id	sid	tid	token	lemma	pos
ejpam-6013	1	1	european	european	PROPN
ejpam-6013	1	2	journal	journal	PROPN
ejpam-6013	1	3	of	of	ADP
ejpam-6013	1	4	pure	pure	ADJ
ejpam-6013	1	5	and	and	CCONJ
ejpam-6013	1	6	applied	applied	ADJ
ejpam-6013	1	7	mathematics	mathematic	NOUN
ejpam-6013	1	8	2025	2025	NUM
ejpam-6013	1	9	,	,	PUNCT
ejpam-6013	1	10	vol	vol	NOUN
ejpam-6013	1	11	.	.	PROPN
ejpam-6013	1	12	18	18	NUM
ejpam-6013	1	13	,	,	PUNCT
ejpam-6013	1	14	issue	issue	NOUN
ejpam-6013	1	15	2	2	NUM
ejpam-6013	1	16	,	,	PUNCT
ejpam-6013	1	17	article	article	NOUN
ejpam-6013	1	18	number	number	NOUN
ejpam-6013	1	19	6013	6013	NUM
ejpam-6013	1	20	issn	issn	PROPN
ejpam-6013	1	21	1307	1307	NUM
ejpam-6013	1	22	-	-	SYM
ejpam-6013	1	23	5543	5543	NUM
ejpam-6013	1	24	–	–	PUNCT
ejpam-6013	1	25	ejpam.com	ejpam.com	X
ejpam-6013	1	26	published	publish	VERB
ejpam-6013	1	27	by	by	ADP
ejpam-6013	1	28	new	new	PROPN
ejpam-6013	1	29	york	york	PROPN
ejpam-6013	1	30	business	business	PROPN
ejpam-6013	1	31	global	global	PROPN
ejpam-6013	1	32	on	on	ADP
ejpam-6013	1	33	r-(τ1	r-(τ1	PROPN
ejpam-6013	1	34	,	,	PUNCT
ejpam-6013	1	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	1	36	functions	function	NOUN
ejpam-6013	1	37	napassanan	napassanan	PROPN
ejpam-6013	1	38	srisarakham1	srisarakham1	PROPN
ejpam-6013	1	39	,	,	PUNCT
ejpam-6013	1	40	supunnee	supunnee	PROPN
ejpam-6013	1	41	sompong2	sompong2	PROPN
ejpam-6013	1	42	,	,	PUNCT
ejpam-6013	1	43	chawalit	chawalit	VERB
ejpam-6013	1	44	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6013	1	45	1	1	NUM
ejpam-6013	1	46	mathematics	mathematic	NOUN
ejpam-6013	1	47	and	and	CCONJ
ejpam-6013	1	48	applied	apply	VERB
ejpam-6013	1	49	mathematics	mathematics	PROPN
ejpam-6013	1	50	research	research	NOUN
ejpam-6013	1	51	unit	unit	NOUN
ejpam-6013	1	52	,	,	PUNCT
ejpam-6013	1	53	department	department	NOUN
ejpam-6013	1	54	of	of	ADP
ejpam-6013	1	55	mathematics	mathematic	NOUN
ejpam-6013	1	56	,	,	PUNCT
ejpam-6013	1	57	faculty	faculty	NOUN
ejpam-6013	1	58	of	of	ADP
ejpam-6013	1	59	science	science	NOUN
ejpam-6013	1	60	,	,	PUNCT
ejpam-6013	1	61	mahasarakham	mahasarakham	PROPN
ejpam-6013	1	62	university	university	PROPN
ejpam-6013	1	63	,	,	PUNCT
ejpam-6013	1	64	maha	maha	PROPN
ejpam-6013	1	65	sarakham	sarakham	PROPN
ejpam-6013	1	66	,	,	PUNCT
ejpam-6013	1	67	44150	44150	NUM
ejpam-6013	1	68	,	,	PUNCT
ejpam-6013	1	69	thailand	thailand	PROPN
ejpam-6013	1	70	2	2	NUM
ejpam-6013	1	71	department	department	NOUN
ejpam-6013	1	72	of	of	ADP
ejpam-6013	1	73	mathematics	mathematic	NOUN
ejpam-6013	1	74	and	and	CCONJ
ejpam-6013	1	75	statistics	statistic	NOUN
ejpam-6013	1	76	,	,	PUNCT
ejpam-6013	1	77	faculty	faculty	NOUN
ejpam-6013	1	78	of	of	ADP
ejpam-6013	1	79	science	science	NOUN
ejpam-6013	1	80	and	and	CCONJ
ejpam-6013	1	81	technology	technology	NOUN
ejpam-6013	1	82	,	,	PUNCT
ejpam-6013	1	83	sakon	sakon	PROPN
ejpam-6013	1	84	nakhon	nakhon	PROPN
ejpam-6013	1	85	rajbhat	rajbhat	PROPN
ejpam-6013	1	86	university	university	PROPN
ejpam-6013	1	87	,	,	PUNCT
ejpam-6013	1	88	sakon	sakon	PROPN
ejpam-6013	1	89	nakhon	nakhon	PROPN
ejpam-6013	1	90	,	,	PUNCT
ejpam-6013	1	91	47000	47000	NUM
ejpam-6013	1	92	,	,	PUNCT
ejpam-6013	1	93	thailand	thailand	PROPN
ejpam-6013	1	94	abstract	abstract	NOUN
ejpam-6013	1	95	.	.	PUNCT
ejpam-6013	2	1	this	this	DET
ejpam-6013	2	2	paper	paper	NOUN
ejpam-6013	2	3	presents	present	VERB
ejpam-6013	2	4	a	a	DET
ejpam-6013	2	5	new	new	ADJ
ejpam-6013	2	6	class	class	NOUN
ejpam-6013	2	7	of	of	ADP
ejpam-6013	2	8	functions	function	NOUN
ejpam-6013	2	9	between	between	ADP
ejpam-6013	2	10	bitopological	bitopological	ADJ
ejpam-6013	2	11	spaces	space	NOUN
ejpam-6013	2	12	called	call	VERB
ejpam-6013	2	13	r(τ1	r(τ1	NOUN
ejpam-6013	2	14	,	,	PUNCT
ejpam-6013	2	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	2	16	functions	function	NOUN
ejpam-6013	2	17	.	.	PUNCT
ejpam-6013	3	1	furthermore	furthermore	ADV
ejpam-6013	3	2	,	,	PUNCT
ejpam-6013	3	3	several	several	ADJ
ejpam-6013	3	4	characterizations	characterization	NOUN
ejpam-6013	3	5	and	and	CCONJ
ejpam-6013	3	6	some	some	DET
ejpam-6013	3	7	properties	property	NOUN
ejpam-6013	3	8	concerning	concern	VERB
ejpam-6013	3	9	r-(τ1	r-(τ1	NOUN
ejpam-6013	3	10	,	,	PUNCT
ejpam-6013	3	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	3	12	functions	function	NOUN
ejpam-6013	3	13	are	be	AUX
ejpam-6013	3	14	established	establish	VERB
ejpam-6013	3	15	.	.	PUNCT
ejpam-6013	4	1	2020	2020	NUM
ejpam-6013	4	2	mathematics	mathematics	PROPN
ejpam-6013	4	3	subject	subject	NOUN
ejpam-6013	4	4	classifications	classification	NOUN
ejpam-6013	4	5	:	:	PUNCT
ejpam-6013	4	6	54c08	54c08	NUM
ejpam-6013	4	7	,	,	PUNCT
ejpam-6013	4	8	54e55	54e55	NUM
ejpam-6013	4	9	key	key	ADJ
ejpam-6013	4	10	words	word	NOUN
ejpam-6013	4	11	and	and	CCONJ
ejpam-6013	4	12	phrases	phrase	NOUN
ejpam-6013	4	13	:	:	PUNCT
ejpam-6013	4	14	τ1τ2	τ1τ2	ADJ
ejpam-6013	4	15	-	-	ADJ
ejpam-6013	4	16	open	open	ADJ
ejpam-6013	4	17	set	set	NOUN
ejpam-6013	4	18	,	,	PUNCT
ejpam-6013	4	19	r-(τ1	r-(τ1	PROPN
ejpam-6013	4	20	,	,	PUNCT
ejpam-6013	4	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	4	22	function	function	NOUN
ejpam-6013	4	23	1	1	NUM
ejpam-6013	4	24	.	.	PUNCT
ejpam-6013	4	25	introduction	introduction	NOUN
ejpam-6013	4	26	the	the	DET
ejpam-6013	4	27	field	field	NOUN
ejpam-6013	4	28	of	of	ADP
ejpam-6013	4	29	the	the	DET
ejpam-6013	4	30	mathematical	mathematical	ADJ
ejpam-6013	4	31	science	science	NOUN
ejpam-6013	4	32	which	which	PRON
ejpam-6013	4	33	goes	go	VERB
ejpam-6013	4	34	under	under	ADP
ejpam-6013	4	35	the	the	DET
ejpam-6013	4	36	name	name	NOUN
ejpam-6013	4	37	of	of	ADP
ejpam-6013	4	38	topology	topology	NOUN
ejpam-6013	4	39	is	be	AUX
ejpam-6013	4	40	concerned	concern	VERB
ejpam-6013	4	41	with	with	ADP
ejpam-6013	4	42	all	all	DET
ejpam-6013	4	43	questions	question	NOUN
ejpam-6013	4	44	directly	directly	ADV
ejpam-6013	4	45	or	or	CCONJ
ejpam-6013	4	46	indirectly	indirectly	ADV
ejpam-6013	4	47	related	relate	VERB
ejpam-6013	4	48	to	to	ADP
ejpam-6013	4	49	continuity	continuity	NOUN
ejpam-6013	4	50	.	.	PUNCT
ejpam-6013	5	1	preopen	preopen	ADJ
ejpam-6013	5	2	sets	set	NOUN
ejpam-6013	5	3	,	,	PUNCT
ejpam-6013	5	4	semiopen	semiopen	ADJ
ejpam-6013	5	5	sets	set	NOUN
ejpam-6013	5	6	,	,	PUNCT
ejpam-6013	5	7	α	α	NOUN
ejpam-6013	5	8	-	-	ADJ
ejpam-6013	5	9	open	open	ADJ
ejpam-6013	5	10	sets	set	NOUN
ejpam-6013	5	11	,	,	PUNCT
ejpam-6013	5	12	β	β	ADJ
ejpam-6013	5	13	-	-	ADJ
ejpam-6013	5	14	open	open	ADJ
ejpam-6013	5	15	sets	set	NOUN
ejpam-6013	5	16	,	,	PUNCT
ejpam-6013	5	17	δ	δ	NOUN
ejpam-6013	5	18	-	-	ADJ
ejpam-6013	5	19	open	open	ADJ
ejpam-6013	5	20	sets	set	NOUN
ejpam-6013	5	21	and	and	CCONJ
ejpam-6013	5	22	θ	θ	ADJ
ejpam-6013	5	23	-	-	ADJ
ejpam-6013	5	24	open	open	ADJ
ejpam-6013	5	25	sets	set	NOUN
ejpam-6013	5	26	play	play	VERB
ejpam-6013	5	27	an	an	DET
ejpam-6013	5	28	important	important	ADJ
ejpam-6013	5	29	role	role	NOUN
ejpam-6013	5	30	in	in	ADP
ejpam-6013	5	31	the	the	DET
ejpam-6013	5	32	research	research	NOUN
ejpam-6013	5	33	of	of	ADP
ejpam-6013	5	34	generalizations	generalization	NOUN
ejpam-6013	5	35	of	of	ADP
ejpam-6013	5	36	continuity	continuity	NOUN
ejpam-6013	5	37	.	.	PUNCT
ejpam-6013	6	1	by	by	ADP
ejpam-6013	6	2	using	use	VERB
ejpam-6013	6	3	these	these	DET
ejpam-6013	6	4	sets	set	NOUN
ejpam-6013	6	5	many	many	ADJ
ejpam-6013	6	6	authors	author	NOUN
ejpam-6013	6	7	introduced	introduce	VERB
ejpam-6013	6	8	and	and	CCONJ
ejpam-6013	6	9	investigated	investigate	VERB
ejpam-6013	6	10	various	various	ADJ
ejpam-6013	6	11	types	type	NOUN
ejpam-6013	6	12	of	of	ADP
ejpam-6013	6	13	continuity	continuity	NOUN
ejpam-6013	6	14	.	.	PUNCT
ejpam-6013	7	1	in	in	ADP
ejpam-6013	7	2	[	[	X
ejpam-6013	7	3	1	1	NUM
ejpam-6013	7	4	]	]	PUNCT
ejpam-6013	7	5	,	,	PUNCT
ejpam-6013	7	6	the	the	DET
ejpam-6013	7	7	present	present	ADJ
ejpam-6013	7	8	authors	author	NOUN
ejpam-6013	7	9	studied	study	VERB
ejpam-6013	7	10	some	some	DET
ejpam-6013	7	11	properties	property	NOUN
ejpam-6013	7	12	of	of	ADP
ejpam-6013	7	13	(	(	PUNCT
ejpam-6013	7	14	λ	λ	PROPN
ejpam-6013	7	15	,	,	PUNCT
ejpam-6013	7	16	sp)-open	sp)-open	ADJ
ejpam-6013	7	17	sets	set	NOUN
ejpam-6013	7	18	,	,	PUNCT
ejpam-6013	7	19	s(λ	s(λ	PROPN
ejpam-6013	7	20	,	,	PUNCT
ejpam-6013	7	21	sp)-open	sp)-open	ADJ
ejpam-6013	7	22	sets	set	NOUN
ejpam-6013	7	23	,	,	PUNCT
ejpam-6013	7	24	p(λ	p(λ	NOUN
ejpam-6013	7	25	,	,	PUNCT
ejpam-6013	7	26	sp)-open	sp)-open	ADJ
ejpam-6013	7	27	sets	set	NOUN
ejpam-6013	7	28	,	,	PUNCT
ejpam-6013	7	29	α(λ	α(λ	PROPN
ejpam-6013	7	30	,	,	PUNCT
ejpam-6013	7	31	sp)-open	sp)-open	ADJ
ejpam-6013	7	32	sets	set	NOUN
ejpam-6013	7	33	and	and	CCONJ
ejpam-6013	7	34	β(λ	β(λ	NOUN
ejpam-6013	7	35	,	,	PUNCT
ejpam-6013	7	36	sp)-open	sp)-open	ADJ
ejpam-6013	7	37	sets	set	NOUN
ejpam-6013	7	38	.	.	PUNCT
ejpam-6013	8	1	viriyapong	viriyapong	VERB
ejpam-6013	8	2	and	and	CCONJ
ejpam-6013	8	3	boonpok	boonpok	VERB
ejpam-6013	8	4	[	[	X
ejpam-6013	8	5	2	2	NUM
ejpam-6013	8	6	]	]	PUNCT
ejpam-6013	8	7	investigated	investigate	VERB
ejpam-6013	8	8	several	several	ADJ
ejpam-6013	8	9	characterizations	characterization	NOUN
ejpam-6013	8	10	of	of	ADP
ejpam-6013	8	11	(	(	PUNCT
ejpam-6013	8	12	λ	λ	PROPN
ejpam-6013	8	13	,	,	PUNCT
ejpam-6013	8	14	sp)-continuous	sp)-continuous	ADJ
ejpam-6013	8	15	functions	function	NOUN
ejpam-6013	8	16	by	by	ADP
ejpam-6013	8	17	utilizing	utilize	VERB
ejpam-6013	8	18	the	the	DET
ejpam-6013	8	19	notions	notion	NOUN
ejpam-6013	8	20	of	of	ADP
ejpam-6013	8	21	(	(	PUNCT
ejpam-6013	8	22	λ	λ	PROPN
ejpam-6013	8	23	,	,	PUNCT
ejpam-6013	8	24	sp)-open	sp)-open	ADJ
ejpam-6013	8	25	sets	set	NOUN
ejpam-6013	8	26	and	and	CCONJ
ejpam-6013	8	27	(	(	PUNCT
ejpam-6013	8	28	λ	λ	PROPN
ejpam-6013	8	29	,	,	PUNCT
ejpam-6013	8	30	sp)-closed	sp)-close	VERB
ejpam-6013	8	31	sets	set	NOUN
ejpam-6013	8	32	.	.	PUNCT
ejpam-6013	9	1	dungthaisong	dungthaisong	NOUN
ejpam-6013	9	2	et	et	PROPN
ejpam-6013	9	3	al	al	PROPN
ejpam-6013	9	4	.	.	PUNCT
ejpam-6013	10	1	[	[	X
ejpam-6013	10	2	3	3	NUM
ejpam-6013	10	3	]	]	PUNCT
ejpam-6013	10	4	introduced	introduce	VERB
ejpam-6013	10	5	and	and	CCONJ
ejpam-6013	10	6	studied	study	VERB
ejpam-6013	10	7	the	the	DET
ejpam-6013	10	8	concept	concept	NOUN
ejpam-6013	10	9	of	of	ADP
ejpam-6013	10	10	g(m	g(m	ADJ
ejpam-6013	10	11	,	,	PUNCT
ejpam-6013	10	12	n)-continuous	n)-continuous	ADJ
ejpam-6013	10	13	functions	function	NOUN
ejpam-6013	10	14	.	.	PUNCT
ejpam-6013	11	1	duangphui	duangphui	NOUN
ejpam-6013	11	2	et	et	PROPN
ejpam-6013	11	3	al	al	PROPN
ejpam-6013	11	4	.	.	PUNCT
ejpam-6013	12	1	[	[	X
ejpam-6013	12	2	4	4	X
ejpam-6013	12	3	]	]	PUNCT
ejpam-6013	12	4	introduced	introduce	VERB
ejpam-6013	12	5	and	and	CCONJ
ejpam-6013	12	6	investigated	investigate	VERB
ejpam-6013	12	7	the	the	DET
ejpam-6013	12	8	notion	notion	NOUN
ejpam-6013	12	9	of	of	ADP
ejpam-6013	12	10	(	(	PUNCT
ejpam-6013	12	11	µ	µ	NOUN
ejpam-6013	12	12	,	,	PUNCT
ejpam-6013	12	13	µ′)(m	µ′)(m	VERB
ejpam-6013	12	14	,	,	PUNCT
ejpam-6013	12	15	n)continuous	n)continuous	ADJ
ejpam-6013	12	16	functions	function	NOUN
ejpam-6013	12	17	.	.	PUNCT
ejpam-6013	13	1	moreover	moreover	ADV
ejpam-6013	13	2	,	,	PUNCT
ejpam-6013	13	3	some	some	DET
ejpam-6013	13	4	characterizations	characterization	NOUN
ejpam-6013	13	5	of	of	ADP
ejpam-6013	13	6	almost	almost	ADV
ejpam-6013	13	7	(	(	PUNCT
ejpam-6013	13	8	λ	λ	PROPN
ejpam-6013	13	9	,	,	PUNCT
ejpam-6013	13	10	p)-continuous	p)-continuous	ADJ
ejpam-6013	13	11	functions	function	NOUN
ejpam-6013	13	12	,	,	PUNCT
ejpam-6013	13	13	almost	almost	ADV
ejpam-6013	13	14	strongly	strongly	ADV
ejpam-6013	13	15	θ(λ	θ(λ	VERB
ejpam-6013	13	16	,	,	PUNCT
ejpam-6013	13	17	p)-continuous	p)-continuous	ADJ
ejpam-6013	13	18	functions	function	NOUN
ejpam-6013	13	19	,	,	PUNCT
ejpam-6013	13	20	weakly	weakly	ADJ
ejpam-6013	13	21	(	(	PUNCT
ejpam-6013	13	22	λ	λ	PROPN
ejpam-6013	13	23	,	,	PUNCT
ejpam-6013	13	24	b)-continuous	b)-continuous	ADJ
ejpam-6013	13	25	functions	function	NOUN
ejpam-6013	13	26	,	,	PUNCT
ejpam-6013	13	27	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-6013	13	28	functions	function	NOUN
ejpam-6013	13	29	,	,	PUNCT
ejpam-6013	13	30	(	(	PUNCT
ejpam-6013	13	31	λ	λ	NOUN
ejpam-6013	13	32	,	,	PUNCT
ejpam-6013	13	33	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-6013	13	34	functions	function	NOUN
ejpam-6013	13	35	,	,	PUNCT
ejpam-6013	13	36	⋆-continuous	⋆-continuous	ADJ
ejpam-6013	13	37	functions	function	NOUN
ejpam-6013	13	38	,	,	PUNCT
ejpam-6013	13	39	θ	θ	PROPN
ejpam-6013	13	40	-	-	ADJ
ejpam-6013	13	41	i	i	VERB
ejpam-6013	13	42	continuous	continuous	ADJ
ejpam-6013	13	43	functions	function	NOUN
ejpam-6013	13	44	,	,	PUNCT
ejpam-6013	13	45	almost	almost	ADV
ejpam-6013	13	46	(	(	PUNCT
ejpam-6013	13	47	g	g	NOUN
ejpam-6013	13	48	,	,	PUNCT
ejpam-6013	13	49	m)-continuous	m)-continuous	ADJ
ejpam-6013	13	50	functions	function	NOUN
ejpam-6013	13	51	,	,	PUNCT
ejpam-6013	13	52	pairwise	pairwise	NOUN
ejpam-6013	13	53	almost	almost	ADV
ejpam-6013	13	54	m	m	VERB
ejpam-6013	13	55	-continuous	-continuous	ADJ
ejpam-6013	13	56	functions	function	NOUN
ejpam-6013	13	57	were	be	AUX
ejpam-6013	13	58	presented	present	VERB
ejpam-6013	13	59	in	in	ADP
ejpam-6013	13	60	[	[	X
ejpam-6013	13	61	5	5	NUM
ejpam-6013	13	62	]	]	PUNCT
ejpam-6013	13	63	,	,	PUNCT
ejpam-6013	13	64	[	[	X
ejpam-6013	13	65	6	6	NUM
ejpam-6013	13	66	]	]	PUNCT
ejpam-6013	13	67	,	,	PUNCT
ejpam-6013	13	68	[	[	X
ejpam-6013	13	69	7	7	NUM
ejpam-6013	13	70	]	]	PUNCT
ejpam-6013	13	71	,	,	PUNCT
ejpam-6013	13	72	[	[	X
ejpam-6013	13	73	8	8	NUM
ejpam-6013	13	74	]	]	PUNCT
ejpam-6013	13	75	,	,	PUNCT
ejpam-6013	14	1	[	[	X
ejpam-6013	14	2	9	9	NUM
ejpam-6013	14	3	]	]	PUNCT
ejpam-6013	14	4	,	,	PUNCT
ejpam-6013	14	5	[	[	X
ejpam-6013	14	6	10	10	NUM
ejpam-6013	14	7	]	]	PUNCT
ejpam-6013	14	8	,	,	PUNCT
ejpam-6013	15	1	[	[	X
ejpam-6013	15	2	11	11	NUM
ejpam-6013	15	3	]	]	PUNCT
ejpam-6013	15	4	,	,	PUNCT
ejpam-6013	15	5	[	[	X
ejpam-6013	15	6	12	12	NUM
ejpam-6013	15	7	]	]	PUNCT
ejpam-6013	15	8	and	and	CCONJ
ejpam-6013	16	1	[	[	X
ejpam-6013	16	2	13	13	NUM
ejpam-6013	16	3	]	]	PUNCT
ejpam-6013	16	4	,	,	PUNCT
ejpam-6013	16	5	respectively	respectively	ADV
ejpam-6013	16	6	.	.	PUNCT
ejpam-6013	17	1	konstadilaki	konstadilaki	NOUN
ejpam-6013	17	2	-	-	PUNCT
ejpam-6013	17	3	savvopoulou	savvopoulou	PROPN
ejpam-6013	17	4	and	and	CCONJ
ejpam-6013	17	5	janković	janković	ADJ
ejpam-6013	18	1	[	[	X
ejpam-6013	18	2	14	14	NUM
ejpam-6013	18	3	]	]	PUNCT
ejpam-6013	18	4	introduced	introduce	VERB
ejpam-6013	18	5	and	and	CCONJ
ejpam-6013	18	6	studied	study	VERB
ejpam-6013	18	7	a	a	DET
ejpam-6013	18	8	strong	strong	ADJ
ejpam-6013	18	9	form	form	NOUN
ejpam-6013	18	10	of	of	ADP
ejpam-6013	18	11	continuity	continuity	NOUN
ejpam-6013	18	12	of	of	ADP
ejpam-6013	18	13	functions	function	NOUN
ejpam-6013	18	14	between	between	ADP
ejpam-6013	18	15	topological	topological	ADJ
ejpam-6013	18	16	spaces	space	NOUN
ejpam-6013	18	17	called	call	VERB
ejpam-6013	18	18	r	r	NOUN
ejpam-6013	18	19	-	-	PUNCT
ejpam-6013	18	20	continuous	continuous	ADJ
ejpam-6013	18	21	functions	function	NOUN
ejpam-6013	18	22	.	.	PUNCT
ejpam-6013	19	1	crossley	crossley	NOUN
ejpam-6013	19	2	and	and	CCONJ
ejpam-6013	19	3	∗corresponding	∗corresponde	VERB
ejpam-6013	19	4	author	author	NOUN
ejpam-6013	19	5	.	.	PUNCT
ejpam-6013	20	1	doi	doi	NOUN
ejpam-6013	20	2	:	:	PUNCT
ejpam-6013	20	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6013	https://doi.org/10.29020/nybg.ejpam.v18i2.6013	PROPN
ejpam-6013	20	4	email	email	NOUN
ejpam-6013	20	5	addresses	address	VERB
ejpam-6013	20	6	:	:	PUNCT
ejpam-6013	20	7	napassanan.sri@msu.ac.th	napassanan.sri@msu.ac.th	PRON
ejpam-6013	20	8	(	(	PUNCT
ejpam-6013	20	9	n.	n.	NOUN
ejpam-6013	20	10	srisarakham	srisarakham	PROPN
ejpam-6013	20	11	)	)	PUNCT
ejpam-6013	20	12	,	,	PUNCT
ejpam-6013	20	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-6013	20	14	(	(	PUNCT
ejpam-6013	20	15	s.	s.	PROPN
ejpam-6013	20	16	sompong	sompong	PROPN
ejpam-6013	20	17	)	)	PUNCT
ejpam-6013	20	18	,	,	PUNCT
ejpam-6013	20	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6013	20	20	(	(	PUNCT
ejpam-6013	20	21	c.	c.	PROPN
ejpam-6013	20	22	boonpok	boonpok	PROPN
ejpam-6013	20	23	)	)	PUNCT
ejpam-6013	20	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6013	21	1	1	1	NUM
ejpam-6013	21	2	copyright	copyright	NOUN
ejpam-6013	21	3	:	:	PUNCT
ejpam-6013	21	4	©	©	PROPN
ejpam-6013	21	5	2025	2025	NUM
ejpam-6013	21	6	the	the	DET
ejpam-6013	21	7	author(s	author(s	NOUN
ejpam-6013	21	8	)	)	PUNCT
ejpam-6013	21	9	.	.	PUNCT
ejpam-6013	22	1	(	(	PUNCT
ejpam-6013	22	2	cc	cc	NOUN
ejpam-6013	22	3	by	by	ADP
ejpam-6013	22	4	-	-	PUNCT
ejpam-6013	22	5	nc	nc	PROPN
ejpam-6013	22	6	4.0	4.0	NUM
ejpam-6013	22	7	)	)	PUNCT
ejpam-6013	22	8	n.	n.	NOUN
ejpam-6013	22	9	srisarakham	srisarakham	PROPN
ejpam-6013	22	10	,	,	PUNCT
ejpam-6013	22	11	s.	s.	PROPN
ejpam-6013	22	12	sompong	sompong	PROPN
ejpam-6013	22	13	,	,	PUNCT
ejpam-6013	22	14	c.	c.	PROPN
ejpam-6013	22	15	boonpok	boonpok	PROPN
ejpam-6013	22	16	/	/	SYM
ejpam-6013	22	17	eur	eur	PROPN
ejpam-6013	22	18	.	.	PUNCT
ejpam-6013	23	1	j.	j.	PROPN
ejpam-6013	23	2	pure	pure	PROPN
ejpam-6013	23	3	appl	appl	PROPN
ejpam-6013	23	4	.	.	PROPN
ejpam-6013	23	5	math	math	PROPN
ejpam-6013	23	6	,	,	PUNCT
ejpam-6013	23	7	18	18	NUM
ejpam-6013	23	8	(	(	PUNCT
ejpam-6013	23	9	2	2	NUM
ejpam-6013	23	10	)	)	PUNCT
ejpam-6013	23	11	(	(	PUNCT
ejpam-6013	23	12	2025	2025	NUM
ejpam-6013	23	13	)	)	PUNCT
ejpam-6013	23	14	,	,	PUNCT
ejpam-6013	23	15	6013	6013	NUM
ejpam-6013	23	16	2	2	NUM
ejpam-6013	23	17	of	of	ADP
ejpam-6013	23	18	12	12	NUM
ejpam-6013	23	19	hildebrand	hildebrand	NOUN
ejpam-6013	24	1	[	[	X
ejpam-6013	24	2	15	15	NUM
ejpam-6013	24	3	]	]	PUNCT
ejpam-6013	24	4	introduced	introduce	VERB
ejpam-6013	24	5	and	and	CCONJ
ejpam-6013	24	6	investigated	investigate	VERB
ejpam-6013	24	7	the	the	DET
ejpam-6013	24	8	concept	concept	NOUN
ejpam-6013	24	9	of	of	ADP
ejpam-6013	24	10	irresolute	irresolute	ADJ
ejpam-6013	24	11	functions	function	NOUN
ejpam-6013	24	12	.	.	PUNCT
ejpam-6013	25	1	reilly	reilly	ADV
ejpam-6013	25	2	and	and	CCONJ
ejpam-6013	25	3	vamanamurthy	vamanamurthy	ADJ
ejpam-6013	26	1	[	[	X
ejpam-6013	26	2	16	16	NUM
ejpam-6013	26	3	]	]	PUNCT
ejpam-6013	26	4	introduced	introduce	VERB
ejpam-6013	26	5	and	and	CCONJ
ejpam-6013	26	6	studied	study	VERB
ejpam-6013	26	7	the	the	DET
ejpam-6013	26	8	notion	notion	NOUN
ejpam-6013	26	9	of	of	ADP
ejpam-6013	26	10	preirresolute	preirresolute	ADJ
ejpam-6013	26	11	functions	function	NOUN
ejpam-6013	26	12	.	.	PUNCT
ejpam-6013	27	1	baker	baker	PROPN
ejpam-6013	28	1	[	[	X
ejpam-6013	28	2	17	17	NUM
ejpam-6013	28	3	]	]	PUNCT
ejpam-6013	28	4	introduced	introduce	VERB
ejpam-6013	28	5	and	and	CCONJ
ejpam-6013	28	6	investigated	investigate	VERB
ejpam-6013	28	7	the	the	DET
ejpam-6013	28	8	concept	concept	NOUN
ejpam-6013	28	9	of	of	ADP
ejpam-6013	28	10	r	r	NOUN
ejpam-6013	28	11	-	-	PUNCT
ejpam-6013	28	12	irrsolute	irrsolute	NOUN
ejpam-6013	28	13	functions	function	NOUN
ejpam-6013	28	14	.	.	PUNCT
ejpam-6013	29	1	furthermore	furthermore	ADV
ejpam-6013	29	2	,	,	PUNCT
ejpam-6013	29	3	the	the	DET
ejpam-6013	29	4	present	present	ADJ
ejpam-6013	29	5	author	author	NOUN
ejpam-6013	29	6	[	[	X
ejpam-6013	29	7	18	18	NUM
ejpam-6013	29	8	]	]	PUNCT
ejpam-6013	29	9	introduced	introduce	VERB
ejpam-6013	29	10	a	a	DET
ejpam-6013	29	11	strong	strong	ADJ
ejpam-6013	29	12	form	form	NOUN
ejpam-6013	29	13	of	of	ADP
ejpam-6013	29	14	preirresolute	preirresolute	ADJ
ejpam-6013	29	15	functions	function	NOUN
ejpam-6013	29	16	called	call	VERB
ejpam-6013	29	17	r	r	NOUN
ejpam-6013	29	18	-	-	PUNCT
ejpam-6013	29	19	preirresolute	preirresolute	ADJ
ejpam-6013	29	20	functions	function	NOUN
ejpam-6013	29	21	.	.	PUNCT
ejpam-6013	30	1	these	these	DET
ejpam-6013	30	2	four	four	NUM
ejpam-6013	30	3	classes	class	NOUN
ejpam-6013	30	4	of	of	ADP
ejpam-6013	30	5	functions	function	NOUN
ejpam-6013	30	6	have	have	VERB
ejpam-6013	30	7	properties	property	NOUN
ejpam-6013	30	8	similar	similar	ADJ
ejpam-6013	30	9	to	to	ADP
ejpam-6013	30	10	the	the	DET
ejpam-6013	30	11	class	class	NOUN
ejpam-6013	30	12	of	of	ADP
ejpam-6013	30	13	r	r	NOUN
ejpam-6013	30	14	-	-	PUNCT
ejpam-6013	30	15	continuous	continuous	ADJ
ejpam-6013	30	16	functions	function	NOUN
ejpam-6013	30	17	.	.	PUNCT
ejpam-6013	31	1	beceren	beceren	NOUN
ejpam-6013	31	2	and	and	CCONJ
ejpam-6013	31	3	noiri	noiri	ADV
ejpam-6013	32	1	[	[	X
ejpam-6013	32	2	19	19	NUM
ejpam-6013	32	3	]	]	PUNCT
ejpam-6013	32	4	introduced	introduce	VERB
ejpam-6013	32	5	and	and	CCONJ
ejpam-6013	32	6	studied	study	VERB
ejpam-6013	32	7	the	the	DET
ejpam-6013	32	8	notions	notion	NOUN
ejpam-6013	32	9	of	of	ADP
ejpam-6013	32	10	new	new	ADJ
ejpam-6013	32	11	classes	class	NOUN
ejpam-6013	32	12	of	of	ADP
ejpam-6013	32	13	functions	function	NOUN
ejpam-6013	32	14	,	,	PUNCT
ejpam-6013	32	15	namely	namely	ADV
ejpam-6013	32	16	α	α	PROPN
ejpam-6013	32	17	-	-	PUNCT
ejpam-6013	32	18	preirresolute	preirresolute	ADJ
ejpam-6013	32	19	functions	function	NOUN
ejpam-6013	32	20	and	and	CCONJ
ejpam-6013	32	21	β	β	NOUN
ejpam-6013	32	22	-	-	PUNCT
ejpam-6013	32	23	preirresolute	preirresolute	ADJ
ejpam-6013	32	24	functions	function	NOUN
ejpam-6013	32	25	.	.	PUNCT
ejpam-6013	33	1	a	a	DET
ejpam-6013	33	2	new	new	ADJ
ejpam-6013	33	3	class	class	NOUN
ejpam-6013	33	4	of	of	ADP
ejpam-6013	33	5	α	α	PROPN
ejpam-6013	33	6	-	-	PUNCT
ejpam-6013	33	7	preirresolute	preirresolute	ADJ
ejpam-6013	33	8	functions	function	NOUN
ejpam-6013	33	9	which	which	PRON
ejpam-6013	33	10	is	be	AUX
ejpam-6013	33	11	stronger	strong	ADJ
ejpam-6013	33	12	than	than	ADP
ejpam-6013	33	13	preirresolute	preirresolute	ADJ
ejpam-6013	33	14	functions	function	NOUN
ejpam-6013	33	15	[	[	X
ejpam-6013	33	16	17	17	NUM
ejpam-6013	33	17	]	]	PUNCT
ejpam-6013	33	18	is	be	AUX
ejpam-6013	33	19	a	a	DET
ejpam-6013	33	20	generalization	generalization	NOUN
ejpam-6013	33	21	of	of	ADP
ejpam-6013	33	22	strongly	strongly	ADV
ejpam-6013	33	23	m	m	VERB
ejpam-6013	33	24	-precontinuous	-precontinuous	ADJ
ejpam-6013	33	25	functions	function	NOUN
ejpam-6013	33	26	[	[	X
ejpam-6013	33	27	20	20	NUM
ejpam-6013	33	28	]	]	PUNCT
ejpam-6013	33	29	.	.	PUNCT
ejpam-6013	34	1	a	a	DET
ejpam-6013	34	2	new	new	ADJ
ejpam-6013	34	3	class	class	NOUN
ejpam-6013	34	4	of	of	ADP
ejpam-6013	34	5	β	β	NOUN
ejpam-6013	34	6	-	-	ADJ
ejpam-6013	34	7	irresolute	irresolute	ADJ
ejpam-6013	34	8	functions	function	NOUN
ejpam-6013	34	9	which	which	PRON
ejpam-6013	34	10	is	be	AUX
ejpam-6013	34	11	stronger	strong	ADJ
ejpam-6013	34	12	than	than	ADP
ejpam-6013	34	13	almost	almost	ADV
ejpam-6013	34	14	α	α	NOUN
ejpam-6013	34	15	-	-	PUNCT
ejpam-6013	34	16	irresolute	irresolute	ADJ
ejpam-6013	34	17	functions	function	NOUN
ejpam-6013	34	18	[	[	X
ejpam-6013	34	19	19	19	NUM
ejpam-6013	34	20	]	]	PUNCT
ejpam-6013	34	21	is	be	AUX
ejpam-6013	34	22	a	a	DET
ejpam-6013	34	23	generalization	generalization	NOUN
ejpam-6013	34	24	of	of	ADP
ejpam-6013	34	25	preirresolute	preirresolute	ADJ
ejpam-6013	34	26	functions	function	NOUN
ejpam-6013	34	27	[	[	X
ejpam-6013	34	28	17	17	NUM
ejpam-6013	34	29	]	]	PUNCT
ejpam-6013	34	30	.	.	PUNCT
ejpam-6013	35	1	noiri	noiri	PROPN
ejpam-6013	35	2	and	and	CCONJ
ejpam-6013	35	3	popa	popa	NOUN
ejpam-6013	35	4	[	[	X
ejpam-6013	35	5	21	21	NUM
ejpam-6013	35	6	]	]	PUNCT
ejpam-6013	35	7	introduced	introduce	VERB
ejpam-6013	35	8	a	a	DET
ejpam-6013	35	9	new	new	ADJ
ejpam-6013	35	10	class	class	NOUN
ejpam-6013	35	11	of	of	ADP
ejpam-6013	35	12	functions	function	NOUN
ejpam-6013	35	13	called	call	VERB
ejpam-6013	35	14	r	r	NOUN
ejpam-6013	35	15	-	-	PUNCT
ejpam-6013	35	16	m	m	NOUN
ejpam-6013	35	17	-continuous	-continuous	ADJ
ejpam-6013	35	18	functions	function	NOUN
ejpam-6013	35	19	as	as	ADP
ejpam-6013	35	20	functions	function	NOUN
ejpam-6013	35	21	defined	define	VERB
ejpam-6013	35	22	between	between	ADP
ejpam-6013	35	23	sets	set	NOUN
ejpam-6013	35	24	satisfying	satisfy	VERB
ejpam-6013	35	25	some	some	DET
ejpam-6013	35	26	minimal	minimal	ADJ
ejpam-6013	35	27	conditions	condition	NOUN
ejpam-6013	35	28	and	and	CCONJ
ejpam-6013	35	29	obtained	obtain	VERB
ejpam-6013	35	30	several	several	ADJ
ejpam-6013	35	31	characterizations	characterization	NOUN
ejpam-6013	35	32	of	of	ADP
ejpam-6013	35	33	r	r	NOUN
ejpam-6013	35	34	-	-	PUNCT
ejpam-6013	35	35	m	m	NOUN
ejpam-6013	35	36	-continuous	-continuous	ADJ
ejpam-6013	35	37	functions	function	NOUN
ejpam-6013	35	38	.	.	PUNCT
ejpam-6013	36	1	noiri	noiri	PROPN
ejpam-6013	36	2	and	and	CCONJ
ejpam-6013	36	3	popa	popa	NOUN
ejpam-6013	36	4	[	[	X
ejpam-6013	36	5	21	21	NUM
ejpam-6013	36	6	]	]	PUNCT
ejpam-6013	36	7	investigated	investigate	VERB
ejpam-6013	36	8	the	the	DET
ejpam-6013	36	9	relationship	relationship	NOUN
ejpam-6013	36	10	between	between	ADP
ejpam-6013	36	11	r	r	NOUN
ejpam-6013	36	12	-	-	PUNCT
ejpam-6013	36	13	m	m	NOUN
ejpam-6013	36	14	-continuity	-continuity	ADJ
ejpam-6013	36	15	and	and	CCONJ
ejpam-6013	36	16	some	some	DET
ejpam-6013	36	17	low	low	ADJ
ejpam-6013	36	18	separation	separation	NOUN
ejpam-6013	36	19	axioms	axiom	NOUN
ejpam-6013	36	20	(	(	PUNCT
ejpam-6013	36	21	m	m	NOUN
ejpam-6013	36	22	-	-	PUNCT
ejpam-6013	36	23	t1	t1	ADJ
ejpam-6013	36	24	,	,	PUNCT
ejpam-6013	36	25	m	m	NOUN
ejpam-6013	36	26	-	-	NOUN
ejpam-6013	36	27	t2	t2	NOUN
ejpam-6013	36	28	,	,	PUNCT
ejpam-6013	36	29	m	m	NOUN
ejpam-6013	36	30	-	-	NOUN
ejpam-6013	36	31	r0	r0	NOUN
ejpam-6013	36	32	)	)	PUNCT
ejpam-6013	36	33	.	.	PUNCT
ejpam-6013	37	1	the	the	DET
ejpam-6013	37	2	notion	notion	NOUN
ejpam-6013	37	3	of	of	ADP
ejpam-6013	37	4	(	(	PUNCT
ejpam-6013	37	5	τ1	τ1	PROPN
ejpam-6013	37	6	,	,	PUNCT
ejpam-6013	37	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	37	8	functions	function	NOUN
ejpam-6013	37	9	was	be	AUX
ejpam-6013	37	10	introduced	introduce	VERB
ejpam-6013	37	11	in	in	ADP
ejpam-6013	37	12	[	[	X
ejpam-6013	37	13	22	22	NUM
ejpam-6013	37	14	]	]	PUNCT
ejpam-6013	37	15	.	.	PUNCT
ejpam-6013	38	1	moreover	moreover	ADV
ejpam-6013	38	2	,	,	PUNCT
ejpam-6013	38	3	several	several	ADJ
ejpam-6013	38	4	characterizations	characterization	NOUN
ejpam-6013	38	5	of	of	ADP
ejpam-6013	38	6	almost	almost	ADV
ejpam-6013	38	7	(	(	PUNCT
ejpam-6013	38	8	τ1	τ1	NOUN
ejpam-6013	38	9	,	,	PUNCT
ejpam-6013	38	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	38	11	functions	function	NOUN
ejpam-6013	38	12	and	and	CCONJ
ejpam-6013	38	13	weakly	weakly	ADJ
ejpam-6013	38	14	(	(	PUNCT
ejpam-6013	38	15	τ1	τ1	NOUN
ejpam-6013	38	16	,	,	PUNCT
ejpam-6013	38	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	38	18	functions	function	NOUN
ejpam-6013	38	19	were	be	AUX
ejpam-6013	38	20	studied	study	VERB
ejpam-6013	38	21	in	in	ADP
ejpam-6013	38	22	[	[	X
ejpam-6013	38	23	23	23	NUM
ejpam-6013	38	24	]	]	PUNCT
ejpam-6013	38	25	and	and	CCONJ
ejpam-6013	38	26	[	[	X
ejpam-6013	38	27	24	24	NUM
ejpam-6013	38	28	]	]	PUNCT
ejpam-6013	38	29	,	,	PUNCT
ejpam-6013	38	30	respectively	respectively	ADV
ejpam-6013	38	31	.	.	PUNCT
ejpam-6013	39	1	in	in	ADP
ejpam-6013	39	2	this	this	DET
ejpam-6013	39	3	paper	paper	NOUN
ejpam-6013	39	4	,	,	PUNCT
ejpam-6013	39	5	we	we	PRON
ejpam-6013	39	6	introduce	introduce	VERB
ejpam-6013	39	7	the	the	DET
ejpam-6013	39	8	concept	concept	NOUN
ejpam-6013	39	9	of	of	ADP
ejpam-6013	39	10	r-(τ1	r-(τ1	PROPN
ejpam-6013	39	11	,	,	PUNCT
ejpam-6013	39	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	39	13	functions	function	NOUN
ejpam-6013	39	14	.	.	PUNCT
ejpam-6013	40	1	we	we	PRON
ejpam-6013	40	2	also	also	ADV
ejpam-6013	40	3	investigate	investigate	VERB
ejpam-6013	40	4	some	some	DET
ejpam-6013	40	5	characterizations	characterization	NOUN
ejpam-6013	40	6	of	of	ADP
ejpam-6013	40	7	r-(τ1	r-(τ1	PROPN
ejpam-6013	40	8	,	,	PUNCT
ejpam-6013	40	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	40	10	functions	function	NOUN
ejpam-6013	40	11	.	.	PUNCT
ejpam-6013	41	1	2	2	X
ejpam-6013	41	2	.	.	NUM
ejpam-6013	41	3	preliminaries	preliminary	NOUN
ejpam-6013	41	4	throughout	throughout	ADP
ejpam-6013	41	5	the	the	DET
ejpam-6013	41	6	present	present	ADJ
ejpam-6013	41	7	paper	paper	NOUN
ejpam-6013	41	8	,	,	PUNCT
ejpam-6013	41	9	spaces	space	NOUN
ejpam-6013	41	10	(	(	PUNCT
ejpam-6013	41	11	x	x	NOUN
ejpam-6013	41	12	,	,	PUNCT
ejpam-6013	41	13	τ1	τ1	NOUN
ejpam-6013	41	14	,	,	PUNCT
ejpam-6013	41	15	τ2	τ2	NOUN
ejpam-6013	41	16	)	)	PUNCT
ejpam-6013	41	17	and	and	CCONJ
ejpam-6013	41	18	(	(	PUNCT
ejpam-6013	41	19	y	y	PROPN
ejpam-6013	41	20	,	,	PUNCT
ejpam-6013	41	21	σ1	σ1	PROPN
ejpam-6013	41	22	,	,	PUNCT
ejpam-6013	41	23	σ2	σ2	NOUN
ejpam-6013	41	24	)	)	PUNCT
ejpam-6013	41	25	(	(	PUNCT
ejpam-6013	41	26	or	or	CCONJ
ejpam-6013	41	27	simply	simply	ADV
ejpam-6013	41	28	x	x	X
ejpam-6013	41	29	and	and	CCONJ
ejpam-6013	41	30	y	y	PROPN
ejpam-6013	41	31	)	)	PUNCT
ejpam-6013	41	32	always	always	ADV
ejpam-6013	41	33	mean	mean	VERB
ejpam-6013	41	34	bitopological	bitopological	ADJ
ejpam-6013	41	35	spaces	space	NOUN
ejpam-6013	41	36	on	on	ADP
ejpam-6013	41	37	which	which	PRON
ejpam-6013	41	38	no	no	DET
ejpam-6013	41	39	separation	separation	NOUN
ejpam-6013	41	40	axioms	axiom	NOUN
ejpam-6013	41	41	are	be	AUX
ejpam-6013	41	42	assumed	assume	VERB
ejpam-6013	41	43	unless	unless	SCONJ
ejpam-6013	41	44	explicitly	explicitly	ADV
ejpam-6013	41	45	stated	state	VERB
ejpam-6013	41	46	.	.	PUNCT
ejpam-6013	42	1	let	let	VERB
ejpam-6013	42	2	a	a	DET
ejpam-6013	42	3	be	be	AUX
ejpam-6013	42	4	a	a	DET
ejpam-6013	42	5	subset	subset	NOUN
ejpam-6013	42	6	of	of	ADP
ejpam-6013	42	7	a	a	DET
ejpam-6013	42	8	bitopological	bitopological	ADJ
ejpam-6013	42	9	space	space	NOUN
ejpam-6013	42	10	(	(	PUNCT
ejpam-6013	42	11	x	x	NOUN
ejpam-6013	42	12	,	,	PUNCT
ejpam-6013	42	13	τ1	τ1	NOUN
ejpam-6013	42	14	,	,	PUNCT
ejpam-6013	42	15	τ2	τ2	NOUN
ejpam-6013	42	16	)	)	PUNCT
ejpam-6013	42	17	.	.	PUNCT
ejpam-6013	43	1	the	the	DET
ejpam-6013	43	2	closure	closure	NOUN
ejpam-6013	43	3	of	of	ADP
ejpam-6013	43	4	a	a	PRON
ejpam-6013	43	5	and	and	CCONJ
ejpam-6013	43	6	the	the	DET
ejpam-6013	43	7	interior	interior	NOUN
ejpam-6013	43	8	of	of	ADP
ejpam-6013	43	9	a	a	PRON
ejpam-6013	43	10	with	with	ADP
ejpam-6013	43	11	respect	respect	NOUN
ejpam-6013	43	12	to	to	ADP
ejpam-6013	43	13	τi	τi	PROPN
ejpam-6013	43	14	are	be	AUX
ejpam-6013	43	15	denoted	denote	VERB
ejpam-6013	43	16	by	by	ADP
ejpam-6013	43	17	τi	τi	NOUN
ejpam-6013	43	18	-	-	PUNCT
ejpam-6013	43	19	cl(a	cl(a	NUM
ejpam-6013	43	20	)	)	PUNCT
ejpam-6013	43	21	and	and	CCONJ
ejpam-6013	43	22	τi	τi	NOUN
ejpam-6013	43	23	-	-	PUNCT
ejpam-6013	43	24	int(a	int(a	NOUN
ejpam-6013	43	25	)	)	PUNCT
ejpam-6013	43	26	,	,	PUNCT
ejpam-6013	43	27	respectively	respectively	ADV
ejpam-6013	43	28	,	,	PUNCT
ejpam-6013	43	29	for	for	ADP
ejpam-6013	43	30	i	i	PROPN
ejpam-6013	43	31	=	=	SYM
ejpam-6013	43	32	1	1	NUM
ejpam-6013	43	33	,	,	PUNCT
ejpam-6013	43	34	2	2	NUM
ejpam-6013	43	35	.	.	X
ejpam-6013	43	36	a	a	DET
ejpam-6013	43	37	subset	subset	NOUN
ejpam-6013	43	38	a	a	PRON
ejpam-6013	43	39	of	of	ADP
ejpam-6013	43	40	a	a	DET
ejpam-6013	43	41	bitopological	bitopological	ADJ
ejpam-6013	43	42	space	space	NOUN
ejpam-6013	43	43	(	(	PUNCT
ejpam-6013	43	44	x	x	NOUN
ejpam-6013	43	45	,	,	PUNCT
ejpam-6013	43	46	τ1	τ1	NOUN
ejpam-6013	43	47	,	,	PUNCT
ejpam-6013	43	48	τ2	τ2	NOUN
ejpam-6013	43	49	)	)	PUNCT
ejpam-6013	43	50	is	be	AUX
ejpam-6013	43	51	called	call	VERB
ejpam-6013	43	52	τ1τ2	τ1τ2	VERB
