id sid tid token lemma pos iajs-3789 1 1 432 432 NUM iajs-3789 1 2 this this DET iajs-3789 1 3 work work NOUN iajs-3789 1 4 is be AUX iajs-3789 1 5 licensed license VERB iajs-3789 1 6 under under ADP iajs-3789 1 7 a a DET iajs-3789 1 8 creative creative ADJ iajs-3789 1 9 commons common NOUN iajs-3789 1 10 attribution attribution NOUN iajs-3789 1 11 4.0 4.0 NUM iajs-3789 1 12 international international ADJ iajs-3789 1 13 license license NOUN iajs-3789 1 14 ihjpas ihjpa NOUN iajs-3789 1 15 . . PUNCT iajs-3789 2 1 37 37 NUM iajs-3789 2 2 ( ( PUNCT iajs-3789 2 3 2 2 NUM iajs-3789 2 4 ) ) PUNCT iajs-3789 2 5 2024 2024 NUM iajs-3789 2 6 ibn ibn PROPN iajs-3789 2 7 al al PROPN iajs-3789 2 8 - - PUNCT iajs-3789 2 9 haitham haitham PROPN iajs-3789 2 10 journal journal PROPN iajs-3789 2 11 for for ADP iajs-3789 2 12 pure pure ADJ iajs-3789 2 13 and and CCONJ iajs-3789 2 14 applied applied ADJ iajs-3789 2 15 sciences sciences PROPN iajs-3789 2 16 journal journal PROPN iajs-3789 2 17 homepage homepage NOUN iajs-3789 2 18 : : PUNCT iajs-3789 2 19 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq NOUN iajs-3789 2 20 pissn pissn ADJ iajs-3789 2 21 : : PUNCT iajs-3789 2 22 1609 1609 NUM iajs-3789 2 23 - - SYM iajs-3789 2 24 4042 4042 NUM iajs-3789 2 25 , , PUNCT iajs-3789 2 26 eissn eissn NOUN iajs-3789 2 27 : : PUNCT iajs-3789 2 28 2521 2521 NUM iajs-3789 2 29 - - SYM iajs-3789 2 30 3407 3407 NUM iajs-3789 2 31 rana rana PROPN iajs-3789 2 32 bahjat bahjat PROPN iajs-3789 2 33 esmaeel esmaeel PROPN iajs-3789 2 34 department department PROPN iajs-3789 2 35 of of ADP iajs-3789 2 36 mathematics mathematics PROPN iajs-3789 2 37 , , PUNCT iajs-3789 2 38 college college NOUN iajs-3789 2 39 of of ADP iajs-3789 2 40 education education NOUN iajs-3789 2 41 for for ADP iajs-3789 2 42 pure pure ADJ iajs-3789 2 43 science science NOUN iajs-3789 2 44 ( ( PUNCT iajs-3789 2 45 ibnalhaitham ibnalhaitham NOUN iajs-3789 2 46 ) ) PUNCT iajs-3789 2 47 , , PUNCT iajs-3789 2 48 university university NOUN iajs-3789 2 49 of of ADP iajs-3789 2 50 baghdad baghdad PROPN iajs-3789 2 51 , , PUNCT iajs-3789 2 52 baghdad baghdad PROPN iajs-3789 2 53 , , PUNCT iajs-3789 2 54 iraq iraq PROPN iajs-3789 2 55 . . PUNCT iajs-3789 3 1 abstract abstract ADJ iajs-3789 3 2 this this DET iajs-3789 3 3 scientific scientific ADJ iajs-3789 3 4 study study NOUN iajs-3789 3 5 aims aim VERB iajs-3789 3 6 to to PART iajs-3789 3 7 introduce introduce VERB iajs-3789 3 8 a a DET iajs-3789 3 9 new new ADJ iajs-3789 3 10 type type NOUN iajs-3789 3 11 of of ADP iajs-3789 3 12 topological topological ADJ iajs-3789 3 13 space space NOUN iajs-3789 3 14 , , PUNCT iajs-3789 3 15 called call VERB iajs-3789 3 16 " " PUNCT iajs-3789 3 17 grill grill ADJ iajs-3789 3 18 π“₯-space π“₯-space PROPN iajs-3789 3 19 " " PUNCT iajs-3789 3 20 , , PUNCT iajs-3789 3 21 with with ADP iajs-3789 3 22 the the DET iajs-3789 3 23 aim aim NOUN iajs-3789 3 24 of of ADP iajs-3789 3 25 contributing contribute VERB iajs-3789 3 26 to to ADP iajs-3789 3 27 scientific scientific ADJ iajs-3789 3 28 knowledge knowledge NOUN iajs-3789 3 29 in in ADP iajs-3789 3 30 this this DET iajs-3789 3 31 field field NOUN iajs-3789 3 32 . . PUNCT iajs-3789 4 1 new new ADJ iajs-3789 4 2 generalizations generalization NOUN iajs-3789 4 3 were be AUX iajs-3789 4 4 developed develop VERB iajs-3789 4 5 for for ADP iajs-3789 4 6 both both CCONJ iajs-3789 4 7 the the DET iajs-3789 4 8 local local ADJ iajs-3789 4 9 function function NOUN iajs-3789 4 10 and and CCONJ iajs-3789 4 11 the the DET iajs-3789 4 12 kuratowski kuratowski ADJ iajs-3789 4 13 closure closure NOUN iajs-3789 4 14 function function NOUN iajs-3789 4 15 in in ADP iajs-3789 4 16 order order NOUN iajs-3789 4 17 to to PART iajs-3789 4 18 generalize generalize VERB iajs-3789 4 19 their their PRON iajs-3789 4 20 concepts concept NOUN iajs-3789 4 21 . . PUNCT iajs-3789 5 1 subsequently subsequently ADV iajs-3789 5 2 , , PUNCT iajs-3789 5 3 these these DET iajs-3789 5 4 generalizations generalization NOUN iajs-3789 5 5 were be AUX iajs-3789 5 6 used use VERB iajs-3789 5 7 to to PART iajs-3789 5 8 define define VERB iajs-3789 5 9 a a DET iajs-3789 5 10 new new ADJ iajs-3789 5 11 topology topology NOUN iajs-3789 5 12 based base VERB iajs-3789 5 13 on on ADP iajs-3789 5 14 the the DET iajs-3789 5 15 concepts concept NOUN iajs-3789 5 16 associated associate VERB iajs-3789 5 17 with with ADP iajs-3789 5 18 the the DET iajs-3789 5 19 grill grill NOUN iajs-3789 5 20 and and CCONJ iajs-3789 5 21 the the DET iajs-3789 5 22 π“₯ π“₯ NOUN iajs-3789 5 23 -space -space NOUN iajs-3789 5 24 . . PUNCT iajs-3789 6 1 the the DET iajs-3789 6 2 set set NOUN iajs-3789 6 3 of of ADP iajs-3789 6 4 π“₯ π“₯ PROPN iajs-3789 6 5 -open -open NOUN iajs-3789 6 6 in in ADP iajs-3789 6 7 π“₯ π“₯ NOUN iajs-3789 6 8 -space -space NOUN iajs-3789 6 9 is be AUX iajs-3789 6 10 defined define VERB iajs-3789 6 11 as as ADP iajs-3789 6 12 the the DET iajs-3789 6 13 sum sum NOUN iajs-3789 6 14 of of ADP iajs-3789 6 15 the the DET iajs-3789 6 16 set set NOUN iajs-3789 6 17 of of ADP iajs-3789 6 18 π“₯ π“₯ NOUN iajs-3789 6 19 -open -open NOUN iajs-3789 6 20 forms form NOUN iajs-3789 6 21 in in ADP iajs-3789 6 22 π“₯ π“₯ NOUN iajs-3789 6 23 -space -space NOUN iajs-3789 6 24 when when SCONJ iajs-3789 6 25 the the DET iajs-3789 6 26 grill grill NOUN iajs-3789 6 27 consists consist VERB iajs-3789 6 28 of of ADP iajs-3789 6 29 the the DET iajs-3789 6 30 subsets subset NOUN iajs-3789 6 31 of of ADP iajs-3789 6 32 p(x p(x PROPN iajs-3789 6 33 ) ) PUNCT iajs-3789 6 34 except except SCONJ iajs-3789 6 35 the the DET iajs-3789 6 36 empty empty ADJ iajs-3789 6 37 set set NOUN iajs-3789 6 38 . . PUNCT iajs-3789 7 1 many many ADJ iajs-3789 7 2 of of ADP iajs-3789 7 3 the the DET iajs-3789 7 4 properties property NOUN iajs-3789 7 5 of of ADP iajs-3789 7 6 this this DET iajs-3789 7 7 new new ADJ iajs-3789 7 8 space space NOUN iajs-3789 7 9 were be AUX iajs-3789 7 10 demonstrated demonstrate VERB iajs-3789 7 11 , , PUNCT iajs-3789 7 12 and and CCONJ iajs-3789 7 13 a a DET iajs-3789 7 14 number number NOUN iajs-3789 7 15 of of ADP iajs-3789 7 16 illustrative illustrative ADJ iajs-3789 7 17 examples example NOUN iajs-3789 7 18 were be AUX iajs-3789 7 19 given give VERB iajs-3789 7 20 . . PUNCT iajs-3789 8 1 keywords keyword NOUN iajs-3789 8 2 : : PUNCT iajs-3789 8 3 grill grill ADJ iajs-3789 8 4 , , PUNCT iajs-3789 8 5 π“₯-open π“₯-open PROPN iajs-3789 8 6 set set NOUN iajs-3789 8 7 , , PUNCT iajs-3789 8 8 π“₯ π“₯ NOUN iajs-3789 8 9 -closed -close VERB iajs-3789 8 10 set set NOUN iajs-3789 8 11 , , PUNCT iajs-3789 8 12 π“₯ π“₯ NOUN iajs-3789 8 13 -interior -interior NOUN iajs-3789 8 14 , , PUNCT iajs-3789 8 15 grill grill ADJ iajs-3789 8 16 π“₯ π“₯ NOUN iajs-3789 8 17 -space -space NOUN iajs-3789 8 18 . . PUNCT iajs-3789 9 1 1 1 X iajs-3789 9 2 . . X iajs-3789 9 3 introduction introduction NOUN iajs-3789 9 4 in in ADP iajs-3789 9 5 1947 1947 NUM iajs-3789 9 6 , , PUNCT iajs-3789 9 7 [ [ X iajs-3789 9 8 1 1 X iajs-3789 9 9 ] ] PUNCT iajs-3789 9 10 introduced introduce VERB iajs-3789 9 11 the the DET iajs-3789 9 12 notion notion NOUN iajs-3789 9 13 of of ADP iajs-3789 9 14 grill grill NOUN iajs-3789 9 15 . . PUNCT iajs-3789 10 1 after after ADP iajs-3789 10 2 that that PRON iajs-3789 10 3 , , PUNCT iajs-3789 10 4 each each PRON iajs-3789 10 5 of of ADP iajs-3789 10 6 the the DET iajs-3789 10 7 researchers researcher NOUN iajs-3789 10 8 , , PUNCT iajs-3789 10 9 [ [ X iajs-3789 10 10 2 2 NUM iajs-3789 10 11 ] ] PUNCT iajs-3789 10 12 , , PUNCT iajs-3789 10 13 [ [ X iajs-3789 10 14 3 3 NUM iajs-3789 10 15 ] ] PUNCT iajs-3789 10 16 studied study VERB iajs-3789 10 17 many many ADJ iajs-3789 10 18 concepts concept NOUN iajs-3789 10 19 for for ADP iajs-3789 10 20 different different ADJ iajs-3789 10 21 types type NOUN iajs-3789 10 22 of of ADP iajs-3789 10 23 spaces space NOUN iajs-3789 10 24 using use VERB iajs-3789 10 25 this this DET iajs-3789 10 26 term term NOUN iajs-3789 10 27 . . PUNCT iajs-3789 11 1 in in ADP iajs-3789 11 2 2007 2007 NUM iajs-3789 11 3 , , PUNCT iajs-3789 11 4 [ [ X iajs-3789 11 5 4 4 X iajs-3789 11 6 ] ] PUNCT iajs-3789 11 7 introduced introduce VERB iajs-3789 11 8 the the DET iajs-3789 11 9 notion notion NOUN iajs-3789 11 10 of of ADP iajs-3789 11 11 grill grill ADJ iajs-3789 11 12 topological topological ADJ iajs-3789 11 13 space space NOUN iajs-3789 11 14 . . PUNCT iajs-3789 12 1 [ [ X iajs-3789 12 2 5 5 NUM iajs-3789 12 3 - - SYM iajs-3789 12 4 7 7 NUM iajs-3789 12 5 ] ] PUNCT iajs-3789 12 6 used use VERB iajs-3789 12 7 the the DET iajs-3789 12 8 studied study VERB iajs-3789 12 9 concept concept NOUN iajs-3789 12 10 to to PART iajs-3789 12 11 introduce introduce VERB iajs-3789 12 12 new new ADJ iajs-3789 12 13 generalized generalized ADJ iajs-3789 12 14 sets set NOUN iajs-3789 12 15 and and CCONJ iajs-3789 12 16 to to PART iajs-3789 12 17 study study VERB iajs-3789 12 18 these these DET iajs-3789 12 19 generalizations generalization NOUN iajs-3789 12 20 and and CCONJ iajs-3789 12 21 their their PRON iajs-3789 12 22 properties property NOUN iajs-3789 12 23 in in ADP iajs-3789 12 24 detail detail NOUN iajs-3789 12 25 . . PUNCT iajs-3789 13 1 in in ADP iajs-3789 13 2 2022 2022 NUM iajs-3789 13 3 , , PUNCT iajs-3789 13 4 [ [ X iajs-3789 13 5 8 8 NUM iajs-3789 13 6 ] ] PUNCT iajs-3789 13 7 introduced introduce VERB iajs-3789 13 8 the the DET iajs-3789 13 9 concept concept NOUN iajs-3789 13 10 of of ADP iajs-3789 13 11 π“₯-open π“₯-open PROPN iajs-3789 13 12 set set NOUN iajs-3789 13 13 for for ADP iajs-3789 13 14 the the DET iajs-3789 13 15 first first ADJ iajs-3789 13 16 time time NOUN iajs-3789 13 17 and and CCONJ iajs-3789 13 18 named name VERB iajs-3789 13 19 the the DET iajs-3789 13 20 ordered order VERB iajs-3789 13 21 pair pair NOUN iajs-3789 13 22 of of ADP iajs-3789 13 23 the the DET iajs-3789 13 24 universal universal ADJ iajs-3789 13 25 set set NOUN iajs-3789 13 26 and and CCONJ iajs-3789 13 27 the the DET iajs-3789 13 28 family family NOUN iajs-3789 13 29 of of ADP iajs-3789 13 30 all all DET iajs-3789 13 31 π“₯-open π“₯-open PROPN iajs-3789 13 32 sets set VERB iajs-3789 13 33 a a DET iajs-3789 13 34 π“₯-space π“₯-space PROPN iajs-3789 13 35 and and CCONJ iajs-3789 13 36 proved prove VERB iajs-3789 13 37 that that SCONJ iajs-3789 13 38 this this DET iajs-3789 13 39 space space NOUN iajs-3789 13 40 represents represent VERB iajs-3789 13 41 a a DET iajs-3789 13 42 topological topological ADJ iajs-3789 13 43 space space NOUN iajs-3789 13 44 under under ADP iajs-3789 13 45 certain certain ADJ iajs-3789 13 46 conditions condition NOUN iajs-3789 13 47 . . PUNCT iajs-3789 14 1 in in ADP iajs-3789 14 2 this this DET iajs-3789 14 3 work work NOUN iajs-3789 14 4 , , PUNCT iajs-3789 14 5 the the DET iajs-3789 14 6 concept concept NOUN iajs-3789 14 7 of of ADP iajs-3789 14 8 the the DET iajs-3789 14 9 grill grill NOUN iajs-3789 14 10 is be AUX iajs-3789 14 11 extending extend VERB iajs-3789 14 12 from from ADP iajs-3789 14 13 the the DET iajs-3789 14 14 grill grill NOUN iajs-3789 14 15 π“₯-space π“₯-space PROPN iajs-3789 14 16 a a DET iajs-3789 14 17 new new ADJ iajs-3789 14 18 kind kind NOUN iajs-3789 14 19 of of ADP iajs-3789 14 20 topological topological ADJ iajs-3789 14 21 space space NOUN iajs-3789 14 22 has have AUX iajs-3789 14 23 been be AUX iajs-3789 14 24 obtained obtain VERB iajs-3789 14 25 . . PUNCT iajs-3789 15 1 grill grill ADJ iajs-3789 15 2 topological topological ADJ iajs-3789 15 3 spaces space NOUN iajs-3789 15 4 are be AUX iajs-3789 15 5 considered consider VERB iajs-3789 15 6 one one NUM iajs-3789 15 7 of of ADP iajs-3789 15 8 the the DET iajs-3789 15 9 most most ADV iajs-3789 15 10 important important ADJ iajs-3789 15 11 areas area NOUN iajs-3789 15 12 in in ADP iajs-3789 15 13 mathematics mathematic NOUN iajs-3789 15 14 , , PUNCT iajs-3789 15 15 and and CCONJ iajs-3789 15 16 there there PRON iajs-3789 15 17 are be VERB iajs-3789 15 18 many many ADJ iajs-3789 15 19 researchers researcher NOUN iajs-3789 15 20 who who PRON iajs-3789 15 21 have have AUX iajs-3789 15 22 studied study VERB iajs-3789 15 23 generalizations generalization NOUN iajs-3789 15 24 of of ADP iajs-3789 15 25 topological topological ADJ iajs-3789 15 26 properties property NOUN iajs-3789 15 27 in in ADP iajs-3789 15 28 this this DET iajs-3789 15 29 context context NOUN iajs-3789 15 30 , , PUNCT iajs-3789 15 31 since since SCONJ iajs-3789 15 32 the the DET iajs-3789 15 33 aim aim NOUN iajs-3789 15 34 is be AUX iajs-3789 15 35 to to PART iajs-3789 15 36 understand understand VERB iajs-3789 15 37 how how SCONJ iajs-3789 15 38 these these DET iajs-3789 15 39 properties property NOUN iajs-3789 15 40 evolve evolve VERB iajs-3789 15 41 and and CCONJ iajs-3789 15 42 change change VERB iajs-3789 15 43 when when SCONJ iajs-3789 15 44 dealing deal VERB iajs-3789 15 45 with with ADP iajs-3789 15 46 generalized generalized ADJ iajs-3789 15 47 topological topological ADJ iajs-3789 15 48 spaces space NOUN iajs-3789 15 49 , , PUNCT iajs-3789 15 50 see see VERB iajs-3789 15 51 [ [ X iajs-3789 15 52 9 9 NUM iajs-3789 15 53 - - SYM iajs-3789 15 54 24 24 NUM iajs-3789 15 55 ] ] PUNCT iajs-3789 15 56 . . PUNCT iajs-3789 16 1 perhaps perhaps ADV iajs-3789 16 2 soft soft ADJ iajs-3789 16 3 topological topological ADJ iajs-3789 16 4 spaces space NOUN iajs-3789 16 5 are be AUX iajs-3789 16 6 one one NUM iajs-3789 16 7 of of ADP iajs-3789 16 8 the the DET iajs-3789 16 9 modern modern ADJ iajs-3789 16 10 spaces space NOUN iajs-3789 16 11 that that SCONJ iajs-3789 16 12 many many ADJ iajs-3789 16 13 researchers researcher NOUN iajs-3789 16 14 are be AUX iajs-3789 16 15 interested interested ADJ iajs-3789 16 16 in in ADP iajs-3789 16 17 studying study VERB iajs-3789 16 18 . . PUNCT iajs-3789 17 1 therefore therefore ADV iajs-3789 17 2 , , PUNCT iajs-3789 17 3 the the DET iajs-3789 17 4 topological topological ADJ iajs-3789 17 5 properties property NOUN iajs-3789 17 6 in in ADP iajs-3789 17 7 soft soft ADJ iajs-3789 17 8 topological topological ADJ iajs-3789 17 9 spaces space NOUN iajs-3789 17 10 have have AUX iajs-3789 17 11 been be AUX iajs-3789 17 12 studied study VERB iajs-3789 17 13 using use VERB iajs-3789 17 14 the the DET iajs-3789 17 15 grill grill NOUN iajs-3789 17 16 concept concept NOUN iajs-3789 17 17 , , PUNCT iajs-3789 17 18 see see VERB iajs-3789 17 19 [ [ X iajs-3789 17 20 2530 2530 NUM iajs-3789 17 21 ] ] PUNCT iajs-3789 17 22 . . PUNCT iajs-3789 18 1 2 2 X iajs-3789 18 2 . . X iajs-3789 18 3 preliminaries preliminary NOUN iajs-3789 18 4 definition definition NOUN iajs-3789 18 5 2.1[8 2.1[8 NOUN iajs-3789 18 6 ] ] X iajs-3789 18 7 let let VERB iajs-3789 18 8 x x PRON iajs-3789 18 9 represent represent VERB iajs-3789 18 10 a a DET iajs-3789 18 11 set set NOUN iajs-3789 18 12 that that PRON iajs-3789 18 13 is be AUX iajs-3789 18 14 not not PART iajs-3789 18 15 empty empty ADJ iajs-3789 18 16 , , PUNCT iajs-3789 18 17 and and CCONJ iajs-3789 18 18 let let VERB iajs-3789 18 19 { { PUNCT iajs-3789 18 20 πœπ‘˜}π‘˜πœ–π½ πœπ‘˜}π‘˜πœ–π½ PROPN iajs-3789 18 21 , , PUNCT iajs-3789 18 22 π‘˜ π‘˜ PRON iajs-3789 18 23 β‰₯ β‰₯ NUM iajs-3789 18 24 2 2 NUM iajs-3789 18 25 be be AUX iajs-3789 18 26 any any DET iajs-3789 18 27 topologies topology NOUN iajs-3789 18 28 on on ADP iajs-3789 18 29 𝐾 𝐾 PROPN iajs-3789 18 30 and and CCONJ iajs-3789 18 31 let let VERB iajs-3789 18 32 the the DET iajs-3789 18 33 family family NOUN iajs-3789 18 34 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 18 35 = = PUNCT iajs-3789 18 36 { { PUNCT iajs-3789 18 37 𝒩 𝒩 PROPN iajs-3789 18 38 βŠ† βŠ† NUM iajs-3789 18 39 𝐾 𝐾 PROPN iajs-3789 18 40 : : PUNCT iajs-3789 18 41 𝒩 𝒩 PROPN iajs-3789 18 42 = = PUNCT iajs-3789 18 43 βˆ… βˆ… NOUN iajs-3789 18 44 π‘œπ‘Ÿ π‘œπ‘Ÿ ADP iajs-3789 18 45 βˆƒ βˆƒ PROPN iajs-3789 18 46 𝒯 𝒯 PROPN iajs-3789 18 47 ∈ ∈ PROPN iajs-3789 18 48 β‹‚ β‹‚ PROPN iajs-3789 18 49 πœπ‘˜π‘˜πœ–π½ πœπ‘˜π‘˜πœ–π½ PROPN iajs-3789 18 50 βˆ‹ βˆ‹ NOUN iajs-3789 18 51 βˆ… βˆ… NOUN iajs-3789 18 52 β‰  β‰  PROPN iajs-3789 18 53 𝒯 𝒯 PROPN iajs-3789 18 54 βŠ† βŠ† NUM iajs-3789 18 55 𝒩 𝒩 PROPN iajs-3789 18 56 } } PUNCT iajs-3789 18 57 satisfying satisfy VERB iajs-3789 18 58 the the DET iajs-3789 18 59 following follow VERB iajs-3789 18 60 axioms axiom NOUN iajs-3789 18 61 : : PUNCT iajs-3789 18 62 received receive VERB iajs-3789 18 63 : : PUNCT iajs-3789 18 64 15 15 NUM iajs-3789 18 65 october october PROPN iajs-3789 18 66 2023 2023 NUM iajs-3789 18 67 accepted accept VERB iajs-3789 18 68 : : PUNCT iajs-3789 18 69 24 24 NUM iajs-3789 18 70 december december PROPN iajs-3789 18 71 2023 2023 NUM iajs-3789 18 72 published publish VERB iajs-3789 18 73 : : PUNCT iajs-3789 18 74 20 20 NUM iajs-3789 18 75 april april PROPN iajs-3789 18 76 2024 2024 NUM iajs-3789 18 77 grill grill NOUN iajs-3789 18 78 π“₯ π“₯ PROPN iajs-3789 18 79 -space -space NOUN iajs-3789 18 80 doi.org/10.30526/37.2.3789 doi.org/10.30526/37.2.3789 PROPN iajs-3789 18 81 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3789 18 82 https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 NOUN iajs-3789 18 83 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 VERB iajs-3789 18 84 https://orcid.org/0000-0002-4743-6034 https://orcid.org/0000-0002-4743-6034 VERB iajs-3789 18 85 mailto:ranamumosa@yahoo.com mailto:ranamumosa@yahoo.com NOUN iajs-3789 18 86 ihjpas ihjpas PROPN iajs-3789 18 87 . . PUNCT iajs-3789 19 1 37 37 NUM iajs-3789 19 2 ( ( PUNCT iajs-3789 19 3 2 2 NUM iajs-3789 19 4 ) ) PUNCT iajs-3789 19 5 2024 2024 NUM iajs-3789 19 6 433 433 NUM iajs-3789 19 7 1 1 NUM iajs-3789 19 8 . . PUNCT iajs-3789 20 1 𝐾 𝐾 NOUN iajs-3789 20 2 , , PUNCT iajs-3789 20 3 βˆ… βˆ… NOUN iajs-3789 20 4 ∈ ∈ PROPN iajs-3789 20 5 𝒱𝑂𝑋. 