id sid tid token lemma pos cana-5898 1 1 communications communication NOUN cana-5898 1 2 on on ADP cana-5898 1 3 applied apply VERB cana-5898 1 4 nonlinear nonlinear ADJ cana-5898 1 5 analysis analysis NOUN cana-5898 1 6 issn issn NOUN cana-5898 1 7 : : PUNCT cana-5898 1 8 1074 1074 NUM cana-5898 1 9 - - PUNCT cana-5898 1 10 133x 133x NUM cana-5898 1 11 vol vol NOUN cana-5898 1 12 31 31 NUM cana-5898 1 13 no no NOUN cana-5898 1 14 . . PUNCT cana-5898 2 1 7s 7 NOUN cana-5898 2 2 ( ( PUNCT cana-5898 2 3 2024 2024 NUM cana-5898 2 4 ) ) PUNCT cana-5898 2 5 790 790 NUM cana-5898 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 2 7 on on ADP cana-5898 2 8 g𝓖-closed g𝓖-close VERB cana-5898 2 9 sets set NOUN cana-5898 2 10 in in ADP cana-5898 2 11 grill grill ADJ cana-5898 2 12 topological topological ADJ cana-5898 2 13 spaces space NOUN cana-5898 2 14 r. r. PROPN cana-5898 2 15 anbarasan1,2 anbarasan1,2 PROPN cana-5898 2 16 and and CCONJ cana-5898 2 17 m. m. NOUN cana-5898 2 18 anitha3 anitha3 PROPN cana-5898 2 19 1 1 NUM cana-5898 2 20 research research NOUN cana-5898 2 21 scholar scholar NOUN cana-5898 2 22 ( ( PUNCT cana-5898 2 23 17231172091001 17231172091001 NUM cana-5898 2 24 ) ) PUNCT cana-5898 2 25 affiliated affiliate VERB cana-5898 2 26 to to ADP cana-5898 2 27 manonmaniam manonmaniam PROPN cana-5898 2 28 sundaranar sundaranar PROPN cana-5898 2 29 university university PROPN cana-5898 2 30 , , PUNCT cana-5898 2 31 rani rani PROPN cana-5898 2 32 anna anna PROPN cana-5898 2 33 government government PROPN cana-5898 2 34 college college PROPN cana-5898 2 35 for for ADP cana-5898 2 36 women woman NOUN cana-5898 2 37 , , PUNCT cana-5898 2 38 tirunelveli tirunelveli PROPN cana-5898 2 39 , , PUNCT cana-5898 2 40 india india PROPN cana-5898 2 41 2 2 NUM cana-5898 2 42 assistant assistant NOUN cana-5898 2 43 professor professor NOUN cana-5898 2 44 , , PUNCT cana-5898 2 45 department department NOUN cana-5898 2 46 of of ADP cana-5898 2 47 mathematics mathematics PROPN cana-5898 2 48 , , PUNCT cana-5898 2 49 psn psn PROPN cana-5898 2 50 college college PROPN cana-5898 2 51 of of ADP cana-5898 2 52 engineering engineering NOUN cana-5898 2 53 and and CCONJ cana-5898 2 54 technology technology NOUN cana-5898 2 55 , , PUNCT cana-5898 2 56 tirunelveli tirunelveli PROPN cana-5898 2 57 , , PUNCT cana-5898 2 58 india india PROPN cana-5898 2 59 . . PUNCT cana-5898 3 1 email email NOUN cana-5898 3 2 : : PUNCT cana-5898 3 3 anbu.arasan1988@gmail.com anbu.arasan1988@gmail.com X cana-5898 3 4 3 3 NUM cana-5898 3 5 associate associate NOUN cana-5898 3 6 professor professor NOUN cana-5898 3 7 , , PUNCT cana-5898 3 8 department department NOUN cana-5898 3 9 of of ADP cana-5898 3 10 mathematics mathematics PROPN cana-5898 3 11 rani rani PROPN cana-5898 3 12 anna anna PROPN cana-5898 3 13 government government PROPN cana-5898 3 14 college college PROPN cana-5898 3 15 for for ADP cana-5898 3 16 women woman NOUN cana-5898 3 17 , , PUNCT cana-5898 3 18 tirunelveli tirunelveli PROPN cana-5898 3 19 , , PUNCT cana-5898 3 20 india india PROPN cana-5898 3 21 email email NOUN cana-5898 3 22 : : PUNCT cana-5898 3 23 drmanitha10@gmail.com drmanitha10@gmail.com PROPN cana-5898 3 24 article article NOUN cana-5898 3 25 history history NOUN cana-5898 3 26 : : PUNCT cana-5898 3 27 received receive VERB cana-5898 3 28 : : PUNCT cana-5898 3 29 02 02 NUM cana-5898 3 30 - - SYM cana-5898 3 31 08 08 NUM cana-5898 3 32 - - PUNCT cana-5898 3 33 2024 2024 NUM cana-5898 3 34 revised revise VERB cana-5898 3 35 : : PUNCT cana-5898 3 36 25 25 NUM cana-5898 3 37 - - PUNCT cana-5898 3 38 09 09 NUM cana-5898 3 39 - - PUNCT cana-5898 3 40 2024 2024 NUM cana-5898 3 41 accepted accept VERB cana-5898 3 42 : : PUNCT cana-5898 3 43 20 20 NUM cana-5898 3 44 - - SYM cana-5898 3 45 10 10 NUM cana-5898 3 46 - - PUNCT cana-5898 3 47 2024 2024 NUM cana-5898 3 48 abstract abstract NOUN cana-5898 3 49 : : PUNCT cana-5898 3 50 in in ADP cana-5898 3 51 this this DET cana-5898 3 52 article article NOUN cana-5898 3 53 , , PUNCT cana-5898 3 54 we we PRON cana-5898 3 55 define define VERB cana-5898 3 56 a a DET cana-5898 3 57 new new ADJ cana-5898 3 58 class class NOUN cana-5898 3 59 of of ADP cana-5898 3 60 gg gg PROPN cana-5898 3 61 - - PUNCT cana-5898 3 62 closed close VERB cana-5898 3 63 sets set NOUN cana-5898 3 64 in in ADP cana-5898 3 65 a a DET cana-5898 3 66 grill grill ADJ cana-5898 3 67 topological topological ADJ cana-5898 3 68 space space NOUN cana-5898 3 69 and and CCONJ cana-5898 3 70 discuss discuss VERB cana-5898 3 71 the the DET cana-5898 3 72 characterizations characterization NOUN cana-5898 3 73 of of ADP cana-5898 3 74 gg gg NOUN cana-5898 3 75 - - PUNCT cana-5898 3 76 closed close VERB cana-5898 3 77 sets set NOUN cana-5898 3 78 and and CCONJ cana-5898 3 79 gg gg NOUN cana-5898 3 80 - - PUNCT cana-5898 3 81 open open ADJ cana-5898 3 82 sets set NOUN cana-5898 3 83 by by ADP cana-5898 3 84 using use VERB cana-5898 3 85 the the DET cana-5898 3 86 map map NOUN cana-5898 3 87 s. s. PROPN cana-5898 3 88 also also ADV cana-5898 3 89 analyze analyze VERB cana-5898 3 90 relationship relationship NOUN cana-5898 3 91 between between ADP cana-5898 3 92 the the DET cana-5898 3 93 gg gg NOUN cana-5898 3 94 - - PUNCT cana-5898 3 95 closed close VERB cana-5898 3 96 sets set NOUN cana-5898 3 97 and and CCONJ cana-5898 3 98 some some PRON cana-5898 3 99 of of ADP cana-5898 3 100 the the DET cana-5898 3 101 generalized generalize VERB cana-5898 3 102 closed closed ADJ cana-5898 3 103 sets set NOUN cana-5898 3 104 . . PUNCT cana-5898 4 1 keywords keyword NOUN cana-5898 4 2 : : PUNCT cana-5898 4 3 gg gg NOUN cana-5898 4 4 - - PUNCT cana-5898 4 5 closed closed ADJ cana-5898 4 6 , , PUNCT cana-5898 4 7 gg gg NOUN cana-5898 4 8 - - PUNCT cana-5898 4 9 open open ADJ cana-5898 4 10 , , PUNCT cana-5898 4 11 s s VERB cana-5898 4 12 - - PUNCT cana-5898 4 13 semiclosed semiclose VERB cana-5898 4 14 , , PUNCT cana-5898 4 15 s s NOUN cana-5898 4 16 - - PUNCT cana-5898 4 17 semi semi ADV cana-5898 4 18 - - ADJ cana-5898 4 19 dense dense ADJ cana-5898 4 20 . . PUNCT cana-5898 5 1 mathematical mathematical ADJ cana-5898 5 2 classification classification NOUN cana-5898 5 3 : : PUNCT cana-5898 5 4 54a05 54a05 NUM cana-5898 5 5 , , PUNCT cana-5898 5 6 54a10 54a10 NUM cana-5898 5 7 , , PUNCT cana-5898 5 8 54d10 54d10 NUM cana-5898 5 9 . . PUNCT cana-5898 6 1 1.introduction 1.introduction NUM cana-5898 6 2 levine[14 levine[14 PROPN cana-5898 6 3 , , PUNCT cana-5898 6 4 15 15 NUM cana-5898 6 5 ] ] PUNCT cana-5898 6 6 introduced introduce VERB cana-5898 6 7 the the DET cana-5898 6 8 concepts concept NOUN cana-5898 6 9 of of ADP cana-5898 6 10 semiopen semiopen ADJ cana-5898 6 11 sets set NOUN cana-5898 6 12 and and CCONJ cana-5898 6 13 generalized generalize VERB cana-5898 6 14 closed closed ADJ cana-5898 6 15 sets set NOUN cana-5898 6 16 in in ADP cana-5898 6 17 topological topological ADJ cana-5898 6 18 spaces space NOUN cana-5898 6 19 . . PUNCT cana-5898 7 1 crossly crossly ADV cana-5898 7 2 et et PROPN cana-5898 7 3 al.[9 al.[9 PROPN cana-5898 7 4 , , PUNCT cana-5898 7 5 10 10 NUM cana-5898 7 6 ] ] PUNCT cana-5898 7 7 described describe VERB cana-5898 7 8 the the DET cana-5898 7 9 concepts concept NOUN cana-5898 7 10 of of ADP cana-5898 7 11 semi semi ADJ cana-5898 7 12 - - ADJ cana-5898 7 13 closure closure NOUN cana-5898 7 14 and and CCONJ cana-5898 7 15 analysed analyse VERB cana-5898 7 16 the the DET cana-5898 7 17 semi semi ADJ cana-5898 7 18 - - ADJ cana-5898 7 19 topological topological ADJ cana-5898 7 20 properties property NOUN cana-5898 7 21 . . PUNCT cana-5898 8 1 chattopadhyay chattopadhyay PROPN cana-5898 8 2 et et PROPN cana-5898 8 3 al.[6 al.[6 PROPN cana-5898 8 4 , , PUNCT cana-5898 8 5 7 7 NUM cana-5898 8 6 ] ] PUNCT cana-5898 8 7 described describe VERB cana-5898 8 8 the the DET cana-5898 8 9 metropic metropic NOUN cana-5898 8 10 spaces space NOUN cana-5898 8 11 and and CCONJ cana-5898 8 12 created create VERB cana-5898 8 13 the the DET cana-5898 8 14 extensions extension NOUN cana-5898 8 15 of of ADP cana-5898 8 16 closure closure NOUN cana-5898 8 17 spaces space NOUN cana-5898 8 18 . . PUNCT cana-5898 9 1 in in ADP cana-5898 9 2 [ [ X cana-5898 9 3 2 2 NUM cana-5898 9 4 , , PUNCT cana-5898 9 5 5 5 NUM cana-5898 9 6 , , PUNCT cana-5898 9 7 13 13 NUM cana-5898 9 8 , , PUNCT cana-5898 9 9 24 24 NUM cana-5898 9 10 , , PUNCT cana-5898 9 11 25 25 NUM cana-5898 9 12 , , PUNCT cana-5898 9 13 26 26 NUM cana-5898 9 14 ] ] PUNCT cana-5898 9 15 , , PUNCT cana-5898 9 16 studied study VERB cana-5898 9 17 the the DET cana-5898 9 18 concepts concept NOUN cana-5898 9 19 of of ADP cana-5898 9 20 generalized generalized ADJ cana-5898 9 21 closed close VERB cana-5898 9 22 sets set NOUN cana-5898 9 23 through through ADP cana-5898 9 24 semiclosed semiclose VERB cana-5898 9 25 sets set NOUN cana-5898 9 26 . . PUNCT cana-5898 10 1 choquet[8 choquet[8 X cana-5898 10 2 ] ] PUNCT cana-5898 10 3 defined define VERB cana-5898 10 4 the the DET cana-5898 10 5 grill grill NOUN cana-5898 10 6 structure structure NOUN cana-5898 10 7 in in ADP cana-5898 10 8 topological topological ADJ cana-5898 10 9 spaces space NOUN cana-5898 10 10 . . PUNCT cana-5898 11 1 roy roy PROPN cana-5898 11 2 et et PROPN cana-5898 11 3 al.[20 al.[20 NOUN cana-5898 11 4 , , PUNCT cana-5898 11 5 21 21 NUM cana-5898 11 6 ] ] PUNCT cana-5898 11 7 developed develop VERB cana-5898 11 8 the the DET cana-5898 11 9 grill grill NOUN cana-5898 11 10 concepts concept NOUN cana-5898 11 11 and and CCONJ cana-5898 11 12 induced induce VERB cana-5898 11 13 τ𝒒 τ𝒒 ADJ cana-5898 11 14 topological topological ADJ cana-5898 11 15 space space NOUN cana-5898 11 16 . . PUNCT cana-5898 12 1 in in ADP cana-5898 12 2 [ [ X cana-5898 12 3 1 1 NUM cana-5898 12 4 , , PUNCT cana-5898 12 5 11 11 NUM cana-5898 12 6 , , PUNCT cana-5898 12 7 16 16 NUM cana-5898 12 8 , , PUNCT cana-5898 12 9 27 27 NUM cana-5898 12 10 ] ] PUNCT cana-5898 12 11 , , PUNCT cana-5898 12 12 initiated initiate VERB cana-5898 12 13 different different ADJ cana-5898 12 14 types type NOUN cana-5898 12 15 of of ADP cana-5898 12 16 grill grill NOUN cana-5898 12 17 sets set NOUN cana-5898 12 18 such such ADJ cana-5898 12 19 as as ADP cana-5898 12 20 𝒒-semiopen 𝒒-semiopen PROPN cana-5898 12 21 sets set NOUN cana-5898 12 22 , , PUNCT cana-5898 12 23 𝒒-open 𝒒-open NOUN cana-5898 12 24 sets set NOUN cana-5898 12 25 and and CCONJ cana-5898 12 26 studied study VERB cana-5898 12 27 the the DET cana-5898 12 28 decomposition decomposition NOUN cana-5898 12 29 of of ADP cana-5898 12 30 continuity continuity NOUN cana-5898 12 31 via via ADP cana-5898 12 32 grill grill NOUN cana-5898 12 33 . . PUNCT cana-5898 13 1 nasef[18 nasef[18 VERB cana-5898 13 2 ] ] PUNCT cana-5898 13 3 , , PUNCT cana-5898 13 4 introduced introduce VERB cana-5898 13 5 οͺs οͺs PROPN cana-5898 13 6 operator operator NOUN cana-5898 13 7 in in ADP cana-5898 13 8 grill grill ADJ cana-5898 13 9 topological topological ADJ cana-5898 13 10 space space NOUN cana-5898 13 11 via via ADP cana-5898 13 12 semiopen semiopen ADJ cana-5898 13 13 sets set NOUN cana-5898 13 14 and and CCONJ cana-5898 13 15 analyzed analyze VERB cana-5898 13 16 the the DET cana-5898 13 17 essential essential ADJ cana-5898 13 18 topological topological ADJ cana-5898 13 19 characterizations characterization NOUN cana-5898 13 20 . . PUNCT cana-5898 14 1 mandal[17 mandal[17 PROPN cana-5898 14 2 ] ] PUNCT cana-5898 14 3 , , PUNCT cana-5898 14 4 generalized generalize VERB cana-5898 14 5 the the DET cana-5898 14 6 closed close VERB cana-5898 14 7 sets set NOUN cana-5898 14 8 in in ADP cana-5898 14 9 grill grill ADJ cana-5898 14 10 topological topological ADJ cana-5898 14 11 space space NOUN cana-5898 14 12 and and CCONJ cana-5898 14 13 saravanakumar saravanakumar PROPN cana-5898 14 14 et et PROPN cana-5898 14 15 al.[22 al.[22 NOUN cana-5898 14 16 , , PUNCT cana-5898 14 17 23 23 NUM cana-5898 14 18 ] ] PUNCT cana-5898 14 19 defined define VERB cana-5898 14 20 𝒒sp 𝒒sp PROPN cana-5898 14 21 -open -open NOUN cana-5898 14 22 sets set NOUN cana-5898 14 23 and and CCONJ cana-5898 14 24 𝒒sΞ± 𝒒sΞ± PROPN cana-5898 14 25 -open -open NOUN cana-5898 14 26 sets set NOUN cana-5898 14 27 through through ADP cana-5898 14 28 semiopen semiopen ADJ cana-5898 14 29 sets set NOUN cana-5898 14 30 mailto:anbu.arasan1988@gmail.com mailto:anbu.arasan1988@gmail.com X cana-5898 14 31 mailto:drmanitha10@gmail.com mailto:drmanitha10@gmail.com PROPN cana-5898 14 32 communications communication NOUN cana-5898 14 33 on on ADP cana-5898 14 34 applied apply VERB cana-5898 14 35 nonlinear nonlinear ADJ cana-5898 14 36 analysis analysis NOUN cana-5898 14 37 issn issn NOUN cana-5898 14 38 : : PUNCT cana-5898 14 39 1074 1074 NUM cana-5898 14 40 - - PUNCT cana-5898 14 41 133x 133x NUM cana-5898 14 42 vol vol NOUN cana-5898 14 43 31 31 NUM cana-5898 14 44 no no NOUN cana-5898 14 45 . . PUNCT cana-5898 15 1 7s 7 NOUN cana-5898 15 2 ( ( PUNCT cana-5898 15 3 2024 2024 NUM cana-5898 15 4 ) ) PUNCT cana-5898 15 5 791 791 NUM cana-5898 15 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 15 7 and and CCONJ cana-5898 15 8 characterized characterize VERB cana-5898 15 9 some some DET cana-5898 15 10 topological topological ADJ cana-5898 15 11 structure structure NOUN cana-5898 15 12 . . PUNCT cana-5898 16 1 anbarasan anbarasan NOUN cana-5898 16 2 et et PROPN cana-5898 16 3 al.[3 al.[3 PROPN cana-5898 16 4 , , PUNCT cana-5898 16 5 4 4 NUM cana-5898 16 6 ] ] PUNCT cana-5898 16 7 studied study VERB cana-5898 16 8 generalized generalize VERB cana-5898 16 9 closed closed ADJ cana-5898 16 10 sets set NOUN cana-5898 16 11 concepts concept NOUN cana-5898 16 12 via via ADP cana-5898 16 13 grill grill NOUN cana-5898 16 14 in in ADP cana-5898 16 15 generalized generalized ADJ cana-5898 16 16 topological topological ADJ cana-5898 16 17 spaces space NOUN cana-5898 16 18 . . PUNCT cana-5898 17 1 in in ADP cana-5898 17 2 this this DET cana-5898 17 3 paper paper NOUN cana-5898 17 4 , , PUNCT cana-5898 17 5 we we PRON cana-5898 17 6 introduced introduce VERB cana-5898 17 7 new new ADJ cana-5898 17 8 grill grill NOUN cana-5898 17 9 closed close VERB cana-5898 17 10 sets set NOUN cana-5898 17 11 namely namely ADV cana-5898 17 12 , , PUNCT cana-5898 17 13 g𝒒-closed g𝒒-close VERB cana-5898 17 14 in in ADP cana-5898 17 15 grill grill ADJ cana-5898 17 16 topological topological ADJ cana-5898 17 17 spaces space NOUN cana-5898 17 18 . . PUNCT cana-5898 18 1 we we PRON cana-5898 18 2 characterized characterize VERB cana-5898 18 3 g𝒒-closed g𝒒-close VERB cana-5898 18 4 sets set NOUN cana-5898 18 5 and and CCONJ cana-5898 18 6 g𝒒-open g𝒒-open NOUN cana-5898 18 7 sets set NOUN cana-5898 18 8 in in ADP cana-5898 18 9 grill grill ADJ cana-5898 18 10 topological topological ADJ cana-5898 18 11 spaces space NOUN cana-5898 18 12 by by ADP cana-5898 18 13 use use VERB cana-5898 18 14 the the DET cana-5898 18 15 mapping mapping NOUN cana-5898 18 16 οͺs οͺs PROPN cana-5898 18 17 and and CCONJ cana-5898 18 18 investigated investigate VERB cana-5898 18 19 some some PRON cana-5898 18 20 of of ADP cana-5898 18 21 their their PRON cana-5898 18 22 properties property NOUN cana-5898 18 23 . . PUNCT cana-5898 19 1 we we PRON cana-5898 19 2 noticed notice VERB cana-5898 19 3 that that SCONJ cana-5898 19 4 the the DET cana-5898 19 5 idea idea NOUN cana-5898 19 6 of of ADP cana-5898 19 7 g𝒒-closed g𝒒-close VERB cana-5898 19 8 sets set NOUN cana-5898 19 9 is be AUX cana-5898 19 10 a a DET cana-5898 19 11 new new ADJ cana-5898 19 12 generalization generalization NOUN cana-5898 19 13 of of ADP cana-5898 19 14 𝒒g 𝒒g PROPN cana-5898 19 15 - - PUNCT cana-5898 19 16 closed close VERB cana-5898 19 17 sets set NOUN cana-5898 19 18 . . PUNCT cana-5898 20 1 also also ADV cana-5898 20 2 , , PUNCT cana-5898 20 3 we we PRON cana-5898 20 4 analyzed analyze VERB cana-5898 20 5 relationship relationship NOUN cana-5898 20 6 between between ADP cana-5898 20 7 this this DET cana-5898 20 8 g𝒒closed g𝒒closed PROPN cana-5898 20 9 sets set NOUN cana-5898 20 10 with with ADP cana-5898 20 11 existence existence NOUN cana-5898 20 12 generalized generalize VERB cana-5898 20 13 closed closed ADJ cana-5898 20 14 sets set NOUN cana-5898 20 15 such such ADJ cana-5898 20 16 as as ADP cana-5898 20 17 g g NOUN cana-5898 20 18 - - PUNCT cana-5898 20 19 closed closed ADJ cana-5898 20 20 , , PUNCT cana-5898 20 21 s*g s*g NOUN cana-5898 20 22 - - PUNCT cana-5898 20 23 closed closed ADJ cana-5898 20 24 , , PUNCT cana-5898 20 25 gs gs NOUN cana-5898 20 26 - - PUNCT cana-5898 20 27 closed closed ADJ cana-5898 20 28 , , PUNCT cana-5898 20 29 τ𝒒closed τ𝒒closed PROPN cana-5898 20 30 , , PUNCT cana-5898 20 31 𝒒g 𝒒g PROPN cana-5898 20 32 - - PUNCT cana-5898 20 33 closed close VERB cana-5898 20 34 etc etc X cana-5898 20 35 . . X cana-5898 21 1 2 2 X cana-5898 21 2 . . NUM cana-5898 21 3 preliminaries preliminary NOUN cana-5898 21 4 in in ADP cana-5898 21 5 a a DET cana-5898 21 6 topological topological ADJ cana-5898 21 7 space space NOUN cana-5898 21 8 x x NOUN cana-5898 21 9 , , PUNCT cana-5898 21 10 a a DET cana-5898 21 11 subset subset NOUN cana-5898 21 12 a a PRON cana-5898 21 13 of of ADP cana-5898 21 14 x x SYM cana-5898 21 15 is be AUX cana-5898 21 16 said say VERB cana-5898 21 17 to to PART cana-5898 21 18 be be AUX cana-5898 21 19 semiopen[14 semiopen[14 PROPN cana-5898 21 20 ] ] PUNCT cana-5898 21 21 ( ( PUNCT cana-5898 21 22 resp resp NOUN cana-5898 21 23 . . PUNCT cana-5898 22 1 -open[19 -open[19 PRON cana-5898 22 2 ] ] X cana-5898 22 3 , , PUNCT cana-5898 22 4 regular regular ADJ cana-5898 22 5 open[12 open[12 NOUN cana-5898 22 6 ] ] PUNCT cana-5898 22 7 ) ) PUNCT cana-5898 22 8 if if SCONJ cana-5898 22 9 a a DET cana-5898 22 10   PROPN cana-5898 22 11 cl(int(a cl(int(a PROPN cana-5898 22 12 ) ) PUNCT cana-5898 22 13 ) ) PUNCT cana-5898 23 1 ( ( PUNCT cana-5898 23 2 resp resp NOUN cana-5898 23 3 . . PUNCT cana-5898 24 1 a a DET cana-5898 24 2   PROPN cana-5898 24 3 int(cl(int(a int(cl(int(a PROPN cana-5898 24 4 ) ) PUNCT cana-5898 24 5 ) ) PUNCT cana-5898 24 6 ) ) PUNCT cana-5898 24 7 , , PUNCT cana-5898 24 8 a a DET cana-5898 24 9 = = X cana-5898 24 10 int(cl(a int(cl(a PROPN cana-5898 24 11 ) ) PUNCT cana-5898 24 12 ) ) PUNCT cana-5898 24 13 ) ) PUNCT cana-5898 24 14 . . PUNCT cana-5898 25 1 the the DET cana-5898 25 2 complement complement NOUN cana-5898 25 3 x x INTJ cana-5898 25 4 – – PUNCT cana-5898 25 5 a a PRON cana-5898 25 6 is be AUX cana-5898 25 7 called call VERB cana-5898 25 8 semiclosed semiclose VERB cana-5898 25 9 ( ( PUNCT cana-5898 25 10 resp resp NOUN cana-5898 25 11 . . PUNCT cana-5898 26 1 -closed -closed ADJ cana-5898 26 2 , , PUNCT cana-5898 26 3 regular regular ADJ cana-5898 26 4 closed closed ADJ cana-5898 26 5 ) ) PUNCT cana-5898 26 6 . . PUNCT cana-5898 27 1 for for ADP cana-5898 27 2 a a DET cana-5898 27 3 subset subset NOUN cana-5898 27 4 a a PRON cana-5898 27 5 of of ADP cana-5898 27 6 x x PRON cana-5898 27 7 , , PUNCT cana-5898 27 8 semiclosure semiclosure NOUN cana-5898 27 9 of of ADP cana-5898 27 10 a a DET cana-5898 27 11 defined define VERB cana-5898 27 12 as as ADP cana-5898 27 13 scl(a)[9 scl(a)[9 PROPN cana-5898 27 14 ] ] PUNCT cana-5898 27 15 = = PUNCT cana-5898 27 16 {f {f NOUN cana-5898 27 17   PROPN cana-5898 27 18 x x X cana-5898 27 19 : : PUNCT cana-5898 27 20 x x X cana-5898 27 21 – – PUNCT cana-5898 27 22 f f PROPN cana-5898 27 23 is be AUX cana-5898 27 24 semiopen semiopen ADJ cana-5898 27 25 and and CCONJ cana-5898 27 26 a a DET cana-5898 27 27   NOUN cana-5898 27 28 f f X cana-5898 27 29 } } PUNCT cana-5898 27 30 ; ; PUNCT cana-5898 27 31 semiinterior semiinterior NOUN cana-5898 27 32 of of ADP cana-5898 27 33 a a DET cana-5898 27 34 defined define VERB cana-5898 27 35 as as ADP cana-5898 27 36 sint(a)[9 sint(a)[9 NOUN cana-5898 27 37 ] ] PUNCT cana-5898 27 38 = = PUNCT cana-5898 28 1 οƒˆ{u οƒˆ{u NOUN cana-5898 28 2   PROPN cana-5898 28 3 x x X cana-5898 28 4 : : PUNCT cana-5898 28 5 u u NOUN cana-5898 28 6 is be AUX cana-5898 28 7 semiopen semiopen ADJ cana-5898 28 8 and and CCONJ cana-5898 28 9 a a DET cana-5898 28 10   PROPN cana-5898 28 11 u u NOUN cana-5898 28 12 } } PUNCT cana-5898 28 13 . . PUNCT cana-5898 29 1 a a DET cana-5898 29 2 subset subset NOUN cana-5898 29 3 a a PRON cana-5898 29 4 of of ADP cana-5898 29 5 x x SYM cana-5898 29 6 is be AUX cana-5898 29 7 said say VERB cana-5898 29 8 to to PART cana-5898 29 9 be be AUX cana-5898 29 10 g g NOUN cana-5898 29 11 - - PUNCT cana-5898 29 12 closed[15 closed[15 NOUN cana-5898 29 13 ] ] PUNCT cana-5898 29 14 ( ( PUNCT cana-5898 29 15 resp resp NOUN cana-5898 29 16 . . PUNCT cana-5898 30 1 s*g s*g PROPN cana-5898 30 2 - - PUNCT cana-5898 30 3 closed[13 closed[13 PROPN cana-5898 30 4 ] ] PUNCT cana-5898 30 5 , , PUNCT cana-5898 30 6 gs gs NOUN cana-5898 30 7 - - PUNCT cana-5898 30 8 closed[2 closed[2 NOUN cana-5898 30 9 ] ] PUNCT cana-5898 30 10 ) ) PUNCT cana-5898 30 11 if if SCONJ cana-5898 30 12 cl(a cl(a NUM cana-5898 30 13 ) ) PUNCT cana-5898 30 14 βŠ† βŠ† NUM cana-5898 30 15 u u NOUN cana-5898 30 16 ( ( PUNCT cana-5898 30 17 resp resp NOUN cana-5898 30 18 . . PUNCT cana-5898 30 19 cl(a cl(a PUNCT cana-5898 30 20 ) ) PUNCT cana-5898 30 21   PROPN cana-5898 30 22 u u PROPN cana-5898 30 23 , , PUNCT cana-5898 30 24 scl(a scl(a PROPN cana-5898 30 25 ) ) PUNCT cana-5898 30 26 βŠ† βŠ† NUM cana-5898 30 27 u u NOUN cana-5898 30 28 ) ) PUNCT cana-5898 30 29 whenever whenever SCONJ cana-5898 30 30 a a DET cana-5898 30 31   PROPN cana-5898 30 32 u u NOUN cana-5898 30 33 and and CCONJ cana-5898 30 34 u u NOUN cana-5898 30 35 is be AUX cana-5898 30 36 open open ADJ cana-5898 30 37 ( ( PUNCT cana-5898 30 38 resp resp NOUN cana-5898 30 39 . . PUNCT cana-5898 31 1 u u NOUN cana-5898 31 2 is be AUX cana-5898 31 3 semi semi ADJ cana-5898 31 4 - - ADJ cana-5898 31 5 open open ADJ cana-5898 31 6 , , PUNCT cana-5898 31 7 u u NOUN cana-5898 31 8 is be AUX cana-5898 31 9 open open ADJ cana-5898 31 10 ) ) PUNCT cana-5898 31 11 in in ADP cana-5898 31 12 x. x. NOUN cana-5898 31 13 a a DET cana-5898 31 14 nonempty nonempty ADJ cana-5898 31 15 collection collection NOUN cana-5898 31 16 𝒒 𝒒 PROPN cana-5898 31 17 of of ADP cana-5898 31 18 subsets subset NOUN cana-5898 31 19 of of ADP cana-5898 31 20 a a DET cana-5898 31 21 topological topological ADJ cana-5898 31 22 space space NOUN cana-5898 31 23 ( ( PUNCT cana-5898 31 24 x x X cana-5898 31 25 , , PUNCT cana-5898 31 26   PROPN cana-5898 31 27 ) ) PUNCT cana-5898 31 28 is be AUX cana-5898 31 29 called call VERB cana-5898 31 30 a a DET cana-5898 31 31 grill[8 grill[8 NOUN cana-5898 31 32 ] ] X cana-5898 31 33 on on ADP cana-5898 31 34 x x SYM cana-5898 31 35 if if SCONJ cana-5898 31 36 ( ( PUNCT cana-5898 31 37 i i NOUN cana-5898 31 38 ) ) PUNCT cana-5898 31 39   PROPN cana-5898 31 40   VERB cana-5898 31 41 𝒒 𝒒 PROPN cana-5898 31 42 , , PUNCT cana-5898 31 43 ( ( PUNCT cana-5898 31 44 ii ii NOUN cana-5898 31 45 ) ) PUNCT cana-5898 31 46 a a DET cana-5898 31 47 οƒŽ οƒŽ NOUN cana-5898 31 48 𝒒 𝒒 PROPN cana-5898 31 49 and and CCONJ cana-5898 31 50 a a DET cana-5898 31 51   PROPN cana-5898 31 52 b b PROPN cana-5898 31 53 implies imply VERB cana-5898 31 54 that that SCONJ cana-5898 31 55 b b X cana-5898 31 56 οƒŽ οƒŽ PROPN cana-5898 31 57 𝒒 𝒒 PROPN cana-5898 31 58 , , PUNCT cana-5898 31 59 ( ( PUNCT cana-5898 31 60 iii iii X cana-5898 31 61 ) ) PUNCT cana-5898 31 62 a a NOUN cana-5898 31 63 , , PUNCT cana-5898 31 64 b b NOUN cana-5898 31 65   PROPN cana-5898 31 66 x x X cana-5898 31 67 and and CCONJ cana-5898 31 68 a a DET cana-5898 31 69 οƒˆ οƒˆ PROPN cana-5898 31 70 b b PROPN cana-5898 31 71 οƒŽ οƒŽ PROPN cana-5898 31 72 𝒒 𝒒 PROPN cana-5898 31 73 implies imply VERB cana-5898 31 74 that that SCONJ cana-5898 31 75 a a DET cana-5898 31 76 οƒŽ οƒŽ NOUN cana-5898 31 77 𝒒 𝒒 PROPN cana-5898 31 78 or or CCONJ cana-5898 31 79 b b PROPN cana-5898 31 80 οƒŽ οƒŽ PROPN cana-5898 31 81 𝒒. 𝒒. PROPN cana-5898 31 82 a a DET cana-5898 31 83 triple triple ADJ cana-5898 31 84 ( ( PUNCT cana-5898 31 85 x x NOUN cana-5898 31 86   NOUN cana-5898 31 87 , , PUNCT cana-5898 31 88 𝒒 𝒒 PROPN cana-5898 31 89 ) ) PUNCT cana-5898 31 90 is be AUX cana-5898 31 91 called call VERB cana-5898 31 92 a a DET cana-5898 31 93 grill grill NOUN cana-5898 31 94 topological topological ADJ cana-5898 31 95 space[20 space[20 NOUN cana-5898 31 96 ] ] PUNCT cana-5898 31 97 . . PUNCT cana-5898 32 1 let let VERB cana-5898 32 2 y y PRON cana-5898 32 3 be be AUX cana-5898 32 4 a a DET cana-5898 32 5 subset subset NOUN cana-5898 32 6 of of ADP cana-5898 32 7 x. x. NOUN cana-5898 32 8 then then ADV cana-5898 32 9 𝒒y 𝒒y PROPN cana-5898 32 10 = = SYM cana-5898 32 11 { { PUNCT cana-5898 32 12 𝒒0 𝒒0 PROPN cana-5898 32 13   PUNCT cana-5898 32 14 y y NOUN cana-5898 32 15 : : PUNCT cana-5898 32 16 𝒒0 𝒒0 PROPN cana-5898 32 17   PROPN cana-5898 32 18 𝒒 𝒒 PROPN cana-5898 32 19 } } PUNCT cana-5898 32 20 is be AUX cana-5898 32 21 a a DET cana-5898 32 22 grill grill NOUN cana-5898 32 23 on on ADP cana-5898 32 24 y y PROPN cana-5898 32 25 and and CCONJ cana-5898 32 26 grill grill ADJ cana-5898 32 27 topological topological ADJ cana-5898 32 28 subspace subspace NOUN cana-5898 32 29 denoted denote VERB cana-5898 32 30 by by ADP cana-5898 32 31 { { PUNCT cana-5898 32 32 y y NOUN cana-5898 32 33 , , PUNCT cana-5898 32 34 y y PROPN cana-5898 32 35 , , PUNCT cana-5898 32 36 𝒒y 𝒒y PROPN cana-5898 32 37 } } PUNCT cana-5898 32 38 . . PUNCT cana-5898 33 1 a a DET cana-5898 33 2 mapping mapping NOUN cana-5898 33 3 οͺ[20 οͺ[20 NOUN cana-5898 33 4 ] ] X cana-5898 33 5 ( ( PUNCT cana-5898 33 6 resp resp NOUN cana-5898 33 7 . . PUNCT cana-5898 34 1 οͺs[18 οͺs[18 NOUN cana-5898 34 2 ] ] PUNCT cana-5898 34 3 ) ) PUNCT cana-5898 34 4 : : PUNCT cana-5898 34 5 p(x p(x PROPN cana-5898 34 6 ) ) PUNCT cana-5898 34 7 β†’ β†’ SYM cana-5898 34 8 p(x p(x PROPN cana-5898 34 9 ) ) PUNCT cana-5898 34 10 is be AUX cana-5898 34 11 defined define VERB cana-5898 34 12 by by ADP cana-5898 34 13 οͺ(a)[20 οͺ(a)[20 X cana-5898 34 14 ] ] PUNCT cana-5898 34 15 ( ( PUNCT cana-5898 34 16 resp resp NOUN cana-5898 34 17 . . PUNCT cana-5898 35 1 οͺs(a)[18 οͺs(a)[18 NOUN cana-5898 35 2 ] ] PUNCT cana-5898 35 3 ) ) PUNCT cana-5898 36 1 = = SYM cana-5898 36 2 { { PUNCT cana-5898 36 3 x x X cana-5898 36 4 οƒŽ οƒŽ NOUN cana-5898 36 5 x x NOUN cana-5898 36 6 : : PUNCT cana-5898 36 7 a a DET cana-5898 36 8   NUM cana-5898 36 9 u u NOUN cana-5898 36 10 οƒŽ οƒŽ NOUN cana-5898 36 11 𝒒 𝒒 PROPN cana-5898 36 12 for for ADP cana-5898 36 13 all all DET cana-5898 36 14 u u PROPN cana-5898 36 15 οƒŽ οƒŽ PROPN cana-5898 36 16 (x (x PROPN cana-5898 36 17 ) ) PUNCT cana-5898 36 18 ( ( PUNCT cana-5898 36 19 resp resp NOUN cana-5898 36 20 . . PUNCT cana-5898 37 1 so(x so(x PROPN cana-5898 37 2 , , PUNCT cana-5898 37 3 x x NOUN cana-5898 37 4 ) ) PUNCT cana-5898 37 5 } } PUNCT cana-5898 37 6 for for ADP cana-5898 37 7 all all DET cana-5898 37 8 a a DET cana-5898 37 9 οƒŽ οƒŽ NOUN cana-5898 37 10 p(x p(x PROPN cana-5898 37 11 ) ) PUNCT cana-5898 37 12 , , PUNCT cana-5898 37 13 where where SCONJ cana-5898 37 14 (x (x NOUN cana-5898 37 15 ) ) PUNCT cana-5898 37 16 ( ( PUNCT cana-5898 37 17 resp resp NOUN cana-5898 37 18 . . PUNCT cana-5898 38 1 so(x so(x PROPN cana-5898 38 2 , , PUNCT cana-5898 38 3 x x NOUN cana-5898 38 4 ) ) PUNCT cana-5898 38 5 ) ) PUNCT cana-5898 39 1 denotes denote VERB cana-5898 39 2 the the DET cana-5898 39 3 collection collection NOUN cana-5898 39 4 of of ADP cana-5898 39 5 all all PRON cana-5898 40 1 open open ADJ cana-5898 40 2 ( ( PUNCT cana-5898 40 3 resp resp NOUN cana-5898 40 4 . . PUNCT cana-5898 40 5 semiopen semiopen PROPN cana-5898 40 6 ) ) PUNCT cana-5898 40 7 neighbourhoods neighbourhood NOUN cana-5898 40 8 of of ADP cana-5898 40 9 x. x. NOUN cana-5898 40 10 a a DET cana-5898 40 11 mapping mapping NOUN cana-5898 40 12 [20 [20 NOUN cana-5898 40 13 ( ( PUNCT cana-5898 40 14 resp resp NOUN cana-5898 40 15 . . PUNCT cana-5898 41 1 s[18 s[18 NOUN cana-5898 41 2 ] ] PUNCT cana-5898 41 3 ) ) PUNCT cana-5898 41 4 : : PUNCT cana-5898 41 5 p(x p(x PROPN cana-5898 41 6 ) ) PUNCT cana-5898 41 7 β†’ β†’ SYM cana-5898 41 8 p(x p(x PROPN cana-5898 41 9 ) ) PUNCT cana-5898 41 10 is be AUX cana-5898 41 11 defined define VERB cana-5898 41 12 by by ADP cana-5898 41 13 (a)[20 (a)[20 PROPN cana-5898 41 14 ] ] X cana-5898 41 15 ( ( PUNCT cana-5898 41 16 resp resp NOUN cana-5898 41 17 . . PUNCT cana-5898 42 1 s(a)[18 s(a)[18 ADV cana-5898 42 2 ] ] X cana-5898 42 3 ) ) PUNCT cana-5898 43 1 = = SYM cana-5898 43 2 a a DET cana-5898 43 3 οƒˆ οƒˆ NOUN cana-5898 43 4 οͺ(a οͺ(a PROPN cana-5898 43 5 ) ) PUNCT cana-5898 43 6 ( ( PUNCT cana-5898 43 7 resp resp NOUN cana-5898 43 8 . . PUNCT cana-5898 44 1 a a DET cana-5898 44 2 οƒˆ οƒˆ NOUN cana-5898 44 3 οͺs(a οͺs(a NOUN cana-5898 44 4 ) ) PUNCT cana-5898 44 5 ) ) PUNCT cana-5898 44 6 for for ADP cana-5898 44 7 all all DET cana-5898 44 8 a a DET cana-5898 44 9 οƒŽ οƒŽ NOUN cana-5898 44 10 p(x p(x PROPN cana-5898 44 11 ) ) PUNCT cana-5898 44 12 . . PUNCT cana-5898 45 1 also also ADV cana-5898 45 2   PRON cana-5898 45 3 ( ( PUNCT cana-5898 45 4 resp resp NOUN cana-5898 45 5 . . PUNCT cana-5898 46 1 s s NUM cana-5898 46 2 ) ) PUNCT cana-5898 46 3 satisfies satisfy VERB cana-5898 46 4 the the DET cana-5898 46 5 kuratowski kuratowski ADJ cana-5898 46 6 closure closure NOUN cana-5898 46 7 axioms axiom NOUN cana-5898 46 8 . . PUNCT cana-5898 47 1 corresponding correspond VERB cana-5898 47 2 to to ADP cana-5898 47 3 a a DET cana-5898 47 4 grill grill NOUN cana-5898 47 5 𝒒 𝒒 NOUN cana-5898 47 6 on on ADP cana-5898 47 7 a a DET cana-5898 47 8 topological topological ADJ cana-5898 47 9 space space NOUN cana-5898 47 10 ( ( PUNCT cana-5898 47 11 x x X cana-5898 47 12 , , PUNCT cana-5898 47 13   PROPN cana-5898 47 14 ) ) PUNCT cana-5898 47 15 , , PUNCT cana-5898 47 16 there there PRON cana-5898 47 17 exists exist VERB cana-5898 47 18 a a DET cana-5898 47 19 unique unique ADJ cana-5898 47 20 topology topology NOUN cana-5898 47 21 τ𝒒[20 τ𝒒[20 NOUN cana-5898 47 22 ] ] X cana-5898 47 23 ( ( PUNCT cana-5898 47 24 resp resp NOUN cana-5898 47 25 . . PUNCT cana-5898 48 1 τ𝒒 τ𝒒 NOUN cana-5898 48 2 s s PART cana-5898 49 1 [ [ X cana-5898 49 2 18 18 NUM cana-5898 49 3 ] ] SYM cana-5898 49 4 ) ) PUNCT cana-5898 50 1 = = SYM cana-5898 50 2 { { PUNCT cana-5898 50 3 u u NOUN cana-5898 50 4   PROPN cana-5898 50 5 x x X cana-5898 50 6 : : PUNCT cana-5898 50 7 (x (x X cana-5898 50 8 – – PUNCT cana-5898 50 9 u u NOUN cana-5898 50 10 ) ) PUNCT cana-5898 50 11 ( ( PUNCT cana-5898 50 12 resp resp NOUN cana-5898 50 13 . . PUNCT cana-5898 51 1 s(x s(x NOUN cana-5898 51 2 – – PUNCT cana-5898 51 3 u u NOUN cana-5898 51 4 ) ) PUNCT cana-5898 51 5 ) ) PUNCT cana-5898 52 1 = = PUNCT cana-5898 52 2 x x X cana-5898 52 3 – – PUNCT cana-5898 52 4 u u NOUN cana-5898 52 5 } } PUNCT cana-5898 52 6 , , PUNCT cana-5898 52 7 where where SCONJ cana-5898 52 8 for for ADP cana-5898 52 9 any any DET cana-5898 52 10 a a DET cana-5898 52 11   PROPN cana-5898 52 12 x x NOUN cana-5898 52 13 , , PUNCT cana-5898 52 14 (a (a PROPN cana-5898 52 15 ) ) PUNCT cana-5898 52 16 ( ( PUNCT cana-5898 52 17 resp resp NOUN cana-5898 52 18 . . PUNCT cana-5898 53 1 s(a s(a NOUN cana-5898 53 2 ) ) PUNCT cana-5898 53 3 ) ) PUNCT cana-5898 54 1 = = PUNCT cana-5898 54 2 a a DET cana-5898 54 3 οƒˆ οƒˆ NOUN cana-5898 54 4 οͺ(a οͺ(a PROPN cana-5898 54 5 ) ) PUNCT cana-5898 54 6 ( ( PUNCT cana-5898 54 7 resp resp NOUN cana-5898 54 8 . . PUNCT cana-5898 55 1 a a DET cana-5898 55 2 οƒˆ οƒˆ NOUN cana-5898 55 3 οͺs(a οͺs(a NOUN cana-5898 55 4 ) ) PUNCT cana-5898 55 5 ) ) PUNCT cana-5898 56 1 = = SYM cana-5898 56 2 τ𝒒cl(a τ𝒒cl(a NUM cana-5898 56 3 ) ) PUNCT cana-5898 56 4 ( ( PUNCT cana-5898 56 5 resp resp NOUN cana-5898 56 6 . . PUNCT cana-5898 57 1 τ𝒒 τ𝒒 NOUN cana-5898 57 2 s s PART cana-5898 57 3 cl(a cl(a NUM cana-5898 57 4 ) ) PUNCT cana-5898 57 5 ) ) PUNCT cana-5898 57 6   PROPN cana-5898 57 7 cl(a cl(a NUM cana-5898 57 8 ) ) PUNCT cana-5898 57 9 ( ( PUNCT cana-5898 57 10 resp resp NOUN cana-5898 57 11 . . PUNCT cana-5898 58 1 scl(a scl(a PROPN cana-5898 58 2 ) ) PUNCT cana-5898 58 3 ) ) PUNCT cana-5898 59 1 and and CCONJ cana-5898 59 2   NOUN cana-5898 59 3   PROPN cana-5898 59 4 τ𝒒 τ𝒒 PROPN cana-5898 59 5 ( ( PUNCT cana-5898 59 6 resp resp NOUN cana-5898 59 7 . . PUNCT cana-5898 59 8 so(x so(x NOUN cana-5898 59 9 ) ) PUNCT cana-5898 59 10   PROPN cana-5898 59 11 τ𝒒 τ𝒒 PROPN cana-5898 59 12 s s PART cana-5898 59 13 ) ) PUNCT cana-5898 59 14 . . PUNCT cana-5898 60 1 a a DET cana-5898 60 2 subset subset NOUN cana-5898 60 3 a a PRON cana-5898 60 4 of of ADP cana-5898 60 5 x x PRON cana-5898 60 6 is be AUX cana-5898 60 7 called call VERB cana-5898 60 8 ( ( PUNCT cana-5898 60 9 i i NOUN cana-5898 60 10 ) ) PUNCT cana-5898 60 11 τ𝒒-closed[20 τ𝒒-closed[20 NOUN cana-5898 60 12 ] ] PUNCT cana-5898 60 13 if if SCONJ cana-5898 60 14 communications communication NOUN cana-5898 60 15 on on ADP cana-5898 60 16 applied apply VERB cana-5898 60 17 nonlinear nonlinear ADJ cana-5898 60 18 analysis analysis NOUN cana-5898 60 19 issn issn NOUN cana-5898 60 20 : : PUNCT cana-5898 60 21 1074 1074 NUM cana-5898 60 22 - - PUNCT cana-5898 60 23 133x 133x NUM cana-5898 60 24 vol vol NOUN cana-5898 60 25 31 31 NUM cana-5898 60 26 no no NOUN cana-5898 60 27 . . PUNCT cana-5898 61 1 7s 7 NOUN cana-5898 61 2 ( ( PUNCT cana-5898 61 3 2024 2024 NUM cana-5898 61 4 ) ) PUNCT cana-5898 61 5 792 792 NUM cana-5898 61 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 61 7 τ𝒒cl(a τ𝒒cl(a NUM cana-5898 61 8 ) ) PUNCT cana-5898 61 9 = = SYM cana-5898 61 10 (a (a NOUN cana-5898 61 11 ) ) PUNCT cana-5898 61 12 = = PUNCT cana-5898 62 1 a a DET cana-5898 62 2 οƒˆ οƒˆ NOUN cana-5898 62 3 οͺ(a οͺ(a PROPN cana-5898 62 4 ) ) PUNCT cana-5898 62 5 ; ; PUNCT cana-5898 63 1 οͺ-dense[20 οͺ-dense[20 X cana-5898 63 2 ] ] PUNCT cana-5898 63 3 if if SCONJ cana-5898 63 4 οͺ(a οͺ(a PROPN cana-5898 63 5 ) ) PUNCT cana-5898 63 6   PROPN cana-5898 63 7 a a PRON cana-5898 63 8 ; ; PUNCT cana-5898 63 9 𝒒g 𝒒g NOUN cana-5898 63 10 - - PUNCT cana-5898 63 11 closed[17 closed[17 NOUN cana-5898 63 12 ] ] PUNCT cana-5898 63 13 if if SCONJ cana-5898 63 14 οͺ(a οͺ(a PROPN cana-5898 63 15 ) ) PUNCT cana-5898 63 16   PROPN cana-5898 63 17 u u PROPN cana-5898 63 18 whenever whenever SCONJ cana-5898 63 19 a a DET cana-5898 63 20   PROPN cana-5898 63 21 u u NOUN cana-5898 63 22 and and CCONJ cana-5898 63 23 u u NOUN cana-5898 63 24 is be AUX cana-5898 63 25 open open ADJ cana-5898 63 26 in in ADP cana-5898 63 27 x. x. NOUN cana-5898 63 28 theorem theorem VERB cana-5898 63 29 2.1.[18 2.1.[18 X cana-5898 63 30 ] ] X cana-5898 63 31 let let VERB cana-5898 63 32 ( ( PUNCT cana-5898 63 33 x x X cana-5898 63 34 , , PUNCT cana-5898 63 35   PROPN cana-5898 63 36 , , PUNCT cana-5898 63 37 𝒒 𝒒 PROPN cana-5898 63 38 ) ) PUNCT cana-5898 63 39 be be VERB cana-5898 63 40 a a DET cana-5898 63 41 grill grill ADJ cana-5898 63 42 topological topological ADJ cana-5898 63 43 space space NOUN cana-5898 63 44 . . PUNCT cana-5898 64 1 then then ADV cana-5898 64 2 , , PUNCT cana-5898 64 3 for for ADP cana-5898 64 4 every every DET cana-5898 64 5 a a PROPN cana-5898 64 6 , , PUNCT cana-5898 64 7 b b NOUN cana-5898 64 8   PROPN cana-5898 64 9 x x PROPN cana-5898 64 10 , , PUNCT cana-5898 64 11 the the DET cana-5898 64 12 following follow VERB cana-5898 64 13 conditions condition NOUN cana-5898 64 14 are be AUX cana-5898 64 15 satisfied satisfied ADJ cana-5898 64 16 : : PUNCT cana-5898 64 17 ( ( PUNCT cana-5898 64 18 i i NOUN cana-5898 64 19 ) ) PUNCT cana-5898 64 20 if if SCONJ cana-5898 64 21 a a DET cana-5898 64 22   PROPN cana-5898 64 23 b b PROPN cana-5898 64 24 , , PUNCT cana-5898 64 25 then then ADV cana-5898 64 26 οͺs(a οͺs(a NOUN cana-5898 64 27 ) ) PUNCT cana-5898 64 28   PROPN cana-5898 64 29 οͺs(b οͺs(b NUM cana-5898 64 30 ) ) PUNCT cana-5898 64 31 ; ; PUNCT cana-5898 64 32 ( ( PUNCT cana-5898 64 33 ii ii NOUN cana-5898 64 34 ) ) PUNCT cana-5898 64 35 οͺs(a οͺs(a NOUN cana-5898 64 36 ) ) PUNCT cana-5898 64 37 = = SYM cana-5898 64 38 scl(οͺs(a scl(οͺs(a NOUN cana-5898 64 39 ) ) PUNCT cana-5898 64 40 ) ) PUNCT cana-5898 65 1   PROPN cana-5898 65 2 scl(a scl(a PROPN cana-5898 65 3 ) ) PUNCT cana-5898 65 4 and and CCONJ cana-5898 65 5 οͺs(a οͺs(a NOUN cana-5898 65 6 ) ) PUNCT cana-5898 65 7 is be AUX cana-5898 65 8 semiclosed semiclose VERB cana-5898 65 9 in in ADP cana-5898 65 10 x x SYM cana-5898 65 11 ; ; PUNCT cana-5898 65 12 ( ( PUNCT cana-5898 65 13 iii iii X cana-5898 65 14 ) ) PUNCT cana-5898 65 15 οͺs(οͺs(a οͺs(οͺs(a NOUN cana-5898 65 16 ) ) PUNCT cana-5898 65 17 ) ) PUNCT cana-5898 66 1   PROPN cana-5898 66 2 οͺs(a οͺs(a PROPN cana-5898 66 3 ) ) PUNCT cana-5898 66 4 ; ; PUNCT cana-5898 66 5 ( ( PUNCT cana-5898 66 6 iv iv X cana-5898 66 7 ) ) PUNCT cana-5898 66 8 οͺs(a οͺs(a NOUN cana-5898 66 9 οƒˆ οƒˆ NOUN cana-5898 66 10 b b PROPN cana-5898 66 11 ) ) PUNCT cana-5898 66 12 = = SYM cana-5898 66 13 οͺs(a οͺs(a NOUN cana-5898 66 14 ) ) PUNCT cana-5898 66 15 οƒˆ οƒˆ NOUN cana-5898 66 16 οͺs(b οͺs(b NOUN cana-5898 66 17 ) ) PUNCT cana-5898 66 18 ; ; PUNCT cana-5898 66 19 ( ( PUNCT cana-5898 66 20 v v NOUN cana-5898 66 21 ) ) PUNCT cana-5898 66 22 if if SCONJ cana-5898 66 23 a a DET cana-5898 66 24   PUNCT cana-5898 66 25 𝒒 𝒒 NOUN cana-5898 66 26 , , PUNCT cana-5898 66 27 then then ADV cana-5898 66 28 οͺs(a οͺs(a NOUN cana-5898 66 29 ) ) PUNCT cana-5898 66 30 = = PUNCT cana-5898 67 1 . . VERB cana-5898 67 2 3 3 X cana-5898 67 3 . . PUNCT cana-5898 67 4 g𝓖-closed g𝓖-close VERB cana-5898 67 5 sets set NOUN cana-5898 67 6 definition definition NOUN cana-5898 67 7 3.1 3.1 NUM cana-5898 67 8 . . PUNCT cana-5898 68 1 let let VERB cana-5898 68 2 ( ( PUNCT cana-5898 68 3 x x X cana-5898 68 4 , , PUNCT cana-5898 68 5   PROPN cana-5898 68 6 , , PUNCT cana-5898 68 7 𝒒 𝒒 PROPN cana-5898 68 8 ) ) PUNCT cana-5898 68 9 be be VERB cana-5898 68 10 a a DET cana-5898 68 11 grill grill ADJ cana-5898 68 12 topological topological ADJ cana-5898 68 13 space space NOUN cana-5898 68 14 and and CCONJ cana-5898 68 15 a a DET cana-5898 68 16 be be AUX cana-5898 68 17 a a DET cana-5898 68 18 subset subset NOUN cana-5898 68 19 of of ADP cana-5898 68 20 x. x. NOUN cana-5898 68 21 then then ADV cana-5898 68 22 a a PRON cana-5898 68 23 is be AUX cana-5898 68 24 said say VERB cana-5898 68 25 to to PART cana-5898 68 26 be be AUX cana-5898 68 27 g𝒒-closed g𝒒-close VERB cana-5898 68 28 if if SCONJ cana-5898 68 29 οͺs(a οͺs(a NOUN cana-5898 68 30 ) ) PUNCT cana-5898 68 31   PROPN cana-5898 68 32 u u PROPN cana-5898 68 33 whenever whenever SCONJ cana-5898 68 34 a a DET cana-5898 68 35   PROPN cana-5898 68 36 u u NOUN cana-5898 68 37 and and CCONJ cana-5898 68 38 u u NOUN cana-5898 68 39 is be AUX cana-5898 68 40 open open ADJ cana-5898 68 41 in in ADP cana-5898 68 42 x. x. NOUN cana-5898 68 43 the the DET cana-5898 68 44 complement complement NOUN cana-5898 68 45 of of ADP cana-5898 68 46 a a DET cana-5898 68 47 g𝒒-closed g𝒒-close VERB cana-5898 68 48 set set NOUN cana-5898 68 49 is be AUX cana-5898 68 50 called call VERB cana-5898 68 51 a a DET cana-5898 68 52 g𝒒-open g𝒒-open NOUN cana-5898 68 53 set set NOUN cana-5898 68 54 . . PUNCT cana-5898 69 1 theorem theorem VERB cana-5898 69 2 3.2 3.2 NUM cana-5898 69 3 . . PUNCT cana-5898 70 1 let let VERB cana-5898 70 2 ( ( PUNCT cana-5898 70 3 x x X cana-5898 70 4 , , PUNCT cana-5898 70 5   PROPN cana-5898 70 6 , , PUNCT cana-5898 70 7 𝒒 𝒒 PROPN cana-5898 70 8 ) ) PUNCT cana-5898 70 9 be be AUX cana-5898 70 10 a a DET cana-5898 70 11 topological topological ADJ cana-5898 70 12 space space NOUN cana-5898 70 13 and and CCONJ cana-5898 70 14 a a DET cana-5898 70 15 be be AUX cana-5898 70 16 a a DET cana-5898 70 17 subset subset NOUN cana-5898 70 18 of of ADP cana-5898 70 19 x. x. NOUN cana-5898 70 20 then then ADV cana-5898 70 21 ( ( PUNCT cana-5898 70 22 i i NOUN cana-5898 70 23 ) ) PUNCT cana-5898 70 24 if if SCONJ cana-5898 70 25 a a PRON cana-5898 70 26 is be AUX cana-5898 70 27 a a DET cana-5898 70 28 τ𝒒-closed τ𝒒-close VERB cana-5898 70 29 set set NOUN cana-5898 70 30 , , PUNCT cana-5898 70 31 then then ADV cana-5898 70 32 a a PRON cana-5898 70 33 is be AUX cana-5898 70 34 𝒒g 𝒒g NOUN cana-5898 70 35 - - PUNCT cana-5898 70 36 closed closed ADJ cana-5898 70 37 ; ; PUNCT cana-5898 70 38 ( ( PUNCT cana-5898 70 39 ii ii NOUN cana-5898 70 40 ) ) PUNCT cana-5898 70 41 if if SCONJ cana-5898 70 42 if if SCONJ cana-5898 70 43 a a PRON cana-5898 70 44 is be AUX cana-5898 70 45 a a DET cana-5898 70 46 g g NOUN cana-5898 70 47 - - PUNCT cana-5898 70 48 closed close VERB cana-5898 70 49 set set NOUN cana-5898 70 50 , , PUNCT cana-5898 70 51 then then ADV cana-5898 70 52 a a PRON cana-5898 70 53 is be AUX cana-5898 70 54 𝒒g 𝒒g NOUN cana-5898 70 55 - - PUNCT cana-5898 70 56 closed closed ADJ cana-5898 70 57 ; ; PUNCT cana-5898 70 58 ( ( PUNCT cana-5898 70 59 iii iii X cana-5898 70 60 ) ) PUNCT cana-5898 70 61 if if SCONJ cana-5898 70 62 a a PRON cana-5898 70 63 is be AUX cana-5898 70 64 a a DET cana-5898 70 65 𝒒g 𝒒g PROPN cana-5898 70 66 - - PUNCT cana-5898 70 67 closed close VERB cana-5898 70 68 set set NOUN cana-5898 70 69 , , PUNCT cana-5898 70 70 then then ADV cana-5898 70 71 a a PRON cana-5898 70 72 is be AUX cana-5898 70 73 g𝒒-closed g𝒒-close VERB cana-5898 70 74 ; ; PUNCT cana-5898 70 75 ( ( PUNCT cana-5898 70 76 iv iv X cana-5898 70 77 ) ) PUNCT cana-5898 70 78 if if SCONJ cana-5898 70 79 a a PRON cana-5898 70 80 is be AUX cana-5898 70 81 a a DET cana-5898 70 82 s*g s*g NOUN cana-5898 70 83 - - PUNCT cana-5898 70 84 closed close VERB cana-5898 70 85 set set NOUN cana-5898 70 86 , , PUNCT cana-5898 70 87 then then ADV cana-5898 70 88 a a PRON cana-5898 70 89 is be AUX cana-5898 70 90 g g NOUN cana-5898 70 91 - - PUNCT cana-5898 70 92 closed closed ADJ cana-5898 70 93 ; ; PUNCT cana-5898 70 94 ( ( PUNCT cana-5898 70 95 v v NOUN cana-5898 70 96 ) ) PUNCT cana-5898 70 97 if if SCONJ cana-5898 70 98 a a PRON cana-5898 70 99 is be AUX cana-5898 70 100 a a DET cana-5898 70 101 g g NOUN cana-5898 70 102 - - PUNCT cana-5898 70 103 closed close VERB cana-5898 70 104 set set NOUN cana-5898 70 105 , , PUNCT cana-5898 70 106 then then ADV cana-5898 70 107 a a PRON cana-5898 70 108 is be AUX cana-5898 70 109 gs gs NOUN cana-5898 70 110 - - PUNCT cana-5898 70 111 closed closed ADJ cana-5898 70 112 ; ; PUNCT cana-5898 70 113 ( ( PUNCT cana-5898 70 114 vi vi X cana-5898 70 115 ) ) PUNCT cana-5898 70 116 if if SCONJ cana-5898 70 117 a a PRON cana-5898 70 118 is be AUX cana-5898 70 119 a a DET cana-5898 70 120 gs gs NOUN cana-5898 70 121 - - PUNCT cana-5898 70 122 closed close VERB cana-5898 70 123 set set NOUN cana-5898 70 124 , , PUNCT cana-5898 70 125 then then ADV cana-5898 70 126 a a PRON cana-5898 70 127 is be AUX cana-5898 70 128 g𝒒-closed g𝒒-close VERB cana-5898 70 129 . . PUNCT cana-5898 71 1 proof proof NOUN cana-5898 71 2 . . PUNCT cana-5898 72 1 ( ( PUNCT cana-5898 72 2 i i NOUN cana-5898 72 3 ) ) PUNCT cana-5898 72 4 and and CCONJ cana-5898 72 5 ( ( PUNCT cana-5898 72 6 ii ii NOUN cana-5898 72 7 ) ) PUNCT cana-5898 72 8 follows follow VERB cana-5898 72 9 from from ADP cana-5898 72 10 the the DET cana-5898 72 11 remark remark NOUN cana-5898 72 12 2.2.