ejpam-6013	43	53	-	-	ADJ
ejpam-6013	43	54	closed	closed	ADJ
ejpam-6013	43	55	[	[	X
ejpam-6013	43	56	25	25	NUM
ejpam-6013	43	57	]	]	PUNCT
ejpam-6013	43	58	if	if	SCONJ
ejpam-6013	43	59	a	a	DET
ejpam-6013	43	60	=	=	NOUN
ejpam-6013	43	61	τ1	τ1	NOUN
ejpam-6013	43	62	-	-	PUNCT
ejpam-6013	43	63	cl(τ2	cl(τ2	NOUN
ejpam-6013	43	64	-	-	PUNCT
ejpam-6013	43	65	cl(a	cl(a	NUM
ejpam-6013	43	66	)	)	PUNCT
ejpam-6013	43	67	)	)	PUNCT
ejpam-6013	43	68	.	.	PUNCT
ejpam-6013	44	1	the	the	DET
ejpam-6013	44	2	complement	complement	NOUN
ejpam-6013	44	3	of	of	ADP
ejpam-6013	44	4	a	a	DET
ejpam-6013	44	5	τ1τ2	τ1τ2	ADJ
ejpam-6013	44	6	-	-	ADJ
ejpam-6013	44	7	closed	closed	ADJ
ejpam-6013	44	8	set	set	NOUN
ejpam-6013	44	9	is	be	AUX
ejpam-6013	44	10	called	call	VERB
ejpam-6013	44	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	44	12	-	-	ADJ
ejpam-6013	44	13	open	open	ADJ
ejpam-6013	44	14	.	.	PUNCT
ejpam-6013	45	1	the	the	DET
ejpam-6013	45	2	intersection	intersection	NOUN
ejpam-6013	45	3	of	of	ADP
ejpam-6013	45	4	all	all	DET
ejpam-6013	45	5	τ1τ2	τ1τ2	ADJ
ejpam-6013	45	6	-	-	ADJ
ejpam-6013	45	7	closed	closed	ADJ
ejpam-6013	45	8	sets	set	NOUN
ejpam-6013	45	9	of	of	ADP
ejpam-6013	45	10	x	x	PUNCT
ejpam-6013	45	11	containing	contain	VERB
ejpam-6013	45	12	a	a	PRON
ejpam-6013	45	13	is	be	AUX
ejpam-6013	45	14	called	call	VERB
ejpam-6013	45	15	the	the	DET
ejpam-6013	45	16	τ1τ2	τ1τ2	NOUN
ejpam-6013	45	17	-	-	NOUN
ejpam-6013	45	18	closure	closure	NOUN
ejpam-6013	45	19	[	[	X
ejpam-6013	45	20	25	25	NUM
ejpam-6013	45	21	]	]	PUNCT
ejpam-6013	45	22	of	of	ADP
ejpam-6013	45	23	a	a	PRON
ejpam-6013	45	24	and	and	CCONJ
ejpam-6013	45	25	is	be	AUX
ejpam-6013	45	26	denoted	denote	VERB
ejpam-6013	45	27	by	by	ADP
ejpam-6013	45	28	τ1τ2	τ1τ2	NOUN
ejpam-6013	45	29	-	-	NUM
ejpam-6013	45	30	cl(a	cl(a	NUM
ejpam-6013	45	31	)	)	PUNCT
ejpam-6013	45	32	.	.	PUNCT
ejpam-6013	46	1	the	the	DET
ejpam-6013	46	2	union	union	NOUN
ejpam-6013	46	3	of	of	ADP
ejpam-6013	46	4	all	all	DET
ejpam-6013	46	5	τ1τ2	τ1τ2	ADJ
ejpam-6013	46	6	-	-	ADJ
ejpam-6013	46	7	open	open	ADJ
ejpam-6013	46	8	sets	set	NOUN
ejpam-6013	46	9	of	of	ADP
ejpam-6013	46	10	x	x	PUNCT
ejpam-6013	46	11	contained	contain	VERB
ejpam-6013	46	12	in	in	ADP
ejpam-6013	46	13	a	a	PRON
ejpam-6013	46	14	is	be	AUX
ejpam-6013	46	15	called	call	VERB
ejpam-6013	46	16	the	the	DET
ejpam-6013	46	17	τ1τ2	τ1τ2	NOUN
ejpam-6013	46	18	-	-	ADJ
ejpam-6013	46	19	interior	interior	ADJ
ejpam-6013	46	20	[	[	X
ejpam-6013	46	21	25	25	NUM
ejpam-6013	46	22	]	]	PUNCT
ejpam-6013	46	23	of	of	ADP
ejpam-6013	46	24	a	a	PRON
ejpam-6013	46	25	and	and	CCONJ
ejpam-6013	46	26	is	be	AUX
ejpam-6013	46	27	denoted	denote	VERB
ejpam-6013	46	28	by	by	ADP
ejpam-6013	46	29	τ1τ2	τ1τ2	NOUN
ejpam-6013	46	30	-	-	ADJ
ejpam-6013	46	31	int(a	int(a	NOUN
ejpam-6013	46	32	)	)	PUNCT
ejpam-6013	46	33	.	.	PUNCT
ejpam-6013	47	1	lemma	lemma	PROPN
ejpam-6013	47	2	1	1	NUM
ejpam-6013	47	3	.	.	PUNCT
ejpam-6013	48	1	[	[	X
ejpam-6013	48	2	25	25	NUM
ejpam-6013	48	3	]	]	PUNCT
ejpam-6013	48	4	let	let	VERB
ejpam-6013	48	5	a	a	PRON
ejpam-6013	48	6	and	and	CCONJ
ejpam-6013	48	7	b	b	NOUN
ejpam-6013	48	8	be	be	AUX
ejpam-6013	48	9	subsets	subset	NOUN
ejpam-6013	48	10	of	of	ADP
ejpam-6013	48	11	a	a	DET
ejpam-6013	48	12	bitopological	bitopological	ADJ
ejpam-6013	48	13	space	space	NOUN
ejpam-6013	48	14	(	(	PUNCT
ejpam-6013	48	15	x	x	NOUN
ejpam-6013	48	16	,	,	PUNCT
ejpam-6013	48	17	τ1	τ1	NOUN
ejpam-6013	48	18	,	,	PUNCT
ejpam-6013	48	19	τ2	τ2	NOUN
ejpam-6013	48	20	)	)	PUNCT
ejpam-6013	48	21	.	.	PUNCT
ejpam-6013	49	1	for	for	ADP
ejpam-6013	49	2	the	the	DET
ejpam-6013	49	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6013	49	4	,	,	PUNCT
ejpam-6013	49	5	the	the	DET
ejpam-6013	49	6	following	follow	VERB
ejpam-6013	49	7	properties	property	NOUN
ejpam-6013	49	8	hold	hold	VERB
ejpam-6013	49	9	:	:	PUNCT
ejpam-6013	49	10	(	(	PUNCT
ejpam-6013	49	11	1	1	X
ejpam-6013	49	12	)	)	PUNCT
ejpam-6013	49	13	a	a	DET
ejpam-6013	49	14	⊆	⊆	NUM
ejpam-6013	49	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	49	16	-	-	NUM
ejpam-6013	49	17	cl(a	cl(a	NUM
ejpam-6013	49	18	)	)	PUNCT
ejpam-6013	49	19	and	and	CCONJ
ejpam-6013	49	20	τ1τ2	τ1τ2	NOUN
ejpam-6013	49	21	-	-	ADJ
ejpam-6013	49	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6013	49	23	-	-	PUNCT
ejpam-6013	49	24	cl(a	cl(a	NUM
ejpam-6013	49	25	)	)	PUNCT
ejpam-6013	49	26	)	)	PUNCT
ejpam-6013	50	1	=	=	PUNCT
ejpam-6013	50	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	50	3	-	-	NUM
ejpam-6013	50	4	cl(a	cl(a	NUM
ejpam-6013	50	5	)	)	PUNCT
ejpam-6013	50	6	.	.	PUNCT
ejpam-6013	51	1	(	(	PUNCT
ejpam-6013	51	2	2	2	X
ejpam-6013	51	3	)	)	PUNCT
ejpam-6013	51	4	if	if	SCONJ
ejpam-6013	51	5	a	a	DET
ejpam-6013	51	6	⊆	⊆	NUM
ejpam-6013	51	7	b	b	NOUN
ejpam-6013	51	8	,	,	PUNCT
ejpam-6013	51	9	then	then	ADV
ejpam-6013	51	10	τ1τ2	τ1τ2	NOUN
ejpam-6013	51	11	-	-	NUM
ejpam-6013	51	12	cl(a	cl(a	NUM
ejpam-6013	51	13	)	)	PUNCT
ejpam-6013	51	14	⊆	⊆	NUM
ejpam-6013	51	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	51	16	-	-	NOUN
ejpam-6013	51	17	cl(b	cl(b	NOUN
ejpam-6013	51	18	)	)	PUNCT
ejpam-6013	51	19	.	.	PUNCT
ejpam-6013	52	1	(	(	PUNCT
ejpam-6013	52	2	3	3	X
ejpam-6013	52	3	)	)	PUNCT
ejpam-6013	52	4	τ1τ2	τ1τ2	NOUN
ejpam-6013	52	5	-	-	NUM
ejpam-6013	52	6	cl(a	cl(a	NUM
ejpam-6013	52	7	)	)	PUNCT
ejpam-6013	52	8	is	be	AUX
ejpam-6013	52	9	τ1τ2	τ1τ2	NOUN
ejpam-6013	52	10	-	-	ADJ
ejpam-6013	52	11	closed	closed	ADJ
ejpam-6013	52	12	.	.	PUNCT
ejpam-6013	53	1	(	(	PUNCT
ejpam-6013	53	2	4	4	X
ejpam-6013	53	3	)	)	PUNCT
ejpam-6013	53	4	a	a	PRON
ejpam-6013	53	5	is	be	AUX
ejpam-6013	53	6	τ1τ2	τ1τ2	NOUN
ejpam-6013	53	7	-	-	ADJ
ejpam-6013	53	8	closed	closed	ADJ
ejpam-6013	53	9	if	if	SCONJ
ejpam-6013	53	10	and	and	CCONJ
ejpam-6013	53	11	only	only	ADV
ejpam-6013	53	12	if	if	SCONJ
ejpam-6013	53	13	a	a	DET
ejpam-6013	53	14	=	=	PUNCT
ejpam-6013	53	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	53	16	-	-	NUM
ejpam-6013	53	17	cl(a	cl(a	NUM
ejpam-6013	53	18	)	)	PUNCT
ejpam-6013	53	19	.	.	PUNCT
ejpam-6013	54	1	(	(	PUNCT
ejpam-6013	54	2	5	5	X
ejpam-6013	54	3	)	)	PUNCT
ejpam-6013	54	4	τ1τ2	τ1τ2	NOUN
ejpam-6013	54	5	-	-	NOUN
ejpam-6013	54	6	cl(x	cl(x	X
ejpam-6013	54	7	−a	−a	NOUN
ejpam-6013	54	8	)	)	PUNCT
ejpam-6013	55	1	=	=	PUNCT
ejpam-6013	55	2	x	x	X
ejpam-6013	56	1	−	−	ADP
ejpam-6013	56	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	56	3	-	-	ADJ
ejpam-6013	56	4	int(a	int(a	NOUN
ejpam-6013	56	5	)	)	PUNCT
ejpam-6013	56	6	.	.	PUNCT
ejpam-6013	57	1	n.	n.	PROPN
ejpam-6013	57	2	srisarakham	srisarakham	PROPN
ejpam-6013	57	3	,	,	PUNCT
ejpam-6013	57	4	s.	s.	PROPN
ejpam-6013	57	5	sompong	sompong	PROPN
ejpam-6013	57	6	,	,	PUNCT
ejpam-6013	57	7	c.	c.	PROPN
ejpam-6013	57	8	boonpok	boonpok	PROPN
ejpam-6013	57	9	/	/	SYM
ejpam-6013	57	10	eur	eur	PROPN
ejpam-6013	57	11	.	.	PUNCT
ejpam-6013	58	1	j.	j.	PROPN
ejpam-6013	58	2	pure	pure	PROPN
ejpam-6013	58	3	appl	appl	PROPN
ejpam-6013	58	4	.	.	PROPN
ejpam-6013	58	5	math	math	PROPN
ejpam-6013	58	6	,	,	PUNCT
ejpam-6013	58	7	18	18	NUM
ejpam-6013	58	8	(	(	PUNCT
ejpam-6013	58	9	2	2	NUM
ejpam-6013	58	10	)	)	PUNCT
ejpam-6013	58	11	(	(	PUNCT
ejpam-6013	58	12	2025	2025	NUM
ejpam-6013	58	13	)	)	PUNCT
ejpam-6013	58	14	,	,	PUNCT
ejpam-6013	58	15	6013	6013	NUM
ejpam-6013	58	16	3	3	NUM
ejpam-6013	58	17	of	of	ADP
ejpam-6013	58	18	12	12	NUM
ejpam-6013	58	19	a	a	DET
ejpam-6013	58	20	subset	subset	NOUN
ejpam-6013	58	21	a	a	PRON
ejpam-6013	58	22	of	of	ADP
ejpam-6013	58	23	a	a	DET
ejpam-6013	58	24	bitopological	bitopological	ADJ
ejpam-6013	58	25	space	space	NOUN
ejpam-6013	58	26	(	(	PUNCT
ejpam-6013	58	27	x	x	NOUN
ejpam-6013	58	28	,	,	PUNCT
ejpam-6013	58	29	τ1	τ1	NOUN
ejpam-6013	58	30	,	,	PUNCT
ejpam-6013	58	31	τ2	τ2	NOUN
ejpam-6013	58	32	)	)	PUNCT
ejpam-6013	58	33	is	be	AUX
ejpam-6013	58	34	said	say	VERB
ejpam-6013	58	35	to	to	PART
ejpam-6013	58	36	be	be	AUX
ejpam-6013	58	37	(	(	PUNCT
ejpam-6013	58	38	τ1	τ1	NOUN
ejpam-6013	58	39	,	,	PUNCT
ejpam-6013	58	40	τ2)r	τ2)r	NOUN
ejpam-6013	58	41	-	-	PUNCT
ejpam-6013	58	42	open	open	NOUN
ejpam-6013	58	43	[	[	X
ejpam-6013	58	44	26	26	NUM
ejpam-6013	58	45	]	]	PUNCT
ejpam-6013	58	46	(	(	PUNCT
ejpam-6013	58	47	resp	resp	NOUN
ejpam-6013	58	48	.	.	PUNCT
ejpam-6013	59	1	(	(	PUNCT
ejpam-6013	59	2	τ1	τ1	NOUN
ejpam-6013	59	3	,	,	PUNCT
ejpam-6013	59	4	τ2)s	τ2)s	NOUN
ejpam-6013	59	5	-	-	PUNCT
ejpam-6013	59	6	open	open	ADJ
ejpam-6013	59	7	[	[	X
ejpam-6013	59	8	27	27	NUM
ejpam-6013	59	9	]	]	NUM
ejpam-6013	59	10	,	,	PUNCT
ejpam-6013	59	11	(	(	PUNCT
ejpam-6013	59	12	τ1	τ1	NOUN
ejpam-6013	59	13	,	,	PUNCT
ejpam-6013	59	14	τ2)p	τ2)p	NOUN
ejpam-6013	59	15	-	-	ADJ
ejpam-6013	59	16	open	open	ADJ
ejpam-6013	59	17	[	[	X
ejpam-6013	59	18	27	27	NUM
ejpam-6013	59	19	]	]	NUM
ejpam-6013	59	20	,	,	PUNCT
ejpam-6013	59	21	(	(	PUNCT
ejpam-6013	59	22	τ1	τ1	NOUN
ejpam-6013	59	23	,	,	PUNCT
ejpam-6013	59	24	τ2)β	τ2)β	ADJ
ejpam-6013	59	25	-	-	PUNCT
ejpam-6013	59	26	open	open	NOUN
ejpam-6013	60	1	[	[	X
ejpam-6013	60	2	27	27	NUM
ejpam-6013	60	3	]	]	SYM
ejpam-6013	60	4	)	)	PUNCT
ejpam-6013	60	5	if	if	SCONJ
ejpam-6013	60	6	a	a	DET
ejpam-6013	60	7	=	=	PUNCT
ejpam-6013	60	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	60	9	-	-	NOUN
ejpam-6013	60	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	60	11	-	-	PUNCT
ejpam-6013	60	12	cl(a	cl(a	NUM
ejpam-6013	60	13	)	)	PUNCT
ejpam-6013	60	14	)	)	PUNCT
ejpam-6013	60	15	(	(	PUNCT
ejpam-6013	60	16	resp	resp	NOUN
ejpam-6013	60	17	.	.	PUNCT
ejpam-6013	61	1	a	a	DET
ejpam-6013	61	2	⊆	⊆	NUM
ejpam-6013	61	3	τ1τ2	τ1τ2	NOUN
ejpam-6013	61	4	-	-	ADJ
ejpam-6013	61	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6013	61	6	-	-	PUNCT
ejpam-6013	61	7	int(a	int(a	NOUN
ejpam-6013	61	8	)	)	PUNCT
ejpam-6013	61	9	)	)	PUNCT
ejpam-6013	61	10	,	,	PUNCT
ejpam-6013	61	11	a	a	DET
ejpam-6013	61	12	⊆	⊆	NUM
ejpam-6013	61	13	τ1τ2	τ1τ2	NOUN
ejpam-6013	61	14	-	-	NOUN
ejpam-6013	61	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	61	16	-	-	PUNCT
ejpam-6013	61	17	cl(a	cl(a	NUM
ejpam-6013	61	18	)	)	PUNCT
ejpam-6013	61	19	)	)	PUNCT
ejpam-6013	61	20	,	,	PUNCT
ejpam-6013	61	21	a	a	DET
ejpam-6013	61	22	⊆	⊆	NUM
ejpam-6013	61	23	τ1τ2	τ1τ2	NOUN
ejpam-6013	61	24	-	-	PUNCT
ejpam-6013	61	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6013	61	26	-	-	PUNCT
ejpam-6013	61	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	61	28	-	-	PUNCT
ejpam-6013	61	29	cl(a	cl(a	NUM
ejpam-6013	61	30	)	)	PUNCT
ejpam-6013	61	31	)	)	PUNCT
ejpam-6013	61	32	)	)	PUNCT
ejpam-6013	61	33	)	)	PUNCT
ejpam-6013	61	34	.	.	PUNCT
ejpam-6013	62	1	the	the	DET
ejpam-6013	62	2	complement	complement	NOUN
ejpam-6013	62	3	of	of	ADP
ejpam-6013	62	4	a	a	DET
ejpam-6013	62	5	(	(	PUNCT
ejpam-6013	62	6	τ1	τ1	NOUN
ejpam-6013	62	7	,	,	PUNCT
ejpam-6013	62	8	τ2)r	τ2)r	NOUN
ejpam-6013	62	9	-	-	PUNCT
ejpam-6013	62	10	open	open	ADJ
ejpam-6013	62	11	(	(	PUNCT
ejpam-6013	62	12	resp	resp	NOUN
ejpam-6013	62	13	.	.	PUNCT
ejpam-6013	63	1	(	(	PUNCT
ejpam-6013	63	2	τ1	τ1	NOUN
ejpam-6013	63	3	,	,	PUNCT
ejpam-6013	63	4	τ2)s	τ2)s	NOUN
ejpam-6013	63	5	-	-	PUNCT
ejpam-6013	63	6	open	open	ADJ
ejpam-6013	63	7	,	,	PUNCT
ejpam-6013	63	8	(	(	PUNCT
ejpam-6013	63	9	τ1	τ1	NOUN
ejpam-6013	63	10	,	,	PUNCT
ejpam-6013	63	11	τ2)p	τ2)p	NOUN
ejpam-6013	63	12	-	-	ADJ
ejpam-6013	63	13	open	open	ADJ
ejpam-6013	63	14	,	,	PUNCT
ejpam-6013	63	15	(	(	PUNCT
ejpam-6013	63	16	τ1	τ1	NOUN
ejpam-6013	63	17	,	,	PUNCT
ejpam-6013	63	18	τ2)β	τ2)β	ADJ
ejpam-6013	63	19	-	-	PUNCT
ejpam-6013	63	20	open	open	ADJ
ejpam-6013	63	21	)	)	PUNCT
ejpam-6013	63	22	set	set	NOUN
ejpam-6013	63	23	is	be	AUX
ejpam-6013	63	24	called	call	VERB
ejpam-6013	63	25	(	(	PUNCT
ejpam-6013	63	26	τ1	τ1	NOUN
ejpam-6013	63	27	,	,	PUNCT
ejpam-6013	63	28	τ2)r	τ2)r	NOUN
ejpam-6013	63	29	-	-	PUNCT
ejpam-6013	63	30	closed	closed	ADJ
ejpam-6013	63	31	(	(	PUNCT
ejpam-6013	63	32	resp	resp	NOUN
ejpam-6013	63	33	.	.	PUNCT
ejpam-6013	64	1	(	(	PUNCT
ejpam-6013	64	2	τ1	τ1	NOUN
ejpam-6013	64	3	,	,	PUNCT
ejpam-6013	64	4	τ2)s	τ2)s	NOUN
ejpam-6013	64	5	-	-	PUNCT
ejpam-6013	64	6	closed	closed	ADJ
ejpam-6013	64	7	,	,	PUNCT
ejpam-6013	64	8	(	(	PUNCT
ejpam-6013	64	9	τ1	τ1	NOUN
ejpam-6013	64	10	,	,	PUNCT
ejpam-6013	64	11	τ2)p	τ2)p	NOUN
ejpam-6013	64	12	-	-	PUNCT
ejpam-6013	64	13	closed	closed	ADJ
ejpam-6013	64	14	,	,	PUNCT
ejpam-6013	64	15	(	(	PUNCT
ejpam-6013	64	16	τ1	τ1	NOUN
ejpam-6013	64	17	,	,	PUNCT
ejpam-6013	64	18	τ2)β	τ2)β	ADJ
ejpam-6013	64	19	-	-	PUNCT
ejpam-6013	64	20	closed	closed	ADJ
ejpam-6013	64	21	)	)	PUNCT
ejpam-6013	64	22	.	.	PUNCT
ejpam-6013	65	1	a	a	DET
ejpam-6013	65	2	subset	subset	NOUN
ejpam-6013	65	3	a	a	PRON
ejpam-6013	65	4	of	of	ADP
ejpam-6013	65	5	a	a	DET
ejpam-6013	65	6	bitopological	bitopological	ADJ
ejpam-6013	65	7	space	space	NOUN
ejpam-6013	65	8	(	(	PUNCT
ejpam-6013	65	9	x	x	NOUN
ejpam-6013	65	10	,	,	PUNCT
ejpam-6013	65	11	τ1	τ1	NOUN
ejpam-6013	65	12	,	,	PUNCT
ejpam-6013	65	13	τ2	τ2	NOUN
ejpam-6013	65	14	)	)	PUNCT
ejpam-6013	65	15	is	be	AUX
ejpam-6013	65	16	said	say	VERB
ejpam-6013	65	17	to	to	PART
ejpam-6013	65	18	be	be	AUX
ejpam-6013	65	19	α(τ1	α(τ1	NOUN
ejpam-6013	65	20	,	,	PUNCT
ejpam-6013	65	21	τ2)-open	τ2)-open	ADJ
ejpam-6013	65	22	[	[	X
ejpam-6013	65	23	28	28	NUM
ejpam-6013	65	24	]	]	X
ejpam-6013	65	25	if	if	SCONJ
ejpam-6013	65	26	a	a	DET
ejpam-6013	65	27	⊆	⊆	NUM
ejpam-6013	65	28	τ1τ2	τ1τ2	NOUN
ejpam-6013	65	29	-	-	PUNCT
ejpam-6013	65	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	65	31	-	-	PUNCT
ejpam-6013	65	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6013	65	33	-	-	PUNCT
ejpam-6013	65	34	int(a	int(a	NOUN
ejpam-6013	65	35	)	)	PUNCT
ejpam-6013	65	36	)	)	PUNCT
ejpam-6013	65	37	)	)	PUNCT
ejpam-6013	65	38	.	.	PUNCT
ejpam-6013	66	1	the	the	DET
ejpam-6013	66	2	complement	complement	NOUN
ejpam-6013	66	3	of	of	ADP
ejpam-6013	66	4	an	an	DET
ejpam-6013	66	5	α(τ1	α(τ1	NOUN
ejpam-6013	66	6	,	,	PUNCT
ejpam-6013	66	7	τ2)-open	τ2)-open	ADJ
ejpam-6013	66	8	set	set	NOUN
ejpam-6013	66	9	is	be	AUX
ejpam-6013	66	10	said	say	VERB
ejpam-6013	66	11	to	to	PART
ejpam-6013	66	12	be	be	AUX
ejpam-6013	66	13	α(τ1	α(τ1	NOUN
ejpam-6013	66	14	,	,	PUNCT
ejpam-6013	66	15	τ2)-closed	τ2)-close	VERB
ejpam-6013	66	16	.	.	PUNCT
ejpam-6013	67	1	let	let	VERB
ejpam-6013	67	2	a	a	DET
ejpam-6013	67	3	be	be	AUX
ejpam-6013	67	4	a	a	DET
ejpam-6013	67	5	subset	subset	NOUN
ejpam-6013	67	6	of	of	ADP
ejpam-6013	67	7	a	a	DET
ejpam-6013	67	8	bitopological	bitopological	ADJ
ejpam-6013	67	9	space	space	NOUN
ejpam-6013	67	10	(	(	PUNCT
ejpam-6013	67	11	x	x	NOUN
ejpam-6013	67	12	,	,	PUNCT
ejpam-6013	67	13	τ1	τ1	NOUN
ejpam-6013	67	14	,	,	PUNCT
ejpam-6013	67	15	τ2	τ2	NOUN
ejpam-6013	67	16	)	)	PUNCT
ejpam-6013	67	17	.	.	PUNCT
ejpam-6013	68	1	a	a	DET
ejpam-6013	68	2	point	point	NOUN
ejpam-6013	68	3	x	x	X
ejpam-6013	68	4	∈	∈	NOUN
ejpam-6013	68	5	x	x	PUNCT
ejpam-6013	68	6	is	be	AUX
ejpam-6013	68	7	called	call	VERB
ejpam-6013	68	8	a	a	DET
ejpam-6013	68	9	(	(	PUNCT
ejpam-6013	68	10	τ1	τ1	NOUN
ejpam-6013	68	11	,	,	PUNCT
ejpam-6013	68	12	τ2)θ	τ2)θ	ADJ
ejpam-6013	68	13	-	-	PUNCT
ejpam-6013	68	14	cluster	cluster	NOUN
ejpam-6013	68	15	point	point	NOUN
ejpam-6013	68	16	[	[	X
ejpam-6013	68	17	26	26	NUM
ejpam-6013	68	18	]	]	PUNCT
ejpam-6013	68	19	of	of	ADP
ejpam-6013	68	20	a	a	DET
ejpam-6013	68	21	if	if	SCONJ
ejpam-6013	68	22	τ1τ2	τ1τ2	NOUN
ejpam-6013	68	23	-	-	NOUN
ejpam-6013	68	24	cl(u	cl(u	NOUN
ejpam-6013	68	25	)	)	PUNCT
ejpam-6013	68	26	∩	∩	NOUN
ejpam-6013	68	27	a	a	DET
ejpam-6013	68	28	̸=	̸=	PROPN
ejpam-6013	68	29	∅	∅	NOUN
ejpam-6013	68	30	for	for	ADP
ejpam-6013	68	31	every	every	DET
ejpam-6013	68	32	τ1τ2	τ1τ2	ADJ
ejpam-6013	68	33	-	-	ADJ
ejpam-6013	68	34	open	open	ADJ
ejpam-6013	68	35	set	set	NOUN
ejpam-6013	68	36	u	u	NOUN
ejpam-6013	68	37	containing	contain	VERB
ejpam-6013	68	38	x.	x.	NOUN
ejpam-6013	68	39	the	the	DET
ejpam-6013	68	40	set	set	NOUN
ejpam-6013	68	41	of	of	ADP
ejpam-6013	68	42	all	all	DET
ejpam-6013	68	43	(	(	PUNCT
ejpam-6013	68	44	τ1	τ1	NOUN
ejpam-6013	68	45	,	,	PUNCT
ejpam-6013	68	46	τ2)θ	τ2)θ	ADJ
ejpam-6013	68	47	-	-	PUNCT
ejpam-6013	68	48	cluster	cluster	NOUN
ejpam-6013	68	49	points	point	NOUN
ejpam-6013	68	50	of	of	ADP
ejpam-6013	68	51	a	a	PRON
ejpam-6013	68	52	is	be	AUX
ejpam-6013	68	53	called	call	VERB
ejpam-6013	68	54	the	the	DET
ejpam-6013	68	55	(	(	PUNCT
ejpam-6013	68	56	τ1	τ1	NOUN
ejpam-6013	68	57	,	,	PUNCT
ejpam-6013	68	58	τ2)θ	τ2)θ	ADJ
ejpam-6013	68	59	-	-	PUNCT
ejpam-6013	68	60	closure	closure	NOUN
ejpam-6013	68	61	[	[	X
ejpam-6013	68	62	26	26	NUM
ejpam-6013	68	63	]	]	PUNCT
ejpam-6013	68	64	of	of	ADP
ejpam-6013	68	65	a	a	PRON
ejpam-6013	68	66	and	and	CCONJ
ejpam-6013	68	67	is	be	AUX
ejpam-6013	68	68	denoted	denote	VERB
ejpam-6013	68	69	by	by	ADP
ejpam-6013	68	70	(	(	PUNCT
ejpam-6013	68	71	τ1	τ1	NOUN
ejpam-6013	68	72	,	,	PUNCT
ejpam-6013	68	73	τ2)θ	τ2)θ	NOUN
ejpam-6013	68	74	-	-	PUNCT
ejpam-6013	68	75	cl(a	cl(a	NUM
ejpam-6013	68	76	)	)	PUNCT
ejpam-6013	68	77	.	.	PUNCT
ejpam-6013	69	1	a	a	DET
ejpam-6013	69	2	subset	subset	NOUN
ejpam-6013	69	3	a	a	PRON
ejpam-6013	69	4	of	of	ADP
ejpam-6013	69	5	a	a	DET
ejpam-6013	69	6	bitopological	bitopological	ADJ
ejpam-6013	69	7	space	space	NOUN
ejpam-6013	69	8	(	(	PUNCT
ejpam-6013	69	9	x	x	NOUN
ejpam-6013	69	10	,	,	PUNCT
ejpam-6013	69	11	τ1	τ1	NOUN
ejpam-6013	69	12	,	,	PUNCT
ejpam-6013	69	13	τ2	τ2	NOUN
ejpam-6013	69	14	)	)	PUNCT
ejpam-6013	69	15	is	be	AUX
ejpam-6013	69	16	said	say	VERB
ejpam-6013	69	17	to	to	PART
ejpam-6013	69	18	be	be	AUX
ejpam-6013	69	19	(	(	PUNCT
ejpam-6013	69	20	τ1	τ1	NOUN
ejpam-6013	69	21	,	,	PUNCT
ejpam-6013	69	22	τ2)θ	τ2)θ	NOUN
ejpam-6013	69	23	-	-	PUNCT
ejpam-6013	69	24	closed	closed	ADJ
ejpam-6013	69	25	[	[	X
ejpam-6013	69	26	26	26	NUM
ejpam-6013	69	27	]	]	X
ejpam-6013	69	28	if	if	SCONJ
ejpam-6013	69	29	a	a	PRON
ejpam-6013	69	30	=	=	X
ejpam-6013	69	31	(	(	PUNCT
ejpam-6013	69	32	τ1	τ1	NOUN
ejpam-6013	69	33	,	,	PUNCT
ejpam-6013	69	34	τ2)θ	τ2)θ	NOUN
ejpam-6013	69	35	-	-	PUNCT
ejpam-6013	69	36	cl(a	cl(a	NUM
ejpam-6013	69	37	)	)	PUNCT
ejpam-6013	69	38	.	.	PUNCT
ejpam-6013	70	1	the	the	DET
ejpam-6013	70	2	complement	complement	NOUN
ejpam-6013	70	3	of	of	ADP
ejpam-6013	70	4	a	a	DET
ejpam-6013	70	5	(	(	PUNCT
ejpam-6013	70	6	τ1	τ1	NOUN
ejpam-6013	70	7	,	,	PUNCT
ejpam-6013	70	8	τ2)θ	τ2)θ	ADJ
ejpam-6013	70	9	-	-	PUNCT
ejpam-6013	70	10	closed	close	VERB
ejpam-6013	70	11	set	set	NOUN
ejpam-6013	70	12	is	be	AUX
ejpam-6013	70	13	said	say	VERB
ejpam-6013	70	14	to	to	PART
ejpam-6013	70	15	be	be	AUX
ejpam-6013	70	16	(	(	PUNCT
ejpam-6013	70	17	τ1	τ1	NOUN
ejpam-6013	70	18	,	,	PUNCT
ejpam-6013	70	19	τ2)θ	τ2)θ	NOUN
ejpam-6013	70	20	-	-	PUNCT
ejpam-6013	70	21	open	open	ADJ
ejpam-6013	70	22	.	.	PUNCT
ejpam-6013	71	1	the	the	DET
ejpam-6013	71	2	union	union	NOUN
ejpam-6013	71	3	of	of	ADP
ejpam-6013	71	4	all	all	DET
ejpam-6013	71	5	(	(	PUNCT
ejpam-6013	71	6	τ1	τ1	NOUN
ejpam-6013	71	7	,	,	PUNCT
ejpam-6013	71	8	τ2)θ	τ2)θ	ADJ
ejpam-6013	71	9	-	-	PUNCT
ejpam-6013	71	10	open	open	ADJ
ejpam-6013	71	11	sets	set	NOUN
ejpam-6013	71	12	contained	contain	VERB
ejpam-6013	71	13	in	in	ADP
ejpam-6013	71	14	a	a	PRON
ejpam-6013	71	15	is	be	AUX
ejpam-6013	71	16	called	call	VERB
ejpam-6013	71	17	the	the	DET
ejpam-6013	71	18	(	(	PUNCT
ejpam-6013	71	19	τ1	τ1	NOUN
ejpam-6013	71	20	,	,	PUNCT
ejpam-6013	71	21	τ2)θ	τ2)θ	ADJ
ejpam-6013	71	22	-	-	PUNCT
ejpam-6013	71	23	interior	interior	NOUN
ejpam-6013	71	24	[	[	X
ejpam-6013	71	25	26	26	NUM
ejpam-6013	71	26	]	]	PUNCT
ejpam-6013	71	27	of	of	ADP
ejpam-6013	71	28	a	a	PRON
ejpam-6013	71	29	and	and	CCONJ
ejpam-6013	71	30	is	be	AUX
ejpam-6013	71	31	denoted	denote	VERB
ejpam-6013	71	32	by	by	ADP
ejpam-6013	71	33	(	(	PUNCT
ejpam-6013	71	34	τ1	τ1	NOUN
ejpam-6013	71	35	,	,	PUNCT
ejpam-6013	71	36	τ2)θ	τ2)θ	NOUN
ejpam-6013	71	37	-	-	PUNCT
ejpam-6013	71	38	int(a	int(a	NOUN
ejpam-6013	71	39	)	)	PUNCT
ejpam-6013	71	40	.	.	PUNCT
ejpam-6013	72	1	lemma	lemma	PROPN
ejpam-6013	72	2	2	2	NUM
ejpam-6013	72	3	.	.	PUNCT
ejpam-6013	73	1	[	[	X
ejpam-6013	73	2	26	26	NUM
ejpam-6013	73	3	]	]	PUNCT
ejpam-6013	73	4	for	for	ADP
ejpam-6013	73	5	a	a	DET
ejpam-6013	73	6	subset	subset	NOUN
ejpam-6013	73	7	a	a	PRON
ejpam-6013	73	8	of	of	ADP
ejpam-6013	73	9	a	a	DET
ejpam-6013	73	10	bitopological	bitopological	ADJ
ejpam-6013	73	11	space	space	NOUN
ejpam-6013	73	12	(	(	PUNCT
ejpam-6013	73	13	x	x	NOUN
ejpam-6013	73	14	,	,	PUNCT
ejpam-6013	73	15	τ1	τ1	NOUN
ejpam-6013	73	16	,	,	PUNCT
ejpam-6013	73	17	τ2	τ2	NOUN
ejpam-6013	73	18	)	)	PUNCT
ejpam-6013	73	19	,	,	PUNCT
ejpam-6013	73	20	the	the	DET
ejpam-6013	73	21	following	follow	VERB
ejpam-6013	73	22	properties	property	NOUN
ejpam-6013	73	23	hold	hold	VERB
ejpam-6013	73	24	:	:	PUNCT
ejpam-6013	73	25	(	(	PUNCT
ejpam-6013	73	26	1	1	X
ejpam-6013	73	27	)	)	PUNCT
ejpam-6013	73	28	if	if	SCONJ
ejpam-6013	73	29	a	a	PRON
ejpam-6013	73	30	is	be	AUX
ejpam-6013	73	31	τ2τ2	τ2τ2	VERB
ejpam-6013	73	32	-	-	VERB
ejpam-6013	73	33	open	open	ADJ
ejpam-6013	73	34	in	in	ADP
ejpam-6013	73	35	x	x	NOUN
ejpam-6013	73	36	,	,	PUNCT
ejpam-6013	73	37	then	then	ADV
ejpam-6013	73	38	τ1τ2	τ1τ2	NOUN
ejpam-6013	73	39	-	-	NUM
ejpam-6013	73	40	cl(a	cl(a	NUM
ejpam-6013	73	41	)	)	PUNCT
ejpam-6013	73	42	=	=	PUNCT
ejpam-6013	73	43	(	(	PUNCT
ejpam-6013	73	44	τ1	τ1	NOUN
ejpam-6013	73	45	,	,	PUNCT
ejpam-6013	73	46	τ2)θ	τ2)θ	NOUN
ejpam-6013	73	47	-	-	PUNCT
ejpam-6013	73	48	cl(a	cl(a	NUM
ejpam-6013	73	49	)	)	PUNCT
ejpam-6013	73	50	.	.	PUNCT
ejpam-6013	74	1	(	(	PUNCT
ejpam-6013	74	2	2	2	X
ejpam-6013	74	3	)	)	PUNCT
ejpam-6013	74	4	(	(	PUNCT
ejpam-6013	74	5	τ1	τ1	NOUN
ejpam-6013	74	6	,	,	PUNCT
ejpam-6013	74	7	τ2)θ	τ2)θ	NOUN
ejpam-6013	74	8	-	-	PUNCT
ejpam-6013	74	9	cl(a	cl(a	NUM
ejpam-6013	74	10	)	)	PUNCT
ejpam-6013	74	11	is	be	AUX
ejpam-6013	74	12	τ1τ2	τ1τ2	NOUN
ejpam-6013	74	13	-	-	ADJ
ejpam-6013	74	14	closed	closed	ADJ
ejpam-6013	74	15	in	in	ADP
ejpam-6013	74	16	x.	x.	NOUN
ejpam-6013	74	17	3	3	NUM
ejpam-6013	74	18	.	.	X
ejpam-6013	74	19	r-(τ1	r-(τ1	PROPN
ejpam-6013	74	20	,	,	PUNCT
ejpam-6013	74	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	74	22	functions	function	NOUN
ejpam-6013	74	23	in	in	ADP
ejpam-6013	74	24	this	this	DET
ejpam-6013	74	25	section	section	NOUN
ejpam-6013	74	26	,	,	PUNCT
ejpam-6013	74	27	we	we	PRON
ejpam-6013	74	28	introduce	introduce	VERB
ejpam-6013	74	29	the	the	DET
ejpam-6013	74	30	concept	concept	NOUN
ejpam-6013	74	31	of	of	ADP
ejpam-6013	74	32	r-(τ1	r-(τ1	PROPN
ejpam-6013	74	33	,	,	PUNCT
ejpam-6013	74	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	74	35	functions	function	NOUN
ejpam-6013	74	36	.	.	PUNCT
ejpam-6013	75	1	furthermore	furthermore	ADV
ejpam-6013	75	2	,	,	PUNCT
ejpam-6013	75	3	several	several	ADJ
ejpam-6013	75	4	characterizations	characterization	NOUN
ejpam-6013	75	5	of	of	ADP
ejpam-6013	75	6	r-(τ1	r-(τ1	PROPN
ejpam-6013	75	7	,	,	PUNCT
ejpam-6013	75	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	75	9	functions	function	NOUN
ejpam-6013	75	10	are	be	AUX
ejpam-6013	75	11	discussed	discuss	VERB
ejpam-6013	75	12	.	.	PUNCT
ejpam-6013	76	1	definition	definition	NOUN
ejpam-6013	76	2	1	1	NUM
ejpam-6013	76	3	.	.	PUNCT
ejpam-6013	77	1	a	a	DET
ejpam-6013	77	2	functions	function	NOUN
ejpam-6013	77	3	f	f	X
ejpam-6013	77	4	:	:	PUNCT
ejpam-6013	77	5	(	(	PUNCT
ejpam-6013	77	6	x	x	NOUN
ejpam-6013	77	7	,	,	PUNCT
ejpam-6013	77	8	τ1	τ1	NOUN
ejpam-6013	77	9	,	,	PUNCT
ejpam-6013	77	10	τ2	τ2	NOUN
ejpam-6013	77	11	)	)	PUNCT
ejpam-6013	77	12	→	→	SYM
ejpam-6013	77	13	(	(	PUNCT
ejpam-6013	77	14	y	y	PROPN
ejpam-6013	77	15	,	,	PUNCT
ejpam-6013	77	16	σ1	σ1	PROPN
ejpam-6013	77	17	,	,	PUNCT
ejpam-6013	77	18	σ2	σ2	PROPN
ejpam-6013	77	19	)	)	PUNCT
ejpam-6013	77	20	is	be	AUX
ejpam-6013	77	21	said	say	VERB
ejpam-6013	77	22	to	to	PART
ejpam-6013	77	23	be	be	AUX
ejpam-6013	77	24	r-(τ1	r-(τ1	PROPN
ejpam-6013	77	25	,	,	PUNCT
ejpam-6013	77	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	77	27	if	if	SCONJ
ejpam-6013	77	28	for	for	ADP
ejpam-6013	77	29	each	each	DET
ejpam-6013	77	30	x	x	SYM
ejpam-6013	77	31	∈	∈	PROPN
ejpam-6013	77	32	x	x	X
ejpam-6013	77	33	and	and	CCONJ
ejpam-6013	77	34	for	for	ADP
ejpam-6013	77	35	each	each	DET
ejpam-6013	77	36	σ1σ2	σ1σ2	VERB
ejpam-6013	77	37	-	-	ADJ
ejpam-6013	77	38	open	open	ADJ
ejpam-6013	77	39	set	set	NOUN
ejpam-6013	77	40	v	v	NOUN
ejpam-6013	77	41	of	of	ADP
ejpam-6013	77	42	y	y	NOUN
ejpam-6013	77	43	containing	contain	VERB
ejpam-6013	77	44	f(x	f(x	PROPN
ejpam-6013	77	45	)	)	PUNCT
ejpam-6013	77	46	,	,	PUNCT
ejpam-6013	77	47	there	there	PRON
ejpam-6013	77	48	exists	exist	VERB
ejpam-6013	77	49	a	a	DET
ejpam-6013	77	50	τ1τ2	τ1τ2	NOUN
ejpam-6013	77	51	-	-	ADJ
ejpam-6013	77	52	open	open	ADJ
ejpam-6013	77	53	set	set	ADJ
ejpam-6013	77	54	u	u	NOUN
ejpam-6013	77	55	of	of	ADP
ejpam-6013	77	56	x	x	PUNCT
ejpam-6013	77	57	containing	contain	VERB
ejpam-6013	77	58	x	x	PUNCT
ejpam-6013	77	59	such	such	ADJ
ejpam-6013	77	60	that	that	SCONJ
ejpam-6013	77	61	σ1σ2	σ1σ2	NOUN
ejpam-6013	77	62	-	-	PUNCT
ejpam-6013	77	63	cl(f(u	cl(f(u	NOUN
ejpam-6013	77	64	)	)	PUNCT
ejpam-6013	77	65	)	)	PUNCT
ejpam-6013	78	1	⊆	⊆	NUM
ejpam-6013	78	2	v	v	NOUN
ejpam-6013	78	3	.	.	PUNCT
ejpam-6013	79	1	definition	definition	NOUN
ejpam-6013	79	2	2	2	NUM
ejpam-6013	79	3	.	.	PUNCT
ejpam-6013	80	1	[	[	X
ejpam-6013	80	2	22	22	NUM
ejpam-6013	80	3	]	]	PUNCT
ejpam-6013	80	4	a	a	DET
ejpam-6013	80	5	function	function	NOUN
ejpam-6013	80	6	f	f	NOUN
ejpam-6013	80	7	:	:	PUNCT
ejpam-6013	80	8	(	(	PUNCT
ejpam-6013	80	9	x	x	NOUN
ejpam-6013	80	10	,	,	PUNCT
ejpam-6013	80	11	τ1	τ1	NOUN
ejpam-6013	80	12	,	,	PUNCT
ejpam-6013	80	13	τ2	τ2	NOUN
ejpam-6013	80	14	)	)	PUNCT
ejpam-6013	80	15	→	→	SYM
ejpam-6013	80	16	(	(	PUNCT
ejpam-6013	80	17	y	y	PROPN
ejpam-6013	80	18	,	,	PUNCT
ejpam-6013	80	19	σ1	σ1	PROPN
ejpam-6013	80	20	,	,	PUNCT
ejpam-6013	80	21	σ2	σ2	PROPN
ejpam-6013	80	22	)	)	PUNCT
ejpam-6013	80	23	is	be	AUX
ejpam-6013	80	24	called	call	VERB
ejpam-6013	80	25	(	(	PUNCT
ejpam-6013	80	26	τ1	τ1	NOUN
ejpam-6013	80	27	,	,	PUNCT
ejpam-6013	80	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	80	29	at	at	ADP
ejpam-6013	80	30	a	a	DET
ejpam-6013	80	31	point	point	NOUN
ejpam-6013	80	32	x	x	SYM
ejpam-6013	80	33	∈	∈	NOUN
ejpam-6013	80	34	x	x	PUNCT
ejpam-6013	80	35	if	if	SCONJ
ejpam-6013	80	36	for	for	ADP
ejpam-6013	80	37	each	each	DET
ejpam-6013	80	38	σ1σ2	σ1σ2	VERB
ejpam-6013	80	39	-	-	ADJ
ejpam-6013	80	40	open	open	ADJ
ejpam-6013	80	41	set	set	NOUN
ejpam-6013	80	42	v	v	NOUN
ejpam-6013	80	43	of	of	ADP
ejpam-6013	80	44	y	y	NOUN
ejpam-6013	80	45	containing	contain	VERB
ejpam-6013	80	46	f(x	f(x	PROPN
ejpam-6013	80	47	)	)	PUNCT
ejpam-6013	80	48	,	,	PUNCT
ejpam-6013	80	49	there	there	PRON
ejpam-6013	80	50	exists	exist	VERB
ejpam-6013	80	51	a	a	DET
ejpam-6013	80	52	τ1τ2	τ1τ2	NOUN
ejpam-6013	80	53	-	-	ADJ
ejpam-6013	80	54	open	open	ADJ
ejpam-6013	80	55	set	set	ADJ
ejpam-6013	80	56	u	u	NOUN
ejpam-6013	80	57	of	of	ADP
ejpam-6013	80	58	x	x	PUNCT
ejpam-6013	80	59	containing	contain	VERB
ejpam-6013	80	60	x	x	PUNCT
ejpam-6013	80	61	such	such	ADJ
ejpam-6013	80	62	that	that	DET
ejpam-6013	80	63	f(u	f(u	PROPN
ejpam-6013	80	64	)	)	PUNCT
ejpam-6013	80	65	⊆	⊆	NUM
ejpam-6013	80	66	v	v	NOUN
ejpam-6013	80	67	.	.	PUNCT
ejpam-6013	81	1	a	a	DET
ejpam-6013	81	2	function	function	NOUN
ejpam-6013	81	3	f	f	NOUN
ejpam-6013	81	4	:	:	PUNCT
ejpam-6013	81	5	(	(	PUNCT
ejpam-6013	81	6	x	x	NOUN
ejpam-6013	81	7	,	,	PUNCT
ejpam-6013	81	8	τ1	τ1	NOUN
ejpam-6013	81	9	,	,	PUNCT
ejpam-6013	81	10	τ2	τ2	NOUN
ejpam-6013	81	11	)	)	PUNCT
ejpam-6013	81	12	→	→	SYM
ejpam-6013	81	13	(	(	PUNCT
ejpam-6013	81	14	y	y	PROPN
ejpam-6013	81	15	,	,	PUNCT
ejpam-6013	81	16	σ1	σ1	PROPN
ejpam-6013	81	17	,	,	PUNCT
ejpam-6013	81	18	σ2	σ2	PROPN
ejpam-6013	81	19	)	)	PUNCT
ejpam-6013	81	20	is	be	AUX
ejpam-6013	81	21	called	call	VERB
ejpam-6013	81	22	(	(	PUNCT
ejpam-6013	81	23	τ1	τ1	NOUN
ejpam-6013	81	24	,	,	PUNCT
ejpam-6013	81	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	81	26	if	if	SCONJ
ejpam-6013	81	27	f	f	PROPN
ejpam-6013	81	28	has	have	VERB
ejpam-6013	81	29	this	this	DET
ejpam-6013	81	30	property	property	NOUN
ejpam-6013	81	31	at	at	ADP
ejpam-6013	81	32	each	each	DET
ejpam-6013	81	33	point	point	NOUN
ejpam-6013	81	34	of	of	ADP
ejpam-6013	81	35	x.	x.	PROPN
ejpam-6013	81	36	lemma	lemma	PROPN
ejpam-6013	82	1	3	3	X
ejpam-6013	82	2	.	.	PUNCT
ejpam-6013	83	1	[	[	X
ejpam-6013	83	2	22	22	NUM
ejpam-6013	83	3	]	]	PUNCT
ejpam-6013	83	4	for	for	ADP
ejpam-6013	83	5	a	a	DET
ejpam-6013	83	6	function	function	NOUN
ejpam-6013	83	7	(	(	PUNCT
ejpam-6013	83	8	x	x	NOUN
ejpam-6013	83	9	,	,	PUNCT
ejpam-6013	83	10	τ1	τ1	NOUN
ejpam-6013	83	11	,	,	PUNCT
ejpam-6013	83	12	τ2	τ2	NOUN
ejpam-6013	83	13	)	)	PUNCT
ejpam-6013	83	14	→	→	SYM
ejpam-6013	83	15	(	(	PUNCT
ejpam-6013	83	16	y	y	PROPN
ejpam-6013	83	17	,	,	PUNCT
ejpam-6013	83	18	σ1	σ1	PROPN
ejpam-6013	83	19	,	,	PUNCT
ejpam-6013	83	20	σ2	σ2	NOUN
ejpam-6013	83	21	)	)	PUNCT
ejpam-6013	83	22	,	,	PUNCT
ejpam-6013	83	23	the	the	DET
ejpam-6013	83	24	following	follow	VERB
ejpam-6013	83	25	properties	property	NOUN
ejpam-6013	83	26	are	be	AUX
ejpam-6013	83	27	equivalent	equivalent	ADJ
ejpam-6013	83	28	:	:	PUNCT
ejpam-6013	83	29	(	(	PUNCT
ejpam-6013	83	30	1	1	X
ejpam-6013	83	31	)	)	PUNCT
ejpam-6013	83	32	f	f	PROPN
ejpam-6013	83	33	is	be	AUX
ejpam-6013	83	34	(	(	PUNCT
ejpam-6013	83	35	τ1	τ1	NOUN
ejpam-6013	83	36	,	,	PUNCT
ejpam-6013	83	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	83	38	;	;	PUNCT
ejpam-6013	83	39	(	(	PUNCT
ejpam-6013	83	40	2	2	X
ejpam-6013	83	41	)	)	PUNCT
ejpam-6013	83	42	f−1(v	f−1(v	NOUN
ejpam-6013	83	43	)	)	PUNCT
ejpam-6013	83	44	is	be	AUX
ejpam-6013	83	45	τ1τ2	τ1τ2	NOUN
ejpam-6013	83	46	-	-	ADJ
ejpam-6013	83	47	open	open	ADJ
ejpam-6013	83	48	in	in	ADP
ejpam-6013	83	49	x	x	PUNCT
ejpam-6013	83	50	for	for	ADP
ejpam-6013	83	51	every	every	DET
ejpam-6013	83	52	σ1σ2	σ1σ2	NOUN
ejpam-6013	83	53	-	-	ADJ
ejpam-6013	83	54	open	open	ADJ
ejpam-6013	83	55	set	set	NOUN
ejpam-6013	83	56	v	v	NOUN
ejpam-6013	83	57	of	of	ADP
ejpam-6013	83	58	y	y	PROPN
ejpam-6013	83	59	;	;	PUNCT
ejpam-6013	83	60	(	(	PUNCT
ejpam-6013	83	61	3	3	X
ejpam-6013	83	62	)	)	PUNCT
ejpam-6013	83	63	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	83	64	-	-	PUNCT
ejpam-6013	83	65	cl(a	cl(a	NUM
ejpam-6013	83	66	)	)	PUNCT
ejpam-6013	83	67	)	)	PUNCT
ejpam-6013	84	1	⊆	⊆	X
ejpam-6013	84	2	σ1σ2	σ1σ2	NUM
ejpam-6013	84	3	-	-	PUNCT
ejpam-6013	84	4	cl(f(a	cl(f(a	NOUN
ejpam-6013	84	5	)	)	PUNCT
ejpam-6013	84	6	)	)	PUNCT
ejpam-6013	84	7	for	for	ADP
ejpam-6013	84	8	every	every	DET
ejpam-6013	84	9	subset	subset	NOUN
ejpam-6013	84	10	a	a	PRON
ejpam-6013	84	11	of	of	ADP
ejpam-6013	84	12	x	x	PRON
ejpam-6013	84	13	;	;	PUNCT
ejpam-6013	84	14	(	(	PUNCT
ejpam-6013	84	15	4	4	X
ejpam-6013	84	16	)	)	PUNCT
ejpam-6013	84	17	τ1τ2	τ1τ2	NOUN
ejpam-6013	84	18	-	-	NOUN
ejpam-6013	84	19	cl(f	cl(f	NOUN
ejpam-6013	84	20	−1(b	−1(b	NOUN
ejpam-6013	84	21	)	)	PUNCT
ejpam-6013	84	22	)	)	PUNCT
ejpam-6013	85	1	⊆	⊆	NUM
ejpam-6013	85	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6013	85	3	-	-	PUNCT
ejpam-6013	85	4	cl(b	cl(b	NOUN
ejpam-6013	85	5	)	)	PUNCT
ejpam-6013	85	6	)	)	PUNCT
ejpam-6013	85	7	for	for	ADP
ejpam-6013	85	8	every	every	DET
ejpam-6013	85	9	subset	subset	NOUN
ejpam-6013	85	10	b	b	PROPN
ejpam-6013	85	11	of	of	ADP
ejpam-6013	85	12	y	y	PROPN
ejpam-6013	85	13	;	;	PUNCT
ejpam-6013	85	14	n.	n.	PROPN
ejpam-6013	85	15	srisarakham	srisarakham	PROPN
ejpam-6013	85	16	,	,	PUNCT
