id	sid	tid	token	lemma	pos
ejpam-6037	1	1	european	european	PROPN
ejpam-6037	1	2	journal	journal	PROPN
ejpam-6037	1	3	of	of	ADP
ejpam-6037	1	4	pure	pure	ADJ
ejpam-6037	1	5	and	and	CCONJ
ejpam-6037	1	6	applied	applied	ADJ
ejpam-6037	1	7	mathematics	mathematic	NOUN
ejpam-6037	1	8	2025	2025	NUM
ejpam-6037	1	9	,	,	PUNCT
ejpam-6037	1	10	vol	vol	NOUN
ejpam-6037	1	11	.	.	PROPN
ejpam-6037	1	12	18	18	NUM
ejpam-6037	1	13	,	,	PUNCT
ejpam-6037	1	14	issue	issue	NOUN
ejpam-6037	1	15	2	2	NUM
ejpam-6037	1	16	,	,	PUNCT
ejpam-6037	1	17	article	article	NOUN
ejpam-6037	1	18	number	number	NOUN
ejpam-6037	1	19	6037	6037	NUM
ejpam-6037	1	20	issn	issn	PROPN
ejpam-6037	1	21	1307	1307	NUM
ejpam-6037	1	22	-	-	SYM
ejpam-6037	1	23	5543	5543	NUM
ejpam-6037	1	24	–	–	PUNCT
ejpam-6037	1	25	ejpam.com	ejpam.com	X
ejpam-6037	1	26	published	publish	VERB
ejpam-6037	1	27	by	by	ADP
ejpam-6037	1	28	new	new	PROPN
ejpam-6037	1	29	york	york	PROPN
ejpam-6037	1	30	business	business	PROPN
ejpam-6037	1	31	global	global	ADJ
ejpam-6037	1	32	characterizations	characterization	NOUN
ejpam-6037	1	33	of	of	ADP
ejpam-6037	1	34	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	1	35	,	,	PUNCT
ejpam-6037	1	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	1	37	functions	function	NOUN
ejpam-6037	1	38	nipaporn	nipaporn	ADV
ejpam-6037	1	39	chutiman1	chutiman1	NOUN
ejpam-6037	1	40	,	,	PUNCT
ejpam-6037	1	41	areeyuth	areeyuth	NOUN
ejpam-6037	1	42	sama	sama	NOUN
ejpam-6037	1	43	-	-	PUNCT
ejpam-6037	1	44	ae2	ae2	PROPN
ejpam-6037	1	45	,	,	PUNCT
ejpam-6037	1	46	chawalit	chawalit	VERB
ejpam-6037	1	47	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6037	1	48	1	1	NUM
ejpam-6037	1	49	mathematics	mathematic	NOUN
ejpam-6037	1	50	and	and	CCONJ
ejpam-6037	1	51	applied	apply	VERB
ejpam-6037	1	52	mathematics	mathematics	PROPN
ejpam-6037	1	53	research	research	NOUN
ejpam-6037	1	54	unit	unit	NOUN
ejpam-6037	1	55	,	,	PUNCT
ejpam-6037	1	56	department	department	NOUN
ejpam-6037	1	57	of	of	ADP
ejpam-6037	1	58	mathematics	mathematic	NOUN
ejpam-6037	1	59	,	,	PUNCT
ejpam-6037	1	60	faculty	faculty	NOUN
ejpam-6037	1	61	of	of	ADP
ejpam-6037	1	62	science	science	NOUN
ejpam-6037	1	63	,	,	PUNCT
ejpam-6037	1	64	mahasarakham	mahasarakham	PROPN
ejpam-6037	1	65	university	university	PROPN
ejpam-6037	1	66	,	,	PUNCT
ejpam-6037	1	67	maha	maha	PROPN
ejpam-6037	1	68	sarakham	sarakham	PROPN
ejpam-6037	1	69	,	,	PUNCT
ejpam-6037	1	70	44150	44150	NUM
ejpam-6037	1	71	,	,	PUNCT
ejpam-6037	1	72	thailand	thailand	PROPN
ejpam-6037	1	73	2	2	NUM
ejpam-6037	1	74	department	department	NOUN
ejpam-6037	1	75	of	of	ADP
ejpam-6037	1	76	mathematics	mathematic	NOUN
ejpam-6037	1	77	and	and	CCONJ
ejpam-6037	1	78	computer	computer	NOUN
ejpam-6037	1	79	science	science	NOUN
ejpam-6037	1	80	,	,	PUNCT
ejpam-6037	1	81	faculty	faculty	NOUN
ejpam-6037	1	82	of	of	ADP
ejpam-6037	1	83	science	science	NOUN
ejpam-6037	1	84	and	and	CCONJ
ejpam-6037	1	85	technology	technology	NOUN
ejpam-6037	1	86	,	,	PUNCT
ejpam-6037	1	87	prince	prince	NOUN
ejpam-6037	1	88	of	of	ADP
ejpam-6037	1	89	songkla	songkla	PROPN
ejpam-6037	1	90	university	university	PROPN
ejpam-6037	1	91	,	,	PUNCT
ejpam-6037	1	92	pattani	pattani	NOUN
ejpam-6037	1	93	campus	campus	NOUN
ejpam-6037	1	94	,	,	PUNCT
ejpam-6037	1	95	pattani	pattani	NOUN
ejpam-6037	1	96	,	,	PUNCT
ejpam-6037	1	97	94000	94000	NUM
ejpam-6037	1	98	,	,	PUNCT
ejpam-6037	1	99	thailand	thailand	PROPN
ejpam-6037	1	100	abstract	abstract	PROPN
ejpam-6037	1	101	.	.	PUNCT
ejpam-6037	2	1	this	this	DET
ejpam-6037	2	2	paper	paper	NOUN
ejpam-6037	2	3	introduces	introduce	VERB
ejpam-6037	2	4	a	a	DET
ejpam-6037	2	5	new	new	ADJ
ejpam-6037	2	6	class	class	NOUN
ejpam-6037	2	7	of	of	ADP
ejpam-6037	2	8	functions	function	NOUN
ejpam-6037	2	9	between	between	ADP
ejpam-6037	2	10	bitopological	bitopological	ADJ
ejpam-6037	2	11	spaces	space	NOUN
ejpam-6037	2	12	,	,	PUNCT
ejpam-6037	2	13	namely	namely	ADV
ejpam-6037	2	14	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	2	15	,	,	PUNCT
ejpam-6037	2	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	2	17	functions	function	NOUN
ejpam-6037	2	18	.	.	PUNCT
ejpam-6037	3	1	furthermore	furthermore	ADV
ejpam-6037	3	2	,	,	PUNCT
ejpam-6037	3	3	several	several	ADJ
ejpam-6037	3	4	characterizations	characterization	NOUN
ejpam-6037	3	5	and	and	CCONJ
ejpam-6037	3	6	some	some	DET
ejpam-6037	3	7	properties	property	NOUN
ejpam-6037	3	8	concerning	concern	VERB
ejpam-6037	3	9	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	3	10	,	,	PUNCT
ejpam-6037	3	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	3	12	functions	function	NOUN
ejpam-6037	3	13	are	be	AUX
ejpam-6037	3	14	investigated	investigate	VERB
ejpam-6037	3	15	.	.	PUNCT
ejpam-6037	4	1	2020	2020	NUM
ejpam-6037	4	2	mathematics	mathematic	NOUN
ejpam-6037	4	3	subject	subject	NOUN
ejpam-6037	4	4	classifications	classification	NOUN
ejpam-6037	4	5	:	:	PUNCT
ejpam-6037	4	6	54c08	54c08	NUM
ejpam-6037	4	7	,	,	PUNCT
ejpam-6037	4	8	54e55	54e55	NUM
ejpam-6037	4	9	key	key	ADJ
ejpam-6037	4	10	words	word	NOUN
ejpam-6037	4	11	and	and	CCONJ
ejpam-6037	4	12	phrases	phrase	NOUN
ejpam-6037	4	13	:	:	PUNCT
ejpam-6037	4	14	τ1τ2	τ1τ2	ADJ
ejpam-6037	4	15	-	-	ADJ
ejpam-6037	4	16	open	open	ADJ
ejpam-6037	4	17	set	set	NOUN
ejpam-6037	4	18	,	,	PUNCT
ejpam-6037	4	19	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	4	20	,	,	PUNCT
ejpam-6037	4	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	4	22	function	function	NOUN
ejpam-6037	4	23	1	1	NUM
ejpam-6037	4	24	.	.	PUNCT
ejpam-6037	4	25	introduction	introduction	NOUN
ejpam-6037	4	26	the	the	DET
ejpam-6037	4	27	field	field	NOUN
ejpam-6037	4	28	of	of	ADP
ejpam-6037	4	29	the	the	DET
ejpam-6037	4	30	mathematical	mathematical	ADJ
ejpam-6037	4	31	science	science	NOUN
ejpam-6037	4	32	which	which	PRON
ejpam-6037	4	33	goes	go	VERB
ejpam-6037	4	34	under	under	ADP
ejpam-6037	4	35	the	the	DET
ejpam-6037	4	36	name	name	NOUN
ejpam-6037	4	37	of	of	ADP
ejpam-6037	4	38	topology	topology	NOUN
ejpam-6037	4	39	is	be	AUX
ejpam-6037	4	40	concerned	concern	VERB
ejpam-6037	4	41	with	with	ADP
ejpam-6037	4	42	all	all	DET
ejpam-6037	4	43	questions	question	NOUN
ejpam-6037	4	44	directly	directly	ADV
ejpam-6037	4	45	or	or	CCONJ
ejpam-6037	4	46	indirectly	indirectly	ADV
ejpam-6037	4	47	related	relate	VERB
ejpam-6037	4	48	to	to	ADP
ejpam-6037	4	49	continuity	continuity	NOUN
ejpam-6037	4	50	.	.	PUNCT
ejpam-6037	5	1	viriyapong	viriyapong	PROPN
ejpam-6037	6	1	and	and	CCONJ
ejpam-6037	6	2	boonpok	boonpok	VERB
ejpam-6037	7	1	[	[	X
ejpam-6037	7	2	1	1	NUM
ejpam-6037	7	3	]	]	PUNCT
ejpam-6037	7	4	investigated	investigate	VERB
ejpam-6037	7	5	some	some	DET
ejpam-6037	7	6	characterizations	characterization	NOUN
ejpam-6037	7	7	of	of	ADP
ejpam-6037	7	8	(	(	PUNCT
ejpam-6037	7	9	λ	λ	PROPN
ejpam-6037	7	10	,	,	PUNCT
ejpam-6037	7	11	sp)-continuous	sp)-continuous	ADJ
ejpam-6037	7	12	functions	function	NOUN
ejpam-6037	7	13	by	by	ADP
ejpam-6037	7	14	utilizing	utilize	VERB
ejpam-6037	7	15	the	the	DET
ejpam-6037	7	16	notions	notion	NOUN
ejpam-6037	7	17	of	of	ADP
ejpam-6037	7	18	(	(	PUNCT
ejpam-6037	7	19	λ	λ	PROPN
ejpam-6037	7	20	,	,	PUNCT
ejpam-6037	7	21	sp)-open	sp)-open	ADJ
ejpam-6037	7	22	sets	set	NOUN
ejpam-6037	7	23	and	and	CCONJ
ejpam-6037	7	24	(	(	PUNCT
ejpam-6037	7	25	λ	λ	PROPN
ejpam-6037	7	26	,	,	PUNCT
ejpam-6037	7	27	sp)-closed	sp)-close	VERB
ejpam-6037	7	28	sets	set	NOUN
ejpam-6037	7	29	due	due	ADP
ejpam-6037	7	30	to	to	ADP
ejpam-6037	7	31	boonpok	boonpok	NOUN
ejpam-6037	7	32	and	and	CCONJ
ejpam-6037	7	33	khampakdee	khampakdee	NOUN
ejpam-6037	7	34	[	[	X
ejpam-6037	7	35	2	2	NUM
ejpam-6037	7	36	]	]	PUNCT
ejpam-6037	7	37	.	.	PUNCT
ejpam-6037	8	1	dungthaisong	dungthaisong	NOUN
ejpam-6037	8	2	et	et	PROPN
ejpam-6037	8	3	al	al	PROPN
ejpam-6037	8	4	.	.	PUNCT
ejpam-6037	9	1	[	[	X
ejpam-6037	9	2	3	3	NUM
ejpam-6037	9	3	]	]	PUNCT
ejpam-6037	9	4	introduced	introduce	VERB
ejpam-6037	9	5	and	and	CCONJ
ejpam-6037	9	6	studied	study	VERB
ejpam-6037	9	7	the	the	DET
ejpam-6037	9	8	concept	concept	NOUN
ejpam-6037	9	9	of	of	ADP
ejpam-6037	9	10	g(m	g(m	ADJ
ejpam-6037	9	11	,	,	PUNCT
ejpam-6037	9	12	n)-continuous	n)-continuous	ADJ
ejpam-6037	9	13	functions	function	NOUN
ejpam-6037	9	14	.	.	PUNCT
ejpam-6037	10	1	duangphui	duangphui	NOUN
ejpam-6037	10	2	et	et	PROPN
ejpam-6037	10	3	al	al	PROPN
ejpam-6037	10	4	.	.	PUNCT
ejpam-6037	11	1	[	[	X
ejpam-6037	11	2	4	4	X
ejpam-6037	11	3	]	]	PUNCT
ejpam-6037	11	4	introduced	introduce	VERB
ejpam-6037	11	5	and	and	CCONJ
ejpam-6037	11	6	investigated	investigate	VERB
ejpam-6037	11	7	the	the	DET
ejpam-6037	11	8	notion	notion	NOUN
ejpam-6037	11	9	of	of	ADP
ejpam-6037	11	10	(	(	PUNCT
ejpam-6037	11	11	µ	µ	NOUN
ejpam-6037	11	12	,	,	PUNCT
ejpam-6037	11	13	µ′)(m	µ′)(m	VERB
ejpam-6037	11	14	,	,	PUNCT
ejpam-6037	11	15	n)continuous	n)continuous	ADJ
ejpam-6037	11	16	functions	function	NOUN
ejpam-6037	11	17	.	.	PUNCT
ejpam-6037	12	1	moreover	moreover	ADV
ejpam-6037	12	2	,	,	PUNCT
ejpam-6037	12	3	several	several	ADJ
ejpam-6037	12	4	characterizations	characterization	NOUN
ejpam-6037	12	5	of	of	ADP
ejpam-6037	12	6	almost	almost	ADV
ejpam-6037	12	7	(	(	PUNCT
ejpam-6037	12	8	λ	λ	PROPN
ejpam-6037	12	9	,	,	PUNCT
ejpam-6037	12	10	p)-continuous	p)-continuous	ADJ
ejpam-6037	12	11	functions	function	NOUN
ejpam-6037	12	12	,	,	PUNCT
ejpam-6037	12	13	strongly	strongly	ADV
ejpam-6037	12	14	θ(λ	θ(λ	PROPN
ejpam-6037	12	15	,	,	PUNCT
ejpam-6037	12	16	p)-continuous	p)-continuous	ADJ
ejpam-6037	12	17	functions	function	NOUN
ejpam-6037	12	18	,	,	PUNCT
ejpam-6037	12	19	almost	almost	ADV
ejpam-6037	12	20	strongly	strongly	ADV
ejpam-6037	12	21	θ(λ	θ(λ	VERB
ejpam-6037	12	22	,	,	PUNCT
ejpam-6037	12	23	p)-continuous	p)-continuous	ADJ
ejpam-6037	12	24	functions	function	NOUN
ejpam-6037	12	25	,	,	PUNCT
ejpam-6037	12	26	θ(λ	θ(λ	PROPN
ejpam-6037	12	27	,	,	PUNCT
ejpam-6037	12	28	p)-continuous	p)-continuous	ADJ
ejpam-6037	12	29	functions	function	NOUN
ejpam-6037	12	30	,	,	PUNCT
ejpam-6037	12	31	weakly	weakly	ADJ
ejpam-6037	12	32	(	(	PUNCT
ejpam-6037	12	33	λ	λ	PROPN
ejpam-6037	12	34	,	,	PUNCT
ejpam-6037	12	35	b)-continuous	b)-continuous	ADJ
ejpam-6037	12	36	functions	function	NOUN
ejpam-6037	12	37	,	,	PUNCT
ejpam-6037	12	38	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-6037	12	39	functions	function	NOUN
ejpam-6037	12	40	,	,	PUNCT
ejpam-6037	12	41	(	(	PUNCT
ejpam-6037	12	42	λ	λ	NOUN
ejpam-6037	12	43	,	,	PUNCT
ejpam-6037	12	44	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-6037	12	45	functions	function	NOUN
ejpam-6037	12	46	,	,	PUNCT
ejpam-6037	12	47	⋆-continuous	⋆-continuous	ADJ
ejpam-6037	12	48	functions	function	NOUN
ejpam-6037	12	49	,	,	PUNCT
ejpam-6037	12	50	θ	θ	PROPN
ejpam-6037	12	51	-	-	ADJ
ejpam-6037	12	52	i	i	NOUN
ejpam-6037	12	53	-continuous	-continuous	ADJ
ejpam-6037	12	54	functions	function	NOUN
ejpam-6037	12	55	,	,	PUNCT
ejpam-6037	12	56	almost	almost	ADV
ejpam-6037	12	57	(	(	PUNCT
ejpam-6037	12	58	g	g	NOUN
ejpam-6037	12	59	,	,	PUNCT
ejpam-6037	12	60	m)-continuous	m)-continuous	ADJ
ejpam-6037	12	61	functions	function	NOUN
ejpam-6037	12	62	,	,	PUNCT
ejpam-6037	12	63	pairwise	pairwise	NOUN
ejpam-6037	12	64	almost	almost	ADV
ejpam-6037	12	65	m	m	VERB
ejpam-6037	12	66	-continuous	-continuous	ADJ
ejpam-6037	12	67	functions	function	NOUN
ejpam-6037	12	68	,	,	PUNCT
ejpam-6037	12	69	faintly	faintly	ADV
ejpam-6037	12	70	(	(	PUNCT
ejpam-6037	12	71	τ1	τ1	NOUN
ejpam-6037	12	72	,	,	PUNCT
ejpam-6037	12	73	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	12	74	functions	function	NOUN
ejpam-6037	12	75	,	,	PUNCT
ejpam-6037	12	76	δ(τ1	δ(τ1	NOUN
ejpam-6037	12	77	,	,	PUNCT
ejpam-6037	12	78	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	12	79	functions	function	NOUN
ejpam-6037	12	80	and	and	CCONJ
ejpam-6037	12	81	almost	almost	ADV
ejpam-6037	12	82	nearly	nearly	ADV
ejpam-6037	12	83	(	(	PUNCT
ejpam-6037	12	84	τ1	τ1	NOUN
ejpam-6037	12	85	,	,	PUNCT
ejpam-6037	12	86	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	12	87	functions	function	NOUN
ejpam-6037	12	88	were	be	AUX
ejpam-6037	12	89	presented	present	VERB
ejpam-6037	12	90	in	in	ADP
ejpam-6037	12	91	[	[	X
ejpam-6037	12	92	5	5	NUM
ejpam-6037	12	93	]	]	PUNCT
ejpam-6037	12	94	,	,	PUNCT
ejpam-6037	12	95	[	[	X
ejpam-6037	12	96	6	6	NUM
ejpam-6037	12	97	]	]	PUNCT
ejpam-6037	12	98	,	,	PUNCT
ejpam-6037	12	99	[	[	X
ejpam-6037	12	100	7	7	NUM
ejpam-6037	12	101	]	]	PUNCT
ejpam-6037	12	102	,	,	PUNCT
ejpam-6037	12	103	[	[	X
ejpam-6037	12	104	8	8	NUM
ejpam-6037	12	105	]	]	PUNCT
ejpam-6037	12	106	,	,	PUNCT
ejpam-6037	13	1	[	[	X
ejpam-6037	13	2	9	9	NUM
ejpam-6037	13	3	]	]	PUNCT
ejpam-6037	13	4	,	,	PUNCT
ejpam-6037	13	5	[	[	X
ejpam-6037	13	6	10	10	NUM
ejpam-6037	13	7	]	]	PUNCT
ejpam-6037	13	8	,	,	PUNCT
ejpam-6037	14	1	[	[	X
ejpam-6037	14	2	11	11	NUM
ejpam-6037	14	3	]	]	PUNCT
ejpam-6037	14	4	,	,	PUNCT
ejpam-6037	14	5	[	[	X
ejpam-6037	14	6	12	12	NUM
ejpam-6037	14	7	]	]	PUNCT
ejpam-6037	14	8	,	,	PUNCT
ejpam-6037	14	9	[	[	X
ejpam-6037	14	10	13	13	NUM
ejpam-6037	14	11	]	]	PUNCT
ejpam-6037	14	12	,	,	PUNCT
ejpam-6037	14	13	[	[	X
ejpam-6037	14	14	14	14	NUM
ejpam-6037	14	15	]	]	PUNCT
ejpam-6037	14	16	,	,	PUNCT
ejpam-6037	14	17	[	[	X
ejpam-6037	14	18	15	15	NUM
ejpam-6037	14	19	]	]	PUNCT
ejpam-6037	14	20	,	,	PUNCT
ejpam-6037	14	21	[	[	X
ejpam-6037	14	22	16	16	NUM
ejpam-6037	14	23	]	]	PUNCT
ejpam-6037	14	24	,	,	PUNCT
ejpam-6037	15	1	[	[	X
ejpam-6037	15	2	17	17	NUM
ejpam-6037	15	3	]	]	PUNCT
ejpam-6037	15	4	and	and	CCONJ
ejpam-6037	15	5	[	[	X
ejpam-6037	15	6	18	18	NUM
ejpam-6037	15	7	]	]	PUNCT
ejpam-6037	15	8	,	,	PUNCT
ejpam-6037	15	9	respectively	respectively	ADV
ejpam-6037	15	10	.	.	PUNCT
ejpam-6037	16	1	the	the	DET
ejpam-6037	16	2	notions	notion	NOUN
ejpam-6037	16	3	of	of	ADP
ejpam-6037	16	4	contra	contra	PROPN
ejpam-6037	16	5	-	-	NOUN
ejpam-6037	16	6	continuity	continuity	NOUN
ejpam-6037	16	7	and	and	CCONJ
ejpam-6037	16	8	strong	strong	ADJ
ejpam-6037	16	9	s	s	NOUN
ejpam-6037	16	10	-	-	NOUN
ejpam-6037	16	11	closedness	closedness	ADJ
ejpam-6037	16	12	in	in	ADP
ejpam-6037	16	13	topological	topological	ADJ
ejpam-6037	16	14	spaces	space	NOUN
ejpam-6037	16	15	were	be	AUX
ejpam-6037	16	16	introduced	introduce	VERB
ejpam-6037	16	17	by	by	ADP
ejpam-6037	16	18	dontchev	dontchev	NOUN
ejpam-6037	16	19	[	[	X
ejpam-6037	16	20	19	19	NUM
ejpam-6037	16	21	]	]	PUNCT
ejpam-6037	16	22	.	.	PUNCT
ejpam-6037	17	1	dontchev	dontchev	PROPN
ejpam-6037	17	2	[	[	X
ejpam-6037	17	3	19	19	NUM
ejpam-6037	17	4	]	]	PUNCT
ejpam-6037	17	5	obtained	obtain	VERB
ejpam-6037	17	6	very	very	ADV
ejpam-6037	17	7	interesting	interesting	ADJ
ejpam-6037	17	8	and	and	CCONJ
ejpam-6037	17	9	important	important	ADJ
ejpam-6037	17	10	results	result	NOUN
ejpam-6037	17	11	concerning	concern	VERB
ejpam-6037	17	12	contra	contra	NOUN
ejpam-6037	17	13	-	-	ADJ
ejpam-6037	17	14	continuity	continuity	NOUN
ejpam-6037	17	15	,	,	PUNCT
ejpam-6037	17	16	compactness	compactness	NOUN
ejpam-6037	17	17	,	,	PUNCT
ejpam-6037	17	18	s	s	NOUN
ejpam-6037	17	19	-	-	PUNCT
ejpam-6037	17	20	closedness	closedness	ADJ
ejpam-6037	17	21	and	and	CCONJ
ejpam-6037	17	22	strong	strong	ADJ
ejpam-6037	17	23	sclosedness	sclosedness	NOUN
ejpam-6037	17	24	.	.	PUNCT
ejpam-6037	18	1	dontchev	dontchev	NOUN
ejpam-6037	18	2	and	and	CCONJ
ejpam-6037	18	3	noiri	noiri	ADV
ejpam-6037	19	1	[	[	X
ejpam-6037	19	2	20	20	NUM
ejpam-6037	19	3	]	]	PUNCT
ejpam-6037	19	4	introduced	introduce	VERB
ejpam-6037	19	5	and	and	CCONJ
ejpam-6037	19	6	studied	study	VERB
ejpam-6037	19	7	the	the	DET
ejpam-6037	19	8	concept	concept	NOUN
ejpam-6037	19	9	of	of	ADP
ejpam-6037	19	10	rc	rc	NOUN
ejpam-6037	19	11	-	-	PUNCT
ejpam-6037	19	12	continuity	continuity	NOUN
ejpam-6037	19	13	∗corresponding	∗corresponde	VERB
ejpam-6037	19	14	author	author	NOUN
ejpam-6037	19	15	.	.	PUNCT
ejpam-6037	20	1	doi	doi	NOUN
ejpam-6037	20	2	:	:	PUNCT
ejpam-6037	20	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6037	https://doi.org/10.29020/nybg.ejpam.v18i2.6037	ADJ
ejpam-6037	20	4	email	email	NOUN
ejpam-6037	20	5	addresses	address	NOUN
ejpam-6037	20	6	:	:	PUNCT
ejpam-6037	20	7	nipaporn.c@msu.ac.th	nipaporn.c@msu.ac.th	PROPN
ejpam-6037	20	8	(	(	PUNCT
ejpam-6037	20	9	n.	n.	NOUN
ejpam-6037	20	10	chutiman	chutiman	NOUN
ejpam-6037	20	11	)	)	PUNCT
ejpam-6037	20	12	,	,	PUNCT
ejpam-6037	20	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6037	20	14	(	(	PUNCT
ejpam-6037	20	15	a.	a.	PROPN
ejpam-6037	20	16	sama	sama	PROPN
ejpam-6037	20	17	-	-	PUNCT
ejpam-6037	20	18	ae	ae	PROPN
ejpam-6037	20	19	)	)	PUNCT
ejpam-6037	20	20	,	,	PUNCT
ejpam-6037	20	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6037	20	22	(	(	PUNCT
ejpam-6037	20	23	c.	c.	PROPN
ejpam-6037	20	24	boonpok	boonpok	PROPN
ejpam-6037	20	25	)	)	PUNCT
ejpam-6037	20	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6037	21	1	1	1	NUM
ejpam-6037	21	2	copyright	copyright	NOUN
ejpam-6037	21	3	:	:	PUNCT
ejpam-6037	21	4	©	©	PROPN
ejpam-6037	21	5	2025	2025	NUM
ejpam-6037	21	6	the	the	DET
ejpam-6037	21	7	author(s	author(s	NOUN
ejpam-6037	21	8	)	)	PUNCT
ejpam-6037	21	9	.	.	PUNCT
ejpam-6037	22	1	(	(	PUNCT
ejpam-6037	22	2	cc	cc	NOUN
ejpam-6037	22	3	by	by	ADP
ejpam-6037	22	4	-	-	PUNCT
ejpam-6037	22	5	nc	nc	PROPN
ejpam-6037	22	6	4.0	4.0	NUM
ejpam-6037	22	7	)	)	PUNCT
ejpam-6037	22	8	n.	n.	NOUN
ejpam-6037	22	9	chutiman	chutiman	NOUN
ejpam-6037	22	10	,	,	PUNCT
ejpam-6037	22	11	a.	a.	PROPN
ejpam-6037	22	12	sama	sama	PROPN
ejpam-6037	22	13	-	-	PUNCT
ejpam-6037	22	14	ae	ae	PROPN
ejpam-6037	22	15	,	,	PUNCT
ejpam-6037	22	16	c.	c.	PROPN
ejpam-6037	22	17	boonpok	boonpok	PROPN
ejpam-6037	22	18	/	/	SYM
ejpam-6037	22	19	eur	eur	PROPN
ejpam-6037	22	20	.	.	PUNCT
ejpam-6037	23	1	j.	j.	PROPN
ejpam-6037	23	2	pure	pure	PROPN
ejpam-6037	23	3	appl	appl	PROPN
ejpam-6037	23	4	.	.	PROPN
ejpam-6037	23	5	math	math	PROPN
ejpam-6037	23	6	,	,	PUNCT
ejpam-6037	23	7	18	18	NUM
ejpam-6037	23	8	(	(	PUNCT
ejpam-6037	23	9	2	2	NUM
ejpam-6037	23	10	)	)	PUNCT
ejpam-6037	23	11	(	(	PUNCT
ejpam-6037	23	12	2025	2025	NUM
ejpam-6037	23	13	)	)	PUNCT
ejpam-6037	23	14	,	,	PUNCT
ejpam-6037	23	15	6037	6037	NUM
ejpam-6037	23	16	2	2	NUM
ejpam-6037	23	17	of	of	ADP
ejpam-6037	23	18	12	12	NUM
ejpam-6037	23	19	between	between	ADP
ejpam-6037	23	20	topological	topological	ADJ
ejpam-6037	23	21	spaces	space	NOUN
ejpam-6037	23	22	which	which	PRON
ejpam-6037	23	23	is	be	AUX
ejpam-6037	23	24	weaker	weak	ADJ
ejpam-6037	23	25	than	than	ADP
ejpam-6037	23	26	contra	contra	NOUN
ejpam-6037	23	27	-	-	NOUN
ejpam-6037	23	28	continuity	continuity	NOUN
ejpam-6037	23	29	.	.	PUNCT
ejpam-6037	24	1	jafari	jafari	PROPN
ejpam-6037	24	2	and	and	CCONJ
ejpam-6037	24	3	noiri	noiri	ADV
ejpam-6037	25	1	[	[	X
ejpam-6037	25	2	21	21	NUM
ejpam-6037	25	3	]	]	PUNCT
ejpam-6037	25	4	introduced	introduce	VERB
ejpam-6037	25	5	and	and	CCONJ
ejpam-6037	25	6	investigated	investigate	VERB
ejpam-6037	25	7	a	a	DET
ejpam-6037	25	8	new	new	ADJ
ejpam-6037	25	9	class	class	NOUN
ejpam-6037	25	10	of	of	ADP
ejpam-6037	25	11	functions	function	NOUN
ejpam-6037	25	12	called	call	VERB
ejpam-6037	25	13	contra	contra	NOUN
ejpam-6037	25	14	-	-	PUNCT
ejpam-6037	25	15	super	super	ADJ
ejpam-6037	25	16	-	-	ADJ
ejpam-6037	25	17	continuous	continuous	ADJ
ejpam-6037	25	18	functions	function	NOUN
ejpam-6037	25	19	which	which	PRON
ejpam-6037	25	20	lies	lie	VERB
ejpam-6037	25	21	between	between	ADP
ejpam-6037	25	22	classes	class	NOUN
ejpam-6037	25	23	of	of	ADP
ejpam-6037	25	24	rc	rc	PROPN
ejpam-6037	25	25	-	-	ADJ
ejpam-6037	25	26	continuous	continuous	ADJ
ejpam-6037	25	27	functions	function	NOUN
ejpam-6037	25	28	and	and	CCONJ
ejpam-6037	25	29	contra	contra	ADJ
ejpam-6037	25	30	-	-	ADJ
ejpam-6037	25	31	continuous	continuous	ADJ
ejpam-6037	25	32	functions	function	NOUN
ejpam-6037	25	33	.	.	PUNCT
ejpam-6037	26	1	jafari	jafari	PROPN
ejpam-6037	26	2	and	and	CCONJ
ejpam-6037	26	3	noiri	noiri	ADV
ejpam-6037	27	1	[	[	X
ejpam-6037	27	2	22	22	NUM
ejpam-6037	27	3	]	]	PUNCT
ejpam-6037	27	4	introduced	introduce	VERB
ejpam-6037	27	5	a	a	DET
ejpam-6037	27	6	new	new	ADJ
ejpam-6037	27	7	class	class	NOUN
ejpam-6037	27	8	of	of	ADP
ejpam-6037	27	9	function	function	NOUN
ejpam-6037	27	10	called	call	VERB
ejpam-6037	27	11	contra	contra	ADJ
ejpam-6037	27	12	-	-	ADJ
ejpam-6037	27	13	precontinuous	precontinuous	ADJ
ejpam-6037	27	14	functions	function	NOUN
ejpam-6037	27	15	which	which	PRON
ejpam-6037	27	16	is	be	AUX
ejpam-6037	27	17	weaker	weak	ADJ
ejpam-6037	27	18	than	than	ADP
ejpam-6037	27	19	contra	contra	ADJ
ejpam-6037	27	20	-	-	ADJ
ejpam-6037	27	21	continuous	continuous	ADJ
ejpam-6037	27	22	functions	function	NOUN
ejpam-6037	27	23	and	and	CCONJ
ejpam-6037	27	24	studied	study	VERB
ejpam-6037	27	25	several	several	ADJ
ejpam-6037	27	26	basic	basic	ADJ
ejpam-6037	27	27	properties	property	NOUN
ejpam-6037	27	28	of	of	ADP
ejpam-6037	27	29	contra	contra	ADJ
ejpam-6037	27	30	-	-	ADJ
ejpam-6037	27	31	precontinuous	precontinuous	ADJ
ejpam-6037	27	32	functions	function	NOUN
ejpam-6037	27	33	.	.	PUNCT
ejpam-6037	28	1	furthermore	furthermore	ADV
ejpam-6037	28	2	,	,	PUNCT
ejpam-6037	28	3	the	the	DET
ejpam-6037	28	4	present	present	ADJ
ejpam-6037	28	5	authors	author	NOUN
ejpam-6037	28	6	[	[	X
ejpam-6037	28	7	22	22	NUM
ejpam-6037	28	8	]	]	PUNCT
ejpam-6037	28	9	defined	define	VERB
ejpam-6037	28	10	contra	contra	PROPN
ejpam-6037	28	11	-	-	ADJ
ejpam-6037	28	12	preclosed	preclose	VERB
ejpam-6037	28	13	graphs	graph	NOUN
ejpam-6037	28	14	and	and	CCONJ
ejpam-6037	28	15	investigated	investigate	VERB
ejpam-6037	28	16	relations	relation	NOUN
ejpam-6037	28	17	between	between	ADP
ejpam-6037	28	18	contra	contra	PROPN
ejpam-6037	28	19	-	-	NOUN
ejpam-6037	28	20	precontinuity	precontinuity	NOUN
ejpam-6037	28	21	and	and	CCONJ
ejpam-6037	28	22	contra	contra	PROPN
ejpam-6037	28	23	-	-	PUNCT
ejpam-6037	28	24	preclosed	preclose	VERB
ejpam-6037	28	25	graphs	graph	NOUN
ejpam-6037	28	26	.	.	PUNCT
ejpam-6037	29	1	ekici	ekici	NOUN
ejpam-6037	30	1	[	[	X
ejpam-6037	30	2	23	23	NUM
ejpam-6037	30	3	]	]	PUNCT
ejpam-6037	30	4	introduced	introduce	VERB
ejpam-6037	30	5	and	and	CCONJ
ejpam-6037	30	6	studied	study	VERB
ejpam-6037	30	7	a	a	DET
ejpam-6037	30	8	new	new	ADJ
ejpam-6037	30	9	class	class	NOUN
ejpam-6037	30	10	of	of	ADP
ejpam-6037	30	11	functions	function	NOUN
ejpam-6037	30	12	called	call	VERB
ejpam-6037	30	13	almost	almost	ADV
ejpam-6037	30	14	contra	contra	ADJ
ejpam-6037	30	15	-	-	ADJ
ejpam-6037	30	16	precontinuous	precontinuous	ADJ
ejpam-6037	30	17	functions	function	NOUN
ejpam-6037	30	18	which	which	PRON
ejpam-6037	30	19	generalize	generalize	VERB
ejpam-6037	30	20	classes	class	NOUN
ejpam-6037	30	21	of	of	ADP
ejpam-6037	30	22	regular	regular	ADJ
ejpam-6037	30	23	set	set	NOUN
ejpam-6037	30	24	-	-	PUNCT
ejpam-6037	30	25	connected	connect	VERB
ejpam-6037	30	26	functions	function	NOUN
ejpam-6037	30	27	[	[	X
ejpam-6037	30	28	24	24	NUM
ejpam-6037	30	29	]	]	PUNCT
ejpam-6037	30	30	,	,	PUNCT
ejpam-6037	30	31	contra	contra	ADJ
ejpam-6037	30	32	-	-	ADJ
ejpam-6037	30	33	precontinuous	precontinuous	ADJ
ejpam-6037	30	34	functions	function	NOUN
ejpam-6037	30	35	[	[	X
ejpam-6037	30	36	22	22	NUM
ejpam-6037	30	37	]	]	PUNCT
ejpam-6037	30	38	,	,	PUNCT
ejpam-6037	30	39	contra	contra	ADJ
ejpam-6037	30	40	-	-	ADJ
ejpam-6037	30	41	continuous	continuous	ADJ
ejpam-6037	30	42	functions	function	NOUN
ejpam-6037	30	43	[	[	X
ejpam-6037	30	44	19	19	NUM
ejpam-6037	30	45	]	]	PUNCT
ejpam-6037	30	46	,	,	PUNCT
ejpam-6037	30	47	almost	almost	ADV
ejpam-6037	30	48	s	s	NOUN
ejpam-6037	30	49	-	-	ADJ
ejpam-6037	30	50	continuous	continuous	ADJ
ejpam-6037	30	51	functions	function	NOUN
ejpam-6037	30	52	[	[	X
ejpam-6037	30	53	25	25	NUM
ejpam-6037	30	54	]	]	PUNCT
ejpam-6037	30	55	and	and	CCONJ
ejpam-6037	30	56	perfectly	perfectly	ADV
ejpam-6037	30	57	continuous	continuous	ADJ
ejpam-6037	30	58	functions	function	NOUN
ejpam-6037	30	59	[	[	X
ejpam-6037	30	60	26	26	NUM
ejpam-6037	30	61	]	]	PUNCT
ejpam-6037	30	62	.	.	PUNCT
ejpam-6037	31	1	al	al	PROPN
ejpam-6037	31	2	-	-	PUNCT
ejpam-6037	31	3	omari	omari	PROPN
ejpam-6037	31	4	and	and	CCONJ
ejpam-6037	31	5	noorani	noorani	ADJ
ejpam-6037	31	6	[	[	X
ejpam-6037	31	7	27	27	NUM
ejpam-6037	31	8	]	]	PUNCT
ejpam-6037	31	9	introduced	introduce	VERB
ejpam-6037	31	10	the	the	DET
ejpam-6037	31	11	concept	concept	NOUN
ejpam-6037	31	12	of	of	ADP
ejpam-6037	31	13	almost	almost	ADV
ejpam-6037	31	14	contra	contra	PROPN
ejpam-6037	31	15	ω	ω	ADJ
ejpam-6037	31	16	-	-	ADJ
ejpam-6037	31	17	continuous	continuous	ADJ
ejpam-6037	31	18	functions	function	NOUN
ejpam-6037	31	19	via	via	ADP
ejpam-6037	31	20	the	the	DET
ejpam-6037	31	21	notion	notion	NOUN
ejpam-6037	31	22	of	of	ADP
ejpam-6037	31	23	ω	ω	VERB
ejpam-6037	31	24	-	-	ADJ
ejpam-6037	31	25	open	open	ADJ
ejpam-6037	31	26	sets	set	NOUN
ejpam-6037	31	27	and	and	CCONJ
ejpam-6037	31	28	investigated	investigate	VERB
ejpam-6037	31	29	several	several	ADJ
ejpam-6037	31	30	characterizations	characterization	NOUN
ejpam-6037	31	31	of	of	ADP
ejpam-6037	31	32	contra	contra	PROPN
ejpam-6037	31	33	ω	ω	PROPN
ejpam-6037	31	34	-	-	ADJ
ejpam-6037	31	35	continuous	continuous	ADJ
ejpam-6037	31	36	functions	function	NOUN
ejpam-6037	31	37	and	and	CCONJ
ejpam-6037	31	38	almost	almost	ADV
ejpam-6037	31	39	contra	contra	PROPN
ejpam-6037	31	40	ω	ω	ADJ
ejpam-6037	31	41	-	-	ADJ
ejpam-6037	31	42	continuous	continuous	ADJ
ejpam-6037	31	43	functions	function	NOUN
ejpam-6037	31	44	.	.	PUNCT
ejpam-6037	32	1	noiri	noiri	PROPN
ejpam-6037	32	2	and	and	CCONJ
ejpam-6037	32	3	popa	popa	NOUN
ejpam-6037	33	1	[	[	X
ejpam-6037	33	2	28	28	NUM
ejpam-6037	33	3	]	]	PUNCT
ejpam-6037	33	4	introduced	introduce	VERB
ejpam-6037	33	5	the	the	PRON
ejpam-6037	33	6	of	of	ADP
ejpam-6037	33	7	contra	contra	PROPN
ejpam-6037	33	8	m	m	PROPN
ejpam-6037	33	9	-	-	ADJ
ejpam-6037	33	10	continuous	continuous	ADJ
ejpam-6037	33	11	functions	function	NOUN
ejpam-6037	33	12	as	as	ADP
ejpam-6037	33	13	functions	function	NOUN
ejpam-6037	33	14	from	from	ADP
ejpam-6037	33	15	a	a	DET
ejpam-6037	33	16	set	set	NOUN
ejpam-6037	33	17	satisfying	satisfy	VERB
ejpam-6037	33	18	some	some	DET
ejpam-6037	33	19	minimal	minimal	ADJ
ejpam-6037	33	20	conditions	condition	NOUN
ejpam-6037	33	21	into	into	ADP
ejpam-6037	33	22	a	a	DET
ejpam-6037	33	23	topological	topological	ADJ
ejpam-6037	33	24	space	space	NOUN
ejpam-6037	33	25	and	and	CCONJ
ejpam-6037	33	26	investigated	investigate	VERB
ejpam-6037	33	27	some	some	DET
ejpam-6037	33	28	characterizations	characterization	NOUN
ejpam-6037	33	29	and	and	CCONJ
ejpam-6037	33	30	the	the	DET
ejpam-6037	33	31	relationships	relationship	NOUN
ejpam-6037	33	32	between	between	ADP
ejpam-6037	33	33	contra	contra	PROPN
ejpam-6037	33	34	m	m	PROPN
ejpam-6037	33	35	-	-	PUNCT
ejpam-6037	33	36	continuity	continuity	NOUN
ejpam-6037	33	37	and	and	CCONJ
ejpam-6037	33	38	other	other	ADJ
ejpam-6037	33	39	related	related	ADJ
ejpam-6037	33	40	generalized	generalized	ADJ
ejpam-6037	33	41	forms	form	NOUN
ejpam-6037	33	42	of	of	ADP
ejpam-6037	33	43	continuity	continuity	NOUN
ejpam-6037	33	44	.	.	PUNCT
ejpam-6037	34	1	it	it	PRON
ejpam-6037	34	2	turns	turn	VERB
ejpam-6037	34	3	out	out	ADP
ejpam-6037	34	4	that	that	SCONJ
ejpam-6037	34	5	the	the	DET
ejpam-6037	34	6	contra	contra	PROPN
ejpam-6037	34	7	m	m	PROPN
ejpam-6037	34	8	-	-	PUNCT
ejpam-6037	34	9	continuity	continuity	NOUN
ejpam-6037	34	10	is	be	AUX
ejpam-6037	34	11	a	a	DET
ejpam-6037	34	12	unified	unified	ADJ
ejpam-6037	34	13	form	form	NOUN
ejpam-6037	34	14	of	of	ADP
ejpam-6037	34	15	several	several	ADJ
ejpam-6037	34	16	modifications	modification	NOUN
ejpam-6037	34	17	of	of	ADP
ejpam-6037	34	18	weak	weak	ADJ
ejpam-6037	34	19	contra	contra	ADJ
ejpam-6037	34	20	-	-	NOUN
ejpam-6037	34	21	continuity	continuity	NOUN
ejpam-6037	34	22	due	due	ADP
ejpam-6037	34	23	to	to	ADP
ejpam-6037	34	24	baker	baker	PROPN
ejpam-6037	35	1	[	[	X
ejpam-6037	35	2	29	29	NUM
ejpam-6037	35	3	]	]	PUNCT
ejpam-6037	35	4	.	.	PUNCT
ejpam-6037	36	1	on	on	ADP
ejpam-6037	36	2	the	the	DET
ejpam-6037	36	3	other	other	ADJ
ejpam-6037	36	4	hand	hand	NOUN
ejpam-6037	36	5	,	,	PUNCT
ejpam-6037	36	6	the	the	DET
ejpam-6037	36	7	present	present	ADJ
ejpam-6037	36	8	authors	author	NOUN
ejpam-6037	36	9	introduced	introduce	VERB
ejpam-6037	36	10	and	and	CCONJ
ejpam-6037	36	11	studied	study	VERB
ejpam-6037	36	12	the	the	DET
ejpam-6037	36	13	notions	notion	NOUN
ejpam-6037	36	14	of	of	ADP
ejpam-6037	36	15	(	(	PUNCT
ejpam-6037	36	16	τ1	τ1	PROPN
ejpam-6037	36	17	,	,	PUNCT
ejpam-6037	36	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	19	functions	function	NOUN
ejpam-6037	36	20	[	[	X
ejpam-6037	36	21	30	30	NUM
ejpam-6037	36	22	]	]	PUNCT
ejpam-6037	36	23	,	,	PUNCT
ejpam-6037	36	24	almost	almost	ADV
ejpam-6037	36	25	(	(	PUNCT
ejpam-6037	36	26	τ1	τ1	NOUN
ejpam-6037	36	27	,	,	PUNCT
ejpam-6037	36	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	29	functions	function	NOUN
ejpam-6037	36	30	[	[	X
ejpam-6037	36	31	31	31	NUM
ejpam-6037	36	32	]	]	PUNCT
ejpam-6037	36	33	,	,	PUNCT
ejpam-6037	36	34	weakly	weakly	ADJ
ejpam-6037	36	35	(	(	PUNCT
ejpam-6037	36	36	τ1	τ1	NOUN
ejpam-6037	36	37	,	,	PUNCT
ejpam-6037	36	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	39	functions	function	NOUN
ejpam-6037	36	40	[	[	X
ejpam-6037	36	41	32	32	NUM
ejpam-6037	36	42	]	]	PUNCT
ejpam-6037	36	43	,	,	PUNCT
ejpam-6037	36	44	quasi	quasi	NOUN
ejpam-6037	36	45	θ(τ1	θ(τ1	NOUN
ejpam-6037	36	46	,	,	PUNCT
ejpam-6037	36	47	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	48	functions	function	NOUN
ejpam-6037	36	49	[	[	X
ejpam-6037	36	50	33	33	NUM
ejpam-6037	36	51	]	]	PUNCT
ejpam-6037	36	52	,	,	PUNCT
ejpam-6037	36	53	almost	almost	ADV
ejpam-6037	36	54	quasi	quasi	NOUN
ejpam-6037	36	55	(	(	PUNCT
ejpam-6037	36	56	τ1	τ1	NOUN
ejpam-6037	36	57	,	,	PUNCT
ejpam-6037	36	58	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	59	functions	function	NOUN
ejpam-6037	36	60	[	[	X
ejpam-6037	36	61	34	34	NUM
ejpam-6037	36	62	]	]	PUNCT
ejpam-6037	36	63	,	,	PUNCT
ejpam-6037	36	64	weakly	weakly	ADJ
ejpam-6037	36	65	quasi	quasi	NOUN
ejpam-6037	36	66	(	(	PUNCT
ejpam-6037	36	67	τ1	τ1	PROPN
ejpam-6037	36	68	,	,	PUNCT
ejpam-6037	36	69	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	70	functions	function	NOUN
ejpam-6037	36	71	[	[	X
ejpam-6037	36	72	35	35	NUM
ejpam-6037	36	73	]	]	X
ejpam-6037	36	74	,	,	PUNCT
ejpam-6037	36	75	almost	almost	ADV
ejpam-6037	36	76	weakly	weakly	ADJ
ejpam-6037	36	77	(	(	PUNCT
ejpam-6037	36	78	τ1	τ1	NOUN
ejpam-6037	36	79	,	,	PUNCT
ejpam-6037	36	80	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	36	81	functions	function	NOUN
ejpam-6037	36	82	[	[	X
ejpam-6037	36	83	36	36	NUM
ejpam-6037	36	84	]	]	PUNCT
ejpam-6037	36	85	and	and	CCONJ
ejpam-6037	36	86	almost	almost	ADV
ejpam-6037	36	87	contra-(λ	contra-(λ	PROPN
ejpam-6037	36	88	,	,	PUNCT
ejpam-6037	36	89	sp)-continuous	sp)-continuous	ADJ
ejpam-6037	36	90	functions	function	NOUN
ejpam-6037	36	91	[	[	X
ejpam-6037	36	92	37	37	NUM
ejpam-6037	36	93	]	]	PUNCT
ejpam-6037	36	94	.	.	PUNCT
ejpam-6037	37	1	in	in	ADP
ejpam-6037	37	2	this	this	DET
ejpam-6037	37	3	paper	paper	NOUN
ejpam-6037	37	4	,	,	PUNCT
ejpam-6037	37	5	we	we	PRON
ejpam-6037	37	6	introduce	introduce	VERB
ejpam-6037	37	7	the	the	DET
ejpam-6037	37	8	concept	concept	NOUN
ejpam-6037	37	9	of	of	ADP
ejpam-6037	37	10	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	37	11	,	,	PUNCT
ejpam-6037	37	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	37	13	functions	function	NOUN
ejpam-6037	37	14	.	.	PUNCT
ejpam-6037	38	1	we	we	PRON
ejpam-6037	38	2	also	also	ADV
ejpam-6037	38	3	investigate	investigate	VERB
ejpam-6037	38	4	some	some	DET
ejpam-6037	38	5	characterizations	characterization	NOUN
ejpam-6037	38	6	of	of	ADP
ejpam-6037	38	7	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	38	8	,	,	PUNCT
ejpam-6037	38	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	38	10	functions	function	NOUN
ejpam-6037	38	11	.	.	PUNCT
ejpam-6037	39	1	2	2	X
ejpam-6037	39	2	.	.	NUM
ejpam-6037	39	3	preliminaries	preliminary	NOUN
ejpam-6037	39	4	throughout	throughout	ADP
ejpam-6037	39	5	the	the	DET
ejpam-6037	39	6	present	present	ADJ
ejpam-6037	39	7	paper	paper	NOUN
ejpam-6037	39	8	,	,	PUNCT
ejpam-6037	39	9	spaces	space	NOUN
ejpam-6037	39	10	(	(	PUNCT
ejpam-6037	39	11	x	x	NOUN
ejpam-6037	39	12	,	,	PUNCT
ejpam-6037	39	13	τ1	τ1	NOUN
ejpam-6037	39	14	,	,	PUNCT
ejpam-6037	39	15	τ2	τ2	NOUN
ejpam-6037	39	16	)	)	PUNCT
ejpam-6037	39	17	and	and	CCONJ
ejpam-6037	39	18	(	(	PUNCT
ejpam-6037	39	19	y	y	PROPN
ejpam-6037	39	20	,	,	PUNCT
ejpam-6037	39	21	σ1	σ1	PROPN
ejpam-6037	39	22	,	,	PUNCT
ejpam-6037	39	23	σ2	σ2	NOUN
ejpam-6037	39	24	)	)	PUNCT
ejpam-6037	39	25	(	(	PUNCT
ejpam-6037	39	26	or	or	CCONJ
ejpam-6037	39	27	simply	simply	ADV
ejpam-6037	39	28	x	x	X
ejpam-6037	39	29	and	and	CCONJ
ejpam-6037	39	30	y	y	PROPN
ejpam-6037	39	31	)	)	PUNCT
ejpam-6037	39	32	always	always	ADV
ejpam-6037	39	33	mean	mean	VERB
