896 317 26 ss ss NOUN ejpam-5896 317 27 - - PUNCT ejpam-5896 317 28 sd sd NOUN ejpam-5896 317 29 - - PUNCT ejpam-5896 317 30 hyperconnected hyperconnecte VERB ejpam-5896 317 31 . . PUNCT ejpam-5896 318 1 definition definition NOUN ejpam-5896 318 2 19 19 NUM ejpam-5896 318 3 . . PUNCT ejpam-5896 319 1 a a DET ejpam-5896 319 2 soft soft ADJ ejpam-5896 319 3 subset subset NOUN ejpam-5896 319 4 ( ( PUNCT ejpam-5896 319 5 f,∆ f,∆ NOUN ejpam-5896 319 6 ) ) PUNCT ejpam-5896 319 7 of of ADP ejpam-5896 319 8 an an DET ejpam-5896 319 9 ssts sst NOUN ejpam-5896 319 10 ( ( PUNCT ejpam-5896 319 11 u u NOUN ejpam-5896 319 12 , , PUNCT ejpam-5896 319 13 µ,∆ µ,∆ NUM ejpam-5896 319 14 ) ) PUNCT ejpam-5896 319 15 is be AUX ejpam-5896 319 16 said say VERB ejpam-5896 319 17 to to PART ejpam-5896 319 18 be be AUX ejpam-5896 319 19 ss ss NOUN ejpam-5896 319 20 - - PUNCT ejpam-5896 319 21 sd sd NOUN ejpam-5896 319 22 - - PUNCT ejpam-5896 319 23 dense dense ADJ ejpam-5896 319 24 if if SCONJ ejpam-5896 319 25 clssd(f,∆ clssd(f,∆ NOUN ejpam-5896 319 26 ) ) PUNCT ejpam-5896 320 1 = = SYM ejpam-5896 320 2 ũ ũ PROPN ejpam-5896 320 3 . . PUNCT ejpam-5896 321 1 theorem theorem NOUN ejpam-5896 321 2 12 12 NUM ejpam-5896 321 3 . . PUNCT ejpam-5896 322 1 an an DET ejpam-5896 322 2 ssts sst NOUN ejpam-5896 322 3 ( ( PUNCT ejpam-5896 322 4 u u NOUN ejpam-5896 322 5 , , PUNCT ejpam-5896 322 6 µ,∆ µ,∆ NUM ejpam-5896 322 7 ) ) PUNCT ejpam-5896 322 8 is be AUX ejpam-5896 322 9 ss ss NOUN ejpam-5896 322 10 - - PUNCT ejpam-5896 322 11 sd sd NOUN ejpam-5896 322 12 - - PUNCT ejpam-5896 322 13 hyperconnected hyperconnecte VERB ejpam-5896 322 14 if if SCONJ ejpam-5896 322 15 and and CCONJ ejpam-5896 322 16 only only ADV ejpam-5896 322 17 if if SCONJ ejpam-5896 322 18 every every DET ejpam-5896 322 19 ss ss NOUN ejpam-5896 322 20 - - PUNCT ejpam-5896 322 21 sd sd NOUN ejpam-5896 322 22 - - PUNCT ejpam-5896 322 23 set set NOUN ejpam-5896 322 24 ( ( PUNCT ejpam-5896 322 25 g,∆ g,∆ X ejpam-5896 322 26 ) ) PUNCT ejpam-5896 322 27 is be AUX ejpam-5896 322 28 ss ss NOUN ejpam-5896 322 29 - - PUNCT ejpam-5896 322 30 sd sd NOUN ejpam-5896 322 31 - - PUNCT ejpam-5896 322 32 dense dense ADJ ejpam-5896 322 33 . . PUNCT ejpam-5896 323 1 proof proof NOUN ejpam-5896 323 2 . . PUNCT ejpam-5896 324 1 necessity necessity NOUN ejpam-5896 324 2 : : PUNCT ejpam-5896 324 3 alternatively alternatively ADV ejpam-5896 324 4 , , PUNCT ejpam-5896 324 5 suppose suppose VERB ejpam-5896 324 6 that that SCONJ ejpam-5896 324 7 there there PRON ejpam-5896 324 8 is be VERB ejpam-5896 324 9 an an DET ejpam-5896 324 10 ss ss NOUN ejpam-5896 324 11 - - PUNCT ejpam-5896 324 12 sd sd NOUN ejpam-5896 324 13 - - PUNCT ejpam-5896 324 14 subset subset NOUN ejpam-5896 324 15 ( ( PUNCT ejpam-5896 324 16 g,∆ g,∆ PROPN ejpam-5896 324 17 ) ) PUNCT ejpam-5896 324 18 of of ADP ejpam-5896 324 19 an an DET ejpam-5896 324 20 ss ss NOUN ejpam-5896 324 21 - - PUNCT ejpam-5896 324 22 sdhyperconnected sdhyperconnecte VERB ejpam-5896 324 23 ssts sst NOUN ejpam-5896 324 24 ( ( PUNCT ejpam-5896 324 25 u u NOUN ejpam-5896 324 26 , , PUNCT ejpam-5896 324 27 µ,∆ µ,∆ NUM ejpam-5896 324 28 ) ) PUNCT ejpam-5896 324 29 such such ADJ ejpam-5896 324 30 that that DET ejpam-5896 324 31 clssd(f,∆ clssd(f,∆ NOUN ejpam-5896 324 32 ) ) PUNCT ejpam-5896 324 33 ̸= ̸= PROPN ejpam-5896 324 34 ũ ũ PROPN ejpam-5896 324 35 , , PUNCT ejpam-5896 324 36 then then ADV ejpam-5896 324 37 there there PRON ejpam-5896 324 38 is be VERB ejpam-5896 324 39 an an DET ejpam-5896 324 40 ss ss ADJ ejpam-5896 324 41 - - PUNCT ejpam-5896 324 42 sc sc NOUN ejpam-5896 324 43 - - PUNCT ejpam-5896 324 44 subset subset NOUN ejpam-5896 324 45 ( ( PUNCT ejpam-5896 324 46 h,∆ h,∆ NOUN ejpam-5896 324 47 ) ) PUNCT ejpam-5896 324 48 of of ADP ejpam-5896 324 49 ũ ũ PROPN ejpam-5896 324 50 such such ADJ ejpam-5896 324 51 that that SCONJ ejpam-5896 324 52 ( ( PUNCT ejpam-5896 324 53 g,∆)⊆̃(h,∆ g,∆)⊆̃(h,∆ NOUN ejpam-5896 324 54 ) ) PUNCT ejpam-5896 324 55 . . PUNCT ejpam-5896 325 1 from from ADP ejpam-5896 325 2 corollary corollary ADJ ejpam-5896 325 3 1 1 NUM ejpam-5896 325 4 , , PUNCT ejpam-5896 325 5 ( ( PUNCT ejpam-5896 325 6 g,∆ g,∆ X ejpam-5896 325 7 ) ) PUNCT ejpam-5896 325 8 is be AUX ejpam-5896 325 9 also also ADV ejpam-5896 325 10 an an DET ejpam-5896 325 11 ss ss ADJ ejpam-5896 325 12 - - PUNCT ejpam-5896 325 13 sc sc NOUN ejpam-5896 325 14 - - PUNCT ejpam-5896 325 15 set set NOUN ejpam-5896 325 16 . . PUNCT ejpam-5896 326 1 hence hence ADV ejpam-5896 326 2 , , PUNCT ejpam-5896 326 3 ( ( PUNCT ejpam-5896 326 4 g,∆ g,∆ X ejpam-5896 326 5 ) ) PUNCT ejpam-5896 326 6 and and CCONJ ejpam-5896 326 7 ( ( PUNCT ejpam-5896 326 8 gc̃,∆ gc̃,∆ NOUN ejpam-5896 326 9 ) ) PUNCT ejpam-5896 326 10 are be AUX ejpam-5896 326 11 ss ss NOUN ejpam-5896 326 12 - - PUNCT ejpam-5896 326 13 sd sd NOUN ejpam-5896 326 14 - - PUNCT ejpam-5896 326 15 sets set NOUN ejpam-5896 326 16 in in ADP ejpam-5896 326 17 which which PRON ejpam-5896 326 18 ( ( PUNCT ejpam-5896 326 19 g,∆)∩̃(gc̃,∆ g,∆)∩̃(gc̃,∆ NOUN ejpam-5896 326 20 ) ) PUNCT ejpam-5896 326 21 = = PUNCT ejpam-5896 327 1 φ̃ φ̃ PROPN ejpam-5896 327 2 , , PUNCT ejpam-5896 327 3 which which PRON ejpam-5896 327 4 contradicts contradict VERB ejpam-5896 327 5 our our PRON ejpam-5896 327 6 hypothesis hypothesis NOUN ejpam-5896 327 7 . . PUNCT ejpam-5896 328 1 sufficient sufficient ADJ ejpam-5896 328 2 : : PUNCT ejpam-5896 328 3 suppose suppose VERB ejpam-5896 328 4 the the DET ejpam-5896 328 5 contrary contrary NOUN ejpam-5896 328 6 that that PRON ejpam-5896 328 7 ũ ũ PROPN ejpam-5896 328 8 is be AUX ejpam-5896 328 9 not not PART ejpam-5896 328 10 ss ss NOUN ejpam-5896 328 11 - - PUNCT ejpam-5896 328 12 sd sd NOUN ejpam-5896 328 13 - - PUNCT ejpam-5896 328 14 hyperconnected hyperconnecte VERB ejpam-5896 328 15 , , PUNCT ejpam-5896 328 16 then then ADV ejpam-5896 328 17 there there PRON ejpam-5896 328 18 are be VERB ejpam-5896 328 19 two two NUM ejpam-5896 328 20 sssd sssd NOUN ejpam-5896 328 21 - - PUNCT ejpam-5896 328 22 subsets subset NOUN ejpam-5896 328 23 ( ( PUNCT ejpam-5896 328 24 a,∆ a,∆ NOUN ejpam-5896 328 25 ) ) PUNCT ejpam-5896 328 26 , , PUNCT ejpam-5896 328 27 ( ( PUNCT ejpam-5896 328 28 b,∆ b,∆ NOUN ejpam-5896 328 29 ) ) PUNCT ejpam-5896 328 30 of of ADP ejpam-5896 328 31 ũ ũ PROPN ejpam-5896 328 32 such such ADJ ejpam-5896 328 33 that that SCONJ ejpam-5896 328 34 ( ( PUNCT ejpam-5896 328 35 a,∆)∩̃(b,∆ a,∆)∩̃(b,∆ PROPN ejpam-5896 328 36 ) ) PUNCT ejpam-5896 328 37 = = SYM ejpam-5896 328 38 φ̃. φ̃. PROPN ejpam-5896 328 39 then then ADV ejpam-5896 328 40 , , PUNCT ejpam-5896 328 41 ( ( PUNCT ejpam-5896 328 42 a,∆)⊆̃(bc̃,∆ a,∆)⊆̃(bc̃,∆ PROPN ejpam-5896 328 43 ) ) PUNCT ejpam-5896 328 44 , , PUNCT ejpam-5896 328 45 ( ( PUNCT ejpam-5896 328 46 bc̃,∆ bc̃,∆ NOUN ejpam-5896 328 47 ) ) PUNCT ejpam-5896 328 48 is be AUX ejpam-5896 328 49 an an DET ejpam-5896 328 50 ss ss ADJ ejpam-5896 328 51 - - PUNCT ejpam-5896 328 52 sc sc NOUN ejpam-5896 328 53 - - PUNCT ejpam-5896 328 54 set set NOUN ejpam-5896 328 55 which which PRON ejpam-5896 328 56 follows follow VERB ejpam-5896 328 57 ( ( PUNCT ejpam-5896 328 58 a,∆ a,∆ NOUN ejpam-5896 328 59 ) ) PUNCT ejpam-5896 328 60 is be AUX ejpam-5896 328 61 also also ADV ejpam-5896 328 62 an an DET ejpam-5896 328 63 ss ss ADJ ejpam-5896 328 64 - - PUNCT ejpam-5896 328 65 sc sc NOUN ejpam-5896 328 66 - - PUNCT ejpam-5896 328 67 set set NOUN ejpam-5896 328 68 ; ; PUNCT ejpam-5896 328 69 from from ADP ejpam-5896 328 70 corollary corollary ADJ ejpam-5896 328 71 1 1 NUM ejpam-5896 328 72 , , PUNCT ejpam-5896 328 73 and and CCONJ ejpam-5896 328 74 hence hence ADV ejpam-5896 328 75 clssd(a,∆ clssd(a,∆ NOUN ejpam-5896 328 76 ) ) PUNCT ejpam-5896 329 1 = = PUNCT ejpam-5896 329 2 ( ( PUNCT ejpam-5896 329 3 a,∆ a,∆ NOUN ejpam-5896 329 4 ) ) PUNCT ejpam-5896 329 5 ̸= ̸= PROPN ejpam-5896 329 6 ũ ũ PROPN ejpam-5896 329 7 . . PUNCT ejpam-5896 330 1 hence hence ADV ejpam-5896 330 2 , , PUNCT ejpam-5896 330 3 ( ( PUNCT ejpam-5896 330 4 a,∆ a,∆ NOUN ejpam-5896 330 5 ) ) PUNCT ejpam-5896 330 6 is be AUX ejpam-5896 330 7 not not PART ejpam-5896 330 8 ss ss NOUN ejpam-5896 330 9 - - PUNCT ejpam-5896 330 10 sd sd NOUN ejpam-5896 330 11 - - PUNCT ejpam-5896 330 12 dense dense NOUN ejpam-5896 330 13 , , PUNCT ejpam-5896 330 14 which which PRON ejpam-5896 330 15 goes go VERB ejpam-5896 330 16 against against ADP ejpam-5896 330 17 what what PRON ejpam-5896 330 18 we we PRON ejpam-5896 330 19 assumed assume VERB ejpam-5896 330 20 . . PUNCT ejpam-5896 331 1 thus thus ADV ejpam-5896 331 2 , , PUNCT ejpam-5896 331 3 ũ ũ PROPN ejpam-5896 331 4 is be AUX ejpam-5896 331 5 an an DET ejpam-5896 331 6 ss ss NOUN ejpam-5896 331 7 - - PUNCT ejpam-5896 331 8 sd sd NOUN ejpam-5896 331 9 - - PUNCT ejpam-5896 331 10 hyperconnected hyperconnecte VERB ejpam-5896 331 11 . . PUNCT ejpam-5896 332 1 corollary corollary ADJ ejpam-5896 332 2 2 2 NUM ejpam-5896 332 3 . . PUNCT ejpam-5896 333 1 an an DET ejpam-5896 333 2 ssts sst NOUN ejpam-5896 333 3 ( ( PUNCT ejpam-5896 333 4 u u NOUN ejpam-5896 333 5 , , PUNCT ejpam-5896 333 6 µ,∆ µ,∆ NUM ejpam-5896 333 7 ) ) PUNCT ejpam-5896 333 8 is be AUX ejpam-5896 333 9 ss ss NOUN ejpam-5896 333 10 - - PUNCT ejpam-5896 333 11 sd sd NOUN ejpam-5896 333 12 - - PUNCT ejpam-5896 333 13 hyperconnected hyperconnecte VERB ejpam-5896 333 14 if if SCONJ ejpam-5896 333 15 and and CCONJ ejpam-5896 333 16 only only ADV ejpam-5896 333 17 if if SCONJ ejpam-5896 333 18 every every DET ejpam-5896 333 19 ss ss NOUN ejpam-5896 333 20 - - ADJ ejpam-5896 333 21 open open ADJ ejpam-5896 333 22 set set NOUN ejpam-5896 333 23 ( ( PUNCT ejpam-5896 333 24 g,∆ g,∆ X ejpam-5896 333 25 ) ) PUNCT ejpam-5896 333 26 is be AUX ejpam-5896 333 27 ss ss NOUN ejpam-5896 333 28 - - PUNCT ejpam-5896 333 29 sd sd NOUN ejpam-5896 333 30 - - PUNCT ejpam-5896 333 31 dense dense ADJ ejpam-5896 333 32 . . PUNCT ejpam-5896 334 1 proof proof NOUN ejpam-5896 334 2 . . PUNCT ejpam-5896 335 1 it it PRON ejpam-5896 335 2 is be AUX ejpam-5896 335 3 evident evident ADJ ejpam-5896 335 4 from from ADP ejpam-5896 335 5 the the DET ejpam-5896 335 6 fact fact NOUN ejpam-5896 335 7 that that SCONJ ejpam-5896 335 8 each each DET ejpam-5896 335 9 ss ss NOUN ejpam-5896 335 10 - - PUNCT ejpam-5896 335 11 open open ADJ ejpam-5896 335 12 set set NOUN ejpam-5896 335 13 is be AUX ejpam-5896 335 14 an an DET ejpam-5896 335 15 ss ss NOUN ejpam-5896 335 16 - - PUNCT ejpam-5896 335 17 sd sd NOUN ejpam-5896 335 18 - - PUNCT ejpam-5896 335 19 set set VERB ejpam-5896 335 20 and and CCONJ ejpam-5896 335 21 theorem theorem VERB ejpam-5896 335 22 12 12 NUM ejpam-5896 335 23 . . PUNCT ejpam-5896 336 1 note note NOUN ejpam-5896 336 2 1 1 NUM ejpam-5896 336 3 . . PUNCT ejpam-5896 337 1 the the DET ejpam-5896 337 2 following follow VERB ejpam-5896 337 3 theorem theorem ADJ ejpam-5896 337 4 introduces introduce NOUN ejpam-5896 337 5 a a DET ejpam-5896 337 6 novel novel ADJ ejpam-5896 337 7 result result NOUN ejpam-5896 337 8 that that PRON ejpam-5896 337 9 is be AUX ejpam-5896 337 10 not not PART ejpam-5896 337 11 satisfied satisfied ADJ ejpam-5896 337 12 by by ADP ejpam-5896 337 13 its its PRON ejpam-5896 337 14 stss stss NOUN ejpam-5896 337 15 and and CCONJ ejpam-5896 337 16 sstss sstss NOUN ejpam-5896 337 17 counterparts counterpart NOUN ejpam-5896 337 18 . . PUNCT ejpam-5896 338 1 theorem theorem VERB ejpam-5896 338 2 13 13 NUM ejpam-5896 338 3 . . PUNCT ejpam-5896 339 1 an an DET ejpam-5896 339 2 ssts sst NOUN ejpam-5896 339 3 ( ( PUNCT ejpam-5896 339 4 u u NOUN ejpam-5896 339 5 , , PUNCT ejpam-5896 339 6 µ,∆ µ,∆ NUM ejpam-5896 339 7 ) ) PUNCT ejpam-5896 339 8 is be AUX ejpam-5896 339 9 ss ss NOUN ejpam-5896 339 10 - - PUNCT ejpam-5896 339 11 sd sd NOUN ejpam-5896 339 12 - - PUNCT ejpam-5896 339 13 hyperconnected hyperconnecte VERB ejpam-5896 339 14 if if SCONJ ejpam-5896 339 15 and and CCONJ ejpam-5896 339 16 only only ADV ejpam-5896 339 17 if if SCONJ ejpam-5896 339 18 it it PRON ejpam-5896 339 19 is be AUX ejpam-5896 339 20 ss ss NOUN ejpam-5896 339 21 - - PUNCT ejpam-5896 339 22 sd sd NOUN ejpam-5896 339 23 - - PUNCT ejpam-5896 339 24 connected connect VERB ejpam-5896 339 25 . . PUNCT ejpam-5896 340 1 proof proof NOUN ejpam-5896 340 2 . . PUNCT ejpam-5896 341 1 suppose suppose VERB ejpam-5896 341 2 the the DET ejpam-5896 341 3 contrary contrary NOUN ejpam-5896 341 4 that that SCONJ ejpam-5896 341 5 ( ( PUNCT ejpam-5896 341 6 u u NOUN ejpam-5896 341 7 , , PUNCT ejpam-5896 341 8 µ,∆ µ,∆ NUM ejpam-5896 341 9 ) ) PUNCT ejpam-5896 341 10 is be AUX ejpam-5896 341 11 ss ss NOUN ejpam-5896 341 12 - - PUNCT ejpam-5896 341 13 sd sd NOUN ejpam-5896 341 14 - - PUNCT ejpam-5896 341 15 disconnected disconnected ADJ ejpam-5896 341 16 . . PUNCT ejpam-5896 342 1 given give VERB ejpam-5896 342 2 theorem theorem NOUN ejpam-5896 342 3 3 3 NUM ejpam-5896 342 4 , , PUNCT ejpam-5896 342 5 there there PRON ejpam-5896 342 6 is be VERB ejpam-5896 342 7 a a DET ejpam-5896 342 8 proper proper ADJ ejpam-5896 342 9 ss ss NOUN ejpam-5896 342 10 - - PUNCT ejpam-5896 342 11 sdc sdc NOUN ejpam-5896 342 12 - - PUNCT ejpam-5896 342 13 subset subset NOUN ejpam-5896 342 14 ( ( PUNCT ejpam-5896 342 15 g,∆ g,∆ PROPN ejpam-5896 342 16 ) ) PUNCT ejpam-5896 342 17 of of ADP ejpam-5896 342 18 ũ ũ PROPN ejpam-5896 342 19 . . PUNCT ejpam-5896 343 1 hence hence ADV ejpam-5896 343 2 , , PUNCT ejpam-5896 343 3 ( ( PUNCT ejpam-5896 343 4 g,∆ g,∆ X ejpam-5896 343 5 ) ) PUNCT ejpam-5896 343 6 and and CCONJ ejpam-5896 343 7 ( ( PUNCT ejpam-5896 343 8 gc̃,∆ gc̃,∆ NOUN ejpam-5896 343 9 ) ) PUNCT ejpam-5896 343 10 are be AUX ejpam-5896 343 11 ss ss NOUN ejpam-5896 343 12 - - PUNCT ejpam-5896 343 13 sd sd NOUN ejpam-5896 343 14 - - PUNCT ejpam-5896 343 15 sets set NOUN ejpam-5896 343 16 in in ADP ejpam-5896 343 17 which which PRON ejpam-5896 343 18 ( ( PUNCT ejpam-5896 343 19 g,∆)∩̃(gc̃,∆ g,∆)∩̃(gc̃,∆ NOUN ejpam-5896 343 20 ) ) PUNCT ejpam-5896 343 21 = = SYM ejpam-5896 343 22 φ̃. φ̃. PROPN ejpam-5896 343 23 thus thus ADV ejpam-5896 343 24 , , PUNCT ejpam-5896 343 25 ũ ũ PROPN ejpam-5896 343 26 is be AUX ejpam-5896 343 27 not not PART ejpam-5896 343 28 ss ss AUX ejpam-5896 343 29 - - PUNCT ejpam-5896 343 30 sd sd NOUN ejpam-5896 343 31 - - PUNCT ejpam-5896 343 32 hyperconnected hyperconnecte VERB ejpam-5896 343 33 . . PUNCT ejpam-5896 344 1 on on ADP ejpam-5896 344 2 the the DET ejpam-5896 344 3 other other ADJ ejpam-5896 344 4 hand hand NOUN ejpam-5896 344 5 , , PUNCT ejpam-5896 344 6 suppose suppose VERB ejpam-5896 344 7 that that SCONJ ejpam-5896 344 8 ( ( PUNCT ejpam-5896 344 9 u u NOUN ejpam-5896 344 10 , , PUNCT ejpam-5896 344 11 µ,∆ µ,∆ NUM ejpam-5896 344 12 ) ) PUNCT ejpam-5896 344 13 is be AUX ejpam-5896 344 14 not not PART ejpam-5896 344 15 ss ss AUX ejpam-5896 344 16 - - PUNCT ejpam-5896 344 17 sd sd NOUN ejpam-5896 344 18 - - PUNCT ejpam-5896 344 19 hyperconnected hyperconnecte VERB ejpam-5896 344 20 , , PUNCT ejpam-5896 344 21 then then ADV ejpam-5896 344 22 there there PRON ejpam-5896 344 23 are be VERB ejpam-5896 344 24 two two NUM ejpam-5896 344 25 disjoint disjoint ADJ ejpam-5896 344 26 ss ss NOUN ejpam-5896 344 27 - - PUNCT ejpam-5896 344 28 sd sd NOUN ejpam-5896 344 29 - - PUNCT ejpam-5896 344 30 subsets subset NOUN ejpam-5896 344 31 ( ( PUNCT ejpam-5896 344 32 g,∆ g,∆ PROPN ejpam-5896 344 33 ) ) PUNCT ejpam-5896 344 34 and and CCONJ ejpam-5896 344 35 ( ( PUNCT ejpam-5896 344 36 h,∆ h,∆ NOUN ejpam-5896 344 37 ) ) PUNCT ejpam-5896 344 38 of of ADP ejpam-5896 344 39 ũ ũ PROPN ejpam-5896 344 40 . . PUNCT ejpam-5896 345 1 it it PRON ejpam-5896 345 2 follows follow VERB ejpam-5896 345 3 that that SCONJ ejpam-5896 345 4 , , PUNCT ejpam-5896 345 5 ( ( PUNCT ejpam-5896 345 6 g,∆)⊆̃(h g,∆)⊆̃(h PROPN ejpam-5896 345 7 c̃,∆ c̃,∆ PROPN ejpam-5896 345 8 ) ) PUNCT ejpam-5896 345 9 , , PUNCT ejpam-5896 345 10 whereas whereas SCONJ ejpam-5896 345 11 ( ( PUNCT ejpam-5896 345 12 h h NOUN ejpam-5896 345 13 c̃,∆ c̃,∆ NOUN ejpam-5896 345 14 ) ) PUNCT ejpam-5896 345 15 is be AUX ejpam-5896 345 16 ss ss PROPN ejpam-5896 345 17 - - ADJ ejpam-5896 345 18 sc sc NOUN ejpam-5896 345 19 - - PUNCT ejpam-5896 345 20 set set NOUN ejpam-5896 345 21 . . PUNCT ejpam-5896 346 1 according accord VERB ejpam-5896 346 2 to to ADP ejpam-5896 346 3 corollary corollary ADJ ejpam-5896 346 4 1 1 NUM ejpam-5896 346 5 , , PUNCT ejpam-5896 346 6 ( ( PUNCT ejpam-5896 346 7 g,∆ g,∆ X ejpam-5896 346 8 ) ) PUNCT ejpam-5896 346 9 is be AUX ejpam-5896 346 10 ss ss NOUN ejpam-5896 346 11 - - PUNCT ejpam-5896 346 12 sd sd NOUN ejpam-5896 346 13 - - PUNCT ejpam-5896 346 14 set set NOUN ejpam-5896 346 15 . . PUNCT ejpam-5896 347 1 this this DET ejpam-5896 347 2 meas mea NOUN ejpam-5896 347 3 that that SCONJ ejpam-5896 347 4 , , PUNCT ejpam-5896 347 5 ( ( PUNCT ejpam-5896 347 6 g,∆ g,∆ X ejpam-5896 347 7 ) ) PUNCT ejpam-5896 347 8 is be AUX ejpam-5896 347 9 ss ss NOUN ejpam-5896 347 10 - - PUNCT ejpam-5896 347 11 sdc sdc NOUN ejpam-5896 347 12 - - PUNCT ejpam-5896 347 13 set set NOUN ejpam-5896 347 14 . . PUNCT ejpam-5896 348 1 thus thus ADV ejpam-5896 348 2 , , PUNCT ejpam-5896 348 3 ( ( PUNCT ejpam-5896 348 4 u u NOUN ejpam-5896 348 5 , , PUNCT ejpam-5896 348 6 µ,∆ µ,∆ NUM ejpam-5896 348 7 ) ) PUNCT ejpam-5896 348 8 is be AUX ejpam-5896 348 9 ss ss NOUN ejpam-5896 348 10 - - PUNCT ejpam-5896 348 11 sd sd NOUN ejpam-5896 348 12 - - PUNCT ejpam-5896 348 13 disconnected disconnected ADJ ejpam-5896 348 14 . . PUNCT ejpam-5896 349 1 proposition proposition NOUN ejpam-5896 349 2 5 5 NUM ejpam-5896 349 3 . . PUNCT ejpam-5896 350 1 if if SCONJ ejpam-5896 350 2 ( ( PUNCT ejpam-5896 350 3 u u NOUN ejpam-5896 350 4 , , PUNCT ejpam-5896 350 5 µ,∆ µ,∆ NUM ejpam-5896 350 6 ) ) PUNCT ejpam-5896 350 7 is be AUX ejpam-5896 350 8 ss ss NOUN ejpam-5896 350 9 - - PUNCT ejpam-5896 350 10 sd sd NOUN ejpam-5896 350 11 - - PUNCT ejpam-5896 350 12 hyperconnected hyperconnecte VERB ejpam-5896 350 13 , , PUNCT ejpam-5896 350 14 then then ADV ejpam-5896 350 15 it it PRON ejpam-5896 350 16 is be AUX ejpam-5896 350 17 ss ss NOUN ejpam-5896 350 18 - - PUNCT ejpam-5896 350 19 hyperconnected hyperconnected ADJ ejpam-5896 350 20 . . PUNCT ejpam-5896 351 1 proof proof NOUN ejpam-5896 351 2 . . PUNCT ejpam-5896 352 1 assume assume VERB ejpam-5896 352 2 the the DET ejpam-5896 352 3 contrary contrary NOUN ejpam-5896 352 4 that that SCONJ ejpam-5896 352 5 ( ( PUNCT ejpam-5896 352 6 u u NOUN ejpam-5896 352 7 , , PUNCT ejpam-5896 352 8 µ,∆ µ,∆ NUM ejpam-5896 352 9 ) ) PUNCT ejpam-5896 352 10 is be AUX ejpam-5896 352 11 not not PART ejpam-5896 352 12 ss ss ADV ejpam-5896 352 13 - - PUNCT ejpam-5896 352 14 hyperconnected hyperconnected ADJ ejpam-5896 352 15 , , PUNCT ejpam-5896 352 16 then then ADV ejpam-5896 352 17 there there PRON ejpam-5896 352 18 are be VERB ejpam-5896 352 19 ( ( PUNCT ejpam-5896 352 20 g,∆ g,∆ PROPN ejpam-5896 352 21 ) ) PUNCT ejpam-5896 352 22 , , PUNCT ejpam-5896 352 23 ( ( PUNCT ejpam-5896 352 24 h,∆ h,∆ X ejpam-5896 352 25 ) ) PUNCT ejpam-5896 352 26 ∈ ∈ PROPN ejpam-5896 352 27 µ µ X ejpam-5896 352 28 in in ADP ejpam-5896 352 29 which which PRON ejpam-5896 352 30 ( ( PUNCT ejpam-5896 352 31 g,∆)∩̃(h,∆ g,∆)∩̃(h,∆ X ejpam-5896 352 32 ) ) PUNCT ejpam-5896 352 33 = = SYM ejpam-5896 352 34 φ̃. φ̃. PROPN ejpam-5896 352 35 hence hence ADV ejpam-5896 352 36 , , PUNCT ejpam-5896 352 37 ( ( PUNCT ejpam-5896 352 38 g,∆ g,∆ PROPN ejpam-5896 352 39 ) ) PUNCT ejpam-5896 352 40 , , PUNCT ejpam-5896 352 41 ( ( PUNCT ejpam-5896 352 42 h,∆ h,∆ X ejpam-5896 352 43 ) ) PUNCT ejpam-5896 352 44 ∈ ∈ NOUN ejpam-5896 352 45 sd(u)∆ sd(u)∆ NOUN ejpam-5896 352 46 and and CCONJ ejpam-5896 352 47 ( ( PUNCT ejpam-5896 352 48 g,∆)∩̃(h,∆ g,∆)∩̃(h,∆ X ejpam-5896 352 49 ) ) PUNCT ejpam-5896 352 50 = = SYM ejpam-5896 352 51 φ̃. φ̃. PROPN ejpam-5896 352 52 therefore therefore ADV ejpam-5896 352 53 , , PUNCT ejpam-5896 352 54 ũ ũ PROPN ejpam-5896 352 55 is be AUX ejpam-5896 352 56 not not PART ejpam-5896 352 57 ss ss AUX ejpam-5896 352 58 - - PUNCT ejpam-5896 352 59 sd sd NOUN ejpam-5896 352 60 - - PUNCT ejpam-5896 352 61 hyperconnected hyperconnecte VERB ejpam-5896 352 62 . . PUNCT ejpam-5896 353 1 remark remark PROPN ejpam-5896 353 2 2 2 NUM ejpam-5896 353 3 . . PUNCT ejpam-5896 354 1 the the DET ejpam-5896 354 2 example example NOUN ejpam-5896 354 3 that that PRON ejpam-5896 354 4 follows follow VERB ejpam-5896 354 5 illustrates illustrate VERB ejpam-5896 354 6 why why SCONJ ejpam-5896 354 7 the the DET ejpam-5896 354 8 opposite opposite NOUN ejpam-5896 354 9 of of ADP ejpam-5896 354 10 theorem theorem ADJ ejpam-5896 354 11 5 5 NUM ejpam-5896 354 12 is be AUX ejpam-5896 354 13 typically typically ADV ejpam-5896 354 14 not not PART ejpam-5896 354 15 true true ADJ ejpam-5896 354 16 . . PUNCT ejpam-5896 355 1 example example NOUN ejpam-5896 356 1 3 3 X ejpam-5896 356 2 . . PUNCT ejpam-5896 356 3 let let VERB ejpam-5896 356 4 u u PRON ejpam-5896 356 5 = = PUNCT ejpam-5896 356 6 { { PUNCT ejpam-5896 356 7 s1 s1 NOUN ejpam-5896 356 8 , , PUNCT ejpam-5896 356 9 s2 s2 PROPN ejpam-5896 356 10 } } PUNCT ejpam-5896 356 11 , , PUNCT ejpam-5896 356 12 ∆ ∆ PROPN ejpam-5896 356 13 = = SYM ejpam-5896 356 14 { { PUNCT ejpam-5896 356 15 γ1 γ1 PROPN ejpam-5896 356 16 , , PUNCT ejpam-5896 356 17 γ2 γ2 PROPN ejpam-5896 356 18 } } PUNCT ejpam-5896 356 19 and and CCONJ ejpam-5896 356 20 consider consider VERB ejpam-5896 356 21 the the DET ejpam-5896 356 22 soft soft ADJ ejpam-5896 356 23 sets set NOUN ejpam-5896 356 24 ( ( PUNCT ejpam-5896 356 25 hi,∆ hi,∆ PROPN ejpam-5896 356 26 ) ) PUNCT ejpam-5896 356 27 , , PUNCT ejpam-5896 356 28 i i PRON ejpam-5896 356 29 = = NOUN ejpam-5896 356 30 1 1 NUM ejpam-5896 356 31 , , PUNCT ejpam-5896 356 32 2 2 NUM ejpam-5896 356 33 , , PUNCT ejpam-5896 356 34 .... .... PUNCT ejpam-5896 356 35 5 5 NUM ejpam-5896 356 36 over over ADP ejpam-5896 356 37 u u NOUN ejpam-5896 356 38 , , PUNCT ejpam-5896 356 39 where where SCONJ ejpam-5896 356 40 : : PUNCT ejpam-5896 356 41 h1(γ1 h1(γ1 NOUN ejpam-5896 356 42 ) ) PUNCT ejpam-5896 356 43 = = SYM ejpam-5896 356 44 { { PUNCT ejpam-5896 356 45 s1 s1 NOUN ejpam-5896 356 46 } } PUNCT ejpam-5896 356 47 , , PUNCT ejpam-5896 356 48 h1(γ2 h1(γ2 NOUN ejpam-5896 356 49 ) ) PUNCT ejpam-5896 356 50 = = SYM ejpam-5896 356 51 { { PUNCT ejpam-5896 356 52 s1 s1 NOUN ejpam-5896 356 53 } } PUNCT ejpam-5896 356 54 . . PUNCT ejpam-5896 357 1 h2(γ1 h2(γ1 NOUN ejpam-5896 357 2 ) ) PUNCT ejpam-5896 357 3 = = SYM ejpam-5896 357 4 u u NOUN ejpam-5896 357 5 , , PUNCT ejpam-5896 357 6 h2(γ2 h2(γ2 NOUN ejpam-5896 357 7 ) ) PUNCT ejpam-5896 357 8 = = SYM ejpam-5896 357 9 { { PUNCT ejpam-5896 357 10 s2 s2 PROPN ejpam-5896 357 11 } } PUNCT ejpam-5896 357 12 . . PUNCT ejpam-5896 358 1 h3(γ1 h3(γ1 NOUN ejpam-5896 358 2 ) ) PUNCT ejpam-5896 358 3 = = PRON ejpam-5896 358 4 { { PUNCT ejpam-5896 358 5 s1 s1 NOUN ejpam-5896 358 6 } } PUNCT ejpam-5896 358 7 , , PUNCT ejpam-5896 358 8 h3(γ2 h3(γ2 NOUN ejpam-5896 358 9 ) ) PUNCT ejpam-5896 358 10 = = PUNCT ejpam-5896 359 1 u. u. PROPN ejpam-5896 359 2 abd abd PROPN ejpam-5896 359 3 el el PROPN ejpam-5896 359 4 - - PROPN ejpam-5896 359 5 latif latif PROPN ejpam-5896 359 6 et et PROPN ejpam-5896 359 7 al al PROPN ejpam-5896 359 8 . . PUNCT ejpam-5896 359 9 / / SYM ejpam-5896 359 10 eur eur PROPN ejpam-5896 359 11 . . PUNCT ejpam-5896 360 1 j. j. PROPN ejpam-5896 360 2 pure pure PROPN ejpam-5896 360 3 appl appl PROPN ejpam-5896 360 4 . . PROPN ejpam-5896 360 5 math math PROPN ejpam-5896 360 6 , , PUNCT ejpam-5896 360 7 18 18 NUM ejpam-5896 360 8 ( ( PUNCT ejpam-5896 360 9 2 2 NUM ejpam-5896 360 10 ) ) PUNCT ejpam-5896 360 11 ( ( PUNCT ejpam-5896 360 12 2025 2025 NUM ejpam-5896 360 13 ) ) PUNCT ejpam-5896 360 14 , , PUNCT ejpam-5896 360 15 5896 5896 NUM ejpam-5896 360 16 10 10 NUM ejpam-5896 360 17 of of ADP ejpam-5896 360 18 20 20 NUM ejpam-5896 360 19 h4(γ1 h4(γ1 NOUN ejpam-5896 360 20 ) ) PUNCT ejpam-5896 360 21 = = SYM ejpam-5896 360 22 u u NOUN ejpam-5896 360 23 , , PUNCT ejpam-5896 360 24 h4(γ2 h4(γ2 NOUN ejpam-5896 360 25 ) ) PUNCT ejpam-5896 360 26 = = SYM ejpam-5896 360 27 { { PUNCT ejpam-5896 360 28 s1 s1 NOUN ejpam-5896 360 29 } } PUNCT ejpam-5896 360 30 . . PUNCT ejpam-5896 361 1 h5(γ1 h5(γ1 NOUN ejpam-5896 361 2 ) ) PUNCT ejpam-5896 362 1 = = PRON ejpam-5896 362 2 { { PUNCT ejpam-5896 362 3 s1 s1 PROPN ejpam-5896 362 4 } } PUNCT ejpam-5896 362 5 , , PUNCT ejpam-5896 362 6 h5(γ2 h5(γ2 NOUN ejpam-5896 362 7 ) ) PUNCT ejpam-5896 362 8 = = SYM ejpam-5896 362 9 { { PUNCT ejpam-5896 362 10 s2 s2 PROPN ejpam-5896 362 11 } } PUNCT ejpam-5896 362 12 . . PUNCT ejpam-5896 363 1 consider consider VERB ejpam-5896 363 2 µ µ X ejpam-5896 363 3 = = SYM ejpam-5896 363 4 { { PUNCT ejpam-5896 363 5 ũ ũ PROPN ejpam-5896 363 6 , , PUNCT ejpam-5896 363 7 φ̃ φ̃ PROPN ejpam-5896 363 8 , , PUNCT ejpam-5896 363 9 ( ( PUNCT ejpam-5896 363 10 hi,∆ hi,∆ PROPN ejpam-5896 363 11 ) ) PUNCT ejpam-5896 363 12 , , PUNCT ejpam-5896 363 13 i i PRON ejpam-5896 363 14 = = NOUN ejpam-5896 363 15 1 1 NUM ejpam-5896 363 16 , , PUNCT ejpam-5896 363 17 2 2 NUM ejpam-5896 363 18 , , PUNCT ejpam-5896 363 19 .... .... PUNCT ejpam-5896 363 20 , , PUNCT ejpam-5896 363 21 5 5 X ejpam-5896 363 22 } } PUNCT ejpam-5896 363 23 is be AUX ejpam-5896 363 24 an an DET ejpam-5896 363 25 ssts sst NOUN ejpam-5896 363 26 on on ADP ejpam-5896 363 27 u u NOUN ejpam-5896 363 28 , , PUNCT ejpam-5896 363 29 then then ADV ejpam-5896 363 30 it it PRON ejpam-5896 363 31 is be AUX ejpam-5896 363 32 easy easy ADJ ejpam-5896 363 33 to to PART ejpam-5896 363 34 check check VERB ejpam-5896 363 35 that that PRON ejpam-5896 363 36 ( ( PUNCT ejpam-5896 363 37 u u NOUN ejpam-5896 363 38 , , PUNCT ejpam-5896 363 39 µ,∆ µ,∆ NUM ejpam-5896 363 40 ) ) PUNCT ejpam-5896 363 41 is be AUX ejpam-5896 363 42 an an DET ejpam-5896 363 43 ss ss NOUN ejpam-5896 363 44 - - PUNCT ejpam-5896 363 45 hyperconnected hyperconnecte VERB ejpam-5896 363 46 . . PUNCT ejpam-5896 364 1 alternatively alternatively ADV ejpam-5896 364 2 , , PUNCT ejpam-5896 364 3 regarding regard VERB ejpam-5896 364 4 the the DET ejpam-5896 364 5 soft soft ADJ ejpam-5896 364 6 sets set NOUN ejpam-5896 364 7 ( ( PUNCT ejpam-5896 364 8 c,∆ c,∆ NOUN ejpam-5896 364 9 ) ) PUNCT ejpam-5896 364 10 , , PUNCT ejpam-5896 364 11 ( ( PUNCT ejpam-5896 364 12 d,∆ d,∆ X ejpam-5896 364 13 ) ) PUNCT ejpam-5896 364 14 where where SCONJ ejpam-5896 364 15 : : PUNCT ejpam-5896 364 16 c(γ1 c(γ1 VERB ejpam-5896 364 17 ) ) PUNCT ejpam-5896 364 18 = = SYM ejpam-5896 364 19 φ φ NUM ejpam-5896 364 20 , , PUNCT ejpam-5896 364 21 c(γ2 c(γ2 NOUN ejpam-5896 364 22 ) ) PUNCT ejpam-5896 364 23 = = SYM ejpam-5896 364 24 u. u. NOUN ejpam-5896 364 25 d(γ1 d(γ1 NOUN ejpam-5896 364 26 ) ) PUNCT ejpam-5896 364 27 = = SYM ejpam-5896 364 28 u u NOUN ejpam-5896 364 29 , , PUNCT ejpam-5896 364 30 d(γ2 d(γ2 NOUN ejpam-5896 364 31 ) ) PUNCT ejpam-5896 364 32 = = SYM ejpam-5896 364 33 φ φ PROPN ejpam-5896 364 34 , , PUNCT ejpam-5896 364 35 we we PRON ejpam-5896 364 36 have have VERB ejpam-5896 364 37 ( ( PUNCT ejpam-5896 364 38 c,∆ c,∆ NOUN ejpam-5896 364 39 ) ) PUNCT ejpam-5896 364 40 , , PUNCT ejpam-5896 364 41 ( ( PUNCT ejpam-5896 364 42 d,∆ d,∆ X ejpam-5896 364 43 ) ) PUNCT ejpam-5896 364 44 ∈ ∈ PROPN ejpam-5896 364 45 sd(u)∆ sd(u)∆ NOUN ejpam-5896 364 46 in in ADP ejpam-5896 364 47 which which PRON ejpam-5896 364 48 ( ( PUNCT ejpam-5896 364 49 c,∆)∩̃(d,∆ c,∆)∩̃(d,∆ X ejpam-5896 364 50 ) ) PUNCT ejpam-5896 364 51 = = SYM ejpam-5896 364 52 φ̃. φ̃. PROPN ejpam-5896 364 53 therefore therefore ADV ejpam-5896 364 54 , , PUNCT ejpam-5896 364 55 u u NOUN ejpam-5896 364 56 is be AUX ejpam-5896 364 57 not not PART ejpam-5896 364 58 ss ss NOUN ejpam-5896 364 59 - - PUNCT ejpam-5896 364 60 sdhyperconnected sdhyperconnecte VERB ejpam-5896 364 61 . . PUNCT ejpam-5896 365 1 corollary corollary ADJ ejpam-5896 365 2 3 3 NUM ejpam-5896 365 3 . . PUNCT ejpam-5896 366 1 the the DET ejpam-5896 366 2 following follow VERB ejpam-5896 366 3 conclusions conclusion NOUN ejpam-5896 366 4 are be AUX ejpam-5896 366 5 valid valid ADJ ejpam-5896 366 6 for for ADP ejpam-5896 366 7 an an DET ejpam-5896 366 8 ssts sst NOUN ejpam-5896 366 9 ( ( PUNCT ejpam-5896 366 10 u u NOUN ejpam-5896 366 11 , , PUNCT ejpam-5896 366 12 µ,∆ µ,∆ NUM ejpam-5896 366 13 ) ) PUNCT ejpam-5896 366 14 from from ADP ejpam-5896 366 15 propositions proposition NOUN ejpam-5896 366 16 3 3 NUM ejpam-5896 366 17 , , PUNCT ejpam-5896 366 18 5 5 NUM ejpam-5896 366 19 , , PUNCT ejpam-5896 366 20 and and CCONJ ejpam-5896 366 21 theorem theorem VERB ejpam-5896 366 22 13 13 NUM ejpam-5896 366 23 , , PUNCT ejpam-5896 366 24 which which PRON ejpam-5896 366 25 are be AUX ejpam-5896 366 26 not not PART ejpam-5896 366 27 reversible reversible ADJ ejpam-5896 366 28 . . PUNCT ejpam-5896 367 1 ss ss AUX ejpam-5896 367 2 - - PUNCT ejpam-5896 367 3 sd sd NOUN ejpam-5896 367 4 - - PUNCT ejpam-5896 367 5 hyperconnected hyperconnecte VERB ejpam-5896 367 6 ⇔ ⇔ PROPN ejpam-5896 367 7 ss ss PROPN ejpam-5896 367 8 - - PUNCT ejpam-5896 367 9 sd sd NOUN ejpam-5896 367 10 - - PUNCT ejpam-5896 367 11 connected connect VERB ejpam-5896 367 12 ⇒ ⇒ NOUN ejpam-5896 367 13 ssl ssl PROPN ejpam-5896 367 14 - - PUNCT ejpam-5896 367 15 sd sd NOUN ejpam-5896 367 16 - - PUNCT ejpam-5896 367 17 connected connect VERB ejpam-5896 367 18 ⇓ ⇓ PROPN ejpam-5896 367 19 ⇓ ⇓ PROPN ejpam-5896 367 20 ⇓ ⇓ PROPN ejpam-5896 367 21 ss ss NOUN ejpam-5896 367 22 - - PUNCT ejpam-5896 367 23 hyperconnected hyperconnecte VERB ejpam-5896 367 24 ⇒ ⇒ NOUN ejpam-5896 367 25 ss ss ADJ ejpam-5896 367 26 - - PUNCT ejpam-5896 367 27 connected connected ADJ ejpam-5896 367 28 ⇒ ⇒ NOUN ejpam-5896 367 29 ssl ssl VERB ejpam-5896 367 30 - - ADJ ejpam-5896 367 31 connected connected ADJ ejpam-5896 367 32 figure figure NOUN ejpam-5896 367 33 1 1 NUM ejpam-5896 367 34 . . PUNCT ejpam-5896 368 1 the the DET ejpam-5896 368 2 relationships relationship NOUN ejpam-5896 368 3 between between ADP ejpam-5896 368 4 different different ADJ ejpam-5896 368 5 types type NOUN ejpam-5896 368 6 of of ADP ejpam-5896 368 7 connectedness connectedness NOUN ejpam-5896 368 8 via via ADP ejpam-5896 368 9 ss ss NOUN ejpam-5896 368 10 - - PUNCT ejpam-5896 368 11 sd sd NOUN ejpam-5896 368 12 - - PUNCT ejpam-5896 368 13 sets set NOUN ejpam-5896 368 14 in in ADP ejpam-5896 368 15 the the DET ejpam-5896 368 16 frame frame NOUN ejpam-5896 368 17 of of ADP ejpam-5896 368 18 sstss sstss NOUN ejpam-5896 368 19 . . PUNCT ejpam-5896 369 1 4 4 X ejpam-5896 369 2 . . X ejpam-5896 369 3 compactness compactness NOUN ejpam-5896 369 4 and and CCONJ ejpam-5896 369 5 lindelöfness lindelöfness X ejpam-5896 369 6 based base VERB ejpam-5896 369 7 on on ADP ejpam-5896 369 8 supra supra PROPN ejpam-5896 369 9 soft soft ADJ ejpam-5896 369 10 sd sd NOUN ejpam-5896 369 11 - - PUNCT ejpam-5896 369 12 sets set NOUN ejpam-5896 369 13 herein herein NOUN ejpam-5896 369 14 , , PUNCT ejpam-5896 369 15 we we PRON ejpam-5896 369 16 define define VERB ejpam-5896 369 17 novel novel ADJ ejpam-5896 369 18 forms form NOUN ejpam-5896 369 19 of of ADP ejpam-5896 369 20 compactness compactness NOUN ejpam-5896 369 21 related relate VERB ejpam-5896 369 22 to to ADP ejpam-5896 369 23 the the DET ejpam-5896 369 24 notions notion NOUN ejpam-5896 369 25 of of ADP ejpam-5896 369 26 ss ss NOUN ejpam-5896 369 27 - - PUNCT ejpam-5896 369 28 sd sd NOUN ejpam-5896 369 29 - - PUNCT ejpam-5896 369 30 sets set NOUN ejpam-5896 369 31 in in ADP ejpam-5896 369 32 the the DET ejpam-5896 369 33 frame frame NOUN ejpam-5896 369 34 of of ADP ejpam-5896 369 35 sstss sstss NOUN ejpam-5896 369 36 , , PUNCT ejpam-5896 369 37 namely namely ADV ejpam-5896 369 38 ss ss NOUN ejpam-5896 369 39 - - PUNCT ejpam-5896 369 40 sd sd NOUN ejpam-5896 369 41 - - PUNCT ejpam-5896 369 42 compactness compactness NOUN ejpam-5896 369 43 and and CCONJ ejpam-5896 369 44 ss ss NOUN ejpam-5896 369 45 - - PUNCT ejpam-5896 369 46 sd sd NOUN ejpam-5896 369 47 - - PUNCT ejpam-5896 369 48 lindelöfness lindelöfness NOUN ejpam-5896 369 49 . . PUNCT ejpam-5896 370 1 we we PRON ejpam-5896 370 2 discuss discuss VERB ejpam-5896 370 3 their their PRON ejpam-5896 370 4 essential essential ADJ ejpam-5896 370 5 properties property NOUN ejpam-5896 370 6 comprehensively comprehensively ADV ejpam-5896 370 7 . . PUNCT ejpam-5896 371 1 specifically specifically ADV ejpam-5896 371 2 , , PUNCT ejpam-5896 371 3 we we PRON ejpam-5896 371 4 show show VERB ejpam-5896 371 5 that that SCONJ ejpam-5896 371 6 the the DET ejpam-5896 371 7 soft soft ADJ ejpam-5896 371 8 intersection intersection NOUN ejpam-5896 371 9 of of ADP ejpam-5896 371 10 an an DET ejpam-5896 371 11 ss ss NOUN ejpam-5896 371 12 - - PUNCT ejpam-5896 371 13 sd sd NOUN ejpam-5896 371 14 - - PUNCT ejpam-5896 371 15 compact compact ADJ ejpam-5896 371 16 ( ( PUNCT ejpam-5896 371 17 lindelöf lindelöf PROPN ejpam-5896 371 18 ) ) PUNCT ejpam-5896 371 19 soft soft ADJ ejpam-5896 371 20 set set NOUN ejpam-5896 371 21 and and CCONJ ejpam-5896 371 22 ss ss NOUN ejpam-5896 371 23 - - ADJ ejpam-5896 371 24 sc sc NOUN ejpam-5896 371 25 - - PUNCT ejpam-5896 371 26 set set NOUN ejpam-5896 371 27 is be AUX ejpam-5896 371 28 an an DET ejpam-5896 371 29 ss ss NOUN ejpam-5896 371 30 - - PUNCT ejpam-5896 371 31 sd sd NOUN ejpam-5896 371 32 - - PUNCT ejpam-5896 371 33 compact compact ADJ ejpam-5896 371 34 ( ( PUNCT ejpam-5896 371 35 lindelöf lindelöf PROPN ejpam-5896 371 36 ) ) PUNCT ejpam-5896 371 37 . . PUNCT ejpam-5896 372 1 moreover moreover ADV ejpam-5896 372 2 , , PUNCT ejpam-5896 372 3 the the DET ejpam-5896 372 4 behaviour behaviour NOUN ejpam-5896 372 5 of of ADP ejpam-5896 372 6 an an DET ejpam-5896 372 7 ss ss NOUN ejpam-5896 372 8 - - PUNCT ejpam-5896 372 9 sd sd NOUN ejpam-5896 372 10 - - PUNCT ejpam-5896 372 11 compact compact ADJ ejpam-5896 372 12 ( ( PUNCT ejpam-5896 372 13 lindelöf lindelöf NOUN ejpam-5896 372 14 ) ) PUNCT ejpam-5896 372 15 ssts sst NOUN ejpam-5896 372 16 with with ADP ejpam-5896 372 17 the the DET ejpam-5896 372 18 sfip sfip NOUN ejpam-5896 372 19 ( ( PUNCT ejpam-5896 372 20 scip scip PROPN ejpam-5896 