id	sid	tid	token	lemma	pos
ejpam-5721	1	1	european	european	PROPN
ejpam-5721	1	2	journal	journal	PROPN
ejpam-5721	1	3	of	of	ADP
ejpam-5721	1	4	pure	pure	ADJ
ejpam-5721	1	5	and	and	CCONJ
ejpam-5721	1	6	applied	applied	ADJ
ejpam-5721	1	7	mathematics	mathematic	NOUN
ejpam-5721	1	8	2025	2025	NUM
ejpam-5721	1	9	,	,	PUNCT
ejpam-5721	1	10	vol	vol	NOUN
ejpam-5721	1	11	.	.	PROPN
ejpam-5721	1	12	18	18	NUM
ejpam-5721	1	13	,	,	PUNCT
ejpam-5721	1	14	issue	issue	NOUN
ejpam-5721	1	15	1	1	NUM
ejpam-5721	1	16	,	,	PUNCT
ejpam-5721	1	17	article	article	NOUN
ejpam-5721	1	18	number	number	NOUN
ejpam-5721	1	19	5721	5721	NUM
ejpam-5721	1	20	issn	issn	PROPN
ejpam-5721	1	21	1307	1307	NUM
ejpam-5721	1	22	-	-	SYM
ejpam-5721	1	23	5543	5543	NUM
ejpam-5721	1	24	–	–	PUNCT
ejpam-5721	1	25	ejpam.com	ejpam.com	X
ejpam-5721	1	26	published	publish	VERB
ejpam-5721	1	27	by	by	ADP
ejpam-5721	1	28	new	new	PROPN
ejpam-5721	1	29	york	york	PROPN
ejpam-5721	1	30	business	business	PROPN
ejpam-5721	1	31	global	global	ADJ
ejpam-5721	1	32	almost	almost	ADV
ejpam-5721	1	33	weakly	weakly	ADJ
ejpam-5721	1	34	(	(	PUNCT
ejpam-5721	1	35	τ1	τ1	NOUN
ejpam-5721	1	36	,	,	PUNCT
ejpam-5721	1	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	1	38	functions	function	NOUN
ejpam-5721	1	39	jeeranunt	jeeranunt	VERB
ejpam-5721	1	40	khampakdee1	khampakdee1	PROPN
ejpam-5721	1	41	,	,	PUNCT
ejpam-5721	1	42	supunnee	supunnee	PROPN
ejpam-5721	1	43	sompong2	sompong2	PROPN
ejpam-5721	1	44	,	,	PUNCT
ejpam-5721	1	45	chawalit	chawalit	VERB
ejpam-5721	1	46	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5721	1	47	1	1	NUM
ejpam-5721	1	48	mathematics	mathematic	NOUN
ejpam-5721	1	49	and	and	CCONJ
ejpam-5721	1	50	applied	apply	VERB
ejpam-5721	1	51	mathematics	mathematics	PROPN
ejpam-5721	1	52	research	research	NOUN
ejpam-5721	1	53	unit	unit	NOUN
ejpam-5721	1	54	,	,	PUNCT
ejpam-5721	1	55	department	department	NOUN
ejpam-5721	1	56	of	of	ADP
ejpam-5721	1	57	mathematics	mathematic	NOUN
ejpam-5721	1	58	,	,	PUNCT
ejpam-5721	1	59	faculty	faculty	NOUN
ejpam-5721	1	60	of	of	ADP
ejpam-5721	1	61	science	science	NOUN
ejpam-5721	1	62	,	,	PUNCT
ejpam-5721	1	63	mahasarakham	mahasarakham	PROPN
ejpam-5721	1	64	university	university	PROPN
ejpam-5721	1	65	,	,	PUNCT
ejpam-5721	1	66	maha	maha	PROPN
ejpam-5721	1	67	sarakham	sarakham	PROPN
ejpam-5721	1	68	,	,	PUNCT
ejpam-5721	1	69	44150	44150	NUM
ejpam-5721	1	70	,	,	PUNCT
ejpam-5721	1	71	thailand	thailand	PROPN
ejpam-5721	1	72	2	2	NUM
ejpam-5721	1	73	department	department	NOUN
ejpam-5721	1	74	of	of	ADP
ejpam-5721	1	75	mathematics	mathematic	NOUN
ejpam-5721	1	76	and	and	CCONJ
ejpam-5721	1	77	statistics	statistic	NOUN
ejpam-5721	1	78	,	,	PUNCT
ejpam-5721	1	79	faculty	faculty	NOUN
ejpam-5721	1	80	of	of	ADP
ejpam-5721	1	81	science	science	NOUN
ejpam-5721	1	82	and	and	CCONJ
ejpam-5721	1	83	technology	technology	NOUN
ejpam-5721	1	84	,	,	PUNCT
ejpam-5721	1	85	sakon	sakon	PROPN
ejpam-5721	1	86	nakhon	nakhon	PROPN
ejpam-5721	1	87	rajbhat	rajbhat	PROPN
ejpam-5721	1	88	university	university	PROPN
ejpam-5721	1	89	,	,	PUNCT
ejpam-5721	1	90	sakon	sakon	PROPN
ejpam-5721	1	91	nakhon	nakhon	PROPN
ejpam-5721	1	92	,	,	PUNCT
ejpam-5721	1	93	47000	47000	NUM
ejpam-5721	1	94	,	,	PUNCT
ejpam-5721	1	95	thailand	thailand	PROPN
ejpam-5721	1	96	abstract	abstract	NOUN
ejpam-5721	1	97	.	.	PUNCT
ejpam-5721	2	1	this	this	DET
ejpam-5721	2	2	paper	paper	NOUN
ejpam-5721	2	3	is	be	AUX
ejpam-5721	2	4	concerned	concern	VERB
ejpam-5721	2	5	with	with	ADP
ejpam-5721	2	6	the	the	DET
ejpam-5721	2	7	concept	concept	NOUN
ejpam-5721	2	8	of	of	ADP
ejpam-5721	2	9	weakly	weakly	ADJ
ejpam-5721	2	10	(	(	PUNCT
ejpam-5721	2	11	τ1	τ1	NOUN
ejpam-5721	2	12	,	,	PUNCT
ejpam-5721	2	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	2	14	functions	function	NOUN
ejpam-5721	2	15	.	.	PUNCT
ejpam-5721	3	1	furthermore	furthermore	ADV
ejpam-5721	3	2	,	,	PUNCT
ejpam-5721	3	3	several	several	ADJ
ejpam-5721	3	4	characterizations	characterization	NOUN
ejpam-5721	3	5	and	and	CCONJ
ejpam-5721	3	6	some	some	DET
ejpam-5721	3	7	properties	property	NOUN
ejpam-5721	3	8	of	of	ADP
ejpam-5721	3	9	almost	almost	ADV
ejpam-5721	3	10	weakly	weakly	ADJ
ejpam-5721	3	11	(	(	PUNCT
ejpam-5721	3	12	τ1	τ1	NOUN
ejpam-5721	3	13	,	,	PUNCT
ejpam-5721	3	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	3	15	functions	function	NOUN
ejpam-5721	3	16	are	be	AUX
ejpam-5721	3	17	investigated	investigate	VERB
ejpam-5721	3	18	.	.	PUNCT
ejpam-5721	4	1	2020	2020	NUM
ejpam-5721	4	2	mathematics	mathematic	NOUN
ejpam-5721	4	3	subject	subject	NOUN
ejpam-5721	4	4	classifications	classification	NOUN
ejpam-5721	4	5	:	:	PUNCT
ejpam-5721	4	6	54c08	54c08	NUM
ejpam-5721	4	7	,	,	PUNCT
ejpam-5721	4	8	54e55	54e55	NUM
ejpam-5721	4	9	key	key	ADJ
ejpam-5721	4	10	words	word	NOUN
ejpam-5721	4	11	and	and	CCONJ
ejpam-5721	4	12	phrases	phrase	NOUN
ejpam-5721	4	13	:	:	PUNCT
ejpam-5721	4	14	τ1τ2	τ1τ2	ADJ
ejpam-5721	4	15	-	-	ADJ
ejpam-5721	4	16	open	open	ADJ
ejpam-5721	4	17	set	set	NOUN
ejpam-5721	4	18	,	,	PUNCT
ejpam-5721	4	19	almost	almost	ADV
ejpam-5721	4	20	weakly	weakly	ADJ
ejpam-5721	4	21	(	(	PUNCT
ejpam-5721	4	22	τ1	τ1	NOUN
ejpam-5721	4	23	,	,	PUNCT
ejpam-5721	4	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	4	25	function	function	NOUN
ejpam-5721	4	26	1	1	NUM
ejpam-5721	4	27	.	.	PUNCT
ejpam-5721	4	28	introduction	introduction	NOUN
ejpam-5721	4	29	topology	topology	NOUN
ejpam-5721	4	30	as	as	ADP
ejpam-5721	4	31	a	a	DET
ejpam-5721	4	32	field	field	NOUN
ejpam-5721	4	33	of	of	ADP
ejpam-5721	4	34	mathematics	mathematic	NOUN
ejpam-5721	4	35	is	be	AUX
ejpam-5721	4	36	concerned	concern	VERB
ejpam-5721	4	37	with	with	ADP
ejpam-5721	4	38	all	all	DET
ejpam-5721	4	39	questions	question	NOUN
ejpam-5721	4	40	directly	directly	ADV
ejpam-5721	4	41	or	or	CCONJ
ejpam-5721	4	42	indirectly	indirectly	ADV
ejpam-5721	4	43	related	relate	VERB
ejpam-5721	4	44	to	to	ADP
ejpam-5721	4	45	continuity	continuity	NOUN
ejpam-5721	4	46	.	.	PUNCT
ejpam-5721	5	1	semi	semi	ADJ
ejpam-5721	5	2	-	-	ADJ
ejpam-5721	5	3	open	open	ADJ
ejpam-5721	5	4	sets	set	NOUN
ejpam-5721	5	5	[	[	X
ejpam-5721	5	6	25	25	NUM
ejpam-5721	5	7	]	]	PUNCT
ejpam-5721	5	8	,	,	PUNCT
ejpam-5721	5	9	preopen	preopen	ADJ
ejpam-5721	5	10	sets	set	NOUN
ejpam-5721	5	11	[	[	X
ejpam-5721	5	12	26	26	NUM
ejpam-5721	5	13	]	]	PUNCT
ejpam-5721	5	14	,	,	PUNCT
ejpam-5721	5	15	α	α	X
ejpam-5721	5	16	-	-	ADJ
ejpam-5721	5	17	open	open	ADJ
ejpam-5721	5	18	sets	set	NOUN
ejpam-5721	5	19	[	[	X
ejpam-5721	5	20	27	27	NUM
ejpam-5721	5	21	]	]	PUNCT
ejpam-5721	5	22	and	and	CCONJ
ejpam-5721	5	23	β	β	X
ejpam-5721	5	24	-	-	ADJ
ejpam-5721	5	25	open	open	ADJ
ejpam-5721	5	26	sets	set	NOUN
ejpam-5721	5	27	[	[	X
ejpam-5721	5	28	20	20	NUM
ejpam-5721	5	29	]	]	PUNCT
ejpam-5721	5	30	play	play	VERB
ejpam-5721	5	31	an	an	DET
ejpam-5721	5	32	important	important	ADJ
ejpam-5721	5	33	role	role	NOUN
ejpam-5721	5	34	in	in	ADP
ejpam-5721	5	35	the	the	DET
ejpam-5721	5	36	researching	researching	NOUN
ejpam-5721	5	37	of	of	ADP
ejpam-5721	5	38	generalizations	generalization	NOUN
ejpam-5721	5	39	of	of	ADP
ejpam-5721	5	40	continuity	continuity	NOUN
ejpam-5721	5	41	in	in	ADP
ejpam-5721	5	42	topological	topological	ADJ
ejpam-5721	5	43	spaces	space	NOUN
ejpam-5721	5	44	.	.	PUNCT
ejpam-5721	6	1	by	by	ADP
ejpam-5721	6	2	using	use	VERB
ejpam-5721	6	3	these	these	DET
ejpam-5721	6	4	sets	set	NOUN
ejpam-5721	6	5	many	many	ADJ
ejpam-5721	6	6	authors	author	NOUN
ejpam-5721	6	7	introduced	introduce	VERB
ejpam-5721	6	8	and	and	CCONJ
ejpam-5721	6	9	studied	study	VERB
ejpam-5721	6	10	various	various	ADJ
ejpam-5721	6	11	types	type	NOUN
ejpam-5721	6	12	of	of	ADP
ejpam-5721	6	13	weak	weak	ADJ
ejpam-5721	6	14	forms	form	NOUN
ejpam-5721	6	15	of	of	ADP
ejpam-5721	6	16	continuity	continuity	NOUN
ejpam-5721	6	17	for	for	ADP
ejpam-5721	6	18	functions	function	NOUN
ejpam-5721	6	19	.	.	PUNCT
ejpam-5721	7	1	levine	levine	PROPN
ejpam-5721	8	1	[	[	X
ejpam-5721	8	2	24	24	NUM
ejpam-5721	8	3	]	]	PUNCT
ejpam-5721	8	4	introduced	introduce	VERB
ejpam-5721	8	5	the	the	DET
ejpam-5721	8	6	concept	concept	NOUN
ejpam-5721	8	7	of	of	ADP
ejpam-5721	8	8	weakly	weakly	ADJ
ejpam-5721	8	9	continuous	continuous	ADJ
ejpam-5721	8	10	functions	function	NOUN
ejpam-5721	8	11	in	in	ADP
ejpam-5721	8	12	topological	topological	ADJ
ejpam-5721	8	13	spaces	space	NOUN
ejpam-5721	8	14	.	.	PUNCT
ejpam-5721	9	1	husain	husain	PROPN
ejpam-5721	10	1	[	[	X
ejpam-5721	10	2	21	21	NUM
ejpam-5721	10	3	]	]	PUNCT
ejpam-5721	10	4	introduced	introduce	VERB
ejpam-5721	10	5	the	the	DET
ejpam-5721	10	6	concept	concept	NOUN
ejpam-5721	10	7	of	of	ADP
ejpam-5721	10	8	almost	almost	ADV
ejpam-5721	10	9	continuous	continuous	ADJ
ejpam-5721	10	10	functions	function	NOUN
ejpam-5721	10	11	.	.	PUNCT
ejpam-5721	11	1	viriyapong	viriyapong	PROPN
ejpam-5721	11	2	and	and	CCONJ
ejpam-5721	11	3	boonpok	boonpok	VERB
ejpam-5721	12	1	[	[	X
ejpam-5721	12	2	36	36	NUM
ejpam-5721	12	3	]	]	PUNCT
ejpam-5721	12	4	investigated	investigate	VERB
ejpam-5721	12	5	some	some	DET
ejpam-5721	12	6	characterizations	characterization	NOUN
ejpam-5721	12	7	of	of	ADP
ejpam-5721	12	8	(	(	PUNCT
ejpam-5721	12	9	λ	λ	PROPN
ejpam-5721	12	10	,	,	PUNCT
ejpam-5721	12	11	sp)-continuous	sp)-continuous	ADJ
ejpam-5721	12	12	functions	function	NOUN
ejpam-5721	12	13	by	by	ADP
ejpam-5721	12	14	utilizing	utilize	VERB
ejpam-5721	12	15	the	the	DET
ejpam-5721	12	16	notions	notion	NOUN
ejpam-5721	12	17	of	of	ADP
ejpam-5721	12	18	(	(	PUNCT
ejpam-5721	12	19	λ	λ	PROPN
ejpam-5721	12	20	,	,	PUNCT
ejpam-5721	12	21	sp)-open	sp)-open	ADJ
ejpam-5721	12	22	sets	set	NOUN
ejpam-5721	12	23	and	and	CCONJ
ejpam-5721	12	24	(	(	PUNCT
ejpam-5721	12	25	λ	λ	PROPN
ejpam-5721	12	26	,	,	PUNCT
ejpam-5721	12	27	sp)closed	sp)close	VERB
ejpam-5721	12	28	sets	set	NOUN
ejpam-5721	12	29	due	due	ADP
ejpam-5721	12	30	to	to	ADP
ejpam-5721	12	31	boonpok	boonpok	NOUN
ejpam-5721	12	32	and	and	CCONJ
ejpam-5721	12	33	khampakdee	khampakdee	NOUN
ejpam-5721	12	34	[	[	X
ejpam-5721	12	35	8	8	NUM
ejpam-5721	12	36	]	]	PUNCT
ejpam-5721	12	37	.	.	PUNCT
ejpam-5721	13	1	dungthaisong	dungthaisong	NOUN
ejpam-5721	13	2	et	et	PROPN
ejpam-5721	13	3	al	al	PROPN
ejpam-5721	13	4	.	.	PUNCT
ejpam-5721	14	1	[	[	X
ejpam-5721	14	2	19	19	NUM
ejpam-5721	14	3	]	]	PUNCT
ejpam-5721	14	4	introduced	introduce	VERB
ejpam-5721	14	5	and	and	CCONJ
ejpam-5721	14	6	studied	study	VERB
ejpam-5721	14	7	the	the	DET
ejpam-5721	14	8	concept	concept	NOUN
ejpam-5721	14	9	of	of	ADP
ejpam-5721	14	10	g(m	g(m	ADJ
ejpam-5721	14	11	,	,	PUNCT
ejpam-5721	14	12	n)-continuous	n)-continuous	ADJ
ejpam-5721	14	13	functions	function	NOUN
ejpam-5721	14	14	.	.	PUNCT
ejpam-5721	15	1	duangphui	duangphui	NOUN
ejpam-5721	15	2	et	et	PROPN
ejpam-5721	15	3	al	al	PROPN
ejpam-5721	15	4	.	.	PUNCT
ejpam-5721	16	1	[	[	X
ejpam-5721	16	2	18	18	NUM
ejpam-5721	16	3	]	]	PUNCT
ejpam-5721	16	4	introduced	introduce	VERB
ejpam-5721	16	5	and	and	CCONJ
ejpam-5721	16	6	investigated	investigate	VERB
ejpam-5721	16	7	the	the	DET
ejpam-5721	16	8	notion	notion	NOUN
ejpam-5721	16	9	of	of	ADP
ejpam-5721	16	10	almost	almost	ADV
ejpam-5721	16	11	(	(	PUNCT
ejpam-5721	16	12	µ	µ	NUM
ejpam-5721	16	13	,	,	PUNCT
ejpam-5721	16	14	µ′)(m	µ′)(m	VERB
ejpam-5721	16	15	,	,	PUNCT
ejpam-5721	16	16	n)-continuous	n)-continuous	ADJ
ejpam-5721	16	17	functions	function	NOUN
ejpam-5721	16	18	.	.	PUNCT
ejpam-5721	17	1	furthermore	furthermore	ADV
ejpam-5721	17	2	,	,	PUNCT
ejpam-5721	17	3	several	several	ADJ
ejpam-5721	17	4	characterizations	characterization	NOUN
ejpam-5721	17	5	of	of	ADP
ejpam-5721	17	6	almost	almost	ADV
ejpam-5721	17	7	(	(	PUNCT
ejpam-5721	17	8	λ	λ	PROPN
ejpam-5721	17	9	,	,	PUNCT
ejpam-5721	17	10	p)-continuous	p)-continuous	ADJ
ejpam-5721	17	11	functions	function	NOUN
ejpam-5721	17	12	,	,	PUNCT
ejpam-5721	17	13	strongly	strongly	ADV
ejpam-5721	17	14	θ(λ	θ(λ	PROPN
ejpam-5721	17	15	,	,	PUNCT
ejpam-5721	17	16	p)-continuous	p)-continuous	ADJ
ejpam-5721	17	17	functions	function	NOUN
ejpam-5721	17	18	,	,	PUNCT
ejpam-5721	17	19	almost	almost	ADV
ejpam-5721	17	20	strongly	strongly	ADV
ejpam-5721	17	21	θ(λ	θ(λ	VERB
ejpam-5721	17	22	,	,	PUNCT
ejpam-5721	17	23	p)-continuous	p)-continuous	ADJ
ejpam-5721	17	24	functions	function	NOUN
ejpam-5721	17	25	,	,	PUNCT
ejpam-5721	17	26	θ(λ	θ(λ	PROPN
ejpam-5721	17	27	,	,	PUNCT
ejpam-5721	17	28	p)-continuous	p)-continuous	ADJ
ejpam-5721	17	29	functions	function	NOUN
ejpam-5721	17	30	,	,	PUNCT
ejpam-5721	17	31	weakly	weakly	ADJ
ejpam-5721	17	32	(	(	PUNCT
ejpam-5721	17	33	λ	λ	PROPN
ejpam-5721	17	34	,	,	PUNCT
ejpam-5721	17	35	b)-continuous	b)-continuous	ADJ
ejpam-5721	17	36	functions	function	NOUN
ejpam-5721	17	37	,	,	PUNCT
ejpam-5721	17	38	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5721	17	39	functions	function	NOUN
ejpam-5721	17	40	,	,	PUNCT
ejpam-5721	17	41	(	(	PUNCT
ejpam-5721	17	42	λ	λ	NOUN
ejpam-5721	17	43	,	,	PUNCT
ejpam-5721	17	44	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5721	17	45	functions	function	NOUN
ejpam-5721	17	46	,	,	PUNCT
ejpam-5721	17	47	⋆-continuous	⋆-continuous	ADJ
ejpam-5721	17	48	functions	function	NOUN
ejpam-5721	17	49	,	,	PUNCT
ejpam-5721	17	50	θ	θ	PROPN
ejpam-5721	17	51	-	-	ADJ
ejpam-5721	17	52	i	i	NOUN
ejpam-5721	17	53	-continuous	-continuous	ADJ
ejpam-5721	17	54	functions	function	NOUN
ejpam-5721	17	55	,	,	PUNCT
ejpam-5721	17	56	almost	almost	ADV
ejpam-5721	17	57	(	(	PUNCT
ejpam-5721	17	58	g	g	NOUN
ejpam-5721	17	59	,	,	PUNCT
ejpam-5721	17	60	m)-continuous	m)-continuous	ADJ
ejpam-5721	17	61	functions	function	NOUN
ejpam-5721	17	62	,	,	PUNCT
ejpam-5721	17	63	pairwise	pairwise	NOUN
ejpam-5721	17	64	almost	almost	ADV
ejpam-5721	17	65	m	m	VERB
ejpam-5721	17	66	-continuous	-continuous	ADJ
ejpam-5721	17	67	functions	function	NOUN
ejpam-5721	17	68	,	,	PUNCT
ejpam-5721	17	69	(	(	PUNCT
ejpam-5721	17	70	τ1	τ1	NOUN
ejpam-5721	17	71	,	,	PUNCT
ejpam-5721	17	72	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	17	73	functions	function	NOUN
ejpam-5721	17	74	,	,	PUNCT
ejpam-5721	17	75	almost	almost	ADV
ejpam-5721	17	76	(	(	PUNCT
ejpam-5721	17	77	τ1	τ1	NOUN
ejpam-5721	17	78	,	,	PUNCT
ejpam-5721	17	79	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	17	80	functions	function	NOUN
ejpam-5721	17	81	and	and	CCONJ
ejpam-5721	17	82	weakly	weakly	ADJ
ejpam-5721	17	83	(	(	PUNCT
ejpam-5721	17	84	τ1	τ1	NOUN
ejpam-5721	17	85	,	,	PUNCT
ejpam-5721	17	86	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	17	87	functions	function	NOUN
ejpam-5721	17	88	were	be	AUX
ejpam-5721	17	89	presented	present	VERB
ejpam-5721	17	90	in	in	ADP
ejpam-5721	17	91	[	[	X
ejpam-5721	17	92	33	33	NUM
ejpam-5721	17	93	]	]	PUNCT
ejpam-5721	17	94	,	,	PUNCT
ejpam-5721	17	95	[	[	X
ejpam-5721	17	96	34	34	NUM
ejpam-5721	17	97	]	]	PUNCT
ejpam-5721	17	98	,	,	PUNCT
ejpam-5721	17	99	∗corresponding	∗corresponde	VERB
ejpam-5721	17	100	author	author	NOUN
ejpam-5721	17	101	.	.	PUNCT
ejpam-5721	18	1	doi	doi	NOUN
ejpam-5721	18	2	:	:	PUNCT
ejpam-5721	18	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5721	https://doi.org/10.29020/nybg.ejpam.v18i1.5721	PROPN
ejpam-5721	18	4	email	email	NOUN
ejpam-5721	18	5	addresses	address	VERB
ejpam-5721	18	6	:	:	PUNCT
ejpam-5721	18	7	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-5721	18	8	(	(	PUNCT
ejpam-5721	18	9	j.	j.	PROPN
ejpam-5721	18	10	khampakdee	khampakdee	PROPN
ejpam-5721	18	11	)	)	PUNCT
ejpam-5721	18	12	,	,	PUNCT
ejpam-5721	18	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-5721	18	14	(	(	PUNCT
ejpam-5721	18	15	s.	s.	PROPN
ejpam-5721	18	16	sompong	sompong	PROPN
ejpam-5721	18	17	)	)	PUNCT
ejpam-5721	18	18	,	,	PUNCT
ejpam-5721	18	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5721	18	20	(	(	PUNCT
ejpam-5721	18	21	c.	c.	PROPN
ejpam-5721	18	22	boonpok	boonpok	PROPN
ejpam-5721	18	23	)	)	PUNCT
ejpam-5721	18	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5721	18	25	1	1	NUM
ejpam-5721	18	26	copyright	copyright	NOUN
ejpam-5721	18	27	:	:	PUNCT
ejpam-5721	19	1	©	©	PROPN
ejpam-5721	19	2	2025	2025	NUM
ejpam-5721	19	3	the	the	DET
ejpam-5721	19	4	author(s	author(s	NOUN
ejpam-5721	19	5	)	)	PUNCT
ejpam-5721	19	6	.	.	PUNCT
ejpam-5721	20	1	(	(	PUNCT
ejpam-5721	20	2	cc	cc	NOUN
ejpam-5721	20	3	by	by	ADP
ejpam-5721	20	4	-	-	PUNCT
ejpam-5721	20	5	nc	nc	PROPN
ejpam-5721	20	6	4.0	4.0	NUM
ejpam-5721	20	7	)	)	PUNCT
ejpam-5721	20	8	j.	j.	PROPN
ejpam-5721	20	9	khampakdee	khampakdee	PROPN
ejpam-5721	20	10	,	,	PUNCT
ejpam-5721	20	11	s.	s.	PROPN
ejpam-5721	20	12	sompong	sompong	PROPN
ejpam-5721	20	13	,	,	PUNCT
ejpam-5721	20	14	c.	c.	PROPN
ejpam-5721	20	15	boonpok	boonpok	PROPN
ejpam-5721	20	16	/	/	SYM
ejpam-5721	20	17	eur	eur	PROPN
ejpam-5721	20	18	.	.	PUNCT
ejpam-5721	21	1	j.	j.	PROPN
ejpam-5721	21	2	pure	pure	PROPN
ejpam-5721	21	3	appl	appl	PROPN
ejpam-5721	21	4	.	.	PROPN
ejpam-5721	21	5	math	math	PROPN
ejpam-5721	21	6	,	,	PUNCT
ejpam-5721	21	7	18	18	NUM
ejpam-5721	21	8	(	(	PUNCT
ejpam-5721	21	9	1	1	NUM
ejpam-5721	21	10	)	)	PUNCT
ejpam-5721	21	11	(	(	PUNCT
ejpam-5721	21	12	2025	2025	NUM
ejpam-5721	21	13	)	)	PUNCT
ejpam-5721	21	14	,	,	PUNCT
ejpam-5721	21	15	5721	5721	NUM
ejpam-5721	21	16	2	2	NUM
ejpam-5721	21	17	of	of	ADP
ejpam-5721	21	18	12	12	NUM
ejpam-5721	22	1	[	[	X
ejpam-5721	22	2	9	9	NUM
ejpam-5721	22	3	]	]	PUNCT
ejpam-5721	22	4	,	,	PUNCT
ejpam-5721	22	5	[	[	X
ejpam-5721	22	6	31	31	NUM
ejpam-5721	22	7	]	]	PUNCT
ejpam-5721	22	8	,	,	PUNCT
ejpam-5721	23	1	[	[	X
ejpam-5721	23	2	12	12	NUM
ejpam-5721	23	3	]	]	PUNCT
ejpam-5721	23	4	,	,	PUNCT
ejpam-5721	24	1	[	[	X
ejpam-5721	24	2	7	7	NUM
ejpam-5721	24	3	]	]	PUNCT
ejpam-5721	24	4	,	,	PUNCT
ejpam-5721	24	5	[	[	X
ejpam-5721	24	6	5	5	NUM
ejpam-5721	24	7	]	]	PUNCT
ejpam-5721	24	8	,	,	PUNCT
ejpam-5721	24	9	[	[	X
ejpam-5721	24	10	6	6	NUM
ejpam-5721	24	11	]	]	PUNCT
ejpam-5721	24	12	,	,	PUNCT
ejpam-5721	24	13	[	[	X
ejpam-5721	24	14	3	3	NUM
ejpam-5721	24	15	]	]	PUNCT
ejpam-5721	24	16	,	,	PUNCT
ejpam-5721	24	17	[	[	X
ejpam-5721	24	18	1	1	NUM
ejpam-5721	24	19	]	]	PUNCT
ejpam-5721	24	20	,	,	PUNCT
ejpam-5721	24	21	[	[	X
ejpam-5721	24	22	2	2	NUM
ejpam-5721	24	23	]	]	PUNCT
ejpam-5721	24	24	,	,	PUNCT
ejpam-5721	24	25	[	[	X
ejpam-5721	24	26	13	13	NUM
ejpam-5721	24	27	]	]	PUNCT
ejpam-5721	24	28	,	,	PUNCT
ejpam-5721	24	29	[	[	X
ejpam-5721	24	30	11	11	NUM
ejpam-5721	24	31	]	]	PUNCT
ejpam-5721	24	32	and	and	CCONJ
ejpam-5721	25	1	[	[	X
ejpam-5721	25	2	10	10	NUM
ejpam-5721	25	3	]	]	PUNCT
ejpam-5721	25	4	,	,	PUNCT
ejpam-5721	25	5	respectively	respectively	ADV
ejpam-5721	25	6	.	.	PUNCT
ejpam-5721	26	1	kong	kong	PROPN
ejpam-5721	26	2	-	-	PUNCT
ejpam-5721	26	3	ied	ied	PROPN
ejpam-5721	26	4	et	et	PROPN
ejpam-5721	26	5	al	al	PROPN
ejpam-5721	26	6	.	.	PUNCT
ejpam-5721	27	1	[	[	X
ejpam-5721	27	2	23	23	NUM
ejpam-5721	27	3	]	]	PUNCT
ejpam-5721	27	4	introduced	introduce	VERB
ejpam-5721	27	5	and	and	CCONJ
ejpam-5721	27	6	investigated	investigate	VERB
ejpam-5721	27	7	the	the	DET
ejpam-5721	27	8	concept	concept	NOUN
ejpam-5721	27	9	of	of	ADP
ejpam-5721	27	10	almost	almost	ADV
ejpam-5721	27	11	quasi	quasi	X
ejpam-5721	27	12	(	(	PUNCT
ejpam-5721	27	13	τ1	τ1	NOUN
ejpam-5721	27	14	,	,	PUNCT
ejpam-5721	27	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	27	16	functions	function	NOUN
ejpam-5721	27	17	.	.	PUNCT
ejpam-5721	28	1	chiangpradit	chiangpradit	NOUN
ejpam-5721	28	2	et	et	PROPN
ejpam-5721	28	3	al	al	PROPN
ejpam-5721	28	4	.	.	PUNCT
ejpam-5721	29	1	[	[	X
ejpam-5721	29	2	16	16	NUM
ejpam-5721	29	3	]	]	PUNCT
ejpam-5721	29	4	introduced	introduce	VERB
ejpam-5721	29	5	and	and	CCONJ
ejpam-5721	29	6	studied	study	VERB
ejpam-5721	29	7	the	the	DET
ejpam-5721	29	8	notion	notion	NOUN
ejpam-5721	29	9	of	of	ADP
ejpam-5721	29	10	weakly	weakly	ADJ
ejpam-5721	29	11	quasi	quasi	NOUN
ejpam-5721	29	12	(	(	PUNCT
ejpam-5721	29	13	τ1	τ1	PROPN
ejpam-5721	29	14	,	,	PUNCT
ejpam-5721	29	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	29	16	functions	function	NOUN
ejpam-5721	29	17	.	.	PUNCT
ejpam-5721	30	1	prachanpol	prachanpol	NOUN
ejpam-5721	30	2	et	et	PROPN
ejpam-5721	30	3	al	al	PROPN
ejpam-5721	30	4	.	.	PUNCT
ejpam-5721	31	1	[	[	X
ejpam-5721	31	2	30	30	NUM
ejpam-5721	31	3	]	]	PUNCT
ejpam-5721	31	4	introduced	introduce	VERB
ejpam-5721	31	5	and	and	CCONJ
ejpam-5721	31	6	investigated	investigate	VERB
ejpam-5721	31	7	the	the	DET
ejpam-5721	31	8	concepts	concept	NOUN
ejpam-5721	31	9	of	of	ADP
ejpam-5721	31	10	almost	almost	ADV
ejpam-5721	31	11	δ(τ1	δ(τ1	NOUN
ejpam-5721	31	12	,	,	PUNCT
ejpam-5721	31	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	31	14	functions	function	NOUN
ejpam-5721	31	15	and	and	CCONJ
ejpam-5721	31	16	weakly	weakly	ADJ
ejpam-5721	31	17	δ(τ1	δ(τ1	NOUN
ejpam-5721	31	18	,	,	PUNCT
ejpam-5721	31	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	31	20	functions	function	NOUN
ejpam-5721	31	21	.	.	PUNCT
ejpam-5721	32	1	janković	janković	PUNCT
ejpam-5721	33	1	[	[	X
ejpam-5721	33	2	22	22	NUM
ejpam-5721	33	3	]	]	PUNCT
ejpam-5721	33	4	defined	define	VERB
ejpam-5721	33	5	almost	almost	ADV
ejpam-5721	33	6	weakly	weakly	ADJ
ejpam-5721	33	7	continuous	continuous	ADJ
ejpam-5721	33	8	functions	function	NOUN
ejpam-5721	33	9	as	as	ADP
ejpam-5721	33	10	a	a	DET
ejpam-5721	33	11	generalization	generalization	NOUN
ejpam-5721	33	12	of	of	ADP
ejpam-5721	33	13	both	both	CCONJ
ejpam-5721	33	14	weakly	weakly	ADJ
ejpam-5721	33	15	continuous	continuous	ADJ
ejpam-5721	33	16	functions	function	NOUN
ejpam-5721	33	17	due	due	ADP
ejpam-5721	33	18	to	to	ADP
ejpam-5721	33	19	levine	levine	PROPN
ejpam-5721	33	20	[	[	X