ejpam-6037	39	34	bitopological	bitopological	ADJ
ejpam-6037	39	35	spaces	space	NOUN
ejpam-6037	39	36	on	on	ADP
ejpam-6037	39	37	which	which	PRON
ejpam-6037	39	38	no	no	DET
ejpam-6037	39	39	separation	separation	NOUN
ejpam-6037	39	40	axioms	axiom	NOUN
ejpam-6037	39	41	are	be	AUX
ejpam-6037	39	42	assumed	assume	VERB
ejpam-6037	39	43	unless	unless	SCONJ
ejpam-6037	39	44	explicitly	explicitly	ADV
ejpam-6037	39	45	stated	state	VERB
ejpam-6037	39	46	.	.	PUNCT
ejpam-6037	40	1	let	let	VERB
ejpam-6037	40	2	a	a	DET
ejpam-6037	40	3	be	be	AUX
ejpam-6037	40	4	a	a	DET
ejpam-6037	40	5	subset	subset	NOUN
ejpam-6037	40	6	of	of	ADP
ejpam-6037	40	7	a	a	DET
ejpam-6037	40	8	bitopological	bitopological	ADJ
ejpam-6037	40	9	space	space	NOUN
ejpam-6037	40	10	(	(	PUNCT
ejpam-6037	40	11	x	x	NOUN
ejpam-6037	40	12	,	,	PUNCT
ejpam-6037	40	13	τ1	τ1	NOUN
ejpam-6037	40	14	,	,	PUNCT
ejpam-6037	40	15	τ2	τ2	NOUN
ejpam-6037	40	16	)	)	PUNCT
ejpam-6037	40	17	.	.	PUNCT
ejpam-6037	41	1	the	the	DET
ejpam-6037	41	2	closure	closure	NOUN
ejpam-6037	41	3	of	of	ADP
ejpam-6037	41	4	a	a	PRON
ejpam-6037	41	5	and	and	CCONJ
ejpam-6037	41	6	the	the	DET
ejpam-6037	41	7	interior	interior	NOUN
ejpam-6037	41	8	of	of	ADP
ejpam-6037	41	9	a	a	PRON
ejpam-6037	41	10	with	with	ADP
ejpam-6037	41	11	respect	respect	NOUN
ejpam-6037	41	12	to	to	ADP
ejpam-6037	41	13	τi	τi	PROPN
ejpam-6037	41	14	are	be	AUX
ejpam-6037	41	15	denoted	denote	VERB
ejpam-6037	41	16	by	by	ADP
ejpam-6037	41	17	τi	τi	NOUN
ejpam-6037	41	18	-	-	PUNCT
ejpam-6037	41	19	cl(a	cl(a	NUM
ejpam-6037	41	20	)	)	PUNCT
ejpam-6037	41	21	and	and	CCONJ
ejpam-6037	41	22	τi	τi	NOUN
ejpam-6037	41	23	-	-	PUNCT
ejpam-6037	41	24	int(a	int(a	NOUN
ejpam-6037	41	25	)	)	PUNCT
ejpam-6037	41	26	,	,	PUNCT
ejpam-6037	41	27	respectively	respectively	ADV
ejpam-6037	41	28	,	,	PUNCT
ejpam-6037	41	29	for	for	ADP
ejpam-6037	41	30	i	i	PROPN
ejpam-6037	41	31	=	=	SYM
ejpam-6037	41	32	1	1	NUM
ejpam-6037	41	33	,	,	PUNCT
ejpam-6037	41	34	2	2	NUM
ejpam-6037	41	35	.	.	X
ejpam-6037	41	36	a	a	DET
ejpam-6037	41	37	subset	subset	NOUN
ejpam-6037	41	38	a	a	PRON
ejpam-6037	41	39	of	of	ADP
ejpam-6037	41	40	a	a	DET
ejpam-6037	41	41	bitopological	bitopological	ADJ
ejpam-6037	41	42	space	space	NOUN
ejpam-6037	41	43	(	(	PUNCT
ejpam-6037	41	44	x	x	NOUN
ejpam-6037	41	45	,	,	PUNCT
ejpam-6037	41	46	τ1	τ1	NOUN
ejpam-6037	41	47	,	,	PUNCT
ejpam-6037	41	48	τ2	τ2	NOUN
ejpam-6037	41	49	)	)	PUNCT
ejpam-6037	41	50	is	be	AUX
ejpam-6037	41	51	called	call	VERB
ejpam-6037	41	52	τ1τ2	τ1τ2	VERB
ejpam-6037	41	53	-	-	ADJ
ejpam-6037	41	54	closed	closed	ADJ
ejpam-6037	41	55	[	[	X
ejpam-6037	41	56	38	38	NUM
ejpam-6037	41	57	]	]	PUNCT
ejpam-6037	41	58	if	if	SCONJ
ejpam-6037	41	59	a	a	DET
ejpam-6037	41	60	=	=	NOUN
ejpam-6037	41	61	τ1	τ1	NOUN
ejpam-6037	41	62	-	-	PUNCT
ejpam-6037	41	63	cl(τ2	cl(τ2	NOUN
ejpam-6037	41	64	-	-	PUNCT
ejpam-6037	41	65	cl(a	cl(a	NUM
ejpam-6037	41	66	)	)	PUNCT
ejpam-6037	41	67	)	)	PUNCT
ejpam-6037	41	68	.	.	PUNCT
ejpam-6037	42	1	the	the	DET
ejpam-6037	42	2	complement	complement	NOUN
ejpam-6037	42	3	of	of	ADP
ejpam-6037	42	4	a	a	DET
ejpam-6037	42	5	τ1τ2	τ1τ2	ADJ
ejpam-6037	42	6	-	-	ADJ
ejpam-6037	42	7	closed	closed	ADJ
ejpam-6037	42	8	set	set	NOUN
ejpam-6037	42	9	is	be	AUX
ejpam-6037	42	10	called	call	VERB
ejpam-6037	42	11	τ1τ2	τ1τ2	NOUN
ejpam-6037	42	12	-	-	ADJ
ejpam-6037	42	13	open	open	ADJ
ejpam-6037	42	14	.	.	PUNCT
ejpam-6037	43	1	the	the	DET
ejpam-6037	43	2	intersection	intersection	NOUN
ejpam-6037	43	3	of	of	ADP
ejpam-6037	43	4	all	all	DET
ejpam-6037	43	5	τ1τ2	τ1τ2	ADJ
ejpam-6037	43	6	-	-	ADJ
ejpam-6037	43	7	closed	closed	ADJ
ejpam-6037	43	8	sets	set	NOUN
ejpam-6037	43	9	of	of	ADP
ejpam-6037	43	10	x	x	PUNCT
ejpam-6037	43	11	containing	contain	VERB
ejpam-6037	43	12	a	a	PRON
ejpam-6037	43	13	is	be	AUX
ejpam-6037	43	14	called	call	VERB
ejpam-6037	43	15	the	the	DET
ejpam-6037	43	16	τ1τ2	τ1τ2	NOUN
ejpam-6037	43	17	-	-	NOUN
ejpam-6037	43	18	closure	closure	NOUN
ejpam-6037	43	19	[	[	X
ejpam-6037	43	20	38	38	NUM
ejpam-6037	43	21	]	]	PUNCT
ejpam-6037	43	22	of	of	ADP
ejpam-6037	43	23	a	a	PRON
ejpam-6037	43	24	and	and	CCONJ
ejpam-6037	43	25	is	be	AUX
ejpam-6037	43	26	denoted	denote	VERB
ejpam-6037	43	27	by	by	ADP
ejpam-6037	43	28	τ1τ2	τ1τ2	NOUN
ejpam-6037	43	29	-	-	NUM
ejpam-6037	43	30	cl(a	cl(a	NUM
ejpam-6037	43	31	)	)	PUNCT
ejpam-6037	43	32	.	.	PUNCT
ejpam-6037	44	1	the	the	DET
ejpam-6037	44	2	union	union	NOUN
ejpam-6037	44	3	of	of	ADP
ejpam-6037	44	4	all	all	DET
ejpam-6037	44	5	τ1τ2	τ1τ2	ADJ
ejpam-6037	44	6	-	-	ADJ
ejpam-6037	44	7	open	open	ADJ
ejpam-6037	44	8	sets	set	NOUN
ejpam-6037	44	9	of	of	ADP
ejpam-6037	44	10	x	x	PUNCT
ejpam-6037	44	11	contained	contain	VERB
ejpam-6037	44	12	in	in	ADP
ejpam-6037	44	13	a	a	PRON
ejpam-6037	44	14	is	be	AUX
ejpam-6037	44	15	called	call	VERB
ejpam-6037	44	16	the	the	DET
ejpam-6037	44	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	44	18	-	-	ADJ
ejpam-6037	44	19	interior	interior	ADJ
ejpam-6037	44	20	[	[	X
ejpam-6037	44	21	38	38	NUM
ejpam-6037	44	22	]	]	PUNCT
ejpam-6037	44	23	of	of	ADP
ejpam-6037	44	24	a	a	PRON
ejpam-6037	44	25	and	and	CCONJ
ejpam-6037	44	26	is	be	AUX
ejpam-6037	44	27	denoted	denote	VERB
ejpam-6037	44	28	by	by	ADP
ejpam-6037	44	29	τ1τ2	τ1τ2	NOUN
ejpam-6037	44	30	-	-	ADJ
ejpam-6037	44	31	int(a	int(a	NOUN
ejpam-6037	44	32	)	)	PUNCT
ejpam-6037	44	33	.	.	PUNCT
ejpam-6037	45	1	lemma	lemma	PROPN
ejpam-6037	45	2	1	1	NUM
ejpam-6037	45	3	.	.	PUNCT
ejpam-6037	46	1	[	[	X
ejpam-6037	46	2	38	38	NUM
ejpam-6037	46	3	]	]	PUNCT
ejpam-6037	46	4	let	let	VERB
ejpam-6037	46	5	a	a	PRON
ejpam-6037	46	6	and	and	CCONJ
ejpam-6037	46	7	b	b	NOUN
ejpam-6037	46	8	be	be	AUX
ejpam-6037	46	9	subsets	subset	NOUN
ejpam-6037	46	10	of	of	ADP
ejpam-6037	46	11	a	a	DET
ejpam-6037	46	12	bitopological	bitopological	ADJ
ejpam-6037	46	13	space	space	NOUN
ejpam-6037	46	14	(	(	PUNCT
ejpam-6037	46	15	x	x	NOUN
ejpam-6037	46	16	,	,	PUNCT
ejpam-6037	46	17	τ1	τ1	NOUN
ejpam-6037	46	18	,	,	PUNCT
ejpam-6037	46	19	τ2	τ2	NOUN
ejpam-6037	46	20	)	)	PUNCT
ejpam-6037	46	21	.	.	PUNCT
ejpam-6037	47	1	for	for	ADP
ejpam-6037	47	2	the	the	DET
ejpam-6037	47	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6037	47	4	,	,	PUNCT
ejpam-6037	47	5	the	the	DET
ejpam-6037	47	6	following	follow	VERB
ejpam-6037	47	7	properties	property	NOUN
ejpam-6037	47	8	hold	hold	VERB
ejpam-6037	47	9	:	:	PUNCT
ejpam-6037	47	10	(	(	PUNCT
ejpam-6037	47	11	1	1	X
ejpam-6037	47	12	)	)	PUNCT
ejpam-6037	47	13	a	a	DET
ejpam-6037	47	14	⊆	⊆	NUM
ejpam-6037	47	15	τ1τ2	τ1τ2	NOUN
ejpam-6037	47	16	-	-	NUM
ejpam-6037	47	17	cl(a	cl(a	NUM
ejpam-6037	47	18	)	)	PUNCT
ejpam-6037	47	19	and	and	CCONJ
ejpam-6037	47	20	τ1τ2	τ1τ2	NOUN
ejpam-6037	47	21	-	-	ADJ
ejpam-6037	47	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6037	47	23	-	-	PUNCT
ejpam-6037	47	24	cl(a	cl(a	NUM
ejpam-6037	47	25	)	)	PUNCT
ejpam-6037	47	26	)	)	PUNCT
ejpam-6037	48	1	=	=	PUNCT
ejpam-6037	48	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	48	3	-	-	NUM
ejpam-6037	48	4	cl(a	cl(a	NUM
ejpam-6037	48	5	)	)	PUNCT
ejpam-6037	48	6	.	.	PUNCT
ejpam-6037	49	1	(	(	PUNCT
ejpam-6037	49	2	2	2	X
ejpam-6037	49	3	)	)	PUNCT
ejpam-6037	49	4	if	if	SCONJ
ejpam-6037	49	5	a	a	DET
ejpam-6037	49	6	⊆	⊆	NUM
ejpam-6037	49	7	b	b	NOUN
ejpam-6037	49	8	,	,	PUNCT
ejpam-6037	49	9	then	then	ADV
ejpam-6037	49	10	τ1τ2	τ1τ2	NOUN
ejpam-6037	49	11	-	-	NUM
ejpam-6037	49	12	cl(a	cl(a	NUM
ejpam-6037	49	13	)	)	PUNCT
ejpam-6037	49	14	⊆	⊆	NUM
ejpam-6037	49	15	τ1τ2	τ1τ2	NOUN
ejpam-6037	49	16	-	-	NOUN
ejpam-6037	49	17	cl(b	cl(b	NOUN
ejpam-6037	49	18	)	)	PUNCT
ejpam-6037	49	19	.	.	PUNCT
ejpam-6037	50	1	n.	n.	PROPN
ejpam-6037	50	2	chutiman	chutiman	PROPN
ejpam-6037	50	3	,	,	PUNCT
ejpam-6037	50	4	a.	a.	PROPN
ejpam-6037	50	5	sama	sama	PROPN
ejpam-6037	50	6	-	-	PUNCT
ejpam-6037	50	7	ae	ae	PROPN
ejpam-6037	50	8	,	,	PUNCT
ejpam-6037	50	9	c.	c.	PROPN
ejpam-6037	50	10	boonpok	boonpok	PROPN
ejpam-6037	50	11	/	/	SYM
ejpam-6037	50	12	eur	eur	PROPN
ejpam-6037	50	13	.	.	PUNCT
ejpam-6037	51	1	j.	j.	PROPN
ejpam-6037	51	2	pure	pure	PROPN
ejpam-6037	51	3	appl	appl	PROPN
ejpam-6037	51	4	.	.	PROPN
ejpam-6037	51	5	math	math	PROPN
ejpam-6037	51	6	,	,	PUNCT
ejpam-6037	51	7	18	18	NUM
ejpam-6037	51	8	(	(	PUNCT
ejpam-6037	51	9	2	2	NUM
ejpam-6037	51	10	)	)	PUNCT
ejpam-6037	51	11	(	(	PUNCT
ejpam-6037	51	12	2025	2025	NUM
ejpam-6037	51	13	)	)	PUNCT
ejpam-6037	51	14	,	,	PUNCT
ejpam-6037	51	15	6037	6037	NUM
ejpam-6037	51	16	3	3	NUM
ejpam-6037	51	17	of	of	ADP
ejpam-6037	51	18	12	12	NUM
ejpam-6037	51	19	(	(	PUNCT
ejpam-6037	51	20	3	3	NUM
ejpam-6037	51	21	)	)	PUNCT
ejpam-6037	51	22	τ1τ2	τ1τ2	NOUN
ejpam-6037	51	23	-	-	NUM
ejpam-6037	51	24	cl(a	cl(a	NUM
ejpam-6037	51	25	)	)	PUNCT
ejpam-6037	51	26	is	be	AUX
ejpam-6037	51	27	τ1τ2	τ1τ2	NOUN
ejpam-6037	51	28	-	-	ADJ
ejpam-6037	51	29	closed	closed	ADJ
ejpam-6037	51	30	.	.	PUNCT
ejpam-6037	52	1	(	(	PUNCT
ejpam-6037	52	2	4	4	X
ejpam-6037	52	3	)	)	PUNCT
ejpam-6037	52	4	a	a	PRON
ejpam-6037	52	5	is	be	AUX
ejpam-6037	52	6	τ1τ2	τ1τ2	NOUN
ejpam-6037	52	7	-	-	ADJ
ejpam-6037	52	8	closed	closed	ADJ
ejpam-6037	52	9	if	if	SCONJ
ejpam-6037	52	10	and	and	CCONJ
ejpam-6037	52	11	only	only	ADV
ejpam-6037	52	12	if	if	SCONJ
ejpam-6037	52	13	a	a	DET
ejpam-6037	52	14	=	=	PUNCT
ejpam-6037	52	15	τ1τ2	τ1τ2	NOUN
ejpam-6037	52	16	-	-	NUM
ejpam-6037	52	17	cl(a	cl(a	NUM
ejpam-6037	52	18	)	)	PUNCT
ejpam-6037	52	19	.	.	PUNCT
ejpam-6037	53	1	(	(	PUNCT
ejpam-6037	53	2	5	5	X
ejpam-6037	53	3	)	)	PUNCT
ejpam-6037	53	4	τ1τ2	τ1τ2	NOUN
ejpam-6037	53	5	-	-	NOUN
ejpam-6037	53	6	cl(x	cl(x	X
ejpam-6037	53	7	−a	−a	NOUN
ejpam-6037	53	8	)	)	PUNCT
ejpam-6037	54	1	=	=	PUNCT
ejpam-6037	54	2	x	x	X
ejpam-6037	55	1	−	−	ADP
ejpam-6037	55	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	55	3	-	-	PUNCT
ejpam-6037	55	4	int(a	int(a	NOUN
ejpam-6037	55	5	)	)	PUNCT
ejpam-6037	55	6	.	.	PUNCT
ejpam-6037	56	1	a	a	DET
ejpam-6037	56	2	subset	subset	NOUN
ejpam-6037	56	3	a	a	PRON
ejpam-6037	56	4	of	of	ADP
ejpam-6037	56	5	a	a	DET
ejpam-6037	56	6	bitopological	bitopological	ADJ
ejpam-6037	56	7	space	space	NOUN
ejpam-6037	56	8	(	(	PUNCT
ejpam-6037	56	9	x	x	NOUN
ejpam-6037	56	10	,	,	PUNCT
ejpam-6037	56	11	τ1	τ1	NOUN
ejpam-6037	56	12	,	,	PUNCT
ejpam-6037	56	13	τ2	τ2	NOUN
ejpam-6037	56	14	)	)	PUNCT
ejpam-6037	56	15	is	be	AUX
ejpam-6037	56	16	said	say	VERB
ejpam-6037	56	17	to	to	PART
ejpam-6037	56	18	be	be	AUX
ejpam-6037	56	19	(	(	PUNCT
ejpam-6037	56	20	τ1	τ1	NOUN
ejpam-6037	56	21	,	,	PUNCT
ejpam-6037	56	22	τ2)r	τ2)r	NOUN
ejpam-6037	56	23	-	-	PUNCT
ejpam-6037	56	24	open	open	NOUN
ejpam-6037	57	1	[	[	X
ejpam-6037	57	2	39	39	NUM
ejpam-6037	57	3	]	]	PUNCT
ejpam-6037	57	4	(	(	PUNCT
ejpam-6037	57	5	resp	resp	NOUN
ejpam-6037	57	6	.	.	PUNCT
ejpam-6037	58	1	(	(	PUNCT
ejpam-6037	58	2	τ1	τ1	NOUN
ejpam-6037	58	3	,	,	PUNCT
ejpam-6037	58	4	τ2)s	τ2)s	NOUN
ejpam-6037	58	5	-	-	PUNCT
ejpam-6037	58	6	open	open	ADJ
ejpam-6037	58	7	[	[	X
ejpam-6037	58	8	40	40	NUM
ejpam-6037	58	9	]	]	PUNCT
ejpam-6037	58	10	,	,	PUNCT
ejpam-6037	58	11	(	(	PUNCT
ejpam-6037	58	12	τ1	τ1	NOUN
ejpam-6037	58	13	,	,	PUNCT
ejpam-6037	58	14	τ2)p	τ2)p	NOUN
ejpam-6037	58	15	-	-	ADJ
ejpam-6037	58	16	open	open	ADJ
ejpam-6037	58	17	[	[	X
ejpam-6037	58	18	40	40	NUM
ejpam-6037	58	19	]	]	PUNCT
ejpam-6037	58	20	,	,	PUNCT
ejpam-6037	58	21	(	(	PUNCT
ejpam-6037	58	22	τ1	τ1	NOUN
ejpam-6037	58	23	,	,	PUNCT
ejpam-6037	58	24	τ2)β	τ2)β	ADJ
ejpam-6037	58	25	-	-	PUNCT
ejpam-6037	58	26	open	open	NOUN
ejpam-6037	59	1	[	[	X
ejpam-6037	59	2	40	40	NUM
ejpam-6037	59	3	]	]	PUNCT
ejpam-6037	59	4	)	)	PUNCT
ejpam-6037	59	5	if	if	SCONJ
ejpam-6037	59	6	a	a	DET
ejpam-6037	59	7	=	=	PUNCT
ejpam-6037	59	8	τ1τ2	τ1τ2	NOUN
ejpam-6037	59	9	-	-	NOUN
ejpam-6037	59	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	59	11	-	-	PUNCT
ejpam-6037	59	12	cl(a	cl(a	NUM
ejpam-6037	59	13	)	)	PUNCT
ejpam-6037	59	14	)	)	PUNCT
ejpam-6037	59	15	(	(	PUNCT
ejpam-6037	59	16	resp	resp	NOUN
ejpam-6037	59	17	.	.	PUNCT
ejpam-6037	60	1	a	a	DET
ejpam-6037	60	2	⊆	⊆	NUM
ejpam-6037	60	3	τ1τ2	τ1τ2	NOUN
ejpam-6037	60	4	-	-	ADJ
ejpam-6037	60	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6037	60	6	-	-	PUNCT
ejpam-6037	60	7	int(a	int(a	NOUN
ejpam-6037	60	8	)	)	PUNCT
ejpam-6037	60	9	)	)	PUNCT
ejpam-6037	60	10	,	,	PUNCT
ejpam-6037	60	11	a	a	DET
ejpam-6037	60	12	⊆	⊆	NUM
ejpam-6037	60	13	τ1τ2	τ1τ2	NOUN
ejpam-6037	60	14	-	-	NOUN
ejpam-6037	60	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	60	16	-	-	PUNCT
ejpam-6037	60	17	cl(a	cl(a	NUM
ejpam-6037	60	18	)	)	PUNCT
ejpam-6037	60	19	)	)	PUNCT
ejpam-6037	60	20	,	,	PUNCT
ejpam-6037	60	21	a	a	DET
ejpam-6037	60	22	⊆	⊆	NUM
ejpam-6037	60	23	τ1τ2	τ1τ2	NOUN
ejpam-6037	60	24	-	-	PUNCT
ejpam-6037	60	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6037	60	26	-	-	PUNCT
ejpam-6037	60	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	60	28	-	-	PUNCT
ejpam-6037	60	29	cl(a	cl(a	NUM
ejpam-6037	60	30	)	)	PUNCT
ejpam-6037	60	31	)	)	PUNCT
ejpam-6037	60	32	)	)	PUNCT
ejpam-6037	60	33	)	)	PUNCT
ejpam-6037	60	34	.	.	PUNCT
ejpam-6037	61	1	the	the	DET
ejpam-6037	61	2	complement	complement	NOUN
ejpam-6037	61	3	of	of	ADP
ejpam-6037	61	4	a	a	DET
ejpam-6037	61	5	(	(	PUNCT
ejpam-6037	61	6	τ1	τ1	NOUN
ejpam-6037	61	7	,	,	PUNCT
ejpam-6037	61	8	τ2)r	τ2)r	NOUN
ejpam-6037	61	9	-	-	PUNCT
ejpam-6037	61	10	open	open	ADJ
ejpam-6037	61	11	(	(	PUNCT
ejpam-6037	61	12	resp	resp	NOUN
ejpam-6037	61	13	.	.	PUNCT
ejpam-6037	62	1	(	(	PUNCT
ejpam-6037	62	2	τ1	τ1	NOUN
ejpam-6037	62	3	,	,	PUNCT
ejpam-6037	62	4	τ2)s	τ2)s	NOUN
ejpam-6037	62	5	-	-	PUNCT
ejpam-6037	62	6	open	open	ADJ
ejpam-6037	62	7	,	,	PUNCT
ejpam-6037	62	8	(	(	PUNCT
ejpam-6037	62	9	τ1	τ1	NOUN
ejpam-6037	62	10	,	,	PUNCT
ejpam-6037	62	11	τ2)p	τ2)p	NOUN
ejpam-6037	62	12	-	-	ADJ
ejpam-6037	62	13	open	open	ADJ
ejpam-6037	62	14	,	,	PUNCT
ejpam-6037	62	15	(	(	PUNCT
ejpam-6037	62	16	τ1	τ1	NOUN
ejpam-6037	62	17	,	,	PUNCT
ejpam-6037	62	18	τ2)β	τ2)β	ADJ
ejpam-6037	62	19	-	-	PUNCT
ejpam-6037	62	20	open	open	ADJ
ejpam-6037	62	21	)	)	PUNCT
ejpam-6037	62	22	set	set	NOUN
ejpam-6037	62	23	is	be	AUX
ejpam-6037	62	24	said	say	VERB
ejpam-6037	62	25	to	to	PART
ejpam-6037	62	26	be	be	AUX
ejpam-6037	62	27	(	(	PUNCT
ejpam-6037	62	28	τ1	τ1	NOUN
ejpam-6037	62	29	,	,	PUNCT
ejpam-6037	62	30	τ2)r	τ2)r	NOUN
ejpam-6037	62	31	-	-	PUNCT
ejpam-6037	62	32	closed	closed	ADJ
ejpam-6037	62	33	(	(	PUNCT
ejpam-6037	62	34	resp	resp	NOUN
ejpam-6037	62	35	.	.	PUNCT
ejpam-6037	63	1	(	(	PUNCT
ejpam-6037	63	2	τ1	τ1	NOUN
ejpam-6037	63	3	,	,	PUNCT
ejpam-6037	63	4	τ2)s	τ2)s	NOUN
ejpam-6037	63	5	-	-	PUNCT
ejpam-6037	63	6	closed	closed	ADJ
ejpam-6037	63	7	,	,	PUNCT
ejpam-6037	63	8	(	(	PUNCT
ejpam-6037	63	9	τ1	τ1	NOUN
ejpam-6037	63	10	,	,	PUNCT
ejpam-6037	63	11	τ2)p	τ2)p	NOUN
ejpam-6037	63	12	-	-	PUNCT
ejpam-6037	63	13	closed	closed	ADJ
ejpam-6037	63	14	,	,	PUNCT
ejpam-6037	63	15	(	(	PUNCT
ejpam-6037	63	16	τ1	τ1	NOUN
ejpam-6037	63	17	,	,	PUNCT
ejpam-6037	63	18	τ2)β	τ2)β	ADJ
ejpam-6037	63	19	-	-	PUNCT
ejpam-6037	63	20	closed	closed	ADJ
ejpam-6037	63	21	)	)	PUNCT
ejpam-6037	63	22	.	.	PUNCT
ejpam-6037	64	1	a	a	DET
ejpam-6037	64	2	subset	subset	NOUN
ejpam-6037	64	3	a	a	PRON
ejpam-6037	64	4	of	of	ADP
ejpam-6037	64	5	a	a	DET
ejpam-6037	64	6	bitopological	bitopological	ADJ
ejpam-6037	64	7	space	space	NOUN
ejpam-6037	64	8	(	(	PUNCT
ejpam-6037	64	9	x	x	NOUN
ejpam-6037	64	10	,	,	PUNCT
ejpam-6037	64	11	τ1	τ1	NOUN
ejpam-6037	64	12	,	,	PUNCT
ejpam-6037	64	13	τ2	τ2	NOUN
ejpam-6037	64	14	)	)	PUNCT
ejpam-6037	64	15	is	be	AUX
ejpam-6037	64	16	said	say	VERB
ejpam-6037	64	17	to	to	PART
ejpam-6037	64	18	be	be	AUX
ejpam-6037	64	19	α(τ1	α(τ1	NOUN
ejpam-6037	64	20	,	,	PUNCT
ejpam-6037	64	21	τ2)-open	τ2)-open	ADJ
ejpam-6037	64	22	[	[	X
ejpam-6037	64	23	41	41	NUM
ejpam-6037	64	24	]	]	X
ejpam-6037	64	25	if	if	SCONJ
ejpam-6037	64	26	a	a	DET
ejpam-6037	64	27	⊆	⊆	NUM
ejpam-6037	64	28	τ1τ2	τ1τ2	NOUN
ejpam-6037	64	29	-	-	PUNCT
ejpam-6037	64	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	64	31	-	-	PUNCT
ejpam-6037	64	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6037	64	33	-	-	PUNCT
ejpam-6037	64	34	int(a	int(a	NOUN
ejpam-6037	64	35	)	)	PUNCT
ejpam-6037	64	36	)	)	PUNCT
ejpam-6037	64	37	)	)	PUNCT
ejpam-6037	64	38	.	.	PUNCT
ejpam-6037	65	1	the	the	DET
ejpam-6037	65	2	complement	complement	NOUN
ejpam-6037	65	3	of	of	ADP
ejpam-6037	65	4	an	an	DET
ejpam-6037	65	5	α(τ1	α(τ1	NOUN
ejpam-6037	65	6	,	,	PUNCT
ejpam-6037	65	7	τ2)-open	τ2)-open	ADJ
ejpam-6037	65	8	set	set	NOUN
ejpam-6037	65	9	is	be	AUX
ejpam-6037	65	10	said	say	VERB
ejpam-6037	65	11	to	to	PART
ejpam-6037	65	12	be	be	AUX
ejpam-6037	65	13	α(τ1	α(τ1	NOUN
ejpam-6037	65	14	,	,	PUNCT
ejpam-6037	65	15	τ2)-closed	τ2)-close	VERB
ejpam-6037	65	16	.	.	PUNCT
ejpam-6037	66	1	let	let	VERB
ejpam-6037	66	2	a	a	DET
ejpam-6037	66	3	be	be	AUX
ejpam-6037	66	4	a	a	DET
ejpam-6037	66	5	subset	subset	NOUN
ejpam-6037	66	6	of	of	ADP
ejpam-6037	66	7	a	a	DET
ejpam-6037	66	8	bitopological	bitopological	ADJ
ejpam-6037	66	9	space	space	NOUN
ejpam-6037	66	10	(	(	PUNCT
ejpam-6037	66	11	x	x	NOUN
ejpam-6037	66	12	,	,	PUNCT
ejpam-6037	66	13	τ1	τ1	NOUN
ejpam-6037	66	14	,	,	PUNCT
ejpam-6037	66	15	τ2	τ2	NOUN
ejpam-6037	66	16	)	)	PUNCT
ejpam-6037	66	17	.	.	PUNCT
ejpam-6037	67	1	the	the	DET
ejpam-6037	67	2	set	set	NOUN
ejpam-6037	67	3	∩{g	∩{g	INTJ
ejpam-6037	67	4	|	|	ADV
ejpam-6037	67	5	a	a	DET
ejpam-6037	67	6	⊆	⊆	NUM
ejpam-6037	67	7	g	g	NOUN
ejpam-6037	67	8	and	and	CCONJ
ejpam-6037	67	9	g	g	PROPN
ejpam-6037	67	10	is	be	AUX
ejpam-6037	67	11	τ1τ2	τ1τ2	VERB
ejpam-6037	67	12	-	-	ADJ
ejpam-6037	67	13	open	open	ADJ
ejpam-6037	67	14	}	}	PUNCT
ejpam-6037	67	15	is	be	AUX
ejpam-6037	67	16	called	call	VERB
ejpam-6037	67	17	the	the	DET
ejpam-6037	67	18	τ1τ2	τ1τ2	NOUN
ejpam-6037	67	19	-	-	NOUN
ejpam-6037	67	20	kernel	kernel	NOUN
ejpam-6037	68	1	[	[	X
ejpam-6037	68	2	38	38	NUM
ejpam-6037	68	3	]	]	PUNCT
ejpam-6037	68	4	of	of	ADP
ejpam-6037	68	5	a	a	PRON
ejpam-6037	68	6	and	and	CCONJ
ejpam-6037	68	7	is	be	AUX
ejpam-6037	68	8	denoted	denote	VERB
ejpam-6037	68	9	by	by	ADP
ejpam-6037	68	10	τ1τ2	τ1τ2	NOUN
ejpam-6037	68	11	-	-	ADJ
ejpam-6037	68	12	ker(a	ker(a	ADJ
ejpam-6037	68	13	)	)	PUNCT
ejpam-6037	68	14	.	.	PUNCT
ejpam-6037	69	1	lemma	lemma	PROPN
ejpam-6037	69	2	2	2	NUM
ejpam-6037	69	3	.	.	PUNCT
ejpam-6037	70	1	[	[	X
ejpam-6037	70	2	38	38	NUM
ejpam-6037	70	3	]	]	PUNCT
ejpam-6037	70	4	for	for	ADP
ejpam-6037	70	5	subsets	subset	NOUN
ejpam-6037	70	6	a	a	DET
ejpam-6037	70	7	,	,	PUNCT
ejpam-6037	70	8	b	b	NOUN
ejpam-6037	70	9	of	of	ADP
ejpam-6037	70	10	a	a	DET
ejpam-6037	70	11	bitopological	bitopological	ADJ
ejpam-6037	70	12	space	space	NOUN
ejpam-6037	70	13	(	(	PUNCT
ejpam-6037	70	14	x	x	NOUN
ejpam-6037	70	15	,	,	PUNCT
ejpam-6037	70	16	τ1	τ1	NOUN
ejpam-6037	70	17	,	,	PUNCT
ejpam-6037	70	18	τ2	τ2	NOUN
ejpam-6037	70	19	)	)	PUNCT
ejpam-6037	70	20	,	,	PUNCT
ejpam-6037	70	21	the	the	DET
ejpam-6037	70	22	following	follow	VERB
ejpam-6037	70	23	properties	property	NOUN
ejpam-6037	70	24	hold	hold	VERB
ejpam-6037	70	25	:	:	PUNCT
ejpam-6037	70	26	(	(	PUNCT
ejpam-6037	70	27	1	1	X
ejpam-6037	70	28	)	)	PUNCT
ejpam-6037	70	29	a	a	DET
ejpam-6037	70	30	⊆	⊆	NUM
ejpam-6037	70	31	τ1τ2	τ1τ2	NOUN
ejpam-6037	70	32	-	-	ADJ
ejpam-6037	70	33	ker(a	ker(a	ADJ
ejpam-6037	70	34	)	)	PUNCT
ejpam-6037	70	35	.	.	PUNCT
ejpam-6037	71	1	(	(	PUNCT
ejpam-6037	71	2	2	2	X
ejpam-6037	71	3	)	)	PUNCT
ejpam-6037	71	4	if	if	SCONJ
ejpam-6037	71	5	a	a	DET
ejpam-6037	71	6	⊆	⊆	NUM
ejpam-6037	71	7	b	b	NOUN
ejpam-6037	71	8	,	,	PUNCT
ejpam-6037	71	9	then	then	ADV
ejpam-6037	71	10	τ1τ2	τ1τ2	NOUN
ejpam-6037	71	11	-	-	ADJ
ejpam-6037	71	12	ker(a	ker(a	ADJ
ejpam-6037	71	13	)	)	PUNCT
ejpam-6037	71	14	⊆	⊆	NUM
ejpam-6037	71	15	τ1τ2	τ1τ2	PROPN
ejpam-6037	71	16	-	-	ADJ
ejpam-6037	71	17	ker(b	ker(b	PROPN
ejpam-6037	71	18	)	)	PUNCT
ejpam-6037	71	19	.	.	PUNCT
ejpam-6037	72	1	(	(	PUNCT
ejpam-6037	72	2	3	3	X
ejpam-6037	72	3	)	)	PUNCT
ejpam-6037	72	4	if	if	SCONJ
ejpam-6037	72	5	a	a	PRON
ejpam-6037	72	6	is	be	AUX
ejpam-6037	72	7	τ1τ2	τ1τ2	NOUN
ejpam-6037	72	8	-	-	ADJ
ejpam-6037	72	9	open	open	ADJ
ejpam-6037	72	10	,	,	PUNCT
ejpam-6037	72	11	then	then	ADV
ejpam-6037	72	12	τ1τ2	τ1τ2	NOUN
ejpam-6037	72	13	-	-	ADJ
ejpam-6037	72	14	ker(a	ker(a	ADJ
ejpam-6037	72	15	)	)	PUNCT
ejpam-6037	72	16	=	=	SYM
ejpam-6037	72	17	a.	a.	NOUN
ejpam-6037	72	18	(	(	PUNCT
ejpam-6037	72	19	4	4	NUM
ejpam-6037	72	20	)	)	PUNCT
ejpam-6037	72	21	x	x	SYM
ejpam-6037	72	22	∈	∈	PROPN
ejpam-6037	72	23	τ1τ2	τ1τ2	NOUN
ejpam-6037	72	24	-	-	ADJ
ejpam-6037	72	25	ker(a	ker(a	ADJ
ejpam-6037	72	26	)	)	PUNCT
ejpam-6037	72	27	if	if	SCONJ
ejpam-6037	72	28	and	and	CCONJ
ejpam-6037	72	29	only	only	ADV
ejpam-6037	72	30	if	if	SCONJ
ejpam-6037	72	31	a	a	DET
ejpam-6037	72	32	∩h	∩h	ADJ
ejpam-6037	72	33	̸=	̸=	PROPN
ejpam-6037	72	34	∅	∅	NOUN
ejpam-6037	72	35	for	for	ADP
ejpam-6037	72	36	every	every	DET
ejpam-6037	72	37	τ1τ2	τ1τ2	ADJ
ejpam-6037	72	38	-	-	ADJ
ejpam-6037	72	39	closed	closed	ADJ
ejpam-6037	72	40	set	set	ADJ
ejpam-6037	72	41	h	h	NOUN
ejpam-6037	72	42	containing	contain	VERB
ejpam-6037	72	43	x.	x.	NOUN
ejpam-6037	72	44	let	let	VERB
ejpam-6037	72	45	a	a	PRON
ejpam-6037	72	46	be	be	AUX
ejpam-6037	72	47	a	a	DET
ejpam-6037	72	48	subset	subset	NOUN
ejpam-6037	72	49	of	of	ADP
ejpam-6037	72	50	a	a	DET
ejpam-6037	72	51	bitopological	bitopological	ADJ
ejpam-6037	72	52	space	space	NOUN
ejpam-6037	72	53	(	(	PUNCT
ejpam-6037	72	54	x	x	NOUN
ejpam-6037	72	55	,	,	PUNCT
ejpam-6037	72	56	τ1	τ1	NOUN
ejpam-6037	72	57	,	,	PUNCT
ejpam-6037	72	58	τ2	τ2	NOUN
ejpam-6037	72	59	)	)	PUNCT
ejpam-6037	72	60	.	.	PUNCT
ejpam-6037	73	1	the	the	DET
ejpam-6037	73	2	intersection	intersection	NOUN
ejpam-6037	73	3	of	of	ADP
ejpam-6037	73	4	all	all	DET
ejpam-6037	73	5	(	(	PUNCT
ejpam-6037	73	6	τ1	τ1	NOUN
ejpam-6037	73	7	,	,	PUNCT
ejpam-6037	73	8	τ2)pclosed	τ2)pclose	VERB
ejpam-6037	73	9	sets	set	NOUN
ejpam-6037	73	10	of	of	ADP
ejpam-6037	73	11	x	x	PUNCT
ejpam-6037	73	12	containing	contain	VERB
ejpam-6037	73	13	a	a	PRON
ejpam-6037	73	14	is	be	AUX
ejpam-6037	73	15	called	call	VERB
ejpam-6037	73	16	the	the	DET
ejpam-6037	73	17	(	(	PUNCT
ejpam-6037	73	18	τ1	τ1	NOUN
ejpam-6037	73	19	,	,	PUNCT
ejpam-6037	73	20	τ2)p	τ2)p	NOUN
ejpam-6037	73	21	-	-	NOUN
ejpam-6037	73	22	closure	closure	NOUN
ejpam-6037	73	23	[	[	X
ejpam-6037	73	24	42	42	NUM
ejpam-6037	73	25	]	]	PUNCT
ejpam-6037	73	26	of	of	ADP
ejpam-6037	73	27	a	a	PRON
ejpam-6037	73	28	and	and	CCONJ
ejpam-6037	73	29	is	be	AUX
ejpam-6037	73	30	denoted	denote	VERB
ejpam-6037	73	31	by	by	ADP
ejpam-6037	73	32	(	(	PUNCT
ejpam-6037	73	33	τ1	τ1	PROPN
ejpam-6037	73	34	,	,	PUNCT
ejpam-6037	73	35	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6037	73	36	)	)	PUNCT
ejpam-6037	73	37	.	.	PUNCT
ejpam-6037	74	1	the	the	DET
ejpam-6037	74	2	union	union	NOUN
ejpam-6037	74	3	of	of	ADP
ejpam-6037	74	4	all	all	DET
ejpam-6037	74	5	(	(	PUNCT
ejpam-6037	74	6	τ1	τ1	NOUN
ejpam-6037	74	7	,	,	PUNCT
ejpam-6037	74	8	τ2)p	τ2)p	ADJ
ejpam-6037	74	9	-	-	PUNCT
ejpam-6037	74	10	open	open	ADJ
ejpam-6037	74	11	sets	set	NOUN
ejpam-6037	74	12	of	of	ADP
ejpam-6037	74	13	x	x	PUNCT
ejpam-6037	74	14	contained	contain	VERB
ejpam-6037	74	15	in	in	ADP
ejpam-6037	74	16	a	a	PRON
ejpam-6037	74	17	is	be	AUX
ejpam-6037	74	18	called	call	VERB
ejpam-6037	74	19	the	the	DET
ejpam-6037	74	20	(	(	PUNCT
ejpam-6037	74	21	τ1	τ1	NOUN
ejpam-6037	74	22	,	,	PUNCT
ejpam-6037	74	23	τ2)p	τ2)p	ADJ
ejpam-6037	74	24	-	-	NOUN
ejpam-6037	74	25	interior	interior	ADJ
ejpam-6037	74	26	[	[	X
ejpam-6037	74	27	42	42	NUM
ejpam-6037	74	28	]	]	PUNCT
ejpam-6037	74	29	of	of	ADP
ejpam-6037	74	30	a	a	PRON
ejpam-6037	74	31	and	and	CCONJ
ejpam-6037	74	32	is	be	AUX
ejpam-6037	74	33	denoted	denote	VERB
ejpam-6037	74	34	by	by	ADP
ejpam-6037	74	35	(	(	PUNCT
ejpam-6037	74	36	τ1	τ1	NOUN
ejpam-6037	74	37	,	,	PUNCT
ejpam-6037	74	38	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6037	74	39	)	)	PUNCT
ejpam-6037	74	40	.	.	PUNCT
ejpam-6037	75	1	lemma	lemma	PROPN
ejpam-6037	75	2	3	3	X
ejpam-6037	75	3	.	.	X
ejpam-6037	76	1	for	for	ADP
ejpam-6037	76	2	a	a	DET
ejpam-6037	76	3	subset	subset	NOUN
ejpam-6037	76	4	a	a	PRON
ejpam-6037	76	5	of	of	ADP
ejpam-6037	76	6	a	a	DET
ejpam-6037	76	7	bitopological	bitopological	ADJ
ejpam-6037	76	8	space	space	NOUN
ejpam-6037	76	9	(	(	PUNCT
ejpam-6037	76	10	x	x	NOUN
ejpam-6037	76	11	,	,	PUNCT
ejpam-6037	76	12	τ1	τ1	NOUN
ejpam-6037	76	13	,	,	PUNCT
ejpam-6037	76	14	τ2	τ2	NOUN
ejpam-6037	76	15	)	)	PUNCT
ejpam-6037	76	16	,	,	PUNCT
ejpam-6037	76	17	the	the	DET
ejpam-6037	76	18	following	follow	VERB
ejpam-6037	76	19	properties	property	NOUN
ejpam-6037	76	20	hold	hold	VERB
ejpam-6037	76	21	:	:	PUNCT
ejpam-6037	76	22	(	(	PUNCT
ejpam-6037	76	23	1	1	X
ejpam-6037	76	24	)	)	PUNCT
ejpam-6037	76	25	(	(	PUNCT
ejpam-6037	76	26	τ1	τ1	NOUN
ejpam-6037	76	27	,	,	PUNCT
ejpam-6037	76	28	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-6037	76	29	)	)	PUNCT
ejpam-6037	76	30	=	=	PUNCT
ejpam-6037	77	1	τ1τ2	τ1τ2	NOUN
ejpam-6037	77	2	-	-	ADJ
ejpam-6037	77	3	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6037	77	4	-	-	PUNCT
ejpam-6037	77	5	int(a	int(a	NOUN
ejpam-6037	77	6	)	)	PUNCT
ejpam-6037	77	7	)	)	PUNCT
ejpam-6037	77	8	∪a	∪a	X
ejpam-6037	78	1	[	[	X
ejpam-6037	78	2	42	42	NUM
ejpam-6037	78	3	]	]	X
ejpam-6037	78	4	;	;	PUNCT
ejpam-6037	78	5	(	(	PUNCT
ejpam-6037	78	6	2	2	X
ejpam-6037	78	7	)	)	PUNCT
ejpam-6037	78	8	(	(	PUNCT
ejpam-6037	78	9	τ1	τ1	NOUN
ejpam-6037	78	10	,	,	PUNCT
ejpam-6037	78	11	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6037	78	12	)	)	PUNCT
ejpam-6037	78	13	=	=	PUNCT
ejpam-6037	79	1	τ1τ2	τ1τ2	NOUN
ejpam-6037	79	2	-	-	NOUN
ejpam-6037	79	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	79	4	-	-	PUNCT
ejpam-6037	79	5	cl(a	cl(a	NUM
ejpam-6037	79	6	)	)	PUNCT
ejpam-6037	79	7	)	)	PUNCT
ejpam-6037	80	1	∩a	∩a	PROPN
ejpam-6037	81	1	[	[	X
ejpam-6037	81	2	36	36	NUM
ejpam-6037	81	3	]	]	PUNCT
ejpam-6037	81	4	.	.	PUNCT
ejpam-6037	82	1	3	3	X
ejpam-6037	82	2	.	.	X
ejpam-6037	82	3	characterizations	characterization	NOUN
ejpam-6037	82	4	of	of	ADP
ejpam-6037	82	5	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	82	6	,	,	PUNCT
ejpam-6037	82	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	82	8	functions	function	NOUN
ejpam-6037	82	9	in	in	ADP
ejpam-6037	82	10	this	this	DET
ejpam-6037	82	11	section	section	NOUN
ejpam-6037	82	12	,	,	PUNCT
ejpam-6037	82	13	we	we	PRON
ejpam-6037	82	14	introduce	introduce	VERB
ejpam-6037	82	15	the	the	DET
ejpam-6037	82	16	concept	concept	NOUN
ejpam-6037	82	17	of	of	ADP
ejpam-6037	82	18	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	82	19	,	,	PUNCT
ejpam-6037	82	20	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	82	21	functions	function	NOUN
ejpam-6037	82	22	.	.	PUNCT
ejpam-6037	83	1	furthermore	furthermore	ADV
ejpam-6037	83	2	,	,	PUNCT
ejpam-6037	83	3	some	some	DET
ejpam-6037	83	4	characterizations	characterization	NOUN
ejpam-6037	83	5	of	of	ADP
ejpam-6037	83	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	83	7	,	,	PUNCT
ejpam-6037	83	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	83	9	functions	function	NOUN
ejpam-6037	83	10	are	be	AUX
ejpam-6037	83	11	discussed	discuss	VERB
ejpam-6037	83	12	.	.	PUNCT
ejpam-6037	84	1	definition	definition	NOUN
ejpam-6037	84	2	1	1	NUM
ejpam-6037	84	3	.	.	PUNCT
ejpam-6037	85	1	a	a	DET
ejpam-6037	85	2	function	function	NOUN
ejpam-6037	85	3	f	f	NOUN
ejpam-6037	85	4	:	:	PUNCT
ejpam-6037	85	5	(	(	PUNCT
ejpam-6037	85	6	x	x	NOUN
ejpam-6037	85	7	,	,	PUNCT
ejpam-6037	85	8	τ1	τ1	NOUN
ejpam-6037	85	9	,	,	PUNCT
ejpam-6037	85	10	τ2	τ2	NOUN
ejpam-6037	85	11	)	)	PUNCT
ejpam-6037	85	12	→	→	SYM
ejpam-6037	85	13	(	(	PUNCT
ejpam-6037	85	14	y	y	PROPN
ejpam-6037	85	15	,	,	PUNCT
ejpam-6037	85	16	σ1	σ1	PROPN
ejpam-6037	85	17	,	,	PUNCT
ejpam-6037	85	18	σ2	σ2	PROPN
ejpam-6037	85	19	)	)	PUNCT
ejpam-6037	85	20	is	be	AUX
ejpam-6037	85	21	said	say	VERB
ejpam-6037	85	22	to	to	PART
ejpam-6037	85	23	be	be	AUX
ejpam-6037	85	24	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	85	25	,	,	PUNCT
ejpam-6037	85	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	85	27	if	if	SCONJ
ejpam-6037	85	28	f−1(v	f−1(v	PROPN
ejpam-6037	85	29	)	)	PUNCT
ejpam-6037	85	30	is	be	AUX
ejpam-6037	85	31	τ1τ2	τ1τ2	NOUN
ejpam-6037	85	32	-	-	ADJ
ejpam-6037	85	33	closed	closed	ADJ
ejpam-6037	85	34	in	in	ADP
ejpam-6037	85	35	x	x	PUNCT
ejpam-6037	85	36	for	for	ADP
ejpam-6037	85	37	every	every	DET
ejpam-6037	85	38	σ1σ2	σ1σ2	NOUN
ejpam-6037	85	39	-	-	ADJ
ejpam-6037	85	40	open	open	ADJ
ejpam-6037	85	41	set	set	NOUN
ejpam-6037	85	42	v	v	NOUN
ejpam-6037	85	43	of	of	ADP
ejpam-6037	85	44	y	y	PROPN
ejpam-6037	85	45	.	.	PUNCT
ejpam-6037	86	1	n.	n.	PROPN
ejpam-6037	86	2	chutiman	chutiman	PROPN
ejpam-6037	86	3	,	,	PUNCT
ejpam-6037	86	4	a.	a.	PROPN
ejpam-6037	86	5	sama	sama	PROPN
ejpam-6037	86	6	-	-	PUNCT
ejpam-6037	86	7	ae	ae	PROPN
ejpam-6037	86	8	,	,	PUNCT
ejpam-6037	86	9	c.	c.	PROPN
ejpam-6037	86	10	boonpok	boonpok	PROPN
ejpam-6037	86	11	/	/	SYM
ejpam-6037	86	12	eur	eur	PROPN
ejpam-6037	86	13	.	.	PUNCT
ejpam-6037	87	1	j.	j.	PROPN
ejpam-6037	87	2	pure	pure	PROPN
ejpam-6037	87	3	appl	appl	PROPN
ejpam-6037	87	4	.	.	PROPN
ejpam-6037	87	5	math	math	PROPN
ejpam-6037	87	6	,	,	PUNCT
ejpam-6037	87	7	18	18	NUM
ejpam-6037	87	8	(	(	PUNCT
ejpam-6037	87	9	2	2	NUM
ejpam-6037	87	10	)	)	PUNCT
ejpam-6037	87	11	(	(	PUNCT
ejpam-6037	87	12	2025	2025	NUM
ejpam-6037	87	13	)	)	PUNCT
ejpam-6037	87	14	,	,	PUNCT
ejpam-6037	87	15	6037	6037	NUM
ejpam-6037	87	16	4	4	NUM
ejpam-6037	87	17	of	of	ADP
ejpam-6037	87	18	12	12	NUM
ejpam-6037	87	19	theorem	theorem	NOUN
ejpam-6037	87	20	1	1	NUM
ejpam-6037	87	21	.	.	X
ejpam-6037	88	1	for	for	ADP
ejpam-6037	88	2	a	a	DET
ejpam-6037	88	3	function	function	NOUN
ejpam-6037	88	4	f	f	NOUN
ejpam-6037	88	5	:	:	PUNCT
ejpam-6037	88	6	(	(	PUNCT
ejpam-6037	88	7	x	x	NOUN
ejpam-6037	88	8	,	,	PUNCT
ejpam-6037	88	9	τ1	τ1	NOUN
ejpam-6037	88	10	,	,	PUNCT
ejpam-6037	88	11	τ2	τ2	NOUN
ejpam-6037	88	12	)	)	PUNCT
ejpam-6037	88	13	→	→	SYM
ejpam-6037	88	14	(	(	PUNCT
ejpam-6037	88	15	y	y	PROPN
ejpam-6037	88	16	,	,	PUNCT
ejpam-6037	88	17	σ1	σ1	PROPN
ejpam-6037	88	18	,	,	PUNCT
ejpam-6037	88	19	σ2	σ2	NOUN
ejpam-6037	88	20	)	)	PUNCT
ejpam-6037	88	21	,	,	PUNCT
ejpam-6037	88	22	the	the	DET
ejpam-6037	88	23	following	follow	VERB
ejpam-6037	88	24	properties	property	NOUN
ejpam-6037	88	25	are	be	AUX
ejpam-6037	88	26	equivalent	equivalent	ADJ
ejpam-6037	88	27	:	:	PUNCT
ejpam-6037	88	28	(	(	PUNCT
ejpam-6037	88	29	1	1	X
ejpam-6037	88	30	)	)	PUNCT
ejpam-6037	88	31	f	f	PROPN
ejpam-6037	88	32	is	be	AUX
ejpam-6037	88	33	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	88	34	,	,	PUNCT
ejpam-6037	88	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	88	36	;	;	PUNCT
ejpam-6037	88	37	(	(	PUNCT
ejpam-6037	88	38	2	2	X
ejpam-6037	88	39	)	)	PUNCT
ejpam-6037	88	40	f−1(k	f−1(k	PROPN
ejpam-6037	88	41	)	)	PUNCT
ejpam-6037	88	42	is	be	AUX
ejpam-6037	88	43	τ1τ2	τ1τ2	NOUN
ejpam-6037	88	44	-	-	ADJ
ejpam-6037	88	45	open	open	ADJ
ejpam-6037	88	46	in	in	ADP
ejpam-6037	88	47	x	x	PUNCT
ejpam-6037	88	48	for	for	ADP
ejpam-6037	88	49	every	every	DET
ejpam-6037	88	50	σ1σ2	σ1σ2	NUM
ejpam-6037	88	51	-	-	PUNCT
ejpam-6037	88	52	closed	closed	ADJ
ejpam-6037	88	53	set	set	NOUN
ejpam-6037	88	54	k	k	PROPN
ejpam-6037	88	55	of	of	ADP
ejpam-6037	88	56	y	y	PROPN
ejpam-6037	88	57	;	;	PUNCT
ejpam-6037	88	58	(	(	PUNCT
ejpam-6037	88	59	3	3	X
ejpam-6037	88	60	)	)	PUNCT
ejpam-6037	88	61	for	for	ADP
ejpam-6037	88	62	each	each	DET
ejpam-6037	88	63	x	x	SYM
ejpam-6037	88	64	∈	∈	PROPN
ejpam-6037	88	65	x	x	X
ejpam-6037	88	66	and	and	CCONJ
ejpam-6037	88	67	each	each	PRON
ejpam-6037	88	68	σ1σ2	σ1σ2	VERB
ejpam-6037	88	69	-	-	PUNCT
ejpam-6037	88	70	closed	closed	ADJ
ejpam-6037	88	71	set	set	NOUN
ejpam-6037	88	72	k	k	PROPN
ejpam-6037	88	73	of	of	ADP
ejpam-6037	88	74	y	y	PROPN
ejpam-6037	88	75	containing	contain	VERB
ejpam-6037	88	76	f(x	f(x	PROPN
ejpam-6037	88	77	)	)	PUNCT
ejpam-6037	88	78	,	,	PUNCT
ejpam-6037	88	79	there	there	PRON
ejpam-6037	88	80	exists	exist	VERB
ejpam-6037	88	81	a	a	DET
ejpam-6037	88	82	τ1τ2	τ1τ2	NOUN
ejpam-6037	88	83	-	-	ADJ
ejpam-6037	88	84	open	open	ADJ
ejpam-6037	88	85	set	set	ADJ
ejpam-6037	88	86	u	u	NOUN
ejpam-6037	88	87	of	of	ADP
ejpam-6037	88	88	x	x	PUNCT
ejpam-6037	88	89	containing	contain	VERB
ejpam-6037	88	90	x	x	PUNCT
ejpam-6037	88	91	such	such	ADJ
ejpam-6037	88	92	that	that	DET
ejpam-6037	88	93	f(u	f(u	PROPN
ejpam-6037	88	94	)	)	PUNCT
ejpam-6037	88	95	⊆	⊆	NUM
ejpam-6037	88	96	k	k	NOUN
ejpam-6037	88	97	;	;	PUNCT
ejpam-6037	88	98	(	(	PUNCT
ejpam-6037	88	99	4	4	X
ejpam-6037	88	100	)	)	PUNCT
ejpam-6037	88	101	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6037	88	102	-	-	PUNCT
ejpam-6037	88	103	cl(a	cl(a	NUM
ejpam-6037	88	104	)	)	PUNCT
ejpam-6037	88	105	)	)	PUNCT
ejpam-6037	89	1	⊆	⊆	X
ejpam-6037	89	2	σ1σ2	σ1σ2	NUM
ejpam-6037	89	3	-	-	PUNCT
ejpam-6037	89	4	ker(f(a	ker(f(a	NOUN
ejpam-6037	89	5	)	)	PUNCT
ejpam-6037	89	6	)	)	PUNCT
ejpam-6037	89	7	for	for	ADP