𝒱𝑂𝑋. X iajs-3789 20 6 2 2 NUM iajs-3789 20 7 . . PUNCT iajs-3789 21 1 ⋃ ⋃ VERB iajs-3789 21 2 π’©π‘–π‘–βˆˆπΌ π’©π‘–π‘–βˆˆπΌ PROPN iajs-3789 21 3 ∈ ∈ PROPN iajs-3789 21 4 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 21 5 βˆ€ βˆ€ X iajs-3789 21 6 { { PUNCT iajs-3789 21 7 𝒩𝑖}π‘–βˆˆπΌ 𝒩𝑖}π‘–βˆˆπΌ X iajs-3789 21 8 ∈ ∈ NOUN iajs-3789 21 9 𝒱𝑂𝑋. 𝒱𝑂𝑋. X iajs-3789 21 10 3 3 NUM iajs-3789 21 11 . . PUNCT iajs-3789 21 12 β‹‚ β‹‚ PROPN iajs-3789 21 13 𝒩𝑖 𝒩𝑖 PROPN iajs-3789 21 14 𝑛 𝑛 PRON iajs-3789 21 15 𝑖=1 𝑖=1 PROPN iajs-3789 21 16 ∈ ∈ PROPN iajs-3789 21 17 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 21 18 βˆ€ βˆ€ PUNCT iajs-3789 21 19 { { PUNCT iajs-3789 21 20 𝒩𝑖}𝑖=1 𝒩𝑖}𝑖=1 VERB iajs-3789 21 21 𝑛 𝑛 DET iajs-3789 21 22 ∈ ∈ PROPN iajs-3789 21 23 𝒱𝑂𝑋. 𝒱𝑂𝑋. X iajs-3789 21 24 then then ADV iajs-3789 21 25 ( ( PUNCT iajs-3789 21 26 𝑋 𝑋 PROPN iajs-3789 21 27 , , PUNCT iajs-3789 21 28 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 21 29 ) ) PUNCT iajs-3789 21 30 is be AUX iajs-3789 21 31 called call VERB iajs-3789 21 32 to to PART iajs-3789 21 33 be be AUX iajs-3789 21 34 π“₯-space π“₯-space PROPN iajs-3789 21 35 and and CCONJ iajs-3789 21 36 the the DET iajs-3789 21 37 elements element NOUN iajs-3789 21 38 of of ADP iajs-3789 21 39 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 21 40 are be AUX iajs-3789 21 41 called call VERB iajs-3789 21 42 𝒱 𝒱 PROPN iajs-3789 21 43 βˆ’ βˆ’ NOUN iajs-3789 21 44 open open ADJ iajs-3789 21 45 sets set NOUN iajs-3789 21 46 and and CCONJ iajs-3789 21 47 the the DET iajs-3789 21 48 complement complement NOUN iajs-3789 21 49 𝒱 𝒱 PROPN iajs-3789 21 50 βˆ’ βˆ’ PROPN iajs-3789 21 51 open open ADJ iajs-3789 21 52 set set NOUN iajs-3789 21 53 is be AUX iajs-3789 21 54 𝒱 𝒱 PROPN iajs-3789 21 55 βˆ’ βˆ’ NOUN iajs-3789 21 56 closed closed ADJ iajs-3789 21 57 set set NOUN iajs-3789 21 58 . . PUNCT iajs-3789 22 1 we we PRON iajs-3789 22 2 denote denote VERB iajs-3789 22 3 the the DET iajs-3789 22 4 set set NOUN iajs-3789 22 5 of of ADP iajs-3789 22 6 all all DET iajs-3789 22 7 𝒱 𝒱 NOUN iajs-3789 22 8 βˆ’ βˆ’ PROPN iajs-3789 22 9 closed close VERB iajs-3789 22 10 of of ADP iajs-3789 22 11 𝑋 𝑋 NOUN iajs-3789 22 12 by by ADP iajs-3789 22 13 𝒱𝐢𝑋. 𝒱𝐢𝑋. NUM iajs-3789 22 14 definition definition NOUN iajs-3789 22 15 2.2 2.2 NUM iajs-3789 22 16 [ [ NOUN iajs-3789 22 17 8 8 NUM iajs-3789 22 18 ] ] PUNCT iajs-3789 22 19 for for ADP iajs-3789 22 20 any any DET iajs-3789 22 21 π“₯-space π“₯-space PROPN iajs-3789 22 22 ( ( PUNCT iajs-3789 22 23 𝐾 𝐾 PROPN iajs-3789 22 24 , , PUNCT iajs-3789 22 25 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 22 26 ) ) PUNCT iajs-3789 22 27 and and CCONJ iajs-3789 22 28 let let VERB iajs-3789 22 29 Ι± Ι± PROPN iajs-3789 22 30 βŠ† βŠ† NUM iajs-3789 22 31 𝑋. 𝑋. PROPN iajs-3789 22 32 then then ADV iajs-3789 22 33 1 1 NUM iajs-3789 22 34 . . PUNCT iajs-3789 23 1 the the DET iajs-3789 23 2 π“₯-closure π“₯-closure PROPN iajs-3789 23 3 of of ADP iajs-3789 23 4 Ι± Ι± PROPN iajs-3789 23 5 is be AUX iajs-3789 23 6 symbolized symbolize VERB iajs-3789 23 7 by by ADP iajs-3789 23 8 𝑐𝑙𝒱(Ι± 𝑐𝑙𝒱(Ι± NOUN iajs-3789 23 9 ) ) PUNCT iajs-3789 23 10 and and CCONJ iajs-3789 23 11 is be AUX iajs-3789 23 12 equal equal ADJ iajs-3789 23 13 𝑐𝑙𝒱(Ι± 𝑐𝑙𝒱(Ι± NOUN iajs-3789 23 14 ) ) PUNCT iajs-3789 23 15 = = SYM iajs-3789 23 16 β‹‚{β„± β‹‚{β„± PROPN iajs-3789 23 17 βŠ† βŠ† NUM iajs-3789 23 18 𝐾 𝐾 PROPN iajs-3789 23 19 : : PUNCT iajs-3789 23 20 β„± β„± PROPN iajs-3789 23 21 𝑖𝑠 𝑖𝑠 CCONJ iajs-3789 23 22 𝒱 𝒱 PROPN iajs-3789 23 23 βˆ’ βˆ’ PROPN iajs-3789 23 24 closed closed ADJ iajs-3789 23 25 and and CCONJ iajs-3789 23 26 Ι± Ι± ADJ iajs-3789 23 27 βŠ† βŠ† NUM iajs-3789 23 28 β„± β„± PROPN iajs-3789 23 29 } } PUNCT iajs-3789 23 30 . . PUNCT iajs-3789 24 1 2 2 X iajs-3789 24 2 . . X iajs-3789 24 3 the the DET iajs-3789 24 4 π“₯-interior π“₯-interior PROPN iajs-3789 24 5 of of ADP iajs-3789 24 6 Ι± Ι± PROPN iajs-3789 24 7 is be AUX iajs-3789 24 8 symbolized symbolize VERB iajs-3789 24 9 by by ADP iajs-3789 24 10 𝑖𝑛𝑑𝒱(Ι± 𝑖𝑛𝑑𝒱(Ι± NOUN iajs-3789 24 11 ) ) PUNCT iajs-3789 24 12 and and CCONJ iajs-3789 24 13 is be AUX iajs-3789 24 14 equal equal ADJ iajs-3789 24 15 𝑖𝑛𝑑𝒱(Ι± 𝑖𝑛𝑑𝒱(Ι± NOUN iajs-3789 24 16 ) ) PUNCT iajs-3789 24 17 = = PUNCT iajs-3789 25 1 ⋃{𝒩 ⋃{𝒩 PROPN iajs-3789 25 2 βŠ† βŠ† NUM iajs-3789 25 3 𝐾 𝐾 PROPN iajs-3789 25 4 : : PUNCT iajs-3789 25 5 𝒩 𝒩 PROPN iajs-3789 25 6 𝑖𝑠 𝑖𝑠 PROPN iajs-3789 25 7 𝒱 𝒱 PROPN iajs-3789 25 8 βˆ’ βˆ’ PROPN iajs-3789 25 9 open open ADJ iajs-3789 25 10 and and CCONJ iajs-3789 25 11 𝒩 𝒩 PROPN iajs-3789 25 12 βŠ† βŠ† NUM iajs-3789 25 13 Ι± Ι± PROPN iajs-3789 25 14 } } PUNCT iajs-3789 25 15 . . PUNCT iajs-3789 26 1 theorem theorem VERB iajs-3789 26 2 2.3 2.3 NUM iajs-3789 26 3 for for ADP iajs-3789 26 4 any any DET iajs-3789 26 5 π“₯-space π“₯-space PROPN iajs-3789 26 6 ( ( PUNCT iajs-3789 26 7 𝐾 𝐾 PROPN iajs-3789 26 8 , , PUNCT iajs-3789 26 9 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 26 10 ) ) PUNCT iajs-3789 26 11 , , PUNCT iajs-3789 26 12 let let VERB iajs-3789 26 13 Ι± Ι± PROPN iajs-3789 26 14 βŠ† βŠ† NUM iajs-3789 26 15 𝐾 𝐾 PROPN iajs-3789 26 16 , , PUNCT iajs-3789 26 17 π‘˜ π‘˜ PRON iajs-3789 26 18 ∈ ∈ PROPN iajs-3789 26 19 𝐾 𝐾 PROPN iajs-3789 26 20 .so .so PUNCT iajs-3789 26 21 π‘˜ π‘˜ PRON iajs-3789 26 22 ∈ ∈ PROPN iajs-3789 26 23 𝑐𝑙𝒱(Ι± 𝑐𝑙𝒱(Ι± NOUN iajs-3789 26 24 ) ) PUNCT iajs-3789 27 1 if if SCONJ iajs-3789 27 2 and and CCONJ iajs-3789 27 3 only only ADV iajs-3789 27 4 if if SCONJ iajs-3789 27 5 𝒩 𝒩 PROPN iajs-3789 27 6 ∩ ∩ NOUN iajs-3789 27 7 Ι± Ι± ADP iajs-3789 27 8 β‰  β‰  PROPN iajs-3789 27 9 βˆ… βˆ… NOUN iajs-3789 27 10 for for ADP iajs-3789 27 11 all all DET iajs-3789 27 12 𝒱 𝒱 NOUN iajs-3789 27 13 βˆ’ βˆ’ NOUN iajs-3789 27 14 open open ADJ iajs-3789 27 15 set set NOUN iajs-3789 27 16 𝒩 𝒩 PROPN iajs-3789 27 17 ; ; PUNCT iajs-3789 27 18 π‘˜ π‘˜ DET iajs-3789 27 19 ∈ ∈ NOUN iajs-3789 27 20 𝒩. 𝒩. ADJ iajs-3789 27 21 proof proof NOUN iajs-3789 27 22 : : PUNCT iajs-3789 27 23 the the PRON iajs-3789 27 24 " " PUNCT iajs-3789 27 25 if if SCONJ iajs-3789 27 26 " " PUNCT iajs-3789 27 27 part part NOUN iajs-3789 27 28 let let VERB iajs-3789 27 29 π‘˜ π‘˜ PRON iajs-3789 27 30 ∈ ∈ PROPN iajs-3789 27 31 cl cl NOUN iajs-3789 27 32 π“₯(Ι± π“₯(Ι± NOUN iajs-3789 27 33 ) ) PUNCT iajs-3789 27 34 = = SYM iajs-3789 27 35 β‹‚{β„± β‹‚{β„± PROPN iajs-3789 27 36 βŠ† βŠ† NUM iajs-3789 27 37 𝐾 𝐾 PROPN iajs-3789 27 38 : : PUNCT iajs-3789 27 39 β„± β„± PROPN iajs-3789 27 40 𝑖𝑠 𝑖𝑠 CCONJ iajs-3789 27 41 𝒱 𝒱 PROPN iajs-3789 27 42 βˆ’ βˆ’ PROPN iajs-3789 27 43 closed closed ADJ iajs-3789 27 44 and and CCONJ iajs-3789 27 45 Ι± Ι± ADJ iajs-3789 27 46 βŠ† βŠ† NUM iajs-3789 27 47 β„± β„± PROPN iajs-3789 27 48 } } PUNCT iajs-3789 27 49 and and CCONJ iajs-3789 27 50 let let VERB iajs-3789 27 51 be be AUX iajs-3789 27 52 there there PRON iajs-3789 27 53 exists exist VERB iajs-3789 27 54 a a DET iajs-3789 27 55 𝒱 𝒱 NOUN iajs-3789 27 56 βˆ’ βˆ’ NOUN iajs-3789 27 57 open open ADJ iajs-3789 27 58 set set NOUN iajs-3789 27 59 𝒩 𝒩 PROPN iajs-3789 27 60 containing contain VERB iajs-3789 27 61 π‘₯ π‘₯ PRON iajs-3789 27 62 such such ADJ iajs-3789 27 63 that that SCONJ iajs-3789 27 64 𝒩 𝒩 PROPN iajs-3789 27 65 ∩ ∩ NOUN iajs-3789 27 66 Ι± Ι± NOUN iajs-3789 27 67 = = NOUN iajs-3789 27 68 βˆ… βˆ… NOUN iajs-3789 27 69 , , PUNCT iajs-3789 27 70 it it PRON iajs-3789 27 71 follows follow VERB iajs-3789 27 72 that that SCONJ iajs-3789 27 73 Ι± Ι± ADP iajs-3789 27 74 βŠ† βŠ† NUM iajs-3789 27 75 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 27 76 which which PRON iajs-3789 27 77 is be AUX iajs-3789 27 78 𝒱 𝒱 PROPN iajs-3789 27 79 βˆ’ βˆ’ NOUN iajs-3789 27 80 closed closed ADJ iajs-3789 27 81 set set VERB iajs-3789 27 82 with with ADP iajs-3789 27 83 π‘₯ π‘₯ PROPN iajs-3789 27 84 βˆ‰ βˆ‰ PROPN iajs-3789 27 85 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 27 86 , , PUNCT iajs-3789 27 87 so so ADV iajs-3789 27 88 π‘₯ π‘₯ PRON iajs-3789 27 89 βˆ‰ βˆ‰ ADJ iajs-3789 27 90 β‹‚{β„± β‹‚{β„± PROPN iajs-3789 27 91 βŠ† βŠ† NUM iajs-3789 27 92 π‘˜ π‘˜ NOUN iajs-3789 27 93 : : PUNCT iajs-3789 27 94 β„± β„± PROPN iajs-3789 27 95 𝑖𝑠 𝑖𝑠 INTJ iajs-3789 27 96 𝒱 𝒱 PROPN iajs-3789 27 97 βˆ’ βˆ’ PROPN iajs-3789 27 98 closed closed ADJ iajs-3789 27 99 and and CCONJ iajs-3789 27 100 Ι± Ι± ADJ iajs-3789 27 101 βŠ† βŠ† NUM iajs-3789 27 102 β„± β„± PROPN iajs-3789 27 103 } } PUNCT iajs-3789 27 104 which which PRON iajs-3789 27 105 is be AUX iajs-3789 27 106 a a DET iajs-3789 27 107 contradiction contradiction NOUN iajs-3789 27 108 . . PUNCT iajs-3789 28 1 the the PRON iajs-3789 28 2 " " PUNCT iajs-3789 28 3 only only ADV iajs-3789 28 4 if if SCONJ iajs-3789 28 5 " " PUNCT iajs-3789 28 6 part part NOUN iajs-3789 28 7 assume assume VERB iajs-3789 28 8 that that SCONJ iajs-3789 28 9 every every DET iajs-3789 28 10 𝒱 𝒱 PROPN iajs-3789 28 11 βˆ’ βˆ’ PROPN iajs-3789 28 12 open open ADJ iajs-3789 28 13 set set NOUN iajs-3789 28 14 𝒩 𝒩 PROPN iajs-3789 28 15 containing contain VERB iajs-3789 28 16 π‘₯ π‘₯ PROPN iajs-3789 28 17 intersects intersect NOUN iajs-3789 28 18 Ι± Ι± ADJ iajs-3789 28 19 and and CCONJ iajs-3789 28 20 suppose suppose VERB iajs-3789 28 21 that that SCONJ iajs-3789 28 22 π‘˜ π‘˜ DET iajs-3789 28 23 βˆ‰ βˆ‰ PROPN iajs-3789 28 24 cl cl NOUN iajs-3789 28 25 π“₯(Ι± π“₯(Ι± PROPN iajs-3789 28 26 ) ) PUNCT iajs-3789 28 27 = = SYM iajs-3789 28 28 β‹‚{β„± β‹‚{β„± PROPN iajs-3789 28 29 βŠ† βŠ† NUM iajs-3789 28 30 𝐾 𝐾 PROPN iajs-3789 28 31 : : PUNCT iajs-3789 28 32 β„± β„± PROPN iajs-3789 28 33 𝑖𝑠 𝑖𝑠 CCONJ iajs-3789 28 34 𝒱 𝒱 PROPN iajs-3789 28 35 βˆ’ βˆ’ PROPN iajs-3789 28 36 closed closed ADJ iajs-3789 28 37 and and CCONJ iajs-3789 28 38 Ι± Ι± ADJ iajs-3789 28 39 βŠ† βŠ† NUM iajs-3789 28 40 β„± β„± PROPN iajs-3789 28 41 } } PUNCT iajs-3789 28 42 , , PUNCT iajs-3789 28 43 then then ADV iajs-3789 28 44 there there PRON iajs-3789 28 45 exists exist VERB iajs-3789 28 46 a a DET iajs-3789 28 47 𝒱 𝒱 PROPN iajs-3789 28 48 βˆ’ βˆ’ PROPN iajs-3789 28 49 closed closed ADJ iajs-3789 28 50 set set VERB iajs-3789 28 51 β„± β„± PROPN iajs-3789 28 52 with with ADP iajs-3789 28 53 Ι± Ι± PROPN iajs-3789 28 54 βŠ† βŠ† NUM iajs-3789 28 55 β„± β„± PROPN iajs-3789 28 56 and and CCONJ iajs-3789 28 57 π‘˜ π‘˜ DET iajs-3789 28 58 βˆ‰ βˆ‰ PROPN iajs-3789 28 59 β„± β„± PROPN iajs-3789 28 60 , , PUNCT iajs-3789 28 61 so so SCONJ iajs-3789 28 62 π‘˜ π‘˜ PRON iajs-3789 28 63 ∈ ∈ PROPN iajs-3789 29 1 ℱ𝑐 ℱ𝑐 PROPN iajs-3789 29 2 that that PRON iajs-3789 29 3 is be AUX iajs-3789 29 4 𝒱 𝒱 PROPN iajs-3789 29 5 βˆ’ βˆ’ NOUN iajs-3789 29 6 open open ADJ iajs-3789 29 7 set set NOUN iajs-3789 29 8 , , PUNCT iajs-3789 29 9 but but CCONJ iajs-3789 29 10 ℱ𝑐 ℱ𝑐 PROPN iajs-3789 29 11 ∩ ∩ NOUN iajs-3789 29 12 Ι± Ι± NOUN iajs-3789 29 13 = = SYM iajs-3789 29 14 βˆ… βˆ… NOUN iajs-3789 29 15 , , PUNCT iajs-3789 29 16 which which PRON iajs-3789 29 17 is be AUX iajs-3789 29 18 contradiction contradiction NOUN iajs-3789 29 19 . . PUNCT iajs-3789 30 1 definition definition NOUN iajs-3789 30 2 2.4 2.4 NUM iajs-3789 30 3 [ [ X iajs-3789 30 4 9 9 NUM iajs-3789 30 5 ] ] PUNCT iajs-3789 30 6 a a DET iajs-3789 30 7 collection collection NOUN iajs-3789 30 8 q q NOUN iajs-3789 30 9 of of ADP iajs-3789 30 10 subsets subset NOUN iajs-3789 30 11 of of ADP iajs-3789 30 12 a a DET iajs-3789 30 13 nonempty nonempty NOUN iajs-3789 30 14 subset subset NOUN iajs-3789 30 15 in in ADP iajs-3789 30 16 a a DET iajs-3789 30 17 topological topological ADJ iajs-3789 30 18 space space NOUN iajs-3789 30 19 ( ( PUNCT iajs-3789 30 20 k k X iajs-3789 30 21 , , PUNCT iajs-3789 30 22 Ο„ Ο„ X iajs-3789 30 23 ) ) PUNCT iajs-3789 30 24 is be AUX iajs-3789 30 25 referred refer VERB iajs-3789 30 26 to to ADP iajs-3789 30 27 as as ADP iajs-3789 30 28 a a DET iajs-3789 30 29 grill grill NOUN iajs-3789 30 30 on on ADP iajs-3789 30 31 k k PROPN iajs-3789 30 32 if if SCONJ iajs-3789 30 33 it it PRON iajs-3789 30 34 fulfils fulfil VERB iajs-3789 30 35 the the DET iajs-3789 30 36 following follow VERB iajs-3789 30 37 conditions condition NOUN iajs-3789 30 38 : : PUNCT iajs-3789 30 39 1 1 X iajs-3789 30 40 . . X iajs-3789 30 41 βˆ… βˆ… NOUN iajs-3789 30 42 βˆ‰ βˆ‰ PROPN iajs-3789 30 43 𝑸 𝑸 PROPN iajs-3789 30 44 . . PUNCT iajs-3789 31 1 2 2 X iajs-3789 31 2 . . X iajs-3789 31 3 Ι± Ι± PROPN iajs-3789 31 4 ∈ ∈ PROPN iajs-3789 31 5 𝑸 𝑸 PROPN iajs-3789 31 6 & & CCONJ iajs-3789 31 7 Ι± Ι± PROPN iajs-3789 31 8 βŠ† βŠ† NUM iajs-3789 31 9 b b PROPN iajs-3789 31 10 therefore therefore ADV iajs-3789 31 11 b b PROPN iajs-3789 31 12 ∈ ∈ PROPN iajs-3789 31 13 𝑸. 𝑸. PROPN iajs-3789 31 14 3 3 NUM iajs-3789 31 15 . . PUNCT iajs-3789 32 1 Ι± Ι± ADJ iajs-3789 32 2 βˆ‰ βˆ‰ PROPN iajs-3789 32 3 𝑸 𝑸 PROPN iajs-3789 32 4 & & CCONJ iajs-3789 32 5 b b PROPN iajs-3789 32 6 βˆ‰ βˆ‰ PROPN iajs-3789 32 7 𝑸 𝑸 PROPN iajs-3789 32 8 therefore therefore ADV iajs-3789 32 9 Ι± Ι± ADP iajs-3789 32 10 ⋃ ⋃ PROPN iajs-3789 32 11 b b PROPN iajs-3789 32 12 βˆ‰ βˆ‰ PROPN iajs-3789 32 13 𝑸 𝑸 PROPN iajs-3789 32 14 . . PUNCT iajs-3789 33 1 a a DET iajs-3789 33 2 topological topological ADJ iajs-3789 33 3 space space NOUN iajs-3789 33 4 ( ( PUNCT iajs-3789 33 5 𝐾 𝐾 PROPN iajs-3789 33 6 , , PUNCT iajs-3789 33 7 𝜏 𝜏 NOUN iajs-3789 33 8 ) ) PUNCT iajs-3789 33 9 with with ADP iajs-3789 33 10 a a DET iajs-3789 33 11 grill grill ADJ iajs-3789 33 12 𝑸 𝑸 NOUN iajs-3789 33 13 on on ADP iajs-3789 33 14 𝐾 𝐾 PROPN iajs-3789 33 15 is be AUX iajs-3789 33 16 named name VERB iajs-3789 33 17 a a DET iajs-3789 33 18 grill grill ADJ iajs-3789 33 19 topological topological ADJ iajs-3789 33 20 space space NOUN iajs-3789 33 21 and and CCONJ iajs-3789 33 22 is be AUX iajs-3789 33 23 symbolize symbolize NOUN iajs-3789 33 24 by by ADP iajs-3789 33 25 ( ( PUNCT iajs-3789 33 26 𝑋 𝑋 PROPN iajs-3789 33 27 , , PUNCT iajs-3789 33 28 𝜏 𝜏 NOUN iajs-3789 33 29 , , PUNCT iajs-3789 33 30 𝑸 𝑸 NOUN iajs-3789 33 31 ) ) PUNCT iajs-3789 33 32 . . PUNCT iajs-3789 34 1 definition definition NOUN iajs-3789 34 2 2.5 2.5 NUM iajs-3789 35 1 [ [ X iajs-3789 35 2 4 4 X iajs-3789 35 3 ] ] PUNCT iajs-3789 35 4 let let VERB iajs-3789 35 5 q q PART iajs-3789 35 6 denote denote VERB iajs-3789 35 7 a a DET iajs-3789 35 8 grill grill NOUN iajs-3789 35 9 on on ADP iajs-3789 35 10 a a DET iajs-3789 35 11 topological topological ADJ iajs-3789 35 12 space space NOUN iajs-3789 35 13 ( ( PUNCT iajs-3789 35 14 𝐾 𝐾 PROPN iajs-3789 35 15 , , PUNCT iajs-3789 35 16 𝜏 𝜏 NOUN iajs-3789 35 17 ) ) PUNCT iajs-3789 35 18 . . PUNCT iajs-3789 36 1 consider consider VERB iajs-3789 36 2 the the DET iajs-3789 36 3 map map NOUN iajs-3789 36 4 πœƒ πœƒ NOUN iajs-3789 36 5 ∢ ∢ NOUN iajs-3789 36 6 𝒫(𝐾 𝒫(𝐾 PROPN iajs-3789 36 7 ) ) PUNCT iajs-3789 37 1 β†’ β†’ SYM iajs-3789 37 2 𝒫(𝐾 𝒫(𝐾 X iajs-3789 37 3 ) ) PUNCT iajs-3789 37 4 such such ADJ iajs-3789 37 5 that that SCONJ iajs-3789 37 6 πœƒ(Ι± πœƒ(Ι± PROPN iajs-3789 37 7 ) ) PUNCT iajs-3789 37 8 = = PRON iajs-3789 37 9 { { PUNCT iajs-3789 37 10 π‘₯ π‘₯ PUNCT iajs-3789 37 11 ∈ ∈ PROPN iajs-3789 37 12 𝐾 𝐾 NOUN iajs-3789 37 13 : : PUNCT iajs-3789 37 14 𝑆⋂ɱ 𝑆⋂ɱ ADJ iajs-3789 37 15 ∈ ∈ PROPN iajs-3789 37 16 𝑸 𝑸 PROPN iajs-3789 37 17 βˆ€π‘† βˆ€π‘† PROPN iajs-3789 37 18 ∈ ∈ PROPN iajs-3789 37 19 𝜏 𝜏 NOUN iajs-3789 37 20 , , PUNCT iajs-3789 37 21 π‘₯πœ–π‘† π‘₯πœ–π‘† ADV iajs-3789 37 22 } } PUNCT iajs-3789 37 23 for for ADP iajs-3789 37 24 each each PRON iajs-3789 37 25 Ι± Ι± PROPN iajs-3789 37 26 βŠ† βŠ† NUM iajs-3789 37 27 𝐾. 