ejpam-5721	33	21	25	25	NUM
ejpam-5721	33	22	]	]	PUNCT
ejpam-5721	33	23	and	and	CCONJ
ejpam-5721	33	24	almost	almost	ADV
ejpam-5721	33	25	continuous	continuous	ADJ
ejpam-5721	33	26	functions	function	NOUN
ejpam-5721	33	27	in	in	ADP
ejpam-5721	33	28	the	the	DET
ejpam-5721	33	29	sense	sense	NOUN
ejpam-5721	33	30	of	of	ADP
ejpam-5721	33	31	husain	husain	NOUN
ejpam-5721	33	32	[	[	X
ejpam-5721	33	33	21	21	NUM
ejpam-5721	33	34	]	]	PUNCT
ejpam-5721	33	35	.	.	PUNCT
ejpam-5721	34	1	noiri	noiri	PROPN
ejpam-5721	34	2	and	and	CCONJ
ejpam-5721	34	3	popa	popa	NOUN
ejpam-5721	34	4	[	[	X
ejpam-5721	34	5	28	28	NUM
ejpam-5721	34	6	,	,	PUNCT
ejpam-5721	34	7	29	29	NUM
ejpam-5721	34	8	]	]	PUNCT
ejpam-5721	34	9	investigated	investigate	VERB
ejpam-5721	34	10	further	further	ADJ
ejpam-5721	34	11	characterizations	characterization	NOUN
ejpam-5721	34	12	of	of	ADP
ejpam-5721	34	13	almost	almost	ADV
ejpam-5721	34	14	weakly	weakly	ADJ
ejpam-5721	34	15	continuous	continuous	ADJ
ejpam-5721	34	16	functions	function	NOUN
ejpam-5721	34	17	.	.	PUNCT
ejpam-5721	35	1	in	in	ADP
ejpam-5721	35	2	this	this	DET
ejpam-5721	35	3	paper	paper	NOUN
ejpam-5721	35	4	,	,	PUNCT
ejpam-5721	35	5	we	we	PRON
ejpam-5721	35	6	introduce	introduce	VERB
ejpam-5721	35	7	the	the	DET
ejpam-5721	35	8	notion	notion	NOUN
ejpam-5721	35	9	of	of	ADP
ejpam-5721	35	10	almost	almost	ADV
ejpam-5721	35	11	weakly	weakly	ADJ
ejpam-5721	35	12	(	(	PUNCT
ejpam-5721	35	13	τ1	τ1	NOUN
ejpam-5721	35	14	,	,	PUNCT
ejpam-5721	35	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	35	16	functions	function	NOUN
ejpam-5721	35	17	.	.	PUNCT
ejpam-5721	36	1	we	we	PRON
ejpam-5721	36	2	also	also	ADV
ejpam-5721	36	3	investigate	investigate	VERB
ejpam-5721	36	4	several	several	ADJ
ejpam-5721	36	5	characterizations	characterization	NOUN
ejpam-5721	36	6	of	of	ADP
ejpam-5721	36	7	almost	almost	ADV
ejpam-5721	36	8	weakly	weakly	ADJ
ejpam-5721	36	9	(	(	PUNCT
ejpam-5721	36	10	τ1	τ1	NOUN
ejpam-5721	36	11	,	,	PUNCT
ejpam-5721	36	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	36	13	functions	function	NOUN
ejpam-5721	36	14	.	.	PUNCT
ejpam-5721	37	1	2	2	X
ejpam-5721	37	2	.	.	NUM
ejpam-5721	37	3	preliminaries	preliminary	NOUN
ejpam-5721	37	4	throughout	throughout	ADP
ejpam-5721	37	5	the	the	DET
ejpam-5721	37	6	present	present	ADJ
ejpam-5721	37	7	paper	paper	NOUN
ejpam-5721	37	8	,	,	PUNCT
ejpam-5721	37	9	spaces	space	NOUN
ejpam-5721	37	10	(	(	PUNCT
ejpam-5721	37	11	x	x	NOUN
ejpam-5721	37	12	,	,	PUNCT
ejpam-5721	37	13	τ1	τ1	NOUN
ejpam-5721	37	14	,	,	PUNCT
ejpam-5721	37	15	τ2	τ2	NOUN
ejpam-5721	37	16	)	)	PUNCT
ejpam-5721	37	17	and	and	CCONJ
ejpam-5721	37	18	(	(	PUNCT
ejpam-5721	37	19	y	y	PROPN
ejpam-5721	37	20	,	,	PUNCT
ejpam-5721	37	21	σ1	σ1	PROPN
ejpam-5721	37	22	,	,	PUNCT
ejpam-5721	37	23	σ2	σ2	NOUN
ejpam-5721	37	24	)	)	PUNCT
ejpam-5721	37	25	(	(	PUNCT
ejpam-5721	37	26	or	or	CCONJ
ejpam-5721	37	27	simply	simply	ADV
ejpam-5721	37	28	x	x	X
ejpam-5721	37	29	and	and	CCONJ
ejpam-5721	37	30	y	y	PROPN
ejpam-5721	37	31	)	)	PUNCT
ejpam-5721	37	32	always	always	ADV
ejpam-5721	37	33	mean	mean	VERB
ejpam-5721	37	34	bitopological	bitopological	ADJ
ejpam-5721	37	35	spaces	space	NOUN
ejpam-5721	37	36	on	on	ADP
ejpam-5721	37	37	which	which	PRON
ejpam-5721	37	38	no	no	DET
ejpam-5721	37	39	separation	separation	NOUN
ejpam-5721	37	40	axioms	axiom	NOUN
ejpam-5721	37	41	are	be	AUX
ejpam-5721	37	42	assumed	assume	VERB
ejpam-5721	37	43	unless	unless	SCONJ
ejpam-5721	37	44	explicitly	explicitly	ADV
ejpam-5721	37	45	stated	state	VERB
ejpam-5721	37	46	.	.	PUNCT
ejpam-5721	38	1	let	let	VERB
ejpam-5721	38	2	a	a	DET
ejpam-5721	38	3	be	be	AUX
ejpam-5721	38	4	a	a	DET
ejpam-5721	38	5	subset	subset	NOUN
ejpam-5721	38	6	of	of	ADP
ejpam-5721	38	7	a	a	DET
ejpam-5721	38	8	bitopological	bitopological	ADJ
ejpam-5721	38	9	space	space	NOUN
ejpam-5721	38	10	(	(	PUNCT
ejpam-5721	38	11	x	x	NOUN
ejpam-5721	38	12	,	,	PUNCT
ejpam-5721	38	13	τ1	τ1	NOUN
ejpam-5721	38	14	,	,	PUNCT
ejpam-5721	38	15	τ2	τ2	NOUN
ejpam-5721	38	16	)	)	PUNCT
ejpam-5721	38	17	.	.	PUNCT
ejpam-5721	39	1	the	the	DET
ejpam-5721	39	2	closure	closure	NOUN
ejpam-5721	39	3	of	of	ADP
ejpam-5721	39	4	a	a	PRON
ejpam-5721	39	5	and	and	CCONJ
ejpam-5721	39	6	the	the	DET
ejpam-5721	39	7	interior	interior	NOUN
ejpam-5721	39	8	of	of	ADP
ejpam-5721	39	9	a	a	PRON
ejpam-5721	39	10	with	with	ADP
ejpam-5721	39	11	respect	respect	NOUN
ejpam-5721	39	12	to	to	ADP
ejpam-5721	39	13	τi	τi	PROPN
ejpam-5721	39	14	are	be	AUX
ejpam-5721	39	15	denoted	denote	VERB
ejpam-5721	39	16	by	by	ADP
ejpam-5721	39	17	τi	τi	NOUN
ejpam-5721	39	18	-	-	PUNCT
ejpam-5721	39	19	cl(a	cl(a	NUM
ejpam-5721	39	20	)	)	PUNCT
ejpam-5721	39	21	and	and	CCONJ
ejpam-5721	39	22	τi	τi	NOUN
ejpam-5721	39	23	-	-	PUNCT
ejpam-5721	39	24	int(a	int(a	NOUN
ejpam-5721	39	25	)	)	PUNCT
ejpam-5721	39	26	,	,	PUNCT
ejpam-5721	39	27	respectively	respectively	ADV
ejpam-5721	39	28	,	,	PUNCT
ejpam-5721	39	29	for	for	ADP
ejpam-5721	39	30	i	i	PROPN
ejpam-5721	39	31	=	=	SYM
ejpam-5721	39	32	1	1	NUM
ejpam-5721	39	33	,	,	PUNCT
ejpam-5721	39	34	2	2	NUM
ejpam-5721	39	35	.	.	X
ejpam-5721	39	36	a	a	DET
ejpam-5721	39	37	subset	subset	NOUN
ejpam-5721	39	38	a	a	PRON
ejpam-5721	39	39	of	of	ADP
ejpam-5721	39	40	a	a	DET
ejpam-5721	39	41	bitopological	bitopological	ADJ
ejpam-5721	39	42	space	space	NOUN
ejpam-5721	39	43	(	(	PUNCT
ejpam-5721	39	44	x	x	NOUN
ejpam-5721	39	45	,	,	PUNCT
ejpam-5721	39	46	τ1	τ1	NOUN
ejpam-5721	39	47	,	,	PUNCT
ejpam-5721	39	48	τ2	τ2	NOUN
ejpam-5721	39	49	)	)	PUNCT
ejpam-5721	39	50	is	be	AUX
ejpam-5721	39	51	called	call	VERB
ejpam-5721	39	52	τ1τ2	τ1τ2	VERB
ejpam-5721	39	53	-	-	ADJ
ejpam-5721	39	54	closed	closed	ADJ
ejpam-5721	39	55	[	[	X
ejpam-5721	39	56	14	14	NUM
ejpam-5721	39	57	]	]	X
ejpam-5721	39	58	if	if	SCONJ
ejpam-5721	39	59	a	a	DET
ejpam-5721	39	60	=	=	NOUN
ejpam-5721	39	61	τ1	τ1	NOUN
ejpam-5721	39	62	-	-	PUNCT
ejpam-5721	39	63	cl(τ2	cl(τ2	NOUN
ejpam-5721	39	64	-	-	PUNCT
ejpam-5721	39	65	cl(a	cl(a	NUM
ejpam-5721	39	66	)	)	PUNCT
ejpam-5721	39	67	)	)	PUNCT
ejpam-5721	39	68	.	.	PUNCT
ejpam-5721	40	1	the	the	DET
ejpam-5721	40	2	complement	complement	NOUN
ejpam-5721	40	3	of	of	ADP
ejpam-5721	40	4	a	a	DET
ejpam-5721	40	5	τ1τ2	τ1τ2	ADJ
ejpam-5721	40	6	-	-	ADJ
ejpam-5721	40	7	closed	closed	ADJ
ejpam-5721	40	8	set	set	NOUN
ejpam-5721	40	9	is	be	AUX
ejpam-5721	40	10	called	call	VERB
ejpam-5721	40	11	τ1τ2	τ1τ2	NOUN
ejpam-5721	40	12	-	-	ADJ
ejpam-5721	40	13	open	open	ADJ
ejpam-5721	40	14	.	.	PUNCT
ejpam-5721	41	1	the	the	DET
ejpam-5721	41	2	intersection	intersection	NOUN
ejpam-5721	41	3	of	of	ADP
ejpam-5721	41	4	all	all	DET
ejpam-5721	41	5	τ1τ2	τ1τ2	ADJ
ejpam-5721	41	6	-	-	ADJ
ejpam-5721	41	7	closed	closed	ADJ
ejpam-5721	41	8	sets	set	NOUN
ejpam-5721	41	9	of	of	ADP
ejpam-5721	41	10	x	x	PUNCT
ejpam-5721	41	11	containing	contain	VERB
ejpam-5721	41	12	a	a	PRON
ejpam-5721	41	13	is	be	AUX
ejpam-5721	41	14	called	call	VERB
ejpam-5721	41	15	the	the	DET
ejpam-5721	41	16	τ1τ2	τ1τ2	NOUN
ejpam-5721	41	17	-	-	NOUN
ejpam-5721	41	18	closure	closure	NOUN
ejpam-5721	41	19	[	[	X
ejpam-5721	41	20	14	14	NUM
ejpam-5721	41	21	]	]	PUNCT
ejpam-5721	41	22	of	of	ADP
ejpam-5721	41	23	a	a	PRON
ejpam-5721	41	24	and	and	CCONJ
ejpam-5721	41	25	is	be	AUX
ejpam-5721	41	26	denoted	denote	VERB
ejpam-5721	41	27	by	by	ADP
ejpam-5721	41	28	τ1τ2	τ1τ2	NOUN
ejpam-5721	41	29	-	-	NUM
ejpam-5721	41	30	cl(a	cl(a	NUM
ejpam-5721	41	31	)	)	PUNCT
ejpam-5721	41	32	.	.	PUNCT
ejpam-5721	42	1	the	the	DET
ejpam-5721	42	2	union	union	NOUN
ejpam-5721	42	3	of	of	ADP
ejpam-5721	42	4	all	all	DET
ejpam-5721	42	5	τ1τ2	τ1τ2	ADJ
ejpam-5721	42	6	-	-	ADJ
ejpam-5721	42	7	open	open	ADJ
ejpam-5721	42	8	sets	set	NOUN
ejpam-5721	42	9	of	of	ADP
ejpam-5721	42	10	x	x	PUNCT
ejpam-5721	42	11	contained	contain	VERB
ejpam-5721	42	12	in	in	ADP
ejpam-5721	42	13	a	a	PRON
ejpam-5721	42	14	is	be	AUX
ejpam-5721	42	15	called	call	VERB
ejpam-5721	42	16	the	the	DET
ejpam-5721	42	17	τ1τ2	τ1τ2	NOUN
ejpam-5721	42	18	-	-	ADJ
ejpam-5721	42	19	interior	interior	ADJ
ejpam-5721	42	20	[	[	X
ejpam-5721	42	21	14	14	NUM
ejpam-5721	42	22	]	]	PUNCT
ejpam-5721	42	23	of	of	ADP
ejpam-5721	42	24	a	a	PRON
ejpam-5721	42	25	and	and	CCONJ
ejpam-5721	42	26	is	be	AUX
ejpam-5721	42	27	denoted	denote	VERB
ejpam-5721	42	28	by	by	ADP
ejpam-5721	42	29	τ1τ2	τ1τ2	NOUN
ejpam-5721	42	30	-	-	ADJ
ejpam-5721	42	31	int(a	int(a	NOUN
ejpam-5721	42	32	)	)	PUNCT
ejpam-5721	42	33	.	.	PUNCT
ejpam-5721	43	1	lemma	lemma	PROPN
ejpam-5721	43	2	1	1	NUM
ejpam-5721	43	3	.	.	PUNCT
ejpam-5721	44	1	[	[	X
ejpam-5721	44	2	14	14	NUM
ejpam-5721	44	3	]	]	PUNCT
ejpam-5721	44	4	let	let	VERB
ejpam-5721	44	5	a	a	PRON
ejpam-5721	44	6	and	and	CCONJ
ejpam-5721	44	7	b	b	NOUN
ejpam-5721	44	8	be	be	AUX
ejpam-5721	44	9	subsets	subset	NOUN
ejpam-5721	44	10	of	of	ADP
ejpam-5721	44	11	a	a	DET
ejpam-5721	44	12	bitopological	bitopological	ADJ
ejpam-5721	44	13	space	space	NOUN
ejpam-5721	44	14	(	(	PUNCT
ejpam-5721	44	15	x	x	NOUN
ejpam-5721	44	16	,	,	PUNCT
ejpam-5721	44	17	τ1	τ1	NOUN
ejpam-5721	44	18	,	,	PUNCT
ejpam-5721	44	19	τ2	τ2	NOUN
ejpam-5721	44	20	)	)	PUNCT
ejpam-5721	44	21	.	.	PUNCT
ejpam-5721	45	1	for	for	ADP
ejpam-5721	45	2	the	the	DET
ejpam-5721	45	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-5721	45	4	,	,	PUNCT
ejpam-5721	45	5	the	the	DET
ejpam-5721	45	6	following	follow	VERB
ejpam-5721	45	7	properties	property	NOUN
ejpam-5721	45	8	hold	hold	VERB
ejpam-5721	45	9	:	:	PUNCT
ejpam-5721	45	10	(	(	PUNCT
ejpam-5721	45	11	1	1	X
ejpam-5721	45	12	)	)	PUNCT
ejpam-5721	45	13	a	a	DET
ejpam-5721	45	14	⊆	⊆	NUM
ejpam-5721	45	15	τ1τ2	τ1τ2	NOUN
ejpam-5721	45	16	-	-	NUM
ejpam-5721	45	17	cl(a	cl(a	NUM
ejpam-5721	45	18	)	)	PUNCT
ejpam-5721	45	19	and	and	CCONJ
ejpam-5721	45	20	τ1τ2	τ1τ2	NOUN
ejpam-5721	45	21	-	-	ADJ
ejpam-5721	45	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	45	23	-	-	PUNCT
ejpam-5721	45	24	cl(a	cl(a	NUM
ejpam-5721	45	25	)	)	PUNCT
ejpam-5721	45	26	)	)	PUNCT
ejpam-5721	46	1	=	=	PUNCT
ejpam-5721	46	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	46	3	-	-	NUM
ejpam-5721	46	4	cl(a	cl(a	NUM
ejpam-5721	46	5	)	)	PUNCT
ejpam-5721	46	6	.	.	PUNCT
ejpam-5721	47	1	(	(	PUNCT
ejpam-5721	47	2	2	2	X
ejpam-5721	47	3	)	)	PUNCT
ejpam-5721	47	4	if	if	SCONJ
ejpam-5721	47	5	a	a	DET
ejpam-5721	47	6	⊆	⊆	NUM
ejpam-5721	47	7	b	b	NOUN
ejpam-5721	47	8	,	,	PUNCT
ejpam-5721	47	9	then	then	ADV
ejpam-5721	47	10	τ1τ2	τ1τ2	NOUN
ejpam-5721	47	11	-	-	NUM
ejpam-5721	47	12	cl(a	cl(a	NUM
ejpam-5721	47	13	)	)	PUNCT
ejpam-5721	47	14	⊆	⊆	NUM
ejpam-5721	47	15	τ1τ2	τ1τ2	NOUN
ejpam-5721	47	16	-	-	NOUN
ejpam-5721	47	17	cl(b	cl(b	NOUN
ejpam-5721	47	18	)	)	PUNCT
ejpam-5721	47	19	.	.	PUNCT
ejpam-5721	48	1	(	(	PUNCT
ejpam-5721	48	2	3	3	X
ejpam-5721	48	3	)	)	PUNCT
ejpam-5721	48	4	τ1τ2	τ1τ2	NOUN
ejpam-5721	48	5	-	-	NUM
ejpam-5721	48	6	cl(a	cl(a	NUM
ejpam-5721	48	7	)	)	PUNCT
ejpam-5721	48	8	is	be	AUX
ejpam-5721	48	9	τ1τ2	τ1τ2	NOUN
ejpam-5721	48	10	-	-	ADJ
ejpam-5721	48	11	closed	closed	ADJ
ejpam-5721	48	12	.	.	PUNCT
ejpam-5721	49	1	(	(	PUNCT
ejpam-5721	49	2	4	4	X
ejpam-5721	49	3	)	)	PUNCT
ejpam-5721	49	4	a	a	PRON
ejpam-5721	49	5	is	be	AUX
ejpam-5721	49	6	τ1τ2	τ1τ2	NOUN
ejpam-5721	49	7	-	-	ADJ
ejpam-5721	49	8	closed	closed	ADJ
ejpam-5721	49	9	if	if	SCONJ
ejpam-5721	49	10	and	and	CCONJ
ejpam-5721	49	11	only	only	ADV
ejpam-5721	49	12	if	if	SCONJ
ejpam-5721	49	13	a	a	DET
ejpam-5721	49	14	=	=	PUNCT
ejpam-5721	49	15	τ1τ2	τ1τ2	NOUN
ejpam-5721	49	16	-	-	NUM
ejpam-5721	49	17	cl(a	cl(a	NUM
ejpam-5721	49	18	)	)	PUNCT
ejpam-5721	49	19	.	.	PUNCT
ejpam-5721	50	1	(	(	PUNCT
ejpam-5721	50	2	5	5	X
ejpam-5721	50	3	)	)	PUNCT
ejpam-5721	50	4	τ1τ2	τ1τ2	NOUN
ejpam-5721	50	5	-	-	NOUN
ejpam-5721	50	6	cl(x	cl(x	X
ejpam-5721	50	7	−a	−a	NOUN
ejpam-5721	50	8	)	)	PUNCT
ejpam-5721	51	1	=	=	PUNCT
ejpam-5721	51	2	x	x	X
ejpam-5721	52	1	−	−	ADP
ejpam-5721	52	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	52	3	-	-	ADJ
ejpam-5721	52	4	int(a	int(a	NOUN
ejpam-5721	52	5	)	)	PUNCT
ejpam-5721	52	6	.	.	PUNCT
ejpam-5721	53	1	lemma	lemma	PROPN
ejpam-5721	53	2	2	2	NUM
ejpam-5721	53	3	.	.	X
ejpam-5721	54	1	for	for	ADP
ejpam-5721	54	2	a	a	DET
ejpam-5721	54	3	subset	subset	NOUN
ejpam-5721	54	4	a	a	PRON
ejpam-5721	54	5	of	of	ADP
ejpam-5721	54	6	a	a	DET
ejpam-5721	54	7	bitopological	bitopological	ADJ
ejpam-5721	54	8	space	space	NOUN
ejpam-5721	54	9	(	(	PUNCT
ejpam-5721	54	10	x	x	NOUN
ejpam-5721	54	11	,	,	PUNCT
ejpam-5721	54	12	τ1	τ1	NOUN
ejpam-5721	54	13	,	,	PUNCT
ejpam-5721	54	14	τ2	τ2	NOUN
ejpam-5721	54	15	)	)	PUNCT
ejpam-5721	54	16	,	,	PUNCT
ejpam-5721	54	17	the	the	DET
ejpam-5721	54	18	following	follow	VERB
ejpam-5721	54	19	properties	property	NOUN
ejpam-5721	54	20	hold	hold	VERB
ejpam-5721	54	21	:	:	PUNCT
ejpam-5721	54	22	(	(	PUNCT
ejpam-5721	54	23	1	1	X
ejpam-5721	54	24	)	)	PUNCT
ejpam-5721	54	25	τ1τ2	τ1τ2	NOUN
ejpam-5721	54	26	-	-	NUM
ejpam-5721	54	27	cl(a	cl(a	NUM
ejpam-5721	54	28	)	)	PUNCT
ejpam-5721	54	29	∩	∩	NOUN
ejpam-5721	54	30	v	v	ADP
ejpam-5721	54	31	⊆	⊆	NUM
ejpam-5721	54	32	τ1τ2	τ1τ2	ADJ
ejpam-5721	54	33	-	-	ADJ
ejpam-5721	54	34	cl(a	cl(a	ADJ
ejpam-5721	54	35	∩	∩	ADJ
ejpam-5721	54	36	v	v	NOUN
ejpam-5721	54	37	)	)	PUNCT
ejpam-5721	54	38	for	for	ADP
ejpam-5721	54	39	every	every	DET
ejpam-5721	54	40	τ1τ2	τ1τ2	ADJ
ejpam-5721	54	41	-	-	ADJ
ejpam-5721	54	42	open	open	ADJ
ejpam-5721	54	43	set	set	NOUN
ejpam-5721	54	44	v	v	NOUN
ejpam-5721	54	45	of	of	ADP
ejpam-5721	54	46	x	x	PRON
ejpam-5721	54	47	;	;	PUNCT
ejpam-5721	54	48	(	(	PUNCT
ejpam-5721	54	49	2	2	X
ejpam-5721	54	50	)	)	PUNCT
ejpam-5721	54	51	τ1τ2	τ1τ2	NOUN
ejpam-5721	54	52	-	-	PUNCT
ejpam-5721	54	53	int(a	int(a	NOUN
ejpam-5721	54	54	∪	∪	NOUN
ejpam-5721	54	55	f	f	NOUN
ejpam-5721	54	56	)	)	PUNCT
ejpam-5721	55	1	⊆	⊆	NUM
ejpam-5721	55	2	τ1τ2	τ1τ2	X
ejpam-5721	55	3	-	-	ADJ
ejpam-5721	55	4	int(a	int(a	NOUN
ejpam-5721	55	5	)	)	PUNCT
ejpam-5721	55	6	∪	∪	NOUN
ejpam-5721	55	7	f	f	PROPN
ejpam-5721	55	8	for	for	ADP
ejpam-5721	55	9	every	every	DET
ejpam-5721	55	10	τ1τ2	τ1τ2	ADJ
ejpam-5721	55	11	-	-	ADJ
ejpam-5721	55	12	closed	closed	ADJ
ejpam-5721	55	13	set	set	ADJ
ejpam-5721	55	14	f	f	PROPN
ejpam-5721	55	15	of	of	ADP
ejpam-5721	55	16	x.	x.	PROPN
ejpam-5721	55	17	a	a	DET
ejpam-5721	55	18	subseta	subseta	NOUN
ejpam-5721	55	19	of	of	ADP
ejpam-5721	55	20	a	a	DET
ejpam-5721	55	21	bitopological	bitopological	ADJ
ejpam-5721	55	22	space	space	NOUN
ejpam-5721	55	23	(	(	PUNCT
ejpam-5721	55	24	x	x	NOUN
ejpam-5721	55	25	,	,	PUNCT
ejpam-5721	55	26	τ1	τ1	NOUN
ejpam-5721	55	27	,	,	PUNCT
ejpam-5721	55	28	τ2	τ2	NOUN
ejpam-5721	55	29	)	)	PUNCT
ejpam-5721	55	30	is	be	AUX
ejpam-5721	55	31	called	call	VERB
ejpam-5721	55	32	(	(	PUNCT
ejpam-5721	55	33	τ1	τ1	NOUN
ejpam-5721	55	34	,	,	PUNCT
ejpam-5721	55	35	τ2)r	τ2)r	NOUN
ejpam-5721	55	36	-	-	PUNCT
ejpam-5721	55	37	open	open	NOUN
ejpam-5721	55	38	[	[	X
ejpam-5721	55	39	35	35	NUM
ejpam-5721	55	40	]	]	X
ejpam-5721	55	41	(	(	PUNCT
ejpam-5721	55	42	resp	resp	NOUN
ejpam-5721	55	43	.	.	PUNCT
ejpam-5721	56	1	(	(	PUNCT
ejpam-5721	56	2	τ1	τ1	NOUN
ejpam-5721	56	3	,	,	PUNCT
ejpam-5721	56	4	τ2)sopen	τ2)sopen	VERB
ejpam-5721	56	5	[	[	X
ejpam-5721	56	6	4	4	NUM
ejpam-5721	56	7	]	]	PUNCT
ejpam-5721	56	8	,	,	PUNCT
ejpam-5721	56	9	(	(	PUNCT
ejpam-5721	56	10	τ1	τ1	NOUN
ejpam-5721	56	11	,	,	PUNCT
ejpam-5721	56	12	τ2)p	τ2)p	NOUN
ejpam-5721	56	13	-	-	ADJ
ejpam-5721	56	14	open	open	ADJ
ejpam-5721	56	15	[	[	X
ejpam-5721	56	16	4	4	NUM
ejpam-5721	56	17	]	]	PUNCT
ejpam-5721	56	18	,	,	PUNCT
ejpam-5721	56	19	(	(	PUNCT
ejpam-5721	56	20	τ1	τ1	NOUN
ejpam-5721	56	21	,	,	PUNCT
ejpam-5721	56	22	τ2)β	τ2)β	ADJ
ejpam-5721	56	23	-	-	PUNCT
ejpam-5721	56	24	open	open	NOUN
ejpam-5721	56	25	[	[	X
ejpam-5721	56	26	4	4	NUM
ejpam-5721	56	27	]	]	PUNCT
ejpam-5721	56	28	)	)	PUNCT
ejpam-5721	56	29	if	if	SCONJ
ejpam-5721	56	30	a	a	DET
ejpam-5721	56	31	=	=	PUNCT
ejpam-5721	56	32	τ1τ2	τ1τ2	NOUN
ejpam-5721	56	33	-	-	NOUN
ejpam-5721	56	34	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	56	35	-	-	PUNCT
ejpam-5721	56	36	cl(a	cl(a	NUM
ejpam-5721	56	37	)	)	PUNCT
ejpam-5721	56	38	)	)	PUNCT
ejpam-5721	57	1	(	(	PUNCT
ejpam-5721	57	2	resp	resp	NOUN
ejpam-5721	57	3	.	.	PUNCT
ejpam-5721	58	1	a	a	DET
ejpam-5721	58	2	⊆	⊆	NUM
ejpam-5721	58	3	τ1τ2	τ1τ2	NOUN
ejpam-5721	58	4	-	-	ADJ
ejpam-5721	58	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	58	6	-	-	PUNCT
ejpam-5721	58	7	int(a	int(a	NOUN
ejpam-5721	58	8	)	)	PUNCT
ejpam-5721	58	9	)	)	PUNCT
ejpam-5721	58	10	,	,	PUNCT
ejpam-5721	58	11	a	a	DET
ejpam-5721	58	12	⊆	⊆	NUM
ejpam-5721	58	13	τ1τ2	τ1τ2	NOUN
ejpam-5721	58	14	-	-	NOUN
ejpam-5721	58	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	58	16	-	-	PUNCT
ejpam-5721	58	17	cl(a	cl(a	NUM
ejpam-5721	58	18	)	)	PUNCT
ejpam-5721	58	19	)	)	PUNCT
ejpam-5721	58	20	,	,	PUNCT
ejpam-5721	58	21	a	a	DET
ejpam-5721	58	22	⊆	⊆	NUM
ejpam-5721	58	23	τ1τ2	τ1τ2	NOUN
ejpam-5721	58	24	-	-	PUNCT
ejpam-5721	58	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	58	26	-	-	PUNCT
ejpam-5721	58	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	58	28	-	-	PUNCT
ejpam-5721	58	29	cl(a	cl(a	NUM
ejpam-5721	58	30	)	)	PUNCT
ejpam-5721	58	31	)	)	PUNCT
ejpam-5721	58	32	)	)	PUNCT
ejpam-5721	58	33	)	)	PUNCT
ejpam-5721	58	34	.	.	PUNCT
ejpam-5721	59	1	the	the	DET
ejpam-5721	59	2	j.	j.	PROPN
ejpam-5721	59	3	khampakdee	khampakdee	PROPN
ejpam-5721	59	4	,	,	PUNCT
ejpam-5721	59	5	s.	s.	PROPN
ejpam-5721	59	6	sompong	sompong	PROPN
ejpam-5721	59	7	,	,	PUNCT
ejpam-5721	59	8	c.	c.	PROPN
ejpam-5721	59	9	boonpok	boonpok	PROPN
ejpam-5721	59	10	/	/	SYM
ejpam-5721	59	11	eur	eur	PROPN
ejpam-5721	59	12	.	.	PUNCT
ejpam-5721	60	1	j.	j.	PROPN
ejpam-5721	60	2	pure	pure	PROPN
ejpam-5721	60	3	appl	appl	PROPN
ejpam-5721	60	4	.	.	PROPN
ejpam-5721	60	5	math	math	PROPN
ejpam-5721	60	6	,	,	PUNCT
ejpam-5721	60	7	18	18	NUM
ejpam-5721	60	8	(	(	PUNCT
ejpam-5721	60	9	1	1	NUM
ejpam-5721	60	10	)	)	PUNCT
ejpam-5721	60	11	(	(	PUNCT
ejpam-5721	60	12	2025	2025	NUM
ejpam-5721	60	13	)	)	PUNCT
ejpam-5721	60	14	,	,	PUNCT
ejpam-5721	60	15	5721	5721	NUM
ejpam-5721	60	16	3	3	NUM
ejpam-5721	60	17	of	of	ADP
ejpam-5721	60	18	12	12	NUM
ejpam-5721	60	19	complement	complement	NOUN
ejpam-5721	60	20	of	of	ADP
ejpam-5721	60	21	a	a	DET
ejpam-5721	60	22	(	(	PUNCT
ejpam-5721	60	23	τ1	τ1	NOUN
ejpam-5721	60	24	,	,	PUNCT
ejpam-5721	60	25	τ2)r	τ2)r	NOUN
ejpam-5721	60	26	-	-	PUNCT
ejpam-5721	60	27	open	open	ADJ
ejpam-5721	60	28	(	(	PUNCT
ejpam-5721	60	29	resp	resp	NOUN
ejpam-5721	60	30	.	.	PUNCT
ejpam-5721	61	1	(	(	PUNCT
ejpam-5721	61	2	τ1	τ1	NOUN
ejpam-5721	61	3	,	,	PUNCT
ejpam-5721	61	4	τ2)s	τ2)s	NOUN
ejpam-5721	61	5	-	-	PUNCT
ejpam-5721	61	6	open	open	ADJ
ejpam-5721	61	7	,	,	PUNCT
ejpam-5721	61	8	(	(	PUNCT
ejpam-5721	61	9	τ1	τ1	NOUN
ejpam-5721	61	10	,	,	PUNCT
ejpam-5721	61	11	τ2)p	τ2)p	NOUN
ejpam-5721	61	12	-	-	ADJ
ejpam-5721	61	13	open	open	ADJ
ejpam-5721	61	14	,	,	PUNCT
ejpam-5721	61	15	(	(	PUNCT
ejpam-5721	61	16	τ1	τ1	NOUN
ejpam-5721	61	17	,	,	PUNCT
ejpam-5721	61	18	τ2)β	τ2)β	ADJ
ejpam-5721	61	19	-	-	PUNCT
ejpam-5721	61	20	open	open	ADJ
ejpam-5721	61	21	)	)	PUNCT
ejpam-5721	61	22	set	set	NOUN
ejpam-5721	61	23	is	be	AUX
ejpam-5721	61	24	called	call	VERB
ejpam-5721	61	25	(	(	PUNCT
ejpam-5721	61	26	τ1	τ1	NOUN
ejpam-5721	61	27	,	,	PUNCT
ejpam-5721	61	28	τ2)r	τ2)r	NOUN
ejpam-5721	61	29	-	-	PUNCT
ejpam-5721	61	30	closed	closed	ADJ
ejpam-5721	61	31	(	(	PUNCT
ejpam-5721	61	32	resp	resp	NOUN
ejpam-5721	61	33	.	.	PUNCT
ejpam-5721	62	1	(	(	PUNCT
ejpam-5721	62	2	τ1	τ1	NOUN
ejpam-5721	62	3	,	,	PUNCT
ejpam-5721	62	4	τ2)s	τ2)s	NOUN
ejpam-5721	62	5	-	-	PUNCT
ejpam-5721	62	6	closed	closed	ADJ
ejpam-5721	62	7	,	,	PUNCT
ejpam-5721	62	8	(	(	PUNCT
ejpam-5721	62	9	τ1	τ1	NOUN
ejpam-5721	62	10	,	,	PUNCT
ejpam-5721	62	11	τ2)p	τ2)p	NOUN
ejpam-5721	62	12	-	-	PUNCT
ejpam-5721	62	13	closed	closed	ADJ
ejpam-5721	62	14	,	,	PUNCT
ejpam-5721	62	15	(	(	PUNCT
ejpam-5721	62	16	τ1	τ1	NOUN
ejpam-5721	62	17	,	,	PUNCT
ejpam-5721	62	18	τ2)β	τ2)β	ADJ
ejpam-5721	62	19	-	-	PUNCT
ejpam-5721	62	20	closed	closed	ADJ
ejpam-5721	62	21	)	)	PUNCT
ejpam-5721	62	22	.	.	PUNCT
ejpam-5721	63	1	a	a	DET
ejpam-5721	63	2	subset	subset	NOUN
ejpam-5721	63	3	a	a	PRON
ejpam-5721	63	4	of	of	ADP
ejpam-5721	63	5	a	a	DET
ejpam-5721	63	6	bitopological	bitopological	ADJ
ejpam-5721	63	7	space	space	NOUN
ejpam-5721	63	8	(	(	PUNCT
ejpam-5721	63	9	x	x	NOUN
ejpam-5721	63	10	,	,	PUNCT
ejpam-5721	63	11	τ1	τ1	NOUN
ejpam-5721	63	12	,	,	PUNCT
ejpam-5721	63	13	τ2	τ2	NOUN
ejpam-5721	63	14	)	)	PUNCT
ejpam-5721	63	15	is	be	AUX
ejpam-5721	63	16	said	say	VERB
ejpam-5721	63	17	to	to	PART
ejpam-5721	63	18	be	be	AUX
ejpam-5721	63	19	α(τ1	α(τ1	NOUN
ejpam-5721	63	20	,	,	PUNCT
ejpam-5721	63	21	τ2)-open	τ2)-open	ADJ
ejpam-5721	63	22	[	[	X
ejpam-5721	63	23	37	37	NUM
ejpam-5721	63	24	]	]	PUNCT
ejpam-5721	63	25	.	.	PUNCT
ejpam-5721	64	1	the	the	DET
ejpam-5721	64	2	complement	complement	NOUN
ejpam-5721	64	3	of	of	ADP
ejpam-5721	64	4	an	an	DET
ejpam-5721	64	5	α(τ1	α(τ1	NOUN
ejpam-5721	64	6	,	,	PUNCT
ejpam-5721	64	7	τ2)-open	τ2)-open	ADJ
ejpam-5721	64	8	set	set	NOUN
ejpam-5721	64	9	is	be	AUX
ejpam-5721	64	10	said	say	VERB
ejpam-5721	64	11	to	to	PART
ejpam-5721	64	12	be	be	AUX
ejpam-5721	64	13	α(τ1	α(τ1	NOUN
ejpam-5721	64	14	,	,	PUNCT
ejpam-5721	64	15	τ2)-closed	τ2)-close	VERB
ejpam-5721	64	16	.	.	PUNCT
ejpam-5721	65	1	let	let	VERB
ejpam-5721	65	2	a	a	DET
ejpam-5721	65	3	be	be	AUX
ejpam-5721	65	4	a	a	DET
ejpam-5721	65	5	subset	subset	NOUN
ejpam-5721	65	6	of	of	ADP
ejpam-5721	65	7	a	a	DET
ejpam-5721	65	8	bitopological	bitopological	ADJ
ejpam-5721	65	9	space	space	NOUN