ejpam-6013	85	17	s.	s.	PROPN
ejpam-6013	85	18	sompong	sompong	PROPN
ejpam-6013	85	19	,	,	PUNCT
ejpam-6013	85	20	c.	c.	PROPN
ejpam-6013	85	21	boonpok	boonpok	PROPN
ejpam-6013	85	22	/	/	SYM
ejpam-6013	85	23	eur	eur	PROPN
ejpam-6013	85	24	.	.	PUNCT
ejpam-6013	86	1	j.	j.	PROPN
ejpam-6013	86	2	pure	pure	PROPN
ejpam-6013	86	3	appl	appl	PROPN
ejpam-6013	86	4	.	.	PROPN
ejpam-6013	86	5	math	math	PROPN
ejpam-6013	86	6	,	,	PUNCT
ejpam-6013	86	7	18	18	NUM
ejpam-6013	86	8	(	(	PUNCT
ejpam-6013	86	9	2	2	NUM
ejpam-6013	86	10	)	)	PUNCT
ejpam-6013	86	11	(	(	PUNCT
ejpam-6013	86	12	2025	2025	NUM
ejpam-6013	86	13	)	)	PUNCT
ejpam-6013	86	14	,	,	PUNCT
ejpam-6013	86	15	6013	6013	NUM
ejpam-6013	86	16	4	4	NUM
ejpam-6013	86	17	of	of	ADP
ejpam-6013	86	18	12	12	NUM
ejpam-6013	86	19	(	(	PUNCT
ejpam-6013	86	20	5	5	NUM
ejpam-6013	86	21	)	)	PUNCT
ejpam-6013	86	22	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6013	86	23	-	-	PUNCT
ejpam-6013	86	24	int(b	int(b	NOUN
ejpam-6013	86	25	)	)	PUNCT
ejpam-6013	86	26	)	)	PUNCT
ejpam-6013	87	1	⊆	⊆	X
ejpam-6013	87	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	87	3	-	-	NUM
ejpam-6013	87	4	int(f	int(f	NOUN
ejpam-6013	87	5	−1(b	−1(b	NOUN
ejpam-6013	87	6	)	)	PUNCT
ejpam-6013	87	7	)	)	PUNCT
ejpam-6013	87	8	for	for	ADP
ejpam-6013	87	9	every	every	DET
ejpam-6013	87	10	subset	subset	NOUN
ejpam-6013	87	11	b	b	PROPN
ejpam-6013	87	12	of	of	ADP
ejpam-6013	87	13	y	y	PROPN
ejpam-6013	87	14	;	;	PUNCT
ejpam-6013	87	15	(	(	PUNCT
ejpam-6013	87	16	6	6	X
ejpam-6013	87	17	)	)	PUNCT
ejpam-6013	87	18	f−1(k	f−1(k	PROPN
ejpam-6013	87	19	)	)	PUNCT
ejpam-6013	87	20	is	be	AUX
ejpam-6013	87	21	τ1τ2	τ1τ2	NOUN
ejpam-6013	87	22	-	-	ADJ
ejpam-6013	87	23	closed	closed	ADJ
ejpam-6013	87	24	in	in	ADP
ejpam-6013	87	25	x	x	PUNCT
ejpam-6013	87	26	for	for	ADP
ejpam-6013	87	27	every	every	DET
ejpam-6013	87	28	σ1σ2	σ1σ2	NUM
ejpam-6013	87	29	-	-	PUNCT
ejpam-6013	87	30	closed	closed	ADJ
ejpam-6013	87	31	set	set	NOUN
ejpam-6013	87	32	k	k	PROPN
ejpam-6013	87	33	of	of	ADP
ejpam-6013	87	34	y	y	PROPN
ejpam-6013	87	35	.	.	PUNCT
ejpam-6013	88	1	lemma	lemma	PROPN
ejpam-6013	88	2	4	4	X
ejpam-6013	88	3	.	.	PUNCT
ejpam-6013	89	1	if	if	SCONJ
ejpam-6013	89	2	a	a	DET
ejpam-6013	89	3	function	function	NOUN
ejpam-6013	89	4	f	f	X
ejpam-6013	89	5	:	:	PUNCT
ejpam-6013	89	6	(	(	PUNCT
ejpam-6013	89	7	x	x	NOUN
ejpam-6013	89	8	,	,	PUNCT
ejpam-6013	89	9	τ1	τ1	NOUN
ejpam-6013	89	10	,	,	PUNCT
ejpam-6013	89	11	τ2	τ2	NOUN
ejpam-6013	89	12	)	)	PUNCT
ejpam-6013	89	13	→	→	SYM
ejpam-6013	89	14	(	(	PUNCT
ejpam-6013	89	15	y	y	PROPN
ejpam-6013	89	16	,	,	PUNCT
ejpam-6013	89	17	σ1	σ1	PROPN
ejpam-6013	89	18	,	,	PUNCT
ejpam-6013	89	19	σ2	σ2	PROPN
ejpam-6013	89	20	)	)	PUNCT
ejpam-6013	89	21	is	be	AUX
ejpam-6013	89	22	r-(τ1	r-(τ1	NOUN
ejpam-6013	89	23	,	,	PUNCT
ejpam-6013	89	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	89	25	,	,	PUNCT
ejpam-6013	89	26	then	then	ADV
ejpam-6013	89	27	f	f	PROPN
ejpam-6013	89	28	is	be	AUX
ejpam-6013	89	29	(	(	PUNCT
ejpam-6013	89	30	τ1	τ1	NOUN
ejpam-6013	89	31	,	,	PUNCT
ejpam-6013	89	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	89	33	.	.	PUNCT
ejpam-6013	90	1	proof	proof	NOUN
ejpam-6013	90	2	.	.	PUNCT
ejpam-6013	91	1	let	let	VERB
ejpam-6013	91	2	x	x	PUNCT
ejpam-6013	91	3	∈	∈	PROPN
ejpam-6013	91	4	x	x	X
ejpam-6013	91	5	and	and	CCONJ
ejpam-6013	91	6	v	v	X
ejpam-6013	91	7	be	be	AUX
ejpam-6013	91	8	any	any	DET
ejpam-6013	91	9	σ1σ2	σ1σ2	NOUN
ejpam-6013	91	10	-	-	ADJ
ejpam-6013	91	11	open	open	ADJ
ejpam-6013	91	12	set	set	NOUN
ejpam-6013	91	13	of	of	ADP
ejpam-6013	91	14	y	y	PROPN
ejpam-6013	91	15	containing	contain	VERB
ejpam-6013	91	16	f(x	f(x	PROPN
ejpam-6013	91	17	)	)	PUNCT
ejpam-6013	91	18	.	.	PUNCT
ejpam-6013	92	1	since	since	SCONJ
ejpam-6013	92	2	f	f	PROPN
ejpam-6013	92	3	is	be	AUX
ejpam-6013	92	4	r-(τ1	r-(τ1	PROPN
ejpam-6013	92	5	,	,	PUNCT
ejpam-6013	92	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	92	7	,	,	PUNCT
ejpam-6013	92	8	there	there	PRON
ejpam-6013	92	9	exists	exist	VERB
ejpam-6013	92	10	a	a	DET
ejpam-6013	92	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	92	12	-	-	ADJ
ejpam-6013	92	13	open	open	ADJ
ejpam-6013	92	14	set	set	ADJ
ejpam-6013	92	15	u	u	NOUN
ejpam-6013	92	16	of	of	ADP
ejpam-6013	92	17	x	x	PUNCT
ejpam-6013	92	18	containing	contain	VERB
ejpam-6013	92	19	x	x	PUNCT
ejpam-6013	92	20	such	such	ADJ
ejpam-6013	92	21	that	that	SCONJ
ejpam-6013	92	22	σ1σ2	σ1σ2	NOUN
ejpam-6013	92	23	-	-	PUNCT
ejpam-6013	92	24	cl(f(u	cl(f(u	NOUN
ejpam-6013	92	25	)	)	PUNCT
ejpam-6013	92	26	)	)	PUNCT
ejpam-6013	93	1	⊆	⊆	NUM
ejpam-6013	93	2	v	v	NOUN
ejpam-6013	93	3	.	.	PUNCT
ejpam-6013	94	1	this	this	PRON
ejpam-6013	94	2	implies	imply	VERB
ejpam-6013	94	3	that	that	SCONJ
ejpam-6013	94	4	f(u	f(u	PROPN
ejpam-6013	94	5	)	)	PUNCT
ejpam-6013	94	6	⊆	⊆	NUM
ejpam-6013	94	7	v	v	NOUN
ejpam-6013	94	8	.	.	PUNCT
ejpam-6013	95	1	thus	thus	ADV
ejpam-6013	95	2	,	,	PUNCT
ejpam-6013	95	3	f	f	PROPN
ejpam-6013	95	4	is	be	AUX
ejpam-6013	95	5	(	(	PUNCT
ejpam-6013	95	6	τ1	τ1	NOUN
ejpam-6013	95	7	,	,	PUNCT
ejpam-6013	95	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	95	9	.	.	PUNCT
ejpam-6013	96	1	theorem	theorem	NOUN
ejpam-6013	96	2	1	1	NUM
ejpam-6013	96	3	.	.	X
ejpam-6013	96	4	for	for	ADP
ejpam-6013	96	5	a	a	DET
ejpam-6013	96	6	function	function	NOUN
ejpam-6013	96	7	f	f	NOUN
ejpam-6013	96	8	:	:	PUNCT
ejpam-6013	96	9	(	(	PUNCT
ejpam-6013	96	10	x	x	NOUN
ejpam-6013	96	11	,	,	PUNCT
ejpam-6013	96	12	τ1	τ1	NOUN
ejpam-6013	96	13	,	,	PUNCT
ejpam-6013	96	14	τ2	τ2	NOUN
ejpam-6013	96	15	)	)	PUNCT
ejpam-6013	96	16	→	→	SYM
ejpam-6013	96	17	(	(	PUNCT
ejpam-6013	96	18	y	y	PROPN
ejpam-6013	96	19	,	,	PUNCT
ejpam-6013	96	20	σ1	σ1	PROPN
ejpam-6013	96	21	,	,	PUNCT
ejpam-6013	96	22	σ2	σ2	NOUN
ejpam-6013	96	23	)	)	PUNCT
ejpam-6013	96	24	,	,	PUNCT
ejpam-6013	96	25	the	the	DET
ejpam-6013	96	26	following	follow	VERB
ejpam-6013	96	27	properties	property	NOUN
ejpam-6013	96	28	are	be	AUX
ejpam-6013	96	29	equivalent	equivalent	ADJ
ejpam-6013	96	30	:	:	PUNCT
ejpam-6013	96	31	(	(	PUNCT
ejpam-6013	96	32	1	1	X
ejpam-6013	96	33	)	)	PUNCT
ejpam-6013	96	34	f	f	PROPN
ejpam-6013	96	35	is	be	AUX
ejpam-6013	96	36	r-(τ1	r-(τ1	PROPN
ejpam-6013	96	37	,	,	PUNCT
ejpam-6013	96	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	96	39	;	;	PUNCT
ejpam-6013	96	40	(	(	PUNCT
ejpam-6013	96	41	2	2	X
ejpam-6013	96	42	)	)	PUNCT
ejpam-6013	96	43	for	for	ADP
ejpam-6013	96	44	each	each	DET
ejpam-6013	96	45	point	point	NOUN
ejpam-6013	96	46	x	x	X
ejpam-6013	96	47	∈	∈	NOUN
ejpam-6013	96	48	x	x	X
ejpam-6013	96	49	and	and	CCONJ
ejpam-6013	96	50	each	each	DET
ejpam-6013	96	51	σ1σ2	σ1σ2	VERB
ejpam-6013	96	52	-	-	ADJ
ejpam-6013	96	53	open	open	ADJ
ejpam-6013	96	54	set	set	NOUN
ejpam-6013	96	55	v	v	NOUN
ejpam-6013	96	56	of	of	ADP
ejpam-6013	96	57	y	y	NOUN
ejpam-6013	96	58	containing	contain	VERB
ejpam-6013	96	59	f(x	f(x	PROPN
ejpam-6013	96	60	)	)	PUNCT
ejpam-6013	96	61	,	,	PUNCT
ejpam-6013	96	62	exists	exist	VERB
ejpam-6013	96	63	a	a	DET
ejpam-6013	96	64	τ1τ2	τ1τ2	NOUN
ejpam-6013	96	65	-	-	ADJ
ejpam-6013	96	66	open	open	ADJ
ejpam-6013	96	67	set	set	ADJ
ejpam-6013	96	68	u	u	NOUN
ejpam-6013	96	69	of	of	ADP
ejpam-6013	96	70	x	x	PUNCT
ejpam-6013	96	71	containing	contain	VERB
ejpam-6013	96	72	x	x	PUNCT
ejpam-6013	96	73	such	such	ADJ
ejpam-6013	96	74	that	that	SCONJ
ejpam-6013	96	75	σ1σ2	σ1σ2	NOUN
ejpam-6013	96	76	-	-	PUNCT
ejpam-6013	96	77	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	96	78	-	-	PUNCT
ejpam-6013	96	79	cl(u	cl(u	NOUN
ejpam-6013	96	80	)	)	PUNCT
ejpam-6013	96	81	)	)	PUNCT
ejpam-6013	96	82	)	)	PUNCT
ejpam-6013	97	1	⊆	⊆	NUM
ejpam-6013	97	2	v	v	NOUN
ejpam-6013	97	3	;	;	PUNCT
ejpam-6013	97	4	(	(	PUNCT
ejpam-6013	97	5	3	3	X
ejpam-6013	97	6	)	)	PUNCT
ejpam-6013	97	7	for	for	ADP
ejpam-6013	97	8	each	each	DET
ejpam-6013	97	9	point	point	NOUN
ejpam-6013	97	10	x	x	X
ejpam-6013	97	11	∈	∈	NOUN
ejpam-6013	97	12	x	x	X
ejpam-6013	97	13	and	and	CCONJ
ejpam-6013	97	14	each	each	DET
ejpam-6013	97	15	σ1σ2	σ1σ2	VERB
ejpam-6013	97	16	-	-	PUNCT
ejpam-6013	97	17	closed	closed	ADJ
ejpam-6013	97	18	set	set	ADJ
ejpam-6013	97	19	f	f	PROPN
ejpam-6013	97	20	of	of	ADP
ejpam-6013	97	21	y	y	PROPN
ejpam-6013	97	22	with	with	ADP
ejpam-6013	97	23	f(x	f(x	PROPN
ejpam-6013	97	24	)	)	PUNCT
ejpam-6013	98	1	̸∈	̸∈	PROPN
ejpam-6013	98	2	f	f	PROPN
ejpam-6013	98	3	,	,	PUNCT
ejpam-6013	98	4	exists	exist	VERB
ejpam-6013	98	5	a	a	DET
ejpam-6013	98	6	τ1τ2open	τ1τ2open	ADJ
ejpam-6013	98	7	set	set	NOUN
ejpam-6013	98	8	u	u	NOUN
ejpam-6013	98	9	of	of	ADP
ejpam-6013	98	10	x	x	PUNCT
ejpam-6013	98	11	containing	contain	VERB
ejpam-6013	98	12	x	x	X
ejpam-6013	98	13	and	and	CCONJ
ejpam-6013	98	14	a	a	DET
ejpam-6013	98	15	σ1σ2	σ1σ2	NUM
ejpam-6013	98	16	-	-	ADJ
ejpam-6013	98	17	open	open	ADJ
ejpam-6013	98	18	set	set	NOUN
ejpam-6013	98	19	v	v	NOUN
ejpam-6013	98	20	of	of	ADP
ejpam-6013	98	21	y	y	PRON
ejpam-6013	98	22	such	such	ADJ
ejpam-6013	98	23	that	that	SCONJ
ejpam-6013	98	24	f	f	PROPN
ejpam-6013	98	25	⊆	⊆	NUM
ejpam-6013	98	26	v	v	NOUN
ejpam-6013	98	27	and	and	CCONJ
ejpam-6013	98	28	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	98	29	-	-	PUNCT
ejpam-6013	98	30	cl(u	cl(u	NOUN
ejpam-6013	98	31	)	)	PUNCT
ejpam-6013	98	32	)	)	PUNCT
ejpam-6013	98	33	∩	∩	NOUN
ejpam-6013	98	34	v	v	NOUN
ejpam-6013	98	35	=	=	SYM
ejpam-6013	98	36	∅	∅	NOUN
ejpam-6013	98	37	;	;	PUNCT
ejpam-6013	98	38	(	(	PUNCT
ejpam-6013	98	39	4	4	X
ejpam-6013	98	40	)	)	PUNCT
ejpam-6013	98	41	for	for	ADP
ejpam-6013	98	42	each	each	DET
ejpam-6013	98	43	point	point	NOUN
ejpam-6013	98	44	x	x	X
ejpam-6013	98	45	∈	∈	NOUN
ejpam-6013	98	46	x	x	X
ejpam-6013	98	47	and	and	CCONJ
ejpam-6013	98	48	each	each	DET
ejpam-6013	98	49	σ1σ2	σ1σ2	VERB
ejpam-6013	98	50	-	-	PUNCT
ejpam-6013	98	51	closed	closed	ADJ
ejpam-6013	98	52	set	set	ADJ
ejpam-6013	98	53	f	f	PROPN
ejpam-6013	98	54	of	of	ADP
ejpam-6013	98	55	y	y	PROPN
ejpam-6013	98	56	with	with	ADP
ejpam-6013	98	57	f(x	f(x	PROPN
ejpam-6013	98	58	)	)	PUNCT
ejpam-6013	98	59	̸∈	̸∈	PROPN
ejpam-6013	98	60	f	f	PROPN
ejpam-6013	98	61	,	,	PUNCT
ejpam-6013	98	62	exists	exist	VERB
ejpam-6013	98	63	a	a	DET
ejpam-6013	98	64	τ1τ2open	τ1τ2open	ADJ
ejpam-6013	98	65	set	set	NOUN
ejpam-6013	98	66	u	u	NOUN
ejpam-6013	98	67	of	of	ADP
ejpam-6013	98	68	x	x	PUNCT
ejpam-6013	98	69	containing	contain	VERB
ejpam-6013	98	70	x	x	X
ejpam-6013	98	71	and	and	CCONJ
ejpam-6013	98	72	a	a	DET
ejpam-6013	98	73	σ1σ2	σ1σ2	NUM
ejpam-6013	98	74	-	-	ADJ
ejpam-6013	98	75	open	open	ADJ
ejpam-6013	98	76	set	set	NOUN
ejpam-6013	98	77	v	v	NOUN
ejpam-6013	98	78	of	of	ADP
ejpam-6013	98	79	y	y	PRON
ejpam-6013	98	80	such	such	ADJ
ejpam-6013	98	81	that	that	SCONJ
ejpam-6013	98	82	f	f	PROPN
ejpam-6013	98	83	⊆	⊆	NUM
ejpam-6013	98	84	v	v	NOUN
ejpam-6013	98	85	and	and	CCONJ
ejpam-6013	98	86	f(u	f(u	PROPN
ejpam-6013	98	87	)	)	PUNCT
ejpam-6013	98	88	∩	∩	NOUN
ejpam-6013	98	89	v	v	NOUN
ejpam-6013	98	90	=	=	PUNCT
ejpam-6013	98	91	∅.	∅.	NOUN
ejpam-6013	98	92	proof	proof	NOUN
ejpam-6013	98	93	.	.	PUNCT
ejpam-6013	99	1	(	(	PUNCT
ejpam-6013	99	2	1	1	X
ejpam-6013	99	3	)	)	PUNCT
ejpam-6013	99	4	⇒	⇒	NOUN
ejpam-6013	99	5	(	(	PUNCT
ejpam-6013	99	6	2	2	NUM
ejpam-6013	99	7	):	):	PUNCT
ejpam-6013	99	8	let	let	VERB
ejpam-6013	99	9	x	x	PUNCT
ejpam-6013	99	10	∈	∈	PROPN
ejpam-6013	99	11	x	x	X
ejpam-6013	99	12	and	and	CCONJ
ejpam-6013	99	13	v	v	X
ejpam-6013	99	14	be	be	AUX
ejpam-6013	99	15	any	any	DET
ejpam-6013	99	16	σ1σ2	σ1σ2	NOUN
ejpam-6013	99	17	-	-	ADJ
ejpam-6013	99	18	open	open	ADJ
ejpam-6013	99	19	set	set	NOUN
ejpam-6013	99	20	of	of	ADP
ejpam-6013	99	21	y	y	PROPN
ejpam-6013	99	22	containing	contain	VERB
ejpam-6013	99	23	f(x	f(x	PROPN
ejpam-6013	99	24	)	)	PUNCT
ejpam-6013	99	25	.	.	PUNCT
ejpam-6013	100	1	since	since	SCONJ
ejpam-6013	100	2	f	f	PROPN
ejpam-6013	100	3	is	be	AUX
ejpam-6013	100	4	r-(τ1	r-(τ1	PROPN
ejpam-6013	100	5	,	,	PUNCT
ejpam-6013	100	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	100	7	,	,	PUNCT
ejpam-6013	100	8	there	there	PRON
ejpam-6013	100	9	exists	exist	VERB
ejpam-6013	100	10	a	a	DET
ejpam-6013	100	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	100	12	-	-	ADJ
ejpam-6013	100	13	open	open	ADJ
ejpam-6013	100	14	set	set	ADJ
ejpam-6013	100	15	u	u	NOUN
ejpam-6013	100	16	of	of	ADP
ejpam-6013	100	17	x	x	PUNCT
ejpam-6013	100	18	containing	contain	VERB
ejpam-6013	100	19	x	x	PUNCT
ejpam-6013	100	20	such	such	ADJ
ejpam-6013	100	21	that	that	SCONJ
ejpam-6013	100	22	σ1σ2	σ1σ2	NOUN
ejpam-6013	100	23	-	-	PUNCT
ejpam-6013	100	24	cl(f(u	cl(f(u	NOUN
ejpam-6013	100	25	)	)	PUNCT
ejpam-6013	100	26	)	)	PUNCT
ejpam-6013	101	1	⊆	⊆	NUM
ejpam-6013	101	2	v	v	NOUN
ejpam-6013	101	3	.	.	PUNCT
ejpam-6013	102	1	by	by	ADP
ejpam-6013	102	2	lemma	lemma	PROPN
ejpam-6013	102	3	4	4	NUM
ejpam-6013	102	4	,	,	PUNCT
ejpam-6013	102	5	we	we	PRON
ejpam-6013	102	6	have	have	VERB
ejpam-6013	102	7	f	f	PROPN
ejpam-6013	102	8	is	be	AUX
ejpam-6013	102	9	(	(	PUNCT
ejpam-6013	102	10	τ1	τ1	NOUN
ejpam-6013	102	11	,	,	PUNCT
ejpam-6013	102	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	102	13	and	and	CCONJ
ejpam-6013	102	14	by	by	ADP
ejpam-6013	102	15	lemma	lemma	PROPN
ejpam-6013	102	16	3	3	NUM
ejpam-6013	102	17	,	,	PUNCT
ejpam-6013	102	18	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	102	19	-	-	PUNCT
ejpam-6013	102	20	cl(u	cl(u	NOUN
ejpam-6013	102	21	)	)	PUNCT
ejpam-6013	102	22	)	)	PUNCT
ejpam-6013	103	1	⊆	⊆	X
ejpam-6013	103	2	σ1σ2	σ1σ2	NUM
ejpam-6013	103	3	-	-	PUNCT
ejpam-6013	103	4	cl(f(u	cl(f(u	NOUN
ejpam-6013	103	5	)	)	PUNCT
ejpam-6013	103	6	)	)	PUNCT
ejpam-6013	104	1	⊆	⊆	NUM
ejpam-6013	104	2	v	v	NOUN
ejpam-6013	104	3	.	.	PUNCT
ejpam-6013	105	1	thus	thus	ADV
ejpam-6013	105	2	,	,	PUNCT
ejpam-6013	105	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	105	4	-	-	PUNCT
ejpam-6013	105	5	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	105	6	-	-	PUNCT
ejpam-6013	105	7	cl(u	cl(u	NOUN
ejpam-6013	105	8	)	)	PUNCT
ejpam-6013	105	9	)	)	PUNCT
ejpam-6013	105	10	)	)	PUNCT
ejpam-6013	106	1	⊆	⊆	X
ejpam-6013	106	2	σ1σ2	σ1σ2	X
ejpam-6013	106	3	-	-	PUNCT
ejpam-6013	106	4	cl(f(u	cl(f(u	NOUN
ejpam-6013	106	5	)	)	PUNCT
ejpam-6013	106	6	)	)	PUNCT
ejpam-6013	107	1	⊆	⊆	NUM
ejpam-6013	107	2	v.	v.	ADP
ejpam-6013	107	3	(	(	PUNCT
ejpam-6013	107	4	2	2	NUM
ejpam-6013	107	5	)	)	PUNCT
ejpam-6013	107	6	⇒	⇒	NOUN
ejpam-6013	107	7	(	(	PUNCT
ejpam-6013	107	8	3	3	NUM
ejpam-6013	107	9	):	):	PUNCT
ejpam-6013	107	10	let	let	VERB
ejpam-6013	107	11	x	x	PUNCT
ejpam-6013	107	12	∈	∈	PROPN
ejpam-6013	107	13	x	x	X
ejpam-6013	107	14	and	and	CCONJ
ejpam-6013	107	15	f	f	PROPN
ejpam-6013	107	16	be	be	AUX
ejpam-6013	107	17	any	any	DET
ejpam-6013	107	18	σ1σ2	σ1σ2	NUM
ejpam-6013	107	19	-	-	PUNCT
ejpam-6013	107	20	closed	closed	ADJ
ejpam-6013	107	21	set	set	NOUN
ejpam-6013	107	22	of	of	ADP
ejpam-6013	107	23	y	y	PROPN
ejpam-6013	107	24	with	with	ADP
ejpam-6013	107	25	f(x	f(x	PROPN
ejpam-6013	107	26	)	)	PUNCT
ejpam-6013	108	1	̸∈	̸∈	PROPN
ejpam-6013	108	2	f	f	PROPN
ejpam-6013	108	3	.	.	PUNCT
ejpam-6013	109	1	then	then	ADV
ejpam-6013	109	2	,	,	PUNCT
ejpam-6013	109	3	we	we	PRON
ejpam-6013	109	4	have	have	VERB
ejpam-6013	109	5	f(x	f(x	PROPN
ejpam-6013	109	6	)	)	PUNCT
ejpam-6013	109	7	∈	∈	PROPN
ejpam-6013	110	1	y	y	NOUN
ejpam-6013	110	2	−	−	PROPN
ejpam-6013	110	3	f	f	PROPN
ejpam-6013	110	4	and	and	CCONJ
ejpam-6013	110	5	y	y	PROPN
ejpam-6013	110	6	−	−	PROPN
ejpam-6013	111	1	f	f	PROPN
ejpam-6013	111	2	is	be	AUX
ejpam-6013	111	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	111	4	-	-	ADJ
ejpam-6013	111	5	open	open	ADJ
ejpam-6013	111	6	in	in	ADP
ejpam-6013	111	7	y	y	PROPN
ejpam-6013	111	8	.	.	PUNCT
ejpam-6013	112	1	by	by	ADP
ejpam-6013	112	2	(	(	PUNCT
ejpam-6013	112	3	2	2	NUM
ejpam-6013	112	4	)	)	PUNCT
ejpam-6013	112	5	,	,	PUNCT
ejpam-6013	112	6	there	there	PRON
ejpam-6013	112	7	exists	exist	VERB
ejpam-6013	112	8	a	a	DET
ejpam-6013	112	9	τ1τ2	τ1τ2	NOUN
ejpam-6013	112	10	-	-	ADJ
ejpam-6013	112	11	open	open	ADJ
ejpam-6013	112	12	set	set	ADJ
ejpam-6013	112	13	u	u	NOUN
ejpam-6013	112	14	of	of	ADP
ejpam-6013	112	15	x	x	PUNCT
ejpam-6013	112	16	containing	contain	VERB
ejpam-6013	112	17	x	x	PUNCT
ejpam-6013	112	18	such	such	ADJ
ejpam-6013	112	19	that	that	SCONJ
ejpam-6013	112	20	σ1σ2	σ1σ2	NOUN
ejpam-6013	112	21	-	-	PUNCT
ejpam-6013	112	22	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	112	23	-	-	PUNCT
ejpam-6013	112	24	cl(u	cl(u	NOUN
ejpam-6013	112	25	)	)	PUNCT
ejpam-6013	112	26	)	)	PUNCT
ejpam-6013	112	27	)	)	PUNCT
ejpam-6013	113	1	⊆	⊆	NUM
ejpam-6013	113	2	y	y	PROPN
ejpam-6013	113	3	−	−	PROPN
ejpam-6013	113	4	f	f	PROPN
ejpam-6013	113	5	.	.	PUNCT
ejpam-6013	114	1	put	put	VERB
ejpam-6013	114	2	v	v	NUM
ejpam-6013	114	3	=	=	SYM
ejpam-6013	114	4	y	y	PROPN
ejpam-6013	114	5	−	−	NUM
ejpam-6013	114	6	σ1σ2	σ1σ2	NOUN
ejpam-6013	114	7	-	-	PUNCT
ejpam-6013	114	8	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	114	9	-	-	PUNCT
ejpam-6013	114	10	cl(u	cl(u	NOUN
ejpam-6013	114	11	)	)	PUNCT
ejpam-6013	114	12	)	)	PUNCT
ejpam-6013	114	13	)	)	PUNCT
ejpam-6013	114	14	.	.	PUNCT
ejpam-6013	115	1	then	then	ADV
ejpam-6013	115	2	,	,	PUNCT
ejpam-6013	115	3	v	v	NOUN
ejpam-6013	115	4	is	be	AUX
ejpam-6013	115	5	σ1σ2	σ1σ2	NOUN
ejpam-6013	115	6	-	-	ADJ
ejpam-6013	115	7	open	open	ADJ
ejpam-6013	115	8	and	and	CCONJ
ejpam-6013	115	9	f	f	NOUN
ejpam-6013	115	10	⊆	⊆	NUM
ejpam-6013	115	11	v	v	NOUN
ejpam-6013	115	12	.	.	PUNCT
ejpam-6013	116	1	furthermore	furthermore	ADV
ejpam-6013	116	2	,	,	PUNCT
ejpam-6013	116	3	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	116	4	-	-	PUNCT
ejpam-6013	116	5	cl(u	cl(u	NOUN
ejpam-6013	116	6	)	)	PUNCT
ejpam-6013	116	7	)	)	PUNCT
ejpam-6013	116	8	∩	∩	NOUN
ejpam-6013	116	9	v	v	X
ejpam-6013	116	10	=	=	SYM
ejpam-6013	116	11	∅.	∅.	X
ejpam-6013	116	12	(	(	PUNCT
ejpam-6013	116	13	3	3	NUM
ejpam-6013	116	14	)	)	PUNCT
ejpam-6013	116	15	⇒	⇒	NOUN
ejpam-6013	116	16	(	(	PUNCT
ejpam-6013	116	17	4	4	NUM
ejpam-6013	116	18	):	):	PUNCT
ejpam-6013	116	19	the	the	DET
ejpam-6013	116	20	proof	proof	NOUN
ejpam-6013	116	21	is	be	AUX
ejpam-6013	116	22	obvious	obvious	ADJ
ejpam-6013	116	23	.	.	PUNCT
ejpam-6013	117	1	(	(	PUNCT
ejpam-6013	117	2	4	4	X
ejpam-6013	117	3	)	)	PUNCT
ejpam-6013	117	4	⇒	⇒	NOUN
ejpam-6013	117	5	(	(	PUNCT
ejpam-6013	117	6	1	1	NUM
ejpam-6013	117	7	):	):	PUNCT
ejpam-6013	117	8	let	let	VERB
ejpam-6013	117	9	x	x	PUNCT
ejpam-6013	117	10	∈	∈	PROPN
ejpam-6013	117	11	x	x	X
ejpam-6013	117	12	and	and	CCONJ
ejpam-6013	117	13	v	v	X
ejpam-6013	117	14	be	be	AUX
ejpam-6013	117	15	any	any	DET
ejpam-6013	117	16	σ1σ2	σ1σ2	NOUN
ejpam-6013	117	17	-	-	ADJ
ejpam-6013	117	18	open	open	ADJ
ejpam-6013	117	19	set	set	NOUN
ejpam-6013	117	20	of	of	ADP
ejpam-6013	117	21	y	y	PROPN
ejpam-6013	117	22	containing	contain	VERB
ejpam-6013	117	23	f(x	f(x	PROPN
ejpam-6013	117	24	)	)	PUNCT
ejpam-6013	117	25	.	.	PUNCT
ejpam-6013	118	1	then	then	ADV
ejpam-6013	118	2	,	,	PUNCT
ejpam-6013	118	3	y	y	PROPN
ejpam-6013	118	4	−	−	PROPN
ejpam-6013	118	5	v	v	NOUN
ejpam-6013	118	6	is	be	AUX
ejpam-6013	118	7	σ1σ2	σ1σ2	NOUN
ejpam-6013	118	8	-	-	ADJ
ejpam-6013	118	9	closed	closed	ADJ
ejpam-6013	118	10	in	in	ADP
ejpam-6013	118	11	y	y	PROPN
ejpam-6013	118	12	and	and	CCONJ
ejpam-6013	118	13	f(x	f(x	PROPN
ejpam-6013	118	14	)	)	PUNCT
ejpam-6013	119	1	̸∈	̸∈	PROPN
ejpam-6013	119	2	y	y	PROPN
ejpam-6013	119	3	−	−	PROPN
ejpam-6013	119	4	v	v	NOUN
ejpam-6013	119	5	.	.	PUNCT
ejpam-6013	120	1	by	by	ADP
ejpam-6013	120	2	(	(	PUNCT
ejpam-6013	120	3	4	4	NUM
ejpam-6013	120	4	)	)	PUNCT
ejpam-6013	120	5	,	,	PUNCT
ejpam-6013	120	6	there	there	PRON
ejpam-6013	120	7	exists	exist	VERB
ejpam-6013	120	8	a	a	DET
ejpam-6013	120	9	τ1τ2	τ1τ2	NOUN
ejpam-6013	120	10	-	-	ADJ
ejpam-6013	120	11	open	open	ADJ
ejpam-6013	120	12	set	set	ADJ
ejpam-6013	120	13	u	u	NOUN
ejpam-6013	120	14	of	of	ADP
ejpam-6013	120	15	x	x	PUNCT
ejpam-6013	120	16	containing	contain	VERB
ejpam-6013	120	17	x	x	X
ejpam-6013	120	18	and	and	CCONJ
ejpam-6013	120	19	a	a	DET
ejpam-6013	120	20	σ1σ2	σ1σ2	NUM
ejpam-6013	120	21	-	-	ADJ
ejpam-6013	120	22	open	open	ADJ
ejpam-6013	120	23	set	set	NOUN
ejpam-6013	120	24	w	w	PROPN
ejpam-6013	120	25	of	of	ADP
ejpam-6013	120	26	y	y	PRON
ejpam-6013	120	27	such	such	ADJ
ejpam-6013	120	28	that	that	SCONJ
ejpam-6013	120	29	y	y	PROPN
ejpam-6013	120	30	−v	−v	VERB
ejpam-6013	120	31	⊆	⊆	NUM
ejpam-6013	120	32	w	w	NOUN
ejpam-6013	120	33	and	and	CCONJ
ejpam-6013	120	34	f(u)∩w	f(u)∩w	ADJ
ejpam-6013	120	35	=	=	PUNCT
ejpam-6013	120	36	∅.	∅.	ADV
ejpam-6013	120	37	since	since	SCONJ
ejpam-6013	120	38	f(u	f(u	PROPN
ejpam-6013	120	39	)	)	PUNCT
ejpam-6013	121	1	⊆	⊆	NUM
ejpam-6013	121	2	y	y	PROPN
ejpam-6013	121	3	−w	−w	ADV
ejpam-6013	121	4	and	and	CCONJ
ejpam-6013	121	5	y	y	PROPN
ejpam-6013	121	6	−w	−w	ADV
ejpam-6013	121	7	is	be	AUX
ejpam-6013	121	8	σ1σ2	σ1σ2	NOUN
ejpam-6013	121	9	-	-	ADJ
ejpam-6013	121	10	closed	closed	ADJ
ejpam-6013	121	11	,	,	PUNCT
ejpam-6013	121	12	σ1σ2	σ1σ2	NOUN
ejpam-6013	121	13	-	-	PUNCT
ejpam-6013	121	14	cl(f(u	cl(f(u	NOUN
ejpam-6013	121	15	)	)	PUNCT
ejpam-6013	121	16	)	)	PUNCT
ejpam-6013	122	1	⊆	⊆	X
ejpam-6013	122	2	σ1σ2	σ1σ2	NUM
ejpam-6013	122	3	-	-	PUNCT
ejpam-6013	122	4	cl(y	cl(y	NOUN
ejpam-6013	122	5	−w	−w	NOUN
ejpam-6013	122	6	)	)	PUNCT
ejpam-6013	122	7	=	=	SYM
ejpam-6013	122	8	y	y	PROPN
ejpam-6013	122	9	−w	−w	ADV
ejpam-6013	122	10	⊆	⊆	NUM
ejpam-6013	122	11	v	v	NOUN
ejpam-6013	122	12	.	.	PUNCT
ejpam-6013	123	1	this	this	PRON
ejpam-6013	123	2	shows	show	VERB
ejpam-6013	123	3	that	that	SCONJ
ejpam-6013	123	4	f	f	PROPN
ejpam-6013	123	5	is	be	AUX
ejpam-6013	123	6	r-(τ1	r-(τ1	PROPN
ejpam-6013	123	7	,	,	PUNCT
ejpam-6013	123	8	τ2)-continuous	τ2)-continuous	PROPN
ejpam-6013	123	9	.	.	PUNCT
ejpam-6013	124	1	n.	n.	PROPN
ejpam-6013	124	2	srisarakham	srisarakham	PROPN
ejpam-6013	124	3	,	,	PUNCT
ejpam-6013	124	4	s.	s.	PROPN
ejpam-6013	124	5	sompong	sompong	PROPN
ejpam-6013	124	6	,	,	PUNCT
ejpam-6013	124	7	c.	c.	PROPN
ejpam-6013	124	8	boonpok	boonpok	PROPN
ejpam-6013	124	9	/	/	SYM
ejpam-6013	124	10	eur	eur	PROPN
ejpam-6013	124	11	.	.	PUNCT
ejpam-6013	125	1	j.	j.	PROPN
ejpam-6013	125	2	pure	pure	PROPN
ejpam-6013	125	3	appl	appl	PROPN
ejpam-6013	125	4	.	.	PROPN
ejpam-6013	125	5	math	math	PROPN
ejpam-6013	125	6	,	,	PUNCT
ejpam-6013	125	7	18	18	NUM
ejpam-6013	125	8	(	(	PUNCT
ejpam-6013	125	9	2	2	NUM
ejpam-6013	125	10	)	)	PUNCT
ejpam-6013	125	11	(	(	PUNCT
ejpam-6013	125	12	2025	2025	NUM
ejpam-6013	125	13	)	)	PUNCT
ejpam-6013	125	14	,	,	PUNCT
ejpam-6013	125	15	6013	6013	NUM
ejpam-6013	125	16	5	5	NUM
ejpam-6013	125	17	of	of	ADP
ejpam-6013	125	18	12	12	NUM
ejpam-6013	125	19	definition	definition	NOUN
ejpam-6013	125	20	3	3	NUM
ejpam-6013	125	21	.	.	PUNCT
ejpam-6013	126	1	[	[	X
ejpam-6013	126	2	29	29	NUM
ejpam-6013	126	3	]	]	PUNCT
ejpam-6013	126	4	a	a	DET
ejpam-6013	126	5	function	function	NOUN
ejpam-6013	126	6	f	f	NOUN
ejpam-6013	126	7	:	:	PUNCT
ejpam-6013	126	8	(	(	PUNCT
ejpam-6013	126	9	x	x	NOUN
ejpam-6013	126	10	,	,	PUNCT
ejpam-6013	126	11	τ1	τ1	NOUN
ejpam-6013	126	12	,	,	PUNCT
ejpam-6013	126	13	τ2	τ2	NOUN
ejpam-6013	126	14	)	)	PUNCT
ejpam-6013	126	15	→	→	SYM
ejpam-6013	126	16	(	(	PUNCT
ejpam-6013	126	17	y	y	PROPN
ejpam-6013	126	18	,	,	PUNCT
ejpam-6013	126	19	σ1	σ1	PROPN
ejpam-6013	126	20	,	,	PUNCT
ejpam-6013	126	21	σ2	σ2	PROPN
ejpam-6013	126	22	)	)	PUNCT
ejpam-6013	126	23	is	be	AUX
ejpam-6013	126	24	said	say	VERB
ejpam-6013	126	25	to	to	PART
ejpam-6013	126	26	be	be	AUX
ejpam-6013	126	27	strongly	strongly	ADV
ejpam-6013	126	28	θ(τ1	θ(τ1	ADJ
ejpam-6013	126	29	,	,	PUNCT
ejpam-6013	126	30	τ2)continuous	τ2)continuous	ADJ
ejpam-6013	126	31	at	at	ADP
ejpam-6013	126	32	a	a	DET
ejpam-6013	126	33	point	point	NOUN
ejpam-6013	126	34	x	x	SYM
ejpam-6013	126	35	∈	∈	NOUN
ejpam-6013	126	36	x	x	PUNCT
ejpam-6013	126	37	if	if	SCONJ
ejpam-6013	126	38	for	for	ADP
ejpam-6013	126	39	each	each	DET
ejpam-6013	126	40	σ1σ2	σ1σ2	VERB
ejpam-6013	126	41	-	-	ADJ
ejpam-6013	126	42	open	open	ADJ
ejpam-6013	126	43	set	set	NOUN
ejpam-6013	126	44	v	v	NOUN
ejpam-6013	126	45	of	of	ADP
ejpam-6013	126	46	y	y	NOUN
ejpam-6013	126	47	containing	contain	VERB
ejpam-6013	126	48	f(x	f(x	PROPN
ejpam-6013	126	49	)	)	PUNCT
ejpam-6013	126	50	,	,	PUNCT
ejpam-6013	126	51	there	there	PRON
ejpam-6013	126	52	exists	exist	VERB
ejpam-6013	126	53	a	a	DET
ejpam-6013	126	54	τ1τ2	τ1τ2	NOUN
ejpam-6013	126	55	-	-	ADJ
ejpam-6013	126	56	open	open	ADJ
ejpam-6013	126	57	set	set	ADJ
ejpam-6013	126	58	u	u	NOUN
ejpam-6013	126	59	of	of	ADP
ejpam-6013	126	60	x	x	PUNCT
ejpam-6013	126	61	containing	contain	VERB
ejpam-6013	126	62	x	x	PUNCT
ejpam-6013	126	63	such	such	ADJ
ejpam-6013	126	64	that	that	SCONJ
ejpam-6013	126	65	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	126	66	-	-	PUNCT
ejpam-6013	126	67	cl(u	cl(u	NOUN
ejpam-6013	126	68	)	)	PUNCT
ejpam-6013	126	69	)	)	PUNCT
ejpam-6013	127	1	⊆	⊆	NUM
ejpam-6013	127	2	v	v	NOUN
ejpam-6013	127	3	.	.	PUNCT
ejpam-6013	128	1	a	a	DET
ejpam-6013	128	2	function	function	NOUN
ejpam-6013	128	3	f	f	NOUN
ejpam-6013	128	4	:	:	PUNCT
ejpam-6013	128	5	(	(	PUNCT
ejpam-6013	128	6	x	x	NOUN
ejpam-6013	128	7	,	,	PUNCT
ejpam-6013	128	8	τ1	τ1	NOUN
ejpam-6013	128	9	,	,	PUNCT
ejpam-6013	128	10	τ2	τ2	NOUN
ejpam-6013	128	11	)	)	PUNCT
ejpam-6013	128	12	→	→	SYM
ejpam-6013	128	13	(	(	PUNCT
ejpam-6013	128	14	y	y	PROPN
ejpam-6013	128	15	,	,	PUNCT
ejpam-6013	128	16	σ1	σ1	PROPN
ejpam-6013	128	17	,	,	PUNCT
ejpam-6013	128	18	σ2	σ2	PROPN
ejpam-6013	128	19	)	)	PUNCT
ejpam-6013	128	20	is	be	AUX
ejpam-6013	128	21	said	say	VERB
ejpam-6013	128	22	to	to	PART
ejpam-6013	128	23	be	be	AUX
ejpam-6013	128	24	strongly	strongly	ADV
ejpam-6013	128	25	θ(τ1	θ(τ1	ADJ
ejpam-6013	128	26	,	,	PUNCT
ejpam-6013	128	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	128	28	if	if	SCONJ
ejpam-6013	128	29	f	f	PROPN
ejpam-6013	128	30	is	be	AUX
ejpam-6013	128	31	strongly	strongly	ADV
ejpam-6013	128	32	θ(τ1	θ(τ1	ADJ
ejpam-6013	128	33	,	,	PUNCT
ejpam-6013	128	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	128	35	at	at	ADP
ejpam-6013	128	36	each	each	DET
ejpam-6013	128	37	point	point	NOUN
ejpam-6013	128	38	x	x	PUNCT
ejpam-6013	128	39	of	of	ADP
ejpam-6013	128	40	x.	x.	PROPN
ejpam-6013	128	41	theorem	theorem	VERB
ejpam-6013	128	42	2	2	NUM
ejpam-6013	128	43	.	.	PUNCT
ejpam-6013	129	1	if	if	SCONJ
ejpam-6013	129	2	a	a	DET
ejpam-6013	129	3	function	function	NOUN
ejpam-6013	129	4	f	f	X
ejpam-6013	129	5	:	:	PUNCT
ejpam-6013	129	6	(	(	PUNCT
ejpam-6013	129	7	x	x	NOUN
ejpam-6013	129	8	,	,	PUNCT
ejpam-6013	129	9	τ1	τ1	NOUN
ejpam-6013	129	10	,	,	PUNCT
ejpam-6013	129	11	τ2	τ2	NOUN
ejpam-6013	129	12	)	)	PUNCT
ejpam-6013	129	13	→	→	SYM
ejpam-6013	129	14	(	(	PUNCT
ejpam-6013	129	15	y	y	PROPN
ejpam-6013	129	16	,	,	PUNCT
ejpam-6013	129	17	σ1	σ1	PROPN
ejpam-6013	129	18	,	,	PUNCT
ejpam-6013	129	19	σ2	σ2	PROPN
ejpam-6013	129	20	)	)	PUNCT
ejpam-6013	129	21	is	be	AUX
ejpam-6013	129	22	r-(τ1	r-(τ1	NOUN
ejpam-6013	129	23	,	,	PUNCT
ejpam-6013	129	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	129	25	,	,	PUNCT
ejpam-6013	129	26	then	then	ADV
ejpam-6013	129	27	f	f	PROPN
ejpam-6013	129	28	is	be	AUX
ejpam-6013	129	29	strongly	strongly	ADV
ejpam-6013	129	30	θ(τ1	θ(τ1	ADJ
ejpam-6013	129	31	,	,	PUNCT
ejpam-6013	129	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	129	33	.	.	PUNCT
ejpam-6013	130	1	proof	proof	NOUN
ejpam-6013	130	2	.	.	PUNCT
ejpam-6013	131	1	it	it	PRON
ejpam-6013	131	2	follows	follow	VERB
ejpam-6013	131	3	from	from	ADP
ejpam-6013	131	4	theorem	theorem	ADJ
ejpam-6013	131	5	1(2	1(2	NUM
ejpam-6013	131	6	)	)	PUNCT
ejpam-6013	131	7	.	.	PUNCT
ejpam-6013	132	1	definition	definition	NOUN
ejpam-6013	132	2	4	4	NUM
ejpam-6013	132	3	.	.	PUNCT
ejpam-6013	133	1	a	a	DET
ejpam-6013	133	2	function	function	NOUN
ejpam-6013	133	3	f	f	NOUN
ejpam-6013	133	4	:	:	PUNCT
ejpam-6013	133	5	(	(	PUNCT
ejpam-6013	133	6	x	x	NOUN
ejpam-6013	133	7	,	,	PUNCT
ejpam-6013	133	8	τ1	τ1	NOUN
ejpam-6013	133	9	,	,	PUNCT
ejpam-6013	133	10	τ2	τ2	NOUN
ejpam-6013	133	11	)	)	PUNCT
ejpam-6013	133	12	→	→	SYM
ejpam-6013	133	13	(	(	PUNCT
ejpam-6013	133	14	y	y	PROPN
ejpam-6013	133	15	,	,	PUNCT
ejpam-6013	133	16	σ1	σ1	PROPN
ejpam-6013	133	17	,	,	PUNCT
ejpam-6013	133	18	σ2	σ2	PROPN
ejpam-6013	133	19	)	)	PUNCT
ejpam-6013	133	20	is	be	AUX
ejpam-6013	133	21	said	say	VERB
ejpam-6013	133	22	to	to	PART
ejpam-6013	133	23	be	be	AUX
ejpam-6013	133	24	weakly	weakly	ADJ
ejpam-6013	133	25	(	(	PUNCT
ejpam-6013	133	26	τ1	τ1	NOUN
ejpam-6013	133	27	,	,	PUNCT
ejpam-6013	133	28	τ2)-closed	τ2)-close	VERB
ejpam-6013	133	29	if	if	SCONJ
ejpam-6013	133	30	for	for	ADP
ejpam-6013	133	31	each	each	DET
ejpam-6013	133	32	τ1τ2	τ1τ2	ADJ
ejpam-6013	133	33	-	-	ADJ
ejpam-6013	133	34	closed	closed	ADJ
ejpam-6013	133	35	set	set	ADJ
ejpam-6013	133	36	f	f	PROPN
ejpam-6013	133	37	of	of	ADP
ejpam-6013	133	38	x	x	PROPN
ejpam-6013	133	39	,	,	PUNCT
ejpam-6013	133	40	σ1σ2	σ1σ2	NOUN
ejpam-6013	133	41	-	-	PUNCT
ejpam-6013	133	42	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	133	43	-	-	PUNCT
ejpam-6013	133	44	int(f	int(f	NOUN
ejpam-6013	133	45	)	)	PUNCT
ejpam-6013	133	46	)	)	PUNCT
ejpam-6013	133	47	)	)	PUNCT
ejpam-6013	134	1	⊆	⊆	NUM
ejpam-6013	134	2	f(f	f(f	PROPN
ejpam-6013	134	3	)	)	PUNCT
ejpam-6013	134	4	.	.	PUNCT
ejpam-6013	135	1	theorem	theorem	NOUN
ejpam-6013	135	2	3	3	NUM
ejpam-6013	135	3	.	.	X
ejpam-6013	135	4	for	for	ADP
ejpam-6013	135	5	a	a	DET
ejpam-6013	135	6	function	function	NOUN
ejpam-6013	135	7	f	f	NOUN
ejpam-6013	135	8	:	:	PUNCT
ejpam-6013	135	9	(	(	PUNCT
ejpam-6013	135	10	x	x	NOUN
ejpam-6013	135	11	,	,	PUNCT
ejpam-6013	135	12	τ1	τ1	NOUN
ejpam-6013	135	13	,	,	PUNCT
ejpam-6013	135	14	τ2	τ2	NOUN
ejpam-6013	135	15	)	)	PUNCT
ejpam-6013	135	16	→	→	SYM
ejpam-6013	135	17	(	(	PUNCT
ejpam-6013	135	18	y	y	PROPN
ejpam-6013	135	19	,	,	PUNCT
ejpam-6013	135	20	σ1	σ1	PROPN
ejpam-6013	135	21	,	,	PUNCT
ejpam-6013	135	22	σ2	σ2	NOUN
ejpam-6013	135	23	)	)	PUNCT
ejpam-6013	135	24	,	,	PUNCT
ejpam-6013	135	25	the	the	DET
ejpam-6013	135	26	following	follow	VERB
ejpam-6013	135	27	properties	property	NOUN
ejpam-6013	135	28	are	be	AUX
ejpam-6013	135	29	equivalent	equivalent	ADJ
ejpam-6013	135	30	:	:	PUNCT
ejpam-6013	135	31	(	(	PUNCT
ejpam-6013	135	32	1	1	X
ejpam-6013	135	33	)	)	PUNCT
ejpam-6013	135	34	f	f	PROPN
ejpam-6013	135	35	is	be	AUX
ejpam-6013	135	36	weakly	weakly	ADJ
ejpam-6013	135	37	(	(	PUNCT
ejpam-6013	135	38	τ1	τ1	NOUN
ejpam-6013	135	39	,	,	PUNCT
ejpam-6013	135	40	τ2)-closed	τ2)-close	VERB
ejpam-6013	135	41	;	;	PUNCT
ejpam-6013	135	42	(	(	PUNCT
ejpam-6013	135	43	2	2	X
ejpam-6013	135	44	)	)	PUNCT
ejpam-6013	135	45	σ1σ2	σ1σ2	NOUN
ejpam-6013	135	46	-	-	PUNCT
ejpam-6013	135	47	cl(f(u	cl(f(u	NOUN
ejpam-6013	135	48	)	)	PUNCT
ejpam-6013	135	49	)	)	PUNCT
ejpam-6013	136	1	⊆	⊆	NUM
ejpam-6013	136	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	136	3	-	-	PUNCT
ejpam-6013	136	4	cl(u	cl(u	NOUN
ejpam-6013	136	5	)	)	PUNCT
ejpam-6013	136	6	)	)	PUNCT
ejpam-6013	136	7	for	for	ADP
ejpam-6013	136	8	each	each	DET
ejpam-6013	136	9	τ1τ2	τ1τ2	ADJ
ejpam-6013	136	10	-	-	ADJ
ejpam-6013	136	11	open	open	ADJ
ejpam-6013	136	12	set	set	ADJ
ejpam-6013	136	13	u	u	NOUN
ejpam-6013	136	14	of	of	ADP
ejpam-6013	136	15	x	x	PRON
ejpam-6013	136	16	;	;	PUNCT
ejpam-6013	136	17	(	(	PUNCT
ejpam-6013	136	18	3	3	X
ejpam-6013	136	19	)	)	PUNCT
ejpam-6013	136	20	for	for	ADP
ejpam-6013	136	21	each	each	DET
ejpam-6013	136	22	subset	subset	NOUN