ejpam-6037	89	8	every	every	DET
ejpam-6037	89	9	subset	subset	NOUN
ejpam-6037	89	10	a	a	PRON
ejpam-6037	89	11	of	of	ADP
ejpam-6037	89	12	x	x	PRON
ejpam-6037	89	13	;	;	PUNCT
ejpam-6037	89	14	(	(	PUNCT
ejpam-6037	89	15	5	5	X
ejpam-6037	89	16	)	)	PUNCT
ejpam-6037	89	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	89	18	-	-	NOUN
ejpam-6037	89	19	cl(f	cl(f	NOUN
ejpam-6037	89	20	−1(b	−1(b	NOUN
ejpam-6037	89	21	)	)	PUNCT
ejpam-6037	89	22	)	)	PUNCT
ejpam-6037	90	1	⊆	⊆	NUM
ejpam-6037	90	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	90	3	-	-	PUNCT
ejpam-6037	90	4	ker(b	ker(b	PROPN
ejpam-6037	90	5	)	)	PUNCT
ejpam-6037	90	6	)	)	PUNCT
ejpam-6037	90	7	for	for	ADP
ejpam-6037	90	8	every	every	DET
ejpam-6037	90	9	subset	subset	NOUN
ejpam-6037	90	10	b	b	PROPN
ejpam-6037	90	11	of	of	ADP
ejpam-6037	90	12	y	y	PROPN
ejpam-6037	90	13	.	.	PUNCT
ejpam-6037	91	1	proof	proof	NOUN
ejpam-6037	91	2	.	.	PUNCT
ejpam-6037	92	1	(	(	PUNCT
ejpam-6037	92	2	1	1	X
ejpam-6037	92	3	)	)	PUNCT
ejpam-6037	92	4	⇒	⇒	NOUN
ejpam-6037	92	5	(	(	PUNCT
ejpam-6037	92	6	2	2	NUM
ejpam-6037	92	7	):	):	PUNCT
ejpam-6037	92	8	the	the	DET
ejpam-6037	92	9	proof	proof	NOUN
ejpam-6037	92	10	is	be	AUX
ejpam-6037	92	11	obvious	obvious	ADJ
ejpam-6037	92	12	.	.	PUNCT
ejpam-6037	93	1	(	(	PUNCT
ejpam-6037	93	2	2	2	X
ejpam-6037	93	3	)	)	PUNCT
ejpam-6037	93	4	⇒	⇒	NOUN
ejpam-6037	93	5	(	(	PUNCT
ejpam-6037	93	6	3	3	NUM
ejpam-6037	93	7	):	):	PUNCT
ejpam-6037	93	8	let	let	VERB
ejpam-6037	93	9	x	x	PUNCT
ejpam-6037	93	10	∈	∈	PROPN
ejpam-6037	93	11	x	x	X
ejpam-6037	93	12	and	and	CCONJ
ejpam-6037	93	13	k	k	PROPN
ejpam-6037	93	14	be	be	AUX
ejpam-6037	93	15	any	any	DET
ejpam-6037	93	16	σ1σ2	σ1σ2	NUM
ejpam-6037	93	17	-	-	PUNCT
ejpam-6037	93	18	closed	closed	ADJ
ejpam-6037	93	19	set	set	NOUN
ejpam-6037	93	20	of	of	ADP
ejpam-6037	93	21	y	y	PROPN
ejpam-6037	93	22	containing	contain	VERB
ejpam-6037	93	23	f(x	f(x	PROPN
ejpam-6037	93	24	)	)	PUNCT
ejpam-6037	93	25	.	.	PUNCT
ejpam-6037	94	1	by	by	ADP
ejpam-6037	94	2	(	(	PUNCT
ejpam-6037	94	3	2	2	NUM
ejpam-6037	94	4	)	)	PUNCT
ejpam-6037	94	5	,	,	PUNCT
ejpam-6037	94	6	f−1(k	f−1(k	PROPN
ejpam-6037	94	7	)	)	PUNCT
ejpam-6037	94	8	is	be	AUX
ejpam-6037	94	9	τ1τ2	τ1τ2	NOUN
ejpam-6037	94	10	-	-	ADJ
ejpam-6037	94	11	open	open	ADJ
ejpam-6037	94	12	in	in	ADP
ejpam-6037	94	13	x.	x.	NOUN
ejpam-6037	94	14	then	then	ADV
ejpam-6037	94	15	,	,	PUNCT
ejpam-6037	94	16	we	we	PRON
ejpam-6037	94	17	have	have	VERB
ejpam-6037	94	18	x	x	PART
ejpam-6037	94	19	∈	∈	PRON
ejpam-6037	94	20	τ1τ2	τ1τ2	NOUN
ejpam-6037	94	21	-	-	NUM
ejpam-6037	94	22	int(f	int(f	PRON
ejpam-6037	94	23	−1(k	−1(k	NOUN
ejpam-6037	94	24	)	)	PUNCT
ejpam-6037	94	25	)	)	PUNCT
ejpam-6037	94	26	and	and	CCONJ
ejpam-6037	94	27	therefore	therefore	ADV
ejpam-6037	94	28	there	there	PRON
ejpam-6037	94	29	exists	exist	VERB
ejpam-6037	94	30	a	a	DET
ejpam-6037	94	31	τ1τ2	τ1τ2	NOUN
ejpam-6037	94	32	-	-	ADJ
ejpam-6037	94	33	open	open	ADJ
ejpam-6037	94	34	set	set	ADJ
ejpam-6037	94	35	u	u	NOUN
ejpam-6037	94	36	of	of	ADP
ejpam-6037	94	37	x	x	PUNCT
ejpam-6037	94	38	containing	contain	VERB
ejpam-6037	94	39	x	x	PUNCT
ejpam-6037	94	40	such	such	ADJ
ejpam-6037	94	41	that	that	SCONJ
ejpam-6037	94	42	u	u	PROPN
ejpam-6037	94	43	⊆	⊆	NUM
ejpam-6037	94	44	f−1(k	f−1(k	PROPN
ejpam-6037	94	45	)	)	PUNCT
ejpam-6037	94	46	.	.	PUNCT
ejpam-6037	95	1	thus	thus	ADV
ejpam-6037	95	2	,	,	PUNCT
ejpam-6037	95	3	f(u	f(u	PROPN
ejpam-6037	95	4	)	)	PUNCT
ejpam-6037	95	5	⊆	⊆	NUM
ejpam-6037	95	6	k.	k.	NOUN
ejpam-6037	95	7	(	(	PUNCT
ejpam-6037	95	8	3	3	X
ejpam-6037	95	9	)	)	PUNCT
ejpam-6037	95	10	⇒	⇒	NOUN
ejpam-6037	95	11	(	(	PUNCT
ejpam-6037	95	12	4	4	NUM
ejpam-6037	95	13	):	):	PUNCT
ejpam-6037	95	14	let	let	VERB
ejpam-6037	95	15	a	a	DET
ejpam-6037	95	16	be	be	AUX
ejpam-6037	95	17	any	any	DET
ejpam-6037	95	18	subset	subset	NOUN
ejpam-6037	95	19	of	of	ADP
ejpam-6037	95	20	x.	x.	NOUN
ejpam-6037	95	21	let	let	VERB
ejpam-6037	95	22	x	x	PUNCT
ejpam-6037	95	23	∈	∈	PROPN
ejpam-6037	95	24	τ1τ2	τ1τ2	NOUN
ejpam-6037	95	25	-	-	NUM
ejpam-6037	95	26	cl(a	cl(a	NUM
ejpam-6037	95	27	)	)	PUNCT
ejpam-6037	95	28	and	and	CCONJ
ejpam-6037	95	29	k	k	PROPN
ejpam-6037	95	30	be	be	AUX
ejpam-6037	95	31	any	any	DET
ejpam-6037	95	32	σ1σ2	σ1σ2	NUM
ejpam-6037	95	33	-	-	PUNCT
ejpam-6037	95	34	closed	closed	ADJ
ejpam-6037	95	35	set	set	NOUN
ejpam-6037	95	36	of	of	ADP
ejpam-6037	95	37	y	y	PROPN
ejpam-6037	95	38	containing	contain	VERB
ejpam-6037	95	39	f(x	f(x	PROPN
ejpam-6037	95	40	)	)	PUNCT
ejpam-6037	95	41	.	.	PUNCT
ejpam-6037	96	1	then	then	ADV
ejpam-6037	96	2	by	by	ADP
ejpam-6037	96	3	(	(	PUNCT
ejpam-6037	96	4	3	3	NUM
ejpam-6037	96	5	)	)	PUNCT
ejpam-6037	96	6	,	,	PUNCT
ejpam-6037	96	7	there	there	PRON
ejpam-6037	96	8	exists	exist	VERB
ejpam-6037	96	9	a	a	DET
ejpam-6037	96	10	τ1τ2	τ1τ2	NOUN
ejpam-6037	96	11	-	-	ADJ
ejpam-6037	96	12	open	open	ADJ
ejpam-6037	96	13	set	set	ADJ
ejpam-6037	96	14	u	u	NOUN
ejpam-6037	96	15	of	of	ADP
ejpam-6037	96	16	x	x	PUNCT
ejpam-6037	96	17	containing	contain	VERB
ejpam-6037	96	18	x	x	PUNCT
ejpam-6037	96	19	such	such	ADJ
ejpam-6037	96	20	that	that	DET
ejpam-6037	96	21	f(u	f(u	PROPN
ejpam-6037	96	22	)	)	PUNCT
ejpam-6037	96	23	⊆	⊆	NUM
ejpam-6037	96	24	k	k	NOUN
ejpam-6037	96	25	;	;	PUNCT
ejpam-6037	96	26	hence	hence	ADV
ejpam-6037	96	27	u	u	NOUN
ejpam-6037	96	28	⊆	⊆	NUM
ejpam-6037	96	29	f−1(k	f−1(k	PROPN
ejpam-6037	96	30	)	)	PUNCT
ejpam-6037	96	31	.	.	PUNCT
ejpam-6037	97	1	since	since	SCONJ
ejpam-6037	97	2	x	x	PROPN
ejpam-6037	97	3	∈	∈	PROPN
ejpam-6037	97	4	τ1τ2	τ1τ2	NOUN
ejpam-6037	97	5	-	-	NUM
ejpam-6037	97	6	cl(a	cl(a	NUM
ejpam-6037	97	7	)	)	PUNCT
ejpam-6037	97	8	,	,	PUNCT
ejpam-6037	97	9	u	u	PROPN
ejpam-6037	97	10	∩	∩	NOUN
ejpam-6037	97	11	a	a	DET
ejpam-6037	97	12	̸=	̸=	PROPN
ejpam-6037	97	13	∅	∅	NOUN
ejpam-6037	97	14	and	and	CCONJ
ejpam-6037	97	15	so	so	ADV
ejpam-6037	97	16	∅	∅	NOUN
ejpam-6037	97	17	=	=	NOUN
ejpam-6037	97	18	̸	̸	ADV
ejpam-6037	97	19	f(u	f(u	ADJ
ejpam-6037	97	20	∩	∩	NOUN
ejpam-6037	97	21	a	a	X
ejpam-6037	97	22	)	)	PUNCT
ejpam-6037	97	23	⊆	⊆	NUM
ejpam-6037	97	24	f(u	f(u	PROPN
ejpam-6037	97	25	)	)	PUNCT
ejpam-6037	97	26	∩	∩	ADJ
ejpam-6037	97	27	f(a	f(a	NOUN
ejpam-6037	97	28	)	)	PUNCT
ejpam-6037	97	29	⊆	⊆	NUM
ejpam-6037	97	30	k	k	PROPN
ejpam-6037	97	31	∩	∩	ADJ
ejpam-6037	97	32	f(a	f(a	NOUN
ejpam-6037	97	33	)	)	PUNCT
ejpam-6037	97	34	.	.	PUNCT
ejpam-6037	98	1	by	by	ADP
ejpam-6037	98	2	lemma	lemma	PROPN
ejpam-6037	98	3	2	2	NUM
ejpam-6037	98	4	,	,	PUNCT
ejpam-6037	98	5	we	we	PRON
ejpam-6037	98	6	have	have	VERB
ejpam-6037	98	7	f(x	f(x	PROPN
ejpam-6037	98	8	)	)	PUNCT
ejpam-6037	98	9	∈	∈	PROPN
ejpam-6037	98	10	σ1σ2	σ1σ2	NOUN
ejpam-6037	98	11	-	-	PUNCT
ejpam-6037	98	12	ker(f(a	ker(f(a	NOUN
ejpam-6037	98	13	)	)	PUNCT
ejpam-6037	98	14	)	)	PUNCT
ejpam-6037	98	15	and	and	CCONJ
ejpam-6037	98	16	hence	hence	ADV
ejpam-6037	98	17	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6037	98	18	-	-	PUNCT
ejpam-6037	98	19	cl(a	cl(a	NUM
ejpam-6037	98	20	)	)	PUNCT
ejpam-6037	98	21	)	)	PUNCT
ejpam-6037	99	1	⊆	⊆	X
ejpam-6037	99	2	σ1σ2	σ1σ2	NUM
ejpam-6037	99	3	-	-	PUNCT
ejpam-6037	99	4	ker(f(a	ker(f(a	NOUN
ejpam-6037	99	5	)	)	PUNCT
ejpam-6037	99	6	)	)	PUNCT
ejpam-6037	99	7	.	.	PUNCT
ejpam-6037	100	1	(	(	PUNCT
ejpam-6037	100	2	4	4	X
ejpam-6037	100	3	)	)	PUNCT
ejpam-6037	100	4	⇒	⇒	NOUN
ejpam-6037	100	5	(	(	PUNCT
ejpam-6037	100	6	5	5	NUM
ejpam-6037	100	7	):	):	PUNCT
ejpam-6037	100	8	let	let	VERB
ejpam-6037	100	9	b	b	X
ejpam-6037	100	10	be	be	AUX
ejpam-6037	100	11	any	any	DET
ejpam-6037	100	12	subset	subset	NOUN
ejpam-6037	100	13	of	of	ADP
ejpam-6037	100	14	y	y	PROPN
ejpam-6037	100	15	.	.	PUNCT
ejpam-6037	101	1	by	by	ADP
ejpam-6037	101	2	(	(	PUNCT
ejpam-6037	101	3	4	4	NUM
ejpam-6037	101	4	)	)	PUNCT
ejpam-6037	101	5	and	and	CCONJ
ejpam-6037	101	6	lemma	lemma	PROPN
ejpam-6037	101	7	2	2	NUM
ejpam-6037	101	8	,	,	PUNCT
ejpam-6037	101	9	we	we	PRON
ejpam-6037	101	10	have	have	VERB
ejpam-6037	101	11	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6037	101	12	-	-	PUNCT
ejpam-6037	101	13	cl(f	cl(f	NOUN
ejpam-6037	101	14	−1(b	−1(b	NOUN
ejpam-6037	101	15	)	)	PUNCT
ejpam-6037	101	16	)	)	PUNCT
ejpam-6037	101	17	)	)	PUNCT
ejpam-6037	102	1	⊆	⊆	X
ejpam-6037	102	2	σ1σ2	σ1σ2	NUM
ejpam-6037	102	3	-	-	PUNCT
ejpam-6037	102	4	ker(f(f	ker(f(f	NOUN
ejpam-6037	102	5	−1(b	−1(b	NOUN
ejpam-6037	102	6	)	)	PUNCT
ejpam-6037	102	7	)	)	PUNCT
ejpam-6037	102	8	)	)	PUNCT
ejpam-6037	103	1	⊆	⊆	X
ejpam-6037	103	2	σ1σ2	σ1σ2	X
ejpam-6037	103	3	-	-	PUNCT
ejpam-6037	103	4	ker(b	ker(b	NOUN
ejpam-6037	103	5	)	)	PUNCT
ejpam-6037	103	6	and	and	CCONJ
ejpam-6037	103	7	hence	hence	ADV
ejpam-6037	103	8	τ1τ2	τ1τ2	NOUN
ejpam-6037	103	9	-	-	NOUN
ejpam-6037	103	10	cl(f	cl(f	NOUN
ejpam-6037	103	11	−1(b	−1(b	NOUN
ejpam-6037	103	12	)	)	PUNCT
ejpam-6037	103	13	)	)	PUNCT
ejpam-6037	103	14	⊆	⊆	NUM
ejpam-6037	103	15	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	103	16	-	-	PUNCT
ejpam-6037	103	17	ker(b	ker(b	PROPN
ejpam-6037	103	18	)	)	PUNCT
ejpam-6037	103	19	)	)	PUNCT
ejpam-6037	103	20	.	.	PUNCT
ejpam-6037	104	1	(	(	PUNCT
ejpam-6037	104	2	5	5	X
ejpam-6037	104	3	)	)	PUNCT
ejpam-6037	104	4	⇒	⇒	NOUN
ejpam-6037	104	5	(	(	PUNCT
ejpam-6037	104	6	1	1	NUM
ejpam-6037	104	7	):	):	PUNCT
ejpam-6037	104	8	let	let	VERB
ejpam-6037	104	9	v	v	PART
ejpam-6037	104	10	be	be	AUX
ejpam-6037	104	11	any	any	DET
ejpam-6037	104	12	σ1σ2	σ1σ2	NOUN
ejpam-6037	104	13	-	-	ADJ
ejpam-6037	104	14	open	open	ADJ
ejpam-6037	104	15	set	set	NOUN
ejpam-6037	104	16	of	of	ADP
ejpam-6037	104	17	y	y	PROPN
ejpam-6037	104	18	.	.	PUNCT
ejpam-6037	105	1	then	then	ADV
ejpam-6037	105	2	by	by	ADP
ejpam-6037	105	3	(	(	PUNCT
ejpam-6037	105	4	5	5	NUM
ejpam-6037	105	5	)	)	PUNCT
ejpam-6037	105	6	and	and	CCONJ
ejpam-6037	105	7	lemma	lemma	PROPN
ejpam-6037	105	8	2	2	NUM
ejpam-6037	105	9	,	,	PUNCT
ejpam-6037	105	10	we	we	PRON
ejpam-6037	105	11	have	have	VERB
ejpam-6037	105	12	τ1τ2	τ1τ2	NOUN
ejpam-6037	105	13	-	-	ADJ
ejpam-6037	105	14	cl(f	cl(f	PRON
ejpam-6037	105	15	−1(v	−1(v	NOUN
ejpam-6037	105	16	)	)	PUNCT
ejpam-6037	105	17	)	)	PUNCT
ejpam-6037	106	1	⊆	⊆	NUM
ejpam-6037	106	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	106	3	-	-	PUNCT
ejpam-6037	106	4	ker(v	ker(v	NOUN
ejpam-6037	106	5	)	)	PUNCT
ejpam-6037	106	6	)	)	PUNCT
ejpam-6037	107	1	=	=	SYM
ejpam-6037	107	2	f−1(v	f−1(v	PROPN
ejpam-6037	107	3	)	)	PUNCT
ejpam-6037	107	4	and	and	CCONJ
ejpam-6037	107	5	so	so	ADV
ejpam-6037	107	6	f−1(v	f−1(v	PROPN
ejpam-6037	107	7	)	)	PUNCT
ejpam-6037	107	8	is	be	AUX
ejpam-6037	107	9	τ1τ2	τ1τ2	NOUN
ejpam-6037	107	10	-	-	ADJ
ejpam-6037	107	11	closed	closed	ADJ
ejpam-6037	107	12	in	in	ADP
ejpam-6037	107	13	x.	x.	NOUN
ejpam-6037	107	14	this	this	PRON
ejpam-6037	107	15	shows	show	VERB
ejpam-6037	107	16	that	that	SCONJ
ejpam-6037	107	17	f	f	PROPN
ejpam-6037	107	18	is	be	AUX
ejpam-6037	107	19	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	107	20	,	,	PUNCT
ejpam-6037	107	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	107	22	.	.	PUNCT
ejpam-6037	108	1	definition	definition	NOUN
ejpam-6037	108	2	2	2	NUM
ejpam-6037	108	3	.	.	PUNCT
ejpam-6037	109	1	[	[	X
ejpam-6037	109	2	32	32	NUM
ejpam-6037	109	3	]	]	PUNCT
ejpam-6037	109	4	a	a	DET
ejpam-6037	109	5	function	function	NOUN
ejpam-6037	109	6	f	f	NOUN
ejpam-6037	109	7	:	:	PUNCT
ejpam-6037	109	8	(	(	PUNCT
ejpam-6037	109	9	x	x	NOUN
ejpam-6037	109	10	,	,	PUNCT
ejpam-6037	109	11	τ1	τ1	NOUN
ejpam-6037	109	12	,	,	PUNCT
ejpam-6037	109	13	τ2	τ2	NOUN
ejpam-6037	109	14	)	)	PUNCT
ejpam-6037	109	15	→	→	SYM
ejpam-6037	109	16	(	(	PUNCT
ejpam-6037	109	17	y	y	PROPN
ejpam-6037	109	18	,	,	PUNCT
ejpam-6037	109	19	σ1	σ1	PROPN
ejpam-6037	109	20	,	,	PUNCT
ejpam-6037	109	21	σ2	σ2	PROPN
ejpam-6037	109	22	)	)	PUNCT
ejpam-6037	109	23	is	be	AUX
ejpam-6037	109	24	said	say	VERB
ejpam-6037	109	25	to	to	PART
ejpam-6037	109	26	be	be	AUX
ejpam-6037	109	27	weakly	weakly	ADJ
ejpam-6037	109	28	(	(	PUNCT
ejpam-6037	109	29	τ1	τ1	NOUN
ejpam-6037	109	30	,	,	PUNCT
ejpam-6037	109	31	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	109	32	at	at	ADP
ejpam-6037	109	33	a	a	DET
ejpam-6037	109	34	point	point	NOUN
ejpam-6037	109	35	x	x	SYM
ejpam-6037	109	36	∈	∈	NOUN
ejpam-6037	109	37	x	x	PUNCT
ejpam-6037	109	38	if	if	SCONJ
ejpam-6037	109	39	for	for	ADP
ejpam-6037	109	40	each	each	DET
ejpam-6037	109	41	σ1σ2	σ1σ2	VERB
ejpam-6037	109	42	-	-	ADJ
ejpam-6037	109	43	open	open	ADJ
ejpam-6037	109	44	set	set	NOUN
ejpam-6037	109	45	v	v	NOUN
ejpam-6037	109	46	of	of	ADP
ejpam-6037	109	47	y	y	NOUN
ejpam-6037	109	48	containing	contain	VERB
ejpam-6037	109	49	f(x	f(x	PROPN
ejpam-6037	109	50	)	)	PUNCT
ejpam-6037	109	51	,	,	PUNCT
ejpam-6037	109	52	there	there	PRON
ejpam-6037	109	53	exists	exist	VERB
ejpam-6037	109	54	a	a	DET
ejpam-6037	109	55	τ1τ2	τ1τ2	NOUN
ejpam-6037	109	56	-	-	ADJ
ejpam-6037	109	57	open	open	ADJ
ejpam-6037	109	58	set	set	ADJ
ejpam-6037	109	59	u	u	NOUN
ejpam-6037	109	60	of	of	ADP
ejpam-6037	109	61	x	x	PUNCT
ejpam-6037	109	62	containing	contain	VERB
ejpam-6037	109	63	x	x	PUNCT
ejpam-6037	109	64	such	such	ADJ
ejpam-6037	109	65	that	that	DET
ejpam-6037	109	66	f(u	f(u	PROPN
ejpam-6037	109	67	)	)	PUNCT
ejpam-6037	109	68	⊆	⊆	NUM
ejpam-6037	109	69	σ1σ2	σ1σ2	NOUN
ejpam-6037	109	70	-	-	NUM
ejpam-6037	109	71	cl(v	cl(v	NOUN
ejpam-6037	109	72	)	)	PUNCT
ejpam-6037	109	73	.	.	PUNCT
ejpam-6037	110	1	a	a	DET
ejpam-6037	110	2	function	function	NOUN
ejpam-6037	110	3	f	f	NOUN
ejpam-6037	110	4	:	:	PUNCT
ejpam-6037	110	5	(	(	PUNCT
ejpam-6037	110	6	x	x	NOUN
ejpam-6037	110	7	,	,	PUNCT
ejpam-6037	110	8	τ1	τ1	NOUN
ejpam-6037	110	9	,	,	PUNCT
ejpam-6037	110	10	τ2	τ2	NOUN
ejpam-6037	110	11	)	)	PUNCT
ejpam-6037	110	12	→	→	SYM
ejpam-6037	110	13	(	(	PUNCT
ejpam-6037	110	14	y	y	PROPN
ejpam-6037	110	15	,	,	PUNCT
ejpam-6037	110	16	σ1	σ1	PROPN
ejpam-6037	110	17	,	,	PUNCT
ejpam-6037	110	18	σ2	σ2	PROPN
ejpam-6037	110	19	)	)	PUNCT
ejpam-6037	110	20	is	be	AUX
ejpam-6037	110	21	said	say	VERB
ejpam-6037	110	22	to	to	PART
ejpam-6037	110	23	be	be	AUX
ejpam-6037	110	24	weakly	weakly	ADJ
ejpam-6037	110	25	(	(	PUNCT
ejpam-6037	110	26	τ1	τ1	NOUN
ejpam-6037	110	27	,	,	PUNCT
ejpam-6037	110	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	110	29	if	if	SCONJ
ejpam-6037	110	30	f	f	PROPN
ejpam-6037	110	31	has	have	VERB
ejpam-6037	110	32	this	this	DET
ejpam-6037	110	33	property	property	NOUN
ejpam-6037	110	34	at	at	ADP
ejpam-6037	110	35	each	each	DET
ejpam-6037	110	36	point	point	NOUN
ejpam-6037	110	37	of	of	ADP
ejpam-6037	110	38	x.	x.	NOUN
ejpam-6037	110	39	theorem	theorem	VERB
ejpam-6037	110	40	2	2	NUM
ejpam-6037	110	41	.	.	PUNCT
ejpam-6037	111	1	if	if	SCONJ
ejpam-6037	111	2	f	f	PROPN
ejpam-6037	111	3	:	:	PUNCT
ejpam-6037	111	4	(	(	PUNCT
ejpam-6037	111	5	x	x	NOUN
ejpam-6037	111	6	,	,	PUNCT
ejpam-6037	111	7	τ1	τ1	NOUN
ejpam-6037	111	8	,	,	PUNCT
ejpam-6037	111	9	τ2	τ2	NOUN
ejpam-6037	111	10	)	)	PUNCT
ejpam-6037	111	11	→	→	SYM
ejpam-6037	111	12	(	(	PUNCT
ejpam-6037	111	13	y	y	PROPN
ejpam-6037	111	14	,	,	PUNCT
ejpam-6037	111	15	σ1	σ1	PROPN
ejpam-6037	111	16	,	,	PUNCT
ejpam-6037	111	17	σ2	σ2	PROPN
ejpam-6037	111	18	)	)	PUNCT
ejpam-6037	111	19	is	be	AUX
ejpam-6037	111	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	111	21	,	,	PUNCT
ejpam-6037	111	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	111	23	,	,	PUNCT
ejpam-6037	111	24	then	then	ADV
ejpam-6037	111	25	f	f	PROPN
ejpam-6037	111	26	is	be	AUX
ejpam-6037	111	27	weakly	weakly	ADJ
ejpam-6037	111	28	(	(	PUNCT
ejpam-6037	111	29	τ1	τ1	NOUN
ejpam-6037	111	30	,	,	PUNCT
ejpam-6037	111	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	111	32	.	.	PUNCT
ejpam-6037	112	1	proof	proof	NOUN
ejpam-6037	112	2	.	.	PUNCT
ejpam-6037	113	1	let	let	VERB
ejpam-6037	113	2	x	x	PUNCT
ejpam-6037	113	3	∈	∈	PROPN
ejpam-6037	113	4	x	x	X
ejpam-6037	113	5	and	and	CCONJ
ejpam-6037	113	6	v	v	X
ejpam-6037	113	7	be	be	AUX
ejpam-6037	113	8	any	any	DET
ejpam-6037	113	9	σ1σ2	σ1σ2	NOUN
ejpam-6037	113	10	-	-	ADJ
ejpam-6037	113	11	open	open	ADJ
ejpam-6037	113	12	set	set	NOUN
ejpam-6037	113	13	of	of	ADP
ejpam-6037	113	14	y	y	PROPN
ejpam-6037	113	15	containing	contain	VERB
ejpam-6037	113	16	f(x	f(x	PROPN
ejpam-6037	113	17	)	)	PUNCT
ejpam-6037	113	18	.	.	PUNCT
ejpam-6037	114	1	then	then	ADV
ejpam-6037	114	2	,	,	PUNCT
ejpam-6037	114	3	σ1σ2	σ1σ2	NOUN
ejpam-6037	114	4	-	-	NUM
ejpam-6037	114	5	cl(v	cl(v	NOUN
ejpam-6037	114	6	)	)	PUNCT
ejpam-6037	114	7	is	be	AUX
ejpam-6037	114	8	a	a	DET
ejpam-6037	114	9	σ1σ2	σ1σ2	NUM
ejpam-6037	114	10	-	-	PUNCT
ejpam-6037	114	11	closed	closed	ADJ
ejpam-6037	114	12	set	set	NOUN
ejpam-6037	114	13	of	of	ADP
ejpam-6037	114	14	y	y	PROPN
ejpam-6037	114	15	containing	contain	VERB
ejpam-6037	114	16	f(x	f(x	PROPN
ejpam-6037	114	17	)	)	PUNCT
ejpam-6037	114	18	.	.	PUNCT
ejpam-6037	115	1	since	since	SCONJ
ejpam-6037	115	2	f	f	PROPN
ejpam-6037	115	3	is	be	AUX
ejpam-6037	115	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	115	5	,	,	PUNCT
ejpam-6037	115	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	115	7	,	,	PUNCT
ejpam-6037	115	8	by	by	ADP
ejpam-6037	115	9	theorem	theorem	NOUN
ejpam-6037	115	10	1	1	NUM
ejpam-6037	115	11	there	there	ADV
ejpam-6037	115	12	exists	exist	VERB
ejpam-6037	115	13	a	a	DET
ejpam-6037	115	14	τ1τ2	τ1τ2	NOUN
ejpam-6037	115	15	-	-	ADJ
ejpam-6037	115	16	open	open	ADJ
ejpam-6037	115	17	set	set	ADJ
ejpam-6037	115	18	u	u	NOUN
ejpam-6037	115	19	of	of	ADP
ejpam-6037	115	20	x	x	PUNCT
ejpam-6037	115	21	containing	contain	VERB
ejpam-6037	115	22	x	x	PUNCT
ejpam-6037	115	23	such	such	ADJ
ejpam-6037	115	24	that	that	DET
ejpam-6037	115	25	f(u	f(u	PROPN
ejpam-6037	115	26	)	)	PUNCT
ejpam-6037	115	27	⊆	⊆	NUM
ejpam-6037	115	28	σ1σ2	σ1σ2	NOUN
ejpam-6037	115	29	-	-	NUM
ejpam-6037	115	30	cl(v	cl(v	NOUN
ejpam-6037	115	31	)	)	PUNCT
ejpam-6037	115	32	.	.	PUNCT
ejpam-6037	116	1	this	this	PRON
ejpam-6037	116	2	shows	show	VERB
ejpam-6037	116	3	that	that	SCONJ
ejpam-6037	116	4	f	f	PROPN
ejpam-6037	116	5	is	be	AUX
ejpam-6037	116	6	weakly	weakly	ADJ
ejpam-6037	116	7	(	(	PUNCT
ejpam-6037	116	8	τ1	τ1	NOUN
ejpam-6037	116	9	,	,	PUNCT
ejpam-6037	116	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	116	11	.	.	PUNCT
ejpam-6037	117	1	n.	n.	PROPN
ejpam-6037	117	2	chutiman	chutiman	PROPN
ejpam-6037	117	3	,	,	PUNCT
ejpam-6037	117	4	a.	a.	PROPN
ejpam-6037	117	5	sama	sama	PROPN
ejpam-6037	117	6	-	-	PUNCT
ejpam-6037	117	7	ae	ae	PROPN
ejpam-6037	117	8	,	,	PUNCT
ejpam-6037	117	9	c.	c.	PROPN
ejpam-6037	117	10	boonpok	boonpok	PROPN
ejpam-6037	117	11	/	/	SYM
ejpam-6037	117	12	eur	eur	PROPN
ejpam-6037	117	13	.	.	PUNCT
ejpam-6037	118	1	j.	j.	PROPN
ejpam-6037	118	2	pure	pure	PROPN
ejpam-6037	118	3	appl	appl	PROPN
ejpam-6037	118	4	.	.	PROPN
ejpam-6037	118	5	math	math	PROPN
ejpam-6037	118	6	,	,	PUNCT
ejpam-6037	118	7	18	18	NUM
ejpam-6037	118	8	(	(	PUNCT
ejpam-6037	118	9	2	2	NUM
ejpam-6037	118	10	)	)	PUNCT
ejpam-6037	118	11	(	(	PUNCT
ejpam-6037	118	12	2025	2025	NUM
ejpam-6037	118	13	)	)	PUNCT
ejpam-6037	118	14	,	,	PUNCT
ejpam-6037	118	15	6037	6037	NUM
ejpam-6037	118	16	5	5	NUM
ejpam-6037	118	17	of	of	ADP
ejpam-6037	118	18	12	12	NUM
ejpam-6037	118	19	definition	definition	NOUN
ejpam-6037	118	20	3	3	NUM
ejpam-6037	118	21	.	.	PUNCT
ejpam-6037	119	1	[	[	X
ejpam-6037	119	2	31	31	NUM
ejpam-6037	119	3	]	]	PUNCT
ejpam-6037	119	4	a	a	DET
ejpam-6037	119	5	function	function	NOUN
ejpam-6037	119	6	f	f	NOUN
ejpam-6037	119	7	:	:	PUNCT
ejpam-6037	119	8	(	(	PUNCT
ejpam-6037	119	9	x	x	NOUN
ejpam-6037	119	10	,	,	PUNCT
ejpam-6037	119	11	τ1	τ1	NOUN
ejpam-6037	119	12	,	,	PUNCT
ejpam-6037	119	13	τ2	τ2	NOUN
ejpam-6037	119	14	)	)	PUNCT
ejpam-6037	119	15	→	→	SYM
ejpam-6037	119	16	(	(	PUNCT
ejpam-6037	119	17	y	y	PROPN
ejpam-6037	119	18	,	,	PUNCT
ejpam-6037	119	19	σ1	σ1	PROPN
ejpam-6037	119	20	,	,	PUNCT
ejpam-6037	119	21	σ2	σ2	PROPN
ejpam-6037	119	22	)	)	PUNCT
ejpam-6037	119	23	is	be	AUX
ejpam-6037	119	24	said	say	VERB
ejpam-6037	119	25	to	to	PART
ejpam-6037	119	26	be	be	AUX
ejpam-6037	119	27	almost	almost	ADV
ejpam-6037	119	28	(	(	PUNCT
ejpam-6037	119	29	τ1	τ1	NOUN
ejpam-6037	119	30	,	,	PUNCT
ejpam-6037	119	31	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	119	32	at	at	ADP
ejpam-6037	119	33	a	a	DET
ejpam-6037	119	34	point	point	NOUN
ejpam-6037	119	35	x	x	SYM
ejpam-6037	119	36	∈	∈	NOUN
ejpam-6037	119	37	x	x	PUNCT
ejpam-6037	119	38	if	if	SCONJ
ejpam-6037	119	39	for	for	ADP
ejpam-6037	119	40	each	each	DET
ejpam-6037	119	41	σ1σ2	σ1σ2	VERB
ejpam-6037	119	42	-	-	ADJ
ejpam-6037	119	43	open	open	ADJ
ejpam-6037	119	44	set	set	NOUN
ejpam-6037	119	45	v	v	NOUN
ejpam-6037	119	46	of	of	ADP
ejpam-6037	119	47	y	y	NOUN
ejpam-6037	119	48	containing	contain	VERB
ejpam-6037	119	49	f(x	f(x	PROPN
ejpam-6037	119	50	)	)	PUNCT
ejpam-6037	119	51	,	,	PUNCT
ejpam-6037	119	52	there	there	PRON
ejpam-6037	119	53	exists	exist	VERB
ejpam-6037	119	54	a	a	DET
ejpam-6037	119	55	τ1τ2	τ1τ2	NOUN
ejpam-6037	119	56	-	-	ADJ
ejpam-6037	119	57	open	open	ADJ
ejpam-6037	119	58	set	set	ADJ
ejpam-6037	119	59	u	u	NOUN
ejpam-6037	119	60	of	of	ADP
ejpam-6037	119	61	x	x	PUNCT
ejpam-6037	119	62	containing	contain	VERB
ejpam-6037	119	63	x	x	PUNCT
ejpam-6037	119	64	such	such	ADJ
ejpam-6037	119	65	that	that	DET
ejpam-6037	119	66	f(u	f(u	PROPN
ejpam-6037	119	67	)	)	PUNCT
ejpam-6037	119	68	⊆	⊆	NUM
ejpam-6037	119	69	σ1σ2	σ1σ2	X
ejpam-6037	119	70	-	-	PUNCT
ejpam-6037	119	71	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	119	72	-	-	PUNCT
ejpam-6037	119	73	cl(v	cl(v	NOUN
ejpam-6037	119	74	)	)	PUNCT
ejpam-6037	119	75	)	)	PUNCT
ejpam-6037	119	76	.	.	PUNCT
ejpam-6037	120	1	a	a	DET
ejpam-6037	120	2	function	function	NOUN
ejpam-6037	120	3	f	f	NOUN
ejpam-6037	120	4	:	:	PUNCT
ejpam-6037	120	5	(	(	PUNCT
ejpam-6037	120	6	x	x	NOUN
ejpam-6037	120	7	,	,	PUNCT
ejpam-6037	120	8	τ1	τ1	NOUN
ejpam-6037	120	9	,	,	PUNCT
ejpam-6037	120	10	τ2	τ2	NOUN
ejpam-6037	120	11	)	)	PUNCT
ejpam-6037	120	12	→	→	SYM
ejpam-6037	120	13	(	(	PUNCT
ejpam-6037	120	14	y	y	PROPN
ejpam-6037	120	15	,	,	PUNCT
ejpam-6037	120	16	σ1	σ1	PROPN
ejpam-6037	120	17	,	,	PUNCT
ejpam-6037	120	18	σ2	σ2	PROPN
ejpam-6037	120	19	)	)	PUNCT
ejpam-6037	120	20	is	be	AUX
ejpam-6037	120	21	said	say	VERB
ejpam-6037	120	22	to	to	PART
ejpam-6037	120	23	be	be	AUX
ejpam-6037	120	24	almost	almost	ADV
ejpam-6037	120	25	(	(	PUNCT
ejpam-6037	120	26	τ1	τ1	NOUN
ejpam-6037	120	27	,	,	PUNCT
ejpam-6037	120	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	120	29	if	if	SCONJ
ejpam-6037	120	30	f	f	PROPN
ejpam-6037	120	31	has	have	VERB
ejpam-6037	120	32	this	this	DET
ejpam-6037	120	33	property	property	NOUN
ejpam-6037	120	34	at	at	ADP
ejpam-6037	120	35	each	each	DET
ejpam-6037	120	36	point	point	NOUN
ejpam-6037	120	37	of	of	ADP
ejpam-6037	120	38	x.	x.	NOUN
ejpam-6037	120	39	definition	definition	NOUN
ejpam-6037	120	40	4	4	NUM
ejpam-6037	120	41	.	.	PUNCT
ejpam-6037	121	1	a	a	DET
ejpam-6037	121	2	function	function	NOUN
ejpam-6037	121	3	f	f	NOUN
ejpam-6037	121	4	:	:	PUNCT
ejpam-6037	121	5	(	(	PUNCT
ejpam-6037	121	6	x	x	NOUN
ejpam-6037	121	7	,	,	PUNCT
ejpam-6037	121	8	τ1	τ1	NOUN
ejpam-6037	121	9	,	,	PUNCT
ejpam-6037	121	10	τ2	τ2	NOUN
ejpam-6037	121	11	)	)	PUNCT
ejpam-6037	121	12	→	→	SYM
ejpam-6037	121	13	(	(	PUNCT
ejpam-6037	121	14	y	y	PROPN
ejpam-6037	121	15	,	,	PUNCT
ejpam-6037	121	16	σ1	σ1	PROPN
ejpam-6037	121	17	,	,	PUNCT
ejpam-6037	121	18	σ2	σ2	PROPN
ejpam-6037	121	19	)	)	PUNCT
ejpam-6037	121	20	is	be	AUX
ejpam-6037	121	21	called	call	VERB
ejpam-6037	121	22	almost	almost	ADV
ejpam-6037	121	23	(	(	PUNCT
ejpam-6037	121	24	τ1	τ1	NOUN
ejpam-6037	121	25	,	,	PUNCT
ejpam-6037	121	26	τ2)-open	τ2)-open	ADJ
ejpam-6037	121	27	if	if	SCONJ
ejpam-6037	121	28	f(u	f(u	PROPN
ejpam-6037	121	29	)	)	PUNCT
ejpam-6037	121	30	⊆	⊆	NUM
ejpam-6037	121	31	σ1σ2	σ1σ2	X
ejpam-6037	121	32	-	-	PUNCT
ejpam-6037	121	33	int(σ1σ2	int(σ1σ2	VERB
ejpam-6037	121	34	-	-	PUNCT
ejpam-6037	121	35	cl(f(u	cl(f(u	NOUN
ejpam-6037	121	36	)	)	PUNCT
ejpam-6037	121	37	)	)	PUNCT
ejpam-6037	121	38	)	)	PUNCT
ejpam-6037	122	1	for	for	ADP
ejpam-6037	122	2	every	every	DET
ejpam-6037	122	3	τ1τ2	τ1τ2	ADJ
ejpam-6037	122	4	-	-	ADJ
ejpam-6037	122	5	open	open	ADJ
ejpam-6037	122	6	set	set	ADJ
ejpam-6037	122	7	u	u	NOUN
ejpam-6037	122	8	of	of	ADP
ejpam-6037	122	9	x.	x.	PROPN
ejpam-6037	122	10	theorem	theorem	VERB
ejpam-6037	122	11	3	3	X
ejpam-6037	122	12	.	.	PUNCT
ejpam-6037	123	1	if	if	SCONJ
ejpam-6037	123	2	f	f	PROPN
ejpam-6037	123	3	:	:	PUNCT
ejpam-6037	123	4	(	(	PUNCT
ejpam-6037	123	5	x	x	NOUN
ejpam-6037	123	6	,	,	PUNCT
ejpam-6037	123	7	τ1	τ1	NOUN
ejpam-6037	123	8	,	,	PUNCT
ejpam-6037	123	9	τ2	τ2	NOUN
ejpam-6037	123	10	)	)	PUNCT
ejpam-6037	123	11	→	→	SYM
ejpam-6037	123	12	(	(	PUNCT
ejpam-6037	123	13	y	y	PROPN
ejpam-6037	123	14	,	,	PUNCT
ejpam-6037	123	15	σ1	σ1	PROPN
ejpam-6037	123	16	,	,	PUNCT
ejpam-6037	123	17	σ2	σ2	PROPN
ejpam-6037	123	18	)	)	PUNCT
ejpam-6037	123	19	is	be	AUX
ejpam-6037	123	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	123	21	,	,	PUNCT
ejpam-6037	123	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	123	23	and	and	CCONJ
ejpam-6037	123	24	almost	almost	ADV
ejpam-6037	123	25	(	(	PUNCT
ejpam-6037	123	26	τ1	τ1	NOUN
ejpam-6037	123	27	,	,	PUNCT
ejpam-6037	123	28	τ2)-open	τ2)-open	ADJ
ejpam-6037	123	29	,	,	PUNCT
ejpam-6037	123	30	then	then	ADV
ejpam-6037	123	31	f	f	PROPN
ejpam-6037	123	32	is	be	AUX
ejpam-6037	123	33	almost	almost	ADV
ejpam-6037	123	34	(	(	PUNCT
ejpam-6037	123	35	τ1	τ1	NOUN
ejpam-6037	123	36	,	,	PUNCT
ejpam-6037	123	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	123	38	.	.	PUNCT
ejpam-6037	124	1	proof	proof	NOUN
ejpam-6037	124	2	.	.	PUNCT
ejpam-6037	125	1	let	let	VERB
ejpam-6037	125	2	x	x	PUNCT
ejpam-6037	125	3	∈	∈	PROPN
ejpam-6037	125	4	x	x	X
ejpam-6037	125	5	and	and	CCONJ
ejpam-6037	125	6	v	v	X
ejpam-6037	125	7	be	be	AUX
ejpam-6037	125	8	any	any	DET
ejpam-6037	125	9	σ1σ2	σ1σ2	NOUN
ejpam-6037	125	10	-	-	ADJ
ejpam-6037	125	11	open	open	ADJ
ejpam-6037	125	12	set	set	NOUN
ejpam-6037	125	13	of	of	ADP
ejpam-6037	125	14	y	y	PROPN
ejpam-6037	125	15	containing	contain	VERB
ejpam-6037	125	16	f(x	f(x	PROPN
ejpam-6037	125	17	)	)	PUNCT
ejpam-6037	125	18	.	.	PUNCT
ejpam-6037	126	1	then	then	ADV
ejpam-6037	126	2	,	,	PUNCT
ejpam-6037	126	3	σ1σ2	σ1σ2	NOUN
ejpam-6037	126	4	-	-	NUM
ejpam-6037	126	5	cl(v	cl(v	NOUN
ejpam-6037	126	6	)	)	PUNCT
ejpam-6037	126	7	is	be	AUX
ejpam-6037	126	8	a	a	DET
ejpam-6037	126	9	σ1σ2	σ1σ2	NUM
ejpam-6037	126	10	-	-	PUNCT
ejpam-6037	126	11	closed	closed	ADJ
ejpam-6037	126	12	set	set	NOUN
ejpam-6037	126	13	of	of	ADP
ejpam-6037	126	14	y	y	PROPN
ejpam-6037	126	15	containing	contain	VERB
ejpam-6037	126	16	f(x	f(x	PROPN
ejpam-6037	126	17	)	)	PUNCT
ejpam-6037	126	18	.	.	PUNCT
ejpam-6037	127	1	since	since	SCONJ
ejpam-6037	127	2	f	f	PROPN
ejpam-6037	127	3	is	be	AUX
ejpam-6037	127	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	127	5	,	,	PUNCT
ejpam-6037	127	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	127	7	,	,	PUNCT
ejpam-6037	127	8	by	by	ADP
ejpam-6037	127	9	theorem	theorem	NOUN
ejpam-6037	127	10	1	1	NUM
ejpam-6037	127	11	there	there	ADV
ejpam-6037	127	12	exists	exist	VERB
ejpam-6037	127	13	a	a	DET
ejpam-6037	127	14	τ1τ2	τ1τ2	NOUN
ejpam-6037	127	15	-	-	ADJ
ejpam-6037	127	16	open	open	ADJ
ejpam-6037	127	17	set	set	ADJ
ejpam-6037	127	18	u	u	NOUN
ejpam-6037	127	19	of	of	ADP
ejpam-6037	127	20	x	x	PUNCT
ejpam-6037	127	21	containing	contain	VERB
ejpam-6037	127	22	x	x	PUNCT
ejpam-6037	127	23	such	such	ADJ
ejpam-6037	127	24	that	that	DET
ejpam-6037	127	25	f(u	f(u	PROPN
ejpam-6037	127	26	)	)	PUNCT
ejpam-6037	127	27	⊆	⊆	NUM
ejpam-6037	127	28	σ1σ2	σ1σ2	NOUN
ejpam-6037	127	29	-	-	NUM
ejpam-6037	127	30	cl(v	cl(v	NOUN
ejpam-6037	127	31	)	)	PUNCT
ejpam-6037	127	32	.	.	PUNCT
ejpam-6037	128	1	since	since	SCONJ
ejpam-6037	128	2	f	f	PROPN
ejpam-6037	128	3	is	be	AUX
ejpam-6037	128	4	almost	almost	ADV
ejpam-6037	128	5	(	(	PUNCT
ejpam-6037	128	6	τ1	τ1	NOUN
ejpam-6037	128	7	,	,	PUNCT
ejpam-6037	128	8	τ2)-open	τ2)-open	ADJ
ejpam-6037	128	9	,	,	PUNCT
ejpam-6037	128	10	we	we	PRON
ejpam-6037	128	11	have	have	AUX
ejpam-6037	128	12	f(u	f(u	PROPN
ejpam-6037	128	13	)	)	PUNCT
ejpam-6037	128	14	⊆	⊆	NUM
ejpam-6037	128	15	σ1σ2	σ1σ2	X
ejpam-6037	128	16	-	-	PUNCT
ejpam-6037	128	17	int(σ1σ2	int(σ1σ2	VERB
ejpam-6037	128	18	-	-	PUNCT
ejpam-6037	128	19	cl(f(u	cl(f(u	NOUN
ejpam-6037	128	20	)	)	PUNCT
ejpam-6037	128	21	)	)	PUNCT
ejpam-6037	128	22	)	)	PUNCT
ejpam-6037	129	1	⊆	⊆	X
ejpam-6037	129	2	σ1σ2	σ1σ2	X
ejpam-6037	129	3	-	-	PUNCT
ejpam-6037	129	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	129	5	-	-	PUNCT
ejpam-6037	129	6	cl(v	cl(v	NOUN
ejpam-6037	129	7	)	)	PUNCT
ejpam-6037	129	8	)	)	PUNCT
ejpam-6037	129	9	and	and	CCONJ
ejpam-6037	129	10	hence	hence	ADV
ejpam-6037	129	11	f	f	PROPN
ejpam-6037	129	12	is	be	AUX
ejpam-6037	129	13	almost	almost	ADV
ejpam-6037	129	14	(	(	PUNCT
ejpam-6037	129	15	τ1	τ1	NOUN
ejpam-6037	129	16	,	,	PUNCT
ejpam-6037	129	17	τ2)-continuous	τ2)-continuous	PROPN
ejpam-6037	129	18	.	.	PUNCT
ejpam-6037	129	19	recall	recall	VERB
ejpam-6037	129	20	that	that	SCONJ
ejpam-6037	129	21	a	a	DET
ejpam-6037	129	22	bitopological	bitopological	ADJ
ejpam-6037	129	23	space	space	NOUN
ejpam-6037	129	24	(	(	PUNCT
ejpam-6037	129	25	x	x	NOUN
ejpam-6037	129	26	,	,	PUNCT
ejpam-6037	129	27	τ1	τ1	NOUN
ejpam-6037	129	28	,	,	PUNCT
ejpam-6037	129	29	τ2	τ2	NOUN
ejpam-6037	129	30	)	)	PUNCT
ejpam-6037	129	31	is	be	AUX
ejpam-6037	129	32	said	say	VERB
ejpam-6037	129	33	to	to	PART
ejpam-6037	129	34	be	be	AUX
ejpam-6037	129	35	almost	almost	ADV
ejpam-6037	129	36	(	(	PUNCT
ejpam-6037	129	37	τ1	τ1	NOUN
ejpam-6037	129	38	,	,	PUNCT
ejpam-6037	129	39	τ2)-regular	τ2)-regular	ADJ
ejpam-6037	129	40	[	[	X
ejpam-6037	129	41	43	43	NUM
ejpam-6037	129	42	]	]	X
ejpam-6037	129	43	if	if	SCONJ
ejpam-6037	129	44	for	for	ADP
ejpam-6037	129	45	each	each	DET
ejpam-6037	129	46	(	(	PUNCT
ejpam-6037	129	47	τ1	τ1	NOUN
ejpam-6037	129	48	,	,	PUNCT
ejpam-6037	129	49	τ2)r	τ2)r	NOUN
ejpam-6037	129	50	-	-	PUNCT
ejpam-6037	129	51	closed	close	VERB
ejpam-6037	129	52	set	set	VERB
ejpam-6037	129	53	f	f	NOUN
ejpam-6037	129	54	and	and	CCONJ
ejpam-6037	129	55	each	each	DET
ejpam-6037	129	56	x	x	PROPN
ejpam-6037	129	57	̸∈	̸∈	PROPN
ejpam-6037	129	58	f	f	PROPN
ejpam-6037	129	59	,	,	PUNCT
ejpam-6037	129	60	there	there	PRON
ejpam-6037	129	61	exist	exist	VERB
ejpam-6037	129	62	disjoint	disjoint	ADJ
ejpam-6037	129	63	τ1τ2	τ1τ2	ADJ
ejpam-6037	129	64	-	-	ADJ
ejpam-6037	129	65	open	open	ADJ
ejpam-6037	129	66	sets	set	NOUN
ejpam-6037	129	67	u	u	NOUN
ejpam-6037	129	68	and	and	CCONJ
ejpam-6037	129	69	v	v	ADP
ejpam-6037	129	70	such	such	ADJ
ejpam-6037	129	71	that	that	SCONJ
ejpam-6037	129	72	x	x	SYM
ejpam-6037	129	73	∈	∈	PROPN
ejpam-6037	129	74	u	u	NOUN
ejpam-6037	129	75	and	and	CCONJ
ejpam-6037	129	76	f	f	PROPN
ejpam-6037	129	77	⊆	⊆	NUM
ejpam-6037	129	78	v	v	NOUN
ejpam-6037	129	79	.	.	PUNCT
ejpam-6037	130	1	lemma	lemma	PROPN
ejpam-6037	130	2	4	4	NUM
ejpam-6037	130	3	.	.	PUNCT
ejpam-6037	131	1	[	[	X
ejpam-6037	131	2	43	43	NUM
ejpam-6037	131	3	]	]	PUNCT
ejpam-6037	131	4	a	a	DET
ejpam-6037	131	5	bitopological	bitopological	ADJ
ejpam-6037	131	6	space	space	NOUN
ejpam-6037	131	7	(	(	PUNCT
ejpam-6037	131	8	x	x	NOUN
ejpam-6037	131	9	,	,	PUNCT
ejpam-6037	131	10	τ1	τ1	NOUN
ejpam-6037	131	11	,	,	PUNCT
ejpam-6037	131	12	τ2	τ2	NOUN
ejpam-6037	131	13	)	)	PUNCT
ejpam-6037	131	14	is	be	AUX
ejpam-6037	131	15	almost	almost	ADV
ejpam-6037	131	16	(	(	PUNCT
ejpam-6037	131	17	τ1	τ1	NOUN
ejpam-6037	131	18	,	,	PUNCT
ejpam-6037	131	19	τ2)-regular	τ2)-regular	ADJ
ejpam-6037	131	20	if	if	SCONJ
ejpam-6037	131	21	and	and	CCONJ
ejpam-6037	131	22	only	only	ADV
ejpam-6037	131	23	if	if	SCONJ
ejpam-6037	131	24	for	for	ADP
ejpam-6037	131	25	each	each	DET
ejpam-6037	131	26	x	x	SYM
ejpam-6037	131	27	∈	∈	PROPN
ejpam-6037	131	28	x	x	X
ejpam-6037	131	29	and	and	CCONJ
ejpam-6037	131	30	each	each	DET
ejpam-6037	131	31	(	(	PUNCT
ejpam-6037	131	32	τ1	τ1	NOUN
ejpam-6037	131	33	,	,	PUNCT
ejpam-6037	131	34	τ2)r	τ2)r	ADJ
ejpam-6037	131	35	-	-	PUNCT
ejpam-6037	131	36	open	open	NOUN
ejpam-6037	131	37	set	set	NOUN
ejpam-6037	131	38	u	u	NOUN
ejpam-6037	131	39	with	with	ADP
ejpam-6037	131	40	x	x	PROPN
ejpam-6037	131	41	∈	∈	PROPN
ejpam-6037	131	42	u	u	NOUN
ejpam-6037	131	43	,	,	PUNCT
ejpam-6037	131	44	there	there	PRON
ejpam-6037	131	45	exists	exist	VERB
ejpam-6037	131	46	a	a	DET
ejpam-6037	131	47	τ1τ2	τ1τ2	NOUN
ejpam-6037	131	48	-	-	ADJ
ejpam-6037	131	49	open	open	ADJ
ejpam-6037	131	50	set	set	VERB
ejpam-6037	131	51	v	v	ADP
ejpam-6037	131	52	such	such	ADJ
ejpam-6037	131	53	that	that	SCONJ
ejpam-6037	131	54	x	x	SYM
ejpam-6037	131	55	∈	∈	NOUN
ejpam-6037	131	56	v	v	ADP
ejpam-6037	131	57	⊆	⊆	NUM
ejpam-6037	131	58	τ1τ2	τ1τ2	NOUN
ejpam-6037	131	59	-	-	NOUN
ejpam-6037	131	60	cl(v	cl(v	X
ejpam-6037	131	61	)	)	PUNCT
ejpam-6037	131	62	⊆	⊆	NUM
ejpam-6037	131	63	u	u	NOUN
ejpam-6037	131	64	.	.	PUNCT
ejpam-6037	132	1	theorem	theorem	VERB
ejpam-6037	132	2	4	4	NUM
ejpam-6037	132	3	.	.	PUNCT
ejpam-6037	133	1	if	if	SCONJ