ejpam-5721	65	10	(	(	PUNCT
ejpam-5721	65	11	x	x	NOUN
ejpam-5721	65	12	,	,	PUNCT
ejpam-5721	65	13	τ1	τ1	NOUN
ejpam-5721	65	14	,	,	PUNCT
ejpam-5721	65	15	τ2	τ2	NOUN
ejpam-5721	65	16	)	)	PUNCT
ejpam-5721	65	17	.	.	PUNCT
ejpam-5721	66	1	the	the	DET
ejpam-5721	66	2	intersection	intersection	NOUN
ejpam-5721	66	3	of	of	ADP
ejpam-5721	66	4	all	all	DET
ejpam-5721	66	5	(	(	PUNCT
ejpam-5721	66	6	τ1	τ1	NOUN
ejpam-5721	66	7	,	,	PUNCT
ejpam-5721	66	8	τ2)p	τ2)p	ADJ
ejpam-5721	66	9	-	-	PUNCT
ejpam-5721	66	10	closed	closed	ADJ
ejpam-5721	66	11	sets	set	NOUN
ejpam-5721	66	12	of	of	ADP
ejpam-5721	66	13	x	x	PUNCT
ejpam-5721	66	14	containing	contain	VERB
ejpam-5721	66	15	a	a	PRON
ejpam-5721	66	16	is	be	AUX
ejpam-5721	66	17	called	call	VERB
ejpam-5721	66	18	the	the	DET
ejpam-5721	66	19	(	(	PUNCT
ejpam-5721	66	20	τ1	τ1	NOUN
ejpam-5721	66	21	,	,	PUNCT
ejpam-5721	66	22	τ2)p	τ2)p	NOUN
ejpam-5721	66	23	-	-	PUNCT
ejpam-5721	66	24	closure	closure	NOUN
ejpam-5721	66	25	of	of	ADP
ejpam-5721	66	26	a	a	PRON
ejpam-5721	66	27	and	and	CCONJ
ejpam-5721	66	28	is	be	AUX
ejpam-5721	66	29	denoted	denote	VERB
ejpam-5721	66	30	by	by	ADP
ejpam-5721	66	31	(	(	PUNCT
ejpam-5721	66	32	τ1	τ1	PROPN
ejpam-5721	66	33	,	,	PUNCT
ejpam-5721	66	34	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	66	35	)	)	PUNCT
ejpam-5721	66	36	.	.	PUNCT
ejpam-5721	67	1	the	the	DET
ejpam-5721	67	2	union	union	NOUN
ejpam-5721	67	3	of	of	ADP
ejpam-5721	67	4	all	all	DET
ejpam-5721	67	5	(	(	PUNCT
ejpam-5721	67	6	τ1	τ1	NOUN
ejpam-5721	67	7	,	,	PUNCT
ejpam-5721	67	8	τ2)p	τ2)p	ADJ
ejpam-5721	67	9	-	-	PUNCT
ejpam-5721	67	10	open	open	ADJ
ejpam-5721	67	11	sets	set	NOUN
ejpam-5721	67	12	of	of	ADP
ejpam-5721	67	13	x	x	PUNCT
ejpam-5721	67	14	contained	contain	VERB
ejpam-5721	67	15	in	in	ADP
ejpam-5721	67	16	a	a	PRON
ejpam-5721	67	17	is	be	AUX
ejpam-5721	67	18	called	call	VERB
ejpam-5721	67	19	the	the	DET
ejpam-5721	67	20	(	(	PUNCT
ejpam-5721	67	21	τ1	τ1	NOUN
ejpam-5721	67	22	,	,	PUNCT
ejpam-5721	67	23	τ2)p	τ2)p	ADJ
ejpam-5721	67	24	-	-	NOUN
ejpam-5721	67	25	interior	interior	NOUN
ejpam-5721	67	26	of	of	ADP
ejpam-5721	67	27	a	a	PRON
ejpam-5721	67	28	and	and	CCONJ
ejpam-5721	67	29	is	be	AUX
ejpam-5721	67	30	denoted	denote	VERB
ejpam-5721	67	31	by	by	ADP
ejpam-5721	67	32	(	(	PUNCT
ejpam-5721	67	33	τ1	τ1	NOUN
ejpam-5721	67	34	,	,	PUNCT
ejpam-5721	67	35	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-5721	67	36	)	)	PUNCT
ejpam-5721	67	37	.	.	PUNCT
ejpam-5721	68	1	lemma	lemma	PROPN
ejpam-5721	68	2	3	3	X
ejpam-5721	68	3	.	.	X
ejpam-5721	69	1	for	for	ADP
ejpam-5721	69	2	a	a	DET
ejpam-5721	69	3	subset	subset	NOUN
ejpam-5721	69	4	a	a	PRON
ejpam-5721	69	5	of	of	ADP
ejpam-5721	69	6	a	a	DET
ejpam-5721	69	7	bitopological	bitopological	ADJ
ejpam-5721	69	8	space	space	NOUN
ejpam-5721	69	9	(	(	PUNCT
ejpam-5721	69	10	x	x	NOUN
ejpam-5721	69	11	,	,	PUNCT
ejpam-5721	69	12	τ1	τ1	NOUN
ejpam-5721	69	13	,	,	PUNCT
ejpam-5721	69	14	τ2	τ2	NOUN
ejpam-5721	69	15	)	)	PUNCT
ejpam-5721	69	16	,	,	PUNCT
ejpam-5721	69	17	the	the	DET
ejpam-5721	69	18	following	follow	VERB
ejpam-5721	69	19	properties	property	NOUN
ejpam-5721	69	20	hold	hold	VERB
ejpam-5721	69	21	:	:	PUNCT
ejpam-5721	69	22	(	(	PUNCT
ejpam-5721	69	23	1	1	X
ejpam-5721	69	24	)	)	PUNCT
ejpam-5721	69	25	(	(	PUNCT
ejpam-5721	69	26	τ1	τ1	NOUN
ejpam-5721	69	27	,	,	PUNCT
ejpam-5721	69	28	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-5721	69	29	)	)	PUNCT
ejpam-5721	69	30	=	=	PUNCT
ejpam-5721	70	1	τ1τ2	τ1τ2	NOUN
ejpam-5721	70	2	-	-	ADJ
ejpam-5721	70	3	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	70	4	-	-	PUNCT
ejpam-5721	70	5	int(a	int(a	NOUN
ejpam-5721	70	6	)	)	PUNCT
ejpam-5721	70	7	)	)	PUNCT
ejpam-5721	70	8	∪a	∪a	X
ejpam-5721	71	1	[	[	X
ejpam-5721	71	2	36	36	NUM
ejpam-5721	71	3	]	]	X
ejpam-5721	71	4	;	;	PUNCT
ejpam-5721	71	5	(	(	PUNCT
ejpam-5721	71	6	2	2	X
ejpam-5721	71	7	)	)	PUNCT
ejpam-5721	71	8	(	(	PUNCT
ejpam-5721	71	9	τ1	τ1	NOUN
ejpam-5721	71	10	,	,	PUNCT
ejpam-5721	71	11	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-5721	71	12	)	)	PUNCT
ejpam-5721	71	13	=	=	PUNCT
ejpam-5721	72	1	τ1τ2	τ1τ2	NOUN
ejpam-5721	72	2	-	-	NOUN
ejpam-5721	72	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	72	4	-	-	PUNCT
ejpam-5721	72	5	cl(a	cl(a	NUM
ejpam-5721	72	6	)	)	PUNCT
ejpam-5721	72	7	)	)	PUNCT
ejpam-5721	73	1	∩a	∩a	PROPN
ejpam-5721	73	2	.	.	PUNCT
ejpam-5721	74	1	let	let	VERB
ejpam-5721	74	2	a	a	DET
ejpam-5721	74	3	be	be	AUX
ejpam-5721	74	4	a	a	DET
ejpam-5721	74	5	subset	subset	NOUN
ejpam-5721	74	6	of	of	ADP
ejpam-5721	74	7	a	a	DET
ejpam-5721	74	8	bitopological	bitopological	ADJ
ejpam-5721	74	9	space	space	NOUN
ejpam-5721	74	10	(	(	PUNCT
ejpam-5721	74	11	x	x	NOUN
ejpam-5721	74	12	,	,	PUNCT
ejpam-5721	74	13	τ1	τ1	NOUN
ejpam-5721	74	14	,	,	PUNCT
ejpam-5721	74	15	τ2	τ2	NOUN
ejpam-5721	74	16	)	)	PUNCT
ejpam-5721	74	17	.	.	PUNCT
ejpam-5721	75	1	a	a	DET
ejpam-5721	75	2	point	point	NOUN
ejpam-5721	75	3	x	x	X
ejpam-5721	75	4	∈	∈	NOUN
ejpam-5721	75	5	x	x	PUNCT
ejpam-5721	75	6	is	be	AUX
ejpam-5721	75	7	called	call	VERB
ejpam-5721	75	8	(	(	PUNCT
ejpam-5721	75	9	τ1	τ1	NOUN
ejpam-5721	75	10	,	,	PUNCT
ejpam-5721	75	11	τ2)θcluster	τ2)θcluster	PROPN
ejpam-5721	75	12	point	point	NOUN
ejpam-5721	75	13	[	[	X
ejpam-5721	75	14	35	35	NUM
ejpam-5721	75	15	]	]	PUNCT
ejpam-5721	75	16	of	of	ADP
ejpam-5721	75	17	a	a	DET
ejpam-5721	75	18	if	if	SCONJ
ejpam-5721	75	19	τ1τ2	τ1τ2	ADJ
ejpam-5721	75	20	-	-	ADJ
ejpam-5721	75	21	cl(u)∩a	cl(u)∩a	ADJ
ejpam-5721	75	22	̸=	̸=	PROPN
ejpam-5721	75	23	∅	∅	NOUN
ejpam-5721	75	24	for	for	ADP
ejpam-5721	75	25	every	every	DET
ejpam-5721	75	26	τ1τ2	τ1τ2	ADJ
ejpam-5721	75	27	-	-	ADJ
ejpam-5721	75	28	open	open	ADJ
ejpam-5721	75	29	set	set	NOUN
ejpam-5721	75	30	u	u	NOUN
ejpam-5721	75	31	containing	contain	VERB
ejpam-5721	75	32	x.	x.	NOUN
ejpam-5721	75	33	the	the	DET
ejpam-5721	75	34	set	set	NOUN
ejpam-5721	75	35	of	of	ADP
ejpam-5721	75	36	all	all	DET
ejpam-5721	75	37	(	(	PUNCT
ejpam-5721	75	38	τ1	τ1	NOUN
ejpam-5721	75	39	,	,	PUNCT
ejpam-5721	75	40	τ2)θ	τ2)θ	ADJ
ejpam-5721	75	41	-	-	PUNCT
ejpam-5721	75	42	cluster	cluster	NOUN
ejpam-5721	75	43	points	point	NOUN
ejpam-5721	75	44	of	of	ADP
ejpam-5721	75	45	a	a	PRON
ejpam-5721	75	46	is	be	AUX
ejpam-5721	75	47	called	call	VERB
ejpam-5721	75	48	the	the	DET
ejpam-5721	75	49	(	(	PUNCT
ejpam-5721	75	50	τ1	τ1	NOUN
ejpam-5721	75	51	,	,	PUNCT
ejpam-5721	75	52	τ2)θ	τ2)θ	ADJ
ejpam-5721	75	53	-	-	PUNCT
ejpam-5721	75	54	closure	closure	NOUN
ejpam-5721	75	55	[	[	X
ejpam-5721	75	56	35	35	NUM
ejpam-5721	75	57	]	]	PUNCT
ejpam-5721	75	58	of	of	ADP
ejpam-5721	75	59	a	a	PRON
ejpam-5721	75	60	and	and	CCONJ
ejpam-5721	75	61	is	be	AUX
ejpam-5721	75	62	denoted	denote	VERB
ejpam-5721	75	63	by	by	ADP
ejpam-5721	75	64	(	(	PUNCT
ejpam-5721	75	65	τ1	τ1	NOUN
ejpam-5721	75	66	,	,	PUNCT
ejpam-5721	75	67	τ2)θ	τ2)θ	NOUN
ejpam-5721	75	68	-	-	PUNCT
ejpam-5721	75	69	cl(a	cl(a	NUM
ejpam-5721	75	70	)	)	PUNCT
ejpam-5721	75	71	.	.	PUNCT
ejpam-5721	76	1	a	a	DET
ejpam-5721	76	2	subset	subset	NOUN
ejpam-5721	76	3	a	a	PRON
ejpam-5721	76	4	of	of	ADP
ejpam-5721	76	5	a	a	DET
ejpam-5721	76	6	bitopological	bitopological	ADJ
ejpam-5721	76	7	space	space	NOUN
ejpam-5721	76	8	(	(	PUNCT
ejpam-5721	76	9	x	x	NOUN
ejpam-5721	76	10	,	,	PUNCT
ejpam-5721	76	11	τ1	τ1	NOUN
ejpam-5721	76	12	,	,	PUNCT
ejpam-5721	76	13	τ2	τ2	NOUN
ejpam-5721	76	14	)	)	PUNCT
ejpam-5721	76	15	is	be	AUX
ejpam-5721	76	16	said	say	VERB
ejpam-5721	76	17	to	to	PART
ejpam-5721	76	18	be	be	AUX
ejpam-5721	76	19	(	(	PUNCT
ejpam-5721	76	20	τ1	τ1	NOUN
ejpam-5721	76	21	,	,	PUNCT
ejpam-5721	76	22	τ2)θ	τ2)θ	NOUN
ejpam-5721	76	23	-	-	PUNCT
ejpam-5721	76	24	closed	closed	ADJ
ejpam-5721	76	25	[	[	X
ejpam-5721	76	26	35	35	NUM
ejpam-5721	76	27	]	]	X
ejpam-5721	76	28	if	if	SCONJ
ejpam-5721	76	29	a	a	PRON
ejpam-5721	76	30	=	=	X
ejpam-5721	76	31	(	(	PUNCT
ejpam-5721	76	32	τ1	τ1	NOUN
ejpam-5721	76	33	,	,	PUNCT
ejpam-5721	76	34	τ2)θ	τ2)θ	NOUN
ejpam-5721	76	35	-	-	PUNCT
ejpam-5721	76	36	cl(a	cl(a	NUM
ejpam-5721	76	37	)	)	PUNCT
ejpam-5721	76	38	.	.	PUNCT
ejpam-5721	77	1	the	the	DET
ejpam-5721	77	2	complement	complement	NOUN
ejpam-5721	77	3	of	of	ADP
ejpam-5721	77	4	a	a	DET
ejpam-5721	77	5	(	(	PUNCT
ejpam-5721	77	6	τ1	τ1	NOUN
ejpam-5721	77	7	,	,	PUNCT
ejpam-5721	77	8	τ2)θ	τ2)θ	ADJ
ejpam-5721	77	9	-	-	PUNCT
ejpam-5721	77	10	closed	close	VERB
ejpam-5721	77	11	set	set	NOUN
ejpam-5721	77	12	is	be	AUX
ejpam-5721	77	13	said	say	VERB
ejpam-5721	77	14	to	to	PART
ejpam-5721	77	15	be	be	AUX
ejpam-5721	77	16	(	(	PUNCT
ejpam-5721	77	17	τ1	τ1	NOUN
ejpam-5721	77	18	,	,	PUNCT
ejpam-5721	77	19	τ2)θopen	τ2)θopen	PROPN
ejpam-5721	77	20	.	.	PUNCT
ejpam-5721	78	1	the	the	DET
ejpam-5721	78	2	union	union	NOUN
ejpam-5721	78	3	of	of	ADP
ejpam-5721	78	4	all	all	DET
ejpam-5721	78	5	(	(	PUNCT
ejpam-5721	78	6	τ1	τ1	NOUN
ejpam-5721	78	7	,	,	PUNCT
ejpam-5721	78	8	τ2)θ	τ2)θ	ADJ
ejpam-5721	78	9	-	-	PUNCT
ejpam-5721	78	10	open	open	ADJ
ejpam-5721	78	11	sets	set	NOUN
ejpam-5721	78	12	contained	contain	VERB
ejpam-5721	78	13	in	in	ADP
ejpam-5721	78	14	a	a	PRON
ejpam-5721	78	15	is	be	AUX
ejpam-5721	78	16	called	call	VERB
ejpam-5721	78	17	the	the	DET
ejpam-5721	78	18	(	(	PUNCT
ejpam-5721	78	19	τ1	τ1	NOUN
ejpam-5721	78	20	,	,	PUNCT
ejpam-5721	78	21	τ2)θ	τ2)θ	ADJ
ejpam-5721	78	22	-	-	PUNCT
ejpam-5721	78	23	interior	interior	NOUN
ejpam-5721	78	24	[	[	X
ejpam-5721	78	25	35	35	NUM
ejpam-5721	78	26	]	]	PUNCT
ejpam-5721	78	27	of	of	ADP
ejpam-5721	78	28	a	a	PRON
ejpam-5721	78	29	and	and	CCONJ
ejpam-5721	78	30	is	be	AUX
ejpam-5721	78	31	denoted	denote	VERB
ejpam-5721	78	32	by	by	ADP
ejpam-5721	78	33	(	(	PUNCT
ejpam-5721	78	34	τ1	τ1	NOUN
ejpam-5721	78	35	,	,	PUNCT
ejpam-5721	78	36	τ2)θ	τ2)θ	NOUN
ejpam-5721	78	37	-	-	PUNCT
ejpam-5721	78	38	int(a	int(a	NOUN
ejpam-5721	78	39	)	)	PUNCT
ejpam-5721	78	40	.	.	PUNCT
ejpam-5721	79	1	lemma	lemma	PROPN
ejpam-5721	79	2	4	4	NUM
ejpam-5721	79	3	.	.	PUNCT
ejpam-5721	80	1	[	[	X
ejpam-5721	80	2	35	35	NUM
ejpam-5721	80	3	]	]	X
ejpam-5721	80	4	for	for	ADP
ejpam-5721	80	5	a	a	DET
ejpam-5721	80	6	subset	subset	NOUN
ejpam-5721	80	7	a	a	PRON
ejpam-5721	80	8	of	of	ADP
ejpam-5721	80	9	a	a	DET
ejpam-5721	80	10	bitopological	bitopological	ADJ
ejpam-5721	80	11	space	space	NOUN
ejpam-5721	80	12	(	(	PUNCT
ejpam-5721	80	13	x	x	NOUN
ejpam-5721	80	14	,	,	PUNCT
ejpam-5721	80	15	τ1	τ1	NOUN
ejpam-5721	80	16	,	,	PUNCT
ejpam-5721	80	17	τ2	τ2	NOUN
ejpam-5721	80	18	)	)	PUNCT
ejpam-5721	80	19	,	,	PUNCT
ejpam-5721	80	20	the	the	DET
ejpam-5721	80	21	following	follow	VERB
ejpam-5721	80	22	properties	property	NOUN
ejpam-5721	80	23	hold	hold	VERB
ejpam-5721	80	24	:	:	PUNCT
ejpam-5721	80	25	(	(	PUNCT
ejpam-5721	80	26	1	1	X
ejpam-5721	80	27	)	)	PUNCT
ejpam-5721	80	28	if	if	SCONJ
ejpam-5721	80	29	a	a	PRON
ejpam-5721	80	30	is	be	AUX
ejpam-5721	80	31	τ2τ2	τ2τ2	VERB
ejpam-5721	80	32	-	-	VERB
ejpam-5721	80	33	open	open	ADJ
ejpam-5721	80	34	in	in	ADP
ejpam-5721	80	35	x	x	NOUN
ejpam-5721	80	36	,	,	PUNCT
ejpam-5721	80	37	then	then	ADV
ejpam-5721	80	38	τ1τ2	τ1τ2	NOUN
ejpam-5721	80	39	-	-	NUM
ejpam-5721	80	40	cl(a	cl(a	NUM
ejpam-5721	80	41	)	)	PUNCT
ejpam-5721	80	42	=	=	PUNCT
ejpam-5721	80	43	(	(	PUNCT
ejpam-5721	80	44	τ1	τ1	NOUN
ejpam-5721	80	45	,	,	PUNCT
ejpam-5721	80	46	τ2)θ	τ2)θ	NOUN
ejpam-5721	80	47	-	-	PUNCT
ejpam-5721	80	48	cl(a	cl(a	NUM
ejpam-5721	80	49	)	)	PUNCT
ejpam-5721	80	50	.	.	PUNCT
ejpam-5721	81	1	(	(	PUNCT
ejpam-5721	81	2	2	2	X
ejpam-5721	81	3	)	)	PUNCT
ejpam-5721	81	4	(	(	PUNCT
ejpam-5721	81	5	τ1	τ1	NOUN
ejpam-5721	81	6	,	,	PUNCT
ejpam-5721	81	7	τ2)θ	τ2)θ	NOUN
ejpam-5721	81	8	-	-	PUNCT
ejpam-5721	81	9	cl(a	cl(a	NUM
ejpam-5721	81	10	)	)	PUNCT
ejpam-5721	81	11	is	be	AUX
ejpam-5721	81	12	τ1τ2	τ1τ2	NOUN
ejpam-5721	81	13	-	-	ADJ
ejpam-5721	81	14	closed	closed	ADJ
ejpam-5721	81	15	in	in	ADP
ejpam-5721	81	16	x.	x.	NOUN
ejpam-5721	81	17	3	3	NUM
ejpam-5721	81	18	.	.	PUNCT
ejpam-5721	81	19	almost	almost	ADV
ejpam-5721	81	20	weakly	weakly	ADJ
ejpam-5721	81	21	(	(	PUNCT
ejpam-5721	81	22	τ1	τ1	NOUN
ejpam-5721	81	23	,	,	PUNCT
ejpam-5721	81	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	81	25	functions	function	NOUN
ejpam-5721	81	26	in	in	ADP
ejpam-5721	81	27	this	this	DET
ejpam-5721	81	28	section	section	NOUN
ejpam-5721	81	29	,	,	PUNCT
ejpam-5721	81	30	we	we	PRON
ejpam-5721	81	31	introduce	introduce	VERB
ejpam-5721	81	32	the	the	DET
ejpam-5721	81	33	notion	notion	NOUN
ejpam-5721	81	34	of	of	ADP
ejpam-5721	81	35	almost	almost	ADV
ejpam-5721	81	36	weakly	weakly	ADJ
ejpam-5721	81	37	(	(	PUNCT
ejpam-5721	81	38	τ1	τ1	NOUN
ejpam-5721	81	39	,	,	PUNCT
ejpam-5721	81	40	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	81	41	functions	function	NOUN
ejpam-5721	81	42	.	.	PUNCT
ejpam-5721	82	1	moreover	moreover	ADV
ejpam-5721	82	2	,	,	PUNCT
ejpam-5721	82	3	several	several	ADJ
ejpam-5721	82	4	characterizations	characterization	NOUN
ejpam-5721	82	5	of	of	ADP
ejpam-5721	82	6	almost	almost	ADV
ejpam-5721	82	7	weakly	weakly	ADJ
ejpam-5721	82	8	(	(	PUNCT
ejpam-5721	82	9	τ1	τ1	NOUN
ejpam-5721	82	10	,	,	PUNCT
ejpam-5721	82	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	82	12	functions	function	NOUN
ejpam-5721	82	13	are	be	AUX
ejpam-5721	82	14	discussed	discuss	VERB
ejpam-5721	82	15	.	.	PUNCT
ejpam-5721	83	1	definition	definition	NOUN
ejpam-5721	83	2	1	1	NUM
ejpam-5721	83	3	.	.	PUNCT
ejpam-5721	84	1	a	a	DET
ejpam-5721	84	2	function	function	NOUN
ejpam-5721	84	3	f	f	NOUN
ejpam-5721	84	4	:	:	PUNCT
ejpam-5721	84	5	(	(	PUNCT
ejpam-5721	84	6	x	x	NOUN
ejpam-5721	84	7	,	,	PUNCT
ejpam-5721	84	8	τ1	τ1	NOUN
ejpam-5721	84	9	,	,	PUNCT
ejpam-5721	84	10	τ2	τ2	NOUN
ejpam-5721	84	11	)	)	PUNCT
ejpam-5721	84	12	→	→	SYM
ejpam-5721	84	13	(	(	PUNCT
ejpam-5721	84	14	y	y	PROPN
ejpam-5721	84	15	,	,	PUNCT
ejpam-5721	84	16	σ1	σ1	PROPN
ejpam-5721	84	17	,	,	PUNCT
ejpam-5721	84	18	σ2	σ2	PROPN
ejpam-5721	84	19	)	)	PUNCT
ejpam-5721	84	20	is	be	AUX
ejpam-5721	84	21	said	say	VERB
ejpam-5721	84	22	to	to	PART
ejpam-5721	84	23	be	be	AUX
ejpam-5721	84	24	almost	almost	ADV
ejpam-5721	84	25	weakly	weakly	ADJ
ejpam-5721	84	26	(	(	PUNCT
ejpam-5721	84	27	τ1	τ1	NOUN
ejpam-5721	84	28	,	,	PUNCT
ejpam-5721	84	29	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	84	30	if	if	SCONJ
ejpam-5721	84	31	for	for	ADP
ejpam-5721	84	32	each	each	DET
ejpam-5721	84	33	x	x	SYM
ejpam-5721	84	34	∈	∈	PROPN
ejpam-5721	84	35	x	x	X
ejpam-5721	84	36	and	and	CCONJ
ejpam-5721	84	37	each	each	DET
ejpam-5721	84	38	σ1σ2	σ1σ2	VERB
ejpam-5721	84	39	-	-	ADJ
ejpam-5721	84	40	open	open	ADJ
ejpam-5721	84	41	set	set	NOUN
ejpam-5721	84	42	v	v	NOUN
ejpam-5721	84	43	of	of	ADP
ejpam-5721	84	44	y	y	NOUN
ejpam-5721	84	45	containing	contain	VERB
ejpam-5721	84	46	f(x	f(x	PROPN
ejpam-5721	84	47	)	)	PUNCT
ejpam-5721	84	48	,	,	PUNCT
ejpam-5721	84	49	x	x	PUNCT
ejpam-5721	84	50	∈	∈	ADP
ejpam-5721	84	51	τ1τ2	τ1τ2	NOUN
ejpam-5721	84	52	-	-	NOUN
ejpam-5721	84	53	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	84	54	-	-	PUNCT
ejpam-5721	84	55	cl(f	cl(f	NOUN
ejpam-5721	84	56	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	84	57	-	-	PUNCT
ejpam-5721	84	58	cl(v	cl(v	NOUN
ejpam-5721	84	59	)	)	PUNCT
ejpam-5721	84	60	)	)	PUNCT
ejpam-5721	84	61	)	)	PUNCT
ejpam-5721	84	62	)	)	PUNCT
ejpam-5721	84	63	.	.	PUNCT
ejpam-5721	85	1	theorem	theorem	NOUN
ejpam-5721	85	2	1	1	NUM
ejpam-5721	85	3	.	.	X
ejpam-5721	85	4	for	for	ADP
ejpam-5721	85	5	a	a	DET
ejpam-5721	85	6	function	function	NOUN
ejpam-5721	85	7	f	f	NOUN
ejpam-5721	85	8	:	:	PUNCT
ejpam-5721	85	9	(	(	PUNCT
ejpam-5721	85	10	x	x	NOUN
ejpam-5721	85	11	,	,	PUNCT
ejpam-5721	85	12	τ1	τ1	NOUN
ejpam-5721	85	13	,	,	PUNCT
ejpam-5721	85	14	τ2	τ2	NOUN
ejpam-5721	85	15	)	)	PUNCT
ejpam-5721	85	16	→	→	SYM
ejpam-5721	85	17	(	(	PUNCT
ejpam-5721	85	18	y	y	PROPN
ejpam-5721	85	19	,	,	PUNCT
ejpam-5721	85	20	σ1	σ1	PROPN
ejpam-5721	85	21	,	,	PUNCT
ejpam-5721	85	22	σ2	σ2	NOUN
ejpam-5721	85	23	)	)	PUNCT
ejpam-5721	85	24	,	,	PUNCT
ejpam-5721	85	25	the	the	DET
ejpam-5721	85	26	following	follow	VERB
ejpam-5721	85	27	properties	property	NOUN
ejpam-5721	85	28	are	be	AUX
ejpam-5721	85	29	equivalent	equivalent	ADJ
ejpam-5721	85	30	:	:	PUNCT
ejpam-5721	85	31	(	(	PUNCT
ejpam-5721	85	32	1	1	X
ejpam-5721	85	33	)	)	PUNCT
ejpam-5721	85	34	f	f	NOUN
ejpam-5721	85	35	is	be	AUX
ejpam-5721	85	36	almost	almost	ADV
ejpam-5721	85	37	weakly	weakly	ADJ
ejpam-5721	85	38	(	(	PUNCT
ejpam-5721	85	39	τ1	τ1	NOUN
ejpam-5721	85	40	,	,	PUNCT
ejpam-5721	85	41	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	85	42	;	;	PUNCT
ejpam-5721	85	43	(	(	PUNCT
ejpam-5721	85	44	2	2	X
ejpam-5721	85	45	)	)	PUNCT
ejpam-5721	85	46	f−1(v	f−1(v	NOUN
ejpam-5721	85	47	)	)	PUNCT
ejpam-5721	86	1	⊆	⊆	X
ejpam-5721	86	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	86	3	-	-	NOUN
ejpam-5721	86	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	86	5	-	-	PUNCT
ejpam-5721	86	6	cl(f	cl(f	NOUN
ejpam-5721	86	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	86	8	-	-	PUNCT
ejpam-5721	86	9	cl(v	cl(v	NOUN
ejpam-5721	86	10	)	)	PUNCT
ejpam-5721	86	11	)	)	PUNCT
ejpam-5721	86	12	)	)	PUNCT
ejpam-5721	86	13	)	)	PUNCT
ejpam-5721	86	14	for	for	ADP
ejpam-5721	86	15	every	every	DET
ejpam-5721	86	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	86	17	-	-	ADJ
ejpam-5721	86	18	open	open	ADJ
ejpam-5721	86	19	set	set	NOUN
ejpam-5721	86	20	v	v	NOUN
ejpam-5721	86	21	of	of	ADP
ejpam-5721	86	22	y	y	PROPN
ejpam-5721	86	23	;	;	PUNCT
ejpam-5721	86	24	j.	j.	PROPN
ejpam-5721	86	25	khampakdee	khampakdee	PROPN
ejpam-5721	86	26	,	,	PUNCT
ejpam-5721	86	27	s.	s.	PROPN
ejpam-5721	86	28	sompong	sompong	PROPN
ejpam-5721	86	29	,	,	PUNCT
ejpam-5721	86	30	c.	c.	PROPN
ejpam-5721	86	31	boonpok	boonpok	PROPN
ejpam-5721	86	32	/	/	SYM
ejpam-5721	86	33	eur	eur	PROPN
ejpam-5721	86	34	.	.	PUNCT
ejpam-5721	87	1	j.	j.	PROPN
ejpam-5721	87	2	pure	pure	PROPN
ejpam-5721	87	3	appl	appl	PROPN
ejpam-5721	87	4	.	.	PROPN
ejpam-5721	87	5	math	math	PROPN
ejpam-5721	87	6	,	,	PUNCT
ejpam-5721	87	7	18	18	NUM
ejpam-5721	87	8	(	(	PUNCT
ejpam-5721	87	9	1	1	NUM
ejpam-5721	87	10	)	)	PUNCT
ejpam-5721	87	11	(	(	PUNCT
ejpam-5721	87	12	2025	2025	NUM
ejpam-5721	87	13	)	)	PUNCT
ejpam-5721	87	14	,	,	PUNCT
ejpam-5721	87	15	5721	5721	NUM
ejpam-5721	87	16	4	4	NUM
ejpam-5721	87	17	of	of	ADP
ejpam-5721	87	18	12	12	NUM
ejpam-5721	87	19	(	(	PUNCT
ejpam-5721	87	20	3	3	NUM
ejpam-5721	87	21	)	)	PUNCT
ejpam-5721	87	22	τ1τ2	τ1τ2	NOUN
ejpam-5721	87	23	-	-	NOUN
ejpam-5721	87	24	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	87	25	-	-	PUNCT
ejpam-5721	87	26	int(f	int(f	PROPN
ejpam-5721	87	27	−1(v	−1(v	NOUN
ejpam-5721	87	28	)	)	PUNCT
ejpam-5721	87	29	)	)	PUNCT
ejpam-5721	87	30	)	)	PUNCT
ejpam-5721	88	1	⊆	⊆	NUM
ejpam-5721	88	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	88	3	-	-	PUNCT
ejpam-5721	88	4	cl(v	cl(v	NOUN
ejpam-5721	88	5	)	)	PUNCT
ejpam-5721	88	6	)	)	PUNCT
ejpam-5721	88	7	for	for	ADP
ejpam-5721	88	8	every	every	DET
ejpam-5721	88	9	σ1σ2	σ1σ2	NOUN
ejpam-5721	88	10	-	-	ADJ
ejpam-5721	88	11	open	open	ADJ
ejpam-5721	88	12	set	set	NOUN
ejpam-5721	88	13	v	v	NOUN
ejpam-5721	88	14	of	of	ADP
ejpam-5721	88	15	y	y	PROPN
ejpam-5721	88	16	;	;	PUNCT
ejpam-5721	88	17	(	(	PUNCT
ejpam-5721	88	18	4	4	X
ejpam-5721	88	19	)	)	PUNCT
ejpam-5721	88	20	(	(	PUNCT
ejpam-5721	88	21	τ1	τ1	NOUN
ejpam-5721	88	22	,	,	PUNCT
ejpam-5721	88	23	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	88	24	−1(v	−1(v	PROPN
ejpam-5721	88	25	)	)	PUNCT
ejpam-5721	88	26	)	)	PUNCT
ejpam-5721	89	1	⊆	⊆	NUM
ejpam-5721	89	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	89	3	-	-	PUNCT
ejpam-5721	89	4	cl(v	cl(v	NOUN
ejpam-5721	89	5	)	)	PUNCT
ejpam-5721	89	6	)	)	PUNCT
ejpam-5721	89	7	for	for	ADP
ejpam-5721	89	8	every	every	DET
ejpam-5721	89	9	σ1σ2	σ1σ2	NOUN
ejpam-5721	89	10	-	-	ADJ
ejpam-5721	89	11	open	open	ADJ
ejpam-5721	89	12	set	set	NOUN
ejpam-5721	89	13	v	v	NOUN
ejpam-5721	89	14	of	of	ADP
ejpam-5721	89	15	y	y	PROPN
ejpam-5721	89	16	;	;	PUNCT
ejpam-5721	89	17	(	(	PUNCT
ejpam-5721	89	18	5	5	X
ejpam-5721	89	19	)	)	PUNCT
ejpam-5721	89	20	f−1(v	f−1(v	NOUN
ejpam-5721	89	21	)	)	PUNCT
ejpam-5721	90	1	⊆	⊆	NUM
ejpam-5721	90	2	(	(	PUNCT
ejpam-5721	90	3	τ1	τ1	NOUN
ejpam-5721	90	4	,	,	PUNCT
ejpam-5721	90	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	90	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	90	7	-	-	PUNCT
ejpam-5721	90	8	cl(v	cl(v	NOUN
ejpam-5721	90	9	)	)	PUNCT
ejpam-5721	90	10	)	)	PUNCT
ejpam-5721	90	11	)	)	PUNCT
ejpam-5721	90	12	for	for	ADP
ejpam-5721	90	13	every	every	DET
ejpam-5721	90	14	σ1σ2	σ1σ2	NOUN
ejpam-5721	90	15	-	-	ADJ
ejpam-5721	90	16	open	open	ADJ
ejpam-5721	90	17	set	set	NOUN
ejpam-5721	90	18	v	v	NOUN
ejpam-5721	90	19	of	of	ADP
ejpam-5721	90	20	y	y	PROPN
ejpam-5721	90	21	;	;	PUNCT
ejpam-5721	90	22	(	(	PUNCT
ejpam-5721	90	23	6	6	NUM
ejpam-5721	90	24	)	)	PUNCT
ejpam-5721	90	25	for	for	ADP
ejpam-5721	90	26	each	each	DET
ejpam-5721	90	27	x	x	SYM
ejpam-5721	90	28	∈	∈	PROPN
ejpam-5721	90	29	x	x	X
ejpam-5721	90	30	and	and	CCONJ
ejpam-5721	90	31	each	each	DET
ejpam-5721	90	32	σ1σ2	σ1σ2	VERB
ejpam-5721	90	33	-	-	ADJ
ejpam-5721	90	34	open	open	ADJ
ejpam-5721	90	35	set	set	NOUN
ejpam-5721	90	36	v	v	NOUN
ejpam-5721	90	37	of	of	ADP
ejpam-5721	90	38	y	y	NOUN
ejpam-5721	90	39	containing	contain	VERB
ejpam-5721	90	40	f(x	f(x	PROPN
ejpam-5721	90	41	)	)	PUNCT
ejpam-5721	90	42	,	,	PUNCT
ejpam-5721	90	43	there	there	PRON
ejpam-5721	90	44	exists	exist	VERB
ejpam-5721	90	45	a	a	DET
ejpam-5721	90	46	(	(	PUNCT
ejpam-5721	90	47	τ1	τ1	NOUN
ejpam-5721	90	48	,	,	PUNCT