372 21 ) ) PUNCT ejpam-5896 372 22 has have AUX ejpam-5896 372 23 been be AUX ejpam-5896 372 24 presented present VERB ejpam-5896 372 25 . . PUNCT ejpam-5896 373 1 also also ADV ejpam-5896 373 2 , , PUNCT ejpam-5896 373 3 we we PRON ejpam-5896 373 4 demonstrate demonstrate VERB ejpam-5896 373 5 that that SCONJ ejpam-5896 373 6 the the DET ejpam-5896 373 7 image image NOUN ejpam-5896 373 8 ( ( PUNCT ejpam-5896 373 9 respectively respectively ADV ejpam-5896 373 10 , , PUNCT ejpam-5896 373 11 pre pre ADJ ejpam-5896 373 12 - - NOUN ejpam-5896 373 13 image image ADJ ejpam-5896 373 14 ) ) PUNCT ejpam-5896 373 15 of of ADP ejpam-5896 373 16 each each DET ejpam-5896 373 17 ss ss NOUN ejpam-5896 373 18 - - PUNCT ejpam-5896 373 19 sd sd NOUN ejpam-5896 373 20 - - PUNCT ejpam-5896 373 21 lindelöf lindelöf NOUN ejpam-5896 373 22 ( ( PUNCT ejpam-5896 373 23 compact compact ADJ ejpam-5896 373 24 ) ) PUNCT ejpam-5896 373 25 is be AUX ejpam-5896 373 26 an an DET ejpam-5896 373 27 ss ss NOUN ejpam-5896 373 28 - - PUNCT ejpam-5896 373 29 lindelöf lindelöf NOUN ejpam-5896 373 30 ( ( PUNCT ejpam-5896 373 31 compact compact ADJ ejpam-5896 373 32 ) ) PUNCT ejpam-5896 373 33 under under ADP ejpam-5896 373 34 a a DET ejpam-5896 373 35 surjective surjective ADJ ejpam-5896 373 36 and and CCONJ ejpam-5896 373 37 ss ss NOUN ejpam-5896 373 38 - - PUNCT ejpam-5896 373 39 sd sd NOUN ejpam-5896 373 40 - - PUNCT ejpam-5896 373 41 continuous continuous ADJ ejpam-5896 373 42 ( ( PUNCT ejpam-5896 373 43 respectively respectively ADV ejpam-5896 373 44 , , PUNCT ejpam-5896 373 45 an an DET ejpam-5896 373 46 injective injective ADJ ejpam-5896 373 47 and and CCONJ ejpam-5896 373 48 ss ss NOUN ejpam-5896 373 49 - - PUNCT ejpam-5896 373 50 sd sd NOUN ejpam-5896 373 51 - - PUNCT ejpam-5896 373 52 open open ADJ ejpam-5896 373 53 ) ) PUNCT ejpam-5896 373 54 map map NOUN ejpam-5896 373 55 . . PUNCT ejpam-5896 374 1 we we PRON ejpam-5896 374 2 discuss discuss VERB ejpam-5896 374 3 relationships relationship NOUN ejpam-5896 374 4 among among ADP ejpam-5896 374 5 supra supra ADJ ejpam-5896 374 6 soft soft ADJ ejpam-5896 374 7 - - PUNCT ejpam-5896 374 8 sd sd NOUN ejpam-5896 374 9 - - PUNCT ejpam-5896 374 10 compact compact ADJ ejpam-5896 374 11 ( ( PUNCT ejpam-5896 374 12 lindelöf lindelöf NOUN ejpam-5896 374 13 ) ) PUNCT ejpam-5896 374 14 ssts sst NOUN ejpam-5896 374 15 and and CCONJ ejpam-5896 374 16 their their PRON ejpam-5896 374 17 parametric parametric ADJ ejpam-5896 374 18 supra supra PROPN ejpam-5896 374 19 topological topological PROPN ejpam-5896 374 20 spaces space NOUN ejpam-5896 374 21 , , PUNCT ejpam-5896 374 22 which which PRON ejpam-5896 374 23 are be AUX ejpam-5896 374 24 defined define VERB ejpam-5896 374 25 on on ADP ejpam-5896 374 26 any any DET ejpam-5896 374 27 universal universal ADJ ejpam-5896 374 28 set set NOUN ejpam-5896 374 29 and and CCONJ ejpam-5896 374 30 finite finite ADJ ejpam-5896 374 31 set set NOUN ejpam-5896 374 32 of of ADP ejpam-5896 374 33 parameters parameter NOUN ejpam-5896 374 34 . . PUNCT ejpam-5896 375 1 furthermore furthermore ADV ejpam-5896 375 2 , , PUNCT ejpam-5896 375 3 we we PRON ejpam-5896 375 4 study study VERB ejpam-5896 375 5 the the DET ejpam-5896 375 6 relationships relationship NOUN ejpam-5896 375 7 with with ADP ejpam-5896 375 8 previous previous ADJ ejpam-5896 375 9 studies study NOUN ejpam-5896 375 10 and and CCONJ ejpam-5896 375 11 add add VERB ejpam-5896 375 12 a a DET ejpam-5896 375 13 topological topological ADJ ejpam-5896 375 14 chart chart NOUN ejpam-5896 375 15 to to PART ejpam-5896 375 16 illustrate illustrate VERB ejpam-5896 375 17 the the DET ejpam-5896 375 18 key key ADJ ejpam-5896 375 19 concepts concept NOUN ejpam-5896 375 20 presented present VERB ejpam-5896 375 21 in in ADP ejpam-5896 375 22 this this DET ejpam-5896 375 23 section section NOUN ejpam-5896 375 24 in in ADP ejpam-5896 375 25 figure figure NOUN ejpam-5896 375 26 2 2 NUM ejpam-5896 375 27 . . PUNCT ejpam-5896 376 1 the the DET ejpam-5896 376 2 arrows arrow NOUN ejpam-5896 376 3 in in ADP ejpam-5896 376 4 this this DET ejpam-5896 376 5 chart chart NOUN ejpam-5896 376 6 are be AUX ejpam-5896 376 7 non non ADJ ejpam-5896 376 8 - - ADJ ejpam-5896 376 9 reversible reversible ADJ ejpam-5896 376 10 , , PUNCT ejpam-5896 376 11 as as SCONJ ejpam-5896 376 12 has have AUX ejpam-5896 376 13 been be AUX ejpam-5896 376 14 confirmed confirm VERB ejpam-5896 376 15 by by ADP ejpam-5896 376 16 concrete concrete ADJ ejpam-5896 376 17 counterexamples counterexample NOUN ejpam-5896 376 18 . . PUNCT ejpam-5896 377 1 definition definition NOUN ejpam-5896 377 2 20 20 NUM ejpam-5896 377 3 . . PUNCT ejpam-5896 378 1 a a DET ejpam-5896 378 2 family family NOUN ejpam-5896 378 3 of of ADP ejpam-5896 378 4 soft soft ADJ ejpam-5896 378 5 sets set NOUN ejpam-5896 378 6 ψ ψ X ejpam-5896 378 7 = = SYM ejpam-5896 378 8 { { PUNCT ejpam-5896 378 9 ( ( PUNCT ejpam-5896 378 10 cϵ,∆ cϵ,∆ ADJ ejpam-5896 378 11 ) ) PUNCT ejpam-5896 378 12 : : PUNCT ejpam-5896 379 1 ϵ ϵ X ejpam-5896 379 2 ∈ ∈ PROPN ejpam-5896 379 3 ε ε PROPN ejpam-5896 379 4 } } PUNCT ejpam-5896 379 5 is be AUX ejpam-5896 379 6 claimed claim VERB ejpam-5896 379 7 to to PART ejpam-5896 379 8 be be AUX ejpam-5896 379 9 a a DET ejpam-5896 379 10 supra supra ADJ ejpam-5896 379 11 soft soft ADJ ejpam-5896 379 12 open open ADJ ejpam-5896 379 13 cover cover NOUN ejpam-5896 379 14 , , PUNCT ejpam-5896 379 15 if if SCONJ ejpam-5896 379 16 all all PRON ejpam-5896 379 17 of of ADP ejpam-5896 379 18 the the DET ejpam-5896 379 19 members member NOUN ejpam-5896 379 20 of of ADP ejpam-5896 379 21 ψ ψ NOUN ejpam-5896 379 22 are be AUX ejpam-5896 379 23 supra supra ADJ ejpam-5896 379 24 open open ADJ ejpam-5896 379 25 soft soft ADJ ejpam-5896 379 26 sets set NOUN ejpam-5896 379 27 . . PUNCT ejpam-5896 379 28 . . PUNCT ejpam-5896 380 1 definition definition NOUN ejpam-5896 380 2 21 21 NUM ejpam-5896 380 3 . . PUNCT ejpam-5896 381 1 a a DET ejpam-5896 381 2 class class NOUN ejpam-5896 381 3 of of ADP ejpam-5896 381 4 ss ss NOUN ejpam-5896 381 5 - - PUNCT ejpam-5896 381 6 sd sd NOUN ejpam-5896 381 7 - - PUNCT ejpam-5896 381 8 subsets subset NOUN ejpam-5896 381 9 ψ ψ NOUN ejpam-5896 381 10 = = SYM ejpam-5896 381 11 { { PUNCT ejpam-5896 381 12 ( ( PUNCT ejpam-5896 381 13 cϵ,∆ cϵ,∆ ADJ ejpam-5896 381 14 ) ) PUNCT ejpam-5896 381 15 : : PUNCT ejpam-5896 382 1 ϵ ϵ X ejpam-5896 382 2 ∈ ∈ PROPN ejpam-5896 382 3 ε ε PROPN ejpam-5896 382 4 } } PUNCT ejpam-5896 382 5 of of ADP ejpam-5896 382 6 an an DET ejpam-5896 382 7 ssts sst NOUN ejpam-5896 382 8 ( ( PUNCT ejpam-5896 382 9 u u NOUN ejpam-5896 382 10 , , PUNCT ejpam-5896 382 11 µ,∆ µ,∆ NUM ejpam-5896 382 12 ) ) PUNCT ejpam-5896 382 13 is be AUX ejpam-5896 382 14 claimed claim VERB ejpam-5896 382 15 to to PART ejpam-5896 382 16 be be AUX ejpam-5896 382 17 an an DET ejpam-5896 382 18 ss ss NOUN ejpam-5896 382 19 - - PUNCT ejpam-5896 382 20 sd sd NOUN ejpam-5896 382 21 - - PUNCT ejpam-5896 382 22 cover cover NOUN ejpam-5896 382 23 of of ADP ejpam-5896 382 24 soft soft ADJ ejpam-5896 382 25 subset subset NOUN ejpam-5896 382 26 ( ( PUNCT ejpam-5896 382 27 g,∆ g,∆ PROPN ejpam-5896 382 28 ) ) PUNCT ejpam-5896 382 29 of of ADP ejpam-5896 382 30 ũ ũ PROPN ejpam-5896 382 31 , , PUNCT ejpam-5896 383 1 if if SCONJ ejpam-5896 383 2 ( ( PUNCT ejpam-5896 383 3 g,∆)⊆̃ψ g,∆)⊆̃ψ NOUN ejpam-5896 383 4 . . PUNCT ejpam-5896 383 5 definition definition NOUN ejpam-5896 383 6 22 22 NUM ejpam-5896 383 7 . . PUNCT ejpam-5896 384 1 a a DET ejpam-5896 384 2 soft soft ADJ ejpam-5896 384 3 subset subset NOUN ejpam-5896 384 4 ( ( PUNCT ejpam-5896 384 5 g,∆ g,∆ PROPN ejpam-5896 384 6 ) ) PUNCT ejpam-5896 384 7 of of ADP ejpam-5896 384 8 an an DET ejpam-5896 384 9 ssts sst NOUN ejpam-5896 384 10 ( ( PUNCT ejpam-5896 384 11 u u NOUN ejpam-5896 384 12 , , PUNCT ejpam-5896 384 13 µ,∆ µ,∆ NUM ejpam-5896 384 14 ) ) PUNCT ejpam-5896 384 15 is be AUX ejpam-5896 384 16 claimed claim VERB ejpam-5896 384 17 to to PART ejpam-5896 384 18 be be AUX ejpam-5896 384 19 ss ss NOUN ejpam-5896 384 20 - - PUNCT ejpam-5896 384 21 sd sd NOUN ejpam-5896 384 22 - - PUNCT ejpam-5896 384 23 compact compact ADJ ejpam-5896 384 24 ( ( PUNCT ejpam-5896 384 25 lindelöf lindelöf PROPN ejpam-5896 384 26 ) ) PUNCT ejpam-5896 384 27 , , PUNCT ejpam-5896 384 28 if if SCONJ ejpam-5896 384 29 every every DET ejpam-5896 384 30 ss ss NOUN ejpam-5896 384 31 - - PUNCT ejpam-5896 384 32 sd sd NOUN ejpam-5896 384 33 - - PUNCT ejpam-5896 384 34 cover cover NOUN ejpam-5896 384 35 { { PUNCT ejpam-5896 384 36 ( ( PUNCT ejpam-5896 384 37 cϵ,∆ cϵ,∆ ADJ ejpam-5896 384 38 ) ) PUNCT ejpam-5896 384 39 : : PUNCT ejpam-5896 385 1 ϵ ϵ X ejpam-5896 385 2 ∈ ∈ PROPN ejpam-5896 385 3 ε ε PROPN ejpam-5896 385 4 } } PUNCT ejpam-5896 385 5 of of ADP ejpam-5896 385 6 ( ( PUNCT ejpam-5896 385 7 g,∆ g,∆ PROPN ejpam-5896 385 8 ) ) PUNCT ejpam-5896 385 9 has have VERB ejpam-5896 385 10 a a DET ejpam-5896 385 11 finite finite NOUN ejpam-5896 385 12 ( ( PUNCT ejpam-5896 385 13 countable countable ADJ ejpam-5896 385 14 ) ) PUNCT ejpam-5896 385 15 subclass subclass NOUN ejpam-5896 385 16 εo εo NOUN ejpam-5896 385 17 of of ADP ejpam-5896 385 18 ε ε PROPN ejpam-5896 385 19 such such ADJ ejpam-5896 385 20 that that SCONJ ejpam-5896 385 21 ( ( PUNCT ejpam-5896 385 22 g,∆)⊆̃ g,∆)⊆̃ PROPN ejpam-5896 385 23 ⋃̃ ⋃̃ PROPN ejpam-5896 385 24 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 385 25 ) ) PUNCT ejpam-5896 385 26 . . PUNCT ejpam-5896 386 1 the the DET ejpam-5896 386 2 space space NOUN ejpam-5896 386 3 ( ( PUNCT ejpam-5896 386 4 u u NOUN ejpam-5896 386 5 , , PUNCT ejpam-5896 386 6 µ,∆ µ,∆ NUM ejpam-5896 386 7 ) ) PUNCT ejpam-5896 386 8 is be AUX ejpam-5896 386 9 claimed claim VERB ejpam-5896 386 10 to to PART ejpam-5896 386 11 be be AUX ejpam-5896 386 12 ss ss NOUN ejpam-5896 386 13 - - PUNCT ejpam-5896 386 14 sd sd NOUN ejpam-5896 386 15 - - PUNCT ejpam-5896 386 16 compact compact ADJ ejpam-5896 386 17 ( ( PUNCT ejpam-5896 386 18 lindelöf lindelöf PROPN ejpam-5896 386 19 ) ) PUNCT ejpam-5896 386 20 if if SCONJ ejpam-5896 386 21 ũ ũ PROPN ejpam-5896 386 22 is be AUX ejpam-5896 386 23 ss ss NOUN ejpam-5896 386 24 - - PUNCT ejpam-5896 386 25 sd sd NOUN ejpam-5896 386 26 - - PUNCT ejpam-5896 386 27 compact compact ADJ ejpam-5896 386 28 ( ( PUNCT ejpam-5896 386 29 lindelöf lindelöf PROPN ejpam-5896 386 30 ) ) PUNCT ejpam-5896 386 31 as as ADP ejpam-5896 386 32 a a DET ejpam-5896 386 33 soft soft ADJ ejpam-5896 386 34 subset subset NOUN ejpam-5896 386 35 . . PUNCT ejpam-5896 387 1 theorem theorem VERB ejpam-5896 387 2 14 14 NUM ejpam-5896 387 3 . . PUNCT ejpam-5896 388 1 if if SCONJ ejpam-5896 388 2 the the DET ejpam-5896 388 3 components component NOUN ejpam-5896 388 4 of of ADP ejpam-5896 388 5 an an DET ejpam-5896 388 6 ss ss NOUN ejpam-5896 388 7 - - PUNCT ejpam-5896 388 8 sd sd NOUN ejpam-5896 388 9 - - PUNCT ejpam-5896 388 10 compact compact ADJ ejpam-5896 388 11 ssts sst NOUN ejpam-5896 388 12 ( ( PUNCT ejpam-5896 388 13 u u NOUN ejpam-5896 388 14 , , PUNCT ejpam-5896 388 15 µ,∆ µ,∆ NUM ejpam-5896 388 16 ) ) PUNCT ejpam-5896 388 17 are be AUX ejpam-5896 388 18 ss ss NOUN ejpam-5896 388 19 - - PUNCT ejpam-5896 388 20 sd sd NOUN ejpam-5896 388 21 - - PUNCT ejpam-5896 388 22 sets set NOUN ejpam-5896 388 23 , , PUNCT ejpam-5896 388 24 then then ADV ejpam-5896 388 25 they they PRON ejpam-5896 388 26 are be AUX ejpam-5896 388 27 finite finite ADJ ejpam-5896 388 28 . . PUNCT ejpam-5896 389 1 abd abd PROPN ejpam-5896 389 2 el el PROPN ejpam-5896 389 3 - - PROPN ejpam-5896 389 4 latif latif PROPN ejpam-5896 389 5 et et PROPN ejpam-5896 389 6 al al PROPN ejpam-5896 389 7 . . PUNCT ejpam-5896 389 8 / / SYM ejpam-5896 389 9 eur eur PROPN ejpam-5896 389 10 . . PUNCT ejpam-5896 390 1 j. j. PROPN ejpam-5896 390 2 pure pure PROPN ejpam-5896 390 3 appl appl PROPN ejpam-5896 390 4 . . PROPN ejpam-5896 390 5 math math PROPN ejpam-5896 390 6 , , PUNCT ejpam-5896 390 7 18 18 NUM ejpam-5896 390 8 ( ( PUNCT ejpam-5896 390 9 2 2 NUM ejpam-5896 390 10 ) ) PUNCT ejpam-5896 390 11 ( ( PUNCT ejpam-5896 390 12 2025 2025 NUM ejpam-5896 390 13 ) ) PUNCT ejpam-5896 390 14 , , PUNCT ejpam-5896 390 15 5896 5896 NUM ejpam-5896 390 16 11 11 NUM ejpam-5896 390 17 of of ADP ejpam-5896 390 18 20 20 NUM ejpam-5896 390 19 proof proof NOUN ejpam-5896 390 20 . . PUNCT ejpam-5896 391 1 let let VERB ejpam-5896 391 2 ( ( PUNCT ejpam-5896 391 3 u u NOUN ejpam-5896 391 4 , , PUNCT ejpam-5896 391 5 µ,∆ µ,∆ NUM ejpam-5896 391 6 ) ) PUNCT ejpam-5896 391 7 be be VERB ejpam-5896 391 8 an an DET ejpam-5896 391 9 ss ss NOUN ejpam-5896 391 10 - - PUNCT ejpam-5896 391 11 sd sd NOUN ejpam-5896 391 12 - - PUNCT ejpam-5896 391 13 compact compact ADJ ejpam-5896 391 14 ssts sst NOUN ejpam-5896 391 15 and and CCONJ ejpam-5896 391 16 assume assume VERB ejpam-5896 391 17 the the DET ejpam-5896 391 18 contrary contrary NOUN ejpam-5896 391 19 that that PRON ejpam-5896 391 20 ũ ũ PROPN ejpam-5896 391 21 has have VERB ejpam-5896 391 22 infinite infinite ADJ ejpam-5896 391 23 number number NOUN ejpam-5896 391 24 of of ADP ejpam-5896 391 25 ss ss NOUN ejpam-5896 391 26 - - PUNCT ejpam-5896 391 27 sd sd NOUN ejpam-5896 391 28 - - PUNCT ejpam-5896 391 29 components component NOUN ejpam-5896 391 30 . . PUNCT ejpam-5896 392 1 then then ADV ejpam-5896 392 2 , , PUNCT ejpam-5896 392 3 c̃s c̃s NOUN ejpam-5896 392 4 ϵsd ϵsd NOUN ejpam-5896 392 5 ( ( PUNCT ejpam-5896 392 6 usγ usγ NOUN ejpam-5896 392 7 , , PUNCT ejpam-5896 392 8 ∆ ∆ X ejpam-5896 392 9 ) ) PUNCT ejpam-5896 392 10 ∈ ∈ PROPN ejpam-5896 392 11 sd(u)∆ sd(u)∆ NOUN ejpam-5896 392 12 for for ADP ejpam-5896 392 13 each each DET ejpam-5896 392 14 sγ∈̃ũ sγ∈̃ũ NOUN ejpam-5896 392 15 . . PUNCT ejpam-5896 393 1 hence hence ADV ejpam-5896 393 2 , , PUNCT ejpam-5896 393 3 the the DET ejpam-5896 393 4 class class NOUN ejpam-5896 393 5 { { PUNCT ejpam-5896 393 6 c̃s c̃s NOUN ejpam-5896 393 7 ϵsd ϵsd NOUN ejpam-5896 393 8 ( ( PUNCT ejpam-5896 393 9 usγ usγ NOUN ejpam-5896 393 10 , , PUNCT ejpam-5896 393 11 ∆ ∆ PROPN ejpam-5896 393 12 ) ) PUNCT ejpam-5896 393 13 : : PUNCT ejpam-5896 393 14 sγ∈̃ũ sγ∈̃ũ NOUN ejpam-5896 393 15 } } PUNCT ejpam-5896 393 16 forms form VERB ejpam-5896 393 17 an an DET ejpam-5896 393 18 ss ss NOUN ejpam-5896 393 19 - - PUNCT ejpam-5896 393 20 sd sd NOUN ejpam-5896 393 21 - - PUNCT ejpam-5896 393 22 cover cover NOUN ejpam-5896 393 23 for for ADP ejpam-5896 393 24 ũ ũ PROPN ejpam-5896 393 25 . . PUNCT ejpam-5896 394 1 since since SCONJ ejpam-5896 394 2 ũ ũ PROPN ejpam-5896 394 3 is be AUX ejpam-5896 394 4 ss ss NOUN ejpam-5896 394 5 - - PUNCT ejpam-5896 394 6 sd sd NOUN ejpam-5896 394 7 - - PUNCT ejpam-5896 394 8 compact compact ADJ ejpam-5896 394 9 , , PUNCT ejpam-5896 394 10 there there PRON ejpam-5896 394 11 is be VERB ejpam-5896 394 12 a a DET ejpam-5896 394 13 finite finite ADJ ejpam-5896 394 14 subclass subclass NOUN ejpam-5896 394 15 εo εo ADP ejpam-5896 394 16 such such ADJ ejpam-5896 394 17 that⋃̃ that⋃̃ PROPN ejpam-5896 394 18 ϵ∈εo{c̃ ϵ∈εo{c̃ PROPN ejpam-5896 394 19 s s PART ejpam-5896 394 20 ϵsd ϵsd PRON ejpam-5896 394 21 ( ( PUNCT ejpam-5896 394 22 usγ usγ NOUN ejpam-5896 394 23 , , PUNCT ejpam-5896 394 24 ∆ ∆ PROPN ejpam-5896 394 25 ) ) PUNCT ejpam-5896 394 26 : : PUNCT ejpam-5896 394 27 sγ∈̃ũ sγ∈̃ũ VERB ejpam-5896 394 28 } } PUNCT ejpam-5896 394 29 = = SYM ejpam-5896 394 30 ũ ũ PROPN ejpam-5896 394 31 . . PUNCT ejpam-5896 395 1 therefore therefore ADV ejpam-5896 395 2 , , PUNCT ejpam-5896 395 3 the the DET ejpam-5896 395 4 cover cover NOUN ejpam-5896 395 5 contains contain VERB ejpam-5896 395 6 some some DET ejpam-5896 395 7 elements element NOUN ejpam-5896 395 8 with with ADP ejpam-5896 395 9 non non ADJ ejpam-5896 395 10 - - ADJ ejpam-5896 395 11 null null ADJ ejpam-5896 395 12 soft soft ADJ ejpam-5896 395 13 intersections intersection NOUN ejpam-5896 395 14 . . PUNCT ejpam-5896 396 1 according accord VERB ejpam-5896 396 2 to to ADP ejpam-5896 396 3 proposition proposition NOUN ejpam-5896 396 4 2 2 NUM ejpam-5896 396 5 , , PUNCT ejpam-5896 396 6 this this PRON ejpam-5896 396 7 is be AUX ejpam-5896 396 8 contradictory contradictory ADJ ejpam-5896 396 9 . . PUNCT ejpam-5896 397 1 thus thus ADV ejpam-5896 397 2 , , PUNCT ejpam-5896 397 3 ũ ũ PROPN ejpam-5896 397 4 has have VERB ejpam-5896 397 5 a a DET ejpam-5896 397 6 finite finite ADJ ejpam-5896 397 7 number number NOUN ejpam-5896 397 8 of of ADP ejpam-5896 397 9 ss ss NOUN ejpam-5896 397 10 - - PUNCT ejpam-5896 397 11 sd sd NOUN ejpam-5896 397 12 - - PUNCT ejpam-5896 397 13 components component NOUN ejpam-5896 397 14 . . PUNCT ejpam-5896 398 1 corollary corollary ADJ ejpam-5896 398 2 4 4 NUM ejpam-5896 398 3 . . PUNCT ejpam-5896 399 1 if if SCONJ ejpam-5896 399 2 the the DET ejpam-5896 399 3 components component NOUN ejpam-5896 399 4 of of ADP ejpam-5896 399 5 an an DET ejpam-5896 399 6 ss ss NOUN ejpam-5896 399 7 - - PUNCT ejpam-5896 399 8 sd sd NOUN ejpam-5896 399 9 - - PUNCT ejpam-5896 399 10 lindelöf lindelöf NOUN ejpam-5896 399 11 ssts sst NOUN ejpam-5896 399 12 ( ( PUNCT ejpam-5896 399 13 u u NOUN ejpam-5896 399 14 , , PUNCT ejpam-5896 399 15 µ,∆ µ,∆ NUM ejpam-5896 399 16 ) ) PUNCT ejpam-5896 399 17 are be AUX ejpam-5896 399 18 ss ss NOUN ejpam-5896 399 19 - - PUNCT ejpam-5896 399 20 sd sd NOUN ejpam-5896 399 21 - - PUNCT ejpam-5896 399 22 sets set NOUN ejpam-5896 399 23 , , PUNCT ejpam-5896 399 24 then then ADV ejpam-5896 399 25 they they PRON ejpam-5896 399 26 are be AUX ejpam-5896 399 27 countable countable ADJ ejpam-5896 399 28 . . PUNCT ejpam-5896 400 1 proof proof NOUN ejpam-5896 400 2 . . PUNCT ejpam-5896 401 1 similarly similarly ADV ejpam-5896 401 2 to to ADP ejpam-5896 401 3 the the DET ejpam-5896 401 4 method method NOUN ejpam-5896 401 5 used use VERB ejpam-5896 401 6 to to PART ejpam-5896 401 7 prove prove VERB ejpam-5896 401 8 theorem theorem ADJ ejpam-5896 401 9 14 14 NUM ejpam-5896 401 10 . . PUNCT ejpam-5896 402 1 proposition proposition NOUN ejpam-5896 402 2 6 6 NUM ejpam-5896 402 3 . . PUNCT ejpam-5896 403 1 [ [ X ejpam-5896 403 2 59 59 NUM ejpam-5896 403 3 ] ] PUNCT ejpam-5896 403 4 every every DET ejpam-5896 403 5 ss ss ADJ ejpam-5896 403 6 - - ADJ ejpam-5896 403 7 compact compact ADJ ejpam-5896 403 8 space space NOUN ejpam-5896 403 9 is be AUX ejpam-5896 403 10 ss ss NOUN ejpam-5896 403 11 - - NOUN ejpam-5896 403 12 lindelöf lindelöf NOUN ejpam-5896 403 13 . . PUNCT ejpam-5896 404 1 proposition proposition NOUN ejpam-5896 404 2 7 7 NUM ejpam-5896 404 3 . . PUNCT ejpam-5896 405 1 ( ( PUNCT ejpam-5896 405 2 1 1 X ejpam-5896 405 3 ) ) PUNCT ejpam-5896 405 4 every every DET ejpam-5896 405 5 ss ss NOUN ejpam-5896 405 6 - - PUNCT ejpam-5896 405 7 sd sd NOUN ejpam-5896 405 8 - - PUNCT ejpam-5896 405 9 compact compact ADJ ejpam-5896 405 10 ( ( PUNCT ejpam-5896 405 11 lindelöf lindelöf NOUN ejpam-5896 405 12 ) ) PUNCT ejpam-5896 405 13 space space NOUN ejpam-5896 405 14 is be AUX ejpam-5896 405 15 ss ss NOUN ejpam-5896 405 16 - - ADJ ejpam-5896 405 17 compact compact ADJ ejpam-5896 405 18 ( ( PUNCT ejpam-5896 405 19 lindelöf lindelöf PROPN ejpam-5896 405 20 ) ) PUNCT ejpam-5896 405 21 . . PUNCT ejpam-5896 406 1 ( ( PUNCT ejpam-5896 406 2 2 2 X ejpam-5896 406 3 ) ) PUNCT ejpam-5896 406 4 every every DET ejpam-5896 406 5 ss ss NOUN ejpam-5896 406 6 - - PUNCT ejpam-5896 406 7 sd sd NOUN ejpam-5896 406 8 - - PUNCT ejpam-5896 406 9 compact compact ADJ ejpam-5896 406 10 space space NOUN ejpam-5896 406 11 is be AUX ejpam-5896 406 12 ss ss NOUN ejpam-5896 406 13 - - PUNCT ejpam-5896 406 14 sd sd NOUN ejpam-5896 406 15 - - PUNCT ejpam-5896 406 16 lindelöf lindelöf NOUN ejpam-5896 406 17 . . PUNCT ejpam-5896 407 1 proof proof NOUN ejpam-5896 407 2 . . PUNCT ejpam-5896 408 1 ( ( PUNCT ejpam-5896 408 2 1 1 X ejpam-5896 408 3 ) ) PUNCT ejpam-5896 408 4 assume assume VERB ejpam-5896 408 5 that that SCONJ ejpam-5896 408 6 { { PUNCT ejpam-5896 408 7 ( ( PUNCT ejpam-5896 408 8 cϵ,∆ cϵ,∆ ADJ ejpam-5896 408 9 ) ) PUNCT ejpam-5896 408 10 : : PUNCT ejpam-5896 408 11 ϵ ϵ X ejpam-5896 408 12 ∈ ∈ PROPN ejpam-5896 408 13 ε ε AUX ejpam-5896 408 14 } } PUNCT ejpam-5896 408 15 be be AUX ejpam-5896 408 16 a a DET ejpam-5896 408 17 supra supra ADJ ejpam-5896 408 18 soft soft ADJ ejpam-5896 408 19 open open ADJ ejpam-5896 408 20 cover cover NOUN ejpam-5896 408 21 of of ADP ejpam-5896 408 22 an an DET ejpam-5896 408 23 ss ss NOUN ejpam-5896 408 24 - - PUNCT ejpam-5896 408 25 sd sd NOUN ejpam-5896 408 26 - - PUNCT ejpam-5896 408 27 compact compact ADJ ejpam-5896 408 28 space space NOUN ejpam-5896 408 29 ( ( PUNCT ejpam-5896 408 30 u u NOUN ejpam-5896 408 31 , , PUNCT ejpam-5896 408 32 µ,∆ µ,∆ NUM ejpam-5896 408 33 ) ) PUNCT ejpam-5896 408 34 , , PUNCT ejpam-5896 408 35 then then ADV ejpam-5896 408 36 ψ ψ X ejpam-5896 408 37 is be AUX ejpam-5896 408 38 an an DET ejpam-5896 408 39 ss ss VERB ejpam-5896 408 40 - - PUNCT ejpam-5896 408 41 sd sd NOUN ejpam-5896 408 42 - - PUNCT ejpam-5896 408 43 cover cover NOUN ejpam-5896 408 44 of of ADP ejpam-5896 408 45 ũ ũ PROPN ejpam-5896 408 46 . . PUNCT ejpam-5896 409 1 since since SCONJ ejpam-5896 409 2 ũ ũ PROPN ejpam-5896 409 3 is be AUX ejpam-5896 409 4 ss ss NOUN ejpam-5896 409 5 - - PUNCT ejpam-5896 409 6 sd sd NOUN ejpam-5896 409 7 - - PUNCT ejpam-5896 409 8 compact compact ADJ ejpam-5896 409 9 , , PUNCT ejpam-5896 409 10 there there PRON ejpam-5896 409 11 is be VERB ejpam-5896 409 12 a a DET ejpam-5896 409 13 finite finite ADJ ejpam-5896 409 14 subclass subclass NOUN ejpam-5896 409 15 εo εo NOUN ejpam-5896 409 16 of of ADP ejpam-5896 409 17 ε ε PROPN ejpam-5896 409 18 such such ADJ ejpam-5896 409 19 that that SCONJ ejpam-5896 409 20 ũ ũ PROPN ejpam-5896 409 21 = = SYM ejpam-5896 409 22 ⋃̃ ⋃̃ PROPN ejpam-5896 409 23 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 409 24 ) ) PUNCT ejpam-5896 409 25 . . PUNCT ejpam-5896 410 1 hence hence ADV ejpam-5896 410 2 , , PUNCT ejpam-5896 410 3 ũ ũ PROPN ejpam-5896 410 4 is be AUX ejpam-5896 410 5 an an DET ejpam-5896 410 6 ss ss NOUN ejpam-5896 410 7 - - ADJ ejpam-5896 410 8 compact compact ADJ ejpam-5896 410 9 . . PUNCT ejpam-5896 411 1 ( ( PUNCT ejpam-5896 411 2 2 2 X ejpam-5896 411 3 ) ) PUNCT ejpam-5896 411 4 follows follow VERB ejpam-5896 411 5 from from ADP ejpam-5896 411 6 the the DET ejpam-5896 411 7 fact fact NOUN ejpam-5896 411 8 that that SCONJ ejpam-5896 411 9 every every DET ejpam-5896 411 10 finite finite ADJ ejpam-5896 411 11 class class NOUN ejpam-5896 411 12 is be AUX ejpam-5896 411 13 countable countable ADJ ejpam-5896 411 14 . . PUNCT ejpam-5896 412 1 remark remark PROPN ejpam-5896 412 2 3 3 NUM ejpam-5896 412 3 . . PUNCT ejpam-5896 413 1 the the DET ejpam-5896 413 2 examples example NOUN ejpam-5896 413 3 below below ADV ejpam-5896 413 4 will will AUX ejpam-5896 413 5 confirm confirm VERB ejpam-5896 413 6 that that SCONJ ejpam-5896 413 7 the the DET ejpam-5896 413 8 opposite opposite NOUN ejpam-5896 413 9 of of ADP ejpam-5896 413 10 proposition proposition NOUN ejpam-5896 413 11 7 7 NUM ejpam-5896 413 12 is be AUX ejpam-5896 413 13 not not PART ejpam-5896 413 14 satisfied satisfied ADJ ejpam-5896 413 15 in in ADP ejpam-5896 413 16 general general ADJ ejpam-5896 413 17 . . PUNCT ejpam-5896 414 1 examples example NOUN ejpam-5896 414 2 1 1 NUM ejpam-5896 414 3 . . PUNCT ejpam-5896 415 1 ( ( PUNCT ejpam-5896 415 2 1 1 X ejpam-5896 415 3 ) ) PUNCT ejpam-5896 415 4 consider consider VERB ejpam-5896 415 5 any any DET ejpam-5896 415 6 real real ADJ ejpam-5896 415 7 number number NOUN ejpam-5896 415 8 e e PROPN ejpam-5896 415 9 ∈ ∈ PROPN ejpam-5896 415 10 r. r. PROPN ejpam-5896 415 11 let let VERB ejpam-5896 415 12 ∆ ∆ PROPN ejpam-5896 415 13 = = PRON ejpam-5896 415 14 { { PUNCT ejpam-5896 415 15 γ1 γ1 PROPN ejpam-5896 415 16 , , PUNCT ejpam-5896 415 17 γ2 γ2 PROPN ejpam-5896 415 18 , , PUNCT ejpam-5896 415 19 γ3 γ3 NOUN ejpam-5896 415 20 , , PUNCT ejpam-5896 415 21 ........ ........ PUNCT ejpam-5896 415 22 } } PUNCT ejpam-5896 415 23 and and CCONJ ejpam-5896 415 24 µ µ X ejpam-5896 415 25 = = SYM ejpam-5896 415 26 { { PUNCT ejpam-5896 415 27 ũ ũ PROPN ejpam-5896 415 28 , , PUNCT ejpam-5896 415 29 φ̃ φ̃ PROPN ejpam-5896 415 30 , , PUNCT ejpam-5896 415 31 ( ( PUNCT ejpam-5896 415 32 y,∆ y,∆ PROPN ejpam-5896 415 33 ) ) PUNCT ejpam-5896 415 34 , , PUNCT ejpam-5896 415 35 ( ( PUNCT ejpam-5896 415 36 z,∆ z,∆ NUM ejpam-5896 415 37 ) ) PUNCT ejpam-5896 415 38 , , PUNCT ejpam-5896 415 39 ( ( PUNCT ejpam-5896 415 40 w,∆ w,∆ NOUN ejpam-5896 415 41 ) ) PUNCT ejpam-5896 415 42 } } PUNCT ejpam-5896 415 43 be be AUX ejpam-5896 415 44 an an DET ejpam-5896 415 45 ssts sst NOUN ejpam-5896 415 46 on on ADP ejpam-5896 415 47 u u NOUN ejpam-5896 415 48 , , PUNCT ejpam-5896 415 49 where where SCONJ ejpam-5896 415 50 : : PUNCT ejpam-5896 415 51 ( ( PUNCT ejpam-5896 415 52 y,∆ y,∆ X ejpam-5896 415 53 ) ) PUNCT ejpam-5896 416 1 = = PRON ejpam-5896 416 2 { { PUNCT ejpam-5896 416 3 ( ( PUNCT ejpam-5896 416 4 γ1 γ1 PROPN ejpam-5896 416 5 , , PUNCT ejpam-5896 416 6 { { PUNCT ejpam-5896 416 7 e e NOUN ejpam-5896 416 8 } } PUNCT ejpam-5896 416 9 ) ) PUNCT ejpam-5896 416 10 , , PUNCT ejpam-5896 416 11 ( ( PUNCT ejpam-5896 416 12 γ2 γ2 ADJ ejpam-5896 416 13 , , PUNCT ejpam-5896 416 14 { { PUNCT ejpam-5896 416 15 e e NOUN ejpam-5896 416 16 } } PUNCT ejpam-5896 416 17 ) ) PUNCT ejpam-5896 416 18 , , PUNCT ejpam-5896 416 19 ( ( PUNCT ejpam-5896 416 20 γ3 γ3 NOUN ejpam-5896 416 21 , , PUNCT ejpam-5896 416 22 { { PUNCT ejpam-5896 416 23 e e NOUN ejpam-5896 416 24 } } PUNCT ejpam-5896 416 25 ) ) PUNCT ejpam-5896 416 26 , , PUNCT ejpam-5896 416 27 ( ( PUNCT ejpam-5896 416 28 γ4 γ4 NOUN ejpam-5896 416 29 , , PUNCT ejpam-5896 416 30 { { PUNCT ejpam-5896 416 31 e e NOUN ejpam-5896 416 32 } } PUNCT ejpam-5896 416 33 ) ) PUNCT ejpam-5896 416 34 , , PUNCT ejpam-5896 416 35 ............ ............ PUNCT ejpam-5896 416 36 } } PUNCT ejpam-5896 416 37 . . PUNCT ejpam-5896 417 1 ( ( PUNCT ejpam-5896 417 2 z,∆ z,∆ NUM ejpam-5896 417 3 ) ) PUNCT ejpam-5896 417 4 = = PRON ejpam-5896 417 5 { { PUNCT ejpam-5896 417 6 ( ( PUNCT ejpam-5896 417 7 γ1 γ1 PROPN ejpam-5896 417 8 , , PUNCT ejpam-5896 417 9 { { PUNCT ejpam-5896 417 10 e e NOUN ejpam-5896 417 11 } } PUNCT ejpam-5896 417 12 ) ) PUNCT ejpam-5896 417 13 , , PUNCT ejpam-5896 417 14 ( ( PUNCT ejpam-5896 417 15 γ2 γ2 PROPN ejpam-5896 417 16 , , PUNCT ejpam-5896 417 17 φ φ NOUN ejpam-5896 417 18 ) ) PUNCT ejpam-5896 417 19 , , PUNCT ejpam-5896 417 20 ( ( PUNCT ejpam-5896 417 21 γ3 γ3 NOUN ejpam-5896 417 22 , , PUNCT ejpam-5896 417 23 { { PUNCT ejpam-5896 417 24 e e NOUN ejpam-5896 417 25 } } PUNCT ejpam-5896 417 26 ) ) PUNCT ejpam-5896 417 27 , , PUNCT ejpam-5896 417 28 ( ( PUNCT ejpam-5896 417 29 γ4 γ4 NOUN ejpam-5896 417 30 , , PUNCT ejpam-5896 417 31 { { PUNCT ejpam-5896 417 32 e e NOUN ejpam-5896 417 33 } } PUNCT ejpam-5896 417 34 ) ) PUNCT ejpam-5896 417 35 , , PUNCT ejpam-5896 417 36 ............ ............ PUNCT ejpam-5896 417 37 } } PUNCT ejpam-5896 417 38 . . PUNCT ejpam-5896 418 1 ( ( PUNCT ejpam-5896 418 2 w,∆ w,∆ NUM ejpam-5896 418 3 ) ) PUNCT ejpam-5896 418 4 = = PRON ejpam-5896 418 5 { { PUNCT ejpam-5896 418 6 ( ( PUNCT ejpam-5896 418 7 γ1 γ1 PROPN ejpam-5896 418 8 , , PUNCT ejpam-5896 418 9 φ φ NOUN ejpam-5896 418 10 ) ) PUNCT ejpam-5896 418 11 , , PUNCT ejpam-5896 418 12 ( ( PUNCT ejpam-5896 418 13 γ2 γ2 ADJ ejpam-5896 418 14 , , PUNCT ejpam-5896 418 15 { { PUNCT ejpam-5896 418 16 e e NOUN ejpam-5896 418 17 } } PUNCT ejpam-5896 418 18 ) ) PUNCT ejpam-5896 418 19 , , PUNCT ejpam-5896 418 20 ( ( PUNCT ejpam-5896 418 21 γ3 γ3 NOUN ejpam-5896 418 22 , , PUNCT ejpam-5896 418 23 { { PUNCT ejpam-5896 418 24 e e NOUN ejpam-5896 418 25 } } PUNCT ejpam-5896 418 26 ) ) PUNCT ejpam-5896 418 27 , , PUNCT ejpam-5896 418 28 ( ( PUNCT ejpam-5896 418 29 γ4 γ4 NOUN ejpam-5896 418 30 , , PUNCT ejpam-5896 418 31 { { PUNCT ejpam-5896 418 32 e e NOUN ejpam-5896 418 33 } } PUNCT ejpam-5896 418 34 ) ) PUNCT ejpam-5896 418 35 , , PUNCT ejpam-5896 418 36 .................... .................... PUNCT ejpam-5896 418 37 } } PUNCT ejpam-5896 418 38 . . PUNCT ejpam-5896 419 1 we we PRON ejpam-5896 419 2 have have VERB ejpam-5896 419 3 that that PRON ejpam-5896 419 4 ũ ũ PROPN ejpam-5896 419 5 is be AUX ejpam-5896 419 6 an an DET ejpam-5896 419 7 ss ss NOUN ejpam-5896 419 8 - - ADJ ejpam-5896 419 9 compact compact ADJ ejpam-5896 419 10 ( ( PUNCT ejpam-5896 419 11 lindelöf lindelöf PROPN ejpam-5896 419 12 ) ) PUNCT ejpam-5896 419 13 . . PUNCT ejpam-5896 420 1 on on ADP ejpam-5896 420 2 the the DET ejpam-5896 420 3 opposite opposite ADJ ejpam-5896 420 4 side side NOUN ejpam-5896 420 5 , , PUNCT ejpam-5896 420 6 the the DET ejpam-5896 420 7 class class NOUN ejpam-5896 420 8 ψ ψ NOUN ejpam-5896 420 9 = = X ejpam-5896 420 10 { { PUNCT ejpam-5896 420 11 ( ( PUNCT ejpam-5896 420 12 h,∆ h,∆ NOUN ejpam-5896 420 13 ) ) PUNCT ejpam-5896 420 14 : : PUNCT ejpam-5896 420 15 h(γi h(γi NUM ejpam-5896 420 16 ) ) PUNCT ejpam-5896 420 17 = = SYM ejpam-5896 420 18 { { PUNCT ejpam-5896 420 19 e e NOUN ejpam-5896 420 20 , , PUNCT ejpam-5896 420 21 k k NOUN ejpam-5896 420 22 } } PUNCT ejpam-5896 420 23 , , PUNCT ejpam-5896 420 24 for for ADP ejpam-5896 420 25 γi γi X ejpam-5896 420 26 ∈ ∈ PROPN ejpam-5896 420 27 ∆ ∆ PROPN ejpam-5896 420 28 and and CCONJ ejpam-5896 420 29 e e NOUN ejpam-5896 420 30 , , PUNCT ejpam-5896 420 31 k k PROPN ejpam-5896 420 32 ∈ ∈ PROPN ejpam-5896 420 33 r r NOUN ejpam-5896 420 34 } } PUNCT ejpam-5896 420 35 forms form VERB ejpam-5896 420 36 an an DET ejpam-5896 420 37 ss ss NOUN ejpam-5896 420 38 - - PUNCT ejpam-5896 420 39 sd sd NOUN ejpam-5896 420 40 - - PUNCT ejpam-5896 420 41 cover cover NOUN ejpam-5896 420 42 for for ADP ejpam-5896 420 43 ũ ũ PROPN ejpam-5896 420 44 . . PUNCT ejpam-5896 421 1 however however ADV ejpam-5896 421 2 , , PUNCT ejpam-5896 421 3 there there PRON ejpam-5896 421 4 is be VERB ejpam-5896 421 5 no no DET ejpam-5896 421 6 finite finite NOUN ejpam-5896 421 7 ( ( PUNCT ejpam-5896 421 8 countable countable ADJ ejpam-5896 421 9 ) ) PUNCT ejpam-5896 421 10 subclass subclass NOUN ejpam-5896 421 11 of of ADP ejpam-5896 421 12 ψ ψ PRON ejpam-5896 421 13 which which PRON ejpam-5896 421 14 cover cover VERB ejpam-5896 421 15 r̃. r̃. NOUN ejpam-5896 421 16 thus thus ADV ejpam-5896 421 17 , , PUNCT ejpam-5896 421 18 r̃ r̃ NOUN ejpam-5896 421 19 is be AUX ejpam-5896 421 20 not not PART ejpam-5896 421 21 ss ss NOUN ejpam-5896 421 22 - - PUNCT ejpam-5896 421 23 sd sd NOUN ejpam-5896 421 24 - - PUNCT ejpam-5896 421 25 compact compact ADJ ejpam-5896 421 26 ( ( PUNCT ejpam-5896 421 27 lindelöf lindelöf PROPN ejpam-5896 421 28 ) ) PUNCT ejpam-5896 421 29 . . PUNCT ejpam-5896 422 1 ( ( PUNCT ejpam-5896 422 2 2 2 X ejpam-5896 422 3 ) ) PUNCT ejpam-5896 422 4 consider consider VERB ejpam-5896 422 5 any any DET ejpam-5896 422 6 natural natural ADJ ejpam-5896 422 7 number number NOUN ejpam-5896 422 8 c c NOUN ejpam-5896 422 9 ∈ ∈ PROPN ejpam-5896 422 10 n n ADV ejpam-5896 422 11 . . PUNCT ejpam-5896 423 1 let let VERB ejpam-5896 423 2 ∆ ∆ PROPN ejpam-5896 423 3 = = PRON ejpam-5896 423 4 { { PUNCT ejpam-5896 423 5 γ1 γ1 PROPN ejpam-5896 423 6 , , PUNCT ejpam-5896 423 7 γ2 γ2 PROPN ejpam-5896 423 8 , , PUNCT ejpam-5896 423 9 γ3 γ3 NOUN ejpam-5896 423 10 , , PUNCT ejpam-5896 423 11 ........ ........ PUNCT ejpam-5896 423 12 , , PUNCT ejpam-5896 423 13 γm γm ADJ ejpam-5896 423 14 , , PUNCT ejpam-5896 423 15 m m VERB ejpam-5896 423 16 ∈ ∈ NOUN ejpam-5896 423 17 n n CCONJ ejpam-5896 423 18 } } PUNCT ejpam-5896 423 19 and and CCONJ ejpam-5896 423 20 µ µ X ejpam-5896 423 21 = = SYM ejpam-5896 423 22 { { PUNCT ejpam-5896 423 23 ũ ũ PROPN ejpam-5896 423 24 , , PUNCT ejpam-5896 423 25 φ̃ φ̃ PROPN ejpam-5896 423 26 , , PUNCT ejpam-5896 423 27 ( ( PUNCT ejpam-5896 423 28 a,∆ a,∆ NOUN ejpam-5896 423 29 ) ) PUNCT ejpam-5896 423 30 , , PUNCT ejpam-5896 423 31 ( ( PUNCT ejpam-5896 423 32 b,∆ b,∆ NOUN ejpam-5896 423 33 ) ) PUNCT ejpam-5896 423 34 , , PUNCT ejpam-5896 423 35 ( ( PUNCT ejpam-5896 423 36 c,∆ c,∆ NOUN ejpam-5896 423 37 ) ) PUNCT ejpam-5896 423 38 } } PUNCT ejpam-5896 423 39 be be AUX ejpam-5896 423 40 an an DET ejpam-5896 423 41 ssts sst NOUN ejpam-5896 423 42 on on ADP ejpam-5896 423 43 u u NOUN ejpam-5896 423 44 , , PUNCT ejpam-5896 423 45 where where SCONJ ejpam-5896 423 46 : : PUNCT ejpam-5896 423 47 ( ( PUNCT ejpam-5896 423 48 a,∆ a,∆ NOUN ejpam-5896 423 49 ) ) PUNCT ejpam-5896 424 1 = = PRON ejpam-5896 424 2 { { PUNCT ejpam-5896 424 3 ( ( PUNCT ejpam-5896 424 4 γ1 γ1 PROPN ejpam-5896 424 5 , , PUNCT ejpam-5896 424 6 { { PUNCT ejpam-5896 424 7 c c NOUN ejpam-5896 424 8 } } PUNCT ejpam-5896 424 9 ) ) PUNCT ejpam-5896 424 10 , , PUNCT ejpam-5896 424 11 ( ( PUNCT ejpam-5896 424 12 γ2 γ2 ADJ ejpam-5896 424 13 , , PUNCT ejpam-5896 424 14 { { PUNCT ejpam-5896 424 15 c c NOUN ejpam-5896 424 16 } } PUNCT ejpam-5896 424 17 ) ) PUNCT ejpam-5896 424 18 , , PUNCT ejpam-5896 424 19 ( ( PUNCT ejpam-5896 424 20 γ3 γ3 NOUN ejpam-5896 424 21 , , PUNCT ejpam-5896 424 22 { { PUNCT ejpam-5896 424 23 c c NOUN ejpam-5896 424 24 } } PUNCT ejpam-5896 424 25 ) ) PUNCT ejpam-5896 424 26 , , PUNCT ejpam-5896 424 27 ( ( PUNCT ejpam-5896 424 28 γ4 γ4 NOUN ejpam-5896 424 29 , , PUNCT ejpam-5896 424 30 { { PUNCT ejpam-5896 424 31 c c NOUN ejpam-5896 424 32 } } PUNCT ejpam-5896 424 33 ) ) PUNCT ejpam-5896 424 34 , , PUNCT ejpam-5896 424 35 ........ ........ PUNCT ejpam-5896 424 36 , , PUNCT ejpam-5896 424 37 ( ( PUNCT ejpam-5896 424 38 γm γm X ejpam-5896 424 39 , , PUNCT ejpam-5896 424 40 { { PUNCT ejpam-5896 424 41 c c NOUN ejpam-5896 424 42 } } PUNCT ejpam-5896 424 43 ) ) PUNCT ejpam-5896 424 44 } } PUNCT ejpam-5896 424 45 . . PUNCT ejpam-5896 425 1 ( ( PUNCT ejpam-5896 425 2 b,∆ b,∆ NOUN ejpam-5896 425 3 ) ) PUNCT ejpam-5896 425 4 = = PRON ejpam-5896 425 5 { { PUNCT ejpam-5896 425 6 ( ( PUNCT ejpam-5896 425 7 γ1 γ1 PROPN ejpam-5896 425 8 , , PUNCT ejpam-5896 425 9 { { PUNCT ejpam-5896 425 10 c c NOUN ejpam-5896 425 11 } } PUNCT ejpam-5896 425 12 ) ) PUNCT ejpam-5896 425 13 , , PUNCT ejpam-5896 425 14 ( ( PUNCT ejpam-5896 425 15 γ2 γ2 PROPN ejpam-5896 425 16 , , PUNCT ejpam-5896 425 17 φ φ NOUN ejpam-5896 425 18 ) ) PUNCT ejpam-5896 425 19 , , PUNCT ejpam-5896 425 20 ( ( PUNCT ejpam-5896 425 21 γ3 γ3 NOUN ejpam-5896 425 22 , , PUNCT ejpam-5896 425 23 { { PUNCT ejpam-5896 425 24 c c NOUN ejpam-5896 425 25 } } PUNCT ejpam-5896 425 26 ) ) PUNCT ejpam-5896 425 27 , , PUNCT ejpam-5896 425 28 ( ( PUNCT ejpam-5896 425 29 γ4 γ4 NOUN ejpam-5896 425 30 , , PUNCT ejpam-5896 425 31 { { PUNCT ejpam-5896 425 32 c c NOUN ejpam-5896 425 33 } } PUNCT ejpam-5896 425 34 ) ) PUNCT ejpam-5896 425 35 , , PUNCT ejpam-5896 425 36 ........ ........ PUNCT ejpam-5896 425 37 , , PUNCT ejpam-5896 425 38 ( ( PUNCT ejpam-5896 425 39 γm γm X ejpam-5896 425 40 , , PUNCT ejpam-5896 425 41 { { PUNCT ejpam-5896 425 42 c c NOUN ejpam-5896 425 43 } } PUNCT ejpam-5896 425 44 ) ) PUNCT ejpam-5896 425 45 } } PUNCT ejpam-5896 425 46 . . PUNCT ejpam-5896 426 1 ( ( PUNCT ejpam-5896 426 2 c,∆ c,∆ X ejpam-5896 426 3 ) ) PUNCT ejpam-5896 426 4 = = PRON ejpam-5896 426 5 { { PUNCT ejpam-5896 426 6 ( ( PUNCT ejpam-5896 426 7 γ1 γ1 PROPN ejpam-5896 426 8 , , PUNCT ejpam-5896 426 9 φ φ NOUN ejpam-5896 426 10 ) ) PUNCT ejpam-5896 426 11 , , PUNCT ejpam-5896 426 12 ( ( PUNCT ejpam-5896 426 13 γ2 γ2 ADJ ejpam-5896 426 14 , , PUNCT ejpam-5896 426 15 { { PUNCT ejpam-5896 426 16 c c NOUN ejpam-5896 426 17 } } PUNCT ejpam-5896 426 18 ) ) PUNCT ejpam-5896 426 19 , , PUNCT ejpam-5896 426 20 ( ( PUNCT ejpam-5896 426 21 γ3 γ3 NOUN ejpam-5896 426 22 , , PUNCT ejpam-5896 426 23 { { PUNCT ejpam-5896 426 24 c c NOUN ejpam-5896 426 25 } } PUNCT ejpam-5896 426 26 ) ) PUNCT ejpam-5896 426 27 , , PUNCT ejpam-5896 426 28 ( ( PUNCT ejpam-5896 426 29 γ4 γ4 NOUN ejpam-5896 426 30 , , PUNCT ejpam-5896 426 31 { { PUNCT ejpam-5896 426 32 c c NOUN ejpam-5896 426 33 } } PUNCT ejpam-5896 426 34 ) ) PUNCT ejpam-5896 426 35 , , PUNCT ejpam-5896 426 36 ........ ........ PUNCT ejpam-5896 426 37 , , PUNCT ejpam-5896 426 38 ( ( PUNCT ejpam-5896 426 39 γm γm X ejpam-5896 426 40 , , PUNCT ejpam-5896 426 41 { { PUNCT ejpam-5896 426 42 c c NOUN ejpam-5896 426 43 } } PUNCT ejpam-5896 426 44 ) ) PUNCT ejpam-5896 426 45 } } PUNCT ejpam-5896 426 46 . . PUNCT ejpam-5896 427 1 since since SCONJ ejpam-5896 427 2 n n NUM ejpam-5896 427 3 and and CCONJ ejpam-5896 427 4 ∆ ∆ PROPN ejpam-5896 427 5 are be AUX ejpam-5896 427 6 countable countable ADJ ejpam-5896 427 7 , , PUNCT ejpam-5896 427 8 it it PRON ejpam-5896 427 9 is be AUX ejpam-5896 427 10 simple simple ADJ ejpam-5896 427 11 to to PART ejpam-5896 427 12 confirm confirm VERB ejpam-5896 427 13 that that SCONJ ejpam-5896 427 14 ñ ñ PROPN ejpam-5896 427 15 is be AUX ejpam-5896 427 16 an an DET ejpam-5896 427 17 ss ss NOUN ejpam-5896 427 18 - - PUNCT ejpam-5896 427 19 sd sd NOUN ejpam-5896 427 20 - - PUNCT ejpam-5896 427 21 lindelöf lindelöf NOUN ejpam-5896 427 22 space space NOUN ejpam-5896 427 23 . . PUNCT ejpam-5896 428 1 on on ADP ejpam-5896 428 2 the the DET ejpam-5896 428 3 opposite opposite ADJ ejpam-5896 428 4 side side NOUN ejpam-5896 428 5 , , PUNCT ejpam-5896 428 6 , , PUNCT ejpam-5896 428 7 the the DET ejpam-5896 428 8 class class NOUN ejpam-5896 428 9 abd abd PROPN ejpam-5896 428 10 el el PROPN ejpam-5896 428 11 - - PROPN ejpam-5896 428 12 latif latif PROPN ejpam-5896 428 13 et et PROPN ejpam-5896 428 14 al al PROPN ejpam-5896 428 15 . . PUNCT ejpam-5896 428 16 / / SYM ejpam-5896 428 17 eur eur PROPN ejpam-5896 428 18 . . PUNCT ejpam-5896 429 1 j. j. PROPN ejpam-5896 429 2 pure pure PROPN ejpam-5896 429 3 appl appl PROPN ejpam-5896 429 4 . . PROPN ejpam-5896 429 5 math math PROPN ejpam-5896 429 6 , , PUNCT ejpam-5896 429 7 18 18 NUM ejpam-5896 429 8 ( ( PUNCT ejpam-5896 429 9 2 2 NUM ejpam-5896 429 10 ) ) PUNCT ejpam-5896 429 11 ( ( PUNCT ejpam-5896 429 12 2025 2025 NUM ejpam-5896 429 13 ) ) PUNCT ejpam-5896 429 14 , , PUNCT ejpam-5896 429 15 5896 5896 NUM ejpam-5896 429 16 12 12 NUM ejpam-5896 429 17 of of ADP ejpam-5896 429 18 20 20 NUM ejpam-5896 429 19 ψ ψ NOUN ejpam-5896 429 20 = = X ejpam-5896 429 21 { { PUNCT ejpam-5896 429 22 ( ( PUNCT ejpam-5896 429 23 h,∆ h,∆ NOUN ejpam-5896 429 24 ) ) PUNCT ejpam-5896 429 25 : : PUNCT ejpam-5896 429 26 h(γi h(γi NUM ejpam-5896 429 27 ) ) PUNCT ejpam-5896 429 28 = = SYM ejpam-5896 429 29 { { PUNCT ejpam-5896 429 30 c c NOUN ejpam-5896 429 31 , , PUNCT ejpam-5896 429 32 k k NOUN ejpam-5896 429 33 } } PUNCT ejpam-5896 429 34 , , PUNCT ejpam-5896 429 35 i i PRON ejpam-5896 429 36 = = NOUN ejpam-5896 429 37 1 1 NUM ejpam-5896 429 38 , , PUNCT ejpam-5896 429 39 2 2 NUM ejpam-5896 429 40 , , PUNCT ejpam-5896 429 41 3 3 NUM ejpam-5896 429 42 , , PUNCT ejpam-5896 429 43 .... .... PUNCT ejpam-5896 429 44 , , PUNCT ejpam-5896 429 45 m m VERB ejpam-5896 429 46 for for ADP ejpam-5896 429 47 γi γi X ejpam-5896 429 48 ∈ ∈ PROPN ejpam-5896 429 49 ∆ ∆ PROPN ejpam-5896 429 50 and and CCONJ ejpam-5896 429 51 m m PROPN ejpam-5896 429 52 , , PUNCT ejpam-5896 429 53 c c X ejpam-5896 429 54 , , PUNCT ejpam-5896 429 55 k k PROPN ejpam-5896 429 56 ∈ ∈ PROPN ejpam-5896 429 57 n n CCONJ ejpam-5896 429 58 } } PUNCT ejpam-5896 429 59 forms form VERB ejpam-5896 429 60 an an DET ejpam-5896 429 61 ss ss NOUN ejpam-5896 429 62 - - PUNCT ejpam-5896 429 63 sd sd NOUN ejpam-5896 429 64 - - PUNCT ejpam-5896 429 65 cover cover NOUN ejpam-5896 429 66 for for ADP ejpam-5896 429 67 ñ ñ PROPN ejpam-5896 429 68 . . PUNCT ejpam-5896 430 1 however however ADV ejpam-5896 430 2 , , PUNCT ejpam-5896 430 3 there there PRON ejpam-5896 430 4 is be VERB ejpam-5896 430 5 no no DET ejpam-5896 430 6 finite finite ADJ ejpam-5896 430 7 subclass subclass NOUN ejpam-5896 430 8 of of ADP ejpam-5896 430 9 ψ ψ PRON ejpam-5896 430 10 which which PRON ejpam-5896 430 11 cover cover VERB ejpam-5896 430 12 ñ ñ PROPN ejpam-5896 430 13 . . PUNCT ejpam-5896 431 1 thus thus ADV ejpam-5896 431 2 , , PUNCT ejpam-5896 431 3 ñ ñ PROPN ejpam-5896 431 4 is be AUX ejpam-5896 431 5 not not PART ejpam-5896 431 6 ss ss NOUN ejpam-5896 431 7 - - PUNCT ejpam-5896 431 8 sd sd NOUN ejpam-5896 431 9 - - PUNCT ejpam-5896 431 10 compact compact ADJ ejpam-5896 431 11 . . PUNCT ejpam-5896 432 1 proposition proposition NOUN ejpam-5896 432 2 8 8 NUM ejpam-5896 432 3 . . PUNCT ejpam-5896 433 1 ( ( PUNCT ejpam-5896 433 2 1 1 X ejpam-5896 433 3 ) ) PUNCT ejpam-5896 433 4 every every DET ejpam-5896 433 5 ssts sst NOUN ejpam-5896 433 6 ( ( PUNCT ejpam-5896 433 7 u u NOUN ejpam-5896 433 8 , , PUNCT ejpam-5896 433 9 µ,∆ µ,∆ NUM ejpam-5896 433 10 ) ) PUNCT ejpam-5896 433 11 defined define VERB ejpam-5896 433 12 on on ADP ejpam-5896 433 13 a a DET ejpam-5896 433 14 finite finite NOUN ejpam-5896 433 15 ( ( PUNCT ejpam-5896 433 16 countable countable ADJ ejpam-5896 433 17 ) ) PUNCT ejpam-5896 433 18 universal universal ADJ ejpam-5896 433 19 set set VERB ejpam-5896 433 20 u u NOUN ejpam-5896 433 21 is be AUX ejpam-5896 433 22 ss ss NOUN ejpam-5896 433 23 - - PUNCT ejpam-5896 433 24 sd sd NOUN ejpam-5896 433 25 - - PUNCT ejpam-5896 433 26 compact compact ADJ ejpam-5896 433 27 ( ( PUNCT ejpam-5896 433 28 lindelöf lindelöf PROPN ejpam-5896 433 29 ) ) PUNCT ejpam-5896 433 30 . . PUNCT ejpam-5896 434 1 ( ( PUNCT ejpam-5896 434 2 2 2 X ejpam-5896 434 3 ) ) PUNCT ejpam-5896 434 4 a a DET ejpam-5896 434 5 finite finite NOUN ejpam-5896 434 6 ( ( PUNCT ejpam-5896 434 7 countable countable ADJ ejpam-5896 434 8 ) ) PUNCT ejpam-5896 434 9 soft soft ADJ ejpam-5896 434 10 union union NOUN ejpam-5896 434 11 of of ADP ejpam-5896 434 12 ss ss NOUN ejpam-5896 434 13 - - PUNCT ejpam-5896 434 14 sd sd NOUN ejpam-5896 434 15 - - PUNCT ejpam-5896 434 16 lindelöf lindelöf NOUN ejpam-5896 434 17 ( ( PUNCT ejpam-5896 434 18 compact compact ADJ ejpam-5896 434 19 ) ) PUNCT ejpam-5896 434 20 subsets subset NOUN ejpam-5896 434 21 of of ADP ejpam-5896 434 22 an an DET ejpam-5896 434 23 ssts sst NOUN ejpam-5896 434 24 ( ( PUNCT ejpam-5896 434 25 u u NOUN ejpam-5896 434 26 , , PUNCT ejpam-5896 434 27 µ,∆ µ,∆ NUM ejpam-5896 434 28 ) ) PUNCT ejpam-5896 434 29 is be AUX ejpam-5896 434 30 ss ss NOUN ejpam-5896 434 31 - - PUNCT ejpam-5896 434 32 sd sd NOUN ejpam-5896 434 33 - - PUNCT ejpam-5896 434 34 compact compact ADJ ejpam-5896 434 35 ( ( PUNCT ejpam-5896 434 36 lindelöf lindelöf PROPN ejpam-5896 434 37 ) ) PUNCT ejpam-5896 434 38 . . PUNCT ejpam-5896 435 1 proof proof NOUN ejpam-5896 435 2 . . PUNCT ejpam-5896 436 1 ( ( PUNCT ejpam-5896 436 2 1 1 X ejpam-5896 436 3 ) ) PUNCT ejpam-5896 436 4 direct direct ADJ ejpam-5896 436 5 from from ADP ejpam-5896 436 6 definition definition NOUN ejpam-5896 436 7 22 22 NUM ejpam-5896 436 8 . . PUNCT ejpam-5896 437 1 ( ( PUNCT ejpam-5896 437 2 2 2 X ejpam-5896 437 3 ) ) PUNCT ejpam-5896 437 4 suppose suppose VERB ejpam-5896 437 5 that that SCONJ ejpam-5896 437 6 ( ( PUNCT ejpam-5896 437 7 j,∆ j,∆ NOUN ejpam-5896 437 8 ) ) PUNCT ejpam-5896 437 9 and and CCONJ ejpam-5896 437 10 ( ( PUNCT ejpam-5896 437 11 k,∆ k,∆ PROPN ejpam-5896 437 12 ) ) PUNCT ejpam-5896 437 13 are be AUX ejpam-5896 437 14 ss ss NOUN ejpam-5896 437 15 - - PUNCT ejpam-5896 437 16 sd sd NOUN ejpam-5896 437 17 - - PUNCT ejpam-5896 437 18 lindelöf lindelöf NOUN ejpam-5896 437 19 subsets subset NOUN ejpam-5896 437 20 of of ADP ejpam-5896 437 21 an an DET ejpam-5896 437 22 ssts sst NOUN ejpam-5896 437 23 ( ( PUNCT ejpam-5896 437 24 u u NOUN ejpam-5896 437 25 , , PUNCT ejpam-5896 437 26 µ,∆ µ,∆ NUM ejpam-5896 437 27 ) ) PUNCT ejpam-5896 437 28 , , PUNCT ejpam-5896 437 29 and and CCONJ ejpam-5896 437 30 ψ ψ X ejpam-5896 437 31 = = SYM ejpam-5896 437 32 { { PUNCT ejpam-5896 437 33 ( ( PUNCT ejpam-5896 437 34 cϵ,∆ cϵ,∆ ADJ ejpam-5896 437 35 ) ) PUNCT ejpam-5896 437 36 : : PUNCT ejpam-5896 437 37 ϵ ϵ X ejpam-5896 437 38 ∈ ∈ PROPN ejpam-5896 437 39 ε ε PROPN ejpam-5896 437 40 } } PUNCT ejpam-5896 437 41 is be AUX ejpam-5896 437 42 ss ss NOUN ejpam-5896 437 43 - - PUNCT ejpam-5896 437 44 sd sd NOUN ejpam-5896 437 45 - - PUNCT ejpam-5896 437 46 cover cover NOUN ejpam-5896 437 47 for for ADP ejpam-5896 437 48 ( ( PUNCT ejpam-5896 437 49 j,∆)∪̃(k,∆ j,∆)∪̃(k,∆ PROPN ejpam-5896 437 50 ) ) PUNCT ejpam-5896 437 51 , , PUNCT ejpam-5896 437 52 then then ADV ejpam-5896 437 53 ψ ψ X ejpam-5896 437 54 is be AUX ejpam-5896 437 55 an an DET ejpam-5896 437 56 ss ss VERB ejpam-5896 437 57 - - PUNCT ejpam-5896 437 58 sd sd NOUN ejpam-5896 437 59 - - PUNCT ejpam-5896 437 60 cover cover NOUN ejpam-5896 437 61 for for ADP ejpam-5896 437 62 ( ( PUNCT ejpam-5896 437 63 j,∆ j,∆ NOUN ejpam-5896 437 64 ) ) PUNCT ejpam-5896 437 65 and and CCONJ ejpam-5896 437 66 ( ( PUNCT ejpam-5896 437 67 k,∆ k,∆ PROPN ejpam-5896 437 68 ) ) PUNCT ejpam-5896 437 69 . . PUNCT ejpam-5896 438 1 according accord VERB ejpam-5896 438 2 to to ADP ejpam-5896 438 3 the the DET ejpam-5896 438 4 hypothesis hypothesis NOUN ejpam-5896 438 5 , , PUNCT ejpam-5896 438 6 there there PRON ejpam-5896 438 7 are be VERB ejpam-5896 438 8 countable countable ADJ ejpam-5896 438 9 subclasses subclass NOUN ejpam-5896 438 10 ε1 ε1 NOUN ejpam-5896 438 11 and and CCONJ ejpam-5896 438 12 ε2 ε2 NOUN ejpam-5896 438 13 of of ADP ejpam-5896 438 14 ε ε PROPN ejpam-5896 438 15 such such ADJ ejpam-5896 438 16 that that SCONJ ejpam-5896 438 17 ( ( PUNCT ejpam-5896 438 18 j,∆)⊆̃ j,∆)⊆̃ X ejpam-5896 438 19 ⋃̃ ⋃̃ PROPN ejpam-5896 438 20 i∈ε1(ci,∆ i∈ε1(ci,∆ PROPN ejpam-5896 438 21 ) ) PUNCT ejpam-5896 438 22 and and CCONJ ejpam-5896 438 23 ( ( PUNCT ejpam-5896 438 24 k,∆)⊆̃ k,∆)⊆̃ PROPN ejpam-5896 438 25 ⋃̃ ⋃̃ PROPN ejpam-5896 438 26 j∈ε2(cj j∈ε2(cj NOUN ejpam-5896 438 27 , , PUNCT ejpam-5896 438 28 ∆ ∆ PROPN ejpam-5896 438 29 ) ) PUNCT ejpam-5896 438 30 . . PUNCT ejpam-5896 439 1 hence hence ADV ejpam-5896 439 2 , , PUNCT ejpam-5896 439 3 ( ( PUNCT ejpam-5896 439 4 j,∆)∪̃(k,∆)⊆̃ j,∆)∪̃(k,∆)⊆̃ PROPN ejpam-5896 439 5 ( ( PUNCT ejpam-5896 439 6 ⋃̃ ⋃̃ PROPN ejpam-5896 439 7 i∈ε1(ci,∆))∪̃ i∈ε1(ci,∆))∪̃ PROPN ejpam-5896 439 8 ( ( PUNCT ejpam-5896 439 9 ⋃̃ ⋃̃ PROPN ejpam-5896 439 10 j∈ε2(cj j∈ε2(cj NOUN ejpam-5896 439 11 , , PUNCT ejpam-5896 439 12 ∆ ∆ PROPN ejpam-5896 439 13 ) ) PUNCT ejpam-5896 439 14 ) ) PUNCT ejpam-5896 439 15 which which PRON ejpam-5896 439 16 follows follow VERB ejpam-5896 439 17 ( ( PUNCT ejpam-5896 439 18 j,∆)∪̃(k,∆)⊆̃ j,∆)∪̃(k,∆)⊆̃ PROPN ejpam-5896 439 19 ⋃̃ ⋃̃ PROPN ejpam-5896 439 20 i∈ε1,j∈ε2 i∈ε1,j∈ε2 PROPN ejpam-5896 439 21 [ [ X ejpam-5896 439 22 ( ( PUNCT ejpam-5896 439 23 ci,∆)∪̃(cj ci,∆)∪̃(cj NOUN ejpam-5896 439 24 , , PUNCT ejpam-5896 439 25 ∆ ∆ PROPN ejpam-5896 439 26 ) ) PUNCT ejpam-5896 439 27 ] ] PUNCT ejpam-5896 440 1 = = SYM ejpam-5896 440 2 ⋃̃ ⋃̃ PROPN ejpam-5896 440 3 k∈ε3(ck,∆ k∈ε3(ck,∆ PROPN ejpam-5896 440 4 ) ) PUNCT ejpam-5896 440 5 , , PUNCT ejpam-5896 440 6 ε3 ε3 PROPN ejpam-5896 440 7 is be AUX ejpam-5896 440 8 a a DET ejpam-5896 440 9 countable countable ADJ ejpam-5896 440 10 subclasses subclass NOUN ejpam-5896 440 11 of of ADP ejpam-5896 440 12 ε ε PROPN ejpam-5896 440 13 . . PUNCT ejpam-5896 441 1 thus thus ADV ejpam-5896 441 2 , , PUNCT ejpam-5896 441 3 ( ( PUNCT ejpam-5896 441 4 j,∆)∪̃(k,∆ j,∆)∪̃(k,∆ PROPN ejpam-5896 441 5 ) ) PUNCT ejpam-5896 441 6 is be AUX ejpam-5896 441 7 ss ss NOUN ejpam-5896 441 8 - - PUNCT ejpam-5896 441 9 sd sd NOUN ejpam-5896 441 10 - - PUNCT ejpam-5896 441 11 lindelöf lindelöf NOUN ejpam-5896 441 12 . . PUNCT ejpam-5896 442 1 when when SCONJ ejpam-5896 442 2 parenthesis parenthesis NOUN ejpam-5896 442 3 are be AUX ejpam-5896 442 4 used use VERB ejpam-5896 442 5 , , PUNCT ejpam-5896 442 6 the the DET ejpam-5896 442 7 case case NOUN ejpam-5896 442 8 can can AUX ejpam-5896 442 9 be be AUX ejpam-5896 442 10 obtained obtain VERB ejpam-5896 442 11 by by ADP ejpam-5896 442 12 a a DET ejpam-5896 442 13 similar similar ADJ ejpam-5896 442 14 way way NOUN ejpam-5896 442 15 . . PUNCT ejpam-5896 443 1 theorem theorem ADJ ejpam-5896 443 2 15 15 NUM ejpam-5896 443 3 . . PUNCT ejpam-5896 444 1 every every DET ejpam-5896 444 2 ss ss PROPN ejpam-5896 444 3 - - ADJ ejpam-5896 444 4 sc sc NOUN ejpam-5896 444 5 - - PUNCT ejpam-5896 444 6 subset subset NOUN ejpam-5896 444 7 of of ADP ejpam-5896 444 8 an an DET ejpam-5896 444 9 ss ss NOUN ejpam-5896 444 10 - - PUNCT ejpam-5896 444 11 sd sd NOUN ejpam-5896 444 12 - - PUNCT ejpam-5896 444 13 compact compact ADJ ejpam-5896 444 14 ( ( PUNCT ejpam-5896 444 15 lindelöf lindelöf NOUN ejpam-5896 444 16 ) ) PUNCT ejpam-5896 444 17 ssts sst NOUN ejpam-5896 444 18 ( ( PUNCT ejpam-5896 444 19 u u NOUN ejpam-5896 444 20 , , PUNCT ejpam-5896 444 21 µ,∆ µ,∆ NUM ejpam-5896 444 22 ) ) PUNCT ejpam-5896 444 23 is be AUX ejpam-5896 444 24 ss ss NOUN ejpam-5896 444 25 - - PUNCT ejpam-5896 444 26 sdcompact sdcompact NOUN ejpam-5896 444 27 ( ( PUNCT ejpam-5896 444 28 lindelöf lindelöf PROPN ejpam-5896 444 29 ) ) PUNCT ejpam-5896 444 30 . . PUNCT ejpam-5896 445 1 proof proof NOUN ejpam-5896 445 2 . . PUNCT ejpam-5896 446 1 let let VERB ejpam-5896 446 2 { { PUNCT ejpam-5896 446 3 ( ( PUNCT ejpam-5896 446 4 cϵ,∆ cϵ,∆ ADJ ejpam-5896 446 5 ) ) PUNCT ejpam-5896 446 6 : : PUNCT ejpam-5896 447 1 ϵ ϵ X ejpam-5896 447 2 ∈ ∈ PROPN ejpam-5896 447 3 ε ε PROPN ejpam-5896 447 4 } } PUNCT ejpam-5896 447 5 is be AUX ejpam-5896 447 6 ss ss NOUN ejpam-5896 447 7 - - PUNCT ejpam-5896 447 8 sd sd NOUN ejpam-5896 447 9 - - PUNCT ejpam-5896 447 10 cover cover NOUN ejpam-5896 447 11 for for ADP ejpam-5896 447 12 an an DET ejpam-5896 447 13 ss ss ADJ ejpam-5896 447 14 - - PUNCT ejpam-5896 447 15 sc sc NOUN ejpam-5896 447 16 - - PUNCT ejpam-5896 447 17 subset subset NOUN ejpam-5896 447 18 ( ( PUNCT ejpam-5896 447 19 s,∆ s,∆ VERB ejpam-5896 447 20 ) ) PUNCT ejpam-5896 447 21 of of ADP ejpam-5896 447 22 an an DET ejpam-5896 447 23 ss ss NOUN ejpam-5896 447 24 - - PUNCT ejpam-5896 447 25 sd sd NOUN ejpam-5896 447 26 - - PUNCT ejpam-5896 447 27 compact compact ADJ ejpam-5896 447 28 ssts sst NOUN ejpam-5896 447 29 ( ( PUNCT ejpam-5896 447 30 u u NOUN ejpam-5896 447 31 , , PUNCT ejpam-5896 447 32 µ,∆ µ,∆ NUM ejpam-5896 447 33 ) ) PUNCT ejpam-5896 447 34 , , PUNCT ejpam-5896 447 35 then then ADV ejpam-5896 447 36 ( ( PUNCT ejpam-5896 447 37 s,∆)⊆̃ s,∆)⊆̃ ADJ ejpam-5896 447 38 ⋃̃ ⋃̃ PROPN ejpam-5896 447 39 ϵ∈ε(cϵ,∆ ϵ∈ε(cϵ,∆ PROPN ejpam-5896 447 40 ) ) PUNCT ejpam-5896 447 41 which which PRON ejpam-5896 447 42 follows follow VERB ejpam-5896 447 43 ũ ũ PROPN ejpam-5896 447 44 = = X ejpam-5896 447 45 ( ( PUNCT ejpam-5896 447 46 s,∆)∪̃(s s,∆)∪̃(s NOUN ejpam-5896 447 47 c̃,∆)⊆̃ c̃,∆)⊆̃ PROPN ejpam-5896 447 48 ⋃̃ ⋃̃ PROPN ejpam-5896 447 49 ϵ∈ε(cϵ,∆)∪̃(s ϵ∈ε(cϵ,∆)∪̃(s PROPN ejpam-5896 447 50 c̃,∆ c̃,∆ NOUN ejpam-5896 447 51 ) ) PUNCT ejpam-5896 447 52 . . PUNCT ejpam-5896 448 1 now now ADV ejpam-5896 448 2 , , PUNCT ejpam-5896 448 3 we we PRON ejpam-5896 448 4 have have VERB ejpam-5896 448 5 that that PRON ejpam-5896 448 6 { { PUNCT ejpam-5896 448 7 ( ( PUNCT ejpam-5896 448 8 cϵ,∆ cϵ,∆ ADJ ejpam-5896 448 9 ) ) PUNCT ejpam-5896 448 10 : : PUNCT ejpam-5896 448 11 ϵ ϵ X ejpam-5896 448 12 ∈ ∈ PROPN ejpam-5896 448 13 ε}∪̃(s ε}∪̃(s PROPN ejpam-5896 448 14 c̃,∆ c̃,∆ NOUN ejpam-5896 448 15 ) ) PUNCT ejpam-5896 448 16 is be AUX ejpam-5896 448 17 an an DET ejpam-5896 448 18 ss ss NOUN ejpam-5896 448 19 - - PUNCT ejpam-5896 448 20 sd sd NOUN ejpam-5896 448 21 - - PUNCT ejpam-5896 448 22 cover cover NOUN ejpam-5896 448 23 of of ADP ejpam-5896 448 24 ũ ũ PROPN ejpam-5896 448 25 . . PUNCT ejpam-5896 449 1 since since SCONJ ejpam-5896 449 2 ũ ũ PROPN ejpam-5896 449 3 is be AUX ejpam-5896 449 4 ss ss NOUN ejpam-5896 449 5 - - PUNCT ejpam-5896 449 6 sd sd NOUN ejpam-5896 449 7 - - PUNCT ejpam-5896 449 8 compact compact ADJ ejpam-5896 449 9 , , PUNCT ejpam-5896 449 10 there there PRON ejpam-5896 449 11 is be VERB ejpam-5896 449 12 a a DET ejpam-5896 449 13 finite finite ADJ ejpam-5896 449 14 subclass subclass NOUN ejpam-5896 449 15 εo εo NOUN ejpam-5896 449 16 of of ADP ejpam-5896 449 17 ε ε PROPN ejpam-5896 449 18 such such ADJ ejpam-5896 449 19 that that SCONJ ejpam-5896 449 20 ( ( PUNCT ejpam-5896 449 21 s,∆)⊆̃ũ s,∆)⊆̃ũ NOUN ejpam-5896 449 22 = = PUNCT ejpam-5896 449 23 ⋃̃ ⋃̃ PROPN ejpam-5896 449 24 ϵ∈εo(cϵ,∆)∪̃(s ϵ∈εo(cϵ,∆)∪̃(s NUM ejpam-5896 449 25 c̃,∆ c̃,∆ PROPN ejpam-5896 449 26 ) ) PUNCT ejpam-5896 449 27 . . PUNCT ejpam-5896 450 1 hence hence ADV ejpam-5896 450 2 , , PUNCT ejpam-5896 450 3 ( ( PUNCT ejpam-5896 450 4 s,∆)⊆̃ s,∆)⊆̃ ADJ ejpam-5896 450 5 ⋃̃ ⋃̃ PROPN ejpam-5896 450 6 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 450 7 ) ) PUNCT ejpam-5896 450 8 . . PUNCT ejpam-5896 451 1 thus thus ADV ejpam-5896 451 2 , , PUNCT ejpam-5896 451 3 ( ( PUNCT ejpam-5896 451 4 s,∆ s,∆ VERB ejpam-5896 451 5 ) ) PUNCT ejpam-5896 451 6 is be AUX ejpam-5896 451 7 an an DET ejpam-5896 451 8 ss ss VERB ejpam-5896 451 9 - - PUNCT ejpam-5896 451 10 sd sd NOUN ejpam-5896 451 11 - - PUNCT ejpam-5896 451 12 compact compact ADJ ejpam-5896 451 13 . . PUNCT ejpam-5896 452 1 the the DET ejpam-5896 452 2 case case NOUN ejpam-5896 452 3 of of ADP ejpam-5896 452 4 ss ss NOUN ejpam-5896 452 5 - - PUNCT ejpam-5896 452 6 sd sd NOUN ejpam-5896 452 7 - - PUNCT ejpam-5896 452 8 lindelöfness lindelöfness NOUN ejpam-5896 452 9 is be AUX ejpam-5896 452 10 similar similar ADJ ejpam-5896 452 11 . . PUNCT ejpam-5896 453 1 proposition proposition NOUN ejpam-5896 453 2 9 9 NUM ejpam-5896 453 3 . . PUNCT ejpam-5896 454 1 the the DET ejpam-5896 454 2 soft soft ADJ ejpam-5896 454 3 intersection intersection NOUN ejpam-5896 454 4 of of ADP ejpam-5896 454 5 an an DET ejpam-5896 454 6 ss ss NOUN ejpam-5896 454 7 - - PUNCT ejpam-5896 454 8 sd sd NOUN ejpam-5896 454 9 - - PUNCT ejpam-5896 454 10 compact compact ADJ ejpam-5896 454 11 ( ( PUNCT ejpam-5896 454 12 lindelöf lindelöf NOUN ejpam-5896 454 13 ) ) PUNCT ejpam-5896 454 14 soft soft ADJ ejpam-5896 454 15 subset subset NOUN ejpam-5896 454 16 ( ( PUNCT ejpam-5896 454 17 a,∆ a,∆ VERB ejpam-5896 454 18 ) ) PUNCT ejpam-5896 454 19 and and CCONJ ejpam-5896 454 20 ss ss PROPN ejpam-5896 454 21 - - PUNCT ejpam-5896 454 22 sc sc NOUN ejpam-5896 454 23 - - PUNCT ejpam-5896 454 24 subset subset NOUN ejpam-5896 454 25 ( ( PUNCT ejpam-5896 454 26 b,∆ b,∆ NOUN ejpam-5896 454 27 ) ) PUNCT ejpam-5896 454 28 of of ADP ejpam-5896 454 29 an an DET ejpam-5896 454 30 ssts sst NOUN ejpam-5896 454 31 ( ( PUNCT ejpam-5896 454 32 u u NOUN ejpam-5896 454 33 , , PUNCT ejpam-5896 454 34 µ,∆ µ,∆ NUM ejpam-5896 454 35 ) ) PUNCT ejpam-5896 454 36 is be AUX ejpam-5896 454 37 ss ss NOUN ejpam-5896 454 38 - - PUNCT ejpam-5896 454 39 sd sd NOUN ejpam-5896 454 40 - - PUNCT ejpam-5896 454 41 compact compact ADJ ejpam-5896 454 42 ( ( PUNCT ejpam-5896 454 43 lindelöf lindelöf PROPN ejpam-5896 454 44 ) ) PUNCT ejpam-5896 454 45 . . PUNCT ejpam-5896 455 1 abd abd PROPN ejpam-5896 455 2 el el PROPN ejpam-5896 455 3 - - PROPN ejpam-5896 455 4 latif latif PROPN ejpam-5896 455 5 et et PROPN ejpam-5896 455 6 al al PROPN ejpam-5896 455 7 . . PUNCT ejpam-5896 455 8 / / SYM ejpam-5896 455 9 eur eur PROPN ejpam-5896 455 10 . . PUNCT ejpam-5896 456 1 j. j. PROPN ejpam-5896 456 2 pure pure PROPN ejpam-5896 456 3 appl appl PROPN ejpam-5896 456 4 . . PROPN ejpam-5896 456 5 math math PROPN ejpam-5896 456 6 , , PUNCT ejpam-5896 456 7 18 18 NUM ejpam-5896 456 8 ( ( PUNCT ejpam-5896 456 9 2 2 NUM ejpam-5896 456 10 ) ) PUNCT ejpam-5896 456 11 ( ( PUNCT ejpam-5896 456 12 2025 2025 NUM ejpam-5896 456 13 ) ) PUNCT ejpam-5896 456 14 , , PUNCT ejpam-5896 456 15 5896 5896 NUM ejpam-5896 456 16 13 13 NUM ejpam-5896 456 17 of of ADP ejpam-5896 456 18 20 20 NUM ejpam-5896 456 19 proof proof NOUN ejpam-5896 456 20 . . PUNCT ejpam-5896 457 1 let let VERB ejpam-5896 457 2 ( ( PUNCT ejpam-5896 457 3 a,∆ a,∆ VERB ejpam-5896 457 4 ) ) PUNCT ejpam-5896 457 5 is be AUX ejpam-5896 457 6 ss ss NOUN ejpam-5896 457 7 - - PUNCT ejpam-5896 457 8 sd sd NOUN ejpam-5896 457 9 - - PUNCT ejpam-5896 457 10 compact compact ADJ ejpam-5896 457 11 , , PUNCT ejpam-5896 457 12 ( ( PUNCT ejpam-5896 457 13 b,∆ b,∆ NOUN ejpam-5896 457 14 ) ) PUNCT ejpam-5896 457 15 is be AUX ejpam-5896 457 16 an an DET ejpam-5896 457 17 ss ss ADJ ejpam-5896 457 18 - - PUNCT ejpam-5896 457 19 sc sc NOUN ejpam-5896 457 20 - - PUNCT ejpam-5896 457 21 set set VERB ejpam-5896 457 22 and and CCONJ ejpam-5896 457 23 { { PUNCT ejpam-5896 457 24 ( ( PUNCT ejpam-5896 457 25 cϵ,∆ cϵ,∆ ADJ ejpam-5896 457 26 ) ) PUNCT ejpam-5896 457 27 : : PUNCT ejpam-5896 458 1 ϵ ϵ X ejpam-5896 458 2 ∈ ∈ PROPN ejpam-5896 458 3 ε ε PROPN ejpam-5896 458 4 } } PUNCT ejpam-5896 458 5 is be AUX ejpam-5896 458 6 ss ss NOUN ejpam-5896 458 7 - - PUNCT ejpam-5896 458 8 sdcover sdcover NOUN ejpam-5896 458 9 of of ADP ejpam-5896 458 10 ( ( PUNCT ejpam-5896 458 11 a,∆)∩̃(b,∆ a,∆)∩̃(b,∆ PROPN ejpam-5896 458 12 ) ) PUNCT ejpam-5896 458 13 , , PUNCT ejpam-5896 458 14 then then ADV ejpam-5896 458 15 ( ( PUNCT ejpam-5896 458 16 a,∆)⊆̃ a,∆)⊆̃ NOUN ejpam-5896 458 17 ⋃̃ ⋃̃ PROPN ejpam-5896 458 18 ϵ∈ε(cϵ,∆)∪̃(bc̃,∆ ϵ∈ε(cϵ,∆)∪̃(bc̃,∆ PROPN ejpam-5896 458 19 ) ) PUNCT ejpam-5896 458 20 . . PUNCT ejpam-5896 459 1 since since SCONJ ejpam-5896 459 2 ( ( PUNCT ejpam-5896 459 3 a,∆ a,∆ NOUN ejpam-5896 459 4 ) ) PUNCT ejpam-5896 459 5 is be AUX ejpam-5896 459 6 ss ss NOUN ejpam-5896 459 7 - - PUNCT ejpam-5896 459 8 sd sd NOUN ejpam-5896 459 9 - - PUNCT ejpam-5896 459 10 compact compact ADJ ejpam-5896 459 11 , , PUNCT ejpam-5896 459 12 there there PRON ejpam-5896 459 13 is be VERB ejpam-5896 459 14 a a DET ejpam-5896 459 15 finite finite ADJ ejpam-5896 459 16 subclass subclass NOUN ejpam-5896 459 17 εo εo NOUN ejpam-5896 459 18 of of ADP ejpam-5896 459 19 ε ε PROPN ejpam-5896 459 20 such such ADJ ejpam-5896 459 21 that that SCONJ ejpam-5896 459 22 ( ( PUNCT ejpam-5896 459 23 a,∆)⊆̃ a,∆)⊆̃ NOUN ejpam-5896 459 24 ⋃̃ ⋃̃ PROPN ejpam-5896 459 25 ϵ∈εo(cϵ,∆)∪̃(bc̃,∆ ϵ∈εo(cϵ,∆)∪̃(bc̃,∆ NUM ejpam-5896 459 26 ) ) PUNCT ejpam-5896 459 27 which which PRON ejpam-5896 459 28 follows follow VERB ejpam-5896 459 29 ( ( PUNCT ejpam-5896 459 30 a,∆)∩̃(b,∆)⊆̃ a,∆)∩̃(b,∆)⊆̃ VERB ejpam-5896 459 31 ⋃̃ ⋃̃ PROPN ejpam-5896 459 32 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 459 33 ) ) PUNCT ejpam-5896 459 34 . . PUNCT ejpam-5896 460 1 therefore therefore ADV ejpam-5896 460 2 , , PUNCT ejpam-5896 460 3 ( ( PUNCT ejpam-5896 460 4 a,∆)∩̃(b,∆ a,∆)∩̃(b,∆ NOUN ejpam-5896 460 5 ) ) PUNCT ejpam-5896 460 6 is be AUX ejpam-5896 460 7 an an DET ejpam-5896 460 8 ss ss VERB ejpam-5896 460 9 - - PUNCT ejpam-5896 460 10 sd sd NOUN ejpam-5896 460 11 - - PUNCT ejpam-5896 460 12 compact compact ADJ ejpam-5896 460 13 . . PUNCT ejpam-5896 461 1 the the DET ejpam-5896 461 2 case case NOUN ejpam-5896 461 3 of of ADP ejpam-5896 461 4 ss ss NOUN ejpam-5896 461 5 - - PUNCT ejpam-5896 461 6 sd sd NOUN ejpam-5896 461 7 - - PUNCT ejpam-5896 461 8 lindelöfness lindelöfness NOUN ejpam-5896 461 9 can can AUX ejpam-5896 461 10 be be AUX ejpam-5896 461 11 achieved achieve VERB ejpam-5896 461 12 similarly similarly ADV ejpam-5896 461 13 . . PUNCT ejpam-5896 462 1 remark remark VERB ejpam-5896 462 2 4 4 NUM ejpam-5896 462 3 . . PUNCT ejpam-5896 463 1 we we PRON ejpam-5896 463 2 will will AUX ejpam-5896 463 3 show show VERB ejpam-5896 463 4 in in ADP ejpam-5896 463 5 the the DET ejpam-5896 463 6 following following ADJ ejpam-5896 463 7 example example NOUN ejpam-5896 463 8 that that SCONJ ejpam-5896 463 9 the the DET ejpam-5896 463 10 contrary contrary NOUN ejpam-5896 463 11 of of ADP ejpam-5896 463 12 proposition proposition NOUN ejpam-5896 463 13 9 9 NUM ejpam-5896 463 14 is be AUX ejpam-5896 463 15 not not PART ejpam-5896 463 16 always always ADV ejpam-5896 463 17 satisfied satisfied ADJ ejpam-5896 463 18 . . PUNCT ejpam-5896 464 1 example example NOUN ejpam-5896 465 1 4 4 NUM ejpam-5896 465 2 . . PUNCT ejpam-5896 465 3 let let VERB ejpam-5896 465 4 u u PRON ejpam-5896 465 5 = = NOUN ejpam-5896 465 6 { { PUNCT ejpam-5896 465 7 u1 u1 NOUN ejpam-5896 465 8 , , PUNCT ejpam-5896 465 9 u2 u2 PROPN ejpam-5896 465 10 } } PUNCT ejpam-5896 465 11 , , PUNCT ejpam-5896 465 12 ∆ ∆ X ejpam-5896 465 13 = = SYM ejpam-5896 465 14 { { PUNCT ejpam-5896 465 15 γ1 γ1 PROPN ejpam-5896 465 16 , , PUNCT ejpam-5896 465 17 γ2 γ2 PROPN ejpam-5896 465 18 } } PUNCT ejpam-5896 465 19 and and CCONJ ejpam-5896 465 20 ( ( PUNCT ejpam-5896 465 21 si,∆ si,∆ PROPN ejpam-5896 465 22 ) ) PUNCT ejpam-5896 465 23 , , PUNCT ejpam-5896 465 24 i i PRON ejpam-5896 465 25 = = NOUN ejpam-5896 465 26 1 1 NUM ejpam-5896 465 27 , , PUNCT ejpam-5896 465 28 2 2 NUM ejpam-5896 465 29 , , PUNCT ejpam-5896 465 30 3 3 NUM ejpam-5896 465 31 , , PUNCT ejpam-5896 465 32 4 4 NUM ejpam-5896 465 33 be be AUX ejpam-5896 465 34 soft soft ADJ ejpam-5896 465 35 sets set NOUN ejpam-5896 465 36 over over ADP ejpam-5896 465 37 u u PROPN ejpam-5896 465 38 , , PUNCT ejpam-5896 465 39 where where SCONJ ejpam-5896 465 40 : : PUNCT ejpam-5896 465 41 s1(γ1 s1(γ1 NOUN ejpam-5896 465 42 ) ) PUNCT ejpam-5896 465 43 = = SYM ejpam-5896 465 44 { { PUNCT ejpam-5896 465 45 u1 u1 NOUN ejpam-5896 465 46 , , PUNCT ejpam-5896 465 47 u2 u2 PROPN ejpam-5896 465 48 } } PUNCT ejpam-5896 465 49 , , PUNCT ejpam-5896 465 50 s1(γ2 s1(γ2 NOUN ejpam-5896 465 51 ) ) PUNCT ejpam-5896 465 52 = = SYM ejpam-5896 465 53 { { PUNCT ejpam-5896 465 54 u2 u2 PROPN ejpam-5896 465 55 } } PUNCT ejpam-5896 465 56 , , PUNCT ejpam-5896 465 57 s2(γ1 s2(γ1 NOUN ejpam-5896 465 58 ) ) PUNCT ejpam-5896 465 59 = = SYM ejpam-5896 465 60 φ φ PROPN ejpam-5896 465 61 , , PUNCT ejpam-5896 465 62 s2(γ2 s2(γ2 NOUN ejpam-5896 465 63 ) ) PUNCT ejpam-5896 465 64 = = SYM ejpam-5896 465 65 { { PUNCT ejpam-5896 465 66 u2 u2 PROPN ejpam-5896 465 67 } } PUNCT ejpam-5896 465 68 , , PUNCT ejpam-5896 465 69 s3(γ1 s3(γ1 NOUN ejpam-5896 465 70 ) ) PUNCT ejpam-5896 465 71 = = SYM ejpam-5896 465 72 { { PUNCT ejpam-5896 465 73 u1 u1 NOUN ejpam-5896 465 74 } } PUNCT ejpam-5896 465 75 , , PUNCT ejpam-5896 465 76 s3(γ2 s3(γ2 NOUN ejpam-5896 465 77 ) ) PUNCT ejpam-5896 465 78 = = SYM ejpam-5896 465 79 { { PUNCT ejpam-5896 465 80 u2 u2 PROPN ejpam-5896 465 81 } } PUNCT ejpam-5896 465 82 , , PUNCT ejpam-5896 465 83 s4(γ1 s4(γ1 NOUN ejpam-5896 465 84 ) ) PUNCT ejpam-5896 465 85 = = SYM ejpam-5896 465 86 { { PUNCT ejpam-5896 465 87 u1 u1 NOUN ejpam-5896 465 88 , , PUNCT ejpam-5896 465 89 u2 u2 PROPN ejpam-5896 465 90 } } PUNCT ejpam-5896 465 91 , , PUNCT ejpam-5896 465 92 s4(γ2 s4(γ2 NOUN ejpam-5896 465 93 ) ) PUNCT ejpam-5896 466 1 = = SYM ejpam-5896 466 2 φ φ PROPN ejpam-5896 466 3 , , PUNCT ejpam-5896 466 4 then then ADV ejpam-5896 466 5 µ µ X ejpam-5896 466 6 = = SYM ejpam-5896 466 7 { { PUNCT ejpam-5896 466 8 ũ ũ PROPN ejpam-5896 466 9 , , PUNCT ejpam-5896 466 10 φ̃ φ̃ PROPN ejpam-5896 466 11 , , PUNCT ejpam-5896 466 12 ( ( PUNCT ejpam-5896 466 13 si,∆ si,∆ PROPN ejpam-5896 466 14 ) ) PUNCT ejpam-5896 466 15 , , PUNCT ejpam-5896 466 16 i i PRON ejpam-5896 466 17 = = NOUN ejpam-5896 466 18 1 1 NUM ejpam-5896 466 19 , , PUNCT ejpam-5896 466 20 2 2 NUM ejpam-5896 466 21 , , PUNCT ejpam-5896 466 22 3 3 NUM ejpam-5896 466 23 , , PUNCT ejpam-5896 466 24 4 4 NUM ejpam-5896 466 25 } } PUNCT ejpam-5896 466 26 defines define VERB ejpam-5896 466 27 an an DET ejpam-5896 466 28 ssts sst NOUN ejpam-5896 466 29 on on ADP ejpam-5896 466 30 u u PROPN ejpam-5896 466 31 . . PUNCT ejpam-5896 467 1 according accord VERB ejpam-5896 467 2 to to ADP ejpam-5896 467 3 proposition proposition NOUN ejpam-5896 467 4 8 8 NUM ejpam-5896 467 5 , , PUNCT ejpam-5896 467 6 it it PRON ejpam-5896 467 7 is be AUX ejpam-5896 467 8 clear clear ADJ ejpam-5896 467 9 that that SCONJ ejpam-5896 467 10 ũ ũ PROPN ejpam-5896 467 11 is be AUX ejpam-5896 467 12 an an DET ejpam-5896 467 13 ss ss VERB ejpam-5896 467 14 - - PUNCT ejpam-5896 467 15 sd sd NOUN ejpam-5896 467 16 - - PUNCT ejpam-5896 467 17 compact compact ADJ ejpam-5896 467 18 ( ( PUNCT ejpam-5896 467 19 lindelöf lindelöf PROPN ejpam-5896 467 20 ) ) PUNCT ejpam-5896 467 21 . . PUNCT ejpam-5896 468 1 also also ADV ejpam-5896 468 2 the the DET ejpam-5896 468 3 soft soft ADJ ejpam-5896 468 4 set set NOUN ejpam-5896 468 5 ( ( PUNCT ejpam-5896 468 6 w,∆ w,∆ NUM ejpam-5896 468 7 ) ) PUNCT ejpam-5896 468 8 where where SCONJ ejpam-5896 468 9 : : PUNCT ejpam-5896 468 10 w w PROPN ejpam-5896 468 11 ( ( PUNCT ejpam-5896 468 12 γ1 γ1 PROPN ejpam-5896 468 13 ) ) PUNCT ejpam-5896 468 14 = = SYM ejpam-5896 468 15 { { PUNCT ejpam-5896 468 16 u1 u1 NOUN ejpam-5896 468 17 } } PUNCT ejpam-5896 468 18 , , PUNCT ejpam-5896 468 19 w w PROPN ejpam-5896 468 20 ( ( PUNCT ejpam-5896 468 21 γ2 γ2 ADJ ejpam-5896 468 22 ) ) PUNCT ejpam-5896 468 23 = = PRON ejpam-5896 468 24 { { PUNCT ejpam-5896 468 25 u2 u2 PROPN ejpam-5896 468 26 } } PUNCT ejpam-5896 468 27 is be AUX ejpam-5896 468 28 ss ss NOUN ejpam-5896 468 29 - - PUNCT ejpam-5896 468 30 sd sd NOUN ejpam-5896 468 31 - - PUNCT ejpam-5896 468 32 compact compact ADJ ejpam-5896 468 33 ( ( PUNCT ejpam-5896 468 34 lindelöf lindelöf PROPN ejpam-5896 468 35 ) ) PUNCT ejpam-5896 468 36 . . PUNCT ejpam-5896 469 1 however however ADV ejpam-5896 469 2 , , PUNCT ejpam-5896 469 3 ũ ũ PROPN ejpam-5896 469 4 ∩̃(w,∆ ∩̃(w,∆ NOUN ejpam-5896 469 5 ) ) PUNCT ejpam-5896 470 1 = = PUNCT ejpam-5896 470 2 ( ( PUNCT ejpam-5896 470 3 w,∆ w,∆ NOUN ejpam-5896 470 4 ) ) PUNCT ejpam-5896 470 5 is be AUX ejpam-5896 470 6 not not PART ejpam-5896 470 7 ss ss PROPN ejpam-5896 470 8 - - ADJ ejpam-5896 470 9 sc sc NOUN ejpam-5896 470 10 - - PUNCT ejpam-5896 470 11 set set NOUN ejpam-5896 470 12 . . PUNCT ejpam-5896 471 1 corollary corollary ADJ ejpam-5896 471 2 5 5 NUM ejpam-5896 471 3 . . PUNCT ejpam-5896 472 1 the the DET ejpam-5896 472 2 soft soft ADJ ejpam-5896 472 3 difference difference NOUN ejpam-5896 472 4 between between ADP ejpam-5896 472 5 an an DET ejpam-5896 472 6 ss ss NOUN ejpam-5896 472 7 - - PUNCT ejpam-5896 472 8 sd sd NOUN ejpam-5896 472 9 - - PUNCT ejpam-5896 472 10 compact compact ADJ ejpam-5896 472 11 ( ( PUNCT ejpam-5896 472 12 lindelöf lindelöf NOUN ejpam-5896 472 13 ) ) PUNCT ejpam-5896 472 14 soft soft ADJ ejpam-5896 472 15 subset subset NOUN ejpam-5896 472 16 ( ( PUNCT ejpam-5896 472 17 a,∆ a,∆ VERB ejpam-5896 472 18 ) ) PUNCT ejpam-5896 472 19 and and CCONJ ejpam-5896 472 20 ss ss NOUN ejpam-5896 472 21 - - PUNCT ejpam-5896 472 22 sd sd NOUN ejpam-5896 472 23 - - PUNCT ejpam-5896 472 24 subset subset NOUN ejpam-5896 472 25 ( ( PUNCT ejpam-5896 472 26 b,∆ b,∆ NOUN ejpam-5896 472 27 ) ) PUNCT ejpam-5896 472 28 of of ADP ejpam-5896 472 29 an an DET ejpam-5896 472 30 ssts sst NOUN ejpam-5896 472 31 ( ( PUNCT ejpam-5896 472 32 u u NOUN ejpam-5896 472 33 , , PUNCT ejpam-5896 472 34 µ,∆ µ,∆ NUM ejpam-5896 472 35 ) ) PUNCT ejpam-5896 472 36 is be AUX ejpam-5896 472 37 ss ss NOUN ejpam-5896 472 38 - - PUNCT ejpam-5896 472 39 sd sd NOUN ejpam-5896 472 40 - - PUNCT ejpam-5896 472 41 compact compact ADJ ejpam-5896 472 42 ( ( PUNCT ejpam-5896 472 43 lindelöf lindelöf PROPN ejpam-5896 472 44 ) ) PUNCT ejpam-5896 472 45 . . PUNCT ejpam-5896 473 1 proof proof NOUN ejpam-5896 473 2 . . PUNCT ejpam-5896 474 1 obvious obvious ADJ ejpam-5896 474 2 from from ADP ejpam-5896 474 3 theorem theorem ADJ ejpam-5896 474 4 9 9 NUM ejpam-5896 474 5 . . PUNCT ejpam-5896 474 6 definition definition NOUN ejpam-5896 474 7 23 23 NUM ejpam-5896 474 8 . . PUNCT ejpam-5896 475 1 [ [ X ejpam-5896 475 2 63 63 NUM ejpam-5896 475 3 ] ] PUNCT ejpam-5896 475 4 a a DET ejpam-5896 475 5 collection collection NOUN ejpam-5896 475 6 ψ ψ NOUN ejpam-5896 475 7 of of ADP ejpam-5896 475 8 soft soft ADJ ejpam-5896 475 9 sets set NOUN ejpam-5896 475 10 has have VERB ejpam-5896 475 11 the the DET ejpam-5896 475 12 soft soft ADJ ejpam-5896 475 13 finite finite NOUN ejpam-5896 475 14 ( ( PUNCT ejpam-5896 475 15 countable countable ADJ ejpam-5896 475 16 ) ) PUNCT ejpam-5896 475 17 intersection intersection NOUN ejpam-5896 475 18 property property NOUN ejpam-5896 475 19 ( ( PUNCT ejpam-5896 475 20 briefly briefly ADV ejpam-5896 475 21 , , PUNCT ejpam-5896 475 22 sfip sfip NOUN ejpam-5896 475 23 ( ( PUNCT ejpam-5896 475 24 scip scip PROPN ejpam-5896 475 25 ) ) PUNCT ejpam-5896 475 26 ) ) PUNCT ejpam-5896 475 27 , , PUNCT ejpam-5896 475 28 if if SCONJ ejpam-5896 475 29 the the DET ejpam-5896 475 30 soft soft ADJ ejpam-5896 475 31 intersection intersection NOUN ejpam-5896 475 32 of of ADP