id sid tid token lemma pos cana-2648 1 1 communications communication NOUN cana-2648 1 2 on on ADP cana-2648 1 3 applied apply VERB cana-2648 1 4 nonlinear nonlinear ADJ cana-2648 1 5 analysis analysis NOUN cana-2648 1 6 issn issn NOUN cana-2648 1 7 : : PUNCT cana-2648 1 8 1074 1074 NUM cana-2648 1 9 - - PUNCT cana-2648 1 10 133x 133x NUM cana-2648 1 11 vol vol NOUN cana-2648 1 12 32 32 NUM cana-2648 1 13 no no NOUN cana-2648 1 14 . . PUNCT cana-2648 2 1 3s 3s NUM cana-2648 2 2 ( ( PUNCT cana-2648 2 3 2025 2025 NUM cana-2648 2 4 ) ) PUNCT cana-2648 2 5 357 357 NUM cana-2648 2 6 https://internationalpubls.com https://internationalpubls.com X cana-2648 2 7 minimal minimal ADJ cana-2648 2 8 and and CCONJ cana-2648 2 9 maximal maximal ADJ cana-2648 2 10 π’ˆπœΌ-continuous π’ˆπœΌ-continuous ADJ cana-2648 2 11 functions function NOUN cana-2648 2 12 in in ADP cana-2648 2 13 topological topological ADJ cana-2648 2 14 spaces space NOUN cana-2648 2 15 d. d. PROPN cana-2648 2 16 subbulakshmi subbulakshmi PROPN cana-2648 2 17 associate associate PROPN cana-2648 2 18 professor professor NOUN cana-2648 2 19 , , PUNCT cana-2648 2 20 department department NOUN cana-2648 2 21 of of ADP cana-2648 2 22 mathematics mathematic NOUN cana-2648 2 23 , , PUNCT cana-2648 2 24 rvs rvs ADJ cana-2648 2 25 college college NOUN cana-2648 2 26 of of ADP cana-2648 2 27 arts art NOUN cana-2648 2 28 and and CCONJ cana-2648 2 29 science science NOUN cana-2648 2 30 , , PUNCT cana-2648 2 31 tamil tamil PROPN cana-2648 2 32 nadu nadu PROPN cana-2648 2 33 , , PUNCT cana-2648 2 34 india india PROPN cana-2648 2 35 , , PUNCT cana-2648 2 36 email email NOUN cana-2648 2 37 : : PUNCT cana-2648 2 38 subbulakshmi169@gmail.com subbulakshmi169@gmail.com X cana-2648 2 39 . . PUNCT cana-2648 3 1 article article NOUN cana-2648 3 2 history history NOUN cana-2648 3 3 : : PUNCT cana-2648 3 4 received receive VERB cana-2648 3 5 : : PUNCT cana-2648 3 6 24 24 NUM cana-2648 3 7 - - PUNCT cana-2648 3 8 09 09 NUM cana-2648 3 9 - - PUNCT cana-2648 3 10 2024 2024 NUM cana-2648 3 11 revised revise VERB cana-2648 3 12 : : PUNCT cana-2648 3 13 05 05 NUM cana-2648 3 14 - - SYM cana-2648 3 15 11 11 NUM cana-2648 3 16 - - PUNCT cana-2648 3 17 2024 2024 NUM cana-2648 3 18 accepted accept VERB cana-2648 3 19 : : PUNCT cana-2648 3 20 20 20 NUM cana-2648 3 21 - - SYM cana-2648 3 22 11 11 NUM cana-2648 3 23 - - PUNCT cana-2648 3 24 2024 2024 NUM cana-2648 3 25 abstract abstract NOUN cana-2648 3 26 : : PUNCT cana-2648 3 27 the the DET cana-2648 3 28 concept concept NOUN cana-2648 3 29 of of ADP cana-2648 3 30 maximal maximal ADJ cana-2648 3 31 and and CCONJ cana-2648 3 32 minimal minimal ADJ cana-2648 3 33 gΞ· gΞ· NOUN cana-2648 3 34 - - PUNCT cana-2648 3 35 continuous continuous ADJ cana-2648 3 36 functions function NOUN cana-2648 3 37 and and CCONJ cana-2648 3 38 some some DET cana-2648 3 39 new new ADJ cana-2648 3 40 results result NOUN cana-2648 3 41 are be AUX cana-2648 3 42 given give VERB cana-2648 3 43 . . PUNCT cana-2648 4 1 keywords keyword NOUN cana-2648 4 2 : : PUNCT cana-2648 4 3 minimal minimal ADJ cana-2648 4 4 gΞ· gΞ· NOUN cana-2648 4 5 - - PUNCT cana-2648 4 6 continuous continuous ADJ cana-2648 4 7 , , PUNCT cana-2648 4 8 maximal maximal ADJ cana-2648 4 9 gΞ· gΞ· NOUN cana-2648 4 10 - - PUNCT cana-2648 4 11 continuous continuous ADJ cana-2648 4 12 functions function NOUN cana-2648 4 13 . . PUNCT cana-2648 5 1 1 1 X cana-2648 5 2 . . X cana-2648 5 3 introduction introduction NOUN cana-2648 5 4 levine levine PROPN cana-2648 5 5 [ [ X cana-2648 5 6 2 2 NUM cana-2648 5 7 ] ] PUNCT cana-2648 5 8 proposed propose VERB cana-2648 5 9 some some DET cana-2648 5 10 properties property NOUN cana-2648 5 11 in in ADP cana-2648 5 12 1963 1963 NUM cana-2648 5 13 , , PUNCT cana-2648 5 14 s s X cana-2648 5 15 - - PUNCT cana-2648 5 16 open open ADJ cana-2648 5 17 sets set NOUN cana-2648 5 18 were be AUX cana-2648 5 19 introduced introduce VERB cana-2648 5 20 into into ADP cana-2648 5 21 topological topological ADJ cana-2648 5 22 spaces space NOUN cana-2648 5 23 . . PUNCT cana-2648 6 1 in in ADP cana-2648 6 2 1984 1984 NUM cana-2648 6 3 , , PUNCT cana-2648 6 4 andrijevic andrijevic VERB cana-2648 6 5 [ [ X cana-2648 6 6 1 1 X cana-2648 6 7 ] ] PUNCT cana-2648 6 8 described describe VERB cana-2648 6 9 some some PRON cana-2648 6 10 of of ADP cana-2648 6 11 the the DET cana-2648 6 12 topological topological ADJ cana-2648 6 13 properties property NOUN cana-2648 6 14 of of ADP cana-2648 6 15 alpha alpha NOUN cana-2648 6 16 sets set NOUN cana-2648 6 17 . . PUNCT cana-2648 7 1 the the DET cana-2648 7 2 concept concept NOUN cana-2648 7 3 of of ADP cana-2648 7 4 generalized generalized ADJ cana-2648 7 5 closed close VERB cana-2648 7 6 sets set NOUN cana-2648 7 7 in in ADP cana-2648 7 8 topological topological ADJ cana-2648 7 9 spaces space NOUN cana-2648 7 10 was be AUX cana-2648 7 11 presented present VERB cana-2648 7 12 by by ADP cana-2648 7 13 norman norman PROPN cana-2648 7 14 levine levine PROPN cana-2648 8 1 [ [ X cana-2648 8 2 3 3 NUM cana-2648 8 3 ] ] PUNCT cana-2648 8 4 . . PUNCT cana-2648 9 1 [ [ X cana-2648 9 2 7 7 NUM cana-2648 9 3 , , PUNCT cana-2648 9 4 8 8 NUM cana-2648 9 5 , , PUNCT cana-2648 9 6 9 9 NUM cana-2648 9 7 ] ] PUNCT cana-2648 9 8 introduced introduce VERB cana-2648 9 9 the the DET cana-2648 9 10 concept concept NOUN cana-2648 9 11 of of ADP cana-2648 9 12 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 9 13 sets set NOUN cana-2648 9 14 and and CCONJ cana-2648 9 15 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 9 16 functions function NOUN cana-2648 9 17 and and CCONJ cana-2648 9 18 their their PRON cana-2648 9 19 various various ADJ cana-2648 9 20 characterizations characterization NOUN cana-2648 9 21 . . PUNCT cana-2648 10 1 nakaoka nakaoka NOUN cana-2648 10 2 and and CCONJ cana-2648 10 3 oda oda PROPN cana-2648 10 4 developed develop VERB cana-2648 10 5 two two NUM cana-2648 10 6 subclasses subclass NOUN cana-2648 10 7 of of ADP cana-2648 10 8 open open ADJ cana-2648 10 9 sets set NOUN cana-2648 10 10 : : PUNCT cana-2648 10 11 maximal maximal ADJ cana-2648 10 12 and and CCONJ cana-2648 10 13 minimal minimal ADJ cana-2648 10 14 open open ADJ cana-2648 10 15 sets set NOUN cana-2648 10 16 [ [ X cana-2648 10 17 4,5,6 4,5,6 NUM cana-2648 10 18 ] ] PUNCT cana-2648 10 19 . . PUNCT cana-2648 11 1 later later ADV cana-2648 11 2 , , PUNCT cana-2648 11 3 numerous numerous ADJ cana-2648 11 4 authors author NOUN cana-2648 11 5 concentrated concentrate VERB cana-2648 11 6 on on ADP cana-2648 11 7 this this DET cana-2648 11 8 subject subject NOUN cana-2648 11 9 , , PUNCT cana-2648 11 10 developing develop VERB cana-2648 11 11 the the DET cana-2648 11 12 concept concept NOUN cana-2648 11 13 of of ADP cana-2648 11 14 minimal minimal ADJ cana-2648 11 15 and and CCONJ cana-2648 11 16 maximal maximal ADJ cana-2648 11 17 open open ADJ cana-2648 11 18 sets set NOUN cana-2648 11 19 . . PUNCT cana-2648 12 1 following follow VERB cana-2648 12 2 these these DET cana-2648 12 3 improvements improvement NOUN cana-2648 12 4 , , PUNCT cana-2648 12 5 we we PRON cana-2648 12 6 investigate investigate VERB cana-2648 12 7 minimal minimal ADJ cana-2648 12 8 and and CCONJ cana-2648 12 9 maximal maximal ADJ cana-2648 12 10 gΞ·continuous gΞ·continuous ADJ cana-2648 12 11 functions function NOUN cana-2648 12 12 . . PUNCT cana-2648 13 1 2 2 X cana-2648 13 2 . . NOUN cana-2648 13 3 minimal minimal ADJ cana-2648 13 4 π’ˆπœΌ-continuous π’ˆπœΌ-continuous ADJ cana-2648 13 5 functions function NOUN cana-2648 13 6 this this DET cana-2648 13 7 section section NOUN cana-2648 13 8 introduces introduce NOUN cana-2648 13 9 and and CCONJ cana-2648 13 10 establishes establish VERB cana-2648 13 11 various various ADJ cana-2648 13 12 properties property NOUN cana-2648 13 13 of of ADP cana-2648 13 14 minimal minimal ADJ cana-2648 13 15 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 13 16 topological topological ADJ cana-2648 13 17 spaces space NOUN cana-2648 13 18 . . PUNCT cana-2648 14 1 definition definition NOUN cana-2648 14 2 2.1[10 2.1[10 NUM cana-2648 14 3 ] ] PUNCT cana-2648 14 4 : : PUNCT cana-2648 14 5 a a DET cana-2648 14 6 minimal minimal ADJ cana-2648 14 7 gΞ· gΞ· NOUN cana-2648 14 8 - - PUNCT cana-2648 14 9 open open ADJ cana-2648 14 10 is be AUX cana-2648 14 11 a a DET cana-2648 14 12 proper proper ADJ cana-2648 14 13 , , PUNCT cana-2648 14 14 nonempty nonempty ADJ cana-2648 14 15 gΞ· gΞ· NOUN cana-2648 14 16 - - PUNCT cana-2648 14 17 open open ADJ cana-2648 14 18 subset subset ADJ cana-2648 14 19 u u NOUN cana-2648 14 20 ' ' PUNCT cana-2648 14 21 of]= of]= NOUN cana-2648 14 22 ( ( PUNCT cana-2648 14 23 𝑋′ 𝑋′ ADJ cana-2648 14 24 , , PUNCT cana-2648 14 25 πœβ€² πœβ€² PROPN cana-2648 14 26 ) ) PUNCT cana-2648 14 27 if if SCONJ cana-2648 14 28 any any PRON cana-2648 14 29 of of ADP cana-2648 14 30 its its PRON cana-2648 14 31 π‘”πœ‚-opens π‘”πœ‚-open NOUN cana-2648 14 32 are be AUX cana-2648 14 33 π‘ˆβ€² π‘ˆβ€² ADJ cana-2648 14 34 is be AUX cana-2648 14 35 πœ‘β€² πœ‘β€² NUM cana-2648 14 36 or or CCONJ cana-2648 14 37 π‘ˆβ€². π‘ˆβ€². NUM cana-2648 14 38 definition definition NOUN cana-2648 14 39 2.2 2.2 NUM cana-2648 14 40 : : PUNCT cana-2648 14 41 if if SCONJ cana-2648 14 42 οΏ½ οΏ½ PROPN cana-2648 14 43 Μ‡ Μ‡ VERB cana-2648 14 44 οΏ½ οΏ½ PROPN cana-2648 14 45 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 14 46 ) ) PUNCT cana-2648 14 47 is be AUX cana-2648 14 48 a a DET cana-2648 14 49 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 14 50 in in ADP cana-2648 14 51 ( ( PUNCT cana-2648 14 52 𝑋′ 𝑋′ ADJ cana-2648 14 53 , , PUNCT cana-2648 14 54 πœβ€² πœβ€² PROPN cana-2648 14 55 ) ) PUNCT cana-2648 14 56 for for ADP cana-2648 14 57 any any DET cana-2648 14 58 minimal minimal ADJ cana-2648 14 59 open open ADJ cana-2648 14 60 𝑀′ 𝑀′ NOUN cana-2648 14 61 in in ADP cana-2648 14 62 ( ( PUNCT cana-2648 14 63 π‘Œβ€² π‘Œβ€² PROPN cana-2648 14 64 , , PUNCT cana-2648 14 65 πœŽβ€² πœŽβ€² NUM cana-2648 14 66 ) ) PUNCT cana-2648 14 67 then then ADV cana-2648 14 68 a a DET cana-2648 14 69 function function NOUN cana-2648 14 70 οΏ½ οΏ½ PROPN cana-2648 14 71 Μ‡ Μ‡ PROPN cana-2648 14 72 οΏ½ οΏ½ PROPN cana-2648 14 73 : : PUNCT cana-2648 14 74 ( ( PUNCT cana-2648 14 75 𝑋′ 𝑋′ X cana-2648 14 76 , , PUNCT cana-2648 14 77 πœβ€² πœβ€² NUM cana-2648 14 78 ) ) PUNCT cana-2648 14 79 β†’ β†’ SYM cana-2648 14 80 ( ( PUNCT cana-2648 14 81 π‘Œβ€² π‘Œβ€² PROPN cana-2648 14 82 , , PUNCT cana-2648 14 83 πœŽβ€² πœŽβ€² NUM cana-2648 14 84 ) ) PUNCT cana-2648 14 85 is be AUX cana-2648 14 86 minimal minimal ADJ cana-2648 14 87 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 14 88 . . PUNCT cana-2648 15 1 example example NOUN cana-2648 15 2 2.3 2.3 NUM cana-2648 15 3 : : PUNCT cana-2648 15 4 let𝑋′ let𝑋′ X cana-2648 15 5 = = PUNCT cana-2648 16 1 π‘Œβ€² π‘Œβ€² PROPN cana-2648 16 2 = = SYM cana-2648 16 3 { { PUNCT cana-2648 16 4 𝑒′ 𝑒′ NOUN cana-2648 16 5 , , PUNCT cana-2648 16 6 𝑓′ 𝑓′ NUM cana-2648 16 7 , , PUNCT cana-2648 16 8 𝑔′ 𝑔′ ADV cana-2648 16 9 } } PUNCT cana-2648 16 10 , , PUNCT cana-2648 16 11 πœβ€² πœβ€² PUNCT cana-2648 16 12 = = PUNCT cana-2648 16 13 { { PUNCT cana-2648 16 14 𝑋′ 𝑋′ NOUN cana-2648 16 15 , , PUNCT cana-2648 16 16 πœ‘β€² πœ‘β€² NUM cana-2648 16 17 , , PUNCT cana-2648 16 18 { { PUNCT cana-2648 16 19 𝑒′ 𝑒′ NOUN cana-2648 16 20 } } PUNCT cana-2648 16 21 , , PUNCT cana-2648 16 22 { { PUNCT cana-2648 16 23 𝑔′ 𝑔′ ADV cana-2648 16 24 } } PUNCT cana-2648 16 25 , , PUNCT cana-2648 16 26 { { PUNCT cana-2648 16 27 𝑒′ 𝑒′ NOUN cana-2648 16 28 , , PUNCT cana-2648 16 29 𝑔′ 𝑔′ ADV cana-2648 16 30 } } PUNCT cana-2648 16 31 } } PUNCT cana-2648 16 32 , , PUNCT cana-2648 16 33 πœŽβ€² πœŽβ€² X cana-2648 16 34 = = PUNCT cana-2648 16 35 { { PUNCT cana-2648 16 36 π‘Œβ€² π‘Œβ€² PROPN cana-2648 16 37 , , PUNCT cana-2648 16 38 πœ‘β€² πœ‘β€² NUM cana-2648 16 39 , , PUNCT cana-2648 16 40 { { PUNCT cana-2648 16 41 𝑒′ 𝑒′ NOUN cana-2648 16 42 } } PUNCT cana-2648 16 43 , , PUNCT cana-2648 16 44 { { PUNCT cana-2648 16 45 𝑓′ 𝑓′ NOUN cana-2648 16 46 } } PUNCT cana-2648 16 47 , , PUNCT cana-2648 16 48 { { PUNCT cana-2648 16 49 𝑒′ 𝑒′ NOUN cana-2648 16 50 , , PUNCT cana-2648 16 51 𝑓′ 𝑓′ NUM cana-2648 16 52 } } PUNCT cana-2648 16 53 } } PUNCT cana-2648 16 54 . . PUNCT cana-2648 17 1 define define VERB cana-2648 17 2 οΏ½ οΏ½ PROPN cana-2648 17 3 Μ‡ Μ‡ PROPN cana-2648 17 4 οΏ½ οΏ½ PROPN cana-2648 17 5 : : PUNCT cana-2648 17 6 ( ( PUNCT cana-2648 17 7 𝑋′ 𝑋′ X cana-2648 17 8 , , PUNCT cana-2648 17 9 πœβ€² πœβ€² NUM cana-2648 17 10 ) ) PUNCT cana-2648 17 11 β†’ β†’ SYM cana-2648 17 12 ( ( PUNCT cana-2648 17 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 17 14 , , PUNCT cana-2648 17 15 πœŽβ€² πœŽβ€² NUM cana-2648 17 16 ) ) PUNCT cana-2648 17 17 by by ADP cana-2648 17 18 οΏ½ οΏ½ PROPN cana-2648 17 19 Μ‡ Μ‡ PROPN cana-2648 17 20 οΏ½ οΏ½ PROPN cana-2648 17 21 (𝑒′ (𝑒′ NUM cana-2648 17 22 ) ) PUNCT cana-2648 17 23 = = SYM cana-2648 17 24 𝑓′ 𝑓′ NUM cana-2648 17 25 , , PUNCT cana-2648 17 26 οΏ½ οΏ½ PROPN cana-2648 17 27 Μ‡ Μ‡ PROPN cana-2648 17 28 οΏ½ οΏ½ PROPN cana-2648 17 29 (𝑓′ (𝑓′ PUNCT cana-2648 17 30 ) ) PUNCT cana-2648 17 31 = = SYM cana-2648 17 32 𝑔′ 𝑔′ NUM cana-2648 17 33 , , PUNCT cana-2648 17 34 οΏ½ οΏ½ PROPN cana-2648 17 35 Μ‡ Μ‡ PROPN cana-2648 17 36 οΏ½ οΏ½ PROPN cana-2648 17 37 (𝑔′ (𝑔′ X cana-2648 17 38 ) ) PUNCT cana-2648 17 39 = = PUNCT cana-2648 18 1 𝑒′. 