ejpam-5721	90	49	τ2)p	τ2)p	ADJ
ejpam-5721	90	50	-	-	PUNCT
ejpam-5721	90	51	open	open	ADJ
ejpam-5721	90	52	set	set	NOUN
ejpam-5721	90	53	u	u	NOUN
ejpam-5721	90	54	of	of	ADP
ejpam-5721	90	55	x	x	PUNCT
ejpam-5721	90	56	containing	contain	VERB
ejpam-5721	90	57	x	x	PUNCT
ejpam-5721	90	58	such	such	ADJ
ejpam-5721	90	59	that	that	DET
ejpam-5721	90	60	f(u	f(u	PROPN
ejpam-5721	90	61	)	)	PUNCT
ejpam-5721	90	62	⊆	⊆	NUM
ejpam-5721	90	63	σ1σ2	σ1σ2	NOUN
ejpam-5721	90	64	-	-	NUM
ejpam-5721	90	65	cl(v	cl(v	NOUN
ejpam-5721	90	66	)	)	PUNCT
ejpam-5721	90	67	.	.	PUNCT
ejpam-5721	91	1	proof	proof	NOUN
ejpam-5721	91	2	.	.	PUNCT
ejpam-5721	92	1	(	(	PUNCT
ejpam-5721	92	2	1	1	X
ejpam-5721	92	3	)	)	PUNCT
ejpam-5721	92	4	⇒	⇒	NOUN
ejpam-5721	92	5	(	(	PUNCT
ejpam-5721	92	6	2	2	NUM
ejpam-5721	92	7	):	):	PUNCT
ejpam-5721	92	8	let	let	VERB
ejpam-5721	92	9	v	v	PART
ejpam-5721	92	10	be	be	AUX
ejpam-5721	92	11	any	any	DET
ejpam-5721	92	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	92	13	-	-	ADJ
ejpam-5721	92	14	open	open	ADJ
ejpam-5721	92	15	set	set	NOUN
ejpam-5721	92	16	of	of	ADP
ejpam-5721	92	17	y	y	PROPN
ejpam-5721	92	18	and	and	CCONJ
ejpam-5721	92	19	x	x	PROPN
ejpam-5721	92	20	∈	∈	PROPN
ejpam-5721	92	21	f−1(v	f−1(v	NOUN
ejpam-5721	92	22	)	)	PUNCT
ejpam-5721	92	23	.	.	PUNCT
ejpam-5721	93	1	then	then	ADV
ejpam-5721	93	2	,	,	PUNCT
ejpam-5721	93	3	we	we	PRON
ejpam-5721	93	4	have	have	VERB
ejpam-5721	93	5	f(x	f(x	PROPN
ejpam-5721	93	6	)	)	PUNCT
ejpam-5721	93	7	∈	∈	PROPN
ejpam-5721	93	8	v	v	NOUN
ejpam-5721	93	9	and	and	CCONJ
ejpam-5721	93	10	by	by	ADP
ejpam-5721	93	11	(	(	PUNCT
ejpam-5721	93	12	1	1	NUM
ejpam-5721	93	13	)	)	PUNCT
ejpam-5721	93	14	,	,	PUNCT
ejpam-5721	93	15	x	x	PUNCT
ejpam-5721	93	16	∈	∈	ADP
ejpam-5721	93	17	τ1τ2	τ1τ2	NOUN
ejpam-5721	93	18	-	-	NOUN
ejpam-5721	93	19	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	93	20	-	-	PUNCT
ejpam-5721	93	21	cl(f	cl(f	NOUN
ejpam-5721	93	22	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	93	23	-	-	PUNCT
ejpam-5721	93	24	cl(v	cl(v	NOUN
ejpam-5721	93	25	)	)	PUNCT
ejpam-5721	93	26	)	)	PUNCT
ejpam-5721	93	27	)	)	PUNCT
ejpam-5721	93	28	)	)	PUNCT
ejpam-5721	93	29	.	.	PUNCT
ejpam-5721	94	1	thus	thus	ADV
ejpam-5721	94	2	,	,	PUNCT
ejpam-5721	94	3	f−1(v	f−1(v	PROPN
ejpam-5721	94	4	)	)	PUNCT
ejpam-5721	95	1	⊆	⊆	X
ejpam-5721	95	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	95	3	-	-	NOUN
ejpam-5721	95	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	95	5	-	-	PUNCT
ejpam-5721	95	6	cl(f	cl(f	NOUN
ejpam-5721	95	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	95	8	-	-	PUNCT
ejpam-5721	95	9	cl(v	cl(v	NOUN
ejpam-5721	95	10	)	)	PUNCT
ejpam-5721	95	11	)	)	PUNCT
ejpam-5721	95	12	)	)	PUNCT
ejpam-5721	95	13	)	)	PUNCT
ejpam-5721	95	14	.	.	PUNCT
ejpam-5721	96	1	(	(	PUNCT
ejpam-5721	96	2	2	2	X
ejpam-5721	96	3	)	)	PUNCT
ejpam-5721	96	4	⇒	⇒	NOUN
ejpam-5721	96	5	(	(	PUNCT
ejpam-5721	96	6	3	3	NUM
ejpam-5721	96	7	):	):	PUNCT
ejpam-5721	96	8	let	let	VERB
ejpam-5721	96	9	v	v	PART
ejpam-5721	96	10	be	be	AUX
ejpam-5721	96	11	any	any	DET
ejpam-5721	96	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	96	13	-	-	ADJ
ejpam-5721	96	14	open	open	ADJ
ejpam-5721	96	15	set	set	NOUN
ejpam-5721	96	16	of	of	ADP
ejpam-5721	96	17	y	y	PROPN
ejpam-5721	96	18	.	.	PUNCT
ejpam-5721	97	1	since	since	SCONJ
ejpam-5721	97	2	y	y	PROPN
ejpam-5721	97	3	−	−	PROPN
ejpam-5721	97	4	σ1σ2	σ1σ2	NOUN
ejpam-5721	97	5	-	-	NUM
ejpam-5721	97	6	cl(v	cl(v	NOUN
ejpam-5721	97	7	)	)	PUNCT
ejpam-5721	97	8	is	be	AUX
ejpam-5721	97	9	σ1σ2	σ1σ2	NOUN
ejpam-5721	97	10	-	-	ADJ
ejpam-5721	97	11	open	open	ADJ
ejpam-5721	97	12	in	in	ADP
ejpam-5721	97	13	y	y	PROPN
ejpam-5721	97	14	and	and	CCONJ
ejpam-5721	97	15	by	by	ADP
ejpam-5721	97	16	(	(	PUNCT
ejpam-5721	97	17	2	2	NUM
ejpam-5721	97	18	)	)	PUNCT
ejpam-5721	97	19	,	,	PUNCT
ejpam-5721	97	20	we	we	PRON
ejpam-5721	97	21	have	have	VERB
ejpam-5721	97	22	x	x	PART
ejpam-5721	97	23	−	−	PRON
ejpam-5721	97	24	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	97	25	-	-	PUNCT
ejpam-5721	97	26	cl(v	cl(v	NOUN
ejpam-5721	97	27	)	)	PUNCT
ejpam-5721	97	28	)	)	PUNCT
ejpam-5721	98	1	=	=	PUNCT
ejpam-5721	98	2	f−1(y	f−1(y	PROPN
ejpam-5721	98	3	−	−	NOUN
ejpam-5721	98	4	σ1σ2	σ1σ2	NOUN
ejpam-5721	98	5	-	-	NUM
ejpam-5721	98	6	cl(v	cl(v	NOUN
ejpam-5721	98	7	)	)	PUNCT
ejpam-5721	98	8	)	)	PUNCT
ejpam-5721	99	1	⊆	⊆	X
ejpam-5721	99	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	99	3	-	-	NOUN
ejpam-5721	99	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	99	5	-	-	PUNCT
ejpam-5721	99	6	cl(f	cl(f	NOUN
ejpam-5721	99	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	99	8	-	-	PUNCT
ejpam-5721	99	9	cl(y	cl(y	NOUN
ejpam-5721	99	10	−	−	NOUN
ejpam-5721	99	11	σ1σ2	σ1σ2	NOUN
ejpam-5721	99	12	-	-	NUM
ejpam-5721	99	13	cl(v	cl(v	NOUN
ejpam-5721	99	14	)	)	PUNCT
ejpam-5721	99	15	)	)	PUNCT
ejpam-5721	99	16	)	)	PUNCT
ejpam-5721	99	17	)	)	PUNCT
ejpam-5721	99	18	)	)	PUNCT
ejpam-5721	100	1	⊆	⊆	X
ejpam-5721	100	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	100	3	-	-	NOUN
ejpam-5721	100	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	100	5	-	-	PUNCT
ejpam-5721	100	6	cl(f	cl(f	NOUN
ejpam-5721	100	7	−1(y	−1(y	VERB
ejpam-5721	100	8	−	−	PROPN
ejpam-5721	100	9	v	v	NOUN
ejpam-5721	100	10	)	)	PUNCT
ejpam-5721	100	11	)	)	PUNCT
ejpam-5721	100	12	)	)	PUNCT
ejpam-5721	101	1	=	=	PUNCT
ejpam-5721	101	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	101	3	-	-	NOUN
ejpam-5721	101	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	101	5	-	-	PUNCT
ejpam-5721	101	6	cl(x	cl(x	NUM
ejpam-5721	101	7	−	−	PROPN
ejpam-5721	101	8	f−1(v	f−1(v	PROPN
ejpam-5721	101	9	)	)	PUNCT
ejpam-5721	101	10	)	)	PUNCT
ejpam-5721	101	11	)	)	PUNCT
ejpam-5721	102	1	=	=	PUNCT
ejpam-5721	102	2	x	x	X
ejpam-5721	103	1	−	−	ADP
ejpam-5721	103	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	103	3	-	-	NUM
ejpam-5721	103	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	103	5	-	-	PUNCT
ejpam-5721	103	6	int(f	int(f	PROPN
ejpam-5721	103	7	−1(v	−1(v	NOUN
ejpam-5721	103	8	)	)	PUNCT
ejpam-5721	103	9	)	)	PUNCT
ejpam-5721	103	10	)	)	PUNCT
ejpam-5721	104	1	and	and	CCONJ
ejpam-5721	104	2	hence	hence	ADV
ejpam-5721	104	3	τ1τ2	τ1τ2	NOUN
ejpam-5721	104	4	-	-	NOUN
ejpam-5721	104	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	104	6	-	-	PUNCT
ejpam-5721	104	7	int(f	int(f	PROPN
ejpam-5721	104	8	−1(v	−1(v	NOUN
ejpam-5721	104	9	)	)	PUNCT
ejpam-5721	104	10	)	)	PUNCT
ejpam-5721	104	11	)	)	PUNCT
ejpam-5721	105	1	⊆	⊆	NUM
ejpam-5721	105	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	105	3	-	-	PUNCT
ejpam-5721	105	4	cl(v	cl(v	NOUN
ejpam-5721	105	5	)	)	PUNCT
ejpam-5721	105	6	)	)	PUNCT
ejpam-5721	105	7	.	.	PUNCT
ejpam-5721	106	1	(	(	PUNCT
ejpam-5721	106	2	3	3	X
ejpam-5721	106	3	)	)	PUNCT
ejpam-5721	106	4	⇒	⇒	NOUN
ejpam-5721	106	5	(	(	PUNCT
ejpam-5721	106	6	4	4	NUM
ejpam-5721	106	7	):	):	PUNCT
ejpam-5721	106	8	let	let	VERB
ejpam-5721	106	9	v	v	PART
ejpam-5721	106	10	be	be	AUX
ejpam-5721	106	11	any	any	DET
ejpam-5721	106	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	106	13	-	-	ADJ
ejpam-5721	106	14	open	open	ADJ
ejpam-5721	106	15	set	set	NOUN
ejpam-5721	106	16	of	of	ADP
ejpam-5721	106	17	y	y	PROPN
ejpam-5721	106	18	.	.	PUNCT
ejpam-5721	107	1	by	by	ADP
ejpam-5721	107	2	(	(	PUNCT
ejpam-5721	107	3	3	3	NUM
ejpam-5721	107	4	)	)	PUNCT
ejpam-5721	107	5	and	and	CCONJ
ejpam-5721	107	6	lemma	lemma	PROPN
ejpam-5721	107	7	3(1	3(1	NUM
ejpam-5721	107	8	)	)	PUNCT
ejpam-5721	107	9	,	,	PUNCT
ejpam-5721	107	10	(	(	PUNCT
ejpam-5721	107	11	τ1	τ1	NOUN
ejpam-5721	107	12	,	,	PUNCT
ejpam-5721	107	13	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	107	14	−1(v	−1(v	X
ejpam-5721	107	15	)	)	PUNCT
ejpam-5721	107	16	)	)	PUNCT
ejpam-5721	107	17	=	=	SYM
ejpam-5721	107	18	f−1(v	f−1(v	PROPN
ejpam-5721	107	19	)	)	PUNCT
ejpam-5721	107	20	∪	∪	ADP
ejpam-5721	107	21	τ1τ2	τ1τ2	NOUN
ejpam-5721	107	22	-	-	ADJ
ejpam-5721	107	23	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	107	24	-	-	PUNCT
ejpam-5721	107	25	int(f	int(f	PROPN
ejpam-5721	107	26	−1(v	−1(v	NOUN
ejpam-5721	107	27	)	)	PUNCT
ejpam-5721	107	28	)	)	PUNCT
ejpam-5721	107	29	)	)	PUNCT
ejpam-5721	108	1	⊆	⊆	NUM
ejpam-5721	108	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	108	3	-	-	PUNCT
ejpam-5721	108	4	cl(v	cl(v	NOUN
ejpam-5721	108	5	)	)	PUNCT
ejpam-5721	108	6	)	)	PUNCT
ejpam-5721	108	7	.	.	PUNCT
ejpam-5721	109	1	(	(	PUNCT
ejpam-5721	109	2	4	4	X
ejpam-5721	109	3	)	)	PUNCT
ejpam-5721	109	4	⇒	⇒	NOUN
ejpam-5721	109	5	(	(	PUNCT
ejpam-5721	109	6	5	5	NUM
ejpam-5721	109	7	):	):	PUNCT
ejpam-5721	109	8	let	let	VERB
ejpam-5721	109	9	v	v	PART
ejpam-5721	109	10	be	be	AUX
ejpam-5721	109	11	any	any	DET
ejpam-5721	109	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	109	13	-	-	ADJ
ejpam-5721	109	14	open	open	ADJ
ejpam-5721	109	15	set	set	NOUN
ejpam-5721	109	16	of	of	ADP
ejpam-5721	109	17	y	y	PROPN
ejpam-5721	109	18	.	.	PUNCT
ejpam-5721	110	1	then	then	ADV
ejpam-5721	110	2	,	,	PUNCT
ejpam-5721	110	3	y	y	PROPN
ejpam-5721	110	4	−	−	NUM
ejpam-5721	110	5	σ1σ2	σ1σ2	NOUN
ejpam-5721	110	6	-	-	NUM
ejpam-5721	110	7	cl(v	cl(v	NOUN
ejpam-5721	110	8	)	)	PUNCT
ejpam-5721	110	9	is	be	AUX
ejpam-5721	110	10	σ1σ2	σ1σ2	NOUN
ejpam-5721	110	11	-	-	ADJ
ejpam-5721	110	12	open	open	ADJ
ejpam-5721	110	13	and	and	CCONJ
ejpam-5721	110	14	by	by	ADP
ejpam-5721	110	15	(	(	PUNCT
ejpam-5721	110	16	4	4	NUM
ejpam-5721	110	17	)	)	PUNCT
ejpam-5721	110	18	,	,	PUNCT
ejpam-5721	110	19	we	we	PRON
ejpam-5721	110	20	have	have	VERB
ejpam-5721	110	21	x	x	X
ejpam-5721	110	22	−	−	PROPN
ejpam-5721	110	23	(	(	PUNCT
ejpam-5721	110	24	τ1	τ1	NOUN
ejpam-5721	110	25	,	,	PUNCT
ejpam-5721	110	26	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	110	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	110	28	-	-	PUNCT
ejpam-5721	110	29	cl(v	cl(v	NOUN
ejpam-5721	110	30	)	)	PUNCT
ejpam-5721	110	31	)	)	PUNCT
ejpam-5721	110	32	)	)	PUNCT
ejpam-5721	111	1	=	=	PRON
ejpam-5721	111	2	(	(	PUNCT
ejpam-5721	111	3	τ1	τ1	PROPN
ejpam-5721	111	4	,	,	PUNCT
ejpam-5721	111	5	τ2)-pcl(x	τ2)-pcl(x	PROPN
ejpam-5721	111	6	−	−	PROPN
ejpam-5721	111	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	111	8	-	-	PUNCT
ejpam-5721	111	9	cl(v	cl(v	NOUN
ejpam-5721	111	10	)	)	PUNCT
ejpam-5721	111	11	)	)	PUNCT
ejpam-5721	111	12	)	)	PUNCT
ejpam-5721	112	1	=	=	PRON
ejpam-5721	112	2	(	(	PUNCT
ejpam-5721	112	3	τ1	τ1	PROPN
ejpam-5721	112	4	,	,	PUNCT
ejpam-5721	112	5	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	112	6	−1(y	−1(y	PUNCT
ejpam-5721	112	7	−	−	PUNCT
ejpam-5721	112	8	σ1σ2	σ1σ2	NOUN
ejpam-5721	112	9	-	-	NUM
ejpam-5721	112	10	cl(v	cl(v	NOUN
ejpam-5721	112	11	)	)	PUNCT
ejpam-5721	112	12	)	)	PUNCT
ejpam-5721	112	13	)	)	PUNCT
ejpam-5721	113	1	⊆	⊆	NUM
ejpam-5721	113	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	113	3	-	-	PUNCT
ejpam-5721	113	4	cl(y	cl(y	NOUN
ejpam-5721	113	5	−	−	NOUN
ejpam-5721	113	6	σ1σ2	σ1σ2	NOUN
ejpam-5721	113	7	-	-	NUM
ejpam-5721	113	8	cl(v	cl(v	NOUN
ejpam-5721	113	9	)	)	PUNCT
ejpam-5721	113	10	)	)	PUNCT
ejpam-5721	113	11	)	)	PUNCT
ejpam-5721	114	1	⊆	⊆	X
ejpam-5721	114	2	f−1(y	f−1(y	NOUN
ejpam-5721	114	3	−	−	PROPN
ejpam-5721	114	4	v	v	NOUN
ejpam-5721	114	5	)	)	PUNCT
ejpam-5721	114	6	=	=	PUNCT
ejpam-5721	114	7	x	x	PUNCT
ejpam-5721	114	8	−	−	PROPN
ejpam-5721	114	9	f−1(v	f−1(v	PROPN
ejpam-5721	114	10	)	)	PUNCT
ejpam-5721	114	11	and	and	CCONJ
ejpam-5721	114	12	hence	hence	ADV
ejpam-5721	114	13	f−1(v	f−1(v	NOUN
ejpam-5721	114	14	)	)	PUNCT
ejpam-5721	115	1	⊆	⊆	NUM
ejpam-5721	115	2	(	(	PUNCT
ejpam-5721	115	3	τ1	τ1	NOUN
ejpam-5721	115	4	,	,	PUNCT
ejpam-5721	115	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	115	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	115	7	-	-	PUNCT
ejpam-5721	115	8	cl(v	cl(v	NOUN
ejpam-5721	115	9	)	)	PUNCT
ejpam-5721	115	10	)	)	PUNCT
ejpam-5721	115	11	)	)	PUNCT
ejpam-5721	115	12	.	.	PUNCT
ejpam-5721	116	1	(	(	PUNCT
ejpam-5721	116	2	5	5	X
ejpam-5721	116	3	)	)	PUNCT
ejpam-5721	116	4	⇒	⇒	NOUN
ejpam-5721	116	5	(	(	PUNCT
ejpam-5721	116	6	6	6	NUM
ejpam-5721	116	7	):	):	PUNCT
ejpam-5721	116	8	let	let	VERB
ejpam-5721	116	9	x	x	PUNCT
ejpam-5721	116	10	∈	∈	PROPN
ejpam-5721	116	11	x	x	X
ejpam-5721	116	12	and	and	CCONJ
ejpam-5721	116	13	v	v	X
ejpam-5721	116	14	be	be	AUX
ejpam-5721	116	15	any	any	DET
ejpam-5721	116	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	116	17	-	-	ADJ
ejpam-5721	116	18	open	open	ADJ
ejpam-5721	116	19	set	set	NOUN
ejpam-5721	116	20	of	of	ADP
ejpam-5721	116	21	y	y	PROPN
ejpam-5721	116	22	containing	contain	VERB
ejpam-5721	116	23	f(x	f(x	PROPN
ejpam-5721	116	24	)	)	PUNCT
ejpam-5721	116	25	.	.	PUNCT
ejpam-5721	117	1	by	by	ADP
ejpam-5721	117	2	(	(	PUNCT
ejpam-5721	117	3	5	5	NUM
ejpam-5721	117	4	)	)	PUNCT
ejpam-5721	117	5	,	,	PUNCT
ejpam-5721	117	6	x	x	PUNCT
ejpam-5721	117	7	∈	∈	PROPN
ejpam-5721	117	8	f−1(v	f−1(v	NOUN
ejpam-5721	117	9	)	)	PUNCT
ejpam-5721	118	1	⊆	⊆	NUM
ejpam-5721	118	2	(	(	PUNCT
ejpam-5721	118	3	τ1	τ1	NOUN
ejpam-5721	118	4	,	,	PUNCT
ejpam-5721	118	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	118	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	118	7	-	-	PUNCT
ejpam-5721	118	8	cl(v	cl(v	NOUN
ejpam-5721	118	9	)	)	PUNCT
ejpam-5721	118	10	)	)	PUNCT
ejpam-5721	118	11	)	)	PUNCT
ejpam-5721	119	1	and	and	CCONJ
ejpam-5721	119	2	there	there	PRON
ejpam-5721	119	3	exists	exist	VERB
ejpam-5721	119	4	a	a	DET
ejpam-5721	119	5	(	(	PUNCT
ejpam-5721	119	6	τ1	τ1	NOUN
ejpam-5721	119	7	,	,	PUNCT
ejpam-5721	119	8	τ2)p	τ2)p	ADJ
ejpam-5721	119	9	-	-	PUNCT
ejpam-5721	119	10	open	open	ADJ
ejpam-5721	119	11	set	set	NOUN
ejpam-5721	119	12	u	u	NOUN
ejpam-5721	119	13	of	of	ADP
ejpam-5721	119	14	x	x	PUNCT
ejpam-5721	119	15	containing	contain	VERB
ejpam-5721	119	16	x	x	PUNCT
ejpam-5721	119	17	such	such	ADJ
ejpam-5721	119	18	that	that	DET
ejpam-5721	119	19	f(u	f(u	PROPN
ejpam-5721	119	20	)	)	PUNCT
ejpam-5721	119	21	⊆	⊆	NUM
ejpam-5721	119	22	σ1σ2	σ1σ2	NOUN
ejpam-5721	119	23	-	-	NUM
ejpam-5721	119	24	cl(v	cl(v	NOUN
ejpam-5721	119	25	)	)	PUNCT
ejpam-5721	119	26	.	.	PUNCT
ejpam-5721	120	1	(	(	PUNCT
ejpam-5721	120	2	6	6	X
ejpam-5721	120	3	)	)	PUNCT
ejpam-5721	120	4	⇒	⇒	NOUN
ejpam-5721	120	5	(	(	PUNCT
ejpam-5721	120	6	1	1	NUM
ejpam-5721	120	7	):	):	PUNCT
ejpam-5721	120	8	let	let	VERB
ejpam-5721	120	9	x	x	PUNCT
ejpam-5721	120	10	∈	∈	PROPN
ejpam-5721	120	11	x	x	X
ejpam-5721	120	12	and	and	CCONJ
ejpam-5721	120	13	v	v	X
ejpam-5721	120	14	be	be	AUX
ejpam-5721	120	15	any	any	DET
ejpam-5721	120	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	120	17	-	-	ADJ
ejpam-5721	120	18	open	open	ADJ
ejpam-5721	120	19	set	set	NOUN
ejpam-5721	120	20	of	of	ADP
ejpam-5721	120	21	y	y	PROPN
ejpam-5721	120	22	containing	contain	VERB
ejpam-5721	120	23	f(x	f(x	PROPN
ejpam-5721	120	24	)	)	PUNCT
ejpam-5721	120	25	.	.	PUNCT
ejpam-5721	121	1	by	by	ADP
ejpam-5721	121	2	(	(	PUNCT
ejpam-5721	121	3	6	6	NUM
ejpam-5721	121	4	)	)	PUNCT
ejpam-5721	121	5	,	,	PUNCT
ejpam-5721	121	6	there	there	PRON
ejpam-5721	121	7	exists	exist	VERB
ejpam-5721	121	8	a	a	DET
ejpam-5721	121	9	(	(	PUNCT
ejpam-5721	121	10	τ1	τ1	NOUN
ejpam-5721	121	11	,	,	PUNCT
ejpam-5721	121	12	τ2)p	τ2)p	ADJ
ejpam-5721	121	13	-	-	PUNCT
ejpam-5721	121	14	open	open	ADJ
ejpam-5721	121	15	set	set	NOUN
ejpam-5721	121	16	u	u	NOUN
ejpam-5721	121	17	of	of	ADP
ejpam-5721	121	18	x	x	PUNCT
ejpam-5721	121	19	containing	contain	VERB
ejpam-5721	121	20	x	x	PUNCT
ejpam-5721	121	21	such	such	ADJ
ejpam-5721	121	22	that	that	DET
ejpam-5721	121	23	f(u	f(u	PROPN
ejpam-5721	121	24	)	)	PUNCT
ejpam-5721	121	25	⊆	⊆	NUM
ejpam-5721	121	26	σ1σ2	σ1σ2	NOUN
ejpam-5721	121	27	-	-	NUM
ejpam-5721	121	28	cl(v	cl(v	NOUN
ejpam-5721	121	29	)	)	PUNCT
ejpam-5721	121	30	;	;	PUNCT
ejpam-5721	121	31	hence	hence	ADV
ejpam-5721	121	32	u	u	NOUN
ejpam-5721	121	33	⊆	⊆	NUM
ejpam-5721	121	34	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	121	35	-	-	PUNCT
ejpam-5721	121	36	cl(v	cl(v	NOUN
ejpam-5721	121	37	)	)	PUNCT
ejpam-5721	121	38	)	)	PUNCT
ejpam-5721	121	39	.	.	PUNCT
ejpam-5721	122	1	thus	thus	ADV
ejpam-5721	122	2	,	,	PUNCT
ejpam-5721	122	3	x	x	PUNCT
ejpam-5721	122	4	∈	∈	PROPN
ejpam-5721	122	5	u	u	NOUN
ejpam-5721	122	6	⊆	⊆	NUM
ejpam-5721	122	7	τ1τ2	τ1τ2	NOUN
ejpam-5721	122	8	-	-	NOUN
ejpam-5721	122	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	122	10	-	-	PUNCT
ejpam-5721	122	11	cl(u	cl(u	NOUN
ejpam-5721	122	12	)	)	PUNCT
ejpam-5721	122	13	)	)	PUNCT
ejpam-5721	123	1	⊆	⊆	X
ejpam-5721	123	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	123	3	-	-	NOUN
ejpam-5721	123	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	123	5	-	-	PUNCT
ejpam-5721	123	6	cl(f	cl(f	NOUN
ejpam-5721	123	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	123	8	-	-	PUNCT
ejpam-5721	123	9	cl(v	cl(v	NOUN
ejpam-5721	123	10	)	)	PUNCT
ejpam-5721	123	11	)	)	PUNCT
ejpam-5721	123	12	)	)	PUNCT
ejpam-5721	123	13	)	)	PUNCT
ejpam-5721	123	14	.	.	PUNCT
ejpam-5721	124	1	j.	j.	PROPN
ejpam-5721	124	2	khampakdee	khampakdee	PROPN
ejpam-5721	124	3	,	,	PUNCT
ejpam-5721	124	4	s.	s.	PROPN
ejpam-5721	124	5	sompong	sompong	PROPN
ejpam-5721	124	6	,	,	PUNCT
ejpam-5721	124	7	c.	c.	PROPN
ejpam-5721	124	8	boonpok	boonpok	PROPN
ejpam-5721	124	9	/	/	SYM
ejpam-5721	124	10	eur	eur	PROPN
ejpam-5721	124	11	.	.	PUNCT
ejpam-5721	125	1	j.	j.	PROPN
ejpam-5721	125	2	pure	pure	PROPN
ejpam-5721	125	3	appl	appl	PROPN
ejpam-5721	125	4	.	.	PROPN
ejpam-5721	125	5	math	math	PROPN
ejpam-5721	125	6	,	,	PUNCT
ejpam-5721	125	7	18	18	NUM
ejpam-5721	125	8	(	(	PUNCT
ejpam-5721	125	9	1	1	NUM
ejpam-5721	125	10	)	)	PUNCT
ejpam-5721	125	11	(	(	PUNCT
ejpam-5721	125	12	2025	2025	NUM
ejpam-5721	125	13	)	)	PUNCT
ejpam-5721	125	14	,	,	PUNCT
ejpam-5721	125	15	5721	5721	NUM
ejpam-5721	125	16	5	5	NUM
ejpam-5721	125	17	of	of	ADP
ejpam-5721	125	18	12	12	NUM
ejpam-5721	125	19	this	this	PRON
ejpam-5721	125	20	shows	show	VERB
ejpam-5721	125	21	that	that	SCONJ
ejpam-5721	125	22	f	f	PROPN
ejpam-5721	125	23	is	be	AUX
ejpam-5721	125	24	almost	almost	ADV
ejpam-5721	125	25	weakly	weakly	ADJ
ejpam-5721	125	26	(	(	PUNCT
ejpam-5721	125	27	τ1	τ1	NOUN
ejpam-5721	125	28	,	,	PUNCT
ejpam-5721	125	29	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	125	30	.	.	PUNCT
ejpam-5721	125	31	theorem	theorem	NOUN
ejpam-5721	125	32	2	2	NUM
ejpam-5721	125	33	.	.	X
ejpam-5721	125	34	for	for	ADP
ejpam-5721	125	35	a	a	DET
ejpam-5721	125	36	function	function	NOUN
ejpam-5721	125	37	f	f	NOUN
ejpam-5721	125	38	:	:	PUNCT
ejpam-5721	125	39	(	(	PUNCT
ejpam-5721	125	40	x	x	NOUN
ejpam-5721	125	41	,	,	PUNCT
ejpam-5721	125	42	τ1	τ1	NOUN
ejpam-5721	125	43	,	,	PUNCT
ejpam-5721	125	44	τ2	τ2	NOUN
ejpam-5721	125	45	)	)	PUNCT
ejpam-5721	125	46	→	→	SYM
ejpam-5721	125	47	(	(	PUNCT
ejpam-5721	125	48	y	y	PROPN
ejpam-5721	125	49	,	,	PUNCT
ejpam-5721	125	50	σ1	σ1	PROPN
ejpam-5721	125	51	,	,	PUNCT
ejpam-5721	125	52	σ2	σ2	NOUN
ejpam-5721	125	53	)	)	PUNCT
ejpam-5721	125	54	,	,	PUNCT
ejpam-5721	125	55	the	the	DET
ejpam-5721	125	56	following	follow	VERB
ejpam-5721	125	57	properties	property	NOUN
ejpam-5721	125	58	are	be	AUX
ejpam-5721	125	59	equivalent	equivalent	ADJ
ejpam-5721	125	60	:	:	PUNCT
ejpam-5721	125	61	(	(	PUNCT
ejpam-5721	125	62	1	1	X
ejpam-5721	125	63	)	)	PUNCT
ejpam-5721	125	64	f	f	NOUN
ejpam-5721	125	65	is	be	AUX
ejpam-5721	125	66	almost	almost	ADV
ejpam-5721	125	67	weakly	weakly	ADJ
ejpam-5721	125	68	(	(	PUNCT
ejpam-5721	125	69	τ1	τ1	NOUN
ejpam-5721	125	70	,	,	PUNCT
ejpam-5721	125	71	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	125	72	;	;	PUNCT
ejpam-5721	125	73	(	(	PUNCT
ejpam-5721	125	74	2	2	X
ejpam-5721	125	75	)	)	PUNCT
ejpam-5721	125	76	τ1τ2	τ1τ2	NOUN
ejpam-5721	125	77	-	-	NOUN
ejpam-5721	125	78	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	125	79	-	-	PUNCT
ejpam-5721	125	80	int(f	int(f	PRON
ejpam-5721	125	81	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	125	82	-	-	PUNCT
ejpam-5721	125	83	int(k	int(k	NOUN
ejpam-5721	125	84	)	)	PUNCT
ejpam-5721	125	85	)	)	PUNCT
ejpam-5721	125	86	)	)	PUNCT
ejpam-5721	125	87	)	)	PUNCT
ejpam-5721	126	1	⊆	⊆	NUM
ejpam-5721	126	2	f−1(k	f−1(k	PROPN
ejpam-5721	126	3	)	)	PUNCT
ejpam-5721	126	4	for	for	ADP
ejpam-5721	126	5	every	every	DET
ejpam-5721	126	6	σ1σ2	σ1σ2	NUM
ejpam-5721	126	7	-	-	PUNCT
ejpam-5721	126	8	closed	closed	ADJ
ejpam-5721	126	9	set	set	NOUN
ejpam-5721	126	10	k	k	PROPN
ejpam-5721	126	11	of	of	ADP
ejpam-5721	126	12	y	y	PROPN
ejpam-5721	126	13	;	;	PUNCT
ejpam-5721	126	14	(	(	PUNCT
ejpam-5721	126	15	3	3	X
ejpam-5721	126	16	)	)	PUNCT
ejpam-5721	126	17	(	(	PUNCT
ejpam-5721	126	18	τ1	τ1	NOUN
ejpam-5721	126	19	,	,	PUNCT
ejpam-5721	126	20	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	126	21	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	126	22	-	-	PUNCT
ejpam-5721	126	23	int(k	int(k	NUM
ejpam-5721	126	24	)	)	PUNCT
ejpam-5721	126	25	)	)	PUNCT
ejpam-5721	126	26	)	)	PUNCT
ejpam-5721	126	27	⊆	⊆	NUM
ejpam-5721	126	28	f−1(k	f−1(k	PROPN
ejpam-5721	126	29	)	)	PUNCT
ejpam-5721	126	30	for	for	ADP
ejpam-5721	126	31	every	every	DET
ejpam-5721	126	32	σ1σ2	σ1σ2	NUM
ejpam-5721	126	33	-	-	PUNCT
ejpam-5721	126	34	closed	closed	ADJ
ejpam-5721	126	35	set	set	NOUN
ejpam-5721	126	36	k	k	PROPN
ejpam-5721	126	37	of	of	ADP
ejpam-5721	126	38	y	y	PROPN
ejpam-5721	126	39	;	;	PUNCT
ejpam-5721	126	40	(	(	PUNCT
ejpam-5721	126	41	4	4	X
ejpam-5721	126	42	)	)	PUNCT
ejpam-5721	126	43	(	(	PUNCT
ejpam-5721	126	44	τ1	τ1	NOUN
ejpam-5721	126	45	,	,	PUNCT
ejpam-5721	126	46	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	126	47	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	126	48	-	-	PUNCT