ejpam-5896 475 33 the the DET ejpam-5896 475 34 finite finite NOUN ejpam-5896 475 35 ( ( PUNCT ejpam-5896 475 36 countable countable ADJ ejpam-5896 475 37 ) ) PUNCT ejpam-5896 475 38 subfamily subfamily ADV ejpam-5896 475 39 of of ADP ejpam-5896 475 40 ψ ψ PROPN ejpam-5896 475 41 is be AUX ejpam-5896 475 42 non non ADJ ejpam-5896 475 43 - - ADJ ejpam-5896 475 44 empty empty ADJ ejpam-5896 475 45 . . PUNCT ejpam-5896 476 1 theorem theorem VERB ejpam-5896 476 2 16 16 NUM ejpam-5896 476 3 . . PUNCT ejpam-5896 477 1 an an DET ejpam-5896 477 2 ssts sst NOUN ejpam-5896 477 3 ( ( PUNCT ejpam-5896 477 4 u u NOUN ejpam-5896 477 5 , , PUNCT ejpam-5896 477 6 µ,∆ µ,∆ NUM ejpam-5896 477 7 ) ) PUNCT ejpam-5896 477 8 is be AUX ejpam-5896 477 9 ss ss NOUN ejpam-5896 477 10 - - PUNCT ejpam-5896 477 11 sd sd NOUN ejpam-5896 477 12 - - PUNCT ejpam-5896 477 13 compact compact ADJ ejpam-5896 477 14 ( ( PUNCT ejpam-5896 477 15 lindelöf lindelöf PROPN ejpam-5896 477 16 ) ) PUNCT ejpam-5896 477 17 if if SCONJ ejpam-5896 477 18 and and CCONJ ejpam-5896 477 19 only only ADV ejpam-5896 477 20 if if SCONJ ejpam-5896 477 21 every every DET ejpam-5896 477 22 family family NOUN ejpam-5896 477 23 of of ADP ejpam-5896 477 24 sssc sssc NOUN ejpam-5896 477 25 - - PUNCT ejpam-5896 477 26 subsets subset NOUN ejpam-5896 477 27 of of ADP ejpam-5896 477 28 ũ ũ PROPN ejpam-5896 477 29 with with ADP ejpam-5896 477 30 the the DET ejpam-5896 477 31 sfip sfip NOUN ejpam-5896 477 32 ( ( PUNCT ejpam-5896 477 33 scip scip PROPN ejpam-5896 477 34 ) ) PUNCT ejpam-5896 477 35 has have VERB ejpam-5896 477 36 a a DET ejpam-5896 477 37 non non ADJ ejpam-5896 477 38 - - ADJ ejpam-5896 477 39 empty empty ADJ ejpam-5896 477 40 intersection intersection NOUN ejpam-5896 477 41 . . PUNCT ejpam-5896 478 1 proof.necessity proof.necessity NOUN ejpam-5896 478 2 : : PUNCT ejpam-5896 478 3 assume assume VERB ejpam-5896 478 4 that that SCONJ ejpam-5896 478 5 { { PUNCT ejpam-5896 478 6 ( ( PUNCT ejpam-5896 478 7 cϵ,∆ cϵ,∆ ADJ ejpam-5896 478 8 ) ) PUNCT ejpam-5896 478 9 : : PUNCT ejpam-5896 478 10 ϵ ϵ X ejpam-5896 478 11 ∈ ∈ PROPN ejpam-5896 478 12 ε ε PROPN ejpam-5896 478 13 } } PUNCT ejpam-5896 478 14 is be AUX ejpam-5896 478 15 a a DET ejpam-5896 478 16 class class NOUN ejpam-5896 478 17 of of ADP ejpam-5896 478 18 ss ss PROPN ejpam-5896 478 19 - - ADJ ejpam-5896 478 20 sc sc NOUN ejpam-5896 478 21 - - PUNCT ejpam-5896 478 22 sets set NOUN ejpam-5896 478 23 with with ADP ejpam-5896 478 24 the the DET ejpam-5896 478 25 sfip sfip NOUN ejpam-5896 478 26 ( ( PUNCT ejpam-5896 478 27 scip scip PROPN ejpam-5896 478 28 ) ) PUNCT ejpam-5896 478 29 , , PUNCT ejpam-5896 478 30 and and CCONJ ejpam-5896 478 31 assume assume VERB ejpam-5896 478 32 contrary contrary ADJ ejpam-5896 479 1 that that DET ejpam-5896 479 2 ⋂̃ ⋂̃ ADV ejpam-5896 479 3 ϵ∈ε(cϵ,∆)=φ̃. ϵ∈ε(cϵ,∆)=φ̃. NOUN ejpam-5896 479 4 it it PRON ejpam-5896 479 5 follows follow VERB ejpam-5896 479 6 that,⋃̃ that,⋃̃ NOUN ejpam-5896 480 1 ϵ∈ε(c ϵ∈ε(c NUM ejpam-5896 480 2 c̃ c̃ PROPN ejpam-5896 480 3 ϵ ϵ PROPN ejpam-5896 480 4 , , PUNCT ejpam-5896 480 5 ∆ ∆ PROPN ejpam-5896 480 6 ) ) PUNCT ejpam-5896 481 1 = = SYM ejpam-5896 481 2 ũ ũ PROPN ejpam-5896 481 3 which which PRON ejpam-5896 481 4 means mean VERB ejpam-5896 481 5 the the DET ejpam-5896 481 6 class class NOUN ejpam-5896 481 7 { { PUNCT ejpam-5896 481 8 ( ( PUNCT ejpam-5896 481 9 c c NOUN ejpam-5896 481 10 c̃ c̃ PROPN ejpam-5896 481 11 ϵ ϵ PROPN ejpam-5896 481 12 , , PUNCT ejpam-5896 481 13 ∆ ∆ PROPN ejpam-5896 481 14 ) ) PUNCT ejpam-5896 481 15 : : PUNCT ejpam-5896 481 16 ϵ ϵ X ejpam-5896 481 17 ∈ ∈ PROPN ejpam-5896 481 18 ε ε PROPN ejpam-5896 481 19 } } PUNCT ejpam-5896 481 20 forms form VERB ejpam-5896 481 21 an an DET ejpam-5896 481 22 ss ss NOUN ejpam-5896 481 23 - - PUNCT ejpam-5896 481 24 sd sd NOUN ejpam-5896 481 25 - - PUNCT ejpam-5896 481 26 cover cover NOUN ejpam-5896 481 27 of of ADP ejpam-5896 481 28 ũ ũ PROPN ejpam-5896 481 29 . . PUNCT ejpam-5896 482 1 since since SCONJ ejpam-5896 482 2 ũ ũ PROPN ejpam-5896 482 3 is be AUX ejpam-5896 482 4 ss ss NOUN ejpam-5896 482 5 - - PUNCT ejpam-5896 482 6 sd sd NOUN ejpam-5896 482 7 - - PUNCT ejpam-5896 482 8 compact compact ADJ ejpam-5896 482 9 , , PUNCT ejpam-5896 482 10 there there PRON ejpam-5896 482 11 is be VERB ejpam-5896 482 12 a a DET ejpam-5896 482 13 finite finite ADJ ejpam-5896 482 14 subclass subclass NOUN ejpam-5896 482 15 εo εo NOUN ejpam-5896 482 16 of of ADP ejpam-5896 482 17 ε ε PROPN ejpam-5896 482 18 which which PRON ejpam-5896 482 19 also also ADV ejpam-5896 482 20 covers cover VERB ejpam-5896 482 21 ũ ũ PROPN ejpam-5896 482 22 . . PUNCT ejpam-5896 483 1 that that PRON ejpam-5896 483 2 ’s ’ VERB ejpam-5896 483 3 is⋃̃ is⋃̃ NOUN ejpam-5896 483 4 ϵ∈εo(c ϵ∈εo(c PROPN ejpam-5896 483 5 c̃ c̃ PROPN ejpam-5896 484 1 ϵ ϵ PROPN ejpam-5896 484 2 , , PUNCT ejpam-5896 484 3 ∆)=ũ ∆)=ũ PROPN ejpam-5896 484 4 . . PUNCT ejpam-5896 485 1 hence hence ADV ejpam-5896 485 2 , , PUNCT ejpam-5896 485 3 ⋂̃ ⋂̃ PROPN ejpam-5896 485 4 ϵ∈εo(cϵ,∆)=φ̃ ϵ∈εo(cϵ,∆)=φ̃ NOUN ejpam-5896 485 5 which which PRON ejpam-5896 485 6 is be AUX ejpam-5896 485 7 a a DET ejpam-5896 485 8 contradiction contradiction NOUN ejpam-5896 485 9 with with ADP ejpam-5896 485 10 the the DET ejpam-5896 485 11 sfip sfip NOUN ejpam-5896 485 12 . . PUNCT ejpam-5896 486 1 sufficient sufficient ADJ ejpam-5896 486 2 : : PUNCT ejpam-5896 486 3 suppose suppose VERB ejpam-5896 486 4 that that SCONJ ejpam-5896 486 5 ψ ψ X ejpam-5896 486 6 = = X ejpam-5896 486 7 { { PUNCT ejpam-5896 486 8 ( ( PUNCT ejpam-5896 486 9 cϵ,∆ cϵ,∆ ADJ ejpam-5896 486 10 ) ) PUNCT ejpam-5896 486 11 : : PUNCT ejpam-5896 487 1 ϵ ϵ X ejpam-5896 487 2 ∈ ∈ PROPN ejpam-5896 487 3 ε ε AUX ejpam-5896 487 4 } } PUNCT ejpam-5896 487 5 be be AUX ejpam-5896 487 6 an an DET ejpam-5896 487 7 ss ss NOUN ejpam-5896 487 8 - - PUNCT ejpam-5896 487 9 sd sd NOUN ejpam-5896 487 10 - - PUNCT ejpam-5896 487 11 cover cover NOUN ejpam-5896 487 12 of of ADP ejpam-5896 487 13 ũ ũ PROPN ejpam-5896 487 14 and and CCONJ ejpam-5896 487 15 assume assume VERB ejpam-5896 487 16 conversely conversely ADV ejpam-5896 487 17 ũ ũ PROPN ejpam-5896 487 18 is be AUX ejpam-5896 487 19 not not PART ejpam-5896 487 20 ss ss NOUN ejpam-5896 487 21 - - PUNCT ejpam-5896 487 22 sd sd NOUN ejpam-5896 487 23 - - PUNCT ejpam-5896 487 24 compact compact ADJ ejpam-5896 487 25 . . PUNCT ejpam-5896 488 1 it it PRON ejpam-5896 488 2 follows follow VERB ejpam-5896 488 3 that that SCONJ ejpam-5896 488 4 , , PUNCT ejpam-5896 488 5 for for ADP ejpam-5896 488 6 every every DET ejpam-5896 488 7 finite finite ADJ ejpam-5896 488 8 subclass subclass NOUN ejpam-5896 488 9 εo εo NOUN ejpam-5896 488 10 of of ADP ejpam-5896 488 11 ε ε PROPN ejpam-5896 488 12 we we PRON ejpam-5896 488 13 have have VERB ejpam-5896 488 14 abd abd PROPN ejpam-5896 488 15 el el PROPN ejpam-5896 488 16 - - PROPN ejpam-5896 488 17 latif latif PROPN ejpam-5896 488 18 et et PROPN ejpam-5896 488 19 al al PROPN ejpam-5896 488 20 . . PUNCT ejpam-5896 488 21 / / SYM ejpam-5896 488 22 eur eur PROPN ejpam-5896 488 23 . . PUNCT ejpam-5896 489 1 j. j. PROPN ejpam-5896 489 2 pure pure PROPN ejpam-5896 489 3 appl appl PROPN ejpam-5896 489 4 . . PROPN ejpam-5896 489 5 math math PROPN ejpam-5896 489 6 , , PUNCT ejpam-5896 489 7 18 18 NUM ejpam-5896 489 8 ( ( PUNCT ejpam-5896 489 9 2 2 NUM ejpam-5896 489 10 ) ) PUNCT ejpam-5896 489 11 ( ( PUNCT ejpam-5896 489 12 2025 2025 NUM ejpam-5896 489 13 ) ) PUNCT ejpam-5896 489 14 , , PUNCT ejpam-5896 489 15 5896 5896 NUM ejpam-5896 489 16 14 14 NUM ejpam-5896 489 17 of of ADP ejpam-5896 489 18 20⋃̃ 20⋃̃ NUM ejpam-5896 489 19 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ ADJ ejpam-5896 489 20 ) ) PUNCT ejpam-5896 489 21 ̸= ̸= PROPN ejpam-5896 489 22 ũ ũ PROPN ejpam-5896 489 23 and and CCONJ ejpam-5896 489 24 so so ADV ejpam-5896 489 25 ⋂̃ ⋂̃ PROPN ejpam-5896 489 26 ϵ∈εo(c ϵ∈εo(c PROPN ejpam-5896 489 27 c̃ c̃ PROPN ejpam-5896 490 1 ϵ ϵ PROPN ejpam-5896 490 2 , , PUNCT ejpam-5896 490 3 ∆ ∆ X ejpam-5896 490 4 ) ) PUNCT ejpam-5896 491 1 ̸= ̸= PROPN ejpam-5896 491 2 φ̃. φ̃. PROPN ejpam-5896 491 3 hence hence ADV ejpam-5896 491 4 , , PUNCT ejpam-5896 491 5 { { PUNCT ejpam-5896 491 6 ( ( PUNCT ejpam-5896 491 7 c c NOUN ejpam-5896 491 8 c̃ c̃ PROPN ejpam-5896 491 9 ϵ ϵ PROPN ejpam-5896 491 10 , , PUNCT ejpam-5896 491 11 ∆ ∆ PROPN ejpam-5896 491 12 ) ) PUNCT ejpam-5896 491 13 : : PUNCT ejpam-5896 491 14 ϵ ϵ X ejpam-5896 491 15 ∈ ∈ PROPN ejpam-5896 491 16 ε ε PROPN ejpam-5896 491 17 } } PUNCT ejpam-5896 491 18 is be AUX ejpam-5896 491 19 a a DET ejpam-5896 491 20 class class NOUN ejpam-5896 491 21 of of ADP ejpam-5896 491 22 ss ss PROPN ejpam-5896 491 23 - - ADJ ejpam-5896 491 24 sc sc NOUN ejpam-5896 491 25 - - PUNCT ejpam-5896 491 26 subsets subset NOUN ejpam-5896 491 27 of of ADP ejpam-5896 491 28 ũ ũ PROPN ejpam-5896 491 29 has have VERB ejpam-5896 491 30 the the DET ejpam-5896 491 31 sfip sfip NOUN ejpam-5896 491 32 . . PUNCT ejpam-5896 492 1 by by ADP ejpam-5896 492 2 assumption,⋂̃ assumption,⋂̃ PROPN ejpam-5896 492 3 ϵ∈ε(c ϵ∈ε(c PUNCT ejpam-5896 493 1 c̃ c̃ PROPN ejpam-5896 493 2 ϵ ϵ PROPN ejpam-5896 493 3 , , PUNCT ejpam-5896 493 4 ∆)̸= ∆)̸= PROPN ejpam-5896 494 1 φ̃ φ̃ PROPN ejpam-5896 494 2 and and CCONJ ejpam-5896 494 3 so so ADV ejpam-5896 494 4 ⋃̃ ⋃̃ PROPN ejpam-5896 494 5 ϵ∈ε(cϵ,∆ ϵ∈ε(cϵ,∆ PROPN ejpam-5896 494 6 ) ) PUNCT ejpam-5896 495 1 ̸= ̸= PROPN ejpam-5896 495 2 ũ ũ PROPN ejpam-5896 495 3 , , PUNCT ejpam-5896 495 4 it it PRON ejpam-5896 495 5 conflicts conflict VERB ejpam-5896 495 6 with with ADP ejpam-5896 495 7 that that DET ejpam-5896 495 8 ψ ψ NOUN ejpam-5896 495 9 is be AUX ejpam-5896 495 10 an an DET ejpam-5896 495 11 ss ss VERB ejpam-5896 495 12 - - PUNCT ejpam-5896 495 13 sd sd NOUN ejpam-5896 495 14 - - PUNCT ejpam-5896 495 15 cover cover NOUN ejpam-5896 495 16 of of ADP ejpam-5896 495 17 ũ ũ PROPN ejpam-5896 495 18 . . PUNCT ejpam-5896 496 1 thus thus ADV ejpam-5896 496 2 , , PUNCT ejpam-5896 496 3 ũ ũ PROPN ejpam-5896 496 4 is be AUX ejpam-5896 496 5 an an DET ejpam-5896 496 6 ss ss VERB ejpam-5896 496 7 - - PUNCT ejpam-5896 496 8 sd sd NOUN ejpam-5896 496 9 - - PUNCT ejpam-5896 496 10 compact compact ADJ ejpam-5896 496 11 . . PUNCT ejpam-5896 497 1 the the DET ejpam-5896 497 2 case case NOUN ejpam-5896 497 3 of of ADP ejpam-5896 497 4 ss ss NOUN ejpam-5896 497 5 - - PUNCT ejpam-5896 497 6 sd sd NOUN ejpam-5896 497 7 - - PUNCT ejpam-5896 497 8 lindelöfness lindelöfness NOUN ejpam-5896 497 9 can can AUX ejpam-5896 497 10 be be AUX ejpam-5896 497 11 obtained obtain VERB ejpam-5896 497 12 by by ADP ejpam-5896 497 13 a a DET ejpam-5896 497 14 similar similar ADJ ejpam-5896 497 15 way way NOUN ejpam-5896 497 16 . . PUNCT ejpam-5896 498 1 theorem theorem VERB ejpam-5896 498 2 17 17 NUM ejpam-5896 498 3 . . PUNCT ejpam-5896 499 1 an an DET ejpam-5896 499 2 ssts sst NOUN ejpam-5896 499 3 ( ( PUNCT ejpam-5896 499 4 u u NOUN ejpam-5896 499 5 , , PUNCT ejpam-5896 499 6 µ,∆ µ,∆ NUM ejpam-5896 499 7 ) ) PUNCT ejpam-5896 499 8 is be AUX ejpam-5896 499 9 ss ss NOUN ejpam-5896 499 10 - - PUNCT ejpam-5896 499 11 sd sd NOUN ejpam-5896 499 12 - - PUNCT ejpam-5896 499 13 compact compact ADJ ejpam-5896 499 14 ( ( PUNCT ejpam-5896 499 15 lindelöf lindelöf PROPN ejpam-5896 499 16 ) ) PUNCT ejpam-5896 500 1 if if SCONJ ejpam-5896 500 2 and and CCONJ ejpam-5896 500 3 only only ADV ejpam-5896 500 4 if if SCONJ ejpam-5896 500 5 every every DET ejpam-5896 500 6 class class NOUN ejpam-5896 500 7 ψ ψ NOUN ejpam-5896 500 8 = = X ejpam-5896 500 9 { { PUNCT ejpam-5896 500 10 ( ( PUNCT ejpam-5896 500 11 cϵ,∆ cϵ,∆ ADJ ejpam-5896 500 12 ) ) PUNCT ejpam-5896 500 13 : : PUNCT ejpam-5896 500 14 ϵ ϵ X ejpam-5896 500 15 ∈ ∈ PROPN ejpam-5896 500 16 ε ε PROPN ejpam-5896 500 17 } } PUNCT ejpam-5896 500 18 of of ADP ejpam-5896 500 19 soft soft ADJ ejpam-5896 500 20 subsets subset NOUN ejpam-5896 500 21 of of ADP ejpam-5896 500 22 ũ ũ PROPN ejpam-5896 500 23 with with ADP ejpam-5896 500 24 the the DET ejpam-5896 500 25 sfip sfip NOUN ejpam-5896 500 26 ( ( PUNCT ejpam-5896 500 27 scip scip PROPN ejpam-5896 500 28 ) ) PUNCT ejpam-5896 500 29 satisfies satisfy VERB ejpam-5896 500 30 ⋂̃ ⋂̃ ADJ ejpam-5896 500 31 ϵ∈ε{clssd(cϵ,∆ ϵ∈ε{clssd(cϵ,∆ ADJ ejpam-5896 500 32 ) ) PUNCT ejpam-5896 500 33 : : PUNCT ejpam-5896 500 34 ( ( PUNCT ejpam-5896 500 35 cϵ,∆)∈ψ}̸= cϵ,∆)∈ψ}̸= PROPN ejpam-5896 500 36 φ̃. φ̃. PROPN ejpam-5896 500 37 proof proof NOUN ejpam-5896 500 38 . . PUNCT ejpam-5896 501 1 we we PRON ejpam-5896 501 2 prove prove VERB ejpam-5896 501 3 the the DET ejpam-5896 501 4 case case NOUN ejpam-5896 501 5 of of ADP ejpam-5896 501 6 ss ss NOUN ejpam-5896 501 7 - - PUNCT ejpam-5896 501 8 sd sd NOUN ejpam-5896 501 9 - - PUNCT ejpam-5896 501 10 compactness compactness NOUN ejpam-5896 501 11 , , PUNCT ejpam-5896 501 12 the the DET ejpam-5896 501 13 case case NOUN ejpam-5896 501 14 that that PRON ejpam-5896 501 15 is be AUX ejpam-5896 501 16 enclosed enclose VERB ejpam-5896 501 17 in in ADP ejpam-5896 501 18 parenthesis parenthesis NOUN ejpam-5896 501 19 can can AUX ejpam-5896 501 20 be be AUX ejpam-5896 501 21 obtained obtain VERB ejpam-5896 501 22 by by ADP ejpam-5896 501 23 the the DET ejpam-5896 501 24 same same ADJ ejpam-5896 501 25 way way NOUN ejpam-5896 501 26 . . PUNCT ejpam-5896 502 1 necessity necessity NOUN ejpam-5896 502 2 : : PUNCT ejpam-5896 502 3 assume assume VERB ejpam-5896 502 4 that that SCONJ ejpam-5896 502 5 ψ ψ X ejpam-5896 502 6 = = X ejpam-5896 502 7 { { PUNCT ejpam-5896 502 8 ( ( PUNCT ejpam-5896 502 9 cϵ,∆ cϵ,∆ ADJ ejpam-5896 502 10 ) ) PUNCT ejpam-5896 502 11 : : PUNCT ejpam-5896 502 12 ϵ ϵ X ejpam-5896 502 13 ∈ ∈ PROPN ejpam-5896 502 14 ε ε PROPN ejpam-5896 502 15 } } PUNCT ejpam-5896 502 16 is be AUX ejpam-5896 502 17 a a DET ejpam-5896 502 18 class class NOUN ejpam-5896 502 19 of of ADP ejpam-5896 502 20 soft soft ADJ ejpam-5896 502 21 sets set NOUN ejpam-5896 502 22 with with ADP ejpam-5896 502 23 the the DET ejpam-5896 502 24 sfip sfip NOUN ejpam-5896 502 25 , , PUNCT ejpam-5896 502 26 and and CCONJ ejpam-5896 502 27 suppose suppose VERB ejpam-5896 502 28 contrary contrary ADJ ejpam-5896 502 29 ⋂̃ ⋂̃ NOUN ejpam-5896 502 30 ϵ∈ε{clssd(cϵ,∆ ϵ∈ε{clssd(cϵ,∆ ADJ ejpam-5896 502 31 ) ) PUNCT ejpam-5896 502 32 : : PUNCT ejpam-5896 502 33 ( ( PUNCT ejpam-5896 502 34 cϵ,∆)∈ψ}=φ̃ cϵ,∆)∈ψ}=φ̃ PROPN ejpam-5896 502 35 , , PUNCT ejpam-5896 502 36 then⋃̃ then⋃̃ VERB ejpam-5896 502 37 ϵ∈ε{(clssd(cϵ,∆))c̃ ϵ∈ε{(clssd(cϵ,∆))c̃ NOUN ejpam-5896 502 38 : : PUNCT ejpam-5896 502 39 ( ( PUNCT ejpam-5896 502 40 cϵ,∆)∈ψ}=ũ cϵ,∆)∈ψ}=ũ X ejpam-5896 502 41 which which PRON ejpam-5896 502 42 means mean VERB ejpam-5896 502 43 { { PUNCT ejpam-5896 502 44 ( ( PUNCT ejpam-5896 502 45 clssd(cϵ,∆))c̃ clssd(cϵ,∆))c̃ NUM ejpam-5896 502 46 : : PUNCT ejpam-5896 502 47 ( ( PUNCT ejpam-5896 502 48 cϵ,∆)∈ψ}is cϵ,∆)∈ψ}is VERB ejpam-5896 502 49 an an DET ejpam-5896 502 50 ss ss NOUN ejpam-5896 502 51 - - PUNCT ejpam-5896 502 52 sd sd NOUN ejpam-5896 502 53 - - PUNCT ejpam-5896 502 54 cover cover NOUN ejpam-5896 502 55 of of ADP ejpam-5896 502 56 ũ ũ PROPN ejpam-5896 502 57 . . PUNCT ejpam-5896 503 1 since since SCONJ ejpam-5896 503 2 ũ ũ PROPN ejpam-5896 503 3 is be AUX ejpam-5896 503 4 ss ss NOUN ejpam-5896 503 5 - - PUNCT ejpam-5896 503 6 sd sd NOUN ejpam-5896 503 7 - - PUNCT ejpam-5896 503 8 compact compact ADJ ejpam-5896 503 9 , , PUNCT ejpam-5896 503 10 there there PRON ejpam-5896 503 11 is be VERB ejpam-5896 503 12 a a DET ejpam-5896 503 13 finite finite ADJ ejpam-5896 503 14 subclass subclass NOUN ejpam-5896 503 15 εo εo NOUN ejpam-5896 503 16 of of ADP ejpam-5896 503 17 ε ε PROPN ejpam-5896 503 18 such such ADJ ejpam-5896 503 19 that⋃̃ that⋃̃ PROPN ejpam-5896 504 1 ϵ∈εo{(cl ϵ∈εo{(cl PROPN ejpam-5896 505 1 s s PART ejpam-5896 505 2 sd(cϵ,∆))c̃ sd(cϵ,∆))c̃ X ejpam-5896 505 3 : : PUNCT ejpam-5896 505 4 ( ( PUNCT ejpam-5896 505 5 cϵ,∆)∈ψ}=ũ cϵ,∆)∈ψ}=ũ PROPN ejpam-5896 505 6 . . PUNCT ejpam-5896 506 1 this this PRON ejpam-5896 506 2 leads lead VERB ejpam-5896 506 3 to⋂̃ to⋂̃ NOUN ejpam-5896 506 4 ϵ∈εo{(cϵ,∆ ϵ∈εo{(cϵ,∆ PUNCT ejpam-5896 506 5 ) ) PUNCT ejpam-5896 506 6 : : PUNCT ejpam-5896 506 7 ( ( PUNCT ejpam-5896 506 8 cϵ,∆ cϵ,∆ ADJ ejpam-5896 506 9 ) ) PUNCT ejpam-5896 506 10 ∈ ∈ PROPN ejpam-5896 506 11 ψ}⊆̃ ψ}⊆̃ PRON ejpam-5896 506 12 ⋂̃ ⋂̃ X ejpam-5896 507 1 ϵ∈εo{cl ϵ∈εo{cl NUM ejpam-5896 507 2 s s NOUN ejpam-5896 507 3 sd(cϵ,∆ sd(cϵ,∆ NOUN ejpam-5896 507 4 ) ) PUNCT ejpam-5896 507 5 : : PUNCT ejpam-5896 507 6 ( ( PUNCT ejpam-5896 507 7 cϵ,∆ cϵ,∆ ADJ ejpam-5896 507 8 ) ) PUNCT ejpam-5896 507 9 ∈ ∈ PROPN ejpam-5896 507 10 ψ ψ NOUN ejpam-5896 507 11 } } PUNCT ejpam-5896 507 12 = = SYM ejpam-5896 507 13 φ̃ φ̃ PROPN ejpam-5896 507 14 , , PUNCT ejpam-5896 507 15 which which PRON ejpam-5896 507 16 is be AUX ejpam-5896 507 17 a a DET ejpam-5896 507 18 contradiction contradiction NOUN ejpam-5896 507 19 with with ADP ejpam-5896 507 20 the the DET ejpam-5896 507 21 sfip sfip NOUN ejpam-5896 507 22 . . PUNCT ejpam-5896 508 1 thus thus ADV ejpam-5896 508 2 , , PUNCT ejpam-5896 508 3 ⋃̃ ⋃̃ PROPN ejpam-5896 508 4 ϵ∈ε{clssd(cϵ,∆ ϵ∈ε{clssd(cϵ,∆ NOUN ejpam-5896 508 5 ) ) PUNCT ejpam-5896 508 6 : : PUNCT ejpam-5896 508 7 ( ( PUNCT ejpam-5896 508 8 cϵ,∆)∈ψ}≠ cϵ,∆)∈ψ}≠ PROPN ejpam-5896 508 9 φ̃. φ̃. PROPN ejpam-5896 508 10 sufficient sufficient ADJ ejpam-5896 508 11 : : PUNCT ejpam-5896 508 12 suppose suppose VERB ejpam-5896 508 13 that that SCONJ ejpam-5896 508 14 ψ ψ X ejpam-5896 508 15 = = X ejpam-5896 508 16 { { PUNCT ejpam-5896 508 17 ( ( PUNCT ejpam-5896 508 18 cϵ,∆ cϵ,∆ ADJ ejpam-5896 508 19 ) ) PUNCT ejpam-5896 508 20 : : PUNCT ejpam-5896 508 21 ϵ ϵ X ejpam-5896 508 22 ∈ ∈ PROPN ejpam-5896 508 23 ε ε AUX ejpam-5896 508 24 } } PUNCT ejpam-5896 508 25 be be AUX ejpam-5896 508 26 an an DET ejpam-5896 508 27 ss ss NOUN ejpam-5896 508 28 - - PUNCT ejpam-5896 508 29 sd sd NOUN ejpam-5896 508 30 - - PUNCT ejpam-5896 508 31 cover cover NOUN ejpam-5896 508 32 for for ADP ejpam-5896 508 33 ũ ũ PROPN ejpam-5896 508 34 and and CCONJ ejpam-5896 508 35 assume assume VERB ejpam-5896 508 36 contrary contrary ADJ ejpam-5896 508 37 that that SCONJ ejpam-5896 508 38 ũ ũ PROPN ejpam-5896 508 39 is be AUX ejpam-5896 508 40 not not PART ejpam-5896 508 41 ss ss NOUN ejpam-5896 508 42 - - PUNCT ejpam-5896 508 43 sd sd NOUN ejpam-5896 508 44 - - PUNCT ejpam-5896 508 45 compact compact ADJ ejpam-5896 508 46 , , PUNCT ejpam-5896 508 47 then then ADV ejpam-5896 508 48 ψ ψ X ejpam-5896 508 49 has have VERB ejpam-5896 508 50 not not PART ejpam-5896 508 51 any any DET ejpam-5896 508 52 finite finite NOUN ejpam-5896 508 53 subcover subcover PROPN ejpam-5896 508 54 which which PRON ejpam-5896 508 55 cover cover VERB ejpam-5896 508 56 ũ ũ PROPN ejpam-5896 508 57 . . PUNCT ejpam-5896 509 1 hence hence ADV ejpam-5896 509 2 , , PUNCT ejpam-5896 509 3 for for ADP ejpam-5896 509 4 each each DET ejpam-5896 509 5 finite finite ADJ ejpam-5896 509 6 subclass subclass NOUN ejpam-5896 509 7 εo εo NOUN ejpam-5896 509 8 of of ADP ejpam-5896 509 9 ε ε PROPN ejpam-5896 509 10 we we PRON ejpam-5896 509 11 have⋃̃ have⋃̃ NOUN ejpam-5896 509 12 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ VERB ejpam-5896 509 13 ) ) PUNCT ejpam-5896 509 14 ̸= ̸= PROPN ejpam-5896 509 15 ũ ũ PROPN ejpam-5896 509 16 which which PRON ejpam-5896 509 17 follows follow VERB ejpam-5896 509 18 ⋂̃ ⋂̃ ADJ ejpam-5896 509 19 ϵ∈εo(c ϵ∈εo(c PROPN ejpam-5896 509 20 c̃ c̃ PROPN ejpam-5896 509 21 ϵ ϵ PROPN ejpam-5896 509 22 , , PUNCT ejpam-5896 509 23 ∆ ∆ X ejpam-5896 509 24 ) ) PUNCT ejpam-5896 509 25 ̸= ̸= PROPN ejpam-5896 509 26 φ̃. φ̃. PROPN ejpam-5896 509 27 therefore therefore ADV ejpam-5896 509 28 , , PUNCT ejpam-5896 509 29 { { PUNCT ejpam-5896 509 30 ( ( PUNCT ejpam-5896 509 31 c c X ejpam-5896 509 32 c̃ c̃ PROPN ejpam-5896 509 33 ϵ ϵ PROPN ejpam-5896 509 34 , , PUNCT ejpam-5896 509 35 ∆ ∆ PROPN ejpam-5896 509 36 ) ) PUNCT ejpam-5896 509 37 : : PUNCT ejpam-5896 509 38 ( ( PUNCT ejpam-5896 509 39 cϵ,∆)∈ψ cϵ,∆)∈ψ NOUN ejpam-5896 509 40 } } PUNCT ejpam-5896 509 41 is be AUX ejpam-5896 509 42 a a DET ejpam-5896 509 43 class class NOUN ejpam-5896 509 44 of of ADP ejpam-5896 509 45 ss ss PROPN ejpam-5896 509 46 - - ADJ ejpam-5896 509 47 sc sc NOUN ejpam-5896 509 48 - - PUNCT ejpam-5896 509 49 sets set NOUN ejpam-5896 509 50 with with ADP ejpam-5896 509 51 sfip sfip NOUN ejpam-5896 509 52 . . PUNCT ejpam-5896 510 1 from from ADP ejpam-5896 510 2 hypothesis,⋂̃ hypothesis,⋂̃ PROPN ejpam-5896 510 3 ϵ∈ε{clssd(c ϵ∈ε{clssd(c PROPN ejpam-5896 510 4 c̃ c̃ PROPN ejpam-5896 510 5 ϵ ϵ PROPN ejpam-5896 510 6 , , PUNCT ejpam-5896 510 7 ∆ ∆ PROPN ejpam-5896 510 8 ) ) PUNCT ejpam-5896 511 1 = = SYM ejpam-5896 511 2 ( ( PUNCT ejpam-5896 511 3 c c X ejpam-5896 511 4 c̃ c̃ PROPN ejpam-5896 511 5 ϵ ϵ PROPN ejpam-5896 511 6 , , PUNCT ejpam-5896 511 7 ∆ ∆ PROPN ejpam-5896 511 8 ) ) PUNCT ejpam-5896 511 9 : : PUNCT ejpam-5896 511 10 ( ( PUNCT ejpam-5896 511 11 cϵ,∆)∈ψ}≠ cϵ,∆)∈ψ}≠ PROPN ejpam-5896 511 12 φ̃. φ̃. PROPN ejpam-5896 511 13 hence,⋃̃ hence,⋃̃ PROPN ejpam-5896 511 14 ϵ∈ε{(cϵ,∆ ϵ∈ε{(cϵ,∆ PROPN ejpam-5896 511 15 ) ) PUNCT ejpam-5896 511 16 : : PUNCT ejpam-5896 511 17 ( ( PUNCT ejpam-5896 511 18 cϵ,∆)∈ψ}≠ cϵ,∆)∈ψ}≠ PROPN ejpam-5896 511 19 ũ ũ PROPN ejpam-5896 511 20 , , PUNCT ejpam-5896 511 21 which which PRON ejpam-5896 511 22 is be AUX ejpam-5896 511 23 in in ADP ejpam-5896 511 24 opposition opposition NOUN ejpam-5896 511 25 to to ADP ejpam-5896 511 26 that that PRON ejpam-5896 511 27 ψ ψ NOUN ejpam-5896 511 28 is be AUX ejpam-5896 511 29 an an DET ejpam-5896 511 30 ss ss VERB ejpam-5896 511 31 - - PUNCT ejpam-5896 511 32 sd sd NOUN ejpam-5896 511 33 - - PUNCT ejpam-5896 511 34 cover cover NOUN ejpam-5896 511 35 for for ADP ejpam-5896 511 36 ũ ũ PROPN ejpam-5896 511 37 . . PUNCT ejpam-5896 512 1 thus thus ADV ejpam-5896 512 2 , , PUNCT ejpam-5896 512 3 ũ ũ PROPN ejpam-5896 512 4 is be AUX ejpam-5896 512 5 an an DET ejpam-5896 512 6 ss ss VERB ejpam-5896 512 7 - - PUNCT ejpam-5896 512 8 sd sd NOUN ejpam-5896 512 9 - - PUNCT ejpam-5896 512 10 compact compact ADJ ejpam-5896 512 11 . . PUNCT ejpam-5896 513 1 theorem theorem VERB ejpam-5896 513 2 18 18 NUM ejpam-5896 513 3 . . PUNCT ejpam-5896 514 1 the the DET ejpam-5896 514 2 image image NOUN ejpam-5896 514 3 of of ADP ejpam-5896 514 4 each each DET ejpam-5896 514 5 ss ss NOUN ejpam-5896 514 6 - - PUNCT ejpam-5896 514 7 sd sd NOUN ejpam-5896 514 8 - - PUNCT ejpam-5896 514 9 lindelöf lindelöf NOUN ejpam-5896 514 10 ( ( PUNCT ejpam-5896 514 11 compact compact ADJ ejpam-5896 514 12 ) ) PUNCT ejpam-5896 514 13 set set NOUN ejpam-5896 514 14 is be AUX ejpam-5896 514 15 ss ss NOUN ejpam-5896 514 16 - - PUNCT ejpam-5896 514 17 lindelöf lindelöf NOUN ejpam-5896 514 18 ( ( PUNCT ejpam-5896 514 19 compact compact ADJ ejpam-5896 514 20 ) ) PUNCT ejpam-5896 514 21 under under ADP ejpam-5896 514 22 a a DET ejpam-5896 514 23 surjective surjective ADJ ejpam-5896 514 24 and and CCONJ ejpam-5896 514 25 ss ss NOUN ejpam-5896 514 26 - - PUNCT ejpam-5896 514 27 sd sd NOUN ejpam-5896 514 28 - - PUNCT ejpam-5896 514 29 continuous continuous ADJ ejpam-5896 514 30 map map NOUN ejpam-5896 514 31 . . PUNCT ejpam-5896 515 1 proof proof NOUN ejpam-5896 515 2 . . PUNCT ejpam-5896 516 1 let let VERB ejpam-5896 516 2 ψsd ψsd VERB ejpam-5896 516 3 : : PUNCT ejpam-5896 516 4 ( ( PUNCT ejpam-5896 516 5 u u NOUN ejpam-5896 516 6 , , PUNCT ejpam-5896 516 7 τ,∆ τ,∆ NOUN ejpam-5896 516 8 ) ) PUNCT ejpam-5896 516 9 → → SYM ejpam-5896 516 10 ( ( PUNCT ejpam-5896 516 11 v v NOUN ejpam-5896 516 12 , , PUNCT ejpam-5896 516 13 σ,∆ σ,∆ NUM ejpam-5896 516 14 ) ) PUNCT ejpam-5896 516 15 be be VERB ejpam-5896 516 16 an an DET ejpam-5896 516 17 ss ss NOUN ejpam-5896 516 18 - - PUNCT ejpam-5896 516 19 sd sd NOUN ejpam-5896 516 20 - - PUNCT ejpam-5896 516 21 continuous continuous ADJ ejpam-5896 516 22 map map NOUN ejpam-5896 516 23 with with ADP ejpam-5896 516 24 µ µ NOUN ejpam-5896 516 25 , , PUNCT ejpam-5896 516 26 µ∗ µ∗ VERB ejpam-5896 516 27 as as ADP ejpam-5896 516 28 associated associated ADJ ejpam-5896 516 29 sstss sstss NOUN ejpam-5896 516 30 with with ADP ejpam-5896 516 31 τ τ PROPN ejpam-5896 516 32 , , PUNCT ejpam-5896 516 33 σ σ PROPN ejpam-5896 516 34 , , PUNCT ejpam-5896 516 35 respectively respectively ADV ejpam-5896 516 36 , , PUNCT ejpam-5896 516 37 and and CCONJ ejpam-5896 516 38 { { PUNCT ejpam-5896 516 39 ( ( PUNCT ejpam-5896 516 40 cϵ,∆ cϵ,∆ ADJ ejpam-5896 516 41 ) ) PUNCT ejpam-5896 516 42 : : PUNCT ejpam-5896 517 1 ϵ ϵ X ejpam-5896 517 2 ∈ ∈ PROPN ejpam-5896 517 3 ε ε PROPN ejpam-5896 517 4 } } PUNCT ejpam-5896 517 5 is be AUX ejpam-5896 517 6 ss ss NOUN ejpam-5896 517 7 - - NOUN ejpam-5896 517 8 cover cover NOUN ejpam-5896 517 9 for for ADP ejpam-5896 517 10 the the DET ejpam-5896 517 11 image image NOUN ejpam-5896 517 12 of of ADP ejpam-5896 517 13 an an DET ejpam-5896 517 14 ss ss NOUN ejpam-5896 517 15 - - PUNCT ejpam-5896 517 16 sd sd NOUN ejpam-5896 517 17 - - PUNCT ejpam-5896 517 18 lindelöf lindelöf NOUN ejpam-5896 517 19 subset subset NOUN ejpam-5896 517 20 ( ( PUNCT ejpam-5896 517 21 k,∆ k,∆ PROPN ejpam-5896 517 22 ) ) PUNCT ejpam-5896 517 23 of of ADP ejpam-5896 517 24 ũ ũ PROPN ejpam-5896 517 25 . . PUNCT ejpam-5896 518 1 it it PRON ejpam-5896 518 2 follows follow VERB ejpam-5896 518 3 that that SCONJ ejpam-5896 518 4 , , PUNCT ejpam-5896 518 5 ψ−1 ψ−1 PROPN ejpam-5896 518 6 sd sd ADP ejpam-5896 518 7 ( ( PUNCT ejpam-5896 518 8 cϵ,∆ cϵ,∆ ADJ ejpam-5896 518 9 ) ) PUNCT ejpam-5896 518 10 ∈ ∈ PROPN ejpam-5896 518 11 sd(u)∆ sd(u)∆ NOUN ejpam-5896 518 12 for for ADP ejpam-5896 518 13 each each PRON ejpam-5896 518 14 ϵ ϵ PROPN ejpam-5896 518 15 ∈ ∈ PROPN ejpam-5896 518 16 ε ε PROPN ejpam-5896 518 17 with with ADP ejpam-5896 518 18 ( ( PUNCT ejpam-5896 518 19 k,∆)⊆̃ k,∆)⊆̃ PROPN ejpam-5896 518 20 ⋃̃ ⋃̃ PROPN ejpam-5896 518 21 ϵ∈ε[ψ ϵ∈ε[ψ PROPN ejpam-5896 518 22 −1 −1 NOUN ejpam-5896 518 23 sd sd ADP ejpam-5896 518 24 ( ( PUNCT ejpam-5896 518 25 cϵ,∆ cϵ,∆ ADJ ejpam-5896 518 26 ) ) PUNCT ejpam-5896 518 27 ] ] PUNCT ejpam-5896 518 28 . . PUNCT ejpam-5896 519 1 since since SCONJ ejpam-5896 519 2 ( ( PUNCT ejpam-5896 519 3 k,∆ k,∆ PROPN ejpam-5896 519 4 ) ) PUNCT ejpam-5896 519 5 is be AUX ejpam-5896 519 6 ss ss NOUN ejpam-5896 519 7 - - PUNCT ejpam-5896 519 8 sd sd NOUN ejpam-5896 519 9 - - PUNCT ejpam-5896 519 10 lindelöf lindelöf NOUN ejpam-5896 519 11 , , PUNCT ejpam-5896 519 12 there there PRON ejpam-5896 519 13 is be VERB ejpam-5896 519 14 a a DET ejpam-5896 519 15 countable countable ADJ ejpam-5896 519 16 subclasses subclass NOUN ejpam-5896 519 17 εo εo NOUN ejpam-5896 519 18 of of ADP ejpam-5896 519 19 ε ε PROPN ejpam-5896 519 20 such such ADJ ejpam-5896 519 21 that that SCONJ ejpam-5896 519 22 ( ( PUNCT ejpam-5896 519 23 k,∆)⊆̃ k,∆)⊆̃ PROPN ejpam-5896 520 1 ⋃̃ ⋃̃ PROPN ejpam-5896 520 2 ϵ∈εo ϵ∈εo PROPN ejpam-5896 520 3 [ [ X ejpam-5896 520 4 ψ ψ X ejpam-5896 520 5 −1 −1 NOUN ejpam-5896 520 6 sd sd ADP ejpam-5896 520 7 ( ( PUNCT ejpam-5896 520 8 cϵ,∆ cϵ,∆ ADJ ejpam-5896 520 9 ) ) PUNCT ejpam-5896 520 10 ] ] PUNCT ejpam-5896 520 11 which which PRON ejpam-5896 520 12 follows follow VERB ejpam-5896 520 13 ψsd(k,∆)⊆̃ ψsd(k,∆)⊆̃ PROPN ejpam-5896 521 1 ⋃̃ ⋃̃ PROPN ejpam-5896 521 2 ϵ∈εoψsd[ψ ϵ∈εoψsd[ψ NOUN ejpam-5896 521 3 −1 −1 NOUN ejpam-5896 521 4 sd sd ADP ejpam-5896 521 5 ( ( PUNCT ejpam-5896 521 6 cϵ,∆ cϵ,∆ ADJ ejpam-5896 521 7 ) ) PUNCT ejpam-5896 521 8 ] ] PUNCT ejpam-5896 522 1 = = SYM ejpam-5896 522 2 ⋃̃ ⋃̃ PROPN ejpam-5896 522 3 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 522 4 ) ) PUNCT ejpam-5896 522 5 , , PUNCT ejpam-5896 522 6 ψsd ψsd PROPN ejpam-5896 522 7 is be AUX ejpam-5896 522 8 surjective surjective ADJ ejpam-5896 522 9 . . PUNCT ejpam-5896 523 1 therefore therefore ADV ejpam-5896 523 2 , , PUNCT ejpam-5896 523 3 ψsd(k,∆ ψsd(k,∆ NOUN ejpam-5896 523 4 ) ) PUNCT ejpam-5896 523 5 is be AUX ejpam-5896 523 6 an an DET ejpam-5896 523 7 ss ss NOUN ejpam-5896 523 8 - - NOUN ejpam-5896 523 9 lindelöf lindelöf NOUN ejpam-5896 523 10 . . PUNCT ejpam-5896 524 1 similarly similarly ADV ejpam-5896 524 2 , , PUNCT ejpam-5896 524 3 one one PRON ejpam-5896 524 4 can can AUX ejpam-5896 524 5 prove prove VERB ejpam-5896 524 6 the the DET ejpam-5896 524 7 case case NOUN ejpam-5896 524 8 of of ADP ejpam-5896 524 9 ss ss NOUN ejpam-5896 524 10 - - PUNCT ejpam-5896 524 11 sd sd NOUN ejpam-5896 524 12 - - PUNCT ejpam-5896 524 13 compactness compactness NOUN ejpam-5896 524 14 . . PUNCT ejpam-5896 525 1 abd abd PROPN ejpam-5896 525 2 el el PROPN ejpam-5896 525 3 - - PROPN ejpam-5896 525 4 latif latif PROPN ejpam-5896 525 5 et et PROPN ejpam-5896 525 6 al al PROPN ejpam-5896 525 7 . . PUNCT ejpam-5896 525 8 / / SYM ejpam-5896 525 9 eur eur PROPN ejpam-5896 525 10 . . PUNCT ejpam-5896 526 1 j. j. PROPN ejpam-5896 526 2 pure pure PROPN ejpam-5896 526 3 appl appl PROPN ejpam-5896 526 4 . . PROPN ejpam-5896 526 5 math math PROPN ejpam-5896 526 6 , , PUNCT ejpam-5896 526 7 18 18 NUM ejpam-5896 526 8 ( ( PUNCT ejpam-5896 526 9 2 2 NUM ejpam-5896 526 10 ) ) PUNCT ejpam-5896 526 11 ( ( PUNCT ejpam-5896 526 12 2025 2025 NUM ejpam-5896 526 13 ) ) PUNCT ejpam-5896 526 14 , , PUNCT ejpam-5896 526 15 5896 5896 NUM ejpam-5896 526 16 15 15 NUM ejpam-5896 526 17 of of ADP ejpam-5896 526 18 20 20 NUM ejpam-5896 526 19 corollary corollary ADJ ejpam-5896 526 20 6 6 NUM ejpam-5896 526 21 . . PUNCT ejpam-5896 527 1 the the DET ejpam-5896 527 2 image image NOUN ejpam-5896 527 3 of of ADP ejpam-5896 527 4 each each DET ejpam-5896 527 5 ss ss NOUN ejpam-5896 527 6 - - PUNCT ejpam-5896 527 7 sd sd NOUN ejpam-5896 527 8 - - PUNCT ejpam-5896 527 9 lindelöf lindelöf NOUN ejpam-5896 527 10 ( ( PUNCT ejpam-5896 527 11 compact compact ADJ ejpam-5896 527 12 ) ) PUNCT ejpam-5896 527 13 set set NOUN ejpam-5896 527 14 is be AUX ejpam-5896 527 15 ss ss NOUN ejpam-5896 527 16 - - PUNCT ejpam-5896 527 17 sd sd NOUN ejpam-5896 527 18 - - PUNCT ejpam-5896 527 19 lindelöf lindelöf NOUN ejpam-5896 527 20 ( ( PUNCT ejpam-5896 527 21 compact compact ADJ ejpam-5896 527 22 ) ) PUNCT ejpam-5896 527 23 under under ADP ejpam-5896 527 24 a a DET ejpam-5896 527 25 surjective surjective ADJ ejpam-5896 527 26 and and CCONJ ejpam-5896 527 27 ss ss NOUN ejpam-5896 527 28 - - PUNCT ejpam-5896 527 29 sd sd NOUN ejpam-5896 527 30 - - PUNCT ejpam-5896 527 31 irresolute irresolute ADJ ejpam-5896 527 32 map map NOUN ejpam-5896 527 33 . . PUNCT ejpam-5896 528 1 proof proof NOUN ejpam-5896 528 2 . . PUNCT ejpam-5896 529 1 immediate immediate ADJ ejpam-5896 529 2 form form NOUN ejpam-5896 529 3 theorem theorem VERB ejpam-5896 529 4 18 18 NUM ejpam-5896 529 5 . . PUNCT ejpam-5896 529 6 definition definition NOUN ejpam-5896 529 7 24 24 NUM ejpam-5896 529 8 . . PUNCT ejpam-5896 530 1 [ [ X ejpam-5896 530 2 57 57 NUM ejpam-5896 530 3 ] ] PUNCT ejpam-5896 530 4 a a DET ejpam-5896 530 5 soft soft ADJ ejpam-5896 530 6 mapping mapping NOUN ejpam-5896 530 7 ψsd ψsd NOUN ejpam-5896 530 8 : : PUNCT ejpam-5896 530 9 ( ( PUNCT ejpam-5896 530 10 u u NOUN ejpam-5896 530 11 , , PUNCT ejpam-5896 530 12 τ,∆ τ,∆ NOUN ejpam-5896 530 13 ) ) PUNCT ejpam-5896 530 14 → → SYM ejpam-5896 530 15 ( ( PUNCT ejpam-5896 530 16 v v NOUN ejpam-5896 530 17 , , PUNCT ejpam-5896 530 18 σ,∆ σ,∆ NUM ejpam-5896 530 19 ) ) PUNCT ejpam-5896 530 20 µ µ NOUN ejpam-5896 530 21 , , PUNCT ejpam-5896 530 22 µ∗ µ∗ VERB ejpam-5896 530 23 as as SCONJ ejpam-5896 530 24 associated associated ADJ ejpam-5896 530 25 sstss sstss NOUN ejpam-5896 530 26 with with ADP ejpam-5896 530 27 τ τ PROPN ejpam-5896 530 28 , , PUNCT ejpam-5896 530 29 σ σ PROPN ejpam-5896 530 30 , , PUNCT ejpam-5896 530 31 respectively respectively ADV ejpam-5896 530 32 , , PUNCT ejpam-5896 530 33 is be AUX ejpam-5896 530 34 claimed claim VERB ejpam-5896 530 35 to to PART ejpam-5896 530 36 be be AUX ejpam-5896 530 37 ss ss NOUN ejpam-5896 530 38 - - PUNCT ejpam-5896 530 39 sd sd NOUN ejpam-5896 530 40 - - PUNCT ejpam-5896 530 41 open open ADJ ejpam-5896 530 42 if if SCONJ ejpam-5896 530 43 ψsd(g,∆ ψsd(g,∆ ADJ ejpam-5896 530 44 ) ) PUNCT ejpam-5896 530 45 ∈ ∈ NOUN ejpam-5896 530 46 sd(v sd(v PUNCT ejpam-5896 530 47 ) ) PUNCT ejpam-5896 530 48 for for ADP ejpam-5896 530 49 each each DET ejpam-5896 530 50 non non ADJ ejpam-5896 530 51 - - ADJ ejpam-5896 530 52 null null ADJ ejpam-5896 530 53 soft soft ADJ ejpam-5896 530 54 open open ADJ ejpam-5896 530 55 subset subset NOUN ejpam-5896 530 56 ( ( PUNCT ejpam-5896 530 57 g,∆ g,∆ PROPN ejpam-5896 530 58 ) ) PUNCT ejpam-5896 530 59 of of ADP ejpam-5896 530 60 ũ ũ PROPN ejpam-5896 530 61 . . PUNCT ejpam-5896 531 1 theorem theorem PROPN ejpam-5896 531 2 19 19 NUM ejpam-5896 531 3 . . PUNCT ejpam-5896 532 1 the the DET ejpam-5896 532 2 pre pre NOUN ejpam-5896 532 3 - - NOUN ejpam-5896 532 4 image image NOUN ejpam-5896 532 5 of of ADP ejpam-5896 532 6 each each DET ejpam-5896 532 7 ss ss NOUN ejpam-5896 532 8 - - PUNCT ejpam-5896 532 9 sd sd NOUN ejpam-5896 532 10 - - PUNCT ejpam-5896 532 11 lindelöf lindelöf NOUN ejpam-5896 532 12 ( ( PUNCT ejpam-5896 532 13 compact compact ADJ ejpam-5896 532 14 ) ) PUNCT ejpam-5896 532 15 set set NOUN ejpam-5896 532 16 is be AUX ejpam-5896 532 17 ss ss NOUN ejpam-5896 532 18 - - PUNCT ejpam-5896 532 19 lindelöf lindelöf NOUN ejpam-5896 532 20 ( ( PUNCT ejpam-5896 532 21 compact compact ADJ ejpam-5896 532 22 ) ) PUNCT ejpam-5896 532 23 under under ADP ejpam-5896 532 24 an an DET ejpam-5896 532 25 injective injective ADJ ejpam-5896 532 26 and and CCONJ ejpam-5896 532 27 ss ss NOUN ejpam-5896 532 28 - - PUNCT ejpam-5896 532 29 sd sd NOUN ejpam-5896 532 30 - - PUNCT ejpam-5896 532 31 open open ADJ ejpam-5896 532 32 map map NOUN ejpam-5896 532 33 . . PUNCT ejpam-5896 533 1 proof proof NOUN ejpam-5896 533 2 . . PUNCT ejpam-5896 534 1 let let VERB ejpam-5896 534 2 ψsd ψsd VERB ejpam-5896 534 3 : : PUNCT ejpam-5896 534 4 ( ( PUNCT ejpam-5896 534 5 u u NOUN ejpam-5896 534 6 , , PUNCT ejpam-5896 534 7 τ,∆ τ,∆ NOUN ejpam-5896 534 8 ) ) PUNCT ejpam-5896 534 9 → → SYM ejpam-5896 534 10 ( ( PUNCT ejpam-5896 534 11 v v NOUN ejpam-5896 534 12 , , PUNCT ejpam-5896 534 13 σ,∆ σ,∆ NUM ejpam-5896 534 14 ) ) PUNCT ejpam-5896 534 15 be be VERB ejpam-5896 534 16 an an DET ejpam-5896 534 17 ss ss NOUN ejpam-5896 534 18 - - PUNCT ejpam-5896 534 19 sd sd NOUN ejpam-5896 534 20 - - PUNCT ejpam-5896 534 21 open open ADJ ejpam-5896 534 22 map map NOUN ejpam-5896 534 23 with with ADP ejpam-5896 534 24 µ µ NOUN ejpam-5896 534 25 , , PUNCT ejpam-5896 534 26 µ∗ µ∗ VERB ejpam-5896 534 27 as as ADP ejpam-5896 534 28 associated associated ADJ ejpam-5896 534 29 sstss sstss NOUN ejpam-5896 534 30 with with ADP ejpam-5896 534 31 τ τ PROPN ejpam-5896 534 32 , , PUNCT ejpam-5896 534 33 σ σ PROPN ejpam-5896 534 34 , , PUNCT ejpam-5896 534 35 respectively respectively ADV ejpam-5896 534 36 , , PUNCT ejpam-5896 534 37 and and CCONJ ejpam-5896 534 38 ( ( PUNCT ejpam-5896 534 39 p,∆ p,∆ ADV ejpam-5896 534 40 ) ) PUNCT ejpam-5896 534 41 is be AUX ejpam-5896 534 42 the the DET ejpam-5896 534 43 ss ss NOUN ejpam-5896 534 44 - - PUNCT ejpam-5896 534 45 sd sd NOUN ejpam-5896 534 46 - - PUNCT ejpam-5896 534 47 lindelöf lindelöf NOUN ejpam-5896 534 48 subset subset NOUN ejpam-5896 534 49 of of ADP ejpam-5896 534 50 ṽ ṽ PROPN ejpam-5896 534 51 . . PUNCT ejpam-5896 535 1 assume assume VERB ejpam-5896 535 2 that that SCONJ ejpam-5896 535 3 { { PUNCT ejpam-5896 535 4 ( ( PUNCT ejpam-5896 535 5 cϵ,∆ cϵ,∆ ADJ ejpam-5896 535 6 ) ) PUNCT ejpam-5896 535 7 : : PUNCT ejpam-5896 535 8 ϵ ϵ X ejpam-5896 535 9 ∈ ∈ PROPN ejpam-5896 535 10 ε ε PROPN ejpam-5896 535 11 } } PUNCT ejpam-5896 535 12 is be AUX ejpam-5896 535 13 an an DET ejpam-5896 535 14 ss ss VERB ejpam-5896 535 15 - - ADJ ejpam-5896 535 16 open open ADJ ejpam-5896 535 17 cover cover NOUN ejpam-5896 535 18 for for ADP ejpam-5896 535 19 ψ−1 ψ−1 PROPN ejpam-5896 535 20 sd sd ADP ejpam-5896 535 21 ( ( PUNCT ejpam-5896 535 22 p,∆ p,∆ NOUN ejpam-5896 535 23 ) ) PUNCT ejpam-5896 535 24 . . PUNCT ejpam-5896 536 1 it it PRON ejpam-5896 536 2 follows follow VERB ejpam-5896 536 3 that that SCONJ ejpam-5896 536 4 , , PUNCT ejpam-5896 536 5 ψsd(cϵ,∆ ψsd(cϵ,∆ ADJ ejpam-5896 536 6 ) ) PUNCT ejpam-5896 536 7 ∈ ∈ NOUN ejpam-5896 536 8 sd(v sd(v PUNCT ejpam-5896 536 9 ) ) PUNCT ejpam-5896 536 10 ∆ ∆ PROPN ejpam-5896 536 11 for for ADP ejpam-5896 536 12 each each DET ejpam-5896 536 13 ϵ ϵ PROPN ejpam-5896 536 14 ∈ ∈ PROPN ejpam-5896 536 15 ε ε PROPN ejpam-5896 536 16 with with ADP ejpam-5896 536 17 ( ( PUNCT ejpam-5896 536 18 p,∆)⊆̃ p,∆)⊆̃ NOUN ejpam-5896 536 19 ⋃̃ ⋃̃ PROPN ejpam-5896 536 20 ϵ∈ε[ψsd(cϵ,∆ ϵ∈ε[ψsd(cϵ,∆ X ejpam-5896 536 21 ) ) PUNCT ejpam-5896 536 22 ] ] PUNCT ejpam-5896 536 23 given give VERB ejpam-5896 536 24 ψsd ψsd PROPN ejpam-5896 536 25 is be AUX ejpam-5896 536 26 injective injective ADJ ejpam-5896 536 27 . . PUNCT ejpam-5896 537 1 since since SCONJ ejpam-5896 537 2 ( ( PUNCT ejpam-5896 537 3 p,∆ p,∆ ADV ejpam-5896 537 4 ) ) PUNCT ejpam-5896 537 5 is be AUX ejpam-5896 537 6 ss ss NOUN ejpam-5896 537 7 - - PUNCT ejpam-5896 537 8 sd sd NOUN ejpam-5896 537 9 - - PUNCT ejpam-5896 537 10 lindelöf lindelöf NOUN ejpam-5896 537 11 , , PUNCT ejpam-5896 537 12 there there PRON ejpam-5896 537 13 is be VERB ejpam-5896 537 14 a a DET ejpam-5896 537 15 countable countable ADJ ejpam-5896 537 16 subclasses subclass NOUN ejpam-5896 537 17 εo εo NOUN ejpam-5896 537 18 of of ADP ejpam-5896 537 19 ε ε PROPN ejpam-5896 537 20 such such ADJ ejpam-5896 537 21 that that SCONJ ejpam-5896 537 22 ( ( PUNCT ejpam-5896 537 23 p,∆)⊆̃ p,∆)⊆̃ NOUN ejpam-5896 537 24 ⋃̃ ⋃̃ PROPN ejpam-5896 537 25 ϵ∈εo ϵ∈εo PROPN ejpam-5896 537 26 [ [ X ejpam-5896 537 27 ψsd(cϵ,∆ ψsd(cϵ,∆ ADJ ejpam-5896 537 28 ) ) PUNCT ejpam-5896 537 29 ] ] PUNCT ejpam-5896 537 30 which which PRON ejpam-5896 537 31 follows follow VERB ejpam-5896 537 32 ψ−1 ψ−1 PROPN ejpam-5896 537 33 sd sd ADP ejpam-5896 537 34 ( ( PUNCT ejpam-5896 537 35 p,∆)⊆̃ p,∆)⊆̃ NOUN ejpam-5896 537 36 ⋃̃ ⋃̃ PROPN ejpam-5896 537 37 ϵ∈εoψ ϵ∈εoψ PROPN ejpam-5896 537 38 −1 −1 NOUN ejpam-5896 537 39 sd sd ADP ejpam-5896 537 40 [ [ X ejpam-5896 537 41 ψsd(cϵ,∆ ψsd(cϵ,∆ X ejpam-5896 537 42 ) ) PUNCT ejpam-5896 537 43 ] ] PUNCT ejpam-5896 538 1 = = SYM ejpam-5896 538 2 ⋃̃ ⋃̃ PROPN ejpam-5896 538 3 ϵ∈εo(cϵ,∆ ϵ∈εo(cϵ,∆ NUM ejpam-5896 538 4 ) ) PUNCT ejpam-5896 538 5 . . PUNCT ejpam-5896 539 1 therefore therefore ADV ejpam-5896 539 2 , , PUNCT ejpam-5896 539 3 ψ−1 ψ−1 PROPN ejpam-5896 539 4 sd sd ADP ejpam-5896 539 5 ( ( PUNCT ejpam-5896 539 6 p,∆ p,∆ ADV ejpam-5896 539 7 ) ) PUNCT ejpam-5896 539 8 is be AUX ejpam-5896 539 9 an an DET ejpam-5896 539 10 ss ss NOUN ejpam-5896 539 11 - - NOUN ejpam-5896 539 12 lindelöf lindelöf NOUN ejpam-5896 539 13 . . PUNCT ejpam-5896 540 1 similarly similarly ADV ejpam-5896 540 2 , , PUNCT ejpam-5896 540 3 the the DET ejpam-5896 540 4 proof proof NOUN ejpam-5896 540 5 of of ADP ejpam-5896 540 6 ss ss NOUN ejpam-5896 540 7 - - PUNCT ejpam-5896 540 8 sd sd NOUN ejpam-5896 540 9 - - PUNCT ejpam-5896 540 10 compactness compactness NOUN ejpam-5896 540 11 can can AUX ejpam-5896 540 12 obtained obtain VERB ejpam-5896 540 13 . . PUNCT ejpam-5896 541 1 corollary corollary ADJ ejpam-5896 541 2 7 7 NUM ejpam-5896 541 3 . . PUNCT ejpam-5896 542 1 the the DET ejpam-5896 542 2 pre pre NOUN ejpam-5896 542 3 - - NOUN ejpam-5896 542 4 image image NOUN ejpam-5896 542 5 of of ADP ejpam-5896 542 6 each each DET ejpam-5896 542 7 ss ss NOUN ejpam-5896 542 8 - - PUNCT ejpam-5896 542 9 sd sd NOUN ejpam-5896 542 10 - - PUNCT ejpam-5896 542 11 lindelöf lindelöf NOUN ejpam-5896 542 12 ( ( PUNCT ejpam-5896 542 13 compact compact ADJ ejpam-5896 542 14 ) ) PUNCT ejpam-5896 542 15 set set NOUN ejpam-5896 542 16 is be AUX ejpam-5896 542 17 ss ss NOUN ejpam-5896 542 18 - - PUNCT ejpam-5896 542 19 sd sd NOUN ejpam-5896 542 20 - - PUNCT ejpam-5896 542 21 lindelöf lindelöf NOUN ejpam-5896 542 22 ( ( PUNCT ejpam-5896 542 23 compact compact ADJ ejpam-5896 542 24 ) ) PUNCT ejpam-5896 542 25 under under ADP ejpam-5896 542 26 an an DET ejpam-5896 542 27 injective injective ADJ ejpam-5896 542 28 and and CCONJ ejpam-5896 542 29 ss∗-sd ss∗-sd ADJ ejpam-5896 542 30 - - ADJ ejpam-5896 542 31 open open ADJ ejpam-5896 542 32 map map NOUN ejpam-5896 542 33 . . PUNCT ejpam-5896 543 1 proof proof NOUN ejpam-5896 543 2 . . PUNCT ejpam-5896 544 1 immediate immediate ADJ ejpam-5896 544 2 form form NOUN ejpam-5896 544 3 theorem theorem VERB ejpam-5896 544 4 19 19 NUM ejpam-5896 544 5 . . PUNCT ejpam-5896 544 6 theorem theorem NOUN ejpam-5896 544 7 20 20 NUM ejpam-5896 544 8 . . PUNCT ejpam-5896 545 1 let let VERB ejpam-5896 545 2 ( ( PUNCT ejpam-5896 545 3 u u NOUN ejpam-5896 545 4 , , PUNCT ejpam-5896 545 5 µ,∆ µ,∆ NUM ejpam-5896 545 6 ) ) PUNCT ejpam-5896 545 7 is be AUX ejpam-5896 545 8 ssts sst NOUN ejpam-5896 545 9 defined define VERB ejpam-5896 545 10 on on ADP ejpam-5896 545 11 any any DET ejpam-5896 545 12 universal universal ADJ ejpam-5896 545 13 set set VERB ejpam-5896 545 14 u u NOUN ejpam-5896 545 15 and and CCONJ ejpam-5896 545 16 finite finite ADJ ejpam-5896 545 17 set set NOUN ejpam-5896 545 18 of of ADP ejpam-5896 545 19 parameters parameter NOUN ejpam-5896 546 1 ∆. ∆. X ejpam-5896 546 2 then then ADV ejpam-5896 546 3 , , PUNCT ejpam-5896 546 4 ( ( PUNCT ejpam-5896 546 5 u u NOUN ejpam-5896 546 6 , , PUNCT ejpam-5896 546 7 µ,∆ µ,∆ NUM ejpam-5896 546 8 ) ) PUNCT ejpam-5896 546 9 is be AUX ejpam-5896 546 10 ss ss NOUN ejpam-5896 546 11 - - PUNCT ejpam-5896 546 12 sd sd NOUN ejpam-5896 546 13 - - PUNCT ejpam-5896 546 14 compact compact ADJ ejpam-5896 546 15 ( ( PUNCT ejpam-5896 546 16 lindelöf lindelöf PROPN ejpam-5896 546 17 ) ) PUNCT ejpam-5896 546 18 if if SCONJ ejpam-5896 546 19 every every DET ejpam-5896 546 20 γ γ PROPN ejpam-5896 546 21 - - PUNCT ejpam-5896 546 22 parameter parameter NOUN ejpam-5896 546 23 supra supra PROPN ejpam-5896 546 24 topological topological ADJ ejpam-5896 546 25 space space NOUN ejpam-5896 546 26 is be AUX ejpam-5896 546 27 supra supra ADJ ejpam-5896 546 28 - - PUNCT ejpam-5896 546 29 sd sd NOUN ejpam-5896 546 30 - - PUNCT ejpam-5896 546 31 compact compact ADJ ejpam-5896 546 32 ( ( PUNCT ejpam-5896 546 33 lindelöf lindelöf PROPN ejpam-5896 546 34 ) ) PUNCT ejpam-5896 546 35 , , PUNCT ejpam-5896 546 36 for for ADP ejpam-5896 546 37 each each DET ejpam-5896 546 38 γ γ X ejpam-5896 546 39 ∈ ∈ PROPN ejpam-5896 546 40 ∆. ∆. NOUN ejpam-5896 546 41 proof proof NOUN ejpam-5896 546 42 . . PUNCT ejpam-5896 547 1 we we PRON ejpam-5896 547 2 prove prove VERB ejpam-5896 547 3 the the DET ejpam-5896 547 4 case case NOUN ejpam-5896 547 5 of of ADP ejpam-5896 547 6 supra supra NOUN ejpam-5896 547 7 - - PUNCT ejpam-5896 547 8 sd sd NOUN ejpam-5896 547 9 - - PUNCT ejpam-5896 547 10 compact compact ADJ ejpam-5896 547 11 , , PUNCT ejpam-5896 547 12 the the DET ejpam-5896 547 13 other other ADJ ejpam-5896 547 14 case case NOUN ejpam-5896 547 15 is be AUX ejpam-5896 547 16 similar similar ADJ ejpam-5896 547 17 . . PUNCT ejpam-5896 548 1 let let VERB ejpam-5896 548 2 ( ( PUNCT ejpam-5896 548 3 u u NOUN ejpam-5896 548 4 , , PUNCT ejpam-5896 548 5 µ,∆ µ,∆ NUM ejpam-5896 548 6 ) ) PUNCT ejpam-5896 548 7 is be AUX ejpam-5896 548 8 ssts sst NOUN ejpam-5896 548 9 over over ADP ejpam-5896 548 10 the the DET ejpam-5896 548 11 set set NOUN ejpam-5896 548 12 of of ADP ejpam-5896 548 13 parameter parameter NOUN ejpam-5896 548 14 ∆ ∆ PROPN ejpam-5896 548 15 = = PRON ejpam-5896 548 16 { { PUNCT ejpam-5896 548 17 γ1 γ1 PROPN ejpam-5896 548 18 , , PUNCT ejpam-5896 548 19 γ2 γ2 PROPN ejpam-5896 548 20 , , PUNCT ejpam-5896 548 21 γ3 γ3 NOUN ejpam-5896 548 22 , , PUNCT ejpam-5896 548 23 .... .... PUNCT ejpam-5896 548 24 , , PUNCT ejpam-5896 548 25 γn γn NUM ejpam-5896 548 26 , , PUNCT ejpam-5896 548 27 n n PROPN ejpam-5896 548 28 ∈ ∈ PROPN ejpam-5896 548 29 n n CCONJ ejpam-5896 548 30 } } PUNCT ejpam-5896 548 31 and and CCONJ ejpam-5896 548 32 the the DET ejpam-5896 548 33 universal universal ADJ ejpam-5896 548 34 set set VERB ejpam-5896 548 35 u u PROPN ejpam-5896 548 36 . . PUNCT ejpam-5896 549 1 assume assume VERB ejpam-5896 549 2 that that SCONJ ejpam-5896 549 3 ( ( PUNCT ejpam-5896 549 4 u u NOUN ejpam-5896 549 5 , , PUNCT ejpam-5896 549 6 µγi µγi PROPN ejpam-5896 549 7 ) ) PUNCT ejpam-5896 549 8 is be AUX ejpam-5896 549 9 the the DET ejpam-5896 549 10 γi γi ADJ ejpam-5896 549 11 - - PUNCT ejpam-5896 549 12 parametric parametric ADJ ejpam-5896 549 13 supra supra PROPN ejpam-5896 549 14 topological topological ADJ ejpam-5896 549 15 spaces space NOUN ejpam-5896 549 16 such such ADJ ejpam-5896 549 17 that that SCONJ ejpam-5896 549 18 ( ( PUNCT ejpam-5896 549 19 u u NOUN ejpam-5896 549 20 , , PUNCT ejpam-5896 549 21 µγi µγi PROPN ejpam-5896 549 22 ) ) PUNCT ejpam-5896 549 23 , , PUNCT ejpam-5896 549 24 i i PRON ejpam-5896 549 25 = = NOUN ejpam-5896 549 26 1 1 NUM ejpam-5896 549 27 , , PUNCT ejpam-5896 549 28 2 2 NUM ejpam-5896 549 29 , , PUNCT ejpam-5896 549 30 3 3 NUM ejpam-5896 549 31 , , PUNCT ejpam-5896 549 32 ... ... PUNCT ejpam-5896 549 33 , , PUNCT ejpam-5896 549 34 n n PRON ejpam-5896 549 35 is be AUX ejpam-5896 549 36 a a DET ejpam-5896 549 37 supra supra NOUN ejpam-5896 549 38 - - PUNCT ejpam-5896 549 39 sd sd NOUN ejpam-5896 549 40 - - PUNCT ejpam-5896 549 41 compact compact ADJ ejpam-5896 549 42 , , PUNCT ejpam-5896 549 43 for for ADP ejpam-5896 549 44 each each DET ejpam-5896 549 45 γ γ X ejpam-5896 549 46 ∈ ∈ NOUN ejpam-5896 549 47 ∆ ∆ PROPN ejpam-5896 549 48 and and CCONJ ejpam-5896 549 49 ψ ψ X ejpam-5896 549 50 = = X ejpam-5896 549 51 { { PUNCT ejpam-5896 549 52 ( ( PUNCT ejpam-5896 549 53 cϵ,∆ cϵ,∆ ADJ ejpam-5896 549 54 ) ) PUNCT ejpam-5896 549 55 : : PUNCT ejpam-5896 550 1 ϵ ϵ X ejpam-5896 550 2 ∈ ∈ PROPN ejpam-5896 550 3 ε ε PROPN ejpam-5896 550 4 } } PUNCT ejpam-5896 550 5 is be AUX ejpam-5896 550 6 ss ss NOUN ejpam-5896 550 7 - - PUNCT ejpam-5896 550 8 sd sd NOUN ejpam-5896 550 9 - - PUNCT ejpam-5896 550 10 cover cover NOUN ejpam-5896 550 11 of of ADP ejpam-5896 550 12 ũ ũ PROPN ejpam-5896 550 13 . . PUNCT ejpam-5896 551 1 since since SCONJ ejpam-5896 551 2 u u NOUN ejpam-5896 551 3 = = PROPN ejpam-5896 551 4 ⋃ ⋃ PROPN ejpam-5896 551 5 ϵ∈ε(cϵ,∆)(γi ϵ∈ε(cϵ,∆)(γi NOUN ejpam-5896 551 6 ) ) PUNCT ejpam-5896 551 7 for for ADP ejpam-5896 551 8 each each DET ejpam-5896 551 9 γi γi X ejpam-5896 551 10 ∈ ∈ PROPN ejpam-5896 551 11 ∆ ∆ PROPN ejpam-5896 551 12 and and CCONJ ejpam-5896 551 13 ( ( PUNCT ejpam-5896 551 14 u u NOUN ejpam-5896 551 15 , , PUNCT ejpam-5896 551 16 µγi µγi ADV ejpam-5896 551 17 ) ) PUNCT ejpam-5896 551 18 is be AUX ejpam-5896 551 19 supra supra ADJ ejpam-5896 551 20 - - PUNCT ejpam-5896 551 21 sd sd NOUN ejpam-5896 551 22 - - PUNCT ejpam-5896 551 23 compact compact ADJ ejpam-5896 551 24 , , PUNCT ejpam-5896 551 25 there there PRON ejpam-5896 551 26 is be VERB ejpam-5896 551 27 a a DET ejpam-5896 551 28 finite finite ADJ ejpam-5896 551 29 subclasses subclass NOUN ejpam-5896 551 30 εi εi VERB ejpam-5896 551 31 of of ADP ejpam-5896 551 32 ε ε PROPN ejpam-5896 551 33 such such ADJ ejpam-5896 551 34 that that SCONJ ejpam-5896 551 35 u u NOUN ejpam-5896 551 36 = = PUNCT ejpam-5896 551 37 ⋃ ⋃ PROPN ejpam-5896 551 38 ϵ∈εi(cϵ,∆)(γi ϵ∈εi(cϵ,∆)(γi NOUN ejpam-5896 551 39 ) ) PUNCT ejpam-5896 551 40 . . PUNCT ejpam-5896 552 1 hence hence ADV ejpam-5896 552 2 , , PUNCT ejpam-5896 552 3 ũ ũ PROPN ejpam-5896 552 4 = = PROPN ejpam-5896 552 5 ⋃̃n ⋃̃n NOUN ejpam-5896 552 6 i=1 i=1 PROPN ejpam-5896 552 7 ⋃̃ ⋃̃ PROPN ejpam-5896 552 8 ϵ∈εi(cϵ,∆ ϵ∈εi(cϵ,∆ PROPN ejpam-5896 552 9 ) ) PUNCT ejpam-5896 552 10 . . PUNCT ejpam-5896 553 1 that that PRON ejpam-5896 553 2 ’s ’ VERB ejpam-5896 553 3 is be AUX ejpam-5896 553 4 , , PUNCT ejpam-5896 553 5 { { PUNCT ejpam-5896 553 6 ( ( PUNCT ejpam-5896 553 7 cϵ,∆ cϵ,∆ ADJ ejpam-5896 553 8 ) ) PUNCT ejpam-5896 553 9 : : PUNCT ejpam-5896 554 1 ϵ ϵ X ejpam-5896 554 2 ∈ ∈ PROPN ejpam-5896 554 3 ∪n ∪n X ejpam-5896 554 4 i=1εi i=1εi X ejpam-5896 554 5 } } PUNCT ejpam-5896 554 6 is be AUX ejpam-5896 554 7 a a DET ejpam-5896 554 8 finite finite ADJ ejpam-5896 554 9 subcover subcover NOUN ejpam-5896 554 10 of of ADP ejpam-5896 554 11 ψ ψ PROPN ejpam-5896 554 12 which which PRON ejpam-5896 554 13 cover cover VERB ejpam-5896 554 14 ũ ũ PROPN ejpam-5896 554 15 . . PUNCT ejpam-5896 555 1 thus thus ADV ejpam-5896 555 2 , , PUNCT ejpam-5896 555 3 ũ ũ PROPN ejpam-5896 555 4 is be AUX ejpam-5896 555 5 ss ss NOUN ejpam-5896 555 6 - - PUNCT ejpam-5896 555 7 sdcompact sdcompact NOUN ejpam-5896 555 8 . . PUNCT ejpam-5896 556 1 remark remark NOUN ejpam-5896 556 2 5 5 NUM ejpam-5896 556 3 . . PUNCT ejpam-5896 557 1 the the DET ejpam-5896 557 2 following following ADJ ejpam-5896 557 3 example example NOUN ejpam-5896 557 4 shall shall AUX ejpam-5896 557 5 show show VERB ejpam-5896 557 6 that that SCONJ ejpam-5896 557 7 the the DET ejpam-5896 557 8 converse converse NOUN ejpam-5896 557 9 of of ADP ejpam-5896 557 10 theorem theorem NOUN ejpam-5896 557 11 20 20 NUM ejpam-5896 557 12 is be AUX ejpam-5896 557 13 not not PART ejpam-5896 557 14 necessarily necessarily ADV ejpam-5896 557 15 satisfied satisfied ADJ ejpam-5896 557 16 in in ADP ejpam-5896 557 17 general general ADJ ejpam-5896 557 18 . . PUNCT ejpam-5896 558 1 abd abd PROPN ejpam-5896 558 2 el el PROPN ejpam-5896 558 3 - - PROPN ejpam-5896 558 4 latif latif PROPN ejpam-5896 558 5 et et PROPN ejpam-5896 558 6 al al PROPN ejpam-5896 558 7 . . PUNCT ejpam-5896 558 8 / / SYM ejpam-5896 558 9 eur eur PROPN ejpam-5896 558 10 . . PUNCT ejpam-5896 559 1 j. j. PROPN ejpam-5896 559 2 pure pure PROPN ejpam-5896 559 3 appl appl PROPN ejpam-5896 559 4 . . PROPN ejpam-5896 559 5 math math PROPN ejpam-5896 559 6 , , PUNCT ejpam-5896 559 7 18 18 NUM ejpam-5896 559 8 ( ( PUNCT ejpam-5896 559 9 2 2 NUM ejpam-5896 559 10 ) ) PUNCT ejpam-5896 559 11 ( ( PUNCT ejpam-5896 559 12 2025 2025 NUM ejpam-5896 559 13 ) ) PUNCT ejpam-5896 559 14 , , PUNCT ejpam-5896 559 15 5896 5896 NUM ejpam-5896 559 16 16 16 NUM ejpam-5896 559 17 of of ADP ejpam-5896 559 18 20 20 NUM ejpam-5896 559 19 example example NOUN ejpam-5896 559 20 5 5 NUM ejpam-5896 559 21 . . PUNCT ejpam-5896 560 1 let let VERB ejpam-5896 560 2 the the DET ejpam-5896 560 3 set set NOUN ejpam-5896 560 4 of of ADP ejpam-5896 560 5 natural natural ADJ ejpam-5896 560 6 numbers number NOUN ejpam-5896 560 7 n n PRON ejpam-5896 560 8 be be VERB ejpam-5896 560 9 the the DET ejpam-5896 560 10 universal universal ADJ ejpam-5896 560 11 set set NOUN ejpam-5896 560 12 and and CCONJ ejpam-5896 560 13 ∆ ∆ PROPN ejpam-5896 560 14 = = SYM ejpam-5896 560 15 { { PUNCT ejpam-5896 560 16 γ1 γ1 PROPN ejpam-5896 560 17 , , PUNCT ejpam-5896 560 18 γ2 γ2 PROPN ejpam-5896 560 19 } } PUNCT ejpam-5896 560 20 . . PUNCT ejpam-5896 561 1 consider consider VERB ejpam-5896 561 2 the the DET ejpam-5896 561 3 classes class NOUN ejpam-5896 561 4 of of ADP ejpam-5896 561 5 soft soft ADJ ejpam-5896 561 6 sets set NOUN ejpam-5896 561 7 { { PUNCT ejpam-5896 561 8 ( ( PUNCT ejpam-5896 561 9 an,∆ an,∆ PROPN ejpam-5896 561 10 ) ) PUNCT ejpam-5896 561 11 : : PUNCT ejpam-5896 562 1 n n X ejpam-5896 562 2 ∈ ∈ PROPN ejpam-5896 562 3 n n CCONJ ejpam-5896 562 4 } } PUNCT ejpam-5896 562 5 and and CCONJ ejpam-5896 562 6 { { PUNCT ejpam-5896 562 7 ( ( PUNCT ejpam-5896 562 8 bn,∆ bn,∆ NOUN ejpam-5896 562 9 ) ) PUNCT ejpam-5896 562 10 : : PUNCT ejpam-5896 563 1 n n X ejpam-5896 563 2 ∈ ∈ PROPN ejpam-5896 563 3 n n CCONJ ejpam-5896 563 4 } } PUNCT ejpam-5896 563 5 , , PUNCT ejpam-5896 563 6 where where SCONJ ejpam-5896 563 7 ( ( PUNCT ejpam-5896 563 8 an,∆ an,∆ PROPN ejpam-5896 563 9 ) ) PUNCT ejpam-5896 563 10 = = PRON ejpam-5896 563 11 { { PUNCT ejpam-5896 563 12 { { PUNCT ejpam-5896 563 13 ( ( PUNCT ejpam-5896 563 14 γ1 γ1 PROPN ejpam-5896 563 15 , , PUNCT ejpam-5896 563 16 { { PUNCT ejpam-5896 563 17 2 2 NUM ejpam-5896 563 18 } } PUNCT ejpam-5896 563 19 ) ) PUNCT ejpam-5896 563 20 , , PUNCT ejpam-5896 563 21 ( ( PUNCT ejpam-5896 563 22 γ2 γ2 ADJ ejpam-5896 563 23 , , PUNCT ejpam-5896 563 24 { { PUNCT ejpam-5896 563 25 2 2 NUM ejpam-5896 563 26 } } PUNCT ejpam-5896 563 27 ) ) PUNCT ejpam-5896 563 28 } } PUNCT ejpam-5896 563 29 , , PUNCT ejpam-5896 563 30 { { PUNCT ejpam-5896 563 31 ( ( PUNCT ejpam-5896 563 32 γ1 γ1 PROPN ejpam-5896 563 33 , , PUNCT ejpam-5896 563 34 { { PUNCT ejpam-5896 563 35 1 1 NUM ejpam-5896 563 36 } } PUNCT ejpam-5896 563 37 ) ) PUNCT ejpam-5896 563 38 , , PUNCT ejpam-5896 563 39 ( ( PUNCT ejpam-5896 563 40 γ2 γ2 PROPN ejpam-5896 563 41 , , PUNCT ejpam-5896 563 42 φ φ NOUN ejpam-5896 563 43 ) ) PUNCT ejpam-5896 563 44 } } PUNCT ejpam-5896 563 45 , , PUNCT ejpam-5896 563 46 { { PUNCT ejpam-5896 563 47 ( ( PUNCT ejpam-5896 563 48 γ1 γ1 PROPN ejpam-5896 563 49 , , PUNCT ejpam-5896 563 50 { { PUNCT ejpam-5896 563 51 1 1 NUM ejpam-5896 563 52 , , PUNCT ejpam-5896 563 53 2 2 NUM ejpam-5896 563 54 } } PUNCT ejpam-5896 563 55 ) ) PUNCT ejpam-5896 563 56 , , PUNCT ejpam-5896 563 57 ( ( PUNCT ejpam-5896 563 58 γ2 γ2 PROPN ejpam-5896 563 59 , , PUNCT ejpam-5896 563 60 φ φ NOUN ejpam-5896 563 61 ) ) PUNCT ejpam-5896 563 62 } } PUNCT ejpam-5896 563 63 , , PUNCT ejpam-5896 563 64 { { PUNCT ejpam-5896 563 65 ( ( PUNCT ejpam-5896 563 66 γ1 γ1 PROPN ejpam-5896 563 67 , , PUNCT ejpam-5896 563 68 { { PUNCT ejpam-5896 563 69 1 1 NUM ejpam-5896 563 70 , , PUNCT ejpam-5896 563 71 2 2 NUM ejpam-5896 563 72 , , PUNCT ejpam-5896 563 73 3 3 NUM ejpam-5896 563 74 } } PUNCT ejpam-5896 563 75 ) ) PUNCT ejpam-5896 563 76 , , PUNCT ejpam-5896 563 77 ( ( PUNCT ejpam-5896 563 78 γ2 γ2 PROPN ejpam-5896 563 79 , , PUNCT ejpam-5896 563 80 φ φ NOUN ejpam-5896 563 81 ) ) PUNCT ejpam-5896 563 82 } } PUNCT ejpam-5896 563 83 , , PUNCT ejpam-5896 563 84 { { PUNCT ejpam-5896 563 85 ( ( PUNCT ejpam-5896 563 86 γ1 γ1 PROPN ejpam-5896 563 87 , , PUNCT ejpam-5896 563 88 { { PUNCT ejpam-5896 563 89 1 1 NUM ejpam-5896 563 90 , , PUNCT ejpam-5896 563 91 2 2 NUM ejpam-5896 563 92 , , PUNCT ejpam-5896 563 93 3 3 NUM ejpam-5896 563 94 , , PUNCT ejpam-5896 563 95 4 4 NUM ejpam-5896 563 96 } } PUNCT ejpam-5896 563 97 ) ) PUNCT ejpam-5896 563 98 , , PUNCT ejpam-5896 563 99 ( ( PUNCT ejpam-5896 563 100 γ2 γ2 PROPN ejpam-5896 563 101 , , PUNCT ejpam-5896 563 102 φ φ NOUN ejpam-5896 563 103 ) ) PUNCT ejpam-5896 563 104 } } PUNCT ejpam-5896 563 105 , , PUNCT ejpam-5896 563 106 ...... ...... PUNCT ejpam-5896 563 107 , , PUNCT ejpam-5896 563 108 { { PUNCT ejpam-5896 563 109 ( ( PUNCT ejpam-5896 563 110 γ1 γ1 PROPN ejpam-5896 563 111 , , PUNCT ejpam-5896 563 112 { { PUNCT ejpam-5896 563 113 1 1 NUM ejpam-5896 563 114 , , PUNCT ejpam-5896 563 115 2 2 NUM ejpam-5896 563 116 , , PUNCT ejpam-5896 563 117 3 3 NUM ejpam-5896 563 118 , , PUNCT ejpam-5896 563 119 4 4 NUM ejpam-5896 563 120 , , PUNCT ejpam-5896 563 121 ..... ..... PUNCT ejpam-5896 563 122 , , PUNCT ejpam-5896 563 123 n n CCONJ ejpam-5896 563 124 } } PUNCT ejpam-5896 563 125 ) ) PUNCT ejpam-5896 563 126 , , PUNCT ejpam-5896 563 127 ( ( PUNCT ejpam-5896 563 128 γ2 γ2 PROPN ejpam-5896 563 129 , , PUNCT ejpam-5896 563 130 φ φ NOUN ejpam-5896 563 131 ) ) PUNCT ejpam-5896 563 132 } } PUNCT ejpam-5896 563 133 } } PUNCT ejpam-5896 563 134 . . PUNCT ejpam-5896 564 1 ( ( PUNCT ejpam-5896 564 2 bn,∆ bn,∆ NOUN ejpam-5896 564 3 ) ) PUNCT ejpam-5896 564 4 = = PRON ejpam-5896 564 5 { { PUNCT ejpam-5896 564 6 { { PUNCT ejpam-5896 564 7 ( ( PUNCT ejpam-5896 564 8 γ1 γ1 PROPN ejpam-5896 564 9 , , PUNCT ejpam-5896 564 10 { { PUNCT ejpam-5896 564 11 1 1 NUM ejpam-5896 564 12 , , PUNCT ejpam-5896 564 13 2 2 NUM ejpam-5896 564 14 } } PUNCT ejpam-5896 564 15 ) ) PUNCT ejpam-5896 564 16 , , PUNCT ejpam-5896 564 17 ( ( PUNCT ejpam-5896 564 18 γ2 γ2 ADJ ejpam-5896 564 19 , , PUNCT ejpam-5896 564 20 { { PUNCT ejpam-5896 564 21 2 2 NUM ejpam-5896 564 22 } } PUNCT ejpam-5896 564 23 ) ) PUNCT ejpam-5896 564 24 } } PUNCT ejpam-5896 564 25 , , PUNCT ejpam-5896 564 26 { { PUNCT ejpam-5896 564 27 ( ( PUNCT ejpam-5896 564 28 γ1 γ1 PROPN ejpam-5896 564 29 , , PUNCT ejpam-5896 564 30 { { PUNCT ejpam-5896 564 31 1 1 NUM ejpam-5896 564 32 , , PUNCT ejpam-5896 564 33 2 2 NUM ejpam-5896 564 34 , , PUNCT ejpam-5896 564 35 3 3 NUM ejpam-5896 564 36 } } PUNCT ejpam-5896 564 37 ) ) PUNCT ejpam-5896 564 38 , , PUNCT ejpam-5896 564 39 ( ( PUNCT ejpam-5896 564 40 γ2 γ2 ADJ ejpam-5896 564 41 , , PUNCT ejpam-5896 564 42 { { PUNCT ejpam-5896 564 43 2 2 NUM ejpam-5896 564 44 } } PUNCT ejpam-5896 564 45 ) ) PUNCT ejpam-5896 564 46 } } PUNCT ejpam-5896 564 47 , , PUNCT ejpam-5896 564 48 { { PUNCT ejpam-5896 564 49 ( ( PUNCT ejpam-5896 564 50 γ1 γ1 PROPN ejpam-5896 564 51 , , PUNCT ejpam-5896 564 52 { { PUNCT ejpam-5896 564 53 1 1 NUM ejpam-5896 564 54 , , PUNCT ejpam-5896 564 55 2 2 NUM ejpam-5896 564 56 , , PUNCT ejpam-5896 564 57 3 3 NUM ejpam-5896 564 58 , , PUNCT ejpam-5896 564 59 4 4 NUM ejpam-5896 564 60 } } PUNCT ejpam-5896 564 61 ) ) PUNCT ejpam-5896 564 62 , , PUNCT ejpam-5896 564 63 ( ( PUNCT ejpam-5896 564 64 γ2 γ2 ADJ ejpam-5896 564 65 , , PUNCT ejpam-5896 564 66 { { PUNCT ejpam-5896 564 67 2 2 NUM ejpam-5896 564 68 } } PUNCT ejpam-5896 564 69 ) ) PUNCT ejpam-5896 564 70 } } PUNCT ejpam-5896 564 71 , , PUNCT ejpam-5896 564 72 ........... ........... PUNCT ejpam-5896 564 73 , , PUNCT ejpam-5896 564 74 { { PUNCT ejpam-5896 564 75 ( ( PUNCT ejpam-5896 564 76 γ1 γ1 PROPN ejpam-5896 564 77 , , PUNCT ejpam-5896 564 78 { { PUNCT ejpam-5896 564 79 1 1 NUM ejpam-5896 564 80 , , PUNCT ejpam-5896 564 81 2 2 NUM ejpam-5896 564 82 , , PUNCT ejpam-5896 564 83 3 3 NUM ejpam-5896 564 84 , , PUNCT ejpam-5896 564 85 4 4 NUM ejpam-5896 564 86 , , PUNCT ejpam-5896 564 87 ..... ..... PUNCT ejpam-5896 564 88 , , PUNCT ejpam-5896 564 89 n n CCONJ ejpam-5896 564 90 } } PUNCT ejpam-5896 564 91 ) ) PUNCT ejpam-5896 564 92 , , PUNCT ejpam-5896 564 93 ( ( PUNCT ejpam-5896 564 94 γ2 γ2 ADJ ejpam-5896 564 95 , , PUNCT ejpam-5896 564 96 { { PUNCT ejpam-5896 564 97 2 2 NUM ejpam-5896 564 98 } } PUNCT ejpam-5896 564 99 ) ) PUNCT ejpam-5896 564 100 } } PUNCT ejpam-5896 564 101 } } PUNCT ejpam-5896 564 102 . . PUNCT ejpam-5896 565 1 then then ADV ejpam-5896 565 2 , , PUNCT ejpam-5896 565 3 µ µ X ejpam-5896 565 4 = = SYM ejpam-5896 565 5 { { PUNCT ejpam-5896 565 6 ñ ñ VERB ejpam-5896 565 7 , , PUNCT ejpam-5896 565 8 φ̃ φ̃ PROPN ejpam-5896 565 9 } } PUNCT ejpam-5896 565 10 ∪ ∪ X ejpam-5896 565 11 { { PUNCT ejpam-5896 565 12 ( ( PUNCT ejpam-5896 565 13 an,∆ an,∆ PROPN ejpam-5896 565 14 ) ) PUNCT ejpam-5896 565 15 : : PUNCT ejpam-5896 565 16 n n X ejpam-5896 565 17 ∈ ∈ PROPN ejpam-5896 565 18 n n CCONJ ejpam-5896 565 19 } } PUNCT ejpam-5896 565 20 ∪ ∪ X ejpam-5896 565 21 { { PUNCT ejpam-5896 565 22 ( ( PUNCT ejpam-5896 565 23 bn,∆ bn,∆ NOUN ejpam-5896 565 24 ) ) PUNCT ejpam-5896 565 25 : : PUNCT ejpam-5896 566 1 n n X ejpam-5896 566 2 ∈ ∈ PROPN ejpam-5896 566 3 n n CCONJ ejpam-5896 566 4 } } PUNCT ejpam-5896 566 5 defines define VERB ejpam-5896 566 6 an an DET ejpam-5896 566 7 ssts sst NOUN ejpam-5896 566 8 on on ADP ejpam-5896 566 9 n. n. NOUN ejpam-5896 566 10 it it PRON ejpam-5896 566 11 is be AUX ejpam-5896 566 12 easy easy ADJ ejpam-5896 566 13 to to PART ejpam-5896 566 14 check check VERB ejpam-5896 566 15 that that SCONJ ejpam-5896 566 16 ñ ñ PROPN ejpam-5896 566 17 is be AUX ejpam-5896 566 18 an an DET ejpam-5896 566 19 ss ss VERB ejpam-5896 566 20 - - PUNCT ejpam-5896 566 21 sd sd NOUN ejpam-5896 566 22 - - PUNCT ejpam-5896 566 23 compact compact ADJ ejpam-5896 566 24 ( ( PUNCT ejpam-5896 566 25 lindelöf lindelöf PROPN ejpam-5896 566 26 ) ) PUNCT ejpam-5896 566 27 . . PUNCT ejpam-5896 567 1 on on ADP ejpam-5896 567 2 the the DET ejpam-5896 567 3 other other ADJ ejpam-5896 567 4 hand hand NOUN ejpam-5896 567 5 , , PUNCT ejpam-5896 567 6 the the DET ejpam-5896 567 7 γ1 γ1 NOUN ejpam-5896 567 8 - - PUNCT ejpam-5896 567 9 parametric parametric ADJ ejpam-5896 567 10 supra supra PROPN ejpam-5896 567 11 topological topological ADJ ejpam-5896 567 12 space space NOUN ejpam-5896 567 13 ( ( PUNCT ejpam-5896 567 14 u u NOUN ejpam-5896 567 15 , , PUNCT ejpam-5896 567 16 µγ1 µγ1 NOUN ejpam-5896 567 17 ) ) PUNCT ejpam-5896 567 18 , , PUNCT ejpam-5896 567 19 where where SCONJ ejpam-5896 567 20 ( ( PUNCT ejpam-5896 567 21 n,µγ1 n,µγ1 ADJ ejpam-5896 567 22 ) ) PUNCT ejpam-5896 567 23 = = PRON ejpam-5896 567 24 { { PUNCT ejpam-5896 567 25 n n CCONJ ejpam-5896 567 26 , , PUNCT ejpam-5896 567 27 φ φ PROPN ejpam-5896 567 28 , , PUNCT ejpam-5896 567 29 { { PUNCT ejpam-5896 567 30 1 1 NUM ejpam-5896 567 31 } } PUNCT ejpam-5896 567 32 , , PUNCT ejpam-5896 567 33 { { PUNCT ejpam-5896 567 34 2 2 NUM ejpam-5896 567 35 } } PUNCT ejpam-5896 567 36 , , PUNCT ejpam-5896 567 37 { { PUNCT ejpam-5896 567 38 1 1 NUM ejpam-5896 567 39 , , PUNCT ejpam-5896 567 40 2 2 NUM ejpam-5896 567 41 } } PUNCT ejpam-5896 567 42 , , PUNCT ejpam-5896 567 43 { { PUNCT ejpam-5896 567 44 1 1 NUM ejpam-5896 567 45 , , PUNCT ejpam-5896 567 46 2 2 NUM ejpam-5896 567 47 , , PUNCT ejpam-5896 567 48 3 3 NUM ejpam-5896 567 49 } } PUNCT ejpam-5896 567 50 , , PUNCT ejpam-5896 567 51 { { PUNCT ejpam-5896 567 52 1 1 NUM ejpam-5896 567 53 , , PUNCT ejpam-5896 567 54 2 2 NUM ejpam-5896 567 55 , , PUNCT ejpam-5896 567 56 3 3 NUM ejpam-5896 567 57 , , PUNCT ejpam-5896 567 58 4 4 NUM ejpam-5896 567 59 } } PUNCT ejpam-5896 567 60 , , PUNCT ejpam-5896 567 61 { { PUNCT ejpam-5896 567 62 1 1 NUM ejpam-5896 567 63 , , PUNCT ejpam-5896 567 64 2 2 NUM ejpam-5896 567 65 , , PUNCT ejpam-5896 567 66 3 3 NUM ejpam-5896 567 67 , , PUNCT ejpam-5896 567 68 4 4 NUM ejpam-5896 567 69 , , PUNCT ejpam-5896 567 70 5 5 NUM ejpam-5896 567 71 } } PUNCT ejpam-5896 567 72 , , PUNCT ejpam-5896 567 73 .......... .......... PUNCT ejpam-5896 567 74 , , PUNCT ejpam-5896 567 75 { { PUNCT ejpam-5896 567 76 1 1 NUM ejpam-5896 567 77 , , PUNCT ejpam-5896 567 78 2 2 NUM ejpam-5896 567 79 , , PUNCT ejpam-5896 567 80 3 3 NUM ejpam-5896 567 81 , , PUNCT ejpam-5896 567 82 4 4 NUM ejpam-5896 567 83 , , PUNCT ejpam-5896 567 84 5 5 NUM ejpam-5896 567 85 , , PUNCT ejpam-5896 567 86 ..... ..... PUNCT ejpam-5896 567 87 , , PUNCT ejpam-5896 567 88 n n CCONJ ejpam-5896 567 89 } } PUNCT ejpam-5896 567 90 , , PUNCT ejpam-5896 567 91 n n PROPN ejpam-5896 567 92 ∈ ∈ PROPN ejpam-5896 567 93 n n CCONJ ejpam-5896 567 94 } } PUNCT ejpam-5896 567 95 is be AUX ejpam-5896 567 96 not not PART ejpam-5896 567 97 supra supra ADJ ejpam-5896 567 98 - - PUNCT ejpam-5896 567 99 sd sd NOUN ejpam-5896 567 100 - - PUNCT ejpam-5896 567 101 compact compact ADJ ejpam-5896 567 102 ( ( PUNCT ejpam-5896 567 103 lindelöf lindelöf PROPN ejpam-5896 567 104 ) ) PUNCT ejpam-5896 567 105 . . PUNCT ejpam-5896 568 1 since since SCONJ ejpam-5896 568 2 the the DET ejpam-5896 568 3 class class NOUN ejpam-5896 568 4 ψ ψ NOUN ejpam-5896 568 5 = = X ejpam-5896 568 6 { { PUNCT ejpam-5896 568 7 { { PUNCT ejpam-5896 568 8 1 1 NUM ejpam-5896 568 9 } } PUNCT ejpam-5896 568 10 , , PUNCT ejpam-5896 568 11 { { PUNCT ejpam-5896 568 12 2 2 NUM ejpam-5896 568 13 } } PUNCT ejpam-5896 568 14 , , PUNCT ejpam-5896 568 15 { { PUNCT ejpam-5896 568 16 1 1 NUM ejpam-5896 568 17 , , PUNCT ejpam-5896 568 18 2 2 NUM ejpam-5896 568 19 } } PUNCT ejpam-5896 568 20 , , PUNCT ejpam-5896 568 21 { { PUNCT ejpam-5896 568 22 1 1 NUM ejpam-5896 568 23 , , PUNCT ejpam-5896 568 24 2 2 NUM ejpam-5896 568 25 , , PUNCT ejpam-5896 568 26 3 3 NUM ejpam-5896 568 27 } } PUNCT ejpam-5896 568 28 , , PUNCT ejpam-5896 568 29 { { PUNCT ejpam-5896 568 30 1 1 NUM ejpam-5896 568 31 , , PUNCT ejpam-5896 568 32 2 2 NUM ejpam-5896 568 33 , , PUNCT ejpam-5896 568 34 3 3 NUM ejpam-5896 568 35 , , PUNCT ejpam-5896 568 36 4 4 NUM ejpam-5896 568 37 } } PUNCT ejpam-5896 568 38 , , PUNCT ejpam-5896 568 39 { { PUNCT ejpam-5896 568 40 1 1 NUM ejpam-5896 568 41 , , PUNCT ejpam-5896 568 42 2 2 NUM ejpam-5896 568 43 , , PUNCT ejpam-5896 568 44 3 3 NUM ejpam-5896 568 45 , , PUNCT ejpam-5896 568 46 4 4 NUM ejpam-5896 568 47 , , PUNCT ejpam-5896 568 48 5 5 NUM ejpam-5896 568 49 } } PUNCT ejpam-5896 568 50 , , PUNCT ejpam-5896 568 51 .......... .......... PUNCT ejpam-5896 568 52 { { PUNCT ejpam-5896 568 53 1 1 NUM ejpam-5896 568 54 , , PUNCT ejpam-5896 568 55 2 2 NUM ejpam-5896 568 56 , , PUNCT ejpam-5896 568 57 3 3 NUM ejpam-5896 568 58 , , PUNCT ejpam-5896 568 59 4 4 NUM ejpam-5896 568 60 , , PUNCT ejpam-5896 568 61 5 5 NUM ejpam-5896 568 62 , , PUNCT ejpam-5896 568 63 6 6 NUM ejpam-5896 568 64 , , PUNCT ejpam-5896 568 65 ..... ..... PUNCT ejpam-5896 568 66 , , PUNCT ejpam-5896 568 67 n n CCONJ ejpam-5896 568 68 } } PUNCT ejpam-5896 568 69 , , PUNCT ejpam-5896 568 70 n n PROPN ejpam-5896 568 71 ∈ ∈ PROPN ejpam-5896 568 72 n n CCONJ ejpam-5896 568 73 } } PUNCT ejpam-5896 568 74 forms form VERB ejpam-5896 568 75 a a DET ejpam-5896 568 76 supra supra NOUN ejpam-5896 568 77 - - PUNCT ejpam-5896 568 78 sd sd NOUN ejpam-5896 568 79 - - PUNCT ejpam-5896 568 80 cover cover NOUN ejpam-5896 568 81 for for ADP ejpam-5896 568 82 ñ ñ PROPN ejpam-5896 568 83 . . PUNCT ejpam-5896 569 1 however however ADV ejpam-5896 569 2 , , PUNCT ejpam-5896 569 3 there there PRON ejpam-5896 569 4 is be VERB ejpam-5896 569 5 no no DET ejpam-5896 569 6 a a DET ejpam-5896 569 7 finite finite NOUN ejpam-5896 569 8 ( ( PUNCT ejpam-5896 569 9 countable countable ADJ ejpam-5896 569 10 ) ) PUNCT ejpam-5896 569 11 subclass subclass NOUN ejpam-5896 569 12 of of ADP ejpam-5896 569 13 ψ ψ PRON ejpam-5896 569 14 which which PRON ejpam-5896 569 15 cover cover VERB ejpam-5896 569 16 ñ ñ VERB ejpam-5896 569 17 . . PUNCT ejpam-5896 570 1 definition definition NOUN ejpam-5896 570 2 25 25 NUM ejpam-5896 570 3 . . PUNCT ejpam-5896 571 1 [ [ X ejpam-5896 571 2 64 64 NUM ejpam-5896 571 3 ] ] PUNCT ejpam-5896 571 4 a a DET ejpam-5896 571 5 soft soft ADJ ejpam-5896 571 6 subset subset NOUN ejpam-5896 571 7 ( ( PUNCT ejpam-5896 571 8 g,∆ g,∆ PROPN ejpam-5896 571 9 ) ) PUNCT ejpam-5896 571 10 of of ADP ejpam-5896 571 11 an an DET ejpam-5896 571 12 ssts sst NOUN ejpam-5896 571 13 ( ( PUNCT ejpam-5896 571 14 u u NOUN ejpam-5896 571 15 , , PUNCT ejpam-5896 571 16 µ,∆ µ,∆ NUM ejpam-5896 571 17 ) ) PUNCT ejpam-5896 571 18 is be AUX ejpam-5896 571 19 claimed claim VERB ejpam-5896 571 20 to to PART ejpam-5896 571 21 be be AUX ejpam-5896 571 22 ss ss NOUN ejpam-5896 571 23 - - PUNCT ejpam-5896 571 24 almost almost ADV ejpam-5896 571 25 compact compact ADJ ejpam-5896 571 26 , , PUNCT ejpam-5896 571 27 if if SCONJ ejpam-5896 571 28 every every DET ejpam-5896 571 29 ss ss NOUN ejpam-5896 571 30 - - ADJ ejpam-5896 571 31 open open ADJ ejpam-5896 571 32 - - PUNCT ejpam-5896 571 33 cover cover NOUN ejpam-5896 571 34 { { PUNCT ejpam-5896 571 35 ( ( PUNCT ejpam-5896 571 36 cϵ,∆ cϵ,∆ ADJ ejpam-5896 571 37 ) ) PUNCT ejpam-5896 571 38 : : PUNCT ejpam-5896 571 39 ϵ ϵ X ejpam-5896 571 40 ∈ ∈ PROPN ejpam-5896 571 41 ε ε PROPN ejpam-5896 571 42 } } PUNCT ejpam-5896 571 43 for for ADP ejpam-5896 571 44 ( ( PUNCT ejpam-5896 571 45 g,∆ g,∆ X ejpam-5896 571 46 ) ) PUNCT ejpam-5896 571 47 has have VERB ejpam-5896 571 48 a a DET ejpam-5896 571 49 finite finite ADJ ejpam-5896 571 50 subclass subclass NOUN ejpam-5896 571 51 εo εo NOUN ejpam-5896 571 52 of of ADP ejpam-5896 571 53 ε ε PROPN ejpam-5896 571 54 such such ADJ ejpam-5896 571 55 that that SCONJ ejpam-5896 571 56 ( ( PUNCT ejpam-5896 571 57 g,∆)⊆̃ g,∆)⊆̃ PROPN ejpam-5896 571 58 ⋃̃ ⋃̃ PROPN ejpam-5896 571 59 ϵ∈εocl ϵ∈εocl PROPN ejpam-5896 571 60 s(cϵ,∆ s(cϵ,∆ NOUN ejpam-5896 571 61 ) ) PUNCT ejpam-5896 571 62 . . PUNCT ejpam-5896 572 1 the the DET ejpam-5896 572 2 space space NOUN ejpam-5896 572 3 ( ( PUNCT ejpam-5896 572 4 u u NOUN ejpam-5896 572 5 , , PUNCT ejpam-5896 572 6 µ,∆ µ,∆ NUM ejpam-5896 572 7 ) ) PUNCT ejpam-5896 572 8 is be AUX ejpam-5896 572 9 claimed claim VERB ejpam-5896 572 10 to to PART ejpam-5896 572 11 be be AUX ejpam-5896 572 12 ss ss VERB ejpam-5896 572 13 - - PUNCT ejpam-5896 572 14 almost almost ADV ejpam-5896 572 15 compact compact ADJ ejpam-5896 572 16 if if SCONJ ejpam-5896 572 17 ũ ũ PROPN ejpam-5896 572 18 is be AUX ejpam-5896 572 19 ss ss NOUN ejpam-5896 572 20 - - PUNCT ejpam-5896 572 21 almost almost ADV ejpam-5896 572 22 compact compact ADJ ejpam-5896 572 23 as as ADP ejpam-5896 572 24 a a DET ejpam-5896 572 25 soft soft ADJ ejpam-5896 572 