𝑒′. NOUN cana-2648 18 2 here here ADV cana-2648 18 3 { { PUNCT cana-2648 18 4 𝑒′ 𝑒′ NOUN cana-2648 18 5 } } PUNCT cana-2648 18 6 , , PUNCT cana-2648 18 7 { { PUNCT cana-2648 18 8 𝑓′ 𝑓′ NOUN cana-2648 18 9 } } PUNCT cana-2648 18 10 are be AUX cana-2648 18 11 minimal minimal ADJ cana-2648 18 12 open open ADJ cana-2648 18 13 in(π‘Œβ€² in(π‘Œβ€² PROPN cana-2648 18 14 , , PUNCT cana-2648 18 15 πœŽβ€² πœŽβ€² NUM cana-2648 18 16 ) ) PUNCT cana-2648 18 17 . . PUNCT cana-2648 19 1 therefore therefore ADV cana-2648 19 2 οΏ½ οΏ½ PROPN cana-2648 19 3 Μ‡ Μ‡ PROPN cana-2648 19 4 οΏ½ οΏ½ PROPN cana-2648 19 5 is be AUX cana-2648 19 6 minimal minimal ADJ cana-2648 19 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 19 8 . . PUNCT cana-2648 20 1 theorem theorem VERB cana-2648 20 2 2.4 2.4 NUM cana-2648 20 3 : : PUNCT cana-2648 20 4 all all DET cana-2648 20 5 π‘”πœ‚-continuonus π‘”πœ‚-continuonus NOUN cana-2648 20 6 is be AUX cana-2648 20 7 minimal minimal ADJ cana-2648 20 8 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 20 9 . . PUNCT cana-2648 21 1 proof proof NOUN cana-2648 21 2 : : PUNCT cana-2648 21 3 let let VERB cana-2648 21 4 𝑀′ 𝑀′ NOUN cana-2648 21 5 be be AUX cana-2648 21 6 a a DET cana-2648 21 7 minimal minimal ADJ cana-2648 21 8 open open NOUN cana-2648 21 9 in in ADP cana-2648 21 10 ( ( PUNCT cana-2648 21 11 π‘Œβ€² π‘Œβ€² NOUN cana-2648 21 12 , , PUNCT cana-2648 21 13 πœŽβ€² πœŽβ€² NUM cana-2648 21 14 ) ) PUNCT cana-2648 21 15 and and CCONJ cana-2648 21 16 οΏ½ οΏ½ PROPN cana-2648 21 17 Μ‡ Μ‡ PROPN cana-2648 21 18 οΏ½ οΏ½ PROPN cana-2648 21 19 : : PUNCT cana-2648 21 20 ( ( PUNCT cana-2648 21 21 𝑋′ 𝑋′ X cana-2648 21 22 , , PUNCT cana-2648 21 23 πœβ€² πœβ€² NUM cana-2648 21 24 ) ) PUNCT cana-2648 21 25 β†’ β†’ SYM cana-2648 21 26 ( ( PUNCT cana-2648 21 27 π‘Œβ€² π‘Œβ€² PROPN cana-2648 21 28 , , PUNCT cana-2648 21 29 πœŽβ€² πœŽβ€² NUM cana-2648 21 30 ) ) PUNCT cana-2648 21 31 be be VERB cana-2648 21 32 a a DET cana-2648 21 33 π‘”πœ‚-contenuous π‘”πœ‚-contenuous ADJ cana-2648 21 34 . . PUNCT cana-2648 22 1 𝑀′ 𝑀′ PROPN cana-2648 22 2 is be AUX cana-2648 22 3 an an DET cana-2648 22 4 open open ADJ cana-2648 22 5 in in ADP cana-2648 22 6 ( ( PUNCT cana-2648 22 7 π‘Œβ€² π‘Œβ€² NOUN cana-2648 22 8 , , PUNCT cana-2648 22 9 πœŽβ€² πœŽβ€² NOUN cana-2648 22 10 ) ) PUNCT cana-2648 22 11 as as ADV cana-2648 22 12 all all PRON cana-2648 22 13 minimal minimal ADJ cana-2648 22 14 opens open VERB cana-2648 22 15 are be AUX cana-2648 22 16 open open ADJ cana-2648 22 17 . . PUNCT cana-2648 23 1 consequently consequently ADV cana-2648 23 2 , , PUNCT cana-2648 23 3 οΏ½ οΏ½ PROPN cana-2648 23 4 Μ‡ Μ‡ PROPN cana-2648 23 5 οΏ½ οΏ½ PROPN cana-2648 23 6 is be AUX cana-2648 23 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 23 8 , , PUNCT cana-2648 23 9 οΏ½ οΏ½ PROPN cana-2648 23 10 Μ‡ Μ‡ PROPN cana-2648 23 11 οΏ½ οΏ½ PROPN cana-2648 23 12 is be AUX cana-2648 23 13 minimal minimal ADJ cana-2648 23 14 π‘”πœ‚continuous π‘”πœ‚continuous ADJ cana-2648 23 15 as as ADP cana-2648 23 16 a a DET cana-2648 23 17 result result NOUN cana-2648 23 18 . . PUNCT cana-2648 24 1 however however ADV cana-2648 24 2 , , PUNCT cana-2648 24 3 the the DET cana-2648 24 4 converse converse NOUN cana-2648 24 5 of of ADP cana-2648 24 6 this this DET cana-2648 24 7 theorem theorem NOUN cana-2648 24 8 is be AUX cana-2648 24 9 not not PART cana-2648 24 10 necessarily necessarily ADV cana-2648 24 11 true true ADJ cana-2648 24 12 , , PUNCT cana-2648 24 13 as as SCONJ cana-2648 24 14 shown show VERB cana-2648 24 15 by by ADP cana-2648 24 16 the the DET cana-2648 24 17 following follow VERB cana-2648 24 18 example example NOUN cana-2648 24 19 . . PUNCT cana-2648 25 1 communications communication NOUN cana-2648 25 2 on on ADP cana-2648 25 3 applied apply VERB cana-2648 25 4 nonlinear nonlinear ADJ cana-2648 25 5 analysis analysis NOUN cana-2648 25 6 issn issn NOUN cana-2648 25 7 : : PUNCT cana-2648 25 8 1074 1074 NUM cana-2648 25 9 - - PUNCT cana-2648 25 10 133x 133x NUM cana-2648 25 11 vol vol NOUN cana-2648 25 12 32 32 NUM cana-2648 25 13 no no NOUN cana-2648 25 14 . . PUNCT cana-2648 26 1 3s 3s NUM cana-2648 26 2 ( ( PUNCT cana-2648 26 3 2025 2025 NUM cana-2648 26 4 ) ) PUNCT cana-2648 26 5 358 358 NUM cana-2648 26 6 https://internationalpubls.com https://internationalpubls.com X cana-2648 26 7 example example NOUN cana-2648 26 8 2.5 2.5 NUM cana-2648 26 9 : : PUNCT cana-2648 26 10 let let VERB cana-2648 26 11 𝑋′ 𝑋′ PROPN cana-2648 26 12 = = X cana-2648 26 13 π‘Œβ€² π‘Œβ€² NOUN cana-2648 26 14 = = SYM cana-2648 26 15 { { PUNCT cana-2648 26 16 𝑒′ 𝑒′ NOUN cana-2648 26 17 , , PUNCT cana-2648 26 18 𝑓′ 𝑓′ NUM cana-2648 26 19 , , PUNCT cana-2648 26 20 𝑔′ 𝑔′ ADV cana-2648 26 21 } } PUNCT cana-2648 26 22 , , PUNCT cana-2648 26 23 πœβ€² πœβ€² PUNCT cana-2648 26 24 = = PUNCT cana-2648 26 25 { { PUNCT cana-2648 26 26 𝑋′ 𝑋′ NOUN cana-2648 26 27 , , PUNCT cana-2648 26 28 πœ‘β€² πœ‘β€² NUM cana-2648 26 29 , , PUNCT cana-2648 26 30 { { PUNCT cana-2648 26 31 𝑒′ 𝑒′ NOUN cana-2648 26 32 } } PUNCT cana-2648 26 33 , , PUNCT cana-2648 26 34 { { PUNCT cana-2648 26 35 𝑔′ 𝑔′ ADV cana-2648 26 36 } } PUNCT cana-2648 26 37 , , PUNCT cana-2648 26 38 { { PUNCT cana-2648 26 39 𝑒′ 𝑒′ NOUN cana-2648 26 40 , , PUNCT cana-2648 26 41 𝑔′ 𝑔′ ADV cana-2648 26 42 } } PUNCT cana-2648 26 43 } } PUNCT cana-2648 26 44 , , PUNCT cana-2648 26 45 πœŽβ€² πœŽβ€² X cana-2648 26 46 = = PUNCT cana-2648 26 47 { { PUNCT cana-2648 26 48 π‘Œβ€² π‘Œβ€² PROPN cana-2648 26 49 , , PUNCT cana-2648 26 50 πœ‘β€² πœ‘β€² NUM cana-2648 26 51 , , PUNCT cana-2648 26 52 { { PUNCT cana-2648 26 53 𝑒′ 𝑒′ NOUN cana-2648 26 54 } } PUNCT cana-2648 26 55 , , PUNCT cana-2648 26 56 { { PUNCT cana-2648 26 57 𝑓′ 𝑓′ NOUN cana-2648 26 58 } } PUNCT cana-2648 26 59 , , PUNCT cana-2648 26 60 { { PUNCT cana-2648 26 61 𝑒′ 𝑒′ NOUN cana-2648 26 62 , , PUNCT cana-2648 26 63 𝑓′ 𝑓′ NUM cana-2648 26 64 } } PUNCT cana-2648 26 65 } } PUNCT cana-2648 26 66 . . PUNCT cana-2648 27 1 define define VERB cana-2648 27 2 οΏ½ οΏ½ PROPN cana-2648 27 3 Μ‡ Μ‡ PROPN cana-2648 27 4 οΏ½ οΏ½ PROPN cana-2648 27 5 : : PUNCT cana-2648 27 6 ( ( PUNCT cana-2648 27 7 𝑋′ 𝑋′ X cana-2648 27 8 , , PUNCT cana-2648 27 9 πœβ€² πœβ€² NUM cana-2648 27 10 ) ) PUNCT cana-2648 27 11 β†’ β†’ SYM cana-2648 27 12 ( ( PUNCT cana-2648 27 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 27 14 , , PUNCT cana-2648 27 15 πœŽβ€² πœŽβ€² NUM cana-2648 27 16 ) ) PUNCT cana-2648 27 17 as as ADP cana-2648 27 18 οΏ½ οΏ½ PROPN cana-2648 27 19 Μ‡ Μ‡ PROPN cana-2648 27 20 οΏ½ οΏ½ PROPN cana-2648 27 21 (𝑒′ (𝑒′ NUM cana-2648 27 22 ) ) PUNCT cana-2648 27 23 = = SYM cana-2648 27 24 𝑓′ 𝑓′ NUM cana-2648 27 25 , , PUNCT cana-2648 27 26 οΏ½ οΏ½ PROPN cana-2648 27 27 Μ‡ Μ‡ PROPN cana-2648 27 28 οΏ½ οΏ½ PROPN cana-2648 27 29 (𝑓′ (𝑓′ PUNCT cana-2648 27 30 ) ) PUNCT cana-2648 27 31 = = SYM cana-2648 27 32 𝑔′ 𝑔′ NUM cana-2648 27 33 , , PUNCT cana-2648 27 34 οΏ½ οΏ½ PROPN cana-2648 27 35 Μ‡ Μ‡ PROPN cana-2648 27 36 οΏ½ οΏ½ PROPN cana-2648 27 37 (𝑔′ (𝑔′ X cana-2648 27 38 ) ) PUNCT cana-2648 27 39 = = SYM cana-2648 28 1 𝑒′. 𝑒′. PROPN cana-2648 28 2 then then ADV cana-2648 28 3 οΏ½ οΏ½ PROPN cana-2648 28 4 Μ‡ Μ‡ PROPN cana-2648 28 5 οΏ½ οΏ½ PROPN cana-2648 28 6 βˆ’1(𝑒′ βˆ’1(𝑒′ NUM cana-2648 28 7 ) ) PUNCT cana-2648 28 8 = = SYM cana-2648 28 9 𝑔′ 𝑔′ NUM cana-2648 28 10 , , PUNCT cana-2648 28 11 οΏ½ οΏ½ PROPN cana-2648 28 12 Μ‡ Μ‡ PROPN cana-2648 28 13 οΏ½ οΏ½ PROPN cana-2648 28 14 βˆ’1(𝑓′ βˆ’1(𝑓′ PROPN cana-2648 28 15 ) ) PUNCT cana-2648 28 16 = = SYM cana-2648 29 1 𝑒′. 𝑒′. PROPN cana-2648 29 2 therefore therefore ADV cana-2648 29 3 οΏ½ οΏ½ PROPN cana-2648 29 4 Μ‡ Μ‡ PROPN cana-2648 29 5 οΏ½ οΏ½ PROPN cana-2648 29 6 is be AUX cana-2648 29 7 minimal minimal ADJ cana-2648 29 8 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 29 9 . . PUNCT cana-2648 30 1 hence hence ADV cana-2648 30 2 οΏ½ οΏ½ PROPN cana-2648 30 3 Μ‡ Μ‡ PROPN cana-2648 30 4 οΏ½ οΏ½ PROPN cana-2648 30 5 is be AUX cana-2648 30 6 not not PART cana-2648 30 7 π‘”πœ‚continuous π‘”πœ‚continuous ADJ cana-2648 30 8 . . PUNCT cana-2648 31 1 theorem theorem VERB cana-2648 31 2 2.6 2.6 NUM cana-2648 31 3 : : PUNCT cana-2648 31 4 if if SCONJ cana-2648 31 5 any any DET cana-2648 31 6 maximal maximal ADJ cana-2648 31 7 closed close VERB cana-2648 31 8 in in ADP cana-2648 31 9 's 's PART cana-2648 31 10 inverse inverse NOUN cana-2648 31 11 image image NOUN cana-2648 31 12 π‘Œβ€² π‘Œβ€² ADV cana-2648 31 13 , , PUNCT cana-2648 31 14 πœŽβ€² πœŽβ€² NUM cana-2648 31 15 ) ) PUNCT cana-2648 31 16 is be AUX cana-2648 31 17 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 31 18 in in ADP cana-2648 31 19 ( ( PUNCT cana-2648 31 20 𝑋′ 𝑋′ ADJ cana-2648 31 21 , , PUNCT cana-2648 31 22 πœβ€² πœβ€² PROPN cana-2648 31 23 ) ) PUNCT cana-2648 31 24 then then ADV cana-2648 31 25 let let VERB cana-2648 31 26 οΏ½ οΏ½ PROPN cana-2648 31 27 Μ‡ Μ‡ VERB cana-2648 31 28 οΏ½ οΏ½ PROPN cana-2648 31 29 : : PUNCT cana-2648 31 30 ( ( PUNCT cana-2648 31 31 𝑋′ 𝑋′ X cana-2648 31 32 , , PUNCT cana-2648 31 33 πœβ€² πœβ€² NUM cana-2648 31 34 ) ) PUNCT cana-2648 31 35 β†’ β†’ SYM cana-2648 31 36 ( ( PUNCT cana-2648 31 37 π‘Œβ€² π‘Œβ€² PROPN cana-2648 31 38 , , PUNCT cana-2648 31 39 πœŽβ€² πœŽβ€² NUM cana-2648 31 40 ) ) PUNCT cana-2648 31 41 be be AUX cana-2648 31 42 minimal minimal ADJ cana-2648 31 43 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 31 44 proof proof NOUN cana-2648 31 45 : : PUNCT cana-2648 31 46 οΏ½ οΏ½ PROPN cana-2648 31 47 Μ‡ Μ‡ PROPN cana-2648 31 48 οΏ½ οΏ½ PROPN cana-2648 31 49 be be AUX cana-2648 31 50 a a DET cana-2648 31 51 minimal minimal ADJ cana-2648 31 52 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 31 53 and and CCONJ cana-2648 31 54 let let VERB cana-2648 31 55 𝑁′ 𝑁′ NOUN cana-2648 31 56 be be AUX cana-2648 31 57 a a DET cana-2648 31 58 maximal maximal ADJ cana-2648 31 59 closed close VERB cana-2648 31 60 in in ADP cana-2648 31 61 ( ( PUNCT cana-2648 31 62 π‘Œβ€² π‘Œβ€² NOUN cana-2648 31 63 , , PUNCT cana-2648 31 64 πœŽβ€² πœŽβ€² NUM cana-2648 31 65 ) ) PUNCT cana-2648 31 66 . . PUNCT cana-2648 32 1 ( ( PUNCT cana-2648 32 2 π‘Œβ€² π‘Œβ€² NOUN cana-2648 32 3 , , PUNCT cana-2648 32 4 πœŽβ€² πœŽβ€² NUM cana-2648 32 5 ) ) PUNCT cana-2648 32 6 βˆ’ βˆ’ PROPN cana-2648 32 7 𝑁′ 𝑁′ PROPN cana-2648 32 8 is be AUX cana-2648 32 9 a a DET cana-2648 32 10 minimal minimal ADJ cana-2648 32 11 open open NOUN cana-2648 32 12 in in ADP cana-2648 32 13 ( ( PUNCT cana-2648 32 14 π‘Œβ€² π‘Œβ€² NOUN cana-2648 32 15 , , PUNCT cana-2648 32 16 πœŽβ€² πœŽβ€² NUM cana-2648 32 17 ) ) PUNCT cana-2648 32 18 . . PUNCT cana-2648 33 1 when when SCONJ cana-2648 33 2 οΏ½ οΏ½ PROPN cana-2648 33 3 Μ‡ Μ‡ PROPN cana-2648 33 4 οΏ½ οΏ½ PROPN cana-2648 33 5 is be AUX cana-2648 33 6 minimal minimal ADJ cana-2648 33 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 33 8 , , PUNCT cana-2648 33 9 οΏ½ οΏ½ PROPN cana-2648 33 10 Μ‡ Μ‡ PROPN cana-2648 33 11 οΏ½ οΏ½ PROPN cana-2648 33 12 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 33 13 , , PUNCT cana-2648 33 14 πœŽβ€² πœŽβ€² NUM cana-2648 33 15 ) ) PUNCT cana-2648 33 16 βˆ’ βˆ’ PROPN cana-2648 33 17 𝑁′ 𝑁′ NOUN cana-2648 33 18 ) ) PUNCT cana-2648 33 19 is be AUX cana-2648 33 20 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 33 21 in in ADP cana-2648 33 22 ( ( PUNCT cana-2648 33 23 𝑋′ 𝑋′ ADJ cana-2648 33 24 , , PUNCT cana-2648 33 25 πœβ€² πœβ€² NUM cana-2648 33 26 ) ) PUNCT cana-2648 33 27 . . PUNCT cana-2648 34 1 so so ADV cana-2648 34 2 οΏ½ οΏ½ PROPN cana-2648 34 3 Μ‡ Μ‡ PROPN cana-2648 34 4 οΏ½ οΏ½ PROPN cana-2648 34 5 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 34 6 , , PUNCT cana-2648 34 7 πœŽβ€² πœŽβ€² NUM cana-2648 34 8 ) ) PUNCT cana-2648 34 9 βˆ’ βˆ’ PROPN cana-2648 34 10 𝑁′ 𝑁′ NOUN cana-2648 34 11 ) ) PUNCT cana-2648 34 12 = = PUNCT cana-2648 34 13 ( ( PUNCT cana-2648 34 14 𝑋′ 𝑋′ NOUN cana-2648 34 15 , , PUNCT cana-2648 34 16 πœβ€² πœβ€² NUM cana-2648 34 17 ) ) PUNCT cana-2648 35 1 βˆ’ βˆ’ PROPN cana-2648 35 2 οΏ½ οΏ½ PROPN cana-2648 35 3 Μ‡ Μ‡ PROPN cana-2648 35 4 οΏ½ οΏ½ PROPN cana-2648 35 5 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 35 6 ) ) PUNCT cana-2648 35 7 is be AUX cana-2648 35 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 35 9 in in ADP cana-2648 35 10 ( ( PUNCT cana-2648 35 11 𝑋′ 𝑋′ ADJ cana-2648 35 12 , , PUNCT cana-2648 35 13 πœβ€² πœβ€² PROPN cana-2648 35 14 ) ) PUNCT cana-2648 35 15 . . PUNCT cana-2648 36 1 οΏ½ οΏ½ PROPN cana-2648 36 2 Μ‡ Μ‡ PROPN cana-2648 36 3 οΏ½ οΏ½ PROPN cana-2648 36 4 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 36 5 ) ) PUNCT cana-2648 36 6 is be AUX cana-2648 36 7 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 36 8 in in ADP cana-2648 36 9 ( ( PUNCT cana-2648 36 10 𝑋′ 𝑋′ ADJ cana-2648 36 11 , , PUNCT cana-2648 36 12 πœβ€² πœβ€² NUM cana-2648 36 13 ) ) PUNCT cana-2648 36 14 . . PUNCT cana-2648 37 1 on on ADP cana-2648 37 2 the the DET cana-2648 37 3 contrary contrary NOUN cana-2648 37 4 , , PUNCT cana-2648 37 5 οΏ½ οΏ½ PROPN cana-2648 37 6 Μ‡ Μ‡ PROPN cana-2648 37 7 οΏ½ οΏ½ PROPN cana-2648 37 8 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 37 9 ) ) PUNCT cana-2648 37 10 is be AUX cana-2648 37 11 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 37 12 in in ADP cana-2648 37 13 ( ( PUNCT cana-2648 37 14 𝑋′ 𝑋′ ADJ cana-2648 37 15 , , PUNCT cana-2648 37 16 πœβ€² πœβ€² NOUN cana-2648 37 17 ) ) PUNCT cana-2648 37 18 for for ADP cana-2648 37 19 all all DET cana-2648 37 20 maximal maximal ADJ cana-2648 37 21 closed closed ADJ cana-2648 37 22 𝑁′ 𝑁′ NOUN cana-2648 37 23 in in ADP cana-2648 37 24 ( ( PUNCT cana-2648 37 25 π‘Œβ€² π‘Œβ€² NOUN cana-2648 37 26 , , PUNCT cana-2648 37 27 πœŽβ€² πœŽβ€² NUM cana-2648 37 28 ) ) PUNCT cana-2648 37 29 . . PUNCT cana-2648 38 1 let let VERB cana-2648 38 2 𝑀′ 𝑀′ NOUN cana-2648 38 3 be be AUX cana-2648 38 4 a a DET cana-2648 38 5 minimal minimal ADJ cana-2648 38 6 open open NOUN cana-2648 38 7 in in ADP cana-2648 38 8 ( ( PUNCT cana-2648 38 9 π‘Œβ€² π‘Œβ€² NOUN cana-2648 38 10 , , PUNCT cana-2648 38 11 πœŽβ€² πœŽβ€² NUM cana-2648 38 12 ) ) PUNCT cana-2648 38 13 . . PUNCT cana-2648 39 1 so so ADV cana-2648 39 2 οΏ½ οΏ½ PROPN cana-2648 39 3 Μ‡ Μ‡ PROPN cana-2648 39 4 οΏ½ οΏ½ PROPN cana-2648 39 5 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 39 6 , , PUNCT cana-2648 39 7 πœŽβ€² πœŽβ€² NUM cana-2648 39 8 ) ) PUNCT cana-2648 40 1 βˆ’ βˆ’ PROPN cana-2648 40 2 𝑀′ 𝑀′ NOUN cana-2648 40 3 ) ) PUNCT cana-2648 40 4 = = PUNCT cana-2648 40 5 ( ( PUNCT cana-2648 40 6 𝑋′ 𝑋′ NOUN cana-2648 40 7 , , PUNCT cana-2648 40 8 πœβ€² πœβ€² NOUN cana-2648 40 9 ) ) PUNCT cana-2648 40 10 βˆ’ βˆ’ PRON cana-2648 40 11 π•’βˆ’1(𝑀′ π•’βˆ’1(𝑀′ NOUN cana-2648 40 12 ) ) PUNCT cana-2648 40 13 is be AUX cana-2648 40 14 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 40 15 in in ADP cana-2648 40 16 ( ( PUNCT cana-2648 40 17 𝑋′ 𝑋′ ADJ cana-2648 40 18 , , PUNCT cana-2648 40 19 πœβ€² πœβ€² NUM cana-2648 40 20 ) ) PUNCT cana-2648 40 21 . . PUNCT cana-2648 41 1 therefore therefore ADV cana-2648 41 2 οΏ½ οΏ½ PROPN cana-2648 41 3 Μ‡ Μ‡ PROPN cana-2648 41 4 οΏ½ οΏ½ PROPN cana-2648 41 5 is be AUX cana-2648 41 6 minimal minimal ADJ cana-2648 41 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 41 8 . . PUNCT cana-2648 42 1 theorem theorem VERB cana-2648 42 2 2.7 2.7 NUM cana-2648 42 3 : : PUNCT cana-2648 42 4 let let VERB cana-2648 42 5 οΏ½ οΏ½ PROPN cana-2648 42 6 Μ‡ Μ‡ VERB cana-2648 42 7 οΏ½ οΏ½ PROPN cana-2648 42 8 : : PUNCT cana-2648 42 9 ( ( PUNCT cana-2648 42 10 𝑋′ 𝑋′ X cana-2648 42 11 , , PUNCT cana-2648 42 12 πœβ€² πœβ€² NUM cana-2648 42 13 ) ) PUNCT cana-2648 42 14 β†’ β†’ SYM cana-2648 42 15 ( ( PUNCT cana-2648 42 16 π‘Œβ€² π‘Œβ€² PROPN cana-2648 42 17 , , PUNCT cana-2648 42 18 πœŽβ€² πœŽβ€² NUM cana-2648 42 19 ) ) PUNCT cana-2648 42 20 is be AUX cana-2648 42 21 minimal minimal ADJ cana-2648 42 22 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 42 23 iff iff NOUN cana-2648 42 24 , , PUNCT cana-2648 42 25 π‘žβ€² π‘žβ€² PROPN cana-2648 42 26 ∈ ∈ PROPN cana-2648 42 27 ( ( PUNCT cana-2648 42 28 𝑋′ 𝑋′ NOUN cana-2648 42 29 , , PUNCT cana-2648 42 30 πœβ€² πœβ€² PROPN cana-2648 42 31 ) ) PUNCT cana-2648 42 32 and and CCONJ cana-2648 42 33 minimal minimal ADJ cana-2648 42 34 open open ADJ cana-2648 42 35 𝑀′ 𝑀′ NOUN cana-2648 42 36 in in ADP cana-2648 42 37 ( ( PUNCT cana-2648 42 38 π‘Œβ€² π‘Œβ€² PROPN cana-2648 42 39 , , PUNCT cana-2648 42 40 πœŽβ€² πœŽβ€² NOUN cana-2648 42 41 ) ) PUNCT cana-2648 42 42 holding hold VERB cana-2648 42 43 οΏ½ οΏ½ PROPN cana-2648 42 44 Μ‡ Μ‡ PROPN cana-2648 42 45 οΏ½ οΏ½ PROPN cana-2648 42 46 (π‘žβ€² (π‘žβ€² NUM cana-2648 42 47 ) ) PUNCT cana-2648 42 48 , , PUNCT cana-2648 42 49 there there PRON cana-2648 42 50 is be VERB cana-2648 42 51 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 42 52 𝑁′ 𝑁′ NOUN cana-2648 42 53 in in ADP cana-2648 42 54 ( ( PUNCT cana-2648 42 55 𝑋′ 𝑋′ ADJ cana-2648 42 56 , , PUNCT cana-2648 42 57 πœβ€² πœβ€² NOUN cana-2648 42 58 ) ) PUNCT cana-2648 42 59 like like ADP cana-2648 42 60 that that DET cana-2648 42 61 π‘žβ€² π‘žβ€² PROPN cana-2648 42 62 ∈ ∈ PROPN cana-2648 42 63 𝑁′ 𝑁′ PROPN cana-2648 42 64 , , PUNCT cana-2648 42 65 οΏ½ οΏ½ PROPN cana-2648 42 66 Μ‡ Μ‡ NOUN cana-2648 42 67 οΏ½ οΏ½ PROPN cana-2648 42 68 (𝑁′ (𝑁′ NOUN cana-2648 42 69 ) ) PUNCT cana-2648 42 70 βŠ‚ βŠ‚ X cana-2648 43 1 𝑀′. 