ejpam-6013	136	23	b	b	PROPN
ejpam-6013	136	24	of	of	ADP
ejpam-6013	136	25	y	y	PROPN
ejpam-6013	136	26	and	and	CCONJ
ejpam-6013	136	27	each	each	DET
ejpam-6013	136	28	τ1τ2	τ1τ2	ADJ
ejpam-6013	136	29	-	-	ADJ
ejpam-6013	136	30	open	open	ADJ
ejpam-6013	136	31	set	set	ADJ
ejpam-6013	136	32	u	u	NOUN
ejpam-6013	136	33	of	of	ADP
ejpam-6013	136	34	x	x	PUNCT
ejpam-6013	136	35	with	with	ADP
ejpam-6013	136	36	f−1(b	f−1(b	PROPN
ejpam-6013	136	37	)	)	PUNCT
ejpam-6013	136	38	⊆	⊆	NUM
ejpam-6013	136	39	u	u	NOUN
ejpam-6013	136	40	,	,	PUNCT
ejpam-6013	136	41	there	there	PRON
ejpam-6013	136	42	exists	exist	VERB
ejpam-6013	136	43	a	a	DET
ejpam-6013	136	44	σ1σ2	σ1σ2	NUM
ejpam-6013	136	45	-	-	ADJ
ejpam-6013	136	46	open	open	ADJ
ejpam-6013	136	47	set	set	NOUN
ejpam-6013	136	48	v	v	NOUN
ejpam-6013	136	49	of	of	ADP
ejpam-6013	136	50	y	y	PRON
ejpam-6013	136	51	such	such	ADJ
ejpam-6013	136	52	that	that	DET
ejpam-6013	136	53	b	b	PROPN
ejpam-6013	136	54	⊆	⊆	NUM
ejpam-6013	136	55	v	v	NOUN
ejpam-6013	136	56	and	and	CCONJ
ejpam-6013	136	57	f−1(v	f−1(v	NOUN
ejpam-6013	136	58	)	)	PUNCT
ejpam-6013	137	1	⊆	⊆	NUM
ejpam-6013	137	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	137	3	-	-	NOUN
ejpam-6013	137	4	cl(u	cl(u	NUM
ejpam-6013	137	5	)	)	PUNCT
ejpam-6013	137	6	;	;	PUNCT
ejpam-6013	137	7	(	(	PUNCT
ejpam-6013	137	8	4	4	X
ejpam-6013	137	9	)	)	PUNCT
ejpam-6013	137	10	for	for	ADP
ejpam-6013	137	11	each	each	DET
ejpam-6013	137	12	y	y	PROPN
ejpam-6013	137	13	∈	∈	PROPN
ejpam-6013	137	14	y	y	PROPN
ejpam-6013	137	15	and	and	CCONJ
ejpam-6013	137	16	each	each	DET
ejpam-6013	137	17	τ1τ2	τ1τ2	ADJ
ejpam-6013	137	18	-	-	ADJ
ejpam-6013	137	19	open	open	ADJ
ejpam-6013	137	20	set	set	ADJ
ejpam-6013	137	21	u	u	NOUN
ejpam-6013	137	22	of	of	ADP
ejpam-6013	137	23	x	x	PUNCT
ejpam-6013	137	24	with	with	ADP
ejpam-6013	137	25	f−1(y	f−1(y	PROPN
ejpam-6013	137	26	)	)	PUNCT
ejpam-6013	137	27	⊆	⊆	NUM
ejpam-6013	137	28	u	u	NOUN
ejpam-6013	137	29	,	,	PUNCT
ejpam-6013	137	30	there	there	PRON
ejpam-6013	137	31	exists	exist	VERB
ejpam-6013	137	32	a	a	DET
ejpam-6013	137	33	σ1σ2	σ1σ2	NUM
ejpam-6013	137	34	-	-	ADJ
ejpam-6013	137	35	open	open	ADJ
ejpam-6013	137	36	set	set	NOUN
ejpam-6013	137	37	v	v	NOUN
ejpam-6013	137	38	of	of	ADP
ejpam-6013	137	39	y	y	PROPN
ejpam-6013	137	40	containing	contain	VERB
ejpam-6013	137	41	y	y	PRON
ejpam-6013	137	42	such	such	ADJ
ejpam-6013	137	43	that	that	PRON
ejpam-6013	137	44	f−1(y	f−1(y	PROPN
ejpam-6013	137	45	)	)	PUNCT
ejpam-6013	138	1	⊆	⊆	NUM
ejpam-6013	138	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	138	3	-	-	NOUN
ejpam-6013	138	4	cl(u	cl(u	NUM
ejpam-6013	138	5	)	)	PUNCT
ejpam-6013	138	6	;	;	PUNCT
ejpam-6013	138	7	(	(	PUNCT
ejpam-6013	138	8	5	5	X
ejpam-6013	138	9	)	)	PUNCT
ejpam-6013	138	10	σ1σ2	σ1σ2	NOUN
ejpam-6013	138	11	-	-	PUNCT
ejpam-6013	138	12	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	138	13	-	-	PUNCT
ejpam-6013	138	14	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	138	15	-	-	PUNCT
ejpam-6013	138	16	cl(a	cl(a	NUM
ejpam-6013	138	17	)	)	PUNCT
ejpam-6013	138	18	)	)	PUNCT
ejpam-6013	138	19	)	)	PUNCT
ejpam-6013	138	20	)	)	PUNCT
ejpam-6013	139	1	⊆	⊆	NUM
ejpam-6013	139	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	139	3	-	-	PUNCT
ejpam-6013	139	4	cl(a	cl(a	NUM
ejpam-6013	139	5	)	)	PUNCT
ejpam-6013	139	6	)	)	PUNCT
ejpam-6013	139	7	for	for	ADP
ejpam-6013	139	8	each	each	PRON
ejpam-6013	139	9	subset	subset	VERB
ejpam-6013	139	10	a	a	PRON
ejpam-6013	139	11	of	of	ADP
ejpam-6013	139	12	x	x	PRON
ejpam-6013	139	13	;	;	PUNCT
ejpam-6013	139	14	(	(	PUNCT
ejpam-6013	139	15	6	6	NUM
ejpam-6013	139	16	)	)	PUNCT
ejpam-6013	139	17	σ1σ2	σ1σ2	NOUN
ejpam-6013	139	18	-	-	PUNCT
ejpam-6013	139	19	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	139	20	-	-	PUNCT
ejpam-6013	139	21	int((τ1	int((τ1	NOUN
ejpam-6013	139	22	,	,	PUNCT
ejpam-6013	139	23	τ2)θ	τ2)θ	NOUN
ejpam-6013	139	24	-	-	PUNCT
ejpam-6013	139	25	cl(a	cl(a	NUM
ejpam-6013	139	26	)	)	PUNCT
ejpam-6013	139	27	)	)	PUNCT
ejpam-6013	139	28	)	)	PUNCT
ejpam-6013	139	29	)	)	PUNCT
ejpam-6013	140	1	⊆	⊆	X
ejpam-6013	140	2	f((τ1	f((τ1	VERB
ejpam-6013	140	3	,	,	PUNCT
ejpam-6013	140	4	τ2)θ	τ2)θ	NOUN
ejpam-6013	140	5	-	-	PUNCT
ejpam-6013	140	6	cl(a	cl(a	NUM
ejpam-6013	140	7	)	)	PUNCT
ejpam-6013	140	8	)	)	PUNCT
ejpam-6013	140	9	for	for	SCONJ
ejpam-6013	140	10	each	each	DET
ejpam-6013	140	11	subset	subset	VERB
ejpam-6013	140	12	a	a	PRON
ejpam-6013	140	13	of	of	ADP
ejpam-6013	140	14	x.	x.	NOUN
ejpam-6013	140	15	proof	proof	NOUN
ejpam-6013	140	16	.	.	PUNCT
ejpam-6013	141	1	(	(	PUNCT
ejpam-6013	141	2	1	1	X
ejpam-6013	141	3	)	)	PUNCT
ejpam-6013	141	4	⇒	⇒	NOUN
ejpam-6013	141	5	(	(	PUNCT
ejpam-6013	141	6	2	2	NUM
ejpam-6013	141	7	):	):	PUNCT
ejpam-6013	141	8	let	let	VERB
ejpam-6013	141	9	u	u	PRON
ejpam-6013	141	10	be	be	AUX
ejpam-6013	141	11	any	any	DET
ejpam-6013	141	12	τ1τ2	τ1τ2	ADJ
ejpam-6013	141	13	-	-	ADJ
ejpam-6013	141	14	open	open	ADJ
ejpam-6013	141	15	set	set	NOUN
ejpam-6013	141	16	of	of	ADP
ejpam-6013	141	17	x.	x.	NOUN
ejpam-6013	141	18	since	since	SCONJ
ejpam-6013	141	19	f	f	PROPN
ejpam-6013	141	20	is	be	AUX
ejpam-6013	141	21	weakly	weakly	ADJ
ejpam-6013	141	22	(	(	PUNCT
ejpam-6013	141	23	τ1	τ1	NOUN
ejpam-6013	141	24	,	,	PUNCT
ejpam-6013	141	25	τ2)-closed	τ2)-closed	PROPN
ejpam-6013	141	26	,	,	PUNCT
ejpam-6013	141	27	we	we	PRON
ejpam-6013	141	28	have	have	VERB
ejpam-6013	141	29	σ1σ2	σ1σ2	NOUN
ejpam-6013	141	30	-	-	PUNCT
ejpam-6013	141	31	cl(f(u	cl(f(u	NOUN
ejpam-6013	141	32	)	)	PUNCT
ejpam-6013	141	33	)	)	PUNCT
ejpam-6013	142	1	⊆	⊆	X
ejpam-6013	142	2	σ1σ2	σ1σ2	NUM
ejpam-6013	142	3	-	-	PUNCT
ejpam-6013	142	4	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	142	5	-	-	PUNCT
ejpam-6013	142	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	142	7	-	-	PUNCT
ejpam-6013	142	8	cl(u	cl(u	NOUN
ejpam-6013	142	9	)	)	PUNCT
ejpam-6013	142	10	)	)	PUNCT
ejpam-6013	142	11	)	)	PUNCT
ejpam-6013	142	12	)	)	PUNCT
ejpam-6013	143	1	⊆	⊆	NUM
ejpam-6013	143	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	143	3	-	-	PUNCT
ejpam-6013	143	4	cl(u	cl(u	NOUN
ejpam-6013	143	5	)	)	PUNCT
ejpam-6013	143	6	)	)	PUNCT
ejpam-6013	143	7	.	.	PUNCT
ejpam-6013	144	1	(	(	PUNCT
ejpam-6013	144	2	2	2	X
ejpam-6013	144	3	)	)	PUNCT
ejpam-6013	144	4	⇒	⇒	NOUN
ejpam-6013	144	5	(	(	PUNCT
ejpam-6013	144	6	3	3	NUM
ejpam-6013	144	7	):	):	PUNCT
ejpam-6013	144	8	let	let	VERB
ejpam-6013	144	9	b	b	X
ejpam-6013	144	10	be	be	AUX
ejpam-6013	144	11	any	any	DET
ejpam-6013	144	12	subset	subset	NOUN
ejpam-6013	144	13	of	of	ADP
ejpam-6013	144	14	y	y	PROPN
ejpam-6013	144	15	and	and	CCONJ
ejpam-6013	144	16	u	u	NOUN
ejpam-6013	144	17	be	be	VERB
ejpam-6013	144	18	any	any	DET
ejpam-6013	144	19	τ1τ2	τ1τ2	ADJ
ejpam-6013	144	20	-	-	ADJ
ejpam-6013	144	21	open	open	ADJ
ejpam-6013	144	22	set	set	ADJ
ejpam-6013	144	23	ofx	ofx	NOUN
ejpam-6013	144	24	with	with	ADP
ejpam-6013	144	25	f−1(b	f−1(b	PROPN
ejpam-6013	144	26	)	)	PUNCT
ejpam-6013	144	27	⊆	⊆	NUM
ejpam-6013	144	28	u	u	NOUN
ejpam-6013	144	29	.	.	PUNCT
ejpam-6013	145	1	since	since	SCONJ
ejpam-6013	145	2	u	u	NOUN
ejpam-6013	145	3	is	be	AUX
ejpam-6013	145	4	τ1τ2	τ1τ2	VERB
ejpam-6013	145	5	-	-	ADJ
ejpam-6013	145	6	open	open	ADJ
ejpam-6013	145	7	,	,	PUNCT
ejpam-6013	145	8	f−1(b	f−1(b	PROPN
ejpam-6013	145	9	)	)	PUNCT
ejpam-6013	145	10	∩	∩	NOUN
ejpam-6013	145	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	145	12	-	-	NOUN
ejpam-6013	145	13	cl(x	cl(x	SYM
ejpam-6013	145	14	−	−	PRON
ejpam-6013	145	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	145	16	-	-	NOUN
ejpam-6013	145	17	cl(u	cl(u	NOUN
ejpam-6013	145	18	)	)	PUNCT
ejpam-6013	145	19	)	)	PUNCT
ejpam-6013	145	20	=	=	SYM
ejpam-6013	145	21	f−1(b	f−1(b	PROPN
ejpam-6013	145	22	)	)	PUNCT
ejpam-6013	145	23	∩	∩	NOUN
ejpam-6013	145	24	(	(	PUNCT
ejpam-6013	145	25	x	x	X
ejpam-6013	145	26	−	−	ADP
ejpam-6013	145	27	τ1τ2	τ1τ2	NOUN
ejpam-6013	145	28	-	-	NOUN
ejpam-6013	145	29	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	145	30	-	-	PUNCT
ejpam-6013	145	31	cl(u	cl(u	NOUN
ejpam-6013	145	32	)	)	PUNCT
ejpam-6013	145	33	)	)	PUNCT
ejpam-6013	145	34	)	)	PUNCT
ejpam-6013	146	1	⊆	⊆	NUM
ejpam-6013	146	2	f−1(b	f−1(b	PROPN
ejpam-6013	146	3	)	)	PUNCT
ejpam-6013	146	4	∩	∩	NOUN
ejpam-6013	146	5	(	(	PUNCT
ejpam-6013	146	6	x	x	X
ejpam-6013	146	7	−	−	NOUN
ejpam-6013	146	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	146	9	-	-	ADJ
ejpam-6013	146	10	int(u	int(u	ADJ
ejpam-6013	146	11	)	)	PUNCT
ejpam-6013	146	12	)	)	PUNCT
ejpam-6013	147	1	=	=	SYM
ejpam-6013	147	2	f−1(b	f−1(b	PROPN
ejpam-6013	147	3	)	)	PUNCT
ejpam-6013	147	4	∩	∩	NOUN
ejpam-6013	147	5	(	(	PUNCT
ejpam-6013	147	6	x	x	SYM
ejpam-6013	147	7	−	−	PROPN
ejpam-6013	147	8	u	u	NOUN
ejpam-6013	147	9	)	)	PUNCT
ejpam-6013	147	10	=	=	PUNCT
ejpam-6013	147	11	∅.	∅.	ADP
ejpam-6013	147	12	thus	thus	ADV
ejpam-6013	147	13	,	,	PUNCT
ejpam-6013	147	14	f−1(b	f−1(b	PROPN
ejpam-6013	147	15	)	)	PUNCT
ejpam-6013	147	16	∩	∩	NOUN
ejpam-6013	147	17	τ1τ2	τ1τ2	NOUN
ejpam-6013	147	18	-	-	NOUN
ejpam-6013	147	19	cl(x	cl(x	SYM
ejpam-6013	147	20	−	−	PRON
ejpam-6013	147	21	τ1τ2	τ1τ2	NOUN
ejpam-6013	147	22	-	-	NOUN
ejpam-6013	147	23	cl(u	cl(u	NOUN
ejpam-6013	147	24	)	)	PUNCT
ejpam-6013	147	25	)	)	PUNCT
ejpam-6013	148	1	=	=	NOUN
ejpam-6013	148	2	∅	∅	NOUN
ejpam-6013	148	3	and	and	CCONJ
ejpam-6013	148	4	hence	hence	ADV
ejpam-6013	148	5	b	b	NOUN
ejpam-6013	148	6	∩	∩	ADJ
ejpam-6013	148	7	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	148	8	-	-	NOUN
ejpam-6013	148	9	cl(x	cl(x	NUM
ejpam-6013	148	10	−	−	PRON
ejpam-6013	148	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	148	12	-	-	NOUN
ejpam-6013	148	13	cl(u	cl(u	NOUN
ejpam-6013	148	14	)	)	PUNCT
ejpam-6013	148	15	)	)	PUNCT
ejpam-6013	148	16	)	)	PUNCT
ejpam-6013	149	1	=	=	PUNCT
ejpam-6013	149	2	∅.	∅.	VERB
ejpam-6013	149	3	on	on	ADP
ejpam-6013	149	4	the	the	DET
ejpam-6013	149	5	other	other	ADJ
ejpam-6013	149	6	hand	hand	NOUN
ejpam-6013	149	7	,	,	PUNCT
ejpam-6013	149	8	we	we	PRON
ejpam-6013	149	9	have	have	VERB
ejpam-6013	149	10	τ1τ2	τ1τ2	NOUN
ejpam-6013	149	11	-	-	NOUN
ejpam-6013	149	12	cl(u	cl(u	NOUN
ejpam-6013	149	13	)	)	PUNCT
ejpam-6013	149	14	is	be	AUX
ejpam-6013	149	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	149	16	-	-	ADJ
ejpam-6013	149	17	closed	closed	ADJ
ejpam-6013	149	18	and	and	CCONJ
ejpam-6013	149	19	x	x	X
ejpam-6013	149	20	−	−	ADP
ejpam-6013	149	21	τ1τ2	τ1τ2	NOUN
ejpam-6013	149	22	-	-	NOUN
ejpam-6013	149	23	cl(u	cl(u	NOUN
ejpam-6013	149	24	)	)	PUNCT
ejpam-6013	149	25	is	be	AUX
ejpam-6013	149	26	τ1τ2	τ1τ2	NOUN
ejpam-6013	149	27	-	-	ADJ
ejpam-6013	149	28	open	open	ADJ
ejpam-6013	149	29	.	.	PUNCT
ejpam-6013	150	1	thus	thus	ADV
ejpam-6013	150	2	by	by	ADP
ejpam-6013	150	3	(	(	PUNCT
ejpam-6013	150	4	2	2	NUM
ejpam-6013	150	5	)	)	PUNCT
ejpam-6013	150	6	,	,	PUNCT
ejpam-6013	150	7	σ1σ2	σ1σ2	X
ejpam-6013	150	8	-	-	PUNCT
ejpam-6013	150	9	cl(f(x	cl(f(x	NOUN
ejpam-6013	150	10	−	−	NOUN
ejpam-6013	150	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	150	12	-	-	NOUN
ejpam-6013	150	13	cl(u	cl(u	NOUN
ejpam-6013	150	14	)	)	PUNCT
ejpam-6013	150	15	)	)	PUNCT
ejpam-6013	150	16	)	)	PUNCT
ejpam-6013	151	1	⊆	⊆	NUM
ejpam-6013	151	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	151	3	-	-	PUNCT
ejpam-6013	151	4	cl(x	cl(x	NOUN
ejpam-6013	151	5	−	−	PRON
ejpam-6013	151	6	τ1τ2	τ1τ2	NOUN
ejpam-6013	151	7	-	-	NOUN
ejpam-6013	151	8	cl(u	cl(u	NOUN
ejpam-6013	151	9	)	)	PUNCT
ejpam-6013	151	10	)	)	PUNCT
ejpam-6013	151	11	)	)	PUNCT
ejpam-6013	151	12	and	and	CCONJ
ejpam-6013	151	13	so	so	ADV
ejpam-6013	151	14	b	b	NOUN
ejpam-6013	151	15	∩	∩	ADJ
ejpam-6013	151	16	σ1σ2	σ1σ2	NOUN
ejpam-6013	151	17	-	-	PUNCT
ejpam-6013	151	18	cl(f(x	cl(f(x	NOUN
ejpam-6013	151	19	−	−	NOUN
ejpam-6013	151	20	τ1τ2	τ1τ2	NOUN
ejpam-6013	151	21	-	-	NOUN
ejpam-6013	151	22	cl(u	cl(u	NOUN
ejpam-6013	151	23	)	)	PUNCT
ejpam-6013	151	24	)	)	PUNCT
ejpam-6013	151	25	)	)	PUNCT
ejpam-6013	152	1	⊆	⊆	NUM
ejpam-6013	152	2	b	b	NOUN
ejpam-6013	152	3	∩	∩	ADJ
ejpam-6013	152	4	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	152	5	-	-	NOUN
ejpam-6013	152	6	cl(x	cl(x	NUM
ejpam-6013	152	7	−	−	PRON
ejpam-6013	152	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	152	9	-	-	NOUN
ejpam-6013	152	10	cl(u	cl(u	NOUN
ejpam-6013	152	11	)	)	PUNCT
ejpam-6013	152	12	)	)	PUNCT
ejpam-6013	152	13	)	)	PUNCT
ejpam-6013	152	14	.	.	PUNCT
ejpam-6013	153	1	n.	n.	PROPN
ejpam-6013	153	2	srisarakham	srisarakham	PROPN
ejpam-6013	153	3	,	,	PUNCT
ejpam-6013	153	4	s.	s.	PROPN
ejpam-6013	153	5	sompong	sompong	PROPN
ejpam-6013	153	6	,	,	PUNCT
ejpam-6013	153	7	c.	c.	PROPN
ejpam-6013	153	8	boonpok	boonpok	PROPN
ejpam-6013	153	9	/	/	SYM
ejpam-6013	153	10	eur	eur	PROPN
ejpam-6013	153	11	.	.	PUNCT
ejpam-6013	154	1	j.	j.	PROPN
ejpam-6013	154	2	pure	pure	PROPN
ejpam-6013	154	3	appl	appl	PROPN
ejpam-6013	154	4	.	.	PROPN
ejpam-6013	154	5	math	math	PROPN
ejpam-6013	154	6	,	,	PUNCT
ejpam-6013	154	7	18	18	NUM
ejpam-6013	154	8	(	(	PUNCT
ejpam-6013	154	9	2	2	NUM
ejpam-6013	154	10	)	)	PUNCT
ejpam-6013	154	11	(	(	PUNCT
ejpam-6013	154	12	2025	2025	NUM
ejpam-6013	154	13	)	)	PUNCT
ejpam-6013	154	14	,	,	PUNCT
ejpam-6013	154	15	6013	6013	NUM
ejpam-6013	154	16	6	6	NUM
ejpam-6013	154	17	of	of	ADP
ejpam-6013	154	18	12	12	NUM
ejpam-6013	154	19	let	let	VERB
ejpam-6013	154	20	v	v	NOUN
ejpam-6013	154	21	=	=	SYM
ejpam-6013	154	22	y	y	NOUN
ejpam-6013	154	23	−	−	NUM
ejpam-6013	154	24	σ1σ2	σ1σ2	NUM
ejpam-6013	154	25	-	-	PUNCT
ejpam-6013	154	26	cl(f(x	cl(f(x	NOUN
ejpam-6013	154	27	−	−	NOUN
ejpam-6013	154	28	τ1τ2	τ1τ2	NOUN
ejpam-6013	154	29	-	-	NOUN
ejpam-6013	154	30	cl(u	cl(u	NOUN
ejpam-6013	154	31	)	)	PUNCT
ejpam-6013	154	32	)	)	PUNCT
ejpam-6013	154	33	)	)	PUNCT
ejpam-6013	154	34	.	.	PUNCT
ejpam-6013	155	1	then	then	ADV
ejpam-6013	155	2	,	,	PUNCT
ejpam-6013	155	3	v	v	NOUN
ejpam-6013	155	4	is	be	AUX
ejpam-6013	155	5	σ1σ2	σ1σ2	NOUN
ejpam-6013	155	6	-	-	ADJ
ejpam-6013	155	7	open	open	ADJ
ejpam-6013	155	8	in	in	ADP
ejpam-6013	155	9	y	y	PROPN
ejpam-6013	155	10	,	,	PUNCT
ejpam-6013	155	11	b	b	PROPN
ejpam-6013	155	12	⊆	⊆	NUM
ejpam-6013	155	13	v	v	NOUN
ejpam-6013	155	14	and	and	CCONJ
ejpam-6013	155	15	f−1(v	f−1(v	NOUN
ejpam-6013	155	16	)	)	PUNCT
ejpam-6013	156	1	=	=	PUNCT
ejpam-6013	156	2	f−1(y	f−1(y	PROPN
ejpam-6013	156	3	−	−	NUM
ejpam-6013	156	4	σ1σ2	σ1σ2	NUM
ejpam-6013	156	5	-	-	PUNCT
ejpam-6013	156	6	cl(f(x	cl(f(x	NOUN
ejpam-6013	156	7	−	−	NOUN
ejpam-6013	156	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	156	9	-	-	NOUN
ejpam-6013	156	10	cl(u	cl(u	NOUN
ejpam-6013	156	11	)	)	PUNCT
ejpam-6013	156	12	)	)	PUNCT
ejpam-6013	156	13	)	)	PUNCT
ejpam-6013	156	14	)	)	PUNCT
ejpam-6013	157	1	⊆	⊆	NUM
ejpam-6013	157	2	x	x	SYM
ejpam-6013	157	3	−	−	NOUN
ejpam-6013	157	4	f−1(f(x	f−1(f(x	NOUN
ejpam-6013	157	5	−	−	ADP
ejpam-6013	157	6	τ1τ2	τ1τ2	NOUN
ejpam-6013	157	7	-	-	NOUN
ejpam-6013	157	8	cl(u	cl(u	NOUN
ejpam-6013	157	9	)	)	PUNCT
ejpam-6013	157	10	)	)	PUNCT
ejpam-6013	157	11	)	)	PUNCT
ejpam-6013	158	1	⊆	⊆	X
ejpam-6013	158	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	158	3	-	-	NOUN
ejpam-6013	158	4	cl(u	cl(u	NUM
ejpam-6013	158	5	)	)	PUNCT
ejpam-6013	158	6	.	.	PUNCT
ejpam-6013	159	1	(	(	PUNCT
ejpam-6013	159	2	3	3	X
ejpam-6013	159	3	)	)	PUNCT
ejpam-6013	159	4	⇒	⇒	NOUN
ejpam-6013	159	5	(	(	PUNCT
ejpam-6013	159	6	4	4	NUM
ejpam-6013	159	7	):	):	PUNCT
ejpam-6013	159	8	the	the	DET
ejpam-6013	159	9	proof	proof	NOUN
ejpam-6013	159	10	is	be	AUX
ejpam-6013	159	11	obvious	obvious	ADJ
ejpam-6013	159	12	.	.	PUNCT
ejpam-6013	160	1	(	(	PUNCT
ejpam-6013	160	2	4	4	X
ejpam-6013	160	3	)	)	PUNCT
ejpam-6013	160	4	⇒	⇒	NOUN
ejpam-6013	160	5	(	(	PUNCT
ejpam-6013	160	6	1	1	NUM
ejpam-6013	160	7	):	):	PUNCT
ejpam-6013	160	8	let	let	VERB
ejpam-6013	160	9	f	f	PRON
ejpam-6013	160	10	be	be	AUX
ejpam-6013	160	11	any	any	DET
ejpam-6013	160	12	τ1τ2	τ1τ2	ADJ
ejpam-6013	160	13	-	-	ADJ
ejpam-6013	160	14	closed	closed	ADJ
ejpam-6013	160	15	set	set	NOUN
ejpam-6013	160	16	of	of	ADP
ejpam-6013	160	17	x	x	PUNCT
ejpam-6013	160	18	and	and	CCONJ
ejpam-6013	160	19	y	y	PROPN
ejpam-6013	160	20	∈	∈	PROPN
ejpam-6013	161	1	y	y	PROPN
ejpam-6013	161	2	−	−	PROPN
ejpam-6013	161	3	f(f	f(f	PROPN
ejpam-6013	161	4	)	)	PUNCT
ejpam-6013	161	5	.	.	PUNCT
ejpam-6013	162	1	since	since	SCONJ
ejpam-6013	162	2	y	y	PROPN
ejpam-6013	162	3	−	−	PROPN
ejpam-6013	162	4	f	f	PROPN
ejpam-6013	162	5	is	be	AUX
ejpam-6013	162	6	σ1σ2	σ1σ2	NOUN
ejpam-6013	162	7	-	-	ADJ
ejpam-6013	162	8	open	open	ADJ
ejpam-6013	162	9	and	and	CCONJ
ejpam-6013	162	10	f−1(y	f−1(y	PROPN
ejpam-6013	162	11	)	)	PUNCT
ejpam-6013	163	1	⊆	⊆	NUM
ejpam-6013	163	2	x	x	SYM
ejpam-6013	163	3	−	−	PROPN
ejpam-6013	163	4	f	f	X
ejpam-6013	163	5	,	,	PUNCT
ejpam-6013	163	6	by	by	ADP
ejpam-6013	163	7	(	(	PUNCT
ejpam-6013	163	8	4	4	X
ejpam-6013	163	9	)	)	PUNCT
ejpam-6013	163	10	there	there	PRON
ejpam-6013	163	11	exists	exist	VERB
ejpam-6013	163	12	a	a	DET
ejpam-6013	163	13	σ1σ2	σ1σ2	NUM
ejpam-6013	163	14	-	-	ADJ
ejpam-6013	163	15	open	open	ADJ
ejpam-6013	163	16	set	set	NOUN
ejpam-6013	163	17	v	v	NOUN
ejpam-6013	163	18	of	of	ADP
ejpam-6013	163	19	y	y	PROPN
ejpam-6013	163	20	with	with	ADP
ejpam-6013	163	21	y	y	PROPN
ejpam-6013	163	22	∈	∈	PROPN
ejpam-6013	163	23	v	v	NOUN
ejpam-6013	163	24	and	and	CCONJ
ejpam-6013	163	25	f−1(v	f−1(v	NOUN
ejpam-6013	163	26	)	)	PUNCT
ejpam-6013	164	1	⊆	⊆	NUM
ejpam-6013	164	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	164	3	-	-	NOUN
ejpam-6013	164	4	cl(x	cl(x	SYM
ejpam-6013	164	5	−	−	PROPN
ejpam-6013	164	6	f	f	NOUN
ejpam-6013	164	7	)	)	PUNCT
ejpam-6013	164	8	=	=	PUNCT
ejpam-6013	165	1	x	x	X
ejpam-6013	165	2	−	−	ADP
ejpam-6013	165	3	τ1τ2	τ1τ2	NOUN
ejpam-6013	165	4	-	-	NUM
ejpam-6013	165	5	int(f	int(f	X
ejpam-6013	165	6	)	)	PUNCT
ejpam-6013	165	7	.	.	PUNCT
ejpam-6013	166	1	thus	thus	ADV
ejpam-6013	166	2	,	,	PUNCT
ejpam-6013	166	3	v	v	ADP
ejpam-6013	166	4	∩	∩	NOUN
ejpam-6013	166	5	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	166	6	-	-	PUNCT
ejpam-6013	166	7	int(f	int(f	PROPN
ejpam-6013	166	8	)	)	PUNCT
ejpam-6013	166	9	)	)	PUNCT
ejpam-6013	167	1	=	=	NOUN
ejpam-6013	167	2	∅	∅	NOUN
ejpam-6013	167	3	and	and	CCONJ
ejpam-6013	167	4	hence	hence	ADV
ejpam-6013	167	5	y	y	PROPN
ejpam-6013	167	6	∈	∈	PROPN
ejpam-6013	167	7	y	y	PROPN
ejpam-6013	167	8	−σ1σ2	−σ1σ2	ADJ
ejpam-6013	167	9	-	-	PUNCT
ejpam-6013	167	10	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	167	11	-	-	PUNCT
ejpam-6013	167	12	int(f	int(f	NOUN
ejpam-6013	167	13	)	)	PUNCT
ejpam-6013	167	14	)	)	PUNCT
ejpam-6013	167	15	)	)	PUNCT
ejpam-6013	167	16	.	.	PUNCT
ejpam-6013	168	1	therefore	therefore	ADV
ejpam-6013	168	2	,	,	PUNCT
ejpam-6013	168	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	168	4	-	-	PUNCT
ejpam-6013	168	5	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	168	6	-	-	PUNCT
ejpam-6013	168	7	int(f	int(f	NOUN
ejpam-6013	168	8	)	)	PUNCT
ejpam-6013	168	9	)	)	PUNCT
ejpam-6013	168	10	)	)	PUNCT
ejpam-6013	169	1	⊆	⊆	NUM
ejpam-6013	169	2	f(f	f(f	PROPN
ejpam-6013	169	3	)	)	PUNCT
ejpam-6013	169	4	which	which	PRON
ejpam-6013	169	5	implies	imply	VERB
ejpam-6013	169	6	that	that	SCONJ
ejpam-6013	169	7	f	f	PROPN
ejpam-6013	169	8	is	be	AUX
ejpam-6013	169	9	weakly	weakly	ADJ
ejpam-6013	169	10	(	(	PUNCT
ejpam-6013	169	11	τ1	τ1	NOUN
ejpam-6013	169	12	,	,	PUNCT
ejpam-6013	169	13	τ2)-closed	τ2)-closed	PROPN
ejpam-6013	169	14	.	.	PUNCT
ejpam-6013	170	1	(	(	PUNCT
ejpam-6013	170	2	1	1	X
ejpam-6013	170	3	)	)	PUNCT
ejpam-6013	170	4	⇒	⇒	NOUN
ejpam-6013	170	5	(	(	PUNCT
ejpam-6013	170	6	5	5	NUM
ejpam-6013	170	7	):	):	PUNCT
ejpam-6013	170	8	let	let	VERB
ejpam-6013	170	9	a	a	PRON
ejpam-6013	170	10	be	be	AUX
ejpam-6013	170	11	any	any	DET
ejpam-6013	170	12	subset	subset	NOUN
ejpam-6013	170	13	of	of	ADP
ejpam-6013	170	14	x.	x.	NOUN
ejpam-6013	170	15	then	then	ADV
ejpam-6013	170	16	,	,	PUNCT
ejpam-6013	170	17	τ1τ2	τ1τ2	NOUN
ejpam-6013	170	18	-	-	NUM
ejpam-6013	170	19	cl(a	cl(a	NUM
ejpam-6013	170	20	)	)	PUNCT
ejpam-6013	170	21	is	be	AUX
ejpam-6013	170	22	τ1τ2	τ1τ2	NOUN
ejpam-6013	170	23	-	-	ADJ
ejpam-6013	170	24	closed	closed	ADJ
ejpam-6013	170	25	and	and	CCONJ
ejpam-6013	170	26	by	by	ADP
ejpam-6013	170	27	(	(	PUNCT
ejpam-6013	170	28	1	1	NUM
ejpam-6013	170	29	)	)	PUNCT
ejpam-6013	170	30	,	,	PUNCT
ejpam-6013	170	31	σ1σ2	σ1σ2	NOUN
ejpam-6013	170	32	-	-	PUNCT
ejpam-6013	170	33	cl(f(τ1τ2	cl(f(τ1τ2	ADJ
ejpam-6013	170	34	-	-	PUNCT
ejpam-6013	170	35	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	170	36	-	-	PUNCT
ejpam-6013	170	37	cl(a	cl(a	NUM
ejpam-6013	170	38	)	)	PUNCT
ejpam-6013	170	39	)	)	PUNCT
ejpam-6013	170	40	)	)	PUNCT
ejpam-6013	170	41	)	)	PUNCT
ejpam-6013	171	1	⊆	⊆	NUM
ejpam-6013	171	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	171	3	-	-	PUNCT
ejpam-6013	171	4	cl(a	cl(a	NUM
ejpam-6013	171	5	)	)	PUNCT
ejpam-6013	171	6	)	)	PUNCT
ejpam-6013	171	7	.	.	PUNCT
ejpam-6013	172	1	(	(	PUNCT
ejpam-6013	172	2	5	5	X
ejpam-6013	172	3	)	)	PUNCT
ejpam-6013	172	4	⇒	⇒	NOUN
ejpam-6013	172	5	(	(	PUNCT
ejpam-6013	172	6	2	2	NUM
ejpam-6013	172	7	):	):	PUNCT
ejpam-6013	172	8	let	let	VERB
ejpam-6013	172	9	u	u	PRON
ejpam-6013	172	10	be	be	AUX
ejpam-6013	172	11	any	any	DET
ejpam-6013	172	12	τ1τ2	τ1τ2	ADJ
ejpam-6013	172	13	-	-	ADJ
ejpam-6013	172	14	open	open	ADJ
ejpam-6013	172	15	set	set	NOUN
ejpam-6013	172	16	of	of	ADP
ejpam-6013	172	17	x.	x.	NOUN
ejpam-6013	172	18	then	then	ADV
ejpam-6013	172	19	by	by	ADP
ejpam-6013	172	20	(	(	PUNCT
ejpam-6013	172	21	5	5	NUM
ejpam-6013	172	22	)	)	PUNCT
ejpam-6013	172	23	,	,	PUNCT
ejpam-6013	172	24	σ1σ2	σ1σ2	X
ejpam-6013	172	25	-	-	PUNCT
ejpam-6013	172	26	cl(f(u	cl(f(u	NOUN
ejpam-6013	172	27	)	)	PUNCT
ejpam-6013	172	28	)	)	PUNCT
ejpam-6013	173	1	⊆	⊆	X
ejpam-6013	173	2	σ1σ2	σ1σ2	NUM
ejpam-6013	173	3	-	-	PUNCT
ejpam-6013	173	4	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	173	5	-	-	PUNCT
ejpam-6013	173	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6013	173	7	-	-	PUNCT
ejpam-6013	173	8	cl(u	cl(u	NOUN
ejpam-6013	173	9	)	)	PUNCT
ejpam-6013	173	10	)	)	PUNCT
ejpam-6013	173	11	)	)	PUNCT
ejpam-6013	173	12	)	)	PUNCT
ejpam-6013	174	1	⊆	⊆	NUM
ejpam-6013	174	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	174	3	-	-	PUNCT
ejpam-6013	174	4	cl(u	cl(u	NOUN
ejpam-6013	174	5	)	)	PUNCT
ejpam-6013	174	6	)	)	PUNCT
ejpam-6013	174	7	.	.	PUNCT
ejpam-6013	175	1	(	(	PUNCT
ejpam-6013	175	2	1	1	X
ejpam-6013	175	3	)	)	PUNCT
ejpam-6013	175	4	⇒	⇒	NOUN
ejpam-6013	175	5	(	(	PUNCT
ejpam-6013	175	6	6	6	NUM
ejpam-6013	175	7	):	):	PUNCT
ejpam-6013	175	8	let	let	VERB
ejpam-6013	175	9	a	a	DET
ejpam-6013	175	10	be	be	AUX
ejpam-6013	175	11	any	any	DET
ejpam-6013	175	12	subset	subset	NOUN
ejpam-6013	175	13	of	of	ADP
ejpam-6013	175	14	x.	x.	NOUN
ejpam-6013	175	15	thus	thus	ADV
ejpam-6013	175	16	by	by	ADP
ejpam-6013	175	17	lemma	lemma	PROPN
ejpam-6013	175	18	2	2	NUM
ejpam-6013	175	19	,	,	PUNCT
ejpam-6013	175	20	(	(	PUNCT
ejpam-6013	175	21	τ1	τ1	NOUN
ejpam-6013	175	22	,	,	PUNCT
ejpam-6013	175	23	τ2)θ	τ2)θ	NOUN
ejpam-6013	175	24	-	-	PUNCT
ejpam-6013	175	25	cl(a	cl(a	NUM
ejpam-6013	175	26	)	)	PUNCT
ejpam-6013	175	27	is	be	AUX
ejpam-6013	175	28	τ1τ2	τ1τ2	NOUN
ejpam-6013	175	29	-	-	ADJ
ejpam-6013	175	30	closed	closed	ADJ
ejpam-6013	175	31	and	and	CCONJ
ejpam-6013	175	32	by	by	ADP
ejpam-6013	175	33	(	(	PUNCT
ejpam-6013	175	34	1	1	NUM
ejpam-6013	175	35	)	)	PUNCT
ejpam-6013	175	36	,	,	PUNCT
ejpam-6013	175	37	we	we	PRON
ejpam-6013	175	38	have	have	VERB
ejpam-6013	175	39	σ1σ2	σ1σ2	NOUN
ejpam-6013	175	40	-	-	PUNCT
ejpam-6013	175	41	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	175	42	-	-	PUNCT
ejpam-6013	175	43	int((τ1	int((τ1	NOUN
ejpam-6013	175	44	,	,	PUNCT
ejpam-6013	175	45	τ2)θ	τ2)θ	NOUN
ejpam-6013	175	46	-	-	PUNCT
ejpam-6013	175	47	cl(a	cl(a	NUM
ejpam-6013	175	48	)	)	PUNCT
ejpam-6013	175	49	)	)	PUNCT
ejpam-6013	175	50	)	)	PUNCT
ejpam-6013	175	51	)	)	PUNCT
ejpam-6013	176	1	⊆	⊆	X
ejpam-6013	176	2	f((τ1	f((τ1	VERB
ejpam-6013	176	3	,	,	PUNCT
ejpam-6013	176	4	τ2)θ	τ2)θ	NOUN
ejpam-6013	176	5	-	-	PUNCT
ejpam-6013	176	6	cl(a	cl(a	NUM
ejpam-6013	176	7	)	)	PUNCT
ejpam-6013	176	8	)	)	PUNCT
ejpam-6013	176	9	.	.	PUNCT
ejpam-6013	177	1	(	(	PUNCT
ejpam-6013	177	2	6	6	X
ejpam-6013	177	3	)	)	PUNCT
ejpam-6013	177	4	⇒	⇒	NOUN
ejpam-6013	177	5	(	(	PUNCT
ejpam-6013	177	6	2	2	NUM
ejpam-6013	177	7	):	):	PUNCT
ejpam-6013	177	8	let	let	VERB
ejpam-6013	177	9	u	u	PRON
ejpam-6013	177	10	be	be	AUX
ejpam-6013	177	11	any	any	DET
ejpam-6013	177	12	τ1τ2	τ1τ2	ADJ
ejpam-6013	177	13	-	-	ADJ
ejpam-6013	177	14	open	open	ADJ
ejpam-6013	177	15	set	set	NOUN
ejpam-6013	177	16	of	of	ADP
ejpam-6013	177	17	x.	x.	NOUN
ejpam-6013	177	18	by	by	ADP
ejpam-6013	177	19	lemma	lemma	PROPN
ejpam-6013	177	20	2	2	NUM
ejpam-6013	177	21	,	,	PUNCT
ejpam-6013	177	22	we	we	PRON
ejpam-6013	177	23	have	have	VERB
ejpam-6013	177	24	τ1τ2	τ1τ2	NOUN
ejpam-6013	177	25	-	-	NOUN
ejpam-6013	177	26	cl(u	cl(u	NOUN
ejpam-6013	177	27	)	)	PUNCT
ejpam-6013	177	28	=	=	PUNCT
ejpam-6013	177	29	(	(	PUNCT
ejpam-6013	177	30	τ1	τ1	NOUN
ejpam-6013	177	31	,	,	PUNCT
ejpam-6013	177	32	τ2)θ	τ2)θ	ADJ
ejpam-6013	177	33	-	-	PUNCT
ejpam-6013	177	34	cl(u	cl(u	NOUN
ejpam-6013	177	35	)	)	PUNCT
ejpam-6013	177	36	and	and	CCONJ
ejpam-6013	177	37	by	by	ADP
ejpam-6013	177	38	(	(	PUNCT
ejpam-6013	177	39	6	6	NUM
ejpam-6013	177	40	)	)	PUNCT
ejpam-6013	177	41	,	,	PUNCT
ejpam-6013	177	42	σ1σ2	σ1σ2	X
ejpam-6013	177	43	-	-	PUNCT
ejpam-6013	177	44	cl(f(u	cl(f(u	NOUN
ejpam-6013	177	45	)	)	PUNCT
ejpam-6013	177	46	)	)	PUNCT
ejpam-6013	178	1	⊆	⊆	X
ejpam-6013	178	2	σ1σ2	σ1σ2	NUM
ejpam-6013	178	3	-	-	PUNCT
ejpam-6013	178	4	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	178	5	-	-	PUNCT
ejpam-6013	178	6	int((τ1	int((τ1	NOUN
ejpam-6013	178	7	,	,	PUNCT
ejpam-6013	178	8	τ2)θ	τ2)θ	ADJ
ejpam-6013	178	9	-	-	PUNCT
ejpam-6013	178	10	cl(u	cl(u	NOUN
ejpam-6013	178	11	)	)	PUNCT
ejpam-6013	178	12	)	)	PUNCT
ejpam-6013	178	13	)	)	PUNCT
ejpam-6013	178	14	)	)	PUNCT
ejpam-6013	179	1	⊆	⊆	X
ejpam-6013	179	2	f((τ1	f((τ1	VERB
ejpam-6013	179	3	,	,	PUNCT
ejpam-6013	179	4	τ2)θ	τ2)θ	ADJ
ejpam-6013	179	5	-	-	PUNCT
ejpam-6013	179	6	cl(u	cl(u	NOUN
ejpam-6013	179	7	)	)	PUNCT
ejpam-6013	179	8	)	)	PUNCT
ejpam-6013	180	1	=	=	PUNCT
ejpam-6013	180	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	180	3	-	-	PUNCT
ejpam-6013	180	4	cl(u	cl(u	NOUN
ejpam-6013	180	5	)	)	PUNCT
ejpam-6013	180	6	)	)	PUNCT
ejpam-6013	180	7	.	.	PUNCT
ejpam-6013	181	1	theorem	theorem	VERB
ejpam-6013	181	2	4	4	NUM
ejpam-6013	181	3	.	.	PUNCT
ejpam-6013	182	1	if	if	SCONJ
ejpam-6013	182	2	a	a	DET
ejpam-6013	182	3	function	function	NOUN
ejpam-6013	182	4	f	f	X
ejpam-6013	182	5	:	:	PUNCT
ejpam-6013	182	6	(	(	PUNCT
ejpam-6013	182	7	x	x	NOUN
ejpam-6013	182	8	,	,	PUNCT
ejpam-6013	182	9	τ1	τ1	NOUN
ejpam-6013	182	10	,	,	PUNCT
ejpam-6013	182	11	τ2	τ2	NOUN
ejpam-6013	182	12	)	)	PUNCT
ejpam-6013	182	13	→	→	SYM
ejpam-6013	182	14	(	(	PUNCT
ejpam-6013	182	15	y	y	PROPN
ejpam-6013	182	16	,	,	PUNCT
ejpam-6013	182	17	σ1	σ1	PROPN
ejpam-6013	182	18	,	,	PUNCT
ejpam-6013	182	19	σ2	σ2	PROPN
ejpam-6013	182	20	)	)	PUNCT
ejpam-6013	182	21	is	be	AUX
ejpam-6013	182	22	strongly	strongly	ADV
ejpam-6013	182	23	θ(τ1	θ(τ1	ADJ
ejpam-6013	182	24	,	,	PUNCT
ejpam-6013	182	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	182	26	and	and	CCONJ
ejpam-6013	182	27	weakly	weakly	ADJ
ejpam-6013	182	28	(	(	PUNCT
ejpam-6013	182	29	τ1	τ1	NOUN
ejpam-6013	182	30	,	,	PUNCT
ejpam-6013	182	31	τ2)-closed	τ2)-closed	PROPN
ejpam-6013	182	32	,	,	PUNCT
ejpam-6013	182	33	then	then	ADV
ejpam-6013	182	34	f	f	PROPN
ejpam-6013	182	35	is	be	AUX
ejpam-6013	182	36	r-(τ1	r-(τ1	PROPN
ejpam-6013	182	37	,	,	PUNCT
ejpam-6013	182	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	182	39	.	.	PUNCT
ejpam-6013	183	1	proof	proof	NOUN
ejpam-6013	183	2	.	.	PUNCT
ejpam-6013	184	1	let	let	VERB
ejpam-6013	184	2	x	x	PUNCT
ejpam-6013	184	3	∈	∈	PROPN
ejpam-6013	184	4	x	x	X
ejpam-6013	184	5	and	and	CCONJ
ejpam-6013	184	6	v	v	X
ejpam-6013	184	7	be	be	AUX
ejpam-6013	184	8	any	any	DET
ejpam-6013	184	9	σ1σ2	σ1σ2	NOUN
ejpam-6013	184	10	-	-	ADJ
ejpam-6013	184	11	open	open	ADJ
ejpam-6013	184	12	set	set	NOUN
ejpam-6013	184	13	of	of	ADP
ejpam-6013	184	14	y	y	PROPN
ejpam-6013	184	15	containing	contain	VERB
ejpam-6013	184	16	f(x	f(x	PROPN
ejpam-6013	184	17	)	)	PUNCT
ejpam-6013	184	18	.	.	PUNCT
ejpam-6013	185	1	since	since	SCONJ
ejpam-6013	185	2	f	f	PROPN
ejpam-6013	185	3	is	be	AUX
ejpam-6013	185	4	strongly	strongly	ADV
ejpam-6013	185	5	θ(τ1	θ(τ1	ADJ
ejpam-6013	185	6	,	,	PUNCT
ejpam-6013	185	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	185	8	,	,	PUNCT
ejpam-6013	185	9	there	there	PRON
ejpam-6013	185	10	exists	exist	VERB
ejpam-6013	185	11	a	a	DET
ejpam-6013	185	12	τ1τ2	τ1τ2	NOUN
ejpam-6013	185	13	-	-	ADJ
ejpam-6013	185	14	open	open	ADJ
ejpam-6013	185	15	set	set	ADJ
ejpam-6013	185	16	u	u	NOUN
ejpam-6013	185	17	of	of	ADP
ejpam-6013	185	18	x	x	PUNCT
ejpam-6013	185	19	containing	contain	VERB
ejpam-6013	185	20	x	x	PUNCT
ejpam-6013	185	21	such	such	ADJ
ejpam-6013	185	22	that	that	SCONJ
ejpam-6013	185	23	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	185	24	-	-	PUNCT
ejpam-6013	185	25	cl(u	cl(u	NOUN
ejpam-6013	185	26	)	)	PUNCT
ejpam-6013	185	27	)	)	PUNCT
ejpam-6013	186	1	⊆	⊆	NUM
ejpam-6013	186	2	v	v	NOUN
ejpam-6013	186	3	.	.	PUNCT
ejpam-6013	187	1	since	since	SCONJ
ejpam-6013	187	2	f	f	PROPN
ejpam-6013	187	3	is	be	AUX
ejpam-6013	187	4	weakly	weakly	ADJ
ejpam-6013	187	5	(	(	PUNCT
ejpam-6013	187	6	τ1	τ1	NOUN
ejpam-6013	187	7	,	,	PUNCT
ejpam-6013	187	8	τ2)-closed	τ2)-close	VERB
ejpam-6013	187	9	,	,	PUNCT
ejpam-6013	187	10	by	by	ADP
ejpam-6013	187	11	theorem	theorem	NOUN
ejpam-6013	187	12	3	3	NUM
ejpam-6013	187	13	we	we	PRON
ejpam-6013	187	14	have	have	VERB
ejpam-6013	187	15	σ1σ2	σ1σ2	NOUN
ejpam-6013	187	16	-	-	PUNCT
ejpam-6013	187	17	cl(f(u	cl(f(u	NOUN
ejpam-6013	187	18	)	)	PUNCT
ejpam-6013	187	19	)	)	PUNCT
ejpam-6013	188	1	⊆	⊆	NUM
ejpam-6013	188	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	188	3	-	-	PUNCT
ejpam-6013	188	4	cl(u	cl(u	NOUN
ejpam-6013	188	5	)	)	PUNCT
ejpam-6013	188	6	)	)	PUNCT
ejpam-6013	189	1	⊆	⊆	NUM
ejpam-6013	189	2	v.	v.	ADP
ejpam-6013	189	3	this	this	PRON
ejpam-6013	189	4	shows	show	VERB
ejpam-6013	189	5	that	that	SCONJ
ejpam-6013	189	6	f	f	PROPN
ejpam-6013	189	7	is	be	AUX
ejpam-6013	189	8	r-(τ1	r-(τ1	PROPN
ejpam-6013	189	9	,	,	PUNCT
ejpam-6013	189	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	189	11	.	.	PUNCT
ejpam-6013	190	1	definition	definition	NOUN
ejpam-6013	190	2	5	5	NUM
ejpam-6013	190	3	.	.	PUNCT
ejpam-6013	191	1	a	a	DET
ejpam-6013	191	2	function	function	NOUN
ejpam-6013	191	3	f	f	NOUN
ejpam-6013	191	4	:	:	PUNCT
ejpam-6013	191	5	(	(	PUNCT
ejpam-6013	191	6	x	x	NOUN
ejpam-6013	191	7	,	,	PUNCT
ejpam-6013	191	8	τ1	τ1	NOUN
ejpam-6013	191	9	,	,	PUNCT
ejpam-6013	191	10	τ2	τ2	NOUN
ejpam-6013	191	11	)	)	PUNCT
ejpam-6013	191	12	→	→	SYM
ejpam-6013	191	13	(	(	PUNCT
ejpam-6013	191	14	y	y	PROPN
ejpam-6013	191	15	,	,	PUNCT
ejpam-6013	191	16	σ1	σ1	PROPN
ejpam-6013	191	17	,	,	PUNCT
ejpam-6013	191	18	σ2	σ2	PROPN
ejpam-6013	191	19	)	)	PUNCT
ejpam-6013	191	20	is	be	AUX
ejpam-6013	191	21	said	say	VERB
ejpam-6013	191	22	to	to	PART
ejpam-6013	191	23	be	be	AUX
ejpam-6013	191	24	contra-(τ1	contra-(τ1	NOUN
ejpam-6013	191	25	,	,	PUNCT
ejpam-6013	191	26	τ2)-open	τ2)-open	ADJ
ejpam-6013	191	27	if	if	SCONJ
ejpam-6013	191	28	f(u	f(u	PROPN
ejpam-6013	191	29	)	)	PUNCT