ejpam-6037	133	2	f	f	PROPN
ejpam-6037	133	3	:	:	PUNCT
ejpam-6037	133	4	(	(	PUNCT
ejpam-6037	133	5	x	x	NOUN
ejpam-6037	133	6	,	,	PUNCT
ejpam-6037	133	7	τ1	τ1	NOUN
ejpam-6037	133	8	,	,	PUNCT
ejpam-6037	133	9	τ2	τ2	NOUN
ejpam-6037	133	10	)	)	PUNCT
ejpam-6037	133	11	→	→	SYM
ejpam-6037	133	12	(	(	PUNCT
ejpam-6037	133	13	y	y	PROPN
ejpam-6037	133	14	,	,	PUNCT
ejpam-6037	133	15	σ1	σ1	PROPN
ejpam-6037	133	16	,	,	PUNCT
ejpam-6037	133	17	σ2	σ2	PROPN
ejpam-6037	133	18	)	)	PUNCT
ejpam-6037	133	19	is	be	AUX
ejpam-6037	133	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	133	21	,	,	PUNCT
ejpam-6037	133	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	133	23	and	and	CCONJ
ejpam-6037	133	24	(	(	PUNCT
ejpam-6037	133	25	y	y	PROPN
ejpam-6037	133	26	,	,	PUNCT
ejpam-6037	133	27	σ1	σ1	PROPN
ejpam-6037	133	28	,	,	PUNCT
ejpam-6037	133	29	σ2	σ2	PROPN
ejpam-6037	133	30	)	)	PUNCT
ejpam-6037	133	31	is	be	AUX
ejpam-6037	133	32	almost	almost	ADV
ejpam-6037	133	33	(	(	PUNCT
ejpam-6037	133	34	σ1	σ1	NOUN
ejpam-6037	133	35	,	,	PUNCT
ejpam-6037	133	36	σ2)-regular	σ2)-regular	ADJ
ejpam-6037	133	37	,	,	PUNCT
ejpam-6037	133	38	then	then	ADV
ejpam-6037	133	39	f	f	PROPN
ejpam-6037	133	40	is	be	AUX
ejpam-6037	133	41	almost	almost	ADV
ejpam-6037	133	42	(	(	PUNCT
ejpam-6037	133	43	τ1	τ1	NOUN
ejpam-6037	133	44	,	,	PUNCT
ejpam-6037	133	45	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	133	46	.	.	PUNCT
ejpam-6037	134	1	proof	proof	NOUN
ejpam-6037	134	2	.	.	PUNCT
ejpam-6037	135	1	let	let	VERB
ejpam-6037	135	2	x	x	PUNCT
ejpam-6037	135	3	∈	∈	PROPN
ejpam-6037	135	4	x	x	X
ejpam-6037	135	5	and	and	CCONJ
ejpam-6037	135	6	v	v	X
ejpam-6037	135	7	be	be	AUX
ejpam-6037	135	8	any	any	DET
ejpam-6037	135	9	σ1σ2	σ1σ2	NOUN
ejpam-6037	135	10	-	-	ADJ
ejpam-6037	135	11	open	open	ADJ
ejpam-6037	135	12	set	set	NOUN
ejpam-6037	135	13	of	of	ADP
ejpam-6037	135	14	y	y	PROPN
ejpam-6037	135	15	containing	contain	VERB
ejpam-6037	135	16	f(x	f(x	PROPN
ejpam-6037	135	17	)	)	PUNCT
ejpam-6037	135	18	.	.	PUNCT
ejpam-6037	136	1	then	then	ADV
ejpam-6037	136	2	,	,	PUNCT
ejpam-6037	136	3	we	we	PRON
ejpam-6037	136	4	have	have	VERB
ejpam-6037	136	5	σ1σ2	σ1σ2	NOUN
ejpam-6037	136	6	-	-	PUNCT
ejpam-6037	136	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	136	8	-	-	PUNCT
ejpam-6037	136	9	cl(v	cl(v	NOUN
ejpam-6037	136	10	)	)	PUNCT
ejpam-6037	136	11	)	)	PUNCT
ejpam-6037	137	1	is	be	AUX
ejpam-6037	137	2	(	(	PUNCT
ejpam-6037	137	3	σ1	σ1	NOUN
ejpam-6037	137	4	,	,	PUNCT
ejpam-6037	137	5	σ2)r	σ2)r	NOUN
ejpam-6037	137	6	-	-	PUNCT
ejpam-6037	137	7	open	open	ADJ
ejpam-6037	137	8	in	in	ADP
ejpam-6037	137	9	y	y	PROPN
ejpam-6037	137	10	.	.	PUNCT
ejpam-6037	138	1	since	since	SCONJ
ejpam-6037	138	2	(	(	PUNCT
ejpam-6037	138	3	y	y	PROPN
ejpam-6037	138	4	,	,	PUNCT
ejpam-6037	138	5	σ1	σ1	PROPN
ejpam-6037	138	6	,	,	PUNCT
ejpam-6037	138	7	σ2	σ2	PROPN
ejpam-6037	138	8	)	)	PUNCT
ejpam-6037	138	9	is	be	AUX
ejpam-6037	138	10	almost	almost	ADV
ejpam-6037	138	11	(	(	PUNCT
ejpam-6037	138	12	σ1	σ1	NOUN
ejpam-6037	138	13	,	,	PUNCT
ejpam-6037	138	14	σ2)-regular	σ2)-regular	ADJ
ejpam-6037	138	15	,	,	PUNCT
ejpam-6037	138	16	by	by	ADP
ejpam-6037	138	17	lemma	lemma	PROPN
ejpam-6037	138	18	4	4	NUM
ejpam-6037	138	19	there	there	ADV
ejpam-6037	138	20	exists	exist	VERB
ejpam-6037	138	21	a	a	DET
ejpam-6037	138	22	σ1σ2	σ1σ2	NUM
ejpam-6037	138	23	-	-	ADJ
ejpam-6037	138	24	open	open	ADJ
ejpam-6037	138	25	set	set	NOUN
ejpam-6037	138	26	w	w	PROPN
ejpam-6037	138	27	of	of	ADP
ejpam-6037	138	28	y	y	PRON
ejpam-6037	138	29	such	such	ADJ
ejpam-6037	138	30	that	that	SCONJ
ejpam-6037	138	31	f(x	f(x	PROPN
ejpam-6037	138	32	)	)	PUNCT
ejpam-6037	138	33	∈	∈	PROPN
ejpam-6037	138	34	w	w	ADP
ejpam-6037	138	35	⊆	⊆	NUM
ejpam-6037	138	36	σ1σ2	σ1σ2	NOUN
ejpam-6037	138	37	-	-	PUNCT
ejpam-6037	138	38	cl(w	cl(w	NOUN
ejpam-6037	138	39	)	)	PUNCT
ejpam-6037	138	40	⊆	⊆	NUM
ejpam-6037	138	41	σ1σ2	σ1σ2	X
ejpam-6037	138	42	-	-	PUNCT
ejpam-6037	138	43	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	138	44	-	-	PUNCT
ejpam-6037	138	45	cl(v	cl(v	NOUN
ejpam-6037	138	46	)	)	PUNCT
ejpam-6037	138	47	)	)	PUNCT
ejpam-6037	138	48	.	.	PUNCT
ejpam-6037	139	1	since	since	SCONJ
ejpam-6037	139	2	f	f	PROPN
ejpam-6037	139	3	is	be	AUX
ejpam-6037	139	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	139	5	,	,	PUNCT
ejpam-6037	139	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	139	7	and	and	CCONJ
ejpam-6037	139	8	σ1σ2	σ1σ2	NOUN
ejpam-6037	139	9	-	-	PUNCT
ejpam-6037	139	10	cl(w	cl(w	NOUN
ejpam-6037	139	11	)	)	PUNCT
ejpam-6037	139	12	is	be	AUX
ejpam-6037	139	13	σ1σ2	σ1σ2	NOUN
ejpam-6037	139	14	-	-	ADJ
ejpam-6037	139	15	closed	closed	ADJ
ejpam-6037	139	16	in	in	ADP
ejpam-6037	139	17	y	y	PROPN
ejpam-6037	139	18	,	,	PUNCT
ejpam-6037	139	19	by	by	ADP
ejpam-6037	139	20	theorem	theorem	NOUN
ejpam-6037	139	21	1	1	NUM
ejpam-6037	139	22	there	there	ADV
ejpam-6037	139	23	exists	exist	VERB
ejpam-6037	139	24	a	a	DET
ejpam-6037	139	25	τ1τ2	τ1τ2	NOUN
ejpam-6037	139	26	-	-	ADJ
ejpam-6037	139	27	open	open	ADJ
ejpam-6037	139	28	set	set	ADJ
ejpam-6037	139	29	u	u	NOUN
ejpam-6037	139	30	of	of	ADP
ejpam-6037	139	31	x	x	PUNCT
ejpam-6037	139	32	containing	contain	VERB
ejpam-6037	139	33	x	x	PUNCT
ejpam-6037	139	34	such	such	ADJ
ejpam-6037	139	35	that	that	DET
ejpam-6037	139	36	f(u	f(u	PROPN
ejpam-6037	139	37	)	)	PUNCT
ejpam-6037	140	1	⊆	⊆	NUM
ejpam-6037	140	2	σ1σ2	σ1σ2	NOUN
ejpam-6037	140	3	-	-	PUNCT
ejpam-6037	140	4	cl(w	cl(w	NOUN
ejpam-6037	140	5	)	)	PUNCT
ejpam-6037	140	6	⊆	⊆	NUM
ejpam-6037	140	7	σ1σ2	σ1σ2	X
ejpam-6037	140	8	-	-	PUNCT
ejpam-6037	140	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	140	10	-	-	PUNCT
ejpam-6037	140	11	cl(v	cl(v	NOUN
ejpam-6037	140	12	)	)	PUNCT
ejpam-6037	140	13	)	)	PUNCT
ejpam-6037	140	14	.	.	PUNCT
ejpam-6037	141	1	thus	thus	ADV
ejpam-6037	141	2	,	,	PUNCT
ejpam-6037	141	3	f	f	PROPN
ejpam-6037	141	4	is	be	AUX
ejpam-6037	141	5	almost	almost	ADV
ejpam-6037	141	6	(	(	PUNCT
ejpam-6037	141	7	τ1	τ1	NOUN
ejpam-6037	141	8	,	,	PUNCT
ejpam-6037	141	9	τ2)-continuous	τ2)-continuous	PROPN
ejpam-6037	141	10	.	.	PUNCT
ejpam-6037	142	1	lemma	lemma	PROPN
ejpam-6037	142	2	5	5	NUM
ejpam-6037	142	3	.	.	PUNCT
ejpam-6037	143	1	[	[	X
ejpam-6037	143	2	31	31	NUM
ejpam-6037	143	3	]	]	PUNCT
ejpam-6037	143	4	for	for	ADP
ejpam-6037	143	5	a	a	DET
ejpam-6037	143	6	function	function	NOUN
ejpam-6037	143	7	(	(	PUNCT
ejpam-6037	143	8	x	x	NOUN
ejpam-6037	143	9	,	,	PUNCT
ejpam-6037	143	10	τ1	τ1	NOUN
ejpam-6037	143	11	,	,	PUNCT
ejpam-6037	143	12	τ2	τ2	NOUN
ejpam-6037	143	13	)	)	PUNCT
ejpam-6037	143	14	→	→	SYM
ejpam-6037	143	15	(	(	PUNCT
ejpam-6037	143	16	y	y	PROPN
ejpam-6037	143	17	,	,	PUNCT
ejpam-6037	143	18	σ1	σ1	PROPN
ejpam-6037	143	19	,	,	PUNCT
ejpam-6037	143	20	σ2	σ2	NOUN
ejpam-6037	143	21	)	)	PUNCT
ejpam-6037	143	22	,	,	PUNCT
ejpam-6037	143	23	the	the	DET
ejpam-6037	143	24	following	follow	VERB
ejpam-6037	143	25	properties	property	NOUN
ejpam-6037	143	26	are	be	AUX
ejpam-6037	143	27	equivalent	equivalent	ADJ
ejpam-6037	143	28	:	:	PUNCT
ejpam-6037	143	29	(	(	PUNCT
ejpam-6037	143	30	1	1	X
ejpam-6037	143	31	)	)	PUNCT
ejpam-6037	143	32	f	f	NOUN
ejpam-6037	143	33	is	be	AUX
ejpam-6037	143	34	almost	almost	ADV
ejpam-6037	143	35	(	(	PUNCT
ejpam-6037	143	36	τ1	τ1	NOUN
ejpam-6037	143	37	,	,	PUNCT
ejpam-6037	143	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	143	39	at	at	ADP
ejpam-6037	143	40	x	x	X
ejpam-6037	143	41	∈	∈	PROPN
ejpam-6037	143	42	x	x	NOUN
ejpam-6037	143	43	;	;	PUNCT
ejpam-6037	143	44	n.	n.	NOUN
ejpam-6037	143	45	chutiman	chutiman	NOUN
ejpam-6037	143	46	,	,	PUNCT
ejpam-6037	143	47	a.	a.	PROPN
ejpam-6037	143	48	sama	sama	PROPN
ejpam-6037	143	49	-	-	PUNCT
ejpam-6037	143	50	ae	ae	PROPN
ejpam-6037	143	51	,	,	PUNCT
ejpam-6037	143	52	c.	c.	PROPN
ejpam-6037	143	53	boonpok	boonpok	PROPN
ejpam-6037	143	54	/	/	SYM
ejpam-6037	143	55	eur	eur	PROPN
ejpam-6037	143	56	.	.	PUNCT
ejpam-6037	144	1	j.	j.	PROPN
ejpam-6037	144	2	pure	pure	PROPN
ejpam-6037	144	3	appl	appl	PROPN
ejpam-6037	144	4	.	.	PROPN
ejpam-6037	144	5	math	math	PROPN
ejpam-6037	144	6	,	,	PUNCT
ejpam-6037	144	7	18	18	NUM
ejpam-6037	144	8	(	(	PUNCT
ejpam-6037	144	9	2	2	NUM
ejpam-6037	144	10	)	)	PUNCT
ejpam-6037	144	11	(	(	PUNCT
ejpam-6037	144	12	2025	2025	NUM
ejpam-6037	144	13	)	)	PUNCT
ejpam-6037	144	14	,	,	PUNCT
ejpam-6037	144	15	6037	6037	NUM
ejpam-6037	144	16	6	6	NUM
ejpam-6037	144	17	of	of	ADP
ejpam-6037	144	18	12	12	NUM
ejpam-6037	144	19	(	(	PUNCT
ejpam-6037	144	20	2	2	NUM
ejpam-6037	144	21	)	)	PUNCT
ejpam-6037	144	22	x	x	SYM
ejpam-6037	144	23	∈	∈	PRON
ejpam-6037	144	24	τ1τ2	τ1τ2	PUNCT
ejpam-6037	144	25	-	-	NUM
ejpam-6037	144	26	int(f	int(f	VERB
ejpam-6037	144	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6037	144	28	-	-	PUNCT
ejpam-6037	144	29	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6037	144	30	-	-	PUNCT
ejpam-6037	144	31	cl(v	cl(v	NOUN
ejpam-6037	144	32	)	)	PUNCT
ejpam-6037	144	33	)	)	PUNCT
ejpam-6037	144	34	)	)	PUNCT
ejpam-6037	144	35	)	)	PUNCT
ejpam-6037	145	1	for	for	ADP
ejpam-6037	145	2	every	every	DET
ejpam-6037	145	3	σ1σ2	σ1σ2	NOUN
ejpam-6037	145	4	-	-	ADJ
ejpam-6037	145	5	open	open	ADJ
ejpam-6037	145	6	set	set	NOUN
ejpam-6037	145	7	v	v	NOUN
ejpam-6037	145	8	of	of	ADP
ejpam-6037	145	9	y	y	NOUN
ejpam-6037	145	10	containing	contain	VERB
ejpam-6037	145	11	f(x	f(x	PROPN
ejpam-6037	145	12	)	)	PUNCT
ejpam-6037	145	13	;	;	PUNCT
ejpam-6037	145	14	(	(	PUNCT
ejpam-6037	145	15	3	3	X
ejpam-6037	145	16	)	)	PUNCT
ejpam-6037	145	17	x	x	SYM
ejpam-6037	145	18	∈	∈	PRON
ejpam-6037	145	19	τ1τ2	τ1τ2	NOUN
ejpam-6037	145	20	-	-	NUM
ejpam-6037	145	21	int(f	int(f	X
ejpam-6037	145	22	−1(v	−1(v	NOUN
ejpam-6037	145	23	)	)	PUNCT
ejpam-6037	145	24	)	)	PUNCT
ejpam-6037	145	25	for	for	ADP
ejpam-6037	145	26	every	every	DET
ejpam-6037	145	27	(	(	PUNCT
ejpam-6037	145	28	σ1	σ1	PROPN
ejpam-6037	145	29	,	,	PUNCT
ejpam-6037	145	30	σ2)r	σ2)r	NOUN
ejpam-6037	145	31	-	-	PUNCT
ejpam-6037	145	32	open	open	ADJ
ejpam-6037	145	33	set	set	VERB
ejpam-6037	145	34	v	v	NOUN
ejpam-6037	145	35	of	of	ADP
ejpam-6037	145	36	y	y	NOUN
ejpam-6037	145	37	containing	contain	VERB
ejpam-6037	145	38	f(x	f(x	PROPN
ejpam-6037	145	39	)	)	PUNCT
ejpam-6037	145	40	;	;	PUNCT
ejpam-6037	145	41	(	(	PUNCT
ejpam-6037	145	42	4	4	X
ejpam-6037	145	43	)	)	PUNCT
ejpam-6037	145	44	for	for	ADP
ejpam-6037	145	45	each	each	DET
ejpam-6037	145	46	(	(	PUNCT
ejpam-6037	145	47	σ1	σ1	PROPN
ejpam-6037	145	48	,	,	PUNCT
ejpam-6037	145	49	σ2)r	σ2)r	NOUN
ejpam-6037	145	50	-	-	PUNCT
ejpam-6037	145	51	open	open	ADJ
ejpam-6037	145	52	set	set	VERB
ejpam-6037	145	53	v	v	NOUN
ejpam-6037	145	54	of	of	ADP
ejpam-6037	145	55	y	y	NOUN
ejpam-6037	145	56	containing	contain	VERB
ejpam-6037	145	57	f(x	f(x	PROPN
ejpam-6037	145	58	)	)	PUNCT
ejpam-6037	145	59	,	,	PUNCT
ejpam-6037	145	60	there	there	PRON
ejpam-6037	145	61	exists	exist	VERB
ejpam-6037	145	62	a	a	DET
ejpam-6037	145	63	τ1τ2	τ1τ2	NOUN
ejpam-6037	145	64	-	-	ADJ
ejpam-6037	145	65	open	open	ADJ
ejpam-6037	145	66	set	set	ADJ
ejpam-6037	145	67	u	u	NOUN
ejpam-6037	145	68	of	of	ADP
ejpam-6037	145	69	x	x	PUNCT
ejpam-6037	145	70	containing	contain	VERB
ejpam-6037	145	71	x	x	PUNCT
ejpam-6037	145	72	such	such	ADJ
ejpam-6037	145	73	that	that	DET
ejpam-6037	145	74	f(u	f(u	PROPN
ejpam-6037	145	75	)	)	PUNCT
ejpam-6037	145	76	⊆	⊆	NUM
ejpam-6037	145	77	v	v	NOUN
ejpam-6037	145	78	.	.	PUNCT
ejpam-6037	146	1	recall	recall	VERB
ejpam-6037	146	2	that	that	SCONJ
ejpam-6037	146	3	a	a	DET
ejpam-6037	146	4	bitopological	bitopological	ADJ
ejpam-6037	146	5	space	space	NOUN
ejpam-6037	146	6	(	(	PUNCT
ejpam-6037	146	7	x	x	NOUN
ejpam-6037	146	8	,	,	PUNCT
ejpam-6037	146	9	τ1	τ1	NOUN
ejpam-6037	146	10	,	,	PUNCT
ejpam-6037	146	11	τ2	τ2	NOUN
ejpam-6037	146	12	)	)	PUNCT
ejpam-6037	146	13	is	be	AUX
ejpam-6037	146	14	said	say	VERB
ejpam-6037	146	15	to	to	PART
ejpam-6037	146	16	be	be	AUX
ejpam-6037	146	17	(	(	PUNCT
ejpam-6037	146	18	τ1	τ1	NOUN
ejpam-6037	146	19	,	,	PUNCT
ejpam-6037	146	20	τ2)-extremally	τ2)-extremally	ADV
ejpam-6037	146	21	disconnected	disconnect	VERB
ejpam-6037	146	22	[	[	X
ejpam-6037	146	23	41	41	NUM
ejpam-6037	146	24	]	]	PUNCT
ejpam-6037	146	25	if	if	SCONJ
ejpam-6037	146	26	the	the	DET
ejpam-6037	146	27	τ1τ2	τ1τ2	NOUN
ejpam-6037	146	28	-	-	NOUN
ejpam-6037	146	29	closure	closure	NOUN
ejpam-6037	146	30	of	of	ADP
ejpam-6037	146	31	every	every	DET
ejpam-6037	146	32	τ1τ2	τ1τ2	NOUN
ejpam-6037	146	33	-	-	ADJ
ejpam-6037	146	34	open	open	ADJ
ejpam-6037	146	35	set	set	ADJ
ejpam-6037	146	36	u	u	NOUN
ejpam-6037	146	37	of	of	ADP
ejpam-6037	146	38	x	x	SYM
ejpam-6037	146	39	is	be	AUX
ejpam-6037	146	40	τ1τ2	τ1τ2	VERB
ejpam-6037	146	41	-	-	ADJ
ejpam-6037	146	42	open	open	ADJ
ejpam-6037	146	43	.	.	PUNCT
ejpam-6037	147	1	lemma	lemma	PROPN
ejpam-6037	147	2	6	6	NUM
ejpam-6037	147	3	.	.	PUNCT
ejpam-6037	148	1	[	[	X
ejpam-6037	148	2	41	41	NUM
ejpam-6037	148	3	]	]	PUNCT
ejpam-6037	148	4	for	for	ADP
ejpam-6037	148	5	a	a	DET
ejpam-6037	148	6	bitopological	bitopological	ADJ
ejpam-6037	148	7	space	space	NOUN
ejpam-6037	148	8	(	(	PUNCT
ejpam-6037	148	9	x	x	NOUN
ejpam-6037	148	10	,	,	PUNCT
ejpam-6037	148	11	τ1	τ1	NOUN
ejpam-6037	148	12	,	,	PUNCT
ejpam-6037	148	13	τ2	τ2	NOUN
ejpam-6037	148	14	)	)	PUNCT
ejpam-6037	148	15	,	,	PUNCT
ejpam-6037	148	16	the	the	DET
ejpam-6037	148	17	following	follow	VERB
ejpam-6037	148	18	properties	property	NOUN
ejpam-6037	148	19	are	be	AUX
ejpam-6037	148	20	equivalent	equivalent	ADJ
ejpam-6037	148	21	:	:	PUNCT
ejpam-6037	148	22	(	(	PUNCT
ejpam-6037	148	23	1	1	X
ejpam-6037	148	24	)	)	PUNCT
ejpam-6037	148	25	(	(	PUNCT
ejpam-6037	148	26	x	x	NOUN
ejpam-6037	148	27	,	,	PUNCT
ejpam-6037	148	28	τ1	τ1	NOUN
ejpam-6037	148	29	,	,	PUNCT
ejpam-6037	148	30	τ2	τ2	NOUN
ejpam-6037	148	31	)	)	PUNCT
ejpam-6037	148	32	is	be	AUX
ejpam-6037	148	33	(	(	PUNCT
ejpam-6037	148	34	τ1	τ1	NOUN
ejpam-6037	148	35	,	,	PUNCT
ejpam-6037	148	36	τ2)-extremally	τ2)-extremally	ADV
ejpam-6037	148	37	disconnected	disconnect	VERB
ejpam-6037	148	38	;	;	PUNCT
ejpam-6037	148	39	(	(	PUNCT
ejpam-6037	148	40	2	2	X
ejpam-6037	148	41	)	)	PUNCT
ejpam-6037	148	42	every	every	DET
ejpam-6037	148	43	(	(	PUNCT
ejpam-6037	148	44	τ1	τ1	NOUN
ejpam-6037	148	45	,	,	PUNCT
ejpam-6037	148	46	τ2)r	τ2)r	ADJ
ejpam-6037	148	47	-	-	PUNCT
ejpam-6037	148	48	open	open	ADJ
ejpam-6037	148	49	set	set	NOUN
ejpam-6037	148	50	of	of	ADP
ejpam-6037	148	51	x	x	PUNCT
ejpam-6037	148	52	is	be	AUX
ejpam-6037	148	53	τ1τ2	τ1τ2	VERB
ejpam-6037	148	54	-	-	ADJ
ejpam-6037	148	55	closed	closed	ADJ
ejpam-6037	148	56	;	;	PUNCT
ejpam-6037	148	57	(	(	PUNCT
ejpam-6037	148	58	3	3	X
ejpam-6037	148	59	)	)	PUNCT
ejpam-6037	148	60	every	every	DET
ejpam-6037	148	61	(	(	PUNCT
ejpam-6037	148	62	τ1	τ1	NOUN
ejpam-6037	148	63	,	,	PUNCT
ejpam-6037	148	64	τ2)r	τ2)r	NOUN
ejpam-6037	148	65	-	-	PUNCT
ejpam-6037	148	66	closed	close	VERB
ejpam-6037	148	67	set	set	NOUN
ejpam-6037	148	68	of	of	ADP
ejpam-6037	148	69	x	x	PUNCT
ejpam-6037	148	70	is	be	AUX
ejpam-6037	148	71	τ1τ2	τ1τ2	VERB
ejpam-6037	148	72	-	-	ADJ
ejpam-6037	148	73	open	open	ADJ
ejpam-6037	148	74	.	.	PUNCT
ejpam-6037	149	1	theorem	theorem	NOUN
ejpam-6037	149	2	5	5	NUM
ejpam-6037	149	3	.	.	PUNCT
ejpam-6037	150	1	if	if	SCONJ
ejpam-6037	150	2	f	f	PROPN
ejpam-6037	150	3	:	:	PUNCT
ejpam-6037	150	4	(	(	PUNCT
ejpam-6037	150	5	x	x	NOUN
ejpam-6037	150	6	,	,	PUNCT
ejpam-6037	150	7	τ1	τ1	NOUN
ejpam-6037	150	8	,	,	PUNCT
ejpam-6037	150	9	τ2	τ2	NOUN
ejpam-6037	150	10	)	)	PUNCT
ejpam-6037	150	11	→	→	SYM
ejpam-6037	150	12	(	(	PUNCT
ejpam-6037	150	13	y	y	PROPN
ejpam-6037	150	14	,	,	PUNCT
ejpam-6037	150	15	σ1	σ1	PROPN
ejpam-6037	150	16	,	,	PUNCT
ejpam-6037	150	17	σ2	σ2	PROPN
ejpam-6037	150	18	)	)	PUNCT
ejpam-6037	150	19	is	be	AUX
ejpam-6037	150	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	150	21	,	,	PUNCT
ejpam-6037	150	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	150	23	and	and	CCONJ
ejpam-6037	150	24	(	(	PUNCT
ejpam-6037	150	25	y	y	PROPN
ejpam-6037	150	26	,	,	PUNCT
ejpam-6037	150	27	σ1	σ1	PROPN
ejpam-6037	150	28	,	,	PUNCT
ejpam-6037	150	29	σ2	σ2	PROPN
ejpam-6037	150	30	)	)	PUNCT
ejpam-6037	150	31	is	be	AUX
ejpam-6037	150	32	(	(	PUNCT
ejpam-6037	150	33	σ1	σ1	NOUN
ejpam-6037	150	34	,	,	PUNCT
ejpam-6037	150	35	σ2)-extremally	σ2)-extremally	ADV
ejpam-6037	150	36	disconnected	disconnect	VERB
ejpam-6037	150	37	,	,	PUNCT
ejpam-6037	150	38	then	then	ADV
ejpam-6037	150	39	f	f	PROPN
ejpam-6037	150	40	is	be	AUX
ejpam-6037	150	41	almost	almost	ADV
ejpam-6037	150	42	(	(	PUNCT
ejpam-6037	150	43	τ1	τ1	NOUN
ejpam-6037	150	44	,	,	PUNCT
ejpam-6037	150	45	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	150	46	.	.	PUNCT
ejpam-6037	151	1	proof	proof	NOUN
ejpam-6037	151	2	.	.	PUNCT
ejpam-6037	152	1	let	let	VERB
ejpam-6037	152	2	x	x	PUNCT
ejpam-6037	152	3	∈	∈	PROPN
ejpam-6037	152	4	x	x	X
ejpam-6037	152	5	and	and	CCONJ
ejpam-6037	152	6	v	v	X
ejpam-6037	152	7	be	be	AUX
ejpam-6037	152	8	any	any	DET
ejpam-6037	152	9	(	(	PUNCT
ejpam-6037	152	10	σ1	σ1	NOUN
ejpam-6037	152	11	,	,	PUNCT
ejpam-6037	152	12	σ2)r	σ2)r	NOUN
ejpam-6037	152	13	-	-	PUNCT
ejpam-6037	152	14	open	open	ADJ
ejpam-6037	152	15	set	set	NOUN
ejpam-6037	152	16	of	of	ADP
ejpam-6037	152	17	y	y	PROPN
ejpam-6037	152	18	containing	contain	VERB
ejpam-6037	152	19	f(x	f(x	PROPN
ejpam-6037	152	20	)	)	PUNCT
ejpam-6037	152	21	.	.	PUNCT
ejpam-6037	153	1	since	since	SCONJ
ejpam-6037	153	2	(	(	PUNCT
ejpam-6037	153	3	y	y	PROPN
ejpam-6037	153	4	,	,	PUNCT
ejpam-6037	153	5	σ1	σ1	PROPN
ejpam-6037	153	6	,	,	PUNCT
ejpam-6037	153	7	σ2	σ2	PROPN
ejpam-6037	153	8	)	)	PUNCT
ejpam-6037	153	9	is	be	AUX
ejpam-6037	153	10	(	(	PUNCT
ejpam-6037	153	11	σ1	σ1	NOUN
ejpam-6037	153	12	,	,	PUNCT
ejpam-6037	153	13	σ2)-extremally	σ2)-extremally	ADV
ejpam-6037	153	14	disconnected	disconnect	VERB
ejpam-6037	153	15	,	,	PUNCT
ejpam-6037	153	16	by	by	ADP
ejpam-6037	153	17	lemma	lemma	PROPN
ejpam-6037	153	18	6	6	NUM
ejpam-6037	153	19	we	we	PRON
ejpam-6037	153	20	have	have	AUX
ejpam-6037	153	21	v	v	NOUN
ejpam-6037	153	22	is	be	AUX
ejpam-6037	153	23	σ1σ2	σ1σ2	NOUN
ejpam-6037	153	24	-	-	PUNCT
ejpam-6037	153	25	clopen	clopen	ADJ
ejpam-6037	153	26	.	.	PUNCT
ejpam-6037	154	1	since	since	SCONJ
ejpam-6037	154	2	f	f	PROPN
ejpam-6037	154	3	is	be	AUX
ejpam-6037	154	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	154	5	,	,	PUNCT
ejpam-6037	154	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	154	7	,	,	PUNCT
ejpam-6037	154	8	by	by	ADP
ejpam-6037	154	9	theorem	theorem	NOUN
ejpam-6037	154	10	1	1	NUM
ejpam-6037	154	11	there	there	ADV
ejpam-6037	154	12	exists	exist	VERB
ejpam-6037	154	13	a	a	DET
ejpam-6037	154	14	τ1τ2	τ1τ2	NOUN
ejpam-6037	154	15	-	-	ADJ
ejpam-6037	154	16	open	open	ADJ
ejpam-6037	154	17	set	set	ADJ
ejpam-6037	154	18	u	u	NOUN
ejpam-6037	154	19	of	of	ADP
ejpam-6037	154	20	x	x	PUNCT
ejpam-6037	154	21	containing	contain	VERB
ejpam-6037	154	22	x	x	PUNCT
ejpam-6037	154	23	such	such	ADJ
ejpam-6037	154	24	that	that	DET
ejpam-6037	154	25	f(u	f(u	PROPN
ejpam-6037	154	26	)	)	PUNCT
ejpam-6037	154	27	⊆	⊆	NUM
ejpam-6037	154	28	v	v	NOUN
ejpam-6037	154	29	.	.	PUNCT
ejpam-6037	155	1	thus	thus	ADV
ejpam-6037	155	2	by	by	ADP
ejpam-6037	155	3	lemma	lemma	PROPN
ejpam-6037	155	4	5	5	NUM
ejpam-6037	155	5	,	,	PUNCT
ejpam-6037	155	6	f	f	PROPN
ejpam-6037	155	7	is	be	AUX
ejpam-6037	155	8	almost	almost	ADV
ejpam-6037	155	9	(	(	PUNCT
ejpam-6037	155	10	τ1	τ1	NOUN
ejpam-6037	155	11	,	,	PUNCT
ejpam-6037	155	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	155	13	.	.	PUNCT
ejpam-6037	156	1	definition	definition	NOUN
ejpam-6037	156	2	5	5	NUM
ejpam-6037	156	3	.	.	PUNCT
ejpam-6037	157	1	[	[	X
ejpam-6037	157	2	30	30	NUM
ejpam-6037	157	3	]	]	X
ejpam-6037	157	4	a	a	DET
ejpam-6037	157	5	function	function	NOUN
ejpam-6037	157	6	f	f	NOUN
ejpam-6037	157	7	:	:	PUNCT
ejpam-6037	157	8	(	(	PUNCT
ejpam-6037	157	9	x	x	NOUN
ejpam-6037	157	10	,	,	PUNCT
ejpam-6037	157	11	τ1	τ1	NOUN
ejpam-6037	157	12	,	,	PUNCT
ejpam-6037	157	13	τ2	τ2	NOUN
ejpam-6037	157	14	)	)	PUNCT
ejpam-6037	157	15	→	→	SYM
ejpam-6037	157	16	(	(	PUNCT
ejpam-6037	157	17	y	y	PROPN
ejpam-6037	157	18	,	,	PUNCT
ejpam-6037	157	19	σ1	σ1	PROPN
ejpam-6037	157	20	,	,	PUNCT
ejpam-6037	157	21	σ2	σ2	PROPN
ejpam-6037	157	22	)	)	PUNCT
ejpam-6037	157	23	is	be	AUX
ejpam-6037	157	24	called	call	VERB
ejpam-6037	157	25	(	(	PUNCT
ejpam-6037	157	26	τ1	τ1	NOUN
ejpam-6037	157	27	,	,	PUNCT
ejpam-6037	157	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	157	29	at	at	ADP
ejpam-6037	157	30	a	a	DET
ejpam-6037	157	31	point	point	NOUN
ejpam-6037	157	32	x	x	SYM
ejpam-6037	157	33	∈	∈	NOUN
ejpam-6037	157	34	x	x	PUNCT
ejpam-6037	157	35	if	if	SCONJ
ejpam-6037	157	36	for	for	ADP
ejpam-6037	157	37	each	each	DET
ejpam-6037	157	38	σ1σ2	σ1σ2	VERB
ejpam-6037	157	39	-	-	ADJ
ejpam-6037	157	40	open	open	ADJ
ejpam-6037	157	41	set	set	NOUN
ejpam-6037	157	42	v	v	NOUN
ejpam-6037	157	43	of	of	ADP
ejpam-6037	157	44	y	y	NOUN
ejpam-6037	157	45	containing	contain	VERB
ejpam-6037	157	46	f(x	f(x	PROPN
ejpam-6037	157	47	)	)	PUNCT
ejpam-6037	157	48	,	,	PUNCT
ejpam-6037	157	49	there	there	PRON
ejpam-6037	157	50	exists	exist	VERB
ejpam-6037	157	51	a	a	DET
ejpam-6037	157	52	τ1τ2	τ1τ2	NOUN
ejpam-6037	157	53	-	-	ADJ
ejpam-6037	157	54	open	open	ADJ
ejpam-6037	157	55	set	set	ADJ
ejpam-6037	157	56	u	u	NOUN
ejpam-6037	157	57	of	of	ADP
ejpam-6037	157	58	x	x	PUNCT
ejpam-6037	157	59	containing	contain	VERB
ejpam-6037	157	60	x	x	PUNCT
ejpam-6037	157	61	such	such	ADJ
ejpam-6037	157	62	that	that	DET
ejpam-6037	157	63	f(u	f(u	PROPN
ejpam-6037	157	64	)	)	PUNCT
ejpam-6037	157	65	⊆	⊆	NUM
ejpam-6037	157	66	v	v	NOUN
ejpam-6037	157	67	.	.	PUNCT
ejpam-6037	158	1	a	a	DET
ejpam-6037	158	2	function	function	NOUN
ejpam-6037	158	3	f	f	NOUN
ejpam-6037	158	4	:	:	PUNCT
ejpam-6037	158	5	(	(	PUNCT
ejpam-6037	158	6	x	x	NOUN
ejpam-6037	158	7	,	,	PUNCT
ejpam-6037	158	8	τ1	τ1	NOUN
ejpam-6037	158	9	,	,	PUNCT
ejpam-6037	158	10	τ2	τ2	NOUN
ejpam-6037	158	11	)	)	PUNCT
ejpam-6037	158	12	→	→	SYM
ejpam-6037	158	13	(	(	PUNCT
ejpam-6037	158	14	y	y	PROPN
ejpam-6037	158	15	,	,	PUNCT
ejpam-6037	158	16	σ1	σ1	PROPN
ejpam-6037	158	17	,	,	PUNCT
ejpam-6037	158	18	σ2	σ2	PROPN
ejpam-6037	158	19	)	)	PUNCT
ejpam-6037	158	20	is	be	AUX
ejpam-6037	158	21	called	call	VERB
ejpam-6037	158	22	(	(	PUNCT
ejpam-6037	158	23	τ1	τ1	NOUN
ejpam-6037	158	24	,	,	PUNCT
ejpam-6037	158	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	158	26	if	if	SCONJ
ejpam-6037	158	27	f	f	PROPN
ejpam-6037	158	28	has	have	VERB
ejpam-6037	158	29	this	this	DET
ejpam-6037	158	30	property	property	NOUN
ejpam-6037	158	31	at	at	ADP
ejpam-6037	158	32	each	each	DET
ejpam-6037	158	33	point	point	NOUN
ejpam-6037	158	34	of	of	ADP
ejpam-6037	158	35	x.	x.	NOUN
ejpam-6037	158	36	lemma	lemma	PROPN
ejpam-6037	158	37	7	7	NUM
ejpam-6037	158	38	.	.	PUNCT
ejpam-6037	159	1	[	[	X
ejpam-6037	159	2	30	30	NUM
ejpam-6037	159	3	]	]	PUNCT
ejpam-6037	159	4	for	for	ADP
ejpam-6037	159	5	a	a	DET
ejpam-6037	159	6	function	function	NOUN
ejpam-6037	159	7	(	(	PUNCT
ejpam-6037	159	8	x	x	NOUN
ejpam-6037	159	9	,	,	PUNCT
ejpam-6037	159	10	τ1	τ1	NOUN
ejpam-6037	159	11	,	,	PUNCT
ejpam-6037	159	12	τ2	τ2	NOUN
ejpam-6037	159	13	)	)	PUNCT
ejpam-6037	159	14	→	→	SYM
ejpam-6037	159	15	(	(	PUNCT
ejpam-6037	159	16	y	y	PROPN
ejpam-6037	159	17	,	,	PUNCT
ejpam-6037	159	18	σ1	σ1	PROPN
ejpam-6037	159	19	,	,	PUNCT
ejpam-6037	159	20	σ2	σ2	NOUN
ejpam-6037	159	21	)	)	PUNCT
ejpam-6037	159	22	,	,	PUNCT
ejpam-6037	159	23	the	the	DET
ejpam-6037	159	24	following	follow	VERB
ejpam-6037	159	25	properties	property	NOUN
ejpam-6037	159	26	are	be	AUX
ejpam-6037	159	27	equivalent	equivalent	ADJ
ejpam-6037	159	28	:	:	PUNCT
ejpam-6037	159	29	(	(	PUNCT
ejpam-6037	159	30	1	1	X
ejpam-6037	159	31	)	)	PUNCT
ejpam-6037	159	32	f	f	PROPN
ejpam-6037	159	33	is	be	AUX
ejpam-6037	159	34	(	(	PUNCT
ejpam-6037	159	35	τ1	τ1	NOUN
ejpam-6037	159	36	,	,	PUNCT
ejpam-6037	159	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	159	38	;	;	PUNCT
ejpam-6037	159	39	(	(	PUNCT
ejpam-6037	159	40	2	2	X
ejpam-6037	159	41	)	)	PUNCT
ejpam-6037	159	42	f−1(v	f−1(v	NOUN
ejpam-6037	159	43	)	)	PUNCT
ejpam-6037	159	44	is	be	AUX
ejpam-6037	159	45	τ1τ2	τ1τ2	NOUN
ejpam-6037	159	46	-	-	ADJ
ejpam-6037	159	47	open	open	ADJ
ejpam-6037	159	48	in	in	ADP
ejpam-6037	159	49	x	x	PUNCT
ejpam-6037	159	50	for	for	ADP
ejpam-6037	159	51	every	every	DET
ejpam-6037	159	52	σ1σ2	σ1σ2	NOUN
ejpam-6037	159	53	-	-	ADJ
ejpam-6037	159	54	open	open	ADJ
ejpam-6037	159	55	set	set	NOUN
ejpam-6037	159	56	v	v	NOUN
ejpam-6037	159	57	of	of	ADP
ejpam-6037	159	58	y	y	PROPN
ejpam-6037	159	59	;	;	PUNCT
ejpam-6037	159	60	(	(	PUNCT
ejpam-6037	159	61	3	3	X
ejpam-6037	159	62	)	)	PUNCT
ejpam-6037	159	63	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6037	159	64	-	-	PUNCT
ejpam-6037	159	65	cl(a	cl(a	NUM
ejpam-6037	159	66	)	)	PUNCT
ejpam-6037	159	67	)	)	PUNCT
ejpam-6037	160	1	⊆	⊆	X
ejpam-6037	160	2	σ1σ2	σ1σ2	NUM
ejpam-6037	160	3	-	-	PUNCT
ejpam-6037	160	4	cl(f(a	cl(f(a	NOUN
ejpam-6037	160	5	)	)	PUNCT
ejpam-6037	160	6	)	)	PUNCT
ejpam-6037	160	7	for	for	ADP
ejpam-6037	160	8	every	every	DET
ejpam-6037	160	9	subset	subset	NOUN
ejpam-6037	160	10	a	a	PRON
ejpam-6037	160	11	of	of	ADP
ejpam-6037	160	12	x	x	PRON
ejpam-6037	160	13	;	;	PUNCT
ejpam-6037	160	14	(	(	PUNCT
ejpam-6037	160	15	4	4	X
ejpam-6037	160	16	)	)	PUNCT
ejpam-6037	160	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	160	18	-	-	NOUN
ejpam-6037	160	19	cl(f	cl(f	NOUN
ejpam-6037	160	20	−1(b	−1(b	NOUN
ejpam-6037	160	21	)	)	PUNCT
ejpam-6037	160	22	)	)	PUNCT
ejpam-6037	161	1	⊆	⊆	NUM
ejpam-6037	161	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	161	3	-	-	PUNCT
ejpam-6037	161	4	cl(b	cl(b	NOUN
ejpam-6037	161	5	)	)	PUNCT
ejpam-6037	161	6	)	)	PUNCT
ejpam-6037	161	7	for	for	ADP
ejpam-6037	161	8	every	every	DET
ejpam-6037	161	9	subset	subset	NOUN
ejpam-6037	161	10	b	b	PROPN
ejpam-6037	161	11	of	of	ADP
ejpam-6037	161	12	y	y	PROPN
ejpam-6037	161	13	;	;	PUNCT
ejpam-6037	161	14	(	(	PUNCT
ejpam-6037	161	15	5	5	X
ejpam-6037	161	16	)	)	PUNCT
ejpam-6037	161	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	161	18	-	-	PUNCT
ejpam-6037	161	19	int(b	int(b	NOUN
ejpam-6037	161	20	)	)	PUNCT
ejpam-6037	161	21	)	)	PUNCT
ejpam-6037	162	1	⊆	⊆	X
ejpam-6037	162	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	162	3	-	-	NUM
ejpam-6037	162	4	int(f	int(f	NOUN
ejpam-6037	162	5	−1(b	−1(b	NOUN
ejpam-6037	162	6	)	)	PUNCT
ejpam-6037	162	7	)	)	PUNCT
ejpam-6037	162	8	for	for	ADP
ejpam-6037	162	9	every	every	DET
ejpam-6037	162	10	subset	subset	NOUN
ejpam-6037	162	11	b	b	PROPN
ejpam-6037	162	12	of	of	ADP
ejpam-6037	162	13	y	y	PROPN
ejpam-6037	162	14	;	;	PUNCT
ejpam-6037	162	15	(	(	PUNCT
ejpam-6037	162	16	6	6	X
ejpam-6037	162	17	)	)	PUNCT
ejpam-6037	162	18	f−1(k	f−1(k	PROPN
ejpam-6037	162	19	)	)	PUNCT
ejpam-6037	162	20	is	be	AUX
ejpam-6037	162	21	τ1τ2	τ1τ2	NOUN
ejpam-6037	162	22	-	-	ADJ
ejpam-6037	162	23	closed	closed	ADJ
ejpam-6037	162	24	in	in	ADP
ejpam-6037	162	25	x	x	PUNCT
ejpam-6037	162	26	for	for	ADP
ejpam-6037	162	27	every	every	DET
ejpam-6037	162	28	σ1σ2	σ1σ2	NUM
ejpam-6037	162	29	-	-	PUNCT
ejpam-6037	162	30	closed	closed	ADJ
ejpam-6037	162	31	set	set	NOUN
ejpam-6037	162	32	k	k	PROPN
ejpam-6037	162	33	of	of	ADP
ejpam-6037	162	34	y	y	PROPN
ejpam-6037	162	35	.	.	PUNCT
ejpam-6037	163	1	definition	definition	NOUN
ejpam-6037	163	2	6	6	NUM
ejpam-6037	163	3	.	.	PUNCT
ejpam-6037	164	1	a	a	DET
ejpam-6037	164	2	function	function	NOUN
ejpam-6037	164	3	f	f	NOUN
ejpam-6037	164	4	:	:	PUNCT
ejpam-6037	164	5	(	(	PUNCT
ejpam-6037	164	6	x	x	NOUN
ejpam-6037	164	7	,	,	PUNCT
ejpam-6037	164	8	τ1	τ1	NOUN
ejpam-6037	164	9	,	,	PUNCT
ejpam-6037	164	10	τ2	τ2	NOUN
ejpam-6037	164	11	)	)	PUNCT
ejpam-6037	164	12	→	→	SYM
ejpam-6037	164	13	(	(	PUNCT
ejpam-6037	164	14	y	y	PROPN
ejpam-6037	164	15	,	,	PUNCT
ejpam-6037	164	16	σ1	σ1	PROPN
ejpam-6037	164	17	,	,	PUNCT
ejpam-6037	164	18	σ2	σ2	PROPN
ejpam-6037	164	19	)	)	PUNCT
ejpam-6037	164	20	is	be	AUX
ejpam-6037	164	21	said	say	VERB
ejpam-6037	164	22	to	to	PART
ejpam-6037	164	23	satisfy	satisfy	VERB
ejpam-6037	164	24	the	the	DET
ejpam-6037	164	25	(	(	PUNCT
ejpam-6037	164	26	τ1	τ1	NOUN
ejpam-6037	164	27	,	,	PUNCT
ejpam-6037	164	28	τ2)interiority	τ2)interiority	NOUN
ejpam-6037	164	29	condition	condition	NOUN
ejpam-6037	164	30	if	if	SCONJ
ejpam-6037	164	31	τ1τ2	τ1τ2	NOUN
ejpam-6037	164	32	-	-	PUNCT
ejpam-6037	164	33	int(f	int(f	DET
ejpam-6037	164	34	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6037	164	35	-	-	PUNCT
ejpam-6037	164	36	cl(v	cl(v	NOUN
ejpam-6037	164	37	)	)	PUNCT
ejpam-6037	164	38	)	)	PUNCT
ejpam-6037	164	39	)	)	PUNCT
ejpam-6037	165	1	⊆	⊆	NUM
ejpam-6037	165	2	f−1(v	f−1(v	NOUN
ejpam-6037	165	3	)	)	PUNCT
ejpam-6037	165	4	for	for	ADP
ejpam-6037	165	5	each	each	DET
ejpam-6037	165	6	σ1σ2	σ1σ2	VERB
ejpam-6037	165	7	-	-	ADJ
ejpam-6037	165	8	open	open	ADJ
ejpam-6037	165	9	set	set	NOUN
ejpam-6037	165	10	v	v	NOUN
ejpam-6037	165	11	of	of	ADP
ejpam-6037	165	12	y	y	PROPN
ejpam-6037	165	13	.	.	PUNCT
ejpam-6037	166	1	n.	n.	PROPN
ejpam-6037	166	2	chutiman	chutiman	PROPN
ejpam-6037	166	3	,	,	PUNCT
ejpam-6037	166	4	a.	a.	PROPN
ejpam-6037	166	5	sama	sama	PROPN
ejpam-6037	166	6	-	-	PUNCT
ejpam-6037	166	7	ae	ae	PROPN
ejpam-6037	166	8	,	,	PUNCT
ejpam-6037	166	9	c.	c.	PROPN
ejpam-6037	166	10	boonpok	boonpok	PROPN
ejpam-6037	166	11	/	/	SYM
ejpam-6037	166	12	eur	eur	PROPN
ejpam-6037	166	13	.	.	PUNCT
ejpam-6037	167	1	j.	j.	PROPN
ejpam-6037	167	2	pure	pure	PROPN
ejpam-6037	167	3	appl	appl	PROPN
ejpam-6037	167	4	.	.	PROPN
ejpam-6037	167	5	math	math	PROPN
ejpam-6037	167	6	,	,	PUNCT
ejpam-6037	167	7	18	18	NUM
ejpam-6037	167	8	(	(	PUNCT
ejpam-6037	167	9	2	2	NUM
ejpam-6037	167	10	)	)	PUNCT
ejpam-6037	167	11	(	(	PUNCT
ejpam-6037	167	12	2025	2025	NUM
ejpam-6037	167	13	)	)	PUNCT
ejpam-6037	167	14	,	,	PUNCT
ejpam-6037	167	15	6037	6037	NUM
ejpam-6037	167	16	7	7	NUM
ejpam-6037	167	17	of	of	ADP
ejpam-6037	167	18	12	12	NUM
ejpam-6037	167	19	theorem	theorem	NOUN
ejpam-6037	167	20	6	6	NUM
ejpam-6037	167	21	.	.	PUNCT
ejpam-6037	168	1	if	if	SCONJ
ejpam-6037	168	2	f	f	PROPN
ejpam-6037	168	3	:	:	PUNCT
ejpam-6037	168	4	(	(	PUNCT
ejpam-6037	168	5	x	x	NOUN
ejpam-6037	168	6	,	,	PUNCT
ejpam-6037	168	7	τ1	τ1	NOUN
ejpam-6037	168	8	,	,	PUNCT
ejpam-6037	168	9	τ2	τ2	NOUN
ejpam-6037	168	10	)	)	PUNCT
ejpam-6037	168	11	→	→	SYM
ejpam-6037	168	12	(	(	PUNCT
ejpam-6037	168	13	y	y	PROPN
ejpam-6037	168	14	,	,	PUNCT
ejpam-6037	168	15	σ1	σ1	PROPN
ejpam-6037	168	16	,	,	PUNCT
ejpam-6037	168	17	σ2	σ2	PROPN
ejpam-6037	168	18	)	)	PUNCT
ejpam-6037	168	19	is	be	AUX
ejpam-6037	168	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	168	21	,	,	PUNCT
ejpam-6037	168	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	168	23	and	and	CCONJ
ejpam-6037	168	24	satisfy	satisfy	VERB
ejpam-6037	168	25	the	the	DET
ejpam-6037	168	26	(	(	PUNCT
ejpam-6037	168	27	τ1	τ1	NOUN
ejpam-6037	168	28	,	,	PUNCT
ejpam-6037	168	29	τ2)-interiority	τ2)-interiority	NOUN
ejpam-6037	168	30	condition	condition	NOUN
ejpam-6037	168	31	,	,	PUNCT
ejpam-6037	168	32	then	then	ADV
ejpam-6037	168	33	f	f	PROPN
ejpam-6037	168	34	is	be	AUX
ejpam-6037	168	35	(	(	PUNCT
ejpam-6037	168	36	τ1	τ1	NOUN
ejpam-6037	168	37	,	,	PUNCT
ejpam-6037	168	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	168	39	.	.	PUNCT
ejpam-6037	169	1	proof	proof	NOUN
ejpam-6037	169	2	.	.	PUNCT
ejpam-6037	170	1	let	let	VERB
ejpam-6037	170	2	v	v	PART
ejpam-6037	170	3	be	be	AUX
ejpam-6037	170	4	any	any	DET
ejpam-6037	170	5	σ1σ2	σ1σ2	NOUN
ejpam-6037	170	6	-	-	ADJ
ejpam-6037	170	7	open	open	ADJ
ejpam-6037	170	8	set	set	NOUN
ejpam-6037	170	9	of	of	ADP
ejpam-6037	170	10	y	y	PROPN
ejpam-6037	170	11	.	.	PUNCT
ejpam-6037	171	1	since	since	SCONJ
ejpam-6037	171	2	f	f	PROPN
ejpam-6037	171	3	is	be	AUX
ejpam-6037	171	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	171	5	,	,	PUNCT
ejpam-6037	171	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	171	7	,	,	PUNCT
ejpam-6037	171	8	by	by	ADP
ejpam-6037	171	9	theorem	theorem	NOUN
ejpam-6037	171	10	1	1	NUM
ejpam-6037	171	11	we	we	PRON
ejpam-6037	171	12	have	have	VERB
ejpam-6037	171	13	f−1(v	f−1(v	NOUN
ejpam-6037	171	14	)	)	PUNCT
ejpam-6037	172	1	⊆	⊆	NUM
ejpam-6037	172	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6037	172	3	-	-	PUNCT
ejpam-6037	172	4	cl(v	cl(v	NOUN
ejpam-6037	172	5	)	)	PUNCT
ejpam-6037	172	6	)	)	PUNCT
ejpam-6037	173	1	=	=	PUNCT
ejpam-6037	173	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	173	3	-	-	PUNCT
ejpam-6037	173	4	int(f	int(f	VERB
ejpam-6037	173	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6037	173	6	-	-	PUNCT
ejpam-6037	173	7	cl(v	cl(v	NOUN
ejpam-6037	173	8	)	)	PUNCT
ejpam-6037	173	9	)	)	PUNCT
ejpam-6037	173	10	)	)	PUNCT
ejpam-6037	174	1	=	=	PUNCT
ejpam-6037	175	1	τ1τ2	τ1τ2	NOUN
ejpam-6037	175	2	-	-	NOUN
ejpam-6037	175	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6037	175	4	-	-	PUNCT
ejpam-6037	175	5	int(f	int(f	PRON