(b 2.2.(b NUM cana-5898 72 13 ) ) PUNCT cana-5898 72 14 and and CCONJ cana-5898 72 15 ( ( PUNCT cana-5898 72 16 d)[17 d)[17 PROPN cana-5898 72 17 ] ] PUNCT cana-5898 72 18 . . PUNCT cana-5898 73 1 ( ( PUNCT cana-5898 73 2 iii iii X cana-5898 73 3 ) ) PUNCT cana-5898 73 4 let let VERB cana-5898 73 5 a a PRON cana-5898 73 6 be be AUX cana-5898 73 7 a a DET cana-5898 73 8 𝒒g 𝒒g PROPN cana-5898 73 9 - - PUNCT cana-5898 73 10 closed close VERB cana-5898 73 11 set set NOUN cana-5898 73 12 such such DET cana-5898 73 13 that that SCONJ cana-5898 73 14 a a DET cana-5898 73 15   PROPN cana-5898 73 16 u u NOUN cana-5898 73 17 and and CCONJ cana-5898 73 18 u u NOUN cana-5898 73 19 οƒŽ οƒŽ PROPN cana-5898 73 20 . . NOUN cana-5898 73 21 then then ADV cana-5898 73 22 by by ADP cana-5898 73 23 assumption assumption NOUN cana-5898 73 24 , , PUNCT cana-5898 73 25 οͺ(a οͺ(a PROPN cana-5898 73 26 ) ) PUNCT cana-5898 73 27   PROPN cana-5898 73 28 u. u. PROPN cana-5898 73 29 since since SCONJ cana-5898 73 30 every every DET cana-5898 73 31 open open ADJ cana-5898 73 32 set set NOUN cana-5898 73 33 is be AUX cana-5898 73 34 semiopen semiopen ADJ cana-5898 73 35 , , PUNCT cana-5898 73 36 we we PRON cana-5898 73 37 have have VERB cana-5898 73 38 that that DET cana-5898 73 39 οͺs(a οͺs(a NOUN cana-5898 73 40 ) ) PUNCT cana-5898 73 41   PROPN cana-5898 73 42 οͺ(a οͺ(a PROPN cana-5898 73 43 ) ) PUNCT cana-5898 73 44 . . PUNCT cana-5898 74 1 therefore therefore ADV cana-5898 74 2 οͺs(a οͺs(a PROPN cana-5898 74 3 ) ) PUNCT cana-5898 74 4   PROPN cana-5898 74 5 u. u. PROPN cana-5898 74 6 hence hence ADV cana-5898 74 7 a a PRON cana-5898 74 8 is be AUX cana-5898 74 9 g𝒒-closed g𝒒-close VERB cana-5898 74 10 . . PUNCT cana-5898 75 1 ( ( PUNCT cana-5898 75 2 iv iv X cana-5898 75 3 ) ) PUNCT cana-5898 75 4 let let VERB cana-5898 75 5 a a PRON cana-5898 75 6 be be AUX cana-5898 75 7 a a DET cana-5898 75 8 s*g s*g NOUN cana-5898 75 9 - - PUNCT cana-5898 75 10 closed close VERB cana-5898 75 11 set set NOUN cana-5898 75 12 such such DET cana-5898 75 13 that that SCONJ cana-5898 75 14 a a DET cana-5898 75 15   PROPN cana-5898 75 16 u u NOUN cana-5898 75 17 and and CCONJ cana-5898 75 18 u u NOUN cana-5898 75 19 οƒŽ οƒŽ PROPN cana-5898 75 20 . . PROPN cana-5898 75 21 since since SCONJ cana-5898 75 22 every every DET cana-5898 75 23 open open ADJ cana-5898 75 24 set set NOUN cana-5898 75 25 is be AUX cana-5898 75 26 semiopen semiopen ADJ cana-5898 75 27 , , PUNCT cana-5898 75 28 a a DET cana-5898 75 29   PROPN cana-5898 75 30 u u NOUN cana-5898 75 31 and and CCONJ cana-5898 75 32 u u NOUN cana-5898 75 33 is be AUX cana-5898 75 34 semiopen semiopen ADJ cana-5898 75 35 . . PUNCT cana-5898 76 1 by by ADP cana-5898 76 2 assumption assumption NOUN cana-5898 76 3 , , PUNCT cana-5898 76 4 cl(a cl(a NUM cana-5898 76 5 ) ) PUNCT cana-5898 76 6 βŠ† βŠ† NUM cana-5898 76 7 u. u. NOUN cana-5898 76 8 then then ADV cana-5898 76 9 by by ADP cana-5898 76 10 definition definition NOUN cana-5898 76 11 of of ADP cana-5898 76 12 g g NOUN cana-5898 76 13 - - PUNCT cana-5898 76 14 closed closed ADJ cana-5898 76 15 , , PUNCT cana-5898 76 16 we we PRON cana-5898 76 17 have have VERB cana-5898 76 18 that that SCONJ cana-5898 76 19 a a PRON cana-5898 76 20 is be AUX cana-5898 76 21 g g NOUN cana-5898 76 22 - - PUNCT cana-5898 76 23 closed closed ADJ cana-5898 76 24 . . PUNCT cana-5898 77 1 communications communication NOUN cana-5898 77 2 on on ADP cana-5898 77 3 applied apply VERB cana-5898 77 4 nonlinear nonlinear ADJ cana-5898 77 5 analysis analysis NOUN cana-5898 77 6 issn issn NOUN cana-5898 77 7 : : PUNCT cana-5898 77 8 1074 1074 NUM cana-5898 77 9 - - PUNCT cana-5898 77 10 133x 133x NUM cana-5898 77 11 vol vol NOUN cana-5898 77 12 31 31 NUM cana-5898 77 13 no no NOUN cana-5898 77 14 . . PUNCT cana-5898 78 1 7s 7 NOUN cana-5898 78 2 ( ( PUNCT cana-5898 78 3 2024 2024 NUM cana-5898 78 4 ) ) PUNCT cana-5898 78 5 793 793 NUM cana-5898 78 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 78 7 ( ( PUNCT cana-5898 78 8 v v NOUN cana-5898 78 9 ) ) PUNCT cana-5898 78 10 let let VERB cana-5898 78 11 a a PRON cana-5898 78 12 be be AUX cana-5898 78 13 a a DET cana-5898 78 14 g g NOUN cana-5898 78 15 - - PUNCT cana-5898 78 16 closed close VERB cana-5898 78 17 set set NOUN cana-5898 78 18 such such DET cana-5898 78 19 that that SCONJ cana-5898 78 20 a a DET cana-5898 78 21   PROPN cana-5898 78 22 u u NOUN cana-5898 78 23 and and CCONJ cana-5898 78 24 u u NOUN cana-5898 78 25 οƒŽ οƒŽ PROPN cana-5898 78 26 . . NOUN cana-5898 78 27 then then ADV cana-5898 78 28 by by ADP cana-5898 78 29 assumption assumption NOUN cana-5898 78 30 , , PUNCT cana-5898 78 31 cl(a cl(a NUM cana-5898 78 32 ) ) PUNCT cana-5898 78 33   PROPN cana-5898 78 34 u. u. PROPN cana-5898 78 35 since since SCONJ cana-5898 78 36 every every DET cana-5898 78 37 closed closed ADJ cana-5898 78 38 set set NOUN cana-5898 78 39 is be AUX cana-5898 78 40 semiclosed semiclose VERB cana-5898 78 41 , , PUNCT cana-5898 78 42 we we PRON cana-5898 78 43 have have AUX cana-5898 78 44 that that DET cana-5898 78 45 scl(a scl(a PROPN cana-5898 78 46 ) ) PUNCT cana-5898 78 47   PROPN cana-5898 78 48 cl(a cl(a NUM cana-5898 78 49 ) ) PUNCT cana-5898 78 50 . . PUNCT cana-5898 79 1 therefore therefore ADV cana-5898 79 2 scl(a scl(a X cana-5898 79 3 ) ) PUNCT cana-5898 80 1   PROPN cana-5898 81 1 u. u. PROPN cana-5898 81 2 hence hence ADV cana-5898 81 3 a a PRON cana-5898 81 4 is be AUX cana-5898 81 5 gs gs NOUN cana-5898 81 6 - - PUNCT cana-5898 81 7 closed closed ADJ cana-5898 81 8 . . PUNCT cana-5898 82 1 ( ( PUNCT cana-5898 82 2 vi vi X cana-5898 82 3 ) ) PUNCT cana-5898 82 4 let let VERB cana-5898 82 5 a a PRON cana-5898 82 6 be be AUX cana-5898 82 7 a a DET cana-5898 82 8 gs gs NOUN cana-5898 82 9 - - PUNCT cana-5898 82 10 closed close VERB cana-5898 82 11 set set NOUN cana-5898 82 12 such such DET cana-5898 82 13 that that SCONJ cana-5898 82 14 a a DET cana-5898 82 15   PROPN cana-5898 82 16 u u NOUN cana-5898 82 17 and and CCONJ cana-5898 82 18 u u NOUN cana-5898 82 19 οƒŽ οƒŽ PROPN cana-5898 82 20 . . NOUN cana-5898 82 21 then then ADV cana-5898 82 22 by by ADP cana-5898 82 23 assumption assumption NOUN cana-5898 82 24 , , PUNCT cana-5898 82 25 scl(a scl(a PROPN cana-5898 82 26 ) ) PUNCT cana-5898 82 27   PROPN cana-5898 82 28 u. u. PROPN cana-5898 82 29 since since SCONJ cana-5898 82 30 οͺs(a οͺs(a NOUN cana-5898 82 31 ) ) PUNCT cana-5898 82 32   PROPN cana-5898 82 33 scl(a scl(a PROPN cana-5898 82 34 ) ) PUNCT cana-5898 82 35 . . PUNCT cana-5898 83 1 therefore therefore ADV cana-5898 83 2 οͺs(a οͺs(a PROPN cana-5898 83 3 ) ) PUNCT cana-5898 83 4   PROPN cana-5898 83 5 u. u. PROPN cana-5898 83 6 hence hence ADV cana-5898 83 7 a a PRON cana-5898 83 8 is be AUX cana-5898 83 9 g𝒒-closed g𝒒-close VERB cana-5898 83 10 . . PUNCT cana-5898 84 1 remark remark PROPN cana-5898 84 2 3.3 3.3 NUM cana-5898 84 3 . . PUNCT cana-5898 85 1 the the DET cana-5898 85 2 following follow VERB cana-5898 85 3 examples example NOUN cana-5898 85 4 shows show VERB cana-5898 85 5 that that SCONJ cana-5898 85 6 the the DET cana-5898 85 7 reverse reverse ADJ cana-5898 85 8 implication implication NOUN cana-5898 85 9 of of ADP cana-5898 85 10 above above ADJ cana-5898 85 11 theorem theorem NOUN cana-5898 85 12 is be AUX cana-5898 85 13 not not PART cana-5898 85 14 true true ADJ cana-5898 85 15 and and CCONJ cana-5898 85 16 the the DET cana-5898 85 17 concepts concept NOUN cana-5898 85 18 of of ADP cana-5898 85 19 some some DET cana-5898 85 20 generalizations generalization NOUN cana-5898 85 21 of of ADP cana-5898 85 22 closed closed ADJ cana-5898 85 23 sets set NOUN cana-5898 85 24 are be AUX cana-5898 85 25 independent independent ADJ cana-5898 85 26 . . PUNCT cana-5898 86 1 ( ( PUNCT cana-5898 86 2 i i NOUN cana-5898 86 3 ) ) PUNCT cana-5898 86 4 let let VERB cana-5898 86 5 x x PUNCT cana-5898 86 6 = = PRON cana-5898 86 7 { { PUNCT cana-5898 86 8 a a PRON cana-5898 86 9 , , PUNCT cana-5898 86 10 b b NOUN cana-5898 86 11 , , PUNCT cana-5898 86 12 c c NOUN cana-5898 86 13 , , PUNCT cana-5898 86 14 d d NOUN cana-5898 86 15 } } PUNCT cana-5898 86 16 , , PUNCT cana-5898 86 17   NOUN cana-5898 86 18 = = X cana-5898 86 19 { { PUNCT cana-5898 86 20   PROPN cana-5898 86 21 , , PUNCT cana-5898 86 22 x x PRON cana-5898 86 23 , , PUNCT cana-5898 86 24 { { PUNCT cana-5898 86 25 a a DET cana-5898 86 26 , , PUNCT cana-5898 86 27 b b NOUN cana-5898 86 28 } } PUNCT cana-5898 86 29 , , PUNCT cana-5898 86 30 { { PUNCT cana-5898 86 31 a a PRON cana-5898 86 32 , , PUNCT cana-5898 86 33 b b NOUN cana-5898 86 34 , , PUNCT cana-5898 86 35 d d NOUN cana-5898 86 36 } } PUNCT cana-5898 86 37 } } PUNCT cana-5898 86 38 and and CCONJ cana-5898 86 39 𝒒 𝒒 PROPN cana-5898 86 40 = = SYM cana-5898 86 41 { { PUNCT cana-5898 86 42 x x NOUN cana-5898 86 43 , , PUNCT cana-5898 86 44 { { PUNCT cana-5898 86 45 a a X cana-5898 86 46 } } PUNCT cana-5898 86 47 , , PUNCT cana-5898 86 48 { { PUNCT cana-5898 86 49 c c X cana-5898 86 50 } } PUNCT cana-5898 86 51 , , PUNCT cana-5898 86 52 { { PUNCT cana-5898 86 53 d d X cana-5898 86 54 } } PUNCT cana-5898 86 55 , , PUNCT cana-5898 86 56 { { PUNCT cana-5898 86 57 a a DET cana-5898 86 58 , , PUNCT cana-5898 86 59 b b NOUN cana-5898 86 60 } } PUNCT cana-5898 86 61 , , PUNCT cana-5898 86 62 { { PUNCT cana-5898 86 63 a a X cana-5898 86 64 , , PUNCT cana-5898 86 65 c c NOUN cana-5898 86 66 } } PUNCT cana-5898 86 67 , , PUNCT cana-5898 86 68 { { PUNCT cana-5898 86 69 a a DET cana-5898 86 70 , , PUNCT cana-5898 86 71 d d NOUN cana-5898 86 72 } } PUNCT cana-5898 86 73 , , PUNCT cana-5898 86 74 { { PUNCT cana-5898 86 75 b b X cana-5898 86 76 , , PUNCT cana-5898 86 77 c c NOUN cana-5898 86 78 } } PUNCT cana-5898 86 79 , , PUNCT cana-5898 86 80 { { PUNCT cana-5898 86 81 b b X cana-5898 86 82 , , PUNCT cana-5898 86 83 d d NOUN cana-5898 86 84 } } PUNCT cana-5898 86 85 , , PUNCT cana-5898 86 86 { { PUNCT cana-5898 86 87 c c X cana-5898 86 88 , , PUNCT cana-5898 86 89 d d NOUN cana-5898 86 90 } } PUNCT cana-5898 86 91 , , PUNCT cana-5898 86 92 { { PUNCT cana-5898 86 93 a a DET cana-5898 86 94 , , PUNCT cana-5898 86 95 b b NOUN cana-5898 86 96 , , PUNCT cana-5898 86 97 c c NOUN cana-5898 86 98 } } PUNCT cana-5898 86 99 , , PUNCT cana-5898 86 100 { { PUNCT cana-5898 86 101 a a DET cana-5898 86 102 , , PUNCT cana-5898 86 103 b b NOUN cana-5898 86 104 , , PUNCT cana-5898 86 105 d d NOUN cana-5898 86 106 } } PUNCT cana-5898 86 107 , , PUNCT cana-5898 86 108 { { PUNCT cana-5898 86 109 a a PRON cana-5898 86 110 , , PUNCT cana-5898 86 111 c c NOUN cana-5898 86 112 , , PUNCT cana-5898 86 113 d d NOUN cana-5898 86 114 } } PUNCT cana-5898 86 115 , , PUNCT cana-5898 86 116 { { PUNCT cana-5898 86 117 b b X cana-5898 86 118 , , PUNCT cana-5898 86 119 c c NOUN cana-5898 86 120 , , PUNCT cana-5898 86 121 d d NOUN cana-5898 86 122 } } PUNCT cana-5898 86 123 } } PUNCT cana-5898 86 124 . . PUNCT cana-5898 87 1 then then ADV cana-5898 87 2 the the DET cana-5898 87 3 set set NOUN cana-5898 87 4 { { PUNCT cana-5898 87 5 a a NOUN cana-5898 87 6 , , PUNCT cana-5898 87 7 c c X cana-5898 87 8 } } PUNCT cana-5898 87 9 is be AUX cana-5898 87 10 𝒒gclosed 𝒒gclose VERB cana-5898 87 11 ( ( PUNCT cana-5898 87 12 resp resp NOUN cana-5898 87 13 . . PUNCT cana-5898 88 1 g g NOUN cana-5898 88 2 - - PUNCT cana-5898 88 3 closed closed ADJ cana-5898 88 4 ) ) PUNCT cana-5898 88 5 , , PUNCT cana-5898 88 6 but but CCONJ cana-5898 88 7 not not PART cana-5898 88 8 τ𝒒-closed τ𝒒-close VERB cana-5898 88 9 ( ( PUNCT cana-5898 88 10 resp resp NOUN cana-5898 88 11 . . PUNCT cana-5898 89 1 s*g s*g PROPN cana-5898 89 2 - - PUNCT cana-5898 89 3 closed closed ADJ cana-5898 89 4 ) ) PUNCT cana-5898 89 5 . . PUNCT cana-5898 90 1 also also ADV cana-5898 90 2 the the DET cana-5898 90 3 set set NOUN cana-5898 90 4 { { PUNCT cana-5898 90 5 d d NOUN cana-5898 90 6 } } PUNCT cana-5898 90 7 is be AUX cana-5898 90 8 𝒒g 𝒒g PROPN cana-5898 90 9 - - PUNCT cana-5898 90 10 closed close VERB cana-5898 90 11 ( ( PUNCT cana-5898 90 12 resp resp NOUN cana-5898 90 13 . . PUNCT cana-5898 91 1 gs gs NOUN cana-5898 91 2 - - PUNCT cana-5898 91 3 closed closed ADJ cana-5898 91 4 ) ) PUNCT cana-5898 91 5 but but CCONJ cana-5898 91 6 not not PART cana-5898 91 7 g g NOUN cana-5898 91 8 - - PUNCT cana-5898 91 9 closed closed ADJ cana-5898 91 10 . . PUNCT cana-5898 92 1 moreover moreover ADV cana-5898 92 2 , , PUNCT cana-5898 92 3 the the DET cana-5898 92 4 set set NOUN cana-5898 92 5 { { PUNCT cana-5898 92 6 b b NOUN cana-5898 92 7 , , PUNCT cana-5898 92 8 d d NOUN cana-5898 92 9 } } PUNCT cana-5898 92 10 is be AUX cana-5898 92 11 g𝒒-closed g𝒒-close VERB cana-5898 92 12 but but CCONJ cana-5898 92 13 it it PRON cana-5898 92 14 is be AUX cana-5898 92 15 not not PART cana-5898 92 16 𝒒g 𝒒g PROPN cana-5898 92 17 - - PUNCT cana-5898 92 18 closed close VERB cana-5898 92 19 ( ( PUNCT cana-5898 92 20 resp resp NOUN cana-5898 92 21 . . PUNCT cana-5898 93 1 gs gs NOUN cana-5898 93 2 - - PUNCT cana-5898 93 3 closed closed ADJ cana-5898 93 4 ) ) PUNCT cana-5898 93 5 . . PUNCT cana-5898 94 1 here here ADV cana-5898 94 2 the the DET cana-5898 94 3 set set NOUN cana-5898 94 4 { { PUNCT cana-5898 94 5 b b NOUN cana-5898 94 6 } } PUNCT cana-5898 94 7 is be AUX cana-5898 94 8 τ𝒒-closed τ𝒒-close VERB cana-5898 94 9 , , PUNCT cana-5898 94 10 but but CCONJ cana-5898 94 11 not not PART cana-5898 94 12 g g NOUN cana-5898 94 13 - - PUNCT cana-5898 94 14 closed closed ADJ cana-5898 94 15 ( ( PUNCT cana-5898 94 16 resp resp NOUN cana-5898 94 17 . . PUNCT cana-5898 95 1 s*g s*g PROPN cana-5898 95 2 - - PUNCT cana-5898 95 3 closed closed ADJ cana-5898 95 4 ) ) PUNCT cana-5898 95 5 . . PUNCT cana-5898 96 1 as as ADV cana-5898 96 2 well well ADV cana-5898 96 3 as as SCONJ cana-5898 96 4 the the DET cana-5898 96 5 set set NOUN cana-5898 96 6 { { PUNCT cana-5898 96 7 a a PRON cana-5898 96 8 , , PUNCT cana-5898 96 9 c c NOUN cana-5898 96 10 , , PUNCT cana-5898 96 11 d d NOUN cana-5898 96 12 } } PUNCT cana-5898 96 13 is be AUX cana-5898 96 14 g g NOUN cana-5898 96 15 - - PUNCT cana-5898 96 16 closed closed ADJ cana-5898 96 17 ( ( PUNCT cana-5898 96 18 resp resp NOUN cana-5898 96 19 . . PUNCT cana-5898 97 1 s*g s*g PROPN cana-5898 97 2 - - PUNCT cana-5898 97 3 closed closed ADJ cana-5898 97 4 ) ) PUNCT cana-5898 97 5 , , PUNCT cana-5898 97 6 but but CCONJ cana-5898 97 7 not not PART cana-5898 97 8 τ𝒒-closed τ𝒒-close VERB cana-5898 97 9 . . PUNCT cana-5898 98 1 ( ( PUNCT cana-5898 98 2 ii ii NOUN cana-5898 98 3 ) ) PUNCT cana-5898 98 4 let let VERB cana-5898 98 5 x x PUNCT cana-5898 98 6 = = PRON cana-5898 98 7 { { PUNCT cana-5898 98 8 a a PRON cana-5898 98 9 , , PUNCT cana-5898 98 10 b b NOUN cana-5898 98 11 , , PUNCT cana-5898 98 12 c c NOUN cana-5898 98 13 , , PUNCT cana-5898 98 14 d d NOUN cana-5898 98 15 } } PUNCT cana-5898 98 16 , , PUNCT cana-5898 98 17   NOUN cana-5898 98 18 = = X cana-5898 98 19 { { PUNCT cana-5898 98 20   PROPN cana-5898 98 21 , , PUNCT cana-5898 98 22 x x PRON cana-5898 98 23 , , PUNCT cana-5898 98 24 { { PUNCT cana-5898 98 25 a a X cana-5898 98 26 } } PUNCT cana-5898 98 27 , , PUNCT cana-5898 98 28 { { PUNCT cana-5898 98 29 b b NOUN cana-5898 98 30 } } PUNCT cana-5898 98 31 , , PUNCT cana-5898 98 32 { { PUNCT cana-5898 98 33 a a PRON cana-5898 98 34 , , PUNCT cana-5898 98 35 b b NOUN cana-5898 98 36 } } PUNCT cana-5898 98 37 } } PUNCT cana-5898 98 38 and and CCONJ cana-5898 98 39 𝒒 𝒒 PROPN cana-5898 98 40 = = SYM cana-5898 98 41 { { PUNCT cana-5898 98 42 x x NOUN cana-5898 98 43 , , PUNCT cana-5898 98 44 { { PUNCT cana-5898 98 45 a a X cana-5898 98 46 } } PUNCT cana-5898 98 47 , , PUNCT cana-5898 98 48 { { PUNCT cana-5898 98 49 d d NOUN cana-5898 98 50 } } PUNCT cana-5898 98 51 , , PUNCT cana-5898 98 52 { { PUNCT cana-5898 98 53 a a DET cana-5898 98 54 , , PUNCT cana-5898 98 55 b b NOUN cana-5898 98 56 } } PUNCT cana-5898 98 57 , , PUNCT cana-5898 98 58 { { PUNCT cana-5898 98 59 a a X cana-5898 98 60 , , PUNCT cana-5898 98 61 c c NOUN cana-5898 98 62 } } PUNCT cana-5898 98 63 , , PUNCT cana-5898 98 64 { { PUNCT cana-5898 98 65 a a DET cana-5898 98 66 , , PUNCT cana-5898 98 67 d d NOUN cana-5898 98 68 } } PUNCT cana-5898 98 69 , , PUNCT cana-5898 98 70 { { PUNCT cana-5898 98 71 b b X cana-5898 98 72 , , PUNCT cana-5898 98 73 d d NOUN cana-5898 98 74 } } PUNCT cana-5898 98 75 , , PUNCT cana-5898 98 76 { { PUNCT cana-5898 98 77 c c X cana-5898 98 78 , , PUNCT cana-5898 98 79 d d NOUN cana-5898 98 80 } } PUNCT cana-5898 98 81 , , PUNCT cana-5898 98 82 { { PUNCT cana-5898 98 83 a a DET cana-5898 98 84 , , PUNCT cana-5898 98 85 b b NOUN cana-5898 98 86 , , PUNCT cana-5898 98 87 c c NOUN cana-5898 98 88 } } PUNCT cana-5898 98 89 , , PUNCT cana-5898 98 90 { { PUNCT cana-5898 98 91 a a DET cana-5898 98 92 , , PUNCT cana-5898 98 93 b b NOUN cana-5898 98 94 , , PUNCT cana-5898 98 95 d d NOUN cana-5898 98 96 } } PUNCT cana-5898 98 97 , , PUNCT cana-5898 98 98 { { PUNCT cana-5898 98 99 a a PRON cana-5898 98 100 , , PUNCT cana-5898 98 101 c c NOUN cana-5898 98 102 , , PUNCT cana-5898 98 103 d d NOUN cana-5898 98 104 } } PUNCT cana-5898 98 105 , , PUNCT cana-5898 98 106 { { PUNCT cana-5898 98 107 b b X cana-5898 98 108 , , PUNCT cana-5898 98 109 c c NOUN cana-5898 98 110 , , PUNCT cana-5898 98 111 d d NOUN cana-5898 98 112 } } PUNCT cana-5898 98 113 } } PUNCT cana-5898 98 114 . . PUNCT cana-5898 99 1 then then ADV cana-5898 99 2 the the DET cana-5898 99 3 set set NOUN cana-5898 99 4 { { PUNCT cana-5898 99 5 a a PRON cana-5898 99 6 } } PUNCT cana-5898 99 7 is be AUX cana-5898 99 8 gs gs NOUN cana-5898 99 9 - - PUNCT cana-5898 99 10 closed closed ADJ cana-5898 99 11 but but CCONJ cana-5898 99 12 not not PART cana-5898 99 13 𝒒g 𝒒g PROPN cana-5898 99 14 - - PUNCT cana-5898 99 15 closed closed ADJ cana-5898 99 16 . . PUNCT cana-5898 100 1 the the DET cana-5898 100 2 following follow VERB cana-5898 100 3 diagram diagram NOUN cana-5898 100 4 shows show VERB cana-5898 100 5 the the DET cana-5898 100 6 relationship relationship NOUN cana-5898 100 7 among among ADP cana-5898 100 8 various various ADJ cana-5898 100 9 generalizations generalization NOUN cana-5898 100 10 of of ADP cana-5898 100 11 closed closed ADJ cana-5898 100 12 sets set NOUN cana-5898 100 13 . . PUNCT cana-5898 101 1 τ𝒒-closed τ𝒒-close VERB cana-5898 101 2 𝒒g 𝒒g PROPN cana-5898 101 3 - - PUNCT cana-5898 101 4 closed close VERB cana-5898 101 5 g𝒒-closed g𝒒-close VERB cana-5898 101 6 s*g s*g NOUN cana-5898 101 7 - - PUNCT cana-5898 101 8 closed close VERB cana-5898 101 