ejpam-6013	191	30	is	be	AUX
ejpam-6013	191	31	σ1σ2	σ1σ2	NOUN
ejpam-6013	191	32	-	-	ADJ
ejpam-6013	191	33	closed	closed	ADJ
ejpam-6013	191	34	in	in	ADP
ejpam-6013	191	35	y	y	PROPN
ejpam-6013	191	36	for	for	ADP
ejpam-6013	191	37	every	every	DET
ejpam-6013	191	38	τ1τ2	τ1τ2	ADJ
ejpam-6013	191	39	-	-	ADJ
ejpam-6013	191	40	open	open	ADJ
ejpam-6013	191	41	set	set	ADJ
ejpam-6013	191	42	u	u	NOUN
ejpam-6013	191	43	of	of	ADP
ejpam-6013	191	44	x.	x.	PROPN
ejpam-6013	191	45	n.	n.	PROPN
ejpam-6013	191	46	srisarakham	srisarakham	PROPN
ejpam-6013	191	47	,	,	PUNCT
ejpam-6013	191	48	s.	s.	PROPN
ejpam-6013	191	49	sompong	sompong	PROPN
ejpam-6013	191	50	,	,	PUNCT
ejpam-6013	191	51	c.	c.	PROPN
ejpam-6013	191	52	boonpok	boonpok	PROPN
ejpam-6013	191	53	/	/	SYM
ejpam-6013	191	54	eur	eur	PROPN
ejpam-6013	191	55	.	.	PUNCT
ejpam-6013	192	1	j.	j.	PROPN
ejpam-6013	192	2	pure	pure	PROPN
ejpam-6013	192	3	appl	appl	PROPN
ejpam-6013	192	4	.	.	PROPN
ejpam-6013	192	5	math	math	PROPN
ejpam-6013	192	6	,	,	PUNCT
ejpam-6013	192	7	18	18	NUM
ejpam-6013	192	8	(	(	PUNCT
ejpam-6013	192	9	2	2	NUM
ejpam-6013	192	10	)	)	PUNCT
ejpam-6013	192	11	(	(	PUNCT
ejpam-6013	192	12	2025	2025	NUM
ejpam-6013	192	13	)	)	PUNCT
ejpam-6013	192	14	,	,	PUNCT
ejpam-6013	192	15	6013	6013	NUM
ejpam-6013	192	16	7	7	NUM
ejpam-6013	192	17	of	of	ADP
ejpam-6013	192	18	12	12	NUM
ejpam-6013	192	19	theorem	theorem	NOUN
ejpam-6013	192	20	5	5	NUM
ejpam-6013	192	21	.	.	PUNCT
ejpam-6013	193	1	if	if	SCONJ
ejpam-6013	193	2	a	a	DET
ejpam-6013	193	3	function	function	NOUN
ejpam-6013	193	4	f	f	X
ejpam-6013	193	5	:	:	PUNCT
ejpam-6013	193	6	(	(	PUNCT
ejpam-6013	193	7	x	x	NOUN
ejpam-6013	193	8	,	,	PUNCT
ejpam-6013	193	9	τ1	τ1	NOUN
ejpam-6013	193	10	,	,	PUNCT
ejpam-6013	193	11	τ2	τ2	NOUN
ejpam-6013	193	12	)	)	PUNCT
ejpam-6013	193	13	→	→	SYM
ejpam-6013	193	14	(	(	PUNCT
ejpam-6013	193	15	y	y	PROPN
ejpam-6013	193	16	,	,	PUNCT
ejpam-6013	193	17	σ1	σ1	PROPN
ejpam-6013	193	18	,	,	PUNCT
ejpam-6013	193	19	σ2	σ2	PROPN
ejpam-6013	193	20	)	)	PUNCT
ejpam-6013	193	21	is	be	AUX
ejpam-6013	193	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6013	193	23	,	,	PUNCT
ejpam-6013	193	24	τ2)-open	τ2)-open	PROPN
ejpam-6013	193	25	,	,	PUNCT
ejpam-6013	193	26	then	then	ADV
ejpam-6013	193	27	f	f	PROPN
ejpam-6013	193	28	is	be	AUX
ejpam-6013	193	29	weakly	weakly	ADJ
ejpam-6013	193	30	(	(	PUNCT
ejpam-6013	193	31	τ1	τ1	NOUN
ejpam-6013	193	32	,	,	PUNCT
ejpam-6013	193	33	τ2)-closed	τ2)-closed	ADJ
ejpam-6013	193	34	.	.	PUNCT
ejpam-6013	194	1	proof	proof	NOUN
ejpam-6013	194	2	.	.	PUNCT
ejpam-6013	195	1	let	let	VERB
ejpam-6013	195	2	f	f	PRON
ejpam-6013	195	3	be	be	AUX
ejpam-6013	195	4	any	any	DET
ejpam-6013	195	5	τ1τ2	τ1τ2	ADJ
ejpam-6013	195	6	-	-	ADJ
ejpam-6013	195	7	closed	closed	ADJ
ejpam-6013	195	8	set	set	NOUN
ejpam-6013	195	9	of	of	ADP
ejpam-6013	195	10	x.	x.	NOUN
ejpam-6013	195	11	then	then	ADV
ejpam-6013	195	12	,	,	PUNCT
ejpam-6013	195	13	τ1τ2	τ1τ2	NOUN
ejpam-6013	195	14	-	-	PUNCT
ejpam-6013	195	15	int(f	int(f	X
ejpam-6013	195	16	)	)	PUNCT
ejpam-6013	195	17	is	be	AUX
ejpam-6013	195	18	a	a	DET
ejpam-6013	195	19	τ1τ2	τ1τ2	ADJ
ejpam-6013	195	20	-	-	ADJ
ejpam-6013	195	21	open	open	ADJ
ejpam-6013	195	22	set	set	NOUN
ejpam-6013	195	23	of	of	ADP
ejpam-6013	195	24	x.	x.	NOUN
ejpam-6013	195	25	since	since	SCONJ
ejpam-6013	195	26	f	f	PROPN
ejpam-6013	195	27	is	be	AUX
ejpam-6013	195	28	contra-(τ1	contra-(τ1	NOUN
ejpam-6013	195	29	,	,	PUNCT
ejpam-6013	195	30	τ2)-open	τ2)-open	PROPN
ejpam-6013	195	31	,	,	PUNCT
ejpam-6013	195	32	we	we	PRON
ejpam-6013	195	33	have	have	VERB
ejpam-6013	195	34	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	195	35	-	-	PUNCT
ejpam-6013	195	36	int(f	int(f	PROPN
ejpam-6013	195	37	)	)	PUNCT
ejpam-6013	195	38	)	)	PUNCT
ejpam-6013	195	39	is	be	AUX
ejpam-6013	195	40	σ1σ2	σ1σ2	NOUN
ejpam-6013	195	41	-	-	ADJ
ejpam-6013	195	42	closed	closed	ADJ
ejpam-6013	195	43	in	in	ADP
ejpam-6013	195	44	y	y	PROPN
ejpam-6013	195	45	and	and	CCONJ
ejpam-6013	195	46	hence	hence	ADV
ejpam-6013	195	47	σ1σ2	σ1σ2	NOUN
ejpam-6013	195	48	-	-	PUNCT
ejpam-6013	195	49	cl(f(τ1τ2	cl(f(τ1τ2	NOUN
ejpam-6013	195	50	-	-	PUNCT
ejpam-6013	195	51	int(f	int(f	NOUN
ejpam-6013	195	52	)	)	PUNCT
ejpam-6013	195	53	)	)	PUNCT
ejpam-6013	195	54	)	)	PUNCT
ejpam-6013	196	1	=	=	PRON
ejpam-6013	196	2	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	196	3	-	-	PUNCT
ejpam-6013	196	4	int(f	int(f	PROPN
ejpam-6013	196	5	)	)	PUNCT
ejpam-6013	196	6	)	)	PUNCT
ejpam-6013	197	1	⊆	⊆	NUM
ejpam-6013	197	2	f(f	f(f	PROPN
ejpam-6013	197	3	)	)	PUNCT
ejpam-6013	197	4	.	.	PUNCT
ejpam-6013	198	1	this	this	PRON
ejpam-6013	198	2	shows	show	VERB
ejpam-6013	198	3	that	that	SCONJ
ejpam-6013	198	4	f	f	PROPN
ejpam-6013	198	5	is	be	AUX
ejpam-6013	198	6	weakly	weakly	ADJ
ejpam-6013	198	7	(	(	PUNCT
ejpam-6013	198	8	τ1	τ1	NOUN
ejpam-6013	198	9	,	,	PUNCT
ejpam-6013	198	10	τ2)closed	τ2)close	VERB
ejpam-6013	198	11	.	.	PUNCT
ejpam-6013	199	1	theorem	theorem	VERB
ejpam-6013	199	2	6	6	NUM
ejpam-6013	199	3	.	.	PUNCT
ejpam-6013	200	1	if	if	SCONJ
ejpam-6013	200	2	a	a	DET
ejpam-6013	200	3	function	function	NOUN
ejpam-6013	200	4	f	f	X
ejpam-6013	200	5	:	:	PUNCT
ejpam-6013	200	6	(	(	PUNCT
ejpam-6013	200	7	x	x	NOUN
ejpam-6013	200	8	,	,	PUNCT
ejpam-6013	200	9	τ1	τ1	NOUN
ejpam-6013	200	10	,	,	PUNCT
ejpam-6013	200	11	τ2	τ2	NOUN
ejpam-6013	200	12	)	)	PUNCT
ejpam-6013	200	13	→	→	SYM
ejpam-6013	200	14	(	(	PUNCT
ejpam-6013	200	15	y	y	PROPN
ejpam-6013	200	16	,	,	PUNCT
ejpam-6013	200	17	σ1	σ1	PROPN
ejpam-6013	200	18	,	,	PUNCT
ejpam-6013	200	19	σ2	σ2	PROPN
ejpam-6013	200	20	)	)	PUNCT
ejpam-6013	200	21	is	be	AUX
ejpam-6013	200	22	(	(	PUNCT
ejpam-6013	200	23	τ1	τ1	NOUN
ejpam-6013	200	24	,	,	PUNCT
ejpam-6013	200	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	200	26	and	and	CCONJ
ejpam-6013	200	27	contra(τ1	contra(τ1	NOUN
ejpam-6013	200	28	,	,	PUNCT
ejpam-6013	200	29	τ2)-open	τ2)-open	PROPN
ejpam-6013	200	30	,	,	PUNCT
ejpam-6013	200	31	then	then	ADV
ejpam-6013	200	32	f	f	PROPN
ejpam-6013	200	33	is	be	AUX
ejpam-6013	200	34	r-(τ1	r-(τ1	PROPN
ejpam-6013	200	35	,	,	PUNCT
ejpam-6013	200	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	200	37	.	.	PUNCT
ejpam-6013	201	1	proof	proof	NOUN
ejpam-6013	201	2	.	.	PUNCT
ejpam-6013	202	1	let	let	VERB
ejpam-6013	202	2	x	x	PUNCT
ejpam-6013	202	3	∈	∈	PROPN
ejpam-6013	202	4	x	x	X
ejpam-6013	202	5	and	and	CCONJ
ejpam-6013	202	6	v	v	X
ejpam-6013	202	7	be	be	AUX
ejpam-6013	202	8	any	any	DET
ejpam-6013	202	9	σ1σ2	σ1σ2	NOUN
ejpam-6013	202	10	-	-	ADJ
ejpam-6013	202	11	open	open	ADJ
ejpam-6013	202	12	set	set	NOUN
ejpam-6013	202	13	of	of	ADP
ejpam-6013	202	14	y	y	PROPN
ejpam-6013	202	15	containing	contain	VERB
ejpam-6013	202	16	f(x	f(x	PROPN
ejpam-6013	202	17	)	)	PUNCT
ejpam-6013	202	18	.	.	PUNCT
ejpam-6013	203	1	since	since	SCONJ
ejpam-6013	203	2	f	f	PROPN
ejpam-6013	203	3	is	be	AUX
ejpam-6013	203	4	(	(	PUNCT
ejpam-6013	203	5	τ1	τ1	NOUN
ejpam-6013	203	6	,	,	PUNCT
ejpam-6013	203	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	203	8	,	,	PUNCT
ejpam-6013	203	9	by	by	ADP
ejpam-6013	203	10	lemma	lemma	PROPN
ejpam-6013	203	11	3	3	NUM
ejpam-6013	203	12	we	we	PRON
ejpam-6013	203	13	have	have	AUX
ejpam-6013	203	14	f−1(v	f−1(v	PROPN
ejpam-6013	203	15	)	)	PUNCT
ejpam-6013	203	16	is	be	AUX
ejpam-6013	203	17	τ1τ2	τ1τ2	NOUN
ejpam-6013	203	18	-	-	ADJ
ejpam-6013	203	19	open	open	ADJ
ejpam-6013	203	20	in	in	ADP
ejpam-6013	203	21	x.	x.	NOUN
ejpam-6013	203	22	since	since	SCONJ
ejpam-6013	203	23	f	f	PROPN
ejpam-6013	203	24	is	be	AUX
ejpam-6013	203	25	contra(τ1	contra(τ1	NOUN
ejpam-6013	203	26	,	,	PUNCT
ejpam-6013	203	27	τ2)-open	τ2)-open	PROPN
ejpam-6013	203	28	,	,	PUNCT
ejpam-6013	203	29	f(f	f(f	PROPN
ejpam-6013	203	30	−1(v	−1(v	PROPN
ejpam-6013	203	31	)	)	PUNCT
ejpam-6013	203	32	)	)	PUNCT
ejpam-6013	204	1	is	be	AUX
ejpam-6013	204	2	σ1σ2	σ1σ2	NOUN
ejpam-6013	204	3	-	-	ADJ
ejpam-6013	204	4	closed	closed	ADJ
ejpam-6013	204	5	in	in	ADP
ejpam-6013	204	6	y	y	PROPN
ejpam-6013	204	7	and	and	CCONJ
ejpam-6013	204	8	σ1σ2	σ1σ2	NOUN
ejpam-6013	204	9	-	-	PUNCT
ejpam-6013	204	10	cl(f(f	cl(f(f	ADJ
ejpam-6013	204	11	−1(v	−1(v	NOUN
ejpam-6013	204	12	)	)	PUNCT
ejpam-6013	204	13	)	)	PUNCT
ejpam-6013	204	14	)	)	PUNCT
ejpam-6013	205	1	=	=	PUNCT
ejpam-6013	205	2	f(f−1(v	f(f−1(v	PROPN
ejpam-6013	205	3	)	)	PUNCT
ejpam-6013	205	4	)	)	PUNCT
ejpam-6013	206	1	⊆	⊆	NUM
ejpam-6013	206	2	v	v	NOUN
ejpam-6013	206	3	.	.	PUNCT
ejpam-6013	207	1	put	put	VERB
ejpam-6013	207	2	u	u	NOUN
ejpam-6013	207	3	=	=	NOUN
ejpam-6013	207	4	f−1(v	f−1(v	PROPN
ejpam-6013	207	5	)	)	PUNCT
ejpam-6013	207	6	.	.	PUNCT
ejpam-6013	208	1	then	then	ADV
ejpam-6013	208	2	,	,	PUNCT
ejpam-6013	208	3	u	u	NOUN
ejpam-6013	208	4	is	be	AUX
ejpam-6013	208	5	a	a	DET
ejpam-6013	208	6	τ1τ2	τ1τ2	ADJ
ejpam-6013	208	7	-	-	ADJ
ejpam-6013	208	8	open	open	ADJ
ejpam-6013	208	9	set	set	NOUN
ejpam-6013	208	10	of	of	ADP
ejpam-6013	208	11	x	x	PUNCT
ejpam-6013	208	12	containing	contain	VERB
ejpam-6013	208	13	x	x	X
ejpam-6013	208	14	and	and	CCONJ
ejpam-6013	208	15	σ1σ2	σ1σ2	NOUN
ejpam-6013	208	16	-	-	PUNCT
ejpam-6013	208	17	cl(f(u	cl(f(u	NOUN
ejpam-6013	208	18	)	)	PUNCT
ejpam-6013	208	19	)	)	PUNCT
ejpam-6013	209	1	⊆	⊆	NUM
ejpam-6013	209	2	v	v	NOUN
ejpam-6013	209	3	.	.	PUNCT
ejpam-6013	210	1	this	this	PRON
ejpam-6013	210	2	shows	show	VERB
ejpam-6013	210	3	that	that	SCONJ
ejpam-6013	210	4	f	f	PROPN
ejpam-6013	210	5	is	be	AUX
ejpam-6013	210	6	r-(τ1	r-(τ1	PROPN
ejpam-6013	210	7	,	,	PUNCT
ejpam-6013	210	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	210	9	.	.	PUNCT
ejpam-6013	211	1	definition	definition	NOUN
ejpam-6013	211	2	6	6	NUM
ejpam-6013	211	3	.	.	PUNCT
ejpam-6013	212	1	[	[	X
ejpam-6013	212	2	24	24	NUM
ejpam-6013	212	3	]	]	PUNCT
ejpam-6013	212	4	a	a	DET
ejpam-6013	212	5	function	function	NOUN
ejpam-6013	212	6	f	f	NOUN
ejpam-6013	212	7	:	:	PUNCT
ejpam-6013	212	8	(	(	PUNCT
ejpam-6013	212	9	x	x	NOUN
ejpam-6013	212	10	,	,	PUNCT
ejpam-6013	212	11	τ1	τ1	NOUN
ejpam-6013	212	12	,	,	PUNCT
ejpam-6013	212	13	τ2	τ2	NOUN
ejpam-6013	212	14	)	)	PUNCT
ejpam-6013	212	15	→	→	SYM
ejpam-6013	212	16	(	(	PUNCT
ejpam-6013	212	17	y	y	PROPN
ejpam-6013	212	18	,	,	PUNCT
ejpam-6013	212	19	σ1	σ1	PROPN
ejpam-6013	212	20	,	,	PUNCT
ejpam-6013	212	21	σ2	σ2	PROPN
ejpam-6013	212	22	)	)	PUNCT
ejpam-6013	212	23	is	be	AUX
ejpam-6013	212	24	said	say	VERB
ejpam-6013	212	25	to	to	PART
ejpam-6013	212	26	be	be	AUX
ejpam-6013	212	27	weakly	weakly	ADJ
ejpam-6013	212	28	(	(	PUNCT
ejpam-6013	212	29	τ1	τ1	NOUN
ejpam-6013	212	30	,	,	PUNCT
ejpam-6013	212	31	τ2)continuous	τ2)continuous	ADJ
ejpam-6013	212	32	at	at	ADP
ejpam-6013	212	33	a	a	DET
ejpam-6013	212	34	point	point	NOUN
ejpam-6013	212	35	x	x	SYM
ejpam-6013	212	36	∈	∈	NOUN
ejpam-6013	212	37	x	x	PUNCT
ejpam-6013	212	38	if	if	SCONJ
ejpam-6013	212	39	for	for	ADP
ejpam-6013	212	40	each	each	DET
ejpam-6013	212	41	σ1σ2	σ1σ2	VERB
ejpam-6013	212	42	-	-	ADJ
ejpam-6013	212	43	open	open	ADJ
ejpam-6013	212	44	set	set	NOUN
ejpam-6013	212	45	v	v	NOUN
ejpam-6013	212	46	of	of	ADP
ejpam-6013	212	47	y	y	NOUN
ejpam-6013	212	48	containing	contain	VERB
ejpam-6013	212	49	f(x	f(x	PROPN
ejpam-6013	212	50	)	)	PUNCT
ejpam-6013	212	51	,	,	PUNCT
ejpam-6013	212	52	there	there	PRON
ejpam-6013	212	53	exists	exist	VERB
ejpam-6013	212	54	a	a	DET
ejpam-6013	212	55	τ1τ2	τ1τ2	NOUN
ejpam-6013	212	56	-	-	ADJ
ejpam-6013	212	57	open	open	ADJ
ejpam-6013	212	58	set	set	ADJ
ejpam-6013	212	59	u	u	NOUN
ejpam-6013	212	60	of	of	ADP
ejpam-6013	212	61	x	x	PUNCT
ejpam-6013	212	62	containing	contain	VERB
ejpam-6013	212	63	x	x	PUNCT
ejpam-6013	212	64	such	such	ADJ
ejpam-6013	212	65	that	that	DET
ejpam-6013	212	66	f(u	f(u	PROPN
ejpam-6013	212	67	)	)	PUNCT
ejpam-6013	212	68	⊆	⊆	NUM
ejpam-6013	212	69	σ1σ2	σ1σ2	NOUN
ejpam-6013	212	70	-	-	NUM
ejpam-6013	212	71	cl(v	cl(v	NOUN
ejpam-6013	212	72	)	)	PUNCT
ejpam-6013	212	73	.	.	PUNCT
ejpam-6013	213	1	a	a	DET
ejpam-6013	213	2	function	function	NOUN
ejpam-6013	213	3	f	f	NOUN
ejpam-6013	213	4	:	:	PUNCT
ejpam-6013	213	5	(	(	PUNCT
ejpam-6013	213	6	x	x	NOUN
ejpam-6013	213	7	,	,	PUNCT
ejpam-6013	213	8	τ1	τ1	NOUN
ejpam-6013	213	9	,	,	PUNCT
ejpam-6013	213	10	τ2	τ2	NOUN
ejpam-6013	213	11	)	)	PUNCT
ejpam-6013	213	12	→	→	SYM
ejpam-6013	213	13	(	(	PUNCT
ejpam-6013	213	14	y	y	PROPN
ejpam-6013	213	15	,	,	PUNCT
ejpam-6013	213	16	σ1	σ1	PROPN
ejpam-6013	213	17	,	,	PUNCT
ejpam-6013	213	18	σ2	σ2	PROPN
ejpam-6013	213	19	)	)	PUNCT
ejpam-6013	213	20	is	be	AUX
ejpam-6013	213	21	said	say	VERB
ejpam-6013	213	22	to	to	PART
ejpam-6013	213	23	be	be	AUX
ejpam-6013	213	24	weakly	weakly	ADJ
ejpam-6013	213	25	(	(	PUNCT
ejpam-6013	213	26	τ1	τ1	NOUN
ejpam-6013	213	27	,	,	PUNCT
ejpam-6013	213	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	213	29	if	if	SCONJ
ejpam-6013	213	30	f	f	PROPN
ejpam-6013	213	31	has	have	VERB
ejpam-6013	213	32	this	this	DET
ejpam-6013	213	33	property	property	NOUN
ejpam-6013	213	34	at	at	ADP
ejpam-6013	213	35	each	each	DET
ejpam-6013	213	36	point	point	NOUN
ejpam-6013	213	37	of	of	ADP
ejpam-6013	213	38	x.	x.	PROPN
ejpam-6013	213	39	lemma	lemma	PROPN
ejpam-6013	213	40	5	5	NUM
ejpam-6013	213	41	.	.	PUNCT
ejpam-6013	214	1	[	[	X
ejpam-6013	214	2	24	24	NUM
ejpam-6013	214	3	]	]	PUNCT
ejpam-6013	214	4	for	for	ADP
ejpam-6013	214	5	a	a	DET
ejpam-6013	214	6	function	function	NOUN
ejpam-6013	214	7	(	(	PUNCT
ejpam-6013	214	8	x	x	NOUN
ejpam-6013	214	9	,	,	PUNCT
ejpam-6013	214	10	τ1	τ1	NOUN
ejpam-6013	214	11	,	,	PUNCT
ejpam-6013	214	12	τ2	τ2	NOUN
ejpam-6013	214	13	)	)	PUNCT
ejpam-6013	214	14	→	→	SYM
ejpam-6013	214	15	(	(	PUNCT
ejpam-6013	214	16	y	y	PROPN
ejpam-6013	214	17	,	,	PUNCT
ejpam-6013	214	18	σ1	σ1	PROPN
ejpam-6013	214	19	,	,	PUNCT
ejpam-6013	214	20	σ2	σ2	NOUN
ejpam-6013	214	21	)	)	PUNCT
ejpam-6013	214	22	,	,	PUNCT
ejpam-6013	214	23	the	the	DET
ejpam-6013	214	24	following	follow	VERB
ejpam-6013	214	25	properties	property	NOUN
ejpam-6013	214	26	are	be	AUX
ejpam-6013	214	27	equivalent	equivalent	ADJ
ejpam-6013	214	28	:	:	PUNCT
ejpam-6013	214	29	(	(	PUNCT
ejpam-6013	214	30	1	1	X
ejpam-6013	214	31	)	)	PUNCT
ejpam-6013	214	32	f	f	PROPN
ejpam-6013	214	33	is	be	AUX
ejpam-6013	214	34	weakly	weakly	ADJ
ejpam-6013	214	35	(	(	PUNCT
ejpam-6013	214	36	τ1	τ1	NOUN
ejpam-6013	214	37	,	,	PUNCT
ejpam-6013	214	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	214	39	;	;	PUNCT
ejpam-6013	214	40	(	(	PUNCT
ejpam-6013	214	41	2	2	X
ejpam-6013	214	42	)	)	PUNCT
ejpam-6013	214	43	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	214	44	-	-	PUNCT
ejpam-6013	214	45	cl(a	cl(a	NUM
ejpam-6013	214	46	)	)	PUNCT
ejpam-6013	214	47	)	)	PUNCT
ejpam-6013	215	1	⊆	⊆	NUM
ejpam-6013	215	2	(	(	PUNCT
ejpam-6013	215	3	σ1	σ1	PROPN
ejpam-6013	215	4	,	,	PUNCT
ejpam-6013	215	5	σ2)θ	σ2)θ	NOUN
ejpam-6013	215	6	-	-	PUNCT
ejpam-6013	215	7	cl(f(a	cl(f(a	NOUN
ejpam-6013	215	8	)	)	PUNCT
ejpam-6013	215	9	)	)	PUNCT
ejpam-6013	215	10	for	for	ADP
ejpam-6013	215	11	every	every	DET
ejpam-6013	215	12	subset	subset	NOUN
ejpam-6013	215	13	a	a	PRON
ejpam-6013	215	14	of	of	ADP
ejpam-6013	215	15	x	x	PRON
ejpam-6013	215	16	;	;	PUNCT
ejpam-6013	215	17	(	(	PUNCT
ejpam-6013	215	18	3	3	X
ejpam-6013	215	19	)	)	PUNCT
ejpam-6013	215	20	τ1τ2	τ1τ2	NOUN
ejpam-6013	215	21	-	-	NOUN
ejpam-6013	215	22	cl(f	cl(f	NOUN
ejpam-6013	215	23	−1(b	−1(b	NOUN
ejpam-6013	215	24	)	)	PUNCT
ejpam-6013	215	25	)	)	PUNCT
ejpam-6013	216	1	⊆	⊆	NUM
ejpam-6013	216	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6013	216	3	,	,	PUNCT
ejpam-6013	216	4	σ2)θ	σ2)θ	NOUN
ejpam-6013	216	5	-	-	PUNCT
ejpam-6013	216	6	cl(b	cl(b	NOUN
ejpam-6013	216	7	)	)	PUNCT
ejpam-6013	216	8	)	)	PUNCT
ejpam-6013	216	9	for	for	ADP
ejpam-6013	216	10	every	every	DET
ejpam-6013	216	11	subset	subset	NOUN
ejpam-6013	216	12	b	b	PROPN
ejpam-6013	216	13	of	of	ADP
ejpam-6013	216	14	y	y	PROPN
ejpam-6013	216	15	.	.	PUNCT
ejpam-6013	217	1	recall	recall	VERB
ejpam-6013	217	2	that	that	SCONJ
ejpam-6013	217	3	a	a	DET
ejpam-6013	217	4	bitopological	bitopological	ADJ
ejpam-6013	217	5	space	space	NOUN
ejpam-6013	217	6	(	(	PUNCT
ejpam-6013	217	7	x	x	NOUN
ejpam-6013	217	8	,	,	PUNCT
ejpam-6013	217	9	τ1	τ1	NOUN
ejpam-6013	217	10	,	,	PUNCT
ejpam-6013	217	11	τ2	τ2	NOUN
ejpam-6013	217	12	)	)	PUNCT
ejpam-6013	217	13	is	be	AUX
ejpam-6013	217	14	said	say	VERB
ejpam-6013	217	15	to	to	PART
ejpam-6013	217	16	be	be	AUX
ejpam-6013	217	17	(	(	PUNCT
ejpam-6013	217	18	τ1	τ1	NOUN
ejpam-6013	217	19	,	,	PUNCT
ejpam-6013	218	1	τ2)-regular	τ2)-regular	ADJ
ejpam-6013	218	2	[	[	X
ejpam-6013	218	3	30	30	NUM
ejpam-6013	218	4	]	]	X
ejpam-6013	218	5	if	if	SCONJ
ejpam-6013	218	6	for	for	ADP
ejpam-6013	218	7	each	each	DET
ejpam-6013	218	8	τ1τ2	τ1τ2	ADJ
ejpam-6013	218	9	-	-	ADJ
ejpam-6013	218	10	closed	closed	ADJ
ejpam-6013	218	11	set	set	VERB
ejpam-6013	218	12	f	f	NOUN
ejpam-6013	218	13	and	and	CCONJ
ejpam-6013	218	14	each	each	DET
ejpam-6013	218	15	x	x	PROPN
ejpam-6013	218	16	̸∈	̸∈	PROPN
ejpam-6013	218	17	f	f	PROPN
ejpam-6013	218	18	,	,	PUNCT
ejpam-6013	218	19	there	there	PRON
ejpam-6013	218	20	exist	exist	VERB
ejpam-6013	218	21	disjoint	disjoint	ADJ
ejpam-6013	218	22	τ1τ2	τ1τ2	ADJ
ejpam-6013	218	23	-	-	ADJ
ejpam-6013	218	24	open	open	ADJ
ejpam-6013	218	25	sets	set	NOUN
ejpam-6013	218	26	u	u	NOUN
ejpam-6013	218	27	and	and	CCONJ
ejpam-6013	218	28	v	v	ADP
ejpam-6013	218	29	such	such	ADJ
ejpam-6013	218	30	that	that	SCONJ
ejpam-6013	218	31	x	x	SYM
ejpam-6013	218	32	∈	∈	PROPN
ejpam-6013	218	33	u	u	NOUN
ejpam-6013	218	34	and	and	CCONJ
ejpam-6013	218	35	f	f	PROPN
ejpam-6013	218	36	⊆	⊆	NUM
ejpam-6013	218	37	v	v	NOUN
ejpam-6013	218	38	.	.	PUNCT
ejpam-6013	219	1	lemma	lemma	PROPN
ejpam-6013	219	2	6	6	NUM
ejpam-6013	219	3	.	.	PUNCT
ejpam-6013	220	1	[	[	X
ejpam-6013	220	2	31	31	NUM
ejpam-6013	220	3	]	]	PUNCT
ejpam-6013	220	4	a	a	DET
ejpam-6013	220	5	bitopological	bitopological	ADJ
ejpam-6013	220	6	space	space	NOUN
ejpam-6013	220	7	(	(	PUNCT
ejpam-6013	220	8	x	x	NOUN
ejpam-6013	220	9	,	,	PUNCT
ejpam-6013	220	10	τ1	τ1	NOUN
ejpam-6013	220	11	,	,	PUNCT
ejpam-6013	220	12	τ2	τ2	NOUN
ejpam-6013	220	13	)	)	PUNCT
ejpam-6013	220	14	is	be	AUX
ejpam-6013	220	15	(	(	PUNCT
ejpam-6013	220	16	τ1	τ1	NOUN
ejpam-6013	220	17	,	,	PUNCT
ejpam-6013	220	18	τ2)-regular	τ2)-regular	ADJ
ejpam-6013	220	19	if	if	SCONJ
ejpam-6013	220	20	and	and	CCONJ
ejpam-6013	220	21	only	only	ADV
ejpam-6013	220	22	if	if	SCONJ
ejpam-6013	220	23	for	for	ADP
ejpam-6013	220	24	each	each	DET
ejpam-6013	220	25	x	x	SYM
ejpam-6013	220	26	∈	∈	PROPN
ejpam-6013	220	27	x	x	X
ejpam-6013	220	28	and	and	CCONJ
ejpam-6013	220	29	each	each	DET
ejpam-6013	220	30	τ1τ2	τ1τ2	ADJ
ejpam-6013	220	31	-	-	ADJ
ejpam-6013	220	32	open	open	ADJ
ejpam-6013	220	33	set	set	NOUN
ejpam-6013	220	34	u	u	NOUN
ejpam-6013	220	35	containing	contain	VERB
ejpam-6013	220	36	x	x	PRON
ejpam-6013	220	37	,	,	PUNCT
ejpam-6013	220	38	there	there	PRON
ejpam-6013	220	39	exists	exist	VERB
ejpam-6013	220	40	a	a	DET
ejpam-6013	220	41	τ1τ2	τ1τ2	NOUN
ejpam-6013	220	42	-	-	ADJ
ejpam-6013	220	43	open	open	ADJ
ejpam-6013	220	44	set	set	VERB
ejpam-6013	220	45	v	v	ADP
ejpam-6013	220	46	such	such	ADJ
ejpam-6013	220	47	that	that	SCONJ
ejpam-6013	220	48	x	x	SYM
ejpam-6013	220	49	∈	∈	NOUN
ejpam-6013	220	50	v	v	ADP
ejpam-6013	220	51	⊆	⊆	NUM
ejpam-6013	220	52	τ1τ2	τ1τ2	NOUN
ejpam-6013	220	53	-	-	NOUN
ejpam-6013	220	54	cl(v	cl(v	X
ejpam-6013	220	55	)	)	PUNCT
ejpam-6013	220	56	⊆	⊆	NUM
ejpam-6013	220	57	u	u	NOUN
ejpam-6013	220	58	.	.	PUNCT
ejpam-6013	221	1	lemma	lemma	PROPN
ejpam-6013	221	2	7	7	NUM
ejpam-6013	221	3	.	.	PUNCT
ejpam-6013	222	1	[	[	X
ejpam-6013	222	2	31	31	NUM
ejpam-6013	222	3	]	]	X
ejpam-6013	222	4	let	let	VERB
ejpam-6013	222	5	(	(	PUNCT
ejpam-6013	222	6	x	x	NOUN
ejpam-6013	222	7	,	,	PUNCT
ejpam-6013	222	8	τ1	τ1	NOUN
ejpam-6013	222	9	,	,	PUNCT
ejpam-6013	222	10	τ2	τ2	PROPN
ejpam-6013	222	11	)	)	PUNCT
ejpam-6013	222	12	be	be	VERB
ejpam-6013	222	13	a	a	DET
ejpam-6013	222	14	(	(	PUNCT
ejpam-6013	222	15	τ1	τ1	NOUN
ejpam-6013	222	16	,	,	PUNCT
ejpam-6013	222	17	τ2)-regular	τ2)-regular	ADJ
ejpam-6013	222	18	space	space	NOUN
ejpam-6013	222	19	.	.	PUNCT
ejpam-6013	223	1	then	then	ADV
ejpam-6013	223	2	,	,	PUNCT
ejpam-6013	223	3	the	the	DET
ejpam-6013	223	4	following	follow	VERB
ejpam-6013	223	5	properties	property	NOUN
ejpam-6013	223	6	hold	hold	VERB
ejpam-6013	223	7	:	:	PUNCT
ejpam-6013	223	8	(	(	PUNCT
ejpam-6013	223	9	1	1	X
ejpam-6013	223	10	)	)	PUNCT
ejpam-6013	223	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	223	12	-	-	NUM
ejpam-6013	223	13	cl(a	cl(a	NUM
ejpam-6013	223	14	)	)	PUNCT
ejpam-6013	223	15	=	=	PUNCT
ejpam-6013	223	16	(	(	PUNCT
ejpam-6013	223	17	τ1	τ1	NOUN
ejpam-6013	223	18	,	,	PUNCT
ejpam-6013	223	19	τ2)θ	τ2)θ	NOUN
ejpam-6013	223	20	-	-	PUNCT
ejpam-6013	223	21	cl(a	cl(a	NUM
ejpam-6013	223	22	)	)	PUNCT
ejpam-6013	223	23	for	for	ADP
ejpam-6013	223	24	every	every	DET
ejpam-6013	223	25	subset	subset	NOUN
ejpam-6013	223	26	a	a	PRON
ejpam-6013	223	27	of	of	ADP
ejpam-6013	223	28	x.	x.	NOUN
ejpam-6013	223	29	(	(	PUNCT
ejpam-6013	223	30	2	2	NUM
ejpam-6013	223	31	)	)	PUNCT
ejpam-6013	223	32	every	every	DET
ejpam-6013	223	33	τ1τ2	τ1τ2	NOUN
ejpam-6013	223	34	-	-	ADJ
ejpam-6013	223	35	open	open	ADJ
ejpam-6013	223	36	set	set	NOUN
ejpam-6013	223	37	is	be	AUX
ejpam-6013	223	38	(	(	PUNCT
ejpam-6013	223	39	τ1	τ1	NOUN
ejpam-6013	223	40	,	,	PUNCT
ejpam-6013	223	41	τ2)θ	τ2)θ	ADJ
ejpam-6013	223	42	-	-	PUNCT
ejpam-6013	223	43	open	open	ADJ
ejpam-6013	223	44	.	.	PUNCT
ejpam-6013	224	1	n.	n.	PROPN
ejpam-6013	224	2	srisarakham	srisarakham	PROPN
ejpam-6013	224	3	,	,	PUNCT
ejpam-6013	224	4	s.	s.	PROPN
ejpam-6013	224	5	sompong	sompong	PROPN
ejpam-6013	224	6	,	,	PUNCT
ejpam-6013	224	7	c.	c.	PROPN
ejpam-6013	224	8	boonpok	boonpok	PROPN
ejpam-6013	224	9	/	/	SYM
ejpam-6013	224	10	eur	eur	PROPN
ejpam-6013	224	11	.	.	PUNCT
ejpam-6013	225	1	j.	j.	PROPN
ejpam-6013	225	2	pure	pure	PROPN
ejpam-6013	225	3	appl	appl	PROPN
ejpam-6013	225	4	.	.	PROPN
ejpam-6013	225	5	math	math	PROPN
ejpam-6013	225	6	,	,	PUNCT
ejpam-6013	225	7	18	18	NUM
ejpam-6013	225	8	(	(	PUNCT
ejpam-6013	225	9	2	2	NUM
ejpam-6013	225	10	)	)	PUNCT
ejpam-6013	225	11	(	(	PUNCT
ejpam-6013	225	12	2025	2025	NUM
ejpam-6013	225	13	)	)	PUNCT
ejpam-6013	225	14	,	,	PUNCT
ejpam-6013	225	15	6013	6013	NUM
ejpam-6013	225	16	8	8	NUM
ejpam-6013	225	17	of	of	ADP
ejpam-6013	225	18	12	12	NUM
ejpam-6013	225	19	theorem	theorem	NOUN
ejpam-6013	225	20	7	7	NUM
ejpam-6013	225	21	.	.	PUNCT
ejpam-6013	226	1	if	if	SCONJ
ejpam-6013	226	2	f	f	PROPN
ejpam-6013	226	3	:	:	PUNCT
ejpam-6013	226	4	(	(	PUNCT
ejpam-6013	226	5	x	x	NOUN
ejpam-6013	226	6	,	,	PUNCT
ejpam-6013	226	7	τ1	τ1	NOUN
ejpam-6013	226	8	,	,	PUNCT
ejpam-6013	226	9	τ2	τ2	NOUN
ejpam-6013	226	10	)	)	PUNCT
ejpam-6013	226	11	→	→	SYM
ejpam-6013	226	12	(	(	PUNCT
ejpam-6013	226	13	y	y	PROPN
ejpam-6013	226	14	,	,	PUNCT
ejpam-6013	226	15	σ1	σ1	PROPN
ejpam-6013	226	16	,	,	PUNCT
ejpam-6013	226	17	σ2	σ2	NOUN
ejpam-6013	226	18	)	)	PUNCT
ejpam-6013	226	19	is	be	AUX
ejpam-6013	226	20	weakly	weakly	ADJ
ejpam-6013	226	21	(	(	PUNCT
ejpam-6013	226	22	τ1	τ1	NOUN
ejpam-6013	226	23	,	,	PUNCT
ejpam-6013	226	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	226	25	and	and	CCONJ
ejpam-6013	226	26	(	(	PUNCT
ejpam-6013	226	27	y	y	PROPN
ejpam-6013	226	28	,	,	PUNCT
ejpam-6013	226	29	σ1	σ1	PROPN
ejpam-6013	226	30	,	,	PUNCT
ejpam-6013	226	31	σ2	σ2	PROPN
ejpam-6013	226	32	)	)	PUNCT
ejpam-6013	226	33	is	be	AUX
ejpam-6013	226	34	(	(	PUNCT
ejpam-6013	226	35	σ1	σ1	NOUN
ejpam-6013	226	36	,	,	PUNCT
ejpam-6013	226	37	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	226	38	,	,	PUNCT
ejpam-6013	226	39	then	then	ADV
ejpam-6013	226	40	f	f	PROPN
ejpam-6013	226	41	is	be	AUX
ejpam-6013	226	42	r-(τ1	r-(τ1	PROPN
ejpam-6013	226	43	,	,	PUNCT
ejpam-6013	226	44	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	226	45	.	.	PUNCT
ejpam-6013	227	1	proof	proof	NOUN
ejpam-6013	227	2	.	.	PUNCT
ejpam-6013	228	1	let	let	VERB
ejpam-6013	228	2	x	x	PUNCT
ejpam-6013	228	3	∈	∈	PROPN
ejpam-6013	228	4	x	x	X
ejpam-6013	228	5	and	and	CCONJ
ejpam-6013	228	6	v	v	X
ejpam-6013	228	7	be	be	AUX
ejpam-6013	228	8	any	any	DET
ejpam-6013	228	9	σ1σ2	σ1σ2	NOUN
ejpam-6013	228	10	-	-	ADJ
ejpam-6013	228	11	open	open	ADJ
ejpam-6013	228	12	set	set	NOUN
ejpam-6013	228	13	of	of	ADP
ejpam-6013	228	14	y	y	PROPN
ejpam-6013	228	15	containing	contain	VERB
ejpam-6013	228	16	f(x	f(x	PROPN
ejpam-6013	228	17	)	)	PUNCT
ejpam-6013	228	18	.	.	PUNCT
ejpam-6013	229	1	since	since	SCONJ
ejpam-6013	229	2	(	(	PUNCT
ejpam-6013	229	3	y	y	PROPN
ejpam-6013	229	4	,	,	PUNCT
ejpam-6013	229	5	σ1	σ1	PROPN
ejpam-6013	229	6	,	,	PUNCT
ejpam-6013	229	7	σ2	σ2	PROPN
ejpam-6013	229	8	)	)	PUNCT
ejpam-6013	229	9	is	be	AUX
ejpam-6013	229	10	(	(	PUNCT
ejpam-6013	229	11	σ1	σ1	NOUN
ejpam-6013	229	12	,	,	PUNCT
ejpam-6013	229	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	229	14	,	,	PUNCT
ejpam-6013	229	15	by	by	ADP
ejpam-6013	229	16	lemma	lemma	PROPN
ejpam-6013	229	17	6	6	NUM
ejpam-6013	229	18	there	there	ADV
ejpam-6013	229	19	exists	exist	VERB
ejpam-6013	229	20	a	a	DET
ejpam-6013	229	21	σ1σ2	σ1σ2	NUM
ejpam-6013	229	22	-	-	ADJ
ejpam-6013	229	23	open	open	ADJ
ejpam-6013	229	24	set	set	NOUN
ejpam-6013	229	25	w	w	PROPN
ejpam-6013	229	26	of	of	ADP
ejpam-6013	229	27	y	y	PRON
ejpam-6013	229	28	such	such	ADJ
ejpam-6013	229	29	that	that	SCONJ
ejpam-6013	229	30	f(x	f(x	PROPN
ejpam-6013	229	31	)	)	PUNCT
ejpam-6013	229	32	∈	∈	PROPN
ejpam-6013	230	1	w	w	ADP
ejpam-6013	230	2	⊆	⊆	NUM
ejpam-6013	230	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	230	4	-	-	PUNCT
ejpam-6013	230	5	cl(w	cl(w	NOUN
ejpam-6013	230	6	)	)	PUNCT
ejpam-6013	231	1	⊆	⊆	PROPN
ejpam-6013	231	2	v.	v.	ADV
ejpam-6013	231	3	since	since	SCONJ
ejpam-6013	231	4	f	f	PROPN
ejpam-6013	231	5	is	be	AUX
ejpam-6013	231	6	weakly	weakly	ADJ
ejpam-6013	231	7	(	(	PUNCT
ejpam-6013	231	8	τ1	τ1	NOUN
ejpam-6013	231	9	,	,	PUNCT
ejpam-6013	231	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	231	11	,	,	PUNCT
ejpam-6013	231	12	there	there	PRON
ejpam-6013	231	13	exists	exist	VERB
ejpam-6013	231	14	a	a	DET
ejpam-6013	231	15	τ1τ2	τ1τ2	NOUN
ejpam-6013	231	16	-	-	ADJ
ejpam-6013	231	17	open	open	ADJ
ejpam-6013	231	18	set	set	ADJ
ejpam-6013	231	19	u	u	NOUN
ejpam-6013	231	20	of	of	ADP
ejpam-6013	231	21	x	x	PUNCT
ejpam-6013	231	22	containing	contain	VERB
ejpam-6013	231	23	x	x	PUNCT
ejpam-6013	231	24	such	such	ADJ
ejpam-6013	231	25	that	that	DET
ejpam-6013	231	26	f(u	f(u	PROPN
ejpam-6013	231	27	)	)	PUNCT
ejpam-6013	232	1	⊆	⊆	NUM
ejpam-6013	232	2	σ1σ2	σ1σ2	NOUN
ejpam-6013	232	3	-	-	PUNCT
ejpam-6013	232	4	cl(w	cl(w	NOUN
ejpam-6013	232	5	)	)	PUNCT
ejpam-6013	232	6	.	.	PUNCT
ejpam-6013	233	1	thus	thus	ADV
ejpam-6013	233	2	,	,	PUNCT
ejpam-6013	233	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	233	4	-	-	PUNCT
ejpam-6013	233	5	cl(f(u	cl(f(u	NOUN
ejpam-6013	233	6	)	)	PUNCT
ejpam-6013	233	7	)	)	PUNCT
ejpam-6013	234	1	⊆	⊆	X
ejpam-6013	234	2	σ1σ2	σ1σ2	NOUN
ejpam-6013	234	3	-	-	PUNCT
ejpam-6013	234	4	cl(w	cl(w	NOUN
ejpam-6013	234	5	)	)	PUNCT
ejpam-6013	234	6	⊆	⊆	NUM
ejpam-6013	234	7	v	v	NOUN
ejpam-6013	234	8	.	.	PUNCT
ejpam-6013	235	1	this	this	PRON
ejpam-6013	235	2	shows	show	VERB
ejpam-6013	235	3	that	that	SCONJ
ejpam-6013	235	4	f	f	PROPN
ejpam-6013	235	5	is	be	AUX
ejpam-6013	235	6	r-(τ1	r-(τ1	PROPN
ejpam-6013	235	7	,	,	PUNCT
ejpam-6013	235	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	235	9	.	.	PUNCT
ejpam-6013	236	1	definition	definition	NOUN
ejpam-6013	236	2	7	7	NUM
ejpam-6013	236	3	.	.	PUNCT
ejpam-6013	237	1	[	[	X
ejpam-6013	237	2	32	32	NUM
ejpam-6013	237	3	]	]	PUNCT
ejpam-6013	237	4	a	a	DET
ejpam-6013	237	5	function	function	NOUN
ejpam-6013	237	6	f	f	NOUN
ejpam-6013	237	7	:	:	PUNCT
ejpam-6013	237	8	(	(	PUNCT
ejpam-6013	237	9	x	x	NOUN
ejpam-6013	237	10	,	,	PUNCT
ejpam-6013	237	11	τ1	τ1	NOUN
ejpam-6013	237	12	,	,	PUNCT
ejpam-6013	237	13	τ2	τ2	NOUN
ejpam-6013	237	14	)	)	PUNCT
ejpam-6013	237	15	→	→	SYM
ejpam-6013	237	16	(	(	PUNCT
ejpam-6013	237	17	y	y	PROPN
ejpam-6013	237	18	,	,	PUNCT
ejpam-6013	237	19	σ1	σ1	PROPN
ejpam-6013	237	20	,	,	PUNCT
ejpam-6013	237	21	σ2	σ2	PROPN
ejpam-6013	237	22	)	)	PUNCT
ejpam-6013	237	23	is	be	AUX
ejpam-6013	237	24	called	call	VERB
ejpam-6013	237	25	faintly	faintly	ADV
ejpam-6013	237	26	(	(	PUNCT
ejpam-6013	237	27	τ1	τ1	NOUN
ejpam-6013	237	28	,	,	PUNCT
ejpam-6013	237	29	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	237	30	at	at	ADP
ejpam-6013	237	31	a	a	DET
ejpam-6013	237	32	point	point	NOUN
ejpam-6013	237	33	x	x	SYM
ejpam-6013	237	34	∈	∈	NOUN
ejpam-6013	237	35	x	x	INTJ
ejpam-6013	237	36	if	if	SCONJ
ejpam-6013	237	37	for	for	SCONJ
ejpam-6013	237	38	each	each	DET
ejpam-6013	237	39	(	(	PUNCT
ejpam-6013	237	40	σ1	σ1	PROPN
ejpam-6013	237	41	,	,	PUNCT
ejpam-6013	237	42	σ2)θ	σ2)θ	NOUN
ejpam-6013	237	43	-	-	PUNCT
ejpam-6013	237	44	open	open	ADJ
ejpam-6013	237	45	set	set	NOUN
ejpam-6013	237	46	v	v	NOUN
ejpam-6013	237	47	of	of	ADP
ejpam-6013	237	48	y	y	NOUN
ejpam-6013	237	49	containing	contain	VERB
ejpam-6013	237	50	f(x	f(x	PROPN
ejpam-6013	237	51	)	)	PUNCT
ejpam-6013	237	52	,	,	PUNCT
ejpam-6013	237	53	there	there	PRON
ejpam-6013	237	54	exists	exist	VERB
ejpam-6013	237	55	a	a	DET
ejpam-6013	237	56	τ1τ2open	τ1τ2open	ADJ
ejpam-6013	237	57	set	set	NOUN
ejpam-6013	237	58	u	u	NOUN
ejpam-6013	237	59	of	of	ADP
ejpam-6013	237	60	x	x	PUNCT
ejpam-6013	237	61	containing	contain	VERB
ejpam-6013	237	62	x	x	PUNCT
ejpam-6013	237	63	such	such	ADJ
ejpam-6013	237	64	that	that	DET
ejpam-6013	237	65	f(u	f(u	PROPN
ejpam-6013	237	66	)	)	PUNCT
ejpam-6013	237	67	⊆	⊆	NUM
ejpam-6013	237	68	v	v	NOUN
ejpam-6013	237	69	.	.	PUNCT
ejpam-6013	238	1	a	a	DET
ejpam-6013	238	2	function	function	NOUN
ejpam-6013	238	3	f	f	NOUN
ejpam-6013	238	4	:	:	PUNCT
ejpam-6013	238	5	(	(	PUNCT
ejpam-6013	238	6	x	x	NOUN
ejpam-6013	238	7	,	,	PUNCT
ejpam-6013	238	8	τ1	τ1	NOUN
ejpam-6013	238	9	,	,	PUNCT
ejpam-6013	238	10	τ2	τ2	NOUN
ejpam-6013	238	11	)	)	PUNCT
ejpam-6013	238	12	→	→	SYM
ejpam-6013	238	13	(	(	PUNCT
ejpam-6013	238	14	y	y	PROPN
ejpam-6013	238	15	,	,	PUNCT
ejpam-6013	238	16	σ1	σ1	PROPN
ejpam-6013	238	17	,	,	PUNCT
ejpam-6013	238	18	σ2	σ2	PROPN
ejpam-6013	238	19	)	)	PUNCT
ejpam-6013	238	20	is	be	AUX
ejpam-6013	238	21	called	call	VERB
ejpam-6013	238	22	faintly	faintly	ADV
ejpam-6013	238	23	(	(	PUNCT
ejpam-6013	238	24	τ1	τ1	NOUN
ejpam-6013	238	25	,	,	PUNCT
ejpam-6013	238	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	238	27	if	if	SCONJ
ejpam-6013	238	28	f	f	PROPN
ejpam-6013	238	29	has	have	VERB
ejpam-6013	238	30	this	this	DET
ejpam-6013	238	31	property	property	NOUN
ejpam-6013	238	32	at	at	ADP
ejpam-6013	238	33	every	every	DET
ejpam-6013	238	34	point	point	NOUN
ejpam-6013	238	35	of	of	ADP
ejpam-6013	238	36	x.	x.	PROPN
ejpam-6013	238	37	lemma	lemma	PROPN
ejpam-6013	238	38	8	8	NUM
ejpam-6013	238	39	.	.	PUNCT
ejpam-6013	239	1	[	[	X
ejpam-6013	239	2	32	32	NUM
ejpam-6013	239	3	]	]	PUNCT
ejpam-6013	239	4	for	for	ADP
ejpam-6013	239	5	a	a	DET
ejpam-6013	239	6	function	function	NOUN
ejpam-6013	239	7	f	f	NOUN
ejpam-6013	239	8	:	:	PUNCT
ejpam-6013	239	9	(	(	PUNCT
ejpam-6013	239	10	x	x	NOUN
ejpam-6013	239	11	,	,	PUNCT
ejpam-6013	239	12	τ1	τ1	NOUN
ejpam-6013	239	13	,	,	PUNCT
ejpam-6013	239	14	τ2	τ2	NOUN
ejpam-6013	239	15	)	)	PUNCT
ejpam-6013	239	16	→	→	SYM
ejpam-6013	239	17	(	(	PUNCT
ejpam-6013	239	18	y	y	PROPN
ejpam-6013	239	19	,	,	PUNCT
