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X iajs-2559 9 3 introduction introduction NOUN iajs-2559 9 4 graph graph NOUN iajs-2559 9 5 theory theory NOUN iajs-2559 9 6 [ [ X iajs-2559 9 7 1 1 NUM iajs-2559 9 8 ] ] PUNCT iajs-2559 9 9 is be AUX iajs-2559 9 10 a a DET iajs-2559 9 11 tool tool NOUN iajs-2559 9 12 for for ADP iajs-2559 9 13 optimization optimization NOUN iajs-2559 9 14 and and CCONJ iajs-2559 9 15 solving solve VERB iajs-2559 9 16 practical practical ADJ iajs-2559 9 17 application application NOUN iajs-2559 9 18 in in ADP iajs-2559 9 19 all all DET iajs-2559 9 20 fields field NOUN iajs-2559 9 21 such such ADJ iajs-2559 9 22 as as ADP iajs-2559 9 23 engineering engineering NOUN iajs-2559 9 24 study study NOUN iajs-2559 9 25 and and CCONJ iajs-2559 9 26 representation representation NOUN iajs-2559 9 27 of of ADP iajs-2559 9 28 economic economic ADJ iajs-2559 9 29 and and CCONJ iajs-2559 9 30 social social ADJ iajs-2559 9 31 networks network NOUN iajs-2559 9 32 , , PUNCT iajs-2559 9 33 complex complex ADJ iajs-2559 9 34 general general ADJ iajs-2559 9 35 systems system NOUN iajs-2559 9 36 , , PUNCT iajs-2559 9 37 information information NOUN iajs-2559 9 38 theory theory NOUN iajs-2559 9 39 and and CCONJ iajs-2559 9 40 others other NOUN iajs-2559 9 41 . . PUNCT iajs-2559 10 1 in in ADP iajs-2559 10 2 particular particular ADJ iajs-2559 10 3 , , PUNCT iajs-2559 10 4 graphs graph NOUN iajs-2559 10 5 are be AUX iajs-2559 10 6 one one NUM iajs-2559 10 7 of of ADP iajs-2559 10 8 the the DET iajs-2559 10 9 prime prime ADJ iajs-2559 10 10 objects object NOUN iajs-2559 10 11 of of ADP iajs-2559 10 12 study study NOUN iajs-2559 10 13 in in ADP iajs-2559 10 14 mathematics mathematic NOUN iajs-2559 10 15 . . 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PUNCT iajs-2559 31 1 for for ADP iajs-2559 31 2 each each DET iajs-2559 31 3 subgraph subgraph NOUN iajs-2559 31 4 𝑄 𝑄 PROPN iajs-2559 31 5 of of ADP iajs-2559 31 6 𝐷 𝐷 PROPN iajs-2559 31 7 , , PUNCT iajs-2559 31 8 we we PRON iajs-2559 31 9 will will AUX iajs-2559 31 10 define define VERB iajs-2559 31 11 operators operator NOUN iajs-2559 31 12 π·βˆ’ π·βˆ’ PROPN iajs-2559 31 13 and and CCONJ iajs-2559 31 14 𝐷+ 𝐷+ AUX iajs-2559 31 15 from from ADP iajs-2559 31 16 𝑃(𝑉(𝐷))to 𝑃(𝑉(𝐷))to VERB iajs-2559 31 17 itself itself PRON iajs-2559 31 18 as as ADP iajs-2559 31 19 the the DET iajs-2559 31 20 following following NOUN iajs-2559 31 21 : : PUNCT iajs-2559 31 22 π·βˆ’(𝑄 π·βˆ’(𝑄 NUM iajs-2559 31 23 ) ) PUNCT iajs-2559 32 1 = = PRON iajs-2559 32 2 { { PUNCT iajs-2559 32 3 ɍ ɍ PROPN iajs-2559 32 4 ∈ ∈ PROPN iajs-2559 32 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 32 6 ): ): PUNCT iajs-2559 32 7 if if SCONJ iajs-2559 32 8 ( ( PUNCT iajs-2559 32 9 ɍ ɍ ADJ iajs-2559 32 10 , , PUNCT iajs-2559 32 11 𝑒 𝑒 ADJ iajs-2559 32 12 ) ) PUNCT iajs-2559 32 13 ∈ ∈ PROPN iajs-2559 32 14 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 32 15 ) ) PUNCT iajs-2559 32 16 , , PUNCT iajs-2559 32 17 then then ADV iajs-2559 32 18 𝑒 𝑒 PROPN iajs-2559 32 19 ∈ ∈ PROPN iajs-2559 32 20 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 32 21 ) ) PUNCT iajs-2559 32 22 } } PUNCT iajs-2559 32 23 , , PUNCT iajs-2559 32 24 𝐷+(𝑄 𝐷+(𝑄 PROPN iajs-2559 32 25 ) ) PUNCT iajs-2559 32 26 = = PRON iajs-2559 32 27 { { PUNCT iajs-2559 32 28 ɍ ɍ PROPN iajs-2559 32 29 ∈ ∈ PROPN iajs-2559 32 30 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 32 31 ): ): PUNCT iajs-2559 32 32 there there PRON iajs-2559 32 33 exists exist VERB iajs-2559 32 34 𝑒 𝑒 PROPN iajs-2559 32 35 ∈ ∈ PROPN iajs-2559 32 36 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 32 37 ) ) PUNCT iajs-2559 32 38 such such ADJ iajs-2559 32 39 that that SCONJ iajs-2559 32 40 ( ( PUNCT iajs-2559 32 41 ɍ ɍ ADJ iajs-2559 32 42 , , PUNCT iajs-2559 32 43 𝑒 𝑒 ADJ iajs-2559 32 44 ) ) PUNCT iajs-2559 32 45 ∈ ∈ PROPN iajs-2559 32 46 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 32 47 ) ) PUNCT iajs-2559 32 48 } } PUNCT iajs-2559 32 49 . . PUNCT iajs-2559 33 1 π·βˆ’(𝑄 π·βˆ’(𝑄 NOUN iajs-2559 33 2 ) ) PUNCT iajs-2559 33 3 is be AUX iajs-2559 33 4 named name VERB iajs-2559 33 5 lower low ADJ iajs-2559 33 6 approximation approximation NOUN iajs-2559 33 7 of of ADP iajs-2559 33 8 𝑄 𝑄 PROPN iajs-2559 33 9 and and CCONJ iajs-2559 33 10 𝐷+(𝑄 𝐷+(𝑄 PROPN iajs-2559 33 11 ) ) PUNCT iajs-2559 33 12 is be AUX iajs-2559 33 13 named name VERB iajs-2559 33 14 upper upper ADJ iajs-2559 33 15 approximation approximation NOUN iajs-2559 33 16 of of ADP iajs-2559 33 17 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 33 18 ) ) PUNCT iajs-2559 33 19 . . PUNCT iajs-2559 34 1 the the DET iajs-2559 34 2 pair pair NOUN iajs-2559 34 3 ( ( PUNCT iajs-2559 34 4 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 34 5 ) ) PUNCT iajs-2559 34 6 , , PUNCT iajs-2559 34 7 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 34 8 ) ) PUNCT iajs-2559 34 9 ) ) PUNCT iajs-2559 34 10 is be AUX iajs-2559 34 11 named name VERB iajs-2559 34 12 generalized generalized ADJ iajs-2559 34 13 rough rough ADJ iajs-2559 34 14 graph graph NOUN iajs-2559 34 15 or or CCONJ iajs-2559 34 16 generalized generalized ADJ iajs-2559 34 17 approximation approximation NOUN iajs-2559 34 18 space space NOUN iajs-2559 34 19 . . PUNCT iajs-2559 35 1 𝑄 𝑄 PRON iajs-2559 35 2 is be AUX iajs-2559 35 3 named name VERB iajs-2559 35 4 generalized generalized ADJ iajs-2559 35 5 exact exact ADJ iajs-2559 35 6 graph graph NOUN iajs-2559 35 7 or or CCONJ iajs-2559 35 8 definable definable ADJ iajs-2559 35 9 graph graph NOUN iajs-2559 35 10 if if SCONJ iajs-2559 35 11 π·βˆ’(𝑄 π·βˆ’(𝑄 NOUN iajs-2559 35 12 ) ) PUNCT iajs-2559 36 1 = = NOUN iajs-2559 36 2 𝐷+(𝑄).while 𝐷+(𝑄).while ADP iajs-2559 36 3 𝑄 𝑄 PROPN iajs-2559 36 4 is be AUX iajs-2559 36 5 called call VERB iajs-2559 36 6 undefinable undefinable ADJ iajs-2559 36 7 graph graph NOUN iajs-2559 36 8 if if SCONJ iajs-2559 36 9 π·βˆ’(𝑄 π·βˆ’(𝑄 NOUN iajs-2559 36 10 ) ) PUNCT iajs-2559 36 11 β‰  β‰  PROPN iajs-2559 36 12 𝐷+(𝑄 𝐷+(𝑄 NOUN iajs-2559 36 13 ) ) PUNCT iajs-2559 36 14 . . PUNCT iajs-2559 37 1 if if SCONJ iajs-2559 37 2 𝐷 𝐷 PROPN iajs-2559 37 3 is be AUX iajs-2559 37 4 an an DET iajs-2559 37 5 equivalence equivalence NOUN iajs-2559 37 6 graph graph NOUN iajs-2559 37 7 , , PUNCT iajs-2559 37 8 a a DET iajs-2559 37 9 generalized generalized ADJ iajs-2559 37 10 rough rough ADJ iajs-2559 37 11 graph graph NOUN iajs-2559 37 12 ( ( PUNCT iajs-2559 37 13 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 37 14 ) ) PUNCT iajs-2559 37 15 , , PUNCT iajs-2559 37 16 𝐸(𝐷))means 𝐸(𝐷))mean VERB iajs-2559 37 17 the the DET iajs-2559 37 18 rough rough ADJ iajs-2559 37 19 graph graph NOUN iajs-2559 37 20 in in ADP iajs-2559 37 21 the the DET iajs-2559 37 22 pawlak pawlak NOUN iajs-2559 37 23 , , PUNCT iajs-2559 37 24 s s VERB iajs-2559 37 25 sense sense NOUN iajs-2559 37 26 . . PUNCT iajs-2559 38 1 definition definition NOUN iajs-2559 38 2 2.1 2.1 NUM iajs-2559 38 3 . . PUNCT iajs-2559 39 1 let let VERB iajs-2559 39 2 𝐷 𝐷 NOUN iajs-2559 39 3 = = SYM iajs-2559 39 4 ( ( PUNCT iajs-2559 39 5 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 39 6 ) ) PUNCT iajs-2559 39 7 , , PUNCT iajs-2559 39 8 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 39 9 ) ) PUNCT iajs-2559 39 10 ) ) PUNCT iajs-2559 40 1 be be AUX iajs-2559 40 2 a a DET iajs-2559 40 3 nonempty nonempty ADJ iajs-2559 40 4 graph graph NOUN iajs-2559 40 5 . . PUNCT iajs-2559 41 1 we we PRON iajs-2559 41 2 define define VERB iajs-2559 41 3 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 41 4 , , PUNCT iajs-2559 41 5 for for ADP iajs-2559 41 6 each each DET iajs-2559 41 7 𝑄 𝑄 PROPN iajs-2559 41 8 βŠ† βŠ† NUM iajs-2559 41 9 𝐷 𝐷 NOUN iajs-2559 41 10 by by ADP iajs-2559 41 11 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 41 12 = = SYM iajs-2559 41 13 { { PUNCT iajs-2559 41 14 𝑄 𝑄 PROPN iajs-2559 41 15 βŠ† βŠ† NUM iajs-2559 41 16 𝐷 𝐷 NOUN iajs-2559 41 17 : : PUNCT iajs-2559 41 18 π·βˆ’π‘„ π·βˆ’π‘„ PROPN iajs-2559 41 19 = = SYM iajs-2559 41 20 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 41 21 ) ) PUNCT iajs-2559 41 22 } } PUNCT iajs-2559 41 23 . . PUNCT iajs-2559 42 1 if if SCONJ iajs-2559 42 2 𝐷 𝐷 PROPN iajs-2559 42 3 is be AUX iajs-2559 42 4 a a DET iajs-2559 42 5 reflexive reflexive ADJ iajs-2559 42 6 graph graph NOUN iajs-2559 42 7 , , PUNCT iajs-2559 42 8 then then ADV iajs-2559 42 9 𝜏𝐷 𝜏𝐷 PROPN iajs-2559 42 10 constitutes constitute VERB iajs-2559 42 11 a a DET iajs-2559 42 12 topology topology NOUN iajs-2559 42 13 on on ADP iajs-2559 42 14 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 42 15 ) ) PUNCT iajs-2559 42 16 , , PUNCT iajs-2559 42 17 𝜏𝐷can 𝜏𝐷can PROPN iajs-2559 42 18 be be AUX iajs-2559 42 19 named name VERB iajs-2559 42 20 the the DET iajs-2559 42 21 topology topology NOUN iajs-2559 42 22 produced produce VERB iajs-2559 42 23 by by ADP iajs-2559 42 24 𝐷. 𝐷. PROPN iajs-2559 42 25 definition definition NOUN iajs-2559 42 26 2.2 2.2 NUM iajs-2559 42 27 . . PUNCT iajs-2559 43 1 if𝐷is if𝐷is VERB iajs-2559 43 2 a a DET iajs-2559 43 3 reflexive reflexive ADJ iajs-2559 43 4 graph graph NOUN iajs-2559 43 5 , , PUNCT iajs-2559 43 6 then then ADV iajs-2559 43 7 ( ( PUNCT iajs-2559 43 8 𝐷 𝐷 NOUN iajs-2559 43 9 , , PUNCT iajs-2559 43 10 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 43 11 ) ) PUNCT iajs-2559 43 12 is be AUX iajs-2559 43 13 named name VERB iajs-2559 43 14 the the DET iajs-2559 43 15 topological topological ADJ iajs-2559 43 16 space space NOUN iajs-2559 43 17 produced produce VERB iajs-2559 43 18 by by ADP iajs-2559 43 19 𝐷. 𝐷. PROPN iajs-2559 43 20 definition definition NOUN iajs-2559 43 21 2.3 2.3 NUM iajs-2559 43 22 . . PUNCT iajs-2559 44 1 let𝐷 let𝐷 INTJ iajs-2559 44 2 be be AUX iajs-2559 44 3 a a DET iajs-2559 44 4 graph graph NOUN iajs-2559 44 5 , , PUNCT iajs-2559 44 6 if if SCONJ iajs-2559 44 7 𝐷 𝐷 PROPN iajs-2559 44 8 is be AUX iajs-2559 44 9 both both PRON iajs-2559 44 10 reflexive reflexive ADJ iajs-2559 44 11 and and CCONJ iajs-2559 44 12 symmetric symmetric ADJ iajs-2559 44 13 graph graph NOUN iajs-2559 44 14 then then ADV iajs-2559 44 15 𝐷 𝐷 PROPN iajs-2559 44 16 is be AUX iajs-2559 44 17 called call VERB iajs-2559 44 18 tolerance tolerance NOUN iajs-2559 44 19 graph graph NOUN iajs-2559 44 20 . . PUNCT iajs-2559 45 1 definition definition NOUN iajs-2559 45 2 2.4 2.4 NUM iajs-2559 45 3 . . PUNCT iajs-2559 46 1 let let VERB iajs-2559 46 2 𝐷𝛼 𝐷𝛼 PROPN iajs-2559 46 3 and and CCONJ iajs-2559 46 4 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 46 5 be be AUX iajs-2559 46 6 two two NUM iajs-2559 46 7 graphs graph NOUN iajs-2559 46 8 on on ADP iajs-2559 46 9 𝑉(𝐷𝛼 𝑉(𝐷𝛼 NUM iajs-2559 46 10 ) ) PUNCT iajs-2559 46 11 = = PUNCT iajs-2559 46 12 𝑉(𝐷𝛽 𝑉(𝐷𝛽 NOUN iajs-2559 46 13 ) ) PUNCT iajs-2559 46 14 = = SYM iajs-2559 46 15 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 46 16 ) ) PUNCT iajs-2559 46 17 . . PUNCT iajs-2559 47 1 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 47 2 is be AUX iajs-2559 47 3 named name VERB iajs-2559 47 4 transmitting transmit VERB iajs-2559 47 5 expression expression NOUN iajs-2559 47 6 of of ADP iajs-2559 47 7 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 47 8 , , PUNCT iajs-2559 47 9 if if SCONJ iajs-2559 47 10 for for ADP iajs-2559 47 11 everyɍ everyɍ NOUN iajs-2559 47 12 , , PUNCT iajs-2559 47 13 𝑒 𝑒 PROPN iajs-2559 47 14 ∈ ∈ PROPN iajs-2559 47 15 𝑉(𝐷),(ɍ 𝑉(𝐷),(ɍ NOUN iajs-2559 47 16 , , PUNCT iajs-2559 47 17 𝑒 𝑒 X iajs-2559 47 18 ) ) PUNCT iajs-2559 47 19 ∈ ∈ NOUN iajs-2559 47 20 𝐸(𝐷𝛽)if 𝐸(𝐷𝛽)if NOUN iajs-2559 47 21 and and CCONJ iajs-2559 47 22 only only ADV iajs-2559 47 23 if if SCONJ iajs-2559 47 24 ( ( PUNCT iajs-2559 47 25 ɍ ɍ ADJ iajs-2559 47 26 , , PUNCT iajs-2559 47 27 𝑒 𝑒 X iajs-2559 47 28 ) ) PUNCT iajs-2559 47 29 ∈ ∈ NOUN iajs-2559 47 30 𝐸(𝐷𝛼 𝐸(𝐷𝛼 NOUN iajs-2559 47 31 ) ) PUNCT iajs-2559 47 32 or or CCONJ iajs-2559 47 33 there there PRON iajs-2559 47 34 exists exist VERB iajs-2559 47 35 { { PUNCT iajs-2559 47 36 𝑣1 𝑣1 PROPN iajs-2559 47 37 , , PUNCT iajs-2559 47 38 𝑣2 𝑣2 PROPN iajs-2559 47 39 , , PUNCT iajs-2559 47 40 𝑣3 𝑣3 ADJ iajs-2559 47 41 , , PUNCT iajs-2559 47 42 … … PUNCT iajs-2559 47 43 , , PUNCT iajs-2559 47 44 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 47 45 } } PUNCT iajs-2559 47 46 βŠ† βŠ† NUM iajs-2559 47 47 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 47 48 ) ) PUNCT iajs-2559 47 49 where(ɍ where(ɍ PROPN iajs-2559 47 50 , , PUNCT iajs-2559 47 51 𝑣1 𝑣1 PROPN iajs-2559 47 52 ) ) PUNCT iajs-2559 47 53 ∈ ∈ PROPN iajs-2559 47 54 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 47 55 ) ) PUNCT iajs-2559 47 56 , , PUNCT iajs-2559 47 57 ( ( PUNCT iajs-2559 47 58 𝑣1 𝑣1 PROPN iajs-2559 47 59 , , PUNCT iajs-2559 47 60 𝑣2 𝑣2 PROPN iajs-2559 47 61 ) ) PUNCT iajs-2559 47 62 ∈ ∈ PROPN iajs-2559 47 63 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 47 64 ) ) PUNCT iajs-2559 47 65 , , PUNCT iajs-2559 47 66 … … PUNCT iajs-2559 47 67 , , PUNCT iajs-2559 47 68 ( ( PUNCT iajs-2559 47 69 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 47 70 , , PUNCT iajs-2559 47 71 𝑒 𝑒 X iajs-2559 47 72 ) ) PUNCT iajs-2559 47 73 ∈ ∈ NOUN iajs-2559 47 74 𝐸(𝐷𝛼 𝐸(𝐷𝛼 NOUN iajs-2559 47 75 ) ) PUNCT iajs-2559 47 76 . . PUNCT iajs-2559 48 1 proposition proposition NOUN iajs-2559 48 2 2.5 2.5 NUM iajs-2559 48 3 . . PUNCT iajs-2559 49 1 let let VERB iajs-2559 49 2 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 49 3 be be AUX iajs-2559 49 4 a a DET iajs-2559 49 5 graph graph NOUN iajs-2559 49 6 and and CCONJ iajs-2559 49 7 𝐷𝛽 𝐷𝛽 VERB iajs-2559 49 8 the the DET iajs-2559 49 9 transmitting transmit VERB iajs-2559 49 10 expression expression NOUN iajs-2559 49 11 of of ADP iajs-2559 49 12 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 49 13 , , PUNCT iajs-2559 49 14 then then ADV iajs-2559 49 15 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 49 16 is be AUX iajs-2559 49 17 a a DET iajs-2559 49 18 transitive transitive ADJ iajs-2559 49 19 graph graph NOUN iajs-2559 49 20 , , PUNCT iajs-2559 49 21 furthermore furthermore ADV iajs-2559 49 22 , , PUNCT iajs-2559 49 23 ( ( PUNCT iajs-2559 49 24 1 1 X iajs-2559 49 25 ) ) PUNCT iajs-2559 49 26 if if SCONJ iajs-2559 49 27 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 49 28 is be AUX iajs-2559 49 29 reflexive reflexive ADJ iajs-2559 49 30 , , PUNCT iajs-2559 49 31 then then ADV iajs-2559 49 32 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 49 33 is be AUX iajs-2559 49 34 also also ADV iajs-2559 49 35 reflexive reflexive ADJ iajs-2559 49 36 , , PUNCT iajs-2559 49 37 ( ( PUNCT iajs-2559 49 38 2 2 X iajs-2559 49 39 ) ) PUNCT iajs-2559 49 40 if if SCONJ iajs-2559 49 41 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 49 42 is be AUX iajs-2559 49 43 symmetric symmetric ADJ iajs-2559 49 44 , , PUNCT iajs-2559 49 45 then then ADV iajs-2559 49 46 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 49 47 is be AUX iajs-2559 49 48 also also ADV iajs-2559 49 49 symmetric symmetric ADJ iajs-2559 49 50 , , PUNCT iajs-2559 49 51 ( ( PUNCT iajs-2559 49 52 3 3 X iajs-2559 49 53 ) ) PUNCT iajs-2559 49 54 if if SCONJ iajs-2559 49 55 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 49 56 is be AUX iajs-2559 49 57 transitive transitive ADJ iajs-2559 49 58 , , PUNCT iajs-2559 49 59 then then ADV iajs-2559 49 60 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 49 61 = = SYM iajs-2559 49 62 𝐷𝛼. 𝐷𝛼. PROPN iajs-2559 49 63 proof proof NOUN iajs-2559 49 64 . . PUNCT iajs-2559 50 1 ( ( PUNCT iajs-2559 50 2 1 1 X iajs-2559 50 3 ) ) PUNCT iajs-2559 50 4 let let VERB iajs-2559 50 5 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 50 6 is be AUX iajs-2559 50 7 reflexive reflexive ADJ iajs-2559 50 8 graph graph NOUN iajs-2559 50 9 , , PUNCT iajs-2559 50 10 then then ADV iajs-2559 50 11 for for ADP iajs-2559 50 12 each each DET iajs-2559 50 13 ɍ ɍ PROPN iajs-2559 50 14 ∈ ∈ PROPN iajs-2559 50 15 𝑉(𝐷),(ɍ 𝑉(𝐷),(ɍ NOUN iajs-2559 50 16 , , PUNCT iajs-2559 50 17 ɍ ɍ NOUN iajs-2559 50 18 ) ) PUNCT iajs-2559 50 19 ∈ ∈ NOUN iajs-2559 50 20 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 50 21 ) ) PUNCT iajs-2559 50 22 , , PUNCT iajs-2559 50 23 since since SCONJ iajs-2559 50 24 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 50 25 is be AUX iajs-2559 50 26 a a DET iajs-2559 50 27 transmitting transmit VERB iajs-2559 50 28 expression expression NOUN iajs-2559 50 29 of of ADP iajs-2559 50 30 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 50 31 , , PUNCT iajs-2559 50 32 then then ADV iajs-2559 50 33 ( ( PUNCT iajs-2559 50 34 ɍ ɍ ADJ iajs-2559 50 35 , , PUNCT iajs-2559 50 36 ɍ ɍ NOUN iajs-2559 50 37 ) ) PUNCT iajs-2559 50 38 ∈ ∈ PROPN iajs-2559 50 39 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 50 40 ) ) PUNCT iajs-2559 50 41 , , PUNCT iajs-2559 50 42 so so ADV iajs-2559 50 43 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 50 44 is be AUX iajs-2559 50 45 a a DET iajs-2559 50 46 reflexive reflexive ADJ iajs-2559 50 47 graph graph NOUN iajs-2559 50 48 . . PUNCT iajs-2559 51 1 107 107 NUM iajs-2559 51 2 ibn ibn PROPN iajs-2559 51 3 al al PROPN iajs-2559 51 4 - - PUNCT iajs-2559 51 5 haitham haitham PROPN iajs-2559 51 6 jour jour X iajs-2559 51 7 . . PROPN iajs-2559 51 8 for for ADP iajs-2559 51 9 pure pure ADJ iajs-2559 51 10 & & CCONJ iajs-2559 51 11 appl appl PROPN iajs-2559 51 12 . . PUNCT iajs-2559 52 1 sci sci PROPN iajs-2559 52 2 . . PROPN iajs-2559 53 1 34 34 NUM iajs-2559 53 2 ( ( PUNCT iajs-2559 53 3 1 1 NUM iajs-2559 53 4 ) ) PUNCT iajs-2559 53 5 2021 2021 NUM iajs-2559 53 6 ( ( PUNCT iajs-2559 53 7 2 2 X iajs-2559 53 8 ) ) PUNCT iajs-2559 53 9 let let VERB iajs-2559 53 10 ( ( PUNCT iajs-2559 53 11 ɍ ɍ ADJ iajs-2559 53 12 , , PUNCT iajs-2559 53 13 𝑒 𝑒 ADJ iajs-2559 53 14 ) ) PUNCT iajs-2559 53 15 ∈ ∈ PROPN iajs-2559 53 16 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 53 17 , , PUNCT iajs-2559 53 18 since since SCONJ iajs-2559 53 19 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 53 20 the the DET iajs-2559 53 21 transmitting transmit VERB iajs-2559 53 22 expression expression NOUN iajs-2559 53 23 of of ADP iajs-2559 53 24 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 53 25 , , PUNCT iajs-2559 53 26 then then ADV iajs-2559 53 27 ( ( PUNCT iajs-2559 53 28 ɍ ɍ X iajs-2559 53 29 , , PUNCT iajs-2559 53 30 𝑒 𝑒 ADJ iajs-2559 53 31 ) ) PUNCT iajs-2559 53 32 ∈ ∈ PROPN iajs-2559 53 33 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 53 34 or or CCONJ iajs-2559 53 35 there there ADV iajs-2559 53 36 exists exist VERB iajs-2559 53 37 { { PUNCT iajs-2559 53 38 𝑣1 𝑣1 PROPN iajs-2559 53 39 , , PUNCT iajs-2559 53 40 𝑣2 𝑣2 PROPN iajs-2559 53 41 , , PUNCT iajs-2559 53 42 𝑣3 𝑣3 ADJ iajs-2559 53 43 , , PUNCT iajs-2559 53 44 … … PUNCT iajs-2559 53 45 , , PUNCT iajs-2559 53 46 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 53 47 } } PUNCT iajs-2559 53 48 βŠ† βŠ† NUM iajs-2559 53 49 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 53 50 ) ) PUNCT iajs-2559 54 1 where where SCONJ iajs-2559 54 2 ( ( PUNCT iajs-2559 54 3 ɍ ɍ ADJ iajs-2559 54 4 , , PUNCT iajs-2559 54 5 𝑣1 𝑣1 NOUN iajs-2559 54 6 ) ) PUNCT iajs-2559 54 7 ∈ ∈ PROPN iajs-2559 54 8 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 9 , , PUNCT iajs-2559 54 10 ( ( PUNCT iajs-2559 54 11 𝑣1 𝑣1 PROPN iajs-2559 54 12 , , PUNCT iajs-2559 54 13 𝑣2 𝑣2 PROPN iajs-2559 54 14 ) ) PUNCT iajs-2559 54 15 ∈ ∈ PROPN iajs-2559 54 16 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 17 , , PUNCT iajs-2559 54 18 … … PUNCT iajs-2559 54 19 , , PUNCT iajs-2559 54 20 ( ( PUNCT iajs-2559 54 21 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 54 22 , , PUNCT iajs-2559 54 23 𝑒 𝑒 NOUN iajs-2559 54 24 ) ) PUNCT iajs-2559 54 25 ∈ ∈ NOUN iajs-2559 54 26 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 27 if if SCONJ iajs-2559 54 28 and and CCONJ iajs-2559 54 29 only only ADV iajs-2559 54 30 if if SCONJ iajs-2559 54 31 ( ( PUNCT iajs-2559 54 32 ɍ ɍ ADJ iajs-2559 54 33 , , PUNCT iajs-2559 54 34 𝑒 𝑒 ADJ iajs-2559 54 35 ) ) PUNCT iajs-2559 54 36 ∈ ∈ PROPN iajs-2559 54 37 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 54 38 , , PUNCT iajs-2559 54 39 because because SCONJ iajs-2559 54 40 𝐷𝛼 𝐷𝛼 PROPN iajs-2559 54 41 is be AUX iajs-2559 54 42 symmetric symmetric ADJ iajs-2559 54 43 , , PUNCT iajs-2559 54 44 so so CCONJ iajs-2559 54 45 ( ( PUNCT iajs-2559 54 46 𝑒 𝑒 X iajs-2559 54 47 , , PUNCT iajs-2559 54 48 ɍ ɍ NOUN iajs-2559 54 49 ) ) PUNCT iajs-2559 54 50 ∈ ∈ PROPN iajs-2559 54 51 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 52 or or CCONJ iajs-2559 54 53 there there ADV iajs-2559 54 54 exists exist VERB iajs-2559 54 55 { { PUNCT iajs-2559 54 56 𝑣1 𝑣1 PROPN iajs-2559 54 57 , , PUNCT iajs-2559 54 58 𝑣2 𝑣2 PROPN iajs-2559 54 59 , , PUNCT iajs-2559 54 60 𝑣3 𝑣3 ADJ iajs-2559 54 61 , , PUNCT iajs-2559 54 62 … … PUNCT iajs-2559 54 63 , , PUNCT iajs-2559 54 64 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 54 65 } } PUNCT iajs-2559 54 66 βŠ† βŠ† NUM iajs-2559 54 67 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 54 68 ) ) PUNCT iajs-2559 54 69 where,(𝑒 where,(𝑒 PROPN iajs-2559 54 70 , , PUNCT iajs-2559 54 71 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 54 72 ) ) PUNCT iajs-2559 54 73 ∈ ∈ PROPN iajs-2559 54 74 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 75 , , PUNCT iajs-2559 54 76 … … PUNCT iajs-2559 54 77 , , PUNCT iajs-2559 54 78 ( ( PUNCT iajs-2559 54 79 𝑣2 𝑣2 NOUN iajs-2559 54 80 , , PUNCT iajs-2559 54 81 𝑣1 𝑣1 PROPN iajs-2559 54 82 ) ) PUNCT iajs-2559 54 83 ∈ ∈ PROPN iajs-2559 54 84 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 85 , , PUNCT iajs-2559 54 86 ( ( PUNCT iajs-2559 54 87 𝑣1 𝑣1 NOUN iajs-2559 54 88 , , PUNCT iajs-2559 54 89 ɍ ɍ NOUN iajs-2559 54 90 ) ) PUNCT iajs-2559 54 91 ∈ ∈ NOUN iajs-2559 54 92 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 54 93 if if SCONJ iajs-2559 54 94 and and CCONJ iajs-2559 54 95 only only ADV iajs-2559 54 96 if if SCONJ iajs-2559 54 97 ( ( PUNCT iajs-2559 54 98 𝑒 𝑒 X iajs-2559 54 99 , , PUNCT iajs-2559 54 100 ɍ ɍ NOUN iajs-2559 54 101 ) ) PUNCT iajs-2559 54 102 ∈ ∈ PROPN iajs-2559 54 103 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 54 104 , , PUNCT iajs-2559 54 105 which which PRON iajs-2559 54 106 implies imply VERB iajs-2559 54 107 to to ADP iajs-2559 54 108 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 54 109 is be AUX iajs-2559 54 110 symmetric symmetric ADJ iajs-2559 54 111 . . PUNCT iajs-2559 55 1 ( ( PUNCT iajs-2559 55 2 3 3 X iajs-2559 55 3 ) ) PUNCT iajs-2559 55 4 let let VERB iajs-2559 55 5 ( ( PUNCT iajs-2559 55 6 ɍ ɍ ADJ iajs-2559 55 7 , , PUNCT iajs-2559 55 8 𝑒 𝑒 ADJ iajs-2559 55 9 ) ) PUNCT iajs-2559 55 10 ∈ ∈ PROPN iajs-2559 55 11 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 55 12 ) ) PUNCT iajs-2559 55 13 , , PUNCT iajs-2559 55 14 we we PRON iajs-2559 55 15 have have VERB iajs-2559 55 16 to to PART iajs-2559 55 17 show show VERB iajs-2559 55 18 that that SCONJ iajs-2559 55 19 ( ( PUNCT iajs-2559 55 20 ɍ ɍ ADJ iajs-2559 55 21 , , PUNCT iajs-2559 55 22 𝑒 𝑒 ADJ iajs-2559 55 23 ) ) PUNCT iajs-2559 55 24 ∈ ∈ NOUN iajs-2559 55 25 𝐸(𝐷𝛼).since 𝐸(𝐷𝛼).since NOUN iajs-2559 56 1 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 56 2 the the DET iajs-2559 56 3 transmitting transmit VERB iajs-2559 56 4 expression expression NOUN iajs-2559 56 5 of of ADP iajs-2559 56 6 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 56 7 , , PUNCT iajs-2559 56 8 then(ɍ then(ɍ NOUN iajs-2559 56 9 , , PUNCT iajs-2559 56 10 𝑒 𝑒 X iajs-2559 56 11 ) ) PUNCT iajs-2559 56 12 ∈ ∈ NOUN iajs-2559 56 13 𝐸(𝐷𝛽)if 𝐸(𝐷𝛽)if NOUN iajs-2559 56 14 and and CCONJ iajs-2559 56 15 only only ADV iajs-2559 56 16 if if SCONJ iajs-2559 56 17 ( ( PUNCT iajs-2559 56 18 ɍ ɍ ADJ iajs-2559 56 19 , , PUNCT iajs-2559 56 20 𝑒 𝑒 ADJ iajs-2559 56 21 ) ) PUNCT iajs-2559 56 22 ∈ ∈ PROPN iajs-2559 56 23 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 56 24 or or CCONJ iajs-2559 56 25 