ejpam-5721	126	49	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	126	50	-	-	PUNCT
ejpam-5721	126	51	cl(b	cl(b	NOUN
ejpam-5721	126	52	)	)	PUNCT
ejpam-5721	126	53	)	)	PUNCT
ejpam-5721	126	54	)	)	PUNCT
ejpam-5721	126	55	)	)	PUNCT
ejpam-5721	127	1	⊆	⊆	NUM
ejpam-5721	127	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	127	3	-	-	PUNCT
ejpam-5721	127	4	cl(b	cl(b	NOUN
ejpam-5721	127	5	)	)	PUNCT
ejpam-5721	127	6	)	)	PUNCT
ejpam-5721	127	7	for	for	ADP
ejpam-5721	127	8	every	every	DET
ejpam-5721	127	9	subset	subset	NOUN
ejpam-5721	127	10	b	b	PROPN
ejpam-5721	127	11	of	of	ADP
ejpam-5721	127	12	y	y	PROPN
ejpam-5721	127	13	;	;	PUNCT
ejpam-5721	127	14	(	(	PUNCT
ejpam-5721	127	15	5	5	X
ejpam-5721	127	16	)	)	PUNCT
ejpam-5721	127	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	127	18	-	-	PUNCT
ejpam-5721	127	19	int(b	int(b	NOUN
ejpam-5721	127	20	)	)	PUNCT
ejpam-5721	127	21	)	)	PUNCT
ejpam-5721	128	1	⊆	⊆	NUM
ejpam-5721	128	2	(	(	PUNCT
ejpam-5721	128	3	τ1	τ1	NOUN
ejpam-5721	128	4	,	,	PUNCT
ejpam-5721	128	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	128	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	128	7	-	-	PUNCT
ejpam-5721	128	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	128	9	-	-	PUNCT
ejpam-5721	128	10	int(b	int(b	NOUN
ejpam-5721	128	11	)	)	PUNCT
ejpam-5721	128	12	)	)	PUNCT
ejpam-5721	128	13	)	)	PUNCT
ejpam-5721	128	14	)	)	PUNCT
ejpam-5721	128	15	for	for	ADP
ejpam-5721	128	16	every	every	DET
ejpam-5721	128	17	subset	subset	NOUN
ejpam-5721	128	18	b	b	PROPN
ejpam-5721	128	19	of	of	ADP
ejpam-5721	128	20	y	y	PROPN
ejpam-5721	128	21	.	.	PUNCT
ejpam-5721	129	1	proof	proof	NOUN
ejpam-5721	129	2	.	.	PUNCT
ejpam-5721	130	1	(	(	PUNCT
ejpam-5721	130	2	1	1	X
ejpam-5721	130	3	)	)	PUNCT
ejpam-5721	130	4	⇒	⇒	NOUN
ejpam-5721	130	5	(	(	PUNCT
ejpam-5721	130	6	2	2	NUM
ejpam-5721	130	7	):	):	PUNCT
ejpam-5721	130	8	let	let	VERB
ejpam-5721	130	9	k	k	PRON
ejpam-5721	130	10	be	be	AUX
ejpam-5721	130	11	any	any	DET
ejpam-5721	130	12	σ1σ2	σ1σ2	NUM
ejpam-5721	130	13	-	-	PUNCT
ejpam-5721	130	14	closed	closed	ADJ
ejpam-5721	130	15	set	set	NOUN
ejpam-5721	130	16	of	of	ADP
ejpam-5721	130	17	y	y	PROPN
ejpam-5721	130	18	.	.	PUNCT
ejpam-5721	131	1	then	then	ADV
ejpam-5721	131	2	,	,	PUNCT
ejpam-5721	131	3	y	y	PROPN
ejpam-5721	131	4	−k	−k	PROPN
ejpam-5721	131	5	is	be	AUX
ejpam-5721	131	6	σ1σ2	σ1σ2	NOUN
ejpam-5721	131	7	-	-	ADJ
ejpam-5721	131	8	open	open	ADJ
ejpam-5721	131	9	in	in	ADP
ejpam-5721	131	10	y	y	PROPN
ejpam-5721	131	11	and	and	CCONJ
ejpam-5721	131	12	by	by	ADP
ejpam-5721	131	13	theorem	theorem	NOUN
ejpam-5721	131	14	1	1	NUM
ejpam-5721	131	15	,	,	PUNCT
ejpam-5721	131	16	we	we	PRON
ejpam-5721	131	17	have	have	VERB
ejpam-5721	131	18	x	x	X
ejpam-5721	131	19	−	−	PROPN
ejpam-5721	131	20	f−1(k	f−1(k	PROPN
ejpam-5721	131	21	)	)	PUNCT
ejpam-5721	131	22	=	=	SYM
ejpam-5721	131	23	f−1(y	f−1(y	PROPN
ejpam-5721	131	24	−k	−k	PROPN
ejpam-5721	131	25	)	)	PUNCT
ejpam-5721	131	26	⊆	⊆	NUM
ejpam-5721	131	27	τ1τ2	τ1τ2	NOUN
ejpam-5721	131	28	-	-	NOUN
ejpam-5721	131	29	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	131	30	-	-	PUNCT
ejpam-5721	131	31	cl(f	cl(f	NOUN
ejpam-5721	131	32	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	131	33	-	-	PUNCT
ejpam-5721	131	34	cl(y	cl(y	NOUN
ejpam-5721	131	35	−k	−k	NOUN
ejpam-5721	131	36	)	)	PUNCT
ejpam-5721	131	37	)	)	PUNCT
ejpam-5721	131	38	)	)	PUNCT
ejpam-5721	131	39	)	)	PUNCT
ejpam-5721	132	1	=	=	PUNCT
ejpam-5721	133	1	τ1τ2	τ1τ2	NOUN
ejpam-5721	133	2	-	-	NOUN
ejpam-5721	133	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	133	4	-	-	PUNCT
ejpam-5721	133	5	cl(f	cl(f	NOUN
ejpam-5721	133	6	−1(y	−1(y	VERB
ejpam-5721	133	7	−	−	PUNCT
ejpam-5721	133	8	σ1σ2	σ1σ2	NUM
ejpam-5721	133	9	-	-	PUNCT
ejpam-5721	133	10	int(k	int(k	NOUN
ejpam-5721	133	11	)	)	PUNCT
ejpam-5721	133	12	)	)	PUNCT
ejpam-5721	133	13	)	)	PUNCT
ejpam-5721	133	14	)	)	PUNCT
ejpam-5721	134	1	=	=	PUNCT
ejpam-5721	134	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	134	3	-	-	NOUN
ejpam-5721	134	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	134	5	-	-	PUNCT
ejpam-5721	134	6	cl(x	cl(x	NUM
ejpam-5721	134	7	−	−	PRON
ejpam-5721	134	8	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	134	9	-	-	PUNCT
ejpam-5721	134	10	int(k	int(k	NOUN
ejpam-5721	134	11	)	)	PUNCT
ejpam-5721	134	12	)	)	PUNCT
ejpam-5721	134	13	)	)	PUNCT
ejpam-5721	134	14	)	)	PUNCT
ejpam-5721	135	1	=	=	PUNCT
ejpam-5721	135	2	x	x	X
ejpam-5721	136	1	−	−	ADP
ejpam-5721	136	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	136	3	-	-	NUM
ejpam-5721	136	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	136	5	-	-	PUNCT
ejpam-5721	136	6	int(f	int(f	PRON
ejpam-5721	136	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	136	8	-	-	PUNCT
ejpam-5721	136	9	int(k	int(k	NOUN
ejpam-5721	136	10	)	)	PUNCT
ejpam-5721	136	11	)	)	PUNCT
ejpam-5721	136	12	)	)	PUNCT
ejpam-5721	136	13	)	)	PUNCT
ejpam-5721	137	1	and	and	CCONJ
ejpam-5721	137	2	hence	hence	ADV
ejpam-5721	137	3	τ1τ2	τ1τ2	NOUN
ejpam-5721	137	4	-	-	NOUN
ejpam-5721	137	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	137	6	-	-	PUNCT
ejpam-5721	137	7	int(f	int(f	PRON
ejpam-5721	137	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	137	9	-	-	PUNCT
ejpam-5721	137	10	int(k	int(k	NOUN
ejpam-5721	137	11	)	)	PUNCT
ejpam-5721	137	12	)	)	PUNCT
ejpam-5721	137	13	)	)	PUNCT
ejpam-5721	137	14	)	)	PUNCT
ejpam-5721	138	1	⊆	⊆	NUM
ejpam-5721	138	2	f−1(k	f−1(k	NOUN
ejpam-5721	138	3	)	)	PUNCT
ejpam-5721	138	4	.	.	PUNCT
ejpam-5721	139	1	(	(	PUNCT
ejpam-5721	139	2	2	2	X
ejpam-5721	139	3	)	)	PUNCT
ejpam-5721	139	4	⇒	⇒	NOUN
ejpam-5721	139	5	(	(	PUNCT
ejpam-5721	139	6	3	3	NUM
ejpam-5721	139	7	):	):	PUNCT
ejpam-5721	139	8	let	let	VERB
ejpam-5721	139	9	k	k	PRON
ejpam-5721	139	10	be	be	AUX
ejpam-5721	139	11	any	any	DET
ejpam-5721	139	12	σ1σ2	σ1σ2	NUM
ejpam-5721	139	13	-	-	PUNCT
ejpam-5721	139	14	closed	closed	ADJ
ejpam-5721	139	15	set	set	NOUN
ejpam-5721	139	16	of	of	ADP
ejpam-5721	139	17	y	y	PROPN
ejpam-5721	139	18	.	.	PUNCT
ejpam-5721	140	1	by	by	ADP
ejpam-5721	140	2	lemma	lemma	PROPN
ejpam-5721	140	3	3(1	3(1	NUM
ejpam-5721	140	4	)	)	PUNCT
ejpam-5721	140	5	,	,	PUNCT
ejpam-5721	140	6	we	we	PRON
ejpam-5721	140	7	have	have	VERB
ejpam-5721	140	8	(	(	PUNCT
ejpam-5721	140	9	τ1	τ1	NOUN
ejpam-5721	140	10	,	,	PUNCT
ejpam-5721	140	11	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	140	12	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	140	13	-	-	PUNCT
ejpam-5721	140	14	int(k	int(k	NUM
ejpam-5721	140	15	)	)	PUNCT
ejpam-5721	140	16	)	)	PUNCT
ejpam-5721	140	17	)	)	PUNCT
ejpam-5721	141	1	=	=	PRON
ejpam-5721	141	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	141	3	-	-	PUNCT
ejpam-5721	141	4	int(k	int(k	NOUN
ejpam-5721	141	5	)	)	PUNCT
ejpam-5721	141	6	)	)	PUNCT
ejpam-5721	141	7	∪	∪	ADP
ejpam-5721	141	8	τ1τ2	τ1τ2	NOUN
ejpam-5721	141	9	-	-	ADJ
ejpam-5721	141	10	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	141	11	-	-	PUNCT
ejpam-5721	141	12	int(f	int(f	PRON
ejpam-5721	141	13	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	141	14	-	-	PUNCT
ejpam-5721	141	15	int(k	int(k	NOUN
ejpam-5721	141	16	)	)	PUNCT
ejpam-5721	141	17	)	)	PUNCT
ejpam-5721	141	18	)	)	PUNCT
ejpam-5721	141	19	)	)	PUNCT
ejpam-5721	142	1	⊆	⊆	NUM
ejpam-5721	142	2	f−1(k	f−1(k	NOUN
ejpam-5721	142	3	)	)	PUNCT
ejpam-5721	142	4	.	.	PUNCT
ejpam-5721	143	1	(	(	PUNCT
ejpam-5721	143	2	3	3	X
ejpam-5721	143	3	)	)	PUNCT
ejpam-5721	143	4	⇒	⇒	NOUN
ejpam-5721	143	5	(	(	PUNCT
ejpam-5721	143	6	4	4	NUM
ejpam-5721	143	7	):	):	PUNCT
ejpam-5721	143	8	this	this	PRON
ejpam-5721	143	9	is	be	AUX
ejpam-5721	143	10	obvious	obvious	ADJ
ejpam-5721	143	11	.	.	PUNCT
ejpam-5721	144	1	(	(	PUNCT
ejpam-5721	144	2	4	4	X
ejpam-5721	144	3	)	)	PUNCT
ejpam-5721	144	4	⇒	⇒	NOUN
ejpam-5721	144	5	(	(	PUNCT
ejpam-5721	144	6	5	5	NUM
ejpam-5721	144	7	):	):	PUNCT
ejpam-5721	144	8	let	let	VERB
ejpam-5721	144	9	b	b	X
ejpam-5721	144	10	be	be	AUX
ejpam-5721	144	11	any	any	DET
ejpam-5721	144	12	subset	subset	NOUN
ejpam-5721	144	13	of	of	ADP
ejpam-5721	144	14	y	y	PROPN
ejpam-5721	144	15	.	.	PUNCT
ejpam-5721	145	1	then	then	ADV
ejpam-5721	145	2	by	by	ADP
ejpam-5721	145	3	(	(	PUNCT
ejpam-5721	145	4	4	4	NUM
ejpam-5721	145	5	)	)	PUNCT
ejpam-5721	145	6	,	,	PUNCT
ejpam-5721	145	7	x	x	PUNCT
ejpam-5721	145	8	−	−	NOUN
ejpam-5721	145	9	(	(	PUNCT
ejpam-5721	145	10	τ1	τ1	PROPN
ejpam-5721	145	11	,	,	PUNCT
ejpam-5721	145	12	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	145	13	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	145	14	-	-	PUNCT
ejpam-5721	145	15	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	145	16	-	-	PUNCT
ejpam-5721	145	17	int(b	int(b	NOUN
ejpam-5721	145	18	)	)	PUNCT
ejpam-5721	145	19	)	)	PUNCT
ejpam-5721	145	20	)	)	PUNCT
ejpam-5721	145	21	)	)	PUNCT
ejpam-5721	146	1	=	=	PRON
ejpam-5721	146	2	(	(	PUNCT
ejpam-5721	146	3	τ1	τ1	PROPN
ejpam-5721	146	4	,	,	PUNCT
ejpam-5721	146	5	τ2)-pcl(x	τ2)-pcl(x	PROPN
ejpam-5721	146	6	−	−	PROPN
ejpam-5721	146	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	146	8	-	-	PUNCT
ejpam-5721	146	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	146	10	-	-	PUNCT
ejpam-5721	146	11	int(b	int(b	NOUN
ejpam-5721	146	12	)	)	PUNCT
ejpam-5721	146	13	)	)	PUNCT
ejpam-5721	146	14	)	)	PUNCT
ejpam-5721	146	15	)	)	PUNCT
ejpam-5721	147	1	=	=	PRON
ejpam-5721	147	2	(	(	PUNCT
ejpam-5721	147	3	τ1	τ1	PROPN
ejpam-5721	147	4	,	,	PUNCT
ejpam-5721	147	5	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	147	6	−1(y	−1(y	PUNCT
ejpam-5721	147	7	−	−	PUNCT
ejpam-5721	147	8	σ1σ2	σ1σ2	ADV
ejpam-5721	147	9	-	-	PUNCT
ejpam-5721	147	10	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	147	11	-	-	PUNCT
ejpam-5721	147	12	int(b	int(b	NOUN
ejpam-5721	147	13	)	)	PUNCT
ejpam-5721	147	14	)	)	PUNCT
ejpam-5721	147	15	)	)	PUNCT
ejpam-5721	147	16	)	)	PUNCT
ejpam-5721	148	1	=	=	PRON
ejpam-5721	148	2	(	(	PUNCT
ejpam-5721	148	3	τ1	τ1	PROPN
ejpam-5721	148	4	,	,	PUNCT
ejpam-5721	148	5	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	148	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	148	7	-	-	PUNCT
ejpam-5721	148	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	148	9	-	-	PUNCT
ejpam-5721	148	10	cl(y	cl(y	NOUN
ejpam-5721	148	11	−b	−b	NOUN
ejpam-5721	148	12	)	)	PUNCT
ejpam-5721	148	13	)	)	PUNCT
ejpam-5721	148	14	)	)	PUNCT
ejpam-5721	148	15	)	)	PUNCT
ejpam-5721	148	16	⊆	⊆	NUM
ejpam-5721	148	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	148	18	-	-	PUNCT
ejpam-5721	148	19	cl(y	cl(y	NOUN
ejpam-5721	148	20	−b	−b	NOUN
ejpam-5721	148	21	)	)	PUNCT
ejpam-5721	148	22	)	)	PUNCT
ejpam-5721	149	1	=	=	PUNCT
ejpam-5721	150	1	x	x	X
ejpam-5721	150	2	−	−	PRON
ejpam-5721	150	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	150	4	-	-	PUNCT
ejpam-5721	150	5	int(b	int(b	NOUN
ejpam-5721	150	6	)	)	PUNCT
ejpam-5721	150	7	)	)	PUNCT
ejpam-5721	150	8	.	.	PUNCT
ejpam-5721	151	1	thus	thus	ADV
ejpam-5721	151	2	,	,	PUNCT
ejpam-5721	151	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	151	4	-	-	PUNCT
ejpam-5721	151	5	int(b	int(b	NOUN
ejpam-5721	151	6	)	)	PUNCT
ejpam-5721	151	7	)	)	PUNCT
ejpam-5721	152	1	⊆	⊆	NUM
ejpam-5721	152	2	(	(	PUNCT
ejpam-5721	152	3	τ1	τ1	NOUN
ejpam-5721	152	4	,	,	PUNCT
ejpam-5721	152	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	152	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	152	7	-	-	PUNCT
ejpam-5721	152	8	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	152	9	-	-	PUNCT
ejpam-5721	152	10	int(b	int(b	NOUN
ejpam-5721	152	11	)	)	PUNCT
ejpam-5721	152	12	)	)	PUNCT
ejpam-5721	152	13	)	)	PUNCT
ejpam-5721	152	14	)	)	PUNCT
ejpam-5721	152	15	.	.	PUNCT
ejpam-5721	153	1	(	(	PUNCT
ejpam-5721	153	2	5	5	X
ejpam-5721	153	3	)	)	PUNCT
ejpam-5721	153	4	⇒	⇒	NOUN
ejpam-5721	153	5	(	(	PUNCT
ejpam-5721	153	6	1	1	NUM
ejpam-5721	153	7	):	):	PUNCT
ejpam-5721	153	8	let	let	VERB
ejpam-5721	153	9	v	v	PART
ejpam-5721	153	10	be	be	AUX
ejpam-5721	153	11	any	any	DET
ejpam-5721	153	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	153	13	-	-	ADJ
ejpam-5721	153	14	open	open	ADJ
ejpam-5721	153	15	set	set	NOUN
ejpam-5721	153	16	of	of	ADP
ejpam-5721	153	17	y	y	PROPN
ejpam-5721	153	18	.	.	PUNCT
ejpam-5721	154	1	by	by	ADP
ejpam-5721	154	2	(	(	PUNCT
ejpam-5721	154	3	5	5	NUM
ejpam-5721	154	4	)	)	PUNCT
ejpam-5721	154	5	,	,	PUNCT
ejpam-5721	154	6	we	we	PRON
ejpam-5721	154	7	have	have	VERB
ejpam-5721	154	8	f−1(v	f−1(v	NOUN
ejpam-5721	154	9	)	)	PUNCT
ejpam-5721	155	1	⊆	⊆	NUM
ejpam-5721	155	2	(	(	PUNCT
ejpam-5721	155	3	τ1	τ1	NOUN
ejpam-5721	155	4	,	,	PUNCT
ejpam-5721	155	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-5721	155	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	155	7	-	-	PUNCT
ejpam-5721	155	8	cl(v	cl(v	NOUN
ejpam-5721	155	9	)	)	PUNCT
ejpam-5721	155	10	)	)	PUNCT
ejpam-5721	155	11	)	)	PUNCT
ejpam-5721	155	12	and	and	CCONJ
ejpam-5721	155	13	hence	hence	ADV
ejpam-5721	155	14	f	f	PROPN
ejpam-5721	155	15	is	be	AUX
ejpam-5721	155	16	almost	almost	ADV
ejpam-5721	155	17	weakly	weakly	ADJ
ejpam-5721	155	18	(	(	PUNCT
ejpam-5721	155	19	τ1	τ1	NOUN
ejpam-5721	155	20	,	,	PUNCT
ejpam-5721	155	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	155	22	by	by	ADP
ejpam-5721	155	23	theorem	theorem	NOUN
ejpam-5721	155	24	1	1	NUM
ejpam-5721	155	25	.	.	PUNCT
ejpam-5721	155	26	j.	j.	PROPN
ejpam-5721	155	27	khampakdee	khampakdee	PROPN
ejpam-5721	155	28	,	,	PUNCT
ejpam-5721	155	29	s.	s.	PROPN
ejpam-5721	155	30	sompong	sompong	PROPN
ejpam-5721	155	31	,	,	PUNCT
ejpam-5721	155	32	c.	c.	PROPN
ejpam-5721	155	33	boonpok	boonpok	PROPN
ejpam-5721	155	34	/	/	SYM
ejpam-5721	155	35	eur	eur	PROPN
ejpam-5721	155	36	.	.	PUNCT
ejpam-5721	156	1	j.	j.	PROPN
ejpam-5721	156	2	pure	pure	PROPN
ejpam-5721	156	3	appl	appl	PROPN
ejpam-5721	156	4	.	.	PROPN
ejpam-5721	156	5	math	math	PROPN
ejpam-5721	156	6	,	,	PUNCT
ejpam-5721	156	7	18	18	NUM
ejpam-5721	156	8	(	(	PUNCT
ejpam-5721	156	9	1	1	NUM
ejpam-5721	156	10	)	)	PUNCT
ejpam-5721	156	11	(	(	PUNCT
ejpam-5721	156	12	2025	2025	NUM
ejpam-5721	156	13	)	)	PUNCT
ejpam-5721	156	14	,	,	PUNCT
ejpam-5721	156	15	5721	5721	NUM
ejpam-5721	156	16	6	6	NUM
ejpam-5721	156	17	of	of	ADP
ejpam-5721	156	18	12	12	NUM
ejpam-5721	156	19	theorem	theorem	NOUN
ejpam-5721	156	20	3	3	NUM
ejpam-5721	156	21	.	.	X
ejpam-5721	157	1	for	for	ADP
ejpam-5721	157	2	a	a	DET
ejpam-5721	157	3	function	function	NOUN
ejpam-5721	157	4	f	f	NOUN
ejpam-5721	157	5	:	:	PUNCT
ejpam-5721	157	6	(	(	PUNCT
ejpam-5721	157	7	x	x	NOUN
ejpam-5721	157	8	,	,	PUNCT
ejpam-5721	157	9	τ1	τ1	NOUN
ejpam-5721	157	10	,	,	PUNCT
ejpam-5721	157	11	τ2	τ2	NOUN
ejpam-5721	157	12	)	)	PUNCT
ejpam-5721	157	13	→	→	SYM
ejpam-5721	157	14	(	(	PUNCT
ejpam-5721	157	15	y	y	PROPN
ejpam-5721	157	16	,	,	PUNCT
ejpam-5721	157	17	σ1	σ1	PROPN
ejpam-5721	157	18	,	,	PUNCT
ejpam-5721	157	19	σ2	σ2	NOUN
ejpam-5721	157	20	)	)	PUNCT
ejpam-5721	157	21	,	,	PUNCT
ejpam-5721	157	22	the	the	DET
ejpam-5721	157	23	following	follow	VERB
ejpam-5721	157	24	properties	property	NOUN
ejpam-5721	157	25	are	be	AUX
ejpam-5721	157	26	equivalent	equivalent	ADJ
ejpam-5721	157	27	:	:	PUNCT
ejpam-5721	157	28	(	(	PUNCT
ejpam-5721	157	29	1	1	X
ejpam-5721	157	30	)	)	PUNCT
ejpam-5721	157	31	f	f	NOUN
ejpam-5721	157	32	is	be	AUX
ejpam-5721	157	33	almost	almost	ADV
ejpam-5721	157	34	weakly	weakly	ADJ
ejpam-5721	157	35	(	(	PUNCT
ejpam-5721	157	36	τ1	τ1	NOUN
ejpam-5721	157	37	,	,	PUNCT
ejpam-5721	157	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	157	39	;	;	PUNCT
ejpam-5721	157	40	(	(	PUNCT
ejpam-5721	157	41	2	2	X
ejpam-5721	157	42	)	)	PUNCT
ejpam-5721	157	43	f((τ1	f((τ1	PROPN
ejpam-5721	157	44	,	,	PUNCT
ejpam-5721	157	45	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	157	46	)	)	PUNCT
ejpam-5721	157	47	)	)	PUNCT
ejpam-5721	158	1	⊆	⊆	NUM
ejpam-5721	158	2	(	(	PUNCT
ejpam-5721	158	3	σ1	σ1	PROPN
ejpam-5721	158	4	,	,	PUNCT
ejpam-5721	158	5	σ2)θ	σ2)θ	NOUN
ejpam-5721	158	6	-	-	PUNCT
ejpam-5721	158	7	cl(f(a	cl(f(a	NOUN
ejpam-5721	158	8	)	)	PUNCT
ejpam-5721	158	9	)	)	PUNCT
ejpam-5721	158	10	for	for	ADP
ejpam-5721	158	11	every	every	DET
ejpam-5721	158	12	subset	subset	NOUN
ejpam-5721	158	13	a	a	PRON
ejpam-5721	158	14	of	of	ADP
ejpam-5721	158	15	x	x	PRON
ejpam-5721	158	16	;	;	PUNCT
ejpam-5721	158	17	(	(	PUNCT
ejpam-5721	158	18	3	3	X
ejpam-5721	158	19	)	)	PUNCT
ejpam-5721	158	20	(	(	PUNCT
ejpam-5721	158	21	τ1	τ1	NOUN
ejpam-5721	158	22	,	,	PUNCT
ejpam-5721	158	23	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	158	24	−1(b	−1(b	NOUN
ejpam-5721	158	25	)	)	PUNCT
ejpam-5721	158	26	)	)	PUNCT
ejpam-5721	159	1	⊆	⊆	NUM
ejpam-5721	159	2	f−1((σ1	f−1((σ1	NOUN
ejpam-5721	159	3	,	,	PUNCT
ejpam-5721	159	4	σ2)θ	σ2)θ	NOUN
ejpam-5721	159	5	-	-	PUNCT
ejpam-5721	159	6	cl(b	cl(b	NOUN
ejpam-5721	159	7	)	)	PUNCT
ejpam-5721	159	8	)	)	PUNCT
ejpam-5721	159	9	for	for	ADP
ejpam-5721	159	10	every	every	DET
ejpam-5721	159	11	subset	subset	NOUN
ejpam-5721	159	12	b	b	PROPN
ejpam-5721	159	13	of	of	ADP
ejpam-5721	159	14	y	y	PROPN
ejpam-5721	159	15	;	;	PUNCT
ejpam-5721	159	16	(	(	PUNCT
ejpam-5721	159	17	4	4	X
ejpam-5721	159	18	)	)	PUNCT
ejpam-5721	159	19	(	(	PUNCT
ejpam-5721	159	20	τ1	τ1	NOUN
ejpam-5721	159	21	,	,	PUNCT
ejpam-5721	159	22	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	159	23	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	159	24	-	-	PUNCT
ejpam-5721	159	25	int((σ1	int((σ1	NUM
ejpam-5721	159	26	,	,	PUNCT
ejpam-5721	159	27	σ2)θ	σ2)θ	ADJ
ejpam-5721	159	28	-	-	PUNCT
ejpam-5721	159	29	cl(b	cl(b	NOUN
ejpam-5721	159	30	)	)	PUNCT
ejpam-5721	159	31	)	)	PUNCT
ejpam-5721	159	32	)	)	PUNCT
ejpam-5721	159	33	)	)	PUNCT
ejpam-5721	160	1	⊆	⊆	NUM
ejpam-5721	160	2	f−1((σ1	f−1((σ1	NOUN
ejpam-5721	160	3	,	,	PUNCT
ejpam-5721	160	4	σ2)θ	σ2)θ	NOUN
ejpam-5721	160	5	-	-	PUNCT
ejpam-5721	160	6	cl(b	cl(b	NOUN
ejpam-5721	160	7	)	)	PUNCT
ejpam-5721	160	8	)	)	PUNCT
ejpam-5721	160	9	for	for	ADP
ejpam-5721	160	10	every	every	DET
ejpam-5721	160	11	subset	subset	NOUN
ejpam-5721	160	12	b	b	PROPN
ejpam-5721	160	13	of	of	ADP
ejpam-5721	160	14	y	y	PROPN
ejpam-5721	160	15	;	;	PUNCT
ejpam-5721	160	16	(	(	PUNCT
ejpam-5721	160	17	5	5	X
ejpam-5721	160	18	)	)	PUNCT
ejpam-5721	160	19	(	(	PUNCT
ejpam-5721	160	20	τ1	τ1	NOUN
ejpam-5721	160	21	,	,	PUNCT
ejpam-5721	160	22	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	160	23	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	160	24	-	-	PUNCT
ejpam-5721	160	25	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	160	26	-	-	PUNCT
ejpam-5721	160	27	cl(v	cl(v	NOUN
ejpam-5721	160	28	)	)	PUNCT
ejpam-5721	160	29	)	)	PUNCT
ejpam-5721	160	30	)	)	PUNCT
ejpam-5721	160	31	)	)	PUNCT
ejpam-5721	161	1	⊆	⊆	NUM
ejpam-5721	161	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	161	3	-	-	PUNCT
ejpam-5721	161	4	cl(v	cl(v	NOUN
ejpam-5721	161	5	)	)	PUNCT
ejpam-5721	161	6	)	)	PUNCT
ejpam-5721	161	7	for	for	ADP
ejpam-5721	161	8	every	every	DET
ejpam-5721	161	9	σ1σ2	σ1σ2	NOUN
ejpam-5721	161	10	-	-	ADJ
ejpam-5721	161	11	open	open	ADJ
ejpam-5721	161	12	set	set	NOUN
ejpam-5721	161	13	v	v	NOUN
ejpam-5721	161	14	of	of	ADP
ejpam-5721	161	15	y	y	PROPN
ejpam-5721	161	16	;	;	PUNCT
ejpam-5721	161	17	(	(	PUNCT
ejpam-5721	161	18	6	6	NUM
ejpam-5721	161	19	)	)	PUNCT
ejpam-5721	161	20	(	(	PUNCT
ejpam-5721	161	21	τ1	τ1	NOUN
ejpam-5721	161	22	,	,	PUNCT
ejpam-5721	161	23	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	161	24	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	161	25	-	-	PUNCT
ejpam-5721	161	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	161	27	-	-	PUNCT
ejpam-5721	161	28	cl(v	cl(v	NOUN
ejpam-5721	161	29	)	)	PUNCT
ejpam-5721	161	30	)	)	PUNCT
ejpam-5721	161	31	)	)	PUNCT
ejpam-5721	161	32	)	)	PUNCT
ejpam-5721	162	1	⊆	⊆	NUM
ejpam-5721	162	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	162	3	-	-	PUNCT
ejpam-5721	162	4	cl(v	cl(v	NOUN
ejpam-5721	162	5	)	)	PUNCT
ejpam-5721	162	6	)	)	PUNCT
ejpam-5721	162	7	for	for	ADP
ejpam-5721	162	8	every	every	DET
ejpam-5721	162	9	(	(	PUNCT
ejpam-5721	162	10	σ1	σ1	PROPN
ejpam-5721	162	11	,	,	PUNCT
ejpam-5721	162	12	σ2)p	σ2)p	NOUN
ejpam-5721	162	13	-	-	PUNCT
ejpam-5721	162	14	open	open	NOUN
ejpam-5721	162	15	set	set	NOUN
ejpam-5721	162	16	v	v	NOUN
ejpam-5721	162	17	of	of	ADP
ejpam-5721	162	18	y	y	PROPN
ejpam-5721	162	19	;	;	PUNCT
ejpam-5721	162	20	(	(	PUNCT
ejpam-5721	162	21	7	7	X
ejpam-5721	162	22	)	)	PUNCT
ejpam-5721	162	23	(	(	PUNCT
ejpam-5721	162	24	τ1	τ1	NOUN
ejpam-5721	162	25	,	,	PUNCT
ejpam-5721	162	26	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	162	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	162	28	-	-	PUNCT
ejpam-5721	162	29	int(k	int(k	NUM
ejpam-5721	162	30	)	)	PUNCT
ejpam-5721	162	31	)	)	PUNCT
ejpam-5721	162	32	)	)	PUNCT
ejpam-5721	163	1	⊆	⊆	NUM
ejpam-5721	163	2	f−1(k	f−1(k	PROPN
ejpam-5721	163	3	)	)	PUNCT
ejpam-5721	163	4	for	for	ADP
ejpam-5721	163	5	every	every	DET
ejpam-5721	163	6	(	(	PUNCT
ejpam-5721	163	7	σ1	σ1	PROPN
ejpam-5721	163	8	,	,	PUNCT
ejpam-5721	163	9	σ2)r	σ2)r	NOUN
ejpam-5721	163	10	-	-	PUNCT
ejpam-5721	163	11	closed	close	VERB
ejpam-5721	163	12	set	set	ADJ
ejpam-5721	163	13	k	k	PROPN
ejpam-5721	163	14	of	of	ADP
ejpam-5721	163	15	y	y	PROPN
ejpam-5721	163	16	.	.	PUNCT
ejpam-5721	164	1	proof	proof	NOUN
ejpam-5721	164	2	.	.	PUNCT
ejpam-5721	165	1	(	(	PUNCT
ejpam-5721	165	2	1	1	X
ejpam-5721	165	3	)	)	PUNCT
ejpam-5721	165	4	⇒	⇒	NOUN
ejpam-5721	165	5	(	(	PUNCT
ejpam-5721	165	6	2	2	NUM
ejpam-5721	165	7	):	):	PUNCT
ejpam-5721	165	8	let	let	VERB
ejpam-5721	165	9	a	a	PRON