ejpam-6013	239	20	σ1	σ1	PROPN
ejpam-6013	239	21	,	,	PUNCT
ejpam-6013	239	22	σ2	σ2	NOUN
ejpam-6013	239	23	)	)	PUNCT
ejpam-6013	239	24	,	,	PUNCT
ejpam-6013	239	25	the	the	DET
ejpam-6013	239	26	following	follow	VERB
ejpam-6013	239	27	properties	property	NOUN
ejpam-6013	239	28	are	be	AUX
ejpam-6013	239	29	equivalent	equivalent	ADJ
ejpam-6013	239	30	:	:	PUNCT
ejpam-6013	239	31	(	(	PUNCT
ejpam-6013	239	32	1	1	X
ejpam-6013	239	33	)	)	PUNCT
ejpam-6013	239	34	f	f	PROPN
ejpam-6013	239	35	is	be	AUX
ejpam-6013	239	36	faintly	faintly	ADV
ejpam-6013	239	37	(	(	PUNCT
ejpam-6013	239	38	τ1	τ1	NOUN
ejpam-6013	239	39	,	,	PUNCT
ejpam-6013	239	40	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	239	41	;	;	PUNCT
ejpam-6013	239	42	(	(	PUNCT
ejpam-6013	239	43	2	2	X
ejpam-6013	239	44	)	)	PUNCT
ejpam-6013	239	45	f−1(v	f−1(v	NOUN
ejpam-6013	239	46	)	)	PUNCT
ejpam-6013	239	47	is	be	AUX
ejpam-6013	239	48	τ1τ2	τ1τ2	NOUN
ejpam-6013	239	49	-	-	ADJ
ejpam-6013	239	50	open	open	ADJ
ejpam-6013	239	51	in	in	ADP
ejpam-6013	239	52	x	x	PUNCT
ejpam-6013	239	53	for	for	ADP
ejpam-6013	239	54	each	each	DET
ejpam-6013	239	55	(	(	PUNCT
ejpam-6013	239	56	σ1	σ1	PROPN
ejpam-6013	239	57	,	,	PUNCT
ejpam-6013	239	58	σ2)θ	σ2)θ	NOUN
ejpam-6013	239	59	-	-	PUNCT
ejpam-6013	239	60	open	open	ADJ
ejpam-6013	239	61	set	set	NOUN
ejpam-6013	239	62	v	v	NOUN
ejpam-6013	239	63	of	of	ADP
ejpam-6013	239	64	y	y	PROPN
ejpam-6013	239	65	;	;	PUNCT
ejpam-6013	239	66	(	(	PUNCT
ejpam-6013	239	67	3	3	X
ejpam-6013	239	68	)	)	PUNCT
ejpam-6013	239	69	f−1(k	f−1(k	PROPN
ejpam-6013	239	70	)	)	PUNCT
ejpam-6013	239	71	is	be	AUX
ejpam-6013	239	72	τ1τ2	τ1τ2	NOUN
ejpam-6013	239	73	-	-	ADJ
ejpam-6013	239	74	closed	closed	ADJ
ejpam-6013	239	75	in	in	ADP
ejpam-6013	239	76	x	x	PUNCT
ejpam-6013	239	77	for	for	ADP
ejpam-6013	239	78	each	each	DET
ejpam-6013	239	79	(	(	PUNCT
ejpam-6013	239	80	σ1	σ1	PROPN
ejpam-6013	239	81	,	,	PUNCT
ejpam-6013	239	82	σ2)θ	σ2)θ	NOUN
ejpam-6013	239	83	-	-	PUNCT
ejpam-6013	239	84	closed	close	VERB
ejpam-6013	239	85	set	set	NOUN
ejpam-6013	239	86	k	k	PROPN
ejpam-6013	239	87	of	of	ADP
ejpam-6013	239	88	y	y	PROPN
ejpam-6013	239	89	.	.	PUNCT
ejpam-6013	240	1	theorem	theorem	ADJ
ejpam-6013	240	2	8	8	NUM
ejpam-6013	240	3	.	.	PUNCT
ejpam-6013	241	1	for	for	ADP
ejpam-6013	241	2	a	a	DET
ejpam-6013	241	3	function	function	NOUN
ejpam-6013	241	4	f	f	NOUN
ejpam-6013	241	5	:	:	PUNCT
ejpam-6013	241	6	(	(	PUNCT
ejpam-6013	241	7	x	x	NOUN
ejpam-6013	241	8	,	,	PUNCT
ejpam-6013	241	9	τ1	τ1	NOUN
ejpam-6013	241	10	,	,	PUNCT
ejpam-6013	241	11	τ2	τ2	NOUN
ejpam-6013	241	12	)	)	PUNCT
ejpam-6013	241	13	→	→	SYM
ejpam-6013	241	14	(	(	PUNCT
ejpam-6013	241	15	y	y	PROPN
ejpam-6013	241	16	,	,	PUNCT
ejpam-6013	241	17	σ1	σ1	PROPN
ejpam-6013	241	18	,	,	PUNCT
ejpam-6013	241	19	σ2	σ2	NOUN
ejpam-6013	241	20	)	)	PUNCT
ejpam-6013	241	21	,	,	PUNCT
ejpam-6013	241	22	where	where	SCONJ
ejpam-6013	241	23	(	(	PUNCT
ejpam-6013	241	24	y	y	PROPN
ejpam-6013	241	25	,	,	PUNCT
ejpam-6013	241	26	σ1	σ1	PROPN
ejpam-6013	241	27	,	,	PUNCT
ejpam-6013	241	28	σ2	σ2	PROPN
ejpam-6013	241	29	)	)	PUNCT
ejpam-6013	241	30	is	be	AUX
ejpam-6013	241	31	(	(	PUNCT
ejpam-6013	241	32	σ1	σ1	PROPN
ejpam-6013	241	33	,	,	PUNCT
ejpam-6013	241	34	σ2)regular	σ2)regular	PROPN
ejpam-6013	241	35	,	,	PUNCT
ejpam-6013	241	36	the	the	DET
ejpam-6013	241	37	following	follow	VERB
ejpam-6013	241	38	properties	property	NOUN
ejpam-6013	241	39	are	be	AUX
ejpam-6013	241	40	equivalent	equivalent	ADJ
ejpam-6013	241	41	:	:	PUNCT
ejpam-6013	241	42	(	(	PUNCT
ejpam-6013	241	43	1	1	X
ejpam-6013	241	44	)	)	PUNCT
ejpam-6013	241	45	f	f	PROPN
ejpam-6013	241	46	is	be	AUX
ejpam-6013	241	47	r-(τ1	r-(τ1	PROPN
ejpam-6013	241	48	,	,	PUNCT
ejpam-6013	241	49	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	241	50	;	;	PUNCT
ejpam-6013	241	51	(	(	PUNCT
ejpam-6013	241	52	2	2	X
ejpam-6013	241	53	)	)	PUNCT
ejpam-6013	241	54	f	f	PROPN
ejpam-6013	241	55	is	be	AUX
ejpam-6013	241	56	strongly	strongly	ADV
ejpam-6013	241	57	θ(τ1	θ(τ1	ADJ
ejpam-6013	241	58	,	,	PUNCT
ejpam-6013	241	59	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	241	60	;	;	PUNCT
ejpam-6013	241	61	(	(	PUNCT
ejpam-6013	241	62	3	3	X
ejpam-6013	241	63	)	)	PUNCT
ejpam-6013	241	64	f	f	PROPN
ejpam-6013	241	65	is	be	AUX
ejpam-6013	241	66	(	(	PUNCT
ejpam-6013	241	67	τ1	τ1	NOUN
ejpam-6013	241	68	,	,	PUNCT
ejpam-6013	241	69	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	241	70	;	;	PUNCT
ejpam-6013	241	71	(	(	PUNCT
ejpam-6013	241	72	4	4	X
ejpam-6013	241	73	)	)	PUNCT
ejpam-6013	241	74	f	f	PROPN
ejpam-6013	241	75	is	be	AUX
ejpam-6013	241	76	weakly	weakly	ADJ
ejpam-6013	241	77	(	(	PUNCT
ejpam-6013	241	78	τ1	τ1	NOUN
ejpam-6013	241	79	,	,	PUNCT
ejpam-6013	241	80	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	241	81	;	;	PUNCT
ejpam-6013	241	82	(	(	PUNCT
ejpam-6013	241	83	5	5	X
ejpam-6013	241	84	)	)	PUNCT
ejpam-6013	241	85	f	f	PROPN
ejpam-6013	241	86	is	be	AUX
ejpam-6013	241	87	faintly	faintly	ADV
ejpam-6013	241	88	(	(	PUNCT
ejpam-6013	241	89	τ1	τ1	NOUN
ejpam-6013	241	90	,	,	PUNCT
ejpam-6013	241	91	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	241	92	.	.	PUNCT
ejpam-6013	242	1	proof	proof	NOUN
ejpam-6013	242	2	.	.	PUNCT
ejpam-6013	243	1	(	(	PUNCT
ejpam-6013	243	2	1	1	X
ejpam-6013	243	3	)	)	PUNCT
ejpam-6013	243	4	⇒	⇒	NOUN
ejpam-6013	243	5	(	(	PUNCT
ejpam-6013	243	6	2	2	NUM
ejpam-6013	243	7	):	):	PUNCT
ejpam-6013	243	8	it	it	PRON
ejpam-6013	243	9	follows	follow	VERB
ejpam-6013	243	10	from	from	ADP
ejpam-6013	243	11	theorem	theorem	ADJ
ejpam-6013	243	12	2	2	NUM
ejpam-6013	243	13	.	.	PUNCT
ejpam-6013	243	14	(	(	PUNCT
ejpam-6013	243	15	2	2	X
ejpam-6013	243	16	)	)	PUNCT
ejpam-6013	243	17	⇒	⇒	NOUN
ejpam-6013	243	18	(	(	PUNCT
ejpam-6013	243	19	3	3	NUM
ejpam-6013	243	20	)	)	PUNCT
ejpam-6013	243	21	and	and	CCONJ
ejpam-6013	243	22	(	(	PUNCT
ejpam-6013	243	23	3	3	X
ejpam-6013	243	24	)	)	PUNCT
ejpam-6013	243	25	⇒	⇒	NOUN
ejpam-6013	243	26	(	(	PUNCT
ejpam-6013	243	27	4	4	NUM
ejpam-6013	243	28	):	):	PUNCT
ejpam-6013	243	29	the	the	DET
ejpam-6013	243	30	proofs	proof	NOUN
ejpam-6013	243	31	are	be	AUX
ejpam-6013	243	32	obvious	obvious	ADJ
ejpam-6013	243	33	.	.	PUNCT
ejpam-6013	244	1	(	(	PUNCT
ejpam-6013	244	2	4	4	X
ejpam-6013	244	3	)	)	PUNCT
ejpam-6013	244	4	⇒	⇒	NOUN
ejpam-6013	244	5	(	(	PUNCT
ejpam-6013	244	6	5	5	NUM
ejpam-6013	244	7	):	):	PUNCT
ejpam-6013	244	8	let	let	VERB
ejpam-6013	244	9	f	f	PRON
ejpam-6013	244	10	be	be	AUX
ejpam-6013	244	11	any	any	DET
ejpam-6013	244	12	θ(τ1	θ(τ1	NOUN
ejpam-6013	244	13	,	,	PUNCT
ejpam-6013	244	14	τ2)-closed	τ2)-closed	ADJ
ejpam-6013	244	15	set	set	NOUN
ejpam-6013	244	16	of	of	ADP
ejpam-6013	244	17	y	y	PROPN
ejpam-6013	244	18	.	.	PUNCT
ejpam-6013	245	1	since	since	SCONJ
ejpam-6013	245	2	f	f	PROPN
ejpam-6013	245	3	is	be	AUX
ejpam-6013	245	4	weakly	weakly	ADJ
ejpam-6013	245	5	(	(	PUNCT
ejpam-6013	245	6	τ1	τ1	NOUN
ejpam-6013	245	7	,	,	PUNCT
ejpam-6013	245	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	245	9	,	,	PUNCT
ejpam-6013	245	10	by	by	ADP
ejpam-6013	245	11	lemma	lemma	PROPN
ejpam-6013	245	12	5	5	NUM
ejpam-6013	245	13	we	we	PRON
ejpam-6013	245	14	have	have	VERB
ejpam-6013	245	15	σ1σ2	σ1σ2	NOUN
ejpam-6013	245	16	-	-	PUNCT
ejpam-6013	245	17	cl(f	cl(f	NOUN
ejpam-6013	245	18	−1(f	−1(f	NUM
ejpam-6013	245	19	)	)	PUNCT
ejpam-6013	245	20	)	)	PUNCT
ejpam-6013	246	1	⊆	⊆	NUM
ejpam-6013	246	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6013	246	3	,	,	PUNCT
ejpam-6013	246	4	σ2)θ	σ2)θ	NOUN
ejpam-6013	246	5	-	-	PUNCT
ejpam-6013	246	6	cl(f	cl(f	PROPN
ejpam-6013	246	7	)	)	PUNCT
ejpam-6013	246	8	)	)	PUNCT
ejpam-6013	247	1	=	=	SYM
ejpam-6013	247	2	f−1(f	f−1(f	PROPN
ejpam-6013	247	3	)	)	PUNCT
ejpam-6013	247	4	and	and	CCONJ
ejpam-6013	247	5	hence	hence	ADV
ejpam-6013	247	6	f−1(f	f−1(f	PROPN
ejpam-6013	247	7	)	)	PUNCT
ejpam-6013	247	8	is	be	AUX
ejpam-6013	247	9	τ1τ2	τ1τ2	NOUN
ejpam-6013	247	10	-	-	ADJ
ejpam-6013	247	11	closed	closed	ADJ
ejpam-6013	247	12	in	in	ADP
ejpam-6013	247	13	x.	x.	NOUN
ejpam-6013	247	14	thus	thus	ADV
ejpam-6013	247	15	by	by	ADP
ejpam-6013	247	16	lemma	lemma	PROPN
ejpam-6013	247	17	8	8	NUM
ejpam-6013	247	18	,	,	PUNCT
ejpam-6013	247	19	f	f	PROPN
ejpam-6013	247	20	is	be	AUX
ejpam-6013	247	21	faintly	faintly	ADV
ejpam-6013	247	22	(	(	PUNCT
ejpam-6013	247	23	τ1	τ1	NOUN
ejpam-6013	247	24	,	,	PUNCT
ejpam-6013	247	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	247	26	.	.	PUNCT
ejpam-6013	248	1	(	(	PUNCT
ejpam-6013	248	2	5	5	X
ejpam-6013	248	3	)	)	PUNCT
ejpam-6013	248	4	⇒	⇒	NOUN
ejpam-6013	248	5	(	(	PUNCT
ejpam-6013	248	6	1	1	NUM
ejpam-6013	248	7	):	):	PUNCT
ejpam-6013	248	8	let	let	VERB
ejpam-6013	248	9	x	x	PUNCT
ejpam-6013	248	10	∈	∈	PROPN
ejpam-6013	248	11	x	x	X
ejpam-6013	248	12	and	and	CCONJ
ejpam-6013	248	13	v	v	X
ejpam-6013	248	14	be	be	AUX
ejpam-6013	248	15	any	any	DET
ejpam-6013	248	16	σ1σ2	σ1σ2	NOUN
ejpam-6013	248	17	-	-	ADJ
ejpam-6013	248	18	open	open	ADJ
ejpam-6013	248	19	set	set	NOUN
ejpam-6013	248	20	of	of	ADP
ejpam-6013	248	21	y	y	PROPN
ejpam-6013	248	22	containing	contain	VERB
ejpam-6013	248	23	f(x	f(x	PROPN
ejpam-6013	248	24	)	)	PUNCT
ejpam-6013	248	25	.	.	PUNCT
ejpam-6013	249	1	since	since	SCONJ
ejpam-6013	249	2	(	(	PUNCT
ejpam-6013	249	3	y	y	PROPN
ejpam-6013	249	4	,	,	PUNCT
ejpam-6013	249	5	σ1	σ1	PROPN
ejpam-6013	249	6	,	,	PUNCT
ejpam-6013	249	7	σ2	σ2	PROPN
ejpam-6013	249	8	)	)	PUNCT
ejpam-6013	249	9	is	be	AUX
ejpam-6013	249	10	(	(	PUNCT
ejpam-6013	249	11	σ1	σ1	NOUN
ejpam-6013	249	12	,	,	PUNCT
ejpam-6013	249	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	249	14	,	,	PUNCT
ejpam-6013	249	15	by	by	ADP
ejpam-6013	249	16	lemma	lemma	PROPN
ejpam-6013	249	17	7	7	NUM
ejpam-6013	249	18	we	we	PRON
ejpam-6013	249	19	have	have	VERB
ejpam-6013	249	20	v	v	NOUN
ejpam-6013	249	21	is	be	AUX
ejpam-6013	249	22	a	a	DET
ejpam-6013	249	23	θ(τ1	θ(τ1	NOUN
ejpam-6013	249	24	,	,	PUNCT
ejpam-6013	249	25	τ2)-open	τ2)-open	ADJ
ejpam-6013	249	26	set	set	NOUN
ejpam-6013	249	27	of	of	ADP
ejpam-6013	249	28	y	y	PROPN
ejpam-6013	249	29	.	.	PUNCT
ejpam-6013	250	1	since	since	SCONJ
ejpam-6013	250	2	f	f	PROPN
ejpam-6013	250	3	is	be	AUX
ejpam-6013	250	4	faintly	faintly	ADV
ejpam-6013	250	5	(	(	PUNCT
ejpam-6013	250	6	τ1	τ1	NOUN
ejpam-6013	250	7	,	,	PUNCT
ejpam-6013	250	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	250	9	,	,	PUNCT
ejpam-6013	250	10	by	by	ADP
ejpam-6013	250	11	lemma	lemma	PROPN
ejpam-6013	250	12	8	8	NUM
ejpam-6013	250	13	we	we	PRON
ejpam-6013	250	14	have	have	VERB
ejpam-6013	250	15	f−1(v	f−1(v	PROPN
ejpam-6013	250	16	)	)	PUNCT
ejpam-6013	250	17	is	be	AUX
ejpam-6013	250	18	τ1τ2	τ1τ2	NOUN
ejpam-6013	250	19	-	-	ADJ
ejpam-6013	250	20	open	open	ADJ
ejpam-6013	250	21	in	in	ADP
ejpam-6013	250	22	x.	x.	NOUN
ejpam-6013	250	23	then	then	ADV
ejpam-6013	250	24	by	by	ADP
ejpam-6013	250	25	lemma	lemma	PROPN
ejpam-6013	250	26	3	3	NUM
ejpam-6013	250	27	,	,	PUNCT
ejpam-6013	250	28	f	f	PROPN
ejpam-6013	250	29	is	be	AUX
ejpam-6013	250	30	(	(	PUNCT
ejpam-6013	250	31	τ1	τ1	NOUN
ejpam-6013	250	32	,	,	PUNCT
ejpam-6013	250	33	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	250	34	.	.	PUNCT
ejpam-6013	251	1	n.	n.	PROPN
ejpam-6013	251	2	srisarakham	srisarakham	PROPN
ejpam-6013	251	3	,	,	PUNCT
ejpam-6013	251	4	s.	s.	PROPN
ejpam-6013	251	5	sompong	sompong	PROPN
ejpam-6013	251	6	,	,	PUNCT
ejpam-6013	251	7	c.	c.	PROPN
ejpam-6013	251	8	boonpok	boonpok	PROPN
ejpam-6013	251	9	/	/	SYM
ejpam-6013	251	10	eur	eur	PROPN
ejpam-6013	251	11	.	.	PUNCT
ejpam-6013	252	1	j.	j.	PROPN
ejpam-6013	252	2	pure	pure	PROPN
ejpam-6013	252	3	appl	appl	PROPN
ejpam-6013	252	4	.	.	PROPN
ejpam-6013	252	5	math	math	PROPN
ejpam-6013	252	6	,	,	PUNCT
ejpam-6013	252	7	18	18	NUM
ejpam-6013	252	8	(	(	PUNCT
ejpam-6013	252	9	2	2	NUM
ejpam-6013	252	10	)	)	PUNCT
ejpam-6013	252	11	(	(	PUNCT
ejpam-6013	252	12	2025	2025	NUM
ejpam-6013	252	13	)	)	PUNCT
ejpam-6013	252	14	,	,	PUNCT
ejpam-6013	252	15	6013	6013	NUM
ejpam-6013	252	16	9	9	NUM
ejpam-6013	252	17	of	of	ADP
ejpam-6013	252	18	12	12	NUM
ejpam-6013	252	19	definition	definition	NOUN
ejpam-6013	252	20	8	8	NUM
ejpam-6013	252	21	.	.	PUNCT
ejpam-6013	253	1	a	a	DET
ejpam-6013	253	2	function	function	NOUN
ejpam-6013	253	3	f	f	NOUN
ejpam-6013	253	4	:	:	PUNCT
ejpam-6013	253	5	(	(	PUNCT
ejpam-6013	253	6	x	x	NOUN
ejpam-6013	253	7	,	,	PUNCT
ejpam-6013	253	8	τ1	τ1	NOUN
ejpam-6013	253	9	,	,	PUNCT
ejpam-6013	253	10	τ2	τ2	NOUN
ejpam-6013	253	11	)	)	PUNCT
ejpam-6013	253	12	→	→	SYM
ejpam-6013	253	13	(	(	PUNCT
ejpam-6013	253	14	y	y	PROPN
ejpam-6013	253	15	,	,	PUNCT
ejpam-6013	253	16	σ1	σ1	PROPN
ejpam-6013	253	17	,	,	PUNCT
ejpam-6013	253	18	σ2	σ2	PROPN
ejpam-6013	253	19	)	)	PUNCT
ejpam-6013	253	20	is	be	AUX
ejpam-6013	253	21	called	call	VERB
ejpam-6013	253	22	(	(	PUNCT
ejpam-6013	253	23	τ1	τ1	NOUN
ejpam-6013	253	24	,	,	PUNCT
ejpam-6013	253	25	τ2)-open	τ2)-open	ADJ
ejpam-6013	253	26	if	if	SCONJ
ejpam-6013	253	27	f(v	f(v	PROPN
ejpam-6013	253	28	)	)	PUNCT
ejpam-6013	253	29	is	be	AUX
ejpam-6013	253	30	σ1σ2	σ1σ2	NOUN
ejpam-6013	253	31	-	-	ADJ
ejpam-6013	253	32	open	open	ADJ
ejpam-6013	253	33	in	in	ADP
ejpam-6013	253	34	y	y	PROPN
ejpam-6013	253	35	for	for	ADP
ejpam-6013	253	36	every	every	DET
ejpam-6013	253	37	τ1τ2	τ1τ2	ADJ
ejpam-6013	253	38	-	-	ADJ
ejpam-6013	253	39	open	open	ADJ
ejpam-6013	253	40	set	set	NOUN
ejpam-6013	253	41	v	v	NOUN
ejpam-6013	253	42	of	of	ADP
ejpam-6013	253	43	x.	x.	NOUN
ejpam-6013	253	44	theorem	theorem	VERB
ejpam-6013	253	45	9	9	NUM
ejpam-6013	253	46	.	.	PUNCT
ejpam-6013	254	1	if	if	SCONJ
ejpam-6013	254	2	f	f	PROPN
ejpam-6013	254	3	:	:	PUNCT
ejpam-6013	254	4	(	(	PUNCT
ejpam-6013	254	5	x	x	NOUN
ejpam-6013	254	6	,	,	PUNCT
ejpam-6013	254	7	τ1	τ1	NOUN
ejpam-6013	254	8	,	,	PUNCT
ejpam-6013	254	9	τ2	τ2	NOUN
ejpam-6013	254	10	)	)	PUNCT
ejpam-6013	254	11	→	→	SYM
ejpam-6013	254	12	(	(	PUNCT
ejpam-6013	254	13	y	y	PROPN
ejpam-6013	254	14	,	,	PUNCT
ejpam-6013	254	15	σ1	σ1	PROPN
ejpam-6013	254	16	,	,	PUNCT
ejpam-6013	254	17	σ2	σ2	PROPN
ejpam-6013	254	18	)	)	PUNCT
ejpam-6013	254	19	is	be	AUX
ejpam-6013	254	20	a	a	DET
ejpam-6013	254	21	r-(τ1	r-(τ1	PROPN
ejpam-6013	254	22	,	,	PUNCT
ejpam-6013	254	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	254	24	and	and	CCONJ
ejpam-6013	254	25	(	(	PUNCT
ejpam-6013	254	26	τ1	τ1	NOUN
ejpam-6013	254	27	,	,	PUNCT
ejpam-6013	254	28	τ2)-open	τ2)-open	ADJ
ejpam-6013	254	29	surjection	surjection	NOUN
ejpam-6013	254	30	,	,	PUNCT
ejpam-6013	254	31	then	then	ADV
ejpam-6013	254	32	(	(	PUNCT
ejpam-6013	254	33	y	y	PROPN
ejpam-6013	254	34	,	,	PUNCT
ejpam-6013	254	35	σ1	σ1	PROPN
ejpam-6013	254	36	,	,	PUNCT
ejpam-6013	254	37	σ2	σ2	PROPN
ejpam-6013	254	38	)	)	PUNCT
ejpam-6013	254	39	is	be	AUX
ejpam-6013	254	40	(	(	PUNCT
ejpam-6013	254	41	σ1	σ1	NOUN
ejpam-6013	254	42	,	,	PUNCT
ejpam-6013	254	43	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	254	44	.	.	PUNCT
ejpam-6013	255	1	proof	proof	NOUN
ejpam-6013	255	2	.	.	PUNCT
ejpam-6013	256	1	let	let	VERB
ejpam-6013	256	2	y	y	PROPN
ejpam-6013	256	3	∈	∈	PROPN
ejpam-6013	256	4	y	y	PROPN
ejpam-6013	256	5	and	and	CCONJ
ejpam-6013	256	6	v	v	NOUN
ejpam-6013	256	7	be	be	AUX
ejpam-6013	256	8	any	any	DET
ejpam-6013	256	9	σ1σ2	σ1σ2	NOUN
ejpam-6013	256	10	-	-	ADJ
ejpam-6013	256	11	open	open	ADJ
ejpam-6013	256	12	set	set	NOUN
ejpam-6013	256	13	of	of	ADP
ejpam-6013	256	14	y	y	PROPN
ejpam-6013	256	15	containing	contain	VERB
ejpam-6013	256	16	y.	y.	NOUN
ejpam-6013	256	17	let	let	VERB
ejpam-6013	256	18	x	x	X
ejpam-6013	256	19	∈	∈	PROPN
ejpam-6013	256	20	f−1(v	f−1(v	PROPN
ejpam-6013	256	21	)	)	PUNCT
ejpam-6013	256	22	.	.	PUNCT
ejpam-6013	257	1	since	since	SCONJ
ejpam-6013	257	2	f	f	PROPN
ejpam-6013	257	3	is	be	AUX
ejpam-6013	257	4	r-(τ1	r-(τ1	PROPN
ejpam-6013	257	5	,	,	PUNCT
ejpam-6013	257	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	257	7	,	,	PUNCT
ejpam-6013	257	8	there	there	PRON
ejpam-6013	257	9	exists	exist	VERB
ejpam-6013	257	10	a	a	DET
ejpam-6013	257	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	257	12	-	-	ADJ
ejpam-6013	257	13	open	open	ADJ
ejpam-6013	257	14	set	set	ADJ
ejpam-6013	257	15	u	u	NOUN
ejpam-6013	257	16	of	of	ADP
ejpam-6013	257	17	x	x	PUNCT
ejpam-6013	257	18	containing	contain	VERB
ejpam-6013	257	19	x	x	PUNCT
ejpam-6013	257	20	such	such	ADJ
ejpam-6013	257	21	that	that	SCONJ
ejpam-6013	257	22	σ1σ2	σ1σ2	NOUN
ejpam-6013	257	23	-	-	PUNCT
ejpam-6013	257	24	cl(f(u	cl(f(u	NOUN
ejpam-6013	257	25	)	)	PUNCT
ejpam-6013	257	26	)	)	PUNCT
ejpam-6013	258	1	⊆	⊆	NUM
ejpam-6013	258	2	v	v	NOUN
ejpam-6013	258	3	.	.	PUNCT
ejpam-6013	259	1	since	since	SCONJ
ejpam-6013	259	2	f	f	PROPN
ejpam-6013	259	3	is	be	AUX
ejpam-6013	259	4	(	(	PUNCT
ejpam-6013	259	5	τ1	τ1	NOUN
ejpam-6013	259	6	,	,	PUNCT
ejpam-6013	259	7	τ2)-open	τ2)-open	ADJ
ejpam-6013	259	8	,	,	PUNCT
ejpam-6013	259	9	we	we	PRON
ejpam-6013	259	10	have	have	AUX
ejpam-6013	259	11	f(u	f(u	PROPN
ejpam-6013	259	12	)	)	PUNCT
ejpam-6013	259	13	is	be	AUX
ejpam-6013	259	14	σ1σ2	σ1σ2	NOUN
ejpam-6013	259	15	-	-	ADJ
ejpam-6013	259	16	open	open	ADJ
ejpam-6013	259	17	in	in	ADP
ejpam-6013	259	18	y	y	PROPN
ejpam-6013	259	19	and	and	CCONJ
ejpam-6013	259	20	hence	hence	ADV
ejpam-6013	259	21	y	y	PROPN
ejpam-6013	259	22	∈	∈	PROPN
ejpam-6013	259	23	f(u	f(u	PROPN
ejpam-6013	259	24	)	)	PUNCT
ejpam-6013	259	25	⊆	⊆	NUM
ejpam-6013	259	26	σ1σ2	σ1σ2	X
ejpam-6013	259	27	-	-	PUNCT
ejpam-6013	259	28	cl(f(u	cl(f(u	NOUN
ejpam-6013	259	29	)	)	PUNCT
ejpam-6013	259	30	)	)	PUNCT
ejpam-6013	260	1	⊆	⊆	NUM
ejpam-6013	260	2	v	v	NOUN
ejpam-6013	260	3	.	.	PUNCT
ejpam-6013	261	1	it	it	PRON
ejpam-6013	261	2	follows	follow	VERB
ejpam-6013	261	3	from	from	ADP
ejpam-6013	261	4	lemma	lemma	PROPN
ejpam-6013	261	5	6	6	NUM
ejpam-6013	261	6	that	that	PRON
ejpam-6013	261	7	(	(	PUNCT
ejpam-6013	261	8	y	y	PROPN
ejpam-6013	261	9	,	,	PUNCT
ejpam-6013	261	10	σ1	σ1	PROPN
ejpam-6013	261	11	,	,	PUNCT
ejpam-6013	261	12	σ2	σ2	PROPN
ejpam-6013	261	13	)	)	PUNCT
ejpam-6013	261	14	is	be	AUX
ejpam-6013	261	15	(	(	PUNCT
ejpam-6013	261	16	σ1	σ1	NOUN
ejpam-6013	261	17	,	,	PUNCT
ejpam-6013	261	18	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	261	19	.	.	PUNCT
ejpam-6013	261	20	corollary	corollary	ADJ
ejpam-6013	261	21	1	1	NUM
ejpam-6013	261	22	.	.	PUNCT
ejpam-6013	262	1	if	if	SCONJ
ejpam-6013	262	2	f	f	PROPN
ejpam-6013	262	3	:	:	PUNCT
ejpam-6013	262	4	(	(	PUNCT
ejpam-6013	262	5	x	x	NOUN
ejpam-6013	262	6	,	,	PUNCT
ejpam-6013	262	7	τ1	τ1	NOUN
ejpam-6013	262	8	,	,	PUNCT
ejpam-6013	262	9	τ2	τ2	NOUN
ejpam-6013	262	10	)	)	PUNCT
ejpam-6013	262	11	→	→	SYM
ejpam-6013	262	12	(	(	PUNCT
ejpam-6013	262	13	y	y	PROPN
ejpam-6013	262	14	,	,	PUNCT
ejpam-6013	262	15	σ1	σ1	PROPN
ejpam-6013	262	16	,	,	PUNCT
ejpam-6013	262	17	σ2	σ2	PROPN
ejpam-6013	262	18	)	)	PUNCT
ejpam-6013	262	19	is	be	AUX
ejpam-6013	262	20	a	a	DET
ejpam-6013	262	21	(	(	PUNCT
ejpam-6013	262	22	τ1	τ1	NOUN
ejpam-6013	262	23	,	,	PUNCT
ejpam-6013	262	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	262	25	and	and	CCONJ
ejpam-6013	262	26	(	(	PUNCT
ejpam-6013	262	27	τ1	τ1	NOUN
ejpam-6013	262	28	,	,	PUNCT
ejpam-6013	262	29	τ2)-open	τ2)-open	ADJ
ejpam-6013	262	30	surjection	surjection	NOUN
ejpam-6013	262	31	,	,	PUNCT
ejpam-6013	262	32	then	then	ADV
ejpam-6013	262	33	f	f	PROPN
ejpam-6013	262	34	is	be	AUX
ejpam-6013	262	35	r-(τ1	r-(τ1	PROPN
ejpam-6013	262	36	,	,	PUNCT
ejpam-6013	262	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	262	38	if	if	SCONJ
ejpam-6013	262	39	and	and	CCONJ
ejpam-6013	262	40	only	only	ADV
ejpam-6013	262	41	if	if	SCONJ
ejpam-6013	262	42	(	(	PUNCT
ejpam-6013	262	43	y	y	PROPN
ejpam-6013	262	44	,	,	PUNCT
ejpam-6013	262	45	σ1	σ1	PROPN
ejpam-6013	262	46	,	,	PUNCT
ejpam-6013	262	47	σ2	σ2	PROPN
ejpam-6013	262	48	)	)	PUNCT
ejpam-6013	262	49	is	be	AUX
ejpam-6013	262	50	(	(	PUNCT
ejpam-6013	262	51	σ1	σ1	NOUN
ejpam-6013	262	52	,	,	PUNCT
ejpam-6013	262	53	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	262	54	.	.	PUNCT
ejpam-6013	262	55	proof	proof	NOUN
ejpam-6013	262	56	.	.	PUNCT
ejpam-6013	263	1	this	this	PRON
ejpam-6013	263	2	is	be	AUX
ejpam-6013	263	3	an	an	DET
ejpam-6013	263	4	immediate	immediate	ADJ
ejpam-6013	263	5	consequence	consequence	NOUN
ejpam-6013	263	6	of	of	ADP
ejpam-6013	263	7	theorem	theorem	ADJ
ejpam-6013	263	8	7	7	NUM
ejpam-6013	263	9	and	and	CCONJ
ejpam-6013	263	10	theorem	theorem	VERB
ejpam-6013	263	11	9	9	NUM
ejpam-6013	263	12	.	.	PUNCT
ejpam-6013	263	13	lemma	lemma	PROPN
ejpam-6013	263	14	9	9	NUM
ejpam-6013	263	15	.	.	PUNCT
ejpam-6013	264	1	[	[	X
ejpam-6013	264	2	33	33	NUM
ejpam-6013	264	3	]	]	PUNCT
ejpam-6013	264	4	let	let	VERB
ejpam-6013	264	5	(	(	PUNCT
ejpam-6013	264	6	x	x	NOUN
ejpam-6013	264	7	,	,	PUNCT
ejpam-6013	264	8	τ1	τ1	NOUN
ejpam-6013	264	9	,	,	PUNCT
ejpam-6013	264	10	τ2	τ2	PROPN
ejpam-6013	264	11	)	)	PUNCT
ejpam-6013	264	12	be	be	AUX
ejpam-6013	264	13	(	(	PUNCT
ejpam-6013	264	14	τ1	τ1	NOUN
ejpam-6013	264	15	,	,	PUNCT
ejpam-6013	264	16	τ2)-regular	τ2)-regular	PROPN
ejpam-6013	264	17	.	.	PUNCT
ejpam-6013	265	1	then	then	ADV
ejpam-6013	265	2	,	,	PUNCT
ejpam-6013	265	3	a	a	DET
ejpam-6013	265	4	function	function	NOUN
ejpam-6013	265	5	f	f	NOUN
ejpam-6013	265	6	:	:	PUNCT
ejpam-6013	265	7	(	(	PUNCT
ejpam-6013	265	8	x	x	NOUN
ejpam-6013	265	9	,	,	PUNCT
ejpam-6013	265	10	τ1	τ1	NOUN
ejpam-6013	265	11	,	,	PUNCT
ejpam-6013	265	12	τ2	τ2	NOUN
ejpam-6013	265	13	)	)	PUNCT
ejpam-6013	265	14	→	→	SYM
ejpam-6013	265	15	(	(	PUNCT
ejpam-6013	265	16	y	y	PROPN
ejpam-6013	265	17	,	,	PUNCT
ejpam-6013	265	18	σ1	σ1	PROPN
ejpam-6013	265	19	,	,	PUNCT
ejpam-6013	265	20	σ2	σ2	PROPN
ejpam-6013	265	21	)	)	PUNCT
ejpam-6013	265	22	is	be	AUX
ejpam-6013	265	23	strongly	strongly	ADV
ejpam-6013	265	24	θ(τ1	θ(τ1	ADJ
ejpam-6013	265	25	,	,	PUNCT
ejpam-6013	265	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	265	27	if	if	SCONJ
ejpam-6013	265	28	and	and	CCONJ
ejpam-6013	265	29	only	only	ADV
ejpam-6013	265	30	if	if	SCONJ
ejpam-6013	265	31	f	f	PROPN
ejpam-6013	265	32	is	be	AUX
ejpam-6013	265	33	(	(	PUNCT
ejpam-6013	265	34	τ1	τ1	NOUN
ejpam-6013	265	35	,	,	PUNCT
ejpam-6013	265	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	265	37	.	.	PUNCT
ejpam-6013	266	1	theorem	theorem	VERB
ejpam-6013	266	2	10	10	NUM
ejpam-6013	266	3	.	.	PUNCT
ejpam-6013	267	1	if	if	SCONJ
ejpam-6013	267	2	f	f	PROPN
ejpam-6013	267	3	:	:	PUNCT
ejpam-6013	267	4	(	(	PUNCT
ejpam-6013	267	5	x	x	NOUN
ejpam-6013	267	6	,	,	PUNCT
ejpam-6013	267	7	τ1	τ1	NOUN
ejpam-6013	267	8	,	,	PUNCT
ejpam-6013	267	9	τ2	τ2	NOUN
ejpam-6013	267	10	)	)	PUNCT
ejpam-6013	267	11	→	→	SYM
ejpam-6013	267	12	(	(	PUNCT
ejpam-6013	267	13	y	y	PROPN
ejpam-6013	267	14	,	,	PUNCT
ejpam-6013	267	15	σ1	σ1	PROPN
ejpam-6013	267	16	,	,	PUNCT
ejpam-6013	267	17	σ2	σ2	PROPN
ejpam-6013	267	18	)	)	PUNCT
ejpam-6013	267	19	is	be	AUX
ejpam-6013	267	20	a	a	DET
ejpam-6013	267	21	(	(	PUNCT
ejpam-6013	267	22	τ1	τ1	NOUN
ejpam-6013	267	23	,	,	PUNCT
ejpam-6013	267	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	267	25	,	,	PUNCT
ejpam-6013	267	26	(	(	PUNCT
ejpam-6013	267	27	τ1	τ1	NOUN
ejpam-6013	267	28	,	,	PUNCT
ejpam-6013	267	29	τ2)-open	τ2)-open	ADJ
ejpam-6013	267	30	and	and	CCONJ
ejpam-6013	267	31	weakly	weakly	ADJ
ejpam-6013	267	32	(	(	PUNCT
ejpam-6013	267	33	τ1	τ1	NOUN
ejpam-6013	267	34	,	,	PUNCT
ejpam-6013	267	35	τ2)-closed	τ2)-closed	ADJ
ejpam-6013	267	36	surjection	surjection	NOUN
ejpam-6013	267	37	and	and	CCONJ
ejpam-6013	267	38	(	(	PUNCT
ejpam-6013	267	39	x	x	NOUN
ejpam-6013	267	40	,	,	PUNCT
ejpam-6013	267	41	τ1	τ1	NOUN
ejpam-6013	267	42	,	,	PUNCT
ejpam-6013	267	43	τ2	τ2	NOUN
ejpam-6013	267	44	)	)	PUNCT
ejpam-6013	267	45	is	be	AUX
ejpam-6013	267	46	(	(	PUNCT
ejpam-6013	267	47	τ1	τ1	NOUN
ejpam-6013	267	48	,	,	PUNCT
ejpam-6013	267	49	τ2)-regular	τ2)-regular	ADJ
ejpam-6013	267	50	,	,	PUNCT
ejpam-6013	267	51	then	then	ADV
ejpam-6013	267	52	(	(	PUNCT
ejpam-6013	267	53	y	y	PROPN
ejpam-6013	267	54	,	,	PUNCT
ejpam-6013	267	55	σ1	σ1	PROPN
ejpam-6013	267	56	,	,	PUNCT
ejpam-6013	267	57	σ2	σ2	PROPN
ejpam-6013	267	58	)	)	PUNCT
ejpam-6013	267	59	is	be	AUX
ejpam-6013	267	60	(	(	PUNCT
ejpam-6013	267	61	σ1	σ1	PROPN
ejpam-6013	267	62	,	,	PUNCT
ejpam-6013	267	63	σ2)regular	σ2)regular	PROPN
ejpam-6013	267	64	.	.	PUNCT
ejpam-6013	268	1	proof	proof	NOUN
ejpam-6013	268	2	.	.	PUNCT
ejpam-6013	269	1	since	since	SCONJ
ejpam-6013	269	2	f	f	PROPN
ejpam-6013	269	3	is	be	AUX
ejpam-6013	269	4	(	(	PUNCT
ejpam-6013	269	5	τ1	τ1	NOUN
ejpam-6013	269	6	,	,	PUNCT
ejpam-6013	269	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	269	8	and	and	CCONJ
ejpam-6013	269	9	(	(	PUNCT
ejpam-6013	269	10	x	x	NOUN
ejpam-6013	269	11	,	,	PUNCT
ejpam-6013	269	12	τ1	τ1	NOUN
ejpam-6013	269	13	,	,	PUNCT
ejpam-6013	269	14	τ2	τ2	NOUN
ejpam-6013	269	15	)	)	PUNCT
ejpam-6013	269	16	is	be	AUX
ejpam-6013	269	17	(	(	PUNCT
ejpam-6013	269	18	τ1	τ1	NOUN
ejpam-6013	269	19	,	,	PUNCT
ejpam-6013	269	20	τ2)-regular	τ2)-regular	ADJ
ejpam-6013	269	21	,	,	PUNCT
ejpam-6013	269	22	by	by	ADP
ejpam-6013	269	23	lemma	lemma	PROPN
ejpam-6013	269	24	9	9	NUM
ejpam-6013	269	25	we	we	PRON
ejpam-6013	269	26	have	have	VERB
ejpam-6013	269	27	f	f	PROPN
ejpam-6013	269	28	is	be	AUX
ejpam-6013	269	29	strongly	strongly	ADV
ejpam-6013	269	30	θ(τ1	θ(τ1	ADJ
ejpam-6013	269	31	,	,	PUNCT
ejpam-6013	269	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	269	33	.	.	PUNCT
ejpam-6013	270	1	furthermore	furthermore	ADV
ejpam-6013	270	2	,	,	PUNCT
ejpam-6013	270	3	since	since	SCONJ
ejpam-6013	270	4	f	f	PROPN
ejpam-6013	270	5	is	be	AUX
ejpam-6013	270	6	weakly	weakly	ADJ
ejpam-6013	270	7	(	(	PUNCT
ejpam-6013	270	8	τ1	τ1	NOUN
ejpam-6013	270	9	,	,	PUNCT
ejpam-6013	270	10	τ2)-closed	τ2)-close	VERB
ejpam-6013	270	11	,	,	PUNCT
ejpam-6013	270	12	by	by	ADP
ejpam-6013	270	13	theorem	theorem	NOUN
ejpam-6013	270	14	4	4	NUM
ejpam-6013	270	15	we	we	PRON
ejpam-6013	270	16	have	have	VERB
ejpam-6013	270	17	f	f	PROPN
ejpam-6013	270	18	is	be	AUX
ejpam-6013	270	19	r-(τ1	r-(τ1	PROPN
ejpam-6013	270	20	,	,	PUNCT
ejpam-6013	270	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	270	22	.	.	PUNCT
ejpam-6013	271	1	it	it	PRON
ejpam-6013	271	2	follows	follow	VERB
ejpam-6013	271	3	from	from	ADP
ejpam-6013	271	4	theorem	theorem	ADJ
ejpam-6013	271	5	9	9	NUM
ejpam-6013	271	6	that	that	PRON
ejpam-6013	271	7	(	(	PUNCT
ejpam-6013	271	8	y	y	PROPN
ejpam-6013	271	9	,	,	PUNCT
ejpam-6013	271	10	σ1	σ1	PROPN
ejpam-6013	271	11	,	,	PUNCT
ejpam-6013	271	12	σ2	σ2	PROPN
ejpam-6013	271	13	)	)	PUNCT
ejpam-6013	271	14	is	be	AUX
ejpam-6013	271	15	(	(	PUNCT
ejpam-6013	271	16	σ1	σ1	NOUN
ejpam-6013	271	17	,	,	PUNCT
ejpam-6013	271	18	σ2)-regular	σ2)-regular	ADJ
ejpam-6013	271	19	.	.	PUNCT
ejpam-6013	271	20	definition	definition	NOUN
ejpam-6013	271	21	9	9	NUM
ejpam-6013	271	22	.	.	PUNCT
ejpam-6013	272	1	[	[	X
ejpam-6013	272	2	29	29	NUM
ejpam-6013	272	3	]	]	PUNCT
ejpam-6013	272	4	a	a	DET
ejpam-6013	272	5	function	function	NOUN
ejpam-6013	272	6	f	f	NOUN
ejpam-6013	272	7	:	:	PUNCT
ejpam-6013	272	8	(	(	PUNCT
ejpam-6013	272	9	x	x	NOUN
ejpam-6013	272	10	,	,	PUNCT
ejpam-6013	272	11	τ1	τ1	NOUN
ejpam-6013	272	12	,	,	PUNCT
ejpam-6013	272	13	τ2	τ2	NOUN
ejpam-6013	272	14	)	)	PUNCT
ejpam-6013	272	15	→	→	SYM
ejpam-6013	272	16	(	(	PUNCT
ejpam-6013	272	17	y	y	PROPN
ejpam-6013	272	18	,	,	PUNCT
ejpam-6013	272	19	σ1	σ1	PROPN
ejpam-6013	272	20	,	,	PUNCT
ejpam-6013	272	21	σ2	σ2	PROPN
ejpam-6013	272	22	)	)	PUNCT
ejpam-6013	272	23	is	be	AUX
ejpam-6013	272	24	said	say	VERB
ejpam-6013	272	25	to	to	PART
ejpam-6013	272	26	have	have	VERB
ejpam-6013	272	27	a	a	DET
ejpam-6013	272	28	strongly	strongly	ADV
ejpam-6013	272	29	θ(τ1	θ(τ1	NOUN
ejpam-6013	272	30	,	,	PUNCT
ejpam-6013	272	31	τ2)-closed	τ2)-close	VERB
ejpam-6013	272	32	graph	graph	NOUN
ejpam-6013	272	33	with	with	ADP
ejpam-6013	272	34	respect	respect	NOUN
ejpam-6013	272	35	to	to	ADP
ejpam-6013	272	36	x	x	PRON
ejpam-6013	272	37	if	if	SCONJ
ejpam-6013	272	38	for	for	ADP
ejpam-6013	272	39	each	each	DET
ejpam-6013	272	40	(	(	PUNCT
ejpam-6013	272	41	x	x	NOUN
ejpam-6013	272	42	,	,	PUNCT
ejpam-6013	272	43	y	y	NOUN
ejpam-6013	272	44	)	)	PUNCT
ejpam-6013	272	45	∈	∈	PROPN
ejpam-6013	272	46	(	(	PUNCT
ejpam-6013	272	47	x	x	SYM
ejpam-6013	272	48	×	×	PROPN
ejpam-6013	272	49	y	y	PROPN
ejpam-6013	272	50	)	)	PUNCT
ejpam-6013	273	1	−	−	PROPN
ejpam-6013	273	2	g(f	g(f	NOUN
ejpam-6013	273	3	)	)	PUNCT
ejpam-6013	273	4	,	,	PUNCT
ejpam-6013	273	5	there	there	PRON
ejpam-6013	273	6	exist	exist	VERB
ejpam-6013	273	7	a	a	DET
ejpam-6013	273	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	273	9	-	-	ADJ
ejpam-6013	273	10	open	open	ADJ
ejpam-6013	273	11	set	set	ADJ
ejpam-6013	273	12	u	u	NOUN
ejpam-6013	273	13	of	of	ADP
ejpam-6013	273	14	x	x	PUNCT
ejpam-6013	273	15	containing	contain	VERB
ejpam-6013	273	16	x	x	X
ejpam-6013	273	17	and	and	CCONJ
ejpam-6013	273	18	a	a	DET
ejpam-6013	273	19	σ1σ2	σ1σ2	NUM
ejpam-6013	273	20	-	-	ADJ
ejpam-6013	273	21	open	open	ADJ
ejpam-6013	273	22	set	set	NOUN
ejpam-6013	273	23	v	v	NOUN
ejpam-6013	273	24	of	of	ADP
ejpam-6013	273	25	y	y	PROPN
ejpam-6013	273	26	containing	contain	VERB
ejpam-6013	273	27	y	y	PRON
ejpam-6013	273	28	such	such	ADJ
ejpam-6013	273	29	that	that	SCONJ
ejpam-6013	274	1	[	[	X
ejpam-6013	274	2	τ1τ2	τ1τ2	NOUN
ejpam-6013	274	3	-	-	ADJ
ejpam-6013	274	4	cl(u)×	cl(u)×	NOUN
ejpam-6013	274	5	v	v	NOUN
ejpam-6013	274	6	]	]	PUNCT
ejpam-6013	274	7	∩g(f	∩g(f	PROPN
ejpam-6013	274	8	)	)	PUNCT
ejpam-6013	274	9	=	=	PUNCT
ejpam-6013	274	10	∅.	∅.	PRON
ejpam-6013	274	11	lemma	lemma	PROPN
ejpam-6013	274	12	10	10	NUM
ejpam-6013	274	13	.	.	PUNCT
ejpam-6013	275	1	[	[	X
ejpam-6013	275	2	29	29	NUM
ejpam-6013	275	3	]	]	PUNCT
ejpam-6013	275	4	a	a	DET
ejpam-6013	275	5	function	function	NOUN
ejpam-6013	275	6	f	f	NOUN
ejpam-6013	275	7	:	:	PUNCT
ejpam-6013	275	8	(	(	PUNCT
ejpam-6013	275	9	x	x	NOUN
ejpam-6013	275	10	,	,	PUNCT
ejpam-6013	275	11	τ1	τ1	NOUN
ejpam-6013	275	12	,	,	PUNCT
ejpam-6013	275	13	τ2	τ2	NOUN
ejpam-6013	275	14	)	)	PUNCT
ejpam-6013	275	15	→	→	SYM
ejpam-6013	275	16	(	(	PUNCT
ejpam-6013	275	17	y	y	PROPN
ejpam-6013	275	18	,	,	PUNCT
ejpam-6013	275	19	σ1	σ1	PROPN
ejpam-6013	275	20	,	,	PUNCT
ejpam-6013	275	21	σ2	σ2	NOUN
ejpam-6013	275	22	)	)	PUNCT
ejpam-6013	275	23	has	have	VERB
ejpam-6013	275	24	a	a	DET
ejpam-6013	275	25	strongly	strongly	ADV
ejpam-6013	275	26	θ(τ1	θ(τ1	NOUN
ejpam-6013	275	27	,	,	PUNCT
ejpam-6013	275	28	τ2)-closed	τ2)-close	VERB
ejpam-6013	275	29	graph	graph	NOUN
ejpam-6013	275	30	with	with	ADP
ejpam-6013	275	31	respect	respect	NOUN
ejpam-6013	275	32	to	to	ADP
ejpam-6013	275	33	x	x	PUNCT
ejpam-6013	275	34	if	if	SCONJ
ejpam-6013	276	1	and	and	CCONJ
ejpam-6013	276	2	only	only	ADV
ejpam-6013	276	3	if	if	SCONJ
ejpam-6013	276	4	for	for	ADP
ejpam-6013	276	5	each	each	DET
ejpam-6013	276	6	(	(	PUNCT
ejpam-6013	276	7	x	x	NOUN
ejpam-6013	276	8	,	,	PUNCT
ejpam-6013	276	9	y	y	NOUN
ejpam-6013	276	10	)	)	PUNCT
ejpam-6013	276	11	∈	∈	PROPN
ejpam-6013	276	12	(	(	PUNCT
ejpam-6013	276	13	x	x	SYM
ejpam-6013	276	14	×	×	PROPN
ejpam-6013	276	15	y	y	PROPN
ejpam-6013	276	16	)	)	PUNCT
ejpam-6013	277	1	−	−	PROPN
ejpam-6013	277	2	g(f	g(f	NOUN
ejpam-6013	277	3	)	)	PUNCT
ejpam-6013	277	4	,	,	PUNCT
ejpam-6013	277	5	there	there	PRON
ejpam-6013	277	6	exist	exist	VERB
ejpam-6013	277	7	a	a	DET
ejpam-6013	277	8	τ1τ2	τ1τ2	NOUN
ejpam-6013	277	9	-	-	ADJ
ejpam-6013	277	10	open	open	ADJ
ejpam-6013	277	11	set	set	ADJ
ejpam-6013	277	12	u	u	NOUN
ejpam-6013	277	13	of	of	ADP