𝐾. PRON iajs-3789 37 28 therefore therefore ADV iajs-3789 37 29 , , PUNCT iajs-3789 37 30 the the DET iajs-3789 37 31 function function NOUN iajs-3789 37 32 Ο‰ Ο‰ PROPN iajs-3789 37 33 ∢ ∢ NOUN iajs-3789 37 34 𝒫(𝐾 𝒫(𝐾 PROPN iajs-3789 37 35 ) ) PUNCT iajs-3789 37 36 β†’ β†’ SYM iajs-3789 37 37 𝒫(𝐾 𝒫(𝐾 PROPN iajs-3789 37 38 ) ) PUNCT iajs-3789 37 39 where where SCONJ iajs-3789 37 40 Ο‰(Ι± Ο‰(Ι± NOUN iajs-3789 37 41 ) ) PUNCT iajs-3789 37 42 = = SYM iajs-3789 37 43 Ι±β‹ƒπœƒ(Ι± Ι±β‹ƒπœƒ(Ι± X iajs-3789 37 44 ) ) PUNCT iajs-3789 37 45 is be AUX iajs-3789 37 46 a a DET iajs-3789 37 47 kuratowski kuratowski NOUN iajs-3789 37 48 ’s ’s PART iajs-3789 37 49 closure closure NOUN iajs-3789 37 50 operator operator NOUN iajs-3789 37 51 and and CCONJ iajs-3789 37 52 the the DET iajs-3789 37 53 topological topological ADJ iajs-3789 37 54 origin origin NOUN iajs-3789 37 55 , , PUNCT iajs-3789 37 56 which which PRON iajs-3789 37 57 is be AUX iajs-3789 37 58 finer fine ADJ iajs-3789 37 59 than than ADP iajs-3789 37 60 𝜏 𝜏 NOUN iajs-3789 37 61 and and CCONJ iajs-3789 37 62 defined define VERB iajs-3789 37 63 as as ADP iajs-3789 37 64 πœπ‘Έ πœπ‘Έ NOUN iajs-3789 37 65 = = PUNCT iajs-3789 37 66 { { PUNCT iajs-3789 37 67 Ι± Ι± PROPN iajs-3789 37 68 βŠ† βŠ† NUM iajs-3789 37 69 x x NOUN iajs-3789 37 70 : : PUNCT iajs-3789 37 71 ψ(𝐾 ψ(𝐾 ADP iajs-3789 37 72 βˆ’ βˆ’ PROPN iajs-3789 37 73 Ι± Ι± X iajs-3789 37 74 ) ) PUNCT iajs-3789 38 1 = = SYM iajs-3789 38 2 𝐾 𝐾 PROPN iajs-3789 38 3 βˆ’ βˆ’ PROPN iajs-3789 38 4 Ι± Ι± PROPN iajs-3789 38 5 } } PUNCT iajs-3789 38 6 . . PUNCT iajs-3789 39 1 3 3 X iajs-3789 39 2 . . X iajs-3789 39 3 closure closure NOUN iajs-3789 39 4 operator operator NOUN iajs-3789 39 5 in in ADP iajs-3789 39 6 grill grill NOUN iajs-3789 39 7 π“₯-spaces π“₯-spaces PROPN iajs-3789 39 8 definition definition NOUN iajs-3789 39 9 3.1 3.1 NUM iajs-3789 39 10 a a DET iajs-3789 39 11 nonempty nonempty ADJ iajs-3789 39 12 family family NOUN iajs-3789 39 13 𝕾 𝕾 NOUN iajs-3789 39 14 of of ADP iajs-3789 39 15 nonempty nonempty ADJ iajs-3789 39 16 sets set NOUN iajs-3789 39 17 of of ADP iajs-3789 39 18 a a DET iajs-3789 39 19 π“₯-space π“₯-space PROPN iajs-3789 39 20 ( ( PUNCT iajs-3789 39 21 𝐾 𝐾 PROPN iajs-3789 39 22 , , PUNCT iajs-3789 39 23 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 39 24 ) ) PUNCT iajs-3789 39 25 is be AUX iajs-3789 39 26 named name VERB iajs-3789 39 27 a a DET iajs-3789 39 28 grill grill NOUN iajs-3789 39 29 on on ADP iajs-3789 39 30 𝐾 𝐾 PROPN iajs-3789 39 31 , , PUNCT iajs-3789 39 32 if if SCONJ iajs-3789 39 33 it it PRON iajs-3789 39 34 satisfies satisfy VERB iajs-3789 39 35 the the DET iajs-3789 39 36 following follow VERB iajs-3789 39 37 conditions condition NOUN iajs-3789 39 38 : : PUNCT iajs-3789 39 39 1 1 X iajs-3789 39 40 . . X iajs-3789 39 41 βˆ… βˆ… NOUN iajs-3789 39 42 βˆ‰ βˆ‰ X iajs-3789 39 43 𝕾. 𝕾. PROPN iajs-3789 39 44 2 2 NUM iajs-3789 39 45 . . PUNCT iajs-3789 39 46 Ι± Ι± NOUN iajs-3789 39 47 ∈ ∈ PROPN iajs-3789 39 48 𝕾 𝕾 NOUN iajs-3789 39 49 ^ ^ PUNCT iajs-3789 39 50 Ι± Ι± PROPN iajs-3789 39 51 βŠ† βŠ† NUM iajs-3789 39 52 b b NOUN iajs-3789 39 53 therefore therefore ADV iajs-3789 39 54 p p PROPN iajs-3789 39 55 ∈ ∈ PROPN iajs-3789 39 56 𝕾. 𝕾. NOUN iajs-3789 39 57 ihjpas ihjpa NOUN iajs-3789 39 58 . . PUNCT iajs-3789 40 1 37 37 NUM iajs-3789 40 2 ( ( PUNCT iajs-3789 40 3 2 2 NUM iajs-3789 40 4 ) ) PUNCT iajs-3789 40 5 2024 2024 NUM iajs-3789 40 6 434 434 NUM iajs-3789 40 7 3 3 NUM iajs-3789 40 8 . . PUNCT iajs-3789 40 9 Ι± Ι± ADJ iajs-3789 40 10 βˆ‰ βˆ‰ ADJ iajs-3789 40 11 𝕾 𝕾 NOUN iajs-3789 40 12 ^ ^ PUNCT iajs-3789 40 13 p p NOUN iajs-3789 40 14 βˆ‰ βˆ‰ PROPN iajs-3789 40 15 𝕾 𝕾 NOUN iajs-3789 40 16 therefore therefore ADV iajs-3789 40 17 Ι± Ι± ADP iajs-3789 40 18 ⋃ ⋃ PROPN iajs-3789 40 19 p p NOUN iajs-3789 40 20 βˆ‰ βˆ‰ ADJ iajs-3789 40 21 𝕾 𝕾 NOUN iajs-3789 40 22 . . PUNCT iajs-3789 41 1 any any DET iajs-3789 41 2 π“₯-space π“₯-space PROPN iajs-3789 41 3 ( ( PUNCT iajs-3789 41 4 𝐾 𝐾 PROPN iajs-3789 41 5 , , PUNCT iajs-3789 41 6 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 41 7 ) ) PUNCT iajs-3789 41 8 with with ADP iajs-3789 41 9 a a DET iajs-3789 41 10 grill grill ADJ iajs-3789 41 11 𝕾 𝕾 NOUN iajs-3789 41 12 on on ADP iajs-3789 41 13 𝐾 𝐾 PROPN iajs-3789 41 14 is be AUX iajs-3789 41 15 named name VERB iajs-3789 41 16 a a DET iajs-3789 41 17 grill grill NOUN iajs-3789 41 18 π“₯-space π“₯-space PROPN iajs-3789 41 19 and and CCONJ iajs-3789 41 20 is be AUX iajs-3789 41 21 symbolize symbolize NOUN iajs-3789 41 22 by by ADP iajs-3789 41 23 ( ( PUNCT iajs-3789 41 24 𝐾 𝐾 PROPN iajs-3789 41 25 , , PUNCT iajs-3789 41 26 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 41 27 , , PUNCT iajs-3789 41 28 𝕾 𝕾 PROPN iajs-3789 41 29 ) ) PUNCT iajs-3789 41 30 . . PUNCT iajs-3789 42 1 definition definition NOUN iajs-3789 42 2 3.2 3.2 NUM iajs-3789 42 3 let let VERB iajs-3789 42 4 𝕾 𝕾 NOUN iajs-3789 42 5 be be AUX iajs-3789 42 6 a a DET iajs-3789 42 7 grill grill NOUN iajs-3789 42 8 on on ADP iajs-3789 42 9 π“₯-space π“₯-space PROPN iajs-3789 42 10 ( ( PUNCT iajs-3789 42 11 𝐾 𝐾 PROPN iajs-3789 42 12 , , PUNCT iajs-3789 42 13 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 42 14 ) ) PUNCT iajs-3789 42 15 . . PUNCT iajs-3789 43 1 the the DET iajs-3789 43 2 map map NOUN iajs-3789 43 3 υ𝕾 υ𝕾 PROPN iajs-3789 43 4 ∢ ∢ PROPN iajs-3789 43 5 𝒫(𝐾 𝒫(𝐾 PROPN iajs-3789 43 6 ) ) PUNCT iajs-3789 43 7 β†’ β†’ SYM iajs-3789 43 8 𝒫(𝐾 𝒫(𝐾 X iajs-3789 43 9 ) ) PUNCT iajs-3789 43 10 such such ADJ iajs-3789 43 11 that that PRON iajs-3789 43 12 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 43 13 ) ) PUNCT iajs-3789 44 1 = = PRON iajs-3789 44 2 { { PUNCT iajs-3789 44 3 π‘₯ π‘₯ PUNCT iajs-3789 44 4 ∈ ∈ PROPN iajs-3789 44 5 𝑋 𝑋 NOUN iajs-3789 44 6 : : PUNCT iajs-3789 44 7 𝑆⋂ɱ 𝑆⋂ɱ ADJ iajs-3789 44 8 ∈ ∈ NOUN iajs-3789 44 9 𝕾 𝕾 NOUN iajs-3789 44 10 βˆ€π‘† βˆ€π‘† PUNCT iajs-3789 44 11 ∈ ∈ PROPN iajs-3789 44 12 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 44 13 , , PUNCT iajs-3789 44 14 π‘₯πœ–π‘† π‘₯πœ–π‘† ADV iajs-3789 44 15 } } PUNCT iajs-3789 44 16 for for ADP iajs-3789 44 17 each each DET iajs-3789 44 18 Ι± Ι± PROPN iajs-3789 44 19 βŠ† βŠ† NUM iajs-3789 44 20 𝐾 𝐾 PROPN iajs-3789 44 21 , , PUNCT iajs-3789 44 22 is be AUX iajs-3789 44 23 named name VERB iajs-3789 44 24 the the DET iajs-3789 44 25 local local ADJ iajs-3789 44 26 map map NOUN iajs-3789 44 27 commitment commitment NOUN iajs-3789 44 28 to to ADP iajs-3789 44 29 a a DET iajs-3789 44 30 grill grill NOUN iajs-3789 44 31 𝕾 𝕾 NOUN iajs-3789 44 32 with with ADP iajs-3789 44 33 the the DET iajs-3789 44 34 topology topology NOUN iajs-3789 44 35 𝒱𝑂𝑋. 𝒱𝑂𝑋. X iajs-3789 44 36 theorem theorem VERB iajs-3789 44 37 3.3 3.3 NUM iajs-3789 44 38 suppose suppose VERB iajs-3789 44 39 that that SCONJ iajs-3789 44 40 ( ( PUNCT iajs-3789 44 41 𝐾 𝐾 PROPN iajs-3789 44 42 , , PUNCT iajs-3789 44 43 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 44 44 ) ) PUNCT iajs-3789 44 45 be be VERB iajs-3789 44 46 a a DET iajs-3789 44 47 π“₯-space π“₯-space PROPN iajs-3789 44 48 . . PUNCT iajs-3789 45 1 so so ADV iajs-3789 45 2 , , PUNCT iajs-3789 45 3 the the DET iajs-3789 45 4 following following NOUN iajs-3789 45 5 are be AUX iajs-3789 45 6 satisfying satisfy VERB iajs-3789 45 7 : : PUNCT iajs-3789 45 8 1 1 X iajs-3789 45 9 . . X iajs-3789 46 1 for for ADP iajs-3789 46 2 any any DET iajs-3789 46 3 grill grill NOUN iajs-3789 46 4 𝕾 𝕾 NOUN iajs-3789 46 5 on on ADP iajs-3789 46 6 𝐾 𝐾 PROPN iajs-3789 46 7 , , PUNCT iajs-3789 46 8 then then ADV iajs-3789 46 9 a a DET iajs-3789 46 10 βŠ† βŠ† NUM iajs-3789 46 11 b b NOUN iajs-3789 46 12 implies imply VERB iajs-3789 46 13 υ𝕾(𝐴 υ𝕾(𝐴 PROPN iajs-3789 46 14 ) ) PUNCT iajs-3789 46 15 βŠ† βŠ† NUM iajs-3789 46 16 υ𝕾(𝐡 υ𝕾(𝐡 NOUN iajs-3789 46 17 ) ) PUNCT iajs-3789 46 18 . . PUNCT iajs-3789 47 1 2 2 X iajs-3789 47 2 . . X iajs-3789 47 3 if if SCONJ iajs-3789 47 4 𝕾1 𝕾1 VERB iajs-3789 47 5 and and CCONJ iajs-3789 47 6 𝕾2 𝕾2 NUM iajs-3789 47 7 are be AUX iajs-3789 47 8 two two NUM iajs-3789 47 9 grilles grille NOUN iajs-3789 47 10 on on ADP iajs-3789 47 11 x x PUNCT iajs-3789 47 12 and and CCONJ iajs-3789 47 13 𝕾1 𝕾1 PROPN iajs-3789 47 14 βŠ† βŠ† NUM iajs-3789 47 15 𝕾2 𝕾2 NUM iajs-3789 47 16 , , PUNCT iajs-3789 47 17 implies imply VERB iajs-3789 47 18 that that SCONJ iajs-3789 47 19 υ𝕾1 υ𝕾1 PROPN iajs-3789 47 20 ( ( PUNCT iajs-3789 47 21 𝐴 𝐴 PROPN iajs-3789 47 22 ) ) PUNCT iajs-3789 47 23 βŠ† βŠ† NUM iajs-3789 47 24 υ𝕾2 υ𝕾2 PROPN iajs-3789 47 25 ( ( PUNCT iajs-3789 47 26 𝐴 𝐴 PROPN iajs-3789 47 27 ) ) PUNCT iajs-3789 47 28 for for ADP iajs-3789 47 29 each each DET iajs-3789 47 30 a a DET iajs-3789 47 31 βŠ† βŠ† NUM iajs-3789 47 32 𝐾. 𝐾. SYM iajs-3789 47 33 3 3 NUM iajs-3789 47 34 . . PUNCT iajs-3789 48 1 whenever whenever SCONJ iajs-3789 48 2 𝕾 𝕾 NOUN iajs-3789 48 3 a a DET iajs-3789 48 4 grill grill NOUN iajs-3789 48 5 from from ADP iajs-3789 48 6 𝐾 𝐾 PROPN iajs-3789 48 7 , , PUNCT iajs-3789 48 8 if if SCONJ iajs-3789 48 9 𝐴 𝐴 PROPN iajs-3789 48 10 βˆ‰ βˆ‰ PROPN iajs-3789 48 11 𝕾 𝕾 PROPN iajs-3789 48 12 , , PUNCT iajs-3789 48 13 implies imply VERB iajs-3789 48 14 that that SCONJ iajs-3789 48 15 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 48 16 ) ) PUNCT iajs-3789 49 1 = = PUNCT iajs-3789 49 2 βˆ…. βˆ…. PRON iajs-3789 49 3 proof proof NOUN iajs-3789 49 4 : : PUNCT iajs-3789 49 5 1 1 X iajs-3789 49 6 . . X iajs-3789 49 7 let let VERB iajs-3789 49 8 x x SYM iajs-3789 49 9 ∈ ∈ PROPN iajs-3789 49 10 υ𝕾(a υ𝕾(a PROPN iajs-3789 49 11 ) ) PUNCT iajs-3789 49 12 , , PUNCT iajs-3789 49 13 so so ADV iajs-3789 50 1 βˆ€s βˆ€s PROPN iajs-3789 50 2 ∈ ∈ PROPN iajs-3789 51 1 𝒱ox 𝒱ox PROPN iajs-3789 51 2 , , PUNCT iajs-3789 51 3 xΟ΅s xΟ΅s PROPN iajs-3789 51 4 , , PUNCT iajs-3789 51 5 we we PRON iajs-3789 51 6 have have VERB iajs-3789 51 7 sβ‹‚a sβ‹‚a NOUN iajs-3789 51 8 ∈ ∈ NOUN iajs-3789 51 9 𝕾. 𝕾. NOUN iajs-3789 51 10 but but CCONJ iajs-3789 51 11 sβ‹‚a sβ‹‚a NOUN iajs-3789 51 12 βŠ† βŠ† NUM iajs-3789 51 13 sβ‹‚b sβ‹‚b NOUN iajs-3789 51 14 since since SCONJ iajs-3789 51 15 a a DET iajs-3789 51 16 βŠ† βŠ† NUM iajs-3789 51 17 b b NOUN iajs-3789 51 18 , , PUNCT iajs-3789 51 19 it it PRON iajs-3789 51 20 follows follow VERB iajs-3789 51 21 from from ADP iajs-3789 51 22 definition definition NOUN iajs-3789 51 23 3.1 3.1 NUM iajs-3789 51 24 that that PRON iajs-3789 51 25 sβ‹‚b sβ‹‚b VERB iajs-3789 51 26 ∈ ∈ NOUN iajs-3789 51 27 𝕾 𝕾 NOUN iajs-3789 51 28 βˆ€s βˆ€s X iajs-3789 51 29 ∈ ∈ PROPN iajs-3789 51 30 𝒱ox 𝒱ox PROPN iajs-3789 51 31 , , PUNCT iajs-3789 51 32 xΟ΅s xΟ΅s PROPN iajs-3789 51 33 , , PUNCT iajs-3789 51 34 then then ADV iajs-3789 51 35 x x SYM iajs-3789 51 36 ∈ ∈ PROPN iajs-3789 51 37 υ𝕾(b υ𝕾(b NUM iajs-3789 51 38 ) ) PUNCT iajs-3789 51 39 . . PUNCT iajs-3789 52 1 hence hence ADV iajs-3789 52 2 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 52 3 ) ) PUNCT iajs-3789 53 1 βŠ† βŠ† NUM iajs-3789 53 2 υ𝕾(𝐡 υ𝕾(𝐡 NOUN iajs-3789 53 3 ) ) PUNCT iajs-3789 53 4 . . PUNCT iajs-3789 54 1 2 2 X iajs-3789 54 2 . . X iajs-3789 54 3 let let VERB iajs-3789 54 4 x x X iajs-3789 54 5 ∈ ∈ PRON iajs-3789 54 6 υ𝕾1 υ𝕾1 NOUN iajs-3789 54 7 ( ( PUNCT iajs-3789 54 8 a a NOUN iajs-3789 54 9 ) ) PUNCT iajs-3789 54 10 , , PUNCT iajs-3789 55 1 so so ADV iajs-3789 55 2 βˆ€s βˆ€s PROPN iajs-3789 55 3 ∈ ∈ PROPN iajs-3789 56 1 𝒱ox 𝒱ox PROPN iajs-3789 56 2 , , PUNCT iajs-3789 56 3 xΟ΅s xΟ΅s PROPN iajs-3789 56 4 , , PUNCT iajs-3789 56 5 we we PRON iajs-3789 56 6 have have VERB iajs-3789 56 7 sβ‹‚a sβ‹‚a NOUN iajs-3789 56 8 ∈ ∈ PROPN iajs-3789 56 9 𝕾1 𝕾1 PROPN iajs-3789 56 10 . . PUNCT iajs-3789 57 1 but but CCONJ iajs-3789 57 2 𝕾1 𝕾1 PRON iajs-3789 57 3 βŠ† βŠ† NUM iajs-3789 57 4 𝕾2 𝕾2 PROPN iajs-3789 57 5 , , PUNCT iajs-3789 57 6 so so SCONJ iajs-3789 57 7 sβ‹‚a sβ‹‚a PROPN iajs-3789 57 8 ∈ ∈ PROPN iajs-3789 57 9 𝕾2 𝕾2 VERB iajs-3789 57 10 βˆ€s βˆ€s X iajs-3789 57 11 ∈ ∈ PROPN iajs-3789 57 12 𝒱ox 𝒱ox PROPN iajs-3789 57 13 , , PUNCT iajs-3789 57 14 xΟ΅s xΟ΅s PROPN iajs-3789 57 15 , , PUNCT iajs-3789 57 16 it it PRON iajs-3789 57 17 follows follow VERB iajs-3789 57 18 from from ADP iajs-3789 57 19 definition definition NOUN iajs-3789 57 20 3.2 3.2 NUM iajs-3789 58 1 that that PRON iajs-3789 58 2 x x PUNCT iajs-3789 58 3 ∈ ∈ ADJ iajs-3789 58 4 υ𝕾2 υ𝕾2 PROPN iajs-3789 58 5 ( ( PUNCT iajs-3789 58 6 𝐴 𝐴 PROPN iajs-3789 58 7 ) ) PUNCT iajs-3789 58 8 . . PUNCT iajs-3789 59 1 hence hence ADV iajs-3789 59 2 υ𝕾1 υ𝕾1 PROPN iajs-3789 59 3 ( ( PUNCT iajs-3789 59 4 𝐴 𝐴 PROPN iajs-3789 59 5 ) ) PUNCT iajs-3789 59 6 βŠ† βŠ† NUM iajs-3789 59 7 υ𝕾2 υ𝕾2 PROPN iajs-3789 59 8 ( ( PUNCT iajs-3789 59 9 𝐴 𝐴 PROPN iajs-3789 59 10 ) ) PUNCT iajs-3789 59 11 . . PUNCT iajs-3789 60 1 3 3 X iajs-3789 60 2 . . X iajs-3789 60 3 suppose suppose VERB iajs-3789 60 4 that that SCONJ iajs-3789 60 5 𝐴 𝐴 PROPN iajs-3789 60 6 βˆ‰ βˆ‰ PROPN iajs-3789 60 7 𝕾 𝕾 PROPN iajs-3789 60 8 , , PUNCT iajs-3789 60 9 and and CCONJ iajs-3789 60 10 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 60 11 ) ) PUNCT iajs-3789 60 12 β‰  β‰  PROPN iajs-3789 60 13 βˆ… βˆ… NOUN iajs-3789 60 14 , , PUNCT iajs-3789 60 15 then then ADV iajs-3789 60 16 there there PRON iajs-3789 60 17 exists exist VERB iajs-3789 60 18 x x X iajs-3789 60 19 ∈ ∈ PROPN iajs-3789 60 20 υ𝕾(a υ𝕾(a PROPN iajs-3789 60 21 ) ) PUNCT iajs-3789 60 22 , , PUNCT iajs-3789 60 23 it it PRON iajs-3789 60 24 follows follow VERB iajs-3789 60 25 from from ADP iajs-3789 60 26 definition definition NOUN iajs-3789 60 27 3.2 3.2 NUM iajs-3789 60 28 that that PRON iajs-3789 60 29 sβ‹‚a sβ‹‚a VERB iajs-3789 60 30 ∈ ∈ NOUN iajs-3789 60 31 𝕾 𝕾 NOUN iajs-3789 60 32 βˆ€s βˆ€s X iajs-3789 60 33 ∈ ∈ PROPN iajs-3789 60 34 𝒱ox 𝒱ox PROPN iajs-3789 60 35 , , PUNCT iajs-3789 60 36 xΟ΅s xΟ΅s PROPN iajs-3789 60 37 . . PUNCT iajs-3789 61 1 but but CCONJ iajs-3789 61 2 sβ‹‚a sβ‹‚a NOUN iajs-3789 61 3 βŠ† βŠ† NUM iajs-3789 61 4 a a PRON iajs-3789 61 5 , , PUNCT iajs-3789 61 6 it it PRON iajs-3789 61 7 follows follow VERB iajs-3789 61 8 from from ADP iajs-3789 61 9 definition definition NOUN iajs-3789 61 10 3.1(2 3.1(2 NUM iajs-3789 61 11 ) ) PUNCT iajs-3789 62 1 that that SCONJ iajs-3789 62 2 a a DET iajs-3789 62 3 ∈ ∈ PROPN iajs-3789 62 4 𝕾 𝕾 NOUN iajs-3789 62 5 , , PUNCT iajs-3789 62 6 which which PRON iajs-3789 62 7 is be AUX iajs-3789 62 8 a a DET iajs-3789 62 9 contradiction contradiction NOUN iajs-3789 62 10 . . PUNCT iajs-3789 63 1 theorem theorem VERB iajs-3789 63 2 3.4 3.4 NUM iajs-3789 63 3 for for ADP iajs-3789 63 4 a a DET iajs-3789 63 5 grill grill NOUN iajs-3789 63 6 π“₯-space π“₯-space PROPN iajs-3789 63 7 ( ( PUNCT iajs-3789 63 8 𝑋 𝑋 PROPN iajs-3789 63 9 , , PUNCT iajs-3789 63 10 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 63 11 , , PUNCT iajs-3789 63 12 𝕾 𝕾 PROPN iajs-3789 63 13 ) ) PUNCT iajs-3789 63 14 . . PUNCT iajs-3789 64 1 and and CCONJ iajs-3789 64 2 for for ADP iajs-3789 64 3 all all DET iajs-3789 64 4 a a PRON iajs-3789 64 5 , , PUNCT iajs-3789 64 6 b b PROPN iajs-3789 64 7 βŠ† βŠ† NUM iajs-3789 64 8 x x NOUN iajs-3789 64 9 , , PUNCT iajs-3789 64 10 the the DET iajs-3789 64 11 following follow VERB iajs-3789 64 12 are be AUX iajs-3789 64 13 true true ADJ iajs-3789 64 14 : : PUNCT iajs-3789 64 15 1 1 X iajs-3789 64 16 . . X iajs-3789 64 17 υ𝕾(𝐴)⋃υ𝕾(𝐡 υ𝕾(𝐴)⋃υ𝕾(𝐡 NOUN iajs-3789 64 18 ) ) PUNCT iajs-3789 64 19 = = SYM iajs-3789 64 20 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NOUN iajs-3789 64 21 ) ) PUNCT iajs-3789 64 22 . . PUNCT iajs-3789 65 1 2 2 X iajs-3789 65 2 . . X iajs-3789 65 3 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 65 4 ) ) PUNCT iajs-3789 65 5 βŠ† βŠ† NUM iajs-3789 65 6 cl cl NOUN iajs-3789 65 7 π“₯(𝐴 π“₯(𝐴 NOUN iajs-3789 65 8 ) ) PUNCT iajs-3789 65 9 . . PUNCT iajs-3789 66 1 3 3 X iajs-3789 66 2 . . X iajs-3789 66 3 υ𝕾(υ𝕾(𝐴 υ𝕾(υ𝕾(𝐴 NOUN iajs-3789 66 4 ) ) PUNCT iajs-3789 66 5 ) ) PUNCT iajs-3789 67 1 βŠ† βŠ† NUM iajs-3789 67 2 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 67 3 ) ) PUNCT iajs-3789 67 4 proof proof NOUN iajs-3789 67 5 : : PUNCT iajs-3789 67 6 1 1 X iajs-3789 67 7 . . X iajs-3789 67 8 let let VERB iajs-3789 67 9 x x PUNCT iajs-3789 67 10 ∈ ∈ PROPN iajs-3789 67 11 υ𝕾(a)⋃υ𝕾(𝐡 υ𝕾(a)⋃υ𝕾(𝐡 NOUN iajs-3789 67 12 ) ) PUNCT iajs-3789 67 13 , , PUNCT iajs-3789 67 14 then then ADV iajs-3789 67 15 x x X iajs-3789 67 16 ∈ ∈ PROPN iajs-3789 67 17 υ𝕾(a υ𝕾(a PROPN iajs-3789 67 18 ) ) PUNCT iajs-3789 67 19 or or CCONJ iajs-3789 67 20 x x PUNCT iajs-3789 67 21 ∈ ∈ PROPN iajs-3789 67 22 υ𝕾(𝐡 υ𝕾(𝐡 PROPN iajs-3789 67 23 ) ) PUNCT iajs-3789 67 24 . . PUNCT iajs-3789 68 1 if if SCONJ iajs-3789 68 2 x x PROPN iajs-3789 68 3 ∈ ∈ PROPN iajs-3789 68 4 υ𝕾(a υ𝕾(a PROPN iajs-3789 68 5 ) ) PUNCT iajs-3789 68 6 , , PUNCT iajs-3789 68 7 so so ADV iajs-3789 68 8 βˆ€s βˆ€s PROPN iajs-3789 68 9 ∈ ∈ PROPN iajs-3789 68 10 𝒱ox 𝒱ox PROPN iajs-3789 68 11 , , PUNCT iajs-3789 68 12 xΟ΅s xΟ΅s PROPN iajs-3789 68 13 , , PUNCT iajs-3789 68 14 we we PRON iajs-3789 68 15 have have VERB iajs-3789 68 16 sβ‹‚a sβ‹‚a NOUN iajs-3789 68 17 ∈ ∈ PROPN iajs-3789 68 18 𝕾. 𝕾. NOUN iajs-3789 68 19 since since SCONJ iajs-3789 68 20 a a DET iajs-3789 68 21 βŠ† βŠ† NUM iajs-3789 68 22 𝐴⋃𝐡 𝐴⋃𝐡 PROPN iajs-3789 68 23 , , PUNCT iajs-3789 68 24 so so ADV iajs-3789 68 25 sβ‹‚a sβ‹‚a NOUN iajs-3789 68 26 βŠ† βŠ† NUM iajs-3789 68 27 sβ‹‚(𝐴⋃𝐡 sβ‹‚(𝐴⋃𝐡 NOUN iajs-3789 68 28 ) ) PUNCT iajs-3789 68 29 . . PUNCT iajs-3789 69 1 from from ADP iajs-3789 69 2 definition definition NOUN iajs-3789 69 3 3.1(2 3.1(2 NUM iajs-3789 69 4 ) ) PUNCT iajs-3789 69 5 we we PRON iajs-3789 69 6 get get VERB iajs-3789 69 7 sβ‹‚(𝐴⋃𝐡 sβ‹‚(𝐴⋃𝐡 NOUN iajs-3789 69 8 ) ) PUNCT iajs-3789 69 9 ∈ ∈ NOUN iajs-3789 69 10 𝕾 𝕾 NOUN iajs-3789 69 11 βˆ€s βˆ€s X iajs-3789 69 12 ∈ ∈ PROPN iajs-3789 69 13 𝒱ox 𝒱ox PROPN iajs-3789 69 14 , , PUNCT iajs-3789 69 15 xΟ΅s xΟ΅s PROPN iajs-3789 69 16 . . PUNCT iajs-3789 70 1 hence hence ADV iajs-3789 70 2 x x SYM iajs-3789 70 3 ∈ ∈ PROPN iajs-3789 70 4 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NOUN iajs-3789 70 5 ) ) PUNCT iajs-3789 70 6 similarly similarly ADV iajs-3789 70 7 , , PUNCT iajs-3789 70 8 we we PRON iajs-3789 70 9 can can AUX iajs-3789 70 10 prove prove VERB iajs-3789 70 11 that that SCONJ iajs-3789 70 12 x x SYM iajs-3789 70 13 ∈ ∈ PROPN iajs-3789 70 14 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NOUN iajs-3789 70 15 ) ) PUNCT iajs-3789 70 16 whenever whenever SCONJ iajs-3789 70 17 x x SYM iajs-3789 70 18 ∈ ∈ PROPN iajs-3789 70 19 υ𝕾(b υ𝕾(b NUM iajs-3789 70 20 ) ) PUNCT iajs-3789 70 21 , , PUNCT iajs-3789 70 22 and and CCONJ iajs-3789 70 23 so so ADV iajs-3789 70 24 υ𝕾(𝐴)⋃υ𝕾(𝐡 υ𝕾(𝐴)⋃υ𝕾(𝐡 NOUN iajs-3789 70 25 ) ) PUNCT iajs-3789 70 26 βŠ† βŠ† NUM iajs-3789 70 27 υ𝕾(𝐴⋃𝐡) υ𝕾(𝐴⋃𝐡) NOUN iajs-3789 70 28 … … PUNCT iajs-3789 70 29 … … SYM iajs-3789 70 30 (1 (1 NUM iajs-3789 70 31 ) ) PUNCT iajs-3789 70 32 . . PUNCT iajs-3789 71 1 now now ADV iajs-3789 71 2 let let VERB iajs-3789 71 3 x x X iajs-3789 71 4 ∈ ∈ PROPN iajs-3789 71 5 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NOUN iajs-3789 71 6 ) ) PUNCT iajs-3789 71 7 , , PUNCT iajs-3789 71 8 so so ADV iajs-3789 71 9 βˆ€s βˆ€s PROPN iajs-3789 71 10 ∈ ∈ PROPN iajs-3789 72 1 𝒱ox 𝒱ox PROPN iajs-3789 72 2 , , PUNCT iajs-3789 72 3 xΟ΅s xΟ΅s PROPN iajs-3789 72 4 , , PUNCT iajs-3789 72 5 we we PRON iajs-3789 72 6 have have VERB iajs-3789 72 7 sβ‹‚(𝐴⋃𝐡 sβ‹‚(𝐴⋃𝐡 NOUN iajs-3789 72 8 ) ) PUNCT iajs-3789 72 9 ∈ ∈ NOUN iajs-3789 72 10 𝕾. 