9 g g NOUN cana-5898 101 10 - - PUNCT cana-5898 101 11 closed close VERB cana-5898 101 12 gs gs NOUN cana-5898 101 13 - - PUNCT cana-5898 101 14 closed closed ADJ cana-5898 101 15 remark remark NOUN cana-5898 101 16 3.4 3.4 NUM cana-5898 101 17 . . PUNCT cana-5898 102 1 in in ADP cana-5898 102 2 a a DET cana-5898 102 3 grill grill ADJ cana-5898 102 4 topological topological ADJ cana-5898 102 5 space space NOUN cana-5898 102 6 ( ( PUNCT cana-5898 102 7 x x X cana-5898 102 8 , , PUNCT cana-5898 102 9   PROPN cana-5898 102 10 , , PUNCT cana-5898 102 11 𝒒 𝒒 PROPN cana-5898 102 12 ) ) PUNCT cana-5898 102 13 , , PUNCT cana-5898 102 14 ( ( PUNCT cana-5898 102 15 i i NOUN cana-5898 102 16 ) ) PUNCT cana-5898 102 17 every every DET cana-5898 102 18 non non ADJ cana-5898 102 19 - - NOUN cana-5898 102 20 member member NOUN cana-5898 102 21 of of ADP cana-5898 102 22 𝒒 𝒒 PROPN cana-5898 102 23 is be AUX cana-5898 102 24 g𝒒-closed g𝒒-close VERB cana-5898 102 25 ; ; PUNCT cana-5898 102 26 ( ( PUNCT cana-5898 102 27 ii ii NOUN cana-5898 102 28 ) ) PUNCT cana-5898 102 29 οͺs οͺs PROPN cana-5898 102 30 is be AUX cana-5898 102 31 g𝒒-closed g𝒒-close VERB cana-5898 102 32 for for ADP cana-5898 102 33 every every DET cana-5898 102 34 subset subset NOUN cana-5898 102 35 a a PRON cana-5898 102 36 of of ADP cana-5898 102 37 x x SYM cana-5898 102 38 ; ; PUNCT cana-5898 102 39 ( ( PUNCT cana-5898 102 40 iii iii X cana-5898 102 41 ) ) PUNCT cana-5898 102 42 if if SCONJ cana-5898 102 43 𝒒 𝒒 PROPN cana-5898 102 44 = = SYM cana-5898 102 45 p(x p(x PROPN cana-5898 102 46 ) ) PUNCT cana-5898 102 47 – – PUNCT cana-5898 102 48 { { PUNCT cana-5898 102 49   NOUN cana-5898 102 50 } } PUNCT cana-5898 102 51 , , PUNCT cana-5898 102 52 then then ADV cana-5898 102 53 οͺs(a οͺs(a NOUN cana-5898 102 54 ) ) PUNCT cana-5898 102 55 = = SYM cana-5898 102 56 scl(a scl(a PROPN cana-5898 102 57 ) ) PUNCT cana-5898 102 58 and and CCONJ cana-5898 102 59 hence hence ADV cana-5898 102 60 g𝒒-closed g𝒒-close VERB cana-5898 102 61 sets set NOUN cana-5898 102 62 coincide coincide VERB cana-5898 102 63 with with ADP cana-5898 102 64 gs gs ADP cana-5898 102 65 closed closed ADJ cana-5898 102 66 sets set NOUN cana-5898 102 67 . . PUNCT cana-5898 103 1 communications communication NOUN cana-5898 103 2 on on ADP cana-5898 103 3 applied apply VERB cana-5898 103 4 nonlinear nonlinear ADJ cana-5898 103 5 analysis analysis NOUN cana-5898 103 6 issn issn NOUN cana-5898 103 7 : : PUNCT cana-5898 103 8 1074 1074 NUM cana-5898 103 9 - - PUNCT cana-5898 103 10 133x 133x NUM cana-5898 103 11 vol vol NOUN cana-5898 103 12 31 31 NUM cana-5898 103 13 no no NOUN cana-5898 103 14 . . PUNCT cana-5898 104 1 7s 7 NOUN cana-5898 104 2 ( ( PUNCT cana-5898 104 3 2024 2024 NUM cana-5898 104 4 ) ) PUNCT cana-5898 104 5 794 794 NUM cana-5898 104 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 104 7 theorem theorem VERB cana-5898 104 8 3.5 3.5 NUM cana-5898 104 9 . . PUNCT cana-5898 105 1 let let VERB cana-5898 105 2 ( ( PUNCT cana-5898 105 3 x x X cana-5898 105 4 , , PUNCT cana-5898 105 5   PROPN cana-5898 105 6 , , PUNCT cana-5898 105 7 𝒒 𝒒 PROPN cana-5898 105 8 ) ) PUNCT cana-5898 105 9 be be VERB cana-5898 105 10 a a DET cana-5898 105 11 grill grill ADJ cana-5898 105 12 topological topological ADJ cana-5898 105 13 space space NOUN cana-5898 105 14 and and CCONJ cana-5898 105 15 a a DET cana-5898 105 16 be be AUX cana-5898 105 17 a a DET cana-5898 105 18 subset subset NOUN cana-5898 105 19 of of ADP cana-5898 105 20 x. x. NOUN cana-5898 105 21 then then ADV cana-5898 105 22 the the DET cana-5898 105 23 following follow VERB cana-5898 105 24 statements statement NOUN cana-5898 105 25 are be AUX cana-5898 105 26 equivalent equivalent ADJ cana-5898 105 27 : : PUNCT cana-5898 105 28 ( ( PUNCT cana-5898 105 29 i i NOUN cana-5898 105 30 ) ) PUNCT cana-5898 105 31 a a PRON cana-5898 105 32 is be AUX cana-5898 105 33 g𝒒-closed g𝒒-close VERB cana-5898 105 34 ; ; PUNCT cana-5898 105 35 ( ( PUNCT cana-5898 105 36 ii ii NOUN cana-5898 105 37 ) ) PUNCT cana-5898 105 38 scl(οͺs(a scl(οͺs(a NOUN cana-5898 105 39 ) ) PUNCT cana-5898 105 40 ) ) PUNCT cana-5898 106 1   PROPN cana-5898 106 2 u u PROPN cana-5898 106 3 for for ADP cana-5898 106 4 every every DET cana-5898 106 5 open open ADJ cana-5898 106 6 set set NOUN cana-5898 106 7 u u NOUN cana-5898 106 8 containing contain VERB cana-5898 106 9 a a DET cana-5898 106 10 ; ; PUNCT cana-5898 106 11 ( ( PUNCT cana-5898 106 12 iii iii NOUN cana-5898 106 13 ) ) PUNCT cana-5898 106 14 for for ADP cana-5898 106 15 all all DET cana-5898 106 16 x x ADJ cana-5898 106 17 οƒŽ οƒŽ NOUN cana-5898 106 18 scl(οͺs(a scl(οͺs(a NOUN cana-5898 106 19 ) ) PUNCT cana-5898 106 20 ) ) PUNCT cana-5898 106 21 , , PUNCT cana-5898 106 22 cl({x cl({x NOUN cana-5898 106 23 } } PUNCT cana-5898 106 24 ) ) PUNCT cana-5898 106 25   PUNCT cana-5898 106 26 a a DET cana-5898 106 27 ο‚Ή ο‚Ή NOUN cana-5898 106 28   NOUN cana-5898 106 29 ; ; PUNCT cana-5898 106 30 ( ( PUNCT cana-5898 106 31 iv iv X cana-5898 106 32 ) ) PUNCT cana-5898 106 33 scl(οͺs(a scl(οͺs(a NOUN cana-5898 106 34 ) ) PUNCT cana-5898 106 35 ) ) PUNCT cana-5898 106 36 – – PUNCT cana-5898 106 37 a a PRON cana-5898 106 38 contains contain VERB cana-5898 106 39 no no DET cana-5898 106 40 non non ADJ cana-5898 106 41 empty empty ADJ cana-5898 106 42 closed closed ADJ cana-5898 106 43 set set NOUN cana-5898 106 44 ; ; PUNCT cana-5898 106 45 ( ( PUNCT cana-5898 106 46 v v NOUN cana-5898 106 47 ) ) PUNCT cana-5898 106 48 οͺs(a οͺs(a NOUN cana-5898 106 49 ) ) PUNCT cana-5898 106 50 – – PUNCT cana-5898 106 51 a a PRON cana-5898 106 52 contains contain VERB cana-5898 106 53 no no DET cana-5898 106 54 non non ADJ cana-5898 106 55 empty empty ADJ cana-5898 106 56 closed closed ADJ cana-5898 106 57 set set NOUN cana-5898 106 58 . . PUNCT cana-5898 107 1 proof proof NOUN cana-5898 107 2 . . PUNCT cana-5898 108 1 ( ( PUNCT cana-5898 108 2 i i NOUN cana-5898 108 3 ) ) PUNCT cana-5898 108 4 β‡’ β‡’ PROPN cana-5898 108 5 ( ( PUNCT cana-5898 108 6 ii ii NOUN cana-5898 108 7 ) ) PUNCT cana-5898 108 8 . . PUNCT cana-5898 109 1 let let VERB cana-5898 109 2 a a DET cana-5898 109 3 be be AUX cana-5898 109 4 a a DET cana-5898 109 5 g𝒒-closed g𝒒-close VERB cana-5898 109 6 set set NOUN cana-5898 109 7 . . PUNCT cana-5898 110 1 then then ADV cana-5898 110 2 clearly clearly ADV cana-5898 110 3 οͺs(a οͺs(a NOUN cana-5898 110 4 ) ) PUNCT cana-5898 110 5 βŠ† βŠ† NUM cana-5898 110 6 u u NOUN cana-5898 110 7 whenever whenever SCONJ cana-5898 110 8 a a DET cana-5898 110 9 βŠ† βŠ† NUM cana-5898 110 10 u u NOUN cana-5898 110 11 and and CCONJ cana-5898 110 12 u u NOUN cana-5898 110 13 is be AUX cana-5898 110 14 open open ADJ cana-5898 110 15 in in ADP cana-5898 110 16 x x PUNCT cana-5898 110 17 and and CCONJ cana-5898 110 18 so so ADV cana-5898 110 19 by by ADP cana-5898 110 20 theorem theorem ADJ cana-5898 110 21 2.1 2.1 NUM cana-5898 110 22 , , PUNCT cana-5898 110 23 scl(οͺs(a scl(οͺs(a NOUN cana-5898 110 24 ) ) PUNCT cana-5898 110 25 ) ) PUNCT cana-5898 111 1 βŠ† βŠ† X cana-5898 111 2 u u NOUN cana-5898 111 3 whenever whenever SCONJ cana-5898 111 4 a a DET cana-5898 111 5 βŠ† βŠ† NUM cana-5898 111 6 u u NOUN cana-5898 111 7 and and CCONJ cana-5898 111 8 u u NOUN cana-5898 111 9 is be AUX cana-5898 111 10 open open ADJ cana-5898 111 11 in in ADP cana-5898 111 12 x. x. NOUN cana-5898 111 13 this this PRON cana-5898 111 14 proves prove VERB cana-5898 111 15 ( ( PUNCT cana-5898 111 16 ii ii NOUN cana-5898 111 17 ) ) PUNCT cana-5898 111 18 . . PUNCT cana-5898 112 1 ( ( PUNCT cana-5898 112 2 ii ii NOUN cana-5898 112 3 ) ) PUNCT cana-5898 112 4 β‡’ β‡’ NOUN cana-5898 112 5 ( ( PUNCT cana-5898 112 6 iii iii NOUN cana-5898 112 7 ) ) PUNCT cana-5898 112 8 . . PUNCT cana-5898 113 1 suppose suppose VERB cana-5898 113 2 x x PUNCT cana-5898 113 3 οƒŽscl(οͺs(a οƒŽscl(οͺs(a NOUN cana-5898 113 4 ) ) PUNCT cana-5898 113 5 ) ) PUNCT cana-5898 113 6 . . PUNCT cana-5898 114 1 if if SCONJ cana-5898 114 2 cl({x cl({x NOUN cana-5898 114 3 } } PUNCT cana-5898 114 4 ) ) PUNCT cana-5898 115 1 ∩ ∩ NOUN cana-5898 115 2 a a DET cana-5898 115 3 = = SYM cana-5898 115 4   PROPN cana-5898 115 5 , , PUNCT cana-5898 115 6 then then ADV cana-5898 115 7 a a DET cana-5898 115 8 βŠ† βŠ† NUM cana-5898 115 9 x x SYM cana-5898 115 10 βˆ’ βˆ’ NOUN cana-5898 115 11 cl({x cl({x NOUN cana-5898 115 12 } } PUNCT cana-5898 115 13 ) ) PUNCT cana-5898 115 14 . . PUNCT cana-5898 116 1 by by ADP cana-5898 116 2 ( ( PUNCT cana-5898 116 3 ii ii NOUN cana-5898 116 4 ) ) PUNCT cana-5898 116 5 , , PUNCT cana-5898 116 6 scl(οͺs(a scl(οͺs(a NOUN cana-5898 116 7 ) ) PUNCT cana-5898 116 8 ) ) PUNCT cana-5898 117 1 βŠ† βŠ† NUM cana-5898 117 2 x x X cana-5898 117 3 βˆ’ βˆ’ NOUN cana-5898 117 4 cl({x cl({x NOUN cana-5898 117 5 } } PUNCT cana-5898 117 6 ) ) PUNCT cana-5898 117 7 . . PUNCT cana-5898 118 1 this this PRON cana-5898 118 2 contradicts contradict VERB cana-5898 118 3 the the DET cana-5898 118 4 fact fact NOUN cana-5898 118 5 that that SCONJ cana-5898 118 6 x x SYM cana-5898 118 7 ∈ ∈ PROPN cana-5898 118 8 scl(οͺs(a scl(οͺs(a NOUN cana-5898 118 9 ) ) PUNCT cana-5898 118 10 ) ) PUNCT cana-5898 118 11 . . PUNCT cana-5898 119 1 hence hence ADV cana-5898 119 2 cl({x cl({x PRON cana-5898 119 3 } } PUNCT cana-5898 119 4 ) ) PUNCT cana-5898 119 5   PUNCT cana-5898 119 6 a a DET cana-5898 119 7 β‰  β‰  PROPN cana-5898 119 8 . . NOUN cana-5898 119 9 this this PRON cana-5898 119 10 proves prove VERB cana-5898 119 11 ( ( PUNCT cana-5898 119 12 iii iii NOUN cana-5898 119 13 ) ) PUNCT cana-5898 119 14 . . PUNCT cana-5898 120 1 ( ( PUNCT cana-5898 120 2 iii iii X cana-5898 120 3 ) ) PUNCT cana-5898 120 4 οƒž οƒž NOUN cana-5898 120 5 ( ( PUNCT cana-5898 120 6 i i NOUN cana-5898 120 7 ) ) PUNCT cana-5898 120 8 . . PUNCT cana-5898 121 1 suppose suppose VERB cana-5898 121 2 that that SCONJ cana-5898 121 3 a a PRON cana-5898 121 4 is be AUX cana-5898 121 5 not not PART cana-5898 121 6 g𝒒-closed g𝒒-close VERB cana-5898 121 7 . . PUNCT cana-5898 122 1 there there PRON cana-5898 122 2 exists exist VERB cana-5898 122 3 an an DET cana-5898 122 4 open open ADJ cana-5898 122 5 set set NOUN cana-5898 122 6 u u PRON cana-5898 122 7 such such ADJ cana-5898 122 8 that that SCONJ cana-5898 122 9 a a DET cana-5898 122 10 βŠ† βŠ† NUM cana-5898 122 11 u u NOUN cana-5898 122 12 and and CCONJ cana-5898 122 13 οͺs(a οͺs(a NOUN cana-5898 122 14 ) ) PUNCT cana-5898 122 15 is be AUX cana-5898 122 16 not not PART cana-5898 122 17 contained contain VERB cana-5898 122 18 in in ADP cana-5898 122 19 u. u. NOUN cana-5898 122 20 then then ADV cana-5898 122 21 , , PUNCT cana-5898 122 22 there there PRON cana-5898 122 23 exists exist VERB cana-5898 122 24 a a DET cana-5898 122 25 point point NOUN cana-5898 122 26 x x SYM cana-5898 122 27 ∈ ∈ PROPN cana-5898 122 28 οͺs(a οͺs(a NOUN cana-5898 122 29 ) ) PUNCT cana-5898 122 30 such such ADJ cana-5898 122 31 that that SCONJ cana-5898 122 32 x x PROPN cana-5898 122 33 βˆ‰ βˆ‰ PROPN cana-5898 122 34 u. u. PROPN cana-5898 122 35 then then ADV cana-5898 122 36 we we PRON cana-5898 122 37 have have VERB cana-5898 122 38 { { PUNCT cana-5898 122 39 x x NOUN cana-5898 122 40 } } PUNCT cana-5898 122 41   ADJ cana-5898 122 42 u u NOUN cana-5898 122 43 = = X cana-5898 122 44   NOUN cana-5898 122 45 and and CCONJ cana-5898 122 46 hence hence ADV cana-5898 122 47 cl({x cl({x ADV cana-5898 122 48 } } PUNCT cana-5898 122 49 ) ) PUNCT cana-5898 122 50 ∩ ∩ NOUN cana-5898 122 51 u u NOUN cana-5898 122 52 = = NOUN cana-5898 122 53 . . NOUN cana-5898 122 54 since since SCONJ cana-5898 122 55 a a DET cana-5898 122 56   PROPN cana-5898 122 57 u u PROPN cana-5898 122 58 , , PUNCT cana-5898 122 59 cl({x cl({x NOUN cana-5898 122 60 } } PUNCT cana-5898 122 61 ) ) PUNCT cana-5898 123 1   PUNCT cana-5898 123 2 a a DET cana-5898 123 3 = = SYM cana-5898 123 4 . . NOUN cana-5898 123 5 by by ADP cana-5898 123 6 theorem theorem ADJ cana-5898 123 7 2.1 2.1 NUM cana-5898 123 8 , , PUNCT cana-5898 123 9 scl(οͺs(a scl(οͺs(a NOUN cana-5898 123 10 ) ) PUNCT cana-5898 123 11 ) ) PUNCT cana-5898 124 1 = = SYM cana-5898 124 2 οͺs(a οͺs(a NOUN cana-5898 124 3 ) ) PUNCT cana-5898 124 4 and and CCONJ cana-5898 124 5 it it PRON cana-5898 124 6 follows follow VERB cana-5898 124 7 that that SCONJ cana-5898 124 8 ( ( PUNCT cana-5898 124 9 iii iii X cana-5898 124 10 ) ) PUNCT cana-5898 124 11 does do AUX cana-5898 124 12 not not PART cana-5898 124 13 hold hold VERB cana-5898 124 14 . . PUNCT cana-5898 125 1 therefore therefore ADV cana-5898 125 2 , , PUNCT cana-5898 125 3 the the DET cana-5898 125 4 proof proof NOUN cana-5898 125 5 completes complete VERB cana-5898 125 6 . . PUNCT cana-5898 126 1 ( ( PUNCT cana-5898 126 2 iii iii X cana-5898 126 3 ) ) PUNCT cana-5898 126 4 οƒž οƒž NOUN cana-5898 126 5 ( ( PUNCT cana-5898 126 6 iv iv NOUN cana-5898 126 7 ) ) PUNCT cana-5898 126 8 . . PUNCT cana-5898 127 1 suppose suppose VERB cana-5898 127 2 f f PROPN cana-5898 127 3 is be AUX cana-5898 127 4 a a DET cana-5898 127 5 closed closed ADJ cana-5898 127 6 set set NOUN cana-5898 127 7 of of ADP cana-5898 127 8 x x PUNCT cana-5898 127 9 contained contain VERB cana-5898 127 10 in in ADP cana-5898 127 11 scl(οͺs(a scl(οͺs(a NOUN cana-5898 127 12 ) ) PUNCT cana-5898 127 13 ) ) PUNCT cana-5898 127 14 – – PUNCT cana-5898 128 1 a a PRON cana-5898 128 2 and and CCONJ cana-5898 128 3 x x SYM cana-5898 128 4 οƒŽ οƒŽ PROPN cana-5898 128 5 f. f. PROPN cana-5898 128 6 since since SCONJ cana-5898 128 7 f f PROPN cana-5898 128 8   PUNCT cana-5898 128 9 a a DET cana-5898 128 10 = = SYM cana-5898 128 11   PROPN cana-5898 128 12 , , PUNCT cana-5898 128 13 we we PRON cana-5898 128 14 have have VERB cana-5898 128 15 cl({x cl({x VERB cana-5898 128 16 } } PUNCT cana-5898 128 17 ) ) PUNCT cana-5898 129 1   PUNCT cana-5898 129 2 a a DET cana-5898 129 3 = = SYM cana-5898 129 4 . . NOUN cana-5898 129 5 again again ADV cana-5898 129 6 , , PUNCT cana-5898 129 7 since since SCONJ cana-5898 129 8 x x PROPN cana-5898 129 9 οƒŽscl(οͺs(a οƒŽscl(οͺs(a NOUN cana-5898 129 10 ) ) PUNCT cana-5898 129 11 ) ) PUNCT cana-5898 129 12 , , PUNCT cana-5898 129 13 by by ADP cana-5898 129 14 ( ( PUNCT cana-5898 129 15 iii iii X cana-5898 129 16 ) ) PUNCT cana-5898 129 17 we we PRON cana-5898 129 18 have have VERB cana-5898 129 19 cl({x cl({x VERB cana-5898 129 20 } } PUNCT cana-5898 129 21 ) ) PUNCT cana-5898 129 22   PUNCT cana-5898 129 23 a a DET cana-5898 129 24 ο‚Ή ο‚Ή NOUN cana-5898 129 25   PROPN cana-5898 129 26 , , PUNCT cana-5898 129 27 a a DET cana-5898 129 28 contradiction contradiction NOUN cana-5898 129 29 . . PUNCT cana-5898 130 1 this this PRON cana-5898 130 2 proves prove VERB cana-5898 130 3 ( ( PUNCT cana-5898 130 4 iv iv NUM cana-5898 130 5 ) ) PUNCT cana-5898 130 6 . . PUNCT cana-5898 131 1 it it PRON cana-5898 131 2 follows follow VERB cana-5898 131 3 from from ADP cana-5898 131 4 theorem theorem ADJ cana-5898 131 5 2.1 2.1 NUM cana-5898 131 6 that that PRON cana-5898 131 7 ( ( PUNCT cana-5898 131 8 iv iv X cana-5898 131 9 ) ) PUNCT cana-5898 131 10 and and CCONJ cana-5898 131 11 ( ( PUNCT cana-5898 131 12 v v NOUN cana-5898 131 13 ) ) PUNCT cana-5898 131 14 are be AUX cana-5898 131 15 equivalent equivalent ADJ cana-5898 131 16 . . PUNCT cana-5898 132 1 theorem theorem VERB cana-5898 132 2 3.6 3.6 NUM cana-5898 132 3 . . PUNCT cana-5898 133 1 let let VERB cana-5898 133 2 ( ( PUNCT cana-5898 133 3 x x X cana-5898 133 4 , , PUNCT cana-5898 133 5   PROPN cana-5898 133 6 , , PUNCT cana-5898 133 7 𝒒 𝒒 PROPN cana-5898 133 8 ) ) PUNCT cana-5898 133 9 be be VERB cana-5898 133 10 a a DET cana-5898 133 11 grill grill ADJ cana-5898 133 12 topological topological ADJ cana-5898 133 13 space space NOUN cana-5898 133 14 and and CCONJ cana-5898 133 15 { { PUNCT cana-5898 133 16 ai ai VERB cana-5898 133 17 : : PUNCT cana-5898 133 18 i i PROPN cana-5898 133 19 ∈ ∈ PROPN cana-5898 133 20 j j PROPN cana-5898 133 21 } } PUNCT cana-5898 133 22 be be AUX cana-5898 133 23 a a DET cana-5898 133 24 locally locally ADV cana-5898 133 25 finite finite ADJ cana-5898 133 26 family family NOUN cana-5898 133 27 of of ADP cana-5898 133 28 sets set NOUN cana-5898 133 29 in in ADP cana-5898 133 30 x. x. NOUN cana-5898 133 31 then then ADV cana-5898 133 32 οƒˆi∈j(οͺ οƒˆi∈j(οͺ PROPN cana-5898 133 33 s(ai s(ai PROPN cana-5898 133 34 ) ) PUNCT cana-5898 133 35 ) ) PUNCT cana-5898 134 1 = = SYM cana-5898 134 2 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai X cana-5898 134 3 ) ) PUNCT cana-5898 134 4 . . PUNCT cana-5898 135 1 proof proof NOUN cana-5898 135 2 . . PUNCT cana-5898 136 1 ai ai VERB cana-5898 136 2   PROPN cana-5898 136 3 οƒˆiοƒŽjai οƒˆiοƒŽjai PROPN cana-5898 136 4 implies imply VERB cana-5898 136 5 οͺs(ai οͺs(ai PROPN cana-5898 136 6 ) ) PUNCT cana-5898 136 7   PROPN cana-5898 136 8 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai NUM cana-5898 136 9 ) ) PUNCT cana-5898 136 10 for for ADP cana-5898 136 11 every every DET cana-5898 136 12 i i PROPN cana-5898 136 13 οƒŽ οƒŽ PROPN cana-5898 136 14 j. j. PROPN cana-5898 137 1 this this PRON cana-5898 137 2 implies imply VERB cana-5898 137 3 οƒˆi∈j(οͺ οƒˆi∈j(οͺ PROPN cana-5898 137 4 s(ai s(ai PROPN cana-5898 137 5 ) ) PUNCT cana-5898 137 6 ) ) PUNCT cana-5898 137 7   PROPN cana-5898 137 8 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai NUM cana-5898 137 9 ) ) PUNCT cana-5898 137 10 . . PUNCT cana-5898 138 1 conversely conversely ADV cana-5898 138 2 , , PUNCT cana-5898 138 3 let let VERB cana-5898 138 4 x x PRON cana-5898 138 5 οƒŽ οƒŽ NOUN cana-5898 138 6 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai NUM cana-5898 138 7 ) ) PUNCT cana-5898 138 8 and and CCONJ cana-5898 138 9 v v X cana-5898 138 10 be be AUX cana-5898 138 11 any any DET cana-5898 138 12 semiopen semiopen ADJ cana-5898 138 13 set set NOUN cana-5898 138 14 of of ADP cana-5898 138 15 x x PUNCT cana-5898 138 16 containing contain VERB cana-5898 138 17 x. x. NOUN cana-5898 138 18 since since SCONJ cana-5898 138 19 { { PUNCT cana-5898 138 20 ai ai VERB cana-5898 138 21 : : PUNCT cana-5898 138 22 i i PROPN cana-5898 138 23 ∈ ∈ PROPN cana-5898 138 24 j j PROPN cana-5898 138 25 } } PUNCT cana-5898 138 26 is be AUX cana-5898 138 27 locally locally ADV cana-5898 138 28 finite finite ADJ cana-5898 138 29 , , PUNCT cana-5898 138 30 there there PRON cana-5898 138 31 exists exist VERB cana-5898 138 32 an an DET cana-5898 138 33 open open ADJ cana-5898 138 34 set set NOUN cana-5898 138 35 u u NOUN cana-5898 138 36 in in ADP cana-5898 138 37 x x PUNCT cana-5898 138 38 containing contain VERB cana-5898 138 39 x x DET cana-5898 138 40 that that SCONJ cana-5898 138 41 communications communication NOUN cana-5898 138 42 on on ADP cana-5898 138 43 applied apply VERB cana-5898 138 44 nonlinear nonlinear ADJ cana-5898 138 45 analysis analysis NOUN cana-5898 138 46 issn issn NOUN cana-5898 138 47 : : PUNCT cana-5898 138 48 1074 1074 NUM cana-5898 138 49 - - PUNCT cana-5898 138 50 133x 133x NUM cana-5898 138 51 vol vol NOUN cana-5898 138 52 31 31 NUM cana-5898 138 53 no no NOUN cana-5898 138 54 . . PUNCT cana-5898 139 1 7s 7 NOUN cana-5898 139 2 ( ( PUNCT cana-5898 139 3 2024 2024 NUM cana-5898 139 4 ) ) PUNCT cana-5898 139 5 795 795 NUM cana-5898 139 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 139 7 intersects intersect NOUN cana-5898 139 8 only only ADV cana-5898 139 9 a a DET cana-5898 139 10 finite finite ADJ cana-5898 139 11 number number NOUN cana-5898 139 12 of of ADP cana-5898 139 13 members member NOUN cana-5898 139 14 , , PUNCT cana-5898 139 15 says say VERB cana-5898 139 16 , , PUNCT cana-5898 139 17 ai1 ai1 PROPN cana-5898 139 18 , , PUNCT cana-5898 139 19 ai2 ai2 NOUN cana-5898 139 20 , , PUNCT cana-5898 139 21 ... ... PUNCT cana-5898 139 22 , , PUNCT cana-5898 139 23 ain ain PROPN cana-5898 139 24 of of ADP cana-5898 139 25 { { PUNCT cana-5898 139 26 ai ai VERB cana-5898 139 27 : : PUNCT cana-5898 139 28 i i PROPN cana-5898 139 29 ∈ ∈ PROPN cana-5898 139 30 j j PROPN cana-5898 139 31 } } PUNCT cana-5898 139 32 . . PUNCT cana-5898 140 1 but but CCONJ cana-5898 140 2 x x X cana-5898 140 3 ∈ ∈ PROPN cana-5898 140 4 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai NOUN cana-5898 140 5 ) ) PUNCT cana-5898 140 6 implies imply VERB cana-5898 140 7 ( ( PUNCT cana-5898 140 8 v v NUM cana-5898 140 9 ∩ ∩ ADJ cana-5898 140 10 u u NOUN cana-5898 140 11 ) ) PUNCT cana-5898 140 12 ∩ ∩ NOUN cana-5898 140 13 ( ( PUNCT cana-5898 140 14 οƒˆiοƒŽjai οƒˆiοƒŽjai X cana-5898 140 15 ) ) PUNCT cana-5898 140 16 = = SYM cana-5898 140 17 οƒˆi∈j((v οƒˆi∈j((v NOUN cana-5898 140 18 ∩ ∩ ADJ cana-5898 140 19 u u NOUN cana-5898 140 20 ) ) PUNCT cana-5898 140 21 ∩ ∩ NOUN cana-5898 140 22 ai ai NOUN cana-5898 140 23 ) ) PUNCT cana-5898 141 1 οƒŽ οƒŽ NOUN cana-5898 141 2 𝒒 𝒒 PROPN cana-5898 141 3 for for ADP cana-5898 141 4 every every DET cana-5898 141 5 v v NOUN cana-5898 141 6 οƒŽ οƒŽ NOUN cana-5898 141 7 so so SCONJ cana-5898 141 8 ( ( PUNCT cana-5898 141 9 x x NOUN cana-5898 141 10 , , PUNCT cana-5898 141 11 x x NOUN cana-5898 141 12 ) ) PUNCT cana-5898 141 13 . . PUNCT cana-5898 142 1 this this PRON cana-5898 142 2 gives give VERB cana-5898 142 3 οƒˆπ‘˜=1 οƒˆπ‘˜=1 NUM cana-5898 142 4 𝑛 𝑛 PROPN cana-5898 142 5 ( ( PUNCT cana-5898 142 6 ( ( PUNCT cana-5898 142 7 v v X cana-5898 142 8 ∩ ∩ ADJ cana-5898 142 9 u u NOUN cana-5898 142 10 ) ) PUNCT cana-5898 142 11 ∩ ∩ ADJ cana-5898 142 12 aik aik NOUN cana-5898 142 13 ) ) PUNCT cana-5898 142 14 οƒŽ οƒŽ NOUN cana-5898 142 15 𝒒 𝒒 PROPN cana-5898 142 16 for for ADP cana-5898 142 17 every every DET cana-5898 142 18 v v NOUN cana-5898 142 19 οƒŽ οƒŽ NOUN cana-5898 143 1 so so SCONJ cana-5898 143 2 ( ( PUNCT cana-5898 143 3 x x NOUN cana-5898 143 4 , , PUNCT cana-5898 143 5 x x NOUN cana-5898 143 6 ) ) PUNCT cana-5898 143 7 . . PUNCT cana-5898 144 1 therefore therefore ADV cana-5898 144 2 , , PUNCT cana-5898 144 3 there there PRON cana-5898 144 4 exists exist VERB cana-5898 144 5 at at ADP cana-5898 144 6 least least ADJ cana-5898 144 7 one one NUM cana-5898 144 8 aij aij PROPN cana-5898 144 9 ∈ ∈ PROPN cana-5898 144 10 { { PUNCT cana-5898 144 11 ai1 ai1 PROPN cana-5898 144 12 , , PUNCT cana-5898 144 13 ai2 ai2 NOUN cana-5898 144 14 , , PUNCT cana-5898 144 15 ... ... PUNCT cana-5898 144 16 , , PUNCT cana-5898 144 17 ain ain PROPN cana-5898 144 18 } } PUNCT cana-5898 144 19 such such ADJ cana-5898 144 20 that that SCONJ cana-5898 144 21 ( ( PUNCT cana-5898 144 22 v v NUM cana-5898 144 23 ∩ ∩ ADJ cana-5898 144 24 u u NOUN cana-5898 144 25 ) ) PUNCT cana-5898 144 26 ∩ ∩ PROPN cana-5898 144 27 aij aij PROPN cana-5898 144 28 οƒŽ οƒŽ PROPN cana-5898 144 29 𝒒 𝒒 PROPN cana-5898 144 30 hence hence ADV cana-5898 144 31 v v ADP cana-5898 144 32 ∩ ∩ X cana-5898 144 33 aij aij PROPN cana-5898 144 34 οƒŽ οƒŽ PROPN cana-5898 144 35 𝒒. 