ejpam-6013	277	14	x	x	PUNCT
ejpam-6013	277	15	containing	contain	VERB
ejpam-6013	277	16	x	x	X
ejpam-6013	277	17	and	and	CCONJ
ejpam-6013	277	18	a	a	DET
ejpam-6013	277	19	σ1σ2	σ1σ2	NUM
ejpam-6013	277	20	-	-	ADJ
ejpam-6013	277	21	open	open	ADJ
ejpam-6013	277	22	set	set	NOUN
ejpam-6013	277	23	v	v	NOUN
ejpam-6013	277	24	of	of	ADP
ejpam-6013	277	25	y	y	PROPN
ejpam-6013	277	26	containing	contain	VERB
ejpam-6013	277	27	y	y	PRON
ejpam-6013	277	28	such	such	ADJ
ejpam-6013	277	29	that	that	SCONJ
ejpam-6013	277	30	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	277	31	-	-	PUNCT
ejpam-6013	277	32	cl(u	cl(u	NOUN
ejpam-6013	277	33	)	)	PUNCT
ejpam-6013	277	34	)	)	PUNCT
ejpam-6013	278	1	∩	∩	NOUN
ejpam-6013	278	2	v	v	AUX
ejpam-6013	278	3	=	=	PUNCT
ejpam-6013	278	4	∅.	∅.	NOUN
ejpam-6013	278	5	recall	recall	VERB
ejpam-6013	278	6	that	that	SCONJ
ejpam-6013	278	7	a	a	DET
ejpam-6013	278	8	bitopological	bitopological	ADJ
ejpam-6013	278	9	space	space	NOUN
ejpam-6013	278	10	(	(	PUNCT
ejpam-6013	278	11	x	x	NOUN
ejpam-6013	278	12	,	,	PUNCT
ejpam-6013	278	13	τ1	τ1	NOUN
ejpam-6013	278	14	,	,	PUNCT
ejpam-6013	278	15	τ2	τ2	NOUN
ejpam-6013	278	16	)	)	PUNCT
ejpam-6013	278	17	is	be	AUX
ejpam-6013	278	18	said	say	VERB
ejpam-6013	278	19	to	to	PART
ejpam-6013	278	20	be	be	AUX
ejpam-6013	278	21	(	(	PUNCT
ejpam-6013	278	22	τ1	τ1	NOUN
ejpam-6013	278	23	,	,	PUNCT
ejpam-6013	278	24	τ2)-t1	τ2)-t1	VERB
ejpam-6013	279	1	[	[	X
ejpam-6013	279	2	34	34	NUM
ejpam-6013	279	3	]	]	X
ejpam-6013	279	4	if	if	SCONJ
ejpam-6013	279	5	for	for	ADP
ejpam-6013	279	6	any	any	DET
ejpam-6013	279	7	pair	pair	NOUN
ejpam-6013	279	8	of	of	ADP
ejpam-6013	279	9	distinct	distinct	ADJ
ejpam-6013	279	10	points	point	NOUN
ejpam-6013	279	11	x	x	X
ejpam-6013	279	12	,	,	PUNCT
ejpam-6013	279	13	y	y	PROPN
ejpam-6013	279	14	in	in	ADP
ejpam-6013	279	15	x	x	SYM
ejpam-6013	279	16	,	,	PUNCT
ejpam-6013	279	17	there	there	PRON
ejpam-6013	279	18	exist	exist	VERB
ejpam-6013	279	19	τ1τ2	τ1τ2	ADJ
ejpam-6013	279	20	-	-	ADJ
ejpam-6013	279	21	open	open	ADJ
ejpam-6013	279	22	sets	set	NOUN
ejpam-6013	279	23	u	u	NOUN
ejpam-6013	279	24	and	and	CCONJ
ejpam-6013	279	25	v	v	NOUN
ejpam-6013	279	26	of	of	ADP
ejpam-6013	279	27	x	x	PUNCT
ejpam-6013	279	28	such	such	ADJ
ejpam-6013	279	29	that	that	SCONJ
ejpam-6013	279	30	x	x	SYM
ejpam-6013	279	31	∈	∈	PROPN
ejpam-6013	279	32	u	u	PROPN
ejpam-6013	279	33	,	,	PUNCT
ejpam-6013	279	34	y	y	PROPN
ejpam-6013	279	35	̸∈	̸∈	PROPN
ejpam-6013	279	36	u	u	PROPN
ejpam-6013	279	37	and	and	CCONJ
ejpam-6013	279	38	y	y	PROPN
ejpam-6013	279	39	∈	∈	PROPN
ejpam-6013	279	40	v	v	NOUN
ejpam-6013	279	41	,	,	PUNCT
ejpam-6013	279	42	x	x	PROPN
ejpam-6013	279	43	̸∈	̸∈	PROPN
ejpam-6013	279	44	v	v	PROPN
ejpam-6013	279	45	.	.	PUNCT
ejpam-6013	280	1	n.	n.	PROPN
ejpam-6013	280	2	srisarakham	srisarakham	PROPN
ejpam-6013	280	3	,	,	PUNCT
ejpam-6013	280	4	s.	s.	PROPN
ejpam-6013	280	5	sompong	sompong	PROPN
ejpam-6013	280	6	,	,	PUNCT
ejpam-6013	280	7	c.	c.	PROPN
ejpam-6013	280	8	boonpok	boonpok	PROPN
ejpam-6013	280	9	/	/	SYM
ejpam-6013	280	10	eur	eur	PROPN
ejpam-6013	280	11	.	.	PUNCT
ejpam-6013	281	1	j.	j.	PROPN
ejpam-6013	281	2	pure	pure	PROPN
ejpam-6013	281	3	appl	appl	PROPN
ejpam-6013	281	4	.	.	PROPN
ejpam-6013	281	5	math	math	PROPN
ejpam-6013	281	6	,	,	PUNCT
ejpam-6013	281	7	18	18	NUM
ejpam-6013	281	8	(	(	PUNCT
ejpam-6013	281	9	2	2	NUM
ejpam-6013	281	10	)	)	PUNCT
ejpam-6013	281	11	(	(	PUNCT
ejpam-6013	281	12	2025	2025	NUM
ejpam-6013	281	13	)	)	PUNCT
ejpam-6013	281	14	,	,	PUNCT
ejpam-6013	281	15	6013	6013	NUM
ejpam-6013	281	16	10	10	NUM
ejpam-6013	281	17	of	of	ADP
ejpam-6013	281	18	12	12	NUM
ejpam-6013	281	19	theorem	theorem	NOUN
ejpam-6013	281	20	11	11	NUM
ejpam-6013	281	21	.	.	PUNCT
ejpam-6013	282	1	if	if	SCONJ
ejpam-6013	282	2	a	a	DET
ejpam-6013	282	3	function	function	NOUN
ejpam-6013	282	4	f	f	X
ejpam-6013	282	5	:	:	PUNCT
ejpam-6013	282	6	(	(	PUNCT
ejpam-6013	282	7	x	x	NOUN
ejpam-6013	282	8	,	,	PUNCT
ejpam-6013	282	9	τ1	τ1	NOUN
ejpam-6013	282	10	,	,	PUNCT
ejpam-6013	282	11	τ2	τ2	NOUN
ejpam-6013	282	12	)	)	PUNCT
ejpam-6013	282	13	→	→	SYM
ejpam-6013	282	14	(	(	PUNCT
ejpam-6013	282	15	y	y	PROPN
ejpam-6013	282	16	,	,	PUNCT
ejpam-6013	282	17	σ1	σ1	PROPN
ejpam-6013	282	18	,	,	PUNCT
ejpam-6013	282	19	σ2	σ2	PROPN
ejpam-6013	282	20	)	)	PUNCT
ejpam-6013	282	21	is	be	AUX
ejpam-6013	282	22	r-(τ1	r-(τ1	PROPN
ejpam-6013	282	23	,	,	PUNCT
ejpam-6013	282	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	282	25	and	and	CCONJ
ejpam-6013	282	26	(	(	PUNCT
ejpam-6013	282	27	y	y	PROPN
ejpam-6013	282	28	,	,	PUNCT
ejpam-6013	282	29	σ1	σ1	PROPN
ejpam-6013	282	30	,	,	PUNCT
ejpam-6013	282	31	σ2	σ2	PROPN
ejpam-6013	282	32	)	)	PUNCT
ejpam-6013	282	33	is	be	AUX
ejpam-6013	282	34	(	(	PUNCT
ejpam-6013	282	35	σ1	σ1	PROPN
ejpam-6013	282	36	,	,	PUNCT
ejpam-6013	282	37	σ2)-t1	σ2)-t1	NOUN
ejpam-6013	282	38	,	,	PUNCT
ejpam-6013	282	39	then	then	ADV
ejpam-6013	282	40	g(f	g(f	PROPN
ejpam-6013	282	41	)	)	PUNCT
ejpam-6013	282	42	is	be	AUX
ejpam-6013	282	43	strongly	strongly	ADV
ejpam-6013	282	44	θ(τ1	θ(τ1	ADJ
ejpam-6013	282	45	,	,	PUNCT
ejpam-6013	282	46	τ2)-closed	τ2)-close	VERB
ejpam-6013	282	47	with	with	ADP
ejpam-6013	282	48	respect	respect	NOUN
ejpam-6013	282	49	to	to	ADP
ejpam-6013	282	50	x.	x.	NOUN
ejpam-6013	282	51	proof	proof	NOUN
ejpam-6013	282	52	.	.	PUNCT
ejpam-6013	283	1	let	let	VERB
ejpam-6013	283	2	(	(	PUNCT
ejpam-6013	283	3	x	x	NOUN
ejpam-6013	283	4	,	,	PUNCT
ejpam-6013	283	5	y	y	NOUN
ejpam-6013	283	6	)	)	PUNCT
ejpam-6013	283	7	∈	∈	PROPN
ejpam-6013	283	8	(	(	PUNCT
ejpam-6013	283	9	x	x	SYM
ejpam-6013	283	10	×	×	PROPN
ejpam-6013	283	11	y	y	PROPN
ejpam-6013	283	12	)	)	PUNCT
ejpam-6013	284	1	−	−	PROPN
ejpam-6013	284	2	g(f	g(f	NOUN
ejpam-6013	284	3	)	)	PUNCT
ejpam-6013	284	4	.	.	PUNCT
ejpam-6013	285	1	then	then	ADV
ejpam-6013	285	2	,	,	PUNCT
ejpam-6013	285	3	y	y	PROPN
ejpam-6013	285	4	̸=	̸=	PROPN
ejpam-6013	285	5	f(x	f(x	PROPN
ejpam-6013	285	6	)	)	PUNCT
ejpam-6013	285	7	.	.	PUNCT
ejpam-6013	286	1	since	since	SCONJ
ejpam-6013	286	2	(	(	PUNCT
ejpam-6013	286	3	y	y	PROPN
ejpam-6013	286	4	,	,	PUNCT
ejpam-6013	286	5	σ1	σ1	PROPN
ejpam-6013	286	6	,	,	PUNCT
ejpam-6013	286	7	σ2	σ2	PROPN
ejpam-6013	286	8	)	)	PUNCT
ejpam-6013	286	9	is	be	AUX
ejpam-6013	286	10	(	(	PUNCT
ejpam-6013	286	11	σ1	σ1	PROPN
ejpam-6013	286	12	,	,	PUNCT
ejpam-6013	286	13	σ2)-t1	σ2)-t1	PROPN
ejpam-6013	286	14	,	,	PUNCT
ejpam-6013	286	15	there	there	PRON
ejpam-6013	286	16	exists	exist	VERB
ejpam-6013	286	17	a	a	DET
ejpam-6013	286	18	σ1σ2	σ1σ2	NUM
ejpam-6013	286	19	-	-	ADJ
ejpam-6013	286	20	open	open	ADJ
ejpam-6013	286	21	set	set	NOUN
ejpam-6013	286	22	v	v	NOUN
ejpam-6013	286	23	of	of	ADP
ejpam-6013	286	24	y	y	PRON
ejpam-6013	286	25	such	such	ADJ
ejpam-6013	286	26	that	that	SCONJ
ejpam-6013	286	27	f(x	f(x	PROPN
ejpam-6013	286	28	)	)	PUNCT
ejpam-6013	286	29	∈	∈	PROPN
ejpam-6013	286	30	v	v	NOUN
ejpam-6013	286	31	and	and	CCONJ
ejpam-6013	286	32	y	y	PROPN
ejpam-6013	286	33	̸∈	̸∈	PROPN
ejpam-6013	286	34	v	v	PROPN
ejpam-6013	286	35	.	.	PUNCT
ejpam-6013	287	1	since	since	SCONJ
ejpam-6013	287	2	f	f	PROPN
ejpam-6013	287	3	is	be	AUX
ejpam-6013	287	4	r-(τ1	r-(τ1	NOUN
ejpam-6013	287	5	,	,	PUNCT
ejpam-6013	287	6	τ2)continuous	τ2)continuous	ADJ
ejpam-6013	287	7	,	,	PUNCT
ejpam-6013	287	8	there	there	PRON
ejpam-6013	287	9	exists	exist	VERB
ejpam-6013	287	10	a	a	DET
ejpam-6013	287	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	287	12	-	-	ADJ
ejpam-6013	287	13	open	open	ADJ
ejpam-6013	287	14	set	set	ADJ
ejpam-6013	287	15	u	u	NOUN
ejpam-6013	287	16	of	of	ADP
ejpam-6013	287	17	x	x	PUNCT
ejpam-6013	287	18	containing	contain	VERB
ejpam-6013	287	19	x	x	PUNCT
ejpam-6013	287	20	such	such	ADJ
ejpam-6013	287	21	that	that	SCONJ
ejpam-6013	287	22	σ1σ2	σ1σ2	NOUN
ejpam-6013	287	23	-	-	PUNCT
ejpam-6013	287	24	cl(f(u	cl(f(u	NOUN
ejpam-6013	287	25	)	)	PUNCT
ejpam-6013	287	26	)	)	PUNCT
ejpam-6013	288	1	⊆	⊆	NUM
ejpam-6013	288	2	v	v	NOUN
ejpam-6013	288	3	.	.	PUNCT
ejpam-6013	289	1	since	since	SCONJ
ejpam-6013	289	2	y	y	PROPN
ejpam-6013	289	3	̸∈	̸∈	PROPN
ejpam-6013	289	4	v	v	PROPN
ejpam-6013	289	5	,	,	PUNCT
ejpam-6013	289	6	we	we	PRON
ejpam-6013	289	7	have	have	VERB
ejpam-6013	289	8	y	y	PROPN
ejpam-6013	289	9	̸∈	̸∈	PROPN
ejpam-6013	289	10	σ1σ2	σ1σ2	X
ejpam-6013	289	11	-	-	PUNCT
ejpam-6013	289	12	cl(f(u	cl(f(u	NOUN
ejpam-6013	289	13	)	)	PUNCT
ejpam-6013	289	14	)	)	PUNCT
ejpam-6013	290	1	and	and	CCONJ
ejpam-6013	290	2	so	so	ADV
ejpam-6013	290	3	y	y	PROPN
ejpam-6013	290	4	∈	∈	PROPN
ejpam-6013	290	5	y	y	NOUN
ejpam-6013	290	6	−	−	VERB
ejpam-6013	290	7	σ1σ2	σ1σ2	NOUN
ejpam-6013	290	8	-	-	PUNCT
ejpam-6013	290	9	cl(f(u	cl(f(u	NOUN
ejpam-6013	290	10	)	)	PUNCT
ejpam-6013	290	11	)	)	PUNCT
ejpam-6013	290	12	.	.	PUNCT
ejpam-6013	291	1	by	by	ADP
ejpam-6013	291	2	lemma	lemma	PROPN
ejpam-6013	291	3	1	1	NUM
ejpam-6013	291	4	,	,	PUNCT
ejpam-6013	291	5	σ1σ2	σ1σ2	X
ejpam-6013	291	6	-	-	PUNCT
ejpam-6013	291	7	cl(f(u	cl(f(u	NOUN
ejpam-6013	291	8	)	)	PUNCT
ejpam-6013	291	9	)	)	PUNCT
ejpam-6013	291	10	is	be	AUX
ejpam-6013	291	11	σ1σ2	σ1σ2	NOUN
ejpam-6013	291	12	-	-	ADJ
ejpam-6013	291	13	closed	closed	ADJ
ejpam-6013	291	14	and	and	CCONJ
ejpam-6013	291	15	y	y	NOUN
ejpam-6013	291	16	−	−	PROPN
ejpam-6013	291	17	σ1σ2	σ1σ2	NOUN
ejpam-6013	291	18	-	-	PUNCT
ejpam-6013	291	19	cl(f(u	cl(f(u	NOUN
ejpam-6013	291	20	)	)	PUNCT
ejpam-6013	291	21	)	)	PUNCT
ejpam-6013	292	1	is	be	AUX
ejpam-6013	292	2	σ1σ2	σ1σ2	NOUN
ejpam-6013	292	3	-	-	ADJ
ejpam-6013	292	4	open	open	ADJ
ejpam-6013	292	5	.	.	PUNCT
ejpam-6013	293	1	since	since	SCONJ
ejpam-6013	293	2	f	f	PROPN
ejpam-6013	293	3	is	be	AUX
ejpam-6013	293	4	r-(τ1	r-(τ1	NOUN
ejpam-6013	293	5	,	,	PUNCT
ejpam-6013	293	6	τ2)continuous	τ2)continuous	ADJ
ejpam-6013	293	7	,	,	PUNCT
ejpam-6013	293	8	f	f	PROPN
ejpam-6013	293	9	is	be	AUX
ejpam-6013	293	10	(	(	PUNCT
ejpam-6013	293	11	τ1	τ1	NOUN
ejpam-6013	293	12	,	,	PUNCT
ejpam-6013	293	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	293	14	and	and	CCONJ
ejpam-6013	293	15	by	by	ADP
ejpam-6013	293	16	lemma	lemma	PROPN
ejpam-6013	293	17	3	3	NUM
ejpam-6013	293	18	,	,	PUNCT
ejpam-6013	293	19	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	293	20	-	-	PUNCT
ejpam-6013	293	21	cl(u	cl(u	NOUN
ejpam-6013	293	22	)	)	PUNCT
ejpam-6013	293	23	)	)	PUNCT
ejpam-6013	294	1	⊆	⊆	X
ejpam-6013	294	2	σ1σ2	σ1σ2	NUM
ejpam-6013	294	3	-	-	PUNCT
ejpam-6013	294	4	cl(f(u	cl(f(u	NOUN
ejpam-6013	294	5	)	)	PUNCT
ejpam-6013	294	6	)	)	PUNCT
ejpam-6013	294	7	.	.	PUNCT
ejpam-6013	295	1	then	then	ADV
ejpam-6013	295	2	,	,	PUNCT
ejpam-6013	295	3	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6013	295	4	-	-	PUNCT
ejpam-6013	295	5	cl(u	cl(u	NOUN
ejpam-6013	295	6	)	)	PUNCT
ejpam-6013	295	7	)	)	PUNCT
ejpam-6013	295	8	∩	∩	NOUN
ejpam-6013	295	9	(	(	PUNCT
ejpam-6013	295	10	y	y	PROPN
ejpam-6013	295	11	−	−	PROPN
ejpam-6013	295	12	σ1σ2	σ1σ2	NUM
ejpam-6013	295	13	-	-	PUNCT
ejpam-6013	295	14	cl(f(u	cl(f(u	NOUN
ejpam-6013	295	15	)	)	PUNCT
ejpam-6013	295	16	)	)	PUNCT
ejpam-6013	295	17	)	)	PUNCT
ejpam-6013	296	1	=	=	PUNCT
ejpam-6013	296	2	∅	∅	NOUN
ejpam-6013	296	3	and	and	CCONJ
ejpam-6013	296	4	by	by	ADP
ejpam-6013	296	5	lemma	lemma	PROPN
ejpam-6013	296	6	10	10	NUM
ejpam-6013	296	7	,	,	PUNCT
ejpam-6013	296	8	g(f	g(f	PROPN
ejpam-6013	296	9	)	)	PUNCT
ejpam-6013	296	10	is	be	AUX
ejpam-6013	296	11	strongly	strongly	ADV
ejpam-6013	296	12	θ(τ1	θ(τ1	ADJ
ejpam-6013	296	13	,	,	PUNCT
ejpam-6013	296	14	τ2)-closed	τ2)-close	VERB
ejpam-6013	296	15	with	with	ADP
ejpam-6013	296	16	respect	respect	NOUN
ejpam-6013	296	17	to	to	ADP
ejpam-6013	296	18	x.	x.	NOUN
ejpam-6013	296	19	recall	recall	VERB
ejpam-6013	296	20	that	that	SCONJ
ejpam-6013	296	21	a	a	DET
ejpam-6013	296	22	bitopological	bitopological	ADJ
ejpam-6013	296	23	space	space	NOUN
ejpam-6013	296	24	(	(	PUNCT
ejpam-6013	296	25	x	x	NOUN
ejpam-6013	296	26	,	,	PUNCT
ejpam-6013	296	27	τ1	τ1	NOUN
ejpam-6013	296	28	,	,	PUNCT
ejpam-6013	296	29	τ2	τ2	NOUN
ejpam-6013	296	30	)	)	PUNCT
ejpam-6013	296	31	is	be	AUX
ejpam-6013	296	32	said	say	VERB
ejpam-6013	296	33	to	to	PART
ejpam-6013	296	34	be	be	AUX
ejpam-6013	296	35	(	(	PUNCT
ejpam-6013	296	36	τ1	τ1	NOUN
ejpam-6013	296	37	,	,	PUNCT
ejpam-6013	296	38	τ2)-t2	τ2)-t2	X
ejpam-6013	297	1	[	[	X
ejpam-6013	297	2	35	35	NUM
ejpam-6013	297	3	]	]	X
ejpam-6013	297	4	if	if	SCONJ
ejpam-6013	297	5	for	for	ADP
ejpam-6013	297	6	any	any	DET
ejpam-6013	297	7	pair	pair	NOUN
ejpam-6013	297	8	of	of	ADP
ejpam-6013	297	9	distinct	distinct	ADJ
ejpam-6013	297	10	points	point	NOUN
ejpam-6013	297	11	x	x	X
ejpam-6013	297	12	,	,	PUNCT
ejpam-6013	297	13	y	y	PROPN
ejpam-6013	297	14	in	in	ADP
ejpam-6013	297	15	x	x	SYM
ejpam-6013	297	16	,	,	PUNCT
ejpam-6013	297	17	there	there	PRON
ejpam-6013	297	18	exist	exist	VERB
ejpam-6013	297	19	disjoint	disjoint	ADJ
ejpam-6013	297	20	τ1τ2	τ1τ2	ADJ
ejpam-6013	297	21	-	-	ADJ
ejpam-6013	297	22	open	open	ADJ
ejpam-6013	297	23	sets	set	NOUN
ejpam-6013	297	24	u	u	NOUN
ejpam-6013	297	25	and	and	CCONJ
ejpam-6013	297	26	v	v	NOUN
ejpam-6013	297	27	of	of	ADP
ejpam-6013	297	28	x	x	PUNCT
ejpam-6013	297	29	containing	contain	VERB
ejpam-6013	297	30	x	x	PROPN
ejpam-6013	297	31	and	and	CCONJ
ejpam-6013	297	32	y	y	PROPN
ejpam-6013	297	33	,	,	PUNCT
ejpam-6013	297	34	respectively	respectively	ADV
ejpam-6013	297	35	.	.	PUNCT
ejpam-6013	298	1	theorem	theorem	VERB
ejpam-6013	298	2	12	12	NUM
ejpam-6013	298	3	.	.	PUNCT
ejpam-6013	299	1	if	if	SCONJ
ejpam-6013	299	2	f	f	PROPN
ejpam-6013	299	3	:	:	PUNCT
ejpam-6013	299	4	(	(	PUNCT
ejpam-6013	299	5	x	x	NOUN
ejpam-6013	299	6	,	,	PUNCT
ejpam-6013	299	7	τ1	τ1	NOUN
ejpam-6013	299	8	,	,	PUNCT
ejpam-6013	299	9	τ2	τ2	NOUN
ejpam-6013	299	10	)	)	PUNCT
ejpam-6013	299	11	→	→	SYM
ejpam-6013	299	12	(	(	PUNCT
ejpam-6013	299	13	y	y	PROPN
ejpam-6013	299	14	,	,	PUNCT
ejpam-6013	299	15	σ1	σ1	PROPN
ejpam-6013	299	16	,	,	PUNCT
ejpam-6013	299	17	σ2	σ2	PROPN
ejpam-6013	299	18	)	)	PUNCT
ejpam-6013	299	19	is	be	AUX
ejpam-6013	299	20	a	a	DET
ejpam-6013	299	21	(	(	PUNCT
ejpam-6013	299	22	τ1	τ1	NOUN
ejpam-6013	299	23	,	,	PUNCT
ejpam-6013	299	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	299	25	injection	injection	NOUN
ejpam-6013	299	26	and	and	CCONJ
ejpam-6013	299	27	(	(	PUNCT
ejpam-6013	299	28	y	y	PROPN
ejpam-6013	299	29	,	,	PUNCT
ejpam-6013	299	30	σ1	σ1	PROPN
ejpam-6013	299	31	,	,	PUNCT
ejpam-6013	299	32	σ2	σ2	PROPN
ejpam-6013	299	33	)	)	PUNCT
ejpam-6013	299	34	is	be	AUX
ejpam-6013	299	35	(	(	PUNCT
ejpam-6013	299	36	σ1	σ1	PROPN
ejpam-6013	299	37	,	,	PUNCT
ejpam-6013	299	38	σ2)-t2	σ2)-t2	PROPN
ejpam-6013	299	39	,	,	PUNCT
ejpam-6013	299	40	then	then	ADV
ejpam-6013	299	41	(	(	PUNCT
ejpam-6013	299	42	x	x	NOUN
ejpam-6013	299	43	,	,	PUNCT
ejpam-6013	299	44	τ1	τ1	NOUN
ejpam-6013	299	45	,	,	PUNCT
ejpam-6013	299	46	τ2	τ2	NOUN
ejpam-6013	299	47	)	)	PUNCT
ejpam-6013	299	48	is	be	AUX
ejpam-6013	299	49	(	(	PUNCT
ejpam-6013	299	50	τ1	τ1	NOUN
ejpam-6013	299	51	,	,	PUNCT
ejpam-6013	299	52	τ2)-t2	τ2)-t2	PROPN
ejpam-6013	299	53	.	.	PUNCT
ejpam-6013	300	1	proof	proof	NOUN
ejpam-6013	300	2	.	.	PUNCT
ejpam-6013	301	1	suppose	suppose	VERB
ejpam-6013	301	2	that	that	SCONJ
ejpam-6013	301	3	(	(	PUNCT
ejpam-6013	301	4	y	y	PROPN
ejpam-6013	301	5	,	,	PUNCT
ejpam-6013	301	6	σ1	σ1	PROPN
ejpam-6013	301	7	,	,	PUNCT
ejpam-6013	301	8	σ2	σ2	PROPN
ejpam-6013	301	9	)	)	PUNCT
ejpam-6013	301	10	is	be	AUX
ejpam-6013	301	11	(	(	PUNCT
ejpam-6013	301	12	σ1	σ1	PROPN
ejpam-6013	301	13	,	,	PUNCT
ejpam-6013	301	14	σ2)-t2	σ2)-t2	PROPN
ejpam-6013	301	15	.	.	PUNCT
ejpam-6013	302	1	let	let	VERB
ejpam-6013	302	2	x	x	PRON
ejpam-6013	302	3	,	,	PUNCT
ejpam-6013	302	4	y	y	PROPN
ejpam-6013	302	5	be	be	VERB
ejpam-6013	302	6	any	any	DET
ejpam-6013	302	7	distinct	distinct	ADJ
ejpam-6013	302	8	points	point	NOUN
ejpam-6013	302	9	of	of	ADP
ejpam-6013	302	10	x.	x.	NOUN
ejpam-6013	302	11	since	since	SCONJ
ejpam-6013	302	12	f	f	PROPN
ejpam-6013	302	13	is	be	AUX
ejpam-6013	302	14	injective	injective	ADJ
ejpam-6013	302	15	,	,	PUNCT
ejpam-6013	302	16	f(x	f(x	PROPN
ejpam-6013	302	17	)	)	PUNCT
ejpam-6013	302	18	̸=	̸=	PROPN
ejpam-6013	302	19	f(y	f(y	NOUN
ejpam-6013	302	20	)	)	PUNCT
ejpam-6013	302	21	.	.	PUNCT
ejpam-6013	303	1	since	since	SCONJ
ejpam-6013	303	2	(	(	PUNCT
ejpam-6013	303	3	y	y	PROPN
ejpam-6013	303	4	,	,	PUNCT
ejpam-6013	303	5	σ1	σ1	PROPN
ejpam-6013	303	6	,	,	PUNCT
ejpam-6013	303	7	σ2	σ2	PROPN
ejpam-6013	303	8	)	)	PUNCT
ejpam-6013	303	9	is	be	AUX
ejpam-6013	303	10	(	(	PUNCT
ejpam-6013	303	11	σ1	σ1	PROPN
ejpam-6013	303	12	,	,	PUNCT
ejpam-6013	303	13	σ2)-t2	σ2)-t2	PROPN
ejpam-6013	303	14	,	,	PUNCT
ejpam-6013	303	15	there	there	PRON
ejpam-6013	303	16	exist	exist	VERB
ejpam-6013	303	17	σ1σ2	σ1σ2	NOUN
ejpam-6013	303	18	-	-	ADJ
ejpam-6013	303	19	open	open	ADJ
ejpam-6013	303	20	sets	set	NOUN
ejpam-6013	303	21	v	v	ADP
ejpam-6013	303	22	and	and	CCONJ
ejpam-6013	303	23	w	w	PROPN
ejpam-6013	303	24	of	of	ADP
ejpam-6013	303	25	y	y	PROPN
ejpam-6013	303	26	containing	contain	VERB
ejpam-6013	303	27	f(x	f(x	PROPN
ejpam-6013	303	28	)	)	PUNCT
ejpam-6013	303	29	and	and	CCONJ
ejpam-6013	303	30	f(y	f(y	NOUN
ejpam-6013	303	31	)	)	PUNCT
ejpam-6013	303	32	,	,	PUNCT
ejpam-6013	303	33	respectively	respectively	ADV
ejpam-6013	303	34	,	,	PUNCT
ejpam-6013	303	35	such	such	ADJ
ejpam-6013	303	36	that	that	DET
ejpam-6013	303	37	v	v	NOUN
ejpam-6013	303	38	∩w	∩w	NOUN
ejpam-6013	303	39	=	=	PUNCT
ejpam-6013	303	40	∅.	∅.	NOUN
ejpam-6013	303	41	since	since	SCONJ
ejpam-6013	303	42	f	f	PROPN
ejpam-6013	303	43	is	be	AUX
ejpam-6013	303	44	(	(	PUNCT
ejpam-6013	303	45	τ1	τ1	NOUN
ejpam-6013	303	46	,	,	PUNCT
ejpam-6013	303	47	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	303	48	,	,	PUNCT
ejpam-6013	303	49	there	there	PRON
ejpam-6013	303	50	exist	exist	VERB
ejpam-6013	303	51	τ1τ2	τ1τ2	ADJ
ejpam-6013	303	52	-	-	ADJ
ejpam-6013	303	53	open	open	ADJ
ejpam-6013	303	54	sets	set	NOUN
ejpam-6013	303	55	u	u	NOUN
ejpam-6013	303	56	and	and	CCONJ
ejpam-6013	303	57	g	g	PROPN
ejpam-6013	303	58	of	of	ADP
ejpam-6013	303	59	x	x	PUNCT
ejpam-6013	303	60	containing	contain	VERB
ejpam-6013	303	61	x	x	PROPN
ejpam-6013	303	62	and	and	CCONJ
ejpam-6013	303	63	y	y	PROPN
ejpam-6013	303	64	,	,	PUNCT
ejpam-6013	303	65	respectively	respectively	ADV
ejpam-6013	303	66	,	,	PUNCT
ejpam-6013	303	67	such	such	ADJ
ejpam-6013	303	68	that	that	DET
ejpam-6013	303	69	f(u	f(u	PROPN
ejpam-6013	303	70	)	)	PUNCT
ejpam-6013	303	71	⊆	⊆	NUM
ejpam-6013	303	72	v	v	NOUN
ejpam-6013	303	73	and	and	CCONJ
ejpam-6013	303	74	f(g	f(g	NOUN
ejpam-6013	303	75	)	)	PUNCT
ejpam-6013	303	76	⊆	⊆	NUM
ejpam-6013	303	77	w	w	NOUN
ejpam-6013	303	78	.	.	PUNCT
ejpam-6013	304	1	it	it	PRON
ejpam-6013	304	2	follows	follow	VERB
ejpam-6013	304	3	that	that	SCONJ
ejpam-6013	304	4	u	u	PROPN
ejpam-6013	304	5	∩	∩	NOUN
ejpam-6013	304	6	g	g	NOUN
ejpam-6013	304	7	=	=	PROPN
ejpam-6013	304	8	∅.	∅.	VERB
ejpam-6013	304	9	thus	thus	ADV
ejpam-6013	304	10	,	,	PUNCT
ejpam-6013	304	11	(	(	PUNCT
ejpam-6013	304	12	x	x	NOUN
ejpam-6013	304	13	,	,	PUNCT
ejpam-6013	304	14	τ1	τ1	NOUN
ejpam-6013	304	15	,	,	PUNCT
ejpam-6013	304	16	τ2	τ2	NOUN
ejpam-6013	304	17	)	)	PUNCT
ejpam-6013	304	18	is	be	AUX
ejpam-6013	304	19	(	(	PUNCT
ejpam-6013	304	20	τ1	τ1	NOUN
ejpam-6013	304	21	,	,	PUNCT
ejpam-6013	304	22	τ2)-t2	τ2)-t2	PROPN
ejpam-6013	304	23	.	.	PUNCT
ejpam-6013	305	1	corollary	corollary	ADJ
ejpam-6013	305	2	2	2	NUM
ejpam-6013	305	3	.	.	PUNCT
ejpam-6013	306	1	if	if	SCONJ
ejpam-6013	306	2	f	f	PROPN
ejpam-6013	306	3	:	:	PUNCT
ejpam-6013	306	4	(	(	PUNCT
ejpam-6013	306	5	x	x	NOUN
ejpam-6013	306	6	,	,	PUNCT
ejpam-6013	306	7	τ1	τ1	NOUN
ejpam-6013	306	8	,	,	PUNCT
ejpam-6013	306	9	τ2	τ2	NOUN
ejpam-6013	306	10	)	)	PUNCT
ejpam-6013	306	11	→	→	SYM
ejpam-6013	306	12	(	(	PUNCT
ejpam-6013	306	13	y	y	PROPN
ejpam-6013	306	14	,	,	PUNCT
ejpam-6013	306	15	σ1	σ1	PROPN
ejpam-6013	306	16	,	,	PUNCT
ejpam-6013	306	17	σ2	σ2	PROPN
ejpam-6013	306	18	)	)	PUNCT
ejpam-6013	306	19	is	be	AUX
ejpam-6013	306	20	a	a	DET
ejpam-6013	306	21	r-(τ1	r-(τ1	PROPN
ejpam-6013	306	22	,	,	PUNCT
ejpam-6013	306	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	306	24	injection	injection	NOUN
ejpam-6013	306	25	and	and	CCONJ
ejpam-6013	306	26	(	(	PUNCT
ejpam-6013	306	27	y	y	PROPN
ejpam-6013	306	28	,	,	PUNCT
ejpam-6013	306	29	σ1	σ1	PROPN
ejpam-6013	306	30	,	,	PUNCT
ejpam-6013	306	31	σ2	σ2	PROPN
ejpam-6013	306	32	)	)	PUNCT
ejpam-6013	306	33	is	be	AUX
ejpam-6013	306	34	(	(	PUNCT
ejpam-6013	306	35	σ1	σ1	PROPN
ejpam-6013	306	36	,	,	PUNCT
ejpam-6013	306	37	σ2)-t2	σ2)-t2	PROPN
ejpam-6013	306	38	,	,	PUNCT
ejpam-6013	306	39	then	then	ADV
ejpam-6013	306	40	(	(	PUNCT
ejpam-6013	306	41	x	x	NOUN
ejpam-6013	306	42	,	,	PUNCT
ejpam-6013	306	43	τ1	τ1	NOUN
ejpam-6013	306	44	,	,	PUNCT
ejpam-6013	306	45	τ2	τ2	NOUN
ejpam-6013	306	46	)	)	PUNCT
ejpam-6013	306	47	is	be	AUX
ejpam-6013	306	48	(	(	PUNCT
ejpam-6013	306	49	τ1	τ1	NOUN
ejpam-6013	306	50	,	,	PUNCT
ejpam-6013	306	51	τ2)-t2	τ2)-t2	PROPN
ejpam-6013	306	52	.	.	PUNCT
ejpam-6013	307	1	proof	proof	NOUN
ejpam-6013	307	2	.	.	PUNCT
ejpam-6013	308	1	this	this	PRON
ejpam-6013	308	2	is	be	AUX
ejpam-6013	308	3	an	an	DET
ejpam-6013	308	4	immediate	immediate	ADJ
ejpam-6013	308	5	consequence	consequence	NOUN
ejpam-6013	308	6	of	of	ADP
ejpam-6013	308	7	lemma	lemma	PROPN
ejpam-6013	308	8	4	4	NUM
ejpam-6013	308	9	and	and	CCONJ
ejpam-6013	308	10	theorem	theorem	VERB
ejpam-6013	308	11	12	12	NUM
ejpam-6013	308	12	.	.	PUNCT
ejpam-6013	309	1	recall	recall	VERB
ejpam-6013	309	2	that	that	SCONJ
ejpam-6013	309	3	a	a	DET
ejpam-6013	309	4	bitopological	bitopological	ADJ
ejpam-6013	309	5	space	space	NOUN
ejpam-6013	309	6	(	(	PUNCT
ejpam-6013	309	7	x	x	NOUN
ejpam-6013	309	8	,	,	PUNCT
ejpam-6013	309	9	τ1	τ1	NOUN
ejpam-6013	309	10	,	,	PUNCT
ejpam-6013	309	11	τ2	τ2	NOUN
ejpam-6013	309	12	)	)	PUNCT
ejpam-6013	309	13	is	be	AUX
ejpam-6013	309	14	said	say	VERB
ejpam-6013	309	15	to	to	PART
ejpam-6013	309	16	be	be	AUX
ejpam-6013	309	17	(	(	PUNCT
ejpam-6013	309	18	τ1	τ1	NOUN
ejpam-6013	309	19	,	,	PUNCT
ejpam-6013	309	20	τ2)-r0	τ2)-r0	X
ejpam-6013	310	1	[	[	X
ejpam-6013	310	2	36	36	NUM
ejpam-6013	310	3	]	]	X
ejpam-6013	310	4	if	if	SCONJ
ejpam-6013	310	5	for	for	ADP
ejpam-6013	310	6	each	each	DET
ejpam-6013	310	7	τ1τ2	τ1τ2	ADJ
ejpam-6013	310	8	-	-	ADJ
ejpam-6013	310	9	open	open	ADJ
ejpam-6013	310	10	set	set	NOUN
ejpam-6013	310	11	u	u	NOUN
ejpam-6013	310	12	and	and	CCONJ
ejpam-6013	310	13	each	each	DET
ejpam-6013	310	14	x	x	SYM
ejpam-6013	310	15	∈	∈	PROPN
ejpam-6013	310	16	u	u	NOUN
ejpam-6013	310	17	,	,	PUNCT
ejpam-6013	310	18	τ1τ2	τ1τ2	NOUN
ejpam-6013	310	19	-	-	PUNCT
ejpam-6013	310	20	cl({x	cl({x	PRON
ejpam-6013	310	21	}	}	PUNCT
ejpam-6013	310	22	)	)	PUNCT
ejpam-6013	311	1	⊆	⊆	NUM
ejpam-6013	311	2	u	u	NOUN
ejpam-6013	311	3	.	.	PUNCT
ejpam-6013	312	1	theorem	theorem	VERB
ejpam-6013	312	2	13	13	NUM
ejpam-6013	312	3	.	.	PUNCT
ejpam-6013	313	1	if	if	SCONJ
ejpam-6013	313	2	f	f	PROPN
ejpam-6013	313	3	:	:	PUNCT
ejpam-6013	313	4	(	(	PUNCT
ejpam-6013	313	5	x	x	NOUN
ejpam-6013	313	6	,	,	PUNCT
ejpam-6013	313	7	τ1	τ1	NOUN
ejpam-6013	313	8	,	,	PUNCT
ejpam-6013	313	9	τ2	τ2	NOUN
ejpam-6013	313	10	)	)	PUNCT
ejpam-6013	313	11	→	→	SYM
ejpam-6013	313	12	(	(	PUNCT
ejpam-6013	313	13	y	y	PROPN
ejpam-6013	313	14	,	,	PUNCT
ejpam-6013	313	15	σ1	σ1	PROPN
ejpam-6013	313	16	,	,	PUNCT
ejpam-6013	313	17	σ2	σ2	PROPN
ejpam-6013	313	18	)	)	PUNCT
ejpam-6013	313	19	is	be	AUX
ejpam-6013	313	20	a	a	DET
ejpam-6013	313	21	r-(τ1	r-(τ1	PROPN
ejpam-6013	313	22	,	,	PUNCT
ejpam-6013	313	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	313	24	surjection	surjection	NOUN
ejpam-6013	313	25	,	,	PUNCT
ejpam-6013	313	26	then	then	ADV
ejpam-6013	313	27	(	(	PUNCT
ejpam-6013	313	28	y	y	PROPN
ejpam-6013	313	29	,	,	PUNCT
ejpam-6013	313	30	σ1	σ1	PROPN
ejpam-6013	313	31	,	,	PUNCT
ejpam-6013	313	32	σ2	σ2	PROPN
ejpam-6013	313	33	)	)	PUNCT
ejpam-6013	313	34	is	be	AUX
ejpam-6013	313	35	(	(	PUNCT
ejpam-6013	313	36	σ1	σ1	PROPN
ejpam-6013	313	37	,	,	PUNCT
ejpam-6013	313	38	σ2)-r0	σ2)-r0	X
ejpam-6013	313	39	.	.	PUNCT
ejpam-6013	314	1	proof	proof	NOUN
ejpam-6013	314	2	.	.	PUNCT
ejpam-6013	315	1	let	let	VERB
ejpam-6013	315	2	v	v	PART
ejpam-6013	315	3	be	be	AUX
ejpam-6013	315	4	any	any	DET
ejpam-6013	315	5	σ1σ2	σ1σ2	NOUN
ejpam-6013	315	6	-	-	ADJ
ejpam-6013	315	7	open	open	ADJ
ejpam-6013	315	8	set	set	NOUN
ejpam-6013	315	9	of	of	ADP
ejpam-6013	315	10	y	y	PROPN
ejpam-6013	315	11	and	and	CCONJ
ejpam-6013	315	12	y	y	PROPN
ejpam-6013	315	13	∈	∈	PROPN
ejpam-6013	315	14	v	v	X
ejpam-6013	315	15	.	.	PUNCT
ejpam-6013	316	1	let	let	VERB
ejpam-6013	316	2	x	x	SYM
ejpam-6013	316	3	∈	∈	PROPN
ejpam-6013	316	4	x	x	X
ejpam-6013	316	5	such	such	ADJ
ejpam-6013	316	6	that	that	SCONJ
ejpam-6013	316	7	y	y	PROPN
ejpam-6013	316	8	=	=	SYM
ejpam-6013	316	9	f(x	f(x	PROPN
ejpam-6013	316	10	)	)	PUNCT
ejpam-6013	316	11	.	.	PUNCT
ejpam-6013	317	1	since	since	SCONJ
ejpam-6013	317	2	f	f	PROPN
ejpam-6013	317	3	is	be	AUX
ejpam-6013	317	4	r-(τ1	r-(τ1	PROPN
ejpam-6013	317	5	,	,	PUNCT
ejpam-6013	317	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	317	7	,	,	PUNCT
ejpam-6013	317	8	there	there	PRON
ejpam-6013	317	9	exists	exist	VERB
ejpam-6013	317	10	a	a	DET
ejpam-6013	317	11	τ1τ2	τ1τ2	NOUN
ejpam-6013	317	12	-	-	ADJ
ejpam-6013	317	13	open	open	ADJ
ejpam-6013	317	14	set	set	ADJ
ejpam-6013	317	15	u	u	NOUN
ejpam-6013	317	16	of	of	ADP
ejpam-6013	317	17	x	x	PUNCT
ejpam-6013	317	18	containing	contain	VERB
ejpam-6013	317	19	x	x	PUNCT
ejpam-6013	317	20	such	such	ADJ
ejpam-6013	317	21	that	that	SCONJ
ejpam-6013	317	22	σ1σ2	σ1σ2	NOUN
ejpam-6013	317	23	-	-	PUNCT
ejpam-6013	317	24	cl(f(u	cl(f(u	NOUN
ejpam-6013	317	25	)	)	PUNCT
ejpam-6013	317	26	)	)	PUNCT
ejpam-6013	318	1	⊆	⊆	NUM
ejpam-6013	318	2	v	v	NOUN
ejpam-6013	318	3	.	.	PUNCT
ejpam-6013	319	1	thus	thus	ADV
ejpam-6013	319	2	,	,	PUNCT
ejpam-6013	319	3	σ1σ2	σ1σ2	NOUN
ejpam-6013	319	4	-	-	NUM
ejpam-6013	319	5	cl({y	cl({y	X
ejpam-6013	319	6	}	}	PUNCT
ejpam-6013	319	7	)	)	PUNCT
ejpam-6013	320	1	=	=	PUNCT
ejpam-6013	320	2	σ1σ2	σ1σ2	X
ejpam-6013	320	3	-	-	PUNCT
ejpam-6013	320	4	cl({f(x	cl({f(x	NOUN
ejpam-6013	320	5	)	)	PUNCT
ejpam-6013	320	6	}	}	PUNCT
ejpam-6013	320	7	)	)	PUNCT
ejpam-6013	321	1	⊆	⊆	X
ejpam-6013	321	2	σ1σ2	σ1σ2	NUM
ejpam-6013	321	3	-	-	PUNCT
ejpam-6013	321	4	cl(f(u	cl(f(u	NOUN
ejpam-6013	321	5	)	)	PUNCT
ejpam-6013	321	6	)	)	PUNCT
ejpam-6013	322	1	⊆	⊆	NUM
ejpam-6013	322	2	v	v	NOUN
ejpam-6013	322	3	and	and	CCONJ
ejpam-6013	322	4	hence	hence	ADV
ejpam-6013	322	5	(	(	PUNCT
ejpam-6013	322	6	y	y	PROPN
ejpam-6013	322	7	,	,	PUNCT
ejpam-6013	322	8	σ1	σ1	PROPN
ejpam-6013	322	9	,	,	PUNCT
ejpam-6013	322	10	σ2	σ2	PROPN
ejpam-6013	322	11	)	)	PUNCT
ejpam-6013	322	12	is	be	AUX
ejpam-6013	322	13	(	(	PUNCT
ejpam-6013	322	14	σ1	σ1	PROPN
ejpam-6013	322	15	,	,	PUNCT
ejpam-6013	322	16	σ2)-r0	σ2)-r0	PROPN
ejpam-6013	322	17	.	.	PUNCT
ejpam-6013	323	1	acknowledgements	acknowledgement	VERB
ejpam-6013	323	2	this	this	DET
ejpam-6013	323	3	research	research	NOUN
ejpam-6013	323	4	project	project	NOUN
ejpam-6013	323	5	was	be	AUX
ejpam-6013	323	6	financially	financially	ADV
ejpam-6013	323	7	supported	support	VERB
ejpam-6013	323	8	by	by	ADP
ejpam-6013	323	9	mahasarakham	mahasarakham	PROPN
ejpam-6013	323	10	university	university	PROPN
ejpam-6013	323	11	.	.	PUNCT
ejpam-6013	324	1	n.	n.	PROPN
ejpam-6013	324	2	srisarakham	srisarakham	PROPN
ejpam-6013	324	3	,	,	PUNCT
ejpam-6013	324	4	s.	s.	PROPN
ejpam-6013	324	5	sompong	sompong	PROPN
ejpam-6013	324	6	,	,	PUNCT
ejpam-6013	324	7	c.	c.	PROPN
ejpam-6013	324	8	boonpok	boonpok	PROPN
ejpam-6013	324	9	/	/	SYM
ejpam-6013	324	10	eur	eur	PROPN
ejpam-6013	324	11	.	.	PUNCT
ejpam-6013	325	1	j.	j.	PROPN
ejpam-6013	325	2	pure	pure	PROPN
ejpam-6013	325	3	appl	appl	PROPN
ejpam-6013	325	4	.	.	PROPN
ejpam-6013	325	5	math	math	PROPN
ejpam-6013	325	6	,	,	PUNCT
ejpam-6013	325	7	18	18	NUM
ejpam-6013	325	8	(	(	PUNCT
ejpam-6013	325	9	2	2	NUM
ejpam-6013	325	10	)	)	PUNCT
ejpam-6013	325	11	(	(	PUNCT
ejpam-6013	325	12	2025	2025	NUM
ejpam-6013	325	13	)	)	PUNCT
ejpam-6013	325	14	,	,	PUNCT
ejpam-6013	325	15	6013	6013	NUM
ejpam-6013	325	16	11	11	NUM
ejpam-6013	325	17	of	of	ADP
ejpam-6013	325	18	12	12	NUM
ejpam-6013	325	19	references	reference	NOUN
ejpam-6013	325	20	[	[	X
ejpam-6013	325	21	1	1	NUM
ejpam-6013	325	22	]	]	PUNCT
ejpam-6013	325	23	c.	c.	PROPN
ejpam-6013	325	24	boonpok	boonpok	PROPN
ejpam-6013	325	25	and	and	CCONJ
ejpam-6013	325	26	j.	j.	PROPN
ejpam-6013	325	27	khampakdee	khampakdee	PROPN
ejpam-6013	325	28	.	.	PUNCT
ejpam-6013	326	1	(	(	PUNCT
ejpam-6013	326	2	λ	λ	NOUN
ejpam-6013	326	3	,	,	PUNCT
ejpam-6013	326	4	sp)-open	sp)-open	ADJ
ejpam-6013	326	5	sets	set	NOUN
ejpam-6013	326	6	in	in	ADP
ejpam-6013	326	7	topological	topological	ADJ
ejpam-6013	326	8	spaces	space	NOUN
ejpam-6013	326	9	.	.	PUNCT
ejpam-6013	327	1	european	european	ADJ
ejpam-6013	327	2	journal	journal	PROPN
ejpam-6013	327	3	of	of	ADP
ejpam-6013	327	4	pure	pure	ADJ
ejpam-6013	327	5	and	and	CCONJ
ejpam-6013	327	6	applied	applied	ADJ
ejpam-6013	327	7	mathematics	mathematic	NOUN
ejpam-6013	327	8	,	,	PUNCT
ejpam-6013	327	9	15(2):572–588	15(2):572–588	NUM
ejpam-6013	327	10	,	,	PUNCT
ejpam-6013	327	11	2022	2022	NUM
ejpam-6013	327	12	.	.	PUNCT
ejpam-6013	328	1	[	[	X
ejpam-6013	328	2	2	2	NUM
ejpam-6013	328	3	]	]	PUNCT
ejpam-6013	328	4	c.	c.	PROPN
ejpam-6013	328	5	viriyapong	viriyapong	PROPN
ejpam-6013	328	6	and	and	CCONJ
ejpam-6013	328	7	c.	c.	PROPN
ejpam-6013	328	8	boonpok	boonpok	PROPN
ejpam-6013	328	9	.	.	PUNCT
ejpam-6013	329	1	(	(	PUNCT
ejpam-6013	329	2	λ	λ	X
ejpam-6013	329	3	,	,	PUNCT
ejpam-6013	329	4	sp)-continuous	sp)-continuous	ADJ
ejpam-6013	329	5	functions	function	NOUN
ejpam-6013	329	6	.	.	PUNCT
ejpam-6013	330	1	wseas	wseas	VERB
ejpam-6013	330	2	transactions	transaction	NOUN
ejpam-6013	330	3	on	on	ADP
ejpam-6013	330	4	mathematics	mathematic	NOUN
ejpam-6013	330	5	,	,	PUNCT
ejpam-6013	330	6	21:380–385	21:380–385	NUM
ejpam-6013	330	7	,	,	PUNCT
ejpam-6013	330	8	2022	2022	NUM
ejpam-6013	330	9	.	.	PUNCT
ejpam-6013	331	1	[	[	X
ejpam-6013	331	2	3	3	X
ejpam-6013	331	3	]	]	PUNCT
ejpam-6013	331	4	t.	t.	NOUN
ejpam-6013	331	5	dungthaisong	dungthaisong	PROPN
ejpam-6013	331	6	,	,	PUNCT
ejpam-6013	331	7	c.	c.	PROPN
ejpam-6013	331	8	boonpok	boonpok	PROPN
ejpam-6013	331	9	,	,	PUNCT
ejpam-6013	331	10	and	and	CCONJ
ejpam-6013	331	11	c.	c.	PROPN
ejpam-6013	331	12	viriyapong	viriyapong	PROPN
ejpam-6013	331	13	.	.	PUNCT
ejpam-6013	332	1	generalized	generalize	VERB
ejpam-6013	332	2	closed	close	VERB
ejpam-6013	332	3	sets	set	NOUN
ejpam-6013	332	4	in	in	ADP
ejpam-6013	332	5	bigeneralized	bigeneralize	VERB
ejpam-6013	332	6	topological	topological	ADJ
ejpam-6013	332	7	spaces	space	NOUN
ejpam-6013	332	8	.	.	PUNCT
ejpam-6013	333	1	international	international	ADJ
ejpam-6013	333	2	journal	journal	PROPN
ejpam-6013	333	3	of	of	ADP
ejpam-6013	333	4	mathematical	mathematical	ADJ
ejpam-6013	333	5	analysis	analysis	NOUN
ejpam-6013	333	6	,	,	PUNCT
ejpam-6013	333	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-6013	333	8	,	,	PUNCT
ejpam-6013	333	9	2011	2011	NUM
ejpam-6013	333	10	.	.	PUNCT
ejpam-6013	334	1	[	[	X
ejpam-6013	334	2	4	4	X
ejpam-6013	334	3	]	]	PUNCT
ejpam-6013	334	4	t.	t.	PROPN
ejpam-6013	334	5	duangphui	duangphui	PROPN
ejpam-6013	334	6	,	,	PUNCT
ejpam-6013	334	7	c.	c.	PROPN
ejpam-6013	334	8	boonpok	boonpok	PROPN
ejpam-6013	334	9	,	,	PUNCT