𝑀′. X cana-2648 43 2 proof proof NOUN cana-2648 43 3 : : PUNCT cana-2648 43 4 𝑀′ 𝑀′ NOUN cana-2648 43 5 be be AUX cana-2648 43 6 minimal minimal ADJ cana-2648 43 7 open open ADJ cana-2648 43 8 in in ADP cana-2648 43 9 ( ( PUNCT cana-2648 43 10 π‘Œβ€² π‘Œβ€² NOUN cana-2648 43 11 , , PUNCT cana-2648 43 12 πœŽβ€² πœŽβ€² NOUN cana-2648 43 13 ) ) PUNCT cana-2648 43 14 holding hold VERB cana-2648 43 15 οΏ½ οΏ½ PROPN cana-2648 43 16 Μ‡ Μ‡ PROPN cana-2648 43 17 οΏ½ οΏ½ PROPN cana-2648 43 18 (π‘žβ€² (π‘žβ€² NUM cana-2648 43 19 ) ) PUNCT cana-2648 43 20 , , PUNCT cana-2648 43 21 π‘žβ€² π‘žβ€² PROPN cana-2648 43 22 ∈ ∈ PROPN cana-2648 43 23 𝑁′ 𝑁′ NOUN cana-2648 43 24 where where SCONJ cana-2648 43 25 𝑁′ 𝑁′ PROPN cana-2648 43 26 is be AUX cana-2648 43 27 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 43 28 in in ADP cana-2648 43 29 ( ( PUNCT cana-2648 43 30 𝑋′ 𝑋′ ADJ cana-2648 43 31 , , PUNCT cana-2648 43 32 πœβ€² πœβ€² NUM cana-2648 43 33 ) ) PUNCT cana-2648 43 34 . . PUNCT cana-2648 44 1 since since SCONJ cana-2648 44 2 οΏ½ οΏ½ PROPN cana-2648 44 3 Μ‡ Μ‡ PROPN cana-2648 44 4 οΏ½ οΏ½ PROPN cana-2648 44 5 is be AUX cana-2648 44 6 minimal minimal ADJ cana-2648 44 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 44 8 , , PUNCT cana-2648 44 9 οΏ½ οΏ½ PROPN cana-2648 44 10 Μ‡ Μ‡ PROPN cana-2648 44 11 οΏ½ οΏ½ PROPN cana-2648 44 12 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 44 13 ) ) PUNCT cana-2648 44 14 is be AUX cana-2648 44 15 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 45 1 in(𝑋′ in(𝑋′ ADJ cana-2648 45 2 , , PUNCT cana-2648 45 3 πœβ€² πœβ€² PROPN cana-2648 45 4 ) ) PUNCT cana-2648 45 5 . . PUNCT cana-2648 46 1 and and CCONJ cana-2648 46 2 𝑁′ 𝑁′ ADJ cana-2648 46 3 = = SYM cana-2648 46 4 οΏ½ οΏ½ PROPN cana-2648 46 5 Μ‡ Μ‡ PROPN cana-2648 46 6 οΏ½ οΏ½ NOUN cana-2648 46 7 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 46 8 ) ) PUNCT cana-2648 46 9 . . PUNCT cana-2648 47 1 therefore therefore ADV cana-2648 47 2 οΏ½ οΏ½ PROPN cana-2648 47 3 Μ‡ Μ‡ PROPN cana-2648 47 4 οΏ½ οΏ½ PROPN cana-2648 47 5 (𝑁′ (𝑁′ NOUN cana-2648 47 6 ) ) PUNCT cana-2648 47 7 βŠ‚ βŠ‚ PROPN cana-2648 47 8 𝑀′. 𝑀′. PROPN cana-2648 47 9 on on ADP cana-2648 47 10 the the DET cana-2648 47 11 contrary contrary NOUN cana-2648 47 12 , , PUNCT cana-2648 47 13 𝑀′ 𝑀′ PROPN cana-2648 47 14 be be AUX cana-2648 47 15 minimal minimal ADJ cana-2648 47 16 open open ADJ cana-2648 47 17 in in ADP cana-2648 47 18 ( ( PUNCT cana-2648 47 19 π‘Œβ€² π‘Œβ€² NOUN cana-2648 47 20 , , PUNCT cana-2648 47 21 πœŽβ€² πœŽβ€² NUM cana-2648 47 22 ) ) PUNCT cana-2648 47 23 . . PUNCT cana-2648 48 1 then then ADV cana-2648 48 2 there there PRON cana-2648 48 3 is be VERB cana-2648 48 4 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 48 5 𝑁′ 𝑁′ NOUN cana-2648 48 6 in in ADP cana-2648 48 7 ( ( PUNCT cana-2648 48 8 𝑋′ 𝑋′ ADJ cana-2648 48 9 , , PUNCT cana-2648 48 10 πœβ€² πœβ€² PROPN cana-2648 48 11 ) ) PUNCT cana-2648 48 12 , , PUNCT cana-2648 48 13 such such ADJ cana-2648 48 14 that that DET cana-2648 48 15 π‘žβ€² π‘žβ€² PROPN cana-2648 48 16 ∈ ∈ PROPN cana-2648 48 17 𝑁′ 𝑁′ PROPN cana-2648 48 18 , , PUNCT cana-2648 48 19 οΏ½ οΏ½ PROPN cana-2648 48 20 Μ‡ Μ‡ PROPN cana-2648 48 21 οΏ½ οΏ½ PROPN cana-2648 48 22 (π‘žβ€² (π‘žβ€² PROPN cana-2648 48 23 ) ) PUNCT cana-2648 48 24 ∈ ∈ PROPN cana-2648 48 25 οΏ½ οΏ½ PROPN cana-2648 48 26 Μ‡ Μ‡ NOUN cana-2648 48 27 οΏ½ οΏ½ PROPN cana-2648 48 28 (𝑁′ (𝑁′ NOUN cana-2648 48 29 ) ) PUNCT cana-2648 48 30 βŠ‚ βŠ‚ PROPN cana-2648 48 31 𝑀′ 𝑀′ PROPN cana-2648 48 32 , , PUNCT cana-2648 48 33 π‘ž π‘ž PROPN cana-2648 48 34 ∈ ∈ PROPN cana-2648 48 35 οΏ½ οΏ½ PROPN cana-2648 48 36 Μ‡ Μ‡ PROPN cana-2648 48 37 οΏ½ οΏ½ PROPN cana-2648 48 38 βˆ’1( βˆ’1( PROPN cana-2648 48 39 οΏ½ οΏ½ PROPN cana-2648 48 40 Μ‡ Μ‡ NOUN cana-2648 48 41 οΏ½ οΏ½ NOUN cana-2648 48 42 (𝑁′ (𝑁′ NOUN cana-2648 48 43 ) ) PUNCT cana-2648 48 44 ) ) PUNCT cana-2648 49 1 βŠ‚ βŠ‚ PROPN cana-2648 49 2 οΏ½ οΏ½ PROPN cana-2648 49 3 Μ‡ Μ‡ PROPN cana-2648 49 4 οΏ½ οΏ½ PROPN cana-2648 49 5 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 49 6 ) ) PUNCT cana-2648 49 7 . . PUNCT cana-2648 50 1 therefore therefore ADV cana-2648 50 2 οΏ½ οΏ½ PROPN cana-2648 50 3 Μ‡ Μ‡ PROPN cana-2648 50 4 οΏ½ οΏ½ PROPN cana-2648 50 5 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 50 6 ) ) PUNCT cana-2648 50 7 is be AUX cana-2648 50 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 50 9 in in ADP cana-2648 50 10 ( ( PUNCT cana-2648 50 11 𝑋′ 𝑋′ ADJ cana-2648 50 12 , , PUNCT cana-2648 50 13 πœβ€² πœβ€² NUM cana-2648 50 14 ) ) PUNCT cana-2648 50 15 . . PUNCT cana-2648 51 1 hence hence ADV cana-2648 51 2 οΏ½ οΏ½ PROPN cana-2648 51 3 Μ‡ Μ‡ PROPN cana-2648 51 4 οΏ½ οΏ½ PROPN cana-2648 51 5 is be AUX cana-2648 51 6 minimal minimal ADJ cana-2648 51 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 51 8 . . PUNCT cana-2648 52 1 theorem theorem VERB cana-2648 52 2 2.8 2.8 NUM cana-2648 52 3 : : PUNCT cana-2648 52 4 let let VERB cana-2648 52 5 𝐡′ 𝐡′ PRON cana-2648 52 6 be be AUX cana-2648 52 7 a a DET cana-2648 52 8 subset subset NOUN cana-2648 52 9 of of ADP cana-2648 52 10 ( ( PUNCT cana-2648 52 11 𝑋′ 𝑋′ ADJ cana-2648 52 12 , , PUNCT cana-2648 52 13 πœβ€² πœβ€² NUM cana-2648 52 14 ) ) PUNCT cana-2648 52 15 . . PUNCT cana-2648 53 1 if if SCONJ cana-2648 53 2 οΏ½ οΏ½ PROPN cana-2648 53 3 Μ‡ Μ‡ PROPN cana-2648 53 4 οΏ½ οΏ½ PROPN cana-2648 53 5 : : PUNCT cana-2648 53 6 ( ( PUNCT cana-2648 53 7 𝑋′ 𝑋′ X cana-2648 53 8 , , PUNCT cana-2648 53 9 πœβ€² πœβ€² NUM cana-2648 53 10 ) ) PUNCT cana-2648 53 11 β†’ β†’ SYM cana-2648 53 12 ( ( PUNCT cana-2648 53 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 53 14 , , PUNCT cana-2648 53 15 πœŽβ€² πœŽβ€² NUM cana-2648 53 16 ) ) PUNCT cana-2648 53 17 is be AUX cana-2648 53 18 minimal minimal ADJ cana-2648 53 19 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 53 20 then then ADV cana-2648 53 21 the the DET cana-2648 53 22 restriction restriction NOUN cana-2648 53 23 function function NOUN cana-2648 53 24 οΏ½ οΏ½ PROPN cana-2648 53 25 Μ‡ Μ‡ PROPN cana-2648 53 26 οΏ½ οΏ½ PROPN cana-2648 53 27 | | ADV cana-2648 53 28 𝐡′ 𝐡′ NUM cana-2648 53 29 : : PUNCT cana-2648 53 30 𝐡′ 𝐡′ PRON cana-2648 53 31 β†’ β†’ SYM cana-2648 53 32 ( ( PUNCT cana-2648 53 33 π‘Œβ€² π‘Œβ€² PROPN cana-2648 53 34 , , PUNCT cana-2648 53 35 πœŽβ€² πœŽβ€² NUM cana-2648 53 36 ) ) PUNCT cana-2648 53 37 is be AUX cana-2648 53 38 minimal minimal ADJ cana-2648 53 39 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 53 40 . . PUNCT cana-2648 54 1 where where SCONJ cana-2648 54 2 𝐡′ 𝐡′ PRON cana-2648 54 3 has have VERB cana-2648 54 4 the the DET cana-2648 54 5 relative relative ADJ cana-2648 54 6 topology topology NOUN cana-2648 54 7 . . PUNCT cana-2648 55 1 proof proof NOUN cana-2648 55 2 : : PUNCT cana-2648 55 3 assume assume VERB cana-2648 55 4 𝐴′ 𝐴′ PROPN cana-2648 55 5 is be AUX cana-2648 55 6 subset subset VERB cana-2648 55 7 of of ADP cana-2648 55 8 ( ( PUNCT cana-2648 55 9 𝑋′ 𝑋′ ADJ cana-2648 55 10 , , PUNCT cana-2648 55 11 πœβ€² πœβ€² PROPN cana-2648 55 12 ) ) PUNCT cana-2648 55 13 and and CCONJ cana-2648 55 14 𝑀′ 𝑀′ PROPN cana-2648 55 15 be be AUX cana-2648 55 16 minimal minimal ADJ cana-2648 55 17 open open ADJ cana-2648 55 18 in in ADP cana-2648 55 19 ( ( PUNCT cana-2648 55 20 π‘Œβ€² π‘Œβ€² NOUN cana-2648 55 21 , , PUNCT cana-2648 55 22 πœŽβ€² πœŽβ€² NUM cana-2648 55 23 ) ) PUNCT cana-2648 55 24 . . PUNCT cana-2648 56 1 when when SCONJ cana-2648 56 2 οΏ½ οΏ½ PROPN cana-2648 56 3 Μ‡ Μ‡ PROPN cana-2648 56 4 οΏ½ οΏ½ PROPN cana-2648 56 5 is be AUX cana-2648 56 6 minimal minimal ADJ cana-2648 56 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 56 8 , , PUNCT cana-2648 56 9 οΏ½ οΏ½ PROPN cana-2648 56 10 Μ‡ Μ‡ PROPN cana-2648 56 11 οΏ½ οΏ½ PROPN cana-2648 56 12 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 56 13 ) ) PUNCT cana-2648 56 14 is be AUX cana-2648 56 15 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 56 16 in in ADP cana-2648 56 17 ( ( PUNCT cana-2648 56 18 𝑋′ 𝑋′ ADJ cana-2648 56 19 , , PUNCT cana-2648 56 20 πœβ€² πœβ€² NUM cana-2648 56 21 ) ) PUNCT cana-2648 56 22 . . PUNCT cana-2648 57 1 so so ADV cana-2648 57 2 ( ( PUNCT cana-2648 57 3 οΏ½ οΏ½ PROPN cana-2648 57 4 Μ‡ Μ‡ PROPN cana-2648 57 5 οΏ½ οΏ½ PROPN cana-2648 57 6 |𝐡′)βˆ’1(𝑀′ |𝐡′)βˆ’1(𝑀′ NUM cana-2648 57 7 ) ) PUNCT cana-2648 57 8 = = PUNCT cana-2648 58 1 𝐡′ 𝐡′ NUM cana-2648 58 2 ∩ ∩ PROPN cana-2648 58 3 οΏ½ οΏ½ PROPN cana-2648 58 4 Μ‡ Μ‡ PROPN cana-2648 58 5 οΏ½ οΏ½ NOUN cana-2648 58 6 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 58 7 ) ) PUNCT cana-2648 58 8 . . PUNCT cana-2648 59 1 hence hence ADV cana-2648 59 2 𝐡′ 𝐡′ NUM cana-2648 59 3 ∩ ∩ PROPN cana-2648 59 4 οΏ½ οΏ½ PROPN cana-2648 59 5 Μ‡ Μ‡ PROPN cana-2648 59 6 οΏ½ οΏ½ PROPN cana-2648 59 7 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 59 8 ) ) PUNCT cana-2648 59 9 is be AUX cana-2648 59 10 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 59 11 in in ADP cana-2648 59 12 𝐡′. 𝐡′. PROPN cana-2648 59 13 thus thus ADV cana-2648 59 14 οΏ½ οΏ½ PROPN cana-2648 59 15 Μ‡ Μ‡ VERB cana-2648 59 16 οΏ½ οΏ½ NOUN cana-2648 59 17 |bβ€² |bβ€² NOUN cana-2648 59 18 is be AUX cana-2648 59 19 minimal minimal ADJ cana-2648 59 20 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 59 21 . . PUNCT cana-2648 60 1 remark remark PROPN cana-2648 60 2 2.9 2.9 NUM cana-2648 60 3 : : PUNCT cana-2648 60 4 minimal minimal ADJ cana-2648 60 5 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 60 6 do do AUX cana-2648 60 7 not not PART cana-2648 60 8 always always ADV cana-2648 60 9 have have VERB cana-2648 60 10 to to PART cana-2648 60 11 be be AUX cana-2648 60 12 minimal minimal ADJ cana-2648 60 13 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 60 14 in in ADP cana-2648 60 15 composition composition NOUN cana-2648 60 16 . . PUNCT cana-2648 61 1 theorem theorem VERB cana-2648 61 2 2.10 2.10 NUM cana-2648 61 3 : : PUNCT cana-2648 61 4 if if SCONJ cana-2648 61 5 οΏ½ οΏ½ PROPN cana-2648 61 6 Μ‡ Μ‡ PROPN cana-2648 61 7 οΏ½ οΏ½ PROPN cana-2648 61 8 : : PUNCT cana-2648 61 9 ( ( PUNCT cana-2648 61 10 𝑋′ 𝑋′ X cana-2648 61 11 , , PUNCT cana-2648 61 12 πœβ€² πœβ€² NUM cana-2648 61 13 ) ) PUNCT cana-2648 61 14 β†’ β†’ SYM cana-2648 61 15 ( ( PUNCT cana-2648 61 16 π‘Œβ€² π‘Œβ€² PROPN cana-2648 61 17 , , PUNCT cana-2648 61 18 πœŽβ€² πœŽβ€² NUM cana-2648 61 19 ) ) PUNCT cana-2648 61 20 and and CCONJ cana-2648 61 21 οΏ½ οΏ½ PROPN cana-2648 61 22 Μ‡ Μ‡ PROPN cana-2648 61 23 οΏ½ οΏ½ PROPN cana-2648 61 24 : : PUNCT cana-2648 61 25 ( ( PUNCT cana-2648 61 26 π‘Œβ€² π‘Œβ€² NOUN cana-2648 61 27 , , PUNCT cana-2648 61 28 πœŽβ€² πœŽβ€² NUM cana-2648 61 29 ) ) PUNCT cana-2648 61 30 β†’ β†’ SYM cana-2648 61 31 ( ( PUNCT cana-2648 61 32 𝑍′ 𝑍′ NOUN cana-2648 61 33 , , PUNCT cana-2648 61 34 ΞΌβ€² ΞΌβ€² NOUN cana-2648 61 35 ) ) PUNCT cana-2648 61 36 be be AUX cana-2648 61 37 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 61 38 and and CCONJ cana-2648 61 39 minimal minimal ADJ cana-2648 61 40 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 61 41 so so SCONJ cana-2648 61 42 οΏ½ οΏ½ PROPN cana-2648 61 43 Μ‡ Μ‡ PROPN cana-2648 61 44 οΏ½ οΏ½ PROPN cana-2648 61 45 o o NOUN cana-2648 61 46 οΏ½ οΏ½ PROPN cana-2648 61 47 Μ‡ Μ‡ PROPN cana-2648 61 48 οΏ½ οΏ½ PROPN cana-2648 61 49 : : PUNCT cana-2648 61 50 ( ( PUNCT cana-2648 61 51 𝑋′ 𝑋′ X cana-2648 61 52 , , PUNCT cana-2648 61 53 πœβ€² πœβ€² NUM cana-2648 61 54 ) ) PUNCT cana-2648 61 55 β†’ β†’ SYM cana-2648 61 56 ( ( PUNCT cana-2648 61 57 𝑍′ 𝑍′ NOUN cana-2648 61 58 , , PUNCT cana-2648 61 59 ΞΌβ€² ΞΌβ€² NOUN cana-2648 61 60 ) ) PUNCT cana-2648 61 61 is be AUX cana-2648 61 62 minimal minimal ADJ cana-2648 61 63 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 61 64 . . PUNCT cana-2648 62 1 proof proof NOUN cana-2648 62 2 : : PUNCT cana-2648 62 3 𝐴′ 𝐴′ PROPN cana-2648 62 4 be be AUX cana-2648 62 5 minimal minimal ADJ cana-2648 62 6 open open ADJ cana-2648 62 7 in(𝑍′ in(𝑍′ NOUN cana-2648 62 8 , , PUNCT cana-2648 62 9 ΞΌβ€² ΞΌβ€² NOUN cana-2648 62 10 ) ) PUNCT cana-2648 62 11 , , PUNCT cana-2648 62 12 when when SCONJ cana-2648 62 13 οΏ½ οΏ½ PROPN cana-2648 62 14 Μ‡ Μ‡ PROPN cana-2648 62 15 οΏ½ οΏ½ PROPN cana-2648 62 16 is be AUX cana-2648 62 17 minimal minimal ADJ cana-2648 62 18 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 62 19 , , PUNCT cana-2648 62 20 οΏ½ οΏ½ PROPN cana-2648 62 21 Μ‡ Μ‡ PROPN cana-2648 62 22 οΏ½ οΏ½ PROPN cana-2648 62 23 βˆ’1(𝐴 βˆ’1(𝐴 NOUN cana-2648 62 24 ) ) PUNCT cana-2648 62 25 is be AUX cana-2648 62 26 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 62 27 in in ADP cana-2648 