ejpam-5721	165	10	be	be	AUX
ejpam-5721	165	11	any	any	DET
ejpam-5721	165	12	subset	subset	ADJ
ejpam-5721	165	13	ofx	ofx	NOUN
ejpam-5721	165	14	.	.	PUNCT
ejpam-5721	166	1	let	let	VERB
ejpam-5721	166	2	x	x	X
ejpam-5721	166	3	∈	∈	PROPN
ejpam-5721	166	4	(	(	PUNCT
ejpam-5721	166	5	τ1	τ1	NOUN
ejpam-5721	166	6	,	,	PUNCT
ejpam-5721	166	7	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	166	8	)	)	PUNCT
ejpam-5721	166	9	and	and	CCONJ
ejpam-5721	166	10	v	v	AUX
ejpam-5721	166	11	be	be	AUX
ejpam-5721	166	12	any	any	DET
ejpam-5721	166	13	σ1σ2open	σ1σ2open	ADJ
ejpam-5721	166	14	set	set	NOUN
ejpam-5721	166	15	of	of	ADP
ejpam-5721	166	16	y	y	PROPN
ejpam-5721	166	17	containing	contain	VERB
ejpam-5721	166	18	f(x	f(x	PROPN
ejpam-5721	166	19	)	)	PUNCT
ejpam-5721	166	20	.	.	PUNCT
ejpam-5721	167	1	since	since	SCONJ
ejpam-5721	167	2	f	f	PROPN
ejpam-5721	167	3	is	be	AUX
ejpam-5721	167	4	almost	almost	ADV
ejpam-5721	167	5	weakly	weakly	ADJ
ejpam-5721	167	6	(	(	PUNCT
ejpam-5721	167	7	τ1	τ1	NOUN
ejpam-5721	167	8	,	,	PUNCT
ejpam-5721	167	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	167	10	,	,	PUNCT
ejpam-5721	167	11	by	by	ADP
ejpam-5721	167	12	theorem	theorem	NOUN
ejpam-5721	167	13	1	1	NUM
ejpam-5721	167	14	there	there	ADV
ejpam-5721	167	15	exists	exist	VERB
ejpam-5721	167	16	a	a	DET
ejpam-5721	167	17	(	(	PUNCT
ejpam-5721	167	18	τ1	τ1	NOUN
ejpam-5721	167	19	,	,	PUNCT
ejpam-5721	167	20	τ2)p	τ2)p	ADJ
ejpam-5721	167	21	-	-	PUNCT
ejpam-5721	167	22	open	open	ADJ
ejpam-5721	167	23	set	set	NOUN
ejpam-5721	167	24	u	u	NOUN
ejpam-5721	167	25	of	of	ADP
ejpam-5721	167	26	x	x	PUNCT
ejpam-5721	167	27	containing	contain	VERB
ejpam-5721	167	28	x	x	PUNCT
ejpam-5721	167	29	such	such	ADJ
ejpam-5721	167	30	that	that	DET
ejpam-5721	167	31	f(u	f(u	PROPN
ejpam-5721	167	32	)	)	PUNCT
ejpam-5721	167	33	⊆	⊆	NUM
ejpam-5721	167	34	σ1σ2	σ1σ2	NOUN
ejpam-5721	167	35	-	-	NUM
ejpam-5721	167	36	cl(v	cl(v	NOUN
ejpam-5721	167	37	)	)	PUNCT
ejpam-5721	167	38	.	.	PUNCT
ejpam-5721	168	1	since	since	SCONJ
ejpam-5721	168	2	x	x	PROPN
ejpam-5721	168	3	∈	∈	PROPN
ejpam-5721	168	4	(	(	PUNCT
ejpam-5721	168	5	τ1	τ1	NOUN
ejpam-5721	168	6	,	,	PUNCT
ejpam-5721	168	7	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	168	8	)	)	PUNCT
ejpam-5721	168	9	,	,	PUNCT
ejpam-5721	168	10	we	we	PRON
ejpam-5721	168	11	have	have	VERB
ejpam-5721	168	12	u	u	NOUN
ejpam-5721	168	13	∩	∩	NOUN
ejpam-5721	168	14	a	a	DET
ejpam-5721	168	15	̸=	̸=	PROPN
ejpam-5721	168	16	∅	∅	NOUN
ejpam-5721	168	17	and	and	CCONJ
ejpam-5721	168	18	hence	hence	ADV
ejpam-5721	168	19	∅	∅	VERB
ejpam-5721	168	20	̸=	̸=	PROPN
ejpam-5721	168	21	f(u	f(u	PROPN
ejpam-5721	168	22	∩	∩	NOUN
ejpam-5721	168	23	a	a	X
ejpam-5721	168	24	)	)	PUNCT
ejpam-5721	168	25	⊆	⊆	NUM
ejpam-5721	168	26	σ1σ2	σ1σ2	NOUN
ejpam-5721	168	27	-	-	PUNCT
ejpam-5721	168	28	cl(v	cl(v	NOUN
ejpam-5721	168	29	)	)	PUNCT
ejpam-5721	168	30	∩	∩	ADJ
ejpam-5721	168	31	f(a	f(a	NOUN
ejpam-5721	168	32	)	)	PUNCT
ejpam-5721	168	33	.	.	PUNCT
ejpam-5721	169	1	therefore	therefore	ADV
ejpam-5721	169	2	,	,	PUNCT
ejpam-5721	169	3	f(x	f(x	PROPN
ejpam-5721	169	4	)	)	PUNCT
ejpam-5721	169	5	∈	∈	PROPN
ejpam-5721	169	6	(	(	PUNCT
ejpam-5721	169	7	σ1	σ1	PROPN
ejpam-5721	169	8	,	,	PUNCT
ejpam-5721	169	9	σ2)θ	σ2)θ	NOUN
ejpam-5721	169	10	-	-	PUNCT
ejpam-5721	169	11	cl(f(a	cl(f(a	NOUN
ejpam-5721	169	12	)	)	PUNCT
ejpam-5721	169	13	)	)	PUNCT
ejpam-5721	169	14	.	.	PUNCT
ejpam-5721	170	1	this	this	PRON
ejpam-5721	170	2	shows	show	VERB
ejpam-5721	170	3	that	that	SCONJ
ejpam-5721	170	4	f((τ1	f((τ1	PROPN
ejpam-5721	170	5	,	,	PUNCT
ejpam-5721	170	6	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	170	7	)	)	PUNCT
ejpam-5721	170	8	)	)	PUNCT
ejpam-5721	171	1	⊆	⊆	NUM
ejpam-5721	171	2	(	(	PUNCT
ejpam-5721	171	3	σ1	σ1	PROPN
ejpam-5721	171	4	,	,	PUNCT
ejpam-5721	171	5	σ2)θ	σ2)θ	NOUN
ejpam-5721	171	6	-	-	PUNCT
ejpam-5721	171	7	cl(f(a	cl(f(a	NOUN
ejpam-5721	171	8	)	)	PUNCT
ejpam-5721	171	9	)	)	PUNCT
ejpam-5721	171	10	.	.	PUNCT
ejpam-5721	172	1	(	(	PUNCT
ejpam-5721	172	2	2	2	X
ejpam-5721	172	3	)	)	PUNCT
ejpam-5721	172	4	⇒	⇒	NOUN
ejpam-5721	172	5	(	(	PUNCT
ejpam-5721	172	6	3	3	NUM
ejpam-5721	172	7	):	):	PUNCT
ejpam-5721	172	8	let	let	VERB
ejpam-5721	172	9	b	b	X
ejpam-5721	172	10	be	be	AUX
ejpam-5721	172	11	any	any	DET
ejpam-5721	172	12	subset	subset	NOUN
ejpam-5721	172	13	of	of	ADP
ejpam-5721	172	14	y	y	PROPN
ejpam-5721	172	15	.	.	PUNCT
ejpam-5721	173	1	then	then	ADV
ejpam-5721	173	2	by	by	ADP
ejpam-5721	173	3	(	(	PUNCT
ejpam-5721	173	4	2	2	NUM
ejpam-5721	173	5	)	)	PUNCT
ejpam-5721	173	6	,	,	PUNCT
ejpam-5721	173	7	we	we	PRON
ejpam-5721	173	8	have	have	AUX
ejpam-5721	173	9	f((τ1	f((τ1	VERB
ejpam-5721	173	10	,	,	PUNCT
ejpam-5721	173	11	τ2)-pcl(f	τ2)-pcl(f	NOUN
ejpam-5721	173	12	−1(b	−1(b	NOUN
ejpam-5721	173	13	)	)	PUNCT
ejpam-5721	173	14	)	)	PUNCT
ejpam-5721	173	15	)	)	PUNCT
ejpam-5721	174	1	⊆	⊆	X
ejpam-5721	174	2	(	(	PUNCT
ejpam-5721	174	3	σ1	σ1	PROPN
ejpam-5721	174	4	,	,	PUNCT
ejpam-5721	174	5	σ2)θ	σ2)θ	ADJ
ejpam-5721	174	6	-	-	PUNCT
ejpam-5721	174	7	cl(f(f	cl(f(f	ADJ
ejpam-5721	174	8	−1(b	−1(b	NOUN
ejpam-5721	174	9	)	)	PUNCT
ejpam-5721	174	10	)	)	PUNCT
ejpam-5721	174	11	)	)	PUNCT
ejpam-5721	175	1	⊆	⊆	X
ejpam-5721	175	2	(	(	PUNCT
ejpam-5721	175	3	σ1	σ1	PROPN
ejpam-5721	175	4	,	,	PUNCT
ejpam-5721	175	5	σ2)θ	σ2)θ	NOUN
ejpam-5721	175	6	-	-	PUNCT
ejpam-5721	175	7	cl(b	cl(b	NOUN
ejpam-5721	175	8	)	)	PUNCT
ejpam-5721	175	9	and	and	CCONJ
ejpam-5721	175	10	hence	hence	ADV
ejpam-5721	175	11	(	(	PUNCT
ejpam-5721	175	12	τ1	τ1	PROPN
ejpam-5721	175	13	,	,	PUNCT
ejpam-5721	175	14	τ2)-pcl(f	τ2)-pcl(f	NOUN
ejpam-5721	175	15	−1(b	−1(b	NOUN
ejpam-5721	175	16	)	)	PUNCT
ejpam-5721	175	17	)	)	PUNCT
ejpam-5721	175	18	⊆	⊆	NUM
ejpam-5721	175	19	f−1((σ1	f−1((σ1	NOUN
ejpam-5721	175	20	,	,	PUNCT
ejpam-5721	175	21	σ2)θ	σ2)θ	NOUN
ejpam-5721	175	22	-	-	PUNCT
ejpam-5721	175	23	cl(b	cl(b	NOUN
ejpam-5721	175	24	)	)	PUNCT
ejpam-5721	175	25	)	)	PUNCT
ejpam-5721	175	26	.	.	PUNCT
ejpam-5721	176	1	(	(	PUNCT
ejpam-5721	176	2	3	3	X
ejpam-5721	176	3	)	)	PUNCT
ejpam-5721	176	4	⇒	⇒	NOUN
ejpam-5721	176	5	(	(	PUNCT
ejpam-5721	176	6	4	4	NUM
ejpam-5721	176	7	):	):	PUNCT
ejpam-5721	176	8	let	let	VERB
ejpam-5721	176	9	b	b	X
ejpam-5721	176	10	be	be	AUX
ejpam-5721	176	11	any	any	DET
ejpam-5721	176	12	subset	subset	NOUN
ejpam-5721	176	13	of	of	ADP
ejpam-5721	176	14	y	y	PROPN
ejpam-5721	176	15	.	.	PUNCT
ejpam-5721	177	1	since	since	SCONJ
ejpam-5721	177	2	(	(	PUNCT
ejpam-5721	177	3	σ1	σ1	PROPN
ejpam-5721	177	4	,	,	PUNCT
ejpam-5721	177	5	σ2)θ	σ2)θ	NOUN
ejpam-5721	177	6	-	-	PUNCT
ejpam-5721	177	7	cl(b	cl(b	NOUN
ejpam-5721	177	8	)	)	PUNCT
ejpam-5721	177	9	is	be	AUX
ejpam-5721	177	10	σ1σ2	σ1σ2	NOUN
ejpam-5721	177	11	-	-	ADJ
ejpam-5721	177	12	closed	closed	ADJ
ejpam-5721	177	13	in	in	ADP
ejpam-5721	177	14	y	y	PROPN
ejpam-5721	177	15	and	and	CCONJ
ejpam-5721	177	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	177	17	-	-	NUM
ejpam-5721	177	18	cl(v	cl(v	X
ejpam-5721	177	19	)	)	PUNCT
ejpam-5721	178	1	=	=	SYM
ejpam-5721	178	2	(	(	PUNCT
ejpam-5721	178	3	σ1	σ1	PROPN
ejpam-5721	178	4	,	,	PUNCT
ejpam-5721	178	5	σ2)θ	σ2)θ	NOUN
ejpam-5721	178	6	-	-	PUNCT
ejpam-5721	178	7	cl(v	cl(v	NOUN
ejpam-5721	178	8	)	)	PUNCT
ejpam-5721	178	9	for	for	ADP
ejpam-5721	178	10	every	every	DET
ejpam-5721	178	11	σ1σ2	σ1σ2	NOUN
ejpam-5721	178	12	-	-	ADJ
ejpam-5721	178	13	open	open	ADJ
ejpam-5721	178	14	set	set	NOUN
ejpam-5721	178	15	v	v	NOUN
ejpam-5721	178	16	of	of	ADP
ejpam-5721	178	17	y	y	PROPN
ejpam-5721	178	18	,	,	PUNCT
ejpam-5721	178	19	by	by	ADP
ejpam-5721	178	20	(	(	PUNCT
ejpam-5721	178	21	3	3	NUM
ejpam-5721	178	22	)	)	PUNCT
ejpam-5721	178	23	(	(	PUNCT
ejpam-5721	178	24	τ1	τ1	NOUN
ejpam-5721	178	25	,	,	PUNCT
ejpam-5721	178	26	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	178	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	178	28	-	-	PUNCT
ejpam-5721	178	29	int((σ1	int((σ1	NUM
ejpam-5721	178	30	,	,	PUNCT
ejpam-5721	178	31	σ2)θ	σ2)θ	ADJ
ejpam-5721	178	32	-	-	PUNCT
ejpam-5721	178	33	cl(b	cl(b	NOUN
ejpam-5721	178	34	)	)	PUNCT
ejpam-5721	178	35	)	)	PUNCT
ejpam-5721	178	36	)	)	PUNCT
ejpam-5721	178	37	)	)	PUNCT
ejpam-5721	179	1	⊆	⊆	NUM
ejpam-5721	179	2	f−1((σ1	f−1((σ1	NOUN
ejpam-5721	179	3	,	,	PUNCT
ejpam-5721	179	4	σ2)θ	σ2)θ	NOUN
ejpam-5721	179	5	-	-	PUNCT
ejpam-5721	179	6	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	179	7	-	-	PUNCT
ejpam-5721	179	8	int((σ1	int((σ1	PROPN
ejpam-5721	179	9	,	,	PUNCT
ejpam-5721	179	10	σ2)θ	σ2)θ	ADJ
ejpam-5721	179	11	-	-	PUNCT
ejpam-5721	179	12	cl(b	cl(b	NOUN
ejpam-5721	179	13	)	)	PUNCT
ejpam-5721	179	14	)	)	PUNCT
ejpam-5721	179	15	)	)	PUNCT
ejpam-5721	179	16	)	)	PUNCT
ejpam-5721	180	1	=	=	PRON
ejpam-5721	180	2	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-5721	180	3	-	-	PUNCT
ejpam-5721	180	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5721	180	5	-	-	PUNCT
ejpam-5721	180	6	int((σ1	int((σ1	PROPN
ejpam-5721	180	7	,	,	PUNCT
ejpam-5721	180	8	σ2)θ	σ2)θ	ADJ
ejpam-5721	180	9	-	-	PUNCT
ejpam-5721	180	10	cl(b	cl(b	NOUN
ejpam-5721	180	11	)	)	PUNCT
ejpam-5721	180	12	)	)	PUNCT
ejpam-5721	180	13	)	)	PUNCT
ejpam-5721	180	14	)	)	PUNCT
ejpam-5721	181	1	⊆	⊆	NUM
ejpam-5721	181	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	181	3	-	-	PUNCT
ejpam-5721	181	4	cl((σ1	cl((σ1	NOUN
ejpam-5721	181	5	,	,	PUNCT
ejpam-5721	181	6	σ2)θ	σ2)θ	ADJ
ejpam-5721	181	7	-	-	PUNCT
ejpam-5721	181	8	cl(b	cl(b	NOUN
ejpam-5721	181	9	)	)	PUNCT
ejpam-5721	181	10	)	)	PUNCT
ejpam-5721	181	11	)	)	PUNCT
ejpam-5721	182	1	=	=	SYM
ejpam-5721	182	2	f−1(σ1	f−1(σ1	NOUN
ejpam-5721	182	3	,	,	PUNCT
ejpam-5721	182	4	σ2)θ	σ2)θ	ADJ
ejpam-5721	182	5	-	-	PUNCT
ejpam-5721	182	6	cl(b	cl(b	NOUN
ejpam-5721	182	7	)	)	PUNCT
ejpam-5721	182	8	)	)	PUNCT
ejpam-5721	182	9	.	.	PUNCT
ejpam-5721	183	1	(	(	PUNCT
ejpam-5721	183	2	4	4	X
ejpam-5721	183	3	)	)	PUNCT
ejpam-5721	183	4	⇒	⇒	NOUN
ejpam-5721	183	5	(	(	PUNCT
ejpam-5721	183	6	5	5	NUM
ejpam-5721	183	7	):	):	PUNCT
ejpam-5721	183	8	this	this	PRON
ejpam-5721	183	9	is	be	AUX
ejpam-5721	183	10	obvious	obvious	ADJ
ejpam-5721	183	11	since	since	SCONJ
ejpam-5721	183	12	(	(	PUNCT
ejpam-5721	183	13	σ1	σ1	PROPN
ejpam-5721	183	14	,	,	PUNCT
ejpam-5721	183	15	σ2)θ	σ2)θ	NOUN
ejpam-5721	183	16	-	-	PUNCT
ejpam-5721	183	17	cl(v	cl(v	NOUN
ejpam-5721	183	18	)	)	PUNCT
ejpam-5721	183	19	=	=	SYM
ejpam-5721	183	20	σ1σ2	σ1σ2	NOUN
ejpam-5721	183	21	-	-	NUM
ejpam-5721	183	22	cl(v	cl(v	NOUN
ejpam-5721	183	23	)	)	PUNCT
ejpam-5721	183	24	for	for	ADP
ejpam-5721	183	25	every	every	DET
ejpam-5721	183	26	σ1σ2	σ1σ2	NOUN
ejpam-5721	183	27	-	-	ADJ
ejpam-5721	183	28	open	open	ADJ
ejpam-5721	183	29	set	set	NOUN
ejpam-5721	183	30	v	v	NOUN
ejpam-5721	183	31	of	of	ADP
ejpam-5721	183	32	y	y	PROPN
ejpam-5721	183	33	.	.	PUNCT
ejpam-5721	184	1	(	(	PUNCT
ejpam-5721	184	2	5	5	X
ejpam-5721	184	3	)	)	PUNCT
ejpam-5721	184	4	⇒	⇒	NOUN
ejpam-5721	184	5	(	(	PUNCT
ejpam-5721	184	6	6	6	NUM
ejpam-5721	184	7	):	):	PUNCT
ejpam-5721	184	8	this	this	PRON
ejpam-5721	184	9	follows	follow	VERB
ejpam-5721	184	10	from	from	ADP
ejpam-5721	184	11	σ1σ2	σ1σ2	SYM
ejpam-5721	184	12	-	-	PUNCT
ejpam-5721	184	13	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5721	184	14	-	-	PUNCT
ejpam-5721	184	15	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	184	16	-	-	PUNCT
ejpam-5721	184	17	cl(v	cl(v	NOUN
ejpam-5721	184	18	)	)	PUNCT
ejpam-5721	184	19	)	)	PUNCT
ejpam-5721	184	20	)	)	PUNCT
ejpam-5721	185	1	=	=	PUNCT
ejpam-5721	185	2	σ1σ2	σ1σ2	NOUN
ejpam-5721	185	3	-	-	NUM
ejpam-5721	185	4	cl(v	cl(v	NOUN
ejpam-5721	185	5	)	)	PUNCT
ejpam-5721	185	6	for	for	ADP
ejpam-5721	185	7	every	every	DET
ejpam-5721	185	8	(	(	PUNCT
ejpam-5721	185	9	σ1	σ1	PROPN
ejpam-5721	185	10	,	,	PUNCT
ejpam-5721	185	11	σ2)p	σ2)p	NOUN
ejpam-5721	185	12	-	-	PUNCT
ejpam-5721	185	13	open	open	NOUN
ejpam-5721	185	14	set	set	NOUN
ejpam-5721	185	15	v	v	NOUN
ejpam-5721	185	16	of	of	ADP
ejpam-5721	185	17	y	y	PROPN
ejpam-5721	185	18	.	.	PUNCT
ejpam-5721	186	1	j.	j.	PROPN
ejpam-5721	186	2	khampakdee	khampakdee	PROPN
ejpam-5721	186	3	,	,	PUNCT
ejpam-5721	186	4	s.	s.	PROPN
ejpam-5721	186	5	sompong	sompong	PROPN
ejpam-5721	186	6	,	,	PUNCT
ejpam-5721	186	7	c.	c.	PROPN
ejpam-5721	186	8	boonpok	boonpok	PROPN
ejpam-5721	186	9	/	/	SYM
ejpam-5721	186	10	eur	eur	PROPN
ejpam-5721	186	11	.	.	PUNCT
ejpam-5721	187	1	j.	j.	PROPN
ejpam-5721	187	2	pure	pure	PROPN
ejpam-5721	187	3	appl	appl	PROPN
ejpam-5721	187	4	.	.	PROPN
ejpam-5721	187	5	math	math	PROPN
ejpam-5721	187	6	,	,	PUNCT
ejpam-5721	187	7	18	18	NUM
ejpam-5721	187	8	(	(	PUNCT
ejpam-5721	187	9	1	1	NUM
ejpam-5721	187	10	)	)	PUNCT
ejpam-5721	187	11	(	(	PUNCT
ejpam-5721	187	12	2025	2025	NUM
ejpam-5721	187	13	)	)	PUNCT
ejpam-5721	187	14	,	,	PUNCT
ejpam-5721	187	15	5721	5721	NUM
ejpam-5721	187	16	7	7	NUM
ejpam-5721	187	17	of	of	ADP
ejpam-5721	187	18	12	12	NUM
ejpam-5721	187	19	(	(	PUNCT
ejpam-5721	187	20	6	6	NUM
ejpam-5721	187	21	)	)	PUNCT
ejpam-5721	187	22	⇒	⇒	NOUN
ejpam-5721	187	23	(	(	PUNCT
ejpam-5721	187	24	7	7	NUM
ejpam-5721	187	25	):	):	PUNCT
ejpam-5721	187	26	let	let	VERB
ejpam-5721	187	27	k	k	PRON
ejpam-5721	187	28	be	be	AUX
ejpam-5721	187	29	any	any	DET
ejpam-5721	187	30	(	(	PUNCT
ejpam-5721	187	31	σ1	σ1	NOUN
ejpam-5721	187	32	,	,	PUNCT
ejpam-5721	187	33	σ2)r	σ2)r	NOUN
ejpam-5721	187	34	-	-	PUNCT
ejpam-5721	187	35	closed	close	VERB
ejpam-5721	187	36	set	set	NOUN
ejpam-5721	187	37	of	of	ADP
ejpam-5721	187	38	y	y	PROPN
ejpam-5721	187	39	.	.	PUNCT
ejpam-5721	188	1	then	then	ADV
ejpam-5721	188	2	,	,	PUNCT
ejpam-5721	188	3	we	we	PRON
ejpam-5721	188	4	have	have	VERB
ejpam-5721	188	5	σ1σ2	σ1σ2	NOUN
ejpam-5721	188	6	-	-	PUNCT
ejpam-5721	188	7	int(k	int(k	NOUN
ejpam-5721	188	8	)	)	PUNCT
ejpam-5721	188	9	is	be	AUX
ejpam-5721	188	10	(	(	PUNCT
ejpam-5721	188	11	σ1	σ1	PROPN
ejpam-5721	188	12	,	,	PUNCT
ejpam-5721	188	13	σ2)p	σ2)p	NOUN
ejpam-5721	188	14	-	-	PUNCT
ejpam-5721	188	15	open	open	ADJ
ejpam-5721	188	16	in	in	ADP
ejpam-5721	188	17	y	y	PROPN
ejpam-5721	188	18	.	.	PUNCT
ejpam-5721	189	1	thus	thus	ADV
ejpam-5721	189	2	by	by	ADP
ejpam-5721	189	3	(	(	PUNCT
ejpam-5721	189	4	6	6	NUM
ejpam-5721	189	5	)	)	PUNCT
ejpam-5721	189	6	,	,	PUNCT
ejpam-5721	189	7	(	(	PUNCT
ejpam-5721	189	8	τ1	τ1	NOUN
ejpam-5721	189	9	,	,	PUNCT
ejpam-5721	189	10	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	189	11	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	189	12	-	-	PUNCT
ejpam-5721	189	13	int(k	int(k	NUM
ejpam-5721	189	14	)	)	PUNCT
ejpam-5721	189	15	)	)	PUNCT
ejpam-5721	189	16	)	)	PUNCT
ejpam-5721	190	1	=	=	PRON
ejpam-5721	190	2	(	(	PUNCT
ejpam-5721	190	3	τ1	τ1	PROPN
ejpam-5721	190	4	,	,	PUNCT
ejpam-5721	190	5	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	190	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	190	7	-	-	PUNCT
ejpam-5721	190	8	int(σ1σ2	int(σ1σ2	ADV
ejpam-5721	190	9	-	-	PUNCT
ejpam-5721	190	10	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5721	190	11	-	-	PUNCT
ejpam-5721	190	12	int(k	int(k	NOUN
ejpam-5721	190	13	)	)	PUNCT
ejpam-5721	190	14	)	)	PUNCT
ejpam-5721	190	15	)	)	PUNCT
ejpam-5721	190	16	)	)	PUNCT
ejpam-5721	190	17	)	)	PUNCT
ejpam-5721	190	18	⊆	⊆	NUM
ejpam-5721	190	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	190	20	-	-	PUNCT
ejpam-5721	190	21	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5721	190	22	-	-	PUNCT
ejpam-5721	190	23	int(k	int(k	NOUN
ejpam-5721	190	24	)	)	PUNCT
ejpam-5721	190	25	)	)	PUNCT
ejpam-5721	190	26	)	)	PUNCT
ejpam-5721	191	1	=	=	PUNCT
ejpam-5721	191	2	f−1(k	f−1(k	PROPN
ejpam-5721	191	3	)	)	PUNCT
ejpam-5721	191	4	.	.	PUNCT
ejpam-5721	192	1	(	(	PUNCT
ejpam-5721	192	2	7	7	X
ejpam-5721	192	3	)	)	PUNCT
ejpam-5721	192	4	⇒	⇒	NOUN
ejpam-5721	192	5	(	(	PUNCT
ejpam-5721	192	6	1	1	NUM
ejpam-5721	192	7	):	):	PUNCT
ejpam-5721	192	8	let	let	VERB
ejpam-5721	192	9	v	v	PART
ejpam-5721	192	10	be	be	AUX
ejpam-5721	192	11	any	any	DET
ejpam-5721	192	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	192	13	-	-	ADJ
ejpam-5721	192	14	open	open	ADJ
ejpam-5721	192	15	set	set	NOUN
ejpam-5721	192	16	of	of	ADP
ejpam-5721	192	17	y	y	PROPN
ejpam-5721	192	18	.	.	PUNCT
ejpam-5721	193	1	then	then	ADV
ejpam-5721	193	2	,	,	PUNCT
ejpam-5721	193	3	σ1σ2	σ1σ2	NOUN
ejpam-5721	193	4	-	-	NUM
ejpam-5721	193	5	cl(v	cl(v	NOUN
ejpam-5721	193	6	)	)	PUNCT
ejpam-5721	193	7	is	be	AUX
ejpam-5721	193	8	(	(	PUNCT
ejpam-5721	193	9	σ1	σ1	NOUN
ejpam-5721	193	10	,	,	PUNCT
ejpam-5721	193	11	σ2)r	σ2)r	NOUN
ejpam-5721	193	12	-	-	PUNCT
ejpam-5721	193	13	closed	closed	ADJ
ejpam-5721	193	14	in	in	ADP
ejpam-5721	193	15	y	y	PROPN
ejpam-5721	193	16	and	and	CCONJ
ejpam-5721	193	17	by	by	ADP
ejpam-5721	193	18	(	(	PUNCT
ejpam-5721	193	19	7	7	NUM
ejpam-5721	193	20	)	)	PUNCT
ejpam-5721	193	21	,	,	PUNCT
ejpam-5721	193	22	we	we	PRON
ejpam-5721	193	23	have	have	VERB
ejpam-5721	193	24	(	(	PUNCT
ejpam-5721	193	25	τ1	τ1	NOUN
ejpam-5721	193	26	,	,	PUNCT
ejpam-5721	193	27	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-5721	193	28	−1(v	−1(v	PROPN
ejpam-5721	193	29	)	)	PUNCT
ejpam-5721	193	30	)	)	PUNCT
ejpam-5721	194	1	⊆	⊆	NUM
ejpam-5721	194	2	(	(	PUNCT
ejpam-5721	194	3	τ1	τ1	NOUN
ejpam-5721	194	4	,	,	PUNCT
ejpam-5721	194	5	τ2)-pcl(f	τ2)-pcl(f	ADJ
ejpam-5721	194	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5721	194	7	-	-	PUNCT
ejpam-5721	194	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	194	9	-	-	PUNCT
ejpam-5721	194	10	cl(v	cl(v	NOUN
ejpam-5721	194	11	)	)	PUNCT
ejpam-5721	194	12	)	)	PUNCT
ejpam-5721	194	13	)	)	PUNCT
ejpam-5721	194	14	)	)	PUNCT
ejpam-5721	195	1	⊆	⊆	NUM
ejpam-5721	195	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5721	195	3	-	-	PUNCT
ejpam-5721	195	4	cl(v	cl(v	NOUN
ejpam-5721	195	5	)	)	PUNCT
ejpam-5721	195	6	)	)	PUNCT
ejpam-5721	195	7	.	.	PUNCT
ejpam-5721	196	1	it	it	PRON
ejpam-5721	196	2	follows	follow	VERB
ejpam-5721	196	3	from	from	ADP
ejpam-5721	196	4	theorem	theorem	ADJ
ejpam-5721	196	5	1	1	NUM
ejpam-5721	196	6	that	that	SCONJ
ejpam-5721	196	7	f	f	PROPN
ejpam-5721	196	8	is	be	AUX
ejpam-5721	196	9	almost	almost	ADV
ejpam-5721	196	10	weakly	weakly	ADJ
ejpam-5721	196	11	(	(	PUNCT
ejpam-5721	196	12	τ1	τ1	NOUN
ejpam-5721	196	13	,	,	PUNCT
ejpam-5721	196	14	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5721	196	15	.	.	PUNCT
ejpam-5721	197	1	lemma	lemma	PROPN
ejpam-5721	197	2	5	5	X
ejpam-5721	197	3	.	.	PUNCT
ejpam-5721	198	1	let	let	AUX
ejpam-5721	198	2	(	(	PUNCT
ejpam-5721	198	3	x	x	NOUN
ejpam-5721	198	4	,	,	PUNCT
ejpam-5721	198	5	τ1	τ1	NOUN
ejpam-5721	198	6	,	,	PUNCT
ejpam-5721	198	7	τ2	τ2	PROPN
ejpam-5721	198	8	)	)	PUNCT
ejpam-5721	198	9	be	be	VERB
ejpam-5721	198	10	a	a	DET
ejpam-5721	198	11	bitopological	bitopological	ADJ
ejpam-5721	198	12	space	space	NOUN
ejpam-5721	198	13	.	.	PUNCT
ejpam-5721	199	1	if	if	SCONJ
ejpam-5721	199	2	a	a	PRON
ejpam-5721	199	3	is	be	AUX
ejpam-5721	199	4	α(τ1	α(τ1	NOUN
ejpam-5721	199	5	,	,	PUNCT
ejpam-5721	199	6	τ2)-open	τ2)-open	ADJ
ejpam-5721	199	7	in	in	ADP
ejpam-5721	199	8	x	x	X
ejpam-5721	199	9	and	and	CCONJ
ejpam-5721	199	10	b	b	PROPN
ejpam-5721	199	11	is	be	AUX
ejpam-5721	199	12	(	(	PUNCT
ejpam-5721	199	13	τ1	τ1	NOUN
ejpam-5721	199	14	,	,	PUNCT
ejpam-5721	199	15	τ2)p	τ2)p	NOUN
ejpam-5721	199	16	-	-	PUNCT
ejpam-5721	199	17	open	open	ADJ
ejpam-5721	199	18	in	in	ADP
ejpam-5721	199	19	x	x	NOUN
ejpam-5721	199	20	,	,	PUNCT
ejpam-5721	199	21	then	then	ADV
ejpam-5721	199	22	a	a	DET
ejpam-5721	199	23	∩b	∩b	NOUN
ejpam-5721	199	24	is	be	AUX
ejpam-5721	199	25	(	(	PUNCT
ejpam-5721	199	26	τ1	τ1	NOUN
ejpam-5721	199	27	,	,	PUNCT
ejpam-5721	199	28	τ2)p	τ2)p	NOUN
ejpam-5721	199	29	-	-	PUNCT
ejpam-5721	199	30	open	open	ADJ
ejpam-5721	199	31	in	in	ADP
ejpam-5721	199	32	x.	x.	NOUN
ejpam-5721	199	33	proof	proof	NOUN
ejpam-5721	199	34	.	.	PUNCT
ejpam-5721	200	1	let	let	VERB
ejpam-5721	200	2	a	a	DET
ejpam-5721	200	3	be	be	AUX
ejpam-5721	200	4	α(τ1	α(τ1	NOUN
ejpam-5721	200	5	,	,	PUNCT
ejpam-5721	200	6	τ2)-open	τ2)-open	ADJ
ejpam-5721	200	7	in	in	ADP
ejpam-5721	200	8	x	x	X
ejpam-5721	200	9	and	and	CCONJ
ejpam-5721	200	10	b	b	NOUN
ejpam-5721	200	11	be	be	AUX
ejpam-5721	200	12	(	(	PUNCT
ejpam-5721	200	13	τ1	τ1	NOUN
ejpam-5721	200	14	,	,	PUNCT
ejpam-5721	200	15	τ2)p	τ2)p	NOUN
ejpam-5721	200	16	-	-	PUNCT
ejpam-5721	200	17	open	open	ADJ
ejpam-5721	200	18	in	in	ADP
ejpam-5721	200	19	x.	x.	NOUN
ejpam-5721	200	20	then	then	ADV
ejpam-5721	200	21	,	,	PUNCT
ejpam-5721	200	22	we	we	PRON
ejpam-5721	200	23	have	have	VERB
ejpam-5721	200	24	a	a	DET
ejpam-5721	200	25	⊆	⊆	NUM
ejpam-5721	200	26	τ1τ2	τ1τ2	NOUN
ejpam-5721	200	27	-	-	PUNCT
ejpam-5721	200	28	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	200	29	-	-	PUNCT
ejpam-5721	200	30	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	200	31	-	-	PUNCT
ejpam-5721	200	32	int(a	int(a	NOUN
ejpam-5721	200	33	)	)	PUNCT
ejpam-5721	200	34	)	)	PUNCT
ejpam-5721	200	35	)	)	PUNCT
ejpam-5721	201	1	and	and	CCONJ
ejpam-5721	201	2	b	b	X
ejpam-5721	201	3	⊆	⊆	NUM
ejpam-5721	201	4	τ1τ2	τ1τ2	NOUN
ejpam-5721	201	5	-	-	NOUN
ejpam-5721	201	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	201	7	-	-	PUNCT
ejpam-5721	201	8	cl(b	cl(b	NOUN
ejpam-5721	201	9	)	)	PUNCT
ejpam-5721	201	10	)	)	PUNCT
ejpam-5721	201	11	.	.	PUNCT
ejpam-5721	202	1	thus	thus	ADV
ejpam-5721	202	2	by	by	ADP
ejpam-5721	202	3	lemma	lemma	PROPN
ejpam-5721	202	4	3(1	3(1	NUM
ejpam-5721	202	5	)	)	PUNCT
ejpam-5721	202	6	,	,	PUNCT
ejpam-5721	202	7	a	a	DET
ejpam-5721	202	8	∩b	∩b	NOUN
ejpam-5721	202	9	⊆	⊆	NUM
ejpam-5721	202	10	τ1τ2	τ1τ2	NOUN
ejpam-5721	202	11	-	-	PUNCT