there there ADV iajs-2559 56 26 exists exist VERB iajs-2559 56 27 { { PUNCT iajs-2559 56 28 𝑣1 𝑣1 PROPN iajs-2559 56 29 , , PUNCT iajs-2559 56 30 𝑣2 𝑣2 PROPN iajs-2559 56 31 , , PUNCT iajs-2559 56 32 𝑣3 𝑣3 ADJ iajs-2559 56 33 , , PUNCT iajs-2559 56 34 … … PUNCT iajs-2559 56 35 , , PUNCT iajs-2559 56 36 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 56 37 } } PUNCT iajs-2559 56 38 βŠ† βŠ† NUM iajs-2559 56 39 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 56 40 ) ) PUNCT iajs-2559 56 41 where where SCONJ iajs-2559 56 42 ( ( PUNCT iajs-2559 56 43 ɍ ɍ ADJ iajs-2559 56 44 , , PUNCT iajs-2559 56 45 𝑣1 𝑣1 NOUN iajs-2559 56 46 ) ) PUNCT iajs-2559 56 47 ∈ ∈ PROPN iajs-2559 56 48 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 56 49 ) ) PUNCT iajs-2559 56 50 , , PUNCT iajs-2559 56 51 ( ( PUNCT iajs-2559 56 52 𝑣1 𝑣1 PROPN iajs-2559 56 53 , , PUNCT iajs-2559 56 54 𝑣2 𝑣2 PROPN iajs-2559 56 55 ) ) PUNCT iajs-2559 56 56 ∈ ∈ PROPN iajs-2559 56 57 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 56 58 ) ) PUNCT iajs-2559 56 59 , , PUNCT iajs-2559 56 60 ( ( PUNCT iajs-2559 56 61 𝑣2 𝑣2 NOUN iajs-2559 56 62 , , PUNCT iajs-2559 56 63 𝑣3 𝑣3 ADJ iajs-2559 56 64 ) ) PUNCT iajs-2559 56 65 ∈ ∈ NOUN iajs-2559 56 66 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 56 67 ) ) PUNCT iajs-2559 56 68 … … PUNCT iajs-2559 56 69 , , PUNCT iajs-2559 56 70 ( ( PUNCT iajs-2559 56 71 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 56 72 , , PUNCT iajs-2559 56 73 𝑒 𝑒 X iajs-2559 56 74 ) ) PUNCT iajs-2559 56 75 ∈ ∈ NOUN iajs-2559 56 76 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 56 77 ) ) PUNCT iajs-2559 56 78 . . PUNCT iajs-2559 57 1 ( ( PUNCT iajs-2559 57 2 i i NOUN iajs-2559 57 3 ) ) PUNCT iajs-2559 57 4 if if SCONJ iajs-2559 57 5 ( ( PUNCT iajs-2559 57 6 ɍ ɍ ADJ iajs-2559 57 7 , , PUNCT iajs-2559 57 8 𝑒 𝑒 X iajs-2559 57 9 ) ) PUNCT iajs-2559 57 10 ∈ ∈ PROPN iajs-2559 57 11 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 57 12 ) ) PUNCT iajs-2559 57 13 the the DET iajs-2559 57 14 prove prove NOUN iajs-2559 57 15 is be AUX iajs-2559 57 16 complete complete ADJ iajs-2559 57 17 . . PUNCT iajs-2559 58 1 ( ( PUNCT iajs-2559 58 2 ii ii NOUN iajs-2559 58 3 ) ) PUNCT iajs-2559 58 4 if if SCONJ iajs-2559 58 5 there there PRON iajs-2559 58 6 exists exist VERB iajs-2559 58 7 { { PUNCT iajs-2559 58 8 𝑣1 𝑣1 PROPN iajs-2559 58 9 , , PUNCT iajs-2559 58 10 𝑣2 𝑣2 PROPN iajs-2559 58 11 , , PUNCT iajs-2559 58 12 𝑣3 𝑣3 ADJ iajs-2559 58 13 , , PUNCT iajs-2559 58 14 … … PUNCT iajs-2559 58 15 , , PUNCT iajs-2559 58 16 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 58 17 } } PUNCT iajs-2559 58 18 βŠ† βŠ† NUM iajs-2559 58 19 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 58 20 ) ) PUNCT iajs-2559 58 21 where where SCONJ iajs-2559 58 22 ( ( PUNCT iajs-2559 58 23 ɍ ɍ ADJ iajs-2559 58 24 , , PUNCT iajs-2559 58 25 𝑣1 𝑣1 NOUN iajs-2559 58 26 ) ) PUNCT iajs-2559 58 27 ∈ ∈ PROPN iajs-2559 58 28 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 58 29 ) ) PUNCT iajs-2559 58 30 , , PUNCT iajs-2559 58 31 ( ( PUNCT iajs-2559 58 32 𝑣1 𝑣1 PROPN iajs-2559 58 33 , , PUNCT iajs-2559 58 34 𝑣2 𝑣2 PROPN iajs-2559 58 35 ) ) PUNCT iajs-2559 58 36 ∈ ∈ PROPN iajs-2559 58 37 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 58 38 ) ) PUNCT iajs-2559 58 39 , , PUNCT iajs-2559 58 40 ( ( PUNCT iajs-2559 58 41 𝑣2 𝑣2 NOUN iajs-2559 58 42 , , PUNCT iajs-2559 58 43 𝑣3 𝑣3 ADJ iajs-2559 58 44 ) ) PUNCT iajs-2559 58 45 ∈ ∈ NOUN iajs-2559 58 46 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 58 47 ) ) PUNCT iajs-2559 58 48 … … PUNCT iajs-2559 58 49 , , PUNCT iajs-2559 58 50 ( ( PUNCT iajs-2559 58 51 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 58 52 , , PUNCT iajs-2559 58 53 𝑒 𝑒 X iajs-2559 58 54 ) ) PUNCT iajs-2559 58 55 ∈ ∈ NOUN iajs-2559 58 56 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 58 57 ) ) PUNCT iajs-2559 58 58 , , PUNCT iajs-2559 58 59 and and CCONJ iajs-2559 58 60 we we PRON iajs-2559 58 61 have have VERB iajs-2559 58 62 that that DET iajs-2559 58 63 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 58 64 is be AUX iajs-2559 58 65 transitive transitive ADJ iajs-2559 58 66 , , PUNCT iajs-2559 58 67 so so CCONJ iajs-2559 58 68 ( ( PUNCT iajs-2559 58 69 ɍ ɍ ADJ iajs-2559 58 70 , , PUNCT iajs-2559 58 71 𝑒 𝑒 X iajs-2559 58 72 ) ) PUNCT iajs-2559 58 73 ∈ ∈ NOUN iajs-2559 58 74 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 58 75 ) ) PUNCT iajs-2559 58 76 , , PUNCT iajs-2559 58 77 which which PRON iajs-2559 58 78 means mean VERB iajs-2559 58 79 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 58 80 = = SYM iajs-2559 58 81 𝐷𝛼. 𝐷𝛼. PROPN iajs-2559 58 82 definition2.6 definition2.6 PROPN iajs-2559 58 83 . . PUNCT iajs-2559 59 1 ( ( PUNCT iajs-2559 59 2 1 1 X iajs-2559 59 3 ) ) PUNCT iajs-2559 59 4 let let AUX iajs-2559 59 5 ( ( PUNCT iajs-2559 59 6 𝐷 𝐷 NOUN iajs-2559 59 7 , , PUNCT iajs-2559 59 8 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 59 9 ) ) PUNCT iajs-2559 59 10 be be VERB iajs-2559 59 11 a a DET iajs-2559 59 12 topological topological ADJ iajs-2559 59 13 space space NOUN iajs-2559 59 14 and and CCONJ iajs-2559 59 15 ℬ𝐷 ℬ𝐷 VERB iajs-2559 59 16 a a DET iajs-2559 59 17 base base NOUN iajs-2559 59 18 of of ADP iajs-2559 59 19 ( ( PUNCT iajs-2559 59 20 𝐷 𝐷 NOUN iajs-2559 59 21 , , PUNCT iajs-2559 59 22 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 59 23 ) ) PUNCT iajs-2559 59 24 , , PUNCT iajs-2559 59 25 where where SCONJ iajs-2559 59 26 ( ( PUNCT iajs-2559 59 27 𝐷 𝐷 NOUN iajs-2559 59 28 , , PUNCT iajs-2559 59 29 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 59 30 ) ) PUNCT iajs-2559 59 31 is be AUX iajs-2559 59 32 induced induce VERB iajs-2559 59 33 by by ADP iajs-2559 59 34 a a DET iajs-2559 59 35 reflexive reflexive ADJ iajs-2559 59 36 graph graph NOUN iajs-2559 59 37 𝐷. 𝐷. NOUN iajs-2559 59 38 then then ADV iajs-2559 59 39 b b X iajs-2559 59 40 ∈ ∈ PROPN iajs-2559 59 41 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 59 42 is be AUX iajs-2559 59 43 called call VERB iajs-2559 59 44 maximal maximal ADJ iajs-2559 59 45 element element NOUN iajs-2559 59 46 of of ADP iajs-2559 59 47 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 59 48 if if SCONJ iajs-2559 59 49 does do AUX iajs-2559 59 50 not not PART iajs-2559 59 51 exist exist VERB iajs-2559 59 52 𝐡𝚀 𝐡𝚀 PROPN iajs-2559 59 53 ∈ ∈ PROPN iajs-2559 59 54 ℬ𝐷\{𝐡 ℬ𝐷\{𝐡 PROPN iajs-2559 59 55 } } PUNCT iajs-2559 59 56 such such ADJ iajs-2559 59 57 that that SCONJ iajs-2559 59 58 𝐡 𝐡 PROPN iajs-2559 59 59 βŠ† βŠ† PROPN iajs-2559 59 60 𝐡𝚀. 𝐡𝚀. PROPN iajs-2559 59 61 ( ( PUNCT iajs-2559 59 62 2)the 2)the DET iajs-2559 59 63 set set NOUN iajs-2559 59 64 of of ADP iajs-2559 59 65 all all DET iajs-2559 59 66 maximal maximal ADJ iajs-2559 59 67 elements element NOUN iajs-2559 59 68 of of ADP iajs-2559 59 69 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 59 70 symbolized symbolize VERB iajs-2559 59 71 by by ADP iajs-2559 59 72 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 59 73 βˆ— βˆ— NOUN iajs-2559 59 74 . . PUNCT iajs-2559 60 1 because because SCONJ iajs-2559 60 2 ⋃ℬ𝐷 ⋃ℬ𝐷 NOUN iajs-2559 60 3 = = SYM iajs-2559 60 4 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 60 5 ) ) PUNCT iajs-2559 60 6 , , PUNCT iajs-2559 60 7 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 60 8 βˆ— βˆ— NOUN iajs-2559 60 9 is be AUX iajs-2559 60 10 referred refer VERB iajs-2559 60 11 to to ADP iajs-2559 60 12 as as ADP iajs-2559 60 13 the the DET iajs-2559 60 14 minimal minimal ADJ iajs-2559 60 15 complete complete ADJ iajs-2559 60 16 cover cover NOUN iajs-2559 60 17 of of ADP iajs-2559 60 18 ( ( PUNCT iajs-2559 60 19 𝐷 𝐷 NOUN iajs-2559 60 20 , , PUNCT iajs-2559 60 21 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 60 22 ) ) PUNCT iajs-2559 60 23 according accord VERB iajs-2559 60 24 to to ADP iajs-2559 60 25 the the DET iajs-2559 60 26 base base NOUN iajs-2559 60 27 ℬ𝐷. ℬ𝐷. PUNCT iajs-2559 60 28 we we PRON iajs-2559 60 29 will will AUX iajs-2559 60 30 define define VERB iajs-2559 60 31 a a DET iajs-2559 60 32 pseudo pseudo NOUN iajs-2559 60 33 - - ADJ iajs-2559 60 34 metric metric ADJ iajs-2559 60 35 map map NOUN iajs-2559 60 36 on on ADP iajs-2559 60 37 graph graph NOUN iajs-2559 60 38 𝐷. 𝐷. NOUN iajs-2559 60 39 definition2.7 definition2.7 NOUN iajs-2559 60 40 . . PUNCT iajs-2559 61 1 let let VERB iajs-2559 61 2 𝐷 𝐷 NOUN iajs-2559 61 3 = = SYM iajs-2559 61 4 ( ( PUNCT iajs-2559 61 5 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 61 6 ) ) PUNCT iajs-2559 61 7 , , PUNCT iajs-2559 61 8 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 61 9 ) ) PUNCT iajs-2559 61 10 ) ) PUNCT iajs-2559 62 1 be be AUX iajs-2559 62 2 a a DET iajs-2559 62 3 nonempty nonempty ADJ iajs-2559 62 4 graph graph NOUN iajs-2559 62 5 , , PUNCT iajs-2559 62 6 then then ADV iajs-2559 62 7 𝑑 𝑑 NOUN iajs-2559 62 8 : : PUNCT iajs-2559 62 9 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 62 10 ) ) PUNCT iajs-2559 62 11 Γ— Γ— NOUN iajs-2559 62 12 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 62 13 ) ) PUNCT iajs-2559 62 14 ⟢ ⟢ NOUN iajs-2559 62 15 [ [ X iajs-2559 62 16 0 0 NUM iajs-2559 62 17 , , PUNCT iajs-2559 62 18 + + NOUN iajs-2559 62 19 ∞ ∞ NOUN iajs-2559 62 20 ) ) PUNCT iajs-2559 62 21 is be AUX iajs-2559 62 22 called call VERB iajs-2559 62 23 pseudo pseudo NOUN iajs-2559 62 24 - - ADJ iajs-2559 62 25 metric metric ADJ iajs-2559 62 26 map map NOUN iajs-2559 62 27 on on ADP iajs-2559 62 28 𝐷 𝐷 PROPN iajs-2559 62 29 , , PUNCT iajs-2559 62 30 if if SCONJ iajs-2559 62 31 for for ADP iajs-2559 62 32 all all DET iajs-2559 62 33 ɍ ɍ ADJ iajs-2559 62 34 , , PUNCT iajs-2559 62 35 𝑣 𝑣 NOUN iajs-2559 62 36 , , PUNCT iajs-2559 62 37 𝑒 𝑒 PROPN iajs-2559 62 38 ∈ ∈ PROPN iajs-2559 62 39 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 62 40 ) ) PUNCT iajs-2559 62 41 , , PUNCT iajs-2559 62 42 ( ( PUNCT iajs-2559 62 43 a a X iajs-2559 62 44 ) ) PUNCT iajs-2559 62 45 𝑑(ɍ 𝑑(ɍ ADJ iajs-2559 62 46 , , PUNCT iajs-2559 62 47 ɍ ɍ NOUN iajs-2559 62 48 ) ) PUNCT iajs-2559 62 49 = = SYM iajs-2559 62 50 0 0 NUM iajs-2559 62 51 , , PUNCT iajs-2559 62 52 ( ( PUNCT iajs-2559 62 53 b b NOUN iajs-2559 62 54 ) ) PUNCT iajs-2559 62 55 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 62 56 , , PUNCT iajs-2559 62 57 𝑣 𝑣 NOUN iajs-2559 62 58 ) ) PUNCT iajs-2559 62 59 = = SYM iajs-2559 62 60 𝑑(𝑣 𝑑(𝑣 NOUN iajs-2559 62 61 , , PUNCT iajs-2559 62 62 ɍ ɍ NOUN iajs-2559 62 63 ) ) PUNCT iajs-2559 62 64 , , PUNCT iajs-2559 62 65 ( ( PUNCT iajs-2559 62 66 c c X iajs-2559 62 67 ) ) PUNCT iajs-2559 62 68 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 62 69 , , PUNCT iajs-2559 62 70 𝑣 𝑣 NOUN iajs-2559 62 71 ) ) PUNCT iajs-2559 62 72 ≀ ≀ NOUN iajs-2559 62 73 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 62 74 , , PUNCT iajs-2559 62 75 𝑒 𝑒 NOUN iajs-2559 62 76 ) ) PUNCT iajs-2559 62 77 + + CCONJ iajs-2559 62 78 𝑑(𝑒 𝑑(𝑒 ADJ iajs-2559 62 79 , , PUNCT iajs-2559 62 80 𝑣 𝑣 NOUN iajs-2559 62 81 ) ) PUNCT iajs-2559 62 82 . . PUNCT iajs-2559 63 1 for for ADP iajs-2559 63 2 each each DET iajs-2559 63 3 ɍ ɍ PROPN iajs-2559 63 4 ∈ ∈ PROPN iajs-2559 63 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 63 6 ) ) PUNCT iajs-2559 63 7 , , PUNCT iajs-2559 63 8 𝑄 𝑄 PROPN iajs-2559 63 9 βŠ† βŠ† NUM iajs-2559 63 10 𝐷 𝐷 PROPN iajs-2559 63 11 , , PUNCT iajs-2559 63 12 πœ– πœ– X iajs-2559 63 13 > > X iajs-2559 63 14 0 0 NUM iajs-2559 63 15 , , PUNCT iajs-2559 63 16 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 63 17 , , PUNCT iajs-2559 63 18 πœ– πœ– X iajs-2559 63 19 ) ) PUNCT iajs-2559 64 1 = = SYM iajs-2559 64 2 { { PUNCT iajs-2559 64 3 𝑣 𝑣 PRON iajs-2559 64 4 ∈ ∈ PROPN iajs-2559 64 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 64 6 ): ): PUNCT iajs-2559 64 7 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 64 8 , , PUNCT iajs-2559 64 9 𝑣 𝑣 NOUN iajs-2559 64 10 ) ) PUNCT iajs-2559 64 11 < < X iajs-2559 64 12 πœ– πœ– X iajs-2559 64 13 } } PUNCT iajs-2559 64 14 , , PUNCT iajs-2559 64 15 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 64 16 , , PUNCT iajs-2559 64 17 𝑄 𝑄 PROPN iajs-2559 64 18 ) ) PUNCT iajs-2559 64 19 = = SYM iajs-2559 65 1 𝑖𝑛𝑓 𝑖𝑛𝑓 NOUN iajs-2559 65 2 { { PUNCT iajs-2559 65 3 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 65 4 , , PUNCT iajs-2559 65 5 𝑣 𝑣 NOUN iajs-2559 65 6 ): ): PUNCT iajs-2559 65 7 𝑣 𝑣 PRON iajs-2559 65 8 ∈ ∈ PROPN iajs-2559 65 9 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 65 10 ) ) PUNCT iajs-2559 65 11 } } PUNCT iajs-2559 65 12 . . PUNCT iajs-2559 66 1 if if SCONJ iajs-2559 66 2 there there PRON iajs-2559 66 3 exists exist VERB iajs-2559 66 4 pseudo pseudo ADJ iajs-2559 66 5 - - ADJ iajs-2559 66 6 metric metric ADJ iajs-2559 66 7 map𝑑on map𝑑on NOUN iajs-2559 66 8 𝐷where{𝐡(ɍ 𝐷where{𝐡(ɍ PROPN iajs-2559 66 9 , , PUNCT iajs-2559 66 10 πœ– πœ– PROPN iajs-2559 66 11 ): ): PUNCT iajs-2559 66 12 ɍ ɍ PROPN iajs-2559 66 13 ∈ ∈ PROPN iajs-2559 66 14 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 66 15 ) ) PUNCT iajs-2559 66 16 , , PUNCT iajs-2559 66 17 πœ– πœ– X iajs-2559 66 18 > > X iajs-2559 66 19 0 0 NUM iajs-2559 66 20 } } PUNCT iajs-2559 66 21 configures configure VERB iajs-2559 66 22 a a DET iajs-2559 66 23 base base NOUN iajs-2559 66 24 of of ADP iajs-2559 66 25 𝐷 𝐷 NOUN iajs-2559 66 26 , , PUNCT iajs-2559 66 27 then then ADV iajs-2559 66 28 a a DET iajs-2559 66 29 topological topological ADJ iajs-2559 66 30 space space NOUN iajs-2559 66 31 ( ( PUNCT iajs-2559 66 32 𝐷 𝐷 NOUN iajs-2559 66 33 , , PUNCT iajs-2559 66 34 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 66 35 ) ) PUNCT iajs-2559 66 36 is be AUX iajs-2559 66 37 referred refer VERB iajs-2559 66 38 to to ADP iajs-2559 66 39 as as ADP iajs-2559 66 40 pseudo pseudo NOUN iajs-2559 66 41 - - ADJ iajs-2559 66 42 metrizable metrizable ADJ iajs-2559 66 43 space space NOUN iajs-2559 66 44 . . PUNCT iajs-2559 67 1 proposition proposition NOUN iajs-2559 67 2 2.8 2.8 NUM iajs-2559 67 3 . . PUNCT iajs-2559 68 1 let let VERB iajs-2559 68 2 𝐷 𝐷 PROPN iajs-2559 68 3 be be AUX iajs-2559 68 4 pseudo pseudo NOUN iajs-2559 68 5 - - ADJ iajs-2559 68 6 metrizable metrizable ADJ iajs-2559 68 7 space space NOUN iajs-2559 68 8 . . PUNCT iajs-2559 69 1 if𝑄 if𝑄 PROPN iajs-2559 69 2 βŠ† βŠ† NUM iajs-2559 69 3 𝐷and 𝐷and PROPN iajs-2559 69 4 𝑑 𝑑 PROPN iajs-2559 69 5 is be AUX iajs-2559 69 6 pseudo pseudo NOUN iajs-2559 69 7 - - ADJ iajs-2559 69 8 metric metric ADJ iajs-2559 69 9 map map NOUN iajs-2559 69 10 on on ADP iajs-2559 69 11 𝐷 𝐷 PROPN iajs-2559 69 12 , , PUNCT iajs-2559 69 13 then then ADV iajs-2559 69 14 ɍ ɍ PROPN iajs-2559 69 15 ∈ ∈ PROPN iajs-2559 69 16 οΏ½ οΏ½ PROPN iajs-2559 69 17 Μ… Μ… NOUN iajs-2559 69 18 οΏ½ οΏ½ PROPN iajs-2559 69 19 if if SCONJ iajs-2559 69 20 and and CCONJ iajs-2559 69 21 only only ADV iajs-2559 69 22 if if SCONJ iajs-2559 69 23 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 69 24 , , PUNCT iajs-2559 69 25 𝑄 𝑄 PROPN iajs-2559 69 26 ) ) PUNCT iajs-2559 69 27 = = SYM iajs-2559 70 1 0 0 X iajs-2559 70 2 . . PUNCT iajs-2559 70 3 proof proof NOUN iajs-2559 70 4 . . PUNCT iajs-2559 71 1 ɍ ɍ PROPN iajs-2559 71 2 ∈ ∈ PROPN iajs-2559 71 3 οΏ½ οΏ½ PROPN iajs-2559 71 4 Μ… Μ… NOUN iajs-2559 71 5 οΏ½ οΏ½ PROPN iajs-2559 71 6 if if SCONJ iajs-2559 71 7 and and CCONJ iajs-2559 71 8 only only ADV iajs-2559 71 9 if if SCONJ iajs-2559 71 10 for for ADP iajs-2559 71 11 each each DET iajs-2559 71 12 πœ– πœ– X iajs-2559 71 13 > > X iajs-2559 71 14 0 0 NUM iajs-2559 71 15 , , PUNCT iajs-2559 71 16 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 71 17 , , PUNCT iajs-2559 71 18 πœ– πœ– NOUN iajs-2559 71 19 ) ) PUNCT iajs-2559 71 20 ∩ ∩ ADJ iajs-2559 71 21 𝑉(𝑄 𝑉(𝑄 NOUN iajs-2559 71 22 ) ) PUNCT iajs-2559 71 23 β‰  β‰  PROPN iajs-2559 71 24 βˆ… βˆ… NOUN iajs-2559 71 25 if if SCONJ iajs-2559 71 26 and and CCONJ iajs-2559 71 27 only only ADV iajs-2559 71 28 if if SCONJ iajs-2559 71 29 for for ADP iajs-2559 71 30 each each DET iajs-2559 71 31 πœ– πœ– X iajs-2559 71 32 > > X iajs-2559 71 33 0 0 NUM iajs-2559 71 34 , , PUNCT iajs-2559 71 35 there there PRON iajs-2559 71 36 exists exist VERB iajs-2559 71 37 𝑒 𝑒 PROPN iajs-2559 71 38 ∈ ∈ PROPN iajs-2559 71 39 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 71 40 ) ) PUNCT iajs-2559 71 41 such such ADJ iajs-2559 71 42 that that SCONJ iajs-2559 71 43 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 71 44 , , PUNCT iajs-2559 71 45 𝑒 𝑒 NOUN iajs-2559 71 46 ) ) PUNCT iajs-2559 71 47 < < X iajs-2559 71 48 πœ– πœ– X iajs-2559 72 1 if if SCONJ iajs-2559 72 2 and and CCONJ iajs-2559 72 3 only only ADV iajs-2559 72 4 if if SCONJ iajs-2559 72 5 𝑖𝑛𝑓 𝑖𝑛𝑓 PROPN iajs-2559 72 6 { { PUNCT iajs-2559 72 7 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 72 8 , , PUNCT iajs-2559 72 9 𝑒 𝑒 X iajs-2559 72 10 ): ): PUNCT iajs-2559 72 11 𝑒 𝑒 PROPN iajs-2559 72 12 ∈ ∈ PROPN iajs-2559 72 13 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 72 14 ) ) PUNCT iajs-2559 72 15 } } PUNCT iajs-2559 72 16 if if SCONJ iajs-2559 72 17 and and CCONJ iajs-2559 72 18 only only ADV iajs-2559 72 19 if if SCONJ iajs-2559 72 20 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 72 21 , , PUNCT iajs-2559 72 22 𝑄 𝑄 PROPN iajs-2559 72 23 ) ) PUNCT iajs-2559 72 24 = = SYM iajs-2559 73 1 0 0 X iajs-2559 73 2 . . NUM iajs-2559 73 3 108 108 NUM iajs-2559 73 4 ibn ibn PROPN iajs-2559 73 5 al al PROPN iajs-2559 73 6 - - PUNCT iajs-2559 73 7 haitham haitham PROPN iajs-2559 73 8 jour jour X iajs-2559 73 9 . . PROPN iajs-2559 74 1 for for ADP iajs-2559 74 2 pure pure ADJ iajs-2559 74 3 & & CCONJ iajs-2559 74 4 appl appl PROPN iajs-2559 74 5 . . PUNCT iajs-2559 75 1 sci sci PROPN iajs-2559 75 2 . . PROPN iajs-2559 76 1 34 34 NUM iajs-2559 76 2 ( ( PUNCT iajs-2559 76 3 1 1 NUM iajs-2559 76 4 ) ) PUNCT iajs-2559 76 5 2021 2021 NUM iajs-2559 76 6 definition definition NOUN iajs-2559 76 7 2.9 2.9 NUM iajs-2559 76 8 . . PUNCT iajs-2559 77 1 let let AUX iajs-2559 77 2 ( ( PUNCT iajs-2559 77 3 𝐷 𝐷 NOUN iajs-2559 77 4 , , PUNCT iajs-2559 77 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 77 6 ) ) PUNCT iajs-2559 77 7 be be VERB iajs-2559 77 8 a a DET iajs-2559 77 9 topological topological ADJ iajs-2559 77 10 space space NOUN iajs-2559 77 11 . . PUNCT iajs-2559 78 1 𝐷 𝐷 NOUN iajs-2559 78 2 is be AUX iajs-2559 78 3 named name VERB iajs-2559 78 4 a a DET iajs-2559 78 5 pseudo pseudo NOUN iajs-2559 78 6 - - ADJ iajs-2559 78 7 discrete discrete ADJ iajs-2559 78 8 space space NOUN iajs-2559 78 9 if if SCONJ iajs-2559 78 10 𝑄 𝑄 PROPN iajs-2559 78 11 βŠ† βŠ† NUM iajs-2559 78 12 𝐷 𝐷 NOUN iajs-2559 78 13 is be AUX iajs-2559 78 14 open open ADJ iajs-2559 78 15 in in ADP iajs-2559 78 16 𝐷 𝐷 PROPN iajs-2559 78 17 if if SCONJ iajs-2559 79 1 and and CCONJ iajs-2559 79 2 only only ADV iajs-2559 79 3 if if SCONJ iajs-2559 79 4 𝑄 𝑄 PRON iajs-2559 79 5 is be AUX iajs-2559 79 6 closed close VERB iajs-2559 79 7 in in ADP iajs-2559 79 8 𝐷. 𝐷. PROPN iajs-2559 79 9 3 3 NUM iajs-2559 79 10 . . PUNCT iajs-2559 80 1 the the DET iajs-2559 80 2 properties property NOUN iajs-2559 80 3 of of ADP iajs-2559 80 4 topological topological ADJ iajs-2559 80 5 spaces space NOUN iajs-2559 80 6 induced induce VERB iajs-2559 80 7 by by ADP iajs-2559 80 8 a a DET iajs-2559 80 9 reflexive reflexive ADJ iajs-2559 80 10 graph graph NOUN iajs-2559 80 11 we we PRON iajs-2559 80 12 will will AUX iajs-2559 80 13 study study VERB iajs-2559 80 14 through through ADP iajs-2559 80 15 this this DET iajs-2559 80 16 part part NOUN iajs-2559 80 17 , , PUNCT iajs-2559 80 18 the the DET iajs-2559 80 19 properties property NOUN iajs-2559 80 20 of of ADP iajs-2559 80 21 the the DET iajs-2559 80 22 topological topological ADJ iajs-2559 80 23 space space NOUN iajs-2559 80 24 ( ( PUNCT iajs-2559 80 25 𝐷 𝐷 NOUN iajs-2559 80 26 , , PUNCT iajs-2559 80 27 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 80 28 ) ) PUNCT iajs-2559 80 29 , , PUNCT iajs-2559 80 30 where where SCONJ iajs-2559 80 31 ( ( PUNCT iajs-2559 80 32 𝐷 𝐷 NOUN iajs-2559 80 33 , , PUNCT iajs-2559 80 34 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 80 35 ) ) PUNCT iajs-2559 80 36 is be AUX iajs-2559 80 37 produced produce VERB iajs-2559 80 38 by by ADP iajs-2559 80 39 a a DET iajs-2559 80 40 reflexive reflexive ADJ iajs-2559 80 41 graph graph NOUN iajs-2559 80 42 𝐷. 𝐷. NOUN iajs-2559 80 43 lemma3.1 lemma3.1 X iajs-2559 80 44 . . PUNCT iajs-2559 81 1 let let VERB iajs-2559 81 2 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 81 3 be be AUX iajs-2559 81 4 a a DET iajs-2559 81 5 reflexive reflexive ADJ iajs-2559 81 6 graph graph NOUN iajs-2559 81 7 and and CCONJ iajs-2559 81 8 𝐷𝛽 𝐷𝛽 VERB iajs-2559 81 9 the the DET iajs-2559 81 10 transmitting transmit VERB iajs-2559 81 11 expression expression NOUN iajs-2559 81 12 of of ADP iajs-2559 81 13 𝐷𝛼 𝐷𝛼 PROPN iajs-2559 81 14 , , PUNCT iajs-2559 81 15 for for ADP iajs-2559 81 16 every every DET iajs-2559 81 17 ɍ ɍ PROPN iajs-2559 81 18 ∈ ∈ PROPN iajs-2559 81 19 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 81 20 ) ) PUNCT iajs-2559 81 21 , , PUNCT iajs-2559 81 22 chose choose VERB iajs-2559 81 23 𝐿ɍ 𝐿ɍ NOUN iajs-2559 81 24 = = PUNCT iajs-2559 81 25 { { PUNCT iajs-2559 81 26 𝑣 𝑣 PRON iajs-2559 81 27 ∈ ∈ PROPN iajs-2559 81 28 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 81 29 ): ): PUNCT iajs-2559 81 30 ( ( PUNCT iajs-2559 81 31 ɍ ɍ X iajs-2559 81 32 , , PUNCT iajs-2559 81 33 𝑣 𝑣 NOUN iajs-2559 81 34 ) ) PUNCT iajs-2559 81 35 ∈ ∈ PROPN iajs-2559 81 36 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 81 37 ) ) PUNCT iajs-2559 81 38 } } PUNCT iajs-2559 81 39 , , PUNCT iajs-2559 81 40 then then ADV iajs-2559 81 41 ( ( PUNCT iajs-2559 81 42 1 1 X iajs-2559 81 43 ) ) PUNCT iajs-2559 81 44 𝐿ɍ 𝐿ɍ PROPN iajs-2559 81 45 ∈ ∈ NOUN iajs-2559 81 46 πœπ·π›Ό πœπ·π›Ό NOUN iajs-2559 81 47 , , PUNCT iajs-2559 81 48 ( ( PUNCT iajs-2559 81 49 2 2 NUM iajs-2559 81 50 ) ) PUNCT iajs-2559 81 51 { { PUNCT iajs-2559 81 52 𝐿ɍ 𝐿ɍ NOUN iajs-2559 81 53 } } PUNCT iajs-2559 81 54 is be AUX iajs-2559 81 55 an an DET iajs-2559 81 56 open open ADJ iajs-2559 81 57 neighborhood neighborhood NOUN iajs-2559 81 58 base base NOUN iajs-2559 81 59 of of ADP iajs-2559 81 60 ɍ ɍ NOUN iajs-2559 81 61 , , PUNCT iajs-2559 81 62 ( ( PUNCT iajs-2559 81 63 3 3 X iajs-2559 81 64 ) ) PUNCT iajs-2559 81 65 𝐿ɍ 𝐿ɍ PROPN iajs-2559 81 66 is be AUX iajs-2559 81 67 compact compact ADJ iajs-2559 81 68 subset subset NOUN iajs-2559 81 69 of of ADP iajs-2559 81 70 ( ( PUNCT iajs-2559 81 71 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 81 72 , , PUNCT iajs-2559 81 73 πœπ·π›Ό πœπ·π›Ό NOUN iajs-2559 81 74 ) ) PUNCT iajs-2559 81 75 , , PUNCT iajs-2559 81 76 ( ( PUNCT iajs-2559 81 77 4 4 X iajs-2559 81 78 ) ) PUNCT iajs-2559 81 79 ℬ𝐷𝛼 ℬ𝐷𝛼 NOUN iajs-2559 82 1 = = PUNCT iajs-2559 82 2 { { PUNCT iajs-2559 82 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 82 4 : : PUNCT iajs-2559 82 5 ɍ ɍ PROPN iajs-2559 82 6 ∈ ∈ PROPN iajs-2559 82 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 82 8 ) ) PUNCT iajs-2559 82 9 } } PUNCT iajs-2559 82 10 is be AUX iajs-2559 82 11 a a DET iajs-2559 82 12 base base NOUN iajs-2559 82 13 for for ADP iajs-2559 82 14 ( ( PUNCT iajs-2559 82 15 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 82 16 , , PUNCT iajs-2559 82 17 πœπ·π›Ό πœπ·π›Ό NOUN iajs-2559 82 18 ) ) PUNCT iajs-2559 82 19 . . PUNCT iajs-2559 83 1 proof proof NOUN iajs-2559 83 2 . . PUNCT iajs-2559 84 1 ( ( PUNCT iajs-2559 84 2 1 1 X iajs-2559 84 3 ) ) PUNCT iajs-2559 84 4 it it PRON iajs-2559 84 5 is be AUX iajs-2559 84 6 sufficient sufficient ADJ iajs-2559 84 7 to to PART iajs-2559 84 8 show show VERB iajs-2559 84 9 that𝐿ɍ that𝐿ɍ NOUN iajs-2559 84 10 βŠ† βŠ† NUM iajs-2559 84 11 𝐼𝑛𝑑(𝐿ɍ 𝐼𝑛𝑑(𝐿ɍ PROPN iajs-2559 84 12 ) ) PUNCT iajs-2559 84 13 . . PUNCT iajs-2559 85 1 let let VERB iajs-2559 85 2 𝑒 𝑒 PRON iajs-2559 85 3 ∈ ∈ PROPN iajs-2559 85 4 𝐿ɍ 𝐿ɍ PROPN iajs-2559 85 5 , , PUNCT iajs-2559 85 6 so so CCONJ iajs-2559 85 7 ( ( PUNCT iajs-2559 85 8 ɍ ɍ ADJ iajs-2559 85 9 , , PUNCT iajs-2559 85 10 𝑒 𝑒 ADJ iajs-2559 85 11 ) ) PUNCT iajs-2559 85 12 ∈ ∈ PROPN iajs-2559 85 13 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 85 14 ) ) PUNCT iajs-2559 85 15 , , PUNCT iajs-2559 85 16 then then ADV iajs-2559 85 17 there there PRON iajs-2559 85 18 exists exist VERB iajs-2559 85 19 { { PUNCT iajs-2559 85 20 𝑣1 𝑣1 PROPN iajs-2559 85 21 , , PUNCT iajs-2559 85 22 𝑣2 𝑣2 PROPN iajs-2559 85 23 , , PUNCT iajs-2559 85 24 𝑣3 𝑣3 ADJ iajs-2559 85 25 , , PUNCT iajs-2559 85 26 … … PUNCT iajs-2559 85 27 , , PUNCT iajs-2559 85 28 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 85 29 } } PUNCT iajs-2559 85 30 βŠ† βŠ† NUM iajs-2559 85 31 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 85 32 ) ) PUNCT iajs-2559 85 33 where where SCONJ iajs-2559 85 34 ( ( PUNCT iajs-2559 85 35 ɍ ɍ ADJ iajs-2559 85 36 , , PUNCT iajs-2559 85 37 𝑣1 𝑣1 NOUN iajs-2559 85 38 ) ) PUNCT iajs-2559 85 39 ∈ ∈ PROPN iajs-2559 85 40 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 85 41 ) ) PUNCT iajs-2559 85 42 , , PUNCT iajs-2559 85 43 ( ( PUNCT iajs-2559 85 44 𝑣1 𝑣1 PROPN iajs-2559 85 45 , , PUNCT iajs-2559 85 46 𝑣2 𝑣2 PROPN iajs-2559 85 47 ) ) PUNCT iajs-2559 85 48 ∈ ∈ PROPN iajs-2559 85 49 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 85 50 ) ) PUNCT iajs-2559 85 51 , , PUNCT iajs-2559 85 52 … … PUNCT iajs-2559 85 53 , , PUNCT iajs-2559 85 54 ( ( PUNCT iajs-2559 85 55 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 85 56 , , PUNCT iajs-2559 85 57 𝑒 𝑒 X iajs-2559 85 58 ) ) PUNCT iajs-2559 85 59 ∈ ∈ NOUN iajs-2559 85 60 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 85 61 ) ) PUNCT iajs-2559 85 62 , , PUNCT iajs-2559 85 63 so so SCONJ iajs-2559 85 64 𝑒 𝑒 X iajs-2559 85 65 ∈ ∈ PROPN iajs-2559 86 1 [ [ X iajs-2559 86 2 𝑣𝑛]𝐷𝛼 𝑣𝑛]𝐷𝛼 NOUN iajs-2559 86 3 . . PUNCT iajs-2559 87 1 for for ADP iajs-2559 87 2 𝑦 𝑦 NOUN iajs-2559 87 3 ∈ ∈ PROPN iajs-2559 87 4 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 87 5 ) ) PUNCT iajs-2559 87 6 such such ADJ iajs-2559 87 7 that that SCONJ iajs-2559 87 8 𝑦 𝑦 NOUN iajs-2559 87 9 ∈ ∈ PROPN iajs-2559 87 10 𝐿ɍ 𝐿ɍ PROPN iajs-2559 87 11 , , PUNCT iajs-2559 87 12 ( ( PUNCT iajs-2559 87 13 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 87 14 , , PUNCT iajs-2559 87 15 𝑦 𝑦 NOUN iajs-2559 87 16 ) ) PUNCT iajs-2559 87 17 ∈ ∈ NOUN iajs-2559 87 18 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 87 19 ) ) PUNCT iajs-2559 87 20 , , PUNCT iajs-2559 87 21 so(ɍ so(ɍ NOUN iajs-2559 87 22 , , PUNCT iajs-2559 87 23 𝑣1 𝑣1 PROPN iajs-2559 87 24 ) ) PUNCT iajs-2559 87 25 ∈ ∈ PROPN iajs-2559 87 26 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 87 27 ) ) PUNCT iajs-2559 87 28 , , PUNCT iajs-2559 87 29 ( ( PUNCT iajs-2559 87 30 𝑣1 𝑣1 PROPN iajs-2559 87 31 , , PUNCT iajs-2559 87 32 𝑣2 𝑣2 PROPN iajs-2559 87 33 ) ) PUNCT iajs-2559 87 34 ∈ ∈ PROPN iajs-2559 87 35 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 87 36 ) ) PUNCT iajs-2559 87 37 , , PUNCT iajs-2559 87 38 … … PUNCT iajs-2559 87 39 , , PUNCT iajs-2559 87 40 ( ( PUNCT iajs-2559 87 41 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 87 42 , , PUNCT iajs-2559 87 43 𝑦 𝑦 NOUN iajs-2559 87 44 ) ) PUNCT iajs-2559 87 45 ∈ ∈ NOUN iajs-2559 87 46 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 87 47 ) ) PUNCT iajs-2559 87 48 , , PUNCT iajs-2559 87 49 then then ADV iajs-2559 87 50 ( ( PUNCT iajs-2559 87 51 ɍ ɍ X iajs-2559 87 52 , , PUNCT iajs-2559 87 53 𝑦 𝑦 NOUN iajs-2559 87 54 ) ) PUNCT iajs-2559 87 55 ∈ ∈ PROPN iajs-2559 87 56 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 87 57 ) ) PUNCT iajs-2559 87 58 , , PUNCT iajs-2559 87 59 so so SCONJ iajs-2559 87 60 𝑦 𝑦 NOUN iajs-2559 87 61 ∈ ∈ NOUN iajs-2559 87 62 [ [ X iajs-2559 87 63 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 87 64 , , PUNCT iajs-2559 87 65 then then ADV iajs-2559 87 66 𝑦 𝑦 X iajs-2559 87 67 ∈ ∈ PROPN iajs-2559 87 68 𝐿ɍ 𝐿ɍ PROPN iajs-2559 87 69 , , PUNCT iajs-2559 87 70 [ [ X iajs-2559 87 71 𝑣𝑛]𝐷𝛼 𝑣𝑛]𝐷𝛼 NOUN iajs-2559 87 72 βŠ† βŠ† NUM iajs-2559 87 73 𝐿ɍ 𝐿ɍ NOUN iajs-2559 87 74 , , PUNCT iajs-2559 87 75 so𝑒 so𝑒 VERB iajs-2559 87 76 ∈ ∈ PROPN iajs-2559 87 77 𝐼𝑛𝑑(𝐿ɍ 𝐼𝑛𝑑(𝐿ɍ PROPN iajs-2559 87 78 ) ) PUNCT iajs-2559 87 79 , , PUNCT iajs-2559 87 80 which which PRON iajs-2559 87 81 implies imply VERB iajs-2559 87 82 to to ADP iajs-2559 87 83 𝐿ɍ 𝐿ɍ PROPN iajs-2559 87 84 βŠ† βŠ† NUM iajs-2559 87 85 𝐼𝑛𝑑(𝐿ɍ 𝐼𝑛𝑑(𝐿ɍ PROPN iajs-2559 87 86 ) ) PUNCT iajs-2559 87 87 . . PUNCT iajs-2559 88 1 hence hence ADV iajs-2559 88 2 , , PUNCT iajs-2559 88 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 88 4 ∈ ∈ NOUN iajs-2559 88 5 πœπ·π›Ό πœπ·π›Ό NOUN iajs-2559 88 6 . . PUNCT iajs-2559 89 1 ( ( PUNCT iajs-2559 89 2 2 2 X iajs-2559 89 3 ) ) PUNCT iajs-2559 89 4 let let VERB iajs-2559 89 5 𝐡 𝐡 NOUN iajs-2559 89 6 ∈ ∈ PROPN iajs-2559 89 7 𝜏𝐷 𝜏𝐷 VERB iajs-2559 89 8 such such ADJ iajs-2559 89 9 that that DET iajs-2559 89 10 ɍ ɍ PROPN iajs-2559 89 11 ∈ ∈ PROPN iajs-2559 89 12 𝐡 𝐡 PROPN iajs-2559 89 13 , , PUNCT iajs-2559 89 14 we we PRON iajs-2559 89 15 will will AUX iajs-2559 89 16 show show VERB iajs-2559 89 17 that that SCONJ iajs-2559 89 18 𝐿ɍ 𝐿ɍ DET iajs-2559 89 19 βŠ† βŠ† NUM iajs-2559 89 20 𝐡. 𝐡. NOUN iajs-2559 89 21 let let VERB iajs-2559 89 22 𝑒 𝑒 PRON iajs-2559 89 23 ∈ ∈ PROPN iajs-2559 89 24 𝐿ɍ 𝐿ɍ PROPN iajs-2559 89 25 , , PUNCT iajs-2559 89 26 then then ADV iajs-2559 89 27 ( ( PUNCT iajs-2559 89 28 ɍ ɍ X iajs-2559 89 29 , , PUNCT iajs-2559 89 30 𝑒 𝑒 X iajs-2559 89 31 ) ) PUNCT iajs-2559 89 32 ∈ ∈ NOUN iajs-2559 89 33 𝐸(𝐷𝛼 𝐸(𝐷𝛼 NOUN iajs-2559 89 34 ) ) PUNCT iajs-2559 89 35 or or CCONJ iajs-2559 89 36 there there PRON iajs-2559 89 37 exists exist VERB iajs-2559 89 38 { { PUNCT iajs-2559 89 39 𝑣1 𝑣1 PROPN iajs-2559 89 40 , , PUNCT iajs-2559 89 41 𝑣2 𝑣2 PROPN iajs-2559 89 42 , , PUNCT iajs-2559 89 43 𝑣3 𝑣3 ADJ iajs-2559 89 44 , , PUNCT iajs-2559 89 45 … … PUNCT iajs-2559 89 46 , , PUNCT iajs-2559 89 47 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 89 48 } } PUNCT iajs-2559 89 49 βŠ† βŠ† NUM iajs-2559 89 50 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 89 51 ) ) PUNCT iajs-2559 89 52 where where SCONJ iajs-2559 89 53 ( ( PUNCT iajs-2559 89 54 ɍ ɍ ADJ iajs-2559 89 55 , , PUNCT iajs-2559 89 56 𝑣1 𝑣1 NOUN iajs-2559 89 57 ) ) PUNCT iajs-2559 89 58 ∈ ∈ PROPN iajs-2559 89 59 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 89 60 ) ) PUNCT iajs-2559 89 61 , , PUNCT iajs-2559 89 62 ( ( PUNCT iajs-2559 89 63 𝑣1 𝑣1 PROPN iajs-2559 89 64 , , PUNCT iajs-2559 89 65 𝑣2 𝑣2 PROPN iajs-2559 89 66 ) ) PUNCT iajs-2559 89 67 ∈ ∈ PROPN iajs-2559 89 68 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 89 69 ) ) PUNCT iajs-2559 89 70 , , PUNCT iajs-2559 89 71 … … PUNCT iajs-2559 89 72 , , PUNCT iajs-2559 89 73 ( ( PUNCT iajs-2559 89 74 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 89 75 , , PUNCT iajs-2559 89 76 𝑒 𝑒 X iajs-2559 89 77 ) ) PUNCT iajs-2559 89 78 ∈ ∈ NOUN iajs-2559 89 79 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 89 80 ) ) PUNCT iajs-2559 89 81 . . PUNCT iajs-2559 90 1 ( ( PUNCT iajs-2559 90 2 i i NOUN iajs-2559 90 3 ) ) PUNCT iajs-2559 90 4 if if SCONJ iajs-2559 90 5 ( ( PUNCT iajs-2559 90 6 ɍ ɍ ADJ iajs-2559 90 7 , , PUNCT iajs-2559 90 8 𝑒 𝑒 X iajs-2559 90 9 ) ) PUNCT iajs-2559 90 10 ∈ ∈ NOUN iajs-2559 90 11 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 90 12 ) ) PUNCT iajs-2559 90 13 , , PUNCT iajs-2559 90 14 then then ADV iajs-2559 90 15 we we PRON iajs-2559 90 16 claim claim VERB iajs-2559 90 17 𝑒 𝑒 PRON iajs-2559 90 18 ∈ ∈ PROPN iajs-2559 90 19 𝐡. 𝐡. PROPN iajs-2559 90 20 for for ADP iajs-2559 90 21 otherwise otherwise ADV iajs-2559 90 22 , , PUNCT iajs-2559 91 1 𝑒 𝑒 PROPN iajs-2559 91 2 ∈ ∈ PROPN iajs-2559 91 3 𝐡𝑐 𝐡𝑐 PROPN iajs-2559 91 4 , , PUNCT iajs-2559 91 5 then then ADV iajs-2559 91 6 , , PUNCT iajs-2559 91 7 𝐿ɍ 𝐿ɍ PROPN iajs-2559 91 8 βŠ† βŠ† NUM iajs-2559 91 9 𝐡𝑐 𝐡𝑐 PROPN iajs-2559 91 10 , , PUNCT iajs-2559 91 11 but but CCONJ iajs-2559 91 12 𝐷𝛼 𝐷𝛼 NOUN iajs-2559 91 13 is be AUX iajs-2559 91 14 reflexive reflexive ADJ iajs-2559 91 15 , , PUNCT iajs-2559 91 16 so so ADV iajs-2559 91 17 ɍ ɍ PROPN iajs-2559 91 18 ∈ ∈ NOUN iajs-2559 91 19 𝐿ɍ 𝐿ɍ PROPN iajs-2559 91 20 and and CCONJ iajs-2559 91 21 𝐿ɍ 𝐿ɍ PRON iajs-2559 91 22 βŠ† βŠ† NUM iajs-2559 91 23 𝐡𝑐 𝐡𝑐 PROPN iajs-2559 91 24 , , PUNCT iajs-2559 91 25 then then ADV iajs-2559 91 26 ɍ ɍ PROPN iajs-2559 91 27 ∈ ∈ PROPN iajs-2559 91 28 𝐡𝑐 𝐡𝑐 PROPN iajs-2559 91 29 , , PUNCT iajs-2559 91 30 which which PRON iajs-2559 91 31 is be AUX iajs-2559 91 32 a a DET iajs-2559 91 33 contradiction contradiction NOUN iajs-2559 91 34 . . PUNCT iajs-2559 92 1 hence hence ADV iajs-2559 92 2 , , PUNCT iajs-2559 92 3 𝑒 𝑒 PROPN iajs-2559 92 4 ∈ ∈ PROPN iajs-2559 92 5 𝐡. 𝐡. PROPN iajs-2559 92 6 ( ( PUNCT iajs-2559 92 7 ii ii NOUN iajs-2559 92 8 ) ) PUNCT iajs-2559 92 9 if if SCONJ iajs-2559 92 10 there there PRON iajs-2559 92 11 exists exist VERB iajs-2559 92 12 { { PUNCT iajs-2559 92 13 𝑣1 𝑣1 PROPN iajs-2559 92 14 , , PUNCT iajs-2559 92 15 𝑣2 𝑣2 PROPN iajs-2559 92 16 , , PUNCT iajs-2559 92 17 𝑣3 𝑣3 ADJ iajs-2559 92 18 , , PUNCT iajs-2559 92 19 … … PUNCT iajs-2559 92 20 , , PUNCT iajs-2559 92 21 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 92 22 } } PUNCT iajs-2559 92 23 βŠ† βŠ† NUM iajs-2559 92 24 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 92 25 ) ) PUNCT iajs-2559 92 26 where where SCONJ iajs-2559 92 27 ( ( PUNCT iajs-2559 92 28 ɍ ɍ ADJ iajs-2559 92 29 , , PUNCT iajs-2559 92 30 𝑣1 𝑣1 NOUN iajs-2559 92 31 ) ) PUNCT iajs-2559 92 32 ∈ ∈ PROPN iajs-2559 92 33 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 92 34 ) ) PUNCT iajs-2559 92 35 , , PUNCT iajs-2559 92 36 ( ( PUNCT iajs-2559 92 37 𝑣1 𝑣1 PROPN iajs-2559 92 38 , , PUNCT iajs-2559 92 39 𝑣2 𝑣2 PROPN iajs-2559 92 40 ) ) PUNCT iajs-2559 92 41 ∈ ∈ PROPN iajs-2559 92 42 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 92 43 ) ) PUNCT iajs-2559 92 44 , , PUNCT iajs-2559 92 45 … … PUNCT iajs-2559 92 46 , , PUNCT iajs-2559 92 47 ( ( PUNCT iajs-2559 92 48 𝑣𝑛 𝑣𝑛 NOUN iajs-2559 92 49 , , PUNCT iajs-2559 92 50 𝑒 𝑒 X iajs-2559 92 51 ) ) PUNCT iajs-2559 92 52 ∈ ∈ NOUN iajs-2559 92 53 𝐸(𝐷𝛼 𝐸(𝐷𝛼 PROPN iajs-2559 92 54 ) ) PUNCT iajs-2559 92 55 , , PUNCT iajs-2559 92 56 then then ADV iajs-2559 92 57 by by ADP iajs-2559 92 58 ( ( PUNCT iajs-2559 92 59 i i NOUN iajs-2559 92 60 ) ) PUNCT iajs-2559 92 61 we we PRON iajs-2559 92 62 get get VERB iajs-2559 92 63 𝑣1 𝑣1 PROPN iajs-2559 92 64 ∈ ∈ PROPN iajs-2559 92 65 𝐡 𝐡 PROPN iajs-2559 92 66 , , PUNCT iajs-2559 92 67 𝑣2 𝑣2 PROPN iajs-2559 92 68 ∈ ∈ PROPN iajs-2559 92 69 𝐡 𝐡 PROPN iajs-2559 92 70 , , PUNCT iajs-2559 92 71 … … PUNCT iajs-2559 92 72 , , PUNCT iajs-2559 92 73 𝑒 𝑒 PROPN iajs-2559 92 74 ∈ ∈ PROPN iajs-2559 92 75 𝐡. 𝐡. NOUN iajs-2559 92 76 so so ADV iajs-2559 92 77 𝐿ɍ 𝐿ɍ ADP iajs-2559 92 78 βŠ† βŠ† NUM iajs-2559 92 79 𝐡 𝐡 PROPN iajs-2559 92 80 , , PUNCT iajs-2559 92 81 which which PRON iajs-2559 92 82 implies imply VERB iajs-2559 92 83 to to PART iajs-2559 92 84 { { PUNCT iajs-2559 92 85 𝐿ɍ}constitutes 𝐿ɍ}constitute VERB iajs-2559 92 86 an an DET iajs-2559 92 87 open open ADJ iajs-2559 92 88 neighborhood neighborhood NOUN iajs-2559 92 89 base base NOUN iajs-2559 92 90 of of ADP iajs-2559 92 91 ɍ ɍ NOUN iajs-2559 92 92 . . PUNCT iajs-2559 93 1 ( ( PUNCT iajs-2559 93 2 3 3 X iajs-2559 93 3 ) ) PUNCT iajs-2559 93 4 let let AUX iajs-2559 93 5 { { PUNCT iajs-2559 93 6 πΎπœ†|πœ† πΎπœ†|πœ† PUNCT iajs-2559 93 7 ∈ ∈ PROPN iajs-2559 93 8 Ξ» Ξ» PROPN iajs-2559 93 9 } } PUNCT iajs-2559 93 10 be be VERB iajs-2559 93 11 an an DET iajs-2559 93 12 open open ADJ iajs-2559 93 13 cover cover NOUN iajs-2559 93 14 of of ADP iajs-2559 93 15 𝐿ɍ 𝐿ɍ PRON iajs-2559 93 16 then then ADV iajs-2559 93 17 ɍ ɍ PROPN iajs-2559 93 18 ∈ ∈ PROPN iajs-2559 93 19 πΎπœ†π‘– πΎπœ†π‘– PROPN iajs-2559 93 20 for for ADP iajs-2559 93 21 some some DET iajs-2559 93 22 πœ†π‘– πœ†π‘– PROPN iajs-2559 93 23 ∈ ∈ PROPN iajs-2559 93 24 Ξ» Ξ» PROPN iajs-2559 93 25 , , PUNCT iajs-2559 93 26 then then ADV iajs-2559 93 27 by by ADP iajs-2559 93 28 ( ( PUNCT iajs-2559 93 29 2 2 X iajs-2559 93 30 ) ) PUNCT iajs-2559 93 31 𝐿ɍ 𝐿ɍ PROPN iajs-2559 93 32 βŠ† βŠ† NUM iajs-2559 93 33 πΎπœ†π‘– πΎπœ†π‘– NOUN iajs-2559 93 34 . . PUNCT iajs-2559 94 1 therefore therefore ADV iajs-2559 94 2 , , PUNCT iajs-2559 94 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 94 4 is be AUX iajs-2559 94 5 a a DET iajs-2559 94 6 compact compact ADJ iajs-2559 94 7 subset subset NOUN iajs-2559 94 8 of of ADP iajs-2559 94 9 ( ( PUNCT iajs-2559 94 10 𝐷 𝐷 PROPN iajs-2559 94 11 , , PUNCT iajs-2559 94 12 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 94 13 ) ) PUNCT iajs-2559 94 14 . . PUNCT iajs-2559 95 1 ( ( PUNCT iajs-2559 95 2 4 4 X iajs-2559 95 3 ) ) PUNCT iajs-2559 95 4 it it PRON iajs-2559 95 5 is be AUX iajs-2559 95 6 obvious obvious ADJ iajs-2559 95 7 by by ADP iajs-2559 95 8 ( ( PUNCT iajs-2559 95 9 2 2 NUM iajs-2559 95 10 ) ) PUNCT iajs-2559 95 11 remark remark NOUN iajs-2559 95 12 3.2 3.2 NUM iajs-2559 95 13 . . PUNCT iajs-2559 96 1 ( ( PUNCT iajs-2559 96 2 1 1 X iajs-2559 96 3 ) ) PUNCT iajs-2559 96 4 let let AUX iajs-2559 96 5 𝐷 𝐷 NOUN iajs-2559 96 6 = = SYM iajs-2559 96 7 ( ( PUNCT iajs-2559 96 8 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 96 9 ) ) PUNCT iajs-2559 96 10 , , PUNCT iajs-2559 96 11 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 96 12 ) ) PUNCT iajs-2559 96 13 ) ) PUNCT iajs-2559 96 14 be be AUX iajs-2559 96 15 a a DET iajs-2559 96 16 graph graph NOUN iajs-2559 96 17 , , PUNCT iajs-2559 96 18 for for ADP iajs-2559 96 19 each each DET iajs-2559 96 20 ɍ ɍ NOUN iajs-2559 96 21 , , PUNCT iajs-2559 96 22 𝑒 𝑒 PROPN iajs-2559 96 23 ∈ ∈ PROPN iajs-2559 96 24 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 96 25 ) ) PUNCT iajs-2559 96 26 , , PUNCT iajs-2559 96 27 if if SCONJ iajs-2559 96 28 ( ( PUNCT iajs-2559 96 29 ɍ ɍ ADJ iajs-2559 96 30 , , PUNCT iajs-2559 96 31 𝑒 𝑒 ADJ iajs-2559 96 32 ) ) PUNCT iajs-2559 96 33 ∈ ∈ PROPN iajs-2559 96 34 𝐸(𝐷)and(𝑒 𝐸(𝐷)and(𝑒 NOUN iajs-2559 96 35 , , PUNCT iajs-2559 96 36 ɍ ɍ NOUN iajs-2559 96 37 ) ) PUNCT iajs-2559 96 38 ∈ ∈ PROPN iajs-2559 96 39 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 96 40 ) ) PUNCT iajs-2559 96 41 , , PUNCT iajs-2559 96 42 then then ADV iajs-2559 96 43 𝐿ɍ 𝐿ɍ PROPN iajs-2559 96 44 = = SYM iajs-2559 96 45 𝐿𝑒. 𝐿𝑒. PROPN iajs-2559 96 46 ( ( PUNCT iajs-2559 96 47 2 2 NUM iajs-2559 96 48 ) ) PUNCT iajs-2559 96 49 for for ADP iajs-2559 96 50 all all DET iajs-2559 96 51 𝐡 𝐡 PROPN iajs-2559 96 52 ∈ ∈ PROPN iajs-2559 96 53 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 96 54 , , PUNCT iajs-2559 96 55 𝐡 𝐡 PROPN iajs-2559 96 56 can can AUX iajs-2559 96 57 not not PART iajs-2559 96 58 be be AUX iajs-2559 96 59 represented represent VERB iajs-2559 96 60 as as ADP iajs-2559 96 61 the the DET iajs-2559 96 62 union union NOUN iajs-2559 96 63 of of ADP iajs-2559 96 64 some some DET iajs-2559 96 65 elements element NOUN iajs-2559 96 66 of of ADP iajs-2559 96 67 ℬ𝐷\{𝐡 ℬ𝐷\{𝐡 PROPN iajs-2559 96 68 } } PUNCT iajs-2559 96 69 . . PUNCT iajs-2559 97 1 otherwise otherwise ADV iajs-2559 97 2 , , PUNCT iajs-2559 97 3 there there PRON iajs-2559 97 4 exists exist VERB iajs-2559 97 5 π’ͺ𝐷 π’ͺ𝐷 PROPN iajs-2559 97 6 βŠ† βŠ† NUM iajs-2559 97 7 ℬ𝐷\{𝐡 ℬ𝐷\{𝐡 PROPN iajs-2559 97 8 } } PUNCT iajs-2559 97 9 such such ADJ iajs-2559 97 10 that that SCONJ iajs-2559 97 11 𝐡 𝐡 NOUN iajs-2559 97 12 = = SYM iajs-2559 97 13 βˆͺ βˆͺ ADJ iajs-2559 97 14 π’ͺ𝐷. π’ͺ𝐷. X iajs-2559 97 15 by by ADP iajs-2559 97 16 𝐡 𝐡 PROPN iajs-2559 97 17 ∈ ∈ PROPN iajs-2559 97 18 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 97 19 , , PUNCT iajs-2559 97 20 there there PRON iajs-2559 97 21 exists exist VERB iajs-2559 97 22 ɍ ɍ PROPN iajs-2559 97 23 ∈ ∈ PROPN iajs-2559 97 24 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 97 25 ) ) PUNCT iajs-2559 97 26 where where SCONJ iajs-2559 97 27 𝐡 𝐡 NOUN iajs-2559 97 28 = = SYM iajs-2559 97 29 𝐿ɍ 𝐿ɍ PROPN iajs-2559 97 30 . . PUNCT iajs-2559 98 1 because because SCONJ iajs-2559 98 2 ɍ ɍ PROPN iajs-2559 98 3 ∈ ∈ PROPN iajs-2559 98 4 𝐡 𝐡 PROPN iajs-2559 98 5 , , PUNCT iajs-2559 98 6 there there PRON iajs-2559 98 7 exists exist VERB iajs-2559 98 8 𝑄 𝑄 PROPN iajs-2559 98 9 ∈ ∈ PROPN iajs-2559 98 10 π’ͺ𝐷 π’ͺ𝐷 PROPN iajs-2559 98 11 such such ADJ iajs-2559 98 12 that that SCONJ iajs-2559 98 13 ɍ ɍ PROPN iajs-2559 98 14 ∈ ∈ NOUN iajs-2559 98 15 𝑄 𝑄 PROPN iajs-2559 98 16 βŠ† βŠ† NUM iajs-2559 98 17 𝐡. 𝐡. PROPN iajs-2559 98 18 by by ADP iajs-2559 98 19 lemma lemma PROPN iajs-2559 98 20 3.1 3.1 NUM iajs-2559 98 21 , , PUNCT iajs-2559 98 22 𝐿ɍ 𝐿ɍ PRON iajs-2559 98 23 βŠ† βŠ† NUM iajs-2559 98 24 𝑄. 𝑄. NOUN iajs-2559 98 25 then then ADV iajs-2559 98 26 𝐡 𝐡 NOUN iajs-2559 98 27 = = SYM iajs-2559 98 28 𝑄 𝑄 PROPN iajs-2559 98 29 , , PUNCT iajs-2559 98 30 so so SCONJ iajs-2559 98 31 we we PRON iajs-2559 98 32 obtain obtain VERB iajs-2559 98 33 a a DET iajs-2559 98 34 contradiction contradiction NOUN iajs-2559 98 35 . . PUNCT iajs-2559 99 1 hence hence ADV iajs-2559 99 2 , , PUNCT iajs-2559 99 3 𝐻 𝐻 PROPN iajs-2559 99 4 can can AUX iajs-2559 99 5 not not PART iajs-2559 99 6 be be AUX iajs-2559 99 7 represented represent VERB iajs-2559 99 8 as as ADP iajs-2559 99 9 the the DET iajs-2559 99 10 union union NOUN iajs-2559 99 11 of of ADP iajs-2559 99 12 some some DET iajs-2559 99 13 elements element NOUN iajs-2559 99 14 of of ADP iajs-2559 99 15 ℬ𝐷\{𝐡 ℬ𝐷\{𝐡 PROPN iajs-2559 99 16 } } PUNCT iajs-2559 99 17 . . PUNCT iajs-2559 100 1 109 109 NUM iajs-2559 100 2 ibn ibn PROPN iajs-2559 100 3 al al PROPN iajs-2559 100 4 - - PUNCT iajs-2559 100 5 haitham haitham PROPN iajs-2559 100 6 jour jour X iajs-2559 100 7 . . PROPN iajs-2559 100 8 for for ADP iajs-2559 100 9 pure pure ADJ iajs-2559 100 10 & & CCONJ iajs-2559 100 11 appl appl PROPN iajs-2559 100 12 . . PUNCT iajs-2559 101 1 sci sci PROPN iajs-2559 101 2 . . PROPN iajs-2559 102 1 34 34 NUM iajs-2559 102 2 ( ( PUNCT iajs-2559 102 3 1 1 NUM iajs-2559 102 4 ) ) PUNCT iajs-2559 102 5 2021 2021 NUM iajs-2559 102 6 ( ( PUNCT iajs-2559 102 7 3 3 X iajs-2559 102 8 ) ) PUNCT iajs-2559 102 9 let let VERB iajs-2559 102 10 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 102 11 form form VERB iajs-2559 102 12 a a DET iajs-2559 102 13 base base NOUN iajs-2559 102 14 for for ADP iajs-2559 102 15 ( ( PUNCT iajs-2559 102 16 𝐷 𝐷 NOUN iajs-2559 102 17 , , PUNCT iajs-2559 102 18 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 102 19 ) ) PUNCT iajs-2559 102 20 . . PUNCT iajs-2559 103 1 then then ADV iajs-2559 103 2 ℬ𝐷 ℬ𝐷 VERB iajs-2559 103 3 βŠ† βŠ† NUM iajs-2559 103 4 ℋ𝐷. ℋ𝐷. NUM iajs-2559 103 5 otherwise otherwise ADV iajs-2559 103 6 , , PUNCT iajs-2559 103 7 there there PRON iajs-2559 103 8 exists exist VERB iajs-2559 103 9 𝐡 𝐡 PROPN iajs-2559 103 10 ∈ ∈ PROPN iajs-2559 103 11 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 103 12 but but CCONJ iajs-2559 103 13 𝐡 𝐡 PROPN iajs-2559 103 14 βˆ‰ βˆ‰ PROPN iajs-2559 103 15 ℋ𝐷. ℋ𝐷. X iajs-2559 103 16 notice notice VERB iajs-2559 103 17 that that SCONJ iajs-2559 103 18 𝐡 𝐡 PROPN iajs-2559 103 19 ∈ ∈ PROPN iajs-2559 103 20 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 103 21 , , PUNCT iajs-2559 103 22 there there PRON iajs-2559 103 23 exists exist VERB iajs-2559 103 24 ɍ ɍ PROPN iajs-2559 103 25 ∈ ∈ PROPN iajs-2559 103 26 𝐡 𝐡 NOUN iajs-2559 103 27 where where SCONJ iajs-2559 103 28 𝐡 𝐡 NOUN iajs-2559 103 29 = = SYM iajs-2559 103 30 𝐿ɍ 𝐿ɍ PROPN iajs-2559 103 31 . . PUNCT iajs-2559 104 1 because because SCONJ iajs-2559 104 2 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 104 3 is be AUX iajs-2559 104 4 a a DET iajs-2559 104 5 base base NOUN iajs-2559 104 6 for for ADP iajs-2559 104 7 ( ( PUNCT iajs-2559 104 8 𝐷 𝐷 NOUN iajs-2559 104 9 , , PUNCT iajs-2559 104 10 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 104 11 ) ) PUNCT iajs-2559 104 12 , , PUNCT iajs-2559 104 13 there there PRON iajs-2559 104 14 exists exist VERB iajs-2559 104 15 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 104 16 𝚀 𝚀 ADP iajs-2559 104 17 βŠ† βŠ† NUM iajs-2559 104 18 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 104 19 such such ADJ iajs-2559 104 20 that that DET iajs-2559 104 21 𝐡 𝐡 NOUN iajs-2559 104 22 = = PUNCT iajs-2559 104 23 βˆͺ βˆͺ VERB iajs-2559 104 24 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 104 25 𝚀 𝚀 X iajs-2559 104 26 . . PUNCT iajs-2559 105 1 thus thus ADV iajs-2559 105 2 ɍ ɍ X iajs-2559 105 3 ∈ ∈ NOUN iajs-2559 105 4 𝐻 𝐻 PROPN iajs-2559 105 5 βŠ† βŠ† NUM iajs-2559 105 6 𝐡 𝐡 NOUN iajs-2559 105 7 for for ADP iajs-2559 105 8 some some DET iajs-2559 105 9 𝐻 𝐻 PROPN iajs-2559 105 10 ∈ ∈ PROPN iajs-2559 105 11 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 105 12 𝚀 𝚀 X iajs-2559 105 13 . . PUNCT iajs-2559 106 1 by by ADP iajs-2559 106 2 using use VERB iajs-2559 106 3 lemma lemma PROPN iajs-2559 106 4 3.1 3.1 NUM iajs-2559 106 5 , , PUNCT iajs-2559 106 6 𝐡 𝐡 PROPN iajs-2559 106 7 βŠ† βŠ† NUM iajs-2559 106 8 𝐻. 