𝕾. NOUN iajs-3789 72 11 distributing distribute VERB iajs-3789 72 12 the the DET iajs-3789 72 13 intercept intercept NOUN iajs-3789 72 14 to to ADP iajs-3789 72 15 the the DET iajs-3789 72 16 union union NOUN iajs-3789 72 17 , , PUNCT iajs-3789 72 18 we we PRON iajs-3789 72 19 get get VERB iajs-3789 72 20 βˆ€s βˆ€s NOUN iajs-3789 72 21 ∈ ∈ PROPN iajs-3789 72 22 𝒱ox 𝒱ox PROPN iajs-3789 72 23 , , PUNCT iajs-3789 72 24 xΟ΅s xΟ΅s PROPN iajs-3789 72 25 , , PUNCT iajs-3789 72 26 ( ( PUNCT iajs-3789 72 27 s⋂𝐴)⋃(s⋂𝐡 s⋂𝐴)⋃(s⋂𝐡 PROPN iajs-3789 72 28 ) ) PUNCT iajs-3789 72 29 ∈ ∈ PROPN iajs-3789 72 30 𝕾 𝕾 PROPN iajs-3789 72 31 , , PUNCT iajs-3789 72 32 it it PRON iajs-3789 72 33 follows follow VERB iajs-3789 72 34 from from ADP iajs-3789 72 35 definition definition NOUN iajs-3789 72 36 3.1(3 3.1(3 NUM iajs-3789 72 37 ) ) PUNCT iajs-3789 72 38 s⋂𝐴 s⋂𝐴 VERB iajs-3789 72 39 ∈ ∈ NOUN iajs-3789 72 40 𝕾 𝕾 NOUN iajs-3789 72 41 or or CCONJ iajs-3789 72 42 s⋂𝐡 s⋂𝐡 ADJ iajs-3789 72 43 ∈ ∈ PROPN iajs-3789 72 44 𝕾 𝕾 NOUN iajs-3789 72 45 , , PUNCT iajs-3789 72 46 if if SCONJ iajs-3789 72 47 s⋂𝐴 s⋂𝐴 NOUN iajs-3789 72 48 ∈ ∈ PROPN iajs-3789 72 49 𝕾 𝕾 PROPN iajs-3789 72 50 , , PUNCT iajs-3789 72 51 then then ADV iajs-3789 72 52 x x SYM iajs-3789 72 53 ∈ ∈ PROPN iajs-3789 72 54 υ𝕾(a υ𝕾(a PROPN iajs-3789 72 55 ) ) PUNCT iajs-3789 72 56 , , PUNCT iajs-3789 72 57 implies imply VERB iajs-3789 72 58 x x X iajs-3789 72 59 ∈ ∈ PROPN iajs-3789 72 60 υ𝕾(a)⋃υ𝕾(𝐡 υ𝕾(a)⋃υ𝕾(𝐡 NOUN iajs-3789 72 61 ) ) PUNCT iajs-3789 72 62 . . PUNCT iajs-3789 73 1 if if SCONJ iajs-3789 73 2 s⋂𝐡 s⋂𝐡 VERB iajs-3789 73 3 ∈ ∈ PROPN iajs-3789 73 4 𝕾 𝕾 PROPN iajs-3789 73 5 , , PUNCT iajs-3789 73 6 then then ADV iajs-3789 73 7 x x SYM iajs-3789 73 8 ∈ ∈ PROPN iajs-3789 73 9 υ𝕾(b υ𝕾(b NUM iajs-3789 73 10 ) ) PUNCT iajs-3789 73 11 , , PUNCT iajs-3789 73 12 implies imply VERB iajs-3789 73 13 x x X iajs-3789 73 14 ∈ ∈ PROPN iajs-3789 73 15 υ𝕾(a)⋃υ𝕾(𝐡 υ𝕾(a)⋃υ𝕾(𝐡 NOUN iajs-3789 73 16 ) ) PUNCT iajs-3789 73 17 . . PUNCT iajs-3789 74 1 thus thus ADV iajs-3789 74 2 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NUM iajs-3789 74 3 ) ) PUNCT iajs-3789 74 4 βŠ† βŠ† NUM iajs-3789 74 5 υ𝕾(𝐴)⋃υ𝕾(𝐡) υ𝕾(𝐴)⋃υ𝕾(𝐡) NOUN iajs-3789 74 6 … … SYM iajs-3789 74 7 … … PUNCT iajs-3789 74 8 (2 (2 NUM iajs-3789 74 9 ) ) PUNCT iajs-3789 74 10 . . PUNCT iajs-3789 75 1 from from ADP iajs-3789 75 2 ( ( PUNCT iajs-3789 75 3 1 1 NUM iajs-3789 75 4 ) ) PUNCT iajs-3789 75 5 and and CCONJ iajs-3789 75 6 ( ( PUNCT iajs-3789 75 7 2 2 NUM iajs-3789 75 8 ) ) PUNCT iajs-3789 75 9 , , PUNCT iajs-3789 75 10 we we PRON iajs-3789 75 11 prove prove VERB iajs-3789 75 12 υ𝕾(𝐴)⋃υ𝕾(𝐡 υ𝕾(𝐴)⋃υ𝕾(𝐡 NOUN iajs-3789 75 13 ) ) PUNCT iajs-3789 75 14 = = SYM iajs-3789 75 15 υ𝕾(𝐴⋃𝐡 υ𝕾(𝐴⋃𝐡 NOUN iajs-3789 75 16 ) ) PUNCT iajs-3789 75 17 . . PUNCT iajs-3789 76 1 2 2 X iajs-3789 76 2 . . X iajs-3789 77 1 if if SCONJ iajs-3789 77 2 π‘₯ π‘₯ PROPN iajs-3789 77 3 βˆ‰ βˆ‰ PROPN iajs-3789 77 4 cl cl NOUN iajs-3789 77 5 π“₯(𝐴 π“₯(𝐴 NOUN iajs-3789 77 6 ) ) PUNCT iajs-3789 78 1 , , PUNCT iajs-3789 78 2 it it PRON iajs-3789 78 3 follows follow VERB iajs-3789 78 4 from from ADP iajs-3789 78 5 theorem theorem ADJ iajs-3789 78 6 2.3 2.3 NUM iajs-3789 78 7 that that PRON iajs-3789 78 8 there there PRON iajs-3789 78 9 exists exist VERB iajs-3789 78 10 a a DET iajs-3789 78 11 𝒱 𝒱 NOUN iajs-3789 78 12 βˆ’ βˆ’ NOUN iajs-3789 78 13 open open ADJ iajs-3789 78 14 set set VERB iajs-3789 78 15 𝑆 𝑆 PROPN iajs-3789 78 16 containing contain VERB iajs-3789 78 17 π‘₯ π‘₯ PRON iajs-3789 78 18 such such ADJ iajs-3789 78 19 that that SCONJ iajs-3789 78 20 𝑆 𝑆 PROPN iajs-3789 78 21 ∩ ∩ NOUN iajs-3789 78 22 a a DET iajs-3789 78 23 = = NOUN iajs-3789 78 24 βˆ… βˆ… NOUN iajs-3789 78 25 , , PUNCT iajs-3789 78 26 it it PRON iajs-3789 78 27 follows follow VERB iajs-3789 78 28 from from ADP iajs-3789 78 29 definition definition NOUN iajs-3789 78 30 3.1(1 3.1(1 NUM iajs-3789 78 31 ) ) PUNCT iajs-3789 78 32 that that SCONJ iajs-3789 78 33 𝑆 𝑆 PROPN iajs-3789 78 34 ∩ ∩ NOUN iajs-3789 78 35 a a DET iajs-3789 78 36 βˆ‰ βˆ‰ PROPN iajs-3789 78 37 𝕾 𝕾 PROPN iajs-3789 78 38 implies imply VERB iajs-3789 78 39 π‘₯ π‘₯ PROPN iajs-3789 78 40 βˆ‰ βˆ‰ PROPN iajs-3789 78 41 υ𝕾(𝐴 υ𝕾(𝐴 PROPN iajs-3789 78 42 ) ) PUNCT iajs-3789 78 43 . . PUNCT iajs-3789 79 1 thus thus ADV iajs-3789 79 2 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 79 3 ) ) PUNCT iajs-3789 79 4 βŠ† βŠ† NUM iajs-3789 79 5 cl cl NOUN iajs-3789 79 6 π“₯(𝐴 π“₯(𝐴 NOUN iajs-3789 79 7 ) ) PUNCT iajs-3789 79 8 . . PUNCT iajs-3789 80 1 3 3 X iajs-3789 80 2 . . X iajs-3789 80 3 let let VERB iajs-3789 80 4 x x X iajs-3789 80 5 ∈ ∈ NOUN iajs-3789 80 6 υ𝕾(υ𝕾(a υ𝕾(υ𝕾(a PROPN iajs-3789 80 7 ) ) PUNCT iajs-3789 80 8 ) ) PUNCT iajs-3789 80 9 , , PUNCT iajs-3789 80 10 so so ADV iajs-3789 80 11 βˆ€s βˆ€s PROPN iajs-3789 80 12 ∈ ∈ PROPN iajs-3789 80 13 𝒱ox 𝒱ox PROPN iajs-3789 80 14 , , PUNCT iajs-3789 80 15 xΟ΅s xΟ΅s PROPN iajs-3789 80 16 , , PUNCT iajs-3789 80 17 we we PRON iajs-3789 80 18 have have VERB iajs-3789 80 19 s⋂υ𝕾(a s⋂υ𝕾(a ADJ iajs-3789 80 20 ) ) PUNCT iajs-3789 80 21 ∈ ∈ PROPN iajs-3789 80 22 𝕾 𝕾 PROPN iajs-3789 80 23 , , PUNCT iajs-3789 80 24 implies imply VERB iajs-3789 80 25 s⋂υ𝕾(a s⋂υ𝕾(a ADJ iajs-3789 80 26 ) ) PUNCT iajs-3789 80 27 β‰  β‰  PROPN iajs-3789 80 28 βˆ… βˆ… NOUN iajs-3789 80 29 , , PUNCT iajs-3789 80 30 let let VERB iajs-3789 80 31 sβˆ— sβˆ— PROPN iajs-3789 80 32 be be AUX iajs-3789 80 33 𝒱 𝒱 PROPN iajs-3789 80 34 βˆ’ βˆ’ NOUN iajs-3789 80 35 open open ADJ iajs-3789 80 36 set set NOUN iajs-3789 80 37 containing contain VERB iajs-3789 80 38 π‘₯ π‘₯ PROPN iajs-3789 80 39 and and CCONJ iajs-3789 80 40 ; ; PUNCT iajs-3789 80 41 𝑦 𝑦 PROPN iajs-3789 80 42 ∈ ∈ PROPN iajs-3789 80 43 sβˆ—β‹‚Ο…π•Ύ(a sβˆ—β‹‚Ο…π•Ύ(a PROPN iajs-3789 80 44 ) ) PUNCT iajs-3789 80 45 , , PUNCT iajs-3789 80 46 therefore therefore ADV iajs-3789 80 47 𝑦 𝑦 PROPN iajs-3789 80 48 ∈ ∈ NOUN iajs-3789 80 49 sβˆ— sβˆ— NOUN iajs-3789 80 50 and and CCONJ iajs-3789 80 51 𝑦 𝑦 NOUN iajs-3789 80 52 ∈ ∈ PROPN iajs-3789 80 53 υ𝕾(a υ𝕾(a PROPN iajs-3789 80 54 ) ) PUNCT iajs-3789 80 55 , , PUNCT iajs-3789 80 56 it it PRON iajs-3789 80 57 follows follow VERB iajs-3789 80 58 ihjpas ihjpa NOUN iajs-3789 80 59 . . PUNCT iajs-3789 81 1 37 37 NUM iajs-3789 81 2 ( ( PUNCT iajs-3789 81 3 2 2 NUM iajs-3789 81 4 ) ) PUNCT iajs-3789 81 5 2024 2024 NUM iajs-3789 81 6 435 435 NUM iajs-3789 81 7 that that SCONJ iajs-3789 81 8 sβˆ—β‹‚a sβˆ—β‹‚a NOUN iajs-3789 81 9 ∈ ∈ PROPN iajs-3789 81 10 𝕾. 𝕾. NOUN iajs-3789 81 11 for for ADP iajs-3789 81 12 each each PRON iajs-3789 81 13 s s X iajs-3789 81 14 ∈ ∈ PROPN iajs-3789 82 1 𝒱ox 𝒱ox PROPN iajs-3789 82 2 , , PUNCT iajs-3789 82 3 xΟ΅s xΟ΅s NOUN iajs-3789 82 4 we we PRON iajs-3789 82 5 can can AUX iajs-3789 82 6 find find VERB iajs-3789 82 7 an an DET iajs-3789 82 8 element element NOUN iajs-3789 82 9 𝑦 𝑦 NOUN iajs-3789 82 10 ∈ ∈ PROPN iajs-3789 82 11 s⋂υ𝕾(a s⋂υ𝕾(a NOUN iajs-3789 82 12 ) ) PUNCT iajs-3789 82 13 which which PRON iajs-3789 82 14 implies imply VERB iajs-3789 82 15 that that SCONJ iajs-3789 82 16 sβ‹‚a sβ‹‚a NOUN iajs-3789 82 17 ∈ ∈ PROPN iajs-3789 82 18 𝕾. 𝕾. NOUN iajs-3789 82 19 hence hence ADV iajs-3789 82 20 x x X iajs-3789 82 21 ∈ ∈ PROPN iajs-3789 82 22 υ𝕾(a υ𝕾(a PROPN iajs-3789 82 23 ) ) PUNCT iajs-3789 82 24 , , PUNCT iajs-3789 82 25 and and CCONJ iajs-3789 82 26 so so ADV iajs-3789 82 27 υ𝕾(υ𝕾(𝐴 υ𝕾(υ𝕾(𝐴 PROPN iajs-3789 82 28 ) ) PUNCT iajs-3789 82 29 ) ) PUNCT iajs-3789 82 30 βŠ† βŠ† NUM iajs-3789 82 31 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 82 32 ) ) PUNCT iajs-3789 82 33 . . PUNCT iajs-3789 83 1 remark remark VERB iajs-3789 83 2 3.5 3.5 NUM iajs-3789 83 3 the the DET iajs-3789 83 4 convers conver NOUN iajs-3789 83 5 does do AUX iajs-3789 83 6 not not PART iajs-3789 83 7 hold hold VERB iajs-3789 83 8 in in ADP iajs-3789 83 9 general general ADJ iajs-3789 83 10 in in ADP iajs-3789 83 11 ( ( PUNCT iajs-3789 83 12 2 2 NUM iajs-3789 83 13 ) ) PUNCT iajs-3789 83 14 of of ADP iajs-3789 83 15 theorem theorem ADJ iajs-3789 83 16 3.4 3.4 NUM iajs-3789 83 17 . . PUNCT iajs-3789 84 1 for for ADP iajs-3789 84 2 example example NOUN iajs-3789 84 3 let let VERB iajs-3789 84 4 x x SYM iajs-3789 84 5 = = PUNCT iajs-3789 84 6 { { PUNCT iajs-3789 84 7 ħ𝟏 ħ𝟏 PROPN iajs-3789 84 8 , , PUNCT iajs-3789 84 9 ħ𝟐 ħ𝟐 NOUN iajs-3789 84 10 , , PUNCT iajs-3789 84 11 Δ§3 Δ§3 NOUN iajs-3789 84 12 , , PUNCT iajs-3789 84 13 Δ§4 Δ§4 NOUN iajs-3789 84 14 } } PUNCT iajs-3789 84 15 , , PUNCT iajs-3789 84 16 and and CCONJ iajs-3789 84 17 let let VERB iajs-3789 84 18 { { PUNCT iajs-3789 84 19 Ο„i}i=1 Ο„i}i=1 NOUN iajs-3789 84 20 3 3 NUM iajs-3789 84 21 be be AUX iajs-3789 84 22 a a DET iajs-3789 84 23 family family NOUN iajs-3789 84 24 of of ADP iajs-3789 84 25 topologies topology NOUN iajs-3789 84 26 defined define VERB iajs-3789 84 27 on on ADP iajs-3789 84 28 x x PUNCT iajs-3789 84 29 as as SCONJ iajs-3789 84 30 follows follow VERB iajs-3789 84 31 : : PUNCT iajs-3789 84 32 Ο„1 Ο„1 NOUN iajs-3789 84 33 = = SYM iajs-3789 84 34 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 84 35 ) ) PUNCT iajs-3789 84 36 , , PUNCT iajs-3789 84 37 Ο„2 Ο„2 NOUN iajs-3789 84 38 = = SYM iajs-3789 84 39 { { PUNCT iajs-3789 84 40 𝑋 𝑋 PROPN iajs-3789 84 41 , , PUNCT iajs-3789 84 42 βˆ… βˆ… NOUN iajs-3789 84 43 , , PUNCT iajs-3789 84 44 { { PUNCT iajs-3789 84 45 κž™ κž™ X iajs-3789 84 46 𝟏 𝟏 PROPN iajs-3789 84 47 } } PUNCT iajs-3789 84 48 , , PUNCT iajs-3789 84 49 { { PUNCT iajs-3789 84 50 κž™ κž™ X iajs-3789 84 51 𝟏 𝟏 NUM iajs-3789 84 52 , , PUNCT iajs-3789 84 53 κž™ κž™ X iajs-3789 84 54 𝟐 𝟐 NUM iajs-3789 84 55 } } PUNCT iajs-3789 84 56 , , PUNCT iajs-3789 84 57 { { PUNCT iajs-3789 84 58 κž™ κž™ X iajs-3789 84 59 3 3 NUM iajs-3789 84 60 } } PUNCT iajs-3789 84 61 , , PUNCT iajs-3789 84 62 { { PUNCT iajs-3789 84 63 κž™ κž™ X iajs-3789 84 64 𝟏 𝟏 NUM iajs-3789 84 65 , , PUNCT iajs-3789 84 66 κž™ κž™ PRON iajs-3789 84 67 3 3 NUM iajs-3789 84 68 } } PUNCT iajs-3789 84 69 , , PUNCT iajs-3789 84 70 { { PUNCT iajs-3789 84 71 κž™ κž™ X iajs-3789 84 72 𝟏 𝟏 NUM iajs-3789 84 73 , , PUNCT iajs-3789 84 74 κž™ κž™ X iajs-3789 84 75 𝟐 𝟐 NUM iajs-3789 84 76 , , PUNCT iajs-3789 84 77 κž™ κž™ PRON iajs-3789 84 78 3 3 NUM iajs-3789 84 79 } } PUNCT iajs-3789 84 80 } } PUNCT iajs-3789 84 81 , , PUNCT iajs-3789 84 82 Ο„3 Ο„3 NOUN iajs-3789 84 83 = = SYM iajs-3789 84 84 { { PUNCT iajs-3789 84 85 𝑋 𝑋 NOUN iajs-3789 84 86 , , PUNCT iajs-3789 84 87 βˆ… βˆ… NOUN iajs-3789 84 88 , , PUNCT iajs-3789 84 89 { { PUNCT iajs-3789 84 90 κž™ κž™ X iajs-3789 84 91 𝟏 𝟏 PROPN iajs-3789 84 92 } } PUNCT iajs-3789 84 93 , , PUNCT iajs-3789 84 94 { { PUNCT iajs-3789 84 95 κž™ κž™ X iajs-3789 84 96 𝟏 𝟏 NUM iajs-3789 84 97 , , PUNCT iajs-3789 84 98 κž™ κž™ X iajs-3789 84 99 𝟐 𝟐 NUM iajs-3789 84 100 } } PUNCT iajs-3789 84 101 , , PUNCT iajs-3789 84 102 { { PUNCT iajs-3789 84 103 κž™ κž™ X iajs-3789 84 104 3 3 NUM iajs-3789 84 105 , , PUNCT iajs-3789 84 106 κž™ κž™ X iajs-3789 84 107 4 4 NUM iajs-3789 84 108 } } PUNCT iajs-3789 84 109 , , PUNCT iajs-3789 84 110 { { PUNCT iajs-3789 84 111 κž™ κž™ X iajs-3789 84 112 𝟏 𝟏 NUM iajs-3789 84 113 , , PUNCT iajs-3789 84 114 κž™ κž™ PROPN iajs-3789 84 115 3 3 NUM iajs-3789 84 116 , , PUNCT iajs-3789 84 117 κž™ κž™ X iajs-3789 84 118 4 4 NUM iajs-3789 84 119 } } PUNCT iajs-3789 84 120 } } PUNCT iajs-3789 84 121 . . PUNCT iajs-3789 85 1 then then ADV iajs-3789 85 2 , , PUNCT iajs-3789 85 3 β‹‚ β‹‚ PROPN iajs-3789 85 4 Ο„i Ο„i VERB iajs-3789 85 5 3 3 NUM iajs-3789 85 6 i=1 i=1 NOUN iajs-3789 86 1 = = PUNCT iajs-3789 86 2 { { PUNCT iajs-3789 86 3 𝑋 𝑋 NOUN iajs-3789 86 4 , , PUNCT iajs-3789 86 5 βˆ… βˆ… NOUN iajs-3789 86 6 , , PUNCT iajs-3789 86 7 { { PUNCT iajs-3789 86 8 κž™ κž™ X iajs-3789 86 9 𝟏 𝟏 PROPN iajs-3789 86 10 } } PUNCT iajs-3789 86 11 , , PUNCT iajs-3789 86 12 { { PUNCT iajs-3789 86 13 κž™ κž™ X iajs-3789 86 14 𝟏 𝟏 NUM iajs-3789 86 15 , , PUNCT iajs-3789 86 16 κž™ κž™ X iajs-3789 86 17 𝟐 𝟐 NUM iajs-3789 86 18 } } PUNCT iajs-3789 86 19 } } PUNCT iajs-3789 86 20 , , PUNCT iajs-3789 86 21 and and CCONJ iajs-3789 86 22 so so ADV iajs-3789 86 23 𝒱ox 𝒱ox ADJ iajs-3789 86 24 = = SYM iajs-3789 86 25 { { PUNCT iajs-3789 86 26 𝑋 𝑋 PROPN iajs-3789 86 27 , , PUNCT iajs-3789 86 28 βˆ… βˆ… NOUN iajs-3789 86 29 , , PUNCT iajs-3789 86 30 { { PUNCT iajs-3789 86 31 κž™ κž™ X iajs-3789 86 32 𝟏 𝟏 PROPN iajs-3789 86 33 } } PUNCT iajs-3789 86 34 , , PUNCT iajs-3789 86 35 { { PUNCT iajs-3789 86 36 κž™ κž™ X iajs-3789 86 37 𝟏 𝟏 NUM iajs-3789 86 38 , , PUNCT iajs-3789 86 39 κž™ κž™ X iajs-3789 86 40 𝟐 𝟐 NUM iajs-3789 86 41 } } PUNCT iajs-3789 86 42 , , PUNCT iajs-3789 86 43 { { PUNCT iajs-3789 86 44 κž™ κž™ X iajs-3789 86 45 𝟏 𝟏 NUM iajs-3789 86 46 , , PUNCT iajs-3789 86 47 κž™ κž™ PRON iajs-3789 86 48 4 4 NUM iajs-3789 86 49 } } PUNCT iajs-3789 86 50 , , PUNCT iajs-3789 86 51 { { PUNCT iajs-3789 86 52 κž™ κž™ X iajs-3789 86 53 𝟏 𝟏 NUM iajs-3789 86 54 , , PUNCT iajs-3789 86 55 } } PUNCT iajs-3789 86 56 , , PUNCT iajs-3789 86 57 { { PUNCT iajs-3789 86 58 κž™ κž™ X iajs-3789 86 59 𝟏 𝟏 NUM iajs-3789 86 60 , , PUNCT iajs-3789 86 61 κž™ κž™ X iajs-3789 86 62 𝟐 𝟐 NUM iajs-3789 86 63 , , PUNCT iajs-3789 86 64 κž™ κž™ PRON iajs-3789 86 65 3 3 NUM iajs-3789 86 66 } } PUNCT iajs-3789 86 67 , , PUNCT iajs-3789 86 68 { { PUNCT iajs-3789 86 69 κž™ κž™ X iajs-3789 86 70 𝟏 𝟏 NUM iajs-3789 86 71 , , PUNCT iajs-3789 86 72 κž™ κž™ PROPN iajs-3789 86 73 3 3 NUM iajs-3789 86 74 , , PUNCT iajs-3789 86 75 κž™ κž™ X iajs-3789 86 76 4 4 NUM iajs-3789 86 77 } } PUNCT iajs-3789 86 78 , , PUNCT iajs-3789 86 79 { { PUNCT iajs-3789 86 80 κž™ κž™ X iajs-3789 86 81 𝟏 𝟏 NUM iajs-3789 86 82 , , PUNCT iajs-3789 86 83 κž™ κž™ X iajs-3789 86 84 𝟐 𝟐 NUM iajs-3789 86 85 , , PUNCT iajs-3789 86 86 κž™ κž™ PRON iajs-3789 86 87 4 4 NUM iajs-3789 86 88 } } PUNCT iajs-3789 86 89 } } PUNCT iajs-3789 86 90 . . PUNCT iajs-3789 87 1 𝒱cx 𝒱cx NOUN iajs-3789 87 2 = = SYM iajs-3789 87 3 { { PUNCT iajs-3789 87 4 βˆ… βˆ… NOUN iajs-3789 87 5 , , PUNCT iajs-3789 87 6 𝑋 𝑋 PROPN iajs-3789 87 7 , , PUNCT iajs-3789 87 8 { { PUNCT iajs-3789 87 9 κž™ κž™ X iajs-3789 87 10 𝟐 𝟐 NUM iajs-3789 87 11 , , PUNCT iajs-3789 87 12 κž™ κž™ PRON iajs-3789 87 13 3 3 NUM iajs-3789 87 14 , , PUNCT iajs-3789 87 15 κž™ κž™ X iajs-3789 87 16 4 4 NUM iajs-3789 87 17 } } PUNCT iajs-3789 87 18 , , PUNCT iajs-3789 87 19 { { PUNCT iajs-3789 87 20 κž™ κž™ X iajs-3789 87 21 3 3 NUM iajs-3789 87 22 , , PUNCT iajs-3789 87 23 κž™ κž™ X iajs-3789 87 24 4 4 NUM iajs-3789 87 25 } } PUNCT iajs-3789 87 26 , , PUNCT iajs-3789 87 27 { { PUNCT iajs-3789 87 28 κž™ κž™ X iajs-3789 87 29 𝟐 𝟐 NUM iajs-3789 87 30 , , PUNCT iajs-3789 87 31 κž™ κž™ PRON iajs-3789 87 32 3 3 NUM iajs-3789 87 33 } } PUNCT iajs-3789 87 34 , , PUNCT iajs-3789 87 35 { { PUNCT iajs-3789 87 36 κž™ κž™ X iajs-3789 87 37 𝟐 𝟐 NUM iajs-3789 87 38 , , PUNCT iajs-3789 87 39 κž™ κž™ PRON iajs-3789 87 40 4 4 NUM iajs-3789 87 41 } } PUNCT iajs-3789 87 42 , , PUNCT iajs-3789 87 43 { { PUNCT iajs-3789 87 44 κž™ κž™ X iajs-3789 87 45 4 4 NUM iajs-3789 87 46 } } PUNCT iajs-3789 87 47 , , PUNCT iajs-3789 87 48 { { PUNCT iajs-3789 87 49 κž™ κž™ X iajs-3789 87 50 𝟐 𝟐 NUM iajs-3789 87 51 } } PUNCT iajs-3789 87 52 , , PUNCT iajs-3789 87 53 { { PUNCT iajs-3789 87 54 κž™ κž™ X iajs-3789 87 55 3 3 NUM iajs-3789 87 56 } } PUNCT iajs-3789 87 57 } } PUNCT iajs-3789 87 58 let let VERB iajs-3789 87 59 𝕾 𝕾 NOUN iajs-3789 87 60 = = PUNCT iajs-3789 87 61 { { PUNCT iajs-3789 87 62 { { PUNCT iajs-3789 87 63 κž™ κž™ X iajs-3789 87 64 𝟏 𝟏 PROPN iajs-3789 87 65 } } PUNCT iajs-3789 87 66 , , PUNCT iajs-3789 87 67 { { PUNCT iajs-3789 87 68 κž™ κž™ X iajs-3789 87 69 𝟏 𝟏 NUM iajs-3789 87 70 , , PUNCT iajs-3789 87 71 κž™ κž™ X iajs-3789 87 72 𝟐 𝟐 NUM iajs-3789 87 73 } } PUNCT iajs-3789 87 74 , , PUNCT iajs-3789 87 75 { { PUNCT iajs-3789 87 76 κž™ κž™ X iajs-3789 87 77 𝟏 𝟏 NUM iajs-3789 87 78 , , PUNCT iajs-3789 87 79 κž™ κž™ PRON iajs-3789 87 80 3 3 NUM iajs-3789 87 81 } } PUNCT iajs-3789 87 82 , , PUNCT iajs-3789 87 83 { { PUNCT iajs-3789 87 84 κž™ κž™ X iajs-3789 87 85 𝟏 𝟏 NUM iajs-3789 87 86 , , PUNCT iajs-3789 87 87 κž™ κž™ PRON iajs-3789 87 88 4 4 NUM iajs-3789 87 89 } } PUNCT iajs-3789 87 90 , , PUNCT iajs-3789 87 91 { { PUNCT iajs-3789 87 92 κž™ κž™ X iajs-3789 87 93 𝟏 𝟏 NUM iajs-3789 87 94 , , PUNCT iajs-3789 87 95 κž™ κž™ X iajs-3789 87 96 𝟐 𝟐 NUM iajs-3789 87 97 , , PUNCT iajs-3789 87 98 κž™ κž™ PRON iajs-3789 87 99 3 3 NUM iajs-3789 87 100 } } PUNCT iajs-3789 87 101 , , PUNCT iajs-3789 87 102 { { PUNCT iajs-3789 87 103 κž™ κž™ X iajs-3789 87 104 𝟏 𝟏 NUM