62 28 ( ( PUNCT cana-2648 62 29 π‘Œβ€² π‘Œβ€² NOUN cana-2648 62 30 , , PUNCT cana-2648 62 31 πœŽβ€² πœŽβ€² NUM cana-2648 62 32 ) ) PUNCT cana-2648 62 33 . . PUNCT cana-2648 63 1 so so ADV cana-2648 63 2 οΏ½ οΏ½ PROPN cana-2648 63 3 Μ‡ Μ‡ PROPN cana-2648 63 4 οΏ½ οΏ½ PROPN cana-2648 63 5 is be AUX cana-2648 63 6 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 63 7 , , PUNCT cana-2648 63 8 οΏ½ οΏ½ PROPN cana-2648 63 9 Μ‡ Μ‡ PROPN cana-2648 63 10 οΏ½ οΏ½ PROPN cana-2648 63 11 βˆ’1( βˆ’1( PROPN cana-2648 63 12 οΏ½ οΏ½ PROPN cana-2648 63 13 Μ‡ Μ‡ NOUN cana-2648 63 14 οΏ½ οΏ½ NOUN cana-2648 63 15 βˆ’1(𝐴 βˆ’1(𝐴 NOUN cana-2648 63 16 ) ) PUNCT cana-2648 63 17 ) ) PUNCT cana-2648 64 1 = = PRON cana-2648 64 2 ( ( PUNCT cana-2648 64 3 οΏ½ οΏ½ PROPN cana-2648 64 4 Μ‡ Μ‡ PROPN cana-2648 64 5 οΏ½ οΏ½ PROPN cana-2648 64 6 o o NOUN cana-2648 64 7 οΏ½ οΏ½ PROPN cana-2648 64 8 Μ‡ Μ‡ PROPN cana-2648 64 9 οΏ½ οΏ½ PROPN cana-2648 64 10 )βˆ’1(𝐴 )βˆ’1(𝐴 PROPN cana-2648 64 11 ) ) PUNCT cana-2648 64 12 is be AUX cana-2648 64 13 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 64 14 in in ADP cana-2648 64 15 ( ( PUNCT cana-2648 64 16 𝑋′ 𝑋′ ADJ cana-2648 64 17 , , PUNCT cana-2648 64 18 πœβ€² πœβ€² NUM cana-2648 64 19 ) ) PUNCT cana-2648 64 20 . . PUNCT cana-2648 65 1 hence hence ADV cana-2648 65 2 οΏ½ οΏ½ PROPN cana-2648 65 3 Μ‡ Μ‡ PROPN cana-2648 65 4 οΏ½ οΏ½ PROPN cana-2648 65 5 o o NOUN cana-2648 65 6 οΏ½ οΏ½ PROPN cana-2648 65 7 Μ‡ Μ‡ PROPN cana-2648 65 8 οΏ½ οΏ½ PROPN cana-2648 65 9 be be AUX cana-2648 65 10 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 65 11 . . PUNCT cana-2648 66 1 communications communication NOUN cana-2648 66 2 on on ADP cana-2648 66 3 applied apply VERB cana-2648 66 4 nonlinear nonlinear ADJ cana-2648 66 5 analysis analysis NOUN cana-2648 66 6 issn issn NOUN cana-2648 66 7 : : PUNCT cana-2648 66 8 1074 1074 NUM cana-2648 66 9 - - PUNCT cana-2648 66 10 133x 133x NUM cana-2648 66 11 vol vol NOUN cana-2648 66 12 32 32 NUM cana-2648 66 13 no no NOUN cana-2648 66 14 . . PUNCT cana-2648 67 1 3s 3s NUM cana-2648 67 2 ( ( PUNCT cana-2648 67 3 2025 2025 NUM cana-2648 67 4 ) ) PUNCT cana-2648 67 5 359 359 NUM cana-2648 67 6 https://internationalpubls.com https://internationalpubls.com X cana-2648 67 7 3 3 X cana-2648 67 8 . . NOUN cana-2648 67 9 maximal maximal ADJ cana-2648 67 10 π’ˆπœΌ-continuous π’ˆπœΌ-continuous ADJ cana-2648 67 11 functions function NOUN cana-2648 67 12 introducing introduce VERB cana-2648 67 13 maximal maximal ADJ cana-2648 67 14 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 67 15 is be AUX cana-2648 67 16 the the DET cana-2648 67 17 goal goal NOUN cana-2648 67 18 of of ADP cana-2648 67 19 this this DET cana-2648 67 20 section section NOUN cana-2648 67 21 . . PUNCT cana-2648 68 1 examples example NOUN cana-2648 68 2 are be AUX cana-2648 68 3 used use VERB cana-2648 68 4 to to PART cana-2648 68 5 obtain obtain VERB cana-2648 68 6 some some DET cana-2648 68 7 attributes attribute NOUN cana-2648 68 8 of of ADP cana-2648 68 9 such such ADJ cana-2648 68 10 functions function NOUN cana-2648 68 11 . . PUNCT cana-2648 69 1 definition definition NOUN cana-2648 69 2 3.1 3.1 NUM cana-2648 69 3 : : PUNCT cana-2648 69 4 a a DET cana-2648 69 5 maximal maximal ADJ cana-2648 69 6 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 69 7 is be AUX cana-2648 69 8 a a DET cana-2648 69 9 proper proper ADJ cana-2648 69 10 nonempty nonempty ADJ cana-2648 69 11 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 69 12 subset subset NOUN cana-2648 69 13 π‘ˆβ€² π‘ˆβ€² ADJ cana-2648 69 14 of of ADP cana-2648 69 15 ( ( PUNCT cana-2648 69 16 𝑋′ 𝑋′ ADJ cana-2648 69 17 , , PUNCT cana-2648 69 18 πœβ€² πœβ€² PROPN cana-2648 69 19 ) ) PUNCT cana-2648 69 20 if if SCONJ cana-2648 69 21 any any PRON cana-2648 69 22 of of ADP cana-2648 69 23 its its PRON cana-2648 69 24 π‘”πœ‚-opens π‘”πœ‚-open NOUN cana-2648 69 25 are be AUX cana-2648 69 26 π‘ˆβ€² π‘ˆβ€² ADJ cana-2648 69 27 is be AUX cana-2648 69 28 either either CCONJ cana-2648 69 29 ( ( PUNCT cana-2648 69 30 𝑋′ 𝑋′ ADJ cana-2648 69 31 , , PUNCT cana-2648 69 32 πœβ€² πœβ€² NUM cana-2648 69 33 ) ) PUNCT cana-2648 69 34 or or CCONJ cana-2648 69 35 π‘ˆβ€². π‘ˆβ€². NUM cana-2648 69 36 definition definition NOUN cana-2648 69 37 3.2 3.2 NUM cana-2648 69 38 : : PUNCT cana-2648 69 39 if if SCONJ cana-2648 69 40 οΏ½ οΏ½ PROPN cana-2648 69 41 Μ‡ Μ‡ VERB cana-2648 69 42 οΏ½ οΏ½ PROPN cana-2648 69 43 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 69 44 ) ) PUNCT cana-2648 69 45 is be AUX cana-2648 69 46 a a DET cana-2648 69 47 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 69 48 in in ADP cana-2648 69 49 ( ( PUNCT cana-2648 69 50 𝑋′ 𝑋′ ADJ cana-2648 69 51 , , PUNCT cana-2648 69 52 πœβ€² πœβ€² PROPN cana-2648 69 53 ) ) PUNCT cana-2648 69 54 for for ADP cana-2648 69 55 any any DET cana-2648 69 56 maximal maximal ADJ cana-2648 69 57 open open ADJ cana-2648 69 58 𝑀′ 𝑀′ NOUN cana-2648 69 59 in in ADP cana-2648 69 60 ( ( PUNCT cana-2648 69 61 π‘Œβ€² π‘Œβ€² PROPN cana-2648 69 62 , , PUNCT cana-2648 69 63 πœŽβ€² πœŽβ€² NUM cana-2648 69 64 ) ) PUNCT cana-2648 69 65 then then ADV cana-2648 69 66 a a DET cana-2648 69 67 function function NOUN cana-2648 69 68 οΏ½ οΏ½ PROPN cana-2648 69 69 Μ‡ Μ‡ PROPN cana-2648 69 70 οΏ½ οΏ½ PROPN cana-2648 69 71 : : PUNCT cana-2648 69 72 ( ( PUNCT cana-2648 69 73 𝑋′ 𝑋′ X cana-2648 69 74 , , PUNCT cana-2648 69 75 πœβ€² πœβ€² NUM cana-2648 69 76 ) ) PUNCT cana-2648 69 77 β†’ β†’ SYM cana-2648 69 78 ( ( PUNCT cana-2648 69 79 π‘Œβ€² π‘Œβ€² PROPN cana-2648 69 80 , , PUNCT cana-2648 69 81 πœŽβ€² πœŽβ€² NUM cana-2648 69 82 ) ) PUNCT cana-2648 69 83 is be AUX cana-2648 69 84 maximal maximal ADJ cana-2648 69 85 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 69 86 . . PUNCT cana-2648 70 1 example example NOUN cana-2648 70 2 3.3 3.3 NUM cana-2648 70 3 : : PUNCT cana-2648 70 4 let let VERB cana-2648 70 5 𝑋′ 𝑋′ PROPN cana-2648 70 6 = = X cana-2648 70 7 π‘Œβ€² π‘Œβ€² NOUN cana-2648 70 8 = = SYM cana-2648 70 9 { { PUNCT cana-2648 70 10 𝑒′ 𝑒′ NOUN cana-2648 70 11 , , PUNCT cana-2648 70 12 𝑓′ 𝑓′ NUM cana-2648 70 13 , , PUNCT cana-2648 70 14 𝑔′ 𝑔′ ADV cana-2648 70 15 } } PUNCT cana-2648 70 16 , , PUNCT cana-2648 70 17 πœβ€² πœβ€² PUNCT cana-2648 70 18 = = PUNCT cana-2648 70 19 { { PUNCT cana-2648 70 20 𝑋′ 𝑋′ NOUN cana-2648 70 21 , , PUNCT cana-2648 70 22 πœ‘β€² πœ‘β€² NUM cana-2648 70 23 , , PUNCT cana-2648 70 24 { { PUNCT cana-2648 70 25 𝑒′ 𝑒′ NOUN cana-2648 70 26 } } PUNCT cana-2648 70 27 , , PUNCT cana-2648 70 28 { { PUNCT cana-2648 70 29 𝑓′ 𝑓′ NOUN cana-2648 70 30 , , PUNCT cana-2648 70 31 𝑔′ 𝑔′ ADV cana-2648 70 32 } } PUNCT cana-2648 70 33 } } PUNCT cana-2648 70 34 , , PUNCT cana-2648 70 35 πœŽβ€² πœŽβ€² X cana-2648 70 36 = = PUNCT cana-2648 70 37 { { PUNCT cana-2648 70 38 π‘Œβ€² π‘Œβ€² PROPN cana-2648 70 39 , , PUNCT cana-2648 70 40 πœ‘β€² πœ‘β€² NUM cana-2648 70 41 , , PUNCT cana-2648 70 42 { { PUNCT cana-2648 70 43 𝑒′ 𝑒′ NOUN cana-2648 70 44 } } PUNCT cana-2648 70 45 , , PUNCT cana-2648 70 46 { { PUNCT cana-2648 70 47 𝑔′ 𝑔′ ADV cana-2648 70 48 } } PUNCT cana-2648 70 49 , , PUNCT cana-2648 70 50 { { PUNCT cana-2648 70 51 𝑒′ 𝑒′ NOUN cana-2648 70 52 , , PUNCT cana-2648 70 53 𝑔′ 𝑔′ ADV cana-2648 70 54 } } PUNCT cana-2648 70 55 } } PUNCT cana-2648 70 56 . . PUNCT cana-2648 71 1 define define VERB cana-2648 71 2 οΏ½ οΏ½ PROPN cana-2648 71 3 Μ‡ Μ‡ PROPN cana-2648 71 4 οΏ½ οΏ½ PROPN cana-2648 71 5 : : PUNCT cana-2648 71 6 ( ( PUNCT cana-2648 71 7 𝑋′ 𝑋′ X cana-2648 71 8 , , PUNCT cana-2648 71 9 πœβ€² πœβ€² NUM cana-2648 71 10 ) ) PUNCT cana-2648 71 11 β†’ β†’ SYM cana-2648 71 12 ( ( PUNCT cana-2648 71 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 71 14 , , PUNCT cana-2648 71 15 πœŽβ€² πœŽβ€² NUM cana-2648 71 16 ) ) PUNCT cana-2648 71 17 by by ADP cana-2648 71 18 οΏ½ οΏ½ PROPN cana-2648 71 19 Μ‡ Μ‡ PROPN cana-2648 71 20 οΏ½ οΏ½ PROPN cana-2648 71 21 (𝑒′ (𝑒′ NUM cana-2648 71 22 ) ) PUNCT cana-2648 71 23 = = SYM cana-2648 71 24 𝑒′ 𝑒′ NOUN cana-2648 71 25 , , PUNCT cana-2648 71 26 οΏ½ οΏ½ PROPN cana-2648 71 27 Μ‡ Μ‡ PROPN cana-2648 71 28 οΏ½ οΏ½ PROPN cana-2648 71 29 (𝑓′ (𝑓′ PUNCT cana-2648 71 30 ) ) PUNCT cana-2648 71 31 = = SYM cana-2648 71 32 𝑔′ 𝑔′ NUM cana-2648 71 33 , , PUNCT cana-2648 71 34 οΏ½ οΏ½ PROPN cana-2648 71 35 Μ‡ Μ‡ PROPN cana-2648 71 36 οΏ½ οΏ½ PROPN cana-2648 71 37 (𝑔′ (𝑔′ X cana-2648 71 38 ) ) PUNCT cana-2648 71 39 = = SYM cana-2648 72 1 𝑓′. 𝑓′. PROPN cana-2648 72 2 here here ADV cana-2648 72 3 { { PUNCT cana-2648 72 4 𝑒′ 𝑒′ NOUN cana-2648 72 5 , , PUNCT cana-2648 72 6 𝑔′ 𝑔′ X cana-2648 72 7 } } PUNCT cana-2648 72 8 is be AUX cana-2648 72 9 maximal maximal ADJ cana-2648 72 10 open open ADJ cana-2648 72 11 in in ADP cana-2648 72 12 ( ( PUNCT cana-2648 72 13 π‘Œβ€² π‘Œβ€² NOUN cana-2648 72 14 , , PUNCT cana-2648 72 15 πœŽβ€² πœŽβ€² NUM cana-2648 72 16 ) ) PUNCT cana-2648 72 17 . . PUNCT cana-2648 73 1 therefore therefore ADV cana-2648 73 2 οΏ½ οΏ½ PROPN cana-2648 73 3 Μ‡ Μ‡ PROPN cana-2648 73 4 οΏ½ οΏ½ PROPN cana-2648 73 5 is be AUX cana-2648 73 6 maximal maximal ADV cana-2648 73 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 73 8 . . PUNCT cana-2648 74 1 theorem theorem VERB cana-2648 74 2 3.4 3.4 NUM cana-2648 74 3 : : PUNCT cana-2648 74 4 all all PRON cana-2648 74 5 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 74 6 is be AUX cana-2648 74 7 maximal maximal ADJ cana-2648 74 8 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 74 9 . . PUNCT cana-2648 75 1 proof proof NOUN cana-2648 75 2 : : PUNCT cana-2648 75 3 let let VERB cana-2648 75 4 𝑀′ 𝑀′ NOUN cana-2648 75 5 be be AUX cana-2648 75 6 a a DET cana-2648 75 7 maximal maximal ADJ cana-2648 75 8 open open NOUN cana-2648 75 9 in in ADP cana-2648 75 10 ( ( PUNCT cana-2648 75 11 π‘Œβ€² π‘Œβ€² NOUN cana-2648 75 12 , , PUNCT cana-2648 75 13 πœŽβ€² πœŽβ€² NUM cana-2648 75 14 ) ) PUNCT cana-2648 75 15 and and CCONJ cana-2648 75 16 οΏ½ οΏ½ PROPN cana-2648 75 17 Μ‡ Μ‡ PROPN cana-2648 75 18 οΏ½ οΏ½ PROPN cana-2648 75 19 : : PUNCT cana-2648 75 20 ( ( PUNCT cana-2648 75 21 𝑋′ 𝑋′ X cana-2648 75 22 , , PUNCT cana-2648 75 23 πœβ€² πœβ€² NUM cana-2648 75 24 ) ) PUNCT cana-2648 75 25 β†’ β†’ SYM cana-2648 75 26 ( ( PUNCT cana-2648 75 27 π‘Œβ€² π‘Œβ€² PROPN cana-2648 75 28 , , PUNCT cana-2648 75 29 πœŽβ€² πœŽβ€² NUM cana-2648 75 30 ) ) PUNCT cana-2648 75 31 be be VERB cana-2648 75 32 a a DET cana-2648 75 33 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 75 34 , , PUNCT cana-2648 75 35 𝑀′ 𝑀′ X cana-2648 75 36 be be AUX cana-2648 75 37 a a DET cana-2648 75 38 maximal maximal ADJ cana-2648 75 39 open open NOUN cana-2648 75 40 in in ADP cana-2648 75 41 ( ( PUNCT cana-2648 75 42 π‘Œβ€² π‘Œβ€² NOUN cana-2648 75 43 , , PUNCT cana-2648 75 44 πœŽβ€² πœŽβ€² NUM cana-2648 75 45 ) ) PUNCT cana-2648 75 46 . . PUNCT cana-2648 76 1 all all DET cana-2648 76 2 maximal maximal ADJ cana-2648 76 3 open open NOUN cana-2648 76 4 is be AUX cana-2648 76 5 an an DET cana-2648 76 6 open open ADJ cana-2648 76 7 when when SCONJ cana-2648 76 8 𝑀′ 𝑀′ PROPN cana-2648 76 9 is be AUX cana-2648 76 10 an an DET cana-2648 76 11 open open ADJ cana-2648 76 12 in in ADP cana-2648 76 13 ( ( PUNCT cana-2648 76 14 π‘Œβ€² π‘Œβ€² NOUN cana-2648 76 15 , , PUNCT cana-2648 76 16 πœŽβ€² πœŽβ€² NUM cana-2648 76 17 ) ) PUNCT cana-2648 76 18 . . PUNCT cana-2648 77 1 so so ADV cana-2648 77 2 οΏ½ οΏ½ PROPN cana-2648 77 3 Μ‡ Μ‡ PROPN cana-2648 77 4 οΏ½ οΏ½ PROPN cana-2648 77 5 is be AUX cana-2648 77 6 π‘”πœ‚continuous π‘”πœ‚continuous ADJ cana-2648 77 7 . . PUNCT cana-2648 78 1 therefore therefore ADV cana-2648 78 2 οΏ½ οΏ½ PROPN cana-2648 78 3 Μ‡ Μ‡ PROPN cana-2648 78 4 οΏ½ οΏ½ PROPN cana-2648 78 5 is be AUX cana-2648 78 6 maximal maximal ADJ cana-2648 78 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 78 8 . . PUNCT cana-2648 79 1 the the DET cana-2648 79 2 converse converse NOUN cana-2648 79 3 of of ADP cana-2648 79 4 the the DET cana-2648 79 5 preceding precede VERB cana-2648 79 6 theorem theorem NOUN cana-2648 79 7 does do AUX cana-2648 79 8 not not PART cana-2648 79 9 necessarily necessarily ADV cana-2648 79 10 have have VERB cana-2648 79 11 to to PART cana-2648 79 12 be be AUX cana-2648 79 13 true true ADJ cana-2648 79 14 , , PUNCT cana-2648 79 15 as as SCONJ cana-2648 79 16 demonstrated demonstrate VERB cana-2648 79 17 by by ADP cana-2648 79 18 the the DET cana-2648 79 19 example example NOUN cana-2648 79 20 that that PRON cana-2648 79 21 follows follow VERB cana-2648 79 22 . . PUNCT cana-2648 80 1 example example NOUN cana-2648 80 2 3.5 3.5 NUM cana-2648 80 3 : : PUNCT cana-2648 80 4 take take VERB cana-2648 80 5 𝑋′ 𝑋′ NOUN cana-2648 80 6 = = X cana-2648 80 7 π‘Œβ€² π‘Œβ€² NOUN cana-2648 80 8 = = SYM cana-2648 80 9 { { PUNCT cana-2648 80 10 𝑒′ 𝑒′ NOUN cana-2648 80 11 , , PUNCT cana-2648 80 12 𝑓′ 𝑓′ NUM cana-2648 80 13 , , PUNCT cana-2648 80 14 𝑔′ 𝑔′ ADV cana-2648 80 15 } } PUNCT cana-2648 80 16 , , PUNCT cana-2648 80 17 πœŽβ€² πœŽβ€² NUM cana-2648 80 18 = = PUNCT cana-2648 80 19 { { PUNCT cana-2648 80 20 π‘Œβ€² π‘Œβ€² PROPN cana-2648 80 21 , , PUNCT cana-2648 80 22 πœ‘β€² πœ‘β€² NUM cana-2648 80 23 , , PUNCT cana-2648 80 24 { { PUNCT cana-2648 80 25 𝑒′ 𝑒′ NOUN cana-2648 80 26 } } PUNCT cana-2648 80 27 , , PUNCT cana-2648 80 28 { { PUNCT cana-2648 80 29 𝑔′ 𝑔′ ADV cana-2648 80 30 } } PUNCT cana-2648 80 31 , , PUNCT cana-2648 80 32 { { PUNCT cana-2648 80 33 𝑒′ 𝑒′ NOUN cana-2648 80 34 , , PUNCT cana-2648 80 35 𝑔′ 𝑔′ ADV cana-2648 80 36 } } PUNCT cana-2648 80 37 } } PUNCT cana-2648 80 38 , , PUNCT cana-2648 80 39 πœβ€² πœβ€² PUNCT cana-2648 80 40 = = SYM cana-2648 80 41 { { PUNCT cana-2648 80 42 𝑋′ 𝑋′ NOUN cana-2648 80 43 , , PUNCT cana-2648 80 44 πœ‘β€² πœ‘β€² NUM cana-2648 80 45 , , PUNCT cana-2648 80 46 { { PUNCT cana-2648 80 47 𝑒′ 𝑒′ NOUN cana-2648 80 48 } } PUNCT cana-2648 80 49 , , PUNCT cana-2648 80 50 { { PUNCT cana-2648 80 51 𝑓′ 𝑓′ NOUN cana-2648 80 52 } } PUNCT cana-2648 80 53 , , PUNCT cana-2648 80 54 { { PUNCT cana-2648 80 55 𝑒′ 𝑒′ NOUN cana-2648 80 56 , , PUNCT cana-2648 80 57 𝑓′ 𝑓′ NUM cana-2648 80 58 } } PUNCT cana-2648 80 59 } } PUNCT cana-2648 80 60 . . PUNCT cana-2648 81 1 assign assign PROPN cana-2648 81 2 οΏ½ οΏ½ PROPN cana-2648 81 3 Μ‡ Μ‡ PROPN cana-2648 81 4 οΏ½ οΏ½ PROPN cana-2648 81 5 : : PUNCT cana-2648 81 6 ( ( PUNCT cana-2648 81 7 𝑋′ 𝑋′ X cana-2648 81 8 , , PUNCT cana-2648 81 9 πœβ€² πœβ€² NUM cana-2648 81 10 ) ) PUNCT cana-2648 81 11 β†’ β†’ SYM cana-2648 81 12 ( ( PUNCT cana-2648 81 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 81 14 , , PUNCT cana-2648 81 15 πœŽβ€² πœŽβ€² NUM cana-2648 81 16 ) ) PUNCT cana-2648 81 17 by by ADP cana-2648 81 18 οΏ½ οΏ½ PROPN cana-2648 81 19 Μ‡ Μ‡ PROPN cana-2648 81 20 οΏ½ οΏ½ PROPN cana-2648 81 21 (𝑒′ (𝑒′ NUM cana-2648 81 22 ) ) PUNCT cana-2648 81 23 = = SYM cana-2648 81 24 𝑓′ 𝑓′ NUM cana-2648 81 25 , , PUNCT cana-2648 81 26 οΏ½ οΏ½ PROPN cana-2648 81 27 Μ‡ Μ‡ PROPN cana-2648 81 28 οΏ½ οΏ½ PROPN cana-2648 81 29 (𝑓′ (𝑓′ PUNCT cana-2648 81 30 ) ) PUNCT cana-2648 81 31 = = SYM cana-2648 81 32 𝑒′ 𝑒′ NOUN cana-2648 81 33 , , PUNCT cana-2648 81 34 οΏ½ οΏ½ PROPN cana-2648 81 35 Μ‡ Μ‡ PROPN cana-2648 81 36 οΏ½ οΏ½ PROPN cana-2648 81 37 (𝑔′ (𝑔′ X cana-2648 81 38 ) ) PUNCT cana-2648 81 39 = = SYM cana-2648 82 1 𝑔′. 