𝐻. PROPN iajs-2559 106 9 so so ADV iajs-2559 106 10 𝐡 𝐡 NOUN iajs-2559 106 11 = = PUNCT iajs-2559 106 12 𝐻 𝐻 PROPN iajs-2559 106 13 ∈ ∈ PROPN iajs-2559 106 14 ℋ𝐷 ℋ𝐷 NOUN iajs-2559 106 15 and and CCONJ iajs-2559 106 16 that that PRON iajs-2559 106 17 means mean VERB iajs-2559 106 18 a a DET iajs-2559 106 19 contradiction contradiction NOUN iajs-2559 106 20 . . PUNCT iajs-2559 107 1 therefore therefore ADV iajs-2559 107 2 , , PUNCT iajs-2559 107 3 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 107 4 βŠ† βŠ† NUM iajs-2559 107 5 ℋ𝐷. ℋ𝐷. X iajs-2559 107 6 theorem theorem VERB iajs-2559 107 7 3.3.let 3.3.let NUM iajs-2559 107 8 ( ( PUNCT iajs-2559 107 9 𝐷 𝐷 PROPN iajs-2559 107 10 , , PUNCT iajs-2559 107 11 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 107 12 ) ) PUNCT iajs-2559 107 13 be be VERB iajs-2559 107 14 a a DET iajs-2559 107 15 topological topological ADJ iajs-2559 107 16 space space NOUN iajs-2559 107 17 generated generate VERB iajs-2559 107 18 by by ADP iajs-2559 107 19 a a DET iajs-2559 107 20 reflexive reflexive ADJ iajs-2559 107 21 graph graph NOUN iajs-2559 107 22 𝐷 𝐷 NOUN iajs-2559 107 23 , , PUNCT iajs-2559 107 24 then then ADV iajs-2559 107 25 ( ( PUNCT iajs-2559 107 26 1 1 X iajs-2559 107 27 ) ) PUNCT iajs-2559 107 28 ( ( PUNCT iajs-2559 107 29 𝐷 𝐷 NOUN iajs-2559 107 30 , , PUNCT iajs-2559 107 31 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 107 32 ) ) PUNCT iajs-2559 107 33 is be AUX iajs-2559 107 34 a a DET iajs-2559 107 35 first first ADJ iajs-2559 107 36 countable countable ADJ iajs-2559 107 37 space space NOUN iajs-2559 107 38 , , PUNCT iajs-2559 107 39 ( ( PUNCT iajs-2559 107 40 2 2 NUM iajs-2559 107 41 ) ) PUNCT iajs-2559 107 42 ( ( PUNCT iajs-2559 107 43 𝐷 𝐷 NOUN iajs-2559 107 44 , , PUNCT iajs-2559 107 45 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 107 46 ) ) PUNCT iajs-2559 107 47 is be AUX iajs-2559 107 48 a a DET iajs-2559 107 49 locally locally ADV iajs-2559 107 50 compact compact ADJ iajs-2559 107 51 space space NOUN iajs-2559 107 52 , , PUNCT iajs-2559 107 53 ( ( PUNCT iajs-2559 107 54 3 3 X iajs-2559 107 55 ) ) PUNCT iajs-2559 107 56 if if SCONJ iajs-2559 107 57 𝐷 𝐷 PROPN iajs-2559 107 58 is be AUX iajs-2559 107 59 countable countable ADJ iajs-2559 107 60 , , PUNCT iajs-2559 107 61 then then ADV iajs-2559 107 62 ( ( PUNCT iajs-2559 107 63 𝐷 𝐷 NOUN iajs-2559 107 64 , , PUNCT iajs-2559 107 65 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 107 66 ) ) PUNCT iajs-2559 107 67 is be AUX iajs-2559 107 68 second second ADJ iajs-2559 107 69 countable countable ADJ iajs-2559 107 70 space space NOUN iajs-2559 107 71 . . PUNCT iajs-2559 108 1 proof proof NOUN iajs-2559 108 2 . . PUNCT iajs-2559 109 1 ( ( PUNCT iajs-2559 109 2 1 1 X iajs-2559 109 3 ) ) PUNCT iajs-2559 109 4 by by ADP iajs-2559 109 5 lemma lemma PROPN iajs-2559 109 6 3.1(2 3.1(2 NUM iajs-2559 109 7 ) ) PUNCT iajs-2559 109 8 { { PUNCT iajs-2559 109 9 𝐿ɍ 𝐿ɍ NOUN iajs-2559 109 10 } } PUNCT iajs-2559 109 11 is be AUX iajs-2559 109 12 an an DET iajs-2559 109 13 open open ADJ iajs-2559 109 14 neighborhood neighborhood NOUN iajs-2559 109 15 base base NOUN iajs-2559 109 16 of of ADP iajs-2559 109 17 ɍ ɍ NOUN iajs-2559 109 18 , , PUNCT iajs-2559 109 19 then then ADV iajs-2559 109 20 ( ( PUNCT iajs-2559 109 21 𝐷 𝐷 NOUN iajs-2559 109 22 , , PUNCT iajs-2559 109 23 𝜏𝐷)is 𝜏𝐷)is ADJ iajs-2559 109 24 a a DET iajs-2559 109 25 first first ADJ iajs-2559 109 26 countable countable ADJ iajs-2559 109 27 space space NOUN iajs-2559 109 28 . . PUNCT iajs-2559 110 1 ( ( PUNCT iajs-2559 110 2 2 2 X iajs-2559 110 3 ) ) PUNCT iajs-2559 110 4 by by ADP iajs-2559 110 5 lemma lemma PROPN iajs-2559 110 6 3.1(3 3.1(3 NUM iajs-2559 110 7 ) ) PUNCT iajs-2559 110 8 , , PUNCT iajs-2559 110 9 we we PRON iajs-2559 110 10 have have VERB iajs-2559 110 11 for for ADP iajs-2559 110 12 each each DET iajs-2559 110 13 ɍ ɍ PROPN iajs-2559 110 14 ∈ ∈ PROPN iajs-2559 110 15 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 110 16 ) ) PUNCT iajs-2559 110 17 , , PUNCT iajs-2559 110 18 ɍ ɍ PRON iajs-2559 110 19 has have VERB iajs-2559 110 20 compact compact ADJ iajs-2559 110 21 neighborhood neighborhood NOUN iajs-2559 110 22 . . PUNCT iajs-2559 111 1 hence hence ADV iajs-2559 111 2 ( ( PUNCT iajs-2559 111 3 𝐷 𝐷 NOUN iajs-2559 111 4 , , PUNCT iajs-2559 111 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 111 6 ) ) PUNCT iajs-2559 111 7 is be AUX iajs-2559 111 8 locally locally ADV iajs-2559 111 9 compact compact ADJ iajs-2559 111 10 space space NOUN iajs-2559 111 11 . . PUNCT iajs-2559 112 1 ( ( PUNCT iajs-2559 112 2 3 3 X iajs-2559 112 3 ) ) PUNCT iajs-2559 112 4 by by ADP iajs-2559 112 5 lemma lemma PROPN iajs-2559 112 6 3.1(4),ℬ𝐷 3.1(4),ℬ𝐷 PROPN iajs-2559 112 7 = = SYM iajs-2559 112 8 { { PUNCT iajs-2559 112 9 𝐿ɍ 𝐿ɍ PROPN iajs-2559 112 10 : : PUNCT iajs-2559 112 11 ɍ ɍ PROPN iajs-2559 112 12 ∈ ∈ PROPN iajs-2559 112 13 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 112 14 ) ) PUNCT iajs-2559 112 15 } } PUNCT iajs-2559 112 16 is be AUX iajs-2559 112 17 a a DET iajs-2559 112 18 base base NOUN iajs-2559 112 19 for for ADP iajs-2559 112 20 ( ( PUNCT iajs-2559 112 21 𝐷 𝐷 NOUN iajs-2559 112 22 , , PUNCT iajs-2559 112 23 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 112 24 ) ) PUNCT iajs-2559 112 25 , , PUNCT iajs-2559 112 26 which which PRON iajs-2559 112 27 implies imply VERB iajs-2559 112 28 to to ADP iajs-2559 112 29 there there PRON iajs-2559 112 30 exists exist VERB iajs-2559 112 31 a a DET iajs-2559 112 32 countable countable ADJ iajs-2559 112 33 base base NOUN iajs-2559 112 34 for for ADP iajs-2559 112 35 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 112 36 , , PUNCT iajs-2559 112 37 so so CCONJ iajs-2559 112 38 ( ( PUNCT iajs-2559 112 39 𝐷 𝐷 NOUN iajs-2559 112 40 , , PUNCT iajs-2559 112 41 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 112 42 ) ) PUNCT iajs-2559 112 43 is be AUX iajs-2559 112 44 a a DET iajs-2559 112 45 second second ADJ iajs-2559 112 46 countable countable ADJ iajs-2559 112 47 space space NOUN iajs-2559 112 48 . . PUNCT iajs-2559 113 1 theorem theorem VERB iajs-2559 113 2 3.4 3.4 NUM iajs-2559 113 3 . . PUNCT iajs-2559 114 1 if if SCONJ iajs-2559 114 2 𝐷 𝐷 NOUN iajs-2559 114 3 = = SYM iajs-2559 114 4 ( ( PUNCT iajs-2559 114 5 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 114 6 ) ) PUNCT iajs-2559 114 7 , , PUNCT iajs-2559 114 8 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 114 9 ) ) PUNCT iajs-2559 114 10 ) ) PUNCT iajs-2559 114 11 is be AUX iajs-2559 114 12 a a DET iajs-2559 114 13 reflexive reflexive ADJ iajs-2559 114 14 graph graph NOUN iajs-2559 114 15 and and CCONJ iajs-2559 114 16 𝐷𝛽 𝐷𝛽 VERB iajs-2559 114 17 the the DET iajs-2559 114 18 transmitting transmit VERB iajs-2559 114 19 expression expression NOUN iajs-2559 114 20 of of ADP iajs-2559 114 21 𝐷 𝐷 NOUN iajs-2559 114 22 , , PUNCT iajs-2559 114 23 then then ADV iajs-2559 114 24 ( ( PUNCT iajs-2559 114 25 𝐷 𝐷 NOUN iajs-2559 114 26 , , PUNCT iajs-2559 114 27 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 114 28 ) ) PUNCT iajs-2559 114 29 = = SYM iajs-2559 114 30 ( ( PUNCT iajs-2559 114 31 𝐷 𝐷 PROPN iajs-2559 114 32 , , PUNCT iajs-2559 114 33 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 114 34 ) ) PUNCT iajs-2559 114 35 . . PUNCT iajs-2559 115 1 proof proof NOUN iajs-2559 115 2 . . PUNCT iajs-2559 116 1 according accord VERB iajs-2559 116 2 to to ADP iajs-2559 116 3 lemma lemma PROPN iajs-2559 116 4 3.1(4 3.1(4 NUM iajs-2559 116 5 ) ) PUNCT iajs-2559 116 6 , , PUNCT iajs-2559 116 7 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 116 8 = = SYM iajs-2559 116 9 { { PUNCT iajs-2559 116 10 𝐿ɍ 𝐿ɍ PROPN iajs-2559 116 11 : : PUNCT iajs-2559 116 12 ɍ ɍ PROPN iajs-2559 116 13 ∈ ∈ PROPN iajs-2559 116 14 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 116 15 ) ) PUNCT iajs-2559 116 16 } } PUNCT iajs-2559 116 17 is be AUX iajs-2559 116 18 a a DET iajs-2559 116 19 base base NOUN iajs-2559 116 20 for for ADP iajs-2559 116 21 ( ( PUNCT iajs-2559 116 22 𝐷 𝐷 NOUN iajs-2559 116 23 , , PUNCT iajs-2559 116 24 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 116 25 ) ) PUNCT iajs-2559 116 26 , , PUNCT iajs-2559 116 27 we we PRON iajs-2559 116 28 will will AUX iajs-2559 116 29 prove prove VERB iajs-2559 116 30 that that SCONJ iajs-2559 116 31 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 116 32 = = PUNCT iajs-2559 116 33 { { PUNCT iajs-2559 116 34 𝐿ɍ 𝐿ɍ PROPN iajs-2559 116 35 : : PUNCT iajs-2559 116 36 ɍ ɍ PROPN iajs-2559 116 37 ∈ ∈ PROPN iajs-2559 116 38 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 116 39 ) ) PUNCT iajs-2559 116 40 } } PUNCT iajs-2559 116 41 is be AUX iajs-2559 116 42 also also ADV iajs-2559 116 43 a a DET iajs-2559 116 44 base base NOUN iajs-2559 116 45 for for ADP iajs-2559 116 46 ( ( PUNCT iajs-2559 116 47 𝐷 𝐷 NOUN iajs-2559 116 48 , , PUNCT iajs-2559 116 49 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 116 50 ) ) PUNCT iajs-2559 116 51 . . PUNCT iajs-2559 117 1 by by ADP iajs-2559 117 2 definition definition NOUN iajs-2559 117 3 𝐿ɍ 𝐿ɍ NOUN iajs-2559 117 4 = = PUNCT iajs-2559 117 5 { { PUNCT iajs-2559 117 6 𝑒 𝑒 X iajs-2559 117 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 117 8 ): ): PUNCT iajs-2559 117 9 ( ( PUNCT iajs-2559 117 10 ɍ ɍ X iajs-2559 117 11 , , PUNCT iajs-2559 117 12 𝑒 𝑒 ADJ iajs-2559 117 13 ) ) PUNCT iajs-2559 117 14 ∈ ∈ PROPN iajs-2559 117 15 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 117 16 } } PUNCT iajs-2559 117 17 ∈ ∈ PROPN iajs-2559 117 18 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 117 19 . . PUNCT iajs-2559 118 1 let let VERB iajs-2559 118 2 ɍ ɍ PRON iajs-2559 118 3 ∈ ∈ NOUN iajs-2559 118 4 𝐾 𝐾 PROPN iajs-2559 118 5 ∈ ∈ PROPN iajs-2559 118 6 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 118 7 , , PUNCT iajs-2559 118 8 for for ADP iajs-2559 118 9 any any DET iajs-2559 118 10 𝑒 𝑒 PRON iajs-2559 118 11 ∈ ∈ PROPN iajs-2559 118 12 𝐿ɍ 𝐿ɍ PROPN iajs-2559 118 13 , , PUNCT iajs-2559 118 14 then then ADV iajs-2559 118 15 ( ( PUNCT iajs-2559 118 16 ɍ ɍ X iajs-2559 118 17 , , PUNCT iajs-2559 118 18 𝑒 𝑒 X iajs-2559 118 19 ) ) PUNCT iajs-2559 118 20 ∈ ∈ PROPN iajs-2559 118 21 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 118 22 , , PUNCT iajs-2559 118 23 since since SCONJ iajs-2559 118 24 ɍ ɍ PROPN iajs-2559 118 25 ∈ ∈ PROPN iajs-2559 118 26 𝐾 𝐾 PROPN iajs-2559 118 27 , , PUNCT iajs-2559 118 28 by by ADP iajs-2559 118 29 lemma lemma PROPN iajs-2559 118 30 3.1(2 3.1(2 NUM iajs-2559 118 31 ) ) PUNCT iajs-2559 118 32 ɍ ɍ PROPN iajs-2559 118 33 ∈ ∈ NOUN iajs-2559 118 34 𝐿ɍ 𝐿ɍ PRON iajs-2559 118 35 βŠ† βŠ† NUM iajs-2559 118 36 𝐾. 𝐾. NOUN iajs-2559 118 37 so so ADV iajs-2559 118 38 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 118 39 is be AUX iajs-2559 118 40 also also ADV iajs-2559 118 41 base base NOUN iajs-2559 118 42 for for ADP iajs-2559 118 43 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 118 44 , , PUNCT iajs-2559 118 45 hence hence ADV iajs-2559 118 46 ( ( PUNCT iajs-2559 118 47 𝐷 𝐷 NOUN iajs-2559 118 48 , , PUNCT iajs-2559 118 49 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 118 50 ) ) PUNCT iajs-2559 118 51 = = SYM iajs-2559 119 1 ( ( PUNCT iajs-2559 119 2 𝐷 𝐷 PROPN iajs-2559 119 3 , , PUNCT iajs-2559 119 4 πœπ·π›½ πœπ·π›½ NOUN iajs-2559 119 5 ) ) PUNCT iajs-2559 119 6 . . PUNCT iajs-2559 120 1 lemma lemma PROPN iajs-2559 120 2 3.5 3.5 NUM iajs-2559 120 3 . . PUNCT iajs-2559 121 1 let let VERB iajs-2559 121 2 ( ( PUNCT iajs-2559 121 3 𝐷 𝐷 NOUN iajs-2559 121 4 , , PUNCT iajs-2559 121 5 𝜏𝐷)be 𝜏𝐷)be NOUN iajs-2559 121 6 atopological atopological ADJ iajs-2559 121 7 space space NOUN iajs-2559 121 8 generated generate VERB iajs-2559 121 9 by by ADP iajs-2559 121 10 a a DET iajs-2559 121 11 reflexive reflexive ADJ iajs-2559 121 12 graph graph NOUN iajs-2559 121 13 𝐷 𝐷 NOUN iajs-2559 121 14 , , PUNCT iajs-2559 121 15 if if SCONJ iajs-2559 121 16 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 121 17 βˆ— βˆ— VERB iajs-2559 121 18 the the DET iajs-2559 121 19 minimal minimal ADJ iajs-2559 121 20 complete complete ADJ iajs-2559 121 21 cover cover NOUN iajs-2559 121 22 of of ADP iajs-2559 121 23 ( ( PUNCT iajs-2559 121 24 𝐷 𝐷 NOUN iajs-2559 121 25 , , PUNCT iajs-2559 121 26 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 121 27 ) ) PUNCT iajs-2559 121 28 according accord VERB iajs-2559 121 29 to to ADP iajs-2559 121 30 the the DET iajs-2559 121 31 base base NOUN iajs-2559 121 32 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 121 33 , , PUNCT iajs-2559 121 34 then then ADV iajs-2559 121 35 for for ADP iajs-2559 121 36 all all PRON iajs-2559 121 37 𝐹 𝐹 PROPN iajs-2559 121 38 ∈ ∈ PROPN iajs-2559 121 39 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 121 40 βˆ— βˆ— NOUN iajs-2559 121 41 , , PUNCT iajs-2559 121 42 ⋃(ℬ𝐷\ ⋃(ℬ𝐷\ PROPN iajs-2559 121 43 { { PUNCT iajs-2559 121 44 𝐹 𝐹 PROPN iajs-2559 121 45 } } PUNCT iajs-2559 121 46 ) ) PUNCT iajs-2559 121 47 β‰  β‰  PROPN iajs-2559 121 48 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 121 49 ) ) PUNCT iajs-2559 121 50 and and CCONJ iajs-2559 121 51 ⋃(ℬ𝐷 ⋃(ℬ𝐷 PROPN iajs-2559 121 52 βˆ— βˆ— NOUN iajs-2559 121 53 \{𝐹 \{𝐹 NOUN iajs-2559 121 54 } } PUNCT iajs-2559 121 55 ) ) PUNCT iajs-2559 122 1 β‰  β‰  PROPN iajs-2559 122 2 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 122 3 ) ) PUNCT iajs-2559 122 4 . . PUNCT iajs-2559 123 1 proof proof NOUN iajs-2559 123 2 . . PUNCT iajs-2559 124 1 suppose suppose VERB iajs-2559 124 2 that that SCONJ iajs-2559 124 3 ⋃(ℬ𝐷\{𝐹 ⋃(ℬ𝐷\{𝐹 PROPN iajs-2559 124 4 } } PUNCT iajs-2559 124 5 ) ) PUNCT iajs-2559 124 6 = = SYM iajs-2559 124 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 124 8 ) ) PUNCT iajs-2559 124 9 , , PUNCT iajs-2559 124 10 then then ADV iajs-2559 124 11 there there PRON iajs-2559 124 12 exists exist VERB iajs-2559 124 13 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 124 14 πœ„ πœ„ ADP iajs-2559 124 15 βŠ† βŠ† NUM iajs-2559 124 16 ℬ𝐷\{𝐹 ℬ𝐷\{𝐹 NOUN iajs-2559 124 17 } } PUNCT iajs-2559 124 18 such such ADJ iajs-2559 124 19 that that SCONJ iajs-2559 124 20 𝐹 𝐹 PROPN iajs-2559 124 21 βŠ† βŠ† NUM iajs-2559 124 22 βˆͺ βˆͺ VERB iajs-2559 124 23 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 124 24 πœ„ πœ„ X iajs-2559 124 25 , , PUNCT iajs-2559 124 26 since since SCONJ iajs-2559 124 27 𝐹 𝐹 PROPN iajs-2559 124 28 ∈ ∈ PROPN iajs-2559 124 29 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 124 30 βˆ— βˆ— NOUN iajs-2559 124 31 βŠ† βŠ† NUM iajs-2559 124 32 ℬ𝐷 ℬ𝐷 NOUN iajs-2559 124 33 , , PUNCT iajs-2559 124 34 there there PRON iajs-2559 124 35 exists exist VERB iajs-2559 124 36 ɍ ɍ PROPN iajs-2559 124 37 ∈ ∈ PROPN iajs-2559 124 38 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 124 39 ) ) PUNCT iajs-2559 124 40 such such ADJ iajs-2559 124 41 that that SCONJ iajs-2559 124 42 𝐹 𝐹 PROPN iajs-2559 124 43 = = SYM iajs-2559 124 44 𝐿ɍ 𝐿ɍ PROPN iajs-2559 124 45 , , PUNCT iajs-2559 124 46 so so SCONJ iajs-2559 124 47 ɍ ɍ PROPN iajs-2559 124 48 ∈ ∈ PROPN iajs-2559 124 49 πΉπœ„ πΉπœ„ NOUN iajs-2559 124 50 for for ADP iajs-2559 124 51 some some DET iajs-2559 124 52 πΉπœ„ πΉπœ„ PROPN iajs-2559 124 53 ∈ ∈ PROPN iajs-2559 124 54 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 124 55 πœ„ πœ„ X iajs-2559 124 56 . . PUNCT iajs-2559 125 1 by by ADP iajs-2559 125 2 lemma lemma PROPN iajs-2559 125 3 3.1(2 3.1(2 NUM iajs-2559 125 4 ) ) PUNCT iajs-2559 125 5 , , PUNCT iajs-2559 125 6 𝐹 𝐹 PROPN iajs-2559 125 7 = = SYM iajs-2559 125 8 𝐿ɍ 𝐿ɍ PRON iajs-2559 125 9 βŠ† βŠ† NUM iajs-2559 125 10 πΉπœ„. πΉπœ„. NOUN iajs-2559 125 11 consequently consequently ADV iajs-2559 125 12 𝐹 𝐹 PRON iajs-2559 125 13 is be AUX iajs-2559 125 14 not not PART iajs-2559 125 15 a a DET iajs-2559 125 16 maximal maximal ADJ iajs-2559 125 17 element element NOUN iajs-2559 125 18 of of ADP iajs-2559 125 19 ℋ𝐺 ℋ𝐺 PROPN iajs-2559 125 20 which which PRON iajs-2559 125 21 implies imply VERB iajs-2559 125 22 a a DET iajs-2559 125 23 contradiction contradiction NOUN iajs-2559 125 24 . . PUNCT iajs-2559 126 1 so so ADV iajs-2559 126 2 , , PUNCT iajs-2559 126 3 ⋃(ℬ𝐷\{𝐹 ⋃(ℬ𝐷\{𝐹 PROPN iajs-2559 126 4 } } PUNCT iajs-2559 126 5 ) ) PUNCT iajs-2559 127 1 β‰  β‰  PROPN iajs-2559 127 2 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 127 3 ) ) PUNCT iajs-2559 127 4 . . PUNCT iajs-2559 128 1 since since SCONJ iajs-2559 128 2 ⋃(ℬ𝐷\{𝐹 ⋃(ℬ𝐷\{𝐹 PROPN iajs-2559 128 3 } } PUNCT iajs-2559 128 4 ) ) PUNCT iajs-2559 128 5 β‰  β‰  PROPN iajs-2559 128 6 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 128 7 ) ) PUNCT iajs-2559 128 8 , , PUNCT iajs-2559 128 9 ⋃(ℬ𝐷 ⋃(ℬ𝐷 PROPN iajs-2559 128 10 βˆ— βˆ— NOUN iajs-2559 128 11 \{𝐹 \{𝐹 NOUN iajs-2559 128 12 } } PUNCT iajs-2559 128 13 ) ) PUNCT iajs-2559 128 14 β‰  β‰  PROPN iajs-2559 128 15 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 128 16 ) ) PUNCT iajs-2559 128 17 . . PUNCT iajs-2559 129 1 lemma lemma PROPN iajs-2559 129 2 3.6.let 3.6.let PROPN iajs-2559 129 3 ( ( PUNCT iajs-2559 129 4 𝐷 𝐷 PROPN iajs-2559 129 5 , , PUNCT iajs-2559 129 6 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 129 7 ) ) PUNCT iajs-2559 129 8 be be VERB iajs-2559 129 9 a a DET iajs-2559 129 10 topological topological ADJ iajs-2559 129 11 space space NOUN iajs-2559 129 12 generated generate VERB iajs-2559 129 13 by by ADP iajs-2559 129 14 a a DET iajs-2559 129 15 reflexive reflexive ADJ iajs-2559 129 16 graph graph NOUN iajs-2559 129 17 𝐷 𝐷 NOUN iajs-2559 129 18 , , PUNCT iajs-2559 129 19 if if SCONJ iajs-2559 129 20 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 129 21 βˆ— βˆ— VERB iajs-2559 129 22 the the DET iajs-2559 129 23 minimal minimal ADJ iajs-2559 129 24 complete complete ADJ iajs-2559 129 25 cover cover NOUN iajs-2559 129 26 of of ADP iajs-2559 129 27 ( ( PUNCT iajs-2559 129 28 𝐷 𝐷 NOUN iajs-2559 129 29 , , PUNCT iajs-2559 129 30 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 129 31 ) ) PUNCT iajs-2559 129 32 according accord VERB iajs-2559 129 33 to to ADP iajs-2559 129 34 the the DET iajs-2559 129 35 base base ADJ iajs-2559 129 36 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 129 37 and and CCONJ iajs-2559 129 38 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 129 39 an an DET iajs-2559 129 40 open open ADJ iajs-2559 129 41 cover cover NOUN iajs-2559 129 42 of of ADP iajs-2559 129 43 ( ( PUNCT iajs-2559 129 44 𝐷 𝐷 NOUN iajs-2559 129 45 , , PUNCT iajs-2559 129 46 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 129 47 ) ) PUNCT iajs-2559 129 48 . . PUNCT iajs-2559 130 1 then then ADV iajs-2559 130 2 for for ADP iajs-2559 130 3 all all DET iajs-2559 130 4 𝐹 𝐹 PROPN iajs-2559 130 5 ∈ ∈ PROPN iajs-2559 130 6 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 130 7 βˆ— βˆ— NOUN iajs-2559 130 8 , , PUNCT iajs-2559 130 9 there there PRON iajs-2559 130 10 exists exist VERB iajs-2559 130 11 𝐻 𝐻 PROPN iajs-2559 130 12 ∈ ∈ PROPN iajs-2559 130 13 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 130 14 where where SCONJ iajs-2559 130 15 𝐹 𝐹 PROPN iajs-2559 130 16 βŠ† βŠ† NUM iajs-2559 130 17 𝐻. 𝐻. PROPN iajs-2559 130 18 proof proof NOUN iajs-2559 130 19 . . PUNCT iajs-2559 131 1 since since SCONJ iajs-2559 131 2 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 131 3 an an DET iajs-2559 131 4 open open ADJ iajs-2559 131 5 cover cover NOUN iajs-2559 131 6 of of ADP iajs-2559 131 7 ( ( PUNCT iajs-2559 131 8 𝐷 𝐷 NOUN iajs-2559 131 9 , , PUNCT iajs-2559 131 10 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 131 11 ) ) PUNCT iajs-2559 131 12 , , PUNCT iajs-2559 131 13 for for ADP iajs-2559 131 14 any any DET iajs-2559 131 15 𝐹 𝐹 PROPN iajs-2559 131 16 ∈ ∈ PROPN iajs-2559 131 17 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 131 18 βˆ— βˆ— NOUN iajs-2559 131 19 , , PUNCT iajs-2559 131 20 there there PRON iajs-2559 131 21 exists exist VERB iajs-2559 131 22 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 131 23 𝚀 𝚀 ADP iajs-2559 131 24 βŠ† βŠ† NUM iajs-2559 131 25 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 131 26 such such ADJ iajs-2559 131 27 that that SCONJ iajs-2559 131 28 𝐹 𝐹 PROPN iajs-2559 131 29 βŠ† βŠ† NUM iajs-2559 131 30 ⋃ ⋃ PROPN iajs-2559 131 31 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 131 32 𝚀 𝚀 X iajs-2559 131 33 . . PUNCT iajs-2559 132 1 because because SCONJ iajs-2559 132 2 𝐹 𝐹 PROPN iajs-2559 132 3 ∈ ∈ PROPN iajs-2559 132 4 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 132 5 βˆ— βˆ— NOUN iajs-2559 132 6 βŠ† βŠ† NUM iajs-2559 132 7 ℬ𝐷 ℬ𝐷 NOUN iajs-2559 132 8 , , PUNCT iajs-2559 132 9 then then ADV iajs-2559 132 10 𝐹 𝐹 PROPN iajs-2559 132 11 = = SYM iajs-2559 132 12 𝐿ɍ 𝐿ɍ PROPN iajs-2559 132 13 for for ADP iajs-2559 132 14 some some DET iajs-2559 132 15 ɍ ɍ PROPN iajs-2559 132 16 ∈ ∈ NOUN iajs-2559 132 17 𝐹 𝐹 PROPN iajs-2559 132 18 , , PUNCT iajs-2559 132 19 so so SCONJ iajs-2559 132 20 there there PRON iajs-2559 132 21 exists exist VERB iajs-2559 132 22 𝐻 𝐻 PROPN iajs-2559 132 23 ∈ ∈ PROPN iajs-2559 132 24 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 132 25 𝚀 𝚀 ADP iajs-2559 132 26 βŠ† βŠ† NUM iajs-2559 132 27 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 132 28 such such ADJ iajs-2559 132 29 that that SCONJ iajs-2559 132 30 ɍ ɍ PROPN iajs-2559 132 31 ∈ ∈ PROPN iajs-2559 132 32 𝐻. 𝐻. PROPN iajs-2559 132 33 by by ADP iajs-2559 132 34 lemma lemma PROPN iajs-2559 132 35 3.1 3.1 NUM iajs-2559 132 36 , , PUNCT iajs-2559 132 37 𝐹 𝐹 PROPN iajs-2559 132 38 βŠ† βŠ† NUM iajs-2559 132 39 𝐻. 𝐻. PROPN iajs-2559 132 40 110 110 NUM iajs-2559 132 41 ibn ibn PROPN iajs-2559 132 42 al al PROPN iajs-2559 132 43 - - PUNCT iajs-2559 132 44 haitham haitham PROPN iajs-2559 132 45 jour jour X iajs-2559 132 46 . . PROPN iajs-2559 133 1 for for ADP iajs-2559 133 2 pure pure ADJ iajs-2559 133 3 & & CCONJ iajs-2559 133 4 appl appl PROPN iajs-2559 133 5 . . PUNCT iajs-2559 134 1 sci sci PROPN iajs-2559 134 2 . . PROPN iajs-2559 135 1 34 34 NUM iajs-2559 135 2 ( ( PUNCT iajs-2559 135 3 1 1 NUM iajs-2559 135 4 ) ) PUNCT iajs-2559 135 5 2021 2021 NUM iajs-2559 135 6 lemma lemma PROPN iajs-2559 135 7 3.7.let 3.7.let NUM iajs-2559 135 8 ( ( PUNCT iajs-2559 135 9 𝐷 𝐷 PROPN iajs-2559 135 10 , , PUNCT iajs-2559 135 11 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 135 12 ) ) PUNCT iajs-2559 135 13 be be VERB iajs-2559 135 14 a a DET iajs-2559 135 15 topological topological ADJ iajs-2559 135 16 space space NOUN iajs-2559 135 17 generated generate VERB iajs-2559 135 18 by by ADP iajs-2559 135 19 a a DET iajs-2559 135 20 reflexive reflexive ADJ iajs-2559 135 21 graph graph NOUN iajs-2559 135 22 𝐷. 𝐷. NOUN iajs-2559 135 23 if if SCONJ iajs-2559 135 24 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 135 25 βˆ— βˆ— NOUN iajs-2559 135 26 is be AUX iajs-2559 135 27 the the DET iajs-2559 135 28 minimal minimal ADJ iajs-2559 135 29 complete complete ADJ iajs-2559 135 30 cover cover NOUN iajs-2559 135 31 of of ADP iajs-2559 135 32 ( ( PUNCT iajs-2559 135 33 𝐷 𝐷 NOUN iajs-2559 135 34 , , PUNCT iajs-2559 135 35 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 135 36 ) ) PUNCT iajs-2559 135 37 according accord VERB iajs-2559 135 38 to to ADP iajs-2559 135 39 the the DET iajs-2559 135 40 base base ADJ iajs-2559 135 41 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 135 42 and and CCONJ iajs-2559 135 43 ℋ𝐷 ℋ𝐷 PROPN iajs-2559 135 44 an an DET iajs-2559 135 45 open open ADJ iajs-2559 135 46 cover cover NOUN iajs-2559 135 47 of of ADP iajs-2559 135 48 ( ( PUNCT iajs-2559 135 49 𝐷 𝐷 NOUN iajs-2559 135 50 , , PUNCT iajs-2559 135 51 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 135 52 ) ) PUNCT iajs-2559 135 53 , , PUNCT iajs-2559 135 54 which which PRON iajs-2559 135 55 is be AUX iajs-2559 135 56 made make VERB iajs-2559 135 57 up up ADP iajs-2559 135 58 of of ADP iajs-2559 135 59 some some DET iajs-2559 135 60 elements element NOUN iajs-2559 135 61 of of ADP iajs-2559 135 62 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 135 63 , , PUNCT iajs-2559 135 64 then then ADV iajs-2559 135 65 ℬ𝐷 ℬ𝐷 NOUN iajs-2559 135 66 βˆ— βˆ— VERB iajs-2559 135 67 βŠ† βŠ† NUM iajs-2559 135 68 π’ͺ𝐷. π’ͺ𝐷. NOUN iajs-2559 135 69 proof proof NOUN iajs-2559 135 70 . . PUNCT iajs-2559 136 1 for for ADP iajs-2559 136 2 each each DET iajs-2559 136 3 𝐡 𝐡 NOUN iajs-2559 136 4 ∈ ∈ PROPN iajs-2559 136 5 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 136 6 βˆ— βˆ— NOUN iajs-2559 136 7 , , PUNCT iajs-2559 136 8 we we PRON iajs-2559 136 9 claim claim VERB iajs-2559 136 10 that that SCONJ iajs-2559 136 11 𝐡 𝐡 PROPN iajs-2559 136 12 ∈ ∈ PROPN iajs-2559 136 13 π’ͺ𝐷. π’ͺ𝐷. VERB iajs-2559 136 14 if if SCONJ iajs-2559 136 15 not not PART iajs-2559 136 16 , , PUNCT iajs-2559 136 17 𝐡 𝐡 NOUN iajs-2559 136 18 βˆ‰ βˆ‰ PROPN iajs-2559 136 19 π’ͺ𝐷. π’ͺ𝐷. VERB iajs-2559 136 20 since since SCONJ iajs-2559 136 21 ⋃ ⋃ ADP iajs-2559 136 22 π’ͺ𝐷 π’ͺ𝐷 PROPN iajs-2559 136 23 = = SYM iajs-2559 136 24 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 136 25 ) ) PUNCT iajs-2559 136 26 , , PUNCT iajs-2559 136 27 ⋃(π’ͺ𝐷\ ⋃(π’ͺ𝐷\ VERB iajs-2559 136 28 { { PUNCT iajs-2559 136 29 𝐡 𝐡 NOUN iajs-2559 136 30 } } PUNCT iajs-2559 136 31 ) ) PUNCT iajs-2559 136 32 = = SYM iajs-2559 136 33 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 136 34 ) ) PUNCT iajs-2559 136 35 , , PUNCT iajs-2559 136 36 so so ADV iajs-2559 136 37 ⋃(ℬ𝐷\{𝐡 ⋃(ℬ𝐷\{𝐡 PROPN iajs-2559 136 38 } } PUNCT iajs-2559 136 39 ) ) PUNCT iajs-2559 136 40 = = SYM iajs-2559 136 41 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 136 42 ) ) PUNCT iajs-2559 136 43 . . PUNCT iajs-2559 137 1 by by ADP iajs-2559 137 2 using use VERB iajs-2559 137 3 lemma3.5,⋃(ℬ𝐷\{𝐡 lemma3.5,⋃(ℬ𝐷\{𝐡 PROPN iajs-2559 137 4 } } PUNCT iajs-2559 137 5 ) ) PUNCT iajs-2559 137 6 β‰  β‰  PROPN iajs-2559 137 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 137 8 ) ) PUNCT iajs-2559 137 9 , , PUNCT iajs-2559 137 10 which which PRON iajs-2559 137 11 implies imply VERB iajs-2559 137 12 a a DET iajs-2559 137 13 contradiction contradiction NOUN iajs-2559 137 14 . . PUNCT iajs-2559 138 1 so so ADV iajs-2559 138 2 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 138 3 βˆ— βˆ— NOUN iajs-2559 138 4 βŠ† βŠ† NUM iajs-2559 138 5 π’ͺ𝐷. π’ͺ𝐷. NUM iajs-2559 138 6 theorem theorem VERB iajs-2559 138 7 3.8.let 3.8.let PROPN iajs-2559 138 8 ( ( PUNCT iajs-2559 138 9 𝐷 𝐷 PROPN iajs-2559 138 10 , , PUNCT iajs-2559 138 11 𝜏𝐷)be 𝜏𝐷)be NOUN iajs-2559 138 12 a a DET iajs-2559 138 13 topological topological ADJ iajs-2559 138 14 space space NOUN iajs-2559 138 15 generated generate VERB iajs-2559 138 16 by by ADP iajs-2559 138 17 a a DET iajs-2559 138 18 reflexive reflexive ADJ iajs-2559 138 19 graph graph NOUN iajs-2559 138 20 𝐷 𝐷 NOUN iajs-2559 138 21 , , PUNCT iajs-2559 138 22 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 138 23 βˆ— βˆ— VERB iajs-2559 138 24 the the DET iajs-2559 138 25 minimal minimal ADJ iajs-2559 138 26 complete complete ADJ iajs-2559 138 27 cover cover NOUN iajs-2559 138 28 of