26 subset subset NOUN ejpam-5896 572 27 . . PUNCT ejpam-5896 573 1 theorem theorem PROPN ejpam-5896 573 2 21 21 NUM ejpam-5896 573 3 . . PUNCT ejpam-5896 574 1 every every DET ejpam-5896 574 2 ss ss NOUN ejpam-5896 574 3 - - PUNCT ejpam-5896 574 4 sd sd NOUN ejpam-5896 574 5 - - PUNCT ejpam-5896 574 6 hyperconnected hyperconnecte VERB ejpam-5896 574 7 ssts sst NOUN ejpam-5896 574 8 is be AUX ejpam-5896 574 9 ss ss NOUN ejpam-5896 574 10 - - PUNCT ejpam-5896 574 11 almost almost ADV ejpam-5896 574 12 compact compact ADJ ejpam-5896 574 13 . . PUNCT ejpam-5896 575 1 proof proof NOUN ejpam-5896 575 2 . . PUNCT ejpam-5896 576 1 assume assume VERB ejpam-5896 576 2 that that SCONJ ejpam-5896 576 3 ψ ψ X ejpam-5896 576 4 = = X ejpam-5896 576 5 { { PUNCT ejpam-5896 576 6 ( ( PUNCT ejpam-5896 576 7 cϵ,∆ cϵ,∆ ADJ ejpam-5896 576 8 ) ) PUNCT ejpam-5896 576 9 : : PUNCT ejpam-5896 576 10 ϵ ϵ X ejpam-5896 576 11 ∈ ∈ PROPN ejpam-5896 576 12 ε ε AUX ejpam-5896 576 13 } } PUNCT ejpam-5896 576 14 be be AUX ejpam-5896 576 15 an an DET ejpam-5896 576 16 ss ss NOUN ejpam-5896 576 17 - - ADJ ejpam-5896 576 18 open open ADJ ejpam-5896 576 19 cover cover NOUN ejpam-5896 576 20 for for ADP ejpam-5896 576 21 an an DET ejpam-5896 576 22 ss ss NOUN ejpam-5896 576 23 - - PUNCT ejpam-5896 576 24 sd sd NOUN ejpam-5896 576 25 - - PUNCT ejpam-5896 576 26 hyperconnected hyperconnecte VERB ejpam-5896 576 27 ssts sst NOUN ejpam-5896 576 28 ( ( PUNCT ejpam-5896 576 29 u u NOUN ejpam-5896 576 30 , , PUNCT ejpam-5896 576 31 µ,∆ µ,∆ NUM ejpam-5896 576 32 ) ) PUNCT ejpam-5896 576 33 , , PUNCT ejpam-5896 576 34 then then ADV ejpam-5896 576 35 clssd(cϵ,∆ clssd(cϵ,∆ ADJ ejpam-5896 576 36 ) ) PUNCT ejpam-5896 577 1 = = SYM ejpam-5896 577 2 ũ ũ PROPN ejpam-5896 577 3 for for ADP ejpam-5896 577 4 each each DET ejpam-5896 577 5 ϵ ϵ PROPN ejpam-5896 577 6 ∈ ∈ PROPN ejpam-5896 577 7 ε ε PROPN ejpam-5896 577 8 , , PUNCT ejpam-5896 577 9 from from ADP ejpam-5896 577 10 corollary corollary ADJ ejpam-5896 577 11 2 2 NUM ejpam-5896 577 12 . . PUNCT ejpam-5896 578 1 therefore therefore ADV ejpam-5896 578 2 , , PUNCT ejpam-5896 578 3 there there PRON ejpam-5896 578 4 is be VERB ejpam-5896 578 5 a a DET ejpam-5896 578 6 finite finite ADJ ejpam-5896 578 7 subclass subclass NOUN ejpam-5896 578 8 εo εo NOUN ejpam-5896 578 9 of of ADP ejpam-5896 578 10 ε ε PROPN ejpam-5896 578 11 such such ADJ ejpam-5896 578 12 that that SCONJ ejpam-5896 578 13 ũ ũ PROPN ejpam-5896 578 14 = = SYM ejpam-5896 578 15 ⋃̃ ⋃̃ PROPN ejpam-5896 578 16 ϵ∈εocl ϵ∈εocl PROPN ejpam-5896 578 17 s s PART ejpam-5896 578 18 sd(cϵ,∆)⊆̃ sd(cϵ,∆)⊆̃ NOUN ejpam-5896 578 19 ⋃̃ ⋃̃ PROPN ejpam-5896 578 20 ϵ∈εocl ϵ∈εocl PROPN ejpam-5896 578 21 s(cϵ,∆ s(cϵ,∆ NOUN ejpam-5896 578 22 ) ) PUNCT ejpam-5896 578 23 . . PUNCT ejpam-5896 579 1 thus thus ADV ejpam-5896 579 2 , , PUNCT ejpam-5896 579 3 ũ ũ PROPN ejpam-5896 579 4 is be AUX ejpam-5896 579 5 an an DET ejpam-5896 579 6 ss ss VERB ejpam-5896 579 7 - - PUNCT ejpam-5896 579 8 almost almost ADV ejpam-5896 579 9 compact compact ADJ ejpam-5896 579 10 . . PUNCT ejpam-5896 580 1 remark remark NOUN ejpam-5896 580 2 6 6 NUM ejpam-5896 580 3 . . PUNCT ejpam-5896 581 1 the the DET ejpam-5896 581 2 following following ADJ ejpam-5896 581 3 example example NOUN ejpam-5896 581 4 shall shall AUX ejpam-5896 581 5 confirm confirm VERB ejpam-5896 581 6 that that SCONJ ejpam-5896 581 7 the the DET ejpam-5896 581 8 converse converse NOUN ejpam-5896 581 9 of of ADP ejpam-5896 581 10 theorem theorem NOUN ejpam-5896 581 11 21 21 NUM ejpam-5896 581 12 is be AUX ejpam-5896 581 13 not not PART ejpam-5896 581 14 satisfied satisfied ADJ ejpam-5896 581 15 in in ADP ejpam-5896 581 16 general general ADJ ejpam-5896 581 17 . . PUNCT ejpam-5896 582 1 example example NOUN ejpam-5896 583 1 6 6 NUM ejpam-5896 583 2 . . PUNCT ejpam-5896 583 3 let let VERB ejpam-5896 583 4 the the DET ejpam-5896 583 5 set set NOUN ejpam-5896 583 6 of of ADP ejpam-5896 583 7 natural natural ADJ ejpam-5896 583 8 numbers number NOUN ejpam-5896 583 9 n n PRON ejpam-5896 583 10 be be VERB ejpam-5896 583 11 the the DET ejpam-5896 583 12 universal universal ADJ ejpam-5896 583 13 set set NOUN ejpam-5896 583 14 and and CCONJ ejpam-5896 583 15 consider consider VERB ejpam-5896 583 16 n1 n1 NOUN ejpam-5896 583 17 , , PUNCT ejpam-5896 583 18 n2 n2 NOUN ejpam-5896 583 19 ∈ ∈ PROPN ejpam-5896 583 20 n n CCONJ ejpam-5896 583 21 any any DET ejpam-5896 583 22 two two NUM ejpam-5896 583 23 natural natural ADJ ejpam-5896 583 24 numbers number NOUN ejpam-5896 583 25 . . PUNCT ejpam-5896 584 1 let let VERB ejpam-5896 584 2 ∆ ∆ PROPN ejpam-5896 584 3 = = PRON ejpam-5896 584 4 { { PUNCT ejpam-5896 584 5 γ1 γ1 PROPN ejpam-5896 584 6 , , PUNCT ejpam-5896 584 7 γ2 γ2 NOUN ejpam-5896 584 8 } } PUNCT ejpam-5896 584 9 and and CCONJ ejpam-5896 584 10 µ µ X ejpam-5896 584 11 = = X ejpam-5896 584 12 { { PUNCT ejpam-5896 584 13 ñ ñ VERB ejpam-5896 584 14 , , PUNCT ejpam-5896 584 15 φ̃ φ̃ PROPN ejpam-5896 584 16 , , PUNCT ejpam-5896 584 17 ( ( PUNCT ejpam-5896 584 18 xi,∆ xi,∆ PROPN ejpam-5896 584 19 ) ) PUNCT ejpam-5896 584 20 , , PUNCT ejpam-5896 584 21 i i PRON ejpam-5896 584 22 = = NOUN ejpam-5896 584 23 1 1 NUM ejpam-5896 584 24 , , PUNCT ejpam-5896 584 25 2 2 NUM ejpam-5896 584 26 , , PUNCT ejpam-5896 584 27 3 3 NUM ejpam-5896 584 28 } } PUNCT ejpam-5896 584 29 be be AUX ejpam-5896 584 30 an an DET ejpam-5896 584 31 ssts sst NOUN ejpam-5896 584 32 on on ADP ejpam-5896 584 33 u u NOUN ejpam-5896 584 34 , , PUNCT ejpam-5896 584 35 where where SCONJ ejpam-5896 584 36 : : PUNCT ejpam-5896 584 37 x1(γ1 x1(γ1 X ejpam-5896 584 38 ) ) PUNCT ejpam-5896 584 39 = = SYM ejpam-5896 584 40 { { PUNCT ejpam-5896 584 41 n1 n1 NOUN ejpam-5896 584 42 } } PUNCT ejpam-5896 584 43 , , PUNCT ejpam-5896 584 44 x1(γ2 x1(γ2 NOUN ejpam-5896 584 45 ) ) PUNCT ejpam-5896 584 46 = = SYM ejpam-5896 584 47 { { PUNCT ejpam-5896 584 48 n2 n2 NOUN ejpam-5896 584 49 } } PUNCT ejpam-5896 584 50 . . PUNCT ejpam-5896 585 1 x2(γ1 x2(γ1 NOUN ejpam-5896 585 2 ) ) PUNCT ejpam-5896 585 3 = = SYM ejpam-5896 585 4 n n PRON ejpam-5896 585 5 − − PROPN ejpam-5896 585 6 { { PUNCT ejpam-5896 585 7 n1 n1 PROPN ejpam-5896 585 8 } } PUNCT ejpam-5896 585 9 , , PUNCT ejpam-5896 585 10 x2(γ2 x2(γ2 PROPN ejpam-5896 585 11 ) ) PUNCT ejpam-5896 585 12 = = SYM ejpam-5896 585 13 n n PRON ejpam-5896 585 14 − − PROPN ejpam-5896 585 15 { { PUNCT ejpam-5896 585 16 n2 n2 NOUN ejpam-5896 585 17 } } PUNCT ejpam-5896 585 18 . . PUNCT ejpam-5896 586 1 x3(γ1 x3(γ1 NOUN ejpam-5896 586 2 ) ) PUNCT ejpam-5896 586 3 = = SYM ejpam-5896 586 4 n n PRON ejpam-5896 586 5 − − PROPN ejpam-5896 586 6 { { PUNCT ejpam-5896 586 7 n1 n1 NOUN ejpam-5896 586 8 } } PUNCT ejpam-5896 586 9 , , PUNCT ejpam-5896 586 10 x3(γ2 x3(γ2 NOUN ejpam-5896 586 11 ) ) PUNCT ejpam-5896 586 12 = = PUNCT ejpam-5896 587 1 n. n. NOUN ejpam-5896 587 2 it it PRON ejpam-5896 587 3 is be AUX ejpam-5896 587 4 easy easy ADJ ejpam-5896 587 5 to to PART ejpam-5896 587 6 check check VERB ejpam-5896 587 7 that that SCONJ ejpam-5896 587 8 ñ ñ PROPN ejpam-5896 587 9 is be AUX ejpam-5896 587 10 an an DET ejpam-5896 587 11 ss ss VERB ejpam-5896 587 12 - - PUNCT ejpam-5896 587 13 almost almost ADV ejpam-5896 587 14 compact compact ADJ ejpam-5896 587 15 space space NOUN ejpam-5896 587 16 . . PUNCT ejpam-5896 588 1 on on ADP ejpam-5896 588 2 the the DET ejpam-5896 588 3 other other ADJ ejpam-5896 588 4 hand hand NOUN ejpam-5896 588 5 , , PUNCT ejpam-5896 588 6 for for ADP ejpam-5896 588 7 the the DET ejpam-5896 588 8 soft soft ADJ ejpam-5896 588 9 sets set NOUN ejpam-5896 588 10 ( ( PUNCT ejpam-5896 588 11 x1,∆ x1,∆ ADJ ejpam-5896 588 12 ) ) PUNCT ejpam-5896 588 13 and and CCONJ ejpam-5896 588 14 ( ( PUNCT ejpam-5896 588 15 x2,∆ x2,∆ PROPN ejpam-5896 588 16 ) ) PUNCT ejpam-5896 588 17 , , PUNCT ejpam-5896 588 18 we we PRON ejpam-5896 588 19 have have VERB ejpam-5896 588 20 ( ( PUNCT ejpam-5896 588 21 x1,∆ x1,∆ ADJ ejpam-5896 588 22 ) ) PUNCT ejpam-5896 588 23 , , PUNCT ejpam-5896 588 24 ( ( PUNCT ejpam-5896 588 25 x2,∆ x2,∆ PROPN ejpam-5896 588 26 ) ) PUNCT ejpam-5896 588 27 ∈ ∈ PROPN ejpam-5896 589 1 sd(n)∆ sd(n)∆ PROPN ejpam-5896 589 2 , , PUNCT ejpam-5896 589 3 whereas whereas SCONJ ejpam-5896 589 4 ( ( PUNCT ejpam-5896 589 5 x1,∆)∩̃(x2,∆ x1,∆)∩̃(x2,∆ PROPN ejpam-5896 589 6 ) ) PUNCT ejpam-5896 589 7 = = SYM ejpam-5896 589 8 φ̃. φ̃. PROPN ejpam-5896 589 9 thus thus ADV ejpam-5896 589 10 , , PUNCT ejpam-5896 589 11 ñ ñ PROPN ejpam-5896 589 12 is be AUX ejpam-5896 589 13 not not PART ejpam-5896 589 14 ss ss AUX ejpam-5896 589 15 - - PUNCT ejpam-5896 589 16 sd sd NOUN ejpam-5896 589 17 - - PUNCT ejpam-5896 589 18 hyperconnected hyperconnecte VERB ejpam-5896 589 19 . . PUNCT ejpam-5896 590 1 corollary corollary ADJ ejpam-5896 590 2 8 8 NUM ejpam-5896 590 3 . . PUNCT ejpam-5896 591 1 let let VERB ejpam-5896 591 2 ( ( PUNCT ejpam-5896 591 3 u u NOUN ejpam-5896 591 4 , , PUNCT ejpam-5896 591 5 µ,∆ µ,∆ NUM ejpam-5896 591 6 ) ) PUNCT ejpam-5896 591 7 be be VERB ejpam-5896 591 8 an an DET ejpam-5896 591 9 ssts sst NOUN ejpam-5896 591 10 , , PUNCT ejpam-5896 591 11 then then ADV ejpam-5896 591 12 the the DET ejpam-5896 591 13 following follow VERB ejpam-5896 591 14 implications implication NOUN ejpam-5896 591 15 hold hold VERB ejpam-5896 591 16 from from ADP ejpam-5896 591 17 proposition proposition NOUN ejpam-5896 591 18 6 6 NUM ejpam-5896 591 19 , , PUNCT ejpam-5896 591 20 proposition proposition NOUN ejpam-5896 591 21 7 7 NUM ejpam-5896 591 22 and and CCONJ ejpam-5896 591 23 theorem theorem VERB ejpam-5896 591 24 21 21 NUM ejpam-5896 591 25 , , PUNCT ejpam-5896 591 26 which which PRON ejpam-5896 591 27 are be AUX ejpam-5896 591 28 not not PART ejpam-5896 591 29 reversible reversible ADJ ejpam-5896 591 30 . . PUNCT ejpam-5896 592 1 ss ss AUX ejpam-5896 592 2 - - PUNCT ejpam-5896 592 3 sd sd NOUN ejpam-5896 592 4 - - PUNCT ejpam-5896 592 5 hyperconnected hyperconnecte VERB ejpam-5896 592 6 ⇓ ⇓ PROPN ejpam-5896 592 7 ss ss PROPN ejpam-5896 592 8 - - PUNCT ejpam-5896 592 9 sd sd NOUN ejpam-5896 592 10 - - PUNCT ejpam-5896 592 11 compact compact ADJ ejpam-5896 592 12 ⇒ ⇒ NOUN ejpam-5896 592 13 ss ss ADJ ejpam-5896 592 14 - - ADJ ejpam-5896 592 15 compact compact ADJ ejpam-5896 592 16 ⇒ ⇒ NOUN ejpam-5896 592 17 ss ss NOUN ejpam-5896 592 18 - - PUNCT ejpam-5896 592 19 almost almost ADV ejpam-5896 592 20 compact compact ADJ ejpam-5896 592 21 ⇓ ⇓ PROPN ejpam-5896 592 22 ⇓ ⇓ PROPN ejpam-5896 592 23 ⇓ ⇓ PROPN ejpam-5896 592 24 ss ss PROPN ejpam-5896 592 25 - - PUNCT ejpam-5896 592 26 sd sd NOUN ejpam-5896 592 27 - - PUNCT ejpam-5896 592 28 lindelöf lindelöf NOUN ejpam-5896 592 29 ⇒ ⇒ PROPN ejpam-5896 592 30 ss ss PROPN ejpam-5896 592 31 - - PUNCT ejpam-5896 592 32 lindelöf lindelöf NOUN ejpam-5896 592 33 ⇒ ⇒ NOUN ejpam-5896 592 34 ss ss PROPN ejpam-5896 592 35 - - PUNCT ejpam-5896 592 36 almost almost ADV ejpam-5896 592 37 lindelöf lindelöf NOUN ejpam-5896 592 38 figure figure NOUN ejpam-5896 592 39 2 2 NUM ejpam-5896 592 40 . . PUNCT ejpam-5896 593 1 the the DET ejpam-5896 593 2 relationships relationship NOUN ejpam-5896 593 3 among among ADP ejpam-5896 593 4 different different ADJ ejpam-5896 593 5 types type NOUN ejpam-5896 593 6 of of ADP ejpam-5896 593 7 compactness compactness NOUN ejpam-5896 593 8 , , PUNCT ejpam-5896 593 9 lindelöfness lindelöfness NOUN ejpam-5896 593 10 and and CCONJ ejpam-5896 593 11 hyperconnectedness hyperconnectedness NOUN ejpam-5896 593 12 via via ADP ejpam-5896 593 13 ss ss NOUN ejpam-5896 593 14 - - PUNCT ejpam-5896 593 15 sd sd NOUN ejpam-5896 593 16 - - PUNCT ejpam-5896 593 17 sets set NOUN ejpam-5896 593 18 in in ADP ejpam-5896 593 19 the the DET ejpam-5896 593 20 frame frame NOUN ejpam-5896 593 21 of of ADP ejpam-5896 593 22 sstss sstss NOUN ejpam-5896 593 23 . . PUNCT ejpam-5896 594 1 abd abd PROPN ejpam-5896 594 2 el el PROPN ejpam-5896 594 3 - - PROPN ejpam-5896 594 4 latif latif PROPN ejpam-5896 594 5 et et PROPN ejpam-5896 594 6 al al PROPN ejpam-5896 594 7 . . PUNCT ejpam-5896 594 8 / / SYM ejpam-5896 594 9 eur eur PROPN ejpam-5896 594 10 . . PUNCT ejpam-5896 595 1 j. j. PROPN ejpam-5896 595 2 pure pure PROPN ejpam-5896 595 3 appl appl PROPN ejpam-5896 595 4 . . PROPN ejpam-5896 595 5 math math PROPN ejpam-5896 595 6 , , PUNCT ejpam-5896 595 7 18 18 NUM ejpam-5896 595 8 ( ( PUNCT ejpam-5896 595 9 2 2 NUM ejpam-5896 595 10 ) ) PUNCT ejpam-5896 595 11 ( ( PUNCT ejpam-5896 595 12 2025 2025 NUM ejpam-5896 595 13 ) ) PUNCT ejpam-5896 595 14 , , PUNCT ejpam-5896 595 15 5896 5896 NUM ejpam-5896 595 16 17 17 NUM ejpam-5896 595 17 of of ADP ejpam-5896 595 18 20 20 NUM ejpam-5896 595 19 5 5 NUM ejpam-5896 595 20 . . PUNCT ejpam-5896 595 21 conclusion conclusion NOUN ejpam-5896 595 22 and and CCONJ ejpam-5896 595 23 future future ADJ ejpam-5896 595 24 work work NOUN ejpam-5896 595 25 this this DET ejpam-5896 595 26 manuscript manuscript NOUN ejpam-5896 595 27 is be AUX ejpam-5896 595 28 devoted devoted ADJ ejpam-5896 595 29 to to ADP ejpam-5896 595 30 investigating investigate VERB ejpam-5896 595 31 more more ADV ejpam-5896 595 32 interesting interesting ADJ ejpam-5896 595 33 properties property NOUN ejpam-5896 595 34 of of ADP ejpam-5896 595 35 the the DET ejpam-5896 595 36 notions notion NOUN ejpam-5896 595 37 of of ADP ejpam-5896 595 38 ss ss NOUN ejpam-5896 595 39 - - PUNCT ejpam-5896 595 40 sd sd NOUN ejpam-5896 595 41 - - PUNCT ejpam-5896 595 42 connectedness connectedness NOUN ejpam-5896 595 43 . . PUNCT ejpam-5896 596 1 specifically specifically ADV ejpam-5896 596 2 , , PUNCT ejpam-5896 596 3 we we PRON ejpam-5896 596 4 use use VERB ejpam-5896 596 5 the the DET ejpam-5896 596 6 notions notion NOUN ejpam-5896 596 7 of of ADP ejpam-5896 596 8 ss ss NOUN ejpam-5896 596 9 - - PUNCT ejpam-5896 596 10 sd sd NOUN ejpam-5896 596 11 - - PUNCT ejpam-5896 596 12 components component NOUN ejpam-5896 596 13 to to PART ejpam-5896 596 14 define define VERB ejpam-5896 596 15 a a DET ejpam-5896 596 16 new new ADJ ejpam-5896 596 17 type type NOUN ejpam-5896 596 18 of of ADP ejpam-5896 596 19 connectedness connectedness NOUN ejpam-5896 596 20 in in ADP ejpam-5896 596 21 sstss sstss PROPN ejpam-5896 596 22 , , PUNCT ejpam-5896 596 23 named name VERB ejpam-5896 596 24 ssl ssl PROPN ejpam-5896 596 25 - - PUNCT ejpam-5896 596 26 sd sd NOUN ejpam-5896 596 27 - - PUNCT ejpam-5896 596 28 connectedness connectedness NOUN ejpam-5896 596 29 . . PUNCT ejpam-5896 597 1 in in ADP ejpam-5896 597 2 addition addition NOUN ejpam-5896 597 3 , , PUNCT ejpam-5896 597 4 we we PRON ejpam-5896 597 5 introduce introduce VERB ejpam-5896 597 6 another another DET ejpam-5896 597 7 type type NOUN ejpam-5896 597 8 , , PUNCT ejpam-5896 597 9 named name VERB ejpam-5896 597 10 ss ss NOUN ejpam-5896 597 11 - - PUNCT ejpam-5896 597 12 sd sd NOUN ejpam-5896 597 13 - - PUNCT ejpam-5896 597 14 hyperconnectedness hyperconnectedness NOUN ejpam-5896 597 15 . . PUNCT ejpam-5896 598 1 we we PRON ejpam-5896 598 2 discovered discover VERB ejpam-5896 598 3 that that SCONJ ejpam-5896 598 4 , , PUNCT ejpam-5896 598 5 in in ADP ejpam-5896 598 6 contrast contrast NOUN ejpam-5896 598 7 to to ADP ejpam-5896 598 8 their their PRON ejpam-5896 598 9 counterparts counterpart NOUN ejpam-5896 598 10 , , PUNCT ejpam-5896 598 11 ss ss NOUN ejpam-5896 598 12 - - PUNCT ejpam-5896 598 13 sd sd NOUN ejpam-5896 598 14 - - PUNCT ejpam-5896 598 15 hyperconnected hyperconnecte VERB ejpam-5896 598 16 spaces space NOUN ejpam-5896 598 17 are be AUX ejpam-5896 598 18 equivalent equivalent ADJ ejpam-5896 598 19 to to ADP ejpam-5896 598 20 ss ss NOUN ejpam-5896 598 21 - - PUNCT ejpam-5896 598 22 sd sd NOUN ejpam-5896 598 23 - - PUNCT ejpam-5896 598 24 connected connect VERB ejpam-5896 598 25 spaces space NOUN ejpam-5896 598 26 . . PUNCT ejpam-5896 599 1 we we PRON ejpam-5896 599 2 discuss discuss VERB ejpam-5896 599 3 their their PRON ejpam-5896 599 4 basic basic ADJ ejpam-5896 599 5 properties property NOUN ejpam-5896 599 6 in in ADP ejpam-5896 599 7 detail detail NOUN ejpam-5896 599 8 . . PUNCT ejpam-5896 600 1 moreover moreover ADV ejpam-5896 600 2 , , PUNCT ejpam-5896 600 3 we we PRON ejpam-5896 600 4 study study VERB ejpam-5896 600 5 another another DET ejpam-5896 600 6 topological topological ADJ ejpam-5896 600 7 property property NOUN ejpam-5896 600 8 in in ADP ejpam-5896 600 9 an an DET ejpam-5896 600 10 ssts sst NOUN ejpam-5896 600 11 named name VERB ejpam-5896 600 12 ss ss NOUN ejpam-5896 600 13 - - PUNCT ejpam-5896 600 14 sdlindelöfness sdlindelöfness NOUN ejpam-5896 600 15 ( ( PUNCT ejpam-5896 600 16 compactness compactness NOUN ejpam-5896 600 17 ) ) PUNCT ejpam-5896 600 18 . . PUNCT ejpam-5896 601 1 the the DET ejpam-5896 601 2 behaviour behaviour NOUN ejpam-5896 601 3 of of ADP ejpam-5896 601 4 an an DET ejpam-5896 601 5 ss ss NOUN ejpam-5896 601 6 - - PUNCT ejpam-5896 601 7 sd sd NOUN ejpam-5896 601 8 - - PUNCT ejpam-5896 601 9 compact compact ADJ ejpam-5896 601 10 ( ( PUNCT ejpam-5896 601 11 lindelöf lindelöf NOUN ejpam-5896 601 12 ) ) PUNCT ejpam-5896 601 13 ssts sst NOUN ejpam-5896 601 14 with with ADP ejpam-5896 601 15 the the DET ejpam-5896 601 16 sfip sfip NOUN ejpam-5896 601 17 ( ( PUNCT ejpam-5896 601 18 scip scip PROPN ejpam-5896 601 19 ) ) PUNCT ejpam-5896 601 20 has have AUX ejpam-5896 601 21 been be AUX ejpam-5896 601 22 discussed discuss VERB ejpam-5896 601 23 . . PUNCT ejpam-5896 602 1 furthermore furthermore ADV ejpam-5896 602 2 , , PUNCT ejpam-5896 602 3 we we PRON ejpam-5896 602 4 show show VERB ejpam-5896 602 5 that that SCONJ ejpam-5896 602 6 the the DET ejpam-5896 602 7 image image NOUN ejpam-5896 602 8 ( ( PUNCT ejpam-5896 602 9 respectively respectively ADV ejpam-5896 602 10 , , PUNCT ejpam-5896 602 11 pre pre ADJ ejpam-5896 602 12 - - NOUN ejpam-5896 602 13 image image ADJ ejpam-5896 602 14 ) ) PUNCT ejpam-5896 602 15 of of ADP ejpam-5896 602 16 each each DET ejpam-5896 602 17 ss ss NOUN ejpam-5896 602 18 - - PUNCT ejpam-5896 602 19 sd sd NOUN ejpam-5896 602 20 - - PUNCT ejpam-5896 602 21 lindelöf lindelöf NOUN ejpam-5896 602 22 ( ( PUNCT ejpam-5896 602 23 compact compact ADJ ejpam-5896 602 24 ) ) PUNCT ejpam-5896 602 25 is be AUX ejpam-5896 602 26 ss ss NOUN ejpam-5896 602 27 - - PUNCT ejpam-5896 602 28 lindelöf lindelöf NOUN ejpam-5896 602 29 ( ( PUNCT ejpam-5896 602 30 compact compact ADJ ejpam-5896 602 31 ) ) PUNCT ejpam-5896 602 32 under under ADP ejpam-5896 602 33 a a DET ejpam-5896 602 34 surjective surjective ADJ ejpam-5896 602 35 and and CCONJ ejpam-5896 602 36 ss ss NOUN ejpam-5896 602 37 - - PUNCT ejpam-5896 602 38 sd sd NOUN ejpam-5896 602 39 - - PUNCT ejpam-5896 602 40 continuous continuous ADJ ejpam-5896 602 41 ( ( PUNCT ejpam-5896 602 42 respectively respectively ADV ejpam-5896 602 43 , , PUNCT ejpam-5896 602 44 an an DET ejpam-5896 602 45 injective injective ADJ ejpam-5896 602 46 and and CCONJ ejpam-5896 602 47 ss ss NOUN ejpam-5896 602 48 - - PUNCT ejpam-5896 602 49 sd sd NOUN ejpam-5896 602 50 - - PUNCT ejpam-5896 602 51 open open ADJ ejpam-5896 602 52 ) ) PUNCT ejpam-5896 602 53 function function NOUN ejpam-5896 602 54 . . PUNCT ejpam-5896 603 1 finally finally ADV ejpam-5896 603 2 , , PUNCT ejpam-5896 603 3 we we PRON ejpam-5896 603 4 provide provide VERB ejpam-5896 603 5 two two NUM ejpam-5896 603 6 topological topological ADJ ejpam-5896 603 7 charts chart NOUN ejpam-5896 603 8 in in ADP ejpam-5896 603 9 figures figure NOUN ejpam-5896 603 10 1 1 NUM ejpam-5896 603 11 and and CCONJ ejpam-5896 603 12 2 2 NUM ejpam-5896 603 13 , , PUNCT ejpam-5896 603 14 to to PART ejpam-5896 603 15 illustrate illustrate VERB ejpam-5896 603 16 the the DET ejpam-5896 603 17 key key ADJ ejpam-5896 603 18 concepts concept NOUN ejpam-5896 603 19 presented present VERB ejpam-5896 603 20 in in ADP ejpam-5896 603 21 this this DET ejpam-5896 603 22 paper paper NOUN ejpam-5896 603 23 , , PUNCT ejpam-5896 603 24 and and CCONJ ejpam-5896 603 25 several several ADJ ejpam-5896 603 26 interesting interesting ADJ ejpam-5896 603 27 examples example NOUN ejpam-5896 603 28 and and CCONJ ejpam-5896 603 29 counterexamples counterexample NOUN ejpam-5896 603 30 have have AUX ejpam-5896 603 31 been be AUX ejpam-5896 603 32 provided provide VERB ejpam-5896 603 33 . . PUNCT ejpam-5896 604 1 our our PRON ejpam-5896 604 2 upcoming upcoming ADJ ejpam-5896 604 3 work work NOUN ejpam-5896 604 4 is be AUX ejpam-5896 604 5 to to PART ejpam-5896 604 6 apply apply VERB ejpam-5896 604 7 the the DET ejpam-5896 604 8 provided provide VERB ejpam-5896 604 9 notions notion NOUN ejpam-5896 604 10 to to ADP ejpam-5896 604 11 decision decision NOUN ejpam-5896 604 12 making making NOUN ejpam-5896 604 13 problem problem NOUN ejpam-5896 604 14 , , PUNCT ejpam-5896 604 15 and and CCONJ ejpam-5896 604 16 introducing introduce VERB ejpam-5896 604 17 more more ADJ ejpam-5896 604 18 types type NOUN ejpam-5896 604 19 of of ADP ejpam-5896 604 20 compactness compactness NOUN ejpam-5896 604 21 and and CCONJ ejpam-5896 604 22 separation separation NOUN ejpam-5896 604 23 axioms axiom NOUN ejpam-5896 604 24 in in ADP ejpam-5896 604 25 an an DET ejpam-5896 604 26 ssts sst NOUN ejpam-5896 604 27 via via ADP ejpam-5896 604 28 ss ss NOUN ejpam-5896 604 29 - - PUNCT ejpam-5896 604 30 sd sd NOUN ejpam-5896 604 31 - - PUNCT ejpam-5896 604 32 sets set NOUN ejpam-5896 604 33 . . PUNCT ejpam-5896 605 1 moreover moreover ADV ejpam-5896 605 2 , , PUNCT ejpam-5896 605 3 we we PRON ejpam-5896 605 4 will will AUX ejpam-5896 605 5 apply apply VERB ejpam-5896 605 6 theses theses DET ejpam-5896 605 7 notions notion NOUN ejpam-5896 605 8 to to ADP ejpam-5896 605 9 the the DET ejpam-5896 605 10 fuzzy fuzzy ADJ ejpam-5896 605 11 supra supra PROPN ejpam-5896 605 12 soft soft ADJ ejpam-5896 605 13 topological topological ADJ ejpam-5896 605 14 spaces space NOUN ejpam-5896 605 15 [ [ X ejpam-5896 605 16 65 65 NUM ejpam-5896 605 17 ] ] PUNCT ejpam-5896 605 18 . . PUNCT ejpam-5896 606 1 acknowledgements acknowledgement NOUN ejpam-5896 606 2 the the DET ejpam-5896 606 3 authors author NOUN ejpam-5896 606 4 extend extend VERB ejpam-5896 606 5 their their PRON ejpam-5896 606 6 appreciation appreciation NOUN ejpam-5896 606 7 to to ADP ejpam-5896 606 8 the the DET ejpam-5896 606 9 deanship deanship NOUN ejpam-5896 606 10 of of ADP ejpam-5896 606 11 scientific scientific ADJ ejpam-5896 606 12 research research NOUN ejpam-5896 606 13 at at ADP ejpam-5896 606 14 northern northern ADJ ejpam-5896 606 15 border border NOUN ejpam-5896 606 16 university university PROPN ejpam-5896 606 17 , , PUNCT ejpam-5896 606 18 arar arar PROPN ejpam-5896 606 19 , , PUNCT ejpam-5896 606 20 ksa ksa PROPN ejpam-5896 606 21 for for ADP ejpam-5896 606 22 funding fund VERB ejpam-5896 606 23 this this DET ejpam-5896 606 24 research research NOUN ejpam-5896 606 25 work work NOUN ejpam-5896 606 26 through through ADP ejpam-5896 606 27 the the DET ejpam-5896 606 28 project project NOUN ejpam-5896 606 29 number number NOUN ejpam-5896 606 30 ” " PUNCT ejpam-5896 606 31 nbuffr-2025 nbuffr-2025 ADJ ejpam-5896 606 32 - - PUNCT ejpam-5896 606 33 2727 2727 NUM ejpam-5896 606 34 - - SYM ejpam-5896 606 35 02 02 NUM ejpam-5896 606 36 ” " PUNCT ejpam-5896 606 37 . . PUNCT ejpam-5896 607 1 also also ADV ejpam-5896 607 2 , , PUNCT ejpam-5896 607 3 this this DET ejpam-5896 607 4 study study NOUN ejpam-5896 607 5 is be AUX ejpam-5896 607 6 supported support VERB ejpam-5896 607 7 via via ADP ejpam-5896 607 8 funding funding NOUN ejpam-5896 607 9 from from ADP ejpam-5896 607 10 prince prince PROPN ejpam-5896 607 11 sattam sattam PROPN ejpam-5896 607 12 bin bin PROPN ejpam-5896 607 13 abdulaziz abdulaziz PROPN ejpam-5896 607 14 university university PROPN ejpam-5896 607 15 project project NOUN ejpam-5896 607 16 number number NOUN ejpam-5896 607 17 ( ( PUNCT ejpam-5896 607 18 psau/2025 psau/2025 NOUN ejpam-5896 607 19 / / SYM ejpam-5896 607 20 r/1446 r/1446 PROPN ejpam-5896 607 21 ) ) PUNCT ejpam-5896 607 22 and and CCONJ ejpam-5896 607 23 this this DET ejpam-5896 607 24 research research NOUN ejpam-5896 607 25 is be AUX ejpam-5896 607 26 funded fund VERB ejpam-5896 607 27 by by ADP ejpam-5896 607 28 zarqa zarqa PROPN ejpam-5896 607 29 university university PROPN ejpam-5896 607 30 jordan jordan PROPN ejpam-5896 607 31 . . PUNCT ejpam-5896 608 1 conflicts conflict NOUN ejpam-5896 608 2 of of ADP ejpam-5896 608 3 interest interest NOUN ejpam-5896 608 4 the the DET ejpam-5896 608 5 authors author NOUN ejpam-5896 608 6 declare declare VERB ejpam-5896 608 7 that that SCONJ ejpam-5896 608 8 they they PRON ejpam-5896 608 9 have have VERB ejpam-5896 608 10 no no DET ejpam-5896 608 11 conflict conflict NOUN ejpam-5896 608 12 of of ADP ejpam-5896 608 13 interest interest NOUN ejpam-5896 608 14 regarding regard VERB ejpam-5896 608 15 the the DET ejpam-5896 608 16 publication publication NOUN ejpam-5896 608 17 of of ADP ejpam-5896 608 18 this this DET ejpam-5896 608 19 paper paper NOUN ejpam-5896 608 20 . . PUNCT ejpam-5896 609 1 references reference NOUN ejpam-5896 609 2 [ [ X ejpam-5896 609 3 1 1 NUM ejpam-5896 609 4 ] ] PUNCT ejpam-5896 609 5 a. a. NOUN ejpam-5896 609 6 s. s. PROPN ejpam-5896 609 7 mashhour mashhour PROPN ejpam-5896 609 8 , , PUNCT ejpam-5896 609 9 a. a. PROPN ejpam-5896 609 10 a. a. PROPN ejpam-5896 609 11 allam allam PROPN ejpam-5896 609 12 , , PUNCT ejpam-5896 609 13 f. f. PROPN ejpam-5896 609 14 s. s. PROPN ejpam-5896 609 15 mahmoud mahmoud PROPN ejpam-5896 609 16 , , PUNCT ejpam-5896 609 17 and and CCONJ ejpam-5896 609 18 f. f. PROPN ejpam-5896 609 19 h. h. PROPN ejpam-5896 609 20 khedr khedr PROPN ejpam-5896 609 21 . . PUNCT ejpam-5896 610 1 on on ADP ejpam-5896 610 2 supra supra PROPN ejpam-5896 610 3 topological topological ADJ ejpam-5896 610 4 spaces space NOUN ejpam-5896 610 5 . . PUNCT ejpam-5896 611 1 indian indian ADJ ejpam-5896 611 2 journal journal PROPN ejpam-5896 611 3 of of ADP ejpam-5896 611 4 pure pure ADJ ejpam-5896 611 5 and and CCONJ ejpam-5896 611 6 applied applied ADJ ejpam-5896 611 7 mathematics mathematic NOUN ejpam-5896 611 8 , , PUNCT ejpam-5896 611 9 14(4):502–510 14(4):502–510 PROPN ejpam-5896 611 10 , , PUNCT ejpam-5896 611 11 1983 1983 NUM ejpam-5896 611 12 . . PUNCT ejpam-5896 612 1 [ [ X ejpam-5896 612 2 2 2 NUM ejpam-5896 612 3 ] ] PUNCT ejpam-5896 612 4 a. a. NOUN ejpam-5896 612 5 m. m. NOUN ejpam-5896 612 6 kozae kozae PROPN ejpam-5896 612 7 , , PUNCT ejpam-5896 612 8 m. m. NOUN ejpam-5896 612 9 shokry shokry PROPN ejpam-5896 612 10 , , PUNCT ejpam-5896 612 11 and and CCONJ ejpam-5896 612 12 m. m. PROPN ejpam-5896 612 13 zidan zidan PROPN ejpam-5896 612 14 . . PUNCT ejpam-5896 613 1 supra supra PROPN ejpam-5896 613 2 topologies topology NOUN ejpam-5896 613 3 for for ADP ejpam-5896 613 4 digital digital ADJ ejpam-5896 613 5 plane plane NOUN ejpam-5896 613 6 . . PUNCT ejpam-5896 614 1 aascit aascit PROPN ejpam-5896 614 2 communications communication NOUN ejpam-5896 614 3 , , PUNCT ejpam-5896 614 4 3:1–10 3:1–10 NUM ejpam-5896 614 5 , , PUNCT ejpam-5896 614 6 2016 2016 NUM ejpam-5896 614 7 . . PUNCT ejpam-5896 615 1 [ [ X ejpam-5896 615 2 3 3 X ejpam-5896 615 3 ] ] PUNCT ejpam-5896 615 4 m. m. PROPN ejpam-5896 615 5 e. e. PROPN ejpam-5896 615 6 el el PROPN ejpam-5896 615 7 - - PROPN ejpam-5896 615 8 shafei shafei PROPN ejpam-5896 615 9 , , PUNCT ejpam-5896 615 10 a. a. NOUN ejpam-5896 615 11 h. h. PROPN ejpam-5896 615 12 zakari zakari PROPN ejpam-5896 615 13 , , PUNCT ejpam-5896 615 14 and and CCONJ ejpam-5896 615 15 t. t. PROPN ejpam-5896 615 16 m. m. PROPN ejpam-5896 615 17 al al PROPN ejpam-5896 615 18 - - PUNCT ejpam-5896 615 19 shami shami PROPN ejpam-5896 615 20 . . PUNCT ejpam-5896 616 1 some some DET ejpam-5896 616 2 applications application NOUN ejpam-5896 616 3 of of ADP ejpam-5896 616 4 supra supra ADJ ejpam-5896 616 5 preopen preopen ADJ ejpam-5896 616 6 sets set NOUN ejpam-5896 616 7 . . PUNCT ejpam-5896 617 1 journal journal NOUN ejpam-5896 617 2 of of ADP ejpam-5896 617 3 mathematics mathematic NOUN ejpam-5896 617 4 , , PUNCT ejpam-5896 617 5 2020:9634206 2020:9634206 NUM ejpam-5896 617 6 , , PUNCT ejpam-5896 617 7 2020 2020 NUM ejpam-5896 617 8 . . PUNCT ejpam-5896 618 1 [ [ X ejpam-5896 618 2 4 4 X ejpam-5896 618 3 ] ] PUNCT ejpam-5896 618 4 t. t. PROPN ejpam-5896 618 5 m. m. PROPN ejpam-5896 618 6 al al PROPN ejpam-5896 618 7 - - PUNCT ejpam-5896 618 8 shami shami PROPN ejpam-5896 618 9 , , PUNCT ejpam-5896 618 10 e. e. PROPN ejpam-5896 618 11 a. a. PROPN ejpam-5896 618 12 abo abo PROPN ejpam-5896 618 13 - - PUNCT ejpam-5896 618 14 tabl tabl NOUN ejpam-5896 618 15 , , PUNCT ejpam-5896 618 16 and and CCONJ ejpam-5896 618 17 b. b. PROPN ejpam-5896 618 18 a. a. PROPN ejpam-5896 618 19 asaad asaad PROPN ejpam-5896 618 20 . . PUNCT ejpam-5896 619 1 investigation investigation NOUN ejpam-5896 619 2 of of ADP ejpam-5896 619 3 limit limit NOUN ejpam-5896 619 4 points point NOUN ejpam-5896 619 5 and and CCONJ ejpam-5896 619 6 separation separation NOUN ejpam-5896 619 7 axioms axiom NOUN ejpam-5896 619 8 using use VERB ejpam-5896 619 9 supra supra PROPN ejpam-5896 619 10 β β NOUN ejpam-5896 619 11 - - ADJ ejpam-5896 619 12 open open ADJ ejpam-5896 619 13 sets set NOUN ejpam-5896 619 14 . . PUNCT ejpam-5896 620 1 missouri missouri PROPN ejpam-5896 620 2 journal journal PROPN ejpam-5896 620 3 of of ADP ejpam-5896 620 4 mathematical mathematical ADJ ejpam-5896 620 5 sciences science NOUN ejpam-5896 620 6 , , PUNCT ejpam-5896 620 7 32(2):171–187 32(2):171–187 PROPN ejpam-5896 620 8 , , PUNCT ejpam-5896 620 9 2020 2020 NUM ejpam-5896 620 10 . . PUNCT ejpam-5896 621 1 [ [ X ejpam-5896 621 2 5 5 X ejpam-5896 621 3 ] ] PUNCT ejpam-5896 621 4 t. t. PROPN ejpam-5896 621 5 m. m. PROPN ejpam-5896 621 6 al al PROPN ejpam-5896 621 7 - - PUNCT ejpam-5896 621 8 shami shami PROPN ejpam-5896 621 9 and and CCONJ ejpam-5896 621 10 i. i. PROPN ejpam-5896 621 11 alshammari alshammari PROPN ejpam-5896 621 12 . . PUNCT ejpam-5896 622 1 rough rough ADJ ejpam-5896 622 2 sets set NOUN ejpam-5896 622 3 models model NOUN ejpam-5896 622 4 inspired inspire VERB ejpam-5896 622 5 by by ADP ejpam-5896 622 6 supra supra ADJ ejpam-5896 622 7 - - PUNCT ejpam-5896 622 8 topology topology NOUN ejpam-5896 622 9 structures structure NOUN ejpam-5896 622 10 . . PUNCT ejpam-5896 623 1 artificial artificial ADJ ejpam-5896 623 2 intelligence intelligence NOUN ejpam-5896 623 3 review review NOUN ejpam-5896 623 4 , , PUNCT ejpam-5896 623 5 56:6855–6883 56:6855–6883 NUM ejpam-5896 623 6 , , PUNCT ejpam-5896 623 7 2023 2023 NUM ejpam-5896 623 8 . . PUNCT ejpam-5896 624 1 [ [ X ejpam-5896 624 2 6 6 NUM ejpam-5896 624 3 ] ] PUNCT ejpam-5896 624 4 d. d. PROPN ejpam-5896 624 5 a. a. PROPN ejpam-5896 624 6 molodtsov molodtsov PROPN ejpam-5896 624 7 . . PUNCT ejpam-5896 625 1 soft soft ADJ ejpam-5896 625 2 set set VERB ejpam-5896 625 3 theory theory NOUN ejpam-5896 625 4 – – PUNCT ejpam-5896 625 5 first first ADJ ejpam-5896 625 6 results result NOUN ejpam-5896 625 7 . . PUNCT ejpam-5896 626 1 computers computer NOUN ejpam-5896 626 2 & & CCONJ ejpam-5896 626 3 mathematics mathematics PROPN ejpam-5896 626 4 with with ADP ejpam-5896 626 5 applications application NOUN ejpam-5896 626 6 , , PUNCT ejpam-5896 626 7 37:19–31 37:19–31 NUM ejpam-5896 626 8 , , PUNCT ejpam-5896 626 9 1999 1999 NUM ejpam-5896 626 10 . . PUNCT ejpam-5896 627 1 [ [ X ejpam-5896 627 2 7 7 X ejpam-5896 627 3 ] ] PUNCT ejpam-5896 627 4 p. p. NOUN ejpam-5896 627 5 k. k. PROPN ejpam-5896 628 1 maji maji PROPN ejpam-5896 628 2 , , PUNCT ejpam-5896 628 3 r. r. PROPN ejpam-5896 628 4 biswas biswas PROPN ejpam-5896 628 5 , , PUNCT ejpam-5896 628 6 and and CCONJ ejpam-5896 628 7 a. a. PROPN ejpam-5896 628 8 r. r. PROPN ejpam-5896 628 9 roy roy PROPN ejpam-5896 628 10 . . PROPN ejpam-5896 628 11 soft soft ADJ ejpam-5896 628 12 set set NOUN ejpam-5896 628 13 theory theory NOUN ejpam-5896 628 14 . . PUNCT ejpam-5896 629 1 computers computer NOUN ejpam-5896 629 2 & & CCONJ ejpam-5896 629 3 mathematics mathematics PROPN ejpam-5896 629 4 with with ADP ejpam-5896 629 5 applications application NOUN ejpam-5896 629 6 , , PUNCT ejpam-5896 629 7 45:555–562 45:555–562 PROPN ejpam-5896 629 8 , , PUNCT ejpam-5896 629 9 2003 2003 NUM ejpam-5896 629 10 . . PUNCT ejpam-5896 630 1 [ [ X ejpam-5896 630 2 8 8 NUM ejpam-5896 630 3 ] ] X ejpam-5896 630 4 f. f. PROPN ejpam-5896 630 5 karaaslan karaaslan PROPN ejpam-5896 630 6 . . PUNCT ejpam-5896 631 1 soft soft ADJ ejpam-5896 631 2 classes class NOUN ejpam-5896 631 3 and and CCONJ ejpam-5896 631 4 soft soft ADJ ejpam-5896 631 5 rough rough ADJ ejpam-5896 631 6 classes class NOUN ejpam-5896 631 7 with with ADP ejpam-5896 631 8 applications application NOUN ejpam-5896 631 9 in in ADP ejpam-5896 631 10 decision decision NOUN ejpam-5896 631 11 making making NOUN ejpam-5896 631 12 . . PUNCT ejpam-5896 632 1 mathematical mathematical ADJ ejpam-5896 632 2 problems problem NOUN ejpam-5896 632 3 in in ADP ejpam-5896 632 4 engineering engineering NOUN ejpam-5896 632 5 , , PUNCT ejpam-5896 632 6 2016:1584528 2016:1584528 NUM ejpam-5896 632 7 , , PUNCT ejpam-5896 632 8 2016 2016 NUM ejpam-5896 632 9 . . PUNCT ejpam-5896 633 1 [ [ X ejpam-5896 633 2 9 9 NUM ejpam-5896 633 3 ] ] PUNCT ejpam-5896 633 4 s. s. PROPN ejpam-5896 633 5 yuksel yuksel PROPN ejpam-5896 633 6 , , PUNCT ejpam-5896 633 7 t. t. PROPN ejpam-5896 633 8 dizman dizman NOUN ejpam-5896 633 9 , , PUNCT ejpam-5896 633 10 g. g. PROPN ejpam-5896 633 11 yildizdan yildizdan PROPN ejpam-5896 633 12 , , PUNCT ejpam-5896 633 13 and and CCONJ ejpam-5896 633 14 u. u. PROPN ejpam-5896 633 15 sert sert PROPN ejpam-5896 633 16 . . PUNCT ejpam-5896 634 1 application application NOUN ejpam-5896 634 2 of of ADP ejpam-5896 634 3 soft soft ADJ ejpam-5896 634 4 sets set NOUN ejpam-5896 634 5 to to PART ejpam-5896 634 6 diagnose diagnose VERB ejpam-5896 634 7 the the DET ejpam-5896 634 8 prostate prostate NOUN ejpam-5896 634 9 cancer cancer NOUN ejpam-5896 634 10 risk risk NOUN ejpam-5896 634 11 . . PUNCT ejpam-5896 635 1 journal journal NOUN ejpam-5896 635 2 of of ADP ejpam-5896 635 3 inequalities