iajs-3789 87 105 , , PUNCT iajs-3789 87 106 κž™ κž™ X iajs-3789 87 107 𝟐 𝟐 NUM iajs-3789 87 108 , , PUNCT iajs-3789 87 109 κž™ κž™ PRON iajs-3789 87 110 4 4 NUM iajs-3789 87 111 } } PUNCT iajs-3789 87 112 , , PUNCT iajs-3789 87 113 { { PUNCT iajs-3789 87 114 κž™ κž™ X iajs-3789 87 115 𝟏 𝟏 NUM iajs-3789 87 116 , , PUNCT iajs-3789 87 117 κž™ κž™ PROPN iajs-3789 87 118 3 3 NUM iajs-3789 87 119 , , PUNCT iajs-3789 87 120 κž™ κž™ X iajs-3789 87 121 4 4 NUM iajs-3789 87 122 } } PUNCT iajs-3789 87 123 , , PUNCT iajs-3789 87 124 𝑋 𝑋 PROPN iajs-3789 87 125 } } PUNCT iajs-3789 87 126 be be AUX iajs-3789 87 127 a a DET iajs-3789 87 128 grill grill NOUN iajs-3789 87 129 on on ADP iajs-3789 87 130 x. x. NOUN iajs-3789 87 131 let let VERB iajs-3789 87 132 𝐴 𝐴 PROPN iajs-3789 87 133 = = PRON iajs-3789 87 134 { { PUNCT iajs-3789 87 135 κž™ κž™ X iajs-3789 87 136 𝟐 𝟐 NUM iajs-3789 87 137 , , PUNCT iajs-3789 87 138 κž™ κž™ PRON iajs-3789 87 139 3 3 NUM iajs-3789 87 140 } } PUNCT iajs-3789 87 141 κž™ κž™ DET iajs-3789 87 142 𝕾 𝕾 NOUN iajs-3789 87 143 ( ( PUNCT iajs-3789 87 144 𝐴 𝐴 PROPN iajs-3789 87 145 ) ) PUNCT iajs-3789 87 146 = = PRON iajs-3789 87 147 { { PUNCT iajs-3789 87 148 π‘₯ π‘₯ PUNCT iajs-3789 87 149 ∈ ∈ PROPN iajs-3789 87 150 𝑋 𝑋 PROPN iajs-3789 87 151 : : PUNCT iajs-3789 87 152 𝑆⋂𝐴 𝑆⋂𝐴 PROPN iajs-3789 87 153 ∈ ∈ PROPN iajs-3789 87 154 𝕾 𝕾 NOUN iajs-3789 87 155 βˆ€π‘† βˆ€π‘† PUNCT iajs-3789 87 156 ∈ ∈ PROPN iajs-3789 87 157 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 87 158 , , PUNCT iajs-3789 87 159 π‘₯πœ–π‘† π‘₯πœ–π‘† ADV iajs-3789 87 160 } } PUNCT iajs-3789 87 161 = = SYM iajs-3789 87 162 βˆ… βˆ… NOUN iajs-3789 87 163 cl cl NOUN iajs-3789 87 164 π“₯(𝐴 π“₯(𝐴 NOUN iajs-3789 87 165 ) ) PUNCT iajs-3789 88 1 = = SYM iajs-3789 88 2 𝐴 𝐴 PROPN iajs-3789 88 3 since since SCONJ iajs-3789 88 4 𝐴 𝐴 PROPN iajs-3789 88 5 is be AUX iajs-3789 88 6 𝒱 𝒱 PROPN iajs-3789 88 7 βˆ’ βˆ’ NOUN iajs-3789 88 8 closed closed ADJ iajs-3789 88 9 . . PUNCT iajs-3789 89 1 remark remark NOUN iajs-3789 89 2 3.6 3.6 NUM iajs-3789 89 3 the the DET iajs-3789 89 4 convers conver NOUN iajs-3789 89 5 does do AUX iajs-3789 89 6 not not PART iajs-3789 89 7 always always ADV iajs-3789 89 8 hold hold VERB iajs-3789 89 9 in in ADP iajs-3789 89 10 ( ( PUNCT iajs-3789 89 11 3 3 NUM iajs-3789 89 12 ) ) PUNCT iajs-3789 89 13 of of ADP iajs-3789 89 14 theorem theorem ADJ iajs-3789 89 15 3.4 3.4 NUM iajs-3789 89 16 . . PUNCT iajs-3789 90 1 for for ADP iajs-3789 90 2 example example NOUN iajs-3789 90 3 let let VERB iajs-3789 90 4 x x PUNCT iajs-3789 90 5 = = PRON iajs-3789 90 6 { { PUNCT iajs-3789 90 7 ꞎ ꞎ NOUN iajs-3789 90 8 𝟏 𝟏 X iajs-3789 90 9 , , PUNCT iajs-3789 90 10 ꞎ ꞎ PROPN iajs-3789 90 11 𝟐 𝟐 NUM iajs-3789 90 12 , , PUNCT iajs-3789 90 13 ꞎ ꞎ NOUN iajs-3789 90 14 3 3 NUM iajs-3789 90 15 , , PUNCT iajs-3789 90 16 ꞎ ꞎ NOUN iajs-3789 90 17 4 4 NUM iajs-3789 90 18 } } PUNCT iajs-3789 90 19 , , PUNCT iajs-3789 90 20 and and CCONJ iajs-3789 90 21 let let VERB iajs-3789 90 22 { { PUNCT iajs-3789 90 23 Ο„i}i=1 Ο„i}i=1 NOUN iajs-3789 90 24 3 3 NUM iajs-3789 90 25 be be AUX iajs-3789 90 26 a a DET iajs-3789 90 27 family family NOUN iajs-3789 90 28 of of ADP iajs-3789 90 29 topologies topology NOUN iajs-3789 90 30 defined define VERB iajs-3789 90 31 on on ADP iajs-3789 90 32 x x PUNCT iajs-3789 90 33 as as SCONJ iajs-3789 90 34 follows follow VERB iajs-3789 90 35 : : PUNCT iajs-3789 90 36 Ο„1 Ο„1 NOUN iajs-3789 90 37 = = SYM iajs-3789 90 38 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 90 39 ) ) PUNCT iajs-3789 90 40 , , PUNCT iajs-3789 90 41 Ο„2 Ο„2 NOUN iajs-3789 90 42 = = SYM iajs-3789 90 43 { { PUNCT iajs-3789 90 44 𝑋 𝑋 PROPN iajs-3789 90 45 , , PUNCT iajs-3789 90 46 βˆ… βˆ… NOUN iajs-3789 90 47 , , PUNCT iajs-3789 90 48 { { PUNCT iajs-3789 90 49 ꞎ ꞎ NOUN iajs-3789 90 50 𝟏 𝟏 NUM iajs-3789 90 51 } } PUNCT iajs-3789 90 52 , , PUNCT iajs-3789 90 53 { { PUNCT iajs-3789 90 54 ꞎ ꞎ NOUN iajs-3789 90 55 𝟏 𝟏 X iajs-3789 90 56 , , PUNCT iajs-3789 90 57 ꞎ ꞎ PROPN iajs-3789 90 58 𝟐 𝟐 NUM iajs-3789 90 59 } } PUNCT iajs-3789 90 60 , , PUNCT iajs-3789 90 61 { { PUNCT iajs-3789 90 62 ꞎ ꞎ NOUN iajs-3789 90 63 3 3 NUM iajs-3789 90 64 } } PUNCT iajs-3789 90 65 , , PUNCT iajs-3789 90 66 { { PUNCT iajs-3789 90 67 ꞎ ꞎ NOUN iajs-3789 90 68 𝟏 𝟏 X iajs-3789 90 69 , , PUNCT iajs-3789 90 70 ꞎ ꞎ NOUN iajs-3789 90 71 3 3 NUM iajs-3789 90 72 } } PUNCT iajs-3789 90 73 , , PUNCT iajs-3789 90 74 { { PUNCT iajs-3789 90 75 ꞎ ꞎ NOUN iajs-3789 90 76 𝟏 𝟏 X iajs-3789 90 77 , , PUNCT iajs-3789 90 78 ꞎ ꞎ PROPN iajs-3789 90 79 𝟐 𝟐 NUM iajs-3789 90 80 , , PUNCT iajs-3789 90 81 ꞎ ꞎ NOUN iajs-3789 90 82 3 3 NUM iajs-3789 90 83 } } PUNCT iajs-3789 90 84 } } PUNCT iajs-3789 90 85 , , PUNCT iajs-3789 90 86 Ο„3 Ο„3 NOUN iajs-3789 90 87 = = SYM iajs-3789 90 88 { { PUNCT iajs-3789 90 89 𝑋 𝑋 NOUN iajs-3789 90 90 , , PUNCT iajs-3789 90 91 βˆ… βˆ… NOUN iajs-3789 90 92 , , PUNCT iajs-3789 90 93 { { PUNCT iajs-3789 90 94 ꞎ ꞎ NOUN iajs-3789 90 95 𝟏 𝟏 X iajs-3789 90 96 , , PUNCT iajs-3789 90 97 ꞎ ꞎ PROPN iajs-3789 90 98 𝟐 𝟐 NUM iajs-3789 90 99 } } PUNCT iajs-3789 90 100 , , PUNCT iajs-3789 90 101 { { PUNCT iajs-3789 90 102 ꞎ ꞎ NOUN iajs-3789 90 103 3 3 NUM iajs-3789 90 104 , , PUNCT iajs-3789 90 105 ꞎ ꞎ NOUN iajs-3789 90 106 4 4 NUM iajs-3789 90 107 } } PUNCT iajs-3789 90 108 } } PUNCT iajs-3789 90 109 . . PUNCT iajs-3789 91 1 then,β‹‚ then,β‹‚ NOUN iajs-3789 91 2 Ο„i Ο„i VERB iajs-3789 91 3 3 3 NUM iajs-3789 91 4 i=1 i=1 NOUN iajs-3789 92 1 = = PUNCT iajs-3789 93 1 { { PUNCT iajs-3789 93 2 𝑋 𝑋 NOUN iajs-3789 93 3 , , PUNCT iajs-3789 93 4 βˆ… βˆ… NOUN iajs-3789 93 5 , , PUNCT iajs-3789 93 6 { { PUNCT iajs-3789 93 7 ꞎ ꞎ NOUN iajs-3789 93 8 𝟏 𝟏 X iajs-3789 93 9 , , PUNCT iajs-3789 93 10 ꞎ ꞎ PROPN iajs-3789 93 11 𝟐 𝟐 NUM iajs-3789 93 12 } } PUNCT iajs-3789 93 13 } } PUNCT iajs-3789 93 14 , , PUNCT iajs-3789 93 15 and and CCONJ iajs-3789 93 16 so so ADV iajs-3789 93 17 𝒱ox 𝒱ox ADJ iajs-3789 93 18 = = SYM iajs-3789 93 19 { { PUNCT iajs-3789 93 20 𝑋 𝑋 PROPN iajs-3789 93 21 , , PUNCT iajs-3789 93 22 βˆ… βˆ… NOUN iajs-3789 93 23 , , PUNCT iajs-3789 93 24 { { PUNCT iajs-3789 93 25 ꞎ ꞎ NOUN iajs-3789 93 26 𝟏 𝟏 X iajs-3789 93 27 , , PUNCT iajs-3789 93 28 ꞎ ꞎ PROPN iajs-3789 93 29 𝟐 𝟐 NUM iajs-3789 93 30 } } PUNCT iajs-3789 93 31 , , PUNCT iajs-3789 93 32 { { PUNCT iajs-3789 93 33 ꞎ ꞎ NOUN iajs-3789 93 34 𝟏 𝟏 X iajs-3789 93 35 , , PUNCT iajs-3789 93 36 ꞎ ꞎ PROPN iajs-3789 93 37 𝟐 𝟐 NUM iajs-3789 93 38 , , PUNCT iajs-3789 93 39 ꞎ ꞎ PRON iajs-3789 93 40 3 3 NUM iajs-3789 93 41 } } PUNCT iajs-3789 93 42 , , PUNCT iajs-3789 93 43 { { PUNCT iajs-3789 93 44 ꞎ ꞎ NOUN iajs-3789 93 45 𝟏 𝟏 X iajs-3789 93 46 , , PUNCT iajs-3789 93 47 ꞎ ꞎ PROPN iajs-3789 93 48 𝟐 𝟐 NUM iajs-3789 93 49 , , PUNCT iajs-3789 93 50 ꞎ ꞎ PRON iajs-3789 93 51 4 4 NUM iajs-3789 93 52 } } PUNCT iajs-3789 93 53 } } PUNCT iajs-3789 93 54 𝕾 𝕾 NOUN iajs-3789 93 55 = = PUNCT iajs-3789 93 56 { { PUNCT iajs-3789 93 57 { { PUNCT iajs-3789 93 58 ꞎ ꞎ NOUN iajs-3789 93 59 𝟏 𝟏 X iajs-3789 93 60 , , PUNCT iajs-3789 93 61 ꞎ ꞎ PROPN iajs-3789 93 62 𝟐 𝟐 NUM iajs-3789 93 63 , , PUNCT iajs-3789 93 64 ꞎ ꞎ PRON iajs-3789 93 65 3 3 NUM iajs-3789 93 66 } } PUNCT iajs-3789 93 67 , , PUNCT iajs-3789 93 68 { { PUNCT iajs-3789 93 69 ꞎ ꞎ NOUN iajs-3789 93 70 𝟏 𝟏 X iajs-3789 93 71 , , PUNCT iajs-3789 93 72 ꞎ ꞎ PROPN iajs-3789 93 73 𝟐 𝟐 NUM iajs-3789 93 74 , , PUNCT iajs-3789 93 75 ꞎ ꞎ PRON iajs-3789 93 76 4 4 NUM iajs-3789 93 77 } } PUNCT iajs-3789 93 78 , , PUNCT iajs-3789 93 79 { { PUNCT iajs-3789 93 80 ꞎ ꞎ NOUN iajs-3789 93 81 𝟏 𝟏 X iajs-3789 93 82 , , PUNCT iajs-3789 93 83 ꞎ ꞎ PROPN iajs-3789 93 84 3 3 NUM iajs-3789 93 85 , , PUNCT iajs-3789 93 86 ꞎ ꞎ PRON iajs-3789 93 87 4 4 NUM iajs-3789 93 88 } } PUNCT iajs-3789 93 89 , , PUNCT iajs-3789 93 90 𝑋 𝑋 PROPN iajs-3789 93 91 } } PUNCT iajs-3789 93 92 let let VERB iajs-3789 93 93 𝐴 𝐴 PROPN iajs-3789 93 94 = = PRON iajs-3789 93 95 { { PUNCT iajs-3789 93 96 ꞎ ꞎ NOUN iajs-3789 93 97 𝟏 𝟏 X iajs-3789 93 98 , , PUNCT iajs-3789 93 99 ꞎ ꞎ PROPN iajs-3789 93 100 𝟐 𝟐 NUM iajs-3789 93 101 , , PUNCT iajs-3789 93 102 ꞎ ꞎ NOUN iajs-3789 93 103 3 3 NUM iajs-3789 93 104 } } PUNCT iajs-3789 93 105 , , PUNCT iajs-3789 93 106 ꞎ ꞎ PRON iajs-3789 93 107 𝕾 𝕾 NOUN iajs-3789 93 108 ( ( PUNCT iajs-3789 93 109 𝐴 𝐴 PROPN iajs-3789 93 110 ) ) PUNCT iajs-3789 93 111 = = PRON iajs-3789 93 112 { { PUNCT iajs-3789 93 113 π‘₯ π‘₯ PUNCT iajs-3789 93 114 ∈ ∈ PROPN iajs-3789 93 115 𝑋 𝑋 PROPN iajs-3789 93 116 : : PUNCT iajs-3789 93 117 𝑆⋂𝐴 𝑆⋂𝐴 PROPN iajs-3789 93 118 ∈ ∈ PROPN iajs-3789 93 119 𝕾 𝕾 NOUN iajs-3789 93 120 βˆ€π‘† βˆ€π‘† PUNCT iajs-3789 93 121 ∈ ∈ PROPN iajs-3789 93 122 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 93 123 , , PUNCT iajs-3789 93 124 π‘₯πœ–π‘† π‘₯πœ–π‘† ADV iajs-3789 93 125 } } PUNCT iajs-3789 93 126 = = SYM iajs-3789 93 127 { { PUNCT iajs-3789 93 128 𝑐 𝑐 NOUN iajs-3789 93 129 , , PUNCT iajs-3789 93 130 𝑑 𝑑 NOUN iajs-3789 93 131 } } PUNCT iajs-3789 93 132 ꞎ ꞎ PRON iajs-3789 93 133 𝕾 𝕾 NOUN iajs-3789 93 134 ( ( PUNCT iajs-3789 93 135 ꞎ ꞎ NOUN iajs-3789 93 136 𝕾 𝕾 NOUN iajs-3789 93 137 ( ( PUNCT iajs-3789 93 138 𝐴 𝐴 PROPN iajs-3789 93 139 ) ) PUNCT iajs-3789 93 140 ) ) PUNCT iajs-3789 94 1 = = PUNCT iajs-3789 94 2 βˆ… βˆ… NOUN iajs-3789 94 3 definition definition NOUN iajs-3789 94 4 3.7 3.7 NUM iajs-3789 94 5 let let VERB iajs-3789 94 6 𝕾 𝕾 NOUN iajs-3789 94 7 be be AUX iajs-3789 94 8 a a DET iajs-3789 94 9 grill grill NOUN iajs-3789 94 10 on on ADP iajs-3789 94 11 π“₯-space π“₯-space PROPN iajs-3789 94 12 ( ( PUNCT iajs-3789 94 13 𝑋 𝑋 PROPN iajs-3789 94 14 , , PUNCT iajs-3789 94 15 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 94 16 ) ) PUNCT iajs-3789 94 17 . . PUNCT iajs-3789 95 1 the the DET iajs-3789 95 2 map map NOUN iajs-3789 95 3 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 95 4 βˆ’ βˆ’ PROPN iajs-3789 95 5 𝑐𝑙 𝑐𝑙 ADP iajs-3789 95 6 ∢ ∢ PROPN iajs-3789 95 7 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 95 8 ) ) PUNCT iajs-3789 95 9 β†’ β†’ SYM iajs-3789 95 10 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 95 11 ) ) PUNCT iajs-3789 95 12 where where SCONJ iajs-3789 95 13 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 95 14 βˆ’ βˆ’ PROPN iajs-3789 95 15 𝑐𝑙(𝐴 𝑐𝑙(𝐴 NOUN iajs-3789 95 16 ) ) PUNCT iajs-3789 95 17 = = SYM iajs-3789 95 18 𝐴⋃l𝕾(𝐴 𝐴⋃l𝕾(𝐴 NOUN iajs-3789 95 19 ) ) PUNCT iajs-3789 95 20 is be AUX iajs-3789 95 21 a a DET iajs-3789 95 22 kuratowski kuratowski NOUN iajs-3789 95 23 ’s ’s PART iajs-3789 95 24 closure closure NOUN iajs-3789 95 25 operator operator NOUN iajs-3789 95 26 and and CCONJ iajs-3789 95 27 hence hence ADV iajs-3789 95 28 induces induce VERB iajs-3789 95 29 a a DET iajs-3789 95 30 topology topology NOUN iajs-3789 95 31 on on ADP iajs-3789 95 32 𝑋 𝑋 PROPN iajs-3789 95 33 defined define VERB iajs-3789 95 34 as as ADP iajs-3789 95 35 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 95 36 = = SYM iajs-3789 95 37 { { PUNCT iajs-3789 95 38 𝐺 𝐺 NOUN iajs-3789 95 39 βŠ† βŠ† NUM iajs-3789 95 40 x x NOUN iajs-3789 95 41 : : PUNCT iajs-3789 95 42 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 95 43 βˆ’ βˆ’ PROPN iajs-3789 95 44 𝑐𝑙(𝑋 𝑐𝑙(𝑋 NOUN iajs-3789 95 45 βˆ’ βˆ’ PROPN iajs-3789 95 46 𝐺 𝐺 NOUN iajs-3789 95 47 ) ) PUNCT iajs-3789 96 1 = = SYM iajs-3789 96 2 𝑋 𝑋 PROPN iajs-3789 96 3 βˆ’ βˆ’ PROPN iajs-3789 96 4 𝐺 𝐺 PROPN iajs-3789 96 5 } } PUNCT iajs-3789 96 6 . . PUNCT iajs-3789 97 1 for for ADP iajs-3789 97 2 example example NOUN iajs-3789 97 3 let let VERB iajs-3789 97 4 x x PUNCT iajs-3789 97 5 = = PRON iajs-3789 97 6 { { PUNCT iajs-3789 97 7 ꞎ ꞎ NOUN iajs-3789 97 8 𝟏 𝟏 X iajs-3789 97 9 , , PUNCT iajs-3789 97 10 ꞎ ꞎ PROPN iajs-3789 97 11 𝟐 𝟐 NUM iajs-3789 97 12 , , PUNCT iajs-3789 97 13 ꞎ ꞎ NOUN iajs-3789 97 14 3 3 NUM iajs-3789 97 15 } } PUNCT iajs-3789 97 16 , , PUNCT iajs-3789 97 17 and and CCONJ iajs-3789 97 18 let let VERB iajs-3789 97 19 { { PUNCT iajs-3789 97 20 Ο„i}i=1 Ο„i}i=1 NOUN iajs-3789 97 21 3 3 NUM iajs-3789 97 22 be be AUX iajs-3789 97 23 a a DET iajs-3789 97 24 family family NOUN iajs-3789 97 25 of of ADP iajs-3789 97 26 topologies topology NOUN iajs-3789 97 27 defined define VERB iajs-3789 97 28 on on ADP iajs-3789 97 29 x x PUNCT iajs-3789 97 30 as as SCONJ iajs-3789 97 31 follows follow VERB iajs-3789 97 32 : : PUNCT iajs-3789 97 33 Ο„1 Ο„1 ADP iajs-3789 97 34 = = SYM iajs-3789 97 35 { { PUNCT iajs-3789 97 36 x x NOUN iajs-3789 97 37 , , PUNCT iajs-3789 97 38 βˆ… βˆ… NOUN iajs-3789 97 39 , , PUNCT iajs-3789 97 40 { { PUNCT iajs-3789 97 41 ꞎ ꞎ NOUN iajs-3789 97 42 𝟏 𝟏 NUM iajs-3789 97 43 } } PUNCT iajs-3789 97 44 , , PUNCT iajs-3789 97 45 { { PUNCT iajs-3789 97 46 ꞎ ꞎ NOUN iajs-3789 97 47 𝟏 𝟏 X iajs-3789 97 48 , , PUNCT iajs-3789 97 49 ꞎ ꞎ PROPN iajs-3789 97 50 𝟐 𝟐 NUM iajs-3789 97 51 } } PUNCT iajs-3789 97 52 } } PUNCT iajs-3789 97 53 , , PUNCT iajs-3789 97 54 Ο„2 Ο„2 NOUN iajs-3789 97 55 = = SYM iajs-3789 97 56 { { PUNCT iajs-3789 97 57 x x NOUN iajs-3789 97 58 , , PUNCT iajs-3789 97 59 βˆ… βˆ… NOUN iajs-3789 97 60 , , PUNCT iajs-3789 97 61 { { PUNCT iajs-3789 97 62 ꞎ ꞎ NOUN iajs-3789 97 63 𝟏 𝟏 NUM iajs-3789 97 64 } } PUNCT iajs-3789 97 65 , , PUNCT iajs-3789 97 66 { { PUNCT iajs-3789 97 67 ꞎ ꞎ NOUN iajs-3789 97 68 𝟏 𝟏 X iajs-3789 97 69 , , PUNCT iajs-3789 97 70 ꞎ ꞎ NOUN iajs-3789 97 71 3 3 NUM iajs-3789 97 72 } } PUNCT iajs-3789 97 73 } } PUNCT iajs-3789 97 74 , , PUNCT iajs-3789 97 75 Ο„3 Ο„3 NOUN iajs-3789 97 76 = = SYM iajs-3789 97 77 { { PUNCT iajs-3789 97 78 x x NOUN iajs-3789 97 79 , , PUNCT iajs-3789 97 80 βˆ… βˆ… NOUN iajs-3789 97 81 , , PUNCT iajs-3789 97 82 { { PUNCT iajs-3789 97 83 ꞎ ꞎ NOUN iajs-3789 97 84 𝟏 𝟏 NUM iajs-3789 97 85 } } PUNCT iajs-3789 97 86 , , PUNCT iajs-3789 97 87 { { PUNCT iajs-3789 97 88 ꞎ ꞎ NOUN iajs-3789 97 89 𝟐 𝟐 NUM iajs-3789 97 90 , , PUNCT iajs-3789 97 91 ꞎ ꞎ NOUN iajs-3789 97 92 3 3 NUM iajs-3789 97 93 } } PUNCT iajs-3789 97 94 } } PUNCT iajs-3789 97 95 , , PUNCT iajs-3789 97 96 then then ADV iajs-3789 97 97 β‹‚ β‹‚ PROPN iajs-3789 97 98 Ο„i Ο„i VERB iajs-3789 97 99 3 3 NUM iajs-3789 97 100 i=1 i=1 NOUN iajs-3789 98 1 = = PUNCT iajs-3789 98 2 { { PUNCT iajs-3789 98 3 x x NOUN iajs-3789 98 4 , , PUNCT iajs-3789 98 5 βˆ… βˆ… NOUN iajs-3789 98 6 , , PUNCT iajs-3789 98 7 { { PUNCT iajs-3789 98 8 ꞎ ꞎ NOUN iajs-3789 98 9 𝟏 𝟏 NUM iajs-3789 98 10 } } PUNCT iajs-3789 98 11 } } PUNCT iajs-3789 98 12 𝒱ox 𝒱ox NOUN iajs-3789 98 13 = = SYM iajs-3789 98 14 { { PUNCT iajs-3789 98 15 x x NOUN iajs-3789 98 16 , , PUNCT iajs-3789 98 17 βˆ… βˆ… NOUN iajs-3789 98 18 , , PUNCT iajs-3789 98 19 { { PUNCT iajs-3789 98 20 ꞎ ꞎ NOUN iajs-3789 98 21 𝟏 𝟏 NUM iajs-3789 98 22 } } PUNCT iajs-3789 98 23 , , PUNCT iajs-3789 98 24 { { PUNCT iajs-3789 98 25 ꞎ ꞎ NOUN iajs-3789 98 26 𝟏 𝟏 X iajs-3789 98 27 , , PUNCT iajs-3789 98 28 ꞎ ꞎ PROPN iajs-3789 98 29 𝟐 𝟐 NUM iajs-3789 98 30 } } PUNCT iajs-3789 98 31 , , PUNCT iajs-3789 98 32 { { PUNCT iajs-3789 98 33 ꞎ ꞎ NOUN iajs-3789 98 34 𝟏 𝟏 X iajs-3789 98 35 , , PUNCT iajs-3789 98 36 ꞎ ꞎ NOUN iajs-3789 98 37 3 3 NUM iajs-3789 98 38 } } PUNCT iajs-3789 98 39 } } PUNCT iajs-3789 98 40 , , PUNCT iajs-3789 98 41 let let VERB iajs-3789 98 42 𝕾 𝕾 NOUN iajs-3789 98 43 = = SYM iajs-3789 98 44 { { PUNCT iajs-3789 98 45 x x NOUN iajs-3789 98 46 , , PUNCT iajs-3789 98 47 { { PUNCT iajs-3789 98 48 ꞎ ꞎ NOUN iajs-3789 98 49 𝟐 𝟐 NUM iajs-3789 98 50 } } PUNCT iajs-3789 98 51 , , PUNCT iajs-3789 98 52 { { PUNCT iajs-3789 98 53 ꞎ ꞎ NOUN iajs-3789 98 54 𝟏 𝟏 X iajs-3789 98 55 , , PUNCT iajs-3789 98 56 ꞎ ꞎ PROPN iajs-3789 98 57 𝟐 𝟐 NUM iajs-3789 98 58 } } PUNCT iajs-3789 98 59 , , PUNCT iajs-3789 98 60 { { PUNCT iajs-3789 98 61 ꞎ ꞎ NOUN iajs-3789 98 62 𝟐 𝟐 NUM iajs-3789 98 63 , , PUNCT iajs-3789 98 64 ꞎ ꞎ NOUN iajs-3789 98 65 3 3 NUM iajs-3789 98 66 } } PUNCT iajs-3789 98 67 } } PUNCT iajs-3789 98 68 be be AUX iajs-3789 98 69 a a DET iajs-3789 98 70 grill grill NOUN iajs-3789 98 71 on on ADP iajs-3789 98 72 x. x. NOUN iajs-3789 98 73 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 98 74 = = SYM iajs-3789 98 75 { { PUNCT iajs-3789 98 76 x x NOUN iajs-3789 98 77 , , PUNCT iajs-3789 98 78 βˆ… βˆ… NOUN iajs-3789 98 79 , , PUNCT iajs-3789 98 80 { { PUNCT iajs-3789 98 81 ꞎ ꞎ NOUN iajs-3789 98 82 𝟏 𝟏 NUM iajs-3789 98 83 } } PUNCT iajs-3789 98 84 , , PUNCT iajs-3789 98 85 { { PUNCT iajs-3789 98 86 ꞎ ꞎ NOUN iajs-3789 98 87 𝟐 𝟐 NUM iajs-3789 98 88 } } PUNCT iajs-3789 98 89 , , PUNCT iajs-3789 98 90 { { PUNCT iajs-3789 98 91 ꞎ ꞎ NOUN iajs-3789 98 92 3 3 NUM iajs-3789 98 93 } } PUNCT iajs-3789 98 94 , , PUNCT iajs-3789 98 95 { { PUNCT iajs-3789 98 96 ꞎ ꞎ NOUN iajs-3789 98 97 𝟏 𝟏 X iajs-3789 98 98 , , PUNCT iajs-3789 99 1 ꞎ ꞎ PROPN iajs-3789 99 2 𝟐 𝟐 NUM iajs-3789 99 3 } } PUNCT iajs-3789 99 4 , , PUNCT iajs-3789 99 5 { { PUNCT iajs-3789 99 6 ꞎ ꞎ NOUN iajs-3789 99 7 𝟏 𝟏 X iajs-3789 99 8 , , PUNCT iajs-3789 99 9 ꞎ ꞎ NOUN iajs-3789 99 10 3 3 NUM iajs-3789 99 11 } } PUNCT iajs-3789 99 12 , , PUNCT iajs-3789 99 13 { { PUNCT iajs-3789 99 14 ꞎ ꞎ NOUN iajs-3789 99 15 𝟐 𝟐 NUM iajs-3789 99 16 , , PUNCT iajs-3789 99 17 ꞎ ꞎ NOUN iajs-3789 99 18 3 3 NUM iajs-3789 99 19 } } PUNCT iajs-3789 99 20 } } PUNCT iajs-3789 99 21 . . PUNCT iajs-3789 100 1 ihjpas ihjpas PROPN iajs-3789 100 2 . . PUNCT iajs-3789 101 1 37 37 NUM iajs-3789 101 2 ( ( PUNCT iajs-3789 101 3 2 2 NUM iajs-3789 101 4 ) ) PUNCT iajs-3789 101 5 2024 2024 NUM