of ADP iajs-2559 138 29 ( ( PUNCT iajs-2559 138 30 𝐷 𝐷 PROPN iajs-2559 138 31 , , PUNCT iajs-2559 138 32 𝜏𝐷)according 𝜏𝐷)accorde VERB iajs-2559 138 33 to to ADP iajs-2559 138 34 the the DET iajs-2559 138 35 base base NOUN iajs-2559 138 36 ℬ𝐷. ℬ𝐷. PUNCT iajs-2559 139 1 thenℋ𝐺 thenℋ𝐺 PROPN iajs-2559 139 2 βˆ— βˆ— NOUN iajs-2559 139 3 is be AUX iajs-2559 139 4 a a DET iajs-2559 139 5 finite finite NOUN iajs-2559 139 6 set set VERB iajs-2559 139 7 if if SCONJ iajs-2559 139 8 and and CCONJ iajs-2559 139 9 only only ADV iajs-2559 139 10 if(𝐷 if(𝐷 ADJ iajs-2559 139 11 , , PUNCT iajs-2559 139 12 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 139 13 ) ) PUNCT iajs-2559 139 14 is be AUX iajs-2559 139 15 compact compact ADJ iajs-2559 139 16 space space NOUN iajs-2559 139 17 . . PUNCT iajs-2559 140 1 proof proof NOUN iajs-2559 140 2 . . PUNCT iajs-2559 141 1 the the PRON iajs-2559 141 2 only only ADJ iajs-2559 141 3 if if SCONJ iajs-2559 141 4 part part NOUN iajs-2559 141 5 clear clear ADJ iajs-2559 141 6 by by ADP iajs-2559 141 7 lemma lemma PROPN iajs-2559 141 8 ( ( PUNCT iajs-2559 141 9 4.6).conversely 4.6).conversely ADV iajs-2559 141 10 , , PUNCT iajs-2559 141 11 suppose suppose VERB iajs-2559 141 12 that that SCONJ iajs-2559 141 13 ( ( PUNCT iajs-2559 141 14 𝐷 𝐷 NOUN iajs-2559 141 15 , , PUNCT iajs-2559 141 16 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 141 17 ) ) PUNCT iajs-2559 141 18 is be AUX iajs-2559 141 19 compact compact ADJ iajs-2559 141 20 , , PUNCT iajs-2559 141 21 asℋ𝐺 asℋ𝐺 PRON iajs-2559 141 22 is be AUX iajs-2559 141 23 an an DET iajs-2559 141 24 open open ADJ iajs-2559 141 25 cover cover NOUN iajs-2559 141 26 of of ADP iajs-2559 141 27 ( ( PUNCT iajs-2559 141 28 𝐷 𝐷 NOUN iajs-2559 141 29 , , PUNCT iajs-2559 141 30 𝜏𝐷 𝜏𝐷 PROPN iajs-2559 141 31 ) ) PUNCT iajs-2559 141 32 then then ADV iajs-2559 141 33 ℋ𝐺 ℋ𝐺 PROPN iajs-2559 141 34 has have VERB iajs-2559 141 35 a a DET iajs-2559 141 36 finite finite ADJ iajs-2559 141 37 subcover subcover PROPN iajs-2559 141 38 ℋ𝐺 ℋ𝐺 PROPN iajs-2559 141 39 πœ„ πœ„ PROPN iajs-2559 141 40 . . PUNCT iajs-2559 142 1 by by ADP iajs-2559 142 2 using use VERB iajs-2559 142 3 lemma lemma PROPN iajs-2559 142 4 3.7 3.7 NUM iajs-2559 142 5 , , PUNCT iajs-2559 142 6 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 142 7 βˆ— βˆ— NOUN iajs-2559 142 8 βŠ† βŠ† NUM iajs-2559 142 9 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 142 10 πœ„ πœ„ X iajs-2559 142 11 , , PUNCT iajs-2559 142 12 thus thus ADV iajs-2559 142 13 |ℬ𝐷 |ℬ𝐷 NUM iajs-2559 142 14 βˆ— βˆ— NOUN iajs-2559 142 15 | | ADV iajs-2559 142 16 ≀ ≀ NUM iajs-2559 142 17 |ℬ𝐷 |ℬ𝐷 PUNCT iajs-2559 143 1 πœ„ πœ„ PRON iajs-2559 143 2 | | NOUN iajs-2559 143 3 . . PUNCT iajs-2559 144 1 hence hence ADV iajs-2559 144 2 ℬ𝐷 ℬ𝐷 PROPN iajs-2559 144 3 βˆ— βˆ— NOUN iajs-2559 144 4 is be AUX iajs-2559 144 5 a a DET iajs-2559 144 6 finite finite ADJ iajs-2559 144 7 set set NOUN iajs-2559 144 8 . . PUNCT iajs-2559 145 1 4 4 X iajs-2559 145 2 . . X iajs-2559 145 3 the the DET iajs-2559 145 4 properties property NOUN iajs-2559 145 5 of of ADP iajs-2559 145 6 topological topological ADJ iajs-2559 145 7 spaces space NOUN iajs-2559 145 8 generated generate VERB iajs-2559 145 9 by by ADP iajs-2559 145 10 a a DET iajs-2559 145 11 tolerance tolerance NOUN iajs-2559 145 12 graph graph NOUN iajs-2559 145 13 through through ADP iajs-2559 145 14 this this DET iajs-2559 145 15 part part NOUN iajs-2559 145 16 , , PUNCT iajs-2559 145 17 we we PRON iajs-2559 145 18 will will AUX iajs-2559 145 19 achieve achieve VERB iajs-2559 145 20 the the DET iajs-2559 145 21 properties property NOUN iajs-2559 145 22 of of ADP iajs-2559 145 23 ( ( PUNCT iajs-2559 145 24 𝐷 𝐷 NOUN iajs-2559 145 25 , , PUNCT iajs-2559 145 26 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 145 27 ) ) PUNCT iajs-2559 145 28 , , PUNCT iajs-2559 145 29 where where SCONJ iajs-2559 145 30 ( ( PUNCT iajs-2559 145 31 𝐷 𝐷 NOUN iajs-2559 145 32 , , PUNCT iajs-2559 145 33 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 145 34 ) ) PUNCT iajs-2559 145 35 is be AUX iajs-2559 145 36 a a DET iajs-2559 145 37 topological topological ADJ iajs-2559 145 38 space space NOUN iajs-2559 145 39 induced induce VERB iajs-2559 145 40 by by ADP iajs-2559 145 41 tolerance tolerance NOUN iajs-2559 145 42 graph graph NOUN iajs-2559 145 43 𝐷. 𝐷. PROPN iajs-2559 145 44 lemma lemma PROPN iajs-2559 145 45 4.1if 4.1if PROPN iajs-2559 145 46 ( ( PUNCT iajs-2559 145 47 𝐷 𝐷 PROPN iajs-2559 145 48 , , PUNCT iajs-2559 145 49 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 145 50 ) ) PUNCT iajs-2559 145 51 is be AUX iajs-2559 145 52 a a DET iajs-2559 145 53 topological topological ADJ iajs-2559 145 54 space space NOUN iajs-2559 145 55 generated generate VERB iajs-2559 145 56 by by ADP iajs-2559 145 57 a a DET iajs-2559 145 58 tolerance tolerance NOUN iajs-2559 145 59 graph graph NOUN iajs-2559 145 60 𝐷 𝐷 NOUN iajs-2559 145 61 , , PUNCT iajs-2559 145 62 then then ADV iajs-2559 145 63 for for ADP iajs-2559 145 64 all all DET iajs-2559 145 65 𝑄 𝑄 PROPN iajs-2559 145 66 βŠ† βŠ† NUM iajs-2559 145 67 𝐷 𝐷 NOUN iajs-2559 145 68 , , PUNCT iajs-2559 145 69 𝑄 𝑄 PROPN iajs-2559 145 70 is be AUX iajs-2559 145 71 open open ADJ iajs-2559 145 72 if if SCONJ iajs-2559 145 73 and and CCONJ iajs-2559 145 74 only only ADV iajs-2559 145 75 if if SCONJ iajs-2559 145 76 𝑄 𝑄 PRON iajs-2559 145 77 is be AUX iajs-2559 145 78 closed closed ADJ iajs-2559 145 79 . . PUNCT iajs-2559 146 1 proof proof NOUN iajs-2559 146 2 . . PUNCT iajs-2559 147 1 𝑄 𝑄 PRON iajs-2559 147 2 is be AUX iajs-2559 147 3 open open ADJ iajs-2559 147 4 ⟺ ⟺ NOUN iajs-2559 147 5 𝑄 𝑄 NOUN iajs-2559 147 6 = = SYM iajs-2559 147 7 𝐼𝑛𝑑(𝑄 𝐼𝑛𝑑(𝑄 NOUN iajs-2559 148 1 ) ) PUNCT iajs-2559 148 2 ⟺ ⟺ NOUN iajs-2559 149 1 𝑄𝑐 𝑄𝑐 PROPN iajs-2559 149 2 = = PROPN iajs-2559 149 3 𝐼𝑛𝑑(𝑄𝑐 𝐼𝑛𝑑(𝑄𝑐 PROPN iajs-2559 149 4 ) ) PUNCT iajs-2559 149 5 ⟺ ⟺ PROPN iajs-2559 150 1 𝑄𝑐 𝑄𝑐 PROPN iajs-2559 150 2 is be AUX iajs-2559 150 3 open open ADJ iajs-2559 150 4 graph graph NOUN iajs-2559 150 5 ⟺ ⟺ PRON iajs-2559 150 6 𝑄 𝑄 PROPN iajs-2559 150 7 is be AUX iajs-2559 150 8 closed close VERB iajs-2559 150 9 . . PUNCT iajs-2559 151 1 theorem theorem VERB iajs-2559 151 2 4.2 4.2 NUM iajs-2559 151 3 . . PUNCT iajs-2559 152 1 if(𝐷 if(𝐷 PROPN iajs-2559 152 2 , , PUNCT iajs-2559 152 3 𝜏𝐷)is 𝜏𝐷)i VERB iajs-2559 152 4 a a DET iajs-2559 152 5 topological topological ADJ iajs-2559 152 6 space space NOUN iajs-2559 152 7 generated generate VERB iajs-2559 152 8 by by ADP iajs-2559 152 9 a a DET iajs-2559 152 10 tolerance tolerance NOUN iajs-2559 152 11 graph graph NOUN iajs-2559 152 12 𝐷. 𝐷. PROPN iajs-2559 152 13 then then ADV iajs-2559 152 14 ( ( PUNCT iajs-2559 152 15 𝐷 𝐷 NOUN iajs-2559 152 16 , , PUNCT iajs-2559 152 17 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 152 18 ) ) PUNCT iajs-2559 152 19 is be AUX iajs-2559 152 20 discrete discrete ADJ iajs-2559 152 21 if if SCONJ iajs-2559 152 22 and and CCONJ iajs-2559 152 23 only only ADV iajs-2559 152 24 if if SCONJ iajs-2559 152 25 ( ( PUNCT iajs-2559 152 26 𝐷 𝐷 NOUN iajs-2559 152 27 , , PUNCT iajs-2559 152 28 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 152 29 ) ) PUNCT iajs-2559 152 30 is be AUX iajs-2559 152 31 𝑇0 𝑇0 NOUN iajs-2559 152 32 βˆ’ βˆ’ NOUN iajs-2559 152 33 π‘ π‘π‘Žπ‘π‘’. π‘ π‘π‘Žπ‘π‘’. VERB iajs-2559 152 34 proof.the proof.the PRON iajs-2559 152 35 only only ADV iajs-2559 152 36 if if SCONJ iajs-2559 152 37 part part NOUN iajs-2559 152 38 is be AUX iajs-2559 152 39 clear clear ADJ iajs-2559 152 40 . . PUNCT iajs-2559 153 1 we we PRON iajs-2559 153 2 are be AUX iajs-2559 153 3 going go VERB iajs-2559 153 4 to to PART iajs-2559 153 5 prove prove VERB iajs-2559 153 6 the the DET iajs-2559 153 7 if if SCONJ iajs-2559 153 8 part part NOUN iajs-2559 153 9 . . PUNCT iajs-2559 154 1 let let VERB iajs-2559 154 2 ( ( PUNCT iajs-2559 154 3 𝐷 𝐷 NOUN iajs-2559 154 4 , , PUNCT iajs-2559 154 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 154 6 ) ) PUNCT iajs-2559 154 7 be be AUX iajs-2559 154 8 𝑇0 𝑇0 NOUN iajs-2559 154 9 βˆ’ βˆ’ NOUN iajs-2559 154 10 π‘ π‘π‘Žπ‘π‘’. π‘ π‘π‘Žπ‘π‘’. NOUN iajs-2559 154 11 depending depend VERB iajs-2559 154 12 on on ADP iajs-2559 154 13 the the DET iajs-2559 154 14 lemma lemma PROPN iajs-2559 154 15 3.1(4 3.1(4 NUM iajs-2559 154 16 ) ) PUNCT iajs-2559 154 17 , , PUNCT iajs-2559 154 18 we we PRON iajs-2559 154 19 have have VERB iajs-2559 154 20 if if SCONJ iajs-2559 154 21 𝐷 𝐷 PROPN iajs-2559 154 22 is be AUX iajs-2559 154 23 reflexive reflexive ADJ iajs-2559 154 24 , , PUNCT iajs-2559 154 25 then then ADV iajs-2559 154 26 { { PUNCT iajs-2559 154 27 𝐿ɍ 𝐿ɍ PROPN iajs-2559 154 28 : : PUNCT iajs-2559 154 29 ɍ ɍ PROPN iajs-2559 154 30 ∈ ∈ PROPN iajs-2559 154 31 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 154 32 ) ) PUNCT iajs-2559 154 33 } } PUNCT iajs-2559 154 34 is be AUX iajs-2559 154 35 a a DET iajs-2559 154 36 base base NOUN iajs-2559 154 37 for for ADP iajs-2559 154 38 ( ( PUNCT iajs-2559 154 39 𝐷 𝐷 NOUN iajs-2559 154 40 , , PUNCT iajs-2559 154 41 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 154 42 ) ) PUNCT iajs-2559 154 43 . . PUNCT iajs-2559 155 1 we we PRON iajs-2559 155 2 claim claim VERB iajs-2559 155 3 that that SCONJ iajs-2559 155 4 𝐿ɍ 𝐿ɍ VERB iajs-2559 155 5 = = X iajs-2559 155 6 { { PUNCT iajs-2559 155 7 ɍ ɍ NOUN iajs-2559 155 8 } } PUNCT iajs-2559 155 9 for for ADP iajs-2559 155 10 any any DET iajs-2559 155 11 ɍ ɍ PROPN iajs-2559 155 12 ∈ ∈ NOUN iajs-2559 155 13 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 155 14 ) ) PUNCT iajs-2559 155 15 . . PUNCT iajs-2559 156 1 suppose suppose VERB iajs-2559 156 2 that that SCONJ iajs-2559 156 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 156 4 β‰  β‰  PROPN iajs-2559 156 5 { { PUNCT iajs-2559 156 6 ɍ ɍ NOUN iajs-2559 156 7 } } PUNCT iajs-2559 156 8 for for ADP iajs-2559 156 9 some some DET iajs-2559 156 10 ɍ ɍ PROPN iajs-2559 156 11 ∈ ∈ PROPN iajs-2559 156 12 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 156 13 ) ) PUNCT iajs-2559 156 14 . . PUNCT iajs-2559 157 1 by by ADP iajs-2559 157 2 proposition proposition NOUN iajs-2559 157 3 2.5 2.5 NUM iajs-2559 157 4 , , PUNCT iajs-2559 157 5 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 157 6 is be AUX iajs-2559 157 7 an an DET iajs-2559 157 8 equivalent equivalent ADJ iajs-2559 157 9 graph graph NOUN iajs-2559 157 10 on on ADP iajs-2559 157 11 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 157 12 ) ) PUNCT iajs-2559 157 13 , , PUNCT iajs-2559 157 14 so so ADV iajs-2559 157 15 𝐿ɍ 𝐿ɍ PRON iajs-2559 157 16 = = PUNCT iajs-2559 158 1 [ [ X iajs-2559 158 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 PROPN iajs-2559 158 3 . . PUNCT iajs-2559 159 1 chose choose VERB iajs-2559 159 2 𝑒 𝑒 PRON iajs-2559 159 3 ∈ ∈ PROPN iajs-2559 160 1 [ [ X iajs-2559 160 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 160 3 such such ADJ iajs-2559 160 4 that that SCONJ iajs-2559 160 5 𝑒 𝑒 PROPN iajs-2559 160 6 β‰  β‰  PROPN iajs-2559 160 7 ɍ ɍ PROPN iajs-2559 160 8 . . PUNCT iajs-2559 160 9 since since SCONJ iajs-2559 160 10 ( ( PUNCT iajs-2559 160 11 𝐷 𝐷 NOUN iajs-2559 160 12 , , PUNCT iajs-2559 160 13 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 160 14 ) ) PUNCT iajs-2559 160 15 is be AUX iajs-2559 160 16 𝑇0 𝑇0 NOUN iajs-2559 160 17 βˆ’ βˆ’ PROPN iajs-2559 160 18 π‘ π‘π‘Žπ‘π‘’ π‘ π‘π‘Žπ‘π‘’ ADV iajs-2559 160 19 , , PUNCT iajs-2559 160 20 there there PRON iajs-2559 160 21 exists exist VERB iajs-2559 160 22 an an DET iajs-2559 160 23 open open ADJ iajs-2559 160 24 subgraph subgraph NOUN iajs-2559 160 25 𝑂 𝑂 PROPN iajs-2559 160 26 where where SCONJ iajs-2559 160 27 ɍ ɍ PROPN iajs-2559 160 28 ∈ ∈ PROPN iajs-2559 160 29 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 160 30 ) ) PUNCT iajs-2559 160 31 and and CCONJ iajs-2559 160 32 𝑒 𝑒 X iajs-2559 160 33 βˆ‰ βˆ‰ PROPN iajs-2559 160 34 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 160 35 ) ) PUNCT iajs-2559 160 36 , , PUNCT iajs-2559 160 37 or or CCONJ iajs-2559 160 38 there there PRON iajs-2559 160 39 exists exist VERB iajs-2559 160 40 an an DET iajs-2559 160 41 open open ADJ iajs-2559 160 42 subgraph subgraph NOUN iajs-2559 160 43 π‘ˆ π‘ˆ PROPN iajs-2559 160 44 where where SCONJ iajs-2559 160 45 𝑒 𝑒 PRON iajs-2559 160 46 ∈ ∈ PROPN iajs-2559 160 47 𝑉(π‘ˆ 𝑉(π‘ˆ NOUN iajs-2559 160 48 ) ) PUNCT iajs-2559 160 49 and and CCONJ iajs-2559 160 50 ɍ ɍ PROPN iajs-2559 160 51 βˆ‰ βˆ‰ PROPN iajs-2559 160 52 𝑉(π‘ˆ 𝑉(π‘ˆ NOUN iajs-2559 160 53 ) ) PUNCT iajs-2559 160 54 . . PUNCT iajs-2559 161 1 if if SCONJ iajs-2559 161 2 there there PRON iajs-2559 161 3 exists exist VERB iajs-2559 161 4 an an DET iajs-2559 161 5 open open ADJ iajs-2559 161 6 subgraph subgraph NOUN iajs-2559 161 7 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 161 8 ) ) PUNCT iajs-2559 161 9 where where SCONJ iajs-2559 161 10 ɍ ɍ PROPN iajs-2559 161 11 ∈ ∈ PROPN iajs-2559 161 12 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 161 13 ) ) PUNCT iajs-2559 161 14 and and CCONJ iajs-2559 161 15 ɍ ɍ PROPN iajs-2559 161 16 βˆ‰ βˆ‰ PROPN iajs-2559 161 17 𝑉(π‘ˆ 𝑉(π‘ˆ NOUN iajs-2559 161 18 ) ) PUNCT iajs-2559 161 19 , , PUNCT iajs-2559 161 20 then then ADV iajs-2559 161 21 ɍ ɍ PROPN iajs-2559 161 22 ∈ ∈ NOUN iajs-2559 161 23 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 161 24 βŠ† βŠ† NUM iajs-2559 161 25 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 161 26 ) ) PUNCT iajs-2559 161 27 for for ADP iajs-2559 161 28 some some DET iajs-2559 161 29 𝑣 𝑣 PRON iajs-2559 161 30 ∈ ∈ PROPN iajs-2559 161 31 𝑉(𝐷)depending 𝑉(𝐷)depende VERB iajs-2559 161 32 on on ADP iajs-2559 161 33 the the DET iajs-2559 161 34 lemma lemma PROPN iajs-2559 161 35 3.1(4 3.1(4 NUM iajs-2559 161 36 ) ) PUNCT iajs-2559 161 37 . . PUNCT iajs-2559 162 1 it it PRON iajs-2559 162 2 follows follow VERB iajs-2559 162 3 𝑒 𝑒 PRON iajs-2559 162 4 βˆ‰ βˆ‰ ADJ iajs-2559 162 5 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 162 6 . . PUNCT iajs-2559 163 1 as as SCONJ iajs-2559 163 2 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 163 3 is be AUX iajs-2559 163 4 an an DET iajs-2559 163 5 equivalence equivalence NOUN iajs-2559 163 6 graph graph NOUN iajs-2559 163 7 on on ADP iajs-2559 163 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 163 9 ) ) PUNCT iajs-2559 163 10 , , PUNCT iajs-2559 164 1 [ [ X iajs-2559 164 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 164 3 = = SYM iajs-2559 164 4 [ [ X iajs-2559 164 5 𝑣]𝐷𝛽 𝑣]𝐷𝛽 PUNCT iajs-2559 164 6 = = SYM iajs-2559 164 7 𝐿𝑣. 𝐿𝑣. PROPN iajs-2559 164 8 thus thus ADV iajs-2559 164 9 𝑒 𝑒 X iajs-2559 164 10 ∈ ∈ NOUN iajs-2559 164 11 [ [ X iajs-2559 164 12 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 164 13 = = SYM iajs-2559 164 14 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 164 15 means mean VERB iajs-2559 164 16 a a DET iajs-2559 164 17 contradiction contradiction NOUN iajs-2559 164 18 . . PUNCT iajs-2559 165 1 similarly similarly ADV iajs-2559 165 2 if if SCONJ iajs-2559 165 3 there there PRON iajs-2559 165 4 exists exist VERB iajs-2559 165 5 an an DET iajs-2559 165 6 open open ADJ iajs-2559 165 7 subgraph subgraph NOUN iajs-2559 165 8 π‘ˆwhere𝑒 π‘ˆwhere𝑒 PROPN iajs-2559 165 9 ∈ ∈ PROPN iajs-2559 165 10 𝑉(π‘ˆ 𝑉(π‘ˆ NOUN iajs-2559 165 11 ) ) PUNCT iajs-2559 165 12 and and CCONJ iajs-2559 165 13 ɍ ɍ PROPN iajs-2559 165 14 βˆ‰ βˆ‰ PROPN iajs-2559 165 15 𝑉(π‘ˆ 𝑉(π‘ˆ NOUN iajs-2559 165 16 ) ) PUNCT iajs-2559 165 17 . . PUNCT iajs-2559 166 1 hence hence ADV iajs-2559 166 2 , , PUNCT iajs-2559 166 3 { { PUNCT iajs-2559 166 4 ɍ ɍ X iajs-2559 166 5 } } PUNCT iajs-2559 166 6 is be AUX iajs-2559 166 7 open open ADJ iajs-2559 166 8 for for ADP iajs-2559 166 9 all all DET iajs-2559 166 10 ɍ ɍ PRON iajs-2559 166 11 ∈ ∈ NOUN iajs-2559 166 12 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 166 13 ) ) PUNCT iajs-2559 166 14 . . PUNCT iajs-2559 167 1 therefore therefore ADV iajs-2559 167 2 , , PUNCT iajs-2559 167 3 all all DET iajs-2559 167 4 subgraphs subgraph NOUN iajs-2559 167 5 of of ADP iajs-2559 167 6 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 167 7 ) ) PUNCT iajs-2559 167 8 are be AUX iajs-2559 167 9 open open ADJ iajs-2559 167 10 which which PRON iajs-2559 167 11 means mean VERB iajs-2559 167 12 that that SCONJ iajs-2559 167 13 ( ( PUNCT iajs-2559 167 14 𝐷 𝐷 NOUN iajs-2559 167 15 , , PUNCT iajs-2559 167 16 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 167 17 ) ) PUNCT iajs-2559 167 18 is be AUX iajs-2559 167 19 discrete discrete ADJ iajs-2559 167 20 . . PUNCT iajs-2559 168 1 theorem theorem PROPN iajs-2559 168 2 4.3.let 4.3.let PROPN iajs-2559 168 3 ( ( PUNCT iajs-2559 168 4 𝐷 𝐷 PROPN iajs-2559 168 5 , , PUNCT iajs-2559 168 6 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 168 7 ) ) PUNCT iajs-2559 168 8 be be VERB iajs-2559 168 9 a a DET iajs-2559 168 10 topological topological ADJ iajs-2559 168 11 space space NOUN iajs-2559 168 12 generated generate VERB iajs-2559 168 13 by by ADP iajs-2559 168 14 a a DET iajs-2559 168 15 tolerance tolerance NOUN iajs-2559 168 16 graph graph NOUN iajs-2559 168 17 𝐷. 𝐷. PROPN iajs-2559 168 18 then then ADV iajs-2559 168 19 , , PUNCT iajs-2559 168 20 the the DET iajs-2559 168 21 statements statement NOUN iajs-2559 168 22 are be AUX iajs-2559 168 23 equivalent equivalent ADJ iajs-2559 168 24 : : PUNCT iajs-2559 168 25 ( ( PUNCT iajs-2559 168 26 1 1 X iajs-2559 168 27 ) ) PUNCT iajs-2559 168 28 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 168 29 ) ) PUNCT iajs-2559 168 30 βˆ• βˆ• NOUN iajs-2559 168 31 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 168 32 ) ) PUNCT iajs-2559 169 1 is be AUX iajs-2559 169 2 countable countable ADJ iajs-2559 169 3 , , PUNCT iajs-2559 169 4 ( ( PUNCT iajs-2559 169 5 2 2 NUM iajs-2559 169 6 ) ) PUNCT iajs-2559 169 7 ( ( PUNCT iajs-2559 169 8 𝐷 𝐷 NOUN iajs-2559 169 9 , , PUNCT iajs-2559 169 10 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 169 11 ) ) PUNCT iajs-2559 169 12 is be AUX iajs-2559 169 13 a a DET iajs-2559 169 14 second second ADJ iajs-2559 169 15 countable countable ADJ iajs-2559 169 16 space space NOUN iajs-2559 169 17 , , PUNCT iajs-2559 169 18 ( ( PUNCT iajs-2559 169 19 3 3 X iajs-2559 169 20 ) ) PUNCT iajs-2559 169 21 ( ( PUNCT iajs-2559 169 22 𝐷 𝐷 NOUN iajs-2559 169 23 , , PUNCT iajs-2559 169 24 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 169 25 ) ) PUNCT iajs-2559 169 26 is be AUX iajs-2559 169 27 a a DET iajs-2559 169 28 separable separable ADJ iajs-2559 169 29 space space NOUN iajs-2559 169 30 , , PUNCT iajs-2559 169 31 111 111 NUM iajs-2559 169 32 ibn ibn PROPN iajs-2559 169 33 al al PROPN iajs-2559 169 34 - - PUNCT iajs-2559 169 35 haitham haitham PROPN iajs-2559 169 36 jour jour X iajs-2559 169 37 . . PROPN iajs-2559 170 1 for for ADP iajs-2559 170 2 pure pure ADJ iajs-2559 170 3 & & CCONJ iajs-2559 170 4 appl appl PROPN iajs-2559 170 5 . . PUNCT iajs-2559 171 1 sci sci PROPN iajs-2559 171 2 . . PROPN iajs-2559 172 1 34 34 NUM iajs-2559 172 2 ( ( PUNCT iajs-2559 172 3 1 1 NUM iajs-2559 172 4 ) ) PUNCT iajs-2559 172 5 2021 2021 NUM iajs-2559 172 6 ( ( PUNCT iajs-2559 172 7 4 4 NUM iajs-2559 172 8 ) ) PUNCT iajs-2559 172 9 ( ( PUNCT iajs-2559 172 10 𝐷 𝐷 NOUN iajs-2559 172 11 , , PUNCT iajs-2559 172 12 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 172 13 ) ) PUNCT iajs-2559 172 14 is be AUX iajs-2559 172 15 a a DET iajs-2559 172 16 lindelΓΆf lindelΓΆf NOUN iajs-2559 172 17 space space NOUN iajs-2559 172 18 . . PUNCT iajs-2559 173 1 proof proof NOUN iajs-2559 173 2 . . PUNCT iajs-2559 174 1 ( ( PUNCT iajs-2559 174 2 1)⟹(2 1)⟹(2 NUM iajs-2559 174 3 ) ) PUNCT iajs-2559 174 4 . . PUNCT iajs-2559 175 1 since since SCONJ iajs-2559 175 2 𝐷𝛽is 𝐷𝛽is PROPN iajs-2559 175 3 an an DET iajs-2559 175 4 equivalence equivalence NOUN iajs-2559 175 5 graph graph NOUN iajs-2559 175 6 on on ADP iajs-2559 175 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 175 8 ) ) PUNCT iajs-2559 175 9 , , PUNCT iajs-2559 175 10 { { PUNCT iajs-2559 175 11 𝐿ɍ 𝐿ɍ ADV iajs-2559 175 12 : : PUNCT iajs-2559 175 13 ɍ ɍ PROPN iajs-2559 175 14 ∈ ∈ PROPN iajs-2559 175 15 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 175 16 ) ) PUNCT iajs-2559 175 17 } } PUNCT iajs-2559 175 18 = = SYM iajs-2559 175 19 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 175 20 ) ) PUNCT iajs-2559 175 21 βˆ• βˆ• NOUN iajs-2559 175 22 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NUM iajs-2559 175 23 ) ) PUNCT iajs-2559 175 24 . . PUNCT iajs-2559 176 1 by by ADP iajs-2559 176 2 lemma lemma PROPN iajs-2559 176 3 3.1(4 3.1(4 NUM iajs-2559 176 4 ) ) PUNCT iajs-2559 176 5 , , PUNCT iajs-2559 176 6 ( ( PUNCT iajs-2559 176 7 𝐷 𝐷 NOUN iajs-2559 176 8 , , PUNCT iajs-2559 176 9 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 176 10 ) ) PUNCT iajs-2559 176 11 is be AUX iajs-2559 176 12 second second ADJ iajs-2559 176 13 countable countable ADJ iajs-2559 176 14 space space NOUN iajs-2559 176 15 . . PUNCT iajs-2559 177 1 ( ( PUNCT iajs-2559 177 2 2)⟹(1 2)⟹(1 NUM iajs-2559 177 3 ) ) PUNCT iajs-2559 177 4 suppose suppose VERB iajs-2559 177 5 that that SCONJ iajs-2559 177 6 ℬ ℬ NOUN iajs-2559 177 7 is be AUX iajs-2559 177 8 a a DET iajs-2559 177 9 countable countable ADJ iajs-2559 177 10 base base NOUN iajs-2559 177 11 for for ADP iajs-2559 177 12 ( ( PUNCT iajs-2559 177 13 𝐷 𝐷 NOUN iajs-2559 177 14 , , PUNCT iajs-2559 177 15 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 177 16 ) ) PUNCT iajs-2559 177 17 , , PUNCT iajs-2559 177 18 then then ADV iajs-2559 177 19 for for ADP iajs-2559 177 20 ɍ ɍ PROPN iajs-2559 177 21 ∈ ∈ PROPN iajs-2559 177 22 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 177 23 ) ) PUNCT iajs-2559 177 24 , , PUNCT iajs-2559 177 25 there there PRON iajs-2559 177 26 exists exist VERB iajs-2559 177 27 𝐡ɍ 𝐡ɍ PROPN iajs-2559 177 28 ∈ ∈ PROPN iajs-2559 177 29 ℬ ℬ NOUN iajs-2559 177 30 such such ADJ iajs-2559 177 31 that that SCONJ iajs-2559 177 32 ɍ ɍ PROPN iajs-2559 177 33 ∈ ∈ NOUN iajs-2559 178 1 𝐡ɍ 𝐡ɍ PROPN iajs-2559 178 2 βŠ† βŠ† NUM iajs-2559 178 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 178 4 . . PUNCT iajs-2559 179 1 by by ADP iajs-2559 179 2 lemma lemma PROPN iajs-2559 179 3 3.1(4 3.1(4 NUM iajs-2559 179 4 ) ) PUNCT iajs-2559 179 5 , , PUNCT iajs-2559 179 6 ɍ ɍ PROPN iajs-2559 179 7 ∈ ∈ NOUN iajs-2559 180 1 𝐿𝑒 𝐿𝑒 NOUN iajs-2559 180 2 βŠ† βŠ† NUM iajs-2559 180 3 𝐡ɍ 𝐡ɍ PROPN iajs-2559 180 4 for for ADP iajs-2559 180 5 some some DET iajs-2559 180 6 𝑒 𝑒 PRON iajs-2559 180 7 ∈ ∈ PROPN iajs-2559 180 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 180 9 ) ) PUNCT iajs-2559 180 10 . . PUNCT iajs-2559 181 1 since since SCONJ iajs-2559 181 2 𝐿ɍ 𝐿ɍ PRON iajs-2559 181 3 = = PUNCT iajs-2559 182 1 [ [ X iajs-2559 182 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 182 3 = = SYM iajs-2559 182 4 [ [ X iajs-2559 182 5 𝑒]𝐷𝛽 𝑒]𝐷𝛽 PROPN iajs-2559 182 6 = = SYM iajs-2559 182 7 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 182 8 , , PUNCT iajs-2559 182 9 𝐡ɍ 𝐡ɍ PROPN iajs-2559 182 10 = = PUNCT iajs-2559 183 1 [ [ X iajs-2559 183 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 PROPN iajs-2559 183 3 , , PUNCT iajs-2559 183 4 we we PRON iajs-2559 183 5 define define VERB iajs-2559 183 6 𝑓 𝑓 DET iajs-2559 183 7 : : PUNCT iajs-2559 183 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 183 9 ) ) PUNCT iajs-2559 183 10 βˆ• βˆ• NOUN iajs-2559 183 11 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 183 12 ) ) PUNCT iajs-2559 183 13 ⟢ ⟢ NOUN iajs-2559 183 14 ℬ ℬ NOUN iajs-2559 183 15 by by ADP iajs-2559 183 16 𝑓 𝑓 PRON iajs-2559 183 17 ( ( PUNCT iajs-2559 183 18 [ [ X iajs-2559 183 19 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 183 20 ) ) PUNCT iajs-2559 184 1 = = SYM iajs-2559 185 1 𝐡ɍ 𝐡ɍ PROPN iajs-2559 185 2 , , PUNCT iajs-2559 185 3 then then ADV iajs-2559 185 4 𝑓 𝑓 PRON iajs-2559 185 5 is be AUX iajs-2559 185 6 injective injective ADJ iajs-2559 185 7 . . PUNCT iajs-2559 186 1 