inequality NOUN ejpam-5896 635 4 and and CCONJ ejpam-5896 635 5 applications application NOUN ejpam-5896 635 6 , , PUNCT ejpam-5896 635 7 2013:229 2013:229 NUM ejpam-5896 635 8 , , PUNCT ejpam-5896 635 9 2013 2013 NUM ejpam-5896 635 10 . . PUNCT ejpam-5896 636 1 abd abd PROPN ejpam-5896 636 2 el el PROPN ejpam-5896 636 3 - - PROPN ejpam-5896 636 4 latif latif PROPN ejpam-5896 636 5 et et PROPN ejpam-5896 636 6 al al PROPN ejpam-5896 636 7 . . PUNCT ejpam-5896 636 8 / / SYM ejpam-5896 636 9 eur eur PROPN ejpam-5896 636 10 . . PUNCT ejpam-5896 637 1 j. j. PROPN ejpam-5896 637 2 pure pure PROPN ejpam-5896 637 3 appl appl PROPN ejpam-5896 637 4 . . PROPN ejpam-5896 637 5 math math PROPN ejpam-5896 637 6 , , PUNCT ejpam-5896 637 7 18 18 NUM ejpam-5896 637 8 ( ( PUNCT ejpam-5896 637 9 2 2 NUM ejpam-5896 637 10 ) ) PUNCT ejpam-5896 637 11 ( ( PUNCT ejpam-5896 637 12 2025 2025 NUM ejpam-5896 637 13 ) ) PUNCT ejpam-5896 637 14 , , PUNCT ejpam-5896 637 15 5896 5896 NUM ejpam-5896 637 16 18 18 NUM ejpam-5896 637 17 of of ADP ejpam-5896 637 18 20 20 NUM ejpam-5896 637 19 [ [ SYM ejpam-5896 637 20 10 10 NUM ejpam-5896 637 21 ] ] X ejpam-5896 637 22 n. n. PROPN ejpam-5896 637 23 çağman çağman PROPN ejpam-5896 637 24 and and CCONJ ejpam-5896 637 25 s. s. PROPN ejpam-5896 637 26 enginoglu enginoglu PROPN ejpam-5896 637 27 . . PUNCT ejpam-5896 638 1 soft soft ADJ ejpam-5896 638 2 matrix matrix NOUN ejpam-5896 638 3 theory theory NOUN ejpam-5896 638 4 and and CCONJ ejpam-5896 638 5 its its PRON ejpam-5896 638 6 decision decision NOUN ejpam-5896 638 7 making making NOUN ejpam-5896 638 8 . . PUNCT ejpam-5896 639 1 computers computer NOUN ejpam-5896 639 2 & & CCONJ ejpam-5896 639 3 mathematics mathematics PROPN ejpam-5896 639 4 with with ADP ejpam-5896 639 5 applications application NOUN ejpam-5896 639 6 , , PUNCT ejpam-5896 639 7 59:3308–3314 59:3308–3314 NUM ejpam-5896 639 8 , , PUNCT ejpam-5896 639 9 2010 2010 NUM ejpam-5896 639 10 . . PUNCT ejpam-5896 640 1 [ [ X ejpam-5896 640 2 11 11 NUM ejpam-5896 640 3 ] ] PUNCT ejpam-5896 640 4 m. m. NOUN ejpam-5896 640 5 shabir shabir PROPN ejpam-5896 640 6 and and CCONJ ejpam-5896 640 7 m. m. PROPN ejpam-5896 640 8 naz naz PROPN ejpam-5896 640 9 . . PUNCT ejpam-5896 641 1 on on ADP ejpam-5896 641 2 soft soft ADJ ejpam-5896 641 3 topological topological ADJ ejpam-5896 641 4 spaces space NOUN ejpam-5896 641 5 . . PUNCT ejpam-5896 642 1 computers computer NOUN ejpam-5896 642 2 & & CCONJ ejpam-5896 642 3 mathematics mathematics PROPN ejpam-5896 642 4 with with ADP ejpam-5896 642 5 applications application NOUN ejpam-5896 642 6 , , PUNCT ejpam-5896 642 7 61:1786–1799 61:1786–1799 NUM ejpam-5896 642 8 , , PUNCT ejpam-5896 642 9 2011 2011 NUM ejpam-5896 642 10 . . PUNCT ejpam-5896 643 1 [ [ X ejpam-5896 643 2 12 12 NUM ejpam-5896 643 3 ] ] X ejpam-5896 643 4 n. n. PROPN ejpam-5896 643 5 çağman çağman PROPN ejpam-5896 643 6 , , PUNCT ejpam-5896 643 7 s. s. PROPN ejpam-5896 643 8 karataş karataş PROPN ejpam-5896 643 9 , , PUNCT ejpam-5896 643 10 and and CCONJ ejpam-5896 643 11 s. s. PROPN ejpam-5896 643 12 enginoglu enginoglu PROPN ejpam-5896 643 13 . . PUNCT ejpam-5896 643 14 soft soft ADJ ejpam-5896 643 15 topology topology NOUN ejpam-5896 643 16 . . PUNCT ejpam-5896 644 1 computers computer NOUN ejpam-5896 644 2 & & CCONJ ejpam-5896 644 3 mathematics mathematics PROPN ejpam-5896 644 4 with with ADP ejpam-5896 644 5 applications application NOUN ejpam-5896 644 6 , , PUNCT ejpam-5896 644 7 62:351–358 62:351–358 PROPN ejpam-5896 644 8 , , PUNCT ejpam-5896 644 9 2011 2011 NUM ejpam-5896 644 10 . . PUNCT ejpam-5896 645 1 [ [ X ejpam-5896 645 2 13 13 NUM ejpam-5896 645 3 ] ] PUNCT ejpam-5896 645 4 zanyar zanyar PROPN ejpam-5896 645 5 a. a. NOUN ejpam-5896 645 6 ameen ameen PROPN ejpam-5896 645 7 and and CCONJ ejpam-5896 645 8 s. s. PROPN ejpam-5896 645 9 al al PROPN ejpam-5896 645 10 ghour ghour PROPN ejpam-5896 645 11 . . PUNCT ejpam-5896 646 1 cluster cluster NOUN ejpam-5896 646 2 soft soft ADJ ejpam-5896 646 3 sets set NOUN ejpam-5896 646 4 and and CCONJ ejpam-5896 646 5 cluster cluster NOUN ejpam-5896 646 6 soft soft ADJ ejpam-5896 646 7 topologies topology NOUN ejpam-5896 646 8 . . PUNCT ejpam-5896 647 1 computational computational ADJ ejpam-5896 647 2 and and CCONJ ejpam-5896 647 3 applied applied ADJ ejpam-5896 647 4 mathematics mathematic NOUN ejpam-5896 647 5 , , PUNCT ejpam-5896 647 6 42:337 42:337 NUM ejpam-5896 647 7 , , PUNCT ejpam-5896 647 8 2023 2023 NUM ejpam-5896 647 9 . . PUNCT ejpam-5896 648 1 [ [ X ejpam-5896 648 2 14 14 NUM ejpam-5896 648 3 ] ] X ejpam-5896 648 4 b. b. PROPN ejpam-5896 648 5 ahmad ahmad PROPN ejpam-5896 648 6 and and CCONJ ejpam-5896 648 7 a. a. PROPN ejpam-5896 648 8 kharal kharal PROPN ejpam-5896 648 9 . . PUNCT ejpam-5896 649 1 mappings mapping NOUN ejpam-5896 649 2 on on ADP ejpam-5896 649 3 soft soft ADJ ejpam-5896 649 4 classes class NOUN ejpam-5896 649 5 . . PUNCT ejpam-5896 650 1 new new ADJ ejpam-5896 650 2 mathematics mathematic NOUN ejpam-5896 650 3 and and CCONJ ejpam-5896 650 4 natural natural ADJ ejpam-5896 650 5 computation computation NOUN ejpam-5896 650 6 , , PUNCT ejpam-5896 650 7 7(3):471–481 7(3):471–481 NUM ejpam-5896 650 8 , , PUNCT ejpam-5896 650 9 2011 2011 NUM ejpam-5896 650 10 . . PUNCT ejpam-5896 651 1 [ [ X ejpam-5896 651 2 15 15 NUM ejpam-5896 651 3 ] ] PUNCT ejpam-5896 651 4 z. z. PROPN ejpam-5896 651 5 a. a. NOUN ejpam-5896 651 6 ameen ameen PROPN ejpam-5896 651 7 and and CCONJ ejpam-5896 651 8 m. m. PROPN ejpam-5896 651 9 h. h. PROPN ejpam-5896 651 10 alqahtani alqahtani PROPN ejpam-5896 651 11 . . PUNCT ejpam-5896 652 1 some some DET ejpam-5896 652 2 classes class NOUN ejpam-5896 652 3 of of ADP ejpam-5896 652 4 soft soft ADJ ejpam-5896 652 5 functions function NOUN ejpam-5896 652 6 defined define VERB ejpam-5896 652 7 by by ADP ejpam-5896 652 8 soft soft ADJ ejpam-5896 652 9 open open ADJ ejpam-5896 652 10 sets set NOUN ejpam-5896 652 11 modulo modulo VERB ejpam-5896 652 12 soft soft ADJ ejpam-5896 652 13 sets set NOUN ejpam-5896 652 14 of of ADP ejpam-5896 652 15 the the DET ejpam-5896 652 16 first first ADJ ejpam-5896 652 17 category category NOUN ejpam-5896 652 18 . . PUNCT ejpam-5896 653 1 mathematics mathematic NOUN ejpam-5896 653 2 , , PUNCT ejpam-5896 653 3 11:4368 11:4368 NUM ejpam-5896 653 4 , , PUNCT ejpam-5896 653 5 2023 2023 NUM ejpam-5896 653 6 . . PUNCT ejpam-5896 654 1 [ [ X ejpam-5896 654 2 16 16 NUM ejpam-5896 654 3 ] ] X ejpam-5896 654 4 i. i. PROPN ejpam-5896 654 5 zorlutuna zorlutuna PROPN ejpam-5896 654 6 and and CCONJ ejpam-5896 654 7 h. h. PROPN ejpam-5896 654 8 çakir çakir PROPN ejpam-5896 654 9 . . PUNCT ejpam-5896 655 1 on on ADP ejpam-5896 655 2 continuity continuity NOUN ejpam-5896 655 3 of of ADP ejpam-5896 655 4 soft soft ADJ ejpam-5896 655 5 mappings mapping NOUN ejpam-5896 655 6 . . PUNCT ejpam-5896 656 1 applied apply VERB ejpam-5896 656 2 mathematics mathematics PROPN ejpam-5896 656 3 & & CCONJ ejpam-5896 656 4 information information NOUN ejpam-5896 656 5 sciences sciences PROPN ejpam-5896 656 6 , , PUNCT ejpam-5896 656 7 9:403–409 9:403–409 NOUN ejpam-5896 656 8 , , PUNCT ejpam-5896 656 9 2015 2015 NUM ejpam-5896 656 10 . . PUNCT ejpam-5896 657 1 [ [ X ejpam-5896 657 2 17 17 NUM ejpam-5896 657 3 ] ] PUNCT ejpam-5896 657 4 s. s. PROPN ejpam-5896 657 5 al al PROPN ejpam-5896 657 6 ghour ghour PROPN ejpam-5896 657 7 . . PUNCT ejpam-5896 658 1 soft soft ADJ ejpam-5896 658 2 ω ω NOUN ejpam-5896 658 3 - - PUNCT ejpam-5896 658 4 continuity continuity NOUN ejpam-5896 658 5 and and CCONJ ejpam-5896 658 6 soft soft ADJ ejpam-5896 658 7 ωs ω NOUN ejpam-5896 658 8 - - NOUN ejpam-5896 658 9 continuity continuity NOUN ejpam-5896 658 10 in in ADP ejpam-5896 658 11 soft soft ADJ ejpam-5896 658 12 topological topological ADJ ejpam-5896 658 13 spaces space NOUN ejpam-5896 658 14 . . PUNCT ejpam-5896 659 1 international international ADJ ejpam-5896 659 2 journal journal NOUN ejpam-5896 659 3 of of ADP ejpam-5896 659 4 fuzzy fuzzy ADJ ejpam-5896 659 5 logic logic NOUN ejpam-5896 659 6 and and CCONJ ejpam-5896 659 7 intelligent intelligent ADJ ejpam-5896 659 8 systems system NOUN ejpam-5896 659 9 , , PUNCT ejpam-5896 659 10 22(2):183–192 22(2):183–192 NOUN ejpam-5896 659 11 , , PUNCT ejpam-5896 659 12 2022 2022 NUM ejpam-5896 659 13 . . PUNCT ejpam-5896 660 1 [ [ X ejpam-5896 660 2 18 18 NUM ejpam-5896 660 3 ] ] PUNCT ejpam-5896 660 4 s. s. PROPN ejpam-5896 660 5 al al PROPN ejpam-5896 660 6 ghour ghour PROPN ejpam-5896 660 7 and and CCONJ ejpam-5896 660 8 b. b. PROPN ejpam-5896 660 9 irshidat irshidat PROPN ejpam-5896 660 10 . . PUNCT ejpam-5896 661 1 on on ADP ejpam-5896 661 2 θω θω ADP ejpam-5896 661 3 continuity continuity NOUN ejpam-5896 661 4 . . PUNCT ejpam-5896 662 1 heliyon heliyon NOUN ejpam-5896 662 2 , , PUNCT ejpam-5896 662 3 6(2):e03349 6(2):e03349 NUM ejpam-5896 662 4 , , PUNCT ejpam-5896 662 5 2020 2020 NUM ejpam-5896 662 6 . . PUNCT ejpam-5896 663 1 [ [ X ejpam-5896 663 2 19 19 NUM ejpam-5896 663 3 ] ] PUNCT ejpam-5896 663 4 s. s. PROPN ejpam-5896 663 5 a. a. PROPN ejpam-5896 663 6 el el PROPN ejpam-5896 663 7 - - PUNCT ejpam-5896 663 8 sheikh sheikh PROPN ejpam-5896 663 9 and and CCONJ ejpam-5896 663 10 a. a. NOUN ejpam-5896 663 11 m. m. PROPN ejpam-5896 663 12 el el PROPN ejpam-5896 663 13 - - PROPN ejpam-5896 663 14 latif latif PROPN ejpam-5896 663 15 . . PUNCT ejpam-5896 664 1 characterization characterization NOUN ejpam-5896 664 2 of of ADP ejpam-5896 664 3 b b NOUN ejpam-5896 664 4 - - PUNCT ejpam-5896 664 5 open open ADJ ejpam-5896 664 6 soft soft ADJ ejpam-5896 664 7 sets set NOUN ejpam-5896 664 8 in in ADP ejpam-5896 664 9 soft soft ADJ ejpam-5896 664 10 topological topological ADJ ejpam-5896 664 11 spaces space NOUN ejpam-5896 664 12 . . PUNCT ejpam-5896 665 1 journal journal NOUN ejpam-5896 665 2 of of ADP ejpam-5896 665 3 new new ADJ ejpam-5896 665 4 theory theory NOUN ejpam-5896 665 5 , , PUNCT ejpam-5896 665 6 2:8–18 2:8–18 NUM ejpam-5896 665 7 , , PUNCT ejpam-5896 665 8 2015 2015 NUM ejpam-5896 665 9 . . PUNCT ejpam-5896 666 1 [ [ X ejpam-5896 666 2 20 20 NUM ejpam-5896 666 3 ] ] PUNCT ejpam-5896 666 4 a. a. NOUN ejpam-5896 666 5 kandil kandil PROPN ejpam-5896 666 6 , , PUNCT ejpam-5896 666 7 o. o. PROPN ejpam-5896 666 8 a. a. PROPN ejpam-5896 666 9 e. e. PROPN ejpam-5896 666 10 tantawy tantawy PROPN ejpam-5896 666 11 , , PUNCT ejpam-5896 666 12 s. s. PROPN ejpam-5896 666 13 a. a. PROPN ejpam-5896 666 14 el el PROPN ejpam-5896 666 15 - - PUNCT ejpam-5896 666 16 sheikh sheikh NOUN ejpam-5896 666 17 , , PUNCT ejpam-5896 666 18 and and CCONJ ejpam-5896 666 19 a. a. NOUN ejpam-5896 666 20 m. m. PROPN ejpam-5896 666 21 abd abd PROPN ejpam-5896 666 22 el el PROPN ejpam-5896 666 23 - - PROPN ejpam-5896 666 24 latif latif PROPN ejpam-5896 666 25 . . PUNCT ejpam-5896 667 1 soft soft ADJ ejpam-5896 667 2 semi semi ADJ ejpam-5896 667 3 separation separation NOUN ejpam-5896 667 4 axioms axiom NOUN ejpam-5896 667 5 and and CCONJ ejpam-5896 667 6 irresolute irresolute ADJ ejpam-5896 667 7 soft soft ADJ ejpam-5896 667 8 functions function NOUN ejpam-5896 667 9 . . PUNCT ejpam-5896 668 1 annals annal NOUN ejpam-5896 668 2 of of ADP ejpam-5896 668 3 fuzzy fuzzy ADJ ejpam-5896 668 4 mathematics mathematic NOUN ejpam-5896 668 5 and and CCONJ ejpam-5896 668 6 informatics informatic NOUN ejpam-5896 668 7 , , PUNCT ejpam-5896 668 8 8(2):305–318 8(2):305–318 NUM ejpam-5896 668 9 , , PUNCT ejpam-5896 668 10 2014 2014 NUM ejpam-5896 668 11 . . PUNCT ejpam-5896 669 1 [ [ X ejpam-5896 669 2 21 21 NUM ejpam-5896 669 3 ] ] X ejpam-5896 669 4 tareq tareq PROPN ejpam-5896 669 5 m. m. PROPN ejpam-5896 669 6 al al PROPN ejpam-5896 669 7 - - PUNCT ejpam-5896 669 8 shami shami PROPN ejpam-5896 669 9 , , PUNCT ejpam-5896 669 10 abdelwaheb abdelwaheb PROPN ejpam-5896 669 11 mhemdi mhemdi PROPN ejpam-5896 669 12 , , PUNCT ejpam-5896 669 13 and and CCONJ ejpam-5896 669 14 radwan radwan VERB ejpam-5896 669 15 abu abu PROPN ejpam-5896 669 16 - - PUNCT ejpam-5896 669 17 gdairi gdairi PROPN ejpam-5896 669 18 . . PUNCT ejpam-5896 670 1 a a DET ejpam-5896 670 2 novel novel ADJ ejpam-5896 670 3 framework framework NOUN ejpam-5896 670 4 for for ADP ejpam-5896 670 5 generalizations generalization NOUN ejpam-5896 670 6 of of ADP ejpam-5896 670 7 soft soft ADJ ejpam-5896 670 8 open open ADJ ejpam-5896 670 9 sets set NOUN ejpam-5896 670 10 and and CCONJ ejpam-5896 670 11 its its PRON ejpam-5896 670 12 applications application NOUN ejpam-5896 670 13 via via ADP ejpam-5896 670 14 soft soft ADJ ejpam-5896 670 15 topologies topology NOUN ejpam-5896 670 16 . . PUNCT ejpam-5896 671 1 mathematics mathematic NOUN ejpam-5896 671 2 , , PUNCT ejpam-5896 671 3 11(4):840 11(4):840 NOUN ejpam-5896 671 4 , , PUNCT ejpam-5896 671 5 2023 2023 NUM ejpam-5896 671 6 . . PUNCT ejpam-5896 672 1 [ [ X ejpam-5896 672 2 22 22 NUM ejpam-5896 672 3 ] ] PUNCT ejpam-5896 672 4 a. a. NOUN ejpam-5896 672 5 kandil kandil PROPN ejpam-5896 672 6 , , PUNCT ejpam-5896 672 7 o. o. PROPN ejpam-5896 672 8 a. a. PROPN ejpam-5896 672 9 e. e. PROPN ejpam-5896 672 10 tantawy tantawy PROPN ejpam-5896 672 11 , , PUNCT ejpam-5896 672 12 s. s. PROPN ejpam-5896 672 13 a. a. PROPN ejpam-5896 672 14 el el PROPN ejpam-5896 672 15 - - PUNCT ejpam-5896 672 16 sheikh sheikh NOUN ejpam-5896 672 17 , , PUNCT ejpam-5896 672 18 and and CCONJ ejpam-5896 672 19 a. a. NOUN ejpam-5896 672 20 m. m. PROPN ejpam-5896 672 21 abd abd PROPN ejpam-5896 672 22 el el PROPN ejpam-5896 672 23 - - PROPN ejpam-5896 672 24 latif latif PROPN ejpam-5896 672 25 . . PUNCT ejpam-5896 673 1 γ γ PROPN ejpam-5896 673 2 - - PUNCT ejpam-5896 673 3 operation operation NOUN ejpam-5896 673 4 and and CCONJ ejpam-5896 673 5 decompositions decomposition NOUN ejpam-5896 673 6 of of ADP ejpam-5896 673 7 some some DET ejpam-5896 673 8 forms form NOUN ejpam-5896 673 9 of of ADP ejpam-5896 673 10 soft soft ADJ ejpam-5896 673 11 continuity continuity NOUN ejpam-5896 673 12 in in ADP ejpam-5896 673 13 soft soft ADJ ejpam-5896 673 14 topological topological ADJ ejpam-5896 673 15 spaces space NOUN ejpam-5896 673 16 . . PUNCT ejpam-5896 674 1 annals annal NOUN ejpam-5896 674 2 of of ADP ejpam-5896 674 3 fuzzy fuzzy ADJ ejpam-5896 674 4 mathematics mathematic NOUN ejpam-5896 674 5 and and CCONJ ejpam-5896 674 6 informatics informatic NOUN ejpam-5896 674 7 , , PUNCT ejpam-5896 674 8 7(2):181–196 7(2):181–196 NUM ejpam-5896 674 9 , , PUNCT ejpam-5896 674 10 2014 2014 NUM ejpam-5896 674 11 . . PUNCT ejpam-5896 675 1 [ [ X ejpam-5896 675 2 23 23 NUM ejpam-5896 675 3 ] ] X ejpam-5896 675 4 tareq tareq PROPN ejpam-5896 675 5 m. m. PROPN ejpam-5896 675 6 al al PROPN ejpam-5896 675 7 - - PUNCT ejpam-5896 675 8 shami shami PROPN ejpam-5896 675 9 . . PUNCT ejpam-5896 676 1 soft soft ADJ ejpam-5896 676 2 somewhere somewhere ADV ejpam-5896 676 3 dense dense ADJ ejpam-5896 676 4 sets set NOUN ejpam-5896 676 5 on on ADP ejpam-5896 676 6 soft soft ADJ ejpam-5896 676 7 topological topological ADJ ejpam-5896 676 8 spaces space NOUN ejpam-5896 676 9 . . PUNCT ejpam-5896 677 1 communications communication NOUN ejpam-5896 677 2 of of ADP ejpam-5896 677 3 the the DET ejpam-5896 677 4 korean korean ADJ ejpam-5896 677 5 mathematical mathematical ADJ ejpam-5896 677 6 society society NOUN ejpam-5896 677 7 , , PUNCT ejpam-5896 677 8 33(2):1341–1356 33(2):1341–1356 NUM ejpam-5896 677 9 , , PUNCT ejpam-5896 677 10 2018 2018 NUM ejpam-5896 677 11 . . PUNCT ejpam-5896 678 1 [ [ X ejpam-5896 678 2 24 24 NUM ejpam-5896 678 3 ] ] PUNCT ejpam-5896 678 4 radwan radwan VERB ejpam-5896 678 5 abu abu PROPN ejpam-5896 678 6 - - PUNCT ejpam-5896 678 7 gdairi gdairi PROPN ejpam-5896 678 8 , , PUNCT ejpam-5896 678 9 a. a. PROPN ejpam-5896 678 10 a. a. PROPN ejpam-5896 678 11 azzam azzam PROPN ejpam-5896 678 12 , , PUNCT ejpam-5896 678 13 and and CCONJ ejpam-5896 678 14 ibrahim ibrahim PROPN ejpam-5896 678 15 noaman noaman PROPN ejpam-5896 678 16 . . PUNCT ejpam-5896 679 1 nearly nearly ADV ejpam-5896 679 2 soft soft ADJ ejpam-5896 679 3 β β ADJ ejpam-5896 679 4 - - ADJ ejpam-5896 679 5 open open ADJ ejpam-5896 679 6 sets set NOUN ejpam-5896 679 7 via via ADP ejpam-5896 679 8 soft soft ADJ ejpam-5896 679 9 ditopological ditopological ADJ ejpam-5896 679 10 spaces space NOUN ejpam-5896 679 11 . . PUNCT ejpam-5896 680 1 european european ADJ ejpam-5896 680 2 journal journal PROPN ejpam-5896 680 3 of of ADP ejpam-5896 680 4 pure pure ADJ ejpam-5896 680 5 and and CCONJ ejpam-5896 680 6 applied applied ADJ ejpam-5896 680 7 mathematics mathematic NOUN ejpam-5896 680 8 , , PUNCT ejpam-5896 680 9 15(1):126–134 15(1):126–134 PROPN ejpam-5896 680 10 , , PUNCT ejpam-5896 680 11 2022 2022 NUM ejpam-5896 680 12 . . PUNCT ejpam-5896 681 1 [ [ X ejpam-5896 681 2 25 25 NUM ejpam-5896 681 3 ] ] X ejpam-5896 681 4 tareq tareq PROPN ejpam-5896 681 5 m. m. PROPN ejpam-5896 681 6 al al PROPN ejpam-5896 681 7 - - PUNCT ejpam-5896 681 8 shami shami PROPN ejpam-5896 681 9 , , PUNCT ejpam-5896 681 10 abdelwaheb abdelwaheb PROPN ejpam-5896 681 11 mhemdi mhemdi PROPN ejpam-5896 681 12 , , PUNCT ejpam-5896 681 13 radwan radwan VERB ejpam-5896 681 14 abu abu PROPN ejpam-5896 681 15 - - PUNCT ejpam-5896 681 16 gdairi gdairi PROPN ejpam-5896 681 17 , , PUNCT ejpam-5896 681 18 and and CCONJ ejpam-5896 681 19 mohammed mohammed PROPN ejpam-5896 681 20 e. e. PROPN ejpam-5896 681 21 elshafei elshafei PROPN ejpam-5896 681 22 . . PUNCT ejpam-5896 682 1 compactness compactness NOUN ejpam-5896 682 2 and and CCONJ ejpam-5896 682 3 connectedness connectedness NOUN ejpam-5896 682 4 via via ADP ejpam-5896 682 5 the the DET ejpam-5896 682 6 class class NOUN ejpam-5896 682 7 of of ADP ejpam-5896 682 8 soft soft ADJ ejpam-5896 682 9 somewhat somewhat ADV ejpam-5896 682 10 open open ADJ ejpam-5896 682 11 sets set NOUN ejpam-5896 682 12 . . PUNCT ejpam-5896 683 1 aims aim VERB ejpam-5896 683 2 mathematics mathematic NOUN ejpam-5896 683 3 , , PUNCT ejpam-5896 683 4 8(1):815–840 8(1):815–840 NUM ejpam-5896 683 5 , , PUNCT ejpam-5896 683 6 2023 2023 NUM ejpam-5896 683 7 . . PUNCT ejpam-5896 684 1 [ [ X ejpam-5896 684 2 26 26 NUM ejpam-5896 684 3 ] ] PUNCT ejpam-5896 684 4 a. a. NOUN ejpam-5896 684 5 kandil kandil PROPN ejpam-5896 684 6 , , PUNCT ejpam-5896 684 7 o. o. PROPN ejpam-5896 684 8 a. a. PROPN ejpam-5896 684 9 e. e. PROPN ejpam-5896 684 10 tantawy tantawy PROPN ejpam-5896 684 11 , , PUNCT ejpam-5896 684 12 s. s. PROPN ejpam-5896 684 13 a. a. PROPN ejpam-5896 684 14 el el PROPN ejpam-5896 684 15 - - PUNCT ejpam-5896 684 16 sheikh sheikh NOUN ejpam-5896 684 17 , , PUNCT ejpam-5896 684 18 and and CCONJ ejpam-5896 684 19 a. a. NOUN ejpam-5896 684 20 m. m. PROPN ejpam-5896 684 21 abd abd PROPN ejpam-5896 684 22 el el PROPN ejpam-5896 684 23 - - PROPN ejpam-5896 684 24 latif latif PROPN ejpam-5896 684 25 . . PUNCT ejpam-5896 685 1 soft soft ADJ ejpam-5896 685 2 ideal ideal ADJ ejpam-5896 685 3 theory theory NOUN ejpam-5896 685 4 , , PUNCT ejpam-5896 685 5 soft soft ADJ ejpam-5896 685 6 local local ADJ ejpam-5896 685 7 function function NOUN ejpam-5896 685 8 and and CCONJ ejpam-5896 685 9 generated generate VERB ejpam-5896 685 10 soft soft ADJ ejpam-5896 685 11 topological topological ADJ ejpam-5896 685 12 spaces space NOUN ejpam-5896 685 13 . . PUNCT ejpam-5896 686 1 applied apply VERB ejpam-5896 686 2 mathematics mathematics PROPN ejpam-5896 686 3 & & CCONJ ejpam-5896 686 4 information information NOUN ejpam-5896 686 5 sciences sciences PROPN ejpam-5896 686 6 , , PUNCT ejpam-5896 686 7 8(4):1595–1603 8(4):1595–1603 NOUN ejpam-5896 686 8 , , PUNCT ejpam-5896 686 9 2014 2014 NUM ejpam-5896 686 10 . . PUNCT ejpam-5896 687 1 [ [ X ejpam-5896 687 2 27 27 NUM ejpam-5896 687 3 ] ] X ejpam-5896 687 4 ahmed ahmed PROPN ejpam-5896 687 5 hadi hadi PROPN ejpam-5896 687 6 hussain hussain PROPN ejpam-5896 687 7 , , PUNCT ejpam-5896 687 8 sameer sameer PROPN ejpam-5896 687 9 annon annon PROPN ejpam-5896 687 10 abbas abbas PROPN ejpam-5896 687 11 , , PUNCT ejpam-5896 687 12 abbas abbas PROPN ejpam-5896 687 13 musleh musleh PROPN ejpam-5896 687 14 salman salman PROPN ejpam-5896 687 15 , , PUNCT ejpam-5896 687 16 and and CCONJ ejpam-5896 687 17 nabeel nabeel PROPN ejpam-5896 687 18 ali ali PROPN ejpam-5896 687 19 hussein hussein PROPN ejpam-5896 687 20 . . PUNCT ejpam-5896 688 1 semi semi VERB ejpam-5896 688 2 soft soft ADJ ejpam-5896 688 3 local local ADJ ejpam-5896 688 4 function function NOUN ejpam-5896 688 5 which which PRON ejpam-5896 688 6 generated generate VERB ejpam-5896 688 7 a a DET ejpam-5896 688 8 new new ADJ ejpam-5896 688 9 topology topology NOUN ejpam-5896 688 10 in in ADP ejpam-5896 688 11 soft soft ADJ ejpam-5896 688 12 ideal ideal ADJ ejpam-5896 688 13 spaces space NOUN ejpam-5896 688 14 . . PUNCT ejpam-5896 689 1 journal journal NOUN ejpam-5896 689 2 of of ADP ejpam-5896 689 3 interdisciplinary interdisciplinary ADJ ejpam-5896 689 4 mathematics mathematic NOUN ejpam-5896 689 5 , , PUNCT ejpam-5896 689 6 22(8):1509–1517 22(8):1509–1517 PROPN ejpam-5896 689 7 , , PUNCT ejpam-5896 689 8 2019 2019 NUM ejpam-5896 689 9 . . PUNCT ejpam-5896 690 1 [ [ X ejpam-5896 690 2 28 28 NUM ejpam-5896 690 3 ] ] X ejpam-5896 690 4 f. f. PROPN ejpam-5896 690 5 gharib gharib PROPN ejpam-5896 690 6 and and CCONJ ejpam-5896 690 7 a. a. NOUN ejpam-5896 690 8 m. m. PROPN ejpam-5896 690 9 abd abd PROPN ejpam-5896 690 10 el el PROPN ejpam-5896 690 11 - - PROPN ejpam-5896 690 12 latif latif PROPN ejpam-5896 690 13 . . PUNCT ejpam-5896 691 1 soft soft ADJ ejpam-5896 691 2 semi semi ADJ ejpam-5896 691 3 local local ADJ ejpam-5896 691 4 functions function NOUN ejpam-5896 691 5 in in ADP ejpam-5896 691 6 soft soft ADJ ejpam-5896 691 7 ideal ideal ADJ ejpam-5896 691 8 topological topological ADJ ejpam-5896 691 9 spaces space NOUN ejpam-5896 691 10 . . PUNCT ejpam-5896 692 1 european european ADJ ejpam-5896 692 2 journal journal PROPN ejpam-5896 692 3 of of ADP ejpam-5896 692 4 pure pure ADJ ejpam-5896 692 5 and and CCONJ ejpam-5896 692 6 applied applied ADJ ejpam-5896 692 7 mathematics mathematic NOUN ejpam-5896 692 8 , , PUNCT ejpam-5896 692 9 12(3):857–869 12(3):857–869 NUM ejpam-5896 692 10 , , PUNCT ejpam-5896 692 11 2019 2019 NUM ejpam-5896 692 12 . . PUNCT ejpam-5896 693 1 [ [ X ejpam-5896 693 2 29 29 NUM ejpam-5896 693 3 ] ] PUNCT ejpam-5896 693 4 a. a. NOUN ejpam-5896 693 5 m. m. PROPN ejpam-5896 693 6 abd abd PROPN ejpam-5896 693 7 el el PROPN ejpam-5896 693 8 - - PROPN ejpam-5896 693 9 latif latif PROPN ejpam-5896 693 10 . . PUNCT ejpam-5896 694 1 generalized generalize VERB ejpam-5896 694 2 soft soft ADJ ejpam-5896 694 3 rough rough ADJ ejpam-5896 694 4 sets set NOUN ejpam-5896 694 5 and and CCONJ ejpam-5896 694 6 generated generate VERB ejpam-5896 694 7 soft soft ADJ ejpam-5896 694 8 ideal ideal NOUN ejpam-5896 694 9 rough rough ADJ ejpam-5896 694 10 topological topological ADJ ejpam-5896 694 11 spaces space NOUN ejpam-5896 694 12 . . PUNCT ejpam-5896 695 1 journal journal NOUN ejpam-5896 695 2 of of ADP ejpam-5896 695 3 intelligent intelligent ADJ ejpam-5896 695 4 and and CCONJ ejpam-5896 695 5 fuzzy fuzzy ADJ ejpam-5896 695 6 systems system NOUN ejpam-5896 695 7 , , PUNCT ejpam-5896 695 8 34:517–524 34:517–524 PROPN ejpam-5896 695 9 , , PUNCT ejpam-5896 695 10 2018 2018 NUM ejpam-5896 695 11 . . PUNCT ejpam-5896 696 1 [ [ X ejpam-5896 696 2 30 30 NUM ejpam-5896 696 3 ] ] X ejpam-5896 696 4 m. m. NOUN ejpam-5896 696 5 akdag akdag PROPN ejpam-5896 696 6 and and CCONJ ejpam-5896 696 7 f. f. PROPN ejpam-5896 696 8 erol erol PROPN ejpam-5896 696 9 . . PUNCT ejpam-5896 697 1 soft soft ADJ ejpam-5896 697 2 i i NOUN ejpam-5896 697 3 - - PUNCT ejpam-5896 697 4 sets set NOUN ejpam-5896 697 5 and and CCONJ ejpam-5896 697 6 soft soft ADJ ejpam-5896 697 7 i i NOUN ejpam-5896 697 8 - - PUNCT ejpam-5896 697 9 continuity continuity NOUN ejpam-5896 697 10 of of ADP ejpam-5896 697 11 functions function NOUN ejpam-5896 697 12 . . PUNCT ejpam-5896 698 1 gazi gazi PROPN ejpam-5896 698 2 university university PROPN ejpam-5896 698 3 journal journal PROPN ejpam-5896 698 4 of of ADP ejpam-5896 698 5 science science NOUN ejpam-5896 698 6 , , PUNCT ejpam-5896 698 7 27:923–932 27:923–932 NUM ejpam-5896 698 8 , , PUNCT ejpam-5896 698 9 2014 2014 NUM ejpam-5896 698 10 . . PUNCT ejpam-5896 699 1 [ [ X ejpam-5896 699 2 31 31 NUM ejpam-5896 699 3 ] ] PUNCT ejpam-5896 699 4 a. a. NOUN ejpam-5896 699 5 kandil kandil PROPN ejpam-5896 699 6 , , PUNCT ejpam-5896 699 7 o. o. PROPN ejpam-5896 699 8 a. a. PROPN ejpam-5896 699 9 e. e. PROPN ejpam-5896 699 10 tantawy tantawy PROPN ejpam-5896 699 11 , , PUNCT ejpam-5896 699 12 s. s. PROPN ejpam-5896 699 13 a. a. PROPN ejpam-5896 699 14 el el PROPN ejpam-5896 699 15 - - PUNCT ejpam-5896 699 16 sheikh sheikh NOUN ejpam-5896 699 17 , , PUNCT ejpam-5896 699 18 and and CCONJ ejpam-5896 699 19 a. a. NOUN ejpam-5896 699 20 m. m. PROPN ejpam-5896 699 21 abd abd PROPN ejpam-5896 699 22 el el PROPN ejpam-5896 699 23 - - PROPN ejpam-5896 699 24 latif latif PROPN ejpam-5896 699 25 . . PUNCT ejpam-5896 700 1 γ γ PROPN ejpam-5896 700 2 - - PUNCT ejpam-5896 700 3 operation operation NOUN ejpam-5896 700 4 and and CCONJ ejpam-5896 700 5 decompositions decomposition NOUN ejpam-5896 700 6 of of ADP ejpam-5896 700 7 some some DET ejpam-5896 700 8 forms form NOUN ejpam-5896 700 9 of of ADP ejpam-5896 700 10 soft soft ADJ ejpam-5896 700 11 continuity continuity NOUN ejpam-5896 700 12 of of ADP ejpam-5896 700 13 soft soft ADJ ejpam-5896 700 14 topological topological ADJ ejpam-5896 700 15 spaces space NOUN ejpam-5896 700 16 via via ADP ejpam-5896 700 17 soft soft ADJ ejpam-5896 700 18 ideal ideal NOUN ejpam-5896 700 19 . . PUNCT ejpam-5896 701 1 annals annal NOUN ejpam-5896 701 2 of of ADP ejpam-5896 701 3 fuzzy fuzzy ADJ ejpam-5896 701 4 mathematics mathematic NOUN ejpam-5896 701 5 and and CCONJ ejpam-5896 701 6 informatics informatic NOUN ejpam-5896 701 7 , , PUNCT ejpam-5896 701 8 9(3):385–402 9(3):385–402 NUM ejpam-5896 701 9 , , PUNCT ejpam-5896 701 10 2015 2015 NUM ejpam-5896 701 11 . . PUNCT ejpam-5896 702 1 [ [ X ejpam-5896 702 2 32 32 NUM ejpam-5896 702 3 ] ] PUNCT ejpam-5896 702 4 a. a. NOUN ejpam-5896 702 5 a. a. PROPN ejpam-5896 702 6 nasef nasef PROPN ejpam-5896 702 7 , , PUNCT ejpam-5896 702 8 m. m. NOUN ejpam-5896 702 9 parimala parimala PROPN ejpam-5896 702 10 , , PUNCT ejpam-5896 702 11 r. r. PROPN ejpam-5896 702 12 jeevitha jeevitha PROPN ejpam-5896 702 13 , , PUNCT ejpam-5896 702 14 and and CCONJ ejpam-5896 702 15 m. m. PROPN ejpam-5896 702 16 k. k. PROPN ejpam-5896 703 1 el el PROPN ejpam-5896 703 2 - - PUNCT ejpam-5896 703 3 sayed say VERB ejpam-5896 703 4 . . PUNCT ejpam-5896 703 5 soft soft ADJ ejpam-5896 703 6 ideal ideal ADJ ejpam-5896 703 7 theory theory NOUN ejpam-5896 703 8 and and CCONJ ejpam-5896 703 9 appliabd appliabd NOUN ejpam-5896 703 10 el el PROPN ejpam-5896 703 11 - - PROPN ejpam-5896 703 12 latif latif PROPN ejpam-5896 703 13 et et PROPN ejpam-5896 703 14 al al PROPN ejpam-5896 703 15 . . PUNCT ejpam-5896 703 16 / / SYM ejpam-5896 703 17 eur eur PROPN ejpam-5896 703 18 . . PUNCT ejpam-5896 704 1 j. j. PROPN ejpam-5896 704 2 pure pure PROPN ejpam-5896 704 3 appl appl PROPN ejpam-5896 704 4 . . PROPN ejpam-5896 704 5 math math PROPN ejpam-5896 704 6 , , PUNCT ejpam-5896 704 7 18 18 NUM ejpam-5896 704 8 ( ( PUNCT ejpam-5896 704 9 2 2 NUM ejpam-5896 704 10 ) ) PUNCT ejpam-5896 704 11 ( ( PUNCT ejpam-5896 704 12 2025 2025 NUM ejpam-5896 704 13 ) ) PUNCT ejpam-5896 704 14 , , PUNCT ejpam-5896 704 15 5896 5896 NUM ejpam-5896 704 16 19 19 NUM ejpam-5896 704 17 of of ADP ejpam-5896 704 18 20 20 NUM ejpam-5896 704 19 cations cation NOUN ejpam-5896 704 20 . . PUNCT ejpam-5896 705 1 international international ADJ ejpam-5896 705 2 journal journal PROPN ejpam-5896 705 3 of of ADP ejpam-5896 705 4 nonlinear nonlinear ADJ ejpam-5896 705 5 analysis analysis NOUN ejpam-5896 705 6 and and CCONJ ejpam-5896 705 7 applications application NOUN ejpam-5896 705 8 , , PUNCT ejpam-5896 705 9 13(2):1335–1342 13(2):1335–1342 NUM ejpam-5896 705 10 , , PUNCT ejpam-5896 705 11 2022 2022 NUM ejpam-5896 705 12 . . PUNCT ejpam-5896 706 1 [ [ X ejpam-5896 706 2 33 33 NUM ejpam-5896 706 3 ] ] PUNCT ejpam-5896 706 4 a. a. NOUN ejpam-5896 706 5 kandil kandil PROPN ejpam-5896 706 6 , , PUNCT ejpam-5896 706 7 o. o. PROPN ejpam-5896 706 8 a. a. PROPN ejpam-5896 706 9 e. e. PROPN ejpam-5896 706 10 tantawy tantawy PROPN ejpam-5896 706 11 , , PUNCT ejpam-5896 706 12 s. s. PROPN ejpam-5896 706 13 a. a. PROPN ejpam-5896 706 14 el el PROPN ejpam-5896 706 15 - - PUNCT ejpam-5896 706 16 sheikh sheikh NOUN ejpam-5896 706 17 , , PUNCT ejpam-5896 706 18 and and CCONJ ejpam-5896 706 19 a. a. NOUN ejpam-5896 706 20 m. m. PROPN ejpam-5896 706 21 abd abd PROPN ejpam-5896 706 22 el el PROPN ejpam-5896 706 23 - - PROPN ejpam-5896 706 24 latif latif PROPN ejpam-5896 706 25 . . PUNCT ejpam-5896 707 1 soft soft ADJ ejpam-5896 707 2 regularity regularity NOUN ejpam-5896 707 3 and and CCONJ ejpam-5896 707 4 normality normality NOUN ejpam-5896 707 5 based base VERB ejpam-5896 707 6 on on ADP ejpam-5896 707 7 semi semi ADJ ejpam-5896 707 8 open open ADJ ejpam-5896 707 9 soft soft ADJ ejpam-5896 707 10 sets set NOUN ejpam-5896 707 11 and and CCONJ ejpam-5896 707 12 soft soft ADJ ejpam-5896 707 13 ideals ideal NOUN ejpam-5896 707 14 . . PUNCT ejpam-5896 708 1 applied apply VERB ejpam-5896 708 2 mathematics mathematics PROPN ejpam-5896 708 3 & & CCONJ ejpam-5896 708 4 information information PROPN ejpam-5896 708 5 sciences sciences PROPN ejpam-5896 708 6 letters letter NOUN ejpam-5896 708 7 , , PUNCT ejpam-5896 708 8 3(2):47–55 3(2):47–55 NUM ejpam-5896 708 9 , , PUNCT ejpam-5896 708 10 2015 2015 NUM ejpam-5896 708 11 . . PUNCT ejpam-5896 709 1 [ [ X ejpam-5896 709 2 34 34 NUM ejpam-5896 709 3 ] ] PUNCT ejpam-5896 709 4 a. a. NOUN ejpam-5896 709 5 kandil kandil PROPN ejpam-5896 709 6 , , PUNCT ejpam-5896 709 7 o. o. PROPN ejpam-5896 709 8 a. a. PROPN ejpam-5896 709 9 e. e. PROPN ejpam-5896 709 10 tantawy tantawy PROPN ejpam-5896 709 11 , , PUNCT ejpam-5896 709 12 s. s. PROPN ejpam-5896 709 13 a. a. PROPN ejpam-5896 709 14 el el PROPN ejpam-5896 709 15 - - PUNCT ejpam-5896 709 16 sheikh sheikh NOUN ejpam-5896 709 17 , , PUNCT ejpam-5896 709 18 and and CCONJ ejpam-5896 709 19 a. a. NOUN ejpam-5896 709 20 m. m. PROPN ejpam-5896 709 21 abd abd PROPN ejpam-5896 709 22 el el PROPN ejpam-5896 709 23 - - PROPN ejpam-5896 709 24 latif latif PROPN ejpam-5896 709 25 . . PUNCT ejpam-5896 710 1 soft soft ADJ ejpam-5896 710 2 semi semi ADJ ejpam-5896 710 3 ( ( PUNCT ejpam-5896 710 4 quasi quasi ADJ ejpam-5896 710 5 ) ) PUNCT ejpam-5896 710 6 hausdorff hausdorff NOUN ejpam-5896 710 7 spaces space NOUN ejpam-5896 710 8 via via ADP ejpam-5896 710 9 soft soft ADJ ejpam-5896 710 10 ideals ideal NOUN ejpam-5896 710 11 . . PUNCT ejpam-5896 711 1 south south ADJ ejpam-5896 711 2 asian asian PROPN ejpam-5896 711 3 journal journal PROPN ejpam-5896 711 4 of of ADP ejpam-5896 711 5 mathematics mathematic NOUN ejpam-5896 711 6 , , PUNCT ejpam-5896 711 7 4(6):265–284 4(6):265–284 NUM ejpam-5896 711 8 , , PUNCT ejpam-5896 711 9 2014 2014 NUM ejpam-5896 711 10 . . PUNCT ejpam-5896 712 1 [ [ X ejpam-5896 712 2 35 35 NUM ejpam-5896 712 3 ] ] X ejpam-5896 712 4 weijian weijian PROPN ejpam-5896 712 5 rong rong PROPN ejpam-5896 712 6 and and CCONJ ejpam-5896 712 7 fucai fucai PROPN ejpam-5896 712 8 lin lin PROPN ejpam-5896 712 9 . . PUNCT ejpam-5896 713 1 soft soft ADJ ejpam-5896 713 2 connected connect VERB ejpam-5896 713 3 spaces space NOUN ejpam-5896 713 4 and and CCONJ ejpam-5896 713 5 soft soft ADJ ejpam-5896 713 6 paracompact paracompact ADJ ejpam-5896 713 7 spaces space NOUN ejpam-5896 713 8 . . PUNCT ejpam-5896 714 1 international international ADJ ejpam-5896 714 2 journal journal PROPN ejpam-5896 714 3 of of ADP ejpam-5896 714 4 applied apply VERB ejpam-5896 714 5 mathematics mathematic NOUN ejpam-5896 714 6 and and CCONJ ejpam-5896 714 7 statistics statistic NOUN ejpam-5896 714 8 , , PUNCT ejpam-5896 714 9 51:667–681 51:667–681 PROPN ejpam-5896 714 10 , , PUNCT ejpam-5896 714 11 2013 2013 NUM ejpam-5896 714 12 . . PUNCT ejpam-5896 715 1 [ [ X ejpam-5896 715 2 36 36 NUM ejpam-5896 715 3 ] ] X ejpam-5896 715 4 h. h. PROPN ejpam-5896 715 5 l. l. PROPN ejpam-5896 715 6 yang yang PROPN ejpam-5896 715 7 , , PUNCT ejpam-5896 715 8 x. x. PROPN ejpam-5896 715 9 liao liao PROPN ejpam-5896 715 10 , , PUNCT ejpam-5896 715 11 and and CCONJ ejpam-5896 715 12 s. s. PROPN ejpam-5896 715 13 g. g. PROPN ejpam-5896 715 14 li li PROPN ejpam-5896 715 15 . . PROPN ejpam-5896 716 1 on on ADP ejpam-5896 716 2 soft soft ADJ ejpam-5896 716 3 continuous continuous ADJ ejpam-5896 716 4 mappings mapping NOUN ejpam-5896 716 5 and and CCONJ ejpam-5896 716 6 soft soft ADJ ejpam-5896 716 7 connectedness connectedness NOUN ejpam-5896 716 8 of of ADP ejpam-5896 716 9 soft soft ADJ ejpam-5896 716 10 topological topological ADJ ejpam-5896 716 11 spaces space NOUN ejpam-5896 716 12 . . PUNCT ejpam-5896 717 1 hacettepe hacettepe PROPN ejpam-5896 717 2 journal journal PROPN ejpam-5896 717 3 of of ADP ejpam-5896 717 4 mathematics mathematic NOUN ejpam-5896 717 5 and and CCONJ ejpam-5896 717 6 statistics statistic NOUN ejpam-5896 717 7 , , PUNCT ejpam-5896 717 8 44:385–398 44:385–398 PROPN ejpam-5896 717 9 , , PUNCT ejpam-5896 717 10 2015 2015 NUM ejpam-5896 717 11 . . PUNCT ejpam-5896 718 1 [ [ X ejpam-5896 718 2 37 37 NUM ejpam-5896 718 3 ] ] PUNCT ejpam-5896 718 4 s. s. PROPN ejpam-5896 718 5 s. s. PROPN ejpam-5896 718 6 thakur thakur PROPN ejpam-5896 718 7 and and CCONJ ejpam-5896 718 8 a. a. NOUN ejpam-5896 718 9 s. s. PROPN ejpam-5896 718 10 rajput rajput PROPN ejpam-5896 718 11 . . PUNCT ejpam-5896 719 1 connectedness connectedness NOUN ejpam-5896 719 2 between between ADP ejpam-5896 719 3 soft soft ADJ ejpam-5896 719 4 sets set NOUN ejpam-5896 719 5 . . PUNCT ejpam-5896 720 1 new new ADJ ejpam-5896 720 2 mathematics mathematic NOUN ejpam-5896 720 3 and and CCONJ ejpam-5896 720 4 natural natural ADJ ejpam-5896 720 5 