𝒒. PROPN cana-5898 144 36 this this PRON cana-5898 144 37 gives give VERB cana-5898 144 38 x x PUNCT cana-5898 144 39 ∈ ∈ PROPN cana-5898 144 40 οͺs(aij οͺs(aij PROPN cana-5898 144 41 ) ) PUNCT cana-5898 144 42 which which PRON cana-5898 144 43 implies imply VERB cana-5898 144 44 x x SYM cana-5898 144 45 ∈ ∈ PROPN cana-5898 144 46 οƒˆπ‘˜=1 οƒˆπ‘˜=1 NUM cana-5898 144 47 𝑛 𝑛 PROPN cana-5898 144 48 ( ( PUNCT cana-5898 144 49 οͺs(aik οͺs(aik NOUN cana-5898 144 50 ) ) PUNCT cana-5898 144 51 ) ) PUNCT cana-5898 144 52 and and CCONJ cana-5898 144 53 hence hence ADV cana-5898 144 54 x x PART cana-5898 144 55 ∈ ∈ PROPN cana-5898 144 56 οƒˆi∈j(οͺ οƒˆi∈j(οͺ NOUN cana-5898 144 57 s(ai s(ai PROPN cana-5898 144 58 ) ) PUNCT cana-5898 144 59 ) ) PUNCT cana-5898 144 60 . . PUNCT cana-5898 145 1 this this PRON cana-5898 145 2 proves prove VERB cana-5898 145 3 that that SCONJ cana-5898 145 4 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai PRON cana-5898 145 5 ) ) PUNCT cana-5898 145 6   PROPN cana-5898 145 7 οƒˆi∈j(οͺ οƒˆi∈j(οͺ PROPN cana-5898 145 8 s(ai s(ai PROPN cana-5898 145 9 ) ) PUNCT cana-5898 145 10 ) ) PUNCT cana-5898 145 11 . . PUNCT cana-5898 146 1 this this PRON cana-5898 146 2 completes complete VERB cana-5898 146 3 the the DET cana-5898 146 4 proof proof NOUN cana-5898 146 5 . . PUNCT cana-5898 147 1 theorem theorem VERB cana-5898 147 2 3.7 3.7 NUM cana-5898 147 3 . . PUNCT cana-5898 148 1 let let AUX cana-5898 148 2 ( ( PUNCT cana-5898 148 3 x x X cana-5898 148 4 , , PUNCT cana-5898 148 5   PROPN cana-5898 148 6 , , PUNCT cana-5898 148 7 𝒒 𝒒 PROPN cana-5898 148 8 ) ) PUNCT cana-5898 148 9 be be VERB cana-5898 148 10 a a DET cana-5898 148 11 grill grill ADJ cana-5898 148 12 topological topological ADJ cana-5898 148 13 space space NOUN cana-5898 148 14 . . PUNCT cana-5898 149 1 if if SCONJ cana-5898 149 2 { { PUNCT cana-5898 149 3 ai ai VERB cana-5898 149 4 : : PUNCT cana-5898 149 5 i i PROPN cana-5898 149 6 ∈ ∈ PROPN cana-5898 149 7 j j PROPN cana-5898 149 8 } } PUNCT cana-5898 149 9 is be AUX cana-5898 149 10 a a DET cana-5898 149 11 locally locally ADV cana-5898 149 12 finite finite ADJ cana-5898 149 13 family family NOUN cana-5898 149 14 of of ADP cana-5898 149 15 sets set NOUN cana-5898 149 16 and and CCONJ cana-5898 149 17 each each DET cana-5898 149 18 ai ai VERB cana-5898 149 19 is be AUX cana-5898 149 20 g𝒒-closed g𝒒-close VERB cana-5898 149 21 , , PUNCT cana-5898 149 22 then then ADV cana-5898 149 23 οƒˆiοƒŽjai οƒˆiοƒŽjai PROPN cana-5898 149 24 is be AUX cana-5898 149 25 g𝒒-closed g𝒒-close VERB cana-5898 149 26 in in ADP cana-5898 149 27 x. x. NOUN cana-5898 149 28 proof proof NOUN cana-5898 149 29 . . PUNCT cana-5898 150 1 let let VERB cana-5898 150 2 οƒˆiοƒŽjai οƒˆiοƒŽjai PROPN cana-5898 150 3 βŠ† βŠ† NUM cana-5898 150 4 u u NOUN cana-5898 150 5 , , PUNCT cana-5898 150 6 where where SCONJ cana-5898 150 7 u u NOUN cana-5898 150 8 is be AUX cana-5898 150 9 open open ADJ cana-5898 150 10 in in ADP cana-5898 150 11 x. x. NOUN cana-5898 150 12 since since SCONJ cana-5898 150 13 ai ai PROPN cana-5898 150 14 is be AUX cana-5898 150 15 g𝒒-closed g𝒒-close VERB cana-5898 150 16 for for ADP cana-5898 150 17 each each DET cana-5898 150 18 i i PROPN cana-5898 150 19 ∈ ∈ PROPN cana-5898 150 20 j j PROPN cana-5898 150 21 , , PUNCT cana-5898 150 22 then then ADV cana-5898 150 23 οͺs(ai οͺs(ai PROPN cana-5898 150 24 ) ) PUNCT cana-5898 150 25 βŠ† βŠ† NUM cana-5898 150 26 u. u. NOUN cana-5898 150 27 hence hence ADV cana-5898 150 28 οƒˆi∈j(οͺ οƒˆi∈j(οͺ PROPN cana-5898 150 29 s(ai s(ai NOUN cana-5898 150 30 ) ) PUNCT cana-5898 150 31 ) ) PUNCT cana-5898 151 1 βŠ† βŠ† X cana-5898 151 2 u. u. NOUN cana-5898 151 3 by by ADP cana-5898 151 4 theorem theorem ADJ cana-5898 151 5 3.6 3.6 NUM cana-5898 151 6 , , PUNCT cana-5898 151 7 οͺs(οƒˆiοƒŽjai οͺs(οƒˆiοƒŽjai NUM cana-5898 151 8 ) ) PUNCT cana-5898 151 9 βŠ† βŠ† NUM cana-5898 151 10 u. u. NOUN cana-5898 151 11 hence hence ADV cana-5898 151 12 οƒˆiοƒŽjai οƒˆiοƒŽjai PROPN cana-5898 151 13 is be AUX cana-5898 151 14 g𝒒-closed g𝒒-close VERB cana-5898 151 15 in in ADP cana-5898 151 16 x. x. NOUN cana-5898 151 17 remark remark PROPN cana-5898 151 18 3.8 3.8 NUM cana-5898 151 19 . . PUNCT cana-5898 152 1 the the DET cana-5898 152 2 following follow VERB cana-5898 152 3 example example NOUN cana-5898 152 4 shows show VERB cana-5898 152 5 that that SCONJ cana-5898 152 6 the the DET cana-5898 152 7 intersection intersection NOUN cana-5898 152 8 of of ADP cana-5898 152 9 two two NUM cana-5898 152 10 g𝒒-closed g𝒒-close VERB cana-5898 152 11 sets set NOUN cana-5898 152 12 need need AUX cana-5898 152 13 not not PART cana-5898 152 14 be be AUX cana-5898 152 15 g𝒒-closed g𝒒-close VERB cana-5898 152 16 . . PUNCT cana-5898 153 1 let let VERB cana-5898 153 2 x x PUNCT cana-5898 153 3 = = PRON cana-5898 153 4 { { PUNCT cana-5898 153 5 a a PRON cana-5898 153 6 , , PUNCT cana-5898 153 7 b b NOUN cana-5898 153 8 , , PUNCT cana-5898 153 9 c c NOUN cana-5898 153 10 , , PUNCT cana-5898 153 11 d d NOUN cana-5898 153 12 } } PUNCT cana-5898 153 13 , , PUNCT cana-5898 153 14   NOUN cana-5898 153 15 = = X cana-5898 153 16 { { PUNCT cana-5898 153 17   PROPN cana-5898 153 18 , , PUNCT cana-5898 153 19 x x PRON cana-5898 153 20 , , PUNCT cana-5898 153 21 { { PUNCT cana-5898 153 22 a a PRON cana-5898 153 23 , , PUNCT cana-5898 153 24 b b NOUN cana-5898 153 25 } } PUNCT cana-5898 153 26 } } PUNCT cana-5898 153 27 and and CCONJ cana-5898 153 28 𝒒 𝒒 PROPN cana-5898 153 29 = = SYM cana-5898 153 30 { { PUNCT cana-5898 153 31 x x NOUN cana-5898 153 32 , , PUNCT cana-5898 153 33 { { PUNCT cana-5898 153 34 a a X cana-5898 153 35 } } PUNCT cana-5898 153 36 , , PUNCT cana-5898 153 37 { { PUNCT cana-5898 153 38 b b NOUN cana-5898 153 39 } } PUNCT cana-5898 153 40 , , PUNCT cana-5898 153 41 { { PUNCT cana-5898 153 42 d d X cana-5898 153 43 } } PUNCT cana-5898 153 44 , , PUNCT cana-5898 153 45 { { PUNCT cana-5898 153 46 a a DET cana-5898 153 47 , , PUNCT cana-5898 153 48 b b NOUN cana-5898 153 49 } } PUNCT cana-5898 153 50 , , PUNCT cana-5898 153 51 { { PUNCT cana-5898 153 52 a a X cana-5898 153 53 , , PUNCT cana-5898 153 54 c c NOUN cana-5898 153 55 } } PUNCT cana-5898 153 56 , , PUNCT cana-5898 153 57 { { PUNCT cana-5898 153 58 a a DET cana-5898 153 59 , , PUNCT cana-5898 153 60 d d NOUN cana-5898 153 61 } } PUNCT cana-5898 153 62 , , PUNCT cana-5898 153 63 { { PUNCT cana-5898 153 64 b b X cana-5898 153 65 , , PUNCT cana-5898 153 66 c c NOUN cana-5898 153 67 } } PUNCT cana-5898 153 68 , , PUNCT cana-5898 153 69 { { PUNCT cana-5898 153 70 b b X cana-5898 153 71 , , PUNCT cana-5898 153 72 d d NOUN cana-5898 153 73 } } PUNCT cana-5898 153 74 , , PUNCT cana-5898 153 75 { { PUNCT cana-5898 153 76 c c X cana-5898 153 77 , , PUNCT cana-5898 153 78 d d NOUN cana-5898 153 79 } } PUNCT cana-5898 153 80 , , PUNCT cana-5898 153 81 { { PUNCT cana-5898 153 82 a a DET cana-5898 153 83 , , PUNCT cana-5898 153 84 b b NOUN cana-5898 153 85 , , PUNCT cana-5898 153 86 c c NOUN cana-5898 153 87 } } PUNCT cana-5898 153 88 , , PUNCT cana-5898 153 89 { { PUNCT cana-5898 153 90 a a DET cana-5898 153 91 , , PUNCT cana-5898 153 92 b b NOUN cana-5898 153 93 , , PUNCT cana-5898 153 94 d d NOUN cana-5898 153 95 } } PUNCT cana-5898 153 96 , , PUNCT cana-5898 153 97 { { PUNCT cana-5898 153 98 a a PRON cana-5898 153 99 , , PUNCT cana-5898 153 100 c c NOUN cana-5898 153 101 , , PUNCT cana-5898 153 102 d d NOUN cana-5898 153 103 } } PUNCT cana-5898 153 104 , , PUNCT cana-5898 153 105 { { PUNCT cana-5898 153 106 b b X cana-5898 153 107 , , PUNCT cana-5898 153 108 c c NOUN cana-5898 153 109 , , PUNCT cana-5898 153 110 d d NOUN cana-5898 153 111 } } PUNCT cana-5898 153 112 } } PUNCT cana-5898 153 113 . . PUNCT cana-5898 154 1 then then ADV cana-5898 154 2 a a DET cana-5898 154 3 = = X cana-5898 154 4 { { PUNCT cana-5898 154 5 a a X cana-5898 154 6 , , PUNCT cana-5898 154 7 c c NOUN cana-5898 154 8 } } PUNCT cana-5898 154 9 and and CCONJ cana-5898 154 10 b b X cana-5898 154 11 = = NOUN cana-5898 154 12 { { PUNCT cana-5898 154 13 a a NOUN cana-5898 154 14 , , PUNCT cana-5898 154 15 d d NOUN cana-5898 154 16 } } PUNCT cana-5898 154 17 are be AUX cana-5898 154 18 g𝒒-closed g𝒒-close VERB cana-5898 154 19 sets set NOUN cana-5898 154 20 , , PUNCT cana-5898 154 21 but but CCONJ cana-5898 154 22 a a DET cana-5898 154 23   NOUN cana-5898 154 24 b b NOUN cana-5898 154 25 = = PUNCT cana-5898 154 26 { { PUNCT cana-5898 154 27 a a PRON cana-5898 154 28 } } PUNCT cana-5898 154 29 is be AUX cana-5898 154 30 not not PART cana-5898 154 31 g𝒒-closed g𝒒-close VERB cana-5898 154 32 . . PUNCT cana-5898 155 1 theorem theorem VERB cana-5898 155 2 3.9 3.9 NUM cana-5898 155 3 . . PUNCT cana-5898 156 1 if if SCONJ cana-5898 156 2 a a PRON cana-5898 156 3 and and CCONJ cana-5898 156 4 b b NOUN cana-5898 156 5 are be AUX cana-5898 156 6 subsets subset NOUN cana-5898 156 7 of of ADP cana-5898 156 8 a a DET cana-5898 156 9 grill grill ADJ cana-5898 156 10 topological topological ADJ cana-5898 156 11 space space NOUN cana-5898 156 12 ( ( PUNCT cana-5898 156 13 x x X cana-5898 156 14 , , PUNCT cana-5898 156 15   PROPN cana-5898 156 16 , , PUNCT cana-5898 156 17 𝒒 𝒒 PROPN cana-5898 156 18 ) ) PUNCT cana-5898 156 19 , , PUNCT cana-5898 156 20 then then ADV cana-5898 156 21 οͺs οͺs PROPN cana-5898 156 22 ( ( PUNCT cana-5898 156 23 a a DET cana-5898 156 24   ADJ cana-5898 156 25 b b NOUN cana-5898 156 26 ) ) PUNCT cana-5898 156 27 βŠ† βŠ† NUM cana-5898 156 28 οͺs(a οͺs(a NOUN cana-5898 156 29 ) ) PUNCT cana-5898 156 30   NOUN cana-5898 156 31 οͺs(b οͺs(b NUM cana-5898 156 32 ) ) PUNCT cana-5898 156 33 . . PUNCT cana-5898 157 1 theorem theorem VERB cana-5898 157 2 3.10 3.10 NUM cana-5898 157 3 . . PUNCT cana-5898 158 1 let let AUX cana-5898 158 2 ( ( PUNCT cana-5898 158 3 x x X cana-5898 158 4 , , PUNCT cana-5898 158 5   PROPN cana-5898 158 6 , , PUNCT cana-5898 158 7 𝒒 𝒒 PROPN cana-5898 158 8 ) ) PUNCT cana-5898 158 9 be be VERB cana-5898 158 10 a a DET cana-5898 158 11 grill grill ADJ cana-5898 158 12 topological topological ADJ cana-5898 158 13 space space NOUN cana-5898 158 14 . . PUNCT cana-5898 159 1 if if SCONJ cana-5898 159 2 a a PRON cana-5898 159 3 is be AUX cana-5898 159 4 g𝒒-closed g𝒒-close VERB cana-5898 159 5 and and CCONJ cana-5898 159 6 b b NOUN cana-5898 159 7 is be AUX cana-5898 159 8 closed close VERB cana-5898 159 9 in in ADP cana-5898 159 10 x x NOUN cana-5898 159 11 , , PUNCT cana-5898 159 12 then then ADV cana-5898 159 13 a a DET cana-5898 159 14 ∩ ∩ ADJ cana-5898 159 15 b b NOUN cana-5898 159 16 is be AUX cana-5898 159 17 g𝒒-closed g𝒒-close VERB cana-5898 159 18 . . PUNCT cana-5898 160 1 proof proof NOUN cana-5898 160 2 . . PUNCT cana-5898 161 1 let let VERB cana-5898 161 2 u u PRON cana-5898 161 3 be be AUX cana-5898 161 4 an an DET cana-5898 161 5 open open ADJ cana-5898 161 6 set set NOUN cana-5898 161 7 in in ADP cana-5898 161 8 x x PUNCT cana-5898 161 9 containing contain VERB cana-5898 161 10 a a DET cana-5898 161 11 ∩ ∩ ADJ cana-5898 161 12 b. b. NOUN cana-5898 161 13 then then ADV cana-5898 161 14 a a DET cana-5898 161 15 βŠ† βŠ† NUM cana-5898 161 16 u u NOUN cana-5898 161 17 οƒˆ(x οƒˆ(x ADV cana-5898 161 18 βˆ’ βˆ’ PROPN cana-5898 161 19 b b NOUN cana-5898 161 20 ) ) PUNCT cana-5898 161 21 . . PUNCT cana-5898 162 1 since since SCONJ cana-5898 162 2 a a PRON cana-5898 162 3 is be AUX cana-5898 162 4 g𝒒-closed g𝒒-close VERB cana-5898 162 5 , , PUNCT cana-5898 162 6 we we PRON cana-5898 162 7 have have VERB cana-5898 162 8 οͺs(a οͺs(a NOUN cana-5898 162 9 ) ) PUNCT cana-5898 162 10 βŠ† βŠ† NUM cana-5898 162 11 u u NOUN cana-5898 162 12 οƒˆ(x οƒˆ(x ADV cana-5898 162 13 βˆ’ βˆ’ PROPN cana-5898 162 14 b b X cana-5898 162 15 ) ) PUNCT cana-5898 162 16 and and CCONJ cana-5898 162 17 b b PROPN cana-5898 162 18 ∩ ∩ ADJ cana-5898 162 19 οͺs(a οͺs(a NOUN cana-5898 162 20 ) ) PUNCT cana-5898 162 21 βŠ† βŠ† NUM cana-5898 162 22 u. u. NOUN cana-5898 162 23 using use VERB cana-5898 162 24 theorem theorem NOUN cana-5898 162 25 3.9 3.9 NUM cana-5898 162 26 , , PUNCT cana-5898 162 27 οͺs(a οͺs(a NOUN cana-5898 162 28   PUNCT cana-5898 162 29 b b NOUN cana-5898 162 30 ) ) PUNCT cana-5898 162 31 βŠ† βŠ† NUM cana-5898 162 32 οͺs(a οͺs(a NOUN cana-5898 162 33 ) ) PUNCT cana-5898 162 34   NOUN cana-5898 162 35 οͺs(b οͺs(b NOUN cana-5898 162 36 ) ) PUNCT cana-5898 162 37 βŠ† βŠ† NUM cana-5898 162 38 οͺs(a οͺs(a NOUN cana-5898 162 39 ) ) PUNCT cana-5898 162 40 ∩ ∩ NOUN cana-5898 162 41 b b X cana-5898 162 42 βŠ† βŠ† NUM cana-5898 162 43 u u NOUN cana-5898 162 44 because because SCONJ cana-5898 162 45 b b PROPN cana-5898 162 46 is be AUX cana-5898 162 47 closed closed ADJ cana-5898 162 48 . . PUNCT cana-5898 163 1 this this PRON cana-5898 163 2 proves prove VERB cana-5898 163 3 that that SCONJ cana-5898 163 4 a a DET cana-5898 163 5 ∩ ∩ ADJ cana-5898 163 6 b b NOUN cana-5898 163 7 is be AUX cana-5898 163 8 g𝒒-closed g𝒒-close VERB cana-5898 163 9 . . PUNCT cana-5898 164 1 communications communication NOUN cana-5898 164 2 on on ADP cana-5898 164 3 applied apply VERB cana-5898 164 4 nonlinear nonlinear ADJ cana-5898 164 5 analysis analysis NOUN cana-5898 164 6 issn issn NOUN cana-5898 164 7 : : PUNCT cana-5898 164 8 1074 1074 NUM cana-5898 164 9 - - PUNCT cana-5898 164 10 133x 133x NUM cana-5898 164 11 vol vol NOUN cana-5898 164 12 31 31 NUM cana-5898 164 13 no no NOUN cana-5898 164 14 . . PUNCT cana-5898 165 1 7s 7 NOUN cana-5898 165 2 ( ( PUNCT cana-5898 165 3 2024 2024 NUM cana-5898 165 4 ) ) PUNCT cana-5898 165 5 796 796 NUM cana-5898 165 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 165 7 definition definition NOUN cana-5898 165 8 3.11 3.11 NUM cana-5898 165 9 . . PUNCT cana-5898 166 1 a a DET cana-5898 166 2 subset subset NOUN cana-5898 166 3 a a PRON cana-5898 166 4 of of ADP cana-5898 166 5 a a DET cana-5898 166 6 grill grill ADJ cana-5898 166 7 topological topological ADJ cana-5898 166 8 space space NOUN cana-5898 166 9 ( ( PUNCT cana-5898 166 10 x x X cana-5898 166 11 , , PUNCT cana-5898 166 12   PROPN cana-5898 166 13 , , PUNCT cana-5898 166 14 𝒒 𝒒 PROPN cana-5898 166 15 ) ) PUNCT cana-5898 166 16 is be AUX cana-5898 166 17 said say VERB cana-5898 166 18 to to PART cana-5898 166 19 be be AUX cana-5898 166 20 οͺs οͺs NOUN cana-5898 166 21 - - PUNCT cana-5898 166 22 semiclosed semiclose VERB cana-5898 166 23 if if SCONJ cana-5898 166 24 οͺs(a οͺs(a NOUN cana-5898 166 25 ) ) PUNCT cana-5898 166 26 βŠ† βŠ† NUM cana-5898 166 27 a. a. NOUN cana-5898 166 28 remark remark NOUN cana-5898 166 29 3.12 3.12 NUM cana-5898 166 30 . . PUNCT cana-5898 167 1 every every DET cana-5898 167 2 πœπ’’-closed πœπ’’-close VERB cana-5898 167 3 set set NOUN cana-5898 167 4 is be AUX cana-5898 167 5 οͺs οͺs NOUN cana-5898 167 6 - - PUNCT cana-5898 167 7 semiclosed semiclose VERB cana-5898 167 8 . . PUNCT cana-5898 168 1 the the DET cana-5898 168 2 converse converse NOUN cana-5898 168 3 is be AUX cana-5898 168 4 not not PART cana-5898 168 5 true true ADJ cana-5898 168 6 . . PUNCT cana-5898 169 1 in in ADP cana-5898 169 2 example(ii example(ii PROPN cana-5898 169 3 ) ) PUNCT cana-5898 169 4 of of ADP cana-5898 169 5 remark remark NOUN cana-5898 169 6 3.3 3.3 NUM cana-5898 169 7 , , PUNCT cana-5898 169 8 a a DET cana-5898 169 9 = = X cana-5898 169 10 { { PUNCT cana-5898 169 11 a a PRON cana-5898 169 12 } } PUNCT cana-5898 169 13 is be AUX cana-5898 169 14 οͺs οͺs NOUN cana-5898 169 15 - - PUNCT cana-5898 169 16 semiclosed semiclose VERB cana-5898 169 17 but but CCONJ cana-5898 169 18 it it PRON cana-5898 169 19 is be AUX cana-5898 169 20 πœπ’’-closed πœπ’’-close VERB cana-5898 169 21 . . PUNCT cana-5898 170 1 definition definition NOUN cana-5898 170 2 3.13 3.13 NUM cana-5898 170 3 . . PUNCT cana-5898 171 1 a a DET cana-5898 171 2 subset subset NOUN cana-5898 171 3 a a PRON cana-5898 171 4 of of ADP cana-5898 171 5 a a DET cana-5898 171 6 grill grill ADJ cana-5898 171 7 topological topological ADJ cana-5898 171 8 space space NOUN cana-5898 171 9 ( ( PUNCT cana-5898 171 10 x x X cana-5898 171 11 , , PUNCT cana-5898 171 12   PROPN cana-5898 171 13 , , PUNCT cana-5898 171 14 𝒒 𝒒 PROPN cana-5898 171 15 ) ) PUNCT cana-5898 171 16 is be AUX cana-5898 171 17 said say VERB cana-5898 171 18 to to PART cana-5898 171 19 be be AUX cana-5898 171 20 οͺs οͺs NOUN cana-5898 171 21 - - PUNCT cana-5898 171 22 semidense semidense NOUN cana-5898 171 23 in in ADP cana-5898 171 24 - - PUNCT cana-5898 171 25 itself itself PRON cana-5898 171 26 if if SCONJ cana-5898 171 27 a a DET cana-5898 171 28 βŠ† βŠ† NUM cana-5898 171 29 οͺs(a οͺs(a NOUN cana-5898 171 30 ) ) PUNCT cana-5898 171 31 . . PUNCT cana-5898 172 1 remark remark PROPN cana-5898 172 2 3.14 3.14 NUM cana-5898 172 3 . . PUNCT cana-5898 173 1 every every DET cana-5898 173 2 οͺs οͺs NOUN cana-5898 173 3 - - PUNCT cana-5898 173 4 semi semi ADV cana-5898 173 5 dense dense ADJ cana-5898 173 6 in in ADP cana-5898 173 7 - - PUNCT cana-5898 173 8 itself itself PRON cana-5898 173 9 set set NOUN cana-5898 173 10 is be AUX cana-5898 173 11 οͺ-dense οͺ-dense NOUN cana-5898 173 12 in in ADP cana-5898 173 13 - - PUNCT cana-5898 173 14 itself itself PRON cana-5898 173 15 . . PUNCT cana-5898 174 1 theorem theorem VERB cana-5898 174 2 3.15 3.15 NUM cana-5898 174 3 . . PUNCT cana-5898 175 1 in in ADP cana-5898 175 2 a a DET cana-5898 175 3 grill grill ADJ cana-5898 175 4 topological topological ADJ cana-5898 175 5 space space NOUN cana-5898 175 6 ( ( PUNCT cana-5898 175 7 x x X cana-5898 175 8 , , PUNCT cana-5898 175 9   PROPN cana-5898 175 10 , , PUNCT cana-5898 175 11 𝒒 𝒒 PROPN cana-5898 175 12 ) ) PUNCT cana-5898 175 13 , , PUNCT cana-5898 175 14 a a DET cana-5898 175 15 g𝒒-closed g𝒒-close VERB cana-5898 175 16 and and CCONJ cana-5898 175 17 οͺs οͺs NOUN cana-5898 175 18 - - PUNCT cana-5898 175 19 semi semi ADJ cana-5898 175 20 - - ADJ cana-5898 175 21 dense dense ADJ cana-5898 175 22 initself initself NOUN cana-5898 175 23 set set VERB cana-5898 175 24 is be AUX cana-5898 175 25 gs gs NOUN cana-5898 175 26 - - PUNCT cana-5898 175 27 closed closed ADJ cana-5898 175 28 . . PUNCT cana-5898 176 1 proof proof NOUN cana-5898 176 2 . . PUNCT cana-5898 177 1 suppose suppose VERB cana-5898 177 2 a a PRON cana-5898 177 3 is be AUX cana-5898 177 4 οͺs οͺs NOUN cana-5898 177 5 - - PUNCT cana-5898 177 6 semi semi ADJ cana-5898 177 7 - - ADJ cana-5898 177 8 dense dense ADJ cana-5898 177 9 in in ADP cana-5898 