𝑔′. PROPN cana-2648 82 2 then then ADV cana-2648 82 3 οΏ½ οΏ½ PROPN cana-2648 82 4 Μ‡ Μ‡ PROPN cana-2648 82 5 οΏ½ οΏ½ NOUN cana-2648 82 6 βˆ’1({𝑒′ βˆ’1({𝑒′ NOUN cana-2648 82 7 , , PUNCT cana-2648 82 8 𝑔′ 𝑔′ ADV cana-2648 82 9 } } PUNCT cana-2648 82 10 ) ) PUNCT cana-2648 83 1 = = SYM cana-2648 83 2 { { PUNCT cana-2648 83 3 𝑓′ 𝑓′ NOUN cana-2648 83 4 , , PUNCT cana-2648 83 5 𝑔′ 𝑔′ X cana-2648 83 6 } } PUNCT cana-2648 83 7 is be AUX cana-2648 83 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 83 9 in in ADP cana-2648 83 10 ( ( PUNCT cana-2648 83 11 𝑋′ 𝑋′ ADJ cana-2648 83 12 , , PUNCT cana-2648 83 13 πœβ€² πœβ€² PROPN cana-2648 83 14 ) ) PUNCT cana-2648 83 15 . . PUNCT cana-2648 84 1 consequently consequently ADV cana-2648 84 2 οΏ½ οΏ½ PROPN cana-2648 84 3 Μ‡ Μ‡ PROPN cana-2648 84 4 οΏ½ οΏ½ PROPN cana-2648 84 5 is be AUX cana-2648 84 6 maximal maximal ADJ cana-2648 84 7 π‘”πœ‚continuous π‘”πœ‚continuous ADJ cana-2648 84 8 . . PUNCT cana-2648 85 1 so so ADV cana-2648 85 2 οΏ½ οΏ½ PROPN cana-2648 85 3 Μ‡ Μ‡ PROPN cana-2648 85 4 οΏ½ οΏ½ PROPN cana-2648 85 5 is be AUX cana-2648 85 6 not not PART cana-2648 85 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 85 8 . . PUNCT cana-2648 86 1 remark remark NOUN cana-2648 86 2 3.6 3.6 NUM cana-2648 87 1 : : PUNCT cana-2648 87 2 there there PRON cana-2648 87 3 is be VERB cana-2648 87 4 independence independence NOUN cana-2648 87 5 between between ADP cana-2648 87 6 maximal maximal ADJ cana-2648 87 7 - - PUNCT cana-2648 87 8 continuous continuous ADJ cana-2648 87 9 and and CCONJ cana-2648 87 10 minimal minimal ADJ cana-2648 87 11 - - ADJ cana-2648 87 12 continuous continuous ADJ cana-2648 87 13 . . PUNCT cana-2648 87 14 example example NOUN cana-2648 87 15 3.7 3.7 NUM cana-2648 87 16 : : PUNCT cana-2648 87 17 take take VERB cana-2648 87 18 𝑋′ 𝑋′ NOUN cana-2648 87 19 = = X cana-2648 87 20 π‘Œβ€² π‘Œβ€² NOUN cana-2648 87 21 = = SYM cana-2648 87 22 { { PUNCT cana-2648 87 23 𝑒′ 𝑒′ NOUN cana-2648 87 24 , , PUNCT cana-2648 87 25 𝑓′ 𝑓′ NUM cana-2648 87 26 , , PUNCT cana-2648 87 27 𝑔′ 𝑔′ ADV cana-2648 87 28 } } PUNCT cana-2648 87 29 , , PUNCT cana-2648 87 30 πœβ€² πœβ€² PUNCT cana-2648 87 31 = = PUNCT cana-2648 87 32 { { PUNCT cana-2648 87 33 𝑋′ 𝑋′ NOUN cana-2648 87 34 , , PUNCT cana-2648 87 35 πœ‘β€² πœ‘β€² NUM cana-2648 87 36 , , PUNCT cana-2648 87 37 { { PUNCT cana-2648 87 38 𝑒′ 𝑒′ NOUN cana-2648 87 39 } } PUNCT cana-2648 87 40 , , PUNCT cana-2648 87 41 { { PUNCT cana-2648 87 42 𝑔′ 𝑔′ ADV cana-2648 87 43 } } PUNCT cana-2648 87 44 , , PUNCT cana-2648 87 45 { { PUNCT cana-2648 87 46 𝑒′ 𝑒′ NOUN cana-2648 87 47 , , PUNCT cana-2648 87 48 𝑔′ 𝑔′ ADV cana-2648 87 49 } } PUNCT cana-2648 87 50 } } PUNCT cana-2648 87 51 , , PUNCT cana-2648 87 52 πœŽβ€² πœŽβ€² X cana-2648 87 53 = = PUNCT cana-2648 87 54 { { PUNCT cana-2648 87 55 π‘Œβ€² π‘Œβ€² PROPN cana-2648 87 56 , , PUNCT cana-2648 87 57 πœ‘β€² πœ‘β€² NUM cana-2648 87 58 , , PUNCT cana-2648 87 59 { { PUNCT cana-2648 87 60 𝑒′ 𝑒′ NOUN cana-2648 87 61 } } PUNCT cana-2648 87 62 , , PUNCT cana-2648 87 63 { { PUNCT cana-2648 87 64 𝑓′ 𝑓′ NOUN cana-2648 87 65 , , PUNCT cana-2648 87 66 𝑔′ 𝑔′ ADV cana-2648 87 67 } } PUNCT cana-2648 87 68 } } PUNCT cana-2648 87 69 . . PUNCT cana-2648 88 1 assign assign PROPN cana-2648 88 2 οΏ½ οΏ½ PROPN cana-2648 88 3 Μ‡ Μ‡ PROPN cana-2648 88 4 οΏ½ οΏ½ PROPN cana-2648 88 5 : : PUNCT cana-2648 88 6 ( ( PUNCT cana-2648 88 7 𝑋′ 𝑋′ X cana-2648 88 8 , , PUNCT cana-2648 88 9 πœβ€² πœβ€² NUM cana-2648 88 10 ) ) PUNCT cana-2648 88 11 β†’ β†’ SYM cana-2648 88 12 ( ( PUNCT cana-2648 88 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 88 14 , , PUNCT cana-2648 88 15 πœŽβ€² πœŽβ€² NUM cana-2648 88 16 ) ) PUNCT cana-2648 88 17 by by ADP cana-2648 88 18 οΏ½ οΏ½ PROPN cana-2648 88 19 Μ‡ Μ‡ PROPN cana-2648 88 20 οΏ½ οΏ½ PROPN cana-2648 88 21 (𝑒 (𝑒 NOUN cana-2648 88 22 ) ) PUNCT cana-2648 89 1 = = SYM cana-2648 89 2 𝑓 𝑓 PROPN cana-2648 89 3 , , PUNCT cana-2648 89 4 οΏ½ οΏ½ PROPN cana-2648 89 5 Μ‡ Μ‡ PROPN cana-2648 89 6 οΏ½ οΏ½ PROPN cana-2648 89 7 (𝑓 (𝑓 NUM cana-2648 89 8 ) ) PUNCT cana-2648 89 9 = = SYM cana-2648 89 10 𝑒 𝑒 PROPN cana-2648 89 11 , , PUNCT cana-2648 89 12 οΏ½ οΏ½ PROPN cana-2648 89 13 Μ‡ Μ‡ PROPN cana-2648 89 14 οΏ½ οΏ½ PROPN cana-2648 89 15 (𝑔 (𝑔 NOUN cana-2648 89 16 ) ) PUNCT cana-2648 89 17 = = VERB cana-2648 90 1 𝑔. 𝑔. NOUN cana-2648 90 2 now now ADV cana-2648 90 3 { { PUNCT cana-2648 90 4 𝑓′ 𝑓′ NOUN cana-2648 90 5 , , PUNCT cana-2648 90 6 𝑔′ 𝑔′ ADV cana-2648 90 7 } } PUNCT cana-2648 90 8 is be AUX cana-2648 90 9 maximal maximal ADJ cana-2648 90 10 open open ADJ cana-2648 90 11 in in ADP cana-2648 90 12 ( ( PUNCT cana-2648 90 13 π‘Œβ€² π‘Œβ€² NOUN cana-2648 90 14 , , PUNCT cana-2648 90 15 πœŽβ€² πœŽβ€² NOUN cana-2648 90 16 ) ) PUNCT cana-2648 90 17 ) ) PUNCT cana-2648 90 18 . . PUNCT cana-2648 91 1 for for ADP cana-2648 91 2 that that DET cana-2648 91 3 reason reason NOUN cana-2648 91 4 οΏ½ οΏ½ PROPN cana-2648 91 5 Μ‡ Μ‡ NOUN cana-2648 91 6 οΏ½ οΏ½ PROPN cana-2648 91 7 is be AUX cana-2648 91 8 maximal maximal ADJ cana-2648 91 9 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 91 10 . . PUNCT cana-2648 92 1 consequently consequently ADV cana-2648 92 2 , , PUNCT cana-2648 92 3 οΏ½ οΏ½ PROPN cana-2648 92 4 Μ‡ Μ‡ PROPN cana-2648 92 5 οΏ½ οΏ½ PROPN cana-2648 92 6 is be AUX cana-2648 92 7 not not PART cana-2648 92 8 minimal minimal ADJ cana-2648 92 9 π‘”πœ‚continuous π‘”πœ‚continuous ADJ cana-2648 92 10 . . PUNCT cana-2648 92 11 example example NOUN cana-2648 92 12 3.8 3.8 NUM cana-2648 92 13 : : PUNCT cana-2648 92 14 take take VERB cana-2648 92 15 𝑋′ 𝑋′ NOUN cana-2648 92 16 = = X cana-2648 92 17 π‘Œβ€² π‘Œβ€² NOUN cana-2648 92 18 = = SYM cana-2648 92 19 { { PUNCT cana-2648 92 20 𝑒′ 𝑒′ NOUN cana-2648 92 21 , , PUNCT cana-2648 92 22 𝑓′ 𝑓′ NUM cana-2648 92 23 , , PUNCT cana-2648 92 24 𝑔′ 𝑔′ ADV cana-2648 92 25 } } PUNCT cana-2648 92 26 , , PUNCT cana-2648 92 27 πœβ€² πœβ€² PUNCT cana-2648 92 28 = = PUNCT cana-2648 92 29 { { PUNCT cana-2648 92 30 𝑋′ 𝑋′ NOUN cana-2648 92 31 , , PUNCT cana-2648 92 32 πœ‘β€² πœ‘β€² NUM cana-2648 92 33 , , PUNCT cana-2648 92 34 { { PUNCT cana-2648 92 35 𝑒′ 𝑒′ NOUN cana-2648 92 36 } } PUNCT cana-2648 92 37 } } PUNCT cana-2648 92 38 , , PUNCT cana-2648 92 39 πœŽβ€² πœŽβ€² X cana-2648 92 40 = = PUNCT cana-2648 92 41 { { PUNCT cana-2648 92 42 π‘Œβ€² π‘Œβ€² PROPN cana-2648 92 43 , , PUNCT cana-2648 92 44 πœ‘β€² πœ‘β€² NUM cana-2648 92 45 , , PUNCT cana-2648 92 46 { { PUNCT cana-2648 92 47 𝑒′ 𝑒′ NOUN cana-2648 92 48 } } PUNCT cana-2648 92 49 , , PUNCT cana-2648 92 50 { { PUNCT cana-2648 92 51 𝑔′ 𝑔′ ADV cana-2648 92 52 } } PUNCT cana-2648 92 53 , , PUNCT cana-2648 92 54 { { PUNCT cana-2648 92 55 𝑒′ 𝑒′ NOUN cana-2648 92 56 , , PUNCT cana-2648 92 57 𝑔′ 𝑔′ ADV cana-2648 92 58 } } PUNCT cana-2648 92 59 } } PUNCT cana-2648 92 60 . . PUNCT cana-2648 93 1 assign assign PROPN cana-2648 93 2 οΏ½ οΏ½ PROPN cana-2648 93 3 Μ‡ Μ‡ PROPN cana-2648 93 4 οΏ½ οΏ½ PROPN cana-2648 93 5 : : PUNCT cana-2648 93 6 ( ( PUNCT cana-2648 93 7 𝑋′ 𝑋′ X cana-2648 93 8 , , PUNCT cana-2648 93 9 πœβ€² πœβ€² NUM cana-2648 93 10 ) ) PUNCT cana-2648 93 11 β†’ β†’ SYM cana-2648 93 12 ( ( PUNCT cana-2648 93 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 93 14 , , PUNCT cana-2648 93 15 πœŽβ€² πœŽβ€² NUM cana-2648 93 16 ) ) PUNCT cana-2648 93 17 by by ADP cana-2648 93 18 οΏ½ οΏ½ PROPN cana-2648 93 19 Μ‡ Μ‡ PROPN cana-2648 93 20 οΏ½ οΏ½ PROPN cana-2648 93 21 (𝑒 (𝑒 NOUN cana-2648 93 22 ) ) PUNCT cana-2648 93 23 = = SYM cana-2648 93 24 𝑓′ 𝑓′ NUM cana-2648 93 25 , , PUNCT cana-2648 93 26 οΏ½ οΏ½ PROPN cana-2648 93 27 Μ‡ Μ‡ PROPN cana-2648 93 28 οΏ½ οΏ½ PROPN cana-2648 93 29 (𝑓′ (𝑓′ PUNCT cana-2648 93 30 ) ) PUNCT cana-2648 93 31 = = SYM cana-2648 93 32 𝑒′ 𝑒′ NOUN cana-2648 93 33 , , PUNCT cana-2648 93 34 οΏ½ οΏ½ PROPN cana-2648 93 35 Μ‡ Μ‡ PROPN cana-2648 93 36 οΏ½ οΏ½ PROPN cana-2648 93 37 (𝑔′ (𝑔′ X cana-2648 93 38 ) ) PUNCT cana-2648 93 39 = = SYM cana-2648 94 1 𝑔′. 𝑔′. PROPN cana-2648 94 2 now now ADV cana-2648 94 3 { { PUNCT cana-2648 94 4 𝑒′ 𝑒′ NOUN cana-2648 94 5 } } PUNCT cana-2648 94 6 , , PUNCT cana-2648 94 7 { { PUNCT cana-2648 94 8 𝑔′ 𝑔′ X cana-2648 94 9 } } PUNCT cana-2648 94 10 is be AUX cana-2648 94 11 minimal minimal ADJ cana-2648 94 12 open open ADJ cana-2648 94 13 in in ADP cana-2648 94 14 ( ( PUNCT cana-2648 94 15 π‘Œβ€² π‘Œβ€² NOUN cana-2648 94 16 , , PUNCT cana-2648 94 17 πœŽβ€² πœŽβ€² NUM cana-2648 94 18 ) ) PUNCT cana-2648 94 19 . . PUNCT cana-2648 95 1 consequently consequently ADV cana-2648 95 2 , , PUNCT cana-2648 95 3 οΏ½ οΏ½ PROPN cana-2648 95 4 Μ‡ Μ‡ PROPN cana-2648 95 5 οΏ½ οΏ½ PROPN cana-2648 95 6 is be AUX cana-2648 95 7 minimal minimal ADJ cana-2648 95 8 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 95 9 and and CCONJ cana-2648 95 10 οΏ½ οΏ½ PROPN cana-2648 95 11 Μ‡ Μ‡ PROPN cana-2648 95 12 οΏ½ οΏ½ PROPN cana-2648 95 13 is be AUX cana-2648 95 14 not not PART cana-2648 95 15 maximal maximal ADJ cana-2648 95 16 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 95 17 . . PUNCT cana-2648 96 1 theorem theorem VERB cana-2648 96 2 3.9 3.9 NUM cana-2648 96 3 : : PUNCT cana-2648 96 4 assign assign PROPN cana-2648 96 5 οΏ½ οΏ½ PROPN cana-2648 96 6 Μ‡ Μ‡ PROPN cana-2648 96 7 οΏ½ οΏ½ PROPN cana-2648 96 8 : : PUNCT cana-2648 96 9 ( ( PUNCT cana-2648 96 10 𝑋′ 𝑋′ X cana-2648 96 11 , , PUNCT cana-2648 96 12 πœβ€² πœβ€² NUM cana-2648 96 13 ) ) PUNCT cana-2648 96 14 β†’ β†’ SYM cana-2648 96 15 ( ( PUNCT cana-2648 96 16 π‘Œβ€² π‘Œβ€² PROPN cana-2648 96 17 , , PUNCT cana-2648 96 18 πœŽβ€² πœŽβ€² NUM cana-2648 96 19 ) ) PUNCT cana-2648 96 20 is be AUX cana-2648 96 21 maximal maximal ADJ cana-2648 96 22 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 96 23 iff iff PROPN cana-2648 96 24 the the DET cana-2648 96 25 inverse inverse ADJ cana-2648 96 26 image image NOUN cana-2648 96 27 of of ADP cana-2648 96 28 each each DET cana-2648 96 29 minimal minimal ADJ cana-2648 96 30 closed close VERB cana-2648 96 31 in in ADP cana-2648 96 32 ( ( PUNCT cana-2648 96 33 π‘Œβ€² π‘Œβ€² PROPN cana-2648 96 34 , , PUNCT cana-2648 96 35 πœŽβ€²)is πœŽβ€²)is PROPN cana-2648 96 36 a a DET cana-2648 96 37 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 96 38 in in ADP cana-2648 96 39 ( ( PUNCT cana-2648 96 40 𝑋′ 𝑋′ ADJ cana-2648 96 41 , , PUNCT cana-2648 96 42 πœβ€² πœβ€² NUM cana-2648 96 43 ) ) PUNCT cana-2648 96 44 . . PUNCT cana-2648 97 1 proof proof NOUN cana-2648 97 2 : : PUNCT cana-2648 97 3 take take VERB cana-2648 97 4 οΏ½ οΏ½ PROPN cana-2648 97 5 Μ‡ Μ‡ PROPN cana-2648 97 6 οΏ½ οΏ½ PROPN cana-2648 97 7 be be AUX cana-2648 97 8 a a DET cana-2648 97 9 maximal maximal ADJ cana-2648 97 10 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 97 11 , , PUNCT cana-2648 97 12 𝑁′ 𝑁′ PROPN cana-2648 97 13 be be AUX cana-2648 97 14 a a DET cana-2648 97 15 minimal minimal ADJ cana-2648 97 16 closed close VERB cana-2648 97 17 in in ADP cana-2648 97 18 ( ( PUNCT cana-2648 97 19 π‘Œβ€² π‘Œβ€² NOUN cana-2648 97 20 , , PUNCT cana-2648 97 21 πœŽβ€² πœŽβ€² NUM cana-2648 97 22 ) ) PUNCT cana-2648 97 23 . . PUNCT cana-2648 98 1 so so ADV cana-2648 98 2 ( ( PUNCT cana-2648 98 3 π‘Œβ€² π‘Œβ€² NOUN cana-2648 98 4 , , PUNCT cana-2648 98 5 πœŽβ€² πœŽβ€² NUM cana-2648 98 6 ) ) PUNCT cana-2648 98 7 βˆ’ βˆ’ PROPN cana-2648 98 8 𝑁′ 𝑁′ PROPN cana-2648 98 9 is be AUX cana-2648 98 10 a a DET cana-2648 98 11 maximal maximal ADJ cana-2648 98 12 open open NOUN cana-2648 98 13 in in ADP cana-2648 98 14 ( ( PUNCT cana-2648 98 15 π‘Œβ€² π‘Œβ€² NOUN cana-2648 98 16 , , PUNCT cana-2648 98 17 πœŽβ€² πœŽβ€² NUM cana-2648 98 18 ) ) PUNCT cana-2648 98 19 . . PUNCT cana-2648 99 1 as as SCONJ cana-2648 99 2 οΏ½ οΏ½ PROPN cana-2648 99 3 Μ‡ Μ‡ PROPN cana-2648 99 4 οΏ½ οΏ½ PROPN cana-2648 99 5 is be AUX cana-2648 99 6 maximal maximal ADJ cana-2648 99 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 99 8 , , PUNCT cana-2648 99 9 οΏ½ οΏ½ PROPN cana-2648 99 10 Μ‡ Μ‡ PROPN cana-2648 99 11 οΏ½ οΏ½ PROPN cana-2648 99 12 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 99 13 , , PUNCT cana-2648 99 14 πœŽβ€² πœŽβ€² NUM cana-2648 99 15 ) ) PUNCT cana-2648 99 16 βˆ’ βˆ’ PROPN cana-2648 99 17 𝑁′ 𝑁′ NOUN cana-2648 99 18 ) ) PUNCT cana-2648 99 19 is be AUX cana-2648 99 20 an an DET cana-2648 99 21 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 99 22 in in ADP cana-2648 99 23 ( ( PUNCT cana-2648 99 24 𝑋′ 𝑋′ ADJ cana-2648 99 25 , , PUNCT cana-2648 99 26 πœβ€² πœβ€² NUM cana-2648 99 27 ) ) PUNCT cana-2648 99 28 . . PUNCT cana-2648 100 1 but but CCONJ cana-2648 100 2 οΏ½ οΏ½ PROPN cana-2648 100 3 Μ‡ Μ‡ PROPN cana-2648 100 4 οΏ½ οΏ½ PROPN cana-2648 100 5 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 100 6 , , PUNCT cana-2648 100 7 πœŽβ€² πœŽβ€² NUM cana-2648 100 8 ) ) PUNCT cana-2648 100 9 βˆ’ βˆ’ PROPN cana-2648 100 10 𝑁′ 𝑁′ NOUN cana-2648 100 11 ) ) PUNCT cana-2648 100 12 = = PUNCT cana-2648 100 13 ( ( PUNCT cana-2648 100 14 𝑋′ 𝑋′ NOUN cana-2648 100 15 , , PUNCT cana-2648 100 16 πœβ€² πœβ€² NUM cana-2648 100 17 ) ) PUNCT cana-2648 101 1 βˆ’ βˆ’ PROPN cana-2648 101 2 οΏ½ οΏ½ PROPN cana-2648 101 3 Μ‡ Μ‡ PROPN cana-2648 101 4 οΏ½ οΏ½ PROPN cana-2648 101 5 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 101 6 ) ) PUNCT cana-2648 101 7 is be AUX cana-2648 101 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 101 9 in in ADP cana-2648 101 10 ( ( PUNCT cana-2648 101 11 𝑋′ 𝑋′ ADJ cana-2648 101 12 , , PUNCT cana-2648 101 13 πœβ€² πœβ€² NUM cana-2648 101 14 ) ) PUNCT cana-2648 101 15 . . PUNCT cana-2648 102 1 consequently consequently ADV cana-2648 102 2 , , PUNCT cana-2648 102 3 οΏ½ οΏ½ PROPN cana-2648 102 4 Μ‡ Μ‡ PROPN cana-2648 102 5 οΏ½ οΏ½ PROPN cana-2648 102 6 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 