computation computation NOUN ejpam-5896 720 6 , , PUNCT ejpam-5896 720 7 14:53–71 14:53–71 NUM ejpam-5896 720 8 , , PUNCT ejpam-5896 720 9 2018 2018 NUM ejpam-5896 720 10 . . PUNCT ejpam-5896 721 1 [ [ X ejpam-5896 721 2 38 38 NUM ejpam-5896 721 3 ] ] PUNCT ejpam-5896 721 4 s. s. PROPN ejpam-5896 721 5 hussain hussain PROPN ejpam-5896 721 6 . . PUNCT ejpam-5896 722 1 binary binary PROPN ejpam-5896 722 2 soft soft ADJ ejpam-5896 722 3 connected connect VERB ejpam-5896 722 4 spaces space NOUN ejpam-5896 722 5 and and CCONJ ejpam-5896 722 6 an an DET ejpam-5896 722 7 application application NOUN ejpam-5896 722 8 of of ADP ejpam-5896 722 9 binary binary ADJ ejpam-5896 722 10 soft soft ADJ ejpam-5896 722 11 sets set NOUN ejpam-5896 722 12 in in ADP ejpam-5896 722 13 decision decision NOUN ejpam-5896 722 14 making making NOUN ejpam-5896 722 15 problem problem NOUN ejpam-5896 722 16 . . PUNCT ejpam-5896 723 1 fuzzy fuzzy ADJ ejpam-5896 723 2 information information NOUN ejpam-5896 723 3 and and CCONJ ejpam-5896 723 4 engineering engineering NOUN ejpam-5896 723 5 , , PUNCT ejpam-5896 723 6 11:506–521 11:506–521 NUM ejpam-5896 723 7 , , PUNCT ejpam-5896 723 8 2019 2019 NUM ejpam-5896 723 9 . . PUNCT ejpam-5896 724 1 [ [ X ejpam-5896 724 2 39 39 NUM ejpam-5896 724 3 ] ] PUNCT ejpam-5896 724 4 a. a. NOUN ejpam-5896 724 5 kandil kandil PROPN ejpam-5896 724 6 , , PUNCT ejpam-5896 724 7 o. o. PROPN ejpam-5896 724 8 a. a. PROPN ejpam-5896 724 9 e. e. PROPN ejpam-5896 724 10 tantawy tantawy PROPN ejpam-5896 724 11 , , PUNCT ejpam-5896 724 12 s. s. PROPN ejpam-5896 724 13 a. a. PROPN ejpam-5896 724 14 el el PROPN ejpam-5896 724 15 - - PUNCT ejpam-5896 724 16 sheikh sheikh NOUN ejpam-5896 724 17 , , PUNCT ejpam-5896 724 18 and and CCONJ ejpam-5896 724 19 a. a. NOUN ejpam-5896 724 20 m. m. PROPN ejpam-5896 724 21 abd abd PROPN ejpam-5896 724 22 el el PROPN ejpam-5896 724 23 - - PROPN ejpam-5896 724 24 latif latif PROPN ejpam-5896 724 25 . . PUNCT ejpam-5896 725 1 soft soft ADJ ejpam-5896 725 2 connectedness connectedness NOUN ejpam-5896 725 3 via via ADP ejpam-5896 725 4 soft soft ADJ ejpam-5896 725 5 ideals ideal NOUN ejpam-5896 725 6 . . PUNCT ejpam-5896 726 1 journal journal NOUN ejpam-5896 726 2 of of ADP ejpam-5896 726 3 new new ADJ ejpam-5896 726 4 results result NOUN ejpam-5896 726 5 in in ADP ejpam-5896 726 6 science science NOUN ejpam-5896 726 7 , , PUNCT ejpam-5896 726 8 4:90–108 4:90–108 NUM ejpam-5896 726 9 , , PUNCT ejpam-5896 726 10 2014 2014 NUM ejpam-5896 726 11 . . PUNCT ejpam-5896 727 1 [ [ X ejpam-5896 727 2 40 40 NUM ejpam-5896 727 3 ] ] PUNCT ejpam-5896 727 4 a. a. NOUN ejpam-5896 727 5 m. m. PROPN ejpam-5896 727 6 abd abd PROPN ejpam-5896 727 7 el el PROPN ejpam-5896 727 8 - - PROPN ejpam-5896 727 9 latif latif PROPN ejpam-5896 727 10 . . PUNCT ejpam-5896 728 1 soft soft ADJ ejpam-5896 728 2 connectedness connectedness NOUN ejpam-5896 728 3 and and CCONJ ejpam-5896 728 4 irresoluteness irresoluteness NOUN ejpam-5896 728 5 via via ADP ejpam-5896 728 6 β β ADJ ejpam-5896 728 7 - - ADJ ejpam-5896 728 8 open open ADJ ejpam-5896 728 9 soft soft ADJ ejpam-5896 728 10 sets set NOUN ejpam-5896 728 11 . . PUNCT ejpam-5896 729 1 afrika afrika ADJ ejpam-5896 729 2 matematika matematika PROPN ejpam-5896 729 3 , , PUNCT ejpam-5896 729 4 28(5–6):805–821 28(5–6):805–821 NUM ejpam-5896 729 5 , , PUNCT ejpam-5896 729 6 2017 2017 NUM ejpam-5896 729 7 . . PUNCT ejpam-5896 730 1 [ [ X ejpam-5896 730 2 41 41 NUM ejpam-5896 730 3 ] ] X ejpam-5896 730 4 samer samer PROPN ejpam-5896 730 5 al al PROPN ejpam-5896 730 6 - - PROPN ejpam-5896 730 7 ghour ghour PROPN ejpam-5896 730 8 and and CCONJ ejpam-5896 730 9 hanan hanan PROPN ejpam-5896 730 10 al al PROPN ejpam-5896 730 11 - - PUNCT ejpam-5896 730 12 saadi saadi NOUN ejpam-5896 730 13 . . PUNCT ejpam-5896 731 1 soft soft ADJ ejpam-5896 731 2 weakly weakly ADJ ejpam-5896 731 3 connected connected ADJ ejpam-5896 731 4 sets set NOUN ejpam-5896 731 5 and and CCONJ ejpam-5896 731 6 soft soft ADJ ejpam-5896 731 7 weakly weakly ADJ ejpam-5896 731 8 connected connected ADJ ejpam-5896 731 9 components component NOUN ejpam-5896 731 10 . . PUNCT ejpam-5896 732 1 aims aim VERB ejpam-5896 732 2 mathematics mathematic NOUN ejpam-5896 732 3 , , PUNCT ejpam-5896 732 4 9(1):1562–1575 9(1):1562–1575 NUM ejpam-5896 732 5 , , PUNCT ejpam-5896 732 6 2023 2023 NUM ejpam-5896 732 7 . . PUNCT ejpam-5896 733 1 [ [ X ejpam-5896 733 2 42 42 NUM ejpam-5896 733 3 ] ] PUNCT ejpam-5896 733 4 a. a. NOUN ejpam-5896 733 5 aygünoglu aygünoglu PUNCT ejpam-5896 733 6 and and CCONJ ejpam-5896 733 7 h. h. PROPN ejpam-5896 733 8 aygün aygün PROPN ejpam-5896 733 9 . . PUNCT ejpam-5896 734 1 some some DET ejpam-5896 734 2 notes note NOUN ejpam-5896 734 3 on on ADP ejpam-5896 734 4 soft soft ADJ ejpam-5896 734 5 topological topological ADJ ejpam-5896 734 6 spaces space NOUN ejpam-5896 734 7 . . PUNCT ejpam-5896 735 1 neural neural ADJ ejpam-5896 735 2 computing computing NOUN ejpam-5896 735 3 and and CCONJ ejpam-5896 735 4 applications application NOUN ejpam-5896 735 5 , , PUNCT ejpam-5896 735 6 21:113–119 21:113–119 PROPN ejpam-5896 735 7 , , PUNCT ejpam-5896 735 8 2012 2012 NUM ejpam-5896 735 9 . . PUNCT ejpam-5896 736 1 [ [ X ejpam-5896 736 2 43 43 NUM ejpam-5896 736 3 ] ] PUNCT ejpam-5896 736 4 t. t. PROPN ejpam-5896 736 5 hida hida PROPN ejpam-5896 736 6 . . PUNCT ejpam-5896 737 1 a a DET ejpam-5896 737 2 comparison comparison NOUN ejpam-5896 737 3 of of ADP ejpam-5896 737 4 two two NUM ejpam-5896 737 5 formulations formulation NOUN ejpam-5896 737 6 of of ADP ejpam-5896 737 7 soft soft ADJ ejpam-5896 737 8 compactness compactness NOUN ejpam-5896 737 9 . . PUNCT ejpam-5896 738 1 annals annal NOUN ejpam-5896 738 2 of of ADP ejpam-5896 738 3 fuzzy fuzzy ADJ ejpam-5896 738 4 mathematics mathematic NOUN ejpam-5896 738 5 and and CCONJ ejpam-5896 738 6 informatics informatic NOUN ejpam-5896 738 7 , , PUNCT ejpam-5896 738 8 8(4):511–524 8(4):511–524 NUM ejpam-5896 738 9 , , PUNCT ejpam-5896 738 10 2014 2014 NUM ejpam-5896 738 11 . . PUNCT ejpam-5896 739 1 [ [ X ejpam-5896 739 2 44 44 NUM ejpam-5896 739 3 ] ] PUNCT ejpam-5896 739 4 a. a. NOUN ejpam-5896 739 5 kandil kandil PROPN ejpam-5896 739 6 , , PUNCT ejpam-5896 739 7 o. o. PROPN ejpam-5896 739 8 a. a. PROPN ejpam-5896 739 9 e. e. PROPN ejpam-5896 739 10 tantawy tantawy PROPN ejpam-5896 739 11 , , PUNCT ejpam-5896 739 12 s. s. PROPN ejpam-5896 739 13 a. a. PROPN ejpam-5896 739 14 el el PROPN ejpam-5896 739 15 - - PUNCT ejpam-5896 739 16 sheikh sheikh NOUN ejpam-5896 739 17 , , PUNCT ejpam-5896 739 18 and and CCONJ ejpam-5896 739 19 a. a. NOUN ejpam-5896 739 20 m. m. PROPN ejpam-5896 739 21 abd abd PROPN ejpam-5896 739 22 el el PROPN ejpam-5896 739 23 - - PROPN ejpam-5896 739 24 latif latif PROPN ejpam-5896 739 25 . . PUNCT ejpam-5896 740 1 soft soft ADJ ejpam-5896 740 2 semi semi ADJ ejpam-5896 740 3 compactness compactness NOUN ejpam-5896 740 4 via via ADP ejpam-5896 740 5 soft soft ADJ ejpam-5896 740 6 ideals ideal NOUN ejpam-5896 740 7 . . PUNCT ejpam-5896 741 1 applied apply VERB ejpam-5896 741 2 mathematics mathematics PROPN ejpam-5896 741 3 & & CCONJ ejpam-5896 741 4 information information NOUN ejpam-5896 741 5 sciences sciences PROPN ejpam-5896 741 6 , , PUNCT ejpam-5896 741 7 8(5):2297–2306 8(5):2297–2306 PROPN ejpam-5896 741 8 , , PUNCT ejpam-5896 741 9 2014 2014 NUM ejpam-5896 741 10 . . PUNCT ejpam-5896 742 1 [ [ X ejpam-5896 742 2 45 45 NUM ejpam-5896 742 3 ] ] X ejpam-5896 742 4 tareq tareq PROPN ejpam-5896 742 5 m. m. PROPN ejpam-5896 742 6 al al PROPN ejpam-5896 742 7 - - PUNCT ejpam-5896 742 8 shami shami PROPN ejpam-5896 742 9 , , PUNCT ejpam-5896 742 10 abdelwaheb abdelwaheb PROPN ejpam-5896 742 11 mhemdi mhemdi PROPN ejpam-5896 742 12 , , PUNCT ejpam-5896 742 13 amani amani PROPN ejpam-5896 742 14 a. a. NOUN ejpam-5896 742 15 rawshdeh rawshdeh PROPN ejpam-5896 742 16 , , PUNCT ejpam-5896 742 17 and and CCONJ ejpam-5896 742 18 heyam heyam PROPN ejpam-5896 742 19 h. h. PROPN ejpam-5896 742 20 al al PROPN ejpam-5896 742 21 - - PUNCT ejpam-5896 742 22 jarrah jarrah PROPN ejpam-5896 742 23 . . PUNCT ejpam-5896 743 1 soft soft ADJ ejpam-5896 743 2 version version NOUN ejpam-5896 743 3 of of ADP ejpam-5896 743 4 compact compact ADJ ejpam-5896 743 5 and and CCONJ ejpam-5896 743 6 lindelöf lindelöf NOUN ejpam-5896 743 7 spaces space VERB ejpam-5896 743 8 using use VERB ejpam-5896 743 9 soft soft ADJ ejpam-5896 743 10 somewhere somewhere ADV ejpam-5896 743 11 dense dense ADJ ejpam-5896 743 12 sets set NOUN ejpam-5896 743 13 . . PUNCT ejpam-5896 744 1 aims aim VERB ejpam-5896 744 2 mathematics mathematic NOUN ejpam-5896 744 3 , , PUNCT ejpam-5896 744 4 6(8):8064–8077 6(8):8064–8077 NOUN ejpam-5896 744 5 , , PUNCT ejpam-5896 744 6 2021 2021 NUM ejpam-5896 744 7 . . PUNCT ejpam-5896 745 1 [ [ X ejpam-5896 745 2 46 46 NUM ejpam-5896 745 3 ] ] PUNCT ejpam-5896 745 4 t. t. PROPN ejpam-5896 745 5 m. m. PROPN ejpam-5896 745 6 al al PROPN ejpam-5896 745 7 - - PUNCT ejpam-5896 745 8 shami shami PROPN ejpam-5896 745 9 and and CCONJ ejpam-5896 745 10 abdelwaheb abdelwaheb PROPN ejpam-5896 745 11 mhemdi mhemdi PROPN ejpam-5896 745 12 . . PUNCT ejpam-5896 745 13 approximation approximation NOUN ejpam-5896 745 14 operators operator NOUN ejpam-5896 745 15 and and CCONJ ejpam-5896 745 16 accuracy accuracy NOUN ejpam-5896 745 17 measures measure NOUN ejpam-5896 745 18 of of ADP ejpam-5896 745 19 rough rough ADJ ejpam-5896 745 20 sets set NOUN ejpam-5896 745 21 from from ADP ejpam-5896 745 22 an an DET ejpam-5896 745 23 infra infra NOUN ejpam-5896 745 24 - - PUNCT ejpam-5896 745 25 topology topology NOUN ejpam-5896 745 26 view view NOUN ejpam-5896 745 27 . . PUNCT ejpam-5896 746 1 soft soft ADJ ejpam-5896 746 2 computing computing NOUN ejpam-5896 746 3 , , PUNCT ejpam-5896 746 4 27:1317–1330 27:1317–1330 NUM ejpam-5896 746 5 , , PUNCT ejpam-5896 746 6 2023 2023 NUM ejpam-5896 746 7 . . PUNCT ejpam-5896 747 1 [ [ X ejpam-5896 747 2 47 47 NUM ejpam-5896 747 3 ] ] PUNCT ejpam-5896 747 4 a. a. NOUN ejpam-5896 747 5 a. a. PROPN ejpam-5896 747 6 azzam azzam PROPN ejpam-5896 747 7 , , PUNCT ejpam-5896 747 8 zanyar zanyar PROPN ejpam-5896 747 9 a. a. NOUN ejpam-5896 747 10 ameen ameen PROPN ejpam-5896 747 11 , , PUNCT ejpam-5896 747 12 tareq tareq PROPN ejpam-5896 747 13 m. m. PROPN ejpam-5896 747 14 al al PROPN ejpam-5896 747 15 - - PUNCT ejpam-5896 747 16 shami shami PROPN ejpam-5896 747 17 , , PUNCT ejpam-5896 747 18 and and CCONJ ejpam-5896 747 19 mohammed mohammed PROPN ejpam-5896 747 20 e. e. PROPN ejpam-5896 747 21 el el PROPN ejpam-5896 747 22 - - PROPN ejpam-5896 747 23 shafei shafei PROPN ejpam-5896 747 24 . . PUNCT ejpam-5896 748 1 generating generate VERB ejpam-5896 748 2 soft soft ADJ ejpam-5896 748 3 topologies topology NOUN ejpam-5896 748 4 via via ADP ejpam-5896 748 5 soft soft ADJ ejpam-5896 748 6 set set ADJ ejpam-5896 748 7 operators operator NOUN ejpam-5896 748 8 . . PUNCT ejpam-5896 749 1 symmetry symmetry NOUN ejpam-5896 749 2 , , PUNCT ejpam-5896 749 3 14(5):914 14(5):914 NUM ejpam-5896 749 4 , , PUNCT ejpam-5896 749 5 2022 2022 NUM ejpam-5896 749 6 . . PUNCT ejpam-5896 750 1 [ [ X ejpam-5896 750 2 48 48 NUM ejpam-5896 750 3 ] ] PUNCT ejpam-5896 750 4 t. t. PROPN ejpam-5896 750 5 m. m. PROPN ejpam-5896 750 6 al al PROPN ejpam-5896 750 7 - - PUNCT ejpam-5896 750 8 shami shami PROPN ejpam-5896 750 9 and and CCONJ ejpam-5896 750 10 t. t. PROPN ejpam-5896 750 11 noiri noiri PROPN ejpam-5896 750 12 . . PUNCT ejpam-5896 751 1 more more ADJ ejpam-5896 751 2 notions notion NOUN ejpam-5896 751 3 and and CCONJ ejpam-5896 751 4 mappings mapping NOUN ejpam-5896 751 5 via via ADP ejpam-5896 751 6 somewhere somewhere ADJ ejpam-5896 751 7 dense dense ADJ ejpam-5896 751 8 sets set NOUN ejpam-5896 751 9 . . PUNCT ejpam-5896 752 1 afrika afrika ADJ ejpam-5896 752 2 matematika matematika PROPN ejpam-5896 752 3 , , PUNCT ejpam-5896 752 4 30(7):1011–1024 30(7):1011–1024 PROPN ejpam-5896 752 5 , , PUNCT ejpam-5896 752 6 2019 2019 NUM ejpam-5896 752 7 . . PUNCT ejpam-5896 753 1 [ [ X ejpam-5896 753 2 49 49 NUM ejpam-5896 753 3 ] ] PUNCT ejpam-5896 753 4 m. m. PROPN ejpam-5896 753 5 e. e. PROPN ejpam-5896 753 6 el el PROPN ejpam-5896 753 7 - - PROPN ejpam-5896 753 8 shafei shafei PROPN ejpam-5896 753 9 and and CCONJ ejpam-5896 753 10 t. t. PROPN ejpam-5896 753 11 m. m. PROPN ejpam-5896 753 12 al al PROPN ejpam-5896 753 13 - - PUNCT ejpam-5896 753 14 shami shami PROPN ejpam-5896 753 15 . . PUNCT ejpam-5896 754 1 some some DET ejpam-5896 754 2 operators operator NOUN ejpam-5896 754 3 of of ADP ejpam-5896 754 4 a a DET ejpam-5896 754 5 soft soft ADJ ejpam-5896 754 6 set set NOUN ejpam-5896 754 7 and and CCONJ ejpam-5896 754 8 soft soft ADJ ejpam-5896 754 9 connected connected ADJ ejpam-5896 754 10 spaces space NOUN ejpam-5896 754 11 using use VERB ejpam-5896 754 12 soft soft ADJ ejpam-5896 754 13 somewhere somewhere ADV ejpam-5896 754 14 dense dense ADJ ejpam-5896 754 15 sets set NOUN ejpam-5896 754 16 . . PUNCT ejpam-5896 755 1 journal journal NOUN ejpam-5896 755 2 of of ADP ejpam-5896 755 3 interdisciplinary interdisciplinary ADJ ejpam-5896 755 4 mathematics mathematic NOUN ejpam-5896 755 5 , , PUNCT ejpam-5896 755 6 24(6):1471–1495 24(6):1471–1495 NUM ejpam-5896 755 7 , , PUNCT ejpam-5896 755 8 2021 2021 NUM ejpam-5896 755 9 . . PUNCT ejpam-5896 756 1 [ [ X ejpam-5896 756 2 50 50 NUM ejpam-5896 756 3 ] ] PUNCT ejpam-5896 756 4 t. t. PROPN ejpam-5896 756 5 h. h. PROPN ejpam-5896 756 6 jassim jassim PROPN ejpam-5896 756 7 . . PUNCT ejpam-5896 757 1 on on ADP ejpam-5896 757 2 supra supra ADJ ejpam-5896 757 3 compactness compactness NOUN ejpam-5896 757 4 in in ADP ejpam-5896 757 5 supra supra PROPN ejpam-5896 757 6 topological topological ADJ ejpam-5896 757 7 spaces space NOUN ejpam-5896 757 8 . . PUNCT ejpam-5896 758 1 department department NOUN ejpam-5896 758 2 of of ADP ejpam-5896 758 3 mathematics mathematics PROPN ejpam-5896 758 4 , , PUNCT ejpam-5896 758 5 college college NOUN ejpam-5896 758 6 of of ADP ejpam-5896 758 7 computer computer NOUN ejpam-5896 758 8 sciences sciences PROPN ejpam-5896 758 9 and and CCONJ ejpam-5896 758 10 mathematics mathematic NOUN ejpam-5896 758 11 , , PUNCT ejpam-5896 758 12 university university NOUN ejpam-5896 758 13 of of ADP ejpam-5896 758 14 tikrit tikrit NOUN ejpam-5896 758 15 , , PUNCT ejpam-5896 758 16 iraq iraq PROPN ejpam-5896 758 17 , , PUNCT ejpam-5896 758 18 pages page NOUN ejpam-5896 758 19 1–3 1–3 NUM ejpam-5896 758 20 , , PUNCT ejpam-5896 758 21 2008 2008 NUM ejpam-5896 758 22 . . PUNCT ejpam-5896 759 1 [ [ X ejpam-5896 759 2 51 51 NUM ejpam-5896 759 3 ] ] PUNCT ejpam-5896 759 4 s. s. PROPN ejpam-5896 759 5 a. a. PROPN ejpam-5896 759 6 el el PROPN ejpam-5896 759 7 - - PUNCT ejpam-5896 759 8 sheikh sheikh PROPN ejpam-5896 759 9 and and CCONJ ejpam-5896 759 10 a. a. NOUN ejpam-5896 759 11 m. m. NOUN ejpam-5896 759 12 abd abd PROPN ejpam-5896 759 13 el el PROPN ejpam-5896 759 14 - - PROPN ejpam-5896 759 15 latif latif PROPN ejpam-5896 759 16 . . PUNCT ejpam-5896 760 1 decompositions decomposition NOUN ejpam-5896 760 2 of of ADP ejpam-5896 760 3 some some DET ejpam-5896 760 4 types type NOUN ejpam-5896 760 5 of of ADP ejpam-5896 760 6 supra supra ADJ ejpam-5896 760 7 soft soft ADJ ejpam-5896 760 8 sets set NOUN ejpam-5896 760 9 and and CCONJ ejpam-5896 760 10 soft soft ADJ ejpam-5896 760 11 continuity continuity NOUN ejpam-5896 760 12 . . PUNCT ejpam-5896 761 1 international international ADJ ejpam-5896 761 2 journal journal PROPN ejpam-5896 761 3 of of ADP ejpam-5896 761 4 mathematical mathematical ADJ ejpam-5896 761 5 trends trend NOUN ejpam-5896 761 6 and and CCONJ ejpam-5896 761 7 technology technology NOUN ejpam-5896 761 8 , , PUNCT ejpam-5896 761 9 9(1):37–56 9(1):37–56 NUM ejpam-5896 761 10 , , PUNCT ejpam-5896 761 11 2014 2014 NUM ejpam-5896 761 12 . . PUNCT ejpam-5896 762 1 [ [ X ejpam-5896 762 2 52 52 NUM ejpam-5896 762 3 ] ] PUNCT ejpam-5896 762 4 a. a. NOUN ejpam-5896 762 5 m. m. PROPN ejpam-5896 762 6 abd abd PROPN ejpam-5896 762 7 el el PROPN ejpam-5896 762 8 - - PROPN ejpam-5896 762 9 latif latif PROPN ejpam-5896 762 10 and and CCONJ ejpam-5896 762 11 s. s. PROPN ejpam-5896 762 12 karataş. karataş. PROPN ejpam-5896 763 1 supra supra PROPN ejpam-5896 763 2 b b PROPN ejpam-5896 763 3 - - PUNCT ejpam-5896 763 4 open open ADJ ejpam-5896 763 5 soft soft ADJ ejpam-5896 763 6 sets set NOUN ejpam-5896 763 7 and and CCONJ ejpam-5896 763 8 supra supra PROPN ejpam-5896 763 9 b b NOUN ejpam-5896 763 10 - - PUNCT ejpam-5896 763 11 soft soft ADJ ejpam-5896 763 12 continuity continuity NOUN ejpam-5896 763 13 on on ADP ejpam-5896 763 14 soft soft ADJ ejpam-5896 763 15 topological topological ADJ ejpam-5896 763 16 spaces space NOUN ejpam-5896 763 17 . . PUNCT ejpam-5896 764 1 journal journal NOUN ejpam-5896 764 2 of of ADP ejpam-5896 764 3 mathematical mathematical ADJ ejpam-5896 764 4 and and CCONJ ejpam-5896 764 5 computational computational ADJ ejpam-5896 764 6 applications application NOUN ejpam-5896 764 7 research research NOUN ejpam-5896 764 8 , , PUNCT ejpam-5896 764 9 5(1):1–18 5(1):1–18 NUM ejpam-5896 764 10 , , PUNCT ejpam-5896 764 11 2015 2015 NUM ejpam-5896 764 12 . . PUNCT ejpam-5896 765 1 [ [ X ejpam-5896 765 2 53 53 NUM ejpam-5896 765 3 ] ] PUNCT ejpam-5896 765 4 a. a. NOUN ejpam-5896 765 5 m. m. PROPN ejpam-5896 765 6 abd abd PROPN ejpam-5896 765 7 el el PROPN ejpam-5896 765 8 - - PROPN ejpam-5896 765 9 latif latif PROPN ejpam-5896 765 10 and and CCONJ ejpam-5896 765 11 mesfer mesfer VERB ejpam-5896 765 12 h. h. PROPN ejpam-5896 765 13 alqahtani alqahtani PROPN ejpam-5896 765 14 . . PUNCT ejpam-5896 766 1 new new ADJ ejpam-5896 766 2 soft soft ADJ ejpam-5896 766 3 operators operator NOUN ejpam-5896 766 4 related relate VERB ejpam-5896 766 5 to to ADP ejpam-5896 766 6 supra supra PROPN ejpam-5896 766 7 soft soft ADJ ejpam-5896 766 8 δi δi NOUN ejpam-5896 766 9 - - PUNCT ejpam-5896 766 10 open open ADJ ejpam-5896 766 11 sets set NOUN ejpam-5896 766 12 and and CCONJ ejpam-5896 766 13 applications application NOUN ejpam-5896 766 14 . . PUNCT ejpam-5896 767 1 aims aim VERB ejpam-5896 767 2 mathematics mathematic NOUN ejpam-5896 767 3 , , PUNCT ejpam-5896 767 4 9(2):3076–3096 9(2):3076–3096 NUM ejpam-5896 767 5 , , PUNCT ejpam-5896 767 6 2024 2024 NUM ejpam-5896 767 7 . . PUNCT ejpam-5896 768 1 [ [ X ejpam-5896 768 2 54 54 NUM ejpam-5896 768 3 ] ] PUNCT ejpam-5896 768 4 alaa alaa PROPN ejpam-5896 768 5 m. m. PROPN ejpam-5896 768 6 abd abd PROPN ejpam-5896 768 7 el el PROPN ejpam-5896 768 8 - - PROPN ejpam-5896 768 9 latif latif PROPN ejpam-5896 768 10 , , PUNCT ejpam-5896 768 11 mesfer mesfer VERB ejpam-5896 768 12 h. h. PROPN ejpam-5896 768 13 alqahtani alqahtani PROPN ejpam-5896 768 14 , , PUNCT ejpam-5896 768 15 and and CCONJ ejpam-5896 768 16 f. f. PROPN ejpam-5896 768 17 a. a. PROPN ejpam-5896 768 18 gharib gharib PROPN ejpam-5896 768 19 . . PUNCT ejpam-5896 769 1 strictly strictly ADV ejpam-5896 769 2 wider wide ADJ ejpam-5896 769 3 class class NOUN ejpam-5896 769 4 of of ADP ejpam-5896 769 5 soft soft ADJ ejpam-5896 769 6 sets set NOUN ejpam-5896 769 7 via via ADP ejpam-5896 769 8 supra supra PROPN ejpam-5896 769 9 soft soft PROPN ejpam-5896 769 10 δ δ PROPN ejpam-5896 769 11 - - PUNCT ejpam-5896 769 12 closure closure NOUN ejpam-5896 769 13 operator operator NOUN ejpam-5896 769 14 . . PUNCT ejpam-5896 770 1 international international ADJ ejpam-5896 770 2 journal journal NOUN ejpam-5896 770 3 of of ADP ejpam-5896 770 4 analysis analysis NOUN ejpam-5896 770 5 and and CCONJ ejpam-5896 770 6 applications application NOUN ejpam-5896 770 7 , , PUNCT ejpam-5896 770 8 abd abd PROPN ejpam-5896 770 9 el el PROPN ejpam-5896 770 10 - - PROPN ejpam-5896 770 11 latif latif PROPN ejpam-5896 770 12 et et PROPN ejpam-5896 770 13 al al PROPN ejpam-5896 770 14 . . PUNCT ejpam-5896 770 15 / / SYM ejpam-5896 770 16 eur eur PROPN ejpam-5896 770 17 . . PUNCT ejpam-5896 771 1 j. j. PROPN ejpam-5896 771 2 pure pure PROPN ejpam-5896 771 3 appl appl PROPN ejpam-5896 771 4 . . PROPN ejpam-5896 771 5 math math PROPN ejpam-5896 771 6 , , PUNCT ejpam-5896 771 7 18 18 NUM ejpam-5896 771 8 ( ( PUNCT ejpam-5896 771 9 2 2 NUM ejpam-5896 771 10 ) ) PUNCT ejpam-5896 771 11 ( ( PUNCT ejpam-5896 771 12 2025 2025 NUM ejpam-5896 771 13 ) ) PUNCT ejpam-5896 771 14 , , PUNCT ejpam-5896 771 15 5896 5896 NUM ejpam-5896 771 16 20 20 NUM ejpam-5896 771 17 of of ADP ejpam-5896 771 18 20 20 NUM ejpam-5896 771 19 22:47 22:47 NUM ejpam-5896 771 20 , , PUNCT ejpam-5896 771 21 2024 2024 NUM ejpam-5896 771 22 . . PUNCT ejpam-5896 772 1 [ [ X ejpam-5896 772 2 55 55 NUM ejpam-5896 772 3 ] ] PUNCT ejpam-5896 772 4 a. a. NOUN ejpam-5896 772 5 m. m. PROPN ejpam-5896 772 6 abd abd PROPN ejpam-5896 772 7 el el PROPN ejpam-5896 772 8 - - PROPN ejpam-5896 772 9 latif latif PROPN ejpam-5896 772 10 , , PUNCT ejpam-5896 772 11 radwan radwan VERB ejpam-5896 772 12 abu abu PROPN ejpam-5896 772 13 - - PUNCT ejpam-5896 772 14 gdairi gdairi PROPN ejpam-5896 772 15 , , PUNCT ejpam-5896 772 16 a. a. PROPN ejpam-5896 772 17 a. a. PROPN ejpam-5896 772 18 azzam azzam PROPN ejpam-5896 772 19 , , PUNCT ejpam-5896 772 20 f. f. PROPN ejpam-5896 772 21 a. a. PROPN ejpam-5896 772 22 gharib gharib PROPN ejpam-5896 772 23 , , PUNCT ejpam-5896 772 24 and and CCONJ ejpam-5896 772 25 khaled khaled PROPN ejpam-5896 772 26 a. a. PROPN ejpam-5896 772 27 aldwoah aldwoah PROPN ejpam-5896 772 28 . . PUNCT ejpam-5896 773 1 supra supra PROPN ejpam-5896 773 2 soft soft ADJ ejpam-5896 773 3 somewhat somewhat ADV ejpam-5896 773 4 open open ADJ ejpam-5896 773 5 sets set NOUN ejpam-5896 773 6 : : PUNCT ejpam-5896 773 7 characterizations characterization NOUN ejpam-5896 773 8 and and CCONJ ejpam-5896 773 9 continuity continuity NOUN ejpam-5896 773 10 . . PUNCT ejpam-5896 774 1 european european ADJ ejpam-5896 774 2 journal journal PROPN ejpam-5896 774 3 of of ADP ejpam-5896 774 4 pure pure ADJ ejpam-5896 774 5 and and CCONJ ejpam-5896 774 6 applied applied ADJ ejpam-5896 774 7 mathematics mathematic NOUN ejpam-5896 774 8 , , PUNCT ejpam-5896 774 9 18(2):5863 18(2):5863 NUM ejpam-5896 774 10 , , PUNCT ejpam-5896 774 11 2025 2025 NUM ejpam-5896 774 12 . . PUNCT ejpam-5896 775 1 [ [ X ejpam-5896 775 2 56 56 NUM ejpam-5896 775 3 ] ] PUNCT ejpam-5896 775 4 a. a. NOUN ejpam-5896 775 5 m. m. PROPN ejpam-5896 775 6 abd abd PROPN ejpam-5896 775 7 el el PROPN ejpam-5896 775 8 - - PROPN ejpam-5896 775 9 latif latif PROPN ejpam-5896 775 10 . . PUNCT ejpam-5896 776 1 novel novel ADJ ejpam-5896 776 2 types type NOUN ejpam-5896 776 3 of of ADP ejpam-5896 776 4 supra supra ADJ ejpam-5896 776 5 soft soft ADJ ejpam-5896 776 6 operators operator NOUN ejpam-5896 776 7 via via ADP ejpam-5896 776 8 supra supra PROPN ejpam-5896 776 9 soft soft ADJ ejpam-5896 776 10 sd sd NOUN ejpam-5896 776 11 - - PUNCT ejpam-5896 776 12 sets set NOUN ejpam-5896 776 13 and and CCONJ ejpam-5896 776 14 applications application NOUN ejpam-5896 776 15 . . PUNCT ejpam-5896 777 1 aims aim VERB ejpam-5896 777 2 mathematics mathematic NOUN ejpam-5896 777 3 , , PUNCT ejpam-5896 777 4 9(3):6586–6602 9(3):6586–6602 PROPN ejpam-5896 777 5 , , PUNCT ejpam-5896 777 6 2024 2024 NUM ejpam-5896 777 7 . . PUNCT ejpam-5896 778 1 [ [ X ejpam-5896 778 2 57 57 NUM ejpam-5896 778 3 ] ] PUNCT ejpam-5896 778 4 a. a. NOUN ejpam-5896 778 5 m. m. PROPN ejpam-5896 778 6 abd abd PROPN ejpam-5896 778 7 el el PROPN ejpam-5896 778 8 - - PROPN ejpam-5896 778 9 latif latif PROPN ejpam-5896 778 10 and and CCONJ ejpam-5896 778 11 mesfer mesfer VERB ejpam-5896 778 12 h. h. PROPN ejpam-5896 778 13 alqahtani alqahtani PROPN ejpam-5896 778 14 . . PUNCT ejpam-5896 779 1 novel novel ADJ ejpam-5896 779 2 categories category NOUN ejpam-5896 779 3 of of ADP ejpam-5896 779 4 supra supra ADJ ejpam-5896 779 5 soft soft ADJ ejpam-5896 779 6 continuous continuous ADJ ejpam-5896 779 7 maps map NOUN ejpam-5896 779 8 via via ADP ejpam-5896 779 9 new new ADJ ejpam-5896 779 10 soft soft ADJ ejpam-5896 779 11 operators operator NOUN ejpam-5896 779 12 . . PUNCT ejpam-5896 780 1 aims aim VERB ejpam-5896 780 2 mathematics mathematic NOUN ejpam-5896 780 3 , , PUNCT ejpam-5896 780 4 9(3):7449–7470 9(3):7449–7470 NUM ejpam-5896 780 5 , , PUNCT ejpam-5896 780 6 2024 2024 NUM ejpam-5896 780 7 . . PUNCT ejpam-5896 781 1 [ [ X ejpam-5896 781 2 58 58 NUM ejpam-5896 781 3 ] ] PUNCT ejpam-5896 781 4 a. a. NOUN ejpam-5896 781 5 m. m. PROPN ejpam-5896 781 6 abd abd PROPN ejpam-5896 781 7 el el PROPN ejpam-5896 781 8 - - PROPN ejpam-5896 781 9 latif latif PROPN ejpam-5896 781 10 . . PUNCT ejpam-5896 782 1 supra supra PROPN ejpam-5896 782 2 soft soft ADJ ejpam-5896 782 3 b b NOUN ejpam-5896 782 4 - - PUNCT ejpam-5896 782 5 connectedness connectedness NOUN ejpam-5896 782 6 ii ii PROPN ejpam-5896 782 7 : : PUNCT ejpam-5896 782 8 some some DET ejpam-5896 782 9 types type NOUN ejpam-5896 782 10 of of ADP ejpam-5896 782 11 supra supra PROPN ejpam-5896 782 12 soft soft ADJ ejpam-5896 782 13 b b NOUN ejpam-5896 782 14 - - NOUN ejpam-5896 782 15 connectedness connectedness NOUN ejpam-5896 782 16 . . PUNCT ejpam-5896 783 1 creative creative ADJ ejpam-5896 783 2 mathematics mathematic NOUN ejpam-5896 783 3 and and CCONJ ejpam-5896 783 4 informatics informatic NOUN ejpam-5896 783 5 , , PUNCT ejpam-5896 783 6 26(1):1–8 26(1):1–8 NUM ejpam-5896 783 7 , , PUNCT ejpam-5896 783 8 2017 2017 NUM ejpam-5896 783 9 . . PUNCT ejpam-5896 784 1 [ [ X ejpam-5896 784 2 59 59 NUM ejpam-5896 784 3 ] ] PUNCT ejpam-5896 784 4 a. a. NOUN ejpam-5896 784 5 m. m. PROPN ejpam-5896 784 6 abd abd PROPN ejpam-5896 784 7 el el PROPN ejpam-5896 784 8 - - PROPN ejpam-5896 784 9 latif latif PROPN ejpam-5896 784 10 . . PUNCT ejpam-5896 785 1 on on ADP ejpam-5896 785 2 soft soft ADJ ejpam-5896 785 3 supra supra ADJ ejpam-5896 785 4 compactness compactness NOUN ejpam-5896 785 5 in in ADP ejpam-5896 785 6 supra supra PROPN ejpam-5896 785 7 soft soft ADJ ejpam-5896 785 8 topological topological ADJ ejpam-5896 785 9 spaces space NOUN ejpam-5896 785 10 . . PUNCT ejpam-5896 786 1 tbilisi tbilisi PROPN ejpam-5896 786 2 mathematical mathematical PROPN ejpam-5896 786 3 journal journal PROPN ejpam-5896 786 4 , , PUNCT ejpam-5896 786 5 11(1):169–178 11(1):169–178 PROPN ejpam-5896 786 6 , , PUNCT ejpam-5896 786 7 2018 2018 NUM ejpam-5896 786 8 . . PUNCT ejpam-5896 787 1 [ [ X ejpam-5896 787 2 60 60 NUM ejpam-5896 787 3 ] ] PUNCT ejpam-5896 787 4 t. t. PROPN ejpam-5896 787 5 m. m. PROPN ejpam-5896 787 6 al al PROPN ejpam-5896 787 7 - - PUNCT ejpam-5896 787 8 shami shami PROPN ejpam-5896 787 9 and and CCONJ ejpam-5896 787 10 m. m. PROPN ejpam-5896 787 11 e. e. PROPN ejpam-5896 787 12 el el PROPN ejpam-5896 787 13 - - PROPN ejpam-5896 787 14 shafei shafei PROPN ejpam-5896 787 15 . . PUNCT ejpam-5896 788 1 on on ADP ejpam-5896 788 2 soft soft ADJ ejpam-5896 788 3 compact compact ADJ ejpam-5896 788 4 and and CCONJ ejpam-5896 788 5 soft soft ADJ ejpam-5896 788 6 lindelöf lindelöf NOUN ejpam-5896 788 7 spaces space NOUN ejpam-5896 788 8 via via ADP ejpam-5896 788 9 soft soft ADJ ejpam-5896 788 10 pre pre ADJ ejpam-5896 788 11 - - ADJ ejpam-5896 788 12 open open ADJ ejpam-5896 788 13 sets set NOUN ejpam-5896 788 14 . . PUNCT ejpam-5896 789 1 annals annal NOUN ejpam-5896 789 2 of of ADP ejpam-5896 789 3 fuzzy fuzzy ADJ ejpam-5896 789 4 mathematics mathematic NOUN ejpam-5896 789 5 and and CCONJ ejpam-5896 789 6 informatics informatic NOUN ejpam-5896 789 7 , , PUNCT ejpam-5896 789 8 17(1):79–100 17(1):79–100 NUM ejpam-5896 789 9 , , PUNCT ejpam-5896 789 10 2019 2019 NUM ejpam-5896 789 11 . . PUNCT ejpam-5896 790 1 [ [ X ejpam-5896 790 2 61 61 NUM ejpam-5896 790 3 ] ] PUNCT ejpam-5896 790 4 m. m. PROPN ejpam-5896 790 5 e. e. PROPN ejpam-5896 790 6 el el PROPN ejpam-5896 790 7 - - PROPN ejpam-5896 790 8 shafei shafei PROPN ejpam-5896 790 9 and and CCONJ ejpam-5896 790 10 t. t. PROPN ejpam-5896 790 11 m. m. PROPN ejpam-5896 790 12 al al PROPN ejpam-5896 790 13 - - PUNCT ejpam-5896 790 14 shami shami PROPN ejpam-5896 790 15 . . PUNCT ejpam-5896 791 1 some some DET ejpam-5896 791 2 operators operator NOUN ejpam-5896 791 3 of of ADP ejpam-5896 791 4 a a DET ejpam-5896 791 5 soft soft ADJ ejpam-5896 791 6 set set NOUN ejpam-5896 791 7 and and CCONJ ejpam-5896 791 8 soft soft ADJ ejpam-5896 791 9 connected connected ADJ ejpam-5896 791 10 spaces space NOUN ejpam-5896 791 11 using use VERB ejpam-5896 791 12 soft soft ADJ ejpam-5896 791 13 somewhere somewhere ADV ejpam-5896 791 14 dense dense ADJ ejpam-5896 791 15 sets set NOUN ejpam-5896 791 16 . . PUNCT ejpam-5896 792 1 journal journal NOUN ejpam-5896 792 2 of of ADP ejpam-5896 792 3 interdisciplinary interdisciplinary ADJ ejpam-5896 792 4 mathematics mathematic NOUN ejpam-5896 792 5 , , PUNCT ejpam-5896 792 6 24(6):1471–1495 24(6):1471–1495 NUM ejpam-5896 792 7 , , PUNCT ejpam-5896 792 8 2021 2021 NUM ejpam-5896 792 9 . . PUNCT ejpam-5896 793 1 [ [ X ejpam-5896 793 2 62 62 NUM ejpam-5896 793 3 ] ] PUNCT ejpam-5896 793 4 a. a. NOUN ejpam-5896 793 5 m. m. PROPN ejpam-5896 793 6 abd abd PROPN ejpam-5896 793 7 el el PROPN ejpam-5896 793 8 - - PROPN ejpam-5896 793 9 latif latif PROPN ejpam-5896 793 10 , , PUNCT ejpam-5896 793 11 a. a. PROPN ejpam-5896 793 12 a. a. PROPN ejpam-5896 793 13 azzam azzam PROPN ejpam-5896 793 14 , , PUNCT ejpam-5896 793 15 radwan radwan VERB ejpam-5896 793 16 abu abu PROPN ejpam-5896 793 17 - - PUNCT ejpam-5896 793 18 gdairi gdairi PROPN ejpam-5896 793 19 , , PUNCT ejpam-5896 793 20 m. m. NOUN ejpam-5896 793 21 aldawood aldawood PROPN ejpam-5896 793 22 , , PUNCT ejpam-5896 793 23 and and CCONJ ejpam-5896 793 24 mesfer mesfer VERB ejpam-5896 793 25 h. h. PROPN ejpam-5896 793 26 alqahtani alqahtani PROPN ejpam-5896 793 27 . . PUNCT ejpam-5896 794 1 new new ADJ ejpam-5896 794 2 versions version NOUN ejpam-5896 794 3 of of ADP ejpam-5896 794 4 maps map NOUN ejpam-5896 794 5 and and CCONJ ejpam-5896 794 6 connected connected ADJ ejpam-5896 794 7 spaces space NOUN ejpam-5896 794 8 via via ADP ejpam-5896 794 9 supra supra PROPN ejpam-5896 794 10 soft soft ADJ ejpam-5896 794 11 sd sd NOUN ejpam-5896 794 12 - - PUNCT ejpam-5896 794 13 operators operator NOUN ejpam-5896 794 14 . . PUNCT ejpam-5896 795 1 plos plos PROPN ejpam-5896 795 2 one one NUM ejpam-5896 795 3 , , PUNCT ejpam-5896 795 4 19(10):e0304042 19(10):e0304042 NUM ejpam-5896 795 5 , , PUNCT ejpam-5896 795 6 2024 2024 NUM ejpam-5896 795 7 . . PUNCT ejpam-5896 796 1 [ [ X ejpam-5896 796 2 63 63 NUM ejpam-5896 796 3 ] ] PUNCT ejpam-5896 796 4 i. i. PROPN ejpam-5896 796 5 zorlutuna zorlutuna PROPN ejpam-5896 796 6 , , PUNCT ejpam-5896 796 7 m. m. NOUN ejpam-5896 796 8 akdag akdag PROPN ejpam-5896 796 9 , , PUNCT ejpam-5896 796 10 w. w. PROPN ejpam-5896 796 11 k. k. PROPN ejpam-5896 796 12 min min PROPN ejpam-5896 796 13 , , PUNCT ejpam-5896 796 14 and and CCONJ ejpam-5896 796 15 s. s. PROPN ejpam-5896 796 16 atmaca atmaca PROPN ejpam-5896 796 17 . . PUNCT ejpam-5896 797 1 remarks remark NOUN ejpam-5896 797 2 on on ADP ejpam-5896 797 3 soft soft ADJ ejpam-5896 797 4 topological topological ADJ ejpam-5896 797 5 spaces space NOUN ejpam-5896 797 6 . . PUNCT ejpam-5896 798 1 annals annal NOUN ejpam-5896 798 2 of of ADP ejpam-5896 798 3 fuzzy fuzzy ADJ ejpam-5896 798 4 mathematics mathematic NOUN ejpam-5896 798 5 and and CCONJ ejpam-5896 798 6 informatics informatic NOUN ejpam-5896 798 7 , , PUNCT ejpam-5896 798 8 3(2):171–185 3(2):171–185 NUM ejpam-5896 798 9 , , PUNCT ejpam-5896 798 10 2012 2012 NUM ejpam-5896 798 11 . . PUNCT ejpam-5896 799 1 [ [ X ejpam-5896 799 2 64 64 NUM ejpam-5896 799 3 ] ] PUNCT ejpam-5896 799 4 m. m. PROPN ejpam-5896 799 5 e. e. PROPN ejpam-5896 799 6 el el PROPN ejpam-5896 799 7 - - PROPN ejpam-5896 799 8 shafei shafei PROPN ejpam-5896 799 9 , , PUNCT ejpam-5896 799 10 m. m. NOUN ejpam-5896 799 11 abo abo NOUN ejpam-5896 799 12 - - PUNCT ejpam-5896 799 13 elhamayel elhamayel NOUN ejpam-5896 799 14 , , PUNCT ejpam-5896 799 15 and and CCONJ ejpam-5896 799 16 t. t. PROPN ejpam-5896 799 17 m. m. PROPN ejpam-5896 799 18 al al PROPN ejpam-5896 799 19 - - PUNCT ejpam-5896 799 20 shami shami PROPN ejpam-5896 799 21 . . PUNCT ejpam-5896 800 1 further further ADJ ejpam-5896 800 2 notions notion NOUN ejpam-5896 800 3 related relate VERB ejpam-5896 800 4 to to ADP ejpam-5896 800 5 new new ADJ ejpam-5896 800 6 operators operator NOUN ejpam-5896 800 7 and and CCONJ ejpam-5896 800 8 compactness compactness NOUN ejpam-5896 800 9 via via ADP ejpam-5896 800 10 supra supra PROPN ejpam-5896 800 11 soft soft ADJ ejpam-5896 800 12 topological topological ADJ ejpam-5896 800 13 spaces space NOUN ejpam-5896 800 14 . . PUNCT ejpam-5896 801 1 international international ADJ ejpam-5896 801 2 journal journal NOUN ejpam-5896 801 3 of of ADP ejpam-5896 801 4 advances advance NOUN ejpam-5896 801 5 in in ADP ejpam-5896 801 6 mathematics mathematic NOUN ejpam-5896 801 7 , , PUNCT ejpam-5896 801 8 1:44–60 1:44–60 NUM ejpam-5896 801 9 , , PUNCT ejpam-5896 801 10 2019 2019 NUM ejpam-5896 801 11 . . PUNCT ejpam-5896 802 1 [ [ X ejpam-5896 802 2 65 65 NUM ejpam-5896 802 3 ] ] X ejpam-5896 802 4 a. a. NOUN ejpam-5896 802 5 m. m. PROPN ejpam-5896 802 6 abd abd PROPN ejpam-5896 802 7 el el PROPN ejpam-5896 802 8 - - PROPN ejpam-5896 802 9 latif latif PROPN ejpam-5896 802 10 . . PUNCT ejpam-5896 803 1 some some DET ejpam-5896 803 2 properties property NOUN ejpam-5896 803 3 of of ADP ejpam-5896 803 4 fuzzy fuzzy ADJ ejpam-5896 803 5 supra supra PROPN ejpam-5896 803 6 soft soft ADJ ejpam-5896 803 7 topological topological ADJ ejpam-5896 803 8 spaces space NOUN ejpam-5896 803 9 . . PUNCT ejpam-5896 804 1 european european ADJ ejpam-5896 804 2 journal journal PROPN ejpam-5896 804 3 of of ADP ejpam-5896 804 4 pure pure ADJ ejpam-5896 804 5 and and CCONJ ejpam-5896 804 6 applied applied ADJ ejpam-5896 804 7 mathematics mathematic NOUN ejpam-5896 804 8 , , PUNCT ejpam-5896 804 9 12(3):999–1017 12(3):999–1017 NUM ejpam-5896 804 10 , , PUNCT ejpam-5896 804 11 2019 2019 NUM ejpam-5896 804 12 . . PUNCT