iajs-3789 101 6 436 436 NUM iajs-3789 101 7 theorem theorem VERB iajs-3789 101 8 3.8 3.8 NUM iajs-3789 101 9 suppose suppose VERB iajs-3789 101 10 that that SCONJ iajs-3789 101 11 ( ( PUNCT iajs-3789 101 12 ꞎ ꞎ NOUN iajs-3789 101 13 , , PUNCT iajs-3789 101 14 π’±π‘‚κžŽ π’±π‘‚κžŽ NOUN iajs-3789 101 15 ) ) PUNCT iajs-3789 101 16 be be AUX iajs-3789 101 17 a a DET iajs-3789 101 18 π“₯-space π“₯-space NOUN iajs-3789 101 19 : : PUNCT iajs-3789 101 20 1 1 NUM iajs-3789 101 21 . . X iajs-3789 101 22 if if SCONJ iajs-3789 101 23 𝕾 𝕾 NOUN iajs-3789 101 24 is be AUX iajs-3789 101 25 any any DET iajs-3789 101 26 grill grill NOUN iajs-3789 101 27 on on ADP iajs-3789 101 28 ꞎ ꞎ PRON iajs-3789 101 29 and and CCONJ iajs-3789 101 30 𝐴 𝐴 PROPN iajs-3789 101 31 βˆ‰ βˆ‰ PROPN iajs-3789 101 32 𝕾 𝕾 NOUN iajs-3789 101 33 then then ADV iajs-3789 101 34 a a PRON iajs-3789 101 35 is be AUX iajs-3789 101 36 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 101 37 βˆ’ βˆ’ NOUN iajs-3789 101 38 closed closed ADJ iajs-3789 101 39 set set VERB iajs-3789 101 40 in in ADP iajs-3789 101 41 ( ( PUNCT iajs-3789 101 42 ꞎ ꞎ NOUN iajs-3789 101 43 , , PUNCT iajs-3789 101 44 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 101 45 ) ) PUNCT iajs-3789 101 46 . . PUNCT iajs-3789 102 1 2 2 X iajs-3789 102 2 . . X iajs-3789 102 3 if if SCONJ iajs-3789 102 4 𝕾1 𝕾1 VERB iajs-3789 102 5 and and CCONJ iajs-3789 102 6 𝕾2 𝕾2 NUM iajs-3789 102 7 are be AUX iajs-3789 102 8 two two NUM iajs-3789 102 9 grilles grille NOUN iajs-3789 102 10 on on ADP iajs-3789 102 11 ꞎ ꞎ NOUN iajs-3789 102 12 with with ADP iajs-3789 102 13 𝕾1 𝕾1 ADP iajs-3789 102 14 βŠ† βŠ† NUM iajs-3789 102 15 𝕾2 𝕾2 NUM iajs-3789 102 16 , , PUNCT iajs-3789 102 17 then then ADV iajs-3789 102 18 𝒱𝕾2 𝒱𝕾2 PROPN iajs-3789 102 19 βŠ† βŠ† NUM iajs-3789 102 20 𝒱𝕾1 𝒱𝕾1 NOUN iajs-3789 102 21 . . PUNCT iajs-3789 103 1 3 3 X iajs-3789 103 2 . . X iajs-3789 103 3 for for ADP iajs-3789 103 4 any any DET iajs-3789 103 5 grill grill NOUN iajs-3789 103 6 𝕾 𝕾 NOUN iajs-3789 103 7 on on ADP iajs-3789 103 8 ꞎ ꞎ NOUN iajs-3789 103 9 and and CCONJ iajs-3789 103 10 any any DET iajs-3789 103 11 subset subset ADJ iajs-3789 103 12 𝐴 𝐴 PROPN iajs-3789 103 13 of of ADP iajs-3789 103 14 π‘˜ π‘˜ PROPN iajs-3789 103 15 , , PUNCT iajs-3789 103 16 υ𝕾(𝐴 υ𝕾(𝐴 PROPN iajs-3789 103 17 ) ) PUNCT iajs-3789 103 18 is be AUX iajs-3789 103 19 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 103 20 βˆ’ βˆ’ PROPN iajs-3789 103 21 closed closed ADJ iajs-3789 103 22 . . PUNCT iajs-3789 104 1 4 4 X iajs-3789 104 2 . . X iajs-3789 104 3 if if SCONJ iajs-3789 104 4 𝐴 𝐴 PROPN iajs-3789 104 5 is be AUX iajs-3789 104 6 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 104 7 βˆ’ βˆ’ PROPN iajs-3789 104 8 closed closed ADJ iajs-3789 104 9 , , PUNCT iajs-3789 104 10 then then ADV iajs-3789 104 11 υ𝕾(a υ𝕾(a NUM iajs-3789 104 12 ) ) PUNCT iajs-3789 104 13 βŠ† βŠ† NUM iajs-3789 104 14 a. a. NOUN iajs-3789 104 15 proof proof NOUN iajs-3789 104 16 : : PUNCT iajs-3789 104 17 1 1 X iajs-3789 104 18 . . X iajs-3789 104 19 since since SCONJ iajs-3789 104 20 𝐴 𝐴 PROPN iajs-3789 104 21 βˆ‰ βˆ‰ PROPN iajs-3789 104 22 𝕾 𝕾 PROPN iajs-3789 104 23 , , PUNCT iajs-3789 104 24 so so ADV iajs-3789 104 25 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 104 26 ) ) PUNCT iajs-3789 104 27 = = NOUN iajs-3789 104 28 βˆ… βˆ… NOUN iajs-3789 104 29 , , PUNCT iajs-3789 104 30 it it PRON iajs-3789 104 31 follows follow VERB iajs-3789 104 32 that that SCONJ iajs-3789 104 33 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 104 34 βˆ’ βˆ’ PROPN iajs-3789 104 35 𝑐𝑙(𝐴 𝑐𝑙(𝐴 NOUN iajs-3789 104 36 ) ) PUNCT iajs-3789 104 37 = = PUNCT iajs-3789 104 38 π΄β‹ƒβˆ… π΄β‹ƒβˆ… PROPN iajs-3789 104 39 = = SYM iajs-3789 104 40 𝐴 𝐴 PROPN iajs-3789 104 41 , , PUNCT iajs-3789 104 42 it it PRON iajs-3789 104 43 means mean VERB iajs-3789 104 44 ꞎ ꞎ PRON iajs-3789 104 45 βˆ’ βˆ’ PROPN iajs-3789 104 46 𝐴 𝐴 PROPN iajs-3789 104 47 ∈ ∈ PROPN iajs-3789 104 48 𝒱𝕾. 𝒱𝕾. PUNCT iajs-3789 104 49 hence hence ADV iajs-3789 104 50 a a PRON iajs-3789 104 51 is be AUX iajs-3789 104 52 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 104 53 βˆ’ βˆ’ NOUN iajs-3789 104 54 closed closed ADJ iajs-3789 104 55 set set VERB iajs-3789 104 56 in in ADP iajs-3789 104 57 ( ( PUNCT iajs-3789 104 58 ꞎ ꞎ NOUN iajs-3789 104 59 , , PUNCT iajs-3789 104 60 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 104 61 ) ) PUNCT iajs-3789 104 62 . . PUNCT iajs-3789 105 1 2 2 X iajs-3789 105 2 . . X iajs-3789 105 3 let let VERB iajs-3789 105 4 𝐺 𝐺 PROPN iajs-3789 105 5 ∈ ∈ PROPN iajs-3789 105 6 𝒱𝕾2 𝒱𝕾2 PROPN iajs-3789 105 7 , , PUNCT iajs-3789 105 8 then then ADV iajs-3789 105 9 𝒱𝕾2 𝒱𝕾2 PROPN iajs-3789 105 10 βˆ’ βˆ’ PROPN iajs-3789 105 11 𝑐𝑙(ꞎ 𝑐𝑙(ꞎ PUNCT iajs-3789 105 12 βˆ’ βˆ’ PROPN iajs-3789 105 13 𝐺 𝐺 PROPN iajs-3789 105 14 ) ) PUNCT iajs-3789 105 15 = = PUNCT iajs-3789 106 1 ꞎ ꞎ PRON iajs-3789 106 2 βˆ’ βˆ’ PROPN iajs-3789 106 3 𝐺 𝐺 PROPN iajs-3789 106 4 , , PUNCT iajs-3789 106 5 and and CCONJ iajs-3789 106 6 so so ADV iajs-3789 106 7 ꞎ ꞎ ADV iajs-3789 106 8 βˆ’ βˆ’ NUM iajs-3789 106 9 𝐺⋃υ𝕾2 𝐺⋃υ𝕾2 X iajs-3789 106 10 ( ( PUNCT iajs-3789 106 11 ꞎ ꞎ NOUN iajs-3789 106 12 βˆ’ βˆ’ PROPN iajs-3789 106 13 𝐺 𝐺 PROPN iajs-3789 106 14 ) ) PUNCT iajs-3789 106 15 = = PUNCT iajs-3789 107 1 ꞎ ꞎ PRON iajs-3789 107 2 βˆ’ βˆ’ PROPN iajs-3789 107 3 𝐺 𝐺 PROPN iajs-3789 107 4 , , PUNCT iajs-3789 107 5 it it PRON iajs-3789 107 6 follows follow VERB iajs-3789 107 7 that that SCONJ iajs-3789 107 8 υ𝕾2 υ𝕾2 PROPN iajs-3789 107 9 ( ( PUNCT iajs-3789 107 10 ꞎ ꞎ NOUN iajs-3789 107 11 βˆ’ βˆ’ PROPN iajs-3789 107 12 𝐺 𝐺 PROPN iajs-3789 107 13 ) ) PUNCT iajs-3789 107 14 βŠ† βŠ† NUM iajs-3789 107 15 ꞎ ꞎ PRON iajs-3789 107 16 βˆ’ βˆ’ PROPN iajs-3789 107 17 𝐺 𝐺 PROPN iajs-3789 107 18 , , PUNCT iajs-3789 107 19 but but CCONJ iajs-3789 107 20 υ𝕾1 υ𝕾1 NOUN iajs-3789 107 21 ( ( PUNCT iajs-3789 107 22 ꞎ ꞎ NOUN iajs-3789 107 23 βˆ’ βˆ’ PROPN iajs-3789 107 24 𝐺 𝐺 PROPN iajs-3789 107 25 ) ) PUNCT iajs-3789 107 26 βŠ† βŠ† NUM iajs-3789 107 27 υ𝕾2 υ𝕾2 PROPN iajs-3789 107 28 ( ( PUNCT iajs-3789 107 29 ꞎ ꞎ NOUN iajs-3789 107 30 βˆ’ βˆ’ PROPN iajs-3789 107 31 𝐺 𝐺 PROPN iajs-3789 107 32 ) ) PUNCT iajs-3789 107 33 , , PUNCT iajs-3789 107 34 so so ADV iajs-3789 107 35 υ𝕾1 υ𝕾1 PROPN iajs-3789 107 36 ( ( PUNCT iajs-3789 107 37 ꞎ ꞎ NOUN iajs-3789 107 38 βˆ’ βˆ’ PROPN iajs-3789 107 39 𝐺 𝐺 PROPN iajs-3789 107 40 ) ) PUNCT iajs-3789 107 41 βŠ† βŠ† NUM iajs-3789 107 42 ꞎ ꞎ PROPN iajs-3789 107 43 βˆ’ βˆ’ PROPN iajs-3789 107 44 𝐺 𝐺 PROPN iajs-3789 107 45 , , PUNCT iajs-3789 107 46 implies imply VERB iajs-3789 107 47 that that SCONJ iajs-3789 108 1 ꞎ ꞎ PRON iajs-3789 108 2 βˆ’ βˆ’ NOUN iajs-3789 108 3 𝐺⋃υ𝕾1 𝐺⋃υ𝕾1 NOUN iajs-3789 108 4 ( ( PUNCT iajs-3789 108 5 ꞎ ꞎ NOUN iajs-3789 108 6 βˆ’ βˆ’ PROPN iajs-3789 108 7 𝐺 𝐺 PROPN iajs-3789 108 8 ) ) PUNCT iajs-3789 108 9 = = PUNCT iajs-3789 109 1 ꞎ ꞎ PRON iajs-3789 109 2 βˆ’ βˆ’ PROPN iajs-3789 109 3 𝐺 𝐺 PROPN iajs-3789 109 4 , , PUNCT iajs-3789 109 5 therefor therefor ADP iajs-3789 109 6 𝐺 𝐺 PROPN iajs-3789 109 7 ∈ ∈ PROPN iajs-3789 109 8 𝒱𝕾1 𝒱𝕾1 PROPN iajs-3789 109 9 . . PUNCT iajs-3789 110 1 thus thus ADV iajs-3789 110 2 𝒱𝕾2 𝒱𝕾2 PROPN iajs-3789 110 3 βŠ† βŠ† NUM iajs-3789 110 4 𝒱𝕾1 𝒱𝕾1 NOUN iajs-3789 110 5 . . PUNCT iajs-3789 111 1 3 3 X iajs-3789 111 2 . . X iajs-3789 111 3 since since SCONJ iajs-3789 111 4 υ𝕾(υ𝕾(a υ𝕾(υ𝕾(a ADJ iajs-3789 111 5 ) ) PUNCT iajs-3789 111 6 ) ) PUNCT iajs-3789 111 7 βŠ† βŠ† NUM iajs-3789 111 8 υ𝕾(a υ𝕾(a NUM iajs-3789 111 9 ) ) PUNCT iajs-3789 111 10 , , PUNCT iajs-3789 111 11 so so ADV iajs-3789 111 12 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 111 13 βˆ’ βˆ’ PROPN iajs-3789 111 14 𝑐𝑙(υ𝕾(a 𝑐𝑙(υ𝕾(a PROPN iajs-3789 111 15 ) ) PUNCT iajs-3789 111 16 ) ) PUNCT iajs-3789 112 1 = = SYM iajs-3789 112 2 υ𝕾(a υ𝕾(a NOUN iajs-3789 112 3 ) ) PUNCT iajs-3789 112 4 βˆͺ βˆͺ ADP iajs-3789 112 5 υ𝕾(υ𝕾(a υ𝕾(υ𝕾(a ADV iajs-3789 112 6 ) ) PUNCT iajs-3789 112 7 ) ) PUNCT iajs-3789 113 1 = = SYM iajs-3789 113 2 υ𝕾(a υ𝕾(a PROPN iajs-3789 113 3 ) ) PUNCT iajs-3789 113 4 . . PUNCT iajs-3789 114 1 hence hence ADV iajs-3789 114 2 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 114 3 ) ) PUNCT iajs-3789 114 4 is be AUX iajs-3789 114 5 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 114 6 βˆ’ βˆ’ PROPN iajs-3789 114 7 closed closed ADJ iajs-3789 114 8 . . PUNCT iajs-3789 115 1 4 4 X iajs-3789 115 2 . . X iajs-3789 115 3 let let VERB iajs-3789 115 4 a a PRON iajs-3789 115 5 is be AUX iajs-3789 115 6 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 115 7 βˆ’ βˆ’ NOUN iajs-3789 115 8 closed closed ADJ iajs-3789 115 9 set set VERB iajs-3789 115 10 in in ADP iajs-3789 115 11 ( ( PUNCT iajs-3789 115 12 ꞎ ꞎ NOUN iajs-3789 115 13 , , PUNCT iajs-3789 115 14 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 115 15 ) ) PUNCT iajs-3789 115 16 , , PUNCT iajs-3789 115 17 then then ADV iajs-3789 115 18 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 115 19 βˆ’ βˆ’ PROPN iajs-3789 115 20 𝑐𝑙(𝐴 𝑐𝑙(𝐴 PROPN iajs-3789 115 21 ) ) PUNCT iajs-3789 115 22 = = SYM iajs-3789 115 23 𝐴⋃υ𝕾(a 𝐴⋃υ𝕾(a ADJ iajs-3789 115 24 ) ) PUNCT iajs-3789 115 25 = = SYM iajs-3789 116 1 𝐴. 𝐴. PROPN iajs-3789 116 2 thus thus ADV iajs-3789 116 3 υ𝕾(a υ𝕾(a NUM iajs-3789 116 4 ) ) PUNCT iajs-3789 116 5 βŠ† βŠ† NUM iajs-3789 116 6 𝐴. 𝐴. NOUN iajs-3789 116 7 theorem theorem VERB iajs-3789 116 8 3.9 3.9 NUM iajs-3789 116 9 for for ADP iajs-3789 116 10 a a DET iajs-3789 116 11 grill grill NOUN iajs-3789 116 12 π“₯-space π“₯-space PROPN iajs-3789 116 13 ( ( PUNCT iajs-3789 116 14 𝑋 𝑋 PROPN iajs-3789 116 15 , , PUNCT iajs-3789 116 16 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 116 17 , , PUNCT iajs-3789 116 18 𝕾 𝕾 PROPN iajs-3789 116 19 ) ) PUNCT iajs-3789 116 20 . . PUNCT iajs-3789 117 1 then then ADV iajs-3789 117 2 the the DET iajs-3789 117 3 collection collection NOUN iajs-3789 117 4 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 117 5 , , PUNCT iajs-3789 117 6 𝒱ok 𝒱ok PROPN iajs-3789 117 7 ) ) PUNCT iajs-3789 117 8 = = PRON iajs-3789 117 9 { { PUNCT iajs-3789 117 10 𝒩 𝒩 PROPN iajs-3789 117 11 βˆ’ βˆ’ PROPN iajs-3789 117 12 𝐴 𝐴 PROPN iajs-3789 117 13 : : PUNCT iajs-3789 117 14 𝒩 𝒩 PROPN iajs-3789 117 15 ∈ ∈ PROPN iajs-3789 117 16 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 117 17 and and CCONJ iajs-3789 117 18 𝐴 𝐴 NOUN iajs-3789 117 19 βˆ‰ βˆ‰ ADJ iajs-3789 117 20 𝕾 𝕾 NOUN iajs-3789 117 21 } } PUNCT iajs-3789 117 22 is be AUX iajs-3789 117 23 a a DET iajs-3789 117 24 basis basis NOUN iajs-3789 117 25 for for ADP iajs-3789 117 26 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 117 27 proof proof NOUN iajs-3789 117 28 : : PUNCT iajs-3789 117 29 let let VERB iajs-3789 117 30 Ι± Ι± PROPN iajs-3789 117 31 ∈ ∈ PROPN iajs-3789 117 32 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 117 33 and and CCONJ iajs-3789 117 34 π‘₯ π‘₯ PRON iajs-3789 117 35 ∈ ∈ PROPN iajs-3789 117 36 Ι± Ι± PROPN iajs-3789 117 37 , , PUNCT iajs-3789 117 38 then then ADV iajs-3789 117 39 π‘₯ π‘₯ PROPN iajs-3789 117 40 βˆ‰ βˆ‰ PROPN iajs-3789 117 41 k k PROPN iajs-3789 118 1 βˆ’ βˆ’ PROPN iajs-3789 118 2 Ι± Ι± PROPN iajs-3789 118 3 , , PUNCT iajs-3789 118 4 but but CCONJ iajs-3789 118 5 k k PROPN iajs-3789 119 1 βˆ’ βˆ’ PROPN iajs-3789 119 2 Ι± Ι± PROPN iajs-3789 119 3 is be AUX iajs-3789 119 4 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 119 5 βˆ’ βˆ’ PROPN iajs-3789 119 6 closed closed ADJ iajs-3789 119 7 set set NOUN iajs-3789 119 8 , , PUNCT iajs-3789 119 9 so so ADV iajs-3789 119 10 υ𝕾(k υ𝕾(k INTJ iajs-3789 119 11 βˆ’ βˆ’ NOUN iajs-3789 119 12 Ι± Ι± NOUN iajs-3789 119 13 ) ) PUNCT iajs-3789 119 14 βŠ† βŠ† NUM iajs-3789 119 15 k k NOUN iajs-3789 119 16 βˆ’ βˆ’ PROPN iajs-3789 119 17 Ι± Ι± PROPN iajs-3789 119 18 , , PUNCT iajs-3789 119 19 it it PRON iajs-3789 119 20 follows follow VERB iajs-3789 119 21 that that SCONJ iajs-3789 119 22 π‘₯ π‘₯ PROPN iajs-3789 119 23 βˆ‰ βˆ‰ X iajs-3789 120 1 υ𝕾(k υ𝕾(k PROPN iajs-3789 120 2 βˆ’ βˆ’ NUM iajs-3789 120 3 Ι± Ι± NOUN iajs-3789 120 4 ) ) PUNCT iajs-3789 120 5 . . PUNCT iajs-3789 121 1 from from ADP iajs-3789 121 2 definition definition NOUN iajs-3789 121 3 3.2 3.2 NUM iajs-3789 121 4 , , PUNCT iajs-3789 121 5 there there PRON iajs-3789 121 6 exists exist VERB iajs-3789 121 7 a a DET iajs-3789 121 8 𝒱 𝒱 NOUN iajs-3789 121 9 βˆ’ βˆ’ NOUN iajs-3789 121 10 open open ADJ iajs-3789 121 11 set set NOUN iajs-3789 121 12 𝒩 𝒩 PROPN iajs-3789 121 13 containing contain VERB iajs-3789 121 14 π‘₯ π‘₯ PRON iajs-3789 121 15 such such ADJ iajs-3789 121 16 that that SCONJ iajs-3789 121 17 𝒩⋂(k 𝒩⋂(k VERB iajs-3789 121 18 βˆ’ βˆ’ PROPN iajs-3789 121 19 Ι± Ι± NOUN iajs-3789 121 20 ) ) PUNCT iajs-3789 121 21 βˆ‰ βˆ‰ PROPN iajs-3789 121 22 𝕾 𝕾 NOUN iajs-3789 121 23 . . PUNCT iajs-3789 122 1 let let VERB iajs-3789 122 2 𝐴 𝐴 PROPN iajs-3789 122 3 = = SYM iajs-3789 122 4 𝒩⋂(k 𝒩⋂(k VERB iajs-3789 122 5 βˆ’ βˆ’ NOUN iajs-3789 122 6 Ι± Ι± NOUN iajs-3789 122 7 ) ) PUNCT iajs-3789 122 8 , , PUNCT iajs-3789 122 9 then then ADV iajs-3789 122 10 π‘₯ π‘₯ PROPN iajs-3789 122 11 ∈ ∈ PROPN iajs-3789 122 12 𝒩 𝒩 PROPN iajs-3789 122 13 βˆ’ βˆ’ PROPN iajs-3789 122 14 𝐴 𝐴 PROPN iajs-3789 122 15 βŠ† βŠ† PROPN iajs-3789 122 16 Ι± Ι± ADP iajs-3789 122 17 such such ADJ iajs-3789 122 18 that that SCONJ iajs-3789 122 19 𝒩 𝒩 PROPN iajs-3789 122 20 ∈ ∈ PROPN iajs-3789 122 21 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 122 22 and and CCONJ iajs-3789 122 23 𝐴 𝐴 PROPN iajs-3789 122 24 βˆ‰ βˆ‰ PROPN iajs-3789 122 25 𝕾. 𝕾. PROPN iajs-3789 122 26 thus thus ADV iajs-3789 122 27 Ι± Ι± PROPN iajs-3789 122 28 is be AUX iajs-3789 122 29 the the DET iajs-3789 122 30 union union NOUN iajs-3789 122 31 of of ADP iajs-3789 122 32 subsets subset NOUN iajs-3789 122 33 in in ADP iajs-3789 122 34 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 122 35 , , PUNCT iajs-3789 122 36 𝒱ok 𝒱ok PROPN iajs-3789 122 37 ) ) PUNCT iajs-3789 122 38 . . PUNCT iajs-3789 123 1 easily easily ADV iajs-3789 123 2 ℬ(𝕾 ℬ(𝕾 X iajs-3789 123 3 , , PUNCT iajs-3789 123 4 𝒱ok 𝒱ok PROPN iajs-3789 123 5 ) ) PUNCT iajs-3789 123 6 is be AUX iajs-3789 123 7 closed close VERB iajs-3789 123 8 when when SCONJ iajs-3789 123 9 the the DET iajs-3789 123 10 intersection intersection NOUN iajs-3789 123 11 are be AUX iajs-3789 123 12 finite finite NOUN iajs-3789 123 13 that that PRON iajs-3789 123 14 is be AUX iajs-3789 123 15 if if SCONJ iajs-3789 123 16 𝒩1 𝒩1 PROPN iajs-3789 123 17 βˆ’ βˆ’ PROPN iajs-3789 123 18 𝐴1 𝐴1 PROPN iajs-3789 123 19 and and CCONJ iajs-3789 123 20 𝒩2 𝒩2 NOUN iajs-3789 123 21 βˆ’ βˆ’ PROPN iajs-3789 123 22 𝐴2 𝐴2 PROPN iajs-3789 123 23 are be AUX iajs-3789 123 24 in in ADP iajs-3789 123 25 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 123 26 , , PUNCT iajs-3789 123 27 𝒱ok 𝒱ok PROPN iajs-3789 123 28 ) ) PUNCT iajs-3789 123 29 , , PUNCT iajs-3789 123 30 then then ADV iajs-3789 123 31 ( ( PUNCT iajs-3789 123 32 𝒩1 𝒩1 PROPN iajs-3789 123 33 βˆ’ βˆ’ PROPN iajs-3789 123 34 𝐴1)β‹‚(𝒩2 𝐴1)β‹‚(𝒩2 PROPN iajs-3789 123 35 βˆ’ βˆ’ PROPN iajs-3789 123 36 𝐴2 𝐴2 PROPN iajs-3789 123 37 ) ) PUNCT iajs-3789 123 38 = = PRON iajs-3789 124 1 ( ( PUNCT iajs-3789 124 2 𝒩1⋂𝒩2 𝒩1⋂𝒩2 NOUN iajs-3789 124 3 ) ) PUNCT iajs-3789 124 4 βˆ’ βˆ’ PROPN iajs-3789 125 1 ( ( PUNCT iajs-3789 125 2 𝐴1 𝐴1 PROPN iajs-3789 125 3 βˆͺ βˆͺ ADP iajs-3789 125 4 𝐴2 𝐴2 PROPN iajs-3789 125 5 ) ) PUNCT iajs-3789 125 6 , , PUNCT iajs-3789 125 7 where where SCONJ iajs-3789 125 8 𝒩1⋂𝒩2 𝒩1⋂𝒩2 ADP iajs-3789 125 9 ∈ ∈ PROPN iajs-3789 125 10 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 125 11 and and CCONJ iajs-3789 125 12 𝐴1 𝐴1 PROPN iajs-3789 125 13 βˆͺ βˆͺ VERB iajs-3789 125 14 𝐴2 𝐴2 PROPN iajs-3789 125 15 βˆ‰ βˆ‰ PROPN iajs-3789 125 16 𝕾. 𝕾. PROPN iajs-3789 125 17 hence hence ADV iajs-3789 125 18 ℬ(𝕾 ℬ(𝕾 NUM iajs-3789 125 19 , , PUNCT iajs-3789 125 20 𝒱ok 𝒱ok PROPN iajs-3789 125 21 ) ) PUNCT iajs-3789 125 22 is be AUX iajs-3789 125 23 a a DET iajs-3789 125 24 basis basis NOUN iajs-3789 125 25 for for ADP iajs-3789 125 26 𝒱𝕾. 