ejpam-6013	334	10	and	and	CCONJ
ejpam-6013	334	11	c.	c.	PROPN
ejpam-6013	334	12	viriyapong	viriyapong	PROPN
ejpam-6013	334	13	.	.	PUNCT
ejpam-6013	335	1	continuous	continuous	ADJ
ejpam-6013	335	2	functions	function	NOUN
ejpam-6013	335	3	on	on	ADP
ejpam-6013	335	4	bigeneralized	bigeneralize	VERB
ejpam-6013	335	5	topological	topological	ADJ
ejpam-6013	335	6	spaces	space	NOUN
ejpam-6013	335	7	.	.	PUNCT
ejpam-6013	336	1	international	international	ADJ
ejpam-6013	336	2	journal	journal	PROPN
ejpam-6013	336	3	of	of	ADP
ejpam-6013	336	4	mathematical	mathematical	ADJ
ejpam-6013	336	5	analysis	analysis	NOUN
ejpam-6013	336	6	,	,	PUNCT
ejpam-6013	336	7	5(24):1165	5(24):1165	NUM
ejpam-6013	336	8	–	–	PUNCT
ejpam-6013	336	9	1174	1174	NUM
ejpam-6013	336	10	,	,	PUNCT
ejpam-6013	336	11	2011	2011	NUM
ejpam-6013	336	12	.	.	PUNCT
ejpam-6013	337	1	[	[	X
ejpam-6013	337	2	5	5	NUM
ejpam-6013	337	3	]	]	X
ejpam-6013	337	4	n.	n.	NOUN
ejpam-6013	337	5	srisarakham	srisarakham	PROPN
ejpam-6013	337	6	and	and	CCONJ
ejpam-6013	337	7	c.	c.	PROPN
ejpam-6013	337	8	boonpok	boonpok	PROPN
ejpam-6013	337	9	.	.	PUNCT
ejpam-6013	338	1	almost	almost	ADV
ejpam-6013	338	2	(	(	PUNCT
ejpam-6013	338	3	λ	λ	NOUN
ejpam-6013	338	4	,	,	PUNCT
ejpam-6013	338	5	p)-continuous	p)-continuous	ADJ
ejpam-6013	338	6	functions	function	NOUN
ejpam-6013	338	7	.	.	PUNCT
ejpam-6013	339	1	international	international	ADJ
ejpam-6013	339	2	journal	journal	PROPN
ejpam-6013	339	3	of	of	ADP
ejpam-6013	339	4	mathematics	mathematic	NOUN
ejpam-6013	339	5	and	and	CCONJ
ejpam-6013	339	6	computer	computer	NOUN
ejpam-6013	339	7	science	science	NOUN
ejpam-6013	339	8	,	,	PUNCT
ejpam-6013	339	9	18(2):255–259	18(2):255–259	NUM
ejpam-6013	339	10	,	,	PUNCT
ejpam-6013	339	11	2023	2023	NUM
ejpam-6013	339	12	.	.	PUNCT
ejpam-6013	340	1	[	[	X
ejpam-6013	340	2	6	6	NUM
ejpam-6013	340	3	]	]	PUNCT
ejpam-6013	340	4	c.	c.	PROPN
ejpam-6013	340	5	boonpok	boonpok	PROPN
ejpam-6013	340	6	and	and	CCONJ
ejpam-6013	340	7	j.	j.	PROPN
ejpam-6013	340	8	khampakdee	khampakdee	PROPN
ejpam-6013	340	9	.	.	PUNCT
ejpam-6013	341	1	almost	almost	ADV
ejpam-6013	341	2	strong	strong	ADJ
ejpam-6013	341	3	θ(λ	θ(λ	PROPN
ejpam-6013	341	4	,	,	PUNCT
ejpam-6013	341	5	p)-continuity	p)-continuity	NOUN
ejpam-6013	341	6	for	for	ADP
ejpam-6013	341	7	functions	function	NOUN
ejpam-6013	341	8	.	.	PUNCT
ejpam-6013	342	1	european	european	ADJ
ejpam-6013	342	2	journal	journal	PROPN
ejpam-6013	342	3	of	of	ADP
ejpam-6013	342	4	pure	pure	ADJ
ejpam-6013	342	5	and	and	CCONJ
ejpam-6013	342	6	applied	applied	ADJ
ejpam-6013	342	7	mathematics	mathematic	NOUN
ejpam-6013	342	8	,	,	PUNCT
ejpam-6013	342	9	17(1):300–309	17(1):300–309	PROPN
ejpam-6013	342	10	,	,	PUNCT
ejpam-6013	342	11	2024	2024	NUM
ejpam-6013	342	12	.	.	PUNCT
ejpam-6013	343	1	[	[	X
ejpam-6013	343	2	7	7	X
ejpam-6013	343	3	]	]	X
ejpam-6013	343	4	c.	c.	PROPN
ejpam-6013	343	5	boonpok	boonpok	PROPN
ejpam-6013	343	6	and	and	CCONJ
ejpam-6013	343	7	n.	n.	PROPN
ejpam-6013	343	8	srisarakham	srisarakham	PROPN
ejpam-6013	343	9	.	.	PUNCT
ejpam-6013	344	1	weak	weak	ADJ
ejpam-6013	344	2	forms	form	NOUN
ejpam-6013	344	3	of	of	ADP
ejpam-6013	344	4	(	(	PUNCT
ejpam-6013	344	5	λ	λ	PROPN
ejpam-6013	344	6	,	,	PUNCT
ejpam-6013	344	7	b)-open	b)-open	VERB
ejpam-6013	344	8	sets	set	NOUN
ejpam-6013	344	9	and	and	CCONJ
ejpam-6013	344	10	weak	weak	ADJ
ejpam-6013	344	11	(	(	PUNCT
ejpam-6013	344	12	λ	λ	NOUN
ejpam-6013	344	13	,	,	PUNCT
ejpam-6013	344	14	b)continuity	b)continuity	NOUN
ejpam-6013	344	15	.	.	PUNCT
ejpam-6013	345	1	european	european	PROPN
ejpam-6013	345	2	journal	journal	PROPN
ejpam-6013	345	3	of	of	ADP
ejpam-6013	345	4	pure	pure	ADJ
ejpam-6013	345	5	and	and	CCONJ
ejpam-6013	345	6	applied	applied	ADJ
ejpam-6013	345	7	mathematics	mathematic	NOUN
ejpam-6013	345	8	,	,	PUNCT
ejpam-6013	345	9	16(1):29–43	16(1):29–43	NUM
ejpam-6013	345	10	,	,	PUNCT
ejpam-6013	345	11	2023	2023	NUM
ejpam-6013	345	12	.	.	PUNCT
ejpam-6013	346	1	[	[	X
ejpam-6013	346	2	8	8	NUM
ejpam-6013	346	3	]	]	X
ejpam-6013	346	4	c.	c.	PROPN
ejpam-6013	346	5	boonpok	boonpok	PROPN
ejpam-6013	346	6	.	.	PUNCT
ejpam-6013	347	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-6013	347	2	.	.	PUNCT
ejpam-6013	348	1	mathematica	mathematica	PROPN
ejpam-6013	348	2	,	,	PUNCT
ejpam-6013	348	3	65(1):31–42	65(1):31–42	NUM
ejpam-6013	348	4	,	,	PUNCT
ejpam-6013	348	5	2023	2023	NUM
ejpam-6013	348	6	.	.	PUNCT
ejpam-6013	349	1	[	[	X
ejpam-6013	349	2	9	9	NUM
ejpam-6013	349	3	]	]	PUNCT
ejpam-6013	349	4	c.	c.	PROPN
ejpam-6013	349	5	boonpok	boonpok	PROPN
ejpam-6013	349	6	.	.	PUNCT
ejpam-6013	350	1	on	on	ADP
ejpam-6013	350	2	some	some	DET
ejpam-6013	350	3	closed	closed	ADJ
ejpam-6013	350	4	sets	set	NOUN
ejpam-6013	350	5	and	and	CCONJ
ejpam-6013	350	6	low	low	ADJ
ejpam-6013	350	7	separation	separation	NOUN
ejpam-6013	350	8	axioms	axiom	NOUN
ejpam-6013	350	9	via	via	ADP
ejpam-6013	350	10	topological	topological	ADJ
ejpam-6013	350	11	ideals	ideal	NOUN
ejpam-6013	350	12	.	.	PUNCT
ejpam-6013	351	1	european	european	ADJ
ejpam-6013	351	2	journal	journal	PROPN
ejpam-6013	351	3	of	of	ADP
ejpam-6013	351	4	pure	pure	ADJ
ejpam-6013	351	5	and	and	CCONJ
ejpam-6013	351	6	applied	applied	ADJ
ejpam-6013	351	7	mathematics	mathematic	NOUN
ejpam-6013	351	8	,	,	PUNCT
ejpam-6013	351	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-6013	351	10	,	,	PUNCT
ejpam-6013	351	11	2022	2022	NUM
ejpam-6013	351	12	.	.	PUNCT
ejpam-6013	352	1	[	[	X
ejpam-6013	352	2	10	10	NUM
ejpam-6013	352	3	]	]	X
ejpam-6013	352	4	c.	c.	PROPN
ejpam-6013	352	5	boonpok	boonpok	PROPN
ejpam-6013	352	6	.	.	PUNCT
ejpam-6013	353	1	on	on	ADP
ejpam-6013	353	2	some	some	DET
ejpam-6013	353	3	spaces	space	NOUN
ejpam-6013	353	4	via	via	ADP
ejpam-6013	353	5	topological	topological	ADJ
ejpam-6013	353	6	ideals	ideal	NOUN
ejpam-6013	353	7	.	.	PUNCT
ejpam-6013	354	1	open	open	ADJ
ejpam-6013	354	2	mathematics	mathematic	NOUN
ejpam-6013	354	3	,	,	PUNCT
ejpam-6013	354	4	21:20230118	21:20230118	NUM
ejpam-6013	354	5	,	,	PUNCT
ejpam-6013	354	6	2023	2023	NUM
ejpam-6013	354	7	.	.	PUNCT
ejpam-6013	355	1	[	[	X
ejpam-6013	355	2	11	11	NUM
ejpam-6013	355	3	]	]	PUNCT
ejpam-6013	355	4	c.	c.	PROPN
ejpam-6013	355	5	boonpok	boonpok	PROPN
ejpam-6013	355	6	.	.	PUNCT
ejpam-6013	356	1	on	on	ADP
ejpam-6013	356	2	characterizations	characterization	NOUN
ejpam-6013	356	3	of	of	ADP
ejpam-6013	356	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-6013	356	5	ideal	ideal	ADJ
ejpam-6013	356	6	topological	topological	ADJ
ejpam-6013	356	7	spaces	space	NOUN
ejpam-6013	356	8	.	.	PUNCT
ejpam-6013	357	1	journal	journal	NOUN
ejpam-6013	357	2	of	of	ADP
ejpam-6013	357	3	mathematics	mathematic	NOUN
ejpam-6013	357	4	,	,	PUNCT
ejpam-6013	357	5	2020:9387601	2020:9387601	NUM
ejpam-6013	357	6	,	,	PUNCT
ejpam-6013	357	7	2020	2020	NUM
ejpam-6013	357	8	.	.	PUNCT
ejpam-6013	358	1	[	[	X
ejpam-6013	358	2	12	12	NUM
ejpam-6013	358	3	]	]	PUNCT
ejpam-6013	358	4	c.	c.	PROPN
ejpam-6013	358	5	boonpok	boonpok	PROPN
ejpam-6013	358	6	.	.	PUNCT
ejpam-6013	359	1	almost	almost	ADV
ejpam-6013	359	2	(	(	PUNCT
ejpam-6013	359	3	g	g	NOUN
ejpam-6013	359	4	,	,	PUNCT
ejpam-6013	359	5	m)-continuous	m)-continuous	ADJ
ejpam-6013	359	6	functions	function	NOUN
ejpam-6013	359	7	.	.	PUNCT
ejpam-6013	360	1	international	international	ADJ
ejpam-6013	360	2	journal	journal	PROPN
ejpam-6013	360	3	of	of	ADP
ejpam-6013	360	4	mathematical	mathematical	ADJ
ejpam-6013	360	5	analysis	analysis	NOUN
ejpam-6013	360	6	,	,	PUNCT
ejpam-6013	360	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-6013	360	8	,	,	PUNCT
ejpam-6013	360	9	2010	2010	NUM
ejpam-6013	360	10	.	.	PUNCT
ejpam-6013	361	1	[	[	X
ejpam-6013	361	2	13	13	NUM
ejpam-6013	361	3	]	]	PUNCT
ejpam-6013	361	4	c.	c.	PROPN
ejpam-6013	361	5	boonpok	boonpok	PROPN
ejpam-6013	361	6	.	.	PUNCT
ejpam-6013	362	1	m	m	VERB
ejpam-6013	362	2	-continuous	-continuous	ADJ
ejpam-6013	362	3	functions	function	NOUN
ejpam-6013	362	4	in	in	ADP
ejpam-6013	362	5	biminimal	biminimal	NOUN
ejpam-6013	362	6	structure	structure	NOUN
ejpam-6013	362	7	spaces	space	NOUN
ejpam-6013	362	8	.	.	PUNCT
ejpam-6013	363	1	far	far	PROPN
ejpam-6013	363	2	east	east	PROPN
ejpam-6013	363	3	journal	journal	PROPN
ejpam-6013	363	4	of	of	ADP
ejpam-6013	363	5	mathematical	mathematical	ADJ
ejpam-6013	363	6	sciences	science	NOUN
ejpam-6013	363	7	,	,	PUNCT
ejpam-6013	363	8	43(1):41–58	43(1):41–58	NUM
ejpam-6013	363	9	,	,	PUNCT
ejpam-6013	363	10	2010	2010	NUM
ejpam-6013	363	11	.	.	PUNCT
ejpam-6013	364	1	[	[	X
ejpam-6013	364	2	14	14	NUM
ejpam-6013	364	3	]	]	X
ejpam-6013	364	4	ch	ch	NOUN
ejpam-6013	364	5	.	.	PUNCT
ejpam-6013	364	6	konstadilaki	konstadilaki	PROPN
ejpam-6013	364	7	-	-	PUNCT
ejpam-6013	364	8	savvopoulou	savvopoulou	PROPN
ejpam-6013	364	9	and	and	CCONJ
ejpam-6013	364	10	d.	d.	PROPN
ejpam-6013	364	11	janković.	janković.	PROPN
ejpam-6013	364	12	r	r	NOUN
ejpam-6013	364	13	-	-	PUNCT
ejpam-6013	364	14	continuous	continuous	ADJ
ejpam-6013	364	15	functions	function	NOUN
ejpam-6013	364	16	.	.	PUNCT
ejpam-6013	365	1	international	international	ADJ
ejpam-6013	365	2	journal	journal	PROPN
ejpam-6013	365	3	of	of	ADP
ejpam-6013	365	4	mathematics	mathematics	PROPN
ejpam-6013	365	5	and	and	CCONJ
ejpam-6013	365	6	mathematical	mathematical	ADJ
ejpam-6013	365	7	sciences	science	NOUN
ejpam-6013	365	8	,	,	PUNCT
ejpam-6013	365	9	15:57–64	15:57–64	NUM
ejpam-6013	365	10	,	,	PUNCT
ejpam-6013	365	11	1992	1992	NUM
ejpam-6013	365	12	.	.	PUNCT
ejpam-6013	366	1	[	[	X
ejpam-6013	366	2	15	15	NUM
ejpam-6013	366	3	]	]	X
ejpam-6013	366	4	s.	s.	PROPN
ejpam-6013	366	5	g.	g.	PROPN
ejpam-6013	366	6	crossley	crossley	PROPN
ejpam-6013	366	7	and	and	CCONJ
ejpam-6013	366	8	s.	s.	PROPN
ejpam-6013	366	9	k.	k.	PROPN
ejpam-6013	366	10	hildebrand	hildebrand	PROPN
ejpam-6013	366	11	.	.	PUNCT
ejpam-6013	367	1	semi	semi	ADJ
ejpam-6013	367	2	-	-	ADJ
ejpam-6013	367	3	topological	topological	ADJ
ejpam-6013	367	4	properties	property	NOUN
ejpam-6013	367	5	.	.	PUNCT
ejpam-6013	368	1	fundamenta	fundamenta	PROPN
ejpam-6013	368	2	mathematicae	mathematicae	PROPN
ejpam-6013	368	3	,	,	PUNCT
ejpam-6013	368	4	74:233–254	74:233–254	PROPN
ejpam-6013	368	5	,	,	PUNCT
ejpam-6013	368	6	1972	1972	NUM
ejpam-6013	368	7	.	.	PUNCT
ejpam-6013	369	1	[	[	X
ejpam-6013	369	2	16	16	NUM
ejpam-6013	369	3	]	]	X
ejpam-6013	369	4	i.	i.	PROPN
ejpam-6013	369	5	l.	l.	PROPN
ejpam-6013	369	6	reilly	reilly	PROPN
ejpam-6013	369	7	and	and	CCONJ
ejpam-6013	369	8	m.	m.	PROPN
ejpam-6013	369	9	k.	k.	PROPN
ejpam-6013	369	10	vamanamurthy	vamanamurthy	PROPN
ejpam-6013	369	11	.	.	PUNCT
ejpam-6013	370	1	on	on	ADP
ejpam-6013	370	2	α	α	NOUN
ejpam-6013	370	3	-	-	NOUN
ejpam-6013	370	4	continuity	continuity	NOUN
ejpam-6013	370	5	in	in	ADP
ejpam-6013	370	6	topological	topological	ADJ
ejpam-6013	370	7	spaces	space	NOUN
ejpam-6013	370	8	.	.	PUNCT
ejpam-6013	371	1	acta	acta	PROPN
ejpam-6013	371	2	mathematica	mathematica	PROPN
ejpam-6013	371	3	hungarica	hungarica	PROPN
ejpam-6013	371	4	,	,	PUNCT
ejpam-6013	371	5	45:27–32	45:27–32	PROPN
ejpam-6013	371	6	,	,	PUNCT
ejpam-6013	371	7	1985	1985	NUM
ejpam-6013	371	8	.	.	PUNCT
ejpam-6013	372	1	[	[	X
ejpam-6013	372	2	17	17	NUM
ejpam-6013	372	3	]	]	X
ejpam-6013	372	4	c.	c.	PROPN
ejpam-6013	372	5	w.	w.	PROPN
ejpam-6013	372	6	baker	baker	PROPN
ejpam-6013	372	7	.	.	PUNCT
ejpam-6013	373	1	r	r	X
ejpam-6013	373	2	-	-	PUNCT
ejpam-6013	373	3	irresolute	irresolute	ADJ
ejpam-6013	373	4	functions	function	NOUN
ejpam-6013	373	5	.	.	PUNCT
ejpam-6013	374	1	acta	acta	PROPN
ejpam-6013	374	2	ciencia	ciencia	PROPN
ejpam-6013	374	3	indica	indica	PROPN
ejpam-6013	374	4	,	,	PUNCT
ejpam-6013	374	5	30:775–780	30:775–780	PROPN
ejpam-6013	374	6	,	,	PUNCT
ejpam-6013	374	7	2004	2004	NUM
ejpam-6013	374	8	.	.	PUNCT
ejpam-6013	375	1	[	[	X
ejpam-6013	375	2	18	18	NUM
ejpam-6013	375	3	]	]	X
ejpam-6013	375	4	c.	c.	PROPN
ejpam-6013	375	5	w.	w.	PROPN
ejpam-6013	375	6	baker	baker	PROPN
ejpam-6013	375	7	.	.	PUNCT
ejpam-6013	376	1	r	r	X
ejpam-6013	376	2	-	-	PUNCT
ejpam-6013	376	3	preirresolute	preirresolute	ADJ
ejpam-6013	376	4	functions	function	NOUN
ejpam-6013	376	5	.	.	PUNCT
ejpam-6013	377	1	carpathian	carpathian	ADJ
ejpam-6013	377	2	journal	journal	PROPN
ejpam-6013	377	3	of	of	ADP
ejpam-6013	377	4	mathematics	mathematic	NOUN
ejpam-6013	377	5	,	,	PUNCT
ejpam-6013	377	6	21:1–6	21:1–6	NUM
ejpam-6013	377	7	,	,	PUNCT
ejpam-6013	377	8	2005	2005	NUM
ejpam-6013	377	9	.	.	PUNCT
ejpam-6013	378	1	[	[	X
ejpam-6013	378	2	19	19	NUM
ejpam-6013	378	3	]	]	X
ejpam-6013	378	4	y.	y.	NOUN
ejpam-6013	378	5	beceren	beceren	PROPN
ejpam-6013	378	6	and	and	CCONJ
ejpam-6013	378	7	t.	t.	PROPN
ejpam-6013	378	8	noiri	noiri	PROPN
ejpam-6013	378	9	.	.	PUNCT
ejpam-6013	379	1	almost	almost	ADV
ejpam-6013	379	2	α	α	NUM
ejpam-6013	379	3	-	-	PUNCT
ejpam-6013	379	4	irresolute	irresolute	ADJ
ejpam-6013	379	5	functions	function	NOUN
ejpam-6013	379	6	.	.	PUNCT
ejpam-6013	380	1	bulletin	bulletin	NOUN
ejpam-6013	380	2	of	of	ADP
ejpam-6013	380	3	the	the	DET
ejpam-6013	380	4	calcutta	calcutta	PROPN
ejpam-6013	380	5	mathematical	mathematical	ADJ
ejpam-6013	380	6	society	society	NOUN
ejpam-6013	380	7	,	,	PUNCT
ejpam-6013	380	8	92:213–218	92:213–218	NUM
ejpam-6013	380	9	,	,	PUNCT
ejpam-6013	380	10	2000	2000	NUM
ejpam-6013	380	11	.	.	PUNCT
ejpam-6013	381	1	[	[	X
ejpam-6013	381	2	20	20	NUM
ejpam-6013	381	3	]	]	PUNCT
ejpam-6013	381	4	m.	m.	NOUN
ejpam-6013	381	5	e.	e.	PROPN
ejpam-6013	381	6	abd	abd	PROPN
ejpam-6013	382	1	el	el	PROPN
ejpam-6013	382	2	-	-	PROPN
ejpam-6013	382	3	monsef	monsef	PROPN
ejpam-6013	382	4	,	,	PUNCT
ejpam-6013	382	5	r.	r.	PROPN
ejpam-6013	382	6	a.	a.	PROPN
ejpam-6013	382	7	mahmoud	mahmoud	PROPN
ejpam-6013	382	8	,	,	PUNCT
ejpam-6013	382	9	and	and	CCONJ
ejpam-6013	382	10	a.	a.	NOUN
ejpam-6013	382	11	a.	a.	NOUN
ejpam-6013	382	12	nasef	nasef	PROPN
ejpam-6013	382	13	.	.	PUNCT
ejpam-6013	383	1	a	a	DET
ejpam-6013	383	2	class	class	NOUN
ejpam-6013	383	3	of	of	ADP
ejpam-6013	383	4	functions	function	NOUN
ejpam-6013	383	5	stronger	strong	ADJ
ejpam-6013	383	6	than	than	ADP
ejpam-6013	383	7	m	m	NOUN
ejpam-6013	383	8	-precontinuity	-precontinuity	ADJ
ejpam-6013	383	9	,	,	PUNCT
ejpam-6013	383	10	preirresolute	preirresolute	ADJ
ejpam-6013	383	11	and	and	CCONJ
ejpam-6013	383	12	a	a	DET
ejpam-6013	383	13	-	-	PUNCT
ejpam-6013	383	14	functions	function	NOUN
ejpam-6013	383	15	.	.	PUNCT
ejpam-6013	384	1	qatar	qatar	PROPN
ejpam-6013	384	2	university	university	PROPN
ejpam-6013	384	3	science	science	NOUN
ejpam-6013	384	4	bulletin	bulletin	NOUN
ejpam-6013	384	5	,	,	PUNCT
ejpam-6013	384	6	10:41–48	10:41–48	NUM
ejpam-6013	384	7	,	,	PUNCT
ejpam-6013	384	8	1990	1990	NUM
ejpam-6013	384	9	.	.	PUNCT
ejpam-6013	385	1	n.	n.	PROPN
ejpam-6013	385	2	srisarakham	srisarakham	PROPN
ejpam-6013	385	3	,	,	PUNCT
ejpam-6013	385	4	s.	s.	PROPN
ejpam-6013	385	5	sompong	sompong	PROPN
ejpam-6013	385	6	,	,	PUNCT
ejpam-6013	385	7	c.	c.	PROPN
ejpam-6013	385	8	boonpok	boonpok	PROPN
ejpam-6013	385	9	/	/	SYM
ejpam-6013	385	10	eur	eur	PROPN
ejpam-6013	385	11	.	.	PUNCT
ejpam-6013	386	1	j.	j.	PROPN
ejpam-6013	386	2	pure	pure	PROPN
ejpam-6013	386	3	appl	appl	PROPN
ejpam-6013	386	4	.	.	PROPN
ejpam-6013	386	5	math	math	PROPN
ejpam-6013	386	6	,	,	PUNCT
ejpam-6013	386	7	18	18	NUM
ejpam-6013	386	8	(	(	PUNCT
ejpam-6013	386	9	2	2	NUM
ejpam-6013	386	10	)	)	PUNCT
ejpam-6013	386	11	(	(	PUNCT
ejpam-6013	386	12	2025	2025	NUM
ejpam-6013	386	13	)	)	PUNCT
ejpam-6013	386	14	,	,	PUNCT
ejpam-6013	386	15	6013	6013	NUM
ejpam-6013	386	16	12	12	NUM
ejpam-6013	386	17	of	of	ADP
ejpam-6013	386	18	12	12	NUM
ejpam-6013	386	19	[	[	X
ejpam-6013	386	20	21	21	NUM
ejpam-6013	386	21	]	]	PUNCT
ejpam-6013	386	22	t.	t.	PROPN
ejpam-6013	386	23	noiri	noiri	PROPN
ejpam-6013	386	24	and	and	CCONJ
ejpam-6013	386	25	v.	v.	ADP
ejpam-6013	386	26	popa	popa	NOUN
ejpam-6013	386	27	.	.	PUNCT
ejpam-6013	387	1	a	a	DET
ejpam-6013	387	2	unified	unified	ADJ
ejpam-6013	387	3	theory	theory	NOUN
ejpam-6013	387	4	of	of	ADP
ejpam-6013	387	5	r	r	NOUN
ejpam-6013	387	6	-	-	PUNCT
ejpam-6013	387	7	continuity	continuity	NOUN
ejpam-6013	387	8	.	.	PUNCT
ejpam-6013	388	1	acta	acta	PROPN
ejpam-6013	388	2	mathematica	mathematica	PROPN
ejpam-6013	388	3	hungarica	hungarica	PROPN
ejpam-6013	388	4	,	,	PUNCT
ejpam-6013	388	5	118:61–74	118:61–74	NUM
ejpam-6013	388	6	,	,	PUNCT
ejpam-6013	388	7	2008	2008	NUM
ejpam-6013	388	8	.	.	PUNCT
ejpam-6013	389	1	[	[	X
ejpam-6013	389	2	22	22	NUM
ejpam-6013	389	3	]	]	PUNCT
ejpam-6013	389	4	c.	c.	PROPN
ejpam-6013	389	5	boonpok	boonpok	PROPN
ejpam-6013	389	6	and	and	CCONJ
ejpam-6013	389	7	n.	n.	PROPN
ejpam-6013	389	8	srisarakham	srisarakham	PROPN
ejpam-6013	389	9	.	.	PUNCT
ejpam-6013	390	1	(	(	PUNCT
ejpam-6013	390	2	τ1	τ1	NOUN
ejpam-6013	390	3	,	,	PUNCT
ejpam-6013	390	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6013	390	5	for	for	ADP
ejpam-6013	390	6	functions	function	NOUN
ejpam-6013	390	7	.	.	PUNCT
ejpam-6013	391	1	asia	asia	PROPN
ejpam-6013	391	2	pacific	pacific	PROPN
ejpam-6013	391	3	journal	journal	PROPN
ejpam-6013	391	4	of	of	ADP
ejpam-6013	391	5	mathematics	mathematic	NOUN
ejpam-6013	391	6	,	,	PUNCT
ejpam-6013	391	7	11:21	11:21	NUM
ejpam-6013	391	8	,	,	PUNCT
ejpam-6013	391	9	2024	2024	NUM
ejpam-6013	391	10	.	.	PUNCT
ejpam-6013	392	1	[	[	X
ejpam-6013	392	2	23	23	NUM
ejpam-6013	392	3	]	]	X
ejpam-6013	392	4	c.	c.	PROPN
ejpam-6013	392	5	boonpok	boonpok	PROPN
ejpam-6013	392	6	and	and	CCONJ
ejpam-6013	392	7	p.	p.	NOUN
ejpam-6013	392	8	pue	pue	NOUN
ejpam-6013	392	9	-	-	PUNCT
ejpam-6013	392	10	on	on	ADP
ejpam-6013	392	11	.	.	PUNCT
ejpam-6013	393	1	characterizations	characterization	NOUN
ejpam-6013	393	2	of	of	ADP
ejpam-6013	393	3	almost	almost	ADV
ejpam-6013	393	4	(	(	PUNCT
ejpam-6013	393	5	τ1	τ1	NOUN
ejpam-6013	393	6	,	,	PUNCT
ejpam-6013	393	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	393	8	functions	function	NOUN
ejpam-6013	393	9	.	.	PUNCT
ejpam-6013	394	1	international	international	ADJ
ejpam-6013	394	2	journal	journal	NOUN
ejpam-6013	394	3	of	of	ADP
ejpam-6013	394	4	analysis	analysis	NOUN
ejpam-6013	394	5	and	and	CCONJ
ejpam-6013	394	6	applications	application	NOUN
ejpam-6013	394	7	,	,	PUNCT
ejpam-6013	394	8	22:33	22:33	NUM
ejpam-6013	394	9	,	,	PUNCT
ejpam-6013	394	10	2024	2024	NUM
ejpam-6013	394	11	.	.	PUNCT
ejpam-6013	395	1	[	[	X
ejpam-6013	395	2	24	24	NUM
ejpam-6013	395	3	]	]	PUNCT
ejpam-6013	395	4	c.	c.	PROPN
ejpam-6013	395	5	boonpok	boonpok	PROPN
ejpam-6013	395	6	and	and	CCONJ
ejpam-6013	395	7	c.	c.	PROPN
ejpam-6013	395	8	klanarong	klanarong	PROPN
ejpam-6013	395	9	.	.	PUNCT
ejpam-6013	396	1	on	on	ADP
ejpam-6013	396	2	weakly	weakly	ADJ
ejpam-6013	396	3	(	(	PUNCT
ejpam-6013	396	4	τ1	τ1	NOUN
ejpam-6013	396	5	,	,	PUNCT
ejpam-6013	396	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	396	7	functions	function	NOUN
ejpam-6013	396	8	.	.	PUNCT
ejpam-6013	397	1	european	european	ADJ
ejpam-6013	397	2	journal	journal	PROPN
ejpam-6013	397	3	of	of	ADP
ejpam-6013	397	4	pure	pure	ADJ
ejpam-6013	397	5	and	and	CCONJ
ejpam-6013	397	6	applied	applied	ADJ
ejpam-6013	397	7	mathematics	mathematic	NOUN
ejpam-6013	397	8	,	,	PUNCT
ejpam-6013	397	9	17(1):416–425	17(1):416–425	NUM
ejpam-6013	397	10	,	,	PUNCT
ejpam-6013	397	11	2024	2024	NUM
ejpam-6013	397	12	.	.	PUNCT
ejpam-6013	398	1	[	[	X
ejpam-6013	398	2	25	25	NUM
ejpam-6013	398	3	]	]	PUNCT
ejpam-6013	398	4	c.	c.	PROPN
ejpam-6013	398	5	boonpok	boonpok	PROPN
ejpam-6013	398	6	,	,	PUNCT
ejpam-6013	398	7	c.	c.	PROPN
ejpam-6013	398	8	viriyapong	viriyapong	PROPN
ejpam-6013	398	9	,	,	PUNCT
ejpam-6013	398	10	and	and	CCONJ
ejpam-6013	398	11	m.	m.	NOUN
ejpam-6013	398	12	thongmoon	thongmoon	NOUN
ejpam-6013	398	13	.	.	PUNCT
ejpam-6013	399	1	on	on	ADP
ejpam-6013	399	2	upper	upper	ADJ
ejpam-6013	399	3	and	and	CCONJ
ejpam-6013	399	4	lower	low	ADJ
ejpam-6013	399	5	(	(	PUNCT
ejpam-6013	399	6	τ1	τ1	NOUN
ejpam-6013	399	7	,	,	PUNCT
ejpam-6013	399	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6013	399	9	multifunctions	multifunction	NOUN
ejpam-6013	399	10	.	.	PUNCT
ejpam-6013	400	1	journal	journal	PROPN
ejpam-6013	400	2	of	of	ADP
ejpam-6013	400	3	mathematics	mathematics	PROPN
ejpam-6013	400	4	and	and	CCONJ
ejpam-6013	400	5	computer	computer	NOUN
ejpam-6013	400	6	science	science	NOUN
ejpam-6013	400	7	,	,	PUNCT
ejpam-6013	400	8	18:282–293	18:282–293	NUM
ejpam-6013	400	9	,	,	PUNCT
ejpam-6013	400	10	2018	2018	NUM
ejpam-6013	400	11	.	.	PUNCT
ejpam-6013	401	1	[	[	X
ejpam-6013	401	2	26	26	NUM
ejpam-6013	401	3	]	]	X
ejpam-6013	401	4	c.	c.	PROPN
ejpam-6013	401	5	viriyapong	viriyapong	PROPN
ejpam-6013	401	6	and	and	CCONJ
ejpam-6013	401	7	c.	c.	PROPN
ejpam-6013	401	8	boonpok	boonpok	PROPN
ejpam-6013	401	9	.	.	PUNCT
ejpam-6013	402	1	(	(	PUNCT
ejpam-6013	402	2	τ1	τ1	NOUN
ejpam-6013	402	3	,	,	PUNCT
ejpam-6013	402	4	τ2)α	τ2)α	NOUN
ejpam-6013	402	5	-	-	PUNCT
ejpam-6013	402	6	continuity	continuity	NOUN
ejpam-6013	402	7	for	for	ADP
ejpam-6013	402	8	multifunctions	multifunction	NOUN
ejpam-6013	402	9	.	.	PUNCT
ejpam-6013	403	1	journal	journal	PROPN
ejpam-6013	403	2	of	of	ADP
ejpam-6013	403	3	mathematics	mathematic	NOUN
ejpam-6013	403	4	,	,	PUNCT
ejpam-6013	403	5	2020:6285763	2020:6285763	NUM
ejpam-6013	403	6	,	,	PUNCT
ejpam-6013	403	7	2020	2020	NUM
ejpam-6013	403	8	.	.	PUNCT
ejpam-6013	404	1	[	[	X
ejpam-6013	404	2	27	27	NUM
ejpam-6013	404	3	]	]	X
ejpam-6013	404	4	c.	c.	PROPN
ejpam-6013	404	5	boonpok	boonpok	PROPN
ejpam-6013	404	6	.	.	PUNCT
ejpam-6013	405	1	(	(	PUNCT
ejpam-6013	405	2	τ1	τ1	NOUN
ejpam-6013	405	3	,	,	PUNCT
ejpam-6013	405	4	τ2)δ	τ2)δ	ADJ
ejpam-6013	405	5	-	-	PUNCT
ejpam-6013	405	6	semicontinuous	semicontinuous	ADJ
ejpam-6013	405	7	multifunctions	multifunction	NOUN
ejpam-6013	405	8	.	.	PUNCT
ejpam-6013	406	1	heliyon	heliyon	NOUN
ejpam-6013	406	2	,	,	PUNCT
ejpam-6013	406	3	6	6	NUM
ejpam-6013	406	4	:	:	SYM
ejpam-6013	406	5	e05367	e05367	PROPN
ejpam-6013	406	6	,	,	PUNCT
ejpam-6013	406	7	2020	2020	NUM
ejpam-6013	406	8	.	.	PUNCT
ejpam-6013	407	1	[	[	X
ejpam-6013	407	2	28	28	NUM
ejpam-6013	407	3	]	]	X
ejpam-6013	407	4	n.	n.	PROPN
ejpam-6013	407	5	viriyapong	viriyapong	PROPN
ejpam-6013	407	6	,	,	PUNCT
ejpam-6013	407	7	s.	s.	PROPN
ejpam-6013	407	8	sompong	sompong	PROPN
ejpam-6013	407	9	,	,	PUNCT
ejpam-6013	407	10	and	and	CCONJ
ejpam-6013	407	11	c.	c.	PROPN
ejpam-6013	407	12	boonpok	boonpok	PROPN
ejpam-6013	407	13	.	.	PUNCT
ejpam-6013	408	1	(	(	PUNCT
ejpam-6013	408	2	τ1	τ1	NOUN
ejpam-6013	408	3	,	,	PUNCT
ejpam-6013	408	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6013	408	5	disconnectedness	disconnectedness	NOUN
ejpam-6013	408	6	in	in	ADP
ejpam-6013	408	7	bitopological	bitopological	ADJ
ejpam-6013	408	8	spaces	space	NOUN
ejpam-6013	408	9	.	.	PUNCT
ejpam-6013	409	1	international	international	ADJ
ejpam-6013	409	2	journal	journal	PROPN
ejpam-6013	409	3	of	of	ADP
ejpam-6013	409	4	mathematics	mathematic	NOUN
ejpam-6013	409	5	and	and	CCONJ
ejpam-6013	409	6	computer	computer	NOUN
ejpam-6013	409	7	science	science	NOUN
ejpam-6013	409	8	,	,	PUNCT
ejpam-6013	409	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6013	409	10	,	,	PUNCT
ejpam-6013	409	11	2024	2024	NUM
ejpam-6013	409	12	.	.	PUNCT
ejpam-6013	410	1	[	[	X
ejpam-6013	410	2	29	29	NUM
ejpam-6013	410	3	]	]	X
ejpam-6013	410	4	p.	p.	NOUN
ejpam-6013	410	5	pue	pue	NOUN
ejpam-6013	410	6	-	-	PUNCT
ejpam-6013	410	7	on	on	ADP
ejpam-6013	410	8	,	,	PUNCT
ejpam-6013	410	9	s.	s.	PROPN
ejpam-6013	410	10	sompong	sompong	PROPN
ejpam-6013	410	11	,	,	PUNCT
ejpam-6013	410	12	and	and	CCONJ
ejpam-6013	410	13	c.	c.	PROPN
ejpam-6013	410	14	boonpok	boonpok	PROPN
ejpam-6013	410	15	.	.	PUNCT
ejpam-6013	411	1	on	on	ADP
ejpam-6013	411	2	strongly	strongly	ADV
ejpam-6013	411	3	θ(τ1	θ(τ1	NOUN
ejpam-6013	411	4	,	,	PUNCT
ejpam-6013	411	5	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6013	411	6	functions	function	NOUN
ejpam-6013	411	7	.	.	PUNCT
ejpam-6013	412	1	(	(	PUNCT
ejpam-6013	412	2	submitted	submit	VERB
ejpam-6013	412	3	)	)	PUNCT
ejpam-6013	412	4	.	.	PUNCT
ejpam-6013	413	1	[	[	X
ejpam-6013	413	2	30	30	NUM
ejpam-6013	413	3	]	]	X
ejpam-6013	413	4	m.	m.	NOUN
ejpam-6013	413	5	chiangpradit	chiangpradit	NOUN
ejpam-6013	413	6	,	,	PUNCT
ejpam-6013	413	7	s.	s.	PROPN
ejpam-6013	413	8	sompong	sompong	PROPN
ejpam-6013	413	9	,	,	PUNCT
ejpam-6013	413	10	and	and	CCONJ
ejpam-6013	413	11	c.	c.	PROPN
ejpam-6013	413	12	boonpok	boonpok	PROPN
ejpam-6013	413	13	.	.	PUNCT
ejpam-6013	414	1	on	on	ADP
ejpam-6013	414	2	characterizations	characterization	NOUN
ejpam-6013	414	3	of	of	ADP
ejpam-6013	414	4	(	(	PUNCT
ejpam-6013	414	5	τ1	τ1	NOUN
ejpam-6013	414	6	,	,	PUNCT
ejpam-6013	414	7	τ2)regular	τ2)regular	ADJ
ejpam-6013	414	8	spaces	space	NOUN
ejpam-6013	414	9	.	.	PUNCT
ejpam-6013	415	1	international	international	ADJ
ejpam-6013	415	2	journal	journal	PROPN
ejpam-6013	415	3	of	of	ADP
ejpam-6013	415	4	mathematics	mathematic	NOUN
ejpam-6013	415	5	and	and	CCONJ
ejpam-6013	415	6	computer	computer	NOUN
ejpam-6013	415	7	science	science	NOUN
ejpam-6013	415	8	,	,	PUNCT
ejpam-6013	415	9	19(4):1229–1334	19(4):1229–1334	NUM
ejpam-6013	415	10	,	,	PUNCT
ejpam-6013	415	11	2024	2024	NUM
ejpam-6013	415	12	.	.	PUNCT
ejpam-6013	416	1	[	[	X
ejpam-6013	416	2	31	31	NUM
ejpam-6013	416	3	]	]	X
ejpam-6013	416	4	c.	c.	PROPN
ejpam-6013	416	5	klanarong	klanarong	PROPN
ejpam-6013	416	6	,	,	PUNCT
ejpam-6013	416	7	s.	s.	PROPN
ejpam-6013	416	8	sompong	sompong	PROPN
ejpam-6013	416	9	,	,	PUNCT
ejpam-6013	416	10	and	and	CCONJ
ejpam-6013	416	11	c.	c.	PROPN
ejpam-6013	416	12	boonpok	boonpok	PROPN
ejpam-6013	416	13	.	.	PUNCT
ejpam-6013	417	1	(	(	PUNCT
ejpam-6013	417	2	τ1	τ1	NOUN
ejpam-6013	417	3	,	,	PUNCT
ejpam-6013	417	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6013	417	5	and	and	CCONJ
ejpam-6013	417	6	(	(	PUNCT
ejpam-6013	417	7	τ1	τ1	NOUN
ejpam-6013	417	8	,	,	PUNCT
ejpam-6013	417	9	τ2)θ	τ2)θ	ADJ
ejpam-6013	417	10	-	-	PUNCT
ejpam-6013	417	11	closed	close	VERB
ejpam-6013	417	12	sets	set	NOUN
ejpam-6013	417	13	.	.	PUNCT
ejpam-6013	418	1	international	international	ADJ
ejpam-6013	418	2	journal	journal	NOUN
ejpam-6013	418	3	of	of	ADP
ejpam-6013	418	4	mathematics	mathematic	NOUN
ejpam-6013	418	5	and	and	CCONJ
ejpam-6013	418	6	computer	computer	NOUN
ejpam-6013	418	7	science	science	NOUN
ejpam-6013	418	8	,	,	PUNCT
ejpam-6013	418	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-6013	418	10	,	,	PUNCT
ejpam-6013	418	11	2024	2024	NUM
ejpam-6013	418	12	.	.	PUNCT
ejpam-6013	419	1	[	[	X
ejpam-6013	419	2	32	32	NUM
ejpam-6013	419	3	]	]	PUNCT
ejpam-6013	419	4	p.	p.	NOUN
ejpam-6013	419	5	pue	pue	NOUN
ejpam-6013	419	6	-	-	PUNCT
ejpam-6013	419	7	on	on	ADP
ejpam-6013	419	8	,	,	PUNCT
ejpam-6013	419	9	s.	s.	PROPN
ejpam-6013	419	10	sompong	sompong	PROPN
ejpam-6013	419	11	,	,	PUNCT
ejpam-6013	419	12	and	and	CCONJ
ejpam-6013	419	13	c.	c.	PROPN
ejpam-6013	419	14	boonpok	boonpok	PROPN
ejpam-6013	419	15	.	.	PUNCT
ejpam-6013	420	1	upper	upper	ADJ
ejpam-6013	420	2	and	and	CCONJ
ejpam-6013	420	3	lower	low	ADJ
ejpam-6013	420	4	faint	faint	ADJ
ejpam-6013	420	5	(	(	PUNCT
ejpam-6013	420	6	τ1	τ1	NOUN
ejpam-6013	420	7	,	,	PUNCT
ejpam-6013	420	8	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6013	420	9	.	.	PUNCT
ejpam-6013	421	1	international	international	ADJ
ejpam-6013	421	2	journal	journal	NOUN
ejpam-6013	421	3	of	of	ADP
ejpam-6013	421	4	analysis	analysis	NOUN
ejpam-6013	421	5	and	and	CCONJ
ejpam-6013	421	6	applications	application	NOUN
ejpam-6013	421	7	,	,	PUNCT
ejpam-6013	421	8	22:169	22:169	NUM
ejpam-6013	421	9	,	,	PUNCT
ejpam-6013	421	10	2024	2024	NUM
ejpam-6013	421	11	.	.	PUNCT
ejpam-6013	422	1	[	[	X
ejpam-6013	422	2	33	33	NUM
ejpam-6013	422	3	]	]	PUNCT
ejpam-6013	422	4	m.	m.	NOUN
ejpam-6013	422	5	thongmoon	thongmoon	NOUN
ejpam-6013	422	6	,	,	PUNCT
ejpam-6013	422	7	s.	s.	PROPN
ejpam-6013	422	8	sompong	sompong	PROPN
ejpam-6013	422	9	,	,	PUNCT
ejpam-6013	422	10	and	and	CCONJ
ejpam-6013	422	11	c.	c.	PROPN
ejpam-6013	422	12	boonpok	boonpok	PROPN
ejpam-6013	422	13	.	.	PUNCT
ejpam-6013	423	1	θ(τ1	θ(τ1	PROPN
ejpam-6013	423	2	,	,	PUNCT
ejpam-6013	423	3	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6013	423	4	for	for	ADP
ejpam-6013	423	5	functions	function	NOUN
ejpam-6013	423	6	.	.	PUNCT
ejpam-6013	424	1	(	(	PUNCT
ejpam-6013	424	2	submitted	submit	VERB
ejpam-6013	424	3	)	)	PUNCT
ejpam-6013	424	4	.	.	PUNCT
ejpam-6013	425	1	[	[	X
ejpam-6013	425	2	34	34	NUM
ejpam-6013	425	3	]	]	PUNCT
ejpam-6013	425	4	m.	m.	NOUN
ejpam-6013	425	5	chiangpradit	chiangpradit	NOUN
ejpam-6013	425	6	,	,	PUNCT
ejpam-6013	425	7	s.	s.	PROPN
ejpam-6013	425	8	sompong	sompong	PROPN
ejpam-6013	425	9	,	,	PUNCT
ejpam-6013	425	10	and	and	CCONJ
ejpam-6013	425	11	c.	c.	PROPN
ejpam-6013	425	12	boonpok	boonpok	PROPN
ejpam-6013	425	13	.	.	PUNCT
ejpam-6013	426	1	λ(τ1,τ2)-sets	λ(τ1,τ2)-set	NOUN
ejpam-6013	426	2	and	and	CCONJ
ejpam-6013	426	3	related	relate	VERB
ejpam-6013	426	4	topological	topological	ADJ
ejpam-6013	426	5	spaces	space	NOUN
ejpam-6013	426	6	.	.	PUNCT
ejpam-6013	427	1	asia	asia	PROPN
ejpam-6013	427	2	pacific	pacific	PROPN
ejpam-6013	427	3	journal	journal	PROPN
ejpam-6013	427	4	of	of	ADP
ejpam-6013	427	5	mathematics	mathematic	NOUN
ejpam-6013	427	6	,	,	PUNCT
ejpam-6013	427	7	11:49	11:49	NUM
ejpam-6013	427	8	,	,	PUNCT
ejpam-6013	427	9	2024	2024	NUM
ejpam-6013	427	10	.	.	PUNCT
ejpam-6013	428	1	[	[	X
ejpam-6013	428	2	35	35	NUM
ejpam-6013	428	3	]	]	X
ejpam-6013	428	4	n.	n.	NOUN
ejpam-6013	428	5	chutiman	chutiman	NOUN
ejpam-6013	428	6	,	,	PUNCT
ejpam-6013	428	7	s.	s.	PROPN
ejpam-6013	428	8	sompong	sompong	PROPN
ejpam-6013	428	9	,	,	PUNCT
ejpam-6013	428	10	and	and	CCONJ
ejpam-6013	428	11	c.	c.	PROPN
ejpam-6013	428	12	boonpok	boonpok	PROPN
ejpam-6013	428	13	.	.	PUNCT
ejpam-6013	429	1	on	on	ADP
ejpam-6013	429	2	some	some	DET
ejpam-6013	429	3	separation	separation	NOUN
ejpam-6013	429	4	axioms	axiom	NOUN
ejpam-6013	429	5	in	in	ADP
ejpam-6013	429	6	bitopological	bitopological	ADJ
ejpam-6013	429	7	spaces	space	NOUN
ejpam-6013	429	8	.	.	PUNCT
ejpam-6013	430	1	asia	asia	PROPN
ejpam-6013	430	2	pacific	pacific	PROPN
ejpam-6013	430	3	journal	journal	PROPN
ejpam-6013	430	4	of	of	ADP
ejpam-6013	430	5	mathematics	mathematic	NOUN
ejpam-6013	430	6	,	,	PUNCT
ejpam-6013	430	7	11:41	11:41	NUM
ejpam-6013	430	8	,	,	PUNCT
ejpam-6013	430	9	2024	2024	NUM
ejpam-6013	430	10	.	.	PUNCT
ejpam-6013	431	1	[	[	X
ejpam-6013	431	2	36	36	NUM
ejpam-6013	431	3	]	]	X
ejpam-6013	431	4	b.	b.	PROPN
ejpam-6013	431	5	kong	kong	PROPN
ejpam-6013	431	6	-	-	PUNCT
ejpam-6013	431	7	ied	ied	PROPN
ejpam-6013	431	8	,	,	PUNCT
ejpam-6013	431	9	s.	s.	PROPN
ejpam-6013	431	10	sompong	sompong	PROPN
ejpam-6013	431	11	,	,	PUNCT
ejpam-6013	431	12	and	and	CCONJ
ejpam-6013	431	13	c.	c.	PROPN
ejpam-6013	431	14	boonpok	boonpok	PROPN
ejpam-6013	431	15	.	.	PUNCT
ejpam-6013	432	1	on	on	ADP
ejpam-6013	432	2	(	(	PUNCT
ejpam-6013	432	3	τ1	τ1	PROPN
ejpam-6013	432	4	,	,	PUNCT
ejpam-6013	432	5	τ2)-r0	τ2)-r0	X
ejpam-6013	432	6	bitopological	bitopological	ADJ
ejpam-6013	432	7	spaces	space	NOUN
ejpam-6013	432	8	.	.	PUNCT
ejpam-6013	433	1	asia	asia	PROPN
ejpam-6013	433	2	pacific	pacific	PROPN
ejpam-6013	433	3	journal	journal	PROPN
ejpam-6013	433	4	of	of	ADP
ejpam-6013	433	5	mathematics	mathematic	NOUN
ejpam-6013	433	6	,	,	PUNCT
ejpam-6013	433	7	11:43	11:43	NUM
ejpam-6013	433	8	,	,	PUNCT
ejpam-6013	433	9	2024	2024	NUM
ejpam-6013	433	10	.	.	PUNCT