102 7 ) ) PUNCT cana-2648 102 8 is be AUX cana-2648 102 9 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 102 10 in in ADP cana-2648 102 11 ( ( PUNCT cana-2648 102 12 𝑋′ 𝑋′ ADJ cana-2648 102 13 , , PUNCT cana-2648 102 14 πœβ€² πœβ€² NUM cana-2648 102 15 ) ) PUNCT cana-2648 102 16 . . PUNCT cana-2648 103 1 on on ADP cana-2648 103 2 the the DET cana-2648 103 3 contrary contrary NOUN cana-2648 103 4 , , PUNCT cana-2648 103 5 οΏ½ οΏ½ PROPN cana-2648 103 6 Μ‡ Μ‡ PROPN cana-2648 103 7 οΏ½ οΏ½ PROPN cana-2648 103 8 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 103 9 ) ) PUNCT cana-2648 103 10 is be AUX cana-2648 103 11 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 103 12 in in ADP cana-2648 103 13 ( ( PUNCT cana-2648 103 14 𝑋′ 𝑋′ ADJ cana-2648 103 15 , , PUNCT cana-2648 103 16 πœβ€² πœβ€² NOUN cana-2648 103 17 ) ) PUNCT cana-2648 103 18 for for ADP cana-2648 103 19 all all DET cana-2648 103 20 minimal minimal ADJ cana-2648 103 21 closed closed ADJ cana-2648 103 22 𝑁′ 𝑁′ NOUN cana-2648 103 23 in in ADP cana-2648 103 24 ( ( PUNCT cana-2648 103 25 π‘Œβ€² π‘Œβ€² NOUN cana-2648 103 26 , , PUNCT cana-2648 103 27 πœŽβ€² πœŽβ€² NUM cana-2648 103 28 ) ) PUNCT cana-2648 103 29 . . PUNCT cana-2648 104 1 let let VERB cana-2648 104 2 𝑀′ 𝑀′ NOUN cana-2648 104 3 be be AUX cana-2648 104 4 maximal maximal ADV cana-2648 104 5 open open ADJ cana-2648 104 6 in in ADP cana-2648 104 7 ( ( PUNCT cana-2648 104 8 π‘Œβ€² π‘Œβ€² NOUN cana-2648 104 9 , , PUNCT cana-2648 104 10 πœŽβ€² πœŽβ€² NUM cana-2648 104 11 ) ) PUNCT cana-2648 104 12 . . PUNCT cana-2648 105 1 so so ADV cana-2648 105 2 ( ( PUNCT cana-2648 105 3 π‘Œβ€² π‘Œβ€² NOUN cana-2648 105 4 , , PUNCT cana-2648 105 5 πœŽβ€² πœŽβ€² NUM cana-2648 105 6 ) ) PUNCT cana-2648 105 7 βˆ’ βˆ’ PROPN cana-2648 105 8 𝑀′ 𝑀′ PROPN cana-2648 105 9 is be AUX cana-2648 105 10 minimal minimal ADJ cana-2648 105 11 closed close VERB cana-2648 105 12 in in ADP cana-2648 105 13 ( ( PUNCT cana-2648 105 14 π‘Œβ€² π‘Œβ€² NOUN cana-2648 105 15 , , PUNCT cana-2648 105 16 πœŽβ€² πœŽβ€² NUM cana-2648 105 17 ) ) PUNCT cana-2648 105 18 . . PUNCT cana-2648 106 1 but but CCONJ cana-2648 106 2 communications communication NOUN cana-2648 106 3 on on ADP cana-2648 106 4 applied apply VERB cana-2648 106 5 nonlinear nonlinear ADJ cana-2648 106 6 analysis analysis NOUN cana-2648 106 7 issn issn NOUN cana-2648 106 8 : : PUNCT cana-2648 106 9 1074 1074 NUM cana-2648 106 10 - - PUNCT cana-2648 106 11 133x 133x NUM cana-2648 106 12 vol vol NOUN cana-2648 106 13 32 32 NUM cana-2648 106 14 no no NOUN cana-2648 106 15 . . PUNCT cana-2648 107 1 3s 3s NUM cana-2648 107 2 ( ( PUNCT cana-2648 107 3 2025 2025 NUM cana-2648 107 4 ) ) PUNCT cana-2648 107 5 360 360 NUM cana-2648 107 6 https://internationalpubls.com https://internationalpubls.com X cana-2648 107 7 οΏ½ οΏ½ PROPN cana-2648 107 8 Μ‡ Μ‡ PROPN cana-2648 107 9 οΏ½ οΏ½ PROPN cana-2648 107 10 βˆ’1((π‘Œβ€² βˆ’1((π‘Œβ€² PROPN cana-2648 107 11 , , PUNCT cana-2648 107 12 πœŽβ€² πœŽβ€² NUM cana-2648 107 13 ) ) PUNCT cana-2648 107 14 βˆ’ βˆ’ PROPN cana-2648 107 15 𝑀′ 𝑀′ NOUN cana-2648 107 16 ) ) PUNCT cana-2648 107 17 = = PUNCT cana-2648 107 18 ( ( PUNCT cana-2648 107 19 𝑋′ 𝑋′ NOUN cana-2648 107 20 , , PUNCT cana-2648 107 21 πœβ€² πœβ€² NUM cana-2648 107 22 ) ) PUNCT cana-2648 107 23 βˆ’ βˆ’ PROPN cana-2648 107 24 οΏ½ οΏ½ PROPN cana-2648 107 25 Μ‡ Μ‡ PROPN cana-2648 107 26 οΏ½ οΏ½ PROPN cana-2648 107 27 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 107 28 ) ) PUNCT cana-2648 107 29 is be AUX cana-2648 107 30 π‘”πœ‚-closed π‘”πœ‚-close VERB cana-2648 107 31 in in ADP cana-2648 107 32 ( ( PUNCT cana-2648 107 33 𝑋′ 𝑋′ ADJ cana-2648 107 34 , , PUNCT cana-2648 107 35 πœβ€² πœβ€² NUM cana-2648 107 36 ) ) PUNCT cana-2648 107 37 . . PUNCT cana-2648 108 1 hence hence ADV cana-2648 108 2 οΏ½ οΏ½ PROPN cana-2648 108 3 Μ‡ Μ‡ PROPN cana-2648 108 4 οΏ½ οΏ½ PROPN cana-2648 108 5 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 108 6 ) ) PUNCT cana-2648 108 7 is be AUX cana-2648 108 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 108 9 in in ADP cana-2648 108 10 ( ( PUNCT cana-2648 108 11 𝑋′ 𝑋′ ADJ cana-2648 108 12 , , PUNCT cana-2648 108 13 πœβ€² πœβ€² PROPN cana-2648 108 14 ) ) PUNCT cana-2648 108 15 . . PUNCT cana-2648 109 1 thus thus ADV cana-2648 109 2 οΏ½ οΏ½ PROPN cana-2648 109 3 Μ‡ Μ‡ NOUN cana-2648 109 4 οΏ½ οΏ½ PROPN cana-2648 109 5 is be AUX cana-2648 109 6 maximal maximal ADV cana-2648 109 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 109 8 . . PUNCT cana-2648 110 1 theorem theorem VERB cana-2648 110 2 3.10 3.10 NUM cana-2648 110 3 : : PUNCT cana-2648 110 4 assign assign PROPN cana-2648 110 5 οΏ½ οΏ½ PROPN cana-2648 110 6 Μ‡ Μ‡ PROPN cana-2648 110 7 οΏ½ οΏ½ PROPN cana-2648 110 8 : : PUNCT cana-2648 110 9 ( ( PUNCT cana-2648 110 10 𝑋′ 𝑋′ X cana-2648 110 11 , , PUNCT cana-2648 110 12 πœβ€² πœβ€² NUM cana-2648 110 13 ) ) PUNCT cana-2648 110 14 β†’ β†’ SYM cana-2648 110 15 ( ( PUNCT cana-2648 110 16 π‘Œβ€² π‘Œβ€² PROPN cana-2648 110 17 , , PUNCT cana-2648 110 18 πœŽβ€² πœŽβ€² NUM cana-2648 110 19 ) ) PUNCT cana-2648 110 20 is be AUX cana-2648 110 21 maximal maximal ADJ cana-2648 110 22 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 110 23 iff iff PROPN cana-2648 110 24 π‘žβ€² π‘žβ€² PROPN cana-2648 110 25 ∈ ∈ PROPN cana-2648 110 26 ( ( PUNCT cana-2648 110 27 𝑋′ 𝑋′ NOUN cana-2648 110 28 , , PUNCT cana-2648 110 29 πœβ€² πœβ€² NOUN cana-2648 110 30 ) ) PUNCT cana-2648 110 31 and and CCONJ cana-2648 110 32 maximal maximal ADJ cana-2648 110 33 open open ADJ cana-2648 110 34 𝑀′ 𝑀′ NOUN cana-2648 110 35 in in ADP cana-2648 110 36 ( ( PUNCT cana-2648 110 37 π‘Œβ€² π‘Œβ€² PROPN cana-2648 110 38 , , PUNCT cana-2648 110 39 πœŽβ€² πœŽβ€² NOUN cana-2648 110 40 ) ) PUNCT cana-2648 110 41 holding hold VERB cana-2648 110 42 οΏ½ οΏ½ PROPN cana-2648 110 43 Μ‡ Μ‡ PROPN cana-2648 110 44 οΏ½ οΏ½ PROPN cana-2648 110 45 (π‘žβ€² (π‘žβ€² NUM cana-2648 110 46 ) ) PUNCT cana-2648 110 47 , , PUNCT cana-2648 110 48 there there PRON cana-2648 110 49 is be VERB cana-2648 110 50 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 110 51 𝑁′ 𝑁′ NOUN cana-2648 110 52 in in ADP cana-2648 110 53 ( ( PUNCT cana-2648 110 54 𝑋′ 𝑋′ ADJ cana-2648 110 55 , , PUNCT cana-2648 110 56 πœβ€² πœβ€² NUM cana-2648 110 57 ) ) PUNCT cana-2648 110 58 in in ADP cana-2648 110 59 order order NOUN cana-2648 111 1 that that SCONJ cana-2648 111 2 π‘žβ€² π‘žβ€² NOUN cana-2648 111 3 ∈ ∈ PROPN cana-2648 111 4 𝑁′ 𝑁′ PROPN cana-2648 111 5 , , PUNCT cana-2648 111 6 οΏ½ οΏ½ PROPN cana-2648 111 7 Μ‡ Μ‡ NOUN cana-2648 111 8 οΏ½ οΏ½ PROPN cana-2648 111 9 (𝑁′ (𝑁′ NOUN cana-2648 111 10 ) ) PUNCT cana-2648 111 11 βŠ‚ βŠ‚ X cana-2648 112 1 𝑀′. 𝑀′. X cana-2648 112 2 proof proof NOUN cana-2648 112 3 : : PUNCT cana-2648 112 4 take take VERB cana-2648 112 5 𝑀′ 𝑀′ NOUN cana-2648 112 6 be be AUX cana-2648 112 7 maximal maximal ADV cana-2648 112 8 open open ADJ cana-2648 112 9 in in ADP cana-2648 112 10 ( ( PUNCT cana-2648 112 11 π‘Œβ€² π‘Œβ€² NOUN cana-2648 112 12 , , PUNCT cana-2648 112 13 πœŽβ€² πœŽβ€² NOUN cana-2648 112 14 ) ) PUNCT cana-2648 112 15 holding hold VERB cana-2648 112 16 οΏ½ οΏ½ PROPN cana-2648 112 17 Μ‡ Μ‡ PROPN cana-2648 112 18 οΏ½ οΏ½ PROPN cana-2648 112 19 (π‘žβ€² (π‘žβ€² NUM cana-2648 112 20 ) ) PUNCT cana-2648 112 21 , , PUNCT cana-2648 112 22 π‘ž π‘ž X cana-2648 112 23 β€² β€² NOUN cana-2648 112 24 ∈ ∈ PROPN cana-2648 112 25 𝑁′ 𝑁′ NOUN cana-2648 112 26 where where SCONJ cana-2648 112 27 𝑁′ 𝑁′ PROPN cana-2648 112 28 is be AUX cana-2648 112 29 an an DET cana-2648 112 30 π‘”πœ‚-open π‘”πœ‚-open NOUN cana-2648 112 31 in in ADP cana-2648 112 32 ( ( PUNCT cana-2648 112 33 𝑋′ 𝑋′ ADJ cana-2648 112 34 , , PUNCT cana-2648 112 35 πœβ€² πœβ€² PROPN cana-2648 112 36 ) ) PUNCT cana-2648 112 37 , , PUNCT cana-2648 112 38 οΏ½ οΏ½ PROPN cana-2648 112 39 Μ‡ Μ‡ PROPN cana-2648 112 40 οΏ½ οΏ½ PROPN cana-2648 112 41 is be AUX cana-2648 112 42 maximal maximal ADJ cana-2648 112 43 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 112 44 and and CCONJ cana-2648 112 45 οΏ½ οΏ½ PROPN cana-2648 112 46 Μ‡ Μ‡ PROPN cana-2648 112 47 οΏ½ οΏ½ PROPN cana-2648 112 48 βˆ’1(𝑁′ βˆ’1(𝑁′ NOUN cana-2648 112 49 ) ) PUNCT cana-2648 112 50 is be AUX cana-2648 112 51 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 112 52 in in ADP cana-2648 112 53 ( ( PUNCT cana-2648 112 54 𝑋′ 𝑋′ ADJ cana-2648 112 55 , , PUNCT cana-2648 112 56 πœβ€² πœβ€² NUM cana-2648 112 57 ) ) PUNCT cana-2648 112 58 . . PUNCT cana-2648 113 1 then then ADV cana-2648 113 2 𝑁′ 𝑁′ X cana-2648 113 3 = = SYM cana-2648 113 4 οΏ½ οΏ½ PROPN cana-2648 113 5 Μ‡ Μ‡ PROPN cana-2648 113 6 οΏ½ οΏ½ NOUN cana-2648 113 7 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 113 8 ) ) PUNCT cana-2648 113 9 . . PUNCT cana-2648 114 1 so so ADV cana-2648 114 2 οΏ½ οΏ½ PROPN cana-2648 114 3 Μ‡ Μ‡ PROPN cana-2648 114 4 οΏ½ οΏ½ PROPN cana-2648 114 5 (𝑁′ (𝑁′ NOUN cana-2648 114 6 ) ) PUNCT cana-2648 115 1 βŠ‚ βŠ‚ PROPN cana-2648 115 2 𝑀′. 𝑀′. PROPN cana-2648 115 3 on on ADP cana-2648 115 4 the the DET cana-2648 115 5 contrary contrary NOUN cana-2648 115 6 , , PUNCT cana-2648 115 7 𝑀′ 𝑀′ PROPN cana-2648 115 8 be be AUX cana-2648 115 9 a a DET cana-2648 115 10 maximal maximal ADJ cana-2648 115 11 open open NOUN cana-2648 115 12 in in ADP cana-2648 115 13 ( ( PUNCT cana-2648 115 14 π‘Œβ€² π‘Œβ€² NOUN cana-2648 115 15 , , PUNCT cana-2648 115 16 πœŽβ€² πœŽβ€² NUM cana-2648 115 17 ) ) PUNCT cana-2648 115 18 . . PUNCT cana-2648 116 1 then then ADV cana-2648 116 2 there there PRON cana-2648 116 3 is be VERB cana-2648 116 4 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 116 5 𝑁′ 𝑁′ NOUN cana-2648 116 6 in in ADP cana-2648 116 7 ( ( PUNCT cana-2648 116 8 𝑋′ 𝑋′ ADJ cana-2648 116 9 , , PUNCT cana-2648 116 10 πœβ€² πœβ€² PROPN cana-2648 116 11 ) ) PUNCT cana-2648 116 12 , , PUNCT cana-2648 116 13 π‘žβ€² π‘žβ€² PROPN cana-2648 116 14 ∈ ∈ PROPN cana-2648 116 15 𝑁′ 𝑁′ PROPN cana-2648 116 16 , , PUNCT cana-2648 116 17 οΏ½ οΏ½ PROPN cana-2648 116 18 Μ‡ Μ‡ PROPN cana-2648 116 19 οΏ½ οΏ½ PROPN cana-2648 116 20 (π‘žβ€² (π‘žβ€² PROPN cana-2648 116 21 ) ) PUNCT cana-2648 116 22 ∈ ∈ PROPN cana-2648 116 23 οΏ½ οΏ½ PROPN cana-2648 116 24 Μ‡ Μ‡ NOUN cana-2648 116 25 οΏ½ οΏ½ PROPN cana-2648 116 26 (𝑁′ (𝑁′ NOUN cana-2648 116 27 ) ) PUNCT cana-2648 117 1 βŠ‚ βŠ‚ PROPN cana-2648 117 2 𝑀′ 𝑀′ X cana-2648 117 3 , , PUNCT cana-2648 117 4 π‘žβ€² π‘žβ€² PROPN cana-2648 117 5 ∈ ∈ PROPN cana-2648 117 6 οΏ½ οΏ½ PROPN cana-2648 117 7 Μ‡ Μ‡ PROPN cana-2648 117 8 οΏ½ οΏ½ PROPN cana-2648 117 9 βˆ’1( βˆ’1( PROPN cana-2648 117 10 οΏ½ οΏ½ PROPN cana-2648 117 11 Μ‡ Μ‡ NOUN cana-2648 117 12 οΏ½ οΏ½ NOUN cana-2648 117 13 (𝑁′ (𝑁′ NOUN cana-2648 117 14 ) ) PUNCT cana-2648 117 15 ) ) PUNCT cana-2648 118 1 βŠ‚ βŠ‚ PROPN cana-2648 118 2 οΏ½ οΏ½ PROPN cana-2648 118 3 Μ‡ Μ‡ PROPN cana-2648 118 4 οΏ½ οΏ½ PROPN cana-2648 118 5 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 118 6 ) ) PUNCT cana-2648 118 7 . . PUNCT cana-2648 119 1 so so ADV cana-2648 119 2 οΏ½ οΏ½ PROPN cana-2648 119 3 Μ‡ Μ‡ PROPN cana-2648 119 4 οΏ½ οΏ½ PROPN cana-2648 119 5 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 119 6 ) ) PUNCT cana-2648 119 7 is be AUX cana-2648 119 8 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 119 9 in in ADP cana-2648 119 10 ( ( PUNCT cana-2648 119 11 𝑋′ 𝑋′ ADJ cana-2648 119 12 , , PUNCT cana-2648 119 13 πœβ€² πœβ€² NUM cana-2648 119 14 ) ) PUNCT cana-2648 119 15 . . PUNCT cana-2648 120 1 consequently consequently ADV cana-2648 120 2 , , PUNCT cana-2648 120 3 οΏ½ οΏ½ PROPN cana-2648 120 4 Μ‡ Μ‡ PROPN cana-2648 120 5 οΏ½ οΏ½ PROPN cana-2648 120 6 is be AUX cana-2648 120 7 maximal maximal ADV cana-2648 120 8 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 120 9 . . PUNCT cana-2648 121 1 theorem theorem VERB cana-2648 121 2 3.11 3.11 NUM cana-2648 121 3 : : PUNCT cana-2648 121 4 take take VERB cana-2648 121 5 𝐡′ 𝐡′ NUM cana-2648 121 6 be be AUX cana-2648 121 7 a a DET cana-2648 121 8 non non ADJ cana-2648 121 9 - - ADJ cana-2648 121 10 empty empty ADJ cana-2648 121 11 subset subset NOUN cana-2648 121 12 of of ADP cana-2648 121 13 ( ( PUNCT cana-2648 121 14 𝑋′ 𝑋′ ADJ cana-2648 121 15 , , PUNCT cana-2648 121 16 πœβ€² πœβ€² PROPN cana-2648 121 17 ) ) PUNCT cana-2648 121 18 and and CCONJ cana-2648 121 19 οΏ½ οΏ½ PROPN cana-2648 121 20 Μ‡ Μ‡ PROPN cana-2648 121 21 οΏ½ οΏ½ PROPN cana-2648 121 22 : : PUNCT cana-2648 121 23 ( ( PUNCT cana-2648 121 24 𝑋′ 𝑋′ X cana-2648 121 25 , , PUNCT cana-2648 121 26 πœβ€² πœβ€² NUM cana-2648 121 27 ) ) PUNCT cana-2648 121 28 β†’ β†’ SYM cana-2648 121 29 ( ( PUNCT cana-2648 121 30 π‘Œβ€² π‘Œβ€² PROPN cana-2648 121 31 , , PUNCT cana-2648 121 32 πœŽβ€² πœŽβ€² NUM cana-2648 121 33 ) ) PUNCT cana-2648 121 34 is be AUX cana-2648 121 35 maximal maximal ADJ cana-2648 121 36 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 121 37 then then ADV cana-2648 121 38 the the DET cana-2648 121 39 restriction restriction NOUN cana-2648 121 40 οΏ½ οΏ½ PROPN cana-2648 121 41 Μ‡ Μ‡ PROPN cana-2648 121 42 οΏ½ οΏ½ PROPN cana-2648 121 43 | | ADV cana-2648 121 44 𝐡′ 𝐡′ NUM cana-2648 121 45 : : PUNCT cana-2648 121 46 𝐡′ 𝐡′ PRON cana-2648 121 47 β†’ β†’ SYM cana-2648 121 48 π‘Œβ€² π‘Œβ€² PROPN cana-2648 121 49 is be AUX cana-2648 121 50 maximal maximal ADJ cana-2648 121 51 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 121 52 . . PUNCT cana-2648 122 1 when when SCONJ cana-2648 122 2 𝐡′ 𝐡′ PROPN cana-2648 122 3 has have VERB cana-2648 122 4 the the DET cana-2648 122 5 relative relative ADJ cana-2648 122 6 topology topology NOUN cana-2648 122 7 . . PUNCT cana-2648 123 1 proof proof NOUN cana-2648 123 2 : : PUNCT cana-2648 123 3 conclude conclude VERB cana-2648 123 4 𝐴′ 𝐴′ PROPN cana-2648 123 5 is be AUX cana-2648 123 6 a a DET cana-2648 123 7 non non ADJ cana-2648 123 8 - - ADJ cana-2648 123 9 empty empty ADJ cana-2648 123 10 subset subset NOUN cana-2648 123 11 of of ADP cana-2648 123 12 a a DET cana-2648 123 13 ( ( PUNCT cana-2648 123 14 𝑋′ 𝑋′ PROPN cana-2648 123 15 , , PUNCT cana-2648 123 16 πœβ€²)and πœβ€²)and X cana-2648 123 17 𝑀′ 𝑀′ X cana-2648 123 18 be be AUX cana-2648 123 19 any any DET cana-2648 123 20 maximal maximal ADJ cana-2648 123 21 open open NOUN cana-2648 123 22 in in ADP cana-2648 123 23 ( ( PUNCT cana-2648 123 24 π‘Œβ€² π‘Œβ€² NOUN cana-2648 123 25 , , PUNCT cana-2648 123 26 πœŽβ€² πœŽβ€² NUM cana-2648 123 27 ) ) PUNCT cana-2648 123 28 . . PUNCT cana-2648 124 1 so so ADV cana-2648 124 2 οΏ½ οΏ½ PROPN cana-2648 124 3 Μ‡ Μ‡ PROPN cana-2648 124 4 οΏ½ οΏ½ PROPN cana-2648 124 5 is be AUX cana-2648 124 6 maximal maximal ADJ cana-2648 124 7 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 124 8 , , PUNCT cana-2648 124 9 οΏ½ οΏ½ PROPN cana-2648 124 10 Μ‡ Μ‡ PROPN cana-2648 124 11 οΏ½ οΏ½ PROPN cana-2648 124 12 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 124 13 ) ) PUNCT cana-2648 124 14 is be AUX cana-2648 124 15 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 124 16 in in ADP cana-2648 124 17 ( ( PUNCT cana-2648 124 18 𝑋′ 𝑋′ ADJ cana-2648 124 19 , , PUNCT cana-2648 124 20 πœβ€² πœβ€² NUM cana-2648 124 21 ) ) PUNCT cana-2648 124 22 . . PUNCT cana-2648 125 1 using use VERB cana-2648 125 2 relative relative ADJ cana-2648 125 3 topology topology NOUN cana-2648 125 4 , , PUNCT cana-2648 125 5 ( ( PUNCT cana-2648 125 6 οΏ½ οΏ½ PROPN cana-2648 125 7 Μ‡ Μ‡ PROPN cana-2648 125 8 οΏ½ οΏ½ PROPN cana-2648 125 9 | | NOUN cana-2648 125 10 𝐡′)βˆ’1(𝑀′ 𝐡′)βˆ’1(𝑀′ PUNCT cana-2648 125 11 ) ) PUNCT cana-2648 126 1 = = PUNCT cana-2648 127 1 𝐡′ 𝐡′ NUM cana-2648 127 2 ∩ ∩ PROPN cana-2648 127 3 οΏ½ οΏ½ PROPN cana-2648 127 4 Μ‡ Μ‡ PROPN cana-2648 127 5 οΏ½ οΏ½ PROPN cana-2648 127 6 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 127 7 ) ) PUNCT cana-2648 127 8 . . PUNCT cana-2648 128 1 so so ADV cana-2648 128 2 𝐡′ 𝐡′ ADJ cana-2648 128 3 ∩ ∩ ADJ cana-2648 128 4 οΏ½ οΏ½ PROPN cana-2648 128 5 Μ‡ Μ‡ PROPN cana-2648 128 6 οΏ½ οΏ½ PROPN cana-2648 128 7 βˆ’1(𝑀′ βˆ’1(𝑀′ NUM cana-2648 128 8 ) ) PUNCT cana-2648 128 9 is be AUX cana-2648 128 10 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 128 11 in in ADP cana-2648 128 12 𝐡′. 