𝒱𝕾. NUM iajs-3789 125 27 theorem theorem VERB iajs-3789 125 28 3.10 3.10 NUM iajs-3789 125 29 . . PUNCT iajs-3789 126 1 from from ADP iajs-3789 126 2 any any DET iajs-3789 126 3 ( ( PUNCT iajs-3789 126 4 𝑋 𝑋 PROPN iajs-3789 126 5 , , PUNCT iajs-3789 126 6 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 126 7 , , PUNCT iajs-3789 126 8 𝕾 𝕾 NOUN iajs-3789 126 9 ) ) PUNCT iajs-3789 126 10 grill grill NOUN iajs-3789 126 11 π“₯-space π“₯-space PROPN iajs-3789 126 12 . . PUNCT iajs-3789 127 1 then then ADV iajs-3789 127 2 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 127 3 βŠ† βŠ† NUM iajs-3789 127 4 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 127 5 , , PUNCT iajs-3789 127 6 𝒱ox 𝒱ox PROPN iajs-3789 127 7 ) ) PUNCT iajs-3789 127 8 βŠ† βŠ† NUM iajs-3789 127 9 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 127 10 and and CCONJ iajs-3789 127 11 𝕾 𝕾 NOUN iajs-3789 127 12 = = SYM iajs-3789 127 13 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 127 14 ) ) PUNCT iajs-3789 127 15 βˆ’ βˆ’ NOUN iajs-3789 127 16 { { PUNCT iajs-3789 127 17 βˆ… βˆ… NOUN iajs-3789 127 18 } } PUNCT iajs-3789 127 19 , , PUNCT iajs-3789 127 20 therefore therefore ADV iajs-3789 127 21 𝒱𝑂𝑋 𝒱𝑂𝑋 PRON iajs-3789 127 22 = = SYM iajs-3789 127 23 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 127 24 , , PUNCT iajs-3789 127 25 𝒱ox 𝒱ox ADJ iajs-3789 127 26 ) ) PUNCT iajs-3789 127 27 = = SYM iajs-3789 127 28 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 128 1 proof proof NOUN iajs-3789 128 2 : : PUNCT iajs-3789 128 3 let let VERB iajs-3789 128 4 𝒩 𝒩 PROPN iajs-3789 128 5 ∈ ∈ PROPN iajs-3789 128 6 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 128 7 , , PUNCT iajs-3789 128 8 implies imply VERB iajs-3789 128 9 𝒩 𝒩 PROPN iajs-3789 128 10 = = SYM iajs-3789 128 11 𝒩 𝒩 PROPN iajs-3789 128 12 βˆ’ βˆ’ PROPN iajs-3789 128 13 βˆ… βˆ… NOUN iajs-3789 128 14 where where SCONJ iajs-3789 128 15 βˆ… βˆ… NOUN iajs-3789 128 16 βˆ‰ βˆ‰ X iajs-3789 128 17 𝕾 𝕾 PROPN iajs-3789 128 18 , , PUNCT iajs-3789 128 19 so so ADV iajs-3789 128 20 𝒩 𝒩 PROPN iajs-3789 128 21 ∈ ∈ PROPN iajs-3789 128 22 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 128 23 , , PUNCT iajs-3789 128 24 𝒱ox 𝒱ox NOUN iajs-3789 128 25 ) ) PUNCT iajs-3789 128 26 . . PUNCT iajs-3789 129 1 thus thus ADV iajs-3789 129 2 𝒱𝑂𝑋 𝒱𝑂𝑋 PRON iajs-3789 129 3 βŠ† βŠ† NUM iajs-3789 129 4 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 129 5 , , PUNCT iajs-3789 129 6 𝒱ox 𝒱ox PROPN iajs-3789 129 7 ) ) PUNCT iajs-3789 129 8 . . PUNCT iajs-3789 130 1 now now ADV iajs-3789 130 2 let let VERB iajs-3789 130 3 𝐺 𝐺 PROPN iajs-3789 130 4 ∈ ∈ PROPN iajs-3789 130 5 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 130 6 , , PUNCT iajs-3789 130 7 𝒱ox 𝒱ox PROPN iajs-3789 130 8 ) ) PUNCT iajs-3789 130 9 , , PUNCT iajs-3789 130 10 then then ADV iajs-3789 130 11 there there PRON iajs-3789 130 12 exists exist VERB iajs-3789 130 13 a a DET iajs-3789 130 14 𝒱 𝒱 NOUN iajs-3789 130 15 βˆ’ βˆ’ NOUN iajs-3789 130 16 open open ADJ iajs-3789 130 17 set set VERB iajs-3789 130 18 𝒩 𝒩 PROPN iajs-3789 130 19 and and CCONJ iajs-3789 130 20 𝐴 𝐴 PROPN iajs-3789 130 21 βˆ‰ βˆ‰ ADJ iajs-3789 130 22 𝕾 𝕾 NOUN iajs-3789 130 23 such such ADJ iajs-3789 130 24 that that DET iajs-3789 130 25 𝐺 𝐺 PROPN iajs-3789 130 26 = = SYM iajs-3789 130 27 𝒩 𝒩 PROPN iajs-3789 130 28 βˆ’ βˆ’ PROPN iajs-3789 130 29 𝐴 𝐴 PROPN iajs-3789 130 30 , , PUNCT iajs-3789 130 31 therefor therefor ADP iajs-3789 130 32 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 130 33 βˆ’ βˆ’ PROPN iajs-3789 130 34 𝑐𝑙(𝐺𝑐 𝑐𝑙(𝐺𝑐 PROPN iajs-3789 130 35 ) ) PUNCT iajs-3789 131 1 = = SYM iajs-3789 131 2 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 131 3 βˆ’ βˆ’ NOUN iajs-3789 131 4 𝑐𝑙((𝒩 𝑐𝑙((𝒩 NOUN iajs-3789 131 5 βˆ’ βˆ’ NOUN iajs-3789 131 6 𝐴)𝑐 𝐴)𝑐 NOUN iajs-3789 131 7 ) ) PUNCT iajs-3789 132 1 = = PUNCT iajs-3789 132 2 ( ( PUNCT iajs-3789 132 3 𝒩 𝒩 PROPN iajs-3789 132 4 βˆ’ βˆ’ PROPN iajs-3789 132 5 𝐴)𝑐⋃υ𝕾((𝒩 𝐴)𝑐⋃υ𝕾((𝒩 NOUN iajs-3789 132 6 βˆ’ βˆ’ PROPN iajs-3789 132 7 𝐴)𝑐 𝐴)𝑐 NOUN iajs-3789 132 8 ) ) PUNCT iajs-3789 133 1 = = SYM iajs-3789 133 2 ( ( PUNCT iajs-3789 133 3 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 133 4 βˆͺ βˆͺ ADJ iajs-3789 133 5 𝐴 𝐴 PROPN iajs-3789 133 6 ) ) PUNCT iajs-3789 133 7 βˆͺ βˆͺ ADP iajs-3789 133 8 υ𝕾(𝒩𝑐 υ𝕾(𝒩𝑐 NOUN iajs-3789 133 9 βˆͺ βˆͺ PROPN iajs-3789 133 10 𝐴 𝐴 PROPN iajs-3789 133 11 ) ) PUNCT iajs-3789 133 12 , , PUNCT iajs-3789 133 13 now now ADV iajs-3789 133 14 by by ADP iajs-3789 133 15 theorem theorem ADJ iajs-3789 133 16 3.4(1 3.4(1 NUM iajs-3789 133 17 ) ) PUNCT iajs-3789 133 18 that that PRON iajs-3789 133 19 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 133 20 βˆ’ βˆ’ PROPN iajs-3789 133 21 𝑐𝑙(𝐺𝑐 𝑐𝑙(𝐺𝑐 PROPN iajs-3789 133 22 ) ) PUNCT iajs-3789 133 23 = = SYM iajs-3789 134 1 ( ( PUNCT iajs-3789 134 2 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 134 3 βˆͺ βˆͺ ADJ iajs-3789 134 4 𝐴 𝐴 PROPN iajs-3789 134 5 ) ) PUNCT iajs-3789 134 6 βˆͺ βˆͺ ADP iajs-3789 134 7 υ𝕾(𝒩𝑐 υ𝕾(𝒩𝑐 NOUN iajs-3789 134 8 ) ) PUNCT iajs-3789 134 9 βˆͺ βˆͺ ADP iajs-3789 134 10 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 134 11 ) ) PUNCT iajs-3789 134 12 . . PUNCT iajs-3789 135 1 but but CCONJ iajs-3789 135 2 𝐴 𝐴 PROPN iajs-3789 135 3 βˆ‰ βˆ‰ PROPN iajs-3789 135 4 𝕾 𝕾 PROPN iajs-3789 135 5 , , PUNCT iajs-3789 135 6 so so ADV iajs-3789 135 7 υ𝕾(𝐴 υ𝕾(𝐴 NOUN iajs-3789 135 8 ) ) PUNCT iajs-3789 136 1 = = PUNCT iajs-3789 136 2 βˆ…. βˆ…. NOUN iajs-3789 136 3 since since SCONJ iajs-3789 136 4 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 136 5 is be AUX iajs-3789 136 6 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 136 7 βˆ’ βˆ’ PROPN iajs-3789 136 8 closed closed ADJ iajs-3789 136 9 , , PUNCT iajs-3789 136 10 then then ADV iajs-3789 136 11 υ𝕾(𝒩𝑐 υ𝕾(𝒩𝑐 NOUN iajs-3789 136 12 ) ) PUNCT iajs-3789 136 13 βŠ† βŠ† NUM iajs-3789 136 14 𝒩𝑐. 𝒩𝑐. NOUN iajs-3789 136 15 so so ADV iajs-3789 136 16 we we PRON iajs-3789 136 17 get get VERB iajs-3789 136 18 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 136 19 βˆ’ βˆ’ PROPN iajs-3789 136 20 𝑐𝑙(𝐺𝑐 𝑐𝑙(𝐺𝑐 PROPN iajs-3789 136 21 ) ) PUNCT iajs-3789 137 1 = = SYM iajs-3789 138 1 ( ( PUNCT iajs-3789 138 2 𝒩𝑐 𝒩𝑐 PROPN iajs-3789 138 3 βˆͺ βˆͺ ADJ iajs-3789 138 4 𝐴 𝐴 PROPN iajs-3789 138 5 ) ) PUNCT iajs-3789 139 1 = = PUNCT iajs-3789 139 2 ( ( PUNCT iajs-3789 139 3 𝒩 𝒩 PROPN iajs-3789 139 4 βˆ’ βˆ’ PROPN iajs-3789 139 5 𝐴)𝑐 𝐴)𝑐 NOUN iajs-3789 140 1 = = SYM iajs-3789 140 2 𝐺𝑐 𝐺𝑐 PROPN iajs-3789 140 3 , , PUNCT iajs-3789 140 4 which which PRON iajs-3789 140 5 implies imply VERB iajs-3789 140 6 that that SCONJ iajs-3789 140 7 𝐺 𝐺 PROPN iajs-3789 140 8 ∈ ∈ PROPN iajs-3789 140 9 𝒱𝕾. 𝒱𝕾. PUNCT iajs-3789 140 10 hence hence ADV iajs-3789 140 11 ℬ(𝕾 ℬ(𝕾 NUM iajs-3789 140 12 , , PUNCT iajs-3789 140 13 𝒱ox 𝒱ox PROPN iajs-3789 140 14 ) ) PUNCT iajs-3789 140 15 βŠ† βŠ† NUM iajs-3789 140 16 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 140 17 if if SCONJ iajs-3789 140 18 𝕾 𝕾 NOUN iajs-3789 140 19 = = SYM iajs-3789 140 20 β„™(𝑋 β„™(𝑋 NOUN iajs-3789 140 21 ) ) PUNCT iajs-3789 140 22 βˆ’ βˆ’ NOUN iajs-3789 140 23 { { PUNCT iajs-3789 140 24 βˆ… βˆ… NOUN iajs-3789 140 25 } } PUNCT iajs-3789 140 26 , , PUNCT iajs-3789 140 27 then then ADV iajs-3789 140 28 we we PRON iajs-3789 140 29 have have VERB iajs-3789 140 30 to to PART iajs-3789 140 31 show show VERB iajs-3789 140 32 that that SCONJ iajs-3789 140 33 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 140 34 βŠ‡ βŠ‡ NOUN iajs-3789 140 35 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 140 36 , , PUNCT iajs-3789 140 37 𝒱ox 𝒱ox ADJ iajs-3789 140 38 ) ) PUNCT iajs-3789 140 39 βŠ‡ βŠ‡ NOUN iajs-3789 140 40 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 140 41 let let VERB iajs-3789 140 42 𝐺 𝐺 PROPN iajs-3789 140 43 ∈ ∈ PROPN iajs-3789 140 44 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 140 45 , , PUNCT iajs-3789 140 46 therefor therefor ADP iajs-3789 140 47 𝐺𝑐 𝐺𝑐 PROPN iajs-3789 140 48 is be AUX iajs-3789 140 49 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 140 50 βˆ’ βˆ’ PROPN iajs-3789 140 51 closed closed ADJ iajs-3789 140 52 , , PUNCT iajs-3789 140 53 then then ADV iajs-3789 140 54 υ𝕾(𝐺𝑐 υ𝕾(𝐺𝑐 X iajs-3789 140 55 ) ) PUNCT iajs-3789 141 1 βŠ† βŠ† X iajs-3789 141 2 𝐺𝑐 𝐺𝑐 PROPN iajs-3789 141 3 and and CCONJ iajs-3789 141 4 so so ADV iajs-3789 141 5 𝐺 𝐺 PROPN iajs-3789 141 6 βŠ† βŠ† PROPN iajs-3789 141 7 𝑋 𝑋 PROPN iajs-3789 141 8 βˆ’ βˆ’ PROPN iajs-3789 141 9 υ𝕾(𝐺𝑐 υ𝕾(𝐺𝑐 PROPN iajs-3789 141 10 ) ) PUNCT iajs-3789 141 11 , , PUNCT iajs-3789 141 12 that that PRON iajs-3789 141 13 means mean VERB iajs-3789 141 14 for for ADP iajs-3789 141 15 each each DET iajs-3789 141 16 π‘₯ π‘₯ PRON iajs-3789 141 17 ∈ ∈ PROPN iajs-3789 141 18 𝐺 𝐺 NOUN iajs-3789 141 19 there there PRON iajs-3789 141 20 exists exist VERB iajs-3789 141 21 𝑆 𝑆 PROPN iajs-3789 141 22 ∈ ∈ PROPN iajs-3789 141 23 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 141 24 such such ADJ iajs-3789 141 25 that that SCONJ iajs-3789 141 26 𝑆⋂𝐺𝑐 𝑆⋂𝐺𝑐 NOUN iajs-3789 141 27 βˆ‰ βˆ‰ PROPN iajs-3789 141 28 𝕾 𝕾 PROPN iajs-3789 141 29 , , PUNCT iajs-3789 141 30 which which PRON iajs-3789 141 31 implies imply VERB iajs-3789 141 32 that that SCONJ iajs-3789 141 33 𝑆⋂𝐺𝑐 𝑆⋂𝐺𝑐 NOUN iajs-3789 141 34 = = SYM iajs-3789 141 35 βˆ… βˆ… NOUN iajs-3789 141 36 , , PUNCT iajs-3789 141 37 then then ADV iajs-3789 141 38 𝑆 𝑆 PROPN iajs-3789 141 39 βŠ† βŠ† NUM iajs-3789 141 40 𝐺 𝐺 PROPN iajs-3789 141 41 , , PUNCT iajs-3789 141 42 it it PRON iajs-3789 141 43 follows follow VERB iajs-3789 141 44 that that SCONJ iajs-3789 141 45 𝐺 𝐺 PROPN iajs-3789 141 46 ∈ ∈ PROPN iajs-3789 141 47 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 141 48 and and CCONJ iajs-3789 141 49 so so ADV iajs-3789 141 50 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 141 51 βŠ‡ βŠ‡ NOUN iajs-3789 141 52 𝒱𝕾. 𝒱𝕾. NOUN iajs-3789 141 53 hence hence ADV iajs-3789 141 54 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 141 55 = = SYM iajs-3789 141 56 𝒱𝕾. 𝒱𝕾. PUNCT iajs-3789 141 57 now now ADV iajs-3789 141 58 let let VERB iajs-3789 141 59 𝒦 𝒦 PROPN iajs-3789 141 60 ∈ ∈ PROPN iajs-3789 141 61 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 141 62 , , PUNCT iajs-3789 141 63 𝒱ox 𝒱ox PROPN iajs-3789 141 64 ) ) PUNCT iajs-3789 141 65 , , PUNCT iajs-3789 141 66 then then ADV iajs-3789 141 67 𝒦 𝒦 PROPN iajs-3789 141 68 = = SYM iajs-3789 141 69 𝒩 𝒩 PROPN iajs-3789 141 70 βˆ’ βˆ’ PROPN iajs-3789 141 71 𝐴 𝐴 PROPN iajs-3789 141 72 βˆ‹ βˆ‹ NOUN iajs-3789 141 73 𝒩 𝒩 PROPN iajs-3789 141 74 ∈ ∈ PROPN iajs-3789 141 75 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 141 76 and and CCONJ iajs-3789 141 77 𝐴 𝐴 PROPN iajs-3789 142 1 βˆ‰ βˆ‰ PROPN iajs-3789 142 2 𝕾 𝕾 PROPN iajs-3789 142 3 , , PUNCT iajs-3789 142 4 but but CCONJ iajs-3789 142 5 𝐴 𝐴 NOUN iajs-3789 142 6 = = NOUN iajs-3789 142 7 βˆ… βˆ… NOUN iajs-3789 142 8 , , PUNCT iajs-3789 142 9 so so SCONJ iajs-3789 142 10 𝒦 𝒦 PROPN iajs-3789 142 11 = = SYM iajs-3789 142 12 𝒩 𝒩 PROPN iajs-3789 142 13 , , PUNCT iajs-3789 142 14 then then ADV iajs-3789 142 15 𝒦 𝒦 PROPN iajs-3789 142 16 ∈ ∈ PROPN iajs-3789 142 17 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 142 18 and and CCONJ iajs-3789 142 19 so so ADV iajs-3789 142 20 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 142 21 βŠ‡ βŠ‡ NOUN iajs-3789 142 22 ℬ(𝕾 ℬ(𝕾 PROPN iajs-3789 142 23 , , PUNCT iajs-3789 142 24 𝒱ox 𝒱ox PROPN iajs-3789 142 25 ) ) PUNCT iajs-3789 142 26 . . PUNCT iajs-3789 143 1 thus thus ADV iajs-3789 143 2 𝒱𝑂𝑋 𝒱𝑂𝑋 PRON iajs-3789 143 3 = = SYM iajs-3789 143 4 ℬ(𝕾 ℬ(𝕾 NOUN iajs-3789 143 5 , , PUNCT iajs-3789 143 6 𝒱ox 𝒱ox PROPN iajs-3789 143 7 ) ) PUNCT iajs-3789 143 8 . . PUNCT iajs-3789 144 1 ihjpas ihjpas PROPN iajs-3789 144 2 . . PUNCT iajs-3789 145 1 37 37 NUM iajs-3789 145 2 ( ( PUNCT iajs-3789 145 3 2 2 NUM iajs-3789 145 4 ) ) PUNCT iajs-3789 145 5 2024 2024 NUM iajs-3789 145 6 437 437 NUM iajs-3789 145 7 corollary corollary NOUN iajs-3789 145 8 3.11 3.11 NUM iajs-3789 145 9 for for ADP iajs-3789 145 10 a a DET iajs-3789 145 11 grill grill NOUN iajs-3789 145 12 π“₯-space π“₯-space PROPN iajs-3789 145 13 ( ( PUNCT iajs-3789 145 14 𝑋 𝑋 PROPN iajs-3789 145 15 , , PUNCT iajs-3789 145 16 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 145 17 , , PUNCT iajs-3789 145 18 𝕾 𝕾 PROPN iajs-3789 145 19 ) ) PUNCT iajs-3789 145 20 . . PUNCT iajs-3789 146 1 if if SCONJ iajs-3789 146 2 𝒩 𝒩 PROPN iajs-3789 146 3 ∈ ∈ PROPN iajs-3789 146 4 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 146 5 , , PUNCT iajs-3789 146 6 then then ADV iajs-3789 146 7 𝒩 𝒩 PROPN iajs-3789 146 8 ∩ ∩ NOUN iajs-3789 146 9 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 146 10 ) ) PUNCT iajs-3789 146 11 = = SYM iajs-3789 146 12 𝒩 𝒩 PROPN iajs-3789 146 13 ∩ ∩ NOUN iajs-3789 146 14 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 146 15 ∩ ∩ PROPN iajs-3789 146 16 Ι± Ι± X iajs-3789 146 17 ) ) PUNCT iajs-3789 146 18 for for ADP iajs-3789 146 19 each each DET iajs-3789 146 20 Ι± Ι± PROPN iajs-3789 146 21 βŠ† βŠ† NUM iajs-3789 146 22 x. x. NOUN iajs-3789 146 23 proof proof NOUN iajs-3789 146 24 : : PUNCT iajs-3789 146 25 let let VERB iajs-3789 146 26 𝒩 𝒩 PROPN iajs-3789 146 27 ∈ ∈ PROPN iajs-3789 146 28 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 146 29 , , PUNCT iajs-3789 146 30 we we PRON iajs-3789 146 31 know know VERB iajs-3789 146 32 that that SCONJ iajs-3789 146 33 𝒩 𝒩 PROPN iajs-3789 146 34 ∩ ∩ NOUN iajs-3789 146 35 Ι± Ι± ADP iajs-3789 146 36 βŠ† βŠ† NUM iajs-3789 146 37 Ι± Ι± NOUN iajs-3789 146 38 , , PUNCT iajs-3789 146 39 it it PRON iajs-3789 146 40 follows follow VERB iajs-3789 146 41 from from ADP iajs-3789 146 42 theorem theorem ADJ iajs-3789 146 43 3.3(1 3.3(1 NUM iajs-3789 146 44 ) ) PUNCT iajs-3789 146 45 that that PRON iajs-3789 146 46 υ𝕾(𝒩 υ𝕾(𝒩 VERB iajs-3789 146 47 ∩ ∩ PROPN iajs-3789 146 48 Ι± Ι± X iajs-3789 146 49 ) ) PUNCT iajs-3789 146 50 βŠ† βŠ† NUM iajs-3789 146 51 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 146 52 ) ) PUNCT iajs-3789 146 53 , , PUNCT iajs-3789 146 54 then then ADV iajs-3789 146 55 𝒩 𝒩 PROPN iajs-3789 146 56 ∩ ∩ NOUN iajs-3789 146 57 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 146 58 ∩ ∩ PROPN iajs-3789 146 59 Ι± Ι± PROPN iajs-3789 146 60 ) ) PUNCT iajs-3789 146 61 βŠ† βŠ† NUM iajs-3789 146 62 𝒩 𝒩 PROPN iajs-3789 146 63 ∩ ∩ NOUN iajs-3789 146 64 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 146 65 ) ) PUNCT iajs-3789 146 66 . . PUNCT iajs-3789 147 1 on on ADP iajs-3789 147 2 the the DET iajs-3789 147 3 other other ADJ iajs-3789 147 4 hand hand NOUN iajs-3789 147 5 , , PUNCT iajs-3789 147 6 let let VERB iajs-3789 147 7 x x PUNCT iajs-3789 147 8 ∈ ∈ PROPN iajs-3789 147 9 𝒩 𝒩 PROPN iajs-3789 147 10 ∩ ∩ NOUN iajs-3789 147 11 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 147 12 ) ) PUNCT iajs-3789 147 13 , , PUNCT iajs-3789 147 14 then then ADV iajs-3789 147 15 x x SYM iajs-3789 147 16 ∈ ∈ PROPN iajs-3789 147 17 𝒩 𝒩 PROPN iajs-3789 147 18 ∧ ∧ PROPN iajs-3789 147 19 x x SYM iajs-3789 147 20 ∈ ∈ PROPN iajs-3789 147 21 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 147 22 ) ) PUNCT iajs-3789 147 23 . . PUNCT iajs-3789 148 1 for for ADP iajs-3789 148 2 each each DET iajs-3789 148 3 𝑆 𝑆 PROPN iajs-3789 148 4 ∈ ∈ PROPN iajs-3789 148 5 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 148 6 βˆ‹ βˆ‹ NOUN iajs-3789 148 7 x x SYM iajs-3789 148 8 ∈ ∈ PROPN iajs-3789 148 9 𝑆 𝑆 PROPN iajs-3789 148 10 , , PUNCT iajs-3789 148 11 we we PRON iajs-3789 148 12 have have VERB iajs-3789 148 13 x x PART iajs-3789 148 14 ∈ ∈ PROPN iajs-3789 148 15 𝒩 𝒩 PROPN iajs-3789 148 16 ∩ ∩ NOUN iajs-3789 148 17 𝑆 𝑆 PROPN iajs-3789 148 18 ∈ ∈ PROPN iajs-3789 148 19 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 148 20 , , PUNCT iajs-3789 148 21 but but CCONJ iajs-3789 148 22 x x X iajs-3789 148 23 ∈ ∈ PROPN iajs-3789 148 24 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 148 25 ) ) PUNCT iajs-3789 148 26 , , PUNCT iajs-3789 148 27 then then ADV iajs-3789 148 28 ( ( PUNCT iajs-3789 148 29 𝒩 𝒩 PROPN iajs-3789 148 30 ∩ ∩ ADJ iajs-3789 148 31 𝑆 𝑆 PROPN iajs-3789 148 32 ) ) PUNCT iajs-3789 148 33 ∩ ∩ NOUN iajs-3789 148 34 Ι± Ι± PROPN iajs-3789 148 35 ∈ ∈ PROPN iajs-3789 148 36 𝕾 𝕾 NOUN iajs-3789 148 37 , , PUNCT iajs-3789 148 38 that that ADV iajs-3789 148 39 is is ADV iajs-3789 148 40 , , PUNCT iajs-3789 148 41 ( ( PUNCT iajs-3789 148 42 𝒩 𝒩 PROPN iajs-3789 148 43 ∩ ∩ ADJ iajs-3789 148 44 Ι± Ι± NOUN iajs-3789 148 45 ) ) PUNCT iajs-3789 148 46 ∩ ∩ NOUN iajs-3789 148 47 𝑆 𝑆 PROPN iajs-3789 148 48 ∈ ∈ PROPN iajs-3789 148 49 𝕾 𝕾 NOUN iajs-3789 148 50 , , PUNCT iajs-3789 148 51 therefor therefor ADP iajs-3789 148 52 x x SYM iajs-3789 148 53 ∈ ∈ PROPN iajs-3789 148 54 υ𝕾(𝒩 υ𝕾(𝒩 NUM iajs-3789 148 55 ∩ ∩ PROPN iajs-3789 148 56 Ι± Ι± X iajs-3789 148 57 ) ) PUNCT iajs-3789 148 58 and and CCONJ iajs-3789 148 59 so so ADV iajs-3789 148 60 x x SYM iajs-3789 148 61 ∈ ∈ PROPN iajs-3789 148 62 𝒩 𝒩 PROPN iajs-3789 148 63 ∩ ∩ NOUN iajs-3789 148 64 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 148 65 ∩ ∩ PROPN iajs-3789 148 66 Ι± Ι± PROPN iajs-3789 148 67 ) ) PUNCT iajs-3789 148 68 , , PUNCT iajs-3789 148 69 which which PRON iajs-3789 148 70 implies imply VERB iajs-3789 148 71 that that SCONJ iajs-3789 148 72 𝒩 𝒩 PROPN iajs-3789 148 73 ∩ ∩ NOUN iajs-3789 148 74 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 148 75 ) ) PUNCT iajs-3789 148 76 βŠ† βŠ† NUM iajs-3789 148 77 𝒩 𝒩 PROPN iajs-3789 148 78 ∩ ∩ NOUN iajs-3789 148 79 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 148 80 ∩ ∩ PROPN iajs-3789 148 81 Ι± Ι± PROPN iajs-3789 148 82 ) ) PUNCT iajs-3789 148 83 . . PUNCT iajs-3789 149 1 thus thus ADV iajs-3789 149 2 𝒩 𝒩 PROPN iajs-3789 149 3 ∩ ∩ NOUN iajs-3789 149 4 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 149 5 ) ) PUNCT iajs-3789 149 6 = = SYM iajs-3789 149 7 𝒩 𝒩 PROPN iajs-3789 149 8 ∩ ∩ NOUN iajs-3789 149 9 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 149 10 ∩ ∩ PROPN iajs-3789 149 11 Ι± Ι± PROPN iajs-3789 149 12 ) ) PUNCT iajs-3789 149 13 . . PUNCT iajs-3789 150 1 corollary corollary ADJ iajs-3789 150 2 3.12 3.12 NUM iajs-3789 150 3 from from ADP iajs-3789 150 4 any any DET iajs-3789 150 5 ( ( PUNCT iajs-3789 150 6 𝑋 𝑋 PROPN iajs-3789 150 7 , , PUNCT iajs-3789 150 8 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 150 9 , , PUNCT iajs-3789 150 10 𝕾 𝕾 NOUN iajs-3789 150 11 ) ) PUNCT iajs-3789 150 12 grill grill NOUN iajs-3789 150 13 π“₯-space π“₯-space PROPN iajs-3789 150 14 . . PUNCT iajs-3789 151 1 if if SCONJ iajs-3789 151 2 𝒱𝑂𝑋 𝒱𝑂𝑋 PRON iajs-3789 151 3 βˆ’ βˆ’ NOUN iajs-3789 151 4 { { PUNCT iajs-3789 151 5 βˆ… βˆ… NOUN iajs-3789 151 6 } } PUNCT iajs-3789 151 7 βŠ† βŠ† NUM iajs-3789 151 8 𝕾 𝕾 NOUN iajs-3789 151 9 , , PUNCT iajs-3789 151 10 implies imply VERB iajs-3789 151 11 𝒩 𝒩 PROPN iajs-3789 151 12 βŠ† βŠ† NUM iajs-3789 151 13 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 151 14 ) ) PUNCT iajs-3789 151 15 for for ADP iajs-3789 151 16 each each DET iajs-3789 151 17 𝒩 𝒩 PROPN iajs-3789 151 18 ∈ ∈ PROPN iajs-3789 151 19 𝒱𝑂𝑋. 