ejpam-6037	175	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6037	175	7	-	-	PUNCT
ejpam-6037	175	8	cl(v	cl(v	NOUN
ejpam-6037	175	9	)	)	PUNCT
ejpam-6037	175	10	)	)	PUNCT
ejpam-6037	175	11	)	)	PUNCT
ejpam-6037	175	12	)	)	PUNCT
ejpam-6037	176	1	⊆	⊆	X
ejpam-6037	176	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	176	3	-	-	NUM
ejpam-6037	176	4	int(f	int(f	PRON
ejpam-6037	176	5	−1(v	−1(v	NOUN
ejpam-6037	176	6	)	)	PUNCT
ejpam-6037	176	7	)	)	PUNCT
ejpam-6037	177	1	⊆	⊆	NUM
ejpam-6037	177	2	f−1(v	f−1(v	NOUN
ejpam-6037	177	3	)	)	PUNCT
ejpam-6037	177	4	and	and	CCONJ
ejpam-6037	177	5	hence	hence	ADV
ejpam-6037	177	6	f−1(v	f−1(v	PROPN
ejpam-6037	177	7	)	)	PUNCT
ejpam-6037	177	8	is	be	AUX
ejpam-6037	177	9	τ1τ2	τ1τ2	NOUN
ejpam-6037	177	10	-	-	ADJ
ejpam-6037	177	11	open	open	ADJ
ejpam-6037	177	12	in	in	ADP
ejpam-6037	177	13	x.	x.	NOUN
ejpam-6037	177	14	by	by	ADP
ejpam-6037	177	15	lemma	lemma	PROPN
ejpam-6037	177	16	7	7	NUM
ejpam-6037	177	17	,	,	PUNCT
ejpam-6037	177	18	f	f	PROPN
ejpam-6037	177	19	is	be	AUX
ejpam-6037	177	20	(	(	PUNCT
ejpam-6037	177	21	τ1	τ1	NOUN
ejpam-6037	177	22	,	,	PUNCT
ejpam-6037	177	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	177	24	.	.	PUNCT
ejpam-6037	178	1	definition	definition	NOUN
ejpam-6037	178	2	7	7	NUM
ejpam-6037	178	3	.	.	PUNCT
ejpam-6037	179	1	a	a	DET
ejpam-6037	179	2	function	function	NOUN
ejpam-6037	179	3	f	f	NOUN
ejpam-6037	179	4	:	:	PUNCT
ejpam-6037	179	5	(	(	PUNCT
ejpam-6037	179	6	x	x	NOUN
ejpam-6037	179	7	,	,	PUNCT
ejpam-6037	179	8	τ1	τ1	NOUN
ejpam-6037	179	9	,	,	PUNCT
ejpam-6037	179	10	τ2	τ2	NOUN
ejpam-6037	179	11	)	)	PUNCT
ejpam-6037	179	12	→	→	SYM
ejpam-6037	179	13	(	(	PUNCT
ejpam-6037	179	14	y	y	PROPN
ejpam-6037	179	15	,	,	PUNCT
ejpam-6037	179	16	σ1	σ1	PROPN
ejpam-6037	179	17	,	,	PUNCT
ejpam-6037	179	18	σ2	σ2	PROPN
ejpam-6037	179	19	)	)	PUNCT
ejpam-6037	179	20	is	be	AUX
ejpam-6037	179	21	said	say	VERB
ejpam-6037	179	22	to	to	PART
ejpam-6037	179	23	have	have	VERB
ejpam-6037	179	24	a	a	DET
ejpam-6037	179	25	contra	contra	PROPN
ejpam-6037	179	26	(	(	PUNCT
ejpam-6037	179	27	τ1	τ1	PROPN
ejpam-6037	179	28	,	,	PUNCT
ejpam-6037	179	29	τ2)closed	τ2)closed	ADJ
ejpam-6037	179	30	graph	graph	NOUN
ejpam-6037	179	31	if	if	SCONJ
ejpam-6037	179	32	for	for	ADP
ejpam-6037	179	33	each	each	DET
ejpam-6037	179	34	(	(	PUNCT
ejpam-6037	179	35	x	x	NOUN
ejpam-6037	179	36	,	,	PUNCT
ejpam-6037	179	37	y	y	NOUN
ejpam-6037	179	38	)	)	PUNCT
ejpam-6037	179	39	∈	∈	PROPN
ejpam-6037	179	40	(	(	PUNCT
ejpam-6037	179	41	x	x	SYM
ejpam-6037	179	42	×	×	PROPN
ejpam-6037	179	43	y	y	PROPN
ejpam-6037	179	44	)	)	PUNCT
ejpam-6037	180	1	−	−	PROPN
ejpam-6037	180	2	g(f	g(f	NOUN
ejpam-6037	180	3	)	)	PUNCT
ejpam-6037	180	4	,	,	PUNCT
ejpam-6037	180	5	there	there	PRON
ejpam-6037	180	6	exist	exist	VERB
ejpam-6037	180	7	a	a	DET
ejpam-6037	180	8	τ1τ2	τ1τ2	NOUN
ejpam-6037	180	9	-	-	ADJ
ejpam-6037	180	10	open	open	ADJ
ejpam-6037	180	11	set	set	ADJ
ejpam-6037	180	12	u	u	NOUN
ejpam-6037	180	13	of	of	ADP
ejpam-6037	180	14	x	x	PUNCT
ejpam-6037	180	15	containing	contain	VERB
ejpam-6037	180	16	x	x	X
ejpam-6037	180	17	and	and	CCONJ
ejpam-6037	180	18	a	a	DET
ejpam-6037	180	19	σ1σ2	σ1σ2	NUM
ejpam-6037	180	20	-	-	PUNCT
ejpam-6037	180	21	closed	closed	ADJ
ejpam-6037	180	22	set	set	ADJ
ejpam-6037	180	23	f	f	PROPN
ejpam-6037	180	24	of	of	ADP
ejpam-6037	180	25	y	y	PROPN
ejpam-6037	180	26	containing	contain	VERB
ejpam-6037	180	27	y	y	PRON
ejpam-6037	180	28	such	such	ADJ
ejpam-6037	180	29	that	that	PRON
ejpam-6037	180	30	(	(	PUNCT
ejpam-6037	180	31	u	u	NOUN
ejpam-6037	180	32	×	×	PROPN
ejpam-6037	180	33	f	f	PROPN
ejpam-6037	180	34	)	)	PUNCT
ejpam-6037	180	35	∩g(f	∩g(f	PROPN
ejpam-6037	180	36	)	)	PUNCT
ejpam-6037	180	37	=	=	PUNCT
ejpam-6037	180	38	∅.	∅.	PRON
ejpam-6037	180	39	lemma	lemma	PROPN
ejpam-6037	180	40	8	8	NUM
ejpam-6037	180	41	.	.	PUNCT
ejpam-6037	181	1	a	a	DET
ejpam-6037	181	2	function	function	NOUN
ejpam-6037	181	3	f	f	NOUN
ejpam-6037	181	4	:	:	PUNCT
ejpam-6037	181	5	(	(	PUNCT
ejpam-6037	181	6	x	x	NOUN
ejpam-6037	181	7	,	,	PUNCT
ejpam-6037	181	8	τ1	τ1	NOUN
ejpam-6037	181	9	,	,	PUNCT
ejpam-6037	181	10	τ2	τ2	NOUN
ejpam-6037	181	11	)	)	PUNCT
ejpam-6037	181	12	→	→	SYM
ejpam-6037	181	13	(	(	PUNCT
ejpam-6037	181	14	y	y	PROPN
ejpam-6037	181	15	,	,	PUNCT
ejpam-6037	181	16	σ1	σ1	PROPN
ejpam-6037	181	17	,	,	PUNCT
ejpam-6037	181	18	σ2	σ2	NOUN
ejpam-6037	181	19	)	)	PUNCT
ejpam-6037	181	20	has	have	VERB
ejpam-6037	181	21	a	a	DET
ejpam-6037	181	22	contra	contra	PROPN
ejpam-6037	181	23	(	(	PUNCT
ejpam-6037	181	24	τ1	τ1	PROPN
ejpam-6037	181	25	,	,	PUNCT
ejpam-6037	181	26	τ2)-closed	τ2)-closed	ADJ
ejpam-6037	181	27	graph	graph	NOUN
ejpam-6037	181	28	if	if	SCONJ
ejpam-6037	181	29	and	and	CCONJ
ejpam-6037	181	30	only	only	ADV
ejpam-6037	181	31	if	if	SCONJ
ejpam-6037	181	32	for	for	ADP
ejpam-6037	181	33	each	each	DET
ejpam-6037	181	34	(	(	PUNCT
ejpam-6037	181	35	x	x	NOUN
ejpam-6037	181	36	,	,	PUNCT
ejpam-6037	181	37	y	y	NOUN
ejpam-6037	181	38	)	)	PUNCT
ejpam-6037	181	39	∈	∈	PROPN
ejpam-6037	181	40	(	(	PUNCT
ejpam-6037	181	41	x	x	SYM
ejpam-6037	181	42	×	×	PROPN
ejpam-6037	181	43	y	y	PROPN
ejpam-6037	181	44	)	)	PUNCT
ejpam-6037	181	45	−g(f	−g(f	NOUN
ejpam-6037	181	46	)	)	PUNCT
ejpam-6037	181	47	,	,	PUNCT
ejpam-6037	181	48	there	there	PRON
ejpam-6037	181	49	exist	exist	VERB
ejpam-6037	181	50	a	a	DET
ejpam-6037	181	51	τ1τ2	τ1τ2	NOUN
ejpam-6037	181	52	-	-	ADJ
ejpam-6037	181	53	open	open	ADJ
ejpam-6037	181	54	set	set	ADJ
ejpam-6037	181	55	u	u	NOUN
ejpam-6037	181	56	of	of	ADP
ejpam-6037	181	57	x	x	PUNCT
ejpam-6037	181	58	containing	contain	VERB
ejpam-6037	181	59	x	x	X
ejpam-6037	181	60	and	and	CCONJ
ejpam-6037	181	61	a	a	DET
ejpam-6037	181	62	σ1σ2	σ1σ2	NUM
ejpam-6037	181	63	-	-	PUNCT
ejpam-6037	181	64	closed	closed	ADJ
ejpam-6037	181	65	set	set	ADJ
ejpam-6037	181	66	f	f	PROPN
ejpam-6037	181	67	of	of	ADP
ejpam-6037	181	68	y	y	PROPN
ejpam-6037	181	69	containing	contain	VERB
ejpam-6037	181	70	y	y	PRON
ejpam-6037	182	1	such	such	ADJ
ejpam-6037	182	2	that	that	DET
ejpam-6037	182	3	f(u	f(u	PROPN
ejpam-6037	182	4	)	)	PUNCT
ejpam-6037	182	5	∩	∩	NOUN
ejpam-6037	182	6	f	f	X
ejpam-6037	183	1	=	=	PUNCT
ejpam-6037	183	2	∅.	∅.	PRON
ejpam-6037	183	3	definition	definition	NOUN
ejpam-6037	183	4	8	8	NUM
ejpam-6037	183	5	.	.	PUNCT
ejpam-6037	184	1	[	[	X
ejpam-6037	184	2	44	44	NUM
ejpam-6037	184	3	]	]	PUNCT
ejpam-6037	184	4	a	a	DET
ejpam-6037	184	5	bitopological	bitopological	ADJ
ejpam-6037	184	6	space	space	NOUN
ejpam-6037	184	7	(	(	PUNCT
ejpam-6037	184	8	x	x	NOUN
ejpam-6037	184	9	,	,	PUNCT
ejpam-6037	184	10	τ1	τ1	NOUN
ejpam-6037	184	11	,	,	PUNCT
ejpam-6037	184	12	τ2	τ2	NOUN
ejpam-6037	184	13	)	)	PUNCT
ejpam-6037	184	14	is	be	AUX
ejpam-6037	184	15	said	say	VERB
ejpam-6037	184	16	to	to	PART
ejpam-6037	184	17	be	be	AUX
ejpam-6037	184	18	τ1τ2	τ1τ2	NOUN
ejpam-6037	184	19	-	-	ADJ
ejpam-6037	184	20	urysohn	urysohn	ADJ
ejpam-6037	184	21	if	if	SCONJ
ejpam-6037	184	22	for	for	ADP
ejpam-6037	184	23	each	each	DET
ejpam-6037	184	24	pair	pair	NOUN
ejpam-6037	184	25	of	of	ADP
ejpam-6037	184	26	distinct	distinct	ADJ
ejpam-6037	184	27	points	point	NOUN
ejpam-6037	184	28	x	x	PUNCT
ejpam-6037	184	29	and	and	CCONJ
ejpam-6037	184	30	y	y	PROPN
ejpam-6037	184	31	in	in	ADP
ejpam-6037	184	32	x	x	SYM
ejpam-6037	184	33	,	,	PUNCT
ejpam-6037	184	34	there	there	PRON
ejpam-6037	184	35	exist	exist	VERB
ejpam-6037	184	36	τ1τ2	τ1τ2	ADJ
ejpam-6037	184	37	-	-	ADJ
ejpam-6037	184	38	open	open	ADJ
ejpam-6037	184	39	sets	set	NOUN
ejpam-6037	184	40	u	u	NOUN
ejpam-6037	184	41	and	and	CCONJ
ejpam-6037	184	42	v	v	ADP
ejpam-6037	184	43	such	such	ADJ
ejpam-6037	184	44	that	that	SCONJ
ejpam-6037	184	45	x	x	SYM
ejpam-6037	184	46	∈	∈	PROPN
ejpam-6037	184	47	u	u	NOUN
ejpam-6037	184	48	,	,	PUNCT
ejpam-6037	184	49	y	y	PROPN
ejpam-6037	184	50	∈	∈	PROPN
ejpam-6037	184	51	v	v	NOUN
ejpam-6037	184	52	and	and	CCONJ
ejpam-6037	184	53	τ1τ2	τ1τ2	NOUN
ejpam-6037	184	54	-	-	NOUN
ejpam-6037	184	55	cl(u	cl(u	NOUN
ejpam-6037	184	56	)	)	PUNCT
ejpam-6037	184	57	∩	∩	NOUN
ejpam-6037	184	58	τ1τ2	τ1τ2	NOUN
ejpam-6037	184	59	-	-	NOUN
ejpam-6037	184	60	cl(v	cl(v	X
ejpam-6037	184	61	)	)	PUNCT
ejpam-6037	184	62	=	=	PUNCT
ejpam-6037	184	63	∅.	∅.	NOUN
ejpam-6037	184	64	theorem	theorem	VERB
ejpam-6037	184	65	7	7	NUM
ejpam-6037	184	66	.	.	PUNCT
ejpam-6037	185	1	if	if	SCONJ
ejpam-6037	185	2	f	f	PROPN
ejpam-6037	185	3	:	:	PUNCT
ejpam-6037	185	4	(	(	PUNCT
ejpam-6037	185	5	x	x	NOUN
ejpam-6037	185	6	,	,	PUNCT
ejpam-6037	185	7	τ1	τ1	NOUN
ejpam-6037	185	8	,	,	PUNCT
ejpam-6037	185	9	τ2	τ2	NOUN
ejpam-6037	185	10	)	)	PUNCT
ejpam-6037	185	11	→	→	SYM
ejpam-6037	185	12	(	(	PUNCT
ejpam-6037	185	13	y	y	PROPN
ejpam-6037	185	14	,	,	PUNCT
ejpam-6037	185	15	σ1	σ1	PROPN
ejpam-6037	185	16	,	,	PUNCT
ejpam-6037	185	17	σ2	σ2	PROPN
ejpam-6037	185	18	)	)	PUNCT
ejpam-6037	185	19	is	be	AUX
ejpam-6037	185	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	185	21	,	,	PUNCT
ejpam-6037	185	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	185	23	and	and	CCONJ
ejpam-6037	185	24	(	(	PUNCT
ejpam-6037	185	25	y	y	PROPN
ejpam-6037	185	26	,	,	PUNCT
ejpam-6037	185	27	σ1	σ1	PROPN
ejpam-6037	185	28	,	,	PUNCT
ejpam-6037	185	29	σ2	σ2	PROPN
ejpam-6037	185	30	)	)	PUNCT
ejpam-6037	185	31	is	be	AUX
ejpam-6037	185	32	σ1σ2	σ1σ2	NOUN
ejpam-6037	185	33	-	-	PUNCT
ejpam-6037	185	34	urysohn	urysohn	ADJ
ejpam-6037	185	35	,	,	PUNCT
ejpam-6037	185	36	then	then	ADV
ejpam-6037	185	37	g(f	g(f	PROPN
ejpam-6037	185	38	)	)	PUNCT
ejpam-6037	186	1	is	be	AUX
ejpam-6037	186	2	contra	contra	PROPN
ejpam-6037	186	3	(	(	PUNCT
ejpam-6037	186	4	τ1	τ1	PROPN
ejpam-6037	186	5	,	,	PUNCT
ejpam-6037	186	6	τ2)-closed	τ2)-closed	ADJ
ejpam-6037	186	7	.	.	PUNCT
ejpam-6037	187	1	proof	proof	NOUN
ejpam-6037	187	2	.	.	PUNCT
ejpam-6037	188	1	let	let	VERB
ejpam-6037	188	2	(	(	PUNCT
ejpam-6037	188	3	x	x	NOUN
ejpam-6037	188	4	,	,	PUNCT
ejpam-6037	188	5	y	y	NOUN
ejpam-6037	188	6	)	)	PUNCT
ejpam-6037	188	7	∈	∈	PROPN
ejpam-6037	188	8	(	(	PUNCT
ejpam-6037	188	9	x×y	x×y	PROPN
ejpam-6037	188	10	)	)	PUNCT
ejpam-6037	188	11	−g(f	−g(f	NOUN
ejpam-6037	188	12	)	)	PUNCT
ejpam-6037	188	13	.	.	PUNCT
ejpam-6037	189	1	then	then	ADV
ejpam-6037	189	2	,	,	PUNCT
ejpam-6037	189	3	y	y	PROPN
ejpam-6037	189	4	̸=	̸=	PROPN
ejpam-6037	189	5	f(x	f(x	PROPN
ejpam-6037	189	6	)	)	PUNCT
ejpam-6037	189	7	.	.	PUNCT
ejpam-6037	190	1	since	since	SCONJ
ejpam-6037	190	2	(	(	PUNCT
ejpam-6037	190	3	y	y	PROPN
ejpam-6037	190	4	,	,	PUNCT
ejpam-6037	190	5	σ1	σ1	PROPN
ejpam-6037	190	6	,	,	PUNCT
ejpam-6037	190	7	σ2	σ2	PROPN
ejpam-6037	190	8	)	)	PUNCT
ejpam-6037	190	9	is	be	AUX
ejpam-6037	190	10	σ1σ2	σ1σ2	NOUN
ejpam-6037	190	11	-	-	PUNCT
ejpam-6037	190	12	urysohn	urysohn	ADJ
ejpam-6037	190	13	,	,	PUNCT
ejpam-6037	190	14	there	there	PRON
ejpam-6037	190	15	exist	exist	VERB
ejpam-6037	190	16	σ1σ2	σ1σ2	NOUN
ejpam-6037	190	17	-	-	ADJ
ejpam-6037	190	18	open	open	ADJ
ejpam-6037	190	19	sets	set	NOUN
ejpam-6037	190	20	v	v	ADP
ejpam-6037	190	21	and	and	CCONJ
ejpam-6037	190	22	w	w	PROPN
ejpam-6037	190	23	of	of	ADP
ejpam-6037	190	24	y	y	PROPN
ejpam-6037	190	25	containing	contain	VERB
ejpam-6037	190	26	y	y	PROPN
ejpam-6037	190	27	and	and	CCONJ
ejpam-6037	190	28	f(x	f(x	PROPN
ejpam-6037	190	29	)	)	PUNCT
ejpam-6037	190	30	,	,	PUNCT
ejpam-6037	190	31	respectively	respectively	ADV
ejpam-6037	190	32	,	,	PUNCT
ejpam-6037	190	33	such	such	ADJ
ejpam-6037	190	34	that	that	SCONJ
ejpam-6037	190	35	σ1σ2	σ1σ2	NOUN
ejpam-6037	190	36	-	-	PUNCT
ejpam-6037	190	37	cl(v	cl(v	NOUN
ejpam-6037	190	38	)	)	PUNCT
ejpam-6037	190	39	∩	∩	NOUN
ejpam-6037	190	40	σ1σ2	σ1σ2	NOUN
ejpam-6037	190	41	-	-	NUM
ejpam-6037	190	42	cl(w	cl(w	NOUN
ejpam-6037	190	43	)	)	PUNCT
ejpam-6037	191	1	=	=	PUNCT
ejpam-6037	191	2	∅.	∅.	NOUN
ejpam-6037	191	3	since	since	SCONJ
ejpam-6037	191	4	f	f	PROPN
ejpam-6037	191	5	is	be	AUX
ejpam-6037	191	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	191	7	,	,	PUNCT
ejpam-6037	191	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	191	9	,	,	PUNCT
ejpam-6037	191	10	by	by	ADP
ejpam-6037	191	11	theorem	theorem	NOUN
ejpam-6037	191	12	1	1	NUM
ejpam-6037	191	13	there	there	ADV
ejpam-6037	191	14	exists	exist	VERB
ejpam-6037	191	15	a	a	DET
ejpam-6037	191	16	τ1τ2	τ1τ2	NOUN
ejpam-6037	191	17	-	-	ADJ
ejpam-6037	191	18	open	open	ADJ
ejpam-6037	191	19	set	set	ADJ
ejpam-6037	191	20	u	u	NOUN
ejpam-6037	191	21	of	of	ADP
ejpam-6037	191	22	x	x	PUNCT
ejpam-6037	191	23	containing	contain	VERB
ejpam-6037	191	24	x	x	PUNCT
ejpam-6037	191	25	such	such	ADJ
ejpam-6037	191	26	that	that	DET
ejpam-6037	191	27	f(u	f(u	PROPN
ejpam-6037	191	28	)	)	PUNCT
ejpam-6037	192	1	⊆	⊆	NUM
ejpam-6037	192	2	σ1σ2	σ1σ2	NOUN
ejpam-6037	192	3	-	-	PUNCT
ejpam-6037	192	4	cl(w	cl(w	NOUN
ejpam-6037	192	5	)	)	PUNCT
ejpam-6037	192	6	.	.	PUNCT
ejpam-6037	193	1	this	this	PRON
ejpam-6037	193	2	implies	imply	VERB
ejpam-6037	193	3	that	that	SCONJ
ejpam-6037	193	4	f(u	f(u	PROPN
ejpam-6037	193	5	)	)	PUNCT
ejpam-6037	193	6	∩	∩	NOUN
ejpam-6037	193	7	σ1σ2	σ1σ2	NOUN
ejpam-6037	193	8	-	-	NUM
ejpam-6037	193	9	cl(v	cl(v	X
ejpam-6037	193	10	)	)	PUNCT
ejpam-6037	193	11	=	=	NOUN
ejpam-6037	193	12	∅	∅	NOUN
ejpam-6037	193	13	and	and	CCONJ
ejpam-6037	193	14	by	by	ADP
ejpam-6037	193	15	lemma	lemma	PROPN
ejpam-6037	193	16	8	8	NUM
ejpam-6037	193	17	,	,	PUNCT
ejpam-6037	193	18	g(f	g(f	PROPN
ejpam-6037	193	19	)	)	PUNCT
ejpam-6037	193	20	is	be	AUX
ejpam-6037	193	21	contra	contra	PROPN
ejpam-6037	193	22	(	(	PUNCT
ejpam-6037	193	23	τ1	τ1	PROPN
ejpam-6037	193	24	,	,	PUNCT
ejpam-6037	193	25	τ2)-closed	τ2)-closed	PROPN
ejpam-6037	193	26	.	.	PUNCT
ejpam-6037	194	1	recall	recall	VERB
ejpam-6037	194	2	that	that	SCONJ
ejpam-6037	194	3	a	a	DET
ejpam-6037	194	4	bitopological	bitopological	ADJ
ejpam-6037	194	5	space	space	NOUN
ejpam-6037	194	6	(	(	PUNCT
ejpam-6037	194	7	x	x	NOUN
ejpam-6037	194	8	,	,	PUNCT
ejpam-6037	194	9	τ1	τ1	NOUN
ejpam-6037	194	10	,	,	PUNCT
ejpam-6037	194	11	τ2	τ2	NOUN
ejpam-6037	194	12	)	)	PUNCT
ejpam-6037	194	13	is	be	AUX
ejpam-6037	194	14	said	say	VERB
ejpam-6037	194	15	to	to	PART
ejpam-6037	194	16	be	be	AUX
ejpam-6037	194	17	(	(	PUNCT
ejpam-6037	194	18	τ1	τ1	NOUN
ejpam-6037	194	19	,	,	PUNCT
ejpam-6037	194	20	τ2)-t1	τ2)-t1	VERB
ejpam-6037	195	1	[	[	X
ejpam-6037	195	2	45	45	NUM
ejpam-6037	195	3	]	]	X
ejpam-6037	195	4	if	if	SCONJ
ejpam-6037	195	5	for	for	ADP
ejpam-6037	195	6	any	any	DET
ejpam-6037	195	7	pair	pair	NOUN
ejpam-6037	195	8	of	of	ADP
ejpam-6037	195	9	distinct	distinct	ADJ
ejpam-6037	195	10	points	point	NOUN
ejpam-6037	195	11	x	x	X
ejpam-6037	195	12	,	,	PUNCT
ejpam-6037	195	13	y	y	PROPN
ejpam-6037	195	14	in	in	ADP
ejpam-6037	195	15	x	x	SYM
ejpam-6037	195	16	,	,	PUNCT
ejpam-6037	195	17	there	there	PRON
ejpam-6037	195	18	exist	exist	VERB
ejpam-6037	195	19	τ1τ2	τ1τ2	ADJ
ejpam-6037	195	20	-	-	ADJ
ejpam-6037	195	21	open	open	ADJ
ejpam-6037	195	22	sets	set	NOUN
ejpam-6037	195	23	u	u	NOUN
ejpam-6037	195	24	and	and	CCONJ
ejpam-6037	195	25	v	v	ADP
ejpam-6037	195	26	such	such	ADJ
ejpam-6037	195	27	that	that	SCONJ
ejpam-6037	195	28	x	x	SYM
ejpam-6037	195	29	∈	∈	PROPN
ejpam-6037	195	30	u	u	PROPN
ejpam-6037	195	31	,	,	PUNCT
ejpam-6037	195	32	y	y	PROPN
ejpam-6037	195	33	̸∈	̸∈	PROPN
ejpam-6037	195	34	u	u	PROPN
ejpam-6037	195	35	and	and	CCONJ
ejpam-6037	195	36	y	y	PROPN
ejpam-6037	195	37	∈	∈	PROPN
ejpam-6037	195	38	v	v	NOUN
ejpam-6037	195	39	,	,	PUNCT
ejpam-6037	195	40	x	x	PROPN
ejpam-6037	195	41	̸∈	̸∈	PROPN
ejpam-6037	195	42	v	v	PROPN
ejpam-6037	195	43	.	.	PUNCT
ejpam-6037	196	1	theorem	theorem	VERB
ejpam-6037	196	2	8	8	NUM
ejpam-6037	196	3	.	.	PUNCT
ejpam-6037	197	1	if	if	SCONJ
ejpam-6037	197	2	f	f	PROPN
ejpam-6037	197	3	:	:	PUNCT
ejpam-6037	197	4	(	(	PUNCT
ejpam-6037	197	5	x	x	NOUN
ejpam-6037	197	6	,	,	PUNCT
ejpam-6037	197	7	τ1	τ1	NOUN
ejpam-6037	197	8	,	,	PUNCT
ejpam-6037	197	9	τ2	τ2	NOUN
ejpam-6037	197	10	)	)	PUNCT
ejpam-6037	197	11	→	→	SYM
ejpam-6037	197	12	(	(	PUNCT
ejpam-6037	197	13	y	y	PROPN
ejpam-6037	197	14	,	,	PUNCT
ejpam-6037	197	15	σ1	σ1	PROPN
ejpam-6037	197	16	,	,	PUNCT
ejpam-6037	197	17	σ2	σ2	PROPN
ejpam-6037	197	18	)	)	PUNCT
ejpam-6037	197	19	is	be	AUX
ejpam-6037	197	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	197	21	,	,	PUNCT
ejpam-6037	197	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	197	23	and	and	CCONJ
ejpam-6037	197	24	(	(	PUNCT
ejpam-6037	197	25	y	y	PROPN
ejpam-6037	197	26	,	,	PUNCT
ejpam-6037	197	27	σ1	σ1	PROPN
ejpam-6037	197	28	,	,	PUNCT
ejpam-6037	197	29	σ2	σ2	PROPN
ejpam-6037	197	30	)	)	PUNCT
ejpam-6037	197	31	is	be	AUX
ejpam-6037	197	32	(	(	PUNCT
ejpam-6037	197	33	σ1	σ1	PROPN
ejpam-6037	197	34	,	,	PUNCT
ejpam-6037	197	35	σ2)-t1	σ2)-t1	NOUN
ejpam-6037	197	36	,	,	PUNCT
ejpam-6037	197	37	then	then	ADV
ejpam-6037	197	38	g(f	g(f	PROPN
ejpam-6037	197	39	)	)	PUNCT
ejpam-6037	198	1	is	be	AUX
ejpam-6037	198	2	contra	contra	PROPN
ejpam-6037	198	3	(	(	PUNCT
ejpam-6037	198	4	τ1	τ1	PROPN
ejpam-6037	198	5	,	,	PUNCT
ejpam-6037	198	6	τ2)-closed	τ2)-closed	ADJ
ejpam-6037	198	7	.	.	PUNCT
ejpam-6037	199	1	proof	proof	NOUN
ejpam-6037	199	2	.	.	PUNCT
ejpam-6037	200	1	let	let	VERB
ejpam-6037	200	2	(	(	PUNCT
ejpam-6037	200	3	x	x	NOUN
ejpam-6037	200	4	,	,	PUNCT
ejpam-6037	200	5	y	y	NOUN
ejpam-6037	200	6	)	)	PUNCT
ejpam-6037	200	7	∈	∈	PROPN
ejpam-6037	200	8	(	(	PUNCT
ejpam-6037	200	9	x	x	SYM
ejpam-6037	200	10	×	×	PROPN
ejpam-6037	200	11	y	y	PROPN
ejpam-6037	200	12	)	)	PUNCT
ejpam-6037	201	1	−	−	PROPN
ejpam-6037	201	2	g(f	g(f	NOUN
ejpam-6037	201	3	)	)	PUNCT
ejpam-6037	201	4	.	.	PUNCT
ejpam-6037	202	1	then	then	ADV
ejpam-6037	202	2	,	,	PUNCT
ejpam-6037	202	3	y	y	PROPN
ejpam-6037	202	4	̸=	̸=	PROPN
ejpam-6037	202	5	f(x	f(x	PROPN
ejpam-6037	202	6	)	)	PUNCT
ejpam-6037	202	7	.	.	PUNCT
ejpam-6037	203	1	since	since	SCONJ
ejpam-6037	203	2	(	(	PUNCT
ejpam-6037	203	3	y	y	PROPN
ejpam-6037	203	4	,	,	PUNCT
ejpam-6037	203	5	σ1	σ1	PROPN
ejpam-6037	203	6	,	,	PUNCT
ejpam-6037	203	7	σ2	σ2	PROPN
ejpam-6037	203	8	)	)	PUNCT
ejpam-6037	203	9	is	be	AUX
ejpam-6037	203	10	(	(	PUNCT
ejpam-6037	203	11	σ1	σ1	PROPN
ejpam-6037	203	12	,	,	PUNCT
ejpam-6037	203	13	σ2)-t1	σ2)-t1	PROPN
ejpam-6037	203	14	,	,	PUNCT
ejpam-6037	203	15	there	there	PRON
ejpam-6037	203	16	exists	exist	VERB
ejpam-6037	203	17	a	a	DET
ejpam-6037	203	18	σ1σ2	σ1σ2	NUM
ejpam-6037	203	19	-	-	ADJ
ejpam-6037	203	20	open	open	ADJ
ejpam-6037	203	21	set	set	NOUN
ejpam-6037	203	22	v	v	NOUN
ejpam-6037	203	23	of	of	ADP
ejpam-6037	203	24	y	y	PRON
ejpam-6037	203	25	such	such	ADJ
ejpam-6037	203	26	that	that	SCONJ
ejpam-6037	203	27	f(x	f(x	PROPN
ejpam-6037	203	28	)	)	PUNCT
ejpam-6037	203	29	∈	∈	PROPN
ejpam-6037	203	30	v	v	NOUN
ejpam-6037	203	31	and	and	CCONJ
ejpam-6037	203	32	y	y	PROPN
ejpam-6037	203	33	̸∈	̸∈	PROPN
ejpam-6037	203	34	v	v	PROPN
ejpam-6037	203	35	.	.	PUNCT
ejpam-6037	204	1	since	since	SCONJ
ejpam-6037	204	2	f	f	PROPN
ejpam-6037	204	3	is	be	AUX
ejpam-6037	204	4	(	(	PUNCT
ejpam-6037	204	5	τ1	τ1	NOUN
ejpam-6037	204	6	,	,	PUNCT
ejpam-6037	204	7	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	204	8	,	,	PUNCT
ejpam-6037	204	9	there	there	PRON
ejpam-6037	204	10	exists	exist	VERB
ejpam-6037	204	11	a	a	DET
ejpam-6037	204	12	τ1τ2	τ1τ2	NOUN
ejpam-6037	204	13	-	-	ADJ
ejpam-6037	204	14	open	open	ADJ
ejpam-6037	204	15	set	set	ADJ
ejpam-6037	204	16	u	u	NOUN
ejpam-6037	204	17	of	of	ADP
ejpam-6037	204	18	x	x	PUNCT
ejpam-6037	204	19	containing	contain	VERB
ejpam-6037	204	20	x	x	PUNCT
ejpam-6037	204	21	such	such	ADJ
ejpam-6037	204	22	that	that	DET
ejpam-6037	204	23	f(u	f(u	PROPN
ejpam-6037	204	24	)	)	PUNCT
ejpam-6037	204	25	⊆	⊆	NUM
ejpam-6037	204	26	v	v	NOUN
ejpam-6037	204	27	.	.	PUNCT
ejpam-6037	205	1	thus	thus	ADV
ejpam-6037	205	2	,	,	PUNCT
ejpam-6037	205	3	f(u	f(u	PROPN
ejpam-6037	205	4	)	)	PUNCT
ejpam-6037	205	5	∩	∩	NOUN
ejpam-6037	205	6	(	(	PUNCT
ejpam-6037	205	7	y	y	PROPN
ejpam-6037	205	8	−	−	PROPN
ejpam-6037	205	9	v	v	NOUN
ejpam-6037	205	10	)	)	PUNCT
ejpam-6037	205	11	=	=	NOUN
ejpam-6037	205	12	∅	∅	NOUN
ejpam-6037	205	13	and	and	CCONJ
ejpam-6037	205	14	y	y	PROPN
ejpam-6037	205	15	−	−	PROPN
ejpam-6037	205	16	v	v	PROPN
ejpam-6037	205	17	is	be	AUX
ejpam-6037	205	18	a	a	DET
ejpam-6037	205	19	σ1σ2	σ1σ2	NUM
ejpam-6037	205	20	-	-	PUNCT
ejpam-6037	205	21	closed	closed	ADJ
ejpam-6037	205	22	set	set	NOUN
ejpam-6037	205	23	of	of	ADP
ejpam-6037	205	24	y	y	PROPN
ejpam-6037	205	25	containing	contain	VERB
ejpam-6037	205	26	y.	y.	NOUN
ejpam-6037	205	27	this	this	PRON
ejpam-6037	205	28	shows	show	VERB
ejpam-6037	205	29	that	that	SCONJ
ejpam-6037	205	30	g(f	g(f	NOUN
ejpam-6037	205	31	)	)	PUNCT
ejpam-6037	205	32	is	be	AUX
ejpam-6037	205	33	contra	contra	PROPN
ejpam-6037	205	34	(	(	PUNCT
ejpam-6037	205	35	τ1	τ1	PROPN
ejpam-6037	205	36	,	,	PUNCT
ejpam-6037	205	37	τ2)-closed	τ2)-closed	PROPN
ejpam-6037	205	38	.	.	PUNCT
ejpam-6037	205	39	n.	n.	PROPN
ejpam-6037	205	40	chutiman	chutiman	PROPN
ejpam-6037	205	41	,	,	PUNCT
ejpam-6037	205	42	a.	a.	PROPN
ejpam-6037	205	43	sama	sama	PROPN
ejpam-6037	205	44	-	-	PUNCT
ejpam-6037	205	45	ae	ae	PROPN
ejpam-6037	205	46	,	,	PUNCT
ejpam-6037	205	47	c.	c.	PROPN
ejpam-6037	205	48	boonpok	boonpok	PROPN
ejpam-6037	205	49	/	/	SYM
ejpam-6037	205	50	eur	eur	PROPN
ejpam-6037	205	51	.	.	PUNCT
ejpam-6037	206	1	j.	j.	PROPN
ejpam-6037	206	2	pure	pure	PROPN
ejpam-6037	206	3	appl	appl	PROPN
ejpam-6037	206	4	.	.	PROPN
ejpam-6037	206	5	math	math	PROPN
ejpam-6037	206	6	,	,	PUNCT
ejpam-6037	206	7	18	18	NUM
ejpam-6037	206	8	(	(	PUNCT
ejpam-6037	206	9	2	2	NUM
ejpam-6037	206	10	)	)	PUNCT
ejpam-6037	206	11	(	(	PUNCT
ejpam-6037	206	12	2025	2025	NUM
ejpam-6037	206	13	)	)	PUNCT
ejpam-6037	206	14	,	,	PUNCT
ejpam-6037	206	15	6037	6037	NUM
ejpam-6037	206	16	8	8	NUM
ejpam-6037	206	17	of	of	ADP
ejpam-6037	206	18	12	12	NUM
ejpam-6037	206	19	definition	definition	NOUN
ejpam-6037	206	20	9	9	NUM
ejpam-6037	206	21	.	.	PUNCT
ejpam-6037	207	1	[	[	X
ejpam-6037	207	2	46	46	NUM
ejpam-6037	207	3	]	]	PUNCT
ejpam-6037	207	4	a	a	DET
ejpam-6037	207	5	bitopological	bitopological	ADJ
ejpam-6037	207	6	space	space	NOUN
ejpam-6037	207	7	(	(	PUNCT
ejpam-6037	207	8	x	x	NOUN
ejpam-6037	207	9	,	,	PUNCT
ejpam-6037	207	10	τ1	τ1	NOUN
ejpam-6037	207	11	,	,	PUNCT
ejpam-6037	207	12	τ2	τ2	NOUN
ejpam-6037	207	13	)	)	PUNCT
ejpam-6037	207	14	is	be	AUX
ejpam-6037	207	15	said	say	VERB
ejpam-6037	207	16	to	to	PART
ejpam-6037	207	17	be	be	AUX
ejpam-6037	207	18	(	(	PUNCT
ejpam-6037	207	19	τ1	τ1	NOUN
ejpam-6037	207	20	,	,	PUNCT
ejpam-6037	207	21	τ2)-t2	τ2)-t2	ADJ
ejpam-6037	207	22	if	if	SCONJ
ejpam-6037	207	23	for	for	ADP
ejpam-6037	207	24	any	any	DET
ejpam-6037	207	25	pair	pair	NOUN
ejpam-6037	207	26	of	of	ADP
ejpam-6037	207	27	distinct	distinct	ADJ
ejpam-6037	207	28	points	point	NOUN
ejpam-6037	207	29	x	x	X
ejpam-6037	207	30	,	,	PUNCT
ejpam-6037	207	31	y	y	PROPN
ejpam-6037	207	32	in	in	ADP
ejpam-6037	207	33	x	x	SYM
ejpam-6037	207	34	,	,	PUNCT
ejpam-6037	207	35	there	there	PRON
ejpam-6037	207	36	exist	exist	VERB
ejpam-6037	207	37	disjoint	disjoint	ADJ
ejpam-6037	207	38	τ1τ2	τ1τ2	ADJ
ejpam-6037	207	39	-	-	ADJ
ejpam-6037	207	40	open	open	ADJ
ejpam-6037	207	41	sets	set	NOUN
ejpam-6037	207	42	u	u	NOUN
ejpam-6037	207	43	and	and	CCONJ
ejpam-6037	207	44	v	v	NOUN
ejpam-6037	207	45	of	of	ADP
ejpam-6037	207	46	x	x	PUNCT
ejpam-6037	207	47	containing	contain	VERB
ejpam-6037	207	48	x	x	PROPN
ejpam-6037	207	49	and	and	CCONJ
ejpam-6037	207	50	y	y	PROPN
ejpam-6037	207	51	,	,	PUNCT
ejpam-6037	207	52	respectively	respectively	ADV
ejpam-6037	207	53	.	.	PUNCT
ejpam-6037	208	1	theorem	theorem	VERB
ejpam-6037	208	2	9	9	NUM
ejpam-6037	208	3	.	.	PUNCT
ejpam-6037	209	1	if	if	SCONJ
ejpam-6037	209	2	f	f	PROPN
ejpam-6037	209	3	:	:	PUNCT
ejpam-6037	209	4	(	(	PUNCT
ejpam-6037	209	5	x	x	NOUN
ejpam-6037	209	6	,	,	PUNCT
ejpam-6037	209	7	τ1	τ1	NOUN
ejpam-6037	209	8	,	,	PUNCT
ejpam-6037	209	9	τ2	τ2	NOUN
ejpam-6037	209	10	)	)	PUNCT
ejpam-6037	209	11	→	→	SYM
ejpam-6037	209	12	(	(	PUNCT
ejpam-6037	209	13	y	y	PROPN
ejpam-6037	209	14	,	,	PUNCT
ejpam-6037	209	15	σ1	σ1	PROPN
ejpam-6037	209	16	,	,	PUNCT
ejpam-6037	209	17	σ2	σ2	PROPN
ejpam-6037	209	18	)	)	PUNCT
ejpam-6037	209	19	is	be	AUX
ejpam-6037	209	20	a	a	DET
ejpam-6037	209	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	209	22	,	,	PUNCT
ejpam-6037	209	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	209	24	injection	injection	NOUN
ejpam-6037	209	25	with	with	ADP
ejpam-6037	209	26	a	a	DET
ejpam-6037	209	27	contra	contra	PROPN
ejpam-6037	209	28	(	(	PUNCT
ejpam-6037	209	29	τ1	τ1	PROPN
ejpam-6037	209	30	,	,	PUNCT
ejpam-6037	209	31	τ2)-closed	τ2)-close	VERB
ejpam-6037	209	32	graph	graph	NOUN
ejpam-6037	209	33	,	,	PUNCT
ejpam-6037	209	34	then	then	ADV
ejpam-6037	209	35	(	(	PUNCT
ejpam-6037	209	36	x	x	NOUN
ejpam-6037	209	37	,	,	PUNCT
ejpam-6037	209	38	τ1	τ1	NOUN
ejpam-6037	209	39	,	,	PUNCT
ejpam-6037	209	40	τ2	τ2	NOUN
ejpam-6037	209	41	)	)	PUNCT
ejpam-6037	209	42	is	be	AUX
ejpam-6037	209	43	(	(	PUNCT
ejpam-6037	209	44	τ1	τ1	NOUN
ejpam-6037	209	45	,	,	PUNCT
ejpam-6037	209	46	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	209	47	.	.	PUNCT
ejpam-6037	210	1	proof	proof	NOUN
ejpam-6037	210	2	.	.	PUNCT
ejpam-6037	211	1	let	let	VERB
ejpam-6037	211	2	x	x	PRON
ejpam-6037	211	3	and	and	CCONJ
ejpam-6037	211	4	y	y	PROPN
ejpam-6037	211	5	be	be	AUX
ejpam-6037	211	6	any	any	DET
ejpam-6037	211	7	distinct	distinct	ADJ
ejpam-6037	211	8	points	point	NOUN
ejpam-6037	211	9	of	of	ADP
ejpam-6037	211	10	x.	x.	NOUN
ejpam-6037	211	11	since	since	SCONJ
ejpam-6037	211	12	f	f	PROPN
ejpam-6037	211	13	is	be	AUX
ejpam-6037	211	14	injective	injective	ADJ
ejpam-6037	211	15	,	,	PUNCT
ejpam-6037	211	16	f(x	f(x	PROPN
ejpam-6037	211	17	)	)	PUNCT
ejpam-6037	211	18	̸=	̸=	PROPN
ejpam-6037	211	19	f(y	f(y	NOUN
ejpam-6037	211	20	)	)	PUNCT
ejpam-6037	211	21	.	.	PUNCT
ejpam-6037	212	1	then	then	ADV
ejpam-6037	212	2	,	,	PUNCT
ejpam-6037	212	3	we	we	PRON
ejpam-6037	212	4	have	have	VERB
ejpam-6037	212	5	(	(	PUNCT
ejpam-6037	212	6	x	x	NOUN
ejpam-6037	212	7	,	,	PUNCT
ejpam-6037	212	8	f(y	f(y	NOUN
ejpam-6037	212	9	)	)	PUNCT
ejpam-6037	212	10	)	)	PUNCT
ejpam-6037	213	1	∈	∈	PROPN
ejpam-6037	213	2	(	(	PUNCT
ejpam-6037	213	3	x×y	x×y	PROPN
ejpam-6037	213	4	)	)	PUNCT
ejpam-6037	213	5	−g(f	−g(f	NOUN
ejpam-6037	213	6	)	)	PUNCT
ejpam-6037	213	7	.	.	PUNCT
ejpam-6037	214	1	since	since	SCONJ
ejpam-6037	214	2	g(f	g(f	PROPN
ejpam-6037	214	3	)	)	PUNCT
ejpam-6037	214	4	is	be	AUX
ejpam-6037	214	5	contra	contra	PROPN
ejpam-6037	214	6	(	(	PUNCT
ejpam-6037	214	7	τ1	τ1	PROPN
ejpam-6037	214	8	,	,	PUNCT
ejpam-6037	214	9	τ2)-closed	τ2)-close	VERB
ejpam-6037	214	10	,	,	PUNCT
ejpam-6037	214	11	by	by	ADP
ejpam-6037	214	12	lemma	lemma	PROPN
ejpam-6037	214	13	8	8	NUM
ejpam-6037	214	14	there	there	PRON
ejpam-6037	214	15	exist	exist	VERB
ejpam-6037	214	16	a	a	DET
ejpam-6037	214	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	214	18	-	-	ADJ
ejpam-6037	214	19	open	open	ADJ
ejpam-6037	214	20	set	set	ADJ
ejpam-6037	214	21	u	u	NOUN
ejpam-6037	214	22	of	of	ADP
ejpam-6037	214	23	x	x	PUNCT
ejpam-6037	214	24	containing	contain	VERB
ejpam-6037	214	25	x	x	X
ejpam-6037	214	26	and	and	CCONJ
ejpam-6037	214	27	a	a	DET
ejpam-6037	214	28	σ1σ2	σ1σ2	NUM
ejpam-6037	214	29	-	-	PUNCT
ejpam-6037	214	30	closed	closed	ADJ
ejpam-6037	214	31	set	set	NOUN
ejpam-6037	214	32	k	k	PROPN
ejpam-6037	214	33	of	of	ADP
ejpam-6037	214	34	y	y	PROPN
ejpam-6037	214	35	containing	contain	VERB
ejpam-6037	214	36	f(y	f(y	NOUN
ejpam-6037	214	37	)	)	PUNCT
ejpam-6037	214	38	such	such	ADJ
ejpam-6037	214	39	that	that	DET
ejpam-6037	214	40	f(u	f(u	PROPN
ejpam-6037	214	41	)	)	PUNCT
ejpam-6037	214	42	∩k	∩k	NOUN
ejpam-6037	215	1	=	=	PUNCT
ejpam-6037	215	2	∅.	∅.	NOUN
ejpam-6037	215	3	since	since	SCONJ
ejpam-6037	215	4	f	f	PROPN
ejpam-6037	215	5	is	be	AUX
ejpam-6037	215	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	215	7	,	,	PUNCT
ejpam-6037	215	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	215	9	,	,	PUNCT
ejpam-6037	215	10	by	by	ADP
ejpam-6037	215	11	theorem	theorem	NOUN
ejpam-6037	215	12	1	1	NUM
ejpam-6037	215	13	there	there	ADV
ejpam-6037	215	14	exists	exist	VERB
ejpam-6037	215	15	a	a	DET
ejpam-6037	215	16	τ1τ2	τ1τ2	ADJ
ejpam-6037	215	17	-	-	ADJ
ejpam-6037	215	18	open	open	ADJ
ejpam-6037	215	19	set	set	ADJ
ejpam-6037	215	20	u0	u0	NOUN
ejpam-6037	215	21	of	of	ADP
ejpam-6037	215	22	x	x	PUNCT
ejpam-6037	215	23	containing	contain	VERB
ejpam-6037	215	24	y	y	PRON
ejpam-6037	215	25	such	such	ADJ
ejpam-6037	215	26	that	that	DET
ejpam-6037	215	27	f(u0	f(u0	NOUN
ejpam-6037	215	28	)	)	PUNCT
ejpam-6037	215	29	⊆	⊆	NUM
ejpam-6037	215	30	k.	k.	PROPN
ejpam-6037	215	31	thus	thus	ADV
ejpam-6037	215	32	,	,	PUNCT
ejpam-6037	215	33	f(u)∩	f(u)∩	PROPN
ejpam-6037	215	34	f(u0	f(u0	NOUN
ejpam-6037	215	35	)	)	PUNCT
ejpam-6037	215	36	=	=	SYM
ejpam-6037	215	37	∅	∅	NOUN
ejpam-6037	215	38	and	and	CCONJ
ejpam-6037	215	39	hence	hence	ADV
ejpam-6037	215	40	u	u	NOUN
ejpam-6037	215	41	∩	∩	ADJ
ejpam-6037	215	42	u0	u0	NOUN
ejpam-6037	215	43	=	=	PROPN
ejpam-6037	215	44	∅.	∅.	NOUN
ejpam-6037	215	45	this	this	PRON
ejpam-6037	215	46	shows	show	VERB
ejpam-6037	215	47	that	that	SCONJ
ejpam-6037	215	48	(	(	PUNCT
ejpam-6037	215	49	x	x	NOUN
ejpam-6037	215	50	,	,	PUNCT
ejpam-6037	215	51	τ1	τ1	NOUN
ejpam-6037	215	52	,	,	PUNCT
ejpam-6037	215	53	τ2	τ2	NOUN
ejpam-6037	215	54	)	)	PUNCT
ejpam-6037	215	55	is	be	AUX
ejpam-6037	215	56	(	(	PUNCT
ejpam-6037	215	57	τ1	τ1	NOUN
ejpam-6037	215	58	,	,	PUNCT
ejpam-6037	215	59	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	215	60	.	.	PUNCT
ejpam-6037	216	1	theorem	theorem	VERB
ejpam-6037	216	2	10	10	NUM
ejpam-6037	216	3	.	.	PUNCT
ejpam-6037	217	1	let	let	AUX
ejpam-6037	217	2	(	(	PUNCT
ejpam-6037	217	3	x	x	NOUN
ejpam-6037	217	4	,	,	PUNCT
ejpam-6037	217	5	τ1	τ1	NOUN
ejpam-6037	217	6	,	,	PUNCT
ejpam-6037	217	7	τ2	τ2	PROPN
ejpam-6037	217	8	)	)	PUNCT
ejpam-6037	217	9	be	be	VERB
ejpam-6037	217	10	a	a	DET
ejpam-6037	217	11	bitopological	bitopological	ADJ
ejpam-6037	217	12	space	space	NOUN
ejpam-6037	217	13	.	.	PUNCT
ejpam-6037	218	1	if	if	SCONJ
ejpam-6037	218	2	for	for	ADP
ejpam-6037	218	3	each	each	DET
ejpam-6037	218	4	pair	pair	NOUN
ejpam-6037	218	5	of	of	ADP
ejpam-6037	218	6	distinct	distinct	ADJ
ejpam-6037	218	7	points	point	NOUN
ejpam-6037	218	8	x	x	PUNCT
ejpam-6037	218	9	and	and	CCONJ
ejpam-6037	218	10	x′	x′	PROPN
ejpam-6037	218	11	in	in	ADP
ejpam-6037	218	12	x	x	SYM
ejpam-6037	218	13	,	,	PUNCT
ejpam-6037	218	14	there	there	PRON
ejpam-6037	218	15	exists	exist	VERB
ejpam-6037	218	16	a	a	DET
ejpam-6037	218	17	function	function	NOUN
ejpam-6037	218	18	f	f	PROPN
ejpam-6037	218	19	of	of	ADP
ejpam-6037	218	20	(	(	PUNCT
ejpam-6037	218	21	x	x	NOUN
ejpam-6037	218	22	,	,	PUNCT
ejpam-6037	218	23	τ1	τ1	NOUN
ejpam-6037	218	24	,	,	PUNCT
ejpam-6037	218	25	τ2	τ2	NOUN
ejpam-6037	218	26	)	)	PUNCT
ejpam-6037	218	27	into	into	ADP
ejpam-6037	218	28	a	a	DET
ejpam-6037	218	29	σ1σ2	σ1σ2	NUM
ejpam-6037	218	30	-	-	PUNCT
ejpam-6037	218	31	urysohn	urysohn	ADJ
ejpam-6037	218	32	space	space	NOUN
ejpam-6037	218	33	(	(	PUNCT
ejpam-6037	218	34	y	y	PROPN
ejpam-6037	218	35	,	,	PUNCT
ejpam-6037	218	36	σ1	σ1	PROPN
ejpam-6037	218	37	,	,	PUNCT
ejpam-6037	218	38	σ2	σ2	NOUN
ejpam-6037	218	39	)	)	PUNCT
ejpam-6037	218	40	such	such	ADJ
ejpam-6037	218	41	that	that	SCONJ
ejpam-6037	218	42	f(x	f(x	PROPN
ejpam-6037	218	43	)	)	PUNCT
ejpam-6037	218	44	̸=	̸=	PROPN
ejpam-6037	218	45	f(x′	f(x′	NUM
ejpam-6037	218	46	)	)	PUNCT
ejpam-6037	218	47	and	and	CCONJ
ejpam-6037	218	48	f	f	PROPN
ejpam-6037	218	49	is	be	AUX
ejpam-6037	218	50	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	218	51	,	,	PUNCT
ejpam-6037	218	52	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	218	53	at	at	ADP
ejpam-6037	218	54	x	x	PROPN
ejpam-6037	218	55	and	and	CCONJ
ejpam-6037	218	56	x′	x′	NUM
ejpam-6037	218	57	,	,	PUNCT
ejpam-6037	218	58	then	then	ADV
ejpam-6037	218	59	(	(	PUNCT
ejpam-6037	218	60	x	x	NOUN
ejpam-6037	218	61	,	,	PUNCT
ejpam-6037	218	62	τ1	τ1	NOUN
ejpam-6037	218	63	,	,	PUNCT
ejpam-6037	218	64	τ2	τ2	NOUN
ejpam-6037	218	65	)	)	PUNCT
ejpam-6037	218	66	is	be	AUX
ejpam-6037	218	67	(	(	PUNCT
ejpam-6037	218	68	τ1	τ1	NOUN
ejpam-6037	218	69	,	,	PUNCT
ejpam-6037	218	70	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	218	71	.	.	PUNCT
ejpam-6037	219	1	proof	proof	NOUN
ejpam-6037	219	2	.	.	PUNCT
ejpam-6037	220	1	let	let	VERB
ejpam-6037	220	2	x	x	PUNCT
ejpam-6037	220	3	and	and	CCONJ
ejpam-6037	220	4	x′	x′	PROPN
ejpam-6037	220	5	be	be	AUX
ejpam-6037	220	6	any	any	DET
ejpam-6037	220	7	distinct	distinct	ADJ
ejpam-6037	220	8	points	point	NOUN
ejpam-6037	220	9	of	of	ADP
ejpam-6037	220	10	x.	x.	NOUN
ejpam-6037	220	11	then	then	ADV
ejpam-6037	220	12	by	by	ADP
ejpam-6037	220	13	the	the	DET
ejpam-6037	220	14	hypothesis	hypothesis	NOUN
ejpam-6037	220	15	,	,	PUNCT
ejpam-6037	220	16	there	there	PRON
ejpam-6037	220	17	exists	exist	VERB
ejpam-6037	220	18	a	a	DET
ejpam-6037	220	19	σ1σ2	σ1σ2	NUM
ejpam-6037	220	20	-	-	PUNCT
ejpam-6037	220	21	urysohn	urysohn	ADJ