𝐡′. PROPN cana-2648 128 13 thus thus ADV cana-2648 128 14 οΏ½ οΏ½ AUX cana-2648 128 15 Μ‡ Μ‡ PROPN cana-2648 128 16 οΏ½ οΏ½ PROPN cana-2648 128 17 | | ADV cana-2648 128 18 𝐡′ 𝐡′ PRON cana-2648 128 19 is be AUX cana-2648 128 20 maximal maximal ADJ cana-2648 128 21 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 128 22 . . PUNCT cana-2648 129 1 remark remark PROPN cana-2648 129 2 3.12 3.12 NUM cana-2648 129 3 : : PUNCT cana-2648 129 4 maximal maximal ADJ cana-2648 129 5 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 129 6 functions function NOUN cana-2648 129 7 do do AUX cana-2648 129 8 not not PART cana-2648 129 9 always always ADV cana-2648 129 10 have have VERB cana-2648 129 11 to to PART cana-2648 129 12 be be AUX cana-2648 129 13 maximal maximal ADV cana-2648 129 14 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 129 15 in in ADP cana-2648 129 16 composition composition NOUN cana-2648 129 17 . . PUNCT cana-2648 130 1 theorem theorem VERB cana-2648 130 2 3.13 3.13 NUM cana-2648 130 3 : : PUNCT cana-2648 130 4 the the DET cana-2648 130 5 maximal maximal ADJ cana-2648 130 6 gΞ· gΞ· NOUN cana-2648 130 7 - - PUNCT cana-2648 130 8 continuousness continuousness NOUN cana-2648 130 9 is be AUX cana-2648 130 10 οΏ½ οΏ½ PROPN cana-2648 130 11 Μ‡ Μ‡ VERB cana-2648 130 12 οΏ½ οΏ½ PROPN cana-2648 130 13 o o NOUN cana-2648 130 14 οΏ½ οΏ½ PROPN cana-2648 130 15 Μ‡ Μ‡ PROPN cana-2648 130 16 οΏ½ οΏ½ PROPN cana-2648 130 17 Μ‡ Μ‡ PROPN cana-2648 130 18 : : PUNCT cana-2648 130 19 ( ( PUNCT cana-2648 130 20 𝑋′ 𝑋′ ADJ cana-2648 130 21 , , PUNCT cana-2648 130 22 πœβ€²)β†’(𝑍′ πœβ€²)β†’(𝑍′ ADJ cana-2648 130 23 , , PUNCT cana-2648 130 24 ΞΌβ€² ΞΌβ€² NOUN cana-2648 130 25 ) ) PUNCT cana-2648 130 26 . . PUNCT cana-2648 131 1 if if SCONJ cana-2648 131 2 οΏ½ οΏ½ PROPN cana-2648 131 3 Μ‡ Μ‡ PROPN cana-2648 131 4 οΏ½ οΏ½ PROPN cana-2648 131 5 : : PUNCT cana-2648 131 6 ( ( PUNCT cana-2648 131 7 𝑋′ 𝑋′ X cana-2648 131 8 , , PUNCT cana-2648 131 9 πœβ€² πœβ€² NUM cana-2648 131 10 ) ) PUNCT cana-2648 131 11 β†’ β†’ SYM cana-2648 131 12 ( ( PUNCT cana-2648 131 13 π‘Œβ€² π‘Œβ€² PROPN cana-2648 131 14 , , PUNCT cana-2648 131 15 πœŽβ€² πœŽβ€² NUM cana-2648 131 16 ) ) PUNCT cana-2648 131 17 is be AUX cana-2648 131 18 gΞ· gΞ· ADV cana-2648 131 19 - - PUNCT cana-2648 131 20 continuous continuous ADJ cana-2648 131 21 and and CCONJ cana-2648 131 22 οΏ½ οΏ½ PROPN cana-2648 131 23 Μ‡ Μ‡ PROPN cana-2648 131 24 οΏ½ οΏ½ PROPN cana-2648 131 25 : : PUNCT cana-2648 131 26 ( ( PUNCT cana-2648 131 27 π‘Œβ€² π‘Œβ€² NOUN cana-2648 131 28 , , PUNCT cana-2648 131 29 πœŽβ€²)β†’(𝑍′ πœŽβ€²)β†’(𝑍′ ADJ cana-2648 131 30 , , PUNCT cana-2648 131 31 ΞΌβ€² ΞΌβ€² NOUN cana-2648 131 32 ) ) PUNCT cana-2648 131 33 is be AUX cana-2648 131 34 maximal maximal ADJ cana-2648 131 35 gΞ· gΞ· ADV cana-2648 131 36 - - PUNCT cana-2648 131 37 continuous continuous ADJ cana-2648 131 38 . . PUNCT cana-2648 132 1 proof proof NOUN cana-2648 132 2 : : PUNCT cana-2648 132 3 since since SCONJ cana-2648 132 4 οΏ½ οΏ½ PROPN cana-2648 132 5 Μ‡ Μ‡ PROPN cana-2648 132 6 οΏ½ οΏ½ PROPN cana-2648 132 7 is be AUX cana-2648 132 8 a a DET cana-2648 132 9 maximal maximal ADJ cana-2648 132 10 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 132 11 , , PUNCT cana-2648 132 12 οΏ½ οΏ½ PROPN cana-2648 132 13 Μ‡ Μ‡ NOUN cana-2648 132 14 οΏ½ οΏ½ NOUN cana-2648 132 15 βˆ’1(𝐴′ βˆ’1(𝐴′ NOUN cana-2648 132 16 ) ) PUNCT cana-2648 132 17 is be AUX cana-2648 132 18 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 132 19 in in ADP cana-2648 132 20 ( ( PUNCT cana-2648 132 21 π‘Œβ€² π‘Œβ€² NOUN cana-2648 132 22 , , PUNCT cana-2648 132 23 πœŽβ€² πœŽβ€² NUM cana-2648 132 24 ) ) PUNCT cana-2648 132 25 , , PUNCT cana-2648 132 26 assuming assume VERB cana-2648 132 27 that that SCONJ cana-2648 132 28 𝐴′ 𝐴′ PROPN cana-2648 132 29 be be AUX cana-2648 132 30 a a DET cana-2648 132 31 maximal maximal ADJ cana-2648 132 32 open open NOUN cana-2648 132 33 in in ADP cana-2648 132 34 ( ( PUNCT cana-2648 132 35 𝑍′ 𝑍′ NOUN cana-2648 132 36 , , PUNCT cana-2648 132 37 ΞΌβ€² ΞΌβ€² NOUN cana-2648 132 38 ) ) PUNCT cana-2648 132 39 . . PUNCT cana-2648 133 1 every every DET cana-2648 133 2 open open NOUN cana-2648 133 3 is be AUX cana-2648 133 4 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 133 5 . . PUNCT cana-2648 134 1 but but CCONJ cana-2648 134 2 οΏ½ οΏ½ PROPN cana-2648 134 3 Μ‡ Μ‡ PROPN cana-2648 134 4 οΏ½ οΏ½ PROPN cana-2648 134 5 is be AUX cana-2648 134 6 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 134 7 , , PUNCT cana-2648 134 8 οΏ½ οΏ½ PROPN cana-2648 134 9 Μ‡ Μ‡ PROPN cana-2648 134 10 οΏ½ οΏ½ PROPN cana-2648 134 11 βˆ’1( βˆ’1( PROPN cana-2648 134 12 οΏ½ οΏ½ PROPN cana-2648 134 13 Μ‡ Μ‡ NOUN cana-2648 134 14 οΏ½ οΏ½ NOUN cana-2648 134 15 βˆ’1(𝐴′ βˆ’1(𝐴′ NOUN cana-2648 134 16 ) ) PUNCT cana-2648 134 17 ) ) PUNCT cana-2648 135 1 = = PRON cana-2648 135 2 ( ( PUNCT cana-2648 135 3 οΏ½ οΏ½ PROPN cana-2648 135 4 Μ‡ Μ‡ PROPN cana-2648 135 5 οΏ½ οΏ½ PROPN cana-2648 135 6 o o NOUN cana-2648 135 7 οΏ½ οΏ½ PROPN cana-2648 135 8 Μ‡ Μ‡ PROPN cana-2648 135 9 οΏ½ οΏ½ NOUN cana-2648 135 10 )βˆ’1(𝐴′ )βˆ’1(𝐴′ PROPN cana-2648 135 11 ) ) PUNCT cana-2648 135 12 is be AUX cana-2648 135 13 π‘”πœ‚-open π‘”πœ‚-open ADJ cana-2648 135 14 in in ADP cana-2648 135 15 ( ( PUNCT cana-2648 135 16 𝑋′ 𝑋′ ADJ cana-2648 135 17 , , PUNCT cana-2648 135 18 πœβ€² πœβ€² NUM cana-2648 135 19 ) ) PUNCT cana-2648 135 20 . . PUNCT cana-2648 136 1 hence hence ADV cana-2648 136 2 οΏ½ οΏ½ PROPN cana-2648 136 3 Μ‡ Μ‡ PROPN cana-2648 136 4 οΏ½ οΏ½ PROPN cana-2648 136 5 o o NOUN cana-2648 136 6 οΏ½ οΏ½ PROPN cana-2648 136 7 Μ‡ Μ‡ PROPN cana-2648 136 8 οΏ½ οΏ½ PROPN cana-2648 136 9 is be AUX cana-2648 136 10 π‘”πœ‚-continuous π‘”πœ‚-continuous ADJ cana-2648 136 11 . . PUNCT cana-2648 137 1 reference reference NOUN cana-2648 137 2 [ [ X cana-2648 137 3 1 1 NUM cana-2648 137 4 ] ] X cana-2648 137 5 o o X cana-2648 137 6 ravi ravi PROPN cana-2648 137 7 , , PUNCT cana-2648 137 8 a a DET cana-2648 137 9 senthil senthil NOUN cana-2648 137 10 kumar kumar PROPN cana-2648 137 11 r r PROPN cana-2648 137 12 & & CCONJ cana-2648 137 13 hamari hamari ADJ cana-2648 137 14 choudhiΜ‡. choudhiΜ‡. NOUN cana-2648 137 15 decompositions decomposition NOUN cana-2648 137 16 of of ADP cana-2648 137 17 Γ― Γ― NOUN cana-2648 137 18 g g NOUN cana-2648 137 19 - - PUNCT cana-2648 137 20 continuity continuity NOUN cana-2648 137 21 via via ADP cana-2648 137 22 idealization idealization NOUN cana-2648 137 23 . . PUNCT cana-2648 138 1 journal journal NOUN cana-2648 138 2 of of ADP cana-2648 138 3 new new ADJ cana-2648 138 4 results result NOUN cana-2648 138 5 in in ADP cana-2648 138 6 science science NOUN cana-2648 138 7 no no NOUN cana-2648 138 8 . . PROPN cana-2648 138 9 7 7 NUM cana-2648 138 10 , , PUNCT cana-2648 138 11 vol vol NOUN cana-2648 138 12 . . PUNCT cana-2648 139 1 3(2014 3(2014 NUM cana-2648 139 2 ) ) PUNCT cana-2648 140 1 ; ; PUNCT cana-2648 140 2 72 72 NUM cana-2648 140 3 - - SYM cana-2648 140 4 80 80 NUM cana-2648 140 5 . . PUNCT cana-2648 141 1 [ [ X cana-2648 141 2 2 2 X cana-2648 141 3 ] ] PUNCT cana-2648 141 4 s. s. PROPN cana-2648 141 5 tharmar tharmar PROPN cana-2648 141 6 and and CCONJ cana-2648 141 7 r. r. PROPN cana-2648 141 8 senthil senthil PROPN cana-2648 141 9 kumar kumar PROPN cana-2648 141 10 . . PROPN cana-2648 141 11 soft soft ADJ cana-2648 141 12 locally locally ADV cana-2648 141 13 closed close VERB cana-2648 141 14 sets set NOUN cana-2648 141 15 in in ADP cana-2648 141 16 soft soft ADJ cana-2648 141 17 ideal ideal ADJ cana-2648 141 18 topological topological ADJ cana-2648 141 19 spaces space NOUN cana-2648 141 20 . . PUNCT cana-2648 142 1 transylvanian transylvanian ADJ cana-2648 142 2 review review NOUN cana-2648 142 3 xxiv xxiv NUM cana-2648 142 4 : : PUNCT cana-2648 142 5 vol vol NOUN cana-2648 142 6 . . PUNCT cana-2648 143 1 10(2016 10(2016 NUM cana-2648 143 2 ) ) PUNCT cana-2648 143 3 , , PUNCT cana-2648 143 4 1593 1593 NUM cana-2648 143 5 - - SYM cana-2648 143 6 1600 1600 NUM cana-2648 143 7 [ [ X cana-2648 143 8 3 3 NUM cana-2648 143 9 ] ] PUNCT cana-2648 143 10 s. s. PROPN cana-2648 143 11 velammal velammal PROPN cana-2648 143 12 b.k.k b.k.k PROPN cana-2648 143 13 . . PUNCT cana-2648 144 1 priyatharsini priyatharsini PROPN cana-2648 144 2 , , PUNCT cana-2648 144 3 r.senthil r.senthil PROPN cana-2648 144 4 kumar kumar PROPN cana-2648 144 5 . . PUNCT cana-2648 145 1 new new ADJ cana-2648 145 2 footprints footprint NOUN cana-2648 145 3 of of ADP cana-2648 145 4 bondage bondage NOUN cana-2648 145 5 number number NOUN cana-2648 145 6 of of ADP cana-2648 145 7 connected connected ADJ cana-2648 145 8 unicyclic unicyclic ADJ cana-2648 145 9 and and CCONJ cana-2648 145 10 line line NOUN cana-2648 145 11 graphs graph NOUN cana-2648 145 12 . . PUNCT cana-2648 146 1 asia asia PROPN cana-2648 146 2 life life PROPN cana-2648 146 3 sciences sciences PROPN cana-2648 146 4 no.2 no.2 PROPN cana-2648 146 5 , , PUNCT cana-2648 146 6 vol.26(2017 vol.26(2017 PROPN cana-2648 146 7 ) ) PUNCT cana-2648 146 8 ; ; PUNCT cana-2648 146 9 321 321 NUM cana-2648 146 10 - - SYM cana-2648 146 11 326 326 NUM cana-2648 146 12 [ [ PUNCT cana-2648 146 13 4 4 NUM cana-2648 146 14 ] ] PUNCT cana-2648 146 15 k. k. PROPN cana-2648 146 16 prabhavathi prabhavathi PROPN cana-2648 146 17 , , PUNCT cana-2648 146 18 r. r. PROPN cana-2648 146 19 senthilkumar senthilkumar PROPN cana-2648 146 20 , , PUNCT cana-2648 147 1 i. i. PROPN cana-2648 147 2 athal athal ADV cana-2648 147 3 , , PUNCT cana-2648 147 4 m. m. NOUN cana-2648 147 5 karthivel karthivel PROPN cana-2648 147 6 . . PUNCT cana-2648 148 1 m m PROPN cana-2648 148 2 - - ADJ cana-2648 148 3 iΟ€g iΟ€g ADV cana-2648 148 4 - - PUNCT cana-2648 148 5 closed close VERB cana-2648 148 6 sets set NOUN cana-2648 148 7 and and CCONJ cana-2648 148 8 m m NOUN cana-2648 148 9 - - ADJ cana-2648 148 10 iΟ€g iΟ€g NOUN cana-2648 148 11 - - PUNCT cana-2648 148 12 continuity continuity NOUN cana-2648 148 13 . . PUNCT cana-2648 149 1 jour jour PROPN cana-2648 149 2 of of ADP cana-2648 149 3 adv adv PROPN cana-2648 149 4 research research NOUN cana-2648 149 5 in in ADP cana-2648 149 6 dynamical dynamical ADJ cana-2648 149 7 & & CCONJ cana-2648 149 8 control control PROPN cana-2648 149 9 systems systems PROPN cana-2648 149 10 vol vol NOUN cana-2648 149 11 . . PUNCT cana-2648 150 1 10 10 NUM cana-2648 150 2 no.4,(2018 no.4,(2018 NOUN cana-2648 150 3 ) ) PUNCT cana-2648 150 4 ; ; PUNCT cana-2648 150 5 112 112 NUM cana-2648 150 6 - - SYM cana-2648 150 7 118 118 NUM cana-2648 150 8 [ [ X cana-2648 150 9 5 5 NUM cana-2648 150 10 ] ] PUNCT cana-2648 150 11 k. k. PROPN cana-2648 150 12 prabhavathi prabhavathi PROPN cana-2648 150 13 , , PUNCT cana-2648 150 14 r. r. PROPN cana-2648 150 15 senthilkumar senthilkumar PROPN cana-2648 150 16 , , PUNCT cana-2648 150 17 i. i. PROPN cana-2648 150 18 athal athal ADV cana-2648 150 19 , , PUNCT cana-2648 150 20 m. m. NOUN cana-2648 150 21 karthivel karthivel PROPN cana-2648 150 22 . . PUNCT cana-2648 151 1 a a DET cana-2648 151 2 note note NOUN cana-2648 151 3 on on ADP cana-2648 151 4 iΞ² iΞ² ADP cana-2648 151 5 * * PUNCT cana-2648 151 6 g g PROPN cana-2648 151 7 closed closed ADJ cana-2648 151 8 sets set NOUN cana-2648 151 9 . . PUNCT cana-2648 152 1 jour jour X cana-2648 152 2 of of ADP cana-2648 152 3 adv adv PROPN cana-2648 152 4 research research NOUN cana-2648 152 5 in in ADP cana-2648 152 6 dynamical dynamical ADJ cana-2648 152 7 & & CCONJ cana-2648 152 8 control control PROPN cana-2648 152 9 systems system NOUN cana-2648 152 10 04 04 NUM cana-2648 152 11 - - PUNCT cana-2648 152 12 special special ADJ cana-2648 152 13 issue issue NOUN cana-2648 152 14 , , PUNCT cana-2648 152 15 vol.11(2019 vol.11(2019 NUM cana-2648 152 16 ) ) PUNCT cana-2648 152 17 ; ; PUNCT cana-2648 152 18 2495 2495 NUM cana-2648 152 19 - - SYM cana-2648 152 20 2502 2502 NUM cana-2648 152 21 [ [ X cana-2648 152 22 6 6 NUM cana-2648 152 23 ] ] X cana-2648 152 24 lavanya lavanya NOUN cana-2648 152 25 , , PUNCT cana-2648 152 26 s. s. PROPN cana-2648 152 27 moghana moghana PROPN cana-2648 152 28 and and CCONJ cana-2648 152 29 mahendran mahendran PROPN cana-2648 152 30 , , PUNCT cana-2648 152 31 k. k. PROPN cana-2648 152 32 and and CCONJ cana-2648 152 33 hemalatha hemalatha PROPN cana-2648 152 34 , , PUNCT cana-2648 152 35 s. s. PROPN cana-2648 152 36 and and CCONJ cana-2648 152 37 senthilkumar senthilkumar PROPN cana-2648 152 38 , , PUNCT cana-2648 152 39 r. r. PROPN cana-2648 152 40 ( ( PUNCT cana-2648 152 41 2019 2019 NUM cana-2648 152 42 ) ) PUNCT cana-2648 152 43 relationship relationship NOUN cana-2648 152 44 between between ADP cana-2648 152 45 service service NOUN cana-2648 152 46 quality quality NOUN cana-2648 152 47 , , PUNCT cana-2648 