𝒱𝑂𝑋. X iajs-3789 151 20 proof proof NOUN iajs-3789 151 21 : : PUNCT iajs-3789 151 22 let let VERB iajs-3789 151 23 𝒩 𝒩 PROPN iajs-3789 151 24 ∈ ∈ PROPN iajs-3789 151 25 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 151 26 , , PUNCT iajs-3789 151 27 if if SCONJ iajs-3789 151 28 𝒩 𝒩 PROPN iajs-3789 151 29 = = PUNCT iajs-3789 151 30 βˆ… βˆ… NOUN iajs-3789 151 31 , , PUNCT iajs-3789 151 32 then then ADV iajs-3789 151 33 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 151 34 ) ) PUNCT iajs-3789 151 35 = = PUNCT iajs-3789 151 36 βˆ…. βˆ…. NOUN iajs-3789 151 37 if if SCONJ iajs-3789 151 38 𝒱𝑂𝑋 𝒱𝑂𝑋 PRON iajs-3789 151 39 βˆ’ βˆ’ NOUN iajs-3789 151 40 { { PUNCT iajs-3789 151 41 βˆ… βˆ… NOUN iajs-3789 151 42 } } PUNCT iajs-3789 151 43 βŠ† βŠ† NUM iajs-3789 151 44 𝕾 𝕾 NOUN iajs-3789 151 45 , , PUNCT iajs-3789 151 46 for for ADP iajs-3789 151 47 each each DET iajs-3789 151 48 𝒩 𝒩 PROPN iajs-3789 151 49 ∈ ∈ PROPN iajs-3789 151 50 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 151 51 , , PUNCT iajs-3789 151 52 we we PRON iajs-3789 151 53 have have VERB iajs-3789 151 54 from from ADP iajs-3789 151 55 corollary corollary ADJ iajs-3789 151 56 3.11 3.11 NUM iajs-3789 151 57 , , PUNCT iajs-3789 151 58 𝒩 𝒩 PROPN iajs-3789 151 59 ∩ ∩ NOUN iajs-3789 151 60 υ𝕾(𝑋 υ𝕾(𝑋 ADV iajs-3789 151 61 ) ) PUNCT iajs-3789 152 1 = = SYM iajs-3789 152 2 𝒩 𝒩 PROPN iajs-3789 152 3 ∩ ∩ NOUN iajs-3789 152 4 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 152 5 ∩ ∩ PROPN iajs-3789 152 6 𝑋 𝑋 PROPN iajs-3789 152 7 ) ) PUNCT iajs-3789 152 8 , , PUNCT iajs-3789 152 9 but but CCONJ iajs-3789 152 10 υ𝕾(𝑋 υ𝕾(𝑋 X iajs-3789 152 11 ) ) PUNCT iajs-3789 152 12 = = SYM iajs-3789 152 13 𝑋 𝑋 PROPN iajs-3789 152 14 , , PUNCT iajs-3789 152 15 it it PRON iajs-3789 152 16 follows follow VERB iajs-3789 152 17 that that SCONJ iajs-3789 152 18 𝒩 𝒩 PROPN iajs-3789 152 19 ∩ ∩ NOUN iajs-3789 152 20 x x ADP iajs-3789 152 21 = = SYM iajs-3789 152 22 𝒩 𝒩 PROPN iajs-3789 152 23 ∩ ∩ NOUN iajs-3789 152 24 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 152 25 ) ) PUNCT iajs-3789 152 26 and and CCONJ iajs-3789 152 27 so so ADV iajs-3789 152 28 𝒩 𝒩 PROPN iajs-3789 152 29 = = SYM iajs-3789 152 30 𝒩 𝒩 PROPN iajs-3789 152 31 ∩ ∩ NOUN iajs-3789 152 32 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 152 33 ) ) PUNCT iajs-3789 152 34 , , PUNCT iajs-3789 152 35 that that PRON iajs-3789 152 36 means mean VERB iajs-3789 152 37 𝒩 𝒩 PROPN iajs-3789 152 38 βŠ† βŠ† NUM iajs-3789 152 39 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 152 40 ) ) PUNCT iajs-3789 152 41 . . PUNCT iajs-3789 153 1 corollary corollary ADJ iajs-3789 153 2 3.13 3.13 NUM iajs-3789 153 3 for for ADP iajs-3789 153 4 a a DET iajs-3789 153 5 grill grill NOUN iajs-3789 153 6 π“₯-space π“₯-space PROPN iajs-3789 153 7 ( ( PUNCT iajs-3789 153 8 𝑋 𝑋 PROPN iajs-3789 153 9 , , PUNCT iajs-3789 153 10 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 153 11 , , PUNCT iajs-3789 153 12 𝕾 𝕾 PROPN iajs-3789 153 13 ) ) PUNCT iajs-3789 153 14 . . PUNCT iajs-3789 154 1 if if SCONJ iajs-3789 154 2 𝒩 𝒩 PROPN iajs-3789 154 3 ∈ ∈ PROPN iajs-3789 154 4 𝒱𝑂𝑋 𝒱𝑂𝑋 NOUN iajs-3789 154 5 and and CCONJ iajs-3789 154 6 Ι± Ι± ADP iajs-3789 154 7 βŠ† βŠ† NUM iajs-3789 154 8 x x NOUN iajs-3789 154 9 , , PUNCT iajs-3789 154 10 then then ADV iajs-3789 154 11 𝒩⋂𝒱𝕾 𝒩⋂𝒱𝕾 VERB iajs-3789 154 12 βˆ’ βˆ’ NOUN iajs-3789 154 13 𝑐𝑙(Ι± 𝑐𝑙(Ι± PUNCT iajs-3789 154 14 ) ) PUNCT iajs-3789 154 15 βŠ† βŠ† NUM iajs-3789 154 16 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 154 17 βˆ’ βˆ’ PROPN iajs-3789 154 18 𝑐𝑙(𝒩 𝑐𝑙(𝒩 PROPN iajs-3789 154 19 ∩ ∩ PROPN iajs-3789 154 20 Ι± Ι± NOUN iajs-3789 154 21 ) ) PUNCT iajs-3789 154 22 . . PUNCT iajs-3789 155 1 proof proof NOUN iajs-3789 155 2 : : PUNCT iajs-3789 155 3 𝒩⋂𝒱𝕾 𝒩⋂𝒱𝕾 NOUN iajs-3789 155 4 βˆ’ βˆ’ NOUN iajs-3789 155 5 𝑐𝑙(Ι± 𝑐𝑙(Ι± PUNCT iajs-3789 155 6 ) ) PUNCT iajs-3789 155 7 = = SYM iajs-3789 155 8 𝒩⋂(ɱ⋃υ𝕾(Ι± 𝒩⋂(ɱ⋃υ𝕾(Ι± X iajs-3789 155 9 ) ) PUNCT iajs-3789 155 10 ) ) PUNCT iajs-3789 155 11 ( ( PUNCT iajs-3789 155 12 by by ADP iajs-3789 155 13 definition definition NOUN iajs-3789 155 14 3.6 3.6 NUM iajs-3789 155 15 ) ) PUNCT iajs-3789 155 16 = = PUNCT iajs-3789 155 17 ( ( PUNCT iajs-3789 155 18 𝒩 𝒩 PROPN iajs-3789 155 19 ∩ ∩ ADJ iajs-3789 155 20 Ι± Ι± NOUN iajs-3789 155 21 ) ) PUNCT iajs-3789 155 22 βˆͺ βˆͺ NOUN iajs-3789 155 23 ( ( PUNCT iajs-3789 155 24 𝒩 𝒩 PROPN iajs-3789 155 25 ∩ ∩ NOUN iajs-3789 155 26 υ𝕾(Ι± υ𝕾(Ι± NUM iajs-3789 155 27 ) ) PUNCT iajs-3789 155 28 ) ) PUNCT iajs-3789 156 1 ( ( PUNCT iajs-3789 156 2 distribution distribution NOUN iajs-3789 156 3 of of ADP iajs-3789 156 4 the the DET iajs-3789 156 5 intersection intersection NOUN iajs-3789 156 6 on on ADP iajs-3789 156 7 the the DET iajs-3789 156 8 union union NOUN iajs-3789 156 9 ) ) PUNCT iajs-3789 157 1 = = PUNCT iajs-3789 157 2 ( ( PUNCT iajs-3789 157 3 𝒩 𝒩 PROPN iajs-3789 157 4 ∩ ∩ ADJ iajs-3789 157 5 Ι± Ι± NOUN iajs-3789 157 6 ) ) PUNCT iajs-3789 157 7 βˆͺ βˆͺ NOUN iajs-3789 157 8 ( ( PUNCT iajs-3789 157 9 𝒩 𝒩 PROPN iajs-3789 157 10 ∩ ∩ NOUN iajs-3789 157 11 υ𝕾(𝒩 υ𝕾(𝒩 NOUN iajs-3789 157 12 ∩ ∩ PROPN iajs-3789 157 13 Ι± Ι± PROPN iajs-3789 157 14 ) ) PUNCT iajs-3789 157 15 ) ) PUNCT iajs-3789 157 16 ( ( PUNCT iajs-3789 157 17 by by ADP iajs-3789 157 18 corollary corollary ADJ iajs-3789 157 19 3.11 3.11 NUM iajs-3789 157 20 ) ) PUNCT iajs-3789 157 21 βŠ† βŠ† NUM iajs-3789 157 22 ( ( PUNCT iajs-3789 157 23 𝒩 𝒩 PROPN iajs-3789 157 24 ∩ ∩ ADJ iajs-3789 157 25 Ι± Ι± NOUN iajs-3789 157 26 ) ) PUNCT iajs-3789 157 27 βˆͺ βˆͺ NOUN iajs-3789 157 28 υ𝕾(𝒩 υ𝕾(𝒩 X iajs-3789 157 29 ∩ ∩ PROPN iajs-3789 157 30 Ι± Ι± ADJ iajs-3789 157 31 ) ) PUNCT iajs-3789 157 32 = = SYM iajs-3789 157 33 𝒱𝕾 𝒱𝕾 PROPN iajs-3789 157 34 βˆ’ βˆ’ PROPN iajs-3789 157 35 𝑐𝑙(𝒩 𝑐𝑙(𝒩 PROPN iajs-3789 157 36 ∩ ∩ PROPN iajs-3789 157 37 Ι± Ι± X iajs-3789 157 38 ) ) PUNCT iajs-3789 157 39 ( ( PUNCT iajs-3789 157 40 by by ADP iajs-3789 157 41 definition definition NOUN iajs-3789 157 42 3.6 3.6 NUM iajs-3789 157 43 ) ) PUNCT iajs-3789 157 44 . . PUNCT iajs-3789 158 1 definition definition NOUN iajs-3789 158 2 3.14 3.14 NUM iajs-3789 158 3 for for ADP iajs-3789 158 4 any any DET iajs-3789 158 5 ( ( PUNCT iajs-3789 158 6 𝑋 𝑋 NOUN iajs-3789 158 7 , , PUNCT iajs-3789 158 8 𝒱𝑂x 𝒱𝑂x NOUN iajs-3789 158 9 , , PUNCT iajs-3789 158 10 𝕾 𝕾 NOUN iajs-3789 158 11 ) ) PUNCT iajs-3789 158 12 , , PUNCT iajs-3789 158 13 and and CCONJ iajs-3789 158 14 Ι± Ι± ADJ iajs-3789 158 15 βŠ†x βŠ†x NOUN iajs-3789 158 16 ; ; PUNCT iajs-3789 158 17 i. i. PROPN iajs-3789 158 18 Ι± Ι± PROPN iajs-3789 158 19 is be AUX iajs-3789 158 20 named name VERB iajs-3789 158 21 π’±π•Ύπ‘π‘Ÿπ‘’π‘œπ‘π‘’π‘› π’±π•Ύπ‘π‘Ÿπ‘’π‘œπ‘π‘’π‘› PROPN iajs-3789 158 22 set set VERB iajs-3789 158 23 if if SCONJ iajs-3789 158 24 Ι± Ι± PROPN iajs-3789 158 25 βŠ† βŠ† NUM iajs-3789 158 26 𝒱𝕾int 𝒱𝕾int PROPN iajs-3789 158 27 𝒱𝕾cl 𝒱𝕾cl PROPN iajs-3789 158 28 ( ( PUNCT iajs-3789 158 29 Ι± Ι± PROPN iajs-3789 158 30 ) ) PUNCT iajs-3789 158 31 . . PUNCT iajs-3789 159 1 ii ii PROPN iajs-3789 159 2 . . PUNCT iajs-3789 160 1 Ι± Ι± PROPN iajs-3789 160 2 is be AUX iajs-3789 160 3 named name VERB iajs-3789 160 4 π’±π•Ύπ‘ π‘’π‘šπ‘–π‘œπ‘π‘’π‘› π’±π•Ύπ‘ π‘’π‘šπ‘–π‘œπ‘π‘’π‘› PROPN iajs-3789 160 5 set set VERB iajs-3789 160 6 if if SCONJ iajs-3789 160 7 Ι± Ι± PROPN iajs-3789 160 8 βŠ† βŠ† NUM iajs-3789 160 9 𝒱𝕾cl 𝒱𝕾cl PROPN iajs-3789 160 10 𝒱𝕾int 𝒱𝕾int PROPN iajs-3789 160 11 ( ( PUNCT iajs-3789 160 12 Ι± Ι± PROPN iajs-3789 160 13 ) ) PUNCT iajs-3789 160 14 . . PUNCT iajs-3789 161 1 iii iii X iajs-3789 161 2 . . PUNCT iajs-3789 162 1 Ι± Ι± PROPN iajs-3789 162 2 is be AUX iajs-3789 162 3 named name VERB iajs-3789 162 4 π’±π•Ύπ›Όπ‘œπ‘π‘’π‘› π’±π•Ύπ›Όπ‘œπ‘π‘’π‘› PROPN iajs-3789 162 5 set set VERB iajs-3789 162 6 if if SCONJ iajs-3789 162 7 Ι± Ι± PROPN iajs-3789 162 8 βŠ† βŠ† NUM iajs-3789 162 9 𝒱𝕾int 𝒱𝕾int PROPN iajs-3789 162 10 𝒱𝕾cl 𝒱𝕾cl PROPN iajs-3789 162 11 𝒱𝕾int 𝒱𝕾int PROPN iajs-3789 162 12 ( ( PUNCT iajs-3789 162 13 Ι± Ι± PROPN iajs-3789 162 14 ) ) PUNCT iajs-3789 162 15 . . PUNCT iajs-3789 163 1 iv iv X iajs-3789 163 2 . . PUNCT iajs-3789 164 1 when when SCONJ iajs-3789 164 2 the the DET iajs-3789 164 3 ( ( PUNCT iajs-3789 164 4 𝒱𝕾int 𝒱𝕾int PROPN iajs-3789 164 5 ( ( PUNCT iajs-3789 164 6 Ι± Ι± PROPN iajs-3789 164 7 ) ) PUNCT iajs-3789 164 8 , , PUNCT iajs-3789 164 9 𝒱𝕾cl 𝒱𝕾cl PROPN iajs-3789 164 10 ( ( PUNCT iajs-3789 164 11 Ι± Ι± PROPN iajs-3789 164 12 ) ) PUNCT iajs-3789 164 13 ) ) PUNCT iajs-3789 164 14 are be AUX iajs-3789 164 15 the the DET iajs-3789 164 16 ( ( PUNCT iajs-3789 164 17 interior interior ADJ iajs-3789 164 18 , , PUNCT iajs-3789 164 19 exterior exterior NOUN iajs-3789 164 20 ) ) PUNCT iajs-3789 164 21 of of ADP iajs-3789 164 22 Ι± Ι± PROPN iajs-3789 164 23 for for ADP iajs-3789 164 24 ( ( PUNCT iajs-3789 164 25 𝑋 𝑋 PROPN iajs-3789 164 26 , , PUNCT iajs-3789 164 27 𝒱𝑂𝑋 𝒱𝑂𝑋 PROPN iajs-3789 164 28 , , PUNCT iajs-3789 164 29 𝕾 𝕾 PROPN iajs-3789 164 30 ) ) PUNCT iajs-3789 164 31 , , PUNCT iajs-3789 164 32 resp resp NOUN iajs-3789 164 33 . . PUNCT iajs-3789 164 34 , , PUNCT iajs-3789 164 35 the the DET iajs-3789 164 36 following follow VERB iajs-3789 164 37 diagram diagram NOUN iajs-3789 164 38 show show VERB iajs-3789 164 39 the the DET iajs-3789 164 40 relationships relationship NOUN iajs-3789 164 41 among among ADP iajs-3789 164 42 the the DET iajs-3789 164 43 above above ADJ iajs-3789 164 44 notions notion NOUN iajs-3789 164 45 : : PUNCT iajs-3789 164 46 diagram diagram NOUN iajs-3789 164 47 -1 -1 PUNCT iajs-3789 164 48 π’±π•Ύπ›Όπ‘œπ‘π‘’π‘› π’±π•Ύπ›Όπ‘œπ‘π‘’π‘› PROPN iajs-3789 164 49 𝒔𝒆𝒕 𝒔𝒆𝒕 NOUN iajs-3789 165 1 π’±π•Ύπ‘π‘Ÿπ‘’π‘œπ‘π‘’π‘› π’±π•Ύπ‘π‘Ÿπ‘’π‘œπ‘π‘’π‘› PROPN iajs-3789 165 2 set set VERB iajs-3789 165 3 π’±π•Ύπ‘ π‘’π‘šπ‘–π‘œπ‘π‘’π‘› π’±π•Ύπ‘ π‘’π‘šπ‘–π‘œπ‘π‘’π‘› PROPN iajs-3789 165 4 ihjpas ihjpa VERB iajs-3789 165 5 . . PUNCT iajs-3789 166 1 37 37 NUM iajs-3789 166 2 ( ( PUNCT iajs-3789 166 3 2 2 NUM iajs-3789 166 4 ) ) PUNCT iajs-3789 166 5 2024 2024 NUM iajs-3789 166 6 438 438 NUM iajs-3789 166 7 4 4 NUM iajs-3789 166 8 . . PUNCT iajs-3789 166 9 conclusions conclusion NOUN iajs-3789 166 10 the the DET iajs-3789 166 11 main main ADJ iajs-3789 166 12 objective objective NOUN iajs-3789 166 13 of of ADP iajs-3789 166 14 this this DET iajs-3789 166 15 research research NOUN iajs-3789 166 16 is be AUX iajs-3789 166 17 to to PART iajs-3789 166 18 define define VERB iajs-3789 166 19 a a DET iajs-3789 166 20 new new ADJ iajs-3789 166 21 type type NOUN iajs-3789 166 22 of of ADP iajs-3789 166 23 topological topological ADJ iajs-3789 166 24 spaces space NOUN iajs-3789 166 25 and and CCONJ iajs-3789 166 26 to to PART iajs-3789 166 27 understand understand VERB iajs-3789 166 28 and and CCONJ iajs-3789 166 29 analyze analyze VERB iajs-3789 166 30 the the DET iajs-3789 166 31 geometric geometric ADJ iajs-3789 166 32 and and CCONJ iajs-3789 166 33 topological topological ADJ iajs-3789 166 34 properties property NOUN iajs-3789 166 35 of of ADP iajs-3789 166 36 these these DET iajs-3789 166 37 spaces space NOUN iajs-3789 166 38 in in ADP iajs-3789 166 39 such such DET iajs-3789 166 40 a a DET iajs-3789 166 41 way way NOUN iajs-3789 166 42 that that PRON iajs-3789 166 43 mathematicians mathematician NOUN iajs-3789 166 44 and and CCONJ iajs-3789 166 45 researchers researcher NOUN iajs-3789 166 46 in in ADP iajs-3789 166 47 the the DET iajs-3789 166 48 field field NOUN iajs-3789 166 49 of of ADP iajs-3789 166 50 topological topological ADJ iajs-3789 166 51 engineering engineering NOUN iajs-3789 166 52 can can AUX iajs-3789 166 53 carry carry VERB iajs-3789 166 54 out out ADP iajs-3789 166 55 deeper deeply ADV iajs-3789 166 56 and and CCONJ iajs-3789 166 57 more more ADV iajs-3789 166 58 effective effective ADJ iajs-3789 166 59 studies study NOUN iajs-3789 166 60 in in ADP iajs-3789 166 61 this this DET iajs-3789 166 62 field field NOUN iajs-3789 166 63 , , PUNCT iajs-3789 166 64 as as ADV iajs-3789 166 65 well well ADV iajs-3789 166 66 as as ADP iajs-3789 166 67 to to PART iajs-3789 166 68 classify classify VERB iajs-3789 166 69 spaces space NOUN iajs-3789 166 70 and and CCONJ iajs-3789 166 71 understand understand VERB iajs-3789 166 72 , , PUNCT iajs-3789 166 73 classify classify VERB iajs-3789 166 74 , , PUNCT iajs-3789 166 75 organize organize VERB iajs-3789 166 76 and and CCONJ iajs-3789 166 77 define define VERB iajs-3789 166 78 different different ADJ iajs-3789 166 79 types type NOUN iajs-3789 166 80 of of ADP iajs-3789 166 81 topological topological ADJ iajs-3789 166 82 spaces space NOUN iajs-3789 166 83 . . PUNCT iajs-3789 167 1 the the DET iajs-3789 167 2 differences difference NOUN iajs-3789 167 3 between between ADP iajs-3789 167 4 them they PRON iajs-3789 167 5 . . PUNCT iajs-3789 168 1 in in ADP iajs-3789 168 2 this this DET iajs-3789 168 3 way way NOUN iajs-3789 168 4 , , PUNCT iajs-3789 168 5 mathematicians mathematician NOUN iajs-3789 168 6 can can AUX iajs-3789 168 7 understand understand VERB iajs-3789 168 8 how how SCONJ iajs-3789 168 9 the the DET iajs-3789 168 10 different different ADJ iajs-3789 168 11 properties property NOUN iajs-3789 168 12 and and CCONJ iajs-3789 168 13 relationships relationship NOUN iajs-3789 168 14 between between ADP iajs-3789 168 15 spaces space NOUN iajs-3789 168 16 are be AUX iajs-3789 168 17 interwoven interweave VERB iajs-3789 168 18 . . PUNCT iajs-3789 169 1 in in ADP iajs-3789 169 2 addition addition NOUN iajs-3789 169 3 to to ADP iajs-3789 169 4 the the DET iajs-3789 169 5 applications application NOUN iajs-3789 169 6 of of ADP iajs-3789 169 7 space space NOUN iajs-3789 169 8 studied study VERB iajs-3789 169 9 in in ADP iajs-3789 169 10 mathematics mathematic NOUN iajs-3789 169 11 and and CCONJ iajs-3789 169 12 other other ADJ iajs-3789 169 13 sciences science NOUN iajs-3789 169 14 , , PUNCT iajs-3789 169 15 it it PRON iajs-3789 169 16 also also ADV iajs-3789 169 17 contributes contribute VERB iajs-3789 169 18 to to ADP iajs-3789 169 19 the the DET iajs-3789 169 20 development development NOUN iajs-3789 169 21 of of ADP iajs-3789 169 22 related related ADJ iajs-3789 169 23 theories theory NOUN iajs-3789 169 24 and and CCONJ iajs-3789 169 25 technologies technology NOUN iajs-3789 169 26 by by ADP iajs-3789 169 27 finding find VERB iajs-3789 169 28 applications application NOUN iajs-3789 169 29 in in ADP iajs-3789 169 30 other other ADJ iajs-3789 169 31 areas area NOUN iajs-3789 169 32 of of ADP iajs-3789 169 33 mathematics mathematic NOUN iajs-3789 169 34 and and CCONJ iajs-3789 169 35 science science NOUN iajs-3789 169 36 , , PUNCT iajs-3789 169 37 which which PRON iajs-3789 169 38 may may AUX iajs-3789 169 39 have have VERB iajs-3789 169 40 implications implication NOUN iajs-3789 169 41 for for ADP iajs-3789 169 42 physics physics NOUN iajs-3789 169 43 , , PUNCT iajs-3789 169 44 computer computer NOUN iajs-3789 169 45 science science NOUN iajs-3789 169 46 and and CCONJ iajs-3789 169 47 data datum NOUN iajs-3789 169 48 science science NOUN iajs-3789 169 49 , , PUNCT iajs-3789 169 50 for for ADP iajs-3789 169 51 example example NOUN iajs-3789 169 52 . . PUNCT iajs-3789 170 1 acknowledgment acknowledgment NOUN iajs-3789 170 2 the the DET iajs-3789 170 3 authors author NOUN iajs-3789 170 4 greatly greatly ADV iajs-3789 170 5 appreciate appreciate VERB iajs-3789 170 6 the the DET iajs-3789 170 7 referees referee NOUN iajs-3789 170 8 for for ADP iajs-3789 170 9 their their PRON iajs-3789 170 10 comments comment NOUN iajs-3789 170 11 and and CCONJ iajs-3789 170 12 suggestions suggestion NOUN iajs-3789 170 13 for for ADP iajs-3789 170 14 improving improve VERB iajs-3789 170 15 the the DET iajs-3789 170 16 paper paper NOUN iajs-3789 170 17 . . PUNCT iajs-3789 171 1 conflict conflict NOUN iajs-3789 171 2 of of ADP iajs-3789 171 3 interest interest NOUN iajs-3789 171 4 β€œ " PUNCT iajs-3789 171 5 conflict conflict NOUN iajs-3789 171 6 of of ADP iajs-3789 171 7 interest interest NOUN iajs-3789 171 8 : : PUNCT iajs-3789 171 9 the the DET iajs-3789 171 10 authors author NOUN iajs-3789 171 11 declare declare VERB iajs-3789 171 12 that that SCONJ iajs-3789 171 13 they they PRON iajs-3789 171 14 have have VERB iajs-3789 171 15 no no DET iajs-3789 171 16 conflicts conflict NOUN iajs-3789 171 17 of of ADP iajs-3789 171 18 interest interest NOUN iajs-3789 171 19 . . PUNCT iajs-3789 171 20 ” " PUNCT iajs-3789 172 1 funding funding NOUN iajs-3789 172 2 : : PUNCT iajs-3789 172 3 none none NOUN iajs-3789 172 4 . . PUNCT iajs-3789 173 1 references reference NOUN iajs-3789 173 2 1 1 NUM iajs-3789 173 3 . . PUNCT iajs-3789 173 4 choquet choquet PROPN iajs-3789 173 5 , , PUNCT iajs-3789 173 6 g. g. 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