so|𝑉(𝐷 so|𝑉(𝐷 NOUN iajs-2559 186 2 ) ) PUNCT iajs-2559 187 1 βˆ• βˆ• NOUN iajs-2559 187 2 𝐸(𝐷𝛽)| 𝐸(𝐷𝛽)| NOUN iajs-2559 187 3 ≀ ≀ NOUN iajs-2559 187 4 |ℬ| |ℬ| NOUN iajs-2559 187 5 . . PUNCT iajs-2559 188 1 hence hence ADV iajs-2559 188 2 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 188 3 ) ) PUNCT iajs-2559 188 4 βˆ• βˆ• NOUN iajs-2559 188 5 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 188 6 ) ) PUNCT iajs-2559 188 7 is be AUX iajs-2559 188 8 countable countable ADJ iajs-2559 188 9 . . PUNCT iajs-2559 189 1 ( ( PUNCT iajs-2559 189 2 2 2 NUM iajs-2559 189 3 ) ) PUNCT iajs-2559 189 4 ⟹ ⟹ NOUN iajs-2559 189 5 ( ( PUNCT iajs-2559 189 6 3 3 NUM iajs-2559 189 7 ) ) PUNCT iajs-2559 189 8 and and CCONJ iajs-2559 189 9 ( ( PUNCT iajs-2559 189 10 2 2 NUM iajs-2559 189 11 ) ) PUNCT iajs-2559 189 12 ⟹ ⟹ NOUN iajs-2559 189 13 ( ( PUNCT iajs-2559 189 14 4 4 X iajs-2559 189 15 ) ) PUNCT iajs-2559 189 16 are be AUX iajs-2559 189 17 clear clear ADJ iajs-2559 189 18 . . PUNCT iajs-2559 190 1 ( ( PUNCT iajs-2559 190 2 3 3 X iajs-2559 190 3 ) ) PUNCT iajs-2559 190 4 ⟹ ⟹ NOUN iajs-2559 190 5 ( ( PUNCT iajs-2559 190 6 2 2 NUM iajs-2559 190 7 ) ) PUNCT iajs-2559 190 8 . . PUNCT iajs-2559 190 9 suppose suppose VERB iajs-2559 190 10 that that SCONJ iajs-2559 190 11 𝐢 𝐢 PROPN iajs-2559 190 12 is be AUX iajs-2559 190 13 a a DET iajs-2559 190 14 countable countable ADJ iajs-2559 190 15 dense dense ADJ iajs-2559 190 16 subgraph subgraph NOUN iajs-2559 190 17 of of ADP iajs-2559 190 18 ( ( PUNCT iajs-2559 190 19 𝐷 𝐷 PROPN iajs-2559 190 20 , , PUNCT iajs-2559 190 21 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 190 22 ) ) PUNCT iajs-2559 190 23 . . PUNCT iajs-2559 191 1 put put VERB iajs-2559 191 2 πœ† πœ† PRON iajs-2559 191 3 = = SYM iajs-2559 191 4 { { PUNCT iajs-2559 191 5 𝐿ɍ 𝐿ɍ PROPN iajs-2559 191 6 : : PUNCT iajs-2559 191 7 ɍ ɍ PROPN iajs-2559 191 8 ∈ ∈ PROPN iajs-2559 191 9 𝑉(𝐢 𝑉(𝐢 NOUN iajs-2559 191 10 ) ) PUNCT iajs-2559 191 11 } } PUNCT iajs-2559 191 12 , , PUNCT iajs-2559 191 13 then then ADV iajs-2559 191 14 πœ† πœ† PRON iajs-2559 191 15 is be AUX iajs-2559 191 16 countable countable ADJ iajs-2559 191 17 . . PUNCT iajs-2559 192 1 by by ADP iajs-2559 192 2 lemma lemma PROPN iajs-2559 192 3 3.1(4 3.1(4 NUM iajs-2559 192 4 ) ) PUNCT iajs-2559 192 5 , , PUNCT iajs-2559 192 6 for for ADP iajs-2559 192 7 all all DET iajs-2559 192 8 ɍ ɍ PRON iajs-2559 192 9 ∈ ∈ NOUN iajs-2559 192 10 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 192 11 ) ) PUNCT iajs-2559 192 12 and and CCONJ iajs-2559 192 13 open open ADJ iajs-2559 192 14 subgraph subgraph NOUN iajs-2559 192 15 𝑂 𝑂 PROPN iajs-2559 192 16 with with ADP iajs-2559 192 17 ɍ ɍ PROPN iajs-2559 192 18 ∈ ∈ PROPN iajs-2559 192 19 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 192 20 ) ) PUNCT iajs-2559 192 21 , , PUNCT iajs-2559 192 22 we we PRON iajs-2559 192 23 have have VERB iajs-2559 192 24 ɍ ɍ PRON iajs-2559 192 25 ∈ ∈ NOUN iajs-2559 192 26 𝐿𝑒 𝐿𝑒 NOUN iajs-2559 192 27 βŠ† βŠ† NUM iajs-2559 192 28 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 192 29 ) ) PUNCT iajs-2559 192 30 for for ADP iajs-2559 192 31 some some DET iajs-2559 192 32 𝑒 𝑒 PRON iajs-2559 192 33 ∈ ∈ PROPN iajs-2559 192 34 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 192 35 ) ) PUNCT iajs-2559 192 36 . . PUNCT iajs-2559 193 1 since since SCONJ iajs-2559 193 2 𝐢 𝐢 PROPN iajs-2559 193 3 is be AUX iajs-2559 193 4 dense dense ADJ iajs-2559 193 5 , , PUNCT iajs-2559 193 6 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 193 7 ∩ ∩ ADJ iajs-2559 193 8 𝑉(𝐢 𝑉(𝐢 NOUN iajs-2559 193 9 ) ) PUNCT iajs-2559 193 10 β‰  β‰  PROPN iajs-2559 193 11 βˆ… βˆ… NOUN iajs-2559 193 12 , , PUNCT iajs-2559 193 13 chose choose VERB iajs-2559 193 14 𝑣 𝑣 PRON iajs-2559 193 15 ∈ ∈ PROPN iajs-2559 193 16 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 193 17 ∩ ∩ ADJ iajs-2559 193 18 𝑉(𝐢 𝑉(𝐢 NOUN iajs-2559 193 19 ) ) PUNCT iajs-2559 193 20 , , PUNCT iajs-2559 193 21 then then ADV iajs-2559 193 22 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 193 23 ∈ ∈ PROPN iajs-2559 193 24 πœ†. πœ†. NOUN iajs-2559 193 25 since since SCONJ iajs-2559 193 26 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 193 27 is be AUX iajs-2559 193 28 an an DET iajs-2559 193 29 equivalence equivalence NOUN iajs-2559 193 30 graph graph NOUN iajs-2559 193 31 on on ADP iajs-2559 193 32 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 193 33 ) ) PUNCT iajs-2559 193 34 , , PUNCT iajs-2559 194 1 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 194 2 = = PUNCT iajs-2559 195 1 [ [ X iajs-2559 195 2 𝑣]𝐷𝛽 𝑣]𝐷𝛽 PUNCT iajs-2559 195 3 = = PUNCT iajs-2559 195 4 [ [ X iajs-2559 195 5 𝑒]𝐷𝛽 𝑒]𝐷𝛽 PROPN iajs-2559 195 6 = = SYM iajs-2559 195 7 𝐿𝑒. 𝐿𝑒. PROPN iajs-2559 195 8 it it PRON iajs-2559 195 9 follows follow VERB iajs-2559 195 10 ɍ ɍ PROPN iajs-2559 195 11 ∈ ∈ PROPN iajs-2559 195 12 𝐿𝑣 𝐿𝑣 PROPN iajs-2559 195 13 βŠ† βŠ† NUM iajs-2559 195 14 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 195 15 ) ) PUNCT iajs-2559 195 16 . . PUNCT iajs-2559 196 1 therefore therefore ADV iajs-2559 196 2 , , PUNCT iajs-2559 196 3 πœ† πœ† PRON iajs-2559 196 4 is be AUX iajs-2559 196 5 a a DET iajs-2559 196 6 base base NOUN iajs-2559 196 7 for for ADP iajs-2559 196 8 ( ( PUNCT iajs-2559 196 9 𝐷 𝐷 NOUN iajs-2559 196 10 , , PUNCT iajs-2559 196 11 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 196 12 ) ) PUNCT iajs-2559 196 13 . . PUNCT iajs-2559 197 1 hence hence ADV iajs-2559 197 2 ( ( PUNCT iajs-2559 197 3 𝐷 𝐷 NOUN iajs-2559 197 4 , , PUNCT iajs-2559 197 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 197 6 ) ) PUNCT iajs-2559 197 7 is be AUX iajs-2559 197 8 a a DET iajs-2559 197 9 second second ADJ iajs-2559 197 10 countable countable ADJ iajs-2559 197 11 space space NOUN iajs-2559 197 12 . . PUNCT iajs-2559 198 1 ( ( PUNCT iajs-2559 198 2 4 4 NUM iajs-2559 198 3 ) ) PUNCT iajs-2559 198 4 ⟹ ⟹ NOUN iajs-2559 198 5 ( ( PUNCT iajs-2559 198 6 2 2 NUM iajs-2559 198 7 ) ) PUNCT iajs-2559 198 8 . . PUNCT iajs-2559 198 9 suppose suppose VERB iajs-2559 198 10 that that SCONJ iajs-2559 198 11 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 198 12 ) ) PUNCT iajs-2559 198 13 βˆ• βˆ• NOUN iajs-2559 198 14 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 198 15 ) ) PUNCT iajs-2559 198 16 is be AUX iajs-2559 198 17 not not PART iajs-2559 198 18 countable countable ADJ iajs-2559 198 19 . . PUNCT iajs-2559 199 1 since since SCONJ iajs-2559 199 2 𝐷𝛽 𝐷𝛽 PROPN iajs-2559 199 3 is be AUX iajs-2559 199 4 an an DET iajs-2559 199 5 equivalence equivalence NOUN iajs-2559 199 6 graph graph NOUN iajs-2559 199 7 on on ADP iajs-2559 199 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 199 9 ) ) PUNCT iajs-2559 199 10 , , PUNCT iajs-2559 199 11 { { PUNCT iajs-2559 199 12 𝐿ɍ 𝐿ɍ ADV iajs-2559 199 13 : : PUNCT iajs-2559 199 14 ɍ ɍ PROPN iajs-2559 199 15 ∈ ∈ PROPN iajs-2559 199 16 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 199 17 ) ) PUNCT iajs-2559 199 18 } } PUNCT iajs-2559 199 19 = = SYM iajs-2559 199 20 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 199 21 ) ) PUNCT iajs-2559 199 22 βˆ• βˆ• NOUN iajs-2559 199 23 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 199 24 ) ) PUNCT iajs-2559 199 25 . . PUNCT iajs-2559 200 1 it it PRON iajs-2559 200 2 is be AUX iajs-2559 200 3 obvious obvious ADJ iajs-2559 200 4 that that SCONJ iajs-2559 200 5 { { PUNCT iajs-2559 200 6 𝐿ɍ 𝐿ɍ NOUN iajs-2559 200 7 : : PUNCT iajs-2559 200 8 ɍ ɍ PROPN iajs-2559 200 9 ∈ ∈ PROPN iajs-2559 200 10 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 200 11 ) ) PUNCT iajs-2559 200 12 } } PUNCT iajs-2559 200 13 is be AUX iajs-2559 200 14 an an DET iajs-2559 200 15 open open ADJ iajs-2559 200 16 cover cover NOUN iajs-2559 200 17 of of ADP iajs-2559 200 18 ( ( PUNCT iajs-2559 200 19 𝐷 𝐷 NOUN iajs-2559 200 20 , , PUNCT iajs-2559 200 21 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 200 22 ) ) PUNCT iajs-2559 200 23 but but CCONJ iajs-2559 200 24 { { PUNCT iajs-2559 200 25 𝐿ɍ 𝐿ɍ NOUN iajs-2559 200 26 : : PUNCT iajs-2559 200 27 ɍ ɍ PROPN iajs-2559 200 28 ∈ ∈ PROPN iajs-2559 200 29 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 200 30 ) ) PUNCT iajs-2559 200 31 } } PUNCT iajs-2559 200 32 does do AUX iajs-2559 200 33 not not PART iajs-2559 200 34 have have VERB iajs-2559 200 35 any any DET iajs-2559 200 36 countable countable ADJ iajs-2559 200 37 subcover subcover NOUN iajs-2559 200 38 hence hence ADV iajs-2559 200 39 we we PRON iajs-2559 200 40 get get VERB iajs-2559 200 41 a a DET iajs-2559 200 42 contradiction contradiction NOUN iajs-2559 200 43 . . PUNCT iajs-2559 201 1 theorem theorem VERB iajs-2559 201 2 5.4 5.4 NUM iajs-2559 201 3 let let NOUN iajs-2559 201 4 ( ( PUNCT iajs-2559 201 5 𝐷 𝐷 NOUN iajs-2559 201 6 , , PUNCT iajs-2559 201 7 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 201 8 ) ) PUNCT iajs-2559 201 9 be be AUX iajs-2559 201 10 a a DET iajs-2559 201 11 topological topological ADJ iajs-2559 201 12 space space NOUN iajs-2559 201 13 generated generate VERB iajs-2559 201 14 by by ADP iajs-2559 201 15 a a DET iajs-2559 201 16 tolerance tolerance NOUN iajs-2559 201 17 graph graph NOUN iajs-2559 201 18 𝐷. 𝐷. PROPN iajs-2559 201 19 then then ADV iajs-2559 201 20 ( ( PUNCT iajs-2559 201 21 𝐷 𝐷 NOUN iajs-2559 201 22 , , PUNCT iajs-2559 201 23 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 201 24 ) ) PUNCT iajs-2559 201 25 is be AUX iajs-2559 201 26 a a DET iajs-2559 201 27 connected connected ADJ iajs-2559 201 28 space space NOUN iajs-2559 202 1 if if SCONJ iajs-2559 202 2 and and CCONJ iajs-2559 202 3 only only ADV iajs-2559 202 4 if if SCONJ iajs-2559 202 5 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 202 6 ) ) PUNCT iajs-2559 202 7 = = SYM iajs-2559 202 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 202 9 ) ) PUNCT iajs-2559 202 10 Γ— Γ— NOUN iajs-2559 202 11 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 202 12 ) ) PUNCT iajs-2559 202 13 . . PUNCT iajs-2559 203 1 proof proof NOUN iajs-2559 203 2 . . PUNCT iajs-2559 204 1 suppose suppose VERB iajs-2559 204 2 that that SCONJ iajs-2559 204 3 ( ( PUNCT iajs-2559 204 4 𝐷 𝐷 NOUN iajs-2559 204 5 , , PUNCT iajs-2559 204 6 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 204 7 ) ) PUNCT iajs-2559 204 8 is be AUX iajs-2559 204 9 connected connect VERB iajs-2559 204 10 , , PUNCT iajs-2559 204 11 if if SCONJ iajs-2559 204 12 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 204 13 ) ) PUNCT iajs-2559 204 14 β‰  β‰  PROPN iajs-2559 204 15 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 204 16 ) ) PUNCT iajs-2559 204 17 Γ— Γ— NOUN iajs-2559 204 18 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 204 19 ) ) PUNCT iajs-2559 204 20 , , PUNCT iajs-2559 204 21 then then ADV iajs-2559 204 22 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 204 23 ) ) PUNCT iajs-2559 204 24 Γ— Γ— NOUN iajs-2559 204 25 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 204 26 ) ) PUNCT iajs-2559 204 27 βˆ• βˆ• NOUN iajs-2559 204 28 𝐸(𝐷𝛽 𝐸(𝐷𝛽 NOUN iajs-2559 204 29 ) ) PUNCT iajs-2559 204 30 β‰  β‰  PROPN iajs-2559 204 31 βˆ…. βˆ…. AUX iajs-2559 204 32 chose choose VERB iajs-2559 204 33 ( ( PUNCT iajs-2559 204 34 ɍ ɍ ADJ iajs-2559 204 35 , , PUNCT iajs-2559 204 36 𝑒 𝑒 ADJ iajs-2559 204 37 ) ) PUNCT iajs-2559 204 38 ∈ ∈ PROPN iajs-2559 204 39 ( ( PUNCT iajs-2559 204 40 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 204 41 ) ) PUNCT iajs-2559 204 42 Γ— Γ— NOUN iajs-2559 204 43 𝑉(𝐷))\𝐸(𝐷𝛽 𝑉(𝐷))\𝐸(𝐷𝛽 PROPN iajs-2559 204 44 ) ) PUNCT iajs-2559 204 45 , , PUNCT iajs-2559 204 46 then then ADV iajs-2559 204 47 𝑒 𝑒 PROPN iajs-2559 204 48 βˆ‰ βˆ‰ X iajs-2559 205 1 [ [ X iajs-2559 205 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 PROPN iajs-2559 205 3 = = SYM iajs-2559 205 4 𝐿ɍ 𝐿ɍ PROPN iajs-2559 205 5 . . PUNCT iajs-2559 206 1 so so ADV iajs-2559 206 2 𝐿ɍ 𝐿ɍ ADJ iajs-2559 206 3 β‰  β‰  PROPN iajs-2559 206 4 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 206 5 ) ) PUNCT iajs-2559 206 6 and and CCONJ iajs-2559 206 7 𝐿ɍ 𝐿ɍ PRON iajs-2559 206 8 β‰  β‰  PROPN iajs-2559 206 9 βˆ…. βˆ…. VERB iajs-2559 206 10 by by ADP iajs-2559 206 11 lemma lemma PROPN iajs-2559 206 12 4.1 4.1 NUM iajs-2559 206 13 , , PUNCT iajs-2559 206 14 𝐿ɍ 𝐿ɍ PROPN iajs-2559 206 15 is be AUX iajs-2559 206 16 both both CCONJ iajs-2559 206 17 open open ADJ iajs-2559 206 18 and and CCONJ iajs-2559 206 19 closed closed ADJ iajs-2559 206 20 , , PUNCT iajs-2559 206 21 so so SCONJ iajs-2559 206 22 we we PRON iajs-2559 206 23 obtain obtain VERB iajs-2559 206 24 a a DET iajs-2559 206 25 contradiction contradiction NOUN iajs-2559 206 26 . . PUNCT iajs-2559 207 1 conversely conversely ADV iajs-2559 207 2 , , PUNCT iajs-2559 207 3 suppose suppose VERB iajs-2559 207 4 that that SCONJ iajs-2559 207 5 𝐸(𝐷𝛽 𝐸(𝐷𝛽 VERB iajs-2559 207 6 ) ) PUNCT iajs-2559 207 7 = = SYM iajs-2559 207 8 𝑉 𝑉 PROPN iajs-2559 207 9 ( ( PUNCT iajs-2559 207 10 𝐷 𝐷 PROPN iajs-2559 207 11 ) ) PUNCT iajs-2559 207 12 Γ— Γ— NOUN iajs-2559 207 13 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 207 14 ) ) PUNCT iajs-2559 207 15 , , PUNCT iajs-2559 207 16 then then ADV iajs-2559 207 17 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 207 18 ) ) PUNCT iajs-2559 207 19 βˆ• βˆ• NOUN iajs-2559 207 20 𝐸(𝐷𝛽 𝐸(𝐷𝛽 X iajs-2559 207 21 ) ) PUNCT iajs-2559 207 22 = = PUNCT iajs-2559 208 1 [ [ X iajs-2559 208 2 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 208 3 ) ) PUNCT iajs-2559 208 4 ] ] PUNCT iajs-2559 208 5 . . PUNCT iajs-2559 209 1 so so ADV iajs-2559 209 2 𝜏𝐷 𝜏𝐷 ADJ iajs-2559 209 3 = = SYM iajs-2559 209 4 { { PUNCT iajs-2559 209 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 209 6 ) ) PUNCT iajs-2559 209 7 , , PUNCT iajs-2559 209 8 βˆ… βˆ… NOUN iajs-2559 209 9 } } PUNCT iajs-2559 209 10 , , PUNCT iajs-2559 209 11 thus thus ADV iajs-2559 209 12 ( ( PUNCT iajs-2559 209 13 𝐷 𝐷 NOUN iajs-2559 209 14 , , PUNCT iajs-2559 209 15 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 209 16 ) ) PUNCT iajs-2559 209 17 is be AUX iajs-2559 209 18 connected connect VERB iajs-2559 209 19 . . PUNCT iajs-2559 210 1 theorem theorem VERB iajs-2559 210 2 4.6 4.6 NUM iajs-2559 210 3 . . PUNCT iajs-2559 211 1 let let AUX iajs-2559 211 2 ( ( PUNCT iajs-2559 211 3 𝐷 𝐷 NOUN iajs-2559 211 4 , , PUNCT iajs-2559 211 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 211 6 ) ) PUNCT iajs-2559 211 7 be be VERB iajs-2559 211 8 a a DET iajs-2559 211 9 topological topological ADJ iajs-2559 211 10 space space NOUN iajs-2559 211 11 generated generate VERB iajs-2559 211 12 by by ADP iajs-2559 211 13 a a DET iajs-2559 211 14 tolerance tolerance NOUN iajs-2559 211 15 graph graph NOUN iajs-2559 211 16 𝐷. 𝐷. PROPN iajs-2559 211 17 then then ADV iajs-2559 211 18 ( ( PUNCT iajs-2559 211 19 1 1 NUM iajs-2559 211 20 ) ) PUNCT iajs-2559 211 21 ( ( PUNCT iajs-2559 211 22 𝐷 𝐷 NOUN iajs-2559 211 23 , , PUNCT iajs-2559 211 24 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 211 25 ) ) PUNCT iajs-2559 211 26 is be AUX iajs-2559 211 27 a a DET iajs-2559 211 28 locally locally ADV iajs-2559 211 29 connected connect VERB iajs-2559 211 30 space space NOUN iajs-2559 211 31 ( ( PUNCT iajs-2559 211 32 2 2 NUM iajs-2559 211 33 ) ) PUNCT iajs-2559 211 34 ( ( PUNCT iajs-2559 211 35 𝐷 𝐷 NOUN iajs-2559 211 36 , , PUNCT iajs-2559 211 37 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 211 38 ) ) PUNCT iajs-2559 211 39 is be AUX iajs-2559 211 40 a a DET iajs-2559 211 41 locally locally ADV iajs-2559 211 42 separable separable ADJ iajs-2559 211 43 space space NOUN iajs-2559 211 44 , , PUNCT iajs-2559 211 45 ( ( PUNCT iajs-2559 211 46 3 3 X iajs-2559 211 47 ) ) PUNCT iajs-2559 211 48 ( ( PUNCT iajs-2559 211 49 𝐷 𝐷 NOUN iajs-2559 211 50 , , PUNCT iajs-2559 211 51 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 211 52 ) ) PUNCT iajs-2559 211 53 is be AUX iajs-2559 211 54 a a DET iajs-2559 211 55 regular regular ADJ iajs-2559 211 56 space space NOUN iajs-2559 211 57 , , PUNCT iajs-2559 211 58 ( ( PUNCT iajs-2559 211 59 4 4 NUM iajs-2559 211 60 ) ) PUNCT iajs-2559 211 61 ( ( PUNCT iajs-2559 211 62 𝐷 𝐷 NOUN iajs-2559 211 63 , , PUNCT iajs-2559 211 64 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 211 65 ) ) PUNCT iajs-2559 211 66 is be AUX iajs-2559 211 67 a a DET iajs-2559 211 68 normal normal ADJ iajs-2559 211 69 space space NOUN iajs-2559 211 70 , , PUNCT iajs-2559 211 71 ( ( PUNCT iajs-2559 211 72 5 5 NUM iajs-2559 211 73 ) ) PUNCT iajs-2559 211 74 ( ( PUNCT iajs-2559 211 75 𝐷 𝐷 NOUN iajs-2559 211 76 , , PUNCT iajs-2559 211 77 𝜏𝐷)is 𝜏𝐷)is ADJ iajs-2559 211 78 a a DET iajs-2559 211 79 pseudo pseudo NOUN iajs-2559 211 80 - - ADJ iajs-2559 211 81 metrizable metrizable ADJ iajs-2559 211 82 space space NOUN iajs-2559 211 83 . . PUNCT iajs-2559 212 1 proof.(1 proof.(1 X iajs-2559 212 2 ) ) PUNCT iajs-2559 212 3 by by ADP iajs-2559 212 4 lemma lemma PROPN iajs-2559 212 5 3.1(2 3.1(2 NUM iajs-2559 212 6 ) ) PUNCT iajs-2559 212 7 every every DET iajs-2559 212 8 open open ADJ iajs-2559 212 9 neighborhood neighborhood NOUN iajs-2559 212 10 of of ADP iajs-2559 212 11 ɍ ɍ PROPN iajs-2559 212 12 contains contain VERB iajs-2559 212 13 𝐿ɍ 𝐿ɍ PROPN iajs-2559 212 14 which which PRON iajs-2559 212 15 is be AUX iajs-2559 212 16 connected connect VERB iajs-2559 212 17 . . PUNCT iajs-2559 213 1 ( ( PUNCT iajs-2559 213 2 2)since 2)since NUM iajs-2559 213 3 { { PUNCT iajs-2559 213 4 𝐿ɍ 𝐿ɍ NOUN iajs-2559 213 5 } } PUNCT iajs-2559 213 6 is be AUX iajs-2559 213 7 an an DET iajs-2559 213 8 open open ADJ iajs-2559 213 9 neighborhood neighborhood NOUN iajs-2559 213 10 base base NOUN iajs-2559 213 11 of of ADP iajs-2559 213 12 ɍ ɍ NOUN iajs-2559 213 13 , , PUNCT iajs-2559 213 14 we we PRON iajs-2559 213 15 just just ADV iajs-2559 213 16 need need VERB iajs-2559 213 17 to to PART iajs-2559 213 18 show show VERB iajs-2559 213 19 that that SCONJ iajs-2559 213 20 𝐿ɍ 𝐿ɍ PRON iajs-2559 213 21 is be AUX iajs-2559 213 22 a a DET iajs-2559 213 23 separable separable ADJ iajs-2559 213 24 subset subset NOUN iajs-2559 213 25 of of ADP iajs-2559 213 26 ( ( PUNCT iajs-2559 213 27 𝐷 𝐷 PROPN iajs-2559 213 28 , , PUNCT iajs-2559 213 29 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 213 30 ) ) PUNCT iajs-2559 213 31 . . PUNCT iajs-2559 214 1 let let VERB iajs-2559 214 2 { { PUNCT iajs-2559 214 3 ɍ}Μ…Μ… ɍ}Μ…Μ… PUNCT iajs-2559 214 4 Μ…Μ… Μ…Μ… PROPN iajs-2559 214 5 be be AUX iajs-2559 214 6 the the DET iajs-2559 214 7 closure closure NOUN iajs-2559 214 8 of of ADP iajs-2559 214 9 { { PUNCT iajs-2559 214 10 ɍ ɍ NOUN iajs-2559 214 11 } } PUNCT iajs-2559 214 12 and and CCONJ iajs-2559 214 13 suppose suppose VERB iajs-2559 214 14 that that SCONJ iajs-2559 214 15 there there PRON iajs-2559 214 16 exists exist VERB iajs-2559 214 17 𝑒 𝑒 PROPN iajs-2559 214 18 ∈ ∈ PROPN iajs-2559 214 19 { { PUNCT iajs-2559 214 20 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 214 21 such such ADJ iajs-2559 214 22 that that SCONJ iajs-2559 214 23 𝑒 𝑒 PROPN iajs-2559 214 24 βˆ‰ βˆ‰ PROPN iajs-2559 214 25 𝐿ɍ 𝐿ɍ PROPN iajs-2559 214 26 , , PUNCT iajs-2559 214 27 so so SCONJ iajs-2559 215 1 [ [ X iajs-2559 215 2 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 215 3 ∩ ∩ NOUN iajs-2559 215 4 [ [ X iajs-2559 215 5 𝑒]𝐷𝛽 𝑒]𝐷𝛽 PROPN iajs-2559 215 6 = = PUNCT iajs-2559 215 7 βˆ…. βˆ…. NOUN iajs-2559 215 8 for for ADP iajs-2559 215 9 an an DET iajs-2559 215 10 open open ADJ iajs-2559 215 11 neighborhood neighborhood NOUN iajs-2559 215 12 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 215 13 of of ADP iajs-2559 215 14 𝑒 𝑒 NOUN iajs-2559 215 15 , , PUNCT iajs-2559 215 16 { { PUNCT iajs-2559 215 17 ɍ ɍ NOUN iajs-2559 215 18 } } PUNCT iajs-2559 215 19 ∩ ∩ NOUN iajs-2559 215 20 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 215 21 = = NOUN iajs-2559 215 22 βˆ… βˆ… NOUN iajs-2559 215 23 , , PUNCT iajs-2559 215 24 so so SCONJ iajs-2559 215 25 𝑒 𝑒 PROPN iajs-2559 215 26 βˆ‰ βˆ‰ X iajs-2559 215 27 { { PUNCT iajs-2559 215 28 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 215 29 112 112 NUM iajs-2559 215 30 ibn ibn PROPN iajs-2559 215 31 al al PROPN iajs-2559 215 32 - - PUNCT iajs-2559 215 33 haitham haitham PROPN iajs-2559 215 34 jour jour X iajs-2559 215 35 . . PROPN iajs-2559 215 36 for for ADP iajs-2559 215 37 pure pure ADJ iajs-2559 215 38 & & CCONJ iajs-2559 215 39 appl appl PROPN iajs-2559 215 40 . . PUNCT iajs-2559 216 1 sci sci PROPN iajs-2559 216 2 . . PROPN iajs-2559 217 1 34 34 NUM iajs-2559 217 2 ( ( PUNCT iajs-2559 217 3 1 1 NUM iajs-2559 217 4 ) ) PUNCT iajs-2559 217 5 2021 2021 NUM iajs-2559 217 6 which which PRON iajs-2559 217 7 is be AUX iajs-2559 217 8 a a DET iajs-2559 217 9 contradiction contradiction NOUN iajs-2559 217 10 , , PUNCT iajs-2559 217 11 hence hence ADV iajs-2559 217 12 , , PUNCT iajs-2559 217 13 𝑒 𝑒 PROPN iajs-2559 217 14 ∈ ∈ PROPN iajs-2559 217 15 𝐿ɍ 𝐿ɍ PRON iajs-2559 217 16 then then ADV iajs-2559 217 17 { { PUNCT iajs-2559 217 18 ɍ}Μ…Μ… ɍ}Μ…Μ… VERB iajs-2559 217 19 Μ…Μ… Μ…Μ… PROPN iajs-2559 217 20 βŠ† βŠ† NUM iajs-2559 217 21 𝐿ɍ 𝐿ɍ PROPN iajs-2559 217 22 . . PUNCT iajs-2559 218 1 on on ADP iajs-2559 218 2 the the DET iajs-2559 218 3 other other ADJ iajs-2559 218 4 hand hand NOUN iajs-2559 218 5 , , PUNCT iajs-2559 218 6 let let VERB iajs-2559 219 1 𝑒 𝑒 PRON iajs-2559 219 2 ∈ ∈ PROPN iajs-2559 219 3 𝐿ɍ 𝐿ɍ PRON iajs-2559 219 4 then then ADV iajs-2559 219 5 𝑒 𝑒 PROPN iajs-2559 219 6 ∈ ∈ NOUN iajs-2559 219 7 [ [ X iajs-2559 219 8 ɍ]𝐷𝛽 ɍ]𝐷𝛽 NOUN iajs-2559 219 9 , , PUNCT iajs-2559 219 10 then then ADV iajs-2559 219 11 𝐿ɍ 𝐿ɍ PROPN iajs-2559 219 12 = = PUNCT iajs-2559 219 13 𝐿𝑒. 𝐿𝑒. PROPN iajs-2559 219 14 suppose suppose VERB iajs-2559 219 15 𝑂 𝑂 PROPN iajs-2559 219 16 is be AUX iajs-2559 219 17 an an DET iajs-2559 219 18 open open ADJ iajs-2559 219 19 neighborhood neighborhood NOUN iajs-2559 219 20 of of ADP iajs-2559 219 21 𝑒 𝑒 NOUN iajs-2559 219 22 , , PUNCT iajs-2559 219 23 so so ADV iajs-2559 219 24 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 219 25 βŠ† βŠ† NUM iajs-2559 219 26 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 219 27 ) ) PUNCT iajs-2559 219 28 then then ADV iajs-2559 219 29 𝐿𝑒 𝐿𝑒 PROPN iajs-2559 219 30 ∩ ∩ ADJ iajs-2559 219 31 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 219 32 ) ) PUNCT iajs-2559 219 33 β‰  β‰  PROPN iajs-2559 219 34 βˆ… βˆ… NOUN iajs-2559 219 35 , , PUNCT iajs-2559 220 1 so so ADV iajs-2559 220 2 𝐿ɍ 𝐿ɍ PROPN iajs-2559 220 3 ∩ ∩ ADJ iajs-2559 220 4 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 220 5 ) ) PUNCT iajs-2559 220 6 β‰  β‰  PROPN iajs-2559 220 7 βˆ… βˆ… NOUN iajs-2559 220 8 , , PUNCT iajs-2559 220 9 then then ADV iajs-2559 220 10 , , PUNCT iajs-2559 220 11 { { PUNCT iajs-2559 220 12 ɍ ɍ NOUN iajs-2559 220 13 } } PUNCT iajs-2559 220 14 ∩ ∩ ADJ iajs-2559 220 15 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 220 16 ) ) PUNCT iajs-2559 220 17 β‰  β‰  PROPN iajs-2559 220 18 βˆ… βˆ… NOUN iajs-2559 220 19 then then ADV iajs-2559 220 20 𝑒 𝑒 PROPN iajs-2559 220 21 ∈ ∈ PROPN iajs-2559 220 22 { { PUNCT iajs-2559 220 23 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 220 24 , , PUNCT iajs-2559 220 25 so so ADV iajs-2559 220 26 𝐿ɍ 𝐿ɍ ADV iajs-2559 220 27 βŠ† βŠ† NUM iajs-2559 220 28 { { PUNCT iajs-2559 220 29 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 220 30 . . PUNCT iajs-2559 221 1 hence hence ADV iajs-2559 221 2 , , PUNCT iajs-2559 221 3 𝐿ɍ 𝐿ɍ PROPN iajs-2559 221 4 = = PRON iajs-2559 221 5 { { PUNCT iajs-2559 221 6 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 221 7 , , PUNCT iajs-2559 221 8 and and CCONJ iajs-2559 221 9 we we PRON iajs-2559 221 10 obtained obtain VERB iajs-2559 221 11 that that SCONJ iajs-2559 221 12 { { PUNCT iajs-2559 221 13 ɍ ɍ X iajs-2559 221 14 } } PUNCT iajs-2559 221 15 is be AUX iajs-2559 221 16 countable countable ADJ iajs-2559 221 17 dense dense ADJ iajs-2559 221 18 subset subset NOUN iajs-2559 221 19 of of ADP iajs-2559 221 20 𝐿ɍ 𝐿ɍ PROPN iajs-2559 221 21 which which PRON iajs-2559 221 22 implies imply VERB iajs-2559 221 23 to to ADP iajs-2559 221 24 𝐿ɍ 𝐿ɍ PROPN iajs-2559 221 25 is be AUX iajs-2559 221 26 separable separable ADJ iajs-2559 221 27 subset subset NOUN iajs-2559 221 28 of of ADP iajs-2559 221 29 ( ( PUNCT iajs-2559 221 30 𝐷 𝐷 PROPN iajs-2559 221 31 , , PUNCT iajs-2559 221 32 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 221 33 ) ) PUNCT iajs-2559 221 34 . . PUNCT iajs-2559 222 1 hence hence ADV iajs-2559 222 2 , , PUNCT iajs-2559 222 3 ( ( PUNCT iajs-2559 222 4 𝐷 𝐷 NOUN iajs-2559 222 5 , , PUNCT iajs-2559 222 6 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 222 7 ) ) PUNCT iajs-2559 222 8 is be AUX iajs-2559 222 9 locally locally ADV iajs-2559 222 10 separable separable ADJ iajs-2559 222 11 space space NOUN iajs-2559 222 12 . . PUNCT iajs-2559 223 1 ( ( PUNCT iajs-2559 223 2 3 3 X iajs-2559 223 3 ) ) PUNCT iajs-2559 223 4 let let VERB iajs-2559 223 5 𝑄 𝑄 PRON iajs-2559 223 6 be be AUX iajs-2559 223 7 closed close VERB iajs-2559 223 8 subgraph subgraph NOUN iajs-2559 223 9 of of ADP iajs-2559 223 10 𝐷 𝐷 PROPN iajs-2559 223 11 and and CCONJ iajs-2559 223 12 ɍ ɍ PROPN iajs-2559 223 13 ∈ ∈ PROPN iajs-2559 223 14 𝑉(𝑄)𝑐 𝑉(𝑄)𝑐 PROPN iajs-2559 223 15 , , PUNCT iajs-2559 223 16 by by ADP iajs-2559 223 17 lemma lemma PROPN iajs-2559 223 18 4.1 4.1 NUM iajs-2559 223 19 𝑄 𝑄 PROPN iajs-2559 223 20 is be AUX iajs-2559 223 21 open open ADJ iajs-2559 223 22 if if SCONJ iajs-2559 223 23 and and CCONJ iajs-2559 223 24 only only ADV iajs-2559 223 25 if if SCONJ iajs-2559 223 26 𝑄 𝑄 PRON iajs-2559 223 27 is be AUX iajs-2559 223 28 closed close VERB iajs-2559 223 29 , , PUNCT iajs-2559 223 30 so𝑄 so𝑄 NOUN iajs-2559 223 31 and and CCONJ iajs-2559 223 32 𝑄𝑐 𝑄𝑐 PROPN iajs-2559 223 33 are be AUX iajs-2559 223 34 two two NUM iajs-2559 223 35 open open ADJ iajs-2559 223 36 disjoint disjoint NOUN iajs-2559 223 37 subgraph subgraph NOUN iajs-2559 223 38 of of ADP iajs-2559 223 39 𝐷 𝐷 NOUN iajs-2559 223 40 such such ADJ iajs-2559 223 41 that that PRON iajs-2559 223 42 𝑉(𝑄 𝑉(𝑄 NOUN iajs-2559 223 43 ) ) PUNCT iajs-2559 223 44 βŠ† βŠ† NUM iajs-2559 223 45 𝑉(𝑄 𝑉(𝑄 NOUN iajs-2559 223 46 ) ) PUNCT iajs-2559 223 47 and and CCONJ iajs-2559 223 48 ɍ ɍ PROPN iajs-2559 223 49 ∈ ∈ PROPN iajs-2559 223 50 𝑉(𝑄)𝑐. 