ejpam-6037	220	22	space	space	NOUN
ejpam-6037	220	23	(	(	PUNCT
ejpam-6037	220	24	y	y	PROPN
ejpam-6037	220	25	,	,	PUNCT
ejpam-6037	220	26	σ1	σ1	PROPN
ejpam-6037	220	27	,	,	PUNCT
ejpam-6037	220	28	σ2	σ2	NOUN
ejpam-6037	220	29	)	)	PUNCT
ejpam-6037	220	30	and	and	CCONJ
ejpam-6037	220	31	a	a	DET
ejpam-6037	220	32	function	function	NOUN
ejpam-6037	220	33	f	f	NOUN
ejpam-6037	220	34	:	:	PUNCT
ejpam-6037	220	35	(	(	PUNCT
ejpam-6037	220	36	x	x	NOUN
ejpam-6037	220	37	,	,	PUNCT
ejpam-6037	220	38	τ1	τ1	NOUN
ejpam-6037	220	39	,	,	PUNCT
ejpam-6037	220	40	τ2	τ2	NOUN
ejpam-6037	220	41	)	)	PUNCT
ejpam-6037	220	42	→	→	SYM
ejpam-6037	220	43	(	(	PUNCT
ejpam-6037	220	44	y	y	PROPN
ejpam-6037	220	45	,	,	PUNCT
ejpam-6037	220	46	σ1	σ1	PROPN
ejpam-6037	220	47	,	,	PUNCT
ejpam-6037	220	48	σ2	σ2	PROPN
ejpam-6037	220	49	)	)	PUNCT
ejpam-6037	220	50	which	which	PRON
ejpam-6037	220	51	satisfies	satisfy	VERB
ejpam-6037	220	52	the	the	DET
ejpam-6037	220	53	conditions	condition	NOUN
ejpam-6037	220	54	of	of	ADP
ejpam-6037	220	55	this	this	DET
ejpam-6037	220	56	theorem	theorem	NOUN
ejpam-6037	220	57	.	.	PUNCT
ejpam-6037	221	1	let	let	VERB
ejpam-6037	221	2	y	y	PROPN
ejpam-6037	221	3	=	=	PUNCT
ejpam-6037	221	4	f(x	f(x	PROPN
ejpam-6037	221	5	)	)	PUNCT
ejpam-6037	221	6	and	and	CCONJ
ejpam-6037	221	7	y′	y′	NOUN
ejpam-6037	221	8	=	=	SYM
ejpam-6037	221	9	f(x′	f(x′	PROPN
ejpam-6037	221	10	)	)	PUNCT
ejpam-6037	221	11	.	.	PUNCT
ejpam-6037	222	1	then	then	ADV
ejpam-6037	222	2	,	,	PUNCT
ejpam-6037	222	3	y	y	PROPN
ejpam-6037	222	4	̸=	̸=	PROPN
ejpam-6037	222	5	y′.	y′.	VERB
ejpam-6037	222	6	since	since	SCONJ
ejpam-6037	222	7	(	(	PUNCT
ejpam-6037	222	8	y	y	PROPN
ejpam-6037	222	9	,	,	PUNCT
ejpam-6037	222	10	σ1	σ1	PROPN
ejpam-6037	222	11	,	,	PUNCT
ejpam-6037	222	12	σ2	σ2	PROPN
ejpam-6037	222	13	)	)	PUNCT
ejpam-6037	222	14	is	be	AUX
ejpam-6037	222	15	σ1σ2	σ1σ2	NOUN
ejpam-6037	222	16	-	-	PUNCT
ejpam-6037	222	17	urysohn	urysohn	ADJ
ejpam-6037	222	18	,	,	PUNCT
ejpam-6037	222	19	there	there	PRON
ejpam-6037	222	20	exist	exist	VERB
ejpam-6037	222	21	σ1σ2	σ1σ2	NOUN
ejpam-6037	222	22	-	-	ADJ
ejpam-6037	222	23	open	open	ADJ
ejpam-6037	222	24	sets	set	NOUN
ejpam-6037	222	25	v	v	ADP
ejpam-6037	222	26	and	and	CCONJ
ejpam-6037	222	27	w	w	PROPN
ejpam-6037	222	28	of	of	ADP
ejpam-6037	222	29	y	y	PROPN
ejpam-6037	222	30	containing	contain	VERB
ejpam-6037	222	31	y	y	PROPN
ejpam-6037	222	32	and	and	CCONJ
ejpam-6037	222	33	y′	y′	NUM
ejpam-6037	222	34	,	,	PUNCT
ejpam-6037	222	35	respectively	respectively	ADV
ejpam-6037	222	36	,	,	PUNCT
ejpam-6037	222	37	such	such	ADJ
ejpam-6037	222	38	that	that	SCONJ
ejpam-6037	222	39	σ1σ2	σ1σ2	NOUN
ejpam-6037	222	40	-	-	NUM
ejpam-6037	222	41	cl(v	cl(v	NOUN
ejpam-6037	222	42	)	)	PUNCT
ejpam-6037	222	43	∩σ1σ2	∩σ1σ2	NOUN
ejpam-6037	222	44	-	-	NUM
ejpam-6037	222	45	cl(w	cl(w	NOUN
ejpam-6037	222	46	)	)	PUNCT
ejpam-6037	223	1	=	=	PUNCT
ejpam-6037	223	2	∅.	∅.	NOUN
ejpam-6037	223	3	since	since	SCONJ
ejpam-6037	223	4	f	f	PROPN
ejpam-6037	223	5	is	be	AUX
ejpam-6037	223	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	223	7	,	,	PUNCT
ejpam-6037	223	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	223	9	at	at	ADP
ejpam-6037	223	10	x	x	PROPN
ejpam-6037	223	11	and	and	CCONJ
ejpam-6037	223	12	x′	x′	NUM
ejpam-6037	223	13	,	,	PUNCT
ejpam-6037	223	14	by	by	ADP
ejpam-6037	223	15	theorem	theorem	NOUN
ejpam-6037	223	16	1	1	NUM
ejpam-6037	223	17	there	there	PRON
ejpam-6037	223	18	exist	exist	VERB
ejpam-6037	223	19	τ1τ2	τ1τ2	ADJ
ejpam-6037	223	20	-	-	ADJ
ejpam-6037	223	21	open	open	ADJ
ejpam-6037	223	22	sets	set	NOUN
ejpam-6037	223	23	u	u	NOUN
ejpam-6037	223	24	and	and	CCONJ
ejpam-6037	223	25	u	u	NOUN
ejpam-6037	223	26	′	′	NOUN
ejpam-6037	223	27	of	of	ADP
ejpam-6037	223	28	x	x	PUNCT
ejpam-6037	223	29	containing	contain	VERB
ejpam-6037	223	30	x	x	PROPN
ejpam-6037	223	31	and	and	CCONJ
ejpam-6037	223	32	x′	x′	NUM
ejpam-6037	223	33	,	,	PUNCT
ejpam-6037	223	34	respectively	respectively	ADV
ejpam-6037	223	35	,	,	PUNCT
ejpam-6037	223	36	such	such	ADJ
ejpam-6037	223	37	that	that	DET
ejpam-6037	223	38	f(u	f(u	PROPN
ejpam-6037	223	39	)	)	PUNCT
ejpam-6037	223	40	⊆	⊆	NUM
ejpam-6037	223	41	σ1σ2	σ1σ2	NOUN
ejpam-6037	223	42	-	-	NUM
ejpam-6037	223	43	cl(v	cl(v	NOUN
ejpam-6037	223	44	)	)	PUNCT
ejpam-6037	223	45	and	and	CCONJ
ejpam-6037	223	46	f(u	f(u	PROPN
ejpam-6037	223	47	′	′	NUM
ejpam-6037	223	48	)	)	PUNCT
ejpam-6037	223	49	⊆	⊆	NUM
ejpam-6037	223	50	σ1σ2	σ1σ2	NOUN
ejpam-6037	223	51	-	-	PUNCT
ejpam-6037	223	52	cl(w	cl(w	NOUN
ejpam-6037	223	53	)	)	PUNCT
ejpam-6037	223	54	.	.	PUNCT
ejpam-6037	224	1	this	this	PRON
ejpam-6037	224	2	implies	imply	VERB
ejpam-6037	224	3	that	that	SCONJ
ejpam-6037	224	4	u	u	PROPN
ejpam-6037	224	5	∩	∩	NOUN
ejpam-6037	224	6	u	u	NOUN
ejpam-6037	224	7	′	′	NOUN
ejpam-6037	224	8	=	=	PUNCT
ejpam-6037	224	9	∅.	∅.	VERB
ejpam-6037	224	10	thus	thus	ADV
ejpam-6037	224	11	,	,	PUNCT
ejpam-6037	224	12	(	(	PUNCT
ejpam-6037	224	13	x	x	NOUN
ejpam-6037	224	14	,	,	PUNCT
ejpam-6037	224	15	τ1	τ1	NOUN
ejpam-6037	224	16	,	,	PUNCT
ejpam-6037	224	17	τ2	τ2	NOUN
ejpam-6037	224	18	)	)	PUNCT
ejpam-6037	224	19	is	be	AUX
ejpam-6037	224	20	(	(	PUNCT
ejpam-6037	224	21	τ1	τ1	NOUN
ejpam-6037	224	22	,	,	PUNCT
ejpam-6037	224	23	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	224	24	.	.	PUNCT
ejpam-6037	225	1	corollary	corollary	ADJ
ejpam-6037	225	2	1	1	NUM
ejpam-6037	225	3	.	.	PUNCT
ejpam-6037	226	1	if	if	SCONJ
ejpam-6037	226	2	f	f	PROPN
ejpam-6037	226	3	:	:	PUNCT
ejpam-6037	226	4	(	(	PUNCT
ejpam-6037	226	5	x	x	NOUN
ejpam-6037	226	6	,	,	PUNCT
ejpam-6037	226	7	τ1	τ1	NOUN
ejpam-6037	226	8	,	,	PUNCT
ejpam-6037	226	9	τ2	τ2	NOUN
ejpam-6037	226	10	)	)	PUNCT
ejpam-6037	226	11	→	→	SYM
ejpam-6037	226	12	(	(	PUNCT
ejpam-6037	226	13	y	y	PROPN
ejpam-6037	226	14	,	,	PUNCT
ejpam-6037	226	15	σ1	σ1	PROPN
ejpam-6037	226	16	,	,	PUNCT
ejpam-6037	226	17	σ2	σ2	PROPN
ejpam-6037	226	18	)	)	PUNCT
ejpam-6037	226	19	is	be	AUX
ejpam-6037	226	20	a	a	DET
ejpam-6037	226	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	226	22	,	,	PUNCT
ejpam-6037	226	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	226	24	injection	injection	NOUN
ejpam-6037	226	25	and	and	CCONJ
ejpam-6037	226	26	(	(	PUNCT
ejpam-6037	226	27	y	y	PROPN
ejpam-6037	226	28	,	,	PUNCT
ejpam-6037	226	29	σ1	σ1	PROPN
ejpam-6037	226	30	,	,	PUNCT
ejpam-6037	226	31	σ2	σ2	PROPN
ejpam-6037	226	32	)	)	PUNCT
ejpam-6037	226	33	is	be	AUX
ejpam-6037	226	34	σ1σ2	σ1σ2	NOUN
ejpam-6037	226	35	-	-	PUNCT
ejpam-6037	226	36	urysohn	urysohn	ADJ
ejpam-6037	226	37	,	,	PUNCT
ejpam-6037	226	38	then	then	ADV
ejpam-6037	226	39	(	(	PUNCT
ejpam-6037	226	40	x	x	NOUN
ejpam-6037	226	41	,	,	PUNCT
ejpam-6037	226	42	τ1	τ1	NOUN
ejpam-6037	226	43	,	,	PUNCT
ejpam-6037	226	44	τ2	τ2	NOUN
ejpam-6037	226	45	)	)	PUNCT
ejpam-6037	226	46	is	be	AUX
ejpam-6037	226	47	(	(	PUNCT
ejpam-6037	226	48	τ1	τ1	NOUN
ejpam-6037	226	49	,	,	PUNCT
ejpam-6037	226	50	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	226	51	.	.	PUNCT
ejpam-6037	227	1	proof	proof	NOUN
ejpam-6037	227	2	.	.	PUNCT
ejpam-6037	228	1	for	for	ADP
ejpam-6037	228	2	each	each	DET
ejpam-6037	228	3	pair	pair	NOUN
ejpam-6037	228	4	of	of	ADP
ejpam-6037	228	5	distinct	distinct	ADJ
ejpam-6037	228	6	points	point	NOUN
ejpam-6037	228	7	x	x	PUNCT
ejpam-6037	228	8	and	and	CCONJ
ejpam-6037	228	9	x′	x′	PROPN
ejpam-6037	228	10	in	in	ADP
ejpam-6037	228	11	x	x	PRON
ejpam-6037	228	12	,	,	PUNCT
ejpam-6037	228	13	f	f	PROPN
ejpam-6037	228	14	is	be	AUX
ejpam-6037	228	15	a	a	DET
ejpam-6037	228	16	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	228	17	,	,	PUNCT
ejpam-6037	228	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	228	19	function	function	NOUN
ejpam-6037	228	20	of	of	ADP
ejpam-6037	228	21	(	(	PUNCT
ejpam-6037	228	22	x	x	NOUN
ejpam-6037	228	23	,	,	PUNCT
ejpam-6037	228	24	τ1	τ1	NOUN
ejpam-6037	228	25	,	,	PUNCT
ejpam-6037	228	26	τ2	τ2	NOUN
ejpam-6037	228	27	)	)	PUNCT
ejpam-6037	228	28	into	into	ADP
ejpam-6037	228	29	a	a	DET
ejpam-6037	228	30	σ1σ2	σ1σ2	NUM
ejpam-6037	228	31	-	-	PUNCT
ejpam-6037	228	32	urysohn	urysohn	ADJ
ejpam-6037	228	33	space	space	NOUN
ejpam-6037	228	34	(	(	PUNCT
ejpam-6037	228	35	y	y	PROPN
ejpam-6037	228	36	,	,	PUNCT
ejpam-6037	228	37	σ1	σ1	PROPN
ejpam-6037	228	38	,	,	PUNCT
ejpam-6037	228	39	σ2	σ2	NOUN
ejpam-6037	228	40	)	)	PUNCT
ejpam-6037	228	41	such	such	ADJ
ejpam-6037	228	42	that	that	SCONJ
ejpam-6037	228	43	f(x	f(x	PROPN
ejpam-6037	228	44	)	)	PUNCT
ejpam-6037	228	45	̸=	̸=	PROPN
ejpam-6037	228	46	f(x′	f(x′	NUM
ejpam-6037	228	47	)	)	PUNCT
ejpam-6037	228	48	because	because	SCONJ
ejpam-6037	228	49	f	f	PROPN
ejpam-6037	228	50	is	be	AUX
ejpam-6037	228	51	injective	injective	ADJ
ejpam-6037	228	52	.	.	PUNCT
ejpam-6037	229	1	thus	thus	ADV
ejpam-6037	229	2	by	by	ADP
ejpam-6037	229	3	theorem	theorem	NOUN
ejpam-6037	229	4	10	10	NUM
ejpam-6037	229	5	,	,	PUNCT
ejpam-6037	229	6	(	(	PUNCT
ejpam-6037	229	7	x	x	NOUN
ejpam-6037	229	8	,	,	PUNCT
ejpam-6037	229	9	τ1	τ1	NOUN
ejpam-6037	229	10	,	,	PUNCT
ejpam-6037	229	11	τ2	τ2	NOUN
ejpam-6037	229	12	)	)	PUNCT
ejpam-6037	229	13	is	be	AUX
ejpam-6037	229	14	(	(	PUNCT
ejpam-6037	229	15	τ1	τ1	NOUN
ejpam-6037	229	16	,	,	PUNCT
ejpam-6037	229	17	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	229	18	.	.	PUNCT
ejpam-6037	230	1	definition	definition	NOUN
ejpam-6037	230	2	10	10	NUM
ejpam-6037	230	3	.	.	PUNCT
ejpam-6037	231	1	a	a	DET
ejpam-6037	231	2	bitopological	bitopological	ADJ
ejpam-6037	231	3	space	space	NOUN
ejpam-6037	231	4	(	(	PUNCT
ejpam-6037	231	5	x	x	NOUN
ejpam-6037	231	6	,	,	PUNCT
ejpam-6037	231	7	τ1	τ1	NOUN
ejpam-6037	231	8	,	,	PUNCT
ejpam-6037	231	9	τ2	τ2	NOUN
ejpam-6037	231	10	)	)	PUNCT
ejpam-6037	231	11	is	be	AUX
ejpam-6037	231	12	said	say	VERB
ejpam-6037	231	13	to	to	PART
ejpam-6037	231	14	be	be	AUX
ejpam-6037	231	15	ultra	ultra	ADJ
ejpam-6037	231	16	-	-	ADJ
ejpam-6037	231	17	τ1τ2	τ1τ2	ADJ
ejpam-6037	231	18	-	-	ADJ
ejpam-6037	231	19	huasdorff	huasdorff	ADJ
ejpam-6037	231	20	if	if	SCONJ
ejpam-6037	231	21	for	for	ADP
ejpam-6037	231	22	each	each	DET
ejpam-6037	231	23	pair	pair	NOUN
ejpam-6037	231	24	of	of	ADP
ejpam-6037	231	25	distinct	distinct	ADJ
ejpam-6037	231	26	points	point	NOUN
ejpam-6037	231	27	x	x	PUNCT
ejpam-6037	231	28	and	and	CCONJ
ejpam-6037	231	29	y	y	PROPN
ejpam-6037	231	30	in	in	ADP
ejpam-6037	231	31	x	x	SYM
ejpam-6037	231	32	,	,	PUNCT
ejpam-6037	231	33	there	there	PRON
ejpam-6037	231	34	exist	exist	VERB
ejpam-6037	231	35	τ1τ2	τ1τ2	ADJ
ejpam-6037	231	36	-	-	ADJ
ejpam-6037	231	37	clopen	clopen	ADJ
ejpam-6037	231	38	sets	set	NOUN
ejpam-6037	231	39	u	u	NOUN
ejpam-6037	231	40	and	and	CCONJ
ejpam-6037	231	41	v	v	NOUN
ejpam-6037	231	42	of	of	ADP
ejpam-6037	231	43	x	x	PUNCT
ejpam-6037	231	44	containing	contain	VERB
ejpam-6037	231	45	x	x	PROPN
ejpam-6037	231	46	and	and	CCONJ
ejpam-6037	231	47	y	y	PROPN
ejpam-6037	231	48	,	,	PUNCT
ejpam-6037	231	49	respectively	respectively	ADV
ejpam-6037	231	50	,	,	PUNCT
ejpam-6037	231	51	such	such	ADJ
ejpam-6037	231	52	that	that	SCONJ
ejpam-6037	231	53	u	u	PROPN
ejpam-6037	231	54	∩	∩	NOUN
ejpam-6037	231	55	v	v	NOUN
ejpam-6037	231	56	=	=	PUNCT
ejpam-6037	231	57	∅.	∅.	NOUN
ejpam-6037	231	58	theorem	theorem	VERB
ejpam-6037	231	59	11	11	NUM
ejpam-6037	231	60	.	.	PUNCT
ejpam-6037	232	1	if	if	SCONJ
ejpam-6037	232	2	f	f	PROPN
ejpam-6037	232	3	:	:	PUNCT
ejpam-6037	232	4	(	(	PUNCT
ejpam-6037	232	5	x	x	NOUN
ejpam-6037	232	6	,	,	PUNCT
ejpam-6037	232	7	τ1	τ1	NOUN
ejpam-6037	232	8	,	,	PUNCT
ejpam-6037	232	9	τ2	τ2	NOUN
ejpam-6037	232	10	)	)	PUNCT
ejpam-6037	232	11	→	→	SYM
ejpam-6037	232	12	(	(	PUNCT
ejpam-6037	232	13	y	y	PROPN
ejpam-6037	232	14	,	,	PUNCT
ejpam-6037	232	15	σ1	σ1	PROPN
ejpam-6037	232	16	,	,	PUNCT
ejpam-6037	232	17	σ2	σ2	PROPN
ejpam-6037	232	18	)	)	PUNCT
ejpam-6037	232	19	is	be	AUX
ejpam-6037	232	20	a	a	DET
ejpam-6037	232	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	232	22	,	,	PUNCT
ejpam-6037	232	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	232	24	injection	injection	NOUN
ejpam-6037	232	25	and	and	CCONJ
ejpam-6037	232	26	(	(	PUNCT
ejpam-6037	232	27	y	y	PROPN
ejpam-6037	232	28	,	,	PUNCT
ejpam-6037	232	29	σ1	σ1	PROPN
ejpam-6037	232	30	,	,	PUNCT
ejpam-6037	232	31	σ2	σ2	PROPN
ejpam-6037	232	32	)	)	PUNCT
ejpam-6037	232	33	is	be	AUX
ejpam-6037	232	34	ultra	ultra	ADJ
ejpam-6037	232	35	-	-	ADJ
ejpam-6037	232	36	σ1σ2	σ1σ2	NOUN
ejpam-6037	232	37	-	-	PUNCT
ejpam-6037	232	38	hausdorff	hausdorff	NOUN
ejpam-6037	232	39	,	,	PUNCT
ejpam-6037	232	40	then	then	ADV
ejpam-6037	232	41	(	(	PUNCT
ejpam-6037	232	42	x	x	NOUN
ejpam-6037	232	43	,	,	PUNCT
ejpam-6037	232	44	τ1	τ1	NOUN
ejpam-6037	232	45	,	,	PUNCT
ejpam-6037	232	46	τ2	τ2	NOUN
ejpam-6037	232	47	)	)	PUNCT
ejpam-6037	232	48	is	be	AUX
ejpam-6037	232	49	(	(	PUNCT
ejpam-6037	232	50	τ1	τ1	NOUN
ejpam-6037	232	51	,	,	PUNCT
ejpam-6037	232	52	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	232	53	.	.	PUNCT
ejpam-6037	233	1	proof	proof	NOUN
ejpam-6037	233	2	.	.	PUNCT
ejpam-6037	234	1	let	let	VERB
ejpam-6037	234	2	x	x	PRON
ejpam-6037	234	3	and	and	CCONJ
ejpam-6037	234	4	y	y	PROPN
ejpam-6037	234	5	be	be	AUX
ejpam-6037	234	6	any	any	DET
ejpam-6037	234	7	distinct	distinct	ADJ
ejpam-6037	234	8	points	point	NOUN
ejpam-6037	234	9	in	in	ADP
ejpam-6037	234	10	x.	x.	NOUN
ejpam-6037	234	11	then	then	ADV
ejpam-6037	234	12	,	,	PUNCT
ejpam-6037	234	13	since	since	SCONJ
ejpam-6037	234	14	f	f	PROPN
ejpam-6037	234	15	is	be	AUX
ejpam-6037	234	16	injective	injective	ADJ
ejpam-6037	234	17	,	,	PUNCT
ejpam-6037	234	18	f(x	f(x	PROPN
ejpam-6037	234	19	)	)	PUNCT
ejpam-6037	234	20	̸=	̸=	PROPN
ejpam-6037	234	21	f(y	f(y	NOUN
ejpam-6037	234	22	)	)	PUNCT
ejpam-6037	234	23	.	.	PUNCT
ejpam-6037	235	1	moreover	moreover	ADV
ejpam-6037	235	2	,	,	PUNCT
ejpam-6037	235	3	since	since	SCONJ
ejpam-6037	235	4	(	(	PUNCT
ejpam-6037	235	5	y	y	PROPN
ejpam-6037	235	6	,	,	PUNCT
ejpam-6037	235	7	σ1	σ1	PROPN
ejpam-6037	235	8	,	,	PUNCT
ejpam-6037	235	9	σ2	σ2	PROPN
ejpam-6037	235	10	)	)	PUNCT
ejpam-6037	235	11	is	be	AUX
ejpam-6037	235	12	ultra	ultra	ADJ
ejpam-6037	235	13	-	-	ADJ
ejpam-6037	235	14	σ1σ2	σ1σ2	NOUN
ejpam-6037	235	15	-	-	PUNCT
ejpam-6037	235	16	hausdorff	hausdorff	NOUN
ejpam-6037	235	17	,	,	PUNCT
ejpam-6037	235	18	there	there	PRON
ejpam-6037	235	19	exist	exist	VERB
ejpam-6037	235	20	σ1σ2	σ1σ2	NOUN
ejpam-6037	235	21	-	-	PUNCT
ejpam-6037	235	22	clopen	clopen	ADJ
ejpam-6037	235	23	sets	set	NOUN
ejpam-6037	235	24	v	v	ADP
ejpam-6037	235	25	and	and	CCONJ
ejpam-6037	235	26	w	w	PROPN
ejpam-6037	235	27	of	of	ADP
ejpam-6037	235	28	y	y	PROPN
ejpam-6037	235	29	containing	contain	VERB
ejpam-6037	235	30	x	x	PROPN
ejpam-6037	235	31	and	and	CCONJ
ejpam-6037	235	32	y	y	PROPN
ejpam-6037	235	33	,	,	PUNCT
ejpam-6037	235	34	respectively	respectively	ADV
ejpam-6037	235	35	,	,	PUNCT
ejpam-6037	236	1	such	such	ADJ
ejpam-6037	236	2	that	that	PRON
ejpam-6037	236	3	v	v	ADP
ejpam-6037	236	4	∩	∩	NOUN
ejpam-6037	236	5	w	w	NOUN
ejpam-6037	236	6	=	=	PUNCT
ejpam-6037	236	7	∅.	∅.	NOUN
ejpam-6037	236	8	since	since	SCONJ
ejpam-6037	236	9	f	f	PROPN
ejpam-6037	236	10	is	be	AUX
ejpam-6037	236	11	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	236	12	,	,	PUNCT
ejpam-6037	236	13	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	236	14	,	,	PUNCT
ejpam-6037	236	15	by	by	ADP
ejpam-6037	236	16	theorem	theorem	NOUN
ejpam-6037	236	17	1	1	NUM
ejpam-6037	236	18	there	there	PRON
ejpam-6037	236	19	exist	exist	VERB
ejpam-6037	236	20	τ1τ2	τ1τ2	ADJ
ejpam-6037	236	21	-	-	ADJ
ejpam-6037	236	22	open	open	ADJ
ejpam-6037	236	23	sets	set	NOUN
ejpam-6037	236	24	u	u	NOUN
ejpam-6037	236	25	and	and	CCONJ
ejpam-6037	236	26	g	g	PROPN
ejpam-6037	236	27	of	of	ADP
ejpam-6037	236	28	x	x	PUNCT
ejpam-6037	236	29	containing	contain	VERB
ejpam-6037	236	30	x	x	PROPN
ejpam-6037	236	31	and	and	CCONJ
ejpam-6037	236	32	y	y	PROPN
ejpam-6037	236	33	,	,	PUNCT
ejpam-6037	236	34	n.	n.	NOUN
ejpam-6037	236	35	chutiman	chutiman	NOUN
ejpam-6037	236	36	,	,	PUNCT
ejpam-6037	236	37	a.	a.	PROPN
ejpam-6037	236	38	sama	sama	PROPN
ejpam-6037	236	39	-	-	PUNCT
ejpam-6037	236	40	ae	ae	PROPN
ejpam-6037	236	41	,	,	PUNCT
ejpam-6037	236	42	c.	c.	PROPN
ejpam-6037	236	43	boonpok	boonpok	PROPN
ejpam-6037	236	44	/	/	SYM
ejpam-6037	236	45	eur	eur	PROPN
ejpam-6037	236	46	.	.	PUNCT
ejpam-6037	237	1	j.	j.	PROPN
ejpam-6037	237	2	pure	pure	PROPN
ejpam-6037	237	3	appl	appl	PROPN
ejpam-6037	237	4	.	.	PROPN
ejpam-6037	237	5	math	math	PROPN
ejpam-6037	237	6	,	,	PUNCT
ejpam-6037	237	7	18	18	NUM
ejpam-6037	237	8	(	(	PUNCT
ejpam-6037	237	9	2	2	NUM
ejpam-6037	237	10	)	)	PUNCT
ejpam-6037	237	11	(	(	PUNCT
ejpam-6037	237	12	2025	2025	NUM
ejpam-6037	237	13	)	)	PUNCT
ejpam-6037	237	14	,	,	PUNCT
ejpam-6037	237	15	6037	6037	NUM
ejpam-6037	237	16	9	9	NUM
ejpam-6037	237	17	of	of	ADP
ejpam-6037	237	18	12	12	NUM
ejpam-6037	237	19	respectively	respectively	ADV
ejpam-6037	237	20	,	,	PUNCT
ejpam-6037	237	21	such	such	ADJ
ejpam-6037	237	22	that	that	DET
ejpam-6037	237	23	f(u	f(u	PROPN
ejpam-6037	237	24	)	)	PUNCT
ejpam-6037	237	25	⊆	⊆	NUM
ejpam-6037	237	26	v	v	NOUN
ejpam-6037	237	27	and	and	CCONJ
ejpam-6037	237	28	f(g	f(g	NOUN
ejpam-6037	237	29	)	)	PUNCT
ejpam-6037	237	30	⊆	⊆	NUM
ejpam-6037	237	31	w	w	NOUN
ejpam-6037	237	32	.	.	PUNCT
ejpam-6037	238	1	thus	thus	ADV
ejpam-6037	238	2	,	,	PUNCT
ejpam-6037	238	3	u	u	NOUN
ejpam-6037	238	4	∩g	∩g	NOUN
ejpam-6037	238	5	=	=	PUNCT
ejpam-6037	238	6	∅	∅	NOUN
ejpam-6037	238	7	and	and	CCONJ
ejpam-6037	238	8	hence	hence	ADV
ejpam-6037	238	9	(	(	PUNCT
ejpam-6037	238	10	x	x	NOUN
ejpam-6037	238	11	,	,	PUNCT
ejpam-6037	238	12	τ1	τ1	NOUN
ejpam-6037	238	13	,	,	PUNCT
ejpam-6037	238	14	τ2	τ2	NOUN
ejpam-6037	238	15	)	)	PUNCT
ejpam-6037	238	16	is	be	AUX
ejpam-6037	238	17	(	(	PUNCT
ejpam-6037	238	18	τ1	τ1	NOUN
ejpam-6037	238	19	,	,	PUNCT
ejpam-6037	238	20	τ2)-t2	τ2)-t2	PROPN
ejpam-6037	238	21	.	.	PUNCT
ejpam-6037	239	1	recall	recall	VERB
ejpam-6037	239	2	that	that	SCONJ
ejpam-6037	239	3	a	a	DET
ejpam-6037	239	4	bitopological	bitopological	ADJ
ejpam-6037	239	5	space	space	NOUN
ejpam-6037	239	6	(	(	PUNCT
ejpam-6037	239	7	x	x	NOUN
ejpam-6037	239	8	,	,	PUNCT
ejpam-6037	239	9	τ1	τ1	NOUN
ejpam-6037	239	10	,	,	PUNCT
ejpam-6037	239	11	τ2	τ2	NOUN
ejpam-6037	239	12	)	)	PUNCT
ejpam-6037	239	13	is	be	AUX
ejpam-6037	239	14	said	say	VERB
ejpam-6037	239	15	to	to	PART
ejpam-6037	239	16	be	be	AUX
ejpam-6037	239	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	239	18	-	-	ADJ
ejpam-6037	239	19	compact	compact	ADJ
ejpam-6037	240	1	[	[	X
ejpam-6037	240	2	38	38	NUM
ejpam-6037	240	3	]	]	PUNCT
ejpam-6037	240	4	if	if	SCONJ
ejpam-6037	240	5	every	every	DET
ejpam-6037	240	6	cover	cover	NOUN
ejpam-6037	240	7	of	of	ADP
ejpam-6037	240	8	x	x	PUNCT
ejpam-6037	240	9	by	by	ADP
ejpam-6037	240	10	τ1τ2	τ1τ2	ADJ
ejpam-6037	240	11	-	-	ADJ
ejpam-6037	240	12	open	open	ADJ
ejpam-6037	240	13	sets	set	NOUN
ejpam-6037	240	14	of	of	ADP
ejpam-6037	240	15	x	x	PUNCT
ejpam-6037	240	16	has	have	VERB
ejpam-6037	240	17	a	a	DET
ejpam-6037	240	18	finite	finite	ADJ
ejpam-6037	240	19	subcover	subcover	PROPN
ejpam-6037	240	20	.	.	PUNCT
ejpam-6037	241	1	a	a	DET
ejpam-6037	241	2	bitopological	bitopological	ADJ
ejpam-6037	241	3	space	space	NOUN
ejpam-6037	241	4	(	(	PUNCT
ejpam-6037	241	5	x	x	NOUN
ejpam-6037	241	6	,	,	PUNCT
ejpam-6037	241	7	τ1	τ1	NOUN
ejpam-6037	241	8	,	,	PUNCT
ejpam-6037	241	9	τ2	τ2	NOUN
ejpam-6037	241	10	)	)	PUNCT
ejpam-6037	241	11	is	be	AUX
ejpam-6037	241	12	said	say	VERB
ejpam-6037	241	13	to	to	PART
ejpam-6037	241	14	be	be	AUX
ejpam-6037	241	15	quasi	quasi	X
ejpam-6037	241	16	(	(	PUNCT
ejpam-6037	241	17	τ1	τ1	NOUN
ejpam-6037	241	18	,	,	PUNCT
ejpam-6037	241	19	τ2)-h	τ2)-h	PUNCT
ejpam-6037	241	20	-closed	-closed	ADJ
ejpam-6037	241	21	[	[	X
ejpam-6037	241	22	47	47	NUM
ejpam-6037	241	23	]	]	PUNCT
ejpam-6037	241	24	if	if	SCONJ
ejpam-6037	241	25	every	every	DET
ejpam-6037	241	26	τ1τ2	τ1τ2	ADJ
ejpam-6037	241	27	-	-	ADJ
ejpam-6037	241	28	open	open	ADJ
ejpam-6037	241	29	cover	cover	NOUN
ejpam-6037	241	30	{	{	PUNCT
ejpam-6037	241	31	uγ	uγ	ADV
ejpam-6037	241	32	|	|	ADV
ejpam-6037	241	33	γ	γ	X
ejpam-6037	241	34	∈	∈	NOUN
ejpam-6037	241	35	∇	∇	X
ejpam-6037	241	36	}	}	PUNCT
ejpam-6037	241	37	,	,	PUNCT
ejpam-6037	241	38	there	there	PRON
ejpam-6037	241	39	exists	exist	VERB
ejpam-6037	241	40	a	a	DET
ejpam-6037	241	41	finite	finite	NOUN
ejpam-6037	241	42	subset	subset	NOUN
ejpam-6037	241	43	∇0	∇0	NUM
ejpam-6037	241	44	of	of	ADP
ejpam-6037	241	45	∇	∇	NOUN
ejpam-6037	241	46	such	such	ADJ
ejpam-6037	241	47	that	that	SCONJ
ejpam-6037	241	48	x	x	X
ejpam-6037	241	49	=	=	PUNCT
ejpam-6037	241	50	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-6037	241	51	-	-	NOUN
ejpam-6037	241	52	cl(uγ	cl(uγ	NOUN
ejpam-6037	241	53	)	)	PUNCT
ejpam-6037	241	54	|	|	ADV
ejpam-6037	241	55	γ	γ	X
ejpam-6037	241	56	∈	∈	NOUN
ejpam-6037	241	57	∇0	∇0	NOUN
ejpam-6037	241	58	}	}	PUNCT
ejpam-6037	241	59	.	.	PUNCT
ejpam-6037	242	1	definition	definition	NOUN
ejpam-6037	242	2	11	11	NUM
ejpam-6037	242	3	.	.	PUNCT
ejpam-6037	243	1	a	a	DET
ejpam-6037	243	2	bitopological	bitopological	ADJ
ejpam-6037	243	3	space	space	NOUN
ejpam-6037	243	4	(	(	PUNCT
ejpam-6037	243	5	x	x	NOUN
ejpam-6037	243	6	,	,	PUNCT
ejpam-6037	243	7	τ1	τ1	NOUN
ejpam-6037	243	8	,	,	PUNCT
ejpam-6037	243	9	τ2	τ2	NOUN
ejpam-6037	243	10	)	)	PUNCT
ejpam-6037	243	11	is	be	AUX
ejpam-6037	243	12	said	say	VERB
ejpam-6037	243	13	to	to	PART
ejpam-6037	243	14	be	be	AUX
ejpam-6037	243	15	strongly	strongly	ADV
ejpam-6037	243	16	s	s	NOUN
ejpam-6037	243	17	-	-	PUNCT
ejpam-6037	243	18	τ1τ2	τ1τ2	ADJ
ejpam-6037	243	19	-	-	ADJ
ejpam-6037	243	20	closed	closed	ADJ
ejpam-6037	243	21	if	if	SCONJ
ejpam-6037	243	22	every	every	DET
ejpam-6037	243	23	cover	cover	NOUN
ejpam-6037	243	24	of	of	ADP
ejpam-6037	243	25	x	x	PUNCT
ejpam-6037	243	26	by	by	ADP
ejpam-6037	243	27	τ1τ2	τ1τ2	ADJ
ejpam-6037	243	28	-	-	ADJ
ejpam-6037	243	29	closed	closed	ADJ
ejpam-6037	243	30	sets	set	NOUN
ejpam-6037	243	31	of	of	ADP
ejpam-6037	243	32	x	x	PUNCT
ejpam-6037	243	33	has	have	VERB
ejpam-6037	243	34	a	a	DET
ejpam-6037	243	35	finite	finite	ADJ
ejpam-6037	243	36	subcover	subcover	PROPN
ejpam-6037	243	37	.	.	PUNCT
ejpam-6037	244	1	definition	definition	NOUN
ejpam-6037	244	2	12	12	NUM
ejpam-6037	244	3	.	.	PUNCT
ejpam-6037	245	1	a	a	DET
ejpam-6037	245	2	bitopological	bitopological	ADJ
ejpam-6037	245	3	space	space	NOUN
ejpam-6037	245	4	(	(	PUNCT
ejpam-6037	245	5	x	x	NOUN
ejpam-6037	245	6	,	,	PUNCT
ejpam-6037	245	7	τ1	τ1	NOUN
ejpam-6037	245	8	,	,	PUNCT
ejpam-6037	245	9	τ2	τ2	NOUN
ejpam-6037	245	10	)	)	PUNCT
ejpam-6037	245	11	is	be	AUX
ejpam-6037	245	12	said	say	VERB
ejpam-6037	245	13	to	to	PART
ejpam-6037	245	14	be	be	AUX
ejpam-6037	245	15	s	s	NOUN
ejpam-6037	245	16	-	-	PUNCT
ejpam-6037	245	17	τ1τ2	τ1τ2	ADJ
ejpam-6037	245	18	-	-	ADJ
ejpam-6037	245	19	closed	closed	ADJ
ejpam-6037	245	20	if	if	SCONJ
ejpam-6037	245	21	every	every	DET
ejpam-6037	245	22	(	(	PUNCT
ejpam-6037	245	23	τ1	τ1	NOUN
ejpam-6037	245	24	,	,	PUNCT
ejpam-6037	245	25	τ2)sopen	τ2)sopen	ADJ
ejpam-6037	245	26	cover	cover	NOUN
ejpam-6037	245	27	{	{	PUNCT
ejpam-6037	245	28	uγ	uγ	ADV
ejpam-6037	245	29	|	|	ADV
ejpam-6037	245	30	γ	γ	X
ejpam-6037	245	31	∈	∈	NOUN
ejpam-6037	245	32	∇	∇	X
ejpam-6037	245	33	}	}	PUNCT
ejpam-6037	245	34	,	,	PUNCT
ejpam-6037	245	35	there	there	PRON
ejpam-6037	245	36	exists	exist	VERB
ejpam-6037	245	37	a	a	DET
ejpam-6037	245	38	finite	finite	NOUN
ejpam-6037	245	39	subset	subset	NOUN
ejpam-6037	245	40	∇0	∇0	NUM
ejpam-6037	245	41	of	of	ADP
ejpam-6037	245	42	∇	∇	NOUN
ejpam-6037	245	43	such	such	ADJ
ejpam-6037	245	44	that	that	SCONJ
ejpam-6037	245	45	x	x	X
ejpam-6037	245	46	=	=	PUNCT
ejpam-6037	245	47	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-6037	245	48	-	-	NOUN
ejpam-6037	245	49	cl(uγ	cl(uγ	NOUN
ejpam-6037	245	50	)	)	PUNCT
ejpam-6037	245	51	|	|	ADV
ejpam-6037	245	52	γ	γ	X
ejpam-6037	245	53	∈	∈	NOUN
ejpam-6037	245	54	∇0	∇0	NOUN
ejpam-6037	245	55	}	}	PUNCT
ejpam-6037	245	56	.	.	PUNCT
ejpam-6037	246	1	theorem	theorem	NOUN
ejpam-6037	246	2	12	12	NUM
ejpam-6037	246	3	.	.	PUNCT
ejpam-6037	247	1	if	if	SCONJ
ejpam-6037	247	2	f	f	PROPN
ejpam-6037	247	3	:	:	PUNCT
ejpam-6037	247	4	(	(	PUNCT
ejpam-6037	247	5	x	x	NOUN
ejpam-6037	247	6	,	,	PUNCT
ejpam-6037	247	7	τ1	τ1	NOUN
ejpam-6037	247	8	,	,	PUNCT
ejpam-6037	247	9	τ2	τ2	NOUN
ejpam-6037	247	10	)	)	PUNCT
ejpam-6037	247	11	→	→	SYM
ejpam-6037	247	12	(	(	PUNCT
ejpam-6037	247	13	y	y	PROPN
ejpam-6037	247	14	,	,	PUNCT
ejpam-6037	247	15	σ1	σ1	PROPN
ejpam-6037	247	16	,	,	PUNCT
ejpam-6037	247	17	σ2	σ2	PROPN
ejpam-6037	247	18	)	)	PUNCT
ejpam-6037	247	19	is	be	AUX
ejpam-6037	247	20	a	a	DET
ejpam-6037	247	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	247	22	,	,	PUNCT
ejpam-6037	247	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	247	24	surjection	surjection	NOUN
ejpam-6037	247	25	and	and	CCONJ
ejpam-6037	247	26	(	(	PUNCT
ejpam-6037	247	27	x	x	NOUN
ejpam-6037	247	28	,	,	PUNCT
ejpam-6037	247	29	τ1	τ1	NOUN
ejpam-6037	247	30	,	,	PUNCT
ejpam-6037	247	31	τ2	τ2	NOUN
ejpam-6037	247	32	)	)	PUNCT
ejpam-6037	247	33	is	be	AUX
ejpam-6037	247	34	τ1τ2	τ1τ2	NOUN
ejpam-6037	247	35	-	-	ADJ
ejpam-6037	247	36	compact	compact	ADJ
ejpam-6037	247	37	,	,	PUNCT
ejpam-6037	247	38	then	then	ADV
ejpam-6037	247	39	(	(	PUNCT
ejpam-6037	247	40	y	y	PROPN
ejpam-6037	247	41	,	,	PUNCT
ejpam-6037	247	42	σ1	σ1	PROPN
ejpam-6037	247	43	,	,	PUNCT
ejpam-6037	247	44	σ2	σ2	PROPN
ejpam-6037	247	45	)	)	PUNCT
ejpam-6037	247	46	is	be	AUX
ejpam-6037	247	47	strongly	strongly	ADV
ejpam-6037	247	48	s	s	NOUN
ejpam-6037	247	49	-	-	PUNCT
ejpam-6037	247	50	σ1σ2	σ1σ2	VERB
ejpam-6037	247	51	-	-	PUNCT
ejpam-6037	247	52	closed	closed	ADJ
ejpam-6037	247	53	.	.	PUNCT
ejpam-6037	248	1	proof	proof	NOUN
ejpam-6037	248	2	.	.	PUNCT
ejpam-6037	249	1	suppose	suppose	VERB
ejpam-6037	249	2	that	that	SCONJ
ejpam-6037	249	3	(	(	PUNCT
ejpam-6037	249	4	x	x	NOUN
ejpam-6037	249	5	,	,	PUNCT
ejpam-6037	249	6	τ1	τ1	NOUN
ejpam-6037	249	7	,	,	PUNCT
ejpam-6037	249	8	τ2	τ2	NOUN
ejpam-6037	249	9	)	)	PUNCT
ejpam-6037	249	10	is	be	AUX
ejpam-6037	249	11	τ1τ2	τ1τ2	NOUN
ejpam-6037	249	12	-	-	ADJ
ejpam-6037	249	13	compact	compact	ADJ
ejpam-6037	249	14	.	.	PUNCT
ejpam-6037	250	1	let	let	VERB
ejpam-6037	250	2	{	{	PUNCT
ejpam-6037	250	3	vγ	vγ	VERB
ejpam-6037	250	4	|	|	ADV
ejpam-6037	250	5	γ	γ	X
ejpam-6037	250	6	∈	∈	PROPN
ejpam-6037	250	7	∇	∇	X
ejpam-6037	250	8	}	}	PUNCT
ejpam-6037	250	9	be	be	AUX
ejpam-6037	250	10	any	any	DET
ejpam-6037	250	11	cover	cover	NOUN
ejpam-6037	250	12	of	of	ADP
ejpam-6037	250	13	y	y	PRON
ejpam-6037	250	14	by	by	ADP
ejpam-6037	250	15	σ1σ2	σ1σ2	NOUN
ejpam-6037	250	16	-	-	PUNCT
ejpam-6037	250	17	closed	closed	ADJ
ejpam-6037	250	18	sets	set	NOUN
ejpam-6037	250	19	of	of	ADP
ejpam-6037	250	20	y	y	PROPN
ejpam-6037	250	21	.	.	PUNCT
ejpam-6037	251	1	for	for	ADP
ejpam-6037	251	2	each	each	DET
ejpam-6037	251	3	x	x	SYM
ejpam-6037	251	4	∈	∈	PROPN
ejpam-6037	251	5	x	x	NOUN
ejpam-6037	251	6	,	,	PUNCT
ejpam-6037	251	7	there	there	PRON
ejpam-6037	251	8	exists	exist	VERB
ejpam-6037	251	9	γ(x	γ(x	NOUN
ejpam-6037	251	10	)	)	PUNCT
ejpam-6037	251	11	∈	∈	PROPN
ejpam-6037	251	12	∇	∇	X
ejpam-6037	251	13	such	such	ADJ
ejpam-6037	251	14	that	that	SCONJ
ejpam-6037	251	15	f(x	f(x	PROPN
ejpam-6037	251	16	)	)	PUNCT
ejpam-6037	251	17	∈	∈	PROPN
ejpam-6037	251	18	vγ(x	vγ(x	NOUN
ejpam-6037	251	19	)	)	PUNCT
ejpam-6037	251	20	.	.	PUNCT
ejpam-6037	252	1	since	since	SCONJ
ejpam-6037	252	2	f	f	PROPN
ejpam-6037	252	3	is	be	AUX
ejpam-6037	252	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	252	5	,	,	PUNCT
ejpam-6037	252	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	252	7	,	,	PUNCT
ejpam-6037	252	8	by	by	ADP
ejpam-6037	252	9	theorem	theorem	NOUN
ejpam-6037	252	10	1	1	NUM
ejpam-6037	252	11	there	there	ADV
ejpam-6037	252	12	exists	exist	VERB
ejpam-6037	252	13	a	a	DET
ejpam-6037	252	14	τ1τ2	τ1τ2	NOUN
ejpam-6037	252	15	-	-	ADJ
ejpam-6037	252	16	open	open	ADJ
ejpam-6037	252	17	set	set	VERB
ejpam-6037	252	18	u(x	u(x	NOUN
ejpam-6037	252	19	)	)	PUNCT
ejpam-6037	252	20	containing	contain	VERB
ejpam-6037	252	21	x	x	PUNCT
ejpam-6037	252	22	such	such	ADJ
ejpam-6037	252	23	that	that	SCONJ
ejpam-6037	252	24	f(u(x	f(u(x	PROPN
ejpam-6037	252	25	)	)	PUNCT
ejpam-6037	252	26	)	)	PUNCT
ejpam-6037	253	1	⊆	⊆	NUM
ejpam-6037	253	2	vγ(x	vγ(x	NOUN
ejpam-6037	253	3	)	)	PUNCT
ejpam-6037	253	4	.	.	PUNCT
ejpam-6037	254	1	the	the	DET
ejpam-6037	254	2	family	family	NOUN
ejpam-6037	254	3	{	{	PUNCT
ejpam-6037	254	4	u(x	u(x	PROPN
ejpam-6037	254	5	)	)	PUNCT
ejpam-6037	254	6	|	|	ADV
ejpam-6037	254	7	x	x	SYM
ejpam-6037	254	8	∈	∈	NOUN
ejpam-6037	254	9	x	x	X
ejpam-6037	254	10	}	}	PUNCT
ejpam-6037	254	11	is	be	AUX
ejpam-6037	254	12	a	a	DET
ejpam-6037	254	13	cover	cover	NOUN
ejpam-6037	254	14	of	of	ADP
ejpam-6037	254	15	x	x	PUNCT
ejpam-6037	254	16	by	by	ADP
ejpam-6037	254	17	τ1τ2	τ1τ2	ADJ
ejpam-6037	254	18	-	-	ADJ
ejpam-6037	254	19	open	open	ADJ
ejpam-6037	254	20	sets	set	NOUN
ejpam-6037	254	21	.	.	PUNCT
ejpam-6037	255	1	since	since	SCONJ
ejpam-6037	255	2	(	(	PUNCT
ejpam-6037	255	3	x	x	NOUN
ejpam-6037	255	4	,	,	PUNCT
ejpam-6037	255	5	τ1	τ1	NOUN
ejpam-6037	255	6	,	,	PUNCT
ejpam-6037	255	7	τ2	τ2	NOUN
ejpam-6037	255	8	)	)	PUNCT
ejpam-6037	255	9	is	be	AUX
ejpam-6037	255	10	τ1τ2	τ1τ2	NOUN
ejpam-6037	255	11	-	-	ADJ
ejpam-6037	255	12	compact	compact	ADJ
ejpam-6037	255	13	,	,	PUNCT
ejpam-6037	255	14	there	there	PRON
ejpam-6037	255	15	exists	exist	VERB
ejpam-6037	255	16	a	a	DET
ejpam-6037	255	17	finite	finite	ADJ
ejpam-6037	255	18	number	number	NOUN
ejpam-6037	255	19	of	of	ADP
ejpam-6037	255	20	pints	pint	NOUN
ejpam-6037	255	21	,	,	PUNCT
ejpam-6037	255	22	say	say	INTJ
ejpam-6037	255	23	,	,	PUNCT
ejpam-6037	255	24	x1	x1	PROPN
ejpam-6037	255	25	,	,	PUNCT
ejpam-6037	255	26	x2	x2	PROPN
ejpam-6037	255	27	,	,	PUNCT
ejpam-6037	255	28	x3	x3	ADJ
ejpam-6037	255	29	,	,	PUNCT
ejpam-6037	255	30	...	...	PUNCT
ejpam-6037	255	31	,	,	PUNCT
ejpam-6037	255	32	xn	xn	PROPN
ejpam-6037	256	1	in	in	ADP
ejpam-6037	256	2	x	x	X
ejpam-6037	256	3	such	such	ADJ
ejpam-6037	256	4	that	that	SCONJ
ejpam-6037	256	5	x	x	PART
ejpam-6037	256	6	=	=	SYM
ejpam-6037	256	7	∪{u(xk	∪{u(xk	X
ejpam-6037	256	8	)	)	PUNCT
ejpam-6037	256	9	|	|	ADV
ejpam-6037	256	10	xk	xk	PROPN
ejpam-6037	256	11	∈	∈	PROPN
ejpam-6037	256	12	x	x	X
ejpam-6037	256	13	;	;	PUNCT
ejpam-6037	256	14	1	1	NUM
ejpam-6037	256	15	≤	≤	NUM
ejpam-6037	256	16	k	k	X
ejpam-6037	256	17	≤	≤	PROPN
ejpam-6037	256	18	n	n	CCONJ
ejpam-6037	256	19	}	}	PUNCT
ejpam-6037	256	20	.	.	PUNCT
ejpam-6037	257	1	thus	thus	ADV
ejpam-6037	257	2	,	,	PUNCT
ejpam-6037	257	3	y	y	PROPN
ejpam-6037	257	4	=	=	SYM
ejpam-6037	257	5	f(x	f(x	PROPN
ejpam-6037	257	6	)	)	PUNCT
ejpam-6037	257	7	=	=	PUNCT
ejpam-6037	258	1	∪{f(u(xk	∪{f(u(xk	X
ejpam-6037	258	2	)	)	PUNCT
ejpam-6037	258	3	)	)	PUNCT
ejpam-6037	259	1	|	|	ADV
ejpam-6037	259	2	xk	xk	PROPN
ejpam-6037	259	3	∈	∈	PROPN
ejpam-6037	259	4	x	x	X
ejpam-6037	259	5	;	;	PUNCT
ejpam-6037	259	6	1	1	NUM
ejpam-6037	259	7	≤	≤	NUM
ejpam-6037	259	8	k	k	X
ejpam-6037	259	9	≤	≤	PROPN
ejpam-6037	259	10	n	n	CCONJ
ejpam-6037	259	11	}	}	PUNCT
ejpam-6037	259	12	⊆	⊆	NUM
ejpam-6037	259	13	∪{vγ(xk	∪{vγ(xk	NOUN
ejpam-6037	259	14	)	)	PUNCT
ejpam-6037	259	15	|	|	ADV
ejpam-6037	259	16	xk	xk	PROPN
ejpam-6037	259	17	∈	∈	PROPN
ejpam-6037	260	1	x	x	X
ejpam-6037	260	2	;	;	PUNCT
ejpam-6037	260	3	1	1	NUM
ejpam-6037	260	4	≤	≤	NUM
ejpam-6037	260	5	k	k	X
ejpam-6037	260	6	≤	≤	PROPN
ejpam-6037	260	7	n	n	CCONJ
ejpam-6037	260	8	}	}	PUNCT
ejpam-6037	260	9	.	.	PUNCT
ejpam-6037	261	1	this	this	PRON
ejpam-6037	261	2	shows	show	VERB
ejpam-6037	261	3	that	that	SCONJ
ejpam-6037	261	4	(	(	PUNCT
ejpam-6037	261	5	y	y	PROPN
ejpam-6037	261	6	,	,	PUNCT
ejpam-6037	261	7	σ1	σ1	PROPN
ejpam-6037	261	8	,	,	PUNCT
ejpam-6037	261	9	σ2	σ2	PROPN
ejpam-6037	261	10	)	)	PUNCT
ejpam-6037	261	11	is	be	AUX
ejpam-6037	261	12	strongly	strongly	ADV
ejpam-6037	261	13	s	s	NOUN
ejpam-6037	261	14	-	-	PUNCT
ejpam-6037	261	15	σ1σ2	σ1σ2	VERB
ejpam-6037	261	16	-	-	PUNCT
ejpam-6037	261	17	closed	closed	ADJ
ejpam-6037	261	18	.	.	PUNCT
ejpam-6037	262	1	corollary	corollary	ADJ
ejpam-6037	262	2	2	2	NUM
ejpam-6037	262	3	.	.	PUNCT
ejpam-6037	263	1	if	if	SCONJ
ejpam-6037	263	2	f	f	PROPN
ejpam-6037	263	3	:	:	PUNCT
ejpam-6037	263	4	(	(	PUNCT
ejpam-6037	263	5	x	x	NOUN
ejpam-6037	263	6	,	,	PUNCT
ejpam-6037	263	7	τ1	τ1	NOUN
ejpam-6037	263	8	,	,	PUNCT
ejpam-6037	263	9	τ2	τ2	NOUN
ejpam-6037	263	10	)	)	PUNCT
ejpam-6037	263	11	→	→	SYM
ejpam-6037	263	12	(	(	PUNCT
ejpam-6037	263	13	y	y	PROPN
ejpam-6037	263	14	,	,	PUNCT
ejpam-6037	263	15	σ1	σ1	PROPN
ejpam-6037	263	16	,	,	PUNCT
ejpam-6037	263	17	σ2	σ2	PROPN
ejpam-6037	263	18	)	)	PUNCT
ejpam-6037	263	19	is	be	AUX
ejpam-6037	263	20	a	a	DET
ejpam-6037	263	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	263	22	,	,	PUNCT
ejpam-6037	263	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	263	24	surjection	surjection	NOUN
ejpam-6037	263	25	and	and	CCONJ