ejpam-5721	202	12	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	202	13	-	-	PUNCT
ejpam-5721	202	14	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	202	15	-	-	PUNCT
ejpam-5721	202	16	int(a	int(a	NOUN
ejpam-5721	202	17	)	)	PUNCT
ejpam-5721	202	18	)	)	PUNCT
ejpam-5721	202	19	)	)	PUNCT
ejpam-5721	203	1	∩	∩	NOUN
ejpam-5721	203	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	203	3	-	-	NOUN
ejpam-5721	203	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	203	5	-	-	PUNCT
ejpam-5721	203	6	cl(b	cl(b	NOUN
ejpam-5721	203	7	)	)	PUNCT
ejpam-5721	203	8	)	)	PUNCT
ejpam-5721	204	1	⊆	⊆	X
ejpam-5721	204	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	204	3	-	-	PUNCT
ejpam-5721	204	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	204	5	-	-	PUNCT
ejpam-5721	204	6	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	204	7	-	-	PUNCT
ejpam-5721	204	8	int(a	int(a	NOUN
ejpam-5721	204	9	)	)	PUNCT
ejpam-5721	204	10	)	)	PUNCT
ejpam-5721	205	1	∩	∩	NOUN
ejpam-5721	205	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	205	3	-	-	NOUN
ejpam-5721	205	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	205	5	-	-	PUNCT
ejpam-5721	205	6	cl(b	cl(b	NOUN
ejpam-5721	205	7	)	)	PUNCT
ejpam-5721	205	8	)	)	PUNCT
ejpam-5721	205	9	)	)	PUNCT
ejpam-5721	206	1	⊆	⊆	X
ejpam-5721	206	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	206	3	-	-	PUNCT
ejpam-5721	206	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	206	5	-	-	PUNCT
ejpam-5721	206	6	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5721	206	7	-	-	PUNCT
ejpam-5721	206	8	int(a	int(a	NOUN
ejpam-5721	206	9	)	)	PUNCT
ejpam-5721	206	10	∩	∩	ADJ
ejpam-5721	206	11	τ1τ2	τ1τ2	NOUN
ejpam-5721	206	12	-	-	NOUN
ejpam-5721	206	13	cl(b	cl(b	NOUN
ejpam-5721	206	14	)	)	PUNCT
ejpam-5721	206	15	)	)	PUNCT
ejpam-5721	206	16	)	)	PUNCT
ejpam-5721	207	1	⊆	⊆	X
ejpam-5721	207	2	τ1τ2	τ1τ2	NOUN
ejpam-5721	207	3	-	-	PUNCT
ejpam-5721	207	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5721	207	5	-	-	PUNCT
ejpam-5721	207	6	cl(a	cl(a	VERB
ejpam-5721	207	7	∩b	∩b	NOUN
ejpam-5721	207	8	)	)	PUNCT
ejpam-5721	207	9	)	)	PUNCT
ejpam-5721	207	10	.	.	PUNCT
ejpam-5721	208	1	this	this	PRON
ejpam-5721	208	2	shows	show	VERB
ejpam-5721	208	3	that	that	SCONJ
ejpam-5721	208	4	a	a	DET
ejpam-5721	208	5	∩b	∩b	NOUN
ejpam-5721	208	6	is	be	AUX
ejpam-5721	208	7	(	(	PUNCT
ejpam-5721	208	8	τ1	τ1	NOUN
ejpam-5721	208	9	,	,	PUNCT
ejpam-5721	208	10	τ2)p	τ2)p	NOUN
ejpam-5721	208	11	-	-	PUNCT
ejpam-5721	208	12	open	open	ADJ
ejpam-5721	208	13	in	in	ADP
ejpam-5721	208	14	x.	x.	NOUN
ejpam-5721	208	15	definition	definition	NOUN
ejpam-5721	208	16	2	2	NUM
ejpam-5721	208	17	.	.	PUNCT
ejpam-5721	209	1	[	[	X
ejpam-5721	209	2	17	17	NUM
ejpam-5721	209	3	]	]	PUNCT
ejpam-5721	209	4	a	a	DET
ejpam-5721	209	5	bitopological	bitopological	ADJ
ejpam-5721	209	6	space	space	NOUN
ejpam-5721	209	7	(	(	PUNCT
ejpam-5721	209	8	x	x	NOUN
ejpam-5721	209	9	,	,	PUNCT
ejpam-5721	209	10	τ1	τ1	NOUN
ejpam-5721	209	11	,	,	PUNCT
ejpam-5721	209	12	τ2	τ2	NOUN
ejpam-5721	209	13	)	)	PUNCT
ejpam-5721	209	14	is	be	AUX
ejpam-5721	209	15	said	say	VERB
ejpam-5721	209	16	to	to	PART
ejpam-5721	209	17	be	be	AUX
ejpam-5721	209	18	(	(	PUNCT
ejpam-5721	209	19	τ1	τ1	NOUN
ejpam-5721	209	20	,	,	PUNCT
ejpam-5721	209	21	τ2)-t2	τ2)-t2	ADJ
ejpam-5721	209	22	if	if	SCONJ
ejpam-5721	209	23	for	for	ADP
ejpam-5721	209	24	any	any	DET
ejpam-5721	209	25	pair	pair	NOUN
ejpam-5721	209	26	of	of	ADP
ejpam-5721	209	27	distinct	distinct	ADJ
ejpam-5721	209	28	points	point	NOUN
ejpam-5721	209	29	x	x	X
ejpam-5721	209	30	,	,	PUNCT
ejpam-5721	209	31	y	y	PROPN
ejpam-5721	209	32	in	in	ADP
ejpam-5721	209	33	x	x	SYM
ejpam-5721	209	34	,	,	PUNCT
ejpam-5721	209	35	there	there	PRON
ejpam-5721	209	36	exist	exist	VERB
ejpam-5721	209	37	disjoint	disjoint	ADJ
ejpam-5721	209	38	τ1τ2	τ1τ2	ADJ
ejpam-5721	209	39	-	-	ADJ
ejpam-5721	209	40	open	open	ADJ
ejpam-5721	209	41	sets	set	NOUN
ejpam-5721	209	42	u	u	NOUN
ejpam-5721	209	43	and	and	CCONJ
ejpam-5721	209	44	v	v	NOUN
ejpam-5721	209	45	of	of	ADP
ejpam-5721	209	46	x	x	PUNCT
ejpam-5721	209	47	containing	contain	VERB
ejpam-5721	209	48	x	x	PROPN
ejpam-5721	209	49	and	and	CCONJ
ejpam-5721	209	50	y	y	PROPN
ejpam-5721	209	51	,	,	PUNCT
ejpam-5721	209	52	respectively	respectively	ADV
ejpam-5721	209	53	.	.	PUNCT
ejpam-5721	210	1	definition	definition	NOUN
ejpam-5721	210	2	3	3	NUM
ejpam-5721	210	3	.	.	PUNCT
ejpam-5721	211	1	a	a	DET
ejpam-5721	211	2	function	function	NOUN
ejpam-5721	211	3	f	f	NOUN
ejpam-5721	211	4	:	:	PUNCT
ejpam-5721	211	5	(	(	PUNCT
ejpam-5721	211	6	x	x	NOUN
ejpam-5721	211	7	,	,	PUNCT
ejpam-5721	211	8	τ1	τ1	NOUN
ejpam-5721	211	9	,	,	PUNCT
ejpam-5721	211	10	τ2	τ2	NOUN
ejpam-5721	211	11	)	)	PUNCT
ejpam-5721	211	12	→	→	SYM
ejpam-5721	211	13	(	(	PUNCT
ejpam-5721	211	14	y	y	PROPN
ejpam-5721	211	15	,	,	PUNCT
ejpam-5721	211	16	σ1	σ1	PROPN
ejpam-5721	211	17	,	,	PUNCT
ejpam-5721	211	18	σ2	σ2	PROPN
ejpam-5721	211	19	)	)	PUNCT
ejpam-5721	211	20	is	be	AUX
ejpam-5721	211	21	said	say	VERB
ejpam-5721	211	22	to	to	PART
ejpam-5721	211	23	be	be	AUX
ejpam-5721	211	24	almost	almost	ADV
ejpam-5721	211	25	α(τ1	α(τ1	NOUN
ejpam-5721	211	26	,	,	PUNCT
ejpam-5721	211	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	211	28	if	if	SCONJ
ejpam-5721	211	29	for	for	ADP
ejpam-5721	211	30	each	each	DET
ejpam-5721	211	31	x	x	SYM
ejpam-5721	211	32	∈	∈	PROPN
ejpam-5721	211	33	x	x	X
ejpam-5721	211	34	and	and	CCONJ
ejpam-5721	211	35	each	each	DET
ejpam-5721	211	36	σ1σ2	σ1σ2	VERB
ejpam-5721	211	37	-	-	ADJ
ejpam-5721	211	38	open	open	ADJ
ejpam-5721	211	39	set	set	NOUN
ejpam-5721	211	40	v	v	NOUN
ejpam-5721	211	41	of	of	ADP
ejpam-5721	211	42	y	y	NOUN
ejpam-5721	211	43	containing	contain	VERB
ejpam-5721	211	44	f(x	f(x	PROPN
ejpam-5721	211	45	)	)	PUNCT
ejpam-5721	211	46	,	,	PUNCT
ejpam-5721	211	47	there	there	PRON
ejpam-5721	211	48	exists	exist	VERB
ejpam-5721	211	49	an	an	DET
ejpam-5721	211	50	α(τ1	α(τ1	NOUN
ejpam-5721	211	51	,	,	PUNCT
ejpam-5721	211	52	τ2)-open	τ2)-open	ADJ
ejpam-5721	211	53	set	set	VERB
ejpam-5721	211	54	u	u	NOUN
ejpam-5721	211	55	of	of	ADP
ejpam-5721	211	56	x	x	SYM
ejpam-5721	211	57	such	such	ADJ
ejpam-5721	211	58	that	that	DET
ejpam-5721	211	59	f(u	f(u	PROPN
ejpam-5721	211	60	)	)	PUNCT
ejpam-5721	211	61	⊆	⊆	NUM
ejpam-5721	211	62	σ1σ2	σ1σ2	X
ejpam-5721	211	63	-	-	PUNCT
ejpam-5721	211	64	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	211	65	-	-	PUNCT
ejpam-5721	211	66	cl(v	cl(v	NOUN
ejpam-5721	211	67	)	)	PUNCT
ejpam-5721	211	68	)	)	PUNCT
ejpam-5721	211	69	.	.	PUNCT
ejpam-5721	212	1	theorem	theorem	ADJ
ejpam-5721	212	2	4	4	NUM
ejpam-5721	212	3	.	.	PUNCT
ejpam-5721	213	1	let	let	AUX
ejpam-5721	213	2	(	(	PUNCT
ejpam-5721	213	3	y	y	PROPN
ejpam-5721	213	4	,	,	PUNCT
ejpam-5721	213	5	σ1	σ1	PROPN
ejpam-5721	213	6	,	,	PUNCT
ejpam-5721	213	7	σ2	σ2	PROPN
ejpam-5721	213	8	)	)	PUNCT
ejpam-5721	213	9	be	be	VERB
ejpam-5721	213	10	a	a	DET
ejpam-5721	213	11	(	(	PUNCT
ejpam-5721	213	12	σ1	σ1	PROPN
ejpam-5721	213	13	,	,	PUNCT
ejpam-5721	213	14	σ2)-t2	σ2)-t2	PROPN
ejpam-5721	213	15	space	space	NOUN
ejpam-5721	213	16	.	.	PUNCT
ejpam-5721	214	1	if	if	SCONJ
ejpam-5721	214	2	f	f	PROPN
ejpam-5721	214	3	:	:	PUNCT
ejpam-5721	214	4	(	(	PUNCT
ejpam-5721	214	5	x	x	NOUN
ejpam-5721	214	6	,	,	PUNCT
ejpam-5721	214	7	τ1	τ1	NOUN
ejpam-5721	214	8	,	,	PUNCT
ejpam-5721	214	9	τ2	τ2	NOUN
ejpam-5721	214	10	)	)	PUNCT
ejpam-5721	214	11	→	→	SYM
ejpam-5721	214	12	(	(	PUNCT
ejpam-5721	214	13	y	y	PROPN
ejpam-5721	214	14	,	,	PUNCT
ejpam-5721	214	15	σ1	σ1	PROPN
ejpam-5721	214	16	,	,	PUNCT
ejpam-5721	214	17	σ2	σ2	PROPN
ejpam-5721	214	18	)	)	PUNCT
ejpam-5721	214	19	is	be	AUX
ejpam-5721	214	20	almost	almost	ADV
ejpam-5721	214	21	α(τ1	α(τ1	NOUN
ejpam-5721	214	22	,	,	PUNCT
ejpam-5721	214	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	214	24	and	and	CCONJ
ejpam-5721	214	25	g	g	NOUN
ejpam-5721	214	26	:	:	PUNCT
ejpam-5721	214	27	(	(	PUNCT
ejpam-5721	214	28	x	x	NOUN
ejpam-5721	214	29	,	,	PUNCT
ejpam-5721	214	30	τ1	τ1	NOUN
ejpam-5721	214	31	,	,	PUNCT
ejpam-5721	214	32	τ2	τ2	NOUN
ejpam-5721	214	33	)	)	PUNCT
ejpam-5721	214	34	→	→	SYM
ejpam-5721	214	35	(	(	PUNCT
ejpam-5721	214	36	y	y	PROPN
ejpam-5721	214	37	,	,	PUNCT
ejpam-5721	214	38	σ1	σ1	PROPN
ejpam-5721	214	39	,	,	PUNCT
ejpam-5721	214	40	σ2	σ2	PROPN
ejpam-5721	214	41	)	)	PUNCT
ejpam-5721	214	42	is	be	AUX
ejpam-5721	214	43	almost	almost	ADV
ejpam-5721	214	44	weakly	weakly	ADJ
ejpam-5721	214	45	(	(	PUNCT
ejpam-5721	214	46	τ1	τ1	NOUN
ejpam-5721	214	47	,	,	PUNCT
ejpam-5721	214	48	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	214	49	,	,	PUNCT
ejpam-5721	214	50	then	then	ADV
ejpam-5721	214	51	the	the	DET
ejpam-5721	214	52	set	set	NOUN
ejpam-5721	214	53	{	{	PUNCT
ejpam-5721	214	54	x	x	SYM
ejpam-5721	214	55	∈	∈	PROPN
ejpam-5721	214	56	x	x	X
ejpam-5721	214	57	|	|	ADV
ejpam-5721	214	58	f(x	f(x	PROPN
ejpam-5721	214	59	)	)	PUNCT
ejpam-5721	214	60	=	=	PUNCT
ejpam-5721	214	61	g(x	g(x	NOUN
ejpam-5721	214	62	)	)	PUNCT
ejpam-5721	214	63	}	}	PUNCT
ejpam-5721	214	64	is	be	AUX
ejpam-5721	214	65	(	(	PUNCT
ejpam-5721	214	66	τ1	τ1	NOUN
ejpam-5721	214	67	,	,	PUNCT
ejpam-5721	214	68	τ2)p	τ2)p	NOUN
ejpam-5721	214	69	-	-	PUNCT
ejpam-5721	214	70	closed	closed	ADJ
ejpam-5721	214	71	in	in	ADP
ejpam-5721	214	72	x.	x.	NOUN
ejpam-5721	214	73	proof	proof	NOUN
ejpam-5721	214	74	.	.	PUNCT
ejpam-5721	215	1	let	let	VERB
ejpam-5721	215	2	a	a	DET
ejpam-5721	215	3	=	=	SYM
ejpam-5721	215	4	{	{	PUNCT
ejpam-5721	215	5	x	x	SYM
ejpam-5721	215	6	∈	∈	PROPN
ejpam-5721	215	7	x	x	X
ejpam-5721	215	8	|	|	ADV
ejpam-5721	215	9	f(x	f(x	PROPN
ejpam-5721	215	10	)	)	PUNCT
ejpam-5721	216	1	=	=	PUNCT
ejpam-5721	216	2	g(x	g(x	NOUN
ejpam-5721	216	3	)	)	PUNCT
ejpam-5721	216	4	}	}	PUNCT
ejpam-5721	216	5	and	and	CCONJ
ejpam-5721	216	6	x	x	PUNCT
ejpam-5721	216	7	∈	∈	NOUN
ejpam-5721	216	8	x	x	PUNCT
ejpam-5721	216	9	−	−	NOUN
ejpam-5721	216	10	a.	a.	NOUN
ejpam-5721	216	11	then	then	ADV
ejpam-5721	216	12	,	,	PUNCT
ejpam-5721	216	13	f(x	f(x	PROPN
ejpam-5721	216	14	)	)	PUNCT
ejpam-5721	216	15	̸=	̸=	PROPN
ejpam-5721	216	16	g(x	g(x	NUM
ejpam-5721	216	17	)	)	PUNCT
ejpam-5721	216	18	and	and	CCONJ
ejpam-5721	216	19	there	there	PRON
ejpam-5721	216	20	exist	exist	VERB
ejpam-5721	216	21	σ1σ2	σ1σ2	NOUN
ejpam-5721	216	22	-	-	ADJ
ejpam-5721	216	23	open	open	ADJ
ejpam-5721	216	24	sets	set	NOUN
ejpam-5721	216	25	v	v	ADP
ejpam-5721	216	26	and	and	CCONJ
ejpam-5721	216	27	v	v	ADP
ejpam-5721	216	28	′	′	NUM
ejpam-5721	216	29	of	of	ADP
ejpam-5721	216	30	y	y	PRON
ejpam-5721	216	31	such	such	ADJ
ejpam-5721	216	32	that	that	SCONJ
ejpam-5721	216	33	f(x	f(x	PROPN
ejpam-5721	216	34	)	)	PUNCT
ejpam-5721	216	35	∈	∈	PROPN
ejpam-5721	216	36	v	v	NOUN
ejpam-5721	216	37	,	,	PUNCT
ejpam-5721	216	38	g(x	g(x	NOUN
ejpam-5721	216	39	)	)	PUNCT
ejpam-5721	216	40	∈	∈	PROPN
ejpam-5721	217	1	v	v	ADP
ejpam-5721	217	2	′	′	NUM
ejpam-5721	217	3	and	and	CCONJ
ejpam-5721	217	4	v	v	ADP
ejpam-5721	217	5	∩	∩	NOUN
ejpam-5721	217	6	v	v	NOUN
ejpam-5721	217	7	′	′	NOUN
ejpam-5721	217	8	=	=	NOUN
ejpam-5721	217	9	∅	∅	NOUN
ejpam-5721	217	10	;	;	PUNCT
ejpam-5721	217	11	hence	hence	ADV
ejpam-5721	217	12	σ1σ2	σ1σ2	ADV
ejpam-5721	217	13	-	-	PUNCT
ejpam-5721	217	14	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	217	15	-	-	PUNCT
ejpam-5721	217	16	cl(v	cl(v	NOUN
ejpam-5721	217	17	)	)	PUNCT
ejpam-5721	217	18	)	)	PUNCT
ejpam-5721	218	1	∩	∩	NOUN
ejpam-5721	218	2	σ1σ2	σ1σ2	NOUN
ejpam-5721	218	3	-	-	PUNCT
ejpam-5721	218	4	cl(v	cl(v	PRON
ejpam-5721	218	5	′	′	NOUN
ejpam-5721	218	6	)	)	PUNCT
ejpam-5721	218	7	=	=	PUNCT
ejpam-5721	218	8	∅.	∅.	NOUN
ejpam-5721	218	9	since	since	SCONJ
ejpam-5721	218	10	f	f	PROPN
ejpam-5721	218	11	is	be	AUX
ejpam-5721	218	12	almost	almost	ADV
ejpam-5721	218	13	α(τ1	α(τ1	NOUN
ejpam-5721	218	14	,	,	PUNCT
ejpam-5721	218	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	218	16	,	,	PUNCT
ejpam-5721	218	17	there	there	PRON
ejpam-5721	218	18	exists	exist	VERB
ejpam-5721	218	19	an	an	DET
ejpam-5721	218	20	α(τ1	α(τ1	NOUN
ejpam-5721	218	21	,	,	PUNCT
ejpam-5721	218	22	τ2)-open	τ2)-open	ADJ
ejpam-5721	218	23	set	set	VERB
ejpam-5721	218	24	u	u	NOUN
ejpam-5721	218	25	of	of	ADP
ejpam-5721	218	26	x	x	SYM
ejpam-5721	218	27	such	such	ADJ
ejpam-5721	218	28	that	that	DET
ejpam-5721	218	29	f(u	f(u	PROPN
ejpam-5721	218	30	)	)	PUNCT
ejpam-5721	218	31	⊆	⊆	NUM
ejpam-5721	218	32	σ1σ2	σ1σ2	X
ejpam-5721	218	33	-	-	PUNCT
ejpam-5721	218	34	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5721	218	35	-	-	PUNCT
ejpam-5721	218	36	cl(v	cl(v	NOUN
ejpam-5721	218	37	)	)	PUNCT
ejpam-5721	218	38	)	)	PUNCT
ejpam-5721	218	39	.	.	PUNCT
ejpam-5721	219	1	since	since	SCONJ
ejpam-5721	219	2	g	g	PROPN
ejpam-5721	219	3	is	be	AUX
ejpam-5721	219	4	almost	almost	ADV
ejpam-5721	219	5	weakly	weakly	ADJ
ejpam-5721	219	6	(	(	PUNCT
ejpam-5721	219	7	τ1	τ1	NOUN
ejpam-5721	219	8	,	,	PUNCT
ejpam-5721	219	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	219	10	,	,	PUNCT
ejpam-5721	219	11	by	by	ADP
ejpam-5721	219	12	theorem	theorem	NOUN
ejpam-5721	219	13	1	1	NUM
ejpam-5721	219	14	there	there	ADV
ejpam-5721	219	15	exists	exist	VERB
ejpam-5721	219	16	a	a	DET
ejpam-5721	219	17	(	(	PUNCT
ejpam-5721	219	18	τ1	τ1	NOUN
ejpam-5721	219	19	,	,	PUNCT
ejpam-5721	219	20	τ2)p	τ2)p	ADJ
ejpam-5721	219	21	-	-	PUNCT
ejpam-5721	219	22	open	open	ADJ
ejpam-5721	219	23	set	set	NOUN
ejpam-5721	219	24	u	u	NOUN
ejpam-5721	219	25	′	′	NOUN
ejpam-5721	219	26	of	of	ADP
ejpam-5721	219	27	x	x	PUNCT
ejpam-5721	219	28	containing	contain	VERB
ejpam-5721	219	29	x	x	PUNCT
ejpam-5721	219	30	such	such	ADJ
ejpam-5721	219	31	that	that	SCONJ
ejpam-5721	219	32	g(u	g(u	PROPN
ejpam-5721	219	33	′	′	NOUN
ejpam-5721	219	34	)	)	PUNCT
ejpam-5721	219	35	⊆	⊆	NUM
ejpam-5721	219	36	σ1σ2	σ1σ2	NUM
ejpam-5721	219	37	-	-	PUNCT
ejpam-5721	219	38	cl(v	cl(v	PRON
ejpam-5721	219	39	′	′	NOUN
ejpam-5721	219	40	)	)	PUNCT
ejpam-5721	219	41	.	.	PUNCT
ejpam-5721	220	1	therefore	therefore	ADV
ejpam-5721	220	2	,	,	PUNCT
ejpam-5721	220	3	f(u	f(u	PROPN
ejpam-5721	220	4	)	)	PUNCT
ejpam-5721	220	5	∩	∩	NOUN
ejpam-5721	220	6	g(u	g(u	PROPN
ejpam-5721	220	7	′	′	NOUN
ejpam-5721	220	8	)	)	PUNCT
ejpam-5721	220	9	=	=	PUNCT
ejpam-5721	220	10	∅.	∅.	X
ejpam-5721	220	11	by	by	ADP
ejpam-5721	220	12	lemma	lemma	PROPN
ejpam-5721	220	13	5	5	NUM
ejpam-5721	220	14	,	,	PUNCT
ejpam-5721	220	15	we	we	PRON
ejpam-5721	220	16	have	have	VERB
ejpam-5721	220	17	u	u	PRON
ejpam-5721	220	18	∩u	∩u	ADJ
ejpam-5721	220	19	′	′	NUM
ejpam-5721	221	1	is	be	AUX
ejpam-5721	221	2	(	(	PUNCT
ejpam-5721	221	3	τ1	τ1	NOUN
ejpam-5721	221	4	,	,	PUNCT
ejpam-5721	221	5	τ2)p	τ2)p	NOUN
ejpam-5721	221	6	-	-	PUNCT
ejpam-5721	221	7	open	open	ADJ
ejpam-5721	221	8	in	in	ADP
ejpam-5721	221	9	x.	x.	NOUN
ejpam-5721	221	10	since	since	SCONJ
ejpam-5721	221	11	(	(	PUNCT
ejpam-5721	221	12	u	u	NOUN
ejpam-5721	221	13	∩u	∩u	NOUN
ejpam-5721	221	14	′)∩a	′)∩a	NOUN
ejpam-5721	221	15	=	=	SYM
ejpam-5721	221	16	∅	∅	NOUN
ejpam-5721	221	17	,	,	PUNCT
ejpam-5721	221	18	x	x	X
ejpam-5721	221	19	∈	∈	NOUN
ejpam-5721	221	20	x	x	X
ejpam-5721	221	21	−	−	PROPN
ejpam-5721	221	22	(	(	PUNCT
ejpam-5721	221	23	τ1	τ1	NOUN
ejpam-5721	221	24	,	,	PUNCT
ejpam-5721	221	25	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-5721	221	26	)	)	PUNCT
ejpam-5721	221	27	.	.	PUNCT
ejpam-5721	222	1	thus	thus	ADV
ejpam-5721	222	2	,	,	PUNCT
ejpam-5721	222	3	a	a	PRON
ejpam-5721	222	4	is	is	NOUN
ejpam-5721	222	5	(	(	PUNCT
ejpam-5721	222	6	τ1	τ1	NOUN
ejpam-5721	222	7	,	,	PUNCT
ejpam-5721	222	8	τ2)p	τ2)p	NOUN
ejpam-5721	222	9	-	-	PUNCT
ejpam-5721	222	10	closed	closed	ADJ
ejpam-5721	222	11	in	in	ADP
ejpam-5721	222	12	x.	x.	PROPN
ejpam-5721	222	13	j.	j.	PROPN
ejpam-5721	222	14	khampakdee	khampakdee	PROPN
ejpam-5721	222	15	,	,	PUNCT
ejpam-5721	222	16	s.	s.	PROPN
ejpam-5721	222	17	sompong	sompong	PROPN
ejpam-5721	222	18	,	,	PUNCT
ejpam-5721	222	19	c.	c.	PROPN
ejpam-5721	222	20	boonpok	boonpok	PROPN
ejpam-5721	222	21	/	/	SYM
ejpam-5721	222	22	eur	eur	PROPN
ejpam-5721	222	23	.	.	PUNCT
ejpam-5721	223	1	j.	j.	PROPN
ejpam-5721	223	2	pure	pure	PROPN
ejpam-5721	223	3	appl	appl	PROPN
ejpam-5721	223	4	.	.	PROPN
ejpam-5721	223	5	math	math	PROPN
ejpam-5721	223	6	,	,	PUNCT
ejpam-5721	223	7	18	18	NUM
ejpam-5721	223	8	(	(	PUNCT
ejpam-5721	223	9	1	1	NUM
ejpam-5721	223	10	)	)	PUNCT
ejpam-5721	223	11	(	(	PUNCT
ejpam-5721	223	12	2025	2025	NUM
ejpam-5721	223	13	)	)	PUNCT
ejpam-5721	223	14	,	,	PUNCT
ejpam-5721	223	15	5721	5721	NUM
ejpam-5721	223	16	8	8	NUM
ejpam-5721	223	17	of	of	ADP
ejpam-5721	223	18	12	12	NUM
ejpam-5721	223	19	definition	definition	NOUN
ejpam-5721	223	20	4	4	NUM
ejpam-5721	223	21	.	.	PUNCT
ejpam-5721	224	1	[	[	X
ejpam-5721	224	2	32	32	NUM
ejpam-5721	224	3	]	]	PUNCT
ejpam-5721	224	4	a	a	DET
ejpam-5721	224	5	bitopological	bitopological	ADJ
ejpam-5721	224	6	space	space	NOUN
ejpam-5721	224	7	(	(	PUNCT
ejpam-5721	224	8	x	x	NOUN
ejpam-5721	224	9	,	,	PUNCT
ejpam-5721	224	10	τ1	τ1	NOUN
ejpam-5721	224	11	,	,	PUNCT
ejpam-5721	224	12	τ2	τ2	NOUN
ejpam-5721	224	13	)	)	PUNCT
ejpam-5721	224	14	is	be	AUX
ejpam-5721	224	15	said	say	VERB
ejpam-5721	224	16	to	to	PART
ejpam-5721	224	17	be	be	AUX
ejpam-5721	224	18	τ1τ2	τ1τ2	NOUN
ejpam-5721	224	19	-	-	ADJ
ejpam-5721	224	20	urysohn	urysohn	ADJ
ejpam-5721	224	21	if	if	SCONJ
ejpam-5721	224	22	for	for	ADP
ejpam-5721	224	23	each	each	DET
ejpam-5721	224	24	pair	pair	NOUN
ejpam-5721	224	25	of	of	ADP
ejpam-5721	224	26	distinct	distinct	ADJ
ejpam-5721	224	27	points	point	NOUN
ejpam-5721	224	28	x	x	PUNCT
ejpam-5721	224	29	and	and	CCONJ
ejpam-5721	224	30	y	y	PROPN
ejpam-5721	224	31	in	in	ADP
ejpam-5721	224	32	x	x	SYM
ejpam-5721	224	33	,	,	PUNCT
ejpam-5721	224	34	there	there	PRON
ejpam-5721	224	35	exist	exist	VERB
ejpam-5721	224	36	τ1τ2	τ1τ2	ADJ
ejpam-5721	224	37	-	-	ADJ
ejpam-5721	224	38	open	open	ADJ
ejpam-5721	224	39	sets	set	NOUN
ejpam-5721	224	40	u	u	NOUN
ejpam-5721	224	41	and	and	CCONJ
ejpam-5721	224	42	v	v	ADP
ejpam-5721	224	43	such	such	ADJ
ejpam-5721	224	44	that	that	SCONJ
ejpam-5721	224	45	x	x	SYM
ejpam-5721	224	46	∈	∈	PROPN
ejpam-5721	224	47	u	u	NOUN
ejpam-5721	224	48	,	,	PUNCT
ejpam-5721	224	49	y	y	PROPN
ejpam-5721	224	50	∈	∈	PROPN
ejpam-5721	224	51	v	v	NOUN
ejpam-5721	224	52	and	and	CCONJ
ejpam-5721	224	53	τ1τ2	τ1τ2	NOUN
ejpam-5721	224	54	-	-	NOUN
ejpam-5721	224	55	cl(u	cl(u	NOUN
ejpam-5721	224	56	)	)	PUNCT
ejpam-5721	224	57	∩	∩	NOUN
ejpam-5721	224	58	τ1τ2	τ1τ2	NOUN
ejpam-5721	224	59	-	-	NOUN
ejpam-5721	224	60	cl(v	cl(v	X
ejpam-5721	224	61	)	)	PUNCT
ejpam-5721	224	62	=	=	PUNCT
ejpam-5721	224	63	∅.	∅.	PRON
ejpam-5721	224	64	definition	definition	NOUN
ejpam-5721	224	65	5	5	NUM
ejpam-5721	224	66	.	.	PUNCT
ejpam-5721	225	1	a	a	DET
ejpam-5721	225	2	function	function	NOUN
ejpam-5721	225	3	f	f	NOUN
ejpam-5721	225	4	:	:	PUNCT
ejpam-5721	225	5	(	(	PUNCT
ejpam-5721	225	6	x	x	NOUN
ejpam-5721	225	7	,	,	PUNCT
ejpam-5721	225	8	τ1	τ1	NOUN
ejpam-5721	225	9	,	,	PUNCT
ejpam-5721	225	10	τ2	τ2	NOUN
ejpam-5721	225	11	)	)	PUNCT
ejpam-5721	225	12	→	→	SYM
ejpam-5721	225	13	(	(	PUNCT
ejpam-5721	225	14	y	y	PROPN
ejpam-5721	225	15	,	,	PUNCT
ejpam-5721	225	16	σ1	σ1	PROPN
ejpam-5721	225	17	,	,	PUNCT
ejpam-5721	225	18	σ2	σ2	PROPN
ejpam-5721	225	19	)	)	PUNCT
ejpam-5721	225	20	is	be	AUX
ejpam-5721	225	21	said	say	VERB
ejpam-5721	225	22	to	to	PART
ejpam-5721	225	23	be	be	AUX
ejpam-5721	225	24	weakly	weakly	ADJ
ejpam-5721	225	25	α(τ1	α(τ1	NOUN
ejpam-5721	225	26	,	,	PUNCT
ejpam-5721	225	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	225	28	if	if	SCONJ
ejpam-5721	225	29	for	for	ADP
ejpam-5721	225	30	each	each	DET
ejpam-5721	225	31	x	x	SYM
ejpam-5721	225	32	∈	∈	PROPN
ejpam-5721	225	33	x	x	X
ejpam-5721	225	34	and	and	CCONJ
ejpam-5721	225	35	each	each	DET
ejpam-5721	225	36	σ1σ2	σ1σ2	VERB
ejpam-5721	225	37	-	-	ADJ
ejpam-5721	225	38	open	open	ADJ
ejpam-5721	225	39	set	set	NOUN
ejpam-5721	225	40	v	v	NOUN
ejpam-5721	225	41	of	of	ADP
ejpam-5721	225	42	y	y	NOUN
ejpam-5721	225	43	containing	contain	VERB
ejpam-5721	225	44	f(x	f(x	PROPN
ejpam-5721	225	45	)	)	PUNCT
ejpam-5721	225	46	,	,	PUNCT
ejpam-5721	225	47	there	there	PRON
ejpam-5721	225	48	exists	exist	VERB
ejpam-5721	225	49	an	an	DET
ejpam-5721	225	50	α(τ1	α(τ1	NOUN
ejpam-5721	225	51	,	,	PUNCT
ejpam-5721	225	52	τ2)-open	τ2)-open	ADJ
ejpam-5721	225	53	set	set	VERB
ejpam-5721	225	54	u	u	NOUN
ejpam-5721	225	55	of	of	ADP
ejpam-5721	225	56	x	x	SYM
ejpam-5721	225	57	such	such	ADJ
ejpam-5721	225	58	that	that	DET
ejpam-5721	225	59	f(u	f(u	PROPN
ejpam-5721	225	60	)	)	PUNCT
ejpam-5721	225	61	⊆	⊆	NUM
ejpam-5721	225	62	σ1σ2	σ1σ2	NOUN
ejpam-5721	225	63	-	-	NUM
ejpam-5721	225	64	cl(v	cl(v	NOUN
ejpam-5721	225	65	)	)	PUNCT
ejpam-5721	225	66	.	.	PUNCT
ejpam-5721	226	1	theorem	theorem	NOUN
ejpam-5721	226	2	5	5	NUM
ejpam-5721	226	3	.	.	PUNCT
ejpam-5721	227	1	let	let	AUX
ejpam-5721	227	2	(	(	PUNCT
ejpam-5721	227	3	y	y	PROPN
ejpam-5721	227	4	,	,	PUNCT
ejpam-5721	227	5	σ1	σ1	PROPN
ejpam-5721	227	6	,	,	PUNCT
ejpam-5721	227	7	σ2	σ2	PROPN
ejpam-5721	227	8	)	)	PUNCT
ejpam-5721	227	9	be	be	VERB
ejpam-5721	227	10	a	a	DET
ejpam-5721	227	11	σ1σ2	σ1σ2	NUM
ejpam-5721	227	12	-	-	PUNCT
ejpam-5721	227	13	urysohn	urysohn	ADJ
ejpam-5721	227	14	space	space	NOUN
ejpam-5721	227	15	.	.	PUNCT
ejpam-5721	228	1	if	if	SCONJ
ejpam-5721	228	2	f	f	PROPN
ejpam-5721	228	3	:	:	PUNCT
ejpam-5721	228	4	(	(	PUNCT
ejpam-5721	228	5	x	x	NOUN
ejpam-5721	228	6	,	,	PUNCT
ejpam-5721	228	7	τ1	τ1	NOUN
ejpam-5721	228	8	,	,	PUNCT
ejpam-5721	228	9	τ2	τ2	NOUN
ejpam-5721	228	10	)	)	PUNCT
ejpam-5721	228	11	→	→	SYM
ejpam-5721	228	12	(	(	PUNCT