177 10 - - PUNCT cana-5898 177 11 itself itself PRON cana-5898 177 12 and and CCONJ cana-5898 177 13 g𝒒-closed g𝒒-close VERB cana-5898 177 14 in in ADP cana-5898 177 15 x. x. NOUN cana-5898 177 16 let let VERB cana-5898 177 17 u u PRON cana-5898 177 18 be be AUX cana-5898 177 19 any any DET cana-5898 177 20 open open ADJ cana-5898 177 21 set set NOUN cana-5898 177 22 containing contain VERB cana-5898 177 23 a. a. NOUN cana-5898 177 24 since since SCONJ cana-5898 177 25 a a PRON cana-5898 177 26 is be AUX cana-5898 177 27 g𝒒-closed g𝒒-close VERB cana-5898 177 28 , , PUNCT cana-5898 177 29 οͺs(a οͺs(a NOUN cana-5898 177 30 ) ) PUNCT cana-5898 177 31   PROPN cana-5898 177 32 u u PROPN cana-5898 177 33 and and CCONJ cana-5898 177 34 by by ADP cana-5898 177 35 theorem theorem ADJ cana-5898 177 36 2.1 2.1 NUM cana-5898 177 37 , , PUNCT cana-5898 177 38 scl(οͺs(a scl(οͺs(a NOUN cana-5898 177 39 ) ) PUNCT cana-5898 177 40 ) ) PUNCT cana-5898 178 1   PROPN cana-5898 178 2 u. u. PROPN cana-5898 178 3 since since SCONJ cana-5898 178 4 a a PRON cana-5898 178 5 is be AUX cana-5898 178 6 οͺs οͺs NOUN cana-5898 178 7 - - PUNCT cana-5898 178 8 semi semi ADJ cana-5898 178 9 - - ADJ cana-5898 178 10 dense dense ADJ cana-5898 178 11 in in ADP cana-5898 178 12 - - PUNCT cana-5898 178 13 itself itself PRON cana-5898 178 14 , , PUNCT cana-5898 178 15 a a DET cana-5898 178 16   PROPN cana-5898 178 17 οͺs(a οͺs(a PROPN cana-5898 178 18 ) ) PUNCT cana-5898 178 19 and and CCONJ cana-5898 178 20 hence hence ADV cana-5898 178 21 scl(a scl(a PROPN cana-5898 178 22 ) ) PUNCT cana-5898 178 23   PROPN cana-5898 178 24 u u PROPN cana-5898 178 25 whenever whenever SCONJ cana-5898 178 26 a a DET cana-5898 178 27   PROPN cana-5898 178 28 u. u. PROPN cana-5898 178 29 this this PRON cana-5898 178 30 proves prove VERB cana-5898 178 31 that that SCONJ cana-5898 178 32 a a PRON cana-5898 178 33 is be AUX cana-5898 178 34 gs gs NOUN cana-5898 178 35 - - PUNCT cana-5898 178 36 closed closed ADJ cana-5898 178 37 . . PUNCT cana-5898 179 1 theorem theorem VERB cana-5898 179 2 3.16 3.16 NUM cana-5898 179 3 . . PUNCT cana-5898 180 1 let let VERB cana-5898 180 2 ( ( PUNCT cana-5898 180 3 x x X cana-5898 180 4 , , PUNCT cana-5898 180 5   PROPN cana-5898 180 6 , , PUNCT cana-5898 180 7 𝒒 𝒒 PROPN cana-5898 180 8 ) ) PUNCT cana-5898 180 9 be be VERB cana-5898 180 10 a a DET cana-5898 180 11 grill grill ADJ cana-5898 180 12 topological topological ADJ cana-5898 180 13 space space NOUN cana-5898 180 14 and and CCONJ cana-5898 180 15 a a DET cana-5898 180 16 be be AUX cana-5898 180 17 a a DET cana-5898 180 18 g𝒒-closed g𝒒-close VERB cana-5898 180 19 subset subset NOUN cana-5898 180 20 of of ADP cana-5898 180 21 x. x. NOUN cana-5898 180 22 if if SCONJ cana-5898 180 23 b b PROPN cana-5898 180 24 is be AUX cana-5898 180 25 a a DET cana-5898 180 26 subset subset NOUN cana-5898 180 27 of of ADP cana-5898 180 28 x x SYM cana-5898 180 29 such such ADJ cana-5898 180 30 that that SCONJ cana-5898 180 31 a a DET cana-5898 180 32   PROPN cana-5898 180 33 b b PROPN cana-5898 180 34   PROPN cana-5898 180 35 οͺs(a οͺs(a PROPN cana-5898 180 36 ) ) PUNCT cana-5898 180 37 , , PUNCT cana-5898 180 38 then then ADV cana-5898 180 39 b b PROPN cana-5898 180 40 is be AUX cana-5898 180 41 g𝒒-closed g𝒒-close VERB cana-5898 180 42 . . PUNCT cana-5898 181 1 proof proof NOUN cana-5898 181 2 . . PUNCT cana-5898 182 1 let let VERB cana-5898 182 2 u u PRON cana-5898 182 3 be be AUX cana-5898 182 4 any any DET cana-5898 182 5 open open ADJ cana-5898 182 6 set set NOUN cana-5898 182 7 of of ADP cana-5898 182 8 x x PUNCT cana-5898 182 9 such such ADJ cana-5898 182 10 that that DET cana-5898 182 11 b b NOUN cana-5898 182 12   PROPN cana-5898 182 13 u. u. PROPN cana-5898 182 14 then then ADV cana-5898 182 15 a a DET cana-5898 182 16   PROPN cana-5898 182 17 u. u. PROPN cana-5898 182 18 since since SCONJ cana-5898 182 19 a a PRON cana-5898 182 20 is be AUX cana-5898 182 21 g𝒒-closed g𝒒-close VERB cana-5898 182 22 , , PUNCT cana-5898 182 23 οͺs(a οͺs(a NOUN cana-5898 182 24 ) ) PUNCT cana-5898 182 25   PROPN cana-5898 182 26 u. u. PROPN cana-5898 182 27 by by ADP cana-5898 182 28 theorem theorem NOUN cana-5898 182 29 2.1 2.1 NUM cana-5898 182 30 , , PUNCT cana-5898 182 31 we we PRON cana-5898 182 32 have have VERB cana-5898 182 33 οͺs(b οͺs(b NUM cana-5898 182 34 ) ) PUNCT cana-5898 182 35   PROPN cana-5898 182 36 οͺs(οͺs(a οͺs(οͺs(a PROPN cana-5898 182 37 ) ) PUNCT cana-5898 182 38 ) ) PUNCT cana-5898 183 1   PROPN cana-5898 183 2 οͺs(a οͺs(a PROPN cana-5898 183 3 ) ) PUNCT cana-5898 183 4   PROPN cana-5898 183 5 u u NOUN cana-5898 183 6 and and CCONJ cana-5898 183 7 hence hence ADV cana-5898 183 8 b b PROPN cana-5898 183 9 is be AUX cana-5898 183 10 g𝒒-closed g𝒒-close VERB cana-5898 183 11 . . PUNCT cana-5898 184 1 theorem theorem VERB cana-5898 184 2 3.17 3.17 NUM cana-5898 184 3 . . PUNCT cana-5898 185 1 let let AUX cana-5898 185 2 ( ( PUNCT cana-5898 185 3 x x X cana-5898 185 4 , , PUNCT cana-5898 185 5   PROPN cana-5898 185 6 , , PUNCT cana-5898 185 7 𝒒 𝒒 PROPN cana-5898 185 8 ) ) PUNCT cana-5898 185 9 be be VERB cana-5898 185 10 a a DET cana-5898 185 11 grill grill ADJ cana-5898 185 12 topological topological ADJ cana-5898 185 13 space space NOUN cana-5898 185 14 and and CCONJ cana-5898 185 15 a a DET cana-5898 185 16   PROPN cana-5898 185 17 y y PROPN cana-5898 185 18   PROPN cana-5898 185 19 x x X cana-5898 185 20 , , PUNCT cana-5898 185 21 where where SCONJ cana-5898 185 22 y y PROPN cana-5898 185 23 is be AUX cana-5898 185 24 Ξ± Ξ± NOUN cana-5898 185 25 - - NOUN cana-5898 185 26 open open ADJ cana-5898 185 27 in in ADP cana-5898 185 28 x. x. PROPN cana-5898 185 29 then then ADV cana-5898 185 30 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ VERB cana-5898 185 31 , , PUNCT cana-5898 185 32 y y ADJ cana-5898 185 33 ) ) PUNCT cana-5898 185 34 ) ) PUNCT cana-5898 186 1 = = SYM cana-5898 186 2 οͺs(a οͺs(a NOUN cana-5898 186 3 ) ) PUNCT cana-5898 186 4 ∩ ∩ PROPN cana-5898 186 5 y. y. NOUN cana-5898 186 6 proof proof PROPN cana-5898 186 7 . . PUNCT cana-5898 187 1 assume assume VERB cana-5898 187 2 that that SCONJ cana-5898 187 3 x x PUNCT cana-5898 187 4 ∈ ∈ NOUN cana-5898 187 5 x x X cana-5898 187 6 βˆ’ βˆ’ PROPN cana-5898 187 7 ( ( PUNCT cana-5898 187 8 οͺs(a οͺs(a NOUN cana-5898 187 9 ) ) PUNCT cana-5898 187 10 ∩ ∩ PROPN cana-5898 187 11 y y PROPN cana-5898 187 12 ) ) PUNCT cana-5898 187 13 . . PUNCT cana-5898 188 1 then then ADV cana-5898 188 2 either either CCONJ cana-5898 188 3 x x SYM cana-5898 188 4 ∈ ∈ PROPN cana-5898 188 5 y y PROPN cana-5898 188 6 or or CCONJ cana-5898 188 7 x x PROPN cana-5898 188 8 βˆ‰ βˆ‰ PROPN cana-5898 188 9 y y PROPN cana-5898 188 10 communications communication NOUN cana-5898 188 11 on on ADP cana-5898 188 12 applied apply VERB cana-5898 188 13 nonlinear nonlinear ADJ cana-5898 188 14 analysis analysis NOUN cana-5898 188 15 issn issn NOUN cana-5898 188 16 : : PUNCT cana-5898 188 17 1074 1074 NUM cana-5898 188 18 - - PUNCT cana-5898 188 19 133x 133x NUM cana-5898 188 20 vol vol NOUN cana-5898 188 21 31 31 NUM cana-5898 188 22 no no NOUN cana-5898 188 23 . . PUNCT cana-5898 189 1 7s 7 NOUN cana-5898 189 2 ( ( PUNCT cana-5898 189 3 2024 2024 NUM cana-5898 189 4 ) ) PUNCT cana-5898 189 5 797 797 NUM cana-5898 189 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 189 7 case(i case(i PROPN cana-5898 189 8 ) ) PUNCT cana-5898 189 9 . . PUNCT cana-5898 190 1 x x X cana-5898 190 2 βˆ‰ βˆ‰ PROPN cana-5898 190 3 y y PROPN cana-5898 190 4 : : PUNCT cana-5898 190 5 since since SCONJ cana-5898 190 6 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ PROPN cana-5898 190 7 , , PUNCT cana-5898 190 8 y y ADJ cana-5898 190 9 ) ) PUNCT cana-5898 190 10 ) ) PUNCT cana-5898 191 1   PROPN cana-5898 191 2 y y PROPN cana-5898 191 3 , , PUNCT cana-5898 191 4 then then ADV cana-5898 191 5 x x PROPN cana-5898 191 6 βˆ‰ βˆ‰ PROPN cana-5898 191 7 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ PROPN cana-5898 191 8 , , PUNCT cana-5898 191 9 y y ADJ cana-5898 191 10 ) ) PUNCT cana-5898 191 11 ) ) PUNCT cana-5898 191 12 . . PUNCT cana-5898 192 1 case(ii case(ii ADJ cana-5898 192 2 ) ) PUNCT cana-5898 192 3 . . PUNCT cana-5898 193 1 x x X cana-5898 193 2 ∈ ∈ PROPN cana-5898 193 3 y y NOUN cana-5898 193 4 : : PUNCT cana-5898 193 5 since since SCONJ cana-5898 193 6 x x PROPN cana-5898 193 7 βˆ‰ βˆ‰ PROPN cana-5898 193 8 οͺs(a οͺs(a PROPN cana-5898 193 9 ) ) PUNCT cana-5898 193 10 , , PUNCT cana-5898 193 11 there there PRON cana-5898 193 12 exists exist VERB cana-5898 193 13 a a DET cana-5898 193 14 semiopen semiopen ADJ cana-5898 193 15 set set VERB cana-5898 193 16 v v NOUN cana-5898 193 17 in in ADP cana-5898 193 18 x x PUNCT cana-5898 193 19 containing contain VERB cana-5898 193 20 x x PUNCT cana-5898 193 21 such such ADJ cana-5898 193 22 that that DET cana-5898 193 23 v v ADP cana-5898 193 24 ∩ ∩ NOUN cana-5898 193 25 a a DET cana-5898 193 26 βˆ‰ βˆ‰ PROPN cana-5898 193 27 𝒒. 𝒒. PROPN cana-5898 193 28 since since SCONJ cana-5898 193 29 x x PROPN cana-5898 193 30 ∈ ∈ PROPN cana-5898 193 31 y y PROPN cana-5898 193 32 and and CCONJ cana-5898 193 33 y y PROPN cana-5898 193 34 is be AUX cana-5898 193 35 Ξ± Ξ± NOUN cana-5898 193 36 - - NOUN cana-5898 193 37 open open ADJ cana-5898 193 38 in in ADP cana-5898 193 39 x x NOUN cana-5898 193 40 , , PUNCT cana-5898 193 41 we we PRON cana-5898 193 42 have have VERB cana-5898 193 43 a a DET cana-5898 193 44 set set ADJ cana-5898 193 45 y y PROPN cana-5898 193 46 ∩ ∩ NOUN cana-5898 193 47 v v ADP cana-5898 193 48 ∈ ∈ PROPN cana-5898 193 49 so so ADV cana-5898 193 50 ( ( PUNCT cana-5898 193 51 y y NOUN cana-5898 193 52 , , PUNCT cana-5898 193 53 y y PROPN cana-5898 193 54 ) ) PUNCT cana-5898 193 55 such such ADJ cana-5898 193 56 that that SCONJ cana-5898 193 57 x x SYM cana-5898 193 58 ∈ ∈ PROPN cana-5898 193 59 y y PROPN cana-5898 193 60 ∩ ∩ PROPN cana-5898 193 61 v v PROPN cana-5898 193 62 and and CCONJ cana-5898 193 63 ( ( PUNCT cana-5898 193 64 y y PROPN cana-5898 193 65 ∩ ∩ ADJ cana-5898 193 66 v v NOUN cana-5898 193 67 ) ) PUNCT cana-5898 193 68 ∩ ∩ NOUN cana-5898 193 69 a a DET cana-5898 193 70 βˆ‰ βˆ‰ PROPN cana-5898 193 71 𝒒 𝒒 PROPN cana-5898 193 72 and and CCONJ cana-5898 193 73 hence hence ADV cana-5898 193 74 ( ( PUNCT cana-5898 193 75 y y PROPN cana-5898 193 76 ∩ ∩ ADJ cana-5898 193 77 v v NOUN cana-5898 193 78 ) ) PUNCT cana-5898 193 79 ∩ ∩ NOUN cana-5898 193 80 a a DET cana-5898 193 81 βˆ‰ βˆ‰ PROPN cana-5898 193 82 π’’π‘Œ. π’’π‘Œ. NOUN cana-5898 193 83 consequently consequently ADV cana-5898 193 84 , , PUNCT cana-5898 193 85 x x PROPN cana-5898 193 86 βˆ‰ βˆ‰ PROPN cana-5898 193 87 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ PROPN cana-5898 193 88 , , PUNCT cana-5898 193 89 y y ADJ cana-5898 193 90 ) ) PUNCT cana-5898 193 91 ) ) PUNCT cana-5898 193 92 . . PUNCT cana-5898 194 1 hence hence ADV cana-5898 194 2 , , PUNCT cana-5898 194 3 we we PRON cana-5898 194 4 get get VERB cana-5898 194 5 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ ADJ cana-5898 194 6 , , PUNCT cana-5898 194 7 y y ADJ cana-5898 194 8 ) ) PUNCT cana-5898 194 9 ) ) PUNCT cana-5898 195 1   PROPN cana-5898 195 2 οͺs(a οͺs(a PROPN cana-5898 195 3 ) ) PUNCT cana-5898 195 4 ∩ ∩ PROPN cana-5898 195 5 y. y. PROPN cana-5898 195 6 to to PART cana-5898 195 7 prove prove VERB cana-5898 195 8 the the DET cana-5898 195 9 reverse reverse ADJ cana-5898 195 10 implication implication NOUN cana-5898 195 11 , , PUNCT cana-5898 195 12 consider consider VERB cana-5898 195 13 x x PROPN cana-5898 195 14 βˆ‰ βˆ‰ PROPN cana-5898 195 15 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ PROPN cana-5898 195 16 , , PUNCT cana-5898 195 17 y y ADJ cana-5898 195 18 ) ) PUNCT cana-5898 195 19 ) ) PUNCT cana-5898 195 20 . . PUNCT cana-5898 196 1 then then ADV cana-5898 196 2 , , PUNCT cana-5898 196 3 for for ADP cana-5898 196 4 some some DET cana-5898 196 5 semiopen semiopen NOUN cana-5898 196 6 set set VERB cana-5898 196 7 v v NOUN cana-5898 196 8 in in ADP cana-5898 196 9 ( ( PUNCT cana-5898 196 10 y y NOUN cana-5898 196 11 , , PUNCT cana-5898 196 12 y y PROPN cana-5898 196 13 ) ) PUNCT cana-5898 196 14 containing contain VERB cana-5898 196 15 x x PUNCT cana-5898 196 16 there there PRON cana-5898 196 17 exists exist VERB cana-5898 196 18 u u NOUN cana-5898 196 19 ∈ ∈ PROPN cana-5898 197 1 so so ADV cana-5898 197 2 ( ( PUNCT cana-5898 197 3 x x NOUN cana-5898 197 4 , , PUNCT cana-5898 197 5 x x NOUN cana-5898 197 6 ) ) PUNCT cana-5898 197 7 such such ADJ cana-5898 197 8 that that DET cana-5898 197 9 v v NOUN cana-5898 197 10 = = SYM cana-5898 197 11 u u NOUN cana-5898 197 12 ∩ ∩ X cana-5898 198 1 y y PROPN cana-5898 199 1 and and CCONJ cana-5898 199 2 we we PRON cana-5898 199 3 have have VERB cana-5898 199 4 ( ( PUNCT cana-5898 199 5 u u PROPN cana-5898 199 6 ∩ ∩ ADJ cana-5898 199 7 y y NOUN cana-5898 199 8 ) ) PUNCT cana-5898 199 9 ∩ ∩ NOUN cana-5898 199 10 a a DET cana-5898 199 11 βˆ‰ βˆ‰ PROPN cana-5898 199 12 π’’π‘Œ. π’’π‘Œ. NOUN cana-5898 199 13 since since SCONJ cana-5898 199 14 a a DET cana-5898 199 15   PROPN cana-5898 199 16 y y PROPN cana-5898 199 17 , , PUNCT cana-5898 199 18 then then ADV cana-5898 199 19 u u PROPN cana-5898 199 20 ∩ ∩ NOUN cana-5898 199 21 a a DET cana-5898 199 22 βˆ‰ βˆ‰ ADJ cana-5898 199 23 π’’π‘Œ π’’π‘Œ PROPN cana-5898 199 24   PROPN cana-5898 199 25 𝒒 𝒒 PROPN cana-5898 199 26 gives give VERB cana-5898 199 27 u u PRON cana-5898 199 28 ∩ ∩ NOUN cana-5898 199 29 a a DET cana-5898 199 30   PUNCT cana-5898 199 31 𝒒 𝒒 NOUN cana-5898 199 32 for for ADP cana-5898 199 33 some some DET cana-5898 199 34 semiopen semiopen ADJ cana-5898 199 35 set set VERB cana-5898 199 36 u u NOUN cana-5898 199 37 in in ADP cana-5898 199 38 ( ( PUNCT cana-5898 199 39 x x NOUN cana-5898 199 40 , , PUNCT cana-5898 199 41 Ο„ Ο„ X cana-5898 199 42 ) ) PUNCT cana-5898 199 43 containing contain VERB cana-5898 199 44 x. x. NOUN cana-5898 199 45 this this PRON cana-5898 199 46 proves prove VERB cana-5898 199 47 x x PUNCT cana-5898 199 48 βˆˆβˆ‰ βˆˆβˆ‰ ADJ cana-5898 199 49 οͺs(a οͺs(a NOUN cana-5898 199 50 ) ) PUNCT cana-5898 199 51 . . PUNCT cana-5898 200 1 this this PRON cana-5898 200 2 completes complete VERB cana-5898 200 3 the the DET cana-5898 200 4 proof proof NOUN cana-5898 200 5 . . PUNCT cana-5898 201 1 theorem theorem VERB cana-5898 201 2 3.18 3.18 NUM cana-5898 201 3 . . PUNCT cana-5898 202 1 let let AUX cana-5898 202 2 ( ( PUNCT cana-5898 202 3 x x X cana-5898 202 4 , , PUNCT cana-5898 202 5   PROPN cana-5898 202 6 , , PUNCT cana-5898 202 7 𝒒 𝒒 PROPN cana-5898 202 8 ) ) PUNCT cana-5898 202 9 be be VERB cana-5898 202 10 a a DET cana-5898 202 11 grill grill ADJ cana-5898 202 12 topological topological ADJ cana-5898 202 13 space space NOUN cana-5898 202 14 and and CCONJ cana-5898 202 15 a a DET cana-5898 202 16   PROPN cana-5898 202 17 y y PROPN cana-5898 202 18   PROPN cana-5898 202 19 x. x. NOUN cana-5898 203 1 if if SCONJ cana-5898 203 2 a a PRON cana-5898 203 3 is be AUX cana-5898 203 4 g𝒒closed g𝒒close VERB cana-5898 203 5 in in ADP cana-5898 203 6 ( ( PUNCT cana-5898 203 7 y y NOUN cana-5898 203 8 , , PUNCT cana-5898 203 9 y y PROPN cana-5898 203 10 , , PUNCT cana-5898 203 11 π’’π‘Œ π’’π‘Œ PROPN cana-5898 203 12 ) ) PUNCT cana-5898 203 13 and and CCONJ cana-5898 203 14 x x X cana-5898 203 15 is be AUX cana-5898 203 16 Ξ± Ξ± NOUN cana-5898 203 17 - - ADJ cana-5898 203 18 open open ADJ cana-5898 203 19 and and CCONJ cana-5898 203 20 οͺs οͺ NOUN cana-5898 203 21 - - PUNCT cana-5898 203 22 semiclosed semiclose VERB cana-5898 203 23 in in ADP cana-5898 203 24 x x NOUN cana-5898 203 25 , , PUNCT cana-5898 203 26 then then ADV cana-5898 203 27 a a PRON cana-5898 203 28 is be AUX cana-5898 203 29 g𝒒-closed g𝒒-close VERB cana-5898 203 30 in in ADP cana-5898 203 31 x. x. NOUN cana-5898 203 32 proof proof NOUN cana-5898 203 33 . . PUNCT cana-5898 204 1 let let VERB cana-5898 204 2 a a DET cana-5898 204 3   PROPN cana-5898 204 4 u u NOUN cana-5898 204 5 and and CCONJ cana-5898 204 6 u u NOUN cana-5898 204 7 be be VERB cana-5898 204 8 open open ADJ cana-5898 204 9 in in ADP cana-5898 204 10 x. x. PROPN cana-5898 204 11 then then ADV cana-5898 204 12 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ VERB cana-5898 204 13 , , PUNCT cana-5898 204 14 y y ADJ cana-5898 204 15 ) ) PUNCT cana-5898 204 16 ) ) PUNCT cana-5898 205 1 = = SYM cana-5898 205 2 οͺs(a οͺs(a NOUN cana-5898 205 3 ) ) PUNCT cana-5898 205 4 ∩ ∩ NOUN cana-5898 205 5 y y PROPN cana-5898 205 6   PROPN cana-5898 205 7 u u PROPN cana-5898 205 8 ∩y ∩y NOUN cana-5898 205 9 . . PUNCT cana-5898 206 1 then then ADV cana-5898 206 2 we we PRON cana-5898 206 3 have have VERB cana-5898 206 4 y y PROPN cana-5898 206 5   PROPN cana-5898 206 6 u u PROPN cana-5898 206 7 οƒˆ(x οƒˆ(x ADV cana-5898 206 8 – – PUNCT cana-5898 206 9 οͺs(a οͺs(a NOUN cana-5898 206 10 ) ) PUNCT cana-5898 206 11 ) ) PUNCT cana-5898 206 12 . . PUNCT cana-5898 207 1 since since SCONJ cana-5898 207 2 y y PROPN cana-5898 207 3 is be AUX cana-5898 207 4 οͺs οͺs PROPN cana-5898 207 5 - - PUNCT cana-5898 207 6 semiclosed semiclose VERB cana-5898 207 7 , , PUNCT cana-5898 207 8 we we PRON cana-5898 207 9 have have VERB cana-5898 207 10 οͺs(a οͺs(a NOUN cana-5898 207 11 ) ) PUNCT cana-5898 207 12   PROPN cana-5898 207 13 οͺs(y οͺs(y PROPN cana-5898 207 14 ) ) PUNCT cana-5898 207 15   PROPN cana-5898 208 1 y y PROPN cana-5898 208 2   PROPN cana-5898 208 3 u u PROPN cana-5898 208 4 οƒˆ(x οƒˆ(x ADV cana-5898 208 5 – – PUNCT cana-5898 208 6 οͺs(a οͺs(a NOUN cana-5898 208 7 ) ) PUNCT cana-5898 208 8 ) ) PUNCT cana-5898 208 9 . . PUNCT cana-5898 209 1 this this PRON cana-5898 209 2 proves prove VERB cana-5898 209 3 that that SCONJ cana-5898 209 4 οͺs(a οͺs(a NOUN cana-5898 209 5 ) ) PUNCT cana-5898 209 6   PROPN cana-5898 209 7 u. u. PROPN cana-5898 209 8 this this PRON cana-5898 209 9 completes complete VERB cana-5898 209 10 the the DET cana-5898 209 11 proof proof NOUN cana-5898 209 12 . . PUNCT cana-5898 210 1 theorem theorem VERB cana-5898 210 2 3.19 3.19 NUM cana-5898 210 3 . . PUNCT cana-5898 211 1 let let AUX cana-5898 211 2 ( ( PUNCT cana-5898 211 3 x x X cana-5898 211 4 , , PUNCT cana-5898 211 5   PROPN cana-5898 211 6 , , PUNCT cana-5898 211 7 𝒒 𝒒 PROPN cana-5898 211 8 ) ) PUNCT cana-5898 211 9 be be VERB cana-5898 211 10 a a DET cana-5898 211 11 grill grill ADJ cana-5898 211 12 topological topological ADJ cana-5898 211 13 space space NOUN cana-5898 211 14 and and CCONJ cana-5898 211 15 a a DET cana-5898 211 16   PROPN cana-5898 211 17 y y PROPN cana-5898 211 18   PROPN cana-5898 211 19 x. x. NOUN cana-5898 212 1 if if SCONJ cana-5898 212 2 a a PRON cana-5898 212 3 is be AUX cana-5898 212 4 g𝒒closed g𝒒close VERB cana-5898 212 5 in in ADP cana-5898 212 6 x x PUNCT cana-5898 212 7 and and CCONJ cana-5898 212 8 y y PROPN cana-5898 212 9 ∈ ∈ PROPN cana-5898 212 10   PROPN cana-5898 212 11 , , PUNCT cana-5898 212 12 then then ADV cana-5898 212 13 a a PRON cana-5898 212 14 is be AUX cana-5898 212 15 g𝒒-closed g𝒒-close VERB cana-5898 212 16 in in ADP cana-5898 212 17 ( ( PUNCT cana-5898 212 18 y y NOUN cana-5898 212 19 , , PUNCT cana-5898 212 20 y y PROPN cana-5898 212 21 , , PUNCT cana-5898 212 22 π’’π‘Œ π’’π‘Œ PROPN cana-5898 212 23 ) ) PUNCT cana-5898 212 24 . . PUNCT cana-5898 213 1 proof proof NOUN cana-5898 213 2 . . PUNCT cana-5898 214 1 let let VERB cana-5898 214 2 u u PRON cana-5898 214 3 be be AUX cana-5898 214 4 an an DET cana-5898 214 5 open open ADJ cana-5898 214 6 subset subset NOUN cana-5898 214 7 of of ADP cana-5898 214 8 ( ( PUNCT cana-5898 214 9 y y NOUN cana-5898 214 10 , , PUNCT cana-5898 214 11 y y PROPN cana-5898 214 12 ) ) PUNCT cana-5898 214 13 such such ADJ cana-5898 214 14 that that SCONJ cana-5898 214 15 a a DET cana-5898 214 16   PROPN cana-5898 214 17 u. u. PROPN cana-5898 214 18 since since SCONJ cana-5898 214 19 y y PROPN cana-5898 214 20 ∈ ∈ PROPN cana-5898 214 21   PROPN cana-5898 214 22 , , PUNCT cana-5898 214 23 then then ADV cana-5898 214 24 u u PROPN cana-5898 214 25 ∈ ∈ PROPN cana-5898 214 26 . . NOUN cana-5898 214 27 thus thus ADV cana-5898 214 28 οͺs(a οͺs(a NOUN cana-5898 214 29 ) ) PUNCT cana-5898 214 30   PROPN cana-5898 214 31 u. u. PROPN cana-5898 214 32 by by ADP cana-5898 214 33 theorem theorem ADJ cana-5898 214 34 3.17 3.17 NUM cana-5898 214 35 , , PUNCT cana-5898 214 36 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ ADJ cana-5898 214 37 , , PUNCT cana-5898 214 38 y y ADJ cana-5898 214 39 ) ) PUNCT cana-5898 214 40 ) ) PUNCT cana-5898 215 1 = = SYM cana-5898 215 2 οͺs(a οͺs(a NOUN