ejpam-6037	263	26	(	(	PUNCT
ejpam-6037	263	27	x	x	NOUN
ejpam-6037	263	28	,	,	PUNCT
ejpam-6037	263	29	τ1	τ1	NOUN
ejpam-6037	263	30	,	,	PUNCT
ejpam-6037	263	31	τ2	τ2	NOUN
ejpam-6037	263	32	)	)	PUNCT
ejpam-6037	263	33	is	be	AUX
ejpam-6037	263	34	τ1τ2	τ1τ2	NOUN
ejpam-6037	263	35	-	-	ADJ
ejpam-6037	263	36	compact	compact	ADJ
ejpam-6037	263	37	,	,	PUNCT
ejpam-6037	263	38	then	then	ADV
ejpam-6037	263	39	(	(	PUNCT
ejpam-6037	263	40	y	y	PROPN
ejpam-6037	263	41	,	,	PUNCT
ejpam-6037	263	42	σ1	σ1	PROPN
ejpam-6037	263	43	,	,	PUNCT
ejpam-6037	263	44	σ2	σ2	PROPN
ejpam-6037	263	45	)	)	PUNCT
ejpam-6037	263	46	is	be	AUX
ejpam-6037	263	47	s	s	NOUN
ejpam-6037	263	48	-	-	PUNCT
ejpam-6037	263	49	σ1σ2	σ1σ2	VERB
ejpam-6037	263	50	-	-	PUNCT
ejpam-6037	263	51	closed	closed	ADJ
ejpam-6037	263	52	and	and	CCONJ
ejpam-6037	263	53	hence	hence	ADV
ejpam-6037	263	54	quasi	quasi	X
ejpam-6037	263	55	(	(	PUNCT
ejpam-6037	263	56	σ1	σ1	PROPN
ejpam-6037	263	57	,	,	PUNCT
ejpam-6037	263	58	σ2)-h	σ2)-h	PROPN
ejpam-6037	263	59	closed	close	VERB
ejpam-6037	263	60	.	.	PUNCT
ejpam-6037	264	1	recall	recall	VERB
ejpam-6037	264	2	that	that	SCONJ
ejpam-6037	264	3	a	a	DET
ejpam-6037	264	4	bitopological	bitopological	ADJ
ejpam-6037	264	5	space	space	NOUN
ejpam-6037	264	6	(	(	PUNCT
ejpam-6037	264	7	x	x	NOUN
ejpam-6037	264	8	,	,	PUNCT
ejpam-6037	264	9	τ1	τ1	NOUN
ejpam-6037	264	10	,	,	PUNCT
ejpam-6037	264	11	τ2	τ2	NOUN
ejpam-6037	264	12	)	)	PUNCT
ejpam-6037	264	13	is	be	AUX
ejpam-6037	264	14	said	say	VERB
ejpam-6037	264	15	to	to	PART
ejpam-6037	264	16	be	be	AUX
ejpam-6037	264	17	τ1τ2	τ1τ2	NOUN
ejpam-6037	264	18	-	-	ADJ
ejpam-6037	264	19	connected	connected	ADJ
ejpam-6037	265	1	[	[	X
ejpam-6037	265	2	38	38	NUM
ejpam-6037	265	3	]	]	PUNCT
ejpam-6037	265	4	if	if	SCONJ
ejpam-6037	265	5	x	x	PRON
ejpam-6037	265	6	can	can	AUX
ejpam-6037	265	7	not	not	PART
ejpam-6037	265	8	be	be	AUX
ejpam-6037	265	9	written	write	VERB
ejpam-6037	265	10	as	as	ADP
ejpam-6037	265	11	the	the	DET
ejpam-6037	265	12	union	union	NOUN
ejpam-6037	265	13	of	of	ADP
ejpam-6037	265	14	two	two	NUM
ejpam-6037	265	15	nonempty	nonempty	ADV
ejpam-6037	265	16	disjoint	disjoint	NOUN
ejpam-6037	265	17	τ1τ2	τ1τ2	ADJ
ejpam-6037	265	18	-	-	ADJ
ejpam-6037	265	19	open	open	ADJ
ejpam-6037	265	20	sets	set	NOUN
ejpam-6037	265	21	.	.	PUNCT
ejpam-6037	266	1	theorem	theorem	VERB
ejpam-6037	266	2	13	13	NUM
ejpam-6037	266	3	.	.	PUNCT
ejpam-6037	267	1	if	if	SCONJ
ejpam-6037	267	2	f	f	PROPN
ejpam-6037	267	3	:	:	PUNCT
ejpam-6037	267	4	(	(	PUNCT
ejpam-6037	267	5	x	x	NOUN
ejpam-6037	267	6	,	,	PUNCT
ejpam-6037	267	7	τ1	τ1	NOUN
ejpam-6037	267	8	,	,	PUNCT
ejpam-6037	267	9	τ2	τ2	NOUN
ejpam-6037	267	10	)	)	PUNCT
ejpam-6037	267	11	→	→	SYM
ejpam-6037	267	12	(	(	PUNCT
ejpam-6037	267	13	y	y	PROPN
ejpam-6037	267	14	,	,	PUNCT
ejpam-6037	267	15	σ1	σ1	PROPN
ejpam-6037	267	16	,	,	PUNCT
ejpam-6037	267	17	σ2	σ2	PROPN
ejpam-6037	267	18	)	)	PUNCT
ejpam-6037	267	19	is	be	AUX
ejpam-6037	267	20	a	a	DET
ejpam-6037	267	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	267	22	,	,	PUNCT
ejpam-6037	267	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	267	24	surjection	surjection	NOUN
ejpam-6037	267	25	and	and	CCONJ
ejpam-6037	267	26	(	(	PUNCT
ejpam-6037	267	27	x	x	NOUN
ejpam-6037	267	28	,	,	PUNCT
ejpam-6037	267	29	τ1	τ1	NOUN
ejpam-6037	267	30	,	,	PUNCT
ejpam-6037	267	31	τ2	τ2	NOUN
ejpam-6037	267	32	)	)	PUNCT
ejpam-6037	267	33	is	be	AUX
ejpam-6037	267	34	τ1τ2	τ1τ2	NOUN
ejpam-6037	267	35	-	-	ADJ
ejpam-6037	267	36	connected	connected	ADJ
ejpam-6037	267	37	,	,	PUNCT
ejpam-6037	267	38	then	then	ADV
ejpam-6037	267	39	(	(	PUNCT
ejpam-6037	267	40	y	y	PROPN
ejpam-6037	267	41	,	,	PUNCT
ejpam-6037	267	42	σ1	σ1	PROPN
ejpam-6037	267	43	,	,	PUNCT
ejpam-6037	267	44	σ2	σ2	PROPN
ejpam-6037	267	45	)	)	PUNCT
ejpam-6037	267	46	is	be	AUX
ejpam-6037	267	47	σ1σ2	σ1σ2	NOUN
ejpam-6037	267	48	-	-	PUNCT
ejpam-6037	267	49	connected	connected	ADJ
ejpam-6037	267	50	.	.	PUNCT
ejpam-6037	268	1	proof	proof	NOUN
ejpam-6037	268	2	.	.	PUNCT
ejpam-6037	269	1	assume	assume	VERB
ejpam-6037	269	2	that	that	SCONJ
ejpam-6037	269	3	(	(	PUNCT
ejpam-6037	269	4	y	y	PROPN
ejpam-6037	269	5	,	,	PUNCT
ejpam-6037	269	6	σ1	σ1	PROPN
ejpam-6037	269	7	,	,	PUNCT
ejpam-6037	269	8	σ2	σ2	PROPN
ejpam-6037	269	9	)	)	PUNCT
ejpam-6037	269	10	is	be	AUX
ejpam-6037	269	11	not	not	PART
ejpam-6037	269	12	σ1σ2	σ1σ2	VERB
ejpam-6037	269	13	-	-	PUNCT
ejpam-6037	269	14	connected	connect	VERB
ejpam-6037	269	15	.	.	PUNCT
ejpam-6037	270	1	then	then	ADV
ejpam-6037	270	2	,	,	PUNCT
ejpam-6037	270	3	there	there	PRON
ejpam-6037	270	4	exist	exist	VERB
ejpam-6037	270	5	σ1σ2	σ1σ2	NOUN
ejpam-6037	270	6	-	-	ADJ
ejpam-6037	270	7	open	open	ADJ
ejpam-6037	270	8	sets	set	NOUN
ejpam-6037	270	9	v	v	ADP
ejpam-6037	270	10	and	and	CCONJ
ejpam-6037	270	11	w	w	NOUN
ejpam-6037	270	12	of	of	ADP
ejpam-6037	270	13	y	y	PRON
ejpam-6037	271	1	such	such	ADJ
ejpam-6037	271	2	that	that	DET
ejpam-6037	271	3	v	v	NOUN
ejpam-6037	271	4	∩w	∩w	NOUN
ejpam-6037	271	5	=	=	NOUN
ejpam-6037	271	6	∅	∅	NOUN
ejpam-6037	271	7	and	and	CCONJ
ejpam-6037	271	8	v	v	ADP
ejpam-6037	271	9	∪w	∪w	PROPN
ejpam-6037	271	10	=	=	SYM
ejpam-6037	271	11	y	y	PROPN
ejpam-6037	271	12	.	.	PUNCT
ejpam-6037	272	1	thus	thus	ADV
ejpam-6037	272	2	,	,	PUNCT
ejpam-6037	272	3	we	we	PRON
ejpam-6037	272	4	have	have	VERB
ejpam-6037	272	5	f−1(v	f−1(v	NOUN
ejpam-6037	272	6	)	)	PUNCT
ejpam-6037	272	7	∩f−1(w	∩f−1(w	X
ejpam-6037	272	8	)	)	PUNCT
ejpam-6037	272	9	=	=	NOUN
ejpam-6037	272	10	∅	∅	NOUN
ejpam-6037	272	11	and	and	CCONJ
ejpam-6037	272	12	f−1(v	f−1(v	NOUN
ejpam-6037	272	13	)	)	PUNCT
ejpam-6037	272	14	∪f−1(w	∪f−1(w	ADV
ejpam-6037	272	15	)	)	PUNCT
ejpam-6037	273	1	=	=	PUNCT
ejpam-6037	273	2	x.	x.	NOUN
ejpam-6037	273	3	since	since	SCONJ
ejpam-6037	273	4	f	f	PROPN
ejpam-6037	273	5	is	be	AUX
ejpam-6037	273	6	surjective	surjective	ADJ
ejpam-6037	273	7	,	,	PUNCT
ejpam-6037	273	8	f−1(v	f−1(v	NOUN
ejpam-6037	273	9	)	)	PUNCT
ejpam-6037	273	10	̸=	̸=	PROPN
ejpam-6037	273	11	∅	∅	NOUN
ejpam-6037	273	12	and	and	CCONJ
ejpam-6037	273	13	f−1(w	f−1(w	ADV
ejpam-6037	273	14	)	)	PUNCT
ejpam-6037	273	15	̸=	̸=	PROPN
ejpam-6037	273	16	∅.	∅.	ADV
ejpam-6037	273	17	since	since	SCONJ
ejpam-6037	273	18	f	f	PROPN
ejpam-6037	273	19	is	be	AUX
ejpam-6037	273	20	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	273	21	,	,	PUNCT
ejpam-6037	273	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	273	23	and	and	CCONJ
ejpam-6037	273	24	v	v	NOUN
ejpam-6037	273	25	,	,	PUNCT
ejpam-6037	273	26	w	w	NOUN
ejpam-6037	273	27	are	be	AUX
ejpam-6037	273	28	σ1σ2	σ1σ2	NOUN
ejpam-6037	273	29	-	-	PUNCT
ejpam-6037	273	30	clopen	clopen	ADJ
ejpam-6037	273	31	sets	set	NOUN
ejpam-6037	273	32	,	,	PUNCT
ejpam-6037	273	33	by	by	ADP
ejpam-6037	273	34	theorem	theorem	NOUN
ejpam-6037	273	35	1	1	NUM
ejpam-6037	273	36	we	we	PRON
ejpam-6037	273	37	have	have	VERB
ejpam-6037	273	38	f−1(v	f−1(v	NOUN
ejpam-6037	273	39	)	)	PUNCT
ejpam-6037	273	40	and	and	CCONJ
ejpam-6037	273	41	f−1(w	f−1(w	PROPN
ejpam-6037	273	42	)	)	PUNCT
ejpam-6037	273	43	are	be	AUX
ejpam-6037	273	44	τ1τ2	τ1τ2	NOUN
ejpam-6037	273	45	-	-	ADJ
ejpam-6037	273	46	open	open	ADJ
ejpam-6037	273	47	in	in	ADP
ejpam-6037	273	48	x.	x.	NOUN
ejpam-6037	273	49	therefore	therefore	ADV
ejpam-6037	273	50	,	,	PUNCT
ejpam-6037	273	51	(	(	PUNCT
ejpam-6037	273	52	x	x	NOUN
ejpam-6037	273	53	,	,	PUNCT
ejpam-6037	273	54	τ1	τ1	NOUN
ejpam-6037	273	55	,	,	PUNCT
ejpam-6037	273	56	τ2	τ2	NOUN
ejpam-6037	273	57	)	)	PUNCT
ejpam-6037	273	58	is	be	AUX
ejpam-6037	273	59	not	not	PART
ejpam-6037	273	60	τ1τ2	τ1τ2	ADJ
ejpam-6037	273	61	-	-	VERB
ejpam-6037	273	62	connected	connected	ADJ
ejpam-6037	273	63	.	.	PUNCT
ejpam-6037	274	1	the	the	DET
ejpam-6037	274	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	274	3	-	-	NOUN
ejpam-6037	274	4	frontier	frontier	NOUN
ejpam-6037	274	5	[	[	X
ejpam-6037	274	6	31	31	NUM
ejpam-6037	274	7	]	]	PUNCT
ejpam-6037	274	8	of	of	ADP
ejpam-6037	274	9	a	a	DET
ejpam-6037	274	10	subset	subset	NOUN
ejpam-6037	274	11	a	a	PRON
ejpam-6037	274	12	of	of	ADP
ejpam-6037	274	13	a	a	DET
ejpam-6037	274	14	bitopological	bitopological	ADJ
ejpam-6037	274	15	space	space	NOUN
ejpam-6037	274	16	(	(	PUNCT
ejpam-6037	274	17	x	x	NOUN
ejpam-6037	274	18	,	,	PUNCT
ejpam-6037	274	19	τ1	τ1	NOUN
ejpam-6037	274	20	,	,	PUNCT
ejpam-6037	274	21	τ2	τ2	PROPN
ejpam-6037	274	22	)	)	PUNCT
ejpam-6037	274	23	,	,	PUNCT
ejpam-6037	274	24	denoted	denote	VERB
ejpam-6037	274	25	by	by	ADP
ejpam-6037	274	26	τ1τ2	τ1τ2	NOUN
ejpam-6037	274	27	-	-	ADJ
ejpam-6037	274	28	fr(a	fr(a	NUM
ejpam-6037	274	29	)	)	PUNCT
ejpam-6037	274	30	,	,	PUNCT
ejpam-6037	274	31	is	be	AUX
ejpam-6037	274	32	defined	define	VERB
ejpam-6037	274	33	by	by	ADP
ejpam-6037	274	34	τ1τ2	τ1τ2	NOUN
ejpam-6037	274	35	-	-	ADJ
ejpam-6037	274	36	fr(a	fr(a	ADJ
ejpam-6037	274	37	)	)	PUNCT
ejpam-6037	275	1	=	=	PUNCT
ejpam-6037	275	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	275	3	-	-	NUM
ejpam-6037	275	4	cl(a	cl(a	NUM
ejpam-6037	275	5	)	)	PUNCT
ejpam-6037	275	6	∩	∩	NOUN
ejpam-6037	275	7	τ1τ2	τ1τ2	NOUN
ejpam-6037	275	8	-	-	ADJ
ejpam-6037	275	9	cl(x	cl(x	SYM
ejpam-6037	275	10	−a	−a	NOUN
ejpam-6037	275	11	)	)	PUNCT
ejpam-6037	275	12	=	=	PUNCT
ejpam-6037	276	1	τ1τ2	τ1τ2	ADJ
ejpam-6037	276	2	-	-	ADJ
ejpam-6037	276	3	cl(a)−	cl(a)−	ADJ
ejpam-6037	276	4	τ1τ2	τ1τ2	NOUN
ejpam-6037	276	5	-	-	ADJ
ejpam-6037	276	6	int(a	int(a	NOUN
ejpam-6037	276	7	)	)	PUNCT
ejpam-6037	276	8	.	.	PUNCT
ejpam-6037	277	1	n.	n.	PROPN
ejpam-6037	277	2	chutiman	chutiman	PROPN
ejpam-6037	277	3	,	,	PUNCT
ejpam-6037	277	4	a.	a.	PROPN
ejpam-6037	277	5	sama	sama	PROPN
ejpam-6037	277	6	-	-	PUNCT
ejpam-6037	277	7	ae	ae	PROPN
ejpam-6037	277	8	,	,	PUNCT
ejpam-6037	277	9	c.	c.	PROPN
ejpam-6037	277	10	boonpok	boonpok	PROPN
ejpam-6037	277	11	/	/	SYM
ejpam-6037	277	12	eur	eur	PROPN
ejpam-6037	277	13	.	.	PUNCT
ejpam-6037	278	1	j.	j.	PROPN
ejpam-6037	278	2	pure	pure	PROPN
ejpam-6037	278	3	appl	appl	PROPN
ejpam-6037	278	4	.	.	PROPN
ejpam-6037	278	5	math	math	PROPN
ejpam-6037	278	6	,	,	PUNCT
ejpam-6037	278	7	18	18	NUM
ejpam-6037	278	8	(	(	PUNCT
ejpam-6037	278	9	2	2	NUM
ejpam-6037	278	10	)	)	PUNCT
ejpam-6037	278	11	(	(	PUNCT
ejpam-6037	278	12	2025	2025	NUM
ejpam-6037	278	13	)	)	PUNCT
ejpam-6037	278	14	,	,	PUNCT
ejpam-6037	278	15	6037	6037	NUM
ejpam-6037	278	16	10	10	NUM
ejpam-6037	278	17	of	of	ADP
ejpam-6037	278	18	12	12	NUM
ejpam-6037	278	19	theorem	theorem	NOUN
ejpam-6037	278	20	14	14	NUM
ejpam-6037	278	21	.	.	PUNCT
ejpam-6037	279	1	the	the	DET
ejpam-6037	279	2	set	set	NOUN
ejpam-6037	279	3	of	of	ADP
ejpam-6037	279	4	all	all	DET
ejpam-6037	279	5	points	point	NOUN
ejpam-6037	279	6	x	x	X
ejpam-6037	279	7	∈	∈	NOUN
ejpam-6037	279	8	x	x	PUNCT
ejpam-6037	279	9	at	at	ADP
ejpam-6037	279	10	which	which	PRON
ejpam-6037	279	11	a	a	DET
ejpam-6037	279	12	function	function	NOUN
ejpam-6037	279	13	f	f	NOUN
ejpam-6037	279	14	:	:	PUNCT
ejpam-6037	279	15	(	(	PUNCT
ejpam-6037	279	16	x	x	NOUN
ejpam-6037	279	17	,	,	PUNCT
ejpam-6037	279	18	τ1	τ1	NOUN
ejpam-6037	279	19	,	,	PUNCT
ejpam-6037	279	20	τ2	τ2	NOUN
ejpam-6037	279	21	)	)	PUNCT
ejpam-6037	279	22	→	→	SYM
ejpam-6037	279	23	(	(	PUNCT
ejpam-6037	279	24	y	y	PROPN
ejpam-6037	279	25	,	,	PUNCT
ejpam-6037	279	26	σ1	σ1	PROPN
ejpam-6037	279	27	,	,	PUNCT
ejpam-6037	279	28	σ2	σ2	PROPN
ejpam-6037	279	29	)	)	PUNCT
ejpam-6037	279	30	is	be	AUX
ejpam-6037	279	31	not	not	PART
ejpam-6037	279	32	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	279	33	,	,	PUNCT
ejpam-6037	279	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	279	35	is	be	AUX
ejpam-6037	279	36	identical	identical	ADJ
ejpam-6037	279	37	with	with	ADP
ejpam-6037	279	38	the	the	DET
ejpam-6037	279	39	union	union	NOUN
ejpam-6037	279	40	of	of	ADP
ejpam-6037	279	41	the	the	DET
ejpam-6037	279	42	τ1τ2	τ1τ2	NOUN
ejpam-6037	279	43	-	-	NOUN
ejpam-6037	279	44	frontier	frontier	NOUN
ejpam-6037	279	45	of	of	ADP
ejpam-6037	279	46	the	the	DET
ejpam-6037	279	47	inverse	inverse	NOUN
ejpam-6037	279	48	images	image	NOUN
ejpam-6037	279	49	of	of	ADP
ejpam-6037	279	50	σ1σ2	σ1σ2	NOUN
ejpam-6037	279	51	-	-	PUNCT
ejpam-6037	279	52	closed	closed	ADJ
ejpam-6037	279	53	sets	set	NOUN
ejpam-6037	279	54	of	of	ADP
ejpam-6037	279	55	y	y	NOUN
ejpam-6037	279	56	containing	contain	VERB
ejpam-6037	279	57	f(x	f(x	PROPN
ejpam-6037	279	58	)	)	PUNCT
ejpam-6037	279	59	.	.	PUNCT
ejpam-6037	280	1	proof	proof	NOUN
ejpam-6037	280	2	.	.	PUNCT
ejpam-6037	281	1	suppose	suppose	VERB
ejpam-6037	281	2	that	that	SCONJ
ejpam-6037	281	3	f	f	PROPN
ejpam-6037	281	4	is	be	AUX
ejpam-6037	281	5	not	not	PART
ejpam-6037	281	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	281	7	,	,	PUNCT
ejpam-6037	281	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	281	9	at	at	ADP
ejpam-6037	281	10	x	x	SYM
ejpam-6037	281	11	∈	∈	PROPN
ejpam-6037	281	12	x.	x.	NOUN
ejpam-6037	281	13	then	then	ADV
ejpam-6037	281	14	,	,	PUNCT
ejpam-6037	281	15	there	there	PRON
ejpam-6037	281	16	exists	exist	VERB
ejpam-6037	281	17	a	a	DET
ejpam-6037	281	18	σ1σ2	σ1σ2	NUM
ejpam-6037	281	19	-	-	PUNCT
ejpam-6037	281	20	closed	closed	ADJ
ejpam-6037	281	21	set	set	NOUN
ejpam-6037	281	22	k	k	PROPN
ejpam-6037	281	23	of	of	ADP
ejpam-6037	281	24	y	y	PROPN
ejpam-6037	281	25	containing	contain	VERB
ejpam-6037	281	26	f(x	f(x	PROPN
ejpam-6037	281	27	)	)	PUNCT
ejpam-6037	281	28	such	such	ADJ
ejpam-6037	281	29	that	that	SCONJ
ejpam-6037	281	30	f(u)∩	f(u)∩	PROPN
ejpam-6037	281	31	(	(	PUNCT
ejpam-6037	281	32	y	y	PROPN
ejpam-6037	281	33	−k	−k	ADJ
ejpam-6037	281	34	)	)	PUNCT
ejpam-6037	281	35	̸=	̸=	NOUN
ejpam-6037	281	36	∅	∅	NOUN
ejpam-6037	281	37	for	for	ADP
ejpam-6037	281	38	every	every	DET
ejpam-6037	281	39	τ1τ2	τ1τ2	ADJ
ejpam-6037	281	40	-	-	ADJ
ejpam-6037	281	41	open	open	ADJ
ejpam-6037	281	42	set	set	ADJ
ejpam-6037	281	43	u	u	NOUN
ejpam-6037	281	44	of	of	ADP
ejpam-6037	281	45	x	x	SYM
ejpam-6037	281	46	containing	contain	VERB
ejpam-6037	281	47	x.	x.	NOUN
ejpam-6037	281	48	thus	thus	ADV
ejpam-6037	281	49	,	,	PUNCT
ejpam-6037	281	50	x	x	SYM
ejpam-6037	281	51	∈	∈	PROPN
ejpam-6037	281	52	τ1τ2	τ1τ2	PROPN
ejpam-6037	281	53	-	-	NOUN
ejpam-6037	281	54	cl(f	cl(f	NOUN
ejpam-6037	281	55	−1(y	−1(y	NOUN
ejpam-6037	281	56	−k	−k	ADJ
ejpam-6037	281	57	)	)	PUNCT
ejpam-6037	281	58	)	)	PUNCT
ejpam-6037	282	1	=	=	PUNCT
ejpam-6037	282	2	τ1τ2	τ1τ2	NOUN
ejpam-6037	282	3	-	-	NOUN
ejpam-6037	282	4	cl(x	cl(x	SYM
ejpam-6037	282	5	−	−	PROPN
ejpam-6037	282	6	f−1(k	f−1(k	PROPN
ejpam-6037	282	7	)	)	PUNCT
ejpam-6037	282	8	)	)	PUNCT
ejpam-6037	282	9	.	.	PUNCT
ejpam-6037	283	1	on	on	ADP
ejpam-6037	283	2	the	the	DET
ejpam-6037	283	3	other	other	ADJ
ejpam-6037	283	4	hand	hand	NOUN
ejpam-6037	283	5	,	,	PUNCT
ejpam-6037	283	6	we	we	PRON
ejpam-6037	283	7	have	have	VERB
ejpam-6037	283	8	x	x	PROPN
ejpam-6037	283	9	∈	∈	PROPN
ejpam-6037	283	10	f−1(k	f−1(k	PROPN
ejpam-6037	283	11	)	)	PUNCT
ejpam-6037	283	12	⊆	⊆	NUM
ejpam-6037	283	13	τ1τ2	τ1τ2	NOUN
ejpam-6037	283	14	-	-	NOUN
ejpam-6037	283	15	cl(f	cl(f	NUM
ejpam-6037	283	16	−1(k	−1(k	NOUN
ejpam-6037	283	17	)	)	PUNCT
ejpam-6037	283	18	)	)	PUNCT
ejpam-6037	283	19	and	and	CCONJ
ejpam-6037	283	20	hence	hence	ADV
ejpam-6037	283	21	x	x	X
ejpam-6037	283	22	∈	∈	PRON
ejpam-6037	283	23	τ1τ2	τ1τ2	NOUN
ejpam-6037	283	24	-	-	ADJ
ejpam-6037	283	25	fr(f	fr(f	NUM
ejpam-6037	283	26	−1(k	−1(k	NOUN
ejpam-6037	283	27	)	)	PUNCT
ejpam-6037	283	28	)	)	PUNCT
ejpam-6037	283	29	.	.	PUNCT
ejpam-6037	284	1	conversely	conversely	ADV
ejpam-6037	284	2	,	,	PUNCT
ejpam-6037	284	3	suppose	suppose	VERB
ejpam-6037	284	4	that	that	SCONJ
ejpam-6037	284	5	f	f	PROPN
ejpam-6037	284	6	is	be	AUX
ejpam-6037	284	7	contra-(τ1	contra-(τ1	NOUN
ejpam-6037	284	8	,	,	PUNCT
ejpam-6037	284	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	284	10	at	at	ADP
ejpam-6037	284	11	x	x	SYM
ejpam-6037	284	12	∈	∈	PROPN
ejpam-6037	284	13	x.	x.	NOUN
ejpam-6037	284	14	let	let	VERB
ejpam-6037	284	15	k	k	X
ejpam-6037	284	16	be	be	AUX
ejpam-6037	284	17	any	any	DET
ejpam-6037	284	18	σ1σ2closed	σ1σ2closed	ADJ
ejpam-6037	284	19	set	set	NOUN
ejpam-6037	284	20	of	of	ADP
ejpam-6037	284	21	y	y	PROPN
ejpam-6037	284	22	containing	contain	VERB
ejpam-6037	284	23	f(x	f(x	PROPN
ejpam-6037	284	24	)	)	PUNCT
ejpam-6037	284	25	.	.	PUNCT
ejpam-6037	285	1	by	by	ADP
ejpam-6037	285	2	theorem	theorem	NOUN
ejpam-6037	285	3	1	1	NUM
ejpam-6037	285	4	,	,	PUNCT
ejpam-6037	285	5	x	x	SYM
ejpam-6037	285	6	∈	∈	PROPN
ejpam-6037	285	7	f−1(k	f−1(k	PROPN
ejpam-6037	285	8	)	)	PUNCT
ejpam-6037	285	9	=	=	PUNCT
ejpam-6037	286	1	τ1τ2	τ1τ2	NOUN
ejpam-6037	286	2	-	-	NUM
ejpam-6037	286	3	int(f	int(f	PRON
ejpam-6037	286	4	−1(k	−1(k	NOUN
ejpam-6037	286	5	)	)	PUNCT
ejpam-6037	286	6	)	)	PUNCT
ejpam-6037	286	7	.	.	PUNCT
ejpam-6037	287	1	thus	thus	ADV
ejpam-6037	287	2	,	,	PUNCT
ejpam-6037	287	3	x	x	PROPN
ejpam-6037	287	4	̸∈	̸∈	PROPN
ejpam-6037	287	5	τ1τ2	τ1τ2	PROPN
ejpam-6037	287	6	-	-	ADJ
ejpam-6037	287	7	fr(f	fr(f	NUM
ejpam-6037	287	8	−1(k	−1(k	NOUN
ejpam-6037	287	9	)	)	PUNCT
ejpam-6037	287	10	)	)	PUNCT
ejpam-6037	287	11	for	for	ADP
ejpam-6037	287	12	every	every	DET
ejpam-6037	287	13	σ1σ2	σ1σ2	NUM
ejpam-6037	287	14	-	-	PUNCT
ejpam-6037	287	15	closed	closed	ADJ
ejpam-6037	287	16	set	set	NOUN
ejpam-6037	287	17	k	k	PROPN
ejpam-6037	287	18	of	of	ADP
ejpam-6037	287	19	y	y	PROPN
ejpam-6037	287	20	containing	contain	VERB
ejpam-6037	287	21	f(x	f(x	PROPN
ejpam-6037	287	22	)	)	PUNCT
ejpam-6037	287	23	.	.	PUNCT
ejpam-6037	288	1	this	this	PRON
ejpam-6037	288	2	completes	complete	VERB
ejpam-6037	288	3	the	the	DET
ejpam-6037	288	4	proof	proof	NOUN
ejpam-6037	288	5	.	.	PUNCT
ejpam-6037	289	1	acknowledgements	acknowledgement	NOUN
ejpam-6037	289	2	this	this	DET
ejpam-6037	289	3	research	research	NOUN
ejpam-6037	289	4	project	project	NOUN
ejpam-6037	289	5	was	be	AUX
ejpam-6037	289	6	financially	financially	ADV
ejpam-6037	289	7	supported	support	VERB
ejpam-6037	289	8	by	by	ADP
ejpam-6037	289	9	mahasarakham	mahasarakham	PROPN
ejpam-6037	289	10	university	university	PROPN
ejpam-6037	289	11	.	.	PUNCT
ejpam-6037	290	1	references	reference	NOUN
ejpam-6037	290	2	[	[	X
ejpam-6037	290	3	1	1	NUM
ejpam-6037	290	4	]	]	PUNCT
ejpam-6037	290	5	c.	c.	PROPN
ejpam-6037	290	6	viriyapong	viriyapong	PROPN
ejpam-6037	290	7	and	and	CCONJ
ejpam-6037	290	8	c.	c.	PROPN
ejpam-6037	290	9	boonpok	boonpok	PROPN
ejpam-6037	290	10	.	.	PUNCT
ejpam-6037	291	1	(	(	PUNCT
ejpam-6037	291	2	λ	λ	X
ejpam-6037	291	3	,	,	PUNCT
ejpam-6037	291	4	sp)-continuous	sp)-continuous	ADJ
ejpam-6037	291	5	functions	function	NOUN
ejpam-6037	291	6	.	.	PUNCT
ejpam-6037	292	1	wseas	wseas	VERB
ejpam-6037	292	2	transactions	transaction	NOUN
ejpam-6037	292	3	on	on	ADP
ejpam-6037	292	4	mathematics	mathematic	NOUN
ejpam-6037	292	5	,	,	PUNCT
ejpam-6037	292	6	21:380–385	21:380–385	NUM
ejpam-6037	292	7	,	,	PUNCT
ejpam-6037	292	8	2022	2022	NUM
ejpam-6037	292	9	.	.	PUNCT
ejpam-6037	293	1	[	[	X
ejpam-6037	293	2	2	2	NUM
ejpam-6037	293	3	]	]	PUNCT
ejpam-6037	293	4	c.	c.	PROPN
ejpam-6037	293	5	boonpok	boonpok	PROPN
ejpam-6037	293	6	and	and	CCONJ
ejpam-6037	293	7	j.	j.	PROPN
ejpam-6037	293	8	khampakdee	khampakdee	PROPN
ejpam-6037	293	9	.	.	PUNCT
ejpam-6037	294	1	(	(	PUNCT
ejpam-6037	294	2	λ	λ	NOUN
ejpam-6037	294	3	,	,	PUNCT
ejpam-6037	294	4	sp)-open	sp)-open	ADJ
ejpam-6037	294	5	sets	set	NOUN
ejpam-6037	294	6	in	in	ADP
ejpam-6037	294	7	topological	topological	ADJ
ejpam-6037	294	8	spaces	space	NOUN
ejpam-6037	294	9	.	.	PUNCT
ejpam-6037	295	1	european	european	ADJ
ejpam-6037	295	2	journal	journal	PROPN
ejpam-6037	295	3	of	of	ADP
ejpam-6037	295	4	pure	pure	ADJ
ejpam-6037	295	5	and	and	CCONJ
ejpam-6037	295	6	applied	applied	ADJ
ejpam-6037	295	7	mathematics	mathematic	NOUN
ejpam-6037	295	8	,	,	PUNCT
ejpam-6037	295	9	15(2):572–588	15(2):572–588	NUM
ejpam-6037	295	10	,	,	PUNCT
ejpam-6037	295	11	2022	2022	NUM
ejpam-6037	295	12	.	.	PUNCT
ejpam-6037	296	1	[	[	X
ejpam-6037	296	2	3	3	X
ejpam-6037	296	3	]	]	PUNCT
ejpam-6037	296	4	t.	t.	NOUN
ejpam-6037	296	5	dungthaisong	dungthaisong	PROPN
ejpam-6037	296	6	,	,	PUNCT
ejpam-6037	296	7	c.	c.	PROPN
ejpam-6037	296	8	boonpok	boonpok	PROPN
ejpam-6037	296	9	,	,	PUNCT
ejpam-6037	296	10	and	and	CCONJ
ejpam-6037	296	11	c.	c.	PROPN
ejpam-6037	296	12	viriyapong	viriyapong	PROPN
ejpam-6037	296	13	.	.	PUNCT
ejpam-6037	297	1	generalized	generalize	VERB
ejpam-6037	297	2	closed	close	VERB
ejpam-6037	297	3	sets	set	NOUN
ejpam-6037	297	4	in	in	ADP
ejpam-6037	297	5	bigeneralized	bigeneralize	VERB
ejpam-6037	297	6	topological	topological	ADJ
ejpam-6037	297	7	spaces	space	NOUN
ejpam-6037	297	8	.	.	PUNCT
ejpam-6037	298	1	international	international	ADJ
ejpam-6037	298	2	journal	journal	PROPN
ejpam-6037	298	3	of	of	ADP
ejpam-6037	298	4	mathematical	mathematical	ADJ
ejpam-6037	298	5	analysis	analysis	NOUN
ejpam-6037	298	6	,	,	PUNCT
ejpam-6037	298	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-6037	298	8	,	,	PUNCT
ejpam-6037	298	9	2011	2011	NUM
ejpam-6037	298	10	.	.	PUNCT
ejpam-6037	299	1	[	[	X
ejpam-6037	299	2	4	4	X
ejpam-6037	299	3	]	]	PUNCT
ejpam-6037	299	4	t.	t.	PROPN
ejpam-6037	299	5	duangphui	duangphui	PROPN
ejpam-6037	299	6	,	,	PUNCT
ejpam-6037	299	7	c.	c.	PROPN
ejpam-6037	299	8	boonpok	boonpok	PROPN
ejpam-6037	299	9	,	,	PUNCT
ejpam-6037	299	10	and	and	CCONJ
ejpam-6037	299	11	c.	c.	PROPN
ejpam-6037	299	12	viriyapong	viriyapong	PROPN
ejpam-6037	299	13	.	.	PUNCT
ejpam-6037	300	1	continuous	continuous	ADJ
ejpam-6037	300	2	functions	function	NOUN
ejpam-6037	300	3	on	on	ADP
ejpam-6037	300	4	bigeneralized	bigeneralize	VERB
ejpam-6037	300	5	topological	topological	ADJ
ejpam-6037	300	6	spaces	space	NOUN
ejpam-6037	300	7	.	.	PUNCT
ejpam-6037	301	1	international	international	ADJ
ejpam-6037	301	2	journal	journal	PROPN
ejpam-6037	301	3	of	of	ADP
ejpam-6037	301	4	mathematical	mathematical	ADJ
ejpam-6037	301	5	analysis	analysis	NOUN
ejpam-6037	301	6	,	,	PUNCT
ejpam-6037	301	7	5(24):1165	5(24):1165	NUM
ejpam-6037	301	8	–	–	PUNCT
ejpam-6037	301	9	1174	1174	NUM
ejpam-6037	301	10	,	,	PUNCT
ejpam-6037	301	11	2011	2011	NUM
ejpam-6037	301	12	.	.	PUNCT
ejpam-6037	302	1	[	[	X
ejpam-6037	302	2	5	5	NUM
ejpam-6037	302	3	]	]	X
ejpam-6037	302	4	n.	n.	NOUN
ejpam-6037	302	5	srisarakham	srisarakham	PROPN
ejpam-6037	302	6	and	and	CCONJ
ejpam-6037	302	7	c.	c.	PROPN
ejpam-6037	302	8	boonpok	boonpok	PROPN
ejpam-6037	302	9	.	.	PUNCT
ejpam-6037	303	1	almost	almost	ADV
ejpam-6037	303	2	(	(	PUNCT
ejpam-6037	303	3	λ	λ	NOUN
ejpam-6037	303	4	,	,	PUNCT
ejpam-6037	303	5	p)-continuous	p)-continuous	ADJ
ejpam-6037	303	6	functions	function	NOUN
ejpam-6037	303	7	.	.	PUNCT
ejpam-6037	304	1	international	international	ADJ
ejpam-6037	304	2	journal	journal	PROPN
ejpam-6037	304	3	of	of	ADP
ejpam-6037	304	4	mathematics	mathematic	NOUN
ejpam-6037	304	5	and	and	CCONJ
ejpam-6037	304	6	computer	computer	NOUN
ejpam-6037	304	7	science	science	NOUN
ejpam-6037	304	8	,	,	PUNCT
ejpam-6037	304	9	18(2):255–259	18(2):255–259	NUM
ejpam-6037	304	10	,	,	PUNCT
ejpam-6037	304	11	2023	2023	NUM
ejpam-6037	304	12	.	.	PUNCT
ejpam-6037	305	1	[	[	X
ejpam-6037	305	2	6	6	NUM
ejpam-6037	305	3	]	]	PUNCT
ejpam-6037	305	4	m.	m.	NOUN
ejpam-6037	305	5	thongmoon	thongmoon	NOUN
ejpam-6037	305	6	and	and	CCONJ
ejpam-6037	305	7	c.	c.	PROPN
ejpam-6037	305	8	boonpok	boonpok	PROPN
ejpam-6037	305	9	.	.	PUNCT
ejpam-6037	306	1	strongly	strongly	ADV
ejpam-6037	306	2	θ(λ	θ(λ	PROPN
ejpam-6037	306	3	,	,	PUNCT
ejpam-6037	306	4	p)-continuous	p)-continuous	ADJ
ejpam-6037	306	5	functions	function	NOUN
ejpam-6037	306	6	.	.	PUNCT
ejpam-6037	307	1	international	international	ADJ
ejpam-6037	307	2	journal	journal	PROPN
ejpam-6037	307	3	of	of	ADP
ejpam-6037	307	4	mathematics	mathematic	NOUN
ejpam-6037	307	5	and	and	CCONJ
ejpam-6037	307	6	computer	computer	NOUN
ejpam-6037	307	7	science	science	NOUN
ejpam-6037	307	8	,	,	PUNCT
ejpam-6037	307	9	19(2):475–479	19(2):475–479	PROPN
ejpam-6037	307	10	,	,	PUNCT
ejpam-6037	307	11	2024	2024	NUM
ejpam-6037	307	12	.	.	PUNCT
ejpam-6037	308	1	[	[	X
ejpam-6037	308	2	7	7	X
ejpam-6037	308	3	]	]	X
ejpam-6037	308	4	c.	c.	PROPN
ejpam-6037	308	5	boonpok	boonpok	PROPN
ejpam-6037	308	6	and	and	CCONJ
ejpam-6037	308	7	j.	j.	PROPN
ejpam-6037	308	8	khampakdee	khampakdee	PROPN
ejpam-6037	308	9	.	.	PUNCT
ejpam-6037	309	1	almost	almost	ADV
ejpam-6037	309	2	strong	strong	ADJ
ejpam-6037	309	3	θ(λ	θ(λ	PROPN
ejpam-6037	309	4	,	,	PUNCT
ejpam-6037	309	5	p)-continuity	p)-continuity	NOUN
ejpam-6037	309	6	for	for	ADP
ejpam-6037	309	7	functions	function	NOUN
ejpam-6037	309	8	.	.	PUNCT
ejpam-6037	310	1	european	european	ADJ
ejpam-6037	310	2	journal	journal	PROPN
ejpam-6037	310	3	of	of	ADP
ejpam-6037	310	4	pure	pure	ADJ
ejpam-6037	310	5	and	and	CCONJ
ejpam-6037	310	6	applied	applied	ADJ
ejpam-6037	310	7	mathematics	mathematic	NOUN
ejpam-6037	310	8	,	,	PUNCT
ejpam-6037	310	9	17(1):300–309	17(1):300–309	PROPN
ejpam-6037	310	10	,	,	PUNCT
ejpam-6037	310	11	2024	2024	NUM
ejpam-6037	310	12	.	.	PUNCT
ejpam-6037	311	1	[	[	X
ejpam-6037	311	2	8	8	NUM
ejpam-6037	311	3	]	]	X
ejpam-6037	311	4	p.	p.	NOUN
ejpam-6037	311	5	pue	pue	NOUN
ejpam-6037	311	6	-	-	PUNCT
ejpam-6037	311	7	on	on	ADP
ejpam-6037	311	8	and	and	CCONJ
ejpam-6037	311	9	c.	c.	PROPN
ejpam-6037	311	10	boonpok	boonpok	PROPN
ejpam-6037	311	11	.	.	PUNCT
ejpam-6037	312	1	θ(λ	θ(λ	PROPN
ejpam-6037	312	2	,	,	PUNCT
ejpam-6037	312	3	p)-continuity	p)-continuity	NOUN
ejpam-6037	312	4	for	for	ADP
ejpam-6037	312	5	functions	function	NOUN
ejpam-6037	312	6	.	.	PUNCT
ejpam-6037	313	1	international	international	ADJ
ejpam-6037	313	2	journal	journal	NOUN
ejpam-6037	313	3	of	of	ADP
ejpam-6037	313	4	mathematics	mathematic	NOUN
ejpam-6037	313	5	and	and	CCONJ
ejpam-6037	313	6	computer	computer	NOUN
ejpam-6037	313	7	science	science	NOUN
ejpam-6037	313	8	,	,	PUNCT
ejpam-6037	313	9	19(2):491–495	19(2):491–495	NUM
ejpam-6037	313	10	,	,	PUNCT
ejpam-6037	313	11	2024	2024	NUM
ejpam-6037	313	12	.	.	PUNCT
ejpam-6037	314	1	[	[	X
ejpam-6037	314	2	9	9	NUM
ejpam-6037	314	3	]	]	PUNCT
ejpam-6037	314	4	c.	c.	NOUN
ejpam-6037	314	5	boonpok	boonpok	PROPN
ejpam-6037	314	6	and	and	CCONJ
ejpam-6037	314	7	n.	n.	PROPN
ejpam-6037	314	8	srisarakham	srisarakham	PROPN
ejpam-6037	314	9	.	.	PUNCT
ejpam-6037	315	1	weak	weak	ADJ
ejpam-6037	315	2	forms	form	NOUN
ejpam-6037	315	3	of	of	ADP
ejpam-6037	315	4	(	(	PUNCT
ejpam-6037	315	5	λ	λ	PROPN
ejpam-6037	315	6	,	,	PUNCT
ejpam-6037	315	7	b)-open	b)-open	VERB
ejpam-6037	315	8	sets	set	NOUN
ejpam-6037	315	9	and	and	CCONJ
ejpam-6037	315	10	weak	weak	ADJ
ejpam-6037	315	11	(	(	PUNCT
ejpam-6037	315	12	λ	λ	NOUN
ejpam-6037	315	13	,	,	PUNCT
ejpam-6037	315	14	b)continuity	b)continuity	NOUN
ejpam-6037	315	15	.	.	PUNCT
ejpam-6037	316	1	european	european	PROPN
ejpam-6037	316	2	journal	journal	PROPN
ejpam-6037	316	3	of	of	ADP
ejpam-6037	316	4	pure	pure	ADJ
ejpam-6037	316	5	and	and	CCONJ
ejpam-6037	316	6	applied	applied	ADJ
ejpam-6037	316	7	mathematics	mathematic	NOUN
ejpam-6037	316	8	,	,	PUNCT
ejpam-6037	316	9	16(1):29–43	16(1):29–43	NUM
ejpam-6037	316	10	,	,	PUNCT
ejpam-6037	316	11	2023	2023	NUM
ejpam-6037	316	12	.	.	PUNCT
ejpam-6037	317	1	[	[	X
ejpam-6037	317	2	10	10	NUM
ejpam-6037	317	3	]	]	X
ejpam-6037	317	4	c.	c.	PROPN
ejpam-6037	317	5	boonpok	boonpok	PROPN
ejpam-6037	317	6	.	.	PUNCT
ejpam-6037	318	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-6037	318	2	.	.	PUNCT
ejpam-6037	319	1	mathematica	mathematica	PROPN
ejpam-6037	319	2	,	,	PUNCT
ejpam-6037	319	3	65(1):31–42	65(1):31–42	NUM
ejpam-6037	319	4	,	,	PUNCT
ejpam-6037	319	5	2023	2023	NUM
ejpam-6037	319	6	.	.	PUNCT
ejpam-6037	320	1	[	[	X
ejpam-6037	320	2	11	11	NUM
ejpam-6037	320	3	]	]	PUNCT
ejpam-6037	320	4	c.	c.	PROPN
ejpam-6037	320	5	boonpok	boonpok	PROPN
ejpam-6037	320	6	.	.	PUNCT
ejpam-6037	321	1	on	on	ADP
ejpam-6037	321	2	some	some	DET
ejpam-6037	321	3	closed	closed	ADJ
ejpam-6037	321	4	sets	set	NOUN
ejpam-6037	321	5	and	and	CCONJ
ejpam-6037	321	6	low	low	ADJ
ejpam-6037	321	7	separation	separation	NOUN
ejpam-6037	321	8	axioms	axiom	NOUN
ejpam-6037	321	9	via	via	ADP
ejpam-6037	321	10	topological	topological	ADJ
ejpam-6037	321	11	ideals	ideal	NOUN
ejpam-6037	321	12	.	.	PUNCT
ejpam-6037	322	1	european	european	ADJ
ejpam-6037	322	2	journal	journal	PROPN
ejpam-6037	322	3	of	of	ADP
ejpam-6037	322	4	pure	pure	ADJ
ejpam-6037	322	5	and	and	CCONJ
ejpam-6037	322	6	applied	applied	ADJ
ejpam-6037	322	7	mathematics	mathematic	NOUN
ejpam-6037	322	8	,	,	PUNCT
ejpam-6037	322	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-6037	322	10	,	,	PUNCT
ejpam-6037	322	11	2022	2022	NUM
ejpam-6037	322	12	.	.	PUNCT
ejpam-6037	323	1	n.	n.	PROPN
ejpam-6037	323	2	chutiman	chutiman	PROPN
ejpam-6037	323	3	,	,	PUNCT
ejpam-6037	323	4	a.	a.	PROPN
ejpam-6037	323	5	sama	sama	PROPN
ejpam-6037	323	6	-	-	PUNCT
ejpam-6037	323	7	ae	ae	PROPN
ejpam-6037	323	8	,	,	PUNCT
ejpam-6037	323	9	c.	c.	PROPN
ejpam-6037	323	10	boonpok	boonpok	PROPN
ejpam-6037	323	11	/	/	SYM
ejpam-6037	323	12	eur	eur	PROPN
ejpam-6037	323	13	.	.	PUNCT
ejpam-6037	324	1	j.	j.	PROPN
ejpam-6037	324	2	pure	pure	PROPN
ejpam-6037	324	3	appl	appl	PROPN
ejpam-6037	324	4	.	.	PROPN
ejpam-6037	324	5	math	math	PROPN
ejpam-6037	324	6	,	,	PUNCT
ejpam-6037	324	7	18	18	NUM
ejpam-6037	324	8	(	(	PUNCT
ejpam-6037	324	9	2	2	NUM
ejpam-6037	324	10	)	)	PUNCT
ejpam-6037	324	11	(	(	PUNCT
ejpam-6037	324	12	2025	2025	NUM
ejpam-6037	324	13	)	)	PUNCT
ejpam-6037	324	14	,	,	PUNCT
ejpam-6037	324	15	6037	6037	NUM
ejpam-6037	324	16	11	11	NUM
ejpam-6037	324	17	of	of	ADP
ejpam-6037	324	18	12	12	NUM
ejpam-6037	324	19	[	[	X
ejpam-6037	324	20	12	12	NUM
ejpam-6037	324	21	]	]	PUNCT
ejpam-6037	324	22	c.	c.	PROPN
ejpam-6037	324	23	boonpok	boonpok	PROPN
ejpam-6037	324	24	.	.	PUNCT
ejpam-6037	325	1	on	on	ADP
ejpam-6037	325	2	some	some	DET
ejpam-6037	325	3	spaces	space	NOUN
ejpam-6037	325	4	via	via	ADP
ejpam-6037	325	5	topological	topological	ADJ
ejpam-6037	325	6	ideals	ideal	NOUN
ejpam-6037	325	7	.	.	PUNCT
ejpam-6037	326	1	open	open	ADJ
ejpam-6037	326	2	mathematics	mathematic	NOUN
ejpam-6037	326	3	,	,	PUNCT
ejpam-6037	326	4	21:20230118	21:20230118	NUM
ejpam-6037	326	5	,	,	PUNCT
ejpam-6037	326	6	2023	2023	NUM
ejpam-6037	326	7	.	.	PUNCT
ejpam-6037	327	1	[	[	X
ejpam-6037	327	2	13	13	NUM
ejpam-6037	327	3	]	]	PUNCT
ejpam-6037	327	4	c.	c.	PROPN
ejpam-6037	327	5	boonpok	boonpok	PROPN
ejpam-6037	327	6	.	.	PUNCT
ejpam-6037	328	1	on	on	ADP
ejpam-6037	328	2	characterizations	characterization	NOUN
ejpam-6037	328	3	of	of	ADP
ejpam-6037	328	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-6037	328	5	ideal	ideal	ADJ
ejpam-6037	328	6	topological	topological	ADJ
ejpam-6037	328	7	spaces	space	NOUN
ejpam-6037	328	8	.	.	PUNCT
ejpam-6037	329	1	journal	journal	NOUN
ejpam-6037	329	2	of	of	ADP
ejpam-6037	329	3	mathematics	mathematic	NOUN
ejpam-6037	329	4	,	,	PUNCT
ejpam-6037	329	5	2020:9387601	2020:9387601	NUM
ejpam-6037	329	6	,	,	PUNCT
ejpam-6037	329	7	2020	2020	NUM
ejpam-6037	329	8	.	.	PUNCT
ejpam-6037	330	1	[	[	X
ejpam-6037	330	2	14	14	NUM
ejpam-6037	330	3	]	]	X
ejpam-6037	330	4	c.	c.	PROPN
ejpam-6037	330	5	boonpok	boonpok	PROPN
ejpam-6037	330	6	.	.	PUNCT
ejpam-6037	331	1	almost	almost	ADV
ejpam-6037	331	2	(	(	PUNCT
ejpam-6037	331	3	g	g	NOUN
ejpam-6037	331	4	,	,	PUNCT
ejpam-6037	331	5	m)-continuous	m)-continuous	ADJ
ejpam-6037	331	6	functions	function	NOUN
ejpam-6037	331	7	.	.	PUNCT
ejpam-6037	332	1	international	international	ADJ
ejpam-6037	332	2	journal	journal	PROPN
ejpam-6037	332	3	of	of	ADP
ejpam-6037	332	4	mathematical	mathematical	ADJ
ejpam-6037	332	5	analysis	analysis	NOUN
ejpam-6037	332	6	,	,	PUNCT
ejpam-6037	332	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-6037	332	8	,	,	PUNCT
ejpam-6037	332	9	2010	2010	NUM
ejpam-6037	332	10	.	.	PUNCT
ejpam-6037	333	1	[	[	X
ejpam-6037	333	2	15	15	NUM
ejpam-6037	333	3	]	]	X
ejpam-6037	333	4	c.	c.	PROPN
ejpam-6037	333	5	boonpok	boonpok	PROPN
ejpam-6037	333	6	.	.	PUNCT
ejpam-6037	334	1	m	m	VERB
ejpam-6037	334	2	-continuous	-continuous	ADJ
ejpam-6037	334	3	functions	function	NOUN
ejpam-6037	334	4	in	in	ADP
ejpam-6037	334	5	biminimal	biminimal	NOUN
ejpam-6037	334	6	structure	structure	NOUN
ejpam-6037	334	7	spaces	space	NOUN
ejpam-6037	334	8	.	.	PUNCT
ejpam-6037	335	1	far	far	PROPN
ejpam-6037	335	2	east	east	PROPN
ejpam-6037	335	3	journal	journal	PROPN
ejpam-6037	335	4	of	of	ADP
ejpam-6037	335	5	mathematical	mathematical	ADJ
ejpam-6037	335	6	sciences	science	NOUN
ejpam-6037	335	7	,	,	PUNCT
ejpam-6037	335	8	43(1):41–58	43(1):41–58	NUM
ejpam-6037	335	9	,	,	PUNCT
ejpam-6037	335	10	2010	2010	NUM
ejpam-6037	335	11	.	.	PUNCT
ejpam-6037	336	1	[	[	X
ejpam-6037	336	2	16	16	NUM
ejpam-6037	336	3	]	]	X
ejpam-6037	336	4	n.	n.	PROPN
ejpam-6037	336	5	srisarakham	srisarakham	PROPN
ejpam-6037	336	6	,	,	PUNCT
ejpam-6037	336	7	a.	a.	PROPN
ejpam-6037	336	8	sama	sama	PROPN
ejpam-6037	336	9	-	-	PUNCT
ejpam-6037	336	10	ae	ae	PROPN
ejpam-6037	336	11	,	,	PUNCT
ejpam-6037	336	12	and	and	CCONJ
ejpam-6037	336	13	c.	c.	PROPN
ejpam-6037	336	14	boonpok	boonpok	PROPN