152 48 customer customer NOUN cana-2648 152 49 satisfaction satisfaction NOUN cana-2648 152 50 and and CCONJ cana-2648 152 51 customer customer NOUN cana-2648 152 52 loyalty loyalty NOUN cana-2648 152 53 in in ADP cana-2648 152 54 retail retail ADJ cana-2648 152 55 outlets outlet NOUN cana-2648 152 56 ; ; PUNCT cana-2648 152 57 a a DET cana-2648 152 58 sem sem NOUN cana-2648 152 59 pls pls NOUN cana-2648 152 60 approach approach NOUN cana-2648 152 61 . . PUNCT cana-2648 153 1 in in ADP cana-2648 153 2 : : PUNCT cana-2648 153 3 current current ADJ cana-2648 153 4 perspective perspective NOUN cana-2648 153 5 to to ADP cana-2648 153 6 economics economic NOUN cana-2648 153 7 and and CCONJ cana-2648 153 8 management management NOUN cana-2648 153 9 vol vol NOUN cana-2648 153 10 . . PUNCT cana-2648 154 1 3 3 NUM cana-2648 154 2 . . X cana-2648 155 1 b b X cana-2648 155 2 p p NOUN cana-2648 155 3 international international ADJ cana-2648 155 4 , , PUNCT cana-2648 155 5 pp pp ADJ cana-2648 155 6 . . PUNCT cana-2648 156 1 44 44 NUM cana-2648 156 2 - - SYM cana-2648 156 3 52 52 NUM cana-2648 156 4 . . PUNCT cana-2648 157 1 communications communication NOUN cana-2648 157 2 on on ADP cana-2648 157 3 applied apply VERB cana-2648 157 4 nonlinear nonlinear ADJ cana-2648 157 5 analysis analysis NOUN cana-2648 157 6 issn issn NOUN cana-2648 157 7 : : PUNCT cana-2648 157 8 1074 1074 NUM cana-2648 157 9 - - PUNCT cana-2648 157 10 133x 133x NUM cana-2648 157 11 vol vol NOUN cana-2648 157 12 32 32 NUM cana-2648 157 13 no no NOUN cana-2648 157 14 . . PUNCT cana-2648 158 1 3s 3s NUM cana-2648 158 2 ( ( PUNCT cana-2648 158 3 2025 2025 NUM cana-2648 158 4 ) ) PUNCT cana-2648 158 5 361 361 NUM cana-2648 158 6 https://internationalpubls.com https://internationalpubls.com X cana-2648 159 1 [ [ X cana-2648 159 2 7 7 X cana-2648 159 3 ] ] X cana-2648 159 4 k k X cana-2648 159 5 prabhavathi prabhavathi PROPN cana-2648 159 6 , , PUNCT cana-2648 159 7 k k PROPN cana-2648 159 8 nirmala nirmala PROPN cana-2648 159 9 , , PUNCT cana-2648 159 10 r r NOUN cana-2648 159 11 senthil senthil PROPN cana-2648 159 12 kumar kumar PROPN cana-2648 159 13 . . PUNCT cana-2648 160 1 weakly weakly ADJ cana-2648 160 2 ( ( PUNCT cana-2648 160 3 1 1 NUM cana-2648 160 4 , , PUNCT cana-2648 160 5 2)-cg 2)-cg NUM cana-2648 160 6 - - PUNCT cana-2648 160 7 closed closed ADJ cana-2648 160 8 sets set NOUN cana-2648 160 9 in in ADP cana-2648 160 10 biotopological biotopological ADJ cana-2648 160 11 spaces space NOUN cana-2648 160 12 . . PUNCT cana-2648 161 1 advances advance NOUN cana-2648 161 2 in in ADP cana-2648 161 3 mathematics mathematic NOUN cana-2648 161 4 : : PUNCT cana-2648 161 5 scientific scientific ADJ cana-2648 161 6 journal journal NOUN cana-2648 161 7 9 9 NUM cana-2648 161 8 issue issue NOUN cana-2648 161 9 11 11 NUM cana-2648 161 10 , , PUNCT cana-2648 161 11 vol.9(2020 vol.9(2020 VERB cana-2648 161 12 ) ) PUNCT cana-2648 161 13 ; ; PUNCT cana-2648 161 14 9341–9344 9341–9344 NUM cana-2648 161 15 . . PUNCT cana-2648 162 1 [ [ X cana-2648 162 2 8 8 NUM cana-2648 162 3 ] ] X cana-2648 162 4 dr.m.peer dr.m.peer PROPN cana-2648 162 5 mohamed mohamed PROPN cana-2648 162 6 & & CCONJ cana-2648 162 7 r r PROPN cana-2648 162 8 senthil senthil PROPN cana-2648 162 9 kumar kumar PROPN cana-2648 162 10 . . PROPN cana-2648 163 1 i_gn i_gn PROPN cana-2648 163 2 -closed -closed PROPN cana-2648 163 3 sets set NOUN cana-2648 163 4 and and CCONJ cana-2648 163 5 its its PRON cana-2648 163 6 properties property NOUN cana-2648 163 7 . . PUNCT cana-2648 164 1 international international ADJ cana-2648 164 2 journal journal NOUN cana-2648 164 3 of of ADP cana-2648 164 4 advanced advanced ADJ cana-2648 164 5 science science NOUN cana-2648 164 6 and and CCONJ cana-2648 164 7 technology technology NOUN cana-2648 164 8 no no NOUN cana-2648 164 9 . . PUNCT cana-2648 165 1 10s 10 NOUN cana-2648 165 2 vol vol NOUN cana-2648 165 3 . . PUNCT cana-2648 166 1 29(2020 29(2020 NUM cana-2648 166 2 ) ) PUNCT cana-2648 167 1 ; ; PUNCT cana-2648 167 2 9006 9006 NUM cana-2648 167 3 - - SYM cana-2648 167 4 9012 9012 NUM cana-2648 167 5 . . PUNCT cana-2648 168 1 [ [ X cana-2648 168 2 9 9 NUM cana-2648 168 3 ] ] PUNCT cana-2648 168 4 beer beer NOUN cana-2648 168 5 mohammed mohammed PROPN cana-2648 168 6 and and CCONJ cana-2648 168 7 r r PROPN cana-2648 168 8 senthil senthil PROPN cana-2648 168 9 kumar kumar PROPN cana-2648 168 10 s s PROPN cana-2648 168 11 krishnakumar krishnakumar PROPN cana-2648 168 12 . . PUNCT cana-2648 169 1 admission admission NOUN cana-2648 169 2 control control NOUN cana-2648 169 3 problem problem NOUN cana-2648 169 4 in in ADP cana-2648 169 5 a a DET cana-2648 169 6 service service NOUN cana-2648 169 7 facility facility NOUN cana-2648 169 8 with with ADP cana-2648 169 9 inventory inventory NOUN cana-2648 169 10 management management NOUN cana-2648 169 11 . . PUNCT cana-2648 170 1 international international ADJ cana-2648 170 2 journal journal PROPN cana-2648 170 3 of of ADP cana-2648 170 4 control control NOUN cana-2648 170 5 and and CCONJ cana-2648 170 6 automation automation NOUN cana-2648 170 7 vol vol NOUN cana-2648 170 8 . . PROPN cana-2648 170 9 13 13 NUM cana-2648 170 10 no.03(2020 no.03(2020 NUM cana-2648 170 11 ) ) PUNCT cana-2648 170 12 ; ; PUNCT cana-2648 170 13 388 388 NUM cana-2648 170 14 - - SYM cana-2648 170 15 396 396 NUM cana-2648 170 16 . . PUNCT cana-2648 171 1 [ [ X cana-2648 171 2 10 10 NUM cana-2648 171 3 ] ] X cana-2648 171 4 k k X cana-2648 171 5 prabhavathi prabhavathi PROPN cana-2648 171 6 , , PUNCT cana-2648 171 7 k k PROPN cana-2648 171 8 nirmala nirmala PROPN cana-2648 171 9 , , PUNCT cana-2648 171 10 p p NOUN cana-2648 171 11 balamurugan balamurugan VERB cana-2648 171 12 , , PUNCT cana-2648 171 13 r r NOUN cana-2648 171 14 senthil senthil PROPN cana-2648 171 15 kumar kumar PROPN cana-2648 171 16 . . PROPN cana-2648 171 17 approximate approximate ADJ cana-2648 171 18 soltuions soltuion NOUN cana-2648 171 19 of of ADP cana-2648 171 20 chemical chemical ADJ cana-2648 171 21 reactiondiffusion reactiondiffusion NOUN cana-2648 171 22 brusselator brusselator NOUN cana-2648 171 23 system system NOUN cana-2648 171 24 using use VERB cana-2648 171 25 new new ADJ cana-2648 171 26 iterative iterative NOUN cana-2648 171 27 method method NOUN cana-2648 171 28 . . PUNCT cana-2648 172 1 solid solid ADJ cana-2648 172 2 state state NOUN cana-2648 172 3 technology technology NOUN cana-2648 172 4 vol vol NOUN cana-2648 172 5 . . PUNCT cana-2648 173 1 63 63 NUM cana-2648 173 2 no no NOUN cana-2648 173 3 . . NOUN cana-2648 173 4 2 2 NUM cana-2648 173 5 ( ( PUNCT cana-2648 173 6 2020 2020 NUM cana-2648 173 7 ) ) PUNCT cana-2648 173 8 ; ; PUNCT cana-2648 174 1 695 695 NUM cana-2648 174 2 - - SYM cana-2648 174 3 701 701 NUM cana-2648 174 4 . . PUNCT cana-2648 175 1 [ [ X cana-2648 175 2 11 11 NUM cana-2648 175 3 ] ] X cana-2648 175 4 d d X cana-2648 175 5 little little ADJ cana-2648 175 6 femilin femilin PROPN cana-2648 175 7 jana jana PROPN cana-2648 175 8 , , PUNCT cana-2648 175 9 r r PROPN cana-2648 175 10 jaya jaya PROPN cana-2648 175 11 , , PUNCT cana-2648 175 12 m m PROPN cana-2648 175 13 arokia arokia NOUN cana-2648 175 14 ranjithkukar ranjithkukar NOUN cana-2648 175 15 , , PUNCT cana-2648 175 16 s s PROPN cana-2648 175 17 krishnakumar krishnakumar PROPN cana-2648 175 18 , , PUNCT cana-2648 175 19 r r PROPN cana-2648 175 20 senthil senthil PROPN cana-2648 175 21 kumar kumar PROPN cana-2648 175 22 . . PUNCT cana-2648 175 23 resolving resolve VERB cana-2648 175 24 sets set NOUN cana-2648 175 25 and and CCONJ cana-2648 175 26 dimension dimension NOUN cana-2648 175 27 in in ADP cana-2648 175 28 special special ADJ cana-2648 175 29 graphs graph NOUN cana-2648 175 30 . . PUNCT cana-2648 176 1 advances advance NOUN cana-2648 176 2 and and CCONJ cana-2648 176 3 application application NOUN cana-2648 176 4 of of ADP cana-2648 176 5 mathematical mathematical ADJ cana-2648 176 6 sciences sciences PROPN cana-2648 176 7 volume volume NOUN cana-2648 176 8 21 21 NUM cana-2648 176 9 , , PUNCT cana-2648 176 10 issue issue NOUN cana-2648 176 11 7(2022 7(2022 NUM cana-2648 176 12 ) ) PUNCT cana-2648 176 13 ; ; PUNCT cana-2648 176 14 3709 3709 NUM cana-2648 176 15 - - SYM cana-2648 176 16 3717 3717 NUM cana-2648 176 17 . . PUNCT cana-2648 177 1 [ [ X cana-2648 177 2 12 12 NUM cana-2648 177 3 ] ] X cana-2648 177 4 r r NOUN cana-2648 177 5 senthil senthil PROPN cana-2648 177 6 kumar kumar PROPN cana-2648 177 7 , , PUNCT cana-2648 177 8 rv rv PROPN cana-2648 177 9 shanmathi shanmathi PROPN cana-2648 177 10 , , PUNCT cana-2648 177 11 g g PROPN cana-2648 177 12 mageswaran mageswaran VERB cana-2648 177 13 , , PUNCT cana-2648 177 14 j j PROPN cana-2648 177 15 manikandan manikandan PROPN cana-2648 177 16 . . PUNCT cana-2648 178 1 power power NOUN cana-2648 178 2 flow flow NOUN cana-2648 178 3 analysis analysis NOUN cana-2648 178 4 of of ADP cana-2648 178 5 transient transient ADJ cana-2648 178 6 stability stability NOUN cana-2648 178 7 in in ADP cana-2648 178 8 microgrids microgrid NOUN cana-2648 178 9 used use VERB cana-2648 178 10 in in ADP cana-2648 178 11 power power NOUN cana-2648 178 12 stations station NOUN cana-2648 178 13 . . PUNCT cana-2648 179 1 2022 2022 NUM cana-2648 179 2 sixth sixth ADJ cana-2648 179 3 international international ADJ cana-2648 179 4 conference conference NOUN cana-2648 179 5 on on ADP cana-2648 179 6 i i PROPN cana-2648 179 7 - - PUNCT cana-2648 179 8 smac smac PROPN cana-2648 179 9 ( ( PUNCT cana-2648 179 10 iot iot PROPN cana-2648 179 11 in in ADP cana-2648 179 12 social social ADJ cana-2648 179 13 , , PUNCT cana-2648 179 14 mobile mobile ADJ cana-2648 179 15 , , PUNCT cana-2648 179 16 analytics analytic NOUN cana-2648 179 17 and and CCONJ cana-2648 179 18 cloud)(i cloud)(i NOUN cana-2648 179 19 - - PUNCT cana-2648 179 20 smac),(2022 smac),(2022 NOUN cana-2648 179 21 ) ) PUNCT cana-2648 179 22 ; ; PUNCT cana-2648 179 23 936 936 NUM cana-2648 179 24 - - SYM cana-2648 179 25 942 942 NUM cana-2648 179 26 . . PUNCT cana-2648 180 1 [ [ X cana-2648 180 2 13 13 NUM cana-2648 180 3 ] ] SYM cana-2648 180 4 y y PROPN cana-2648 180 5 rosemathy rosemathy NOUN cana-2648 180 6 , , PUNCT cana-2648 180 7 k k PROPN cana-2648 180 8 alli alli PROPN cana-2648 180 9 , , PUNCT cana-2648 180 10 t t PROPN cana-2648 180 11 thanigasalam thanigasalam PROPN cana-2648 180 12 , , PUNCT cana-2648 180 13 e e PROPN cana-2648 180 14 rajesh rajesh PROPN cana-2648 180 15 , , PUNCT cana-2648 180 16 r r PROPN cana-2648 180 17 senthil senthil PROPN cana-2648 180 18 kumar kumar PROPN cana-2648 180 19 . . PROPN cana-2648 181 1 on on ADP cana-2648 181 2 soft soft ADJ cana-2648 181 3 sigΞ΄s sigΞ΄s NOUN cana-2648 181 4 - - PUNCT cana-2648 181 5 closed close VERB cana-2648 181 6 sets set NOUN cana-2648 181 7 . . PUNCT cana-2648 182 1 e3s e3s PROPN cana-2648 182 2 web web NOUN cana-2648 182 3 of of ADP cana-2648 182 4 conferences conference NOUN cana-2648 182 5 376 376 NUM cana-2648 182 6 , , PUNCT cana-2648 182 7 01112 01112 NUM cana-2648 182 8 ( ( PUNCT cana-2648 182 9 2023 2023 NUM cana-2648 182 10 ) ) PUNCT cana-2648 182 11 . . PUNCT cana-2648 183 1 [ [ X cana-2648 183 2 14 14 NUM cana-2648 183 3 ] ] X cana-2648 183 4 r r NOUN cana-2648 183 5 senthil senthil PROPN cana-2648 183 6 kumar kumar PROPN cana-2648 183 7 , , PUNCT cana-2648 183 8 bvs bvs PROPN cana-2648 183 9 acharyulu acharyulu ADJ cana-2648 183 10 , , PUNCT cana-2648 183 11 pk pk NOUN cana-2648 183 12 dhal dhal NOUN cana-2648 183 13 , , PUNCT cana-2648 183 14 richa richa PROPN cana-2648 183 15 adlakha adlakha PROPN cana-2648 183 16 , , PUNCT cana-2648 183 17 sonu sonu PROPN cana-2648 183 18 kumar kumar PROPN cana-2648 183 19 , , PUNCT cana-2648 183 20 c c PROPN cana-2648 183 21 saravanan saravanan PROPN cana-2648 183 22 , , PUNCT cana-2648 183 23 krishna krishna PROPN cana-2648 183 24 bikram bikram PROPN cana-2648 183 25 shah shah PROPN cana-2648 183 26 . . PUNCT cana-2648 184 1 optimization optimization NOUN cana-2648 184 2 technique technique NOUN cana-2648 184 3 for for ADP cana-2648 184 4 renewable renewable ADJ cana-2648 184 5 energy energy NOUN cana-2648 184 6 storage storage NOUN cana-2648 184 7 systems system NOUN cana-2648 184 8 for for ADP cana-2648 184 9 power power NOUN cana-2648 184 10 quality quality NOUN cana-2648 184 11 analysis analysis NOUN cana-2648 184 12 with with ADP cana-2648 184 13 connected connected ADJ cana-2648 184 14 grid grid NOUN cana-2648 184 15 . . PUNCT cana-2648 185 1 international international ADJ cana-2648 185 2 transactions transaction NOUN cana-2648 185 3 on on ADP cana-2648 185 4 electrical electrical ADJ cana-2648 185 5 energy energy NOUN cana-2648 185 6 systems system NOUN cana-2648 185 7 volume volume NOUN cana-2648 185 8 2023 2023 NUM cana-2648 185 9 , , PUNCT cana-2648 185 10 article article NOUN cana-2648 185 11 i i PROPN cana-2648 185 12 d d PROPN cana-2648 185 13 4675421 4675421 NUM cana-2648 185 14 . . PUNCT cana-2648 186 1 [ [ X cana-2648 186 2 15 15 NUM cana-2648 186 3 ] ] X cana-2648 186 4 senthil senthil PROPN cana-2648 186 5 kumar kumar PROPN cana-2648 186 6 r4 r4 PROPN cana-2648 186 7 and and CCONJ cana-2648 186 8 tharmar tharmar PROPN cana-2648 186 9 s4 s4 PROPN cana-2648 186 10 rajeev rajeev PROPN cana-2648 186 11 gandhi gandhi PROPN cana-2648 186 12 s1 s1 PROPN cana-2648 186 13 , , PUNCT cana-2648 186 14 prabhavathi prabhavathi PROPN cana-2648 186 15 k2 k2 PROPN cana-2648 186 16 * * PROPN cana-2648 186 17 , , PUNCT cana-2648 186 18 veerasivaji veerasivaji PROPN cana-2648 186 19 r3 r3 PROPN cana-2648 186 20 . . PUNCT cana-2648 187 1 efficient efficient ADJ cana-2648 187 2 domination domination NOUN cana-2648 187 3 in in ADP cana-2648 187 4 fuzzy fuzzy ADJ cana-2648 187 5 graphs graph NOUN cana-2648 187 6 and and CCONJ cana-2648 187 7 intuitionistic intuitionistic ADJ cana-2648 187 8 fuzzy fuzzy ADJ cana-2648 187 9 graphs graph NOUN cana-2648 187 10 in in ADP cana-2648 187 11 strong strong ADJ cana-2648 187 12 and and CCONJ cana-2648 187 13 weak weak ADJ cana-2648 187 14 forms form NOUN cana-2648 187 15 . . PUNCT cana-2648 188 1 e3s e3s PROPN cana-2648 188 2 web web NOUN cana-2648 188 3 of of ADP cana-2648 188 4 conferences conference NOUN cana-2648 188 5 399 399 NUM cana-2648 188 6 , , PUNCT cana-2648 188 7 04026 04026 NUM cana-2648 188 8 ( ( PUNCT cana-2648 188 9 2023 2023 NUM cana-2648 188 10 ) ) PUNCT