𝑉(𝑄)𝑐. NUM iajs-2559 223 51 hence hence ADV iajs-2559 223 52 ( ( PUNCT iajs-2559 223 53 𝐷 𝐷 NOUN iajs-2559 223 54 , , PUNCT iajs-2559 223 55 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 223 56 ) ) PUNCT iajs-2559 223 57 is be AUX iajs-2559 223 58 a a DET iajs-2559 223 59 regular regular ADJ iajs-2559 223 60 space space NOUN iajs-2559 223 61 . . PUNCT iajs-2559 224 1 ( ( PUNCT iajs-2559 224 2 4 4 X iajs-2559 224 3 ) ) PUNCT iajs-2559 224 4 let let VERB iajs-2559 224 5 𝑄 𝑄 PRON iajs-2559 224 6 , , PUNCT iajs-2559 224 7 𝑀 𝑀 PROPN iajs-2559 224 8 are be AUX iajs-2559 224 9 two two NUM iajs-2559 224 10 disjoint disjoint NOUN iajs-2559 224 11 closed closed ADJ iajs-2559 224 12 subgraphs subgraph NOUN iajs-2559 224 13 of of ADP iajs-2559 224 14 𝐷 𝐷 NOUN iajs-2559 224 15 , , PUNCT iajs-2559 224 16 then then ADV iajs-2559 224 17 by by ADP iajs-2559 224 18 lemma lemma PROPN iajs-2559 224 19 4.1 4.1 NUM iajs-2559 224 20 they they PRON iajs-2559 224 21 are be AUX iajs-2559 224 22 also also ADV iajs-2559 224 23 disjoint disjoint NOUN iajs-2559 224 24 closed closed ADJ iajs-2559 224 25 subgraphs subgraph NOUN iajs-2559 224 26 of of ADP iajs-2559 224 27 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 224 28 ) ) PUNCT iajs-2559 224 29 . . PUNCT iajs-2559 225 1 but but CCONJ iajs-2559 225 2 we we PRON iajs-2559 225 3 have have VERB iajs-2559 225 4 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 225 5 ) ) PUNCT iajs-2559 225 6 βŠ† βŠ† NUM iajs-2559 225 7 𝑉(𝑄)and 𝑉(𝑄)and SYM iajs-2559 225 8 𝑉(𝑀 𝑉(𝑀 ADJ iajs-2559 225 9 ) ) PUNCT iajs-2559 225 10 βŠ† βŠ† NUM iajs-2559 225 11 𝑉(𝑀 𝑉(𝑀 ADJ iajs-2559 225 12 ) ) PUNCT iajs-2559 225 13 . . PUNCT iajs-2559 226 1 hence hence ADV iajs-2559 226 2 ( ( PUNCT iajs-2559 226 3 𝐷 𝐷 NOUN iajs-2559 226 4 , , PUNCT iajs-2559 226 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 226 6 ) ) PUNCT iajs-2559 226 7 is be AUX iajs-2559 226 8 a a DET iajs-2559 226 9 normal normal ADJ iajs-2559 226 10 space space NOUN iajs-2559 226 11 . . PUNCT iajs-2559 227 1 ( ( PUNCT iajs-2559 227 2 5 5 NUM iajs-2559 227 3 ) ) PUNCT iajs-2559 227 4 since since SCONJ iajs-2559 227 5 there there PRON iajs-2559 227 6 exists exist VERB iajs-2559 227 7 the the DET iajs-2559 227 8 trivial trivial ADJ iajs-2559 227 9 pseudo pseudo NOUN iajs-2559 227 10 - - ADJ iajs-2559 227 11 metrizable metrizable ADJ iajs-2559 227 12 map map NOUN iajs-2559 227 13 𝑑 𝑑 AUX iajs-2559 227 14 induced induce VERB iajs-2559 227 15 by by ADP iajs-2559 227 16 the the DET iajs-2559 227 17 pseudo pseudo NOUN iajs-2559 227 18 - - ADJ iajs-2559 227 19 metrizable metrizable ADJ iajs-2559 227 20 space space NOUN iajs-2559 227 21 , , PUNCT iajs-2559 227 22 where where SCONJ iajs-2559 227 23 𝑑 𝑑 VERB iajs-2559 227 24 : : PUNCT iajs-2559 227 25 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 227 26 ) ) PUNCT iajs-2559 227 27 Γ— Γ— NOUN iajs-2559 227 28 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 227 29 ) ) PUNCT iajs-2559 227 30 ⟢ ⟢ NOUN iajs-2559 227 31 [ [ X iajs-2559 227 32 0 0 NUM iajs-2559 227 33 , , PUNCT iajs-2559 227 34 ∞ ∞ PROPN iajs-2559 227 35 ) ) PUNCT iajs-2559 227 36 , , PUNCT iajs-2559 227 37 such such ADJ iajs-2559 227 38 that that SCONJ iajs-2559 227 39 𝑑 𝑑 PRON iajs-2559 227 40 = = X iajs-2559 227 41 { { PUNCT iajs-2559 227 42 1 1 NUM iajs-2559 227 43 𝑖𝑓 𝑖𝑓 NUM iajs-2559 227 44 ɍ ɍ PROPN iajs-2559 227 45 = = X iajs-2559 227 46 𝑒 𝑒 X iajs-2559 227 47 0 0 NUM iajs-2559 227 48 𝑖𝑓 𝑖𝑓 NUM iajs-2559 227 49 ɍ ɍ PROPN iajs-2559 227 50 β‰  β‰  PROPN iajs-2559 227 51 𝑒 𝑒 NOUN iajs-2559 227 52 for for ADP iajs-2559 227 53 any any DET iajs-2559 227 54 ɍ ɍ PROPN iajs-2559 227 55 ∈ ∈ NOUN iajs-2559 227 56 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 227 57 ) ) PUNCT iajs-2559 227 58 and and CCONJ iajs-2559 227 59 πœ– πœ– X iajs-2559 227 60 > > X iajs-2559 227 61 0 0 NUM iajs-2559 227 62 , , PUNCT iajs-2559 227 63 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 227 64 , , PUNCT iajs-2559 227 65 πœ– πœ– X iajs-2559 227 66 ) ) PUNCT iajs-2559 227 67 = = SYM iajs-2559 227 68 { { PUNCT iajs-2559 227 69 { { PUNCT iajs-2559 227 70 ɍ ɍ NOUN iajs-2559 227 71 } } PUNCT iajs-2559 227 72 π‘–π‘“πœ– π‘–π‘“πœ– ADV iajs-2559 227 73 < < X iajs-2559 227 74 1 1 NUM iajs-2559 227 75 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 227 76 ) ) PUNCT iajs-2559 227 77 𝑖𝑓 𝑖𝑓 ADP iajs-2559 228 1 πœ– πœ– X iajs-2559 228 2 β‰₯ β‰₯ NUM iajs-2559 228 3 1 1 NUM iajs-2559 228 4 then then ADV iajs-2559 228 5 , , PUNCT iajs-2559 228 6 { { PUNCT iajs-2559 228 7 ɍ ɍ NOUN iajs-2559 228 8 } } PUNCT iajs-2559 228 9 ∈ ∈ PROPN iajs-2559 228 10 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 228 11 , , PUNCT iajs-2559 228 12 so so CCONJ iajs-2559 228 13 ( ( PUNCT iajs-2559 228 14 𝐷 𝐷 NOUN iajs-2559 228 15 , , PUNCT iajs-2559 228 16 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 228 17 ) ) PUNCT iajs-2559 228 18 is be AUX iajs-2559 228 19 pseudo pseudo NOUN iajs-2559 228 20 - - NOUN iajs-2559 228 21 discrete discrete ADJ iajs-2559 228 22 , , PUNCT iajs-2559 228 23 then then ADV iajs-2559 228 24 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 228 25 , , PUNCT iajs-2559 228 26 1 1 NUM iajs-2559 228 27 ) ) PUNCT iajs-2559 228 28 = = PRON iajs-2559 228 29 { { PUNCT iajs-2559 228 30 𝑒 𝑒 PRON iajs-2559 228 31 ∈ ∈ PROPN iajs-2559 228 32 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 228 33 ): ): PUNCT iajs-2559 228 34 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 228 35 , , PUNCT iajs-2559 228 36 𝑒 𝑒 NOUN iajs-2559 228 37 ) ) PUNCT iajs-2559 228 38 < < X iajs-2559 228 39 1 1 NUM iajs-2559 228 40 } } PUNCT iajs-2559 228 41 = = PRON iajs-2559 228 42 { { PUNCT iajs-2559 228 43 𝑒 𝑒 PART iajs-2559 228 44 ∈ ∈ PROPN iajs-2559 228 45 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 228 46 ): ): PUNCT iajs-2559 228 47 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 228 48 , , PUNCT iajs-2559 228 49 𝑒 𝑒 NOUN iajs-2559 228 50 ) ) PUNCT iajs-2559 228 51 = = SYM iajs-2559 228 52 0 0 X iajs-2559 228 53 } } PUNCT iajs-2559 228 54 = = SYM iajs-2559 228 55 { { PUNCT iajs-2559 228 56 ɍ ɍ NOUN iajs-2559 228 57 } } PUNCT iajs-2559 228 58 . . PUNCT iajs-2559 229 1 thus thus ADV iajs-2559 229 2 { { PUNCT iajs-2559 229 3 𝐡(ɍ 𝐡(ɍ NUM iajs-2559 229 4 , , PUNCT iajs-2559 229 5 πœ– πœ– PROPN iajs-2559 229 6 ): ): PUNCT iajs-2559 229 7 ɍ ɍ PROPN iajs-2559 229 8 ∈ ∈ NOUN iajs-2559 229 9 𝑉(𝐷)π‘Žπ‘›π‘‘ 𝑉(𝐷)π‘Žπ‘›π‘‘ PUNCT iajs-2559 229 10 πœ– πœ– X iajs-2559 229 11 > > X iajs-2559 229 12 0 0 NUM iajs-2559 229 13 } } PUNCT iajs-2559 229 14 forms form VERB iajs-2559 229 15 a a DET iajs-2559 229 16 base base NOUN iajs-2559 229 17 for for ADP iajs-2559 229 18 ( ( PUNCT iajs-2559 229 19 𝐷 𝐷 NOUN iajs-2559 229 20 , , PUNCT iajs-2559 229 21 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 229 22 ) ) PUNCT iajs-2559 229 23 . . PUNCT iajs-2559 230 1 hence hence ADV iajs-2559 230 2 ( ( PUNCT iajs-2559 230 3 𝐷 𝐷 NOUN iajs-2559 230 4 , , PUNCT iajs-2559 230 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 230 6 ) ) PUNCT iajs-2559 230 7 is be AUX iajs-2559 230 8 pseudo pseudo NOUN iajs-2559 230 9 - - ADJ iajs-2559 230 10 metrizable metrizable ADJ iajs-2559 230 11 . . PUNCT iajs-2559 231 1 5 5 X iajs-2559 231 2 . . X iajs-2559 231 3 approximation approximation NOUN iajs-2559 231 4 spaces space NOUN iajs-2559 231 5 on on ADP iajs-2559 231 6 digraph digraph NOUN iajs-2559 231 7 we we PRON iajs-2559 231 8 will will AUX iajs-2559 231 9 present present VERB iajs-2559 231 10 the the DET iajs-2559 231 11 concept concept NOUN iajs-2559 231 12 of of ADP iajs-2559 231 13 approximation approximation NOUN iajs-2559 231 14 spaces space NOUN iajs-2559 231 15 in in ADP iajs-2559 231 16 this this DET iajs-2559 231 17 part part NOUN iajs-2559 231 18 ; ; PUNCT iajs-2559 231 19 furthermore furthermore ADV iajs-2559 231 20 , , PUNCT iajs-2559 231 21 we we PRON iajs-2559 231 22 will will AUX iajs-2559 231 23 get get VERB iajs-2559 231 24 their their PRON iajs-2559 231 25 characterizations characterization NOUN iajs-2559 231 26 and and CCONJ iajs-2559 231 27 properties property NOUN iajs-2559 231 28 . . PUNCT iajs-2559 232 1 definition definition NOUN iajs-2559 232 2 5.1 5.1 NUM iajs-2559 232 3 . . PUNCT iajs-2559 233 1 let let AUX iajs-2559 233 2 ( ( PUNCT iajs-2559 233 3 𝐷 𝐷 NOUN iajs-2559 233 4 , , PUNCT iajs-2559 233 5 𝜌 𝜌 AUX iajs-2559 233 6 ) ) PUNCT iajs-2559 233 7 be be VERB iajs-2559 233 8 a a DET iajs-2559 233 9 topological topological ADJ iajs-2559 233 10 space space NOUN iajs-2559 233 11 , , PUNCT iajs-2559 233 12 then then ADV iajs-2559 233 13 ( ( PUNCT iajs-2559 233 14 𝐷 𝐷 PROPN iajs-2559 233 15 , , PUNCT iajs-2559 233 16 𝜌 𝜌 NOUN iajs-2559 233 17 ) ) PUNCT iajs-2559 233 18 is be AUX iajs-2559 233 19 called call VERB iajs-2559 233 20 an an DET iajs-2559 233 21 approximation approximation NOUN iajs-2559 233 22 space space NOUN iajs-2559 233 23 if if SCONJ iajs-2559 233 24 there there PRON iajs-2559 233 25 exists exist VERB iajs-2559 233 26 an an DET iajs-2559 233 27 equivalence equivalence NOUN iajs-2559 233 28 graph graph NOUN iajs-2559 233 29 𝐷 𝐷 PROPN iajs-2559 233 30 = = SYM iajs-2559 233 31 ( ( PUNCT iajs-2559 233 32 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 233 33 ) ) PUNCT iajs-2559 233 34 , , PUNCT iajs-2559 233 35 𝐸(𝐷))such 𝐸(𝐷))such NOUN iajs-2559 233 36 that that SCONJ iajs-2559 233 37 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 233 38 = = SYM iajs-2559 233 39 𝜌. 𝜌. NOUN iajs-2559 233 40 according accord VERB iajs-2559 233 41 to to ADP iajs-2559 233 42 lemma lemma PROPN iajs-2559 233 43 4.1 4.1 NUM iajs-2559 233 44 , , PUNCT iajs-2559 233 45 we we PRON iajs-2559 233 46 get get VERB iajs-2559 233 47 that that PRON iajs-2559 233 48 approximating approximate VERB iajs-2559 233 49 spaces space NOUN iajs-2559 233 50 are be AUX iajs-2559 233 51 pseudo pseudo NOUN iajs-2559 233 52 - - ADJ iajs-2559 233 53 discrete discrete ADJ iajs-2559 233 54 spaces space NOUN iajs-2559 233 55 . . PUNCT iajs-2559 234 1 but but CCONJ iajs-2559 234 2 the the DET iajs-2559 234 3 question question NOUN iajs-2559 234 4 , , PUNCT iajs-2559 234 5 would would AUX iajs-2559 234 6 pseudo pseudo VERB iajs-2559 234 7 - - NOUN iajs-2559 234 8 discrete discrete ADJ iajs-2559 234 9 spaces space NOUN iajs-2559 234 10 are be AUX iajs-2559 234 11 approximating approximate VERB iajs-2559 234 12 space space NOUN iajs-2559 234 13 ? ? PUNCT iajs-2559 235 1 this this DET iajs-2559 235 2 problem problem NOUN iajs-2559 235 3 is be AUX iajs-2559 235 4 certainly certainly ADV iajs-2559 235 5 answered answer VERB iajs-2559 235 6 by by ADP iajs-2559 235 7 the the DET iajs-2559 235 8 following follow VERB iajs-2559 235 9 theorem theorem PROPN iajs-2559 235 10 . . PUNCT iajs-2559 235 11 theorem theorem VERB iajs-2559 235 12 5.2 5.2 NUM iajs-2559 235 13 . . PUNCT iajs-2559 236 1 6.2 6.2 NUM iajs-2559 236 2 if if SCONJ iajs-2559 236 3 ( ( PUNCT iajs-2559 236 4 𝐷 𝐷 NOUN iajs-2559 236 5 , , PUNCT iajs-2559 236 6 𝜌 𝜌 NOUN iajs-2559 236 7 ) ) PUNCT iajs-2559 236 8 is be AUX iajs-2559 236 9 a a DET iajs-2559 236 10 topological topological ADJ iajs-2559 236 11 space space NOUN iajs-2559 236 12 , , PUNCT iajs-2559 236 13 we we PRON iajs-2559 236 14 have have VERB iajs-2559 236 15 the the DET iajs-2559 236 16 next next ADJ iajs-2559 236 17 equivalence equivalence NOUN iajs-2559 236 18 : : PUNCT iajs-2559 236 19 ( ( PUNCT iajs-2559 236 20 1 1 X iajs-2559 236 21 ) ) PUNCT iajs-2559 236 22 ( ( PUNCT iajs-2559 236 23 𝐷 𝐷 NOUN iajs-2559 236 24 , , PUNCT iajs-2559 236 25 𝜌 𝜌 NOUN iajs-2559 236 26 ) ) PUNCT iajs-2559 236 27 is be AUX iajs-2559 236 28 an an DET iajs-2559 236 29 approximating approximate VERB iajs-2559 236 30 space space NOUN iajs-2559 236 31 , , PUNCT iajs-2559 236 32 ( ( PUNCT iajs-2559 236 33 2 2 NUM iajs-2559 236 34 ) ) PUNCT iajs-2559 236 35 ( ( PUNCT iajs-2559 236 36 𝐷 𝐷 NOUN iajs-2559 236 37 , , PUNCT iajs-2559 236 38 𝜌 𝜌 NOUN iajs-2559 236 39 ) ) PUNCT iajs-2559 236 40 is be AUX iajs-2559 236 41 both both DET iajs-2559 236 42 pseudo pseudo NOUN iajs-2559 236 43 - - ADJ iajs-2559 236 44 metrizable metrizable ADJ iajs-2559 236 45 and and CCONJ iajs-2559 236 46 pseudo pseudo NOUN iajs-2559 236 47 - - NOUN iajs-2559 236 48 discrete discrete ADJ iajs-2559 236 49 , , PUNCT iajs-2559 236 50 ( ( PUNCT iajs-2559 236 51 3 3 NUM iajs-2559 236 52 ) ) PUNCT iajs-2559 236 53 ( ( PUNCT iajs-2559 236 54 𝐷 𝐷 NOUN iajs-2559 236 55 , , PUNCT iajs-2559 236 56 𝜌 𝜌 NOUN iajs-2559 236 57 ) ) PUNCT iajs-2559 236 58 is be AUX iajs-2559 236 59 pseudo pseudo NOUN iajs-2559 236 60 - - ADJ iajs-2559 236 61 discrete discrete ADJ iajs-2559 236 62 space space NOUN iajs-2559 236 63 . . PUNCT iajs-2559 237 1 proof proof NOUN iajs-2559 237 2 . . PUNCT iajs-2559 238 1 ( ( PUNCT iajs-2559 238 2 1 1 X iajs-2559 238 3 ) ) PUNCT iajs-2559 238 4 ⟹ ⟹ NOUN iajs-2559 238 5 ( ( PUNCT iajs-2559 238 6 2 2 NUM iajs-2559 238 7 ) ) PUNCT iajs-2559 238 8 . . PUNCT iajs-2559 239 1 it it PRON iajs-2559 239 2 holds hold VERB iajs-2559 239 3 depending depend VERB iajs-2559 239 4 on on ADP iajs-2559 239 5 lemma lemma PROPN iajs-2559 239 6 4.1 4.1 NUM iajs-2559 239 7 and and CCONJ iajs-2559 239 8 theorem theorem VERB iajs-2559 239 9 4.6 4.6 NUM iajs-2559 239 10 . . NOUN iajs-2559 239 11 113 113 NUM iajs-2559 239 12 ibn ibn PROPN iajs-2559 239 13 al al PROPN iajs-2559 239 14 - - PUNCT iajs-2559 239 15 haitham haitham PROPN iajs-2559 239 16 jour jour X iajs-2559 239 17 . . PROPN iajs-2559 240 1 for for ADP iajs-2559 240 2 pure pure ADJ iajs-2559 240 3 & & CCONJ iajs-2559 240 4 appl appl PROPN iajs-2559 240 5 . . PUNCT iajs-2559 241 1 sci sci PROPN iajs-2559 241 2 . . PROPN iajs-2559 242 1 34 34 NUM iajs-2559 242 2 ( ( PUNCT iajs-2559 242 3 1 1 NUM iajs-2559 242 4 ) ) PUNCT iajs-2559 242 5 2021 2021 NUM iajs-2559 242 6 ( ( PUNCT iajs-2559 242 7 2 2 NUM iajs-2559 242 8 ) ) PUNCT iajs-2559 242 9 ⟹ ⟹ NOUN iajs-2559 242 10 ( ( PUNCT iajs-2559 242 11 1 1 NUM iajs-2559 242 12 ) ) PUNCT iajs-2559 242 13 . . PUNCT iajs-2559 243 1 let(𝐷 let(𝐷 PROPN iajs-2559 243 2 , , PUNCT iajs-2559 243 3 𝜌)be 𝜌)be PROPN iajs-2559 243 4 both both DET iajs-2559 243 5 pseudo pseudo NOUN iajs-2559 243 6 - - ADJ iajs-2559 243 7 metrizable metrizable ADJ iajs-2559 243 8 and and CCONJ iajs-2559 243 9 pseudo pseudo NOUN iajs-2559 243 10 - - NOUN iajs-2559 243 11 discrete discrete ADJ iajs-2559 243 12 , , PUNCT iajs-2559 243 13 then then ADV iajs-2559 243 14 there there PRON iajs-2559 243 15 exists exist VERB iajs-2559 243 16 a a DET iajs-2559 243 17 pseudo pseudo NOUN iajs-2559 243 18 - - ADJ iajs-2559 243 19 metric metric ADJ iajs-2559 243 20 map map NOUN iajs-2559 243 21 𝑑 𝑑 PROPN iajs-2559 243 22 on on ADP iajs-2559 243 23 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 243 24 ) ) PUNCT iajs-2559 243 25 where where SCONJ iajs-2559 243 26 { { PUNCT iajs-2559 243 27 𝐡(ɍ 𝐡(ɍ NOUN iajs-2559 243 28 , , PUNCT iajs-2559 243 29 πœ– πœ– PROPN iajs-2559 243 30 ): ): PUNCT iajs-2559 243 31 ɍ ɍ PROPN iajs-2559 243 32 ∈ ∈ NOUN iajs-2559 243 33 𝑉(𝐷)π‘Žπ‘›π‘‘ 𝑉(𝐷)π‘Žπ‘›π‘‘ PUNCT iajs-2559 243 34 πœ– πœ– X iajs-2559 243 35 > > X iajs-2559 243 36 0 0 NUM iajs-2559 243 37 } } PUNCT iajs-2559 243 38 is be AUX iajs-2559 243 39 a a DET iajs-2559 243 40 base base NOUN iajs-2559 243 41 for for ADP iajs-2559 243 42 ( ( PUNCT iajs-2559 243 43 𝐷 𝐷 NOUN iajs-2559 243 44 , , PUNCT iajs-2559 243 45 𝜌 𝜌 NOUN iajs-2559 243 46 ) ) PUNCT iajs-2559 243 47 . . PUNCT iajs-2559 244 1 we we PRON iajs-2559 244 2 define define VERB iajs-2559 244 3 a a DET iajs-2559 244 4 graph graph NOUN iajs-2559 244 5 𝐷 𝐷 NOUN iajs-2559 244 6 on on ADP iajs-2559 244 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 244 8 ) ) PUNCT iajs-2559 244 9 as as ADP iajs-2559 244 10 thereinafter thereinafter NOUN iajs-2559 244 11 : : PUNCT iajs-2559 244 12 for for ADP iajs-2559 244 13 all all DET iajs-2559 244 14 ɍ ɍ ADJ iajs-2559 244 15 , , PUNCT iajs-2559 244 16 𝑒 𝑒 PROPN iajs-2559 244 17 ∈ ∈ PROPN iajs-2559 244 18 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 244 19 ) ) PUNCT iajs-2559 244 20 , , PUNCT iajs-2559 244 21 ( ( PUNCT iajs-2559 244 22 ɍ ɍ X iajs-2559 244 23 , , PUNCT iajs-2559 244 24 𝑒 𝑒 ADJ iajs-2559 244 25 ) ) PUNCT iajs-2559 244 26 ∈ ∈ PROPN iajs-2559 244 27 𝐸(𝐷 𝐸(𝐷 NOUN iajs-2559 244 28 ) ) PUNCT iajs-2559 244 29 if if SCONJ iajs-2559 244 30 and and CCONJ iajs-2559 244 31 only only ADV iajs-2559 244 32 if if SCONJ iajs-2559 244 33 𝑑(ɍ 𝑑(ɍ ADJ iajs-2559 244 34 , , PUNCT iajs-2559 244 35 𝑒 𝑒 NOUN iajs-2559 244 36 ) ) PUNCT iajs-2559 244 37 = = SYM iajs-2559 244 38 0 0 X iajs-2559 244 39 . . PUNCT iajs-2559 245 1 since since SCONJ iajs-2559 245 2 𝑑 𝑑 PROPN iajs-2559 245 3 is be AUX iajs-2559 245 4 pseudo pseudo NOUN iajs-2559 245 5 - - ADJ iajs-2559 245 6 metric metric ADJ iajs-2559 245 7 on on ADP iajs-2559 245 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 245 9 ) ) PUNCT iajs-2559 245 10 , , PUNCT iajs-2559 245 11 so so CCONJ iajs-2559 245 12 𝐷 𝐷 PROPN iajs-2559 245 13 is be AUX iajs-2559 245 14 an an DET iajs-2559 245 15 equivalence equivalence NOUN iajs-2559 245 16 graph graph NOUN iajs-2559 245 17 . . PUNCT iajs-2559 246 1 we we PRON iajs-2559 246 2 will will AUX iajs-2559 246 3 prove prove VERB iajs-2559 246 4 that that PRON iajs-2559 246 5 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 246 6 = = SYM iajs-2559 246 7 𝜌. 𝜌. NOUN iajs-2559 246 8 let let VERB iajs-2559 246 9 𝑄 𝑄 PRON iajs-2559 246 10 ∈ ∈ PROPN iajs-2559 246 11 𝜌 𝜌 X iajs-2559 246 12 , , PUNCT iajs-2559 246 13 by by ADP iajs-2559 246 14 proposition proposition NOUN iajs-2559 246 15 2.8 2.8 NUM iajs-2559 246 16 , , PUNCT iajs-2559 246 17 οΏ½ οΏ½ NOUN iajs-2559 246 18 Μ… Μ… NOUN iajs-2559 246 19 οΏ½ οΏ½ NOUN iajs-2559 246 20 = = SYM iajs-2559 246 21 { { PUNCT iajs-2559 246 22 ɍ ɍ PROPN iajs-2559 246 23 ∈ ∈ PROPN iajs-2559 246 24 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 246 25 ): ): PUNCT iajs-2559 246 26 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 246 27 , , PUNCT iajs-2559 246 28 𝑄 𝑄 PROPN iajs-2559 246 29 ) ) PUNCT iajs-2559 246 30 = = PUNCT iajs-2559 246 31 0 0 NUM iajs-2559 246 32 } } PUNCT iajs-2559 246 33 . . PUNCT iajs-2559 247 1 since since SCONJ iajs-2559 247 2 ( ( PUNCT iajs-2559 247 3 𝐷 𝐷 NOUN iajs-2559 247 4 , , PUNCT iajs-2559 247 5 𝜌)is 𝜌)is PROPN iajs-2559 247 6 pseudo pseudo NOUN iajs-2559 247 7 - - NOUN iajs-2559 247 8 discrete discrete NOUN iajs-2559 247 9 , , PUNCT iajs-2559 247 10 𝑄 𝑄 PRON iajs-2559 247 11 is be AUX iajs-2559 247 12 closed close VERB iajs-2559 247 13 in in ADP iajs-2559 247 14 ( ( PUNCT iajs-2559 247 15 𝐷 𝐷 NOUN iajs-2559 247 16 , , PUNCT iajs-2559 247 17 𝜌 𝜌 NOUN iajs-2559 247 18 ) ) PUNCT iajs-2559 247 19 , , PUNCT iajs-2559 247 20 so so CCONJ iajs-2559 247 21 𝑉(𝑄 𝑉(𝑄 VERB iajs-2559 247 22 ) ) PUNCT iajs-2559 248 1 = = PRON iajs-2559 248 2 { { PUNCT iajs-2559 248 3 ɍ ɍ PROPN iajs-2559 248 4 ∈ ∈ PROPN iajs-2559 248 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 248 6 ): ): PUNCT iajs-2559 248 7 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 248 8 , , PUNCT iajs-2559 248 9 𝑄 𝑄 PROPN iajs-2559 248 10 ) ) PUNCT iajs-2559 248 11 = = PUNCT iajs-2559 248 12 0 0 NUM iajs-2559 248 13 } } PUNCT iajs-2559 248 14 . . PUNCT iajs-2559 249 1 it it PRON iajs-2559 249 2 is be AUX iajs-2559 249 3 obvious obvious ADJ iajs-2559 249 4 that that SCONJ iajs-2559 249 5 𝑉(𝑄 𝑉(𝑄 VERB iajs-2559 249 6 ) ) PUNCT iajs-2559 249 7 βŠ† βŠ† NUM iajs-2559 249 8 ⋃{[ɍ]𝐷 ⋃{[ɍ]𝐷 NUM iajs-2559 249 9 : : PUNCT iajs-2559 249 10 ɍ ɍ PROPN iajs-2559 249 11 ∈ ∈ PROPN iajs-2559 249 12 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 249 13 ) ) PUNCT iajs-2559 249 14 } } PUNCT iajs-2559 249 15 . . PUNCT iajs-2559 250 1 if if SCONJ iajs-2559 250 2 𝑒 𝑒 PRON iajs-2559 250 3 ∈ ∈ PROPN iajs-2559 250 4 [ [ X iajs-2559 250 5 ɍ]𝐷 ɍ]𝐷 ADJ iajs-2559 250 6 with with ADP iajs-2559 250 7 ɍ ɍ PROPN iajs-2559 250 8 ∈ ∈ PROPN iajs-2559 250 9 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 250 10 ) ) PUNCT iajs-2559 250 11 , , PUNCT iajs-2559 250 12 then then ADV iajs-2559 250 13 , , PUNCT iajs-2559 250 14 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 250 15 , , PUNCT iajs-2559 250 16 𝑄 𝑄 PROPN iajs-2559 250 17 ) ) PUNCT iajs-2559 250 18 ≀ ≀ NOUN iajs-2559 250 19 𝑑(𝑒 𝑑(𝑒 ADJ iajs-2559 250 20 , , PUNCT iajs-2559 250 21 ɍ ɍ NOUN iajs-2559 250 22 ) ) PUNCT iajs-2559 250 23 = = SYM iajs-2559 251 1 0 0 X iajs-2559 251 2 . . PUNCT iajs-2559 252 1 so so ADV iajs-2559 252 2 𝑒 𝑒 X iajs-2559 252 3 ∈ ∈ PROPN iajs-2559 252 4 { { PUNCT iajs-2559 252 5 ɍ ɍ PROPN iajs-2559 252 6 ∈ ∈ PROPN iajs-2559 252 7 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 252 8 ): ): PUNCT iajs-2559 252 9 𝑑(ɍ 𝑑(ɍ NOUN iajs-2559 252 10 , , PUNCT iajs-2559 252 11 𝑄 𝑄 PROPN iajs-2559 252 12 ) ) PUNCT iajs-2559 252 13 = = SYM iajs-2559 252 14 0 0 X iajs-2559 252 15 } } PUNCT iajs-2559 252 16 = = SYM iajs-2559 252 17 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 252 18 ) ) PUNCT iajs-2559 252 19 , , PUNCT iajs-2559 252 20 that that PRON iajs-2559 252 21 is be AUX iajs-2559 252 22 mean𝑉(𝑄 mean𝑉(𝑄 PROPN iajs-2559 252 23 ) ) PUNCT iajs-2559 252 24 βŠ‡ βŠ‡ NOUN iajs-2559 252 25 ⋃{[ɍ]𝐷 ⋃{[ɍ]𝐷 X iajs-2559 252 26 : : PUNCT iajs-2559 252 27 ɍ ɍ PROPN iajs-2559 252 28 ∈ ∈ PROPN iajs-2559 252 29 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 252 30 ) ) PUNCT iajs-2559 252 31 } } PUNCT iajs-2559 252 32 . . PUNCT iajs-2559 253 1 thus thus ADV iajs-2559 253 2 𝑉(𝑄 𝑉(𝑄 VERB iajs-2559 253 3 ) ) PUNCT iajs-2559 253 4 = = SYM iajs-2559 253 5 ⋃{[ɍ]𝐷 ⋃{[ɍ]𝐷 X iajs-2559 253 6 : : PUNCT iajs-2559 253 7 ɍ ɍ PROPN iajs-2559 253 8 ∈ ∈ PROPN iajs-2559 253 9 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 253 10 ) ) PUNCT iajs-2559 253 11 } } PUNCT iajs-2559 253 12 , , PUNCT iajs-2559 253 13 it it PRON iajs-2559 253 14 follows follow VERB iajs-2559 253 15 that that SCONJ iajs-2559 253 16 𝑉(𝑄 𝑉(𝑄 NOUN iajs-2559 253 17 ) ) PUNCT iajs-2559 253 18 ∈ ∈ PROPN iajs-2559 253 19 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 253 20 , , PUNCT iajs-2559 253 21 so𝜌 so𝜌 NOUN iajs-2559 253 22 βŠ† βŠ† NUM iajs-2559 253 23 𝜏𝐷. 