ejpam-6037	336	15	.	.	PUNCT
ejpam-6037	337	1	characterizations	characterization	NOUN
ejpam-6037	337	2	of	of	ADP
ejpam-6037	337	3	faintly	faintly	ADV
ejpam-6037	337	4	(	(	PUNCT
ejpam-6037	337	5	τ1	τ1	PROPN
ejpam-6037	337	6	,	,	PUNCT
ejpam-6037	337	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	337	8	functions	function	NOUN
ejpam-6037	337	9	.	.	PUNCT
ejpam-6037	338	1	european	european	ADJ
ejpam-6037	338	2	journal	journal	PROPN
ejpam-6037	338	3	of	of	ADP
ejpam-6037	338	4	pure	pure	ADJ
ejpam-6037	338	5	and	and	CCONJ
ejpam-6037	338	6	applied	applied	ADJ
ejpam-6037	338	7	mathematics	mathematic	NOUN
ejpam-6037	338	8	,	,	PUNCT
ejpam-6037	338	9	17(4):2753–2762	17(4):2753–2762	NUM
ejpam-6037	338	10	,	,	PUNCT
ejpam-6037	338	11	2024	2024	NUM
ejpam-6037	338	12	.	.	PUNCT
ejpam-6037	339	1	[	[	X
ejpam-6037	339	2	17	17	NUM
ejpam-6037	339	3	]	]	PUNCT
ejpam-6037	339	4	c.	c.	NOUN
ejpam-6037	339	5	prachanpol	prachanpol	NOUN
ejpam-6037	339	6	,	,	PUNCT
ejpam-6037	339	7	c.	c.	PROPN
ejpam-6037	339	8	boonpok	boonpok	PROPN
ejpam-6037	339	9	,	,	PUNCT
ejpam-6037	339	10	and	and	CCONJ
ejpam-6037	339	11	c.	c.	PROPN
ejpam-6037	339	12	viriyapong	viriyapong	PROPN
ejpam-6037	339	13	.	.	PUNCT
ejpam-6037	340	1	δ(τ1	δ(τ1	PROPN
ejpam-6037	340	2	,	,	PUNCT
ejpam-6037	340	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	340	4	functions	function	NOUN
ejpam-6037	340	5	.	.	PUNCT
ejpam-6037	341	1	european	european	ADJ
ejpam-6037	341	2	journal	journal	PROPN
ejpam-6037	341	3	of	of	ADP
ejpam-6037	341	4	pure	pure	ADJ
ejpam-6037	341	5	and	and	CCONJ
ejpam-6037	341	6	applied	applied	ADJ
ejpam-6037	341	7	mathematics	mathematic	NOUN
ejpam-6037	341	8	,	,	PUNCT
ejpam-6037	341	9	17(4):3730–3742	17(4):3730–3742	NUM
ejpam-6037	341	10	,	,	PUNCT
ejpam-6037	341	11	2024	2024	NUM
ejpam-6037	341	12	.	.	PUNCT
ejpam-6037	342	1	[	[	X
ejpam-6037	342	2	18	18	NUM
ejpam-6037	342	3	]	]	X
ejpam-6037	342	4	b.	b.	PROPN
ejpam-6037	342	5	kong	kong	PROPN
ejpam-6037	342	6	-	-	PUNCT
ejpam-6037	342	7	ied	ied	PROPN
ejpam-6037	342	8	,	,	PUNCT
ejpam-6037	342	9	a.	a.	PROPN
ejpam-6037	342	10	sama	sama	PROPN
ejpam-6037	342	11	-	-	PUNCT
ejpam-6037	342	12	ae	ae	PROPN
ejpam-6037	342	13	,	,	PUNCT
ejpam-6037	342	14	and	and	CCONJ
ejpam-6037	342	15	c.	c.	PROPN
ejpam-6037	342	16	boonpok	boonpok	PROPN
ejpam-6037	342	17	.	.	PUNCT
ejpam-6037	343	1	almost	almost	ADV
ejpam-6037	343	2	nearly	nearly	ADV
ejpam-6037	343	3	(	(	PUNCT
ejpam-6037	343	4	τ1	τ1	NOUN
ejpam-6037	343	5	,	,	PUNCT
ejpam-6037	343	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	343	7	functions	function	NOUN
ejpam-6037	343	8	.	.	PUNCT
ejpam-6037	344	1	international	international	ADJ
ejpam-6037	344	2	journal	journal	NOUN
ejpam-6037	344	3	of	of	ADP
ejpam-6037	344	4	analysis	analysis	NOUN
ejpam-6037	344	5	and	and	CCONJ
ejpam-6037	344	6	applications	application	NOUN
ejpam-6037	344	7	,	,	PUNCT
ejpam-6037	344	8	23:14	23:14	NUM
ejpam-6037	344	9	,	,	PUNCT
ejpam-6037	344	10	2025	2025	NUM
ejpam-6037	344	11	.	.	PUNCT
ejpam-6037	345	1	[	[	X
ejpam-6037	345	2	19	19	NUM
ejpam-6037	345	3	]	]	PUNCT
ejpam-6037	345	4	j.	j.	PROPN
ejpam-6037	345	5	dontchev	dontchev	PROPN
ejpam-6037	345	6	.	.	PUNCT
ejpam-6037	346	1	contra	contra	ADJ
ejpam-6037	346	2	-	-	ADJ
ejpam-6037	346	3	continuous	continuous	ADJ
ejpam-6037	346	4	functions	function	NOUN
ejpam-6037	346	5	and	and	CCONJ
ejpam-6037	346	6	strongly	strongly	ADV
ejpam-6037	346	7	s	s	NOUN
ejpam-6037	346	8	-	-	PUNCT
ejpam-6037	346	9	closed	closed	ADJ
ejpam-6037	346	10	spaces	space	NOUN
ejpam-6037	346	11	.	.	PUNCT
ejpam-6037	347	1	international	international	ADJ
ejpam-6037	347	2	journal	journal	PROPN
ejpam-6037	347	3	of	of	ADP
ejpam-6037	347	4	mathematics	mathematics	PROPN
ejpam-6037	347	5	and	and	CCONJ
ejpam-6037	347	6	mathematical	mathematical	ADJ
ejpam-6037	347	7	sciences	science	NOUN
ejpam-6037	347	8	,	,	PUNCT
ejpam-6037	347	9	19:303–310	19:303–310	PROPN
ejpam-6037	347	10	,	,	PUNCT
ejpam-6037	347	11	1966	1966	NUM
ejpam-6037	347	12	.	.	PUNCT
ejpam-6037	348	1	[	[	X
ejpam-6037	348	2	20	20	NUM
ejpam-6037	348	3	]	]	PUNCT
ejpam-6037	348	4	j.	j.	PROPN
ejpam-6037	348	5	dontchev	dontchev	PROPN
ejpam-6037	348	6	and	and	CCONJ
ejpam-6037	348	7	t.	t.	PROPN
ejpam-6037	348	8	noiri	noiri	PROPN
ejpam-6037	348	9	.	.	PUNCT
ejpam-6037	349	1	contra	contra	ADJ
ejpam-6037	349	2	-	-	ADJ
ejpam-6037	349	3	semicontinuous	semicontinuous	ADJ
ejpam-6037	349	4	functions	function	NOUN
ejpam-6037	349	5	.	.	PUNCT
ejpam-6037	350	1	mathematica	mathematica	PROPN
ejpam-6037	350	2	pannonica	pannonica	PROPN
ejpam-6037	350	3	,	,	PUNCT
ejpam-6037	350	4	10:159–168	10:159–168	NOUN
ejpam-6037	350	5	,	,	PUNCT
ejpam-6037	350	6	1999	1999	NUM
ejpam-6037	350	7	.	.	PUNCT
ejpam-6037	351	1	[	[	X
ejpam-6037	351	2	21	21	NUM
ejpam-6037	351	3	]	]	X
ejpam-6037	351	4	s.	s.	PROPN
ejpam-6037	351	5	jafari	jafari	PROPN
ejpam-6037	351	6	and	and	CCONJ
ejpam-6037	351	7	t.	t.	PROPN
ejpam-6037	351	8	noiri	noiri	PROPN
ejpam-6037	351	9	.	.	PUNCT
ejpam-6037	352	1	contra	contra	PROPN
ejpam-6037	352	2	-	-	ADJ
ejpam-6037	352	3	super	super	ADJ
ejpam-6037	352	4	-	-	ADJ
ejpam-6037	352	5	continuous	continuous	ADJ
ejpam-6037	352	6	functions	function	NOUN
ejpam-6037	352	7	.	.	PUNCT
ejpam-6037	353	1	annales	annales	PROPN
ejpam-6037	353	2	universitatis	universitatis	PROPN
ejpam-6037	353	3	scientiarum	scientiarum	PROPN
ejpam-6037	353	4	budapestinensis	budapestinensis	PROPN
ejpam-6037	353	5	de	de	PROPN
ejpam-6037	353	6	rolando	rolando	PROPN
ejpam-6037	353	7	eötvös	eötvös	PROPN
ejpam-6037	353	8	,	,	PUNCT
ejpam-6037	353	9	sectio	sectio	PROPN
ejpam-6037	353	10	mathematica	mathematica	PROPN
ejpam-6037	353	11	,	,	PUNCT
ejpam-6037	353	12	42:27–34	42:27–34	PROPN
ejpam-6037	353	13	,	,	PUNCT
ejpam-6037	353	14	1999	1999	NUM
ejpam-6037	353	15	.	.	PUNCT
ejpam-6037	354	1	[	[	X
ejpam-6037	354	2	22	22	NUM
ejpam-6037	354	3	]	]	PUNCT
ejpam-6037	354	4	s.	s.	PROPN
ejpam-6037	354	5	jafari	jafari	PROPN
ejpam-6037	354	6	and	and	CCONJ
ejpam-6037	354	7	t.	t.	PROPN
ejpam-6037	354	8	noiri	noiri	PROPN
ejpam-6037	354	9	.	.	PUNCT
ejpam-6037	355	1	on	on	ADP
ejpam-6037	355	2	contra	contra	ADJ
ejpam-6037	355	3	-	-	ADJ
ejpam-6037	355	4	precontinuous	precontinuous	ADJ
ejpam-6037	355	5	functions	function	NOUN
ejpam-6037	355	6	.	.	PUNCT
ejpam-6037	356	1	bulletin	bulletin	NOUN
ejpam-6037	356	2	of	of	ADP
ejpam-6037	356	3	the	the	DET
ejpam-6037	356	4	malaysian	malaysian	PROPN
ejpam-6037	356	5	mathematical	mathematical	PROPN
ejpam-6037	356	6	sciences	sciences	PROPN
ejpam-6037	356	7	society	society	NOUN
ejpam-6037	356	8	,	,	PUNCT
ejpam-6037	356	9	25:115–128	25:115–128	PROPN
ejpam-6037	356	10	,	,	PUNCT
ejpam-6037	356	11	2002	2002	NUM
ejpam-6037	356	12	.	.	PUNCT
ejpam-6037	357	1	[	[	X
ejpam-6037	357	2	23	23	NUM
ejpam-6037	357	3	]	]	X
ejpam-6037	357	4	e.	e.	PROPN
ejpam-6037	357	5	ekici	ekici	PROPN
ejpam-6037	357	6	.	.	PUNCT
ejpam-6037	358	1	almost	almost	ADV
ejpam-6037	358	2	contra	contra	ADJ
ejpam-6037	358	3	-	-	ADJ
ejpam-6037	358	4	precontinuous	precontinuous	ADJ
ejpam-6037	358	5	functions	function	NOUN
ejpam-6037	358	6	.	.	PUNCT
ejpam-6037	359	1	bulletin	bulletin	NOUN
ejpam-6037	359	2	of	of	ADP
ejpam-6037	359	3	the	the	DET
ejpam-6037	359	4	malaysian	malaysian	PROPN
ejpam-6037	359	5	mathematical	mathematical	PROPN
ejpam-6037	359	6	sciences	sciences	PROPN
ejpam-6037	359	7	society	society	NOUN
ejpam-6037	359	8	,	,	PUNCT
ejpam-6037	359	9	27:53–65	27:53–65	NUM
ejpam-6037	359	10	,	,	PUNCT
ejpam-6037	359	11	2004	2004	NUM
ejpam-6037	359	12	.	.	PUNCT
ejpam-6037	360	1	[	[	X
ejpam-6037	360	2	24	24	NUM
ejpam-6037	360	3	]	]	PUNCT
ejpam-6037	360	4	j.	j.	PROPN
ejpam-6037	360	5	dontchev	dontchev	PROPN
ejpam-6037	360	6	,	,	PUNCT
ejpam-6037	360	7	m.	m.	NOUN
ejpam-6037	360	8	ganster	ganster	NOUN
ejpam-6037	360	9	,	,	PUNCT
ejpam-6037	360	10	and	and	CCONJ
ejpam-6037	360	11	i.	i.	PROPN
ejpam-6037	360	12	reilly	reilly	PROPN
ejpam-6037	360	13	.	.	PUNCT
ejpam-6037	361	1	more	more	ADV
ejpam-6037	361	2	on	on	ADP
ejpam-6037	361	3	almost	almost	ADV
ejpam-6037	361	4	s	s	NOUN
ejpam-6037	361	5	-	-	NOUN
ejpam-6037	361	6	continuity	continuity	NOUN
ejpam-6037	361	7	.	.	PUNCT
ejpam-6037	362	1	indian	indian	ADJ
ejpam-6037	362	2	journal	journal	PROPN
ejpam-6037	362	3	of	of	ADP
ejpam-6037	362	4	mathematics	mathematics	PROPN
ejpam-6037	362	5	,	,	PUNCT
ejpam-6037	362	6	41:139–146	41:139–146	PROPN
ejpam-6037	362	7	,	,	PUNCT
ejpam-6037	362	8	1999	1999	NUM
ejpam-6037	362	9	.	.	PUNCT
ejpam-6037	363	1	[	[	X
ejpam-6037	363	2	25	25	NUM
ejpam-6037	363	3	]	]	PUNCT
ejpam-6037	363	4	t.	t.	PROPN
ejpam-6037	363	5	noiri	noiri	PROPN
ejpam-6037	363	6	,	,	PUNCT
ejpam-6037	363	7	b.	b.	PROPN
ejpam-6037	363	8	ahmad	ahmad	PROPN
ejpam-6037	363	9	,	,	PUNCT
ejpam-6037	363	10	and	and	CCONJ
ejpam-6037	363	11	m.	m.	PROPN
ejpam-6037	363	12	khan	khan	PROPN
ejpam-6037	363	13	.	.	PUNCT
ejpam-6037	364	1	almost	almost	ADV
ejpam-6037	364	2	s	s	NOUN
ejpam-6037	364	3	-	-	PUNCT
ejpam-6037	364	4	continuous	continuous	ADJ
ejpam-6037	364	5	functions	function	NOUN
ejpam-6037	364	6	.	.	PUNCT
ejpam-6037	365	1	kyungpook	kyungpook	PROPN
ejpam-6037	365	2	mathematical	mathematical	PROPN
ejpam-6037	365	3	journal	journal	PROPN
ejpam-6037	365	4	,	,	PUNCT
ejpam-6037	365	5	35:311–322	35:311–322	PROPN
ejpam-6037	365	6	,	,	PUNCT
ejpam-6037	365	7	1995	1995	NUM
ejpam-6037	365	8	.	.	PUNCT
ejpam-6037	366	1	[	[	X
ejpam-6037	366	2	26	26	NUM
ejpam-6037	366	3	]	]	PUNCT
ejpam-6037	366	4	t.	t.	PROPN
ejpam-6037	366	5	noiri	noiri	PROPN
ejpam-6037	366	6	.	.	PUNCT
ejpam-6037	367	1	super	super	ADJ
ejpam-6037	367	2	-	-	NOUN
ejpam-6037	367	3	continuity	continuity	NOUN
ejpam-6037	367	4	and	and	CCONJ
ejpam-6037	367	5	some	some	DET
ejpam-6037	367	6	strong	strong	ADJ
ejpam-6037	367	7	forms	form	NOUN
ejpam-6037	367	8	of	of	ADP
ejpam-6037	367	9	continuity	continuity	NOUN
ejpam-6037	367	10	.	.	PUNCT
ejpam-6037	368	1	indian	indian	ADJ
ejpam-6037	368	2	journal	journal	PROPN
ejpam-6037	368	3	of	of	ADP
ejpam-6037	368	4	pure	pure	ADJ
ejpam-6037	368	5	and	and	CCONJ
ejpam-6037	368	6	applied	applied	ADJ
ejpam-6037	368	7	mathematics	mathematic	NOUN
ejpam-6037	368	8	,	,	PUNCT
ejpam-6037	368	9	15:241–250	15:241–250	NUM
ejpam-6037	368	10	,	,	PUNCT
ejpam-6037	368	11	1984	1984	NUM
ejpam-6037	368	12	.	.	PUNCT
ejpam-6037	369	1	[	[	X
ejpam-6037	369	2	27	27	NUM
ejpam-6037	369	3	]	]	PUNCT
ejpam-6037	369	4	a.	a.	PROPN
ejpam-6037	369	5	al	al	PROPN
ejpam-6037	369	6	-	-	PUNCT
ejpam-6037	369	7	omari	omari	PROPN
ejpam-6037	369	8	and	and	CCONJ
ejpam-6037	369	9	m.	m.	PROPN
ejpam-6037	369	10	s.	s.	PROPN
ejpam-6037	369	11	m.	m.	PROPN
ejpam-6037	369	12	noorani	noorani	PROPN
ejpam-6037	369	13	.	.	PUNCT
ejpam-6037	370	1	contra	contra	PROPN
ejpam-6037	370	2	ω	ω	PROPN
ejpam-6037	370	3	-	-	ADJ
ejpam-6037	370	4	continuous	continuous	ADJ
ejpam-6037	370	5	and	and	CCONJ
ejpam-6037	370	6	almost	almost	ADV
ejpam-6037	370	7	contra	contra	PROPN
ejpam-6037	370	8	ωcontinuous	ωcontinuous	ADJ
ejpam-6037	370	9	functions	function	NOUN
ejpam-6037	370	10	.	.	PUNCT
ejpam-6037	371	1	international	international	ADJ
ejpam-6037	371	2	journal	journal	PROPN
ejpam-6037	371	3	of	of	ADP
ejpam-6037	371	4	mathematics	mathematics	PROPN
ejpam-6037	371	5	and	and	CCONJ
ejpam-6037	371	6	mathematical	mathematical	ADJ
ejpam-6037	371	7	sciences	science	NOUN
ejpam-6037	371	8	,	,	PUNCT
ejpam-6037	371	9	2007:40469	2007:40469	NUM
ejpam-6037	371	10	,	,	PUNCT
ejpam-6037	371	11	2007	2007	NUM
ejpam-6037	371	12	.	.	PUNCT
ejpam-6037	372	1	[	[	X
ejpam-6037	372	2	28	28	NUM
ejpam-6037	372	3	]	]	X
ejpam-6037	372	4	t.	t.	PROPN
ejpam-6037	372	5	noiri	noiri	PROPN
ejpam-6037	372	6	and	and	CCONJ
ejpam-6037	372	7	s.	s.	PROPN
ejpam-6037	372	8	jafari	jafari	PROPN
ejpam-6037	372	9	.	.	PUNCT
ejpam-6037	373	1	some	some	DET
ejpam-6037	373	2	properties	property	NOUN
ejpam-6037	373	3	of	of	ADP
ejpam-6037	373	4	almost	almost	ADV
ejpam-6037	373	5	contra	contra	ADJ
ejpam-6037	373	6	-	-	ADJ
ejpam-6037	373	7	precontinuous	precontinuous	ADJ
ejpam-6037	373	8	functions	function	NOUN
ejpam-6037	373	9	.	.	PUNCT
ejpam-6037	374	1	bulletin	bulletin	NOUN
ejpam-6037	374	2	of	of	ADP
ejpam-6037	374	3	the	the	DET
ejpam-6037	374	4	malaysian	malaysian	PROPN
ejpam-6037	374	5	mathematical	mathematical	PROPN
ejpam-6037	374	6	sciences	sciences	PROPN
ejpam-6037	374	7	society	society	NOUN
ejpam-6037	374	8	,	,	PUNCT
ejpam-6037	374	9	28:107–116	28:107–116	PROPN
ejpam-6037	374	10	,	,	PUNCT
ejpam-6037	374	11	2005	2005	NUM
ejpam-6037	374	12	.	.	PUNCT
ejpam-6037	375	1	[	[	X
ejpam-6037	375	2	29	29	NUM
ejpam-6037	375	3	]	]	X
ejpam-6037	375	4	c.	c.	PROPN
ejpam-6037	375	5	w.	w.	PROPN
ejpam-6037	375	6	baker	baker	PROPN
ejpam-6037	375	7	.	.	PUNCT
ejpam-6037	376	1	weakly	weakly	ADJ
ejpam-6037	376	2	contra	contra	ADJ
ejpam-6037	376	3	-	-	ADJ
ejpam-6037	376	4	continuous	continuous	ADJ
ejpam-6037	376	5	functions	function	NOUN
ejpam-6037	376	6	.	.	PUNCT
ejpam-6037	377	1	international	international	ADJ
ejpam-6037	377	2	journal	journal	NOUN
ejpam-6037	377	3	of	of	ADP
ejpam-6037	377	4	pure	pure	ADJ
ejpam-6037	377	5	and	and	CCONJ
ejpam-6037	377	6	applied	applied	ADJ
ejpam-6037	377	7	mathematics	mathematic	NOUN
ejpam-6037	377	8	,	,	PUNCT
ejpam-6037	377	9	40:265–271	40:265–271	PROPN
ejpam-6037	377	10	,	,	PUNCT
ejpam-6037	377	11	2007	2007	NUM
ejpam-6037	377	12	.	.	PUNCT
ejpam-6037	378	1	[	[	X
ejpam-6037	378	2	30	30	NUM
ejpam-6037	378	3	]	]	X
ejpam-6037	378	4	c.	c.	PROPN
ejpam-6037	378	5	boonpok	boonpok	PROPN
ejpam-6037	378	6	and	and	CCONJ
ejpam-6037	378	7	n.	n.	PROPN
ejpam-6037	378	8	srisarakham	srisarakham	PROPN
ejpam-6037	378	9	.	.	PUNCT
ejpam-6037	379	1	(	(	PUNCT
ejpam-6037	379	2	τ1	τ1	NOUN
ejpam-6037	379	3	,	,	PUNCT
ejpam-6037	379	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6037	379	5	for	for	ADP
ejpam-6037	379	6	functions	function	NOUN
ejpam-6037	379	7	.	.	PUNCT
ejpam-6037	380	1	asia	asia	PROPN
ejpam-6037	380	2	pacific	pacific	PROPN
ejpam-6037	380	3	journal	journal	PROPN
ejpam-6037	380	4	of	of	ADP
ejpam-6037	380	5	mathematics	mathematic	NOUN
ejpam-6037	380	6	,	,	PUNCT
ejpam-6037	380	7	11:21	11:21	NUM
ejpam-6037	380	8	,	,	PUNCT
ejpam-6037	380	9	2024	2024	NUM
ejpam-6037	380	10	.	.	PUNCT
ejpam-6037	381	1	[	[	X
ejpam-6037	381	2	31	31	NUM
ejpam-6037	381	3	]	]	PUNCT
ejpam-6037	381	4	c.	c.	PROPN
ejpam-6037	381	5	boonpok	boonpok	PROPN
ejpam-6037	381	6	and	and	CCONJ
ejpam-6037	381	7	p.	p.	NOUN
ejpam-6037	381	8	pue	pue	NOUN
ejpam-6037	381	9	-	-	PUNCT
ejpam-6037	381	10	on	on	ADP
ejpam-6037	381	11	.	.	PUNCT
ejpam-6037	382	1	characterizations	characterization	NOUN
ejpam-6037	382	2	of	of	ADP
ejpam-6037	382	3	almost	almost	ADV
ejpam-6037	382	4	(	(	PUNCT
ejpam-6037	382	5	τ1	τ1	NOUN
ejpam-6037	382	6	,	,	PUNCT
ejpam-6037	382	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	382	8	functions	function	NOUN
ejpam-6037	382	9	.	.	PUNCT
ejpam-6037	383	1	international	international	ADJ
ejpam-6037	383	2	journal	journal	NOUN
ejpam-6037	383	3	of	of	ADP
ejpam-6037	383	4	analysis	analysis	NOUN
ejpam-6037	383	5	and	and	CCONJ
ejpam-6037	383	6	applications	application	NOUN
ejpam-6037	383	7	,	,	PUNCT
ejpam-6037	383	8	22:33	22:33	NUM
ejpam-6037	383	9	,	,	PUNCT
ejpam-6037	383	10	2024	2024	NUM
ejpam-6037	383	11	.	.	PUNCT
ejpam-6037	384	1	[	[	X
ejpam-6037	384	2	32	32	NUM
ejpam-6037	384	3	]	]	PUNCT
ejpam-6037	384	4	c.	c.	PROPN
ejpam-6037	384	5	boonpok	boonpok	PROPN
ejpam-6037	384	6	and	and	CCONJ
ejpam-6037	384	7	c.	c.	PROPN
ejpam-6037	384	8	klanarong	klanarong	PROPN
ejpam-6037	384	9	.	.	PUNCT
ejpam-6037	385	1	on	on	ADP
ejpam-6037	385	2	weakly	weakly	ADJ
ejpam-6037	385	3	(	(	PUNCT
ejpam-6037	385	4	τ1	τ1	NOUN
ejpam-6037	385	5	,	,	PUNCT
ejpam-6037	385	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	385	7	functions	function	NOUN
ejpam-6037	385	8	.	.	PUNCT
ejpam-6037	386	1	european	european	PROPN
ejpam-6037	386	2	n.	n.	PROPN
ejpam-6037	386	3	chutiman	chutiman	PROPN
ejpam-6037	386	4	,	,	PUNCT
ejpam-6037	386	5	a.	a.	PROPN
ejpam-6037	386	6	sama	sama	PROPN
ejpam-6037	386	7	-	-	PUNCT
ejpam-6037	386	8	ae	ae	PROPN
ejpam-6037	386	9	,	,	PUNCT
ejpam-6037	386	10	c.	c.	PROPN
ejpam-6037	386	11	boonpok	boonpok	PROPN
ejpam-6037	386	12	/	/	SYM
ejpam-6037	386	13	eur	eur	PROPN
ejpam-6037	386	14	.	.	PUNCT
ejpam-6037	387	1	j.	j.	PROPN
ejpam-6037	387	2	pure	pure	PROPN
ejpam-6037	387	3	appl	appl	PROPN
ejpam-6037	387	4	.	.	PROPN
ejpam-6037	387	5	math	math	PROPN
ejpam-6037	387	6	,	,	PUNCT
ejpam-6037	387	7	18	18	NUM
ejpam-6037	387	8	(	(	PUNCT
ejpam-6037	387	9	2	2	NUM
ejpam-6037	387	10	)	)	PUNCT
ejpam-6037	387	11	(	(	PUNCT
ejpam-6037	387	12	2025	2025	NUM
ejpam-6037	387	13	)	)	PUNCT
ejpam-6037	387	14	,	,	PUNCT
ejpam-6037	387	15	6037	6037	NUM
ejpam-6037	387	16	12	12	NUM
ejpam-6037	387	17	of	of	ADP
ejpam-6037	387	18	12	12	NUM
ejpam-6037	387	19	journal	journal	NOUN
ejpam-6037	387	20	of	of	ADP
ejpam-6037	387	21	pure	pure	ADJ
ejpam-6037	387	22	and	and	CCONJ
ejpam-6037	387	23	applied	applied	ADJ
ejpam-6037	387	24	mathematics	mathematic	NOUN
ejpam-6037	387	25	,	,	PUNCT
ejpam-6037	387	26	17(1):416–425	17(1):416–425	NUM
ejpam-6037	387	27	,	,	PUNCT
ejpam-6037	387	28	2024	2024	NUM
ejpam-6037	387	29	.	.	PUNCT
ejpam-6037	388	1	[	[	X
ejpam-6037	388	2	33	33	NUM
ejpam-6037	388	3	]	]	X
ejpam-6037	388	4	n.	n.	PROPN
ejpam-6037	388	5	srisarakham	srisarakham	PROPN
ejpam-6037	388	6	,	,	PUNCT
ejpam-6037	388	7	s.	s.	PROPN
ejpam-6037	388	8	sompong	sompong	PROPN
ejpam-6037	388	9	,	,	PUNCT
ejpam-6037	388	10	and	and	CCONJ
ejpam-6037	388	11	c.	c.	PROPN
ejpam-6037	388	12	boonpok	boonpok	PROPN
ejpam-6037	388	13	.	.	PUNCT
ejpam-6037	389	1	quasi	quasi	PROPN
ejpam-6037	389	2	θ(τ1	θ(τ1	PROPN
ejpam-6037	389	3	,	,	PUNCT
ejpam-6037	389	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	389	5	functions	function	NOUN
ejpam-6037	389	6	.	.	PUNCT
ejpam-6037	390	1	european	european	ADJ
ejpam-6037	390	2	journal	journal	PROPN
ejpam-6037	390	3	of	of	ADP
ejpam-6037	390	4	pure	pure	ADJ
ejpam-6037	390	5	and	and	CCONJ
ejpam-6037	390	6	applied	applied	ADJ
ejpam-6037	390	7	mathematics	mathematic	NOUN
ejpam-6037	390	8	,	,	PUNCT
ejpam-6037	390	9	18(1):5722	18(1):5722	NUM
ejpam-6037	390	10	,	,	PUNCT
ejpam-6037	390	11	2025	2025	NUM
ejpam-6037	390	12	.	.	PUNCT
ejpam-6037	391	1	[	[	X
ejpam-6037	391	2	34	34	NUM
ejpam-6037	391	3	]	]	X
ejpam-6037	391	4	b.	b.	PROPN
ejpam-6037	391	5	kong	kong	PROPN
ejpam-6037	391	6	-	-	PUNCT
ejpam-6037	391	7	ied	ied	PROPN
ejpam-6037	391	8	,	,	PUNCT
ejpam-6037	391	9	s.	s.	PROPN
ejpam-6037	391	10	sompong	sompong	PROPN
ejpam-6037	391	11	,	,	PUNCT
ejpam-6037	391	12	and	and	CCONJ
ejpam-6037	391	13	c.	c.	PROPN
ejpam-6037	391	14	boonpok	boonpok	PROPN
ejpam-6037	391	15	.	.	PUNCT
ejpam-6037	392	1	almost	almost	ADV
ejpam-6037	392	2	quasi	quasi	X
ejpam-6037	392	3	(	(	PUNCT
ejpam-6037	392	4	τ1	τ1	NOUN
ejpam-6037	392	5	,	,	PUNCT
ejpam-6037	392	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	392	7	functions	function	NOUN
ejpam-6037	392	8	.	.	PUNCT
ejpam-6037	393	1	asia	asia	PROPN
ejpam-6037	393	2	pacific	pacific	PROPN
ejpam-6037	393	3	journal	journal	PROPN
ejpam-6037	393	4	of	of	ADP
ejpam-6037	393	5	mathematics	mathematic	NOUN
ejpam-6037	393	6	,	,	PUNCT
ejpam-6037	393	7	11:64	11:64	NUM
ejpam-6037	393	8	,	,	PUNCT
ejpam-6037	393	9	2024	2024	NUM
ejpam-6037	393	10	.	.	PUNCT
ejpam-6037	394	1	[	[	X
ejpam-6037	394	2	35	35	NUM
ejpam-6037	394	3	]	]	PUNCT
ejpam-6037	394	4	m.	m.	NOUN
ejpam-6037	394	5	chiangpradit	chiangpradit	NOUN
ejpam-6037	394	6	,	,	PUNCT
ejpam-6037	394	7	s.	s.	PROPN
ejpam-6037	394	8	sompong	sompong	PROPN
ejpam-6037	394	9	,	,	PUNCT
ejpam-6037	394	10	and	and	CCONJ
ejpam-6037	394	11	c.	c.	PROPN
ejpam-6037	394	12	boonpok	boonpok	PROPN
ejpam-6037	394	13	.	.	PUNCT
ejpam-6037	395	1	weakly	weakly	ADJ
ejpam-6037	395	2	quasi	quasi	NOUN
ejpam-6037	395	3	(	(	PUNCT
ejpam-6037	395	4	τ1	τ1	PROPN
ejpam-6037	395	5	,	,	PUNCT
ejpam-6037	395	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	395	7	functions	function	NOUN
ejpam-6037	395	8	.	.	PUNCT
ejpam-6037	396	1	international	international	ADJ
ejpam-6037	396	2	journal	journal	NOUN
ejpam-6037	396	3	of	of	ADP
ejpam-6037	396	4	analysis	analysis	NOUN
ejpam-6037	396	5	and	and	CCONJ
ejpam-6037	396	6	applications	application	NOUN
ejpam-6037	396	7	,	,	PUNCT
ejpam-6037	396	8	22:125	22:125	NUM
ejpam-6037	396	9	,	,	PUNCT
ejpam-6037	396	10	2024	2024	NUM
ejpam-6037	396	11	.	.	PUNCT
ejpam-6037	397	1	[	[	X
ejpam-6037	397	2	36	36	NUM
ejpam-6037	397	3	]	]	X
ejpam-6037	397	4	j.	j.	PROPN
ejpam-6037	397	5	khampakdee	khampakdee	PROPN
ejpam-6037	397	6	,	,	PUNCT
ejpam-6037	397	7	s.	s.	PROPN
ejpam-6037	397	8	sompong	sompong	PROPN
ejpam-6037	397	9	,	,	PUNCT
ejpam-6037	397	10	and	and	CCONJ
ejpam-6037	397	11	c.	c.	PROPN
ejpam-6037	397	12	boonpok	boonpok	PROPN
ejpam-6037	397	13	.	.	PUNCT
ejpam-6037	398	1	almost	almost	ADV
ejpam-6037	398	2	weakly	weakly	ADJ
ejpam-6037	398	3	(	(	PUNCT
ejpam-6037	398	4	τ1	τ1	NOUN
ejpam-6037	398	5	,	,	PUNCT
ejpam-6037	398	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6037	398	7	functions	function	NOUN
ejpam-6037	398	8	.	.	PUNCT
ejpam-6037	399	1	european	european	ADJ
ejpam-6037	399	2	journal	journal	PROPN
ejpam-6037	399	3	of	of	ADP
ejpam-6037	399	4	pure	pure	ADJ
ejpam-6037	399	5	and	and	CCONJ
ejpam-6037	399	6	applied	applied	ADJ
ejpam-6037	399	7	mathematics	mathematic	NOUN
ejpam-6037	399	8	,	,	PUNCT
ejpam-6037	399	9	18(1):5721	18(1):5721	NUM
ejpam-6037	399	10	,	,	PUNCT
ejpam-6037	399	11	2025	2025	NUM
ejpam-6037	399	12	.	.	PUNCT
ejpam-6037	400	1	[	[	X
ejpam-6037	400	2	37	37	NUM
ejpam-6037	400	3	]	]	PUNCT
ejpam-6037	400	4	c.	c.	PROPN
ejpam-6037	400	5	boonpok	boonpok	PROPN
ejpam-6037	400	6	and	and	CCONJ
ejpam-6037	400	7	j.	j.	PROPN
ejpam-6037	400	8	khampakdee	khampakdee	PROPN
ejpam-6037	400	9	.	.	PUNCT
ejpam-6037	401	1	upper	upper	ADJ
ejpam-6037	401	2	and	and	CCONJ
ejpam-6037	401	3	lower	low	ADJ
ejpam-6037	401	4	almost	almost	ADV
ejpam-6037	401	5	contra-(λ	contra-(λ	PROPN
ejpam-6037	401	6	,	,	PUNCT
ejpam-6037	401	7	sp)-continuity	sp)-continuity	NOUN
ejpam-6037	401	8	.	.	PUNCT
ejpam-6037	402	1	european	european	PROPN
ejpam-6037	402	2	journal	journal	PROPN
ejpam-6037	402	3	of	of	ADP
ejpam-6037	402	4	pure	pure	ADJ
ejpam-6037	402	5	and	and	CCONJ
ejpam-6037	402	6	applied	applied	ADJ
ejpam-6037	402	7	mathematics	mathematic	NOUN
ejpam-6037	402	8	,	,	PUNCT
ejpam-6037	402	9	16(1):156–168	16(1):156–168	PROPN
ejpam-6037	402	10	,	,	PUNCT
ejpam-6037	402	11	2023	2023	NUM
ejpam-6037	402	12	.	.	PUNCT
ejpam-6037	403	1	[	[	X
ejpam-6037	403	2	38	38	NUM
ejpam-6037	403	3	]	]	PUNCT
ejpam-6037	403	4	c.	c.	PROPN
ejpam-6037	403	5	boonpok	boonpok	PROPN
ejpam-6037	403	6	,	,	PUNCT
ejpam-6037	403	7	c.	c.	PROPN
ejpam-6037	403	8	viriyapong	viriyapong	PROPN
ejpam-6037	403	9	,	,	PUNCT
ejpam-6037	403	10	and	and	CCONJ
ejpam-6037	403	11	m.	m.	NOUN
ejpam-6037	403	12	thongmoon	thongmoon	NOUN
ejpam-6037	403	13	.	.	PUNCT
ejpam-6037	404	1	on	on	ADP
ejpam-6037	404	2	upper	upper	ADJ
ejpam-6037	404	3	and	and	CCONJ
ejpam-6037	404	4	lower	low	ADJ
ejpam-6037	404	5	(	(	PUNCT
ejpam-6037	404	6	τ1	τ1	NOUN
ejpam-6037	404	7	,	,	PUNCT
ejpam-6037	404	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6037	404	9	multifunctions	multifunction	NOUN
ejpam-6037	404	10	.	.	PUNCT
ejpam-6037	405	1	journal	journal	PROPN
ejpam-6037	405	2	of	of	ADP
ejpam-6037	405	3	mathematics	mathematics	PROPN
ejpam-6037	405	4	and	and	CCONJ
ejpam-6037	405	5	computer	computer	NOUN
ejpam-6037	405	6	science	science	NOUN
ejpam-6037	405	7	,	,	PUNCT
ejpam-6037	405	8	18:282–293	18:282–293	NUM
ejpam-6037	405	9	,	,	PUNCT
ejpam-6037	405	10	2018	2018	NUM
ejpam-6037	405	11	.	.	PUNCT
ejpam-6037	406	1	[	[	X
ejpam-6037	406	2	39	39	NUM
ejpam-6037	406	3	]	]	PUNCT
ejpam-6037	406	4	c.	c.	PROPN
ejpam-6037	406	5	viriyapong	viriyapong	PROPN
ejpam-6037	406	6	and	and	CCONJ
ejpam-6037	406	7	c.	c.	PROPN
ejpam-6037	406	8	boonpok	boonpok	PROPN
ejpam-6037	406	9	.	.	PUNCT
ejpam-6037	407	1	(	(	PUNCT
ejpam-6037	407	2	τ1	τ1	NOUN
ejpam-6037	407	3	,	,	PUNCT
ejpam-6037	407	4	τ2)α	τ2)α	NOUN
ejpam-6037	407	5	-	-	PUNCT
ejpam-6037	407	6	continuity	continuity	NOUN
ejpam-6037	407	7	for	for	ADP
ejpam-6037	407	8	multifunctions	multifunction	NOUN
ejpam-6037	407	9	.	.	PUNCT
ejpam-6037	408	1	journal	journal	PROPN
ejpam-6037	408	2	of	of	ADP
ejpam-6037	408	3	mathematics	mathematic	NOUN
ejpam-6037	408	4	,	,	PUNCT
ejpam-6037	408	5	2020:6285763	2020:6285763	NUM
ejpam-6037	408	6	,	,	PUNCT
ejpam-6037	408	7	2020	2020	NUM
ejpam-6037	408	8	.	.	PUNCT
ejpam-6037	409	1	[	[	X
ejpam-6037	409	2	40	40	NUM
ejpam-6037	409	3	]	]	PUNCT
ejpam-6037	409	4	c.	c.	PROPN
ejpam-6037	409	5	boonpok	boonpok	PROPN
ejpam-6037	409	6	.	.	PUNCT
ejpam-6037	410	1	(	(	PUNCT
ejpam-6037	410	2	τ1	τ1	NOUN
ejpam-6037	410	3	,	,	PUNCT
ejpam-6037	410	4	τ2)δ	τ2)δ	ADJ
ejpam-6037	410	5	-	-	PUNCT
ejpam-6037	410	6	semicontinuous	semicontinuous	ADJ
ejpam-6037	410	7	multifunctions	multifunction	NOUN
ejpam-6037	410	8	.	.	PUNCT
ejpam-6037	411	1	heliyon	heliyon	NOUN
ejpam-6037	411	2	,	,	PUNCT
ejpam-6037	411	3	6	6	NUM
ejpam-6037	411	4	:	:	SYM
ejpam-6037	411	5	e05367	e05367	PROPN
ejpam-6037	411	6	,	,	PUNCT
ejpam-6037	411	7	2020	2020	NUM
ejpam-6037	411	8	.	.	PUNCT
ejpam-6037	412	1	[	[	X
ejpam-6037	412	2	41	41	NUM
ejpam-6037	412	3	]	]	X
ejpam-6037	412	4	n.	n.	PROPN
ejpam-6037	412	5	viriyapong	viriyapong	PROPN
ejpam-6037	412	6	,	,	PUNCT
ejpam-6037	412	7	s.	s.	PROPN
ejpam-6037	412	8	sompong	sompong	PROPN
ejpam-6037	412	9	,	,	PUNCT
ejpam-6037	412	10	and	and	CCONJ
ejpam-6037	412	11	c.	c.	PROPN
ejpam-6037	412	12	boonpok	boonpok	PROPN
ejpam-6037	412	13	.	.	PUNCT
ejpam-6037	413	1	(	(	PUNCT
ejpam-6037	413	2	τ1	τ1	NOUN
ejpam-6037	413	3	,	,	PUNCT
ejpam-6037	413	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6037	413	5	disconnectedness	disconnectedness	NOUN
ejpam-6037	413	6	in	in	ADP
ejpam-6037	413	7	bitopological	bitopological	ADJ
ejpam-6037	413	8	spaces	space	NOUN
ejpam-6037	413	9	.	.	PUNCT
ejpam-6037	414	1	international	international	ADJ
ejpam-6037	414	2	journal	journal	PROPN
ejpam-6037	414	3	of	of	ADP
ejpam-6037	414	4	mathematics	mathematic	NOUN
ejpam-6037	414	5	and	and	CCONJ
ejpam-6037	414	6	computer	computer	NOUN
ejpam-6037	414	7	science	science	NOUN
ejpam-6037	414	8	,	,	PUNCT
ejpam-6037	414	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6037	414	10	,	,	PUNCT
ejpam-6037	414	11	2024	2024	NUM
ejpam-6037	414	12	.	.	PUNCT
ejpam-6037	415	1	[	[	X
ejpam-6037	415	2	42	42	NUM
ejpam-6037	415	3	]	]	X
ejpam-6037	415	4	n.	n.	PROPN
ejpam-6037	415	5	viriyapong	viriyapong	PROPN
ejpam-6037	415	6	,	,	PUNCT
ejpam-6037	415	7	s.	s.	PROPN
ejpam-6037	415	8	sompong	sompong	PROPN
ejpam-6037	415	9	,	,	PUNCT
ejpam-6037	415	10	and	and	CCONJ
ejpam-6037	415	11	c.	c.	PROPN
ejpam-6037	415	12	boonpok	boonpok	PROPN
ejpam-6037	415	13	.	.	PUNCT
ejpam-6037	416	1	upper	upper	ADJ
ejpam-6037	416	2	and	and	CCONJ
ejpam-6037	416	3	lower	low	ADJ
ejpam-6037	416	4	s-(τ1	s-(τ1	NOUN
ejpam-6037	416	5	,	,	PUNCT
ejpam-6037	416	6	τ2)p	τ2)p	ADJ
ejpam-6037	416	7	-	-	PUNCT
ejpam-6037	416	8	continuous	continuous	ADJ
ejpam-6037	416	9	multifunctions	multifunction	NOUN
ejpam-6037	416	10	.	.	PUNCT
ejpam-6037	417	1	european	european	ADJ
ejpam-6037	417	2	journal	journal	PROPN
ejpam-6037	417	3	of	of	ADP
ejpam-6037	417	4	pure	pure	ADJ
ejpam-6037	417	5	and	and	CCONJ
ejpam-6037	417	6	applied	applied	ADJ
ejpam-6037	417	7	mathematics	mathematic	NOUN
ejpam-6037	417	8	,	,	PUNCT
ejpam-6037	417	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6037	417	10	,	,	PUNCT
ejpam-6037	417	11	2024	2024	NUM
ejpam-6037	417	12	.	.	PUNCT
ejpam-6037	418	1	[	[	X
ejpam-6037	418	2	43	43	NUM
ejpam-6037	418	3	]	]	X
ejpam-6037	418	4	n.	n.	NOUN
ejpam-6037	418	5	chutiman	chutiman	NOUN
ejpam-6037	418	6	,	,	PUNCT
ejpam-6037	418	7	s.	s.	PROPN
ejpam-6037	418	8	sompong	sompong	PROPN
ejpam-6037	418	9	,	,	PUNCT
ejpam-6037	418	10	and	and	CCONJ
ejpam-6037	418	11	c.	c.	PROPN
ejpam-6037	418	12	boonpok	boonpok	PROPN
ejpam-6037	418	13	.	.	PUNCT
ejpam-6037	419	1	on	on	ADP
ejpam-6037	419	2	almost	almost	ADV
ejpam-6037	419	3	(	(	PUNCT
ejpam-6037	419	4	τ1	τ1	NOUN
ejpam-6037	419	5	,	,	PUNCT
ejpam-6037	419	6	τ2)-regular	τ2)-regular	ADJ
ejpam-6037	419	7	spaces	space	NOUN
ejpam-6037	419	8	.	.	PUNCT
ejpam-6037	420	1	international	international	ADJ
ejpam-6037	420	2	journal	journal	PROPN
ejpam-6037	420	3	of	of	ADP
ejpam-6037	420	4	mathematics	mathematic	NOUN
ejpam-6037	420	5	and	and	CCONJ
ejpam-6037	420	6	computer	computer	NOUN
ejpam-6037	420	7	science	science	NOUN
ejpam-6037	420	8	,	,	PUNCT
ejpam-6037	420	9	19(4):1363–1368	19(4):1363–1368	NUM
ejpam-6037	420	10	,	,	PUNCT
ejpam-6037	420	11	2024	2024	NUM
ejpam-6037	420	12	.	.	PUNCT
ejpam-6037	421	1	[	[	X
ejpam-6037	421	2	44	44	NUM
ejpam-6037	421	3	]	]	PUNCT
ejpam-6037	421	4	p.	p.	NOUN
ejpam-6037	421	5	pue	pue	NOUN
ejpam-6037	421	6	-	-	PUNCT
ejpam-6037	421	7	on	on	ADP
ejpam-6037	421	8	,	,	PUNCT
ejpam-6037	421	9	a.	a.	PROPN
ejpam-6037	421	10	sama	sama	PROPN
ejpam-6037	421	11	-	-	PUNCT
ejpam-6037	421	12	ae	ae	PROPN
ejpam-6037	421	13	,	,	PUNCT
ejpam-6037	421	14	and	and	CCONJ
ejpam-6037	421	15	c.	c.	PROPN
ejpam-6037	421	16	boonpok	boonpok	PROPN
ejpam-6037	421	17	.	.	PUNCT
ejpam-6037	422	1	characterizations	characterization	NOUN
ejpam-6037	422	2	of	of	ADP
ejpam-6037	422	3	quasi	quasi	NOUN
ejpam-6037	422	4	θ(τ1	θ(τ1	NOUN
ejpam-6037	422	5	,	,	PUNCT
ejpam-6037	422	6	τ2)continuous	τ2)continuous	ADJ
ejpam-6037	422	7	multifunctions	multifunction	NOUN
ejpam-6037	422	8	.	.	PUNCT
ejpam-6037	423	1	international	international	ADJ
ejpam-6037	423	2	journal	journal	NOUN
ejpam-6037	423	3	of	of	ADP
ejpam-6037	423	4	analysis	analysis	NOUN
ejpam-6037	423	5	and	and	CCONJ
ejpam-6037	423	6	applications	application	NOUN
ejpam-6037	423	7	,	,	PUNCT
ejpam-6037	423	8	23:59	23:59	NUM
ejpam-6037	423	9	,	,	PUNCT
ejpam-6037	423	10	2025	2025	NUM
ejpam-6037	423	11	.	.	PUNCT
ejpam-6037	424	1	[	[	X
ejpam-6037	424	2	45	45	NUM
ejpam-6037	424	3	]	]	PUNCT
ejpam-6037	424	4	m.	m.	NOUN
ejpam-6037	424	5	chiangpradit	chiangpradit	NOUN
ejpam-6037	424	6	,	,	PUNCT
ejpam-6037	424	7	s.	s.	PROPN
ejpam-6037	424	8	sompong	sompong	PROPN
ejpam-6037	424	9	,	,	PUNCT
ejpam-6037	424	10	and	and	CCONJ
ejpam-6037	424	11	c.	c.	PROPN
ejpam-6037	424	12	boonpok	boonpok	PROPN
ejpam-6037	424	13	.	.	PUNCT
ejpam-6037	425	1	λ(τ1,τ2)-sets	λ(τ1,τ2)-set	NOUN
ejpam-6037	425	2	and	and	CCONJ
ejpam-6037	425	3	related	relate	VERB
ejpam-6037	425	4	topological	topological	ADJ
ejpam-6037	425	5	spaces	space	NOUN
ejpam-6037	425	6	.	.	PUNCT
ejpam-6037	426	1	asia	asia	PROPN
ejpam-6037	426	2	pacific	pacific	PROPN
ejpam-6037	426	3	journal	journal	PROPN
ejpam-6037	426	4	of	of	ADP
ejpam-6037	426	5	mathematics	mathematic	NOUN
ejpam-6037	426	6	,	,	PUNCT
ejpam-6037	426	7	11:49	11:49	NUM
ejpam-6037	426	8	,	,	PUNCT
ejpam-6037	426	9	2024	2024	NUM
ejpam-6037	426	10	.	.	PUNCT
ejpam-6037	427	1	[	[	X
ejpam-6037	427	2	46	46	NUM
ejpam-6037	427	3	]	]	X
ejpam-6037	427	4	n.	n.	NOUN
ejpam-6037	427	5	chutiman	chutiman	NOUN
ejpam-6037	427	6	,	,	PUNCT
ejpam-6037	427	7	s.	s.	PROPN
ejpam-6037	427	8	sompong	sompong	PROPN
ejpam-6037	427	9	,	,	PUNCT
ejpam-6037	427	10	and	and	CCONJ
ejpam-6037	427	11	c.	c.	PROPN
ejpam-6037	427	12	boonpok	boonpok	PROPN
ejpam-6037	427	13	.	.	PUNCT
ejpam-6037	428	1	on	on	ADP
ejpam-6037	428	2	some	some	DET
ejpam-6037	428	3	separation	separation	NOUN
ejpam-6037	428	4	axioms	axiom	NOUN
ejpam-6037	428	5	in	in	ADP
ejpam-6037	428	6	bitopological	bitopological	ADJ
ejpam-6037	428	7	spaces	space	NOUN
ejpam-6037	428	8	.	.	PUNCT
ejpam-6037	429	1	asia	asia	PROPN
ejpam-6037	429	2	pacific	pacific	PROPN
ejpam-6037	429	3	journal	journal	PROPN
ejpam-6037	429	4	of	of	ADP
ejpam-6037	429	5	mathematics	mathematic	NOUN
ejpam-6037	429	6	,	,	PUNCT
ejpam-6037	429	7	11:41	11:41	NUM
ejpam-6037	429	8	,	,	PUNCT
ejpam-6037	429	9	2024	2024	NUM
ejpam-6037	429	10	.	.	PUNCT
ejpam-6037	430	1	[	[	X
ejpam-6037	430	2	47	47	NUM
ejpam-6037	430	3	]	]	PUNCT
ejpam-6037	430	4	m.	m.	NOUN
ejpam-6037	430	5	thongmoon	thongmoon	NOUN
ejpam-6037	430	6	,	,	PUNCT
ejpam-6037	430	7	s.	s.	PROPN
ejpam-6037	430	8	sompong	sompong	PROPN
ejpam-6037	430	9	,	,	PUNCT
ejpam-6037	430	10	and	and	CCONJ
ejpam-6037	430	11	c.	c.	PROPN
ejpam-6037	430	12	boonpok	boonpok	PROPN
ejpam-6037	430	13	.	.	PUNCT
ejpam-6037	431	1	upper	upper	ADJ
ejpam-6037	431	2	and	and	CCONJ
ejpam-6037	431	3	lower	low	ADJ
ejpam-6037	431	4	weak	weak	ADJ
ejpam-6037	431	5	(	(	PUNCT
ejpam-6037	431	6	τ1	τ1	NOUN
ejpam-6037	431	7	,	,	PUNCT
ejpam-6037	431	8	τ2)continuity	τ2)continuity	PROPN
ejpam-6037	431	9	.	.	PUNCT
ejpam-6037	432	1	european	european	PROPN
ejpam-6037	432	2	journal	journal	PROPN
ejpam-6037	432	3	of	of	ADP
ejpam-6037	432	4	pure	pure	ADJ
ejpam-6037	432	5	and	and	CCONJ
ejpam-6037	432	6	applied	applied	ADJ
ejpam-6037	432	7	mathematics	mathematic	NOUN
ejpam-6037	432	8	,	,	PUNCT
ejpam-6037	432	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-6037	432	10	,	,	PUNCT
ejpam-6037	432	11	2024	2024	NUM
ejpam-6037	432	12	.	.	PUNCT