cana-5898 215 3 ) ) PUNCT cana-5898 215 4 ∩ ∩ PROPN cana-5898 215 5 y y PROPN cana-5898 215 6   PROPN cana-5898 215 7 u u PROPN cana-5898 215 8 ∩ ∩ NOUN cana-5898 215 9 y= y= NUM cana-5898 215 10 u u NOUN cana-5898 215 11 and and CCONJ cana-5898 215 12 we we PRON cana-5898 215 13 have have VERB cana-5898 215 14 οͺs(a(π’’π‘Œ οͺs(a(π’’π‘Œ VERB cana-5898 215 15 , , PUNCT cana-5898 215 16 y y ADJ cana-5898 215 17 ) ) PUNCT cana-5898 215 18 ) ) PUNCT cana-5898 216 1   PROPN cana-5898 216 2 u. u. PROPN cana-5898 216 3 hence hence ADV cana-5898 216 4 a a PRON cana-5898 216 5 is be AUX cana-5898 216 6 g𝒒-closed g𝒒-close VERB cana-5898 216 7 in in ADP cana-5898 216 8 ( ( PUNCT cana-5898 216 9 y y NOUN cana-5898 216 10 , , PUNCT cana-5898 216 11 y y PROPN cana-5898 216 12 , , PUNCT cana-5898 216 13 π’’π‘Œ π’’π‘Œ PROPN cana-5898 216 14 ) ) PUNCT cana-5898 216 15 . . PUNCT cana-5898 217 1 corollary corollary ADJ cana-5898 217 2 3.19 3.19 NUM cana-5898 217 3 . . PUNCT cana-5898 218 1 let let VERB cana-5898 218 2 ( ( PUNCT cana-5898 218 3 x x X cana-5898 218 4 , , PUNCT cana-5898 218 5   PROPN cana-5898 218 6 , , PUNCT cana-5898 218 7 𝒒 𝒒 PROPN cana-5898 218 8 ) ) PUNCT cana-5898 218 9 be be VERB cana-5898 218 10 a a DET cana-5898 218 11 grill grill ADJ cana-5898 218 12 topological topological ADJ cana-5898 218 13 space space NOUN cana-5898 218 14 and and CCONJ cana-5898 218 15 a a DET cana-5898 218 16   PROPN cana-5898 218 17 y y PROPN cana-5898 218 18   PROPN cana-5898 218 19 x x X cana-5898 218 20 , , PUNCT cana-5898 218 21 where where SCONJ cana-5898 218 22 y y PROPN cana-5898 218 23 is be AUX cana-5898 218 24 a a DET cana-5898 218 25 regular regular ADJ cana-5898 218 26 open open ADJ cana-5898 218 27 subset subset NOUN cana-5898 218 28 of of ADP cana-5898 218 29 x. x. NOUN cana-5898 218 30 then then ADV cana-5898 218 31 a a PRON cana-5898 218 32 is be AUX cana-5898 218 33 g𝒒-closed g𝒒-close VERB cana-5898 218 34 in in ADP cana-5898 218 35 ( ( PUNCT cana-5898 218 36 y y NOUN cana-5898 218 37 , , PUNCT cana-5898 218 38 y y PROPN cana-5898 218 39 , , PUNCT cana-5898 218 40 π’’π‘Œ π’’π‘Œ PROPN cana-5898 218 41 ) ) PUNCT cana-5898 219 1 if if SCONJ cana-5898 219 2 and and CCONJ cana-5898 219 3 only only ADV cana-5898 219 4 if if SCONJ cana-5898 219 5 a a PRON cana-5898 219 6 is be AUX cana-5898 219 7 g𝒒closed g𝒒close VERB cana-5898 219 8 in in ADP cana-5898 219 9 x. x. NOUN cana-5898 219 10 theorem theorem VERB cana-5898 219 11 3.20 3.20 NUM cana-5898 219 12 . . PUNCT cana-5898 220 1 let let VERB cana-5898 220 2 ( ( PUNCT cana-5898 220 3 x x X cana-5898 220 4 , , PUNCT cana-5898 220 5   PROPN cana-5898 220 6 , , PUNCT cana-5898 220 7 𝒒 𝒒 PROPN cana-5898 220 8 ) ) PUNCT cana-5898 220 9 be be VERB cana-5898 220 10 a a DET cana-5898 220 11 grill grill ADJ cana-5898 220 12 topological topological ADJ cana-5898 220 13 space space NOUN cana-5898 220 14 and and CCONJ cana-5898 220 15 a a DET cana-5898 220 16   PROPN cana-5898 220 17 x. x. NOUN cana-5898 220 18 if if SCONJ cana-5898 220 19 a a PRON cana-5898 220 20 is be AUX cana-5898 220 21 g𝒒-closed g𝒒-close VERB cana-5898 220 22 , , PUNCT cana-5898 220 23 then then ADV cana-5898 220 24 a a DET cana-5898 220 25 βˆͺ βˆͺ X cana-5898 220 26 ( ( PUNCT cana-5898 220 27 x x NOUN cana-5898 220 28 βˆ’ βˆ’ PROPN cana-5898 220 29 οͺs(a οͺs(a NOUN cana-5898 220 30 ) ) PUNCT cana-5898 220 31 ) ) PUNCT cana-5898 220 32 is be AUX cana-5898 220 33 g𝒒-closed g𝒒-close VERB cana-5898 220 34 . . PUNCT cana-5898 221 1 communications communication NOUN cana-5898 221 2 on on ADP cana-5898 221 3 applied apply VERB cana-5898 221 4 nonlinear nonlinear ADJ cana-5898 221 5 analysis analysis NOUN cana-5898 221 6 issn issn NOUN cana-5898 221 7 : : PUNCT cana-5898 221 8 1074 1074 NUM cana-5898 221 9 - - PUNCT cana-5898 221 10 133x 133x NUM cana-5898 221 11 vol vol NOUN cana-5898 221 12 31 31 NUM cana-5898 221 13 no no NOUN cana-5898 221 14 . . PUNCT cana-5898 222 1 7s 7 NOUN cana-5898 222 2 ( ( PUNCT cana-5898 222 3 2024 2024 NUM cana-5898 222 4 ) ) PUNCT cana-5898 222 5 798 798 NUM cana-5898 222 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 222 7 proof proof NOUN cana-5898 222 8 . . PUNCT cana-5898 222 9 suppose suppose VERB cana-5898 222 10 a a PRON cana-5898 222 11 is be AUX cana-5898 222 12 g𝒒-closed g𝒒-close VERB cana-5898 222 13 . . PUNCT cana-5898 223 1 let let VERB cana-5898 223 2 u u PRON cana-5898 223 3 be be AUX cana-5898 223 4 an an DET cana-5898 223 5 open open ADJ cana-5898 223 6 set set NOUN cana-5898 223 7 such such ADJ cana-5898 223 8 that that SCONJ cana-5898 223 9 a a DET cana-5898 223 10 βˆͺ βˆͺ X cana-5898 223 11 ( ( PUNCT cana-5898 223 12 x x X cana-5898 223 13 – – PUNCT cana-5898 223 14 οͺs(a οͺs(a NOUN cana-5898 223 15 ) ) PUNCT cana-5898 223 16 ) ) PUNCT cana-5898 224 1   PROPN cana-5898 224 2 u. u. PROPN cana-5898 224 3 then then ADV cana-5898 224 4 x x X cana-5898 224 5 βˆ’ βˆ’ PUNCT cana-5898 224 6 u u NOUN cana-5898 224 7   PROPN cana-5898 224 8 x x X cana-5898 224 9 βˆ’ βˆ’ PROPN cana-5898 224 10 ( ( PUNCT cana-5898 224 11 a a DET cana-5898 224 12 βˆͺ βˆͺ X cana-5898 224 13 ( ( PUNCT cana-5898 224 14 x x X cana-5898 224 15 – – PUNCT cana-5898 224 16 οͺs(a οͺs(a NOUN cana-5898 224 17 ) ) PUNCT cana-5898 224 18 ) ) PUNCT cana-5898 224 19 ) ) PUNCT cana-5898 225 1 = = SYM cana-5898 225 2 οͺs(a οͺs(a NOUN cana-5898 225 3 ) ) PUNCT cana-5898 225 4 βˆ’ βˆ’ NOUN cana-5898 225 5 a. a. NOUN cana-5898 225 6 since since SCONJ cana-5898 225 7 a a PRON cana-5898 225 8 is be AUX cana-5898 225 9 g𝒒-closed g𝒒-close VERB cana-5898 225 10 , , PUNCT cana-5898 225 11 by by ADP cana-5898 225 12 theorem theorem NOUN cana-5898 225 13 3.5 3.5 NUM cana-5898 225 14 , , PUNCT cana-5898 225 15 it it PRON cana-5898 225 16 follows follow VERB cana-5898 225 17 that that SCONJ cana-5898 225 18 οͺs(a οͺs(a NOUN cana-5898 225 19 ) ) PUNCT cana-5898 225 20 βˆ’ βˆ’ NOUN cana-5898 226 1 a a PRON cana-5898 226 2 contains contain VERB cana-5898 226 3 no no DET cana-5898 226 4 non non ADJ cana-5898 226 5 - - ADJ cana-5898 226 6 empty empty ADJ cana-5898 226 7 closed closed ADJ cana-5898 226 8 set set NOUN cana-5898 226 9 . . PUNCT cana-5898 227 1 this this PRON cana-5898 227 2 implies imply VERB cana-5898 227 3 x x X cana-5898 227 4 βˆ’ βˆ’ NOUN cana-5898 227 5 u u NOUN cana-5898 227 6 = = NOUN cana-5898 227 7   NOUN cana-5898 227 8 or or CCONJ cana-5898 227 9 x x X cana-5898 227 10 = = PUNCT cana-5898 227 11 u. u. PROPN cana-5898 227 12 thus thus ADV cana-5898 227 13 , , PUNCT cana-5898 227 14 x x PRON cana-5898 227 15 is be AUX cana-5898 227 16 the the DET cana-5898 227 17 only only ADJ cana-5898 227 18 open open ADJ cana-5898 227 19 set set NOUN cana-5898 227 20 containing contain VERB cana-5898 227 21 a a DET cana-5898 227 22 βˆͺ βˆͺ NOUN cana-5898 227 23 ( ( PUNCT cana-5898 227 24 x x NOUN cana-5898 227 25 βˆ’ βˆ’ PROPN cana-5898 227 26 οͺs(a οͺs(a NOUN cana-5898 227 27 ) ) PUNCT cana-5898 227 28 ) ) PUNCT cana-5898 227 29 . . PUNCT cana-5898 228 1 this this PRON cana-5898 228 2 gives give VERB cana-5898 228 3 οͺs(a οͺs(a NOUN cana-5898 228 4 οƒˆ οƒˆ NOUN cana-5898 228 5 ( ( PUNCT cana-5898 228 6 x x X cana-5898 228 7 – – PUNCT cana-5898 228 8 οͺs(a οͺs(a NOUN cana-5898 228 9 ) ) PUNCT cana-5898 228 10 ) ) PUNCT cana-5898 228 11 ) ) PUNCT cana-5898 229 1   PROPN cana-5898 229 2 x. x. NOUN cana-5898 229 3 this this PRON cana-5898 229 4 proves prove VERB cana-5898 229 5 a a DET cana-5898 229 6 βˆͺ βˆͺ ADJ cana-5898 229 7 ( ( PUNCT cana-5898 229 8 x x NOUN cana-5898 229 9 βˆ’ βˆ’ PROPN cana-5898 229 10 οͺs(a οͺs(a NOUN cana-5898 229 11 ) ) PUNCT cana-5898 229 12 ) ) PUNCT cana-5898 229 13 is be AUX cana-5898 229 14 g𝒒-closed g𝒒-close VERB cana-5898 229 15 . . PUNCT cana-5898 230 1 theorem theorem VERB cana-5898 230 2 3.21 3.21 NUM cana-5898 230 3 . . PUNCT cana-5898 231 1 let let AUX cana-5898 231 2 ( ( PUNCT cana-5898 231 3 x x X cana-5898 231 4 , , PUNCT cana-5898 231 5   PROPN cana-5898 231 6 , , PUNCT cana-5898 231 7 𝒒 𝒒 PROPN cana-5898 231 8 ) ) PUNCT cana-5898 231 9 be be VERB cana-5898 231 10 a a DET cana-5898 231 11 grill grill ADJ cana-5898 231 12 topological topological ADJ cana-5898 231 13 space space NOUN cana-5898 231 14 . . PUNCT cana-5898 232 1 then then ADV cana-5898 232 2 a a DET cana-5898 232 3 οƒˆ οƒˆ NOUN cana-5898 232 4 ( ( PUNCT cana-5898 232 5 x x PROPN cana-5898 232 6 – – PUNCT cana-5898 232 7 οͺs(a οͺs(a NOUN cana-5898 232 8 ) ) PUNCT cana-5898 232 9 ) ) PUNCT cana-5898 232 10 is be AUX cana-5898 232 11 g𝒒closed g𝒒close VERB cana-5898 232 12 if if SCONJ cana-5898 232 13 and and CCONJ cana-5898 232 14 only only ADV cana-5898 232 15 if if SCONJ cana-5898 232 16 οͺs(a οͺs(a NOUN cana-5898 232 17 ) ) PUNCT cana-5898 232 18 – – PUNCT cana-5898 232 19 a a PRON cana-5898 232 20 is be AUX cana-5898 232 21 g𝒒-open g𝒒-open NOUN cana-5898 232 22 . . PUNCT cana-5898 233 1 proof proof NOUN cana-5898 233 2 . . PUNCT cana-5898 234 1 since since SCONJ cana-5898 234 2 x x X cana-5898 234 3 βˆ’ βˆ’ PROPN cana-5898 234 4 ( ( PUNCT cana-5898 234 5 οͺs(a οͺs(a NOUN cana-5898 234 6 ) ) PUNCT cana-5898 234 7 βˆ’ βˆ’ PROPN cana-5898 234 8 a a X cana-5898 234 9 ) ) PUNCT cana-5898 234 10 = = SYM cana-5898 234 11 a a DET cana-5898 234 12 οƒˆ οƒˆ NOUN cana-5898 234 13 ( ( PUNCT cana-5898 234 14 x x PROPN cana-5898 234 15 – – PUNCT cana-5898 234 16 οͺs(a οͺs(a NOUN cana-5898 234 17 ) ) PUNCT cana-5898 234 18 ) ) PUNCT cana-5898 234 19 , , PUNCT cana-5898 234 20 the the DET cana-5898 234 21 proof proof NOUN cana-5898 234 22 follows follow VERB cana-5898 234 23 immediately immediately ADV cana-5898 234 24 . . PUNCT cana-5898 235 1 theorem theorem VERB cana-5898 235 2 3.22 3.22 NUM cana-5898 235 3 . . PUNCT cana-5898 236 1 let let AUX cana-5898 236 2 ( ( PUNCT cana-5898 236 3 x x X cana-5898 236 4 , , PUNCT cana-5898 236 5   PROPN cana-5898 236 6 , , PUNCT cana-5898 236 7 𝒒 𝒒 PROPN cana-5898 236 8 ) ) PUNCT cana-5898 236 9 be be VERB cana-5898 236 10 a a DET cana-5898 236 11 grill grill ADJ cana-5898 236 12 topological topological ADJ cana-5898 236 13 space space NOUN cana-5898 236 14 . . PUNCT cana-5898 237 1 then then ADV cana-5898 237 2 every every DET cana-5898 237 3 subset subset NOUN cana-5898 237 4 of of ADP cana-5898 237 5 x x PUNCT cana-5898 237 6 is be AUX cana-5898 237 7 g𝒒closed g𝒒close VERB cana-5898 237 8 if if SCONJ cana-5898 237 9 every every DET cana-5898 237 10 open open ADJ cana-5898 237 11 set set NOUN cana-5898 237 12 is be AUX cana-5898 237 13 οͺs οͺs NOUN cana-5898 237 14 - - PUNCT cana-5898 237 15 semiclosed semiclose VERB cana-5898 237 16 . . PUNCT cana-5898 238 1 proof proof NOUN cana-5898 238 2 . . PUNCT cana-5898 239 1 suppose suppose VERB cana-5898 239 2 that that SCONJ cana-5898 239 3 every every DET cana-5898 239 4 open open ADJ cana-5898 239 5 set set NOUN cana-5898 239 6 is be AUX cana-5898 239 7 οͺs οͺs NOUN cana-5898 239 8 - - PUNCT cana-5898 239 9 semiclosed semiclose VERB cana-5898 239 10 . . PUNCT cana-5898 240 1 if if SCONJ cana-5898 240 2 a a DET cana-5898 240 3   PROPN cana-5898 240 4 x x X cana-5898 240 5 and and CCONJ cana-5898 240 6 u u NOUN cana-5898 240 7 is be AUX cana-5898 240 8 an an DET cana-5898 240 9 open open ADJ cana-5898 240 10 set set NOUN cana-5898 240 11 such such ADJ cana-5898 240 12 that that SCONJ cana-5898 240 13 a a DET cana-5898 240 14   PROPN cana-5898 240 15 u u NOUN cana-5898 240 16 , , PUNCT cana-5898 240 17 then then ADV cana-5898 240 18 οͺs(a οͺs(a PROPN cana-5898 240 19 ) ) PUNCT cana-5898 240 20   PROPN cana-5898 240 21 οͺs(u οͺs(u PROPN cana-5898 240 22 ) ) PUNCT cana-5898 240 23   PROPN cana-5898 240 24 u u NOUN cana-5898 240 25 and and CCONJ cana-5898 240 26 οͺs(a οͺs(a NOUN cana-5898 240 27 ) ) PUNCT cana-5898 240 28   PROPN cana-5898 240 29 u. u. PROPN cana-5898 240 30 this this PRON cana-5898 240 31 proves prove VERB cana-5898 240 32 that that SCONJ cana-5898 240 33 a a PRON cana-5898 240 34 is be AUX cana-5898 240 35 g𝒒-closed g𝒒-close VERB cana-5898 240 36 . . PUNCT cana-5898 241 1 references reference NOUN cana-5898 241 2 [ [ X cana-5898 241 3 1 1 NUM cana-5898 241 4 ] ] PUNCT cana-5898 241 5 a. a. PROPN cana-5898 241 6 al al PROPN cana-5898 241 7 - - PUNCT cana-5898 241 8 omari omari PROPN cana-5898 241 9 and and CCONJ cana-5898 241 10 t. t. PROPN cana-5898 241 11 noiri noiri PROPN cana-5898 241 12 , , PUNCT cana-5898 241 13 decomposition decomposition NOUN cana-5898 241 14 of of ADP cana-5898 241 15 continuity continuity NOUN cana-5898 241 16 via via ADP cana-5898 241 17 grills grill NOUN cana-5898 241 18 , , PUNCT cana-5898 241 19 jordan jordan PROPN cana-5898 241 20 j. j. PROPN cana-5898 241 21 mat mat PROPN cana-5898 241 22 . . PROPN cana-5898 241 23 stat stat PROPN cana-5898 241 24 . . PUNCT cana-5898 241 25 , , PUNCT cana-5898 241 26 4 4 NUM cana-5898 241 27 ( ( PUNCT cana-5898 241 28 2011 2011 NUM cana-5898 241 29 ) ) PUNCT cana-5898 241 30 , , PUNCT cana-5898 241 31 33 33 NUM cana-5898 241 32 - - SYM cana-5898 241 33 46 46 NUM cana-5898 241 34 . . PUNCT cana-5898 242 1 [ [ X cana-5898 242 2 2 2 NUM cana-5898 242 3 ] ] X cana-5898 242 4 s.p s.p PROPN cana-5898 242 5 . . PROPN cana-5898 242 6 arya arya PROPN cana-5898 242 7 and and CCONJ cana-5898 242 8 t. t. PROPN cana-5898 242 9 nour nour PROPN cana-5898 242 10 . . PUNCT cana-5898 243 1 characterizations characterization NOUN cana-5898 243 2 of of ADP cana-5898 243 3 s s NOUN cana-5898 243 4 - - ADJ cana-5898 243 5 normal normal ADJ cana-5898 243 6 spaces space NOUN cana-5898 243 7 , , PUNCT cana-5898 243 8 indian indian PROPN cana-5898 243 9 j. j. PROPN cana-5898 243 10 pure pure PROPN cana-5898 243 11 . . PUNCT cana-5898 244 1 appl appl PROPN cana-5898 244 2 . . PROPN cana-5898 244 3 math math PROPN cana-5898 244 4 . . PUNCT cana-5898 244 5 , , PUNCT cana-5898 244 6 1990 1990 NUM cana-5898 244 7 , , PUNCT cana-5898 244 8 21(8 21(8 NUM cana-5898 244 9 ): ): PUNCT cana-5898 244 10 717 717 NUM cana-5898 244 11 - - SYM cana-5898 244 12 719 719 NUM cana-5898 244 13 . . PUNCT cana-5898 245 1 [ [ X cana-5898 245 2 3 3 NUM cana-5898 245 3 ] ] X cana-5898 245 4 r. r. PROPN cana-5898 245 5 anbarasan anbarasan PROPN cana-5898 245 6 , , PUNCT cana-5898 245 7 m. m. PROPN cana-5898 245 8 anitha anitha PROPN cana-5898 245 9 , , PUNCT cana-5898 245 10 𝒒 𝒒 PROPN cana-5898 245 11 οΏ½ οΏ½ PROPN cana-5898 245 12 Μ‚ Μ‚ SYM cana-5898 245 13 οΏ½ οΏ½ NOUN cana-5898 245 14 -closed -close VERB cana-5898 245 15 sets set NOUN cana-5898 245 16 in in ADP cana-5898 245 17 grill grill ADJ cana-5898 245 18 generalized generalize VERB cana-5898 245 19 topological topological ADJ cana-5898 245 20 spaces space NOUN cana-5898 245 21 , , PUNCT cana-5898 245 22 j. j. PROPN cana-5898 245 23 visual visual PROPN cana-5898 245 24 perf perf NOUN cana-5898 245 25 . . PUNCT cana-5898 246 1 arts art NOUN cana-5898 246 2 , , PUNCT cana-5898 246 3 5(1 5(1 NUM cana-5898 246 4 ) ) PUNCT cana-5898 246 5 ( ( PUNCT cana-5898 246 6 2024 2024 NUM cana-5898 246 7 ) ) PUNCT cana-5898 246 8 , , PUNCT cana-5898 246 9 2453 2453 NUM cana-5898 246 10 - - SYM cana-5898 246 11 2460 2460 NUM cana-5898 246 12 . . PUNCT cana-5898 247 1 [ [ X cana-5898 247 2 4 4 NUM cana-5898 247 3 ] ] X cana-5898 247 4 r. r. PROPN cana-5898 247 5 anbarasan anbarasan PROPN cana-5898 247 6 , , PUNCT cana-5898 247 7 m. m. NOUN cana-5898 247 8 anitha anitha PROPN cana-5898 247 9 and and CCONJ cana-5898 247 10 d. d. PROPN cana-5898 247 11 saravanakumar saravanakumar PROPN cana-5898 247 12 , , PUNCT cana-5898 247 13 -semi -semi ADP cana-5898 247 14 approaches approach NOUN cana-5898 247 15 on on ADP cana-5898 247 16 -preopen -preopen NOUN cana-5898 247 17 sets set NOUN cana-5898 247 18 via via ADP cana-5898 247 19 grill grill NOUN cana-5898 247 20 , , PUNCT cana-5898 247 21 tanz tanz PROPN cana-5898 247 22 j. j. PROPN cana-5898 247 23 , , PUNCT cana-5898 247 24 ( ( PUNCT cana-5898 247 25 accpted accpte VERB cana-5898 247 26 ) ) PUNCT cana-5898 247 27 . . PUNCT cana-5898 248 1 [ [ X cana-5898 248 2 5 5 X cana-5898 248 3 ] ] PUNCT cana-5898 248 4 p. p. NOUN cana-5898 248 5 bhattacharyya bhattacharyya PROPN cana-5898 248 6 and and CCONJ cana-5898 248 7 b. b. PROPN cana-5898 248 8 k. k. PROPN cana-5898 248 9 lahiri lahiri PROPN cana-5898 248 10 , , PUNCT cana-5898 248 11 semi semi ADJ cana-5898 248 12 - - ADJ cana-5898 248 13 generalized generalized ADJ cana-5898 248 14 closed closed ADJ cana-5898 248 15 sets set NOUN cana-5898 248 16 in in ADP cana-5898 248 17 topology topology NOUN cana-5898 248 18 , , PUNCT cana-5898 248 19 indian indian PROPN cana-5898 248 20 . . PUNCT cana-5898 249 1 j j PROPN cana-5898 249 2 , , PUNCT cana-5898 249 3 math math NOUN cana-5898 249 4 . . PUNCT cana-5898 249 5 , , PUNCT cana-5898 249 6 29 29 NUM cana-5898 249 7 ( ( PUNCT cana-5898 249 8 1987 1987 NUM cana-5898 249 9 ) ) PUNCT cana-5898 249 10 , , PUNCT cana-5898 249 11 376 376 NUM cana-5898 249 12 - - SYM cana-5898 249 13 382 382 NUM cana-5898 249 14 . . PUNCT cana-5898 250 1 [ [ X cana-5898 250 2 6 6 NUM cana-5898 250 3 ] ] PUNCT cana-5898 250 4 k. k. PROPN cana-5898 250 5 c. c. PROPN cana-5898 250 6 chattopadhyay chattopadhyay PROPN cana-5898 250 7 , , PUNCT cana-5898 250 8 o. o. PROPN cana-5898 250 9 njastad njastad PROPN cana-5898 250 10 and and CCONJ cana-5898 250 11 w. w. PROPN cana-5898 250 12 j. j. PROPN cana-5898 250 13 thron thron PROPN cana-5898 250 14 , , PUNCT cana-5898 250 15 metropic metropic NOUN cana-5898 250 16 spaces space NOUN cana-5898 250 17 and and CCONJ cana-5898 250 18 extensions extension NOUN cana-5898 250 19 of of ADP cana-5898 250 20 closure closure NOUN cana-5898 250 21 spaces space NOUN cana-5898 250 22 , , PUNCT cana-5898 250 23 can can AUX cana-5898 250 24 . . PUNCT cana-5898 251 1 j. j. PROPN cana-5898 251 2 math math PROPN cana-5898 251 3 . . PUNCT cana-5898 251 4 , , PUNCT cana-5898 251 5 4 4 NUM cana-5898 251 6 ( ( PUNCT cana-5898 251 7 1983 1983 NUM cana-5898 251 8 ) ) PUNCT cana-5898 251 9 , , PUNCT cana-5898 251 10 613 613 NUM cana-5898 251 11 - - SYM cana-5898 251 12 629 629 NUM cana-5898 251 13 . . PUNCT cana-5898 252 1 [ [ X cana-5898 252 2 7 7 X cana-5898 252 3 ] ] PUNCT cana-5898 252 4 k. k. PROPN cana-5898 252 5 c. c. PROPN cana-5898 252 6 chattopadhyay chattopadhyay PROPN cana-5898 252 7 and and CCONJ cana-5898 252 8 w. w. PROPN cana-5898 252 9 j. j. PROPN cana-5898 252 10 thron thron PROPN cana-5898 252 11 , , PUNCT cana-5898 252 12 extensions extension NOUN cana-5898 252 13 of of ADP cana-5898 252 14 closure closure NOUN cana-5898 252 15 spaces space NOUN cana-5898 252 16 , , PUNCT cana-5898 252 17 can can AUX cana-5898 252 18 . . PUNCT cana-5898 253 1 j. j. PROPN cana-5898 253 2 math math PROPN cana-5898 253 3 . . PROPN cana-5898 253 4 , , PUNCT cana-5898 253 5 6 6 NUM cana-5898 253 6 ( ( PUNCT cana-5898 253 7 1977 1977 NUM cana-5898 253 8 ) ) PUNCT cana-5898 253 9 , , PUNCT cana-5898 253 10 communications communication NOUN cana-5898 253 11 on on ADP cana-5898 253 12 applied apply VERB cana-5898 253 13 nonlinear nonlinear ADJ cana-5898 253 14 analysis analysis NOUN cana-5898 253 15 issn issn NOUN cana-5898 253 16 : : PUNCT cana-5898 253 17 1074 1074 NUM cana-5898 253 18 - - PUNCT cana-5898 253 19 133x 133x NUM cana-5898 253 20 vol vol NOUN cana-5898 253 21 31 31 NUM cana-5898 253 22 no no NOUN cana-5898 253 23 . . PUNCT cana-5898 254 1 7s 7 NOUN cana-5898 254 2 ( ( PUNCT cana-5898 254 3 2024 2024 NUM cana-5898 254 4 ) ) PUNCT cana-5898 254 5 799 799 NUM cana-5898 255 1 https://internationalpubls.com https://internationalpubls.com X cana-5898 255 2 1277 1277 NUM cana-5898 255 3 - - SYM cana-5898 255 4 1286 1286 NUM cana-5898 255 5 . . PUNCT cana-5898 256 1 [ [ X cana-5898 256 2 8 8 NUM cana-5898 256 3 ] ] X cana-5898 256 4 g. g. NOUN cana-5898 256 5 choquet choquet PROPN cana-5898 256 6 , , PUNCT cana-5898 256 7 sur sur PROPN cana-5898 256 8 les les PROPN cana-5898 256 9 notions notions X cana-5898 256 10 de de X cana-5898 256 11 filtre filtre NOUN cana-5898 256 12 et et NOUN cana-5898 256 13 grille grille NOUN cana-5898 256 14 , , PUNCT cana-5898 256 15 comptes compte VERB cana-5898 256 16 rendus rendus PROPN cana-5898 256 17 acad acad PROPN cana-5898 256 18 . . PUNCT cana-5898 257 1 sci sci PROPN cana-5898 257 2 . . PROPN cana-5898 257 3 paris paris PROPN cana-5898 257 4 , , PUNCT cana-5898 257 5 224 224 NUM cana-5898 257 6 ( ( PUNCT cana-5898 257 7 1947 1947 NUM cana-5898 257 8 ) ) PUNCT cana-5898 257 9 , , PUNCT cana-5898 257 10 171 171 NUM cana-5898 257 11 - - SYM cana-5898 257 12 173 173 NUM cana-5898 257 13 . . PUNCT cana-5898 258 1 [ [ X cana-5898 258 2 9 9 NUM cana-5898 258 3 ] ] PUNCT cana-5898 258 4 s. s. PROPN