𝜏𝐷. PROPN iajs-2559 253 24 on on ADP iajs-2559 253 25 the the DET iajs-2559 253 26 other other ADJ iajs-2559 253 27 side side NOUN iajs-2559 253 28 , , PUNCT iajs-2559 253 29 let let VERB iajs-2559 253 30 ɍ ɍ PRON iajs-2559 253 31 ∈ ∈ PROPN iajs-2559 253 32 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 253 33 ) ) PUNCT iajs-2559 253 34 , , PUNCT iajs-2559 253 35 by by ADP iajs-2559 253 36 proposition proposition NOUN iajs-2559 253 37 2.8 2.8 NUM iajs-2559 253 38 , , PUNCT iajs-2559 253 39 { { PUNCT iajs-2559 253 40 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 253 41 Μ…Μ… Μ…Μ… PROPN iajs-2559 253 42 = = SYM iajs-2559 253 43 { { PUNCT iajs-2559 253 44 𝑒 𝑒 PRON iajs-2559 253 45 ∈ ∈ PROPN iajs-2559 253 46 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 253 47 ): ): PUNCT iajs-2559 253 48 𝑑(𝑒 𝑑(𝑒 ADJ iajs-2559 253 49 , , PUNCT iajs-2559 253 50 ɍ ɍ NOUN iajs-2559 253 51 ) ) PUNCT iajs-2559 253 52 = = SYM iajs-2559 254 1 0 0 NUM iajs-2559 254 2 } } PUNCT iajs-2559 254 3 , , PUNCT iajs-2559 254 4 then then ADV iajs-2559 254 5 { { PUNCT iajs-2559 254 6 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 254 7 Μ…Μ… Μ…Μ… PROPN iajs-2559 254 8 = = SYM iajs-2559 254 9 { { PUNCT iajs-2559 254 10 ɍ}𝐷. ɍ}𝐷. NOUN iajs-2559 254 11 now now ADV iajs-2559 254 12 [ [ PUNCT iajs-2559 254 13 ɍ]𝐷 ɍ]𝐷 ADJ iajs-2559 254 14 is be AUX iajs-2559 254 15 closed close VERB iajs-2559 254 16 in in ADP iajs-2559 254 17 ( ( PUNCT iajs-2559 254 18 𝐷 𝐷 NOUN iajs-2559 254 19 , , PUNCT iajs-2559 254 20 𝜌 𝜌 NOUN iajs-2559 254 21 ) ) PUNCT iajs-2559 254 22 , , PUNCT iajs-2559 254 23 since since SCONJ iajs-2559 254 24 ( ( PUNCT iajs-2559 254 25 𝐷 𝐷 NOUN iajs-2559 254 26 , , PUNCT iajs-2559 254 27 𝜌)is 𝜌)is PROPN iajs-2559 254 28 pseudo pseudo NOUN iajs-2559 254 29 - - PUNCT iajs-2559 254 30 discrete,[ɍ]𝐷 discrete,[ɍ]𝐷 PROPN iajs-2559 254 31 ∈ ∈ PROPN iajs-2559 254 32 𝜌. 𝜌. NOUN iajs-2559 254 33 since since SCONJ iajs-2559 254 34 { { PUNCT iajs-2559 254 35 [ [ X iajs-2559 254 36 ɍ]𝐷 ɍ]𝐷 ADV iajs-2559 254 37 : : PUNCT iajs-2559 254 38 ɍ ɍ PROPN iajs-2559 254 39 ∈ ∈ PROPN iajs-2559 254 40 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 254 41 ) ) PUNCT iajs-2559 254 42 } } PUNCT iajs-2559 254 43 is be AUX iajs-2559 254 44 a a DET iajs-2559 254 45 base base NOUN iajs-2559 254 46 for for ADP iajs-2559 254 47 ( ( PUNCT iajs-2559 254 48 𝐷 𝐷 NOUN iajs-2559 254 49 , , PUNCT iajs-2559 254 50 𝜏𝐷 𝜏𝐷 NOUN iajs-2559 254 51 ) ) PUNCT iajs-2559 254 52 , , PUNCT iajs-2559 254 53 𝜏𝐷 𝜏𝐷 VERB iajs-2559 254 54 βŠ† βŠ† NUM iajs-2559 254 55 𝜌. 𝜌. NOUN iajs-2559 254 56 hence hence ADV iajs-2559 254 57 𝜏𝐷 𝜏𝐷 ADJ iajs-2559 254 58 = = SYM iajs-2559 254 59 𝜌. 𝜌. NOUN iajs-2559 254 60 this this PRON iajs-2559 254 61 means mean VERB iajs-2559 254 62 that that SCONJ iajs-2559 254 63 ( ( PUNCT iajs-2559 254 64 𝐷 𝐷 NOUN iajs-2559 254 65 , , PUNCT iajs-2559 254 66 𝜌 𝜌 NOUN iajs-2559 254 67 ) ) PUNCT iajs-2559 254 68 are be AUX iajs-2559 254 69 an an DET iajs-2559 254 70 approximation approximation NOUN iajs-2559 254 71 space space NOUN iajs-2559 254 72 . . PUNCT iajs-2559 255 1 ( ( PUNCT iajs-2559 255 2 2 2 NUM iajs-2559 255 3 ) ) PUNCT iajs-2559 255 4 ⟹ ⟹ NOUN iajs-2559 255 5 ( ( PUNCT iajs-2559 255 6 3 3 NUM iajs-2559 255 7 ) ) PUNCT iajs-2559 255 8 . . PUNCT iajs-2559 256 1 clear clear ADJ iajs-2559 256 2 . . PUNCT iajs-2559 257 1 ( ( PUNCT iajs-2559 257 2 3 3 X iajs-2559 257 3 ) ) PUNCT iajs-2559 257 4 ⟹ ⟹ NOUN iajs-2559 257 5 ( ( PUNCT iajs-2559 257 6 2 2 NUM iajs-2559 257 7 ) ) PUNCT iajs-2559 257 8 . . PUNCT iajs-2559 258 1 let(𝐷 let(𝐷 PROPN iajs-2559 258 2 , , PUNCT iajs-2559 258 3 𝜌)be 𝜌)be ADJ iajs-2559 258 4 pseudo pseudo NOUN iajs-2559 258 5 - - NOUN iajs-2559 258 6 discrete discrete NOUN iajs-2559 258 7 . . PUNCT iajs-2559 259 1 for for ADP iajs-2559 259 2 eachɍ eachɍ NOUN iajs-2559 259 3 ∈ ∈ PROPN iajs-2559 259 4 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 259 5 ) ) PUNCT iajs-2559 259 6 , , PUNCT iajs-2559 259 7 𝐢(ɍ 𝐢(ɍ PRON iajs-2559 259 8 ) ) PUNCT iajs-2559 259 9 denoted denote VERB iajs-2559 259 10 a a DET iajs-2559 259 11 connected connected ADJ iajs-2559 259 12 component component NOUN iajs-2559 259 13 with with ADP iajs-2559 259 14 ɍ ɍ PROPN iajs-2559 259 15 ∈ ∈ PROPN iajs-2559 259 16 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 259 17 ) ) PUNCT iajs-2559 259 18 , , PUNCT iajs-2559 259 19 then then ADV iajs-2559 259 20 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 259 21 ) ) PUNCT iajs-2559 259 22 is be AUX iajs-2559 259 23 closed close VERB iajs-2559 259 24 in in ADP iajs-2559 259 25 ( ( PUNCT iajs-2559 259 26 𝐷 𝐷 NOUN iajs-2559 259 27 , , PUNCT iajs-2559 259 28 𝜌 𝜌 NOUN iajs-2559 259 29 ) ) PUNCT iajs-2559 259 30 . . PUNCT iajs-2559 260 1 so so ADV iajs-2559 260 2 { { PUNCT iajs-2559 260 3 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 260 4 Μ…Μ… Μ…Μ… PROPN iajs-2559 260 5 βŠ† βŠ† NUM iajs-2559 260 6 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 260 7 ) ) PUNCT iajs-2559 260 8 . . PUNCT iajs-2559 261 1 let let VERB iajs-2559 261 2 𝑒 𝑒 PRON iajs-2559 261 3 ∈ ∈ PROPN iajs-2559 261 4 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 261 5 ) ) PUNCT iajs-2559 261 6 , , PUNCT iajs-2559 261 7 since since SCONJ iajs-2559 261 8 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 261 9 ) ) PUNCT iajs-2559 261 10 is be AUX iajs-2559 261 11 a a DET iajs-2559 261 12 connected connected ADJ iajs-2559 261 13 component component NOUN iajs-2559 261 14 with with ADP iajs-2559 261 15 ɍ ɍ PROPN iajs-2559 261 16 ∈ ∈ PROPN iajs-2559 261 17 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 261 18 ) ) PUNCT iajs-2559 261 19 , , PUNCT iajs-2559 261 20 there there PRON iajs-2559 261 21 exists exist VERB iajs-2559 261 22 a a DET iajs-2559 261 23 connected connected ADJ iajs-2559 261 24 subgraph subgraph NOUN iajs-2559 261 25 𝑄 𝑄 PROPN iajs-2559 261 26 of of ADP iajs-2559 261 27 𝐷whereɍ 𝐷whereɍ PROPN iajs-2559 261 28 , , PUNCT iajs-2559 261 29 𝑒 𝑒 PROPN iajs-2559 261 30 ∈ ∈ PROPN iajs-2559 261 31 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 261 32 ) ) PUNCT iajs-2559 261 33 . . PUNCT iajs-2559 262 1 since since SCONJ iajs-2559 262 2 𝐷 𝐷 PROPN iajs-2559 262 3 is be AUX iajs-2559 262 4 pseudo pseudo NOUN iajs-2559 262 5 - - NOUN iajs-2559 262 6 discrete discrete ADJ iajs-2559 262 7 , , PUNCT iajs-2559 262 8 { { PUNCT iajs-2559 262 9 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 262 10 Μ…Μ… Μ…Μ… PROPN iajs-2559 262 11 is be AUX iajs-2559 262 12 both both CCONJ iajs-2559 262 13 open open ADJ iajs-2559 262 14 and and CCONJ iajs-2559 262 15 closed close VERB iajs-2559 262 16 in in ADP iajs-2559 262 17 ( ( PUNCT iajs-2559 262 18 𝐷 𝐷 NOUN iajs-2559 262 19 , , PUNCT iajs-2559 262 20 𝜌 𝜌 NOUN iajs-2559 262 21 ) ) PUNCT iajs-2559 262 22 . . PUNCT iajs-2559 263 1 note note VERB iajs-2559 263 2 that that SCONJ iajs-2559 263 3 { { PUNCT iajs-2559 263 4 ɍ}Μ…Μ… ɍ}Μ…Μ… PROPN iajs-2559 263 5 Μ…Μ… Μ…Μ… PROPN iajs-2559 263 6 ∩ ∩ NOUN iajs-2559 263 7 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 263 8 ) ) PUNCT iajs-2559 263 9 is be AUX iajs-2559 263 10 both both CCONJ iajs-2559 263 11 open open ADJ iajs-2559 263 12 and and CCONJ iajs-2559 263 13 closed close VERB iajs-2559 263 14 in in ADP iajs-2559 263 15 the the DET iajs-2559 263 16 subspace subspace NOUN iajs-2559 263 17 𝑄 𝑄 PROPN iajs-2559 263 18 and and CCONJ iajs-2559 263 19 𝑄 𝑄 PROPN iajs-2559 263 20 is be AUX iajs-2559 263 21 connected connect VERB iajs-2559 263 22 . . PUNCT iajs-2559 264 1 then then ADV iajs-2559 264 2 { { PUNCT iajs-2559 264 3 ɍ}Μ…Μ… ɍ}Μ…Μ… PROPN iajs-2559 264 4 Μ…Μ… Μ…Μ… PROPN iajs-2559 264 5 ∩ ∩ PROPN iajs-2559 264 6 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 264 7 ) ) PUNCT iajs-2559 264 8 = = SYM iajs-2559 264 9 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 264 10 ) ) PUNCT iajs-2559 264 11 , , PUNCT iajs-2559 264 12 so so SCONJ iajs-2559 264 13 𝑒 𝑒 PROPN iajs-2559 264 14 ∈ ∈ PROPN iajs-2559 264 15 { { PUNCT iajs-2559 264 16 ɍ}Μ…Μ…Μ…Μ… ɍ}Μ…Μ…Μ…Μ… PROPN iajs-2559 264 17 . . PUNCT iajs-2559 265 1 this this PRON iajs-2559 265 2 indicates indicate VERB iajs-2559 265 3 that that SCONJ iajs-2559 265 4 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 265 5 ) ) PUNCT iajs-2559 265 6 βŠ† βŠ† NUM iajs-2559 265 7 { { PUNCT iajs-2559 265 8 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 265 9 Μ…Μ… Μ…Μ… PROPN iajs-2559 265 10 . . PUNCT iajs-2559 266 1 thus thus ADV iajs-2559 266 2 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 266 3 ) ) PUNCT iajs-2559 266 4 = = PRON iajs-2559 266 5 { { PUNCT iajs-2559 266 6 ɍ}Μ…Μ… ɍ}Μ…Μ… X iajs-2559 266 7 Μ…Μ… Μ…Μ… PROPN iajs-2559 266 8 . . PUNCT iajs-2559 267 1 we we PRON iajs-2559 267 2 define define VERB iajs-2559 267 3 𝑑 𝑑 NOUN iajs-2559 267 4 : : PUNCT iajs-2559 267 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 267 6 ) ) PUNCT iajs-2559 267 7 Γ— Γ— NOUN iajs-2559 267 8 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 267 9 ) ) PUNCT iajs-2559 267 10 ⟢ ⟢ NOUN iajs-2559 267 11 [ [ X iajs-2559 267 12 0 0 NUM iajs-2559 267 13 , , PUNCT iajs-2559 267 14 ∞ ∞ PROPN iajs-2559 267 15 ) ) PUNCT iajs-2559 267 16 as as SCONJ iajs-2559 267 17 follows follow VERB iajs-2559 267 18 : : PUNCT iajs-2559 267 19 𝑑(ɍ 𝑑(ɍ NUM iajs-2559 267 20 , , PUNCT iajs-2559 267 21 𝑒 𝑒 NOUN iajs-2559 267 22 ) ) PUNCT iajs-2559 267 23 = = SYM iajs-2559 267 24 { { PUNCT iajs-2559 267 25 0 0 NUM iajs-2559 267 26 𝑖𝑓 𝑖𝑓 NUM iajs-2559 267 27 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 267 28 ) ) PUNCT iajs-2559 267 29 = = SYM iajs-2559 268 1 𝐢(𝑒 𝐢(𝑒 NOUN iajs-2559 268 2 ) ) PUNCT iajs-2559 268 3 , , PUNCT iajs-2559 268 4 1 1 NUM iajs-2559 268 5 𝑖𝑓 𝑖𝑓 ADP iajs-2559 268 6 𝐢(ɍ 𝐢(ɍ NUM iajs-2559 268 7 ) ) PUNCT iajs-2559 268 8 β‰  β‰  PROPN iajs-2559 268 9 𝐢(𝑒 𝐢(𝑒 NUM iajs-2559 268 10 ) ) PUNCT iajs-2559 268 11 . . PUNCT iajs-2559 269 1 the the DET iajs-2559 269 2 assumption assumption NOUN iajs-2559 269 3 that that SCONJ iajs-2559 269 4 𝑑 𝑑 NOUN iajs-2559 269 5 is be AUX iajs-2559 269 6 pseudo pseudo NOUN iajs-2559 269 7 - - ADJ iajs-2559 269 8 metric metric ADJ iajs-2559 269 9 on on ADV iajs-2559 269 10 𝑉(𝐷)can 𝑉(𝐷)can PUNCT iajs-2559 269 11 be be AUX iajs-2559 269 12 easily easily ADV iajs-2559 269 13 proved prove VERB iajs-2559 269 14 . . PUNCT iajs-2559 270 1 for for ADP iajs-2559 270 2 any any DET iajs-2559 270 3 ɍ ɍ PROPN iajs-2559 270 4 ∈ ∈ NOUN iajs-2559 270 5 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 270 6 ) ) PUNCT iajs-2559 270 7 and and CCONJ iajs-2559 270 8 πœ– πœ– X iajs-2559 270 9 > > X iajs-2559 270 10 0 0 NUM iajs-2559 270 11 , , PUNCT iajs-2559 270 12 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 270 13 , , PUNCT iajs-2559 270 14 πœ– πœ– X iajs-2559 270 15 ) ) PUNCT iajs-2559 270 16 = = SYM iajs-2559 270 17 { { PUNCT iajs-2559 270 18 { { PUNCT iajs-2559 270 19 ɍ}Μ…Μ… ɍ}Μ…Μ… PROPN iajs-2559 270 20 Μ…Μ… Μ…Μ… PROPN iajs-2559 270 21 𝑖𝑓 𝑖𝑓 X iajs-2559 270 22 πœ– πœ– PROPN iajs-2559 270 23 ≀ ≀ NUM iajs-2559 270 24 1 1 NUM iajs-2559 270 25 , , PUNCT iajs-2559 270 26 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 270 27 ) ) PUNCT iajs-2559 270 28 𝑖𝑓 𝑖𝑓 ADP iajs-2559 270 29 πœ– πœ– X iajs-2559 270 30 > > X iajs-2559 270 31 1 1 NUM iajs-2559 270 32 . . PUNCT iajs-2559 271 1 then then ADV iajs-2559 271 2 𝐡(ɍ 𝐡(ɍ NUM iajs-2559 271 3 , , PUNCT iajs-2559 271 4 πœ–)will πœ–)will X iajs-2559 271 5 be be AUX iajs-2559 271 6 closed close VERB iajs-2559 271 7 in in ADP iajs-2559 271 8 ( ( PUNCT iajs-2559 271 9 𝐷 𝐷 NOUN iajs-2559 271 10 , , PUNCT iajs-2559 271 11 𝜌 𝜌 NOUN iajs-2559 271 12 ) ) PUNCT iajs-2559 271 13 . . PUNCT iajs-2559 272 1 because𝐷 because𝐷 PROPN iajs-2559 272 2 is be AUX iajs-2559 272 3 pseudo pseudo NOUN iajs-2559 272 4 - - NOUN iajs-2559 272 5 discrete discrete ADJ iajs-2559 272 6 , , PUNCT iajs-2559 272 7 𝐡(ɍ 𝐡(ɍ PROPN iajs-2559 272 8 , , PUNCT iajs-2559 272 9 πœ– πœ– NOUN iajs-2559 272 10 ) ) PUNCT iajs-2559 272 11 ∈ ∈ PROPN iajs-2559 272 12 𝜌. 𝜌. NOUN iajs-2559 272 13 let let VERB iajs-2559 272 14 ɍ ɍ PRON iajs-2559 272 15 ∈ ∈ NOUN iajs-2559 272 16 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 272 17 ) ) PUNCT iajs-2559 272 18 and and CCONJ iajs-2559 272 19 𝑉(𝑂 𝑉(𝑂 NOUN iajs-2559 272 20 ) ) PUNCT iajs-2559 272 21 ∈ ∈ PROPN iajs-2559 273 1 𝜌 𝜌 X iajs-2559 273 2 with with ADP iajs-2559 273 3 ɍ ɍ PROPN iajs-2559 273 4 ∈ ∈ PROPN iajs-2559 273 5 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 273 6 ) ) PUNCT iajs-2559 273 7 . . PUNCT iajs-2559 274 1 since since SCONJ iajs-2559 274 2 𝐷is 𝐷is PROPN iajs-2559 274 3 pseudo pseudo NOUN iajs-2559 274 4 - - NOUN iajs-2559 274 5 discrete discrete ADJ iajs-2559 274 6 , , PUNCT iajs-2559 274 7 𝑉(𝑄)is 𝑉(𝑄)i NOUN iajs-2559 274 8 closed close VERB iajs-2559 274 9 in in ADP iajs-2559 274 10 ( ( PUNCT iajs-2559 274 11 𝐷 𝐷 NOUN iajs-2559 274 12 , , PUNCT iajs-2559 274 13 𝜌 𝜌 NOUN iajs-2559 274 14 ) ) PUNCT iajs-2559 274 15 . . PUNCT iajs-2559 275 1 so{ɍ}Μ…Μ… so{ɍ}Μ…Μ… PROPN iajs-2559 275 2 Μ…Μ… Μ…Μ… PROPN iajs-2559 275 3 βŠ† βŠ† NUM iajs-2559 275 4 𝑉(𝑄 𝑉(𝑄 PROPN iajs-2559 275 5 ) ) PUNCT iajs-2559 275 6 . . PUNCT iajs-2559 276 1 by by ADP iajs-2559 276 2 proposition proposition NOUN iajs-2559 276 3 2.8,{ɍ}Μ…Μ… 2.8,{ɍ}Μ…Μ… NUM iajs-2559 276 4 Μ…Μ… Μ…Μ… NOUN iajs-2559 276 5 = = PRON iajs-2559 276 6 { { PUNCT iajs-2559 276 7 𝑒 𝑒 PRON iajs-2559 276 8 ∈ ∈ PROPN iajs-2559 276 9 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 276 10 ): ): PUNCT iajs-2559 276 11 𝑑(ɍ 𝑑(ɍ PROPN iajs-2559 276 12 , , PUNCT iajs-2559 276 13 𝑒 𝑒 NOUN iajs-2559 276 14 ) ) PUNCT iajs-2559 276 15 = = SYM iajs-2559 276 16 0 0 NUM iajs-2559 276 17 } } PUNCT iajs-2559 276 18 . . PUNCT iajs-2559 277 1 then then ADV iajs-2559 277 2 . . PUNCT iajs-2559 278 1 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 278 2 ) ) PUNCT iajs-2559 278 3 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-2559 279 1 πœ– πœ– PROPN iajs-2559 279 2 > > X iajs-2559 279 3 0 0 NUM iajs-2559 279 4 } } PUNCT iajs-2559 279 5 is be AUX iajs-2559 279 6 a a DET iajs-2559 279 7 base base NOUN iajs-2559 279 8 for for ADP iajs-2559 279 9 ( ( PUNCT iajs-2559 279 10 𝐷 𝐷 NOUN iajs-2559 279 11 , , PUNCT iajs-2559 279 12 𝜌 𝜌 NOUN iajs-2559 279 13 ) ) PUNCT iajs-2559 279 14 . . PUNCT iajs-2559 280 1 therefore(𝐷 therefore(𝐷 NOUN iajs-2559 280 2 , , PUNCT iajs-2559 280 3 𝜌 𝜌 X iajs-2559 280 4 ) ) PUNCT iajs-2559 280 5 is be AUX iajs-2559 280 6 pseudo pseudo NOUN iajs-2559 280 7 - - ADJ iajs-2559 280 8 metrizable metrizable ADJ iajs-2559 280 9 . . PUNCT iajs-2559 281 1 corollary corollary ADJ iajs-2559 281 2 5.3 5.3 NUM iajs-2559 281 3 . . PUNCT iajs-2559 282 1 discrete discrete ADJ iajs-2559 282 2 spaces space NOUN iajs-2559 282 3 are be AUX iajs-2559 282 4 approximating approximate VERB iajs-2559 282 5 spaces space NOUN iajs-2559 282 6 . . PUNCT iajs-2559 283 1 theorem theorem VERB iajs-2559 283 2 5.4 5.4 NUM iajs-2559 283 3 . . PUNCT iajs-2559 284 1 quotient quotient NOUN iajs-2559 284 2 maps map NOUN iajs-2559 284 3 preserve preserve VERB iajs-2559 284 4 approximating approximate VERB iajs-2559 284 5 spaces space NOUN iajs-2559 284 6 . . PUNCT iajs-2559 285 1 114 114 NUM iajs-2559 285 2 ibn ibn PROPN iajs-2559 285 3 al al PROPN iajs-2559 285 4 - - PUNCT iajs-2559 285 5 haitham haitham PROPN iajs-2559 285 6 jour jour X iajs-2559 285 7 . . PROPN iajs-2559 285 8 for for ADP iajs-2559 285 9 pure pure ADJ iajs-2559 285 10 & & CCONJ iajs-2559 285 11 appl appl PROPN iajs-2559 285 12 . . PUNCT iajs-2559 286 1 sci sci PROPN iajs-2559 286 2 . . PROPN iajs-2559 287 1 34 34 NUM iajs-2559 287 2 ( ( PUNCT iajs-2559 287 3 1 1 NUM iajs-2559 287 4 ) ) PUNCT iajs-2559 287 5 2021 2021 NUM iajs-2559 287 6 proof proof NOUN iajs-2559 287 7 . . PUNCT iajs-2559 287 8 suppose suppose VERB iajs-2559 287 9 that that SCONJ iajs-2559 287 10 the the DET iajs-2559 287 11 image image NOUN iajs-2559 287 12 of of ADP iajs-2559 287 13 an an DET iajs-2559 287 14 approximating approximate VERB iajs-2559 287 15 space space NOUN iajs-2559 287 16 𝐷under 𝐷under ADP iajs-2559 287 17 a a DET iajs-2559 287 18 quotient quotient NOUN iajs-2559 287 19 map map NOUN iajs-2559 287 20 𝑓 𝑓 NOUN iajs-2559 287 21 is be AUX iajs-2559 287 22 𝐷′. 𝐷′. NOUN iajs-2559 287 23 we we PRON iajs-2559 287 24 have have VERB iajs-2559 287 25 to to PART iajs-2559 287 26 show show VERB iajs-2559 287 27 that that SCONJ iajs-2559 287 28 𝐷′ 𝐷′ PROPN iajs-2559 287 29 is be AUX iajs-2559 287 30 an an DET iajs-2559 287 31 approximating approximate VERB iajs-2559 287 32 space space NOUN iajs-2559 287 33 . . PUNCT iajs-2559 288 1 since since SCONJ iajs-2559 288 2 𝑓is 𝑓is PROPN iajs-2559 288 3 a a DET iajs-2559 288 4 quotient quotient NOUN iajs-2559 288 5 map map NOUN iajs-2559 288 6 , , PUNCT iajs-2559 288 7 𝑁 𝑁 PROPN iajs-2559 288 8 βŠ† βŠ† NUM iajs-2559 288 9 𝐷′ 𝐷′ NOUN iajs-2559 288 10 is be AUX iajs-2559 288 11 open open ADJ iajs-2559 288 12 in in ADP iajs-2559 288 13 𝐷′ 𝐷′ PROPN iajs-2559 288 14 if if SCONJ iajs-2559 289 1 and and CCONJ iajs-2559 289 2 only only ADV iajs-2559 289 3 if if SCONJ iajs-2559 289 4 π‘“βˆ’1(𝑁 π‘“βˆ’1(𝑁 NOUN iajs-2559 289 5 ) ) PUNCT iajs-2559 289 6 is be AUX iajs-2559 289 7 open open ADJ iajs-2559 289 8 in in ADP iajs-2559 289 9 𝐷. 𝐷. NOUN iajs-2559 289 10 by by ADP iajs-2559 289 11 using use VERB iajs-2559 289 12 theorem theorem VERB iajs-2559 289 13 , , PUNCT iajs-2559 289 14 5.2 5.2 NUM iajs-2559 289 15 , , PUNCT iajs-2559 289 16 π‘“βˆ’1(𝑁 π‘“βˆ’1(𝑁 NUM iajs-2559 289 17 ) ) PUNCT iajs-2559 290 1 is be AUX iajs-2559 290 2 open open ADJ iajs-2559 290 3 in in ADP iajs-2559 290 4 𝐷 𝐷 PROPN iajs-2559 290 5 if if SCONJ iajs-2559 290 6 and and CCONJ iajs-2559 290 7 only only ADV iajs-2559 290 8 if if SCONJ iajs-2559 290 9 π‘“βˆ’1(𝑁 π‘“βˆ’1(𝑁 NOUN iajs-2559 290 10 ) ) PUNCT iajs-2559 290 11 is be AUX iajs-2559 290 12 closed close VERB iajs-2559 290 13 in in ADP iajs-2559 290 14 𝐷. 𝐷. PROPN iajs-2559 290 15 since𝑓 since𝑓 NOUN iajs-2559 290 16 is be AUX iajs-2559 290 17 a a DET iajs-2559 290 18 quotient quotient NOUN iajs-2559 290 19 map map NOUN iajs-2559 290 20 , , PUNCT iajs-2559 290 21 then then ADV iajs-2559 290 22 , , PUNCT iajs-2559 290 23 π‘“βˆ’1(𝑁 π‘“βˆ’1(𝑁 NUM iajs-2559 290 24 ) ) PUNCT iajs-2559 290 25 is be AUX iajs-2559 290 26 closed close VERB iajs-2559 290 27 if if SCONJ iajs-2559 290 28 and and CCONJ iajs-2559 290 29 only only ADV iajs-2559 290 30 if if SCONJ iajs-2559 290 31 𝑁 𝑁 PROPN iajs-2559 290 32 βŠ† βŠ† NUM iajs-2559 290 33 𝐷′ 𝐷′ NOUN iajs-2559 290 34 is be AUX iajs-2559 290 35 closed close VERB iajs-2559 290 36 in in ADP iajs-2559 290 37 𝐷′. 𝐷′. NOUN iajs-2559 290 38 so𝑁 so𝑁 VERB iajs-2559 290 39 βŠ† βŠ† NUM iajs-2559 290 40 𝐷′is 𝐷′is PROPN iajs-2559 290 41 open open ADJ iajs-2559 290 42 in in ADP iajs-2559 290 43 𝐷′ 𝐷′ PROPN iajs-2559 291 1 if if SCONJ iajs-2559 291 2 and and CCONJ iajs-2559 291 3 only only ADV iajs-2559 291 4 if if SCONJ iajs-2559 291 5 𝑁 𝑁 PROPN iajs-2559 291 6 is be AUX iajs-2559 291 7 closed closed ADJ iajs-2559 291 8 in𝐷′. in𝐷′. NOUN iajs-2559 291 9 according accord VERB iajs-2559 291 10 to to ADP iajs-2559 291 11 theorem theorem ADJ iajs-2559 291 12 5.2,𝐷′ 5.2,𝐷′ PROPN iajs-2559 291 13 is be AUX iajs-2559 291 14 an an DET iajs-2559 291 15 approximating approximate VERB iajs-2559 291 16 space space NOUN iajs-2559 291 17 . . PUNCT iajs-2559 292 1 corollary5.5 corollary5.5 NOUN iajs-2559 292 2 . . PUNCT iajs-2559 293 1 continuous continuous ADJ iajs-2559 293 2 maps map NOUN iajs-2559 293 3 do do AUX iajs-2559 293 4 not not PART iajs-2559 293 5 preserve preserve VERB iajs-2559 293 6 an an DET iajs-2559 293 7 approximating approximate VERB iajs-2559 293 8 space space NOUN iajs-2559 293 9 . . PUNCT iajs-2559 294 1 we we PRON iajs-2559 294 2 will will AUX iajs-2559 294 3 explicate explicate VERB iajs-2559 294 4 corollary corollary ADJ iajs-2559 294 5 5.5 5.5 NUM iajs-2559 294 6 . . PUNCT iajs-2559 295 1 in in ADP iajs-2559 295 2 the the DET iajs-2559 295 3 next next ADJ iajs-2559 295 4 example example NOUN iajs-2559 295 5 . . PUNCT iajs-2559 295 6 example example NOUN iajs-2559 296 1 5.6.suppose 5.6.suppose NUM iajs-2559 296 2 that𝑉(𝐷 that𝑉(𝐷 NOUN iajs-2559 296 3 ) ) PUNCT iajs-2559 296 4 is be AUX iajs-2559 296 5 a a DET iajs-2559 296 6 real real ADJ iajs-2559 296 7 numbers number NOUN iajs-2559 296 8 set set VERB iajs-2559 296 9 r r NOUN iajs-2559 296 10 given give VERB iajs-2559 296 11 with with ADP iajs-2559 296 12 the the DET iajs-2559 296 13 usual usual ADJ iajs-2559 296 14 discrete discrete ADJ iajs-2559 296 15 topology topology NOUN iajs-2559 296 16 and and CCONJ iajs-2559 296 17 𝑉(𝐷′ 𝑉(𝐷′ NOUN iajs-2559 296 18 ) ) PUNCT iajs-2559 296 19 is be AUX iajs-2559 296 20 a a DET iajs-2559 296 21 real real ADJ iajs-2559 296 22 numbers number NOUN iajs-2559 296 23 𝑅given 𝑅given PROPN iajs-2559 296 24 with with ADP iajs-2559 296 25 the the DET iajs-2559 296 26 usual usual ADJ iajs-2559 296 27 euclidean euclidean ADJ iajs-2559 296 28 topology topology NOUN iajs-2559 296 29 , , PUNCT iajs-2559 296 30 let let VERB iajs-2559 296 31 𝑓 𝑓 PRON iajs-2559 296 32 : : PUNCT iajs-2559 296 33 𝑉(𝐷 𝑉(𝐷 NOUN iajs-2559 296 34 ) ) PUNCT iajs-2559 296 35 ⟢ ⟢ NOUN iajs-2559 296 36 𝑉(𝐷′ 𝑉(𝐷′ NOUN iajs-2559 296 37 ) ) PUNCT iajs-2559 296 38 be be VERB iajs-2559 296 39 the the DET iajs-2559 296 40 identity identity NOUN iajs-2559 296 41 map map NOUN iajs-2559 296 42 , , PUNCT iajs-2559 296 43 it it PRON iajs-2559 296 44 is be AUX iajs-2559 296 45 obvious obvious ADJ iajs-2559 296 46 that that SCONJ iajs-2559 296 47 𝑓 𝑓 PRON iajs-2559 296 48 is be AUX iajs-2559 296 49 continuous continuous ADJ iajs-2559 296 50 map map NOUN iajs-2559 296 51 . . PUNCT iajs-2559 297 1 according accord VERB iajs-2559 297 2 to to ADP iajs-2559 297 3 the the DET iajs-2559 297 4 corollary corollary ADJ iajs-2559 297 5 5.3 5.3 NUM iajs-2559 297 6 , , PUNCT iajs-2559 297 7 𝑉(𝐷 𝑉(𝐷 PROPN iajs-2559 297 8 ) ) PUNCT iajs-2559 297 9 is be AUX iajs-2559 297 10 an an DET iajs-2559 297 11 approximating approximate VERB iajs-2559 297 12 space space NOUN iajs-2559 297 13 . . PUNCT iajs-2559 298 1 but but CCONJ iajs-2559 298 2 𝑉(𝐷′ 𝑉(𝐷′ X iajs-2559 298 3 ) ) PUNCT iajs-2559 298 4 is be AUX iajs-2559 298 5 not not PART iajs-2559 298 6 an an DET iajs-2559 298 7 approximating approximate VERB iajs-2559 298 8 space space NOUN iajs-2559 298 9 . . PUNCT iajs-2559 299 1 therefore therefore ADV iajs-2559 299 2 , , PUNCT iajs-2559 299 3 continuous continuous ADJ iajs-2559 299 4 maps map NOUN iajs-2559 299 5 do do AUX iajs-2559 299 6 not not PART iajs-2559 299 7 preserve preserve VERB iajs-2559 299 8 approximating approximate VERB iajs-2559 299 9 spaces space NOUN iajs-2559 299 10 . . PUNCT iajs-2559 300 1 3.conclusion 3.conclusion NUM iajs-2559 300 2 we we PRON iajs-2559 300 3 offer offer VERB iajs-2559 300 4 the the DET iajs-2559 300 5 topological topological ADJ iajs-2559 300 6 space space NOUN iajs-2559 300 7 generated generate VERB iajs-2559 300 8 by by ADP iajs-2559 300 9 a a DET iajs-2559 300 10 reflexive reflexive ADJ iajs-2559 300 11 graph graph NOUN iajs-2559 300 12 and and CCONJ iajs-2559 300 13 tolerance tolerance NOUN iajs-2559 300 14 graph graph NOUN iajs-2559 300 15 consecutively consecutively ADV iajs-2559 300 16 and and CCONJ iajs-2559 300 17 discussed discuss VERB iajs-2559 300 18 the the DET iajs-2559 300 19 topological topological ADJ iajs-2559 300 20 structure structure NOUN iajs-2559 300 21 of of ADP iajs-2559 300 22 generalized generalized ADJ iajs-2559 300 23 rough rough ADJ iajs-2559 300 24 graph graph NOUN iajs-2559 300 25 . . 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