cana-5898 258 5 g. g. PROPN cana-5898 258 6 crossly crossly ADV cana-5898 258 7 and and CCONJ cana-5898 258 8 s. s. PROPN cana-5898 258 9 k. k. PROPN cana-5898 258 10 hildebrand hildebrand PROPN cana-5898 258 11 , , PUNCT cana-5898 258 12 semi semi ADJ cana-5898 258 13 - - ADJ cana-5898 258 14 closure closure ADJ cana-5898 258 15 , , PUNCT cana-5898 258 16 texas texas PROPN cana-5898 258 17 j. j. PROPN cana-5898 258 18 sci sci PROPN cana-5898 258 19 . . PROPN cana-5898 258 20 , , PUNCT cana-5898 258 21 22 22 NUM cana-5898 258 22 ( ( PUNCT cana-5898 258 23 1971 1971 NUM cana-5898 258 24 ) ) PUNCT cana-5898 258 25 , , PUNCT cana-5898 258 26 99 99 NUM cana-5898 258 27 - - SYM cana-5898 258 28 112 112 NUM cana-5898 258 29 . . PUNCT cana-5898 259 1 [ [ X cana-5898 259 2 10 10 NUM cana-5898 259 3 ] ] X cana-5898 259 4 s. s. PROPN cana-5898 259 5 g. g. PROPN cana-5898 259 6 crossly crossly ADV cana-5898 259 7 and and CCONJ cana-5898 259 8 s. s. PROPN cana-5898 259 9 k. k. PROPN cana-5898 259 10 hildebrand hildebrand PROPN cana-5898 259 11 , , PUNCT cana-5898 259 12 semi semi ADJ cana-5898 259 13 - - ADJ cana-5898 259 14 topological topological ADJ cana-5898 259 15 properties property NOUN cana-5898 259 16 , , PUNCT cana-5898 259 17 fund fund NOUN cana-5898 259 18 . . PUNCT cana-5898 260 1 math math NOUN cana-5898 260 2 . . PUNCT cana-5898 261 1 , , PUNCT cana-5898 261 2 74 74 NUM cana-5898 261 3 ( ( PUNCT cana-5898 261 4 1974 1974 NUM cana-5898 261 5 ) ) PUNCT cana-5898 261 6 , , PUNCT cana-5898 261 7 233 233 NUM cana-5898 261 8 - - SYM cana-5898 261 9 254 254 NUM cana-5898 261 10 . . PUNCT cana-5898 262 1 [ [ X cana-5898 262 2 11 11 NUM cana-5898 262 3 ] ] PUNCT cana-5898 262 4 e. e. PROPN cana-5898 262 5 hatir hatir PROPN cana-5898 262 6 and and CCONJ cana-5898 262 7 s. s. PROPN cana-5898 262 8 jafari jafari PROPN cana-5898 262 9 , , PUNCT cana-5898 262 10 on on ADP cana-5898 262 11 some some DET cana-5898 262 12 new new ADJ cana-5898 262 13 classes class NOUN cana-5898 262 14 of of ADP cana-5898 262 15 sets set NOUN cana-5898 262 16 and and CCONJ cana-5898 262 17 a a DET cana-5898 262 18 new new ADJ cana-5898 262 19 decomposition decomposition NOUN cana-5898 262 20 of of ADP cana-5898 262 21 continuity continuity NOUN cana-5898 262 22 via via ADP cana-5898 262 23 grills grill NOUN cana-5898 262 24 , , PUNCT cana-5898 262 25 j. j. PROPN cana-5898 262 26 adv adv PROPN cana-5898 262 27 . . PUNCT cana-5898 262 28 math math PROPN cana-5898 262 29 . . PUNCT cana-5898 263 1 studies study NOUN cana-5898 263 2 , , PUNCT cana-5898 263 3 3(1 3(1 NUM cana-5898 263 4 ) ) PUNCT cana-5898 263 5 ( ( PUNCT cana-5898 263 6 2010 2010 NUM cana-5898 263 7 ) ) PUNCT cana-5898 263 8 , , PUNCT cana-5898 263 9 33 33 NUM cana-5898 263 10 - - SYM cana-5898 263 11 40 40 NUM cana-5898 263 12 . . PUNCT cana-5898 264 1 [ [ X cana-5898 264 2 12 12 NUM cana-5898 264 3 ] ] PUNCT cana-5898 264 4 r. r. PROPN cana-5898 264 5 c. c. PROPN cana-5898 264 6 jain jain PROPN cana-5898 264 7 , , PUNCT cana-5898 264 8 the the DET cana-5898 264 9 role role NOUN cana-5898 264 10 of of ADP cana-5898 264 11 regularly regularly ADV cana-5898 264 12 open open ADJ cana-5898 264 13 sets set NOUN cana-5898 264 14 in in ADP cana-5898 264 15 general general ADJ cana-5898 264 16 topology topology NOUN cana-5898 264 17 , , PUNCT cana-5898 264 18 ph.d ph.d PROPN cana-5898 264 19 . . PUNCT cana-5898 265 1 thesis thesis PROPN cana-5898 265 2 , , PUNCT cana-5898 265 3 meerut meerut PROPN cana-5898 265 4 univ univ PROPN cana-5898 265 5 . . PROPN cana-5898 265 6 , , PUNCT cana-5898 265 7 ( ( PUNCT cana-5898 265 8 meerut meerut PROPN cana-5898 265 9 , , PUNCT cana-5898 265 10 india india PROPN cana-5898 265 11 1980 1980 NUM cana-5898 265 12 ) ) PUNCT cana-5898 265 13 . . PUNCT cana-5898 266 1 [ [ X cana-5898 266 2 13 13 NUM cana-5898 266 3 ] ] PUNCT cana-5898 266 4 m. m. NOUN cana-5898 266 5 khan khan PROPN cana-5898 266 6 , , PUNCT cana-5898 266 7 t. t. PROPN cana-5898 266 8 noiri noiri PROPN cana-5898 266 9 and and CCONJ cana-5898 266 10 m. m. NOUN cana-5898 266 11 hussain hussain PROPN cana-5898 266 12 . . PUNCT cana-5898 267 1 on on ADP cana-5898 267 2 s*g s*g NOUN cana-5898 267 3 - - PUNCT cana-5898 267 4 closed close VERB cana-5898 267 5 sets set NOUN cana-5898 267 6 and and CCONJ cana-5898 267 7 s*-normal s*-normal ADJ cana-5898 267 8 spaces space NOUN cana-5898 267 9 , , PUNCT cana-5898 267 10 j. j. PROPN cana-5898 267 11 natur natur PROPN cana-5898 267 12 . . PUNCT cana-5898 268 1 sci sci PROPN cana-5898 268 2 . . PUNCT cana-5898 268 3 math math PROPN cana-5898 268 4 . . PUNCT cana-5898 268 5 , , PUNCT cana-5898 268 6 2016 2016 NUM cana-5898 268 7 , , PUNCT cana-5898 268 8 48 48 NUM cana-5898 268 9 . . PUNCT cana-5898 269 1 [ [ X cana-5898 269 2 14 14 NUM cana-5898 269 3 ] ] X cana-5898 269 4 n. n. PROPN cana-5898 269 5 levine levine PROPN cana-5898 269 6 . . PUNCT cana-5898 270 1 semi semi ADJ cana-5898 270 2 - - ADJ cana-5898 270 3 open open ADJ cana-5898 270 4 sets set NOUN cana-5898 270 5 and and CCONJ cana-5898 270 6 semi semi ADJ cana-5898 270 7 - - NOUN cana-5898 270 8 continuity continuity NOUN cana-5898 270 9 in in ADP cana-5898 270 10 topological topological ADJ cana-5898 270 11 spaces space NOUN cana-5898 270 12 , , PUNCT cana-5898 270 13 amer amer PROPN cana-5898 270 14 . . PROPN cana-5898 270 15 math math PROPN cana-5898 270 16 . . PUNCT cana-5898 271 1 monthly monthly ADJ cana-5898 271 2 , , PUNCT cana-5898 271 3 1963 1963 NUM cana-5898 271 4 , , PUNCT cana-5898 271 5 70 70 NUM cana-5898 271 6 : : SYM cana-5898 271 7 36 36 NUM cana-5898 271 8 - - SYM cana-5898 271 9 41 41 NUM cana-5898 271 10 . . PUNCT cana-5898 272 1 [ [ X cana-5898 272 2 15 15 NUM cana-5898 272 3 ] ] X cana-5898 272 4 n. n. PROPN cana-5898 272 5 levine levine PROPN cana-5898 272 6 . . PUNCT cana-5898 273 1 generalized generalize VERB cana-5898 273 2 closed closed ADJ cana-5898 273 3 sets set NOUN cana-5898 273 4 in in ADP cana-5898 273 5 topology topology NOUN cana-5898 273 6 , , PUNCT cana-5898 273 7 rend rend VERB cana-5898 273 8 . . PUNCT cana-5898 274 1 circ circ PROPN cana-5898 274 2 . . PUNCT cana-5898 275 1 mat mat PROPN cana-5898 275 2 . . PUNCT cana-5898 275 3 palermo palermo PROPN cana-5898 275 4 , , PUNCT cana-5898 275 5 1970 1970 NUM cana-5898 275 6 , , PUNCT cana-5898 275 7 19(2 19(2 NUM cana-5898 275 8 ): ): PUNCT cana-5898 275 9 89 89 NUM cana-5898 275 10 - - SYM cana-5898 275 11 96 96 NUM cana-5898 275 12 . . PUNCT cana-5898 276 1 [ [ X cana-5898 276 2 16 16 NUM cana-5898 276 3 ] ] X cana-5898 276 4 d. d. PROPN cana-5898 276 5 mandal mandal PROPN cana-5898 276 6 and and CCONJ cana-5898 276 7 m. m. PROPN cana-5898 276 8 n. n. PROPN cana-5898 276 9 mukerjee mukerjee PROPN cana-5898 276 10 , , PUNCT cana-5898 276 11 on on ADP cana-5898 276 12 a a DET cana-5898 276 13 class class NOUN cana-5898 276 14 of of ADP cana-5898 276 15 sets set NOUN cana-5898 276 16 via via ADP cana-5898 276 17 grill grill NOUN cana-5898 276 18 : : PUNCT cana-5898 276 19 a a DET cana-5898 276 20 decomposition decomposition NOUN cana-5898 276 21 of of ADP cana-5898 276 22 continuity continuity NOUN cana-5898 276 23 , , PUNCT cana-5898 276 24 an an PROPN cana-5898 276 25 . . PUNCT cana-5898 276 26 st st PROPN cana-5898 276 27 . . PROPN cana-5898 276 28 univ univ PROPN cana-5898 276 29 . . PUNCT cana-5898 277 1 ovidius ovidius PROPN cana-5898 277 2 constanta constanta PROPN cana-5898 277 3 , , PUNCT cana-5898 277 4 20 20 NUM cana-5898 277 5 ( ( PUNCT cana-5898 277 6 2012 2012 NUM cana-5898 277 7 ) ) PUNCT cana-5898 277 8 , , PUNCT cana-5898 277 9 307 307 NUM cana-5898 277 10 - - SYM cana-5898 277 11 316 316 NUM cana-5898 277 12 . . PUNCT cana-5898 278 1 [ [ X cana-5898 278 2 17 17 NUM cana-5898 278 3 ] ] X cana-5898 278 4 d. d. PROPN cana-5898 278 5 mandal mandal PROPN cana-5898 278 6 and and CCONJ cana-5898 278 7 m. m. PROPN cana-5898 278 8 n. n. PROPN cana-5898 278 9 mukerjee mukerjee PROPN cana-5898 278 10 , , PUNCT cana-5898 278 11 on on ADP cana-5898 278 12 a a DET cana-5898 278 13 type type NOUN cana-5898 278 14 of of ADP cana-5898 278 15 generalized generalized ADJ cana-5898 278 16 closed closed ADJ cana-5898 278 17 sets set NOUN cana-5898 278 18 , , PUNCT cana-5898 278 19 bol bol NOUN cana-5898 278 20 . . PUNCT cana-5898 279 1 soc soc PROPN cana-5898 279 2 . . PUNCT cana-5898 280 1 paran paran PROPN cana-5898 280 2 . . PUNCT cana-5898 281 1 mat mat PROPN cana-5898 281 2 . . PROPN cana-5898 281 3 , , PUNCT cana-5898 281 4 30 30 NUM cana-5898 281 5 ( ( PUNCT cana-5898 281 6 2012 2012 NUM cana-5898 281 7 ) ) PUNCT cana-5898 281 8 , , PUNCT cana-5898 281 9 67 67 NUM cana-5898 281 10 - - SYM cana-5898 281 11 76 76 NUM cana-5898 281 12 . . PUNCT cana-5898 282 1 [ [ X cana-5898 282 2 18 18 NUM cana-5898 282 3 ] ] PUNCT cana-5898 282 4 a. a. NOUN cana-5898 282 5 a. a. NOUN cana-5898 282 6 nasef nasef PROPN cana-5898 282 7 and and CCONJ cana-5898 282 8 a. a. NOUN cana-5898 282 9 a. a. PROPN cana-5898 282 10 azzam azzam PROPN cana-5898 282 11 , , PUNCT cana-5898 282 12 some some DET cana-5898 282 13 topological topological ADJ cana-5898 282 14 operators operator NOUN cana-5898 282 15 via via ADP cana-5898 282 16 grills grill NOUN cana-5898 282 17 , , PUNCT cana-5898 282 18 j. j. PROPN cana-5898 282 19 linear linear PROPN cana-5898 282 20 . . PUNCT cana-5898 283 1 top top PROPN cana-5898 283 2 . . PUNCT cana-5898 284 1 alg alg PROPN cana-5898 284 2 . . PROPN cana-5898 284 3 , , PUNCT cana-5898 284 4 5 5 NUM cana-5898 284 5 ( ( PUNCT cana-5898 284 6 2016 2016 NUM cana-5898 284 7 ) ) PUNCT cana-5898 284 8 , , PUNCT cana-5898 284 9 199 199 NUM cana-5898 284 10 - - SYM cana-5898 284 11 204 204 NUM cana-5898 284 12 . . PUNCT cana-5898 285 1 [ [ X cana-5898 285 2 19 19 NUM cana-5898 285 3 ] ] X cana-5898 285 4 o. o. PROPN cana-5898 285 5 njastad njastad PROPN cana-5898 285 6 . . PUNCT cana-5898 286 1 on on ADP cana-5898 286 2 some some DET cana-5898 286 3 classes class NOUN cana-5898 286 4 of of ADP cana-5898 286 5 nearly nearly ADV cana-5898 286 6 open open ADJ cana-5898 286 7 sets set NOUN cana-5898 286 8 , , PUNCT cana-5898 286 9 pacific pacific PROPN cana-5898 286 10 j. j. PROPN cana-5898 286 11 math math PROPN cana-5898 286 12 . . PUNCT cana-5898 286 13 , , PUNCT cana-5898 286 14 1965 1965 NUM cana-5898 286 15 , , PUNCT cana-5898 286 16 15 15 NUM cana-5898 286 17 : : PUNCT cana-5898 286 18 961 961 NUM cana-5898 286 19 - - SYM cana-5898 286 20 970 970 NUM cana-5898 286 21 . . PUNCT cana-5898 287 1 [ [ X cana-5898 287 2 20 20 NUM cana-5898 287 3 ] ] PUNCT cana-5898 287 4 b. b. PROPN cana-5898 287 5 roy roy PROPN cana-5898 287 6 and and CCONJ cana-5898 287 7 m. m. PROPN cana-5898 287 8 n. n. PROPN cana-5898 287 9 mukherjee mukherjee PROPN cana-5898 287 10 , , PUNCT cana-5898 287 11 on on ADP cana-5898 287 12 a a DET cana-5898 287 13 typical typical ADJ cana-5898 287 14 topology topology NOUN cana-5898 287 15 induced induce VERB cana-5898 287 16 by by ADP cana-5898 287 17 a a DET cana-5898 287 18 grill grill NOUN cana-5898 287 19 , , PUNCT cana-5898 287 20 soochow soochow PROPN cana-5898 287 21 j. j. PROPN cana-5898 287 22 math math PROPN cana-5898 287 23 . . PUNCT cana-5898 287 24 , , PUNCT cana-5898 287 25 33(4 33(4 NUM cana-5898 287 26 ) ) PUNCT cana-5898 287 27 ( ( PUNCT cana-5898 287 28 2007 2007 NUM cana-5898 287 29 ) ) PUNCT cana-5898 287 30 , , PUNCT cana-5898 287 31 771 771 NUM cana-5898 287 32 - - SYM cana-5898 287 33 786 786 NUM cana-5898 287 34 . . PUNCT cana-5898 288 1 [ [ X cana-5898 288 2 21 21 NUM cana-5898 288 3 ] ] X cana-5898 288 4 b. b. PROPN cana-5898 288 5 roy roy PROPN cana-5898 288 6 and and CCONJ cana-5898 288 7 m. m. PROPN cana-5898 288 8 n. n. PROPN cana-5898 288 9 mukherjee mukherjee PROPN cana-5898 288 10 , , PUNCT cana-5898 288 11 concerning concern VERB cana-5898 288 12 topologies topology NOUN cana-5898 288 13 induced induce VERB cana-5898 288 14 by by ADP cana-5898 288 15 principal principal ADJ cana-5898 288 16 grills grill NOUN cana-5898 288 17 , , PUNCT cana-5898 288 18 an an PROPN cana-5898 288 19 . . NOUN cana-5898 288 20 stiint stiint PROPN cana-5898 288 21 . . PUNCT cana-5898 289 1 univ univ PROPN cana-5898 289 2 . . PUNCT cana-5898 290 1 al al PROPN cana-5898 290 2 . . PROPN cana-5898 290 3 i. i. PROPN cana-5898 290 4 cuza cuza PROPN cana-5898 290 5 iasi iasi PROPN cana-5898 290 6 . . PUNCT cana-5898 291 1 mat mat PROPN cana-5898 291 2 . . PUNCT cana-5898 292 1 ( ( PUNCT cana-5898 292 2 n.s n.s PROPN cana-5898 292 3 . . PROPN cana-5898 292 4 ) ) PUNCT cana-5898 292 5 , , PUNCT cana-5898 292 6 55(2 55(2 NUM cana-5898 292 7 ) ) PUNCT cana-5898 292 8 ( ( PUNCT cana-5898 292 9 2009 2009 NUM cana-5898 292 10 ) ) PUNCT cana-5898 292 11 , , PUNCT cana-5898 292 12 285 285 NUM cana-5898 292 13 - - SYM cana-5898 292 14 294 294 NUM cana-5898 292 15 . . PUNCT cana-5898 293 1 [ [ X cana-5898 293 2 22 22 NUM cana-5898 293 3 ] ] X cana-5898 293 4 d. d. PROPN cana-5898 293 5 saravanakumar saravanakumar PROPN cana-5898 293 6 and and CCONJ cana-5898 293 7 n. n. PROPN cana-5898 293 8 kalaivani kalaivani PROPN cana-5898 293 9 , , PUNCT cana-5898 293 10 on on ADP cana-5898 293 11 grill grill NOUN cana-5898 293 12 sp sp NOUN cana-5898 293 13 - - PUNCT cana-5898 293 14 open open ADJ cana-5898 293 15 set set NOUN cana-5898 293 16 in in ADP cana-5898 293 17 grill grill ADJ cana-5898 293 18 topological topological ADJ cana-5898 293 19 spaces space NOUN cana-5898 293 20 , , PUNCT cana-5898 293 21 j. j. PROPN cana-5898 293 22 new new PROPN cana-5898 293 23 theory theory NOUN cana-5898 293 24 , , PUNCT cana-5898 293 25 23(4 23(4 NOUN cana-5898 293 26 ) ) PUNCT cana-5898 293 27 ( ( PUNCT cana-5898 293 28 2018 2018 NUM cana-5898 293 29 ) ) PUNCT cana-5898 293 30 , , PUNCT cana-5898 293 31 85 85 NUM cana-5898 293 32 - - SYM cana-5898 293 33 92 92 NUM cana-5898 293 34 . . PUNCT cana-5898 294 1 [ [ X cana-5898 294 2 23 23 NUM cana-5898 294 3 ] ] X cana-5898 294 4 d. d. PROPN cana-5898 294 5 saravanakumar saravanakumar PROPN cana-5898 294 6 and and CCONJ cana-5898 294 7 n. n. PROPN cana-5898 294 8 kalaivani kalaivani PROPN cana-5898 294 9 , , PUNCT cana-5898 294 10 𝒒𝑠𝛼 𝒒𝑠𝛼 PROPN cana-5898 294 11 -open -open NOUN cana-5898 294 12 sets set NOUN cana-5898 294 13 in in ADP cana-5898 294 14 grill grill ADJ cana-5898 294 15 topological topological ADJ cana-5898 294 16 spaces space NOUN cana-5898 294 17 , , PUNCT cana-5898 294 18 math math NOUN cana-5898 294 19 . . PUNCT cana-5898 294 20 bilten bilten PROPN cana-5898 294 21 , , PUNCT cana-5898 294 22 44(1 44(1 PROPN cana-5898 294 23 ) ) PUNCT cana-5898 294 24 ( ( PUNCT cana-5898 294 25 2020 2020 NUM cana-5898 294 26 ) ) PUNCT cana-5898 294 27 , , PUNCT cana-5898 294 28 79 79 NUM cana-5898 294 29 - - SYM cana-5898 294 30 90 90 NUM cana-5898 294 31 . . PUNCT cana-5898 295 1 [ [ X cana-5898 295 2 24 24 NUM cana-5898 295 3 ] ] PUNCT cana-5898 295 4 d. d. PROPN cana-5898 295 5 saravanakumar saravanakumar PROPN cana-5898 295 6 , , PUNCT cana-5898 295 7 n. n. PROPN cana-5898 295 8 kalaivani kalaivani PROPN cana-5898 295 9 and and CCONJ cana-5898 295 10 g. g. PROPN cana-5898 295 11 sai sai PROPN cana-5898 295 12 sundara sundara PROPN cana-5898 295 13 krishnan krishnan PROPN cana-5898 295 14 , , PUNCT cana-5898 295 15 on on ADP cana-5898 295 16 *-pre *-pre PROPN cana-5898 295 17 - - PUNCT cana-5898 295 18 regular-𝑇1 regular-𝑇1 ADJ cana-5898 295 19 1 1 NUM cana-5898 295 20 communications communication NOUN cana-5898 295 21 on on ADP cana-5898 295 22 applied apply VERB cana-5898 295 23 nonlinear nonlinear ADJ cana-5898 295 24 analysis analysis NOUN cana-5898 295 25 issn issn NOUN cana-5898 295 26 : : PUNCT cana-5898 295 27 1074 1074 NUM cana-5898 295 28 - - PUNCT cana-5898 295 29 133x 133x NUM cana-5898 295 30 vol vol NOUN cana-5898 295 31 31 31 NUM cana-5898 295 32 no no NOUN cana-5898 295 33 . . PUNCT cana-5898 296 1 7s 7 NOUN cana-5898 296 2 ( ( PUNCT cana-5898 296 3 2024 2024 NUM cana-5898 296 4 ) ) PUNCT cana-5898 296 5 800 800 NUM cana-5898 296 6 https://internationalpubls.com https://internationalpubls.com X cana-5898 296 7 spaces space NOUN cana-5898 296 8 associated associate VERB cana-5898 296 9 with with ADP cana-5898 296 10 operations operation NOUN cana-5898 296 11 separation separation NOUN cana-5898 296 12 axioms axiom NOUN cana-5898 296 13 , , PUNCT cana-5898 296 14 jour jour X cana-5898 296 15 . . PROPN cana-5898 296 16 interdisc interdisc PROPN cana-5898 296 17 . . PUNCT cana-5898 297 1 math math NOUN cana-5898 297 2 . . PUNCT cana-5898 298 1 , , PUNCT cana-5898 298 2 17 17 NUM cana-5898 298 3 ( ( PUNCT cana-5898 298 4 2014 2014 NUM cana-5898 298 5 ) ) PUNCT cana-5898 298 6 , , PUNCT cana-5898 298 7 485 485 NUM cana-5898 298 8 - - SYM cana-5898 298 9 498 498 NUM cana-5898 298 10 . . PUNCT cana-5898 299 1 [ [ X cana-5898 299 2 25 25 NUM cana-5898 299 3 ] ] PUNCT cana-5898 299 4 d. d. PROPN cana-5898 299 5 saravanakumar saravanakumar PROPN cana-5898 299 6 , , PUNCT cana-5898 299 7 n. n. PROPN cana-5898 299 8 kalaivani kalaivani PROPN cana-5898 299 9 and and CCONJ cana-5898 299 10 g. g. PROPN cana-5898 299 11 sai sai PROPN cana-5898 299 12 sundara sundara PROPN cana-5898 299 13 krishnan krishnan PROPN cana-5898 299 14 , , PUNCT cana-5898 299 15 πœ‡-open πœ‡-open PROPN cana-5898 299 16 sets set NOUN cana-5898 299 17 in in ADP cana-5898 299 18 generalized generalized ADJ cana-5898 299 19 topological topological ADJ cana-5898 299 20 spaces space NOUN cana-5898 299 21 , , PUNCT cana-5898 299 22 malaya malaya PROPN cana-5898 299 23 j. j. PROPN cana-5898 299 24 mat mat PROPN cana-5898 299 25 . . PROPN cana-5898 299 26 , , PUNCT cana-5898 299 27 3 3 NUM cana-5898 299 28 ( ( PUNCT cana-5898 299 29 2015 2015 NUM cana-5898 299 30 ) ) PUNCT cana-5898 299 31 , , PUNCT cana-5898 299 32 268 268 NUM cana-5898 299 33 - - SYM cana-5898 299 34 276 276 NUM cana-5898 299 35 . . PUNCT cana-5898 300 1 [ [ X cana-5898 300 2 26 26 NUM cana-5898 300 3 ] ] X cana-5898 300 4 d. d. PROPN cana-5898 300 5 saravanakumar saravanakumar PROPN cana-5898 300 6 and and CCONJ cana-5898 300 7 g. g. PROPN cana-5898 300 8 sai sai PROPN cana-5898 300 9 sundara sundara PROPN cana-5898 300 10 krishnan krishnan PROPN cana-5898 300 11 , , PUNCT cana-5898 300 12 generalized generalized ADJ cana-5898 300 13 mappings mapping NOUN cana-5898 300 14 via via ADP cana-5898 300 15 new new ADJ cana-5898 300 16 closed close VERB cana-5898 300 17 sets set NOUN cana-5898 300 18 , , PUNCT cana-5898 300 19 int int NOUN cana-5898 300 20 . . PUNCT cana-5898 301 1 j. j. PROPN cana-5898 301 2 mat mat PROPN cana-5898 301 3 . . PUNCT cana-5898 301 4 sci sci PROPN cana-5898 301 5 . . PUNCT cana-5898 301 6 appl appl PROPN cana-5898 301 7 . . PROPN cana-5898 301 8 , , PUNCT cana-5898 301 9 2 2 NUM cana-5898 301 10 ( ( PUNCT cana-5898 301 11 2012 2012 NUM cana-5898 301 12 ) ) PUNCT cana-5898 301 13 , , PUNCT cana-5898 301 14 127 127 NUM cana-5898 301 15 - - SYM cana-5898 301 16 137 137 NUM cana-5898 301 17 . . PUNCT cana-5898 302 1 [ [ X cana-5898 302 2 27 27 NUM cana-5898 302 3 ] ] X cana-5898 302 4 w. w. PROPN cana-5898 302 5 j. j. PROPN cana-5898 302 6 thron thron PROPN cana-5898 302 7 , , PUNCT cana-5898 302 8 proximity proximity NOUN cana-5898 302 9 structures structure NOUN cana-5898 302 10 and and CCONJ cana-5898 302 11 grills grill NOUN cana-5898 302 12 , , PUNCT cana-5898 302 13 math math NOUN cana-5898 302 14 . . PUNCT cana-5898 303 1 ann ann PROPN cana-5898 303 2 . . PROPN cana-5898 303 3 , , PUNCT cana-5898 303 4 206 206 NUM cana-5898 303 5 ( ( PUNCT cana-5898 303 6 1973 1973 NUM cana-5898 303 7 ) ) PUNCT cana-5898 303 8 , , PUNCT cana-5898 303 9 35 35 NUM cana-5898 303 10 - - SYM cana-5898 303 11 62 62 NUM cana-5898 303 12 . . PUNCT cana-5898 304 1 [ [ X cana-5898 304 2 28 28 NUM cana-5898 304 3 ] ] X cana-5898 304 4 n. n. NOUN cana-5898 304 5 v. v. ADP cana-5898 304 6 velicko velicko ADJ cana-5898 304 7 , , PUNCT cana-5898 304 8 h h NOUN cana-5898 304 9 - - PUNCT cana-5898 304 10 closed closed ADJ cana-5898 304 11 topological topological ADJ cana-5898 304 12 spaces space NOUN cana-5898 304 13 , , PUNCT cana-5898 304 14 amer amer PROPN cana-5898 304 15 . . PROPN cana-5898 304 16 math math PROPN cana-5898 304 17 . . PUNCT cana-5898 305 1 soc soc PROPN cana-5898 305 2 . . PUNCT cana-5898 306 1 transl transl PROPN cana-5898 306 2 . . PUNCT cana-5898 306 3 , , PUNCT cana-5898 306 4 78 78 NUM cana-5898 306 5 ( ( PUNCT cana-5898 306 6 1968 1968 NUM cana-5898 306 7 ) ) PUNCT cana-5898 306 8 . . PUNCT cana-5898 307 1 102 102 NUM cana-5898 307 2 - - SYM cana-5898 307 3 118 118 NUM cana-5898 307 4 . . PUNCT