ejpam-5721	228	13	y	y	PROPN
ejpam-5721	228	14	,	,	PUNCT
ejpam-5721	228	15	σ1	σ1	PROPN
ejpam-5721	228	16	,	,	PUNCT
ejpam-5721	228	17	σ2	σ2	NOUN
ejpam-5721	228	18	)	)	PUNCT
ejpam-5721	228	19	is	be	AUX
ejpam-5721	228	20	weakly	weakly	ADJ
ejpam-5721	228	21	α(τ1	α(τ1	NOUN
ejpam-5721	228	22	,	,	PUNCT
ejpam-5721	228	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	228	24	and	and	CCONJ
ejpam-5721	228	25	g	g	NOUN
ejpam-5721	228	26	:	:	PUNCT
ejpam-5721	228	27	(	(	PUNCT
ejpam-5721	228	28	x	x	NOUN
ejpam-5721	228	29	,	,	PUNCT
ejpam-5721	228	30	τ1	τ1	NOUN
ejpam-5721	228	31	,	,	PUNCT
ejpam-5721	228	32	τ2	τ2	NOUN
ejpam-5721	228	33	)	)	PUNCT
ejpam-5721	228	34	→	→	SYM
ejpam-5721	228	35	(	(	PUNCT
ejpam-5721	228	36	y	y	PROPN
ejpam-5721	228	37	,	,	PUNCT
ejpam-5721	228	38	σ1	σ1	PROPN
ejpam-5721	228	39	,	,	PUNCT
ejpam-5721	228	40	σ2	σ2	PROPN
ejpam-5721	228	41	)	)	PUNCT
ejpam-5721	228	42	is	be	AUX
ejpam-5721	228	43	almost	almost	ADV
ejpam-5721	228	44	weakly	weakly	ADJ
ejpam-5721	228	45	(	(	PUNCT
ejpam-5721	228	46	τ1	τ1	NOUN
ejpam-5721	228	47	,	,	PUNCT
ejpam-5721	228	48	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	228	49	,	,	PUNCT
ejpam-5721	228	50	then	then	ADV
ejpam-5721	228	51	the	the	DET
ejpam-5721	228	52	set	set	NOUN
ejpam-5721	228	53	{	{	PUNCT
ejpam-5721	228	54	x	x	SYM
ejpam-5721	228	55	∈	∈	PROPN
ejpam-5721	228	56	x	x	X
ejpam-5721	228	57	|	|	ADV
ejpam-5721	228	58	f(x	f(x	PROPN
ejpam-5721	228	59	)	)	PUNCT
ejpam-5721	229	1	=	=	PUNCT
ejpam-5721	229	2	g(x	g(x	NOUN
ejpam-5721	229	3	)	)	PUNCT
ejpam-5721	229	4	}	}	PUNCT
ejpam-5721	229	5	is	be	AUX
ejpam-5721	229	6	(	(	PUNCT
ejpam-5721	229	7	τ1	τ1	NOUN
ejpam-5721	229	8	,	,	PUNCT
ejpam-5721	229	9	τ2)p	τ2)p	NOUN
ejpam-5721	229	10	-	-	PUNCT
ejpam-5721	229	11	closed	closed	ADJ
ejpam-5721	229	12	in	in	ADP
ejpam-5721	229	13	x.	x.	NOUN
ejpam-5721	229	14	proof	proof	NOUN
ejpam-5721	229	15	.	.	PUNCT
ejpam-5721	230	1	the	the	DET
ejpam-5721	230	2	proof	proof	NOUN
ejpam-5721	230	3	is	be	AUX
ejpam-5721	230	4	quite	quite	ADV
ejpam-5721	230	5	similar	similar	ADJ
ejpam-5721	230	6	to	to	ADP
ejpam-5721	230	7	that	that	PRON
ejpam-5721	230	8	of	of	ADP
ejpam-5721	230	9	theorem	theorem	ADJ
ejpam-5721	230	10	4	4	NUM
ejpam-5721	230	11	.	.	PUNCT
ejpam-5721	230	12	definition	definition	NOUN
ejpam-5721	230	13	6	6	NUM
ejpam-5721	230	14	.	.	PUNCT
ejpam-5721	231	1	a	a	DET
ejpam-5721	231	2	bitopological	bitopological	ADJ
ejpam-5721	231	3	space	space	NOUN
ejpam-5721	231	4	(	(	PUNCT
ejpam-5721	231	5	x	x	NOUN
ejpam-5721	231	6	,	,	PUNCT
ejpam-5721	231	7	τ1	τ1	NOUN
ejpam-5721	231	8	,	,	PUNCT
ejpam-5721	231	9	τ2	τ2	NOUN
ejpam-5721	231	10	)	)	PUNCT
ejpam-5721	231	11	is	be	AUX
ejpam-5721	231	12	said	say	VERB
ejpam-5721	231	13	to	to	PART
ejpam-5721	231	14	be	be	AUX
ejpam-5721	231	15	(	(	PUNCT
ejpam-5721	231	16	τ1	τ1	NOUN
ejpam-5721	231	17	,	,	PUNCT
ejpam-5721	231	18	τ2)p	τ2)p	ADJ
ejpam-5721	231	19	-	-	PUNCT
ejpam-5721	231	20	hausdorff	hausdorff	NOUN
ejpam-5721	231	21	if	if	SCONJ
ejpam-5721	231	22	for	for	ADP
ejpam-5721	231	23	each	each	DET
ejpam-5721	231	24	distinct	distinct	ADJ
ejpam-5721	231	25	points	point	NOUN
ejpam-5721	231	26	x	x	NOUN
ejpam-5721	231	27	,	,	PUNCT
ejpam-5721	231	28	y	y	PROPN
ejpam-5721	231	29	∈	∈	PROPN
ejpam-5721	231	30	x	x	PRON
ejpam-5721	231	31	,	,	PUNCT
ejpam-5721	231	32	there	there	PRON
ejpam-5721	231	33	exist	exist	VERB
ejpam-5721	231	34	(	(	PUNCT
ejpam-5721	231	35	τ1	τ1	NOUN
ejpam-5721	231	36	,	,	PUNCT
ejpam-5721	231	37	τ2)p	τ2)p	ADJ
ejpam-5721	231	38	-	-	PUNCT
ejpam-5721	231	39	open	open	ADJ
ejpam-5721	231	40	sets	set	VERB
ejpam-5721	231	41	u	u	NOUN
ejpam-5721	231	42	and	and	CCONJ
ejpam-5721	231	43	v	v	NOUN
ejpam-5721	231	44	of	of	ADP
ejpam-5721	231	45	x	x	PUNCT
ejpam-5721	231	46	containing	contain	VERB
ejpam-5721	231	47	x	x	PROPN
ejpam-5721	231	48	and	and	CCONJ
ejpam-5721	231	49	y	y	PROPN
ejpam-5721	231	50	,	,	PUNCT
ejpam-5721	231	51	respectively	respectively	ADV
ejpam-5721	231	52	,	,	PUNCT
ejpam-5721	232	1	such	such	ADJ
ejpam-5721	232	2	that	that	SCONJ
ejpam-5721	232	3	u	u	PROPN
ejpam-5721	232	4	∩	∩	NOUN
ejpam-5721	232	5	v	v	NOUN
ejpam-5721	232	6	=	=	PUNCT
ejpam-5721	232	7	∅.	∅.	NOUN
ejpam-5721	232	8	theorem	theorem	VERB
ejpam-5721	232	9	6	6	NUM
ejpam-5721	232	10	.	.	PUNCT
ejpam-5721	233	1	if	if	SCONJ
ejpam-5721	233	2	f	f	PROPN
ejpam-5721	233	3	:	:	PUNCT
ejpam-5721	233	4	(	(	PUNCT
ejpam-5721	233	5	x	x	NOUN
ejpam-5721	233	6	,	,	PUNCT
ejpam-5721	233	7	τ1	τ1	NOUN
ejpam-5721	233	8	,	,	PUNCT
ejpam-5721	233	9	τ2	τ2	NOUN
ejpam-5721	233	10	)	)	PUNCT
ejpam-5721	233	11	→	→	SYM
ejpam-5721	233	12	(	(	PUNCT
ejpam-5721	233	13	y	y	PROPN
ejpam-5721	233	14	,	,	PUNCT
ejpam-5721	233	15	σ1	σ1	PROPN
ejpam-5721	233	16	,	,	PUNCT
ejpam-5721	233	17	σ2	σ2	PROPN
ejpam-5721	233	18	)	)	PUNCT
ejpam-5721	233	19	is	be	AUX
ejpam-5721	233	20	an	an	DET
ejpam-5721	233	21	almost	almost	ADV
ejpam-5721	233	22	weakly	weakly	ADJ
ejpam-5721	233	23	(	(	PUNCT
ejpam-5721	233	24	τ1	τ1	NOUN
ejpam-5721	233	25	,	,	PUNCT
ejpam-5721	233	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	233	27	injection	injection	NOUN
ejpam-5721	233	28	and	and	CCONJ
ejpam-5721	233	29	(	(	PUNCT
ejpam-5721	233	30	y	y	PROPN
ejpam-5721	233	31	,	,	PUNCT
ejpam-5721	233	32	σ1	σ1	PROPN
ejpam-5721	233	33	,	,	PUNCT
ejpam-5721	233	34	σ2	σ2	PROPN
ejpam-5721	233	35	)	)	PUNCT
ejpam-5721	233	36	is	be	AUX
ejpam-5721	233	37	σ1σ2	σ1σ2	NOUN
ejpam-5721	233	38	-	-	PUNCT
ejpam-5721	233	39	urysohn	urysohn	ADJ
ejpam-5721	233	40	,	,	PUNCT
ejpam-5721	233	41	then	then	ADV
ejpam-5721	233	42	(	(	PUNCT
ejpam-5721	233	43	x	x	NOUN
ejpam-5721	233	44	,	,	PUNCT
ejpam-5721	233	45	τ1	τ1	NOUN
ejpam-5721	233	46	,	,	PUNCT
ejpam-5721	233	47	τ2	τ2	NOUN
ejpam-5721	233	48	)	)	PUNCT
ejpam-5721	233	49	is	be	AUX
ejpam-5721	233	50	(	(	PUNCT
ejpam-5721	233	51	τ1	τ1	NOUN
ejpam-5721	233	52	,	,	PUNCT
ejpam-5721	233	53	τ2)p	τ2)p	ADJ
ejpam-5721	233	54	-	-	PUNCT
ejpam-5721	233	55	hausdorff	hausdorff	NOUN
ejpam-5721	233	56	.	.	PUNCT
ejpam-5721	234	1	proof	proof	NOUN
ejpam-5721	234	2	.	.	PUNCT
ejpam-5721	235	1	since	since	SCONJ
ejpam-5721	235	2	f	f	PROPN
ejpam-5721	235	3	is	be	AUX
ejpam-5721	235	4	injective	injective	ADJ
ejpam-5721	235	5	,	,	PUNCT
ejpam-5721	235	6	then	then	ADV
ejpam-5721	235	7	f(x	f(x	PROPN
ejpam-5721	235	8	)	)	PUNCT
ejpam-5721	235	9	̸=	̸=	PROPN
ejpam-5721	235	10	f(y	f(y	NOUN
ejpam-5721	235	11	)	)	PUNCT
ejpam-5721	235	12	for	for	ADP
ejpam-5721	235	13	any	any	DET
ejpam-5721	235	14	distinct	distinct	ADJ
ejpam-5721	235	15	points	point	NOUN
ejpam-5721	235	16	x	x	PUNCT
ejpam-5721	235	17	and	and	CCONJ
ejpam-5721	235	18	y	y	PROPN
ejpam-5721	235	19	in	in	ADP
ejpam-5721	235	20	x.	x.	NOUN
ejpam-5721	235	21	since	since	SCONJ
ejpam-5721	235	22	(	(	PUNCT
ejpam-5721	235	23	y	y	PROPN
ejpam-5721	235	24	,	,	PUNCT
ejpam-5721	235	25	σ1	σ1	PROPN
ejpam-5721	235	26	,	,	PUNCT
ejpam-5721	235	27	σ2	σ2	PROPN
ejpam-5721	235	28	)	)	PUNCT
ejpam-5721	235	29	is	be	AUX
ejpam-5721	235	30	σ1σ2	σ1σ2	NOUN
ejpam-5721	235	31	-	-	PUNCT
ejpam-5721	235	32	urysohn	urysohn	ADJ
ejpam-5721	235	33	,	,	PUNCT
ejpam-5721	235	34	there	there	PRON
ejpam-5721	235	35	exist	exist	VERB
ejpam-5721	235	36	σ1	σ1	NOUN
ejpam-5721	235	37	,	,	PUNCT
ejpam-5721	235	38	σ2	σ2	NOUN
ejpam-5721	235	39	-	-	PUNCT
ejpam-5721	235	40	open	open	ADJ
ejpam-5721	235	41	sets	set	NOUN
ejpam-5721	235	42	v	v	ADP
ejpam-5721	235	43	and	and	CCONJ
ejpam-5721	235	44	v	v	ADP
ejpam-5721	235	45	′	′	NUM
ejpam-5721	235	46	of	of	ADP
ejpam-5721	235	47	y	y	PRON
ejpam-5721	235	48	such	such	ADJ
ejpam-5721	235	49	that	that	SCONJ
ejpam-5721	235	50	f(x	f(x	PROPN
ejpam-5721	235	51	)	)	PUNCT
ejpam-5721	235	52	∈	∈	PROPN
ejpam-5721	235	53	v	v	NOUN
ejpam-5721	235	54	,	,	PUNCT
ejpam-5721	235	55	f(y	f(y	NOUN
ejpam-5721	235	56	)	)	PUNCT
ejpam-5721	235	57	∈	∈	PROPN
ejpam-5721	235	58	v	v	ADP
ejpam-5721	235	59	′	′	NUM
ejpam-5721	235	60	and	and	CCONJ
ejpam-5721	235	61	σ1σ2	σ1σ2	NOUN
ejpam-5721	235	62	-	-	NUM
ejpam-5721	235	63	cl(v	cl(v	NOUN
ejpam-5721	235	64	)	)	PUNCT
ejpam-5721	235	65	∩	∩	NOUN
ejpam-5721	235	66	σ1σ2	σ1σ2	NOUN
ejpam-5721	235	67	-	-	PUNCT
ejpam-5721	235	68	cl(v	cl(v	PRON
ejpam-5721	235	69	′	′	NOUN
ejpam-5721	235	70	)	)	PUNCT
ejpam-5721	236	1	=	=	PUNCT
ejpam-5721	236	2	∅.	∅.	NOUN
ejpam-5721	236	3	since	since	SCONJ
ejpam-5721	236	4	f	f	PROPN
ejpam-5721	236	5	is	be	AUX
ejpam-5721	236	6	almost	almost	ADV
ejpam-5721	236	7	weakly	weakly	ADJ
ejpam-5721	236	8	(	(	PUNCT
ejpam-5721	236	9	τ1	τ1	NOUN
ejpam-5721	236	10	,	,	PUNCT
ejpam-5721	236	11	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	236	12	,	,	PUNCT
ejpam-5721	236	13	there	there	PRON
ejpam-5721	236	14	exist	exist	VERB
ejpam-5721	236	15	(	(	PUNCT
ejpam-5721	236	16	τ1	τ1	NOUN
ejpam-5721	236	17	,	,	PUNCT
ejpam-5721	236	18	τ2)p	τ2)p	ADJ
ejpam-5721	236	19	-	-	PUNCT
ejpam-5721	236	20	open	open	ADJ
ejpam-5721	236	21	sets	set	VERB
ejpam-5721	236	22	u	u	NOUN
ejpam-5721	236	23	and	and	CCONJ
ejpam-5721	236	24	u	u	NOUN
ejpam-5721	236	25	′	′	NOUN
ejpam-5721	236	26	of	of	ADP
ejpam-5721	236	27	x	x	PUNCT
ejpam-5721	236	28	containing	contain	VERB
ejpam-5721	236	29	x	x	PROPN
ejpam-5721	236	30	and	and	CCONJ
ejpam-5721	236	31	y	y	PROPN
ejpam-5721	236	32	,	,	PUNCT
ejpam-5721	236	33	respectively	respectively	ADV
ejpam-5721	236	34	,	,	PUNCT
ejpam-5721	236	35	such	such	ADJ
ejpam-5721	236	36	that	that	DET
ejpam-5721	236	37	f(u	f(u	PROPN
ejpam-5721	236	38	)	)	PUNCT
ejpam-5721	236	39	⊆	⊆	NUM
ejpam-5721	236	40	σ1σ2	σ1σ2	NOUN
ejpam-5721	236	41	-	-	NUM
ejpam-5721	236	42	cl(v	cl(v	NOUN
ejpam-5721	236	43	)	)	PUNCT
ejpam-5721	236	44	and	and	CCONJ
ejpam-5721	236	45	f(u	f(u	PROPN
ejpam-5721	236	46	′	′	NUM
ejpam-5721	236	47	)	)	PUNCT
ejpam-5721	237	1	⊆	⊆	NUM
ejpam-5721	237	2	σ1σ2	σ1σ2	NUM
ejpam-5721	237	3	-	-	PUNCT
ejpam-5721	237	4	cl(v	cl(v	PRON
ejpam-5721	237	5	′	′	NOUN
ejpam-5721	237	6	)	)	PUNCT
ejpam-5721	237	7	.	.	PUNCT
ejpam-5721	238	1	this	this	PRON
ejpam-5721	238	2	implies	imply	VERB
ejpam-5721	238	3	that	that	SCONJ
ejpam-5721	238	4	u	u	PROPN
ejpam-5721	238	5	∩	∩	NOUN
ejpam-5721	238	6	u	u	NOUN
ejpam-5721	238	7	′	′	NOUN
ejpam-5721	238	8	=	=	PUNCT
ejpam-5721	238	9	∅.	∅.	VERB
ejpam-5721	238	10	thus	thus	ADV
ejpam-5721	238	11	,	,	PUNCT
ejpam-5721	238	12	(	(	PUNCT
ejpam-5721	238	13	x	x	NOUN
ejpam-5721	238	14	,	,	PUNCT
ejpam-5721	238	15	τ1	τ1	NOUN
ejpam-5721	238	16	,	,	PUNCT
ejpam-5721	238	17	τ2	τ2	NOUN
ejpam-5721	238	18	)	)	PUNCT
ejpam-5721	238	19	is	be	AUX
ejpam-5721	238	20	(	(	PUNCT
ejpam-5721	238	21	τ1	τ1	NOUN
ejpam-5721	238	22	,	,	PUNCT
ejpam-5721	238	23	τ2)p	τ2)p	ADJ
ejpam-5721	238	24	-	-	PUNCT
ejpam-5721	238	25	hausdorff	hausdorff	NOUN
ejpam-5721	238	26	.	.	PUNCT
ejpam-5721	239	1	definition	definition	NOUN
ejpam-5721	239	2	7	7	NUM
ejpam-5721	239	3	.	.	PUNCT
ejpam-5721	240	1	[	[	X
ejpam-5721	240	2	15	15	NUM
ejpam-5721	240	3	]	]	X
ejpam-5721	240	4	a	a	DET
ejpam-5721	240	5	bitopological	bitopological	ADJ
ejpam-5721	240	6	space	space	NOUN
ejpam-5721	240	7	(	(	PUNCT
ejpam-5721	240	8	x	x	NOUN
ejpam-5721	240	9	,	,	PUNCT
ejpam-5721	240	10	τ1	τ1	NOUN
ejpam-5721	240	11	,	,	PUNCT
ejpam-5721	240	12	τ2	τ2	NOUN
ejpam-5721	240	13	)	)	PUNCT
ejpam-5721	240	14	is	be	AUX
ejpam-5721	240	15	said	say	VERB
ejpam-5721	240	16	to	to	PART
ejpam-5721	240	17	be	be	AUX
ejpam-5721	240	18	(	(	PUNCT
ejpam-5721	240	19	τ1	τ1	NOUN
ejpam-5721	240	20	,	,	PUNCT
ejpam-5721	240	21	τ2)-regular	τ2)-regular	ADJ
ejpam-5721	240	22	if	if	SCONJ
ejpam-5721	240	23	for	for	ADP
ejpam-5721	240	24	each	each	DET
ejpam-5721	240	25	τ1τ2	τ1τ2	ADJ
ejpam-5721	240	26	-	-	ADJ
ejpam-5721	240	27	closed	closed	ADJ
ejpam-5721	240	28	set	set	VERB
ejpam-5721	240	29	f	f	NOUN
ejpam-5721	240	30	and	and	CCONJ
ejpam-5721	240	31	each	each	DET
ejpam-5721	240	32	point	point	NOUN
ejpam-5721	240	33	x	x	X
ejpam-5721	240	34	∈	∈	NOUN
ejpam-5721	240	35	x	x	X
ejpam-5721	240	36	−	−	PROPN
ejpam-5721	240	37	f	f	NOUN
ejpam-5721	240	38	,	,	PUNCT
ejpam-5721	240	39	there	there	PRON
ejpam-5721	240	40	exist	exist	VERB
ejpam-5721	240	41	disjoint	disjoint	ADJ
ejpam-5721	240	42	τ1τ2	τ1τ2	ADJ
ejpam-5721	240	43	-	-	ADJ
ejpam-5721	240	44	open	open	ADJ
ejpam-5721	240	45	sets	set	NOUN
ejpam-5721	240	46	u	u	NOUN
ejpam-5721	240	47	and	and	CCONJ
ejpam-5721	240	48	v	v	ADP
ejpam-5721	240	49	such	such	ADJ
ejpam-5721	240	50	that	that	SCONJ
ejpam-5721	240	51	x	x	SYM
ejpam-5721	240	52	∈	∈	PROPN
ejpam-5721	240	53	u	u	NOUN
ejpam-5721	240	54	and	and	CCONJ
ejpam-5721	240	55	f	f	PROPN
ejpam-5721	240	56	⊆	⊆	NUM
ejpam-5721	240	57	v	v	NOUN
ejpam-5721	240	58	.	.	PUNCT
ejpam-5721	241	1	definition	definition	NOUN
ejpam-5721	241	2	8	8	NUM
ejpam-5721	241	3	.	.	PUNCT
ejpam-5721	242	1	a	a	DET
ejpam-5721	242	2	function	function	NOUN
ejpam-5721	242	3	f	f	NOUN
ejpam-5721	242	4	:	:	PUNCT
ejpam-5721	242	5	(	(	PUNCT
ejpam-5721	242	6	x	x	NOUN
ejpam-5721	242	7	,	,	PUNCT
ejpam-5721	242	8	τ1	τ1	NOUN
ejpam-5721	242	9	,	,	PUNCT
ejpam-5721	242	10	τ2	τ2	NOUN
ejpam-5721	242	11	)	)	PUNCT
ejpam-5721	242	12	→	→	SYM
ejpam-5721	242	13	(	(	PUNCT
ejpam-5721	242	14	y	y	PROPN
ejpam-5721	242	15	,	,	PUNCT
ejpam-5721	242	16	σ1	σ1	PROPN
ejpam-5721	242	17	,	,	PUNCT
ejpam-5721	242	18	σ2	σ2	PROPN
ejpam-5721	242	19	)	)	PUNCT
ejpam-5721	242	20	is	be	AUX
ejpam-5721	242	21	said	say	VERB
ejpam-5721	242	22	to	to	PART
ejpam-5721	242	23	be	be	AUX
ejpam-5721	242	24	(	(	PUNCT
ejpam-5721	242	25	τ1	τ1	NOUN
ejpam-5721	242	26	,	,	PUNCT
ejpam-5721	242	27	τ2)p	τ2)p	ADJ
ejpam-5721	242	28	-	-	ADJ
ejpam-5721	242	29	continuous	continuous	ADJ
ejpam-5721	242	30	if	if	SCONJ
ejpam-5721	242	31	for	for	ADP
ejpam-5721	242	32	each	each	DET
ejpam-5721	242	33	x	x	SYM
ejpam-5721	242	34	∈	∈	PROPN
ejpam-5721	242	35	x	x	X
ejpam-5721	242	36	and	and	CCONJ
ejpam-5721	242	37	each	each	DET
ejpam-5721	242	38	σ1σ2	σ1σ2	VERB
ejpam-5721	242	39	-	-	ADJ
ejpam-5721	242	40	open	open	ADJ
ejpam-5721	242	41	set	set	NOUN
ejpam-5721	242	42	v	v	NOUN
ejpam-5721	242	43	of	of	ADP
ejpam-5721	242	44	y	y	NOUN
ejpam-5721	242	45	containing	contain	VERB
ejpam-5721	242	46	f(x	f(x	PROPN
ejpam-5721	242	47	)	)	PUNCT
ejpam-5721	242	48	,	,	PUNCT
ejpam-5721	242	49	there	there	PRON
ejpam-5721	242	50	exists	exist	VERB
ejpam-5721	242	51	a	a	DET
ejpam-5721	242	52	(	(	PUNCT
ejpam-5721	242	53	τ1τ2)p	τ1τ2)p	NOUN
ejpam-5721	242	54	-	-	PUNCT
ejpam-5721	242	55	open	open	ADJ
ejpam-5721	242	56	set	set	NOUN
ejpam-5721	242	57	u	u	NOUN
ejpam-5721	242	58	of	of	ADP
ejpam-5721	242	59	x	x	SYM
ejpam-5721	242	60	such	such	ADJ
ejpam-5721	242	61	that	that	DET
ejpam-5721	242	62	f(u	f(u	PROPN
ejpam-5721	242	63	)	)	PUNCT
ejpam-5721	242	64	⊆	⊆	NUM
ejpam-5721	242	65	v	v	NOUN
ejpam-5721	242	66	.	.	PUNCT
ejpam-5721	243	1	theorem	theorem	ADJ
ejpam-5721	243	2	7	7	NUM
ejpam-5721	243	3	.	.	PUNCT
ejpam-5721	244	1	let	let	AUX
ejpam-5721	244	2	(	(	PUNCT
ejpam-5721	244	3	y	y	PROPN
ejpam-5721	244	4	,	,	PUNCT
ejpam-5721	244	5	σ1	σ1	PROPN
ejpam-5721	244	6	,	,	PUNCT
ejpam-5721	244	7	σ2	σ2	PROPN
ejpam-5721	244	8	)	)	PUNCT
ejpam-5721	244	9	be	be	VERB
ejpam-5721	244	10	a	a	DET
ejpam-5721	244	11	(	(	PUNCT
ejpam-5721	244	12	σ1	σ1	NOUN
ejpam-5721	244	13	,	,	PUNCT
ejpam-5721	244	14	σ2)-regular	σ2)-regular	ADJ
ejpam-5721	244	15	space	space	NOUN
ejpam-5721	244	16	.	.	PUNCT
ejpam-5721	245	1	then	then	ADV
ejpam-5721	245	2	a	a	DET
ejpam-5721	245	3	function	function	NOUN
ejpam-5721	245	4	f	f	NOUN
ejpam-5721	245	5	:	:	PUNCT
ejpam-5721	245	6	(	(	PUNCT
ejpam-5721	245	7	x	x	NOUN
ejpam-5721	245	8	,	,	PUNCT
ejpam-5721	245	9	τ1	τ1	NOUN
ejpam-5721	245	10	,	,	PUNCT
ejpam-5721	245	11	τ2	τ2	NOUN
ejpam-5721	245	12	)	)	PUNCT
ejpam-5721	245	13	→	→	SYM
ejpam-5721	245	14	(	(	PUNCT
ejpam-5721	245	15	y	y	PROPN
ejpam-5721	245	16	,	,	PUNCT
ejpam-5721	245	17	σ1	σ1	PROPN
ejpam-5721	245	18	,	,	PUNCT
ejpam-5721	245	19	σ2	σ2	PROPN
ejpam-5721	245	20	)	)	PUNCT
ejpam-5721	245	21	is	be	AUX
ejpam-5721	245	22	(	(	PUNCT
ejpam-5721	245	23	τ1	τ1	NOUN
ejpam-5721	245	24	,	,	PUNCT
ejpam-5721	245	25	τ2)p	τ2)p	ADJ
ejpam-5721	245	26	-	-	ADJ
ejpam-5721	245	27	continuous	continuous	ADJ
ejpam-5721	245	28	if	if	SCONJ
ejpam-5721	245	29	and	and	CCONJ
ejpam-5721	245	30	only	only	ADV
ejpam-5721	245	31	if	if	SCONJ
ejpam-5721	245	32	f	f	PROPN
ejpam-5721	245	33	is	be	AUX
ejpam-5721	245	34	almost	almost	ADV
ejpam-5721	245	35	weakly	weakly	ADJ
ejpam-5721	245	36	(	(	PUNCT
ejpam-5721	245	37	τ1	τ1	NOUN
ejpam-5721	245	38	,	,	PUNCT
ejpam-5721	245	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	245	40	.	.	PUNCT
ejpam-5721	246	1	proof	proof	NOUN
ejpam-5721	246	2	.	.	PUNCT
ejpam-5721	247	1	we	we	PRON
ejpam-5721	247	2	prove	prove	VERB
ejpam-5721	247	3	only	only	ADV
ejpam-5721	247	4	the	the	DET
ejpam-5721	247	5	sufficiency	sufficiency	NOUN
ejpam-5721	247	6	since	since	SCONJ
ejpam-5721	247	7	the	the	DET
ejpam-5721	247	8	necessity	necessity	NOUN
ejpam-5721	247	9	is	be	AUX
ejpam-5721	247	10	evident	evident	ADJ
ejpam-5721	247	11	.	.	PUNCT
ejpam-5721	248	1	let	let	VERB
ejpam-5721	248	2	x	x	PUNCT
ejpam-5721	248	3	∈	∈	PROPN
ejpam-5721	248	4	x	x	X
ejpam-5721	248	5	and	and	CCONJ
ejpam-5721	248	6	w	w	PROPN
ejpam-5721	248	7	be	be	AUX
ejpam-5721	248	8	any	any	DET
ejpam-5721	248	9	σ1σ2	σ1σ2	NOUN
ejpam-5721	248	10	-	-	ADJ
ejpam-5721	248	11	open	open	ADJ
ejpam-5721	248	12	set	set	NOUN
ejpam-5721	248	13	of	of	ADP
ejpam-5721	248	14	y	y	PROPN
ejpam-5721	248	15	containing	contain	VERB
ejpam-5721	248	16	f(x	f(x	PROPN
ejpam-5721	248	17	)	)	PUNCT
ejpam-5721	248	18	.	.	PUNCT
ejpam-5721	249	1	by	by	ADP
ejpam-5721	249	2	the	the	DET
ejpam-5721	249	3	regularity	regularity	NOUN
ejpam-5721	249	4	of	of	ADP
ejpam-5721	249	5	(	(	PUNCT
ejpam-5721	249	6	y	y	PROPN
ejpam-5721	249	7	,	,	PUNCT
ejpam-5721	249	8	σ1	σ1	PROPN
ejpam-5721	249	9	,	,	PUNCT
ejpam-5721	249	10	σ2	σ2	NOUN
ejpam-5721	249	11	)	)	PUNCT
ejpam-5721	249	12	,	,	PUNCT
ejpam-5721	249	13	there	there	PRON
ejpam-5721	249	14	exists	exist	VERB
ejpam-5721	249	15	a	a	DET
ejpam-5721	249	16	σ1σ2	σ1σ2	NUM
ejpam-5721	249	17	-	-	ADJ
ejpam-5721	249	18	open	open	ADJ
ejpam-5721	249	19	set	set	NOUN
ejpam-5721	249	20	v	v	NOUN
ejpam-5721	249	21	of	of	ADP
ejpam-5721	249	22	y	y	PRON
ejpam-5721	249	23	such	such	ADJ
ejpam-5721	249	24	that	that	SCONJ
ejpam-5721	249	25	f(x	f(x	PROPN
ejpam-5721	249	26	)	)	PUNCT
ejpam-5721	249	27	∈	∈	PROPN
ejpam-5721	249	28	v	v	NOUN
ejpam-5721	249	29	and	and	CCONJ
ejpam-5721	249	30	σ1σ2	σ1σ2	NOUN
ejpam-5721	249	31	-	-	NUM
ejpam-5721	249	32	cl(v	cl(v	NOUN
ejpam-5721	249	33	)	)	PUNCT
ejpam-5721	249	34	⊆	⊆	NUM
ejpam-5721	249	35	w	w	NOUN
ejpam-5721	249	36	.	.	PUNCT
ejpam-5721	250	1	since	since	SCONJ
ejpam-5721	250	2	f	f	PROPN
ejpam-5721	250	3	is	be	AUX
ejpam-5721	250	4	almost	almost	ADV
ejpam-5721	250	5	weakly	weakly	ADJ
ejpam-5721	250	6	(	(	PUNCT
ejpam-5721	250	7	τ1	τ1	NOUN
ejpam-5721	250	8	,	,	PUNCT
ejpam-5721	250	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	250	10	,	,	PUNCT
ejpam-5721	250	11	there	there	PRON
ejpam-5721	250	12	exists	exist	VERB
ejpam-5721	250	13	a	a	DET
ejpam-5721	250	14	(	(	PUNCT
ejpam-5721	250	15	τ1	τ1	NOUN
ejpam-5721	250	16	,	,	PUNCT
ejpam-5721	250	17	τ2)p	τ2)p	ADJ
ejpam-5721	250	18	-	-	PUNCT
ejpam-5721	250	19	open	open	ADJ
ejpam-5721	250	20	set	set	NOUN
ejpam-5721	250	21	u	u	NOUN
ejpam-5721	250	22	of	of	ADP
ejpam-5721	250	23	x	x	PUNCT
ejpam-5721	250	24	containing	contain	VERB
ejpam-5721	250	25	x	x	PUNCT
ejpam-5721	250	26	such	such	ADJ
ejpam-5721	250	27	that	that	DET
ejpam-5721	250	28	f(u	f(u	PROPN
ejpam-5721	250	29	)	)	PUNCT
ejpam-5721	250	30	⊆	⊆	NUM
ejpam-5721	250	31	σ1σ2	σ1σ2	NOUN
ejpam-5721	250	32	-	-	NUM
ejpam-5721	250	33	cl(v	cl(v	NOUN
ejpam-5721	250	34	)	)	PUNCT
ejpam-5721	250	35	.	.	PUNCT
ejpam-5721	251	1	this	this	PRON
ejpam-5721	251	2	implies	imply	VERB
ejpam-5721	251	3	that	that	SCONJ
ejpam-5721	251	4	f(u	f(u	PROPN
ejpam-5721	251	5	)	)	PUNCT
ejpam-5721	251	6	⊆	⊆	NUM
ejpam-5721	251	7	w	w	NOUN
ejpam-5721	251	8	and	and	CCONJ
ejpam-5721	251	9	hence	hence	ADV
ejpam-5721	251	10	f	f	PROPN
ejpam-5721	251	11	is	be	AUX
ejpam-5721	251	12	(	(	PUNCT
ejpam-5721	251	13	τ1	τ1	NOUN
ejpam-5721	251	14	,	,	PUNCT
ejpam-5721	251	15	τ2)p	τ2)p	ADJ
ejpam-5721	251	16	-	-	NOUN
ejpam-5721	251	17	continuous	continuous	ADJ
ejpam-5721	251	18	.	.	PUNCT
ejpam-5721	252	1	j.	j.	PROPN
ejpam-5721	252	2	khampakdee	khampakdee	PROPN
ejpam-5721	252	3	,	,	PUNCT
ejpam-5721	252	4	s.	s.	PROPN
ejpam-5721	252	5	sompong	sompong	PROPN
ejpam-5721	252	6	,	,	PUNCT
ejpam-5721	252	7	c.	c.	PROPN
ejpam-5721	252	8	boonpok	boonpok	PROPN
ejpam-5721	252	9	/	/	SYM
ejpam-5721	252	10	eur	eur	PROPN
ejpam-5721	252	11	.	.	PUNCT
ejpam-5721	253	1	j.	j.	PROPN
ejpam-5721	253	2	pure	pure	PROPN
ejpam-5721	253	3	appl	appl	PROPN
ejpam-5721	253	4	.	.	PROPN
ejpam-5721	253	5	math	math	PROPN
ejpam-5721	253	6	,	,	PUNCT
ejpam-5721	253	7	18	18	NUM
ejpam-5721	253	8	(	(	PUNCT
ejpam-5721	253	9	1	1	NUM
ejpam-5721	253	10	)	)	PUNCT
ejpam-5721	253	11	(	(	PUNCT
ejpam-5721	253	12	2025	2025	NUM
ejpam-5721	253	13	)	)	PUNCT
ejpam-5721	253	14	,	,	PUNCT
ejpam-5721	253	15	5721	5721	NUM
ejpam-5721	253	16	9	9	NUM
ejpam-5721	253	17	of	of	ADP
ejpam-5721	253	18	12	12	NUM
ejpam-5721	253	19	recall	recall	NOUN
ejpam-5721	253	20	that	that	SCONJ
ejpam-5721	253	21	a	a	DET
ejpam-5721	253	22	bitopological	bitopological	ADJ
ejpam-5721	253	23	space	space	NOUN
ejpam-5721	253	24	(	(	PUNCT
ejpam-5721	253	25	x	x	NOUN
ejpam-5721	253	26	,	,	PUNCT
ejpam-5721	253	27	τ1	τ1	NOUN
ejpam-5721	253	28	,	,	PUNCT
ejpam-5721	253	29	τ2	τ2	NOUN
ejpam-5721	253	30	)	)	PUNCT
ejpam-5721	253	31	is	be	AUX
ejpam-5721	253	32	said	say	VERB
ejpam-5721	253	33	to	to	PART
ejpam-5721	253	34	be	be	AUX
ejpam-5721	253	35	quasi	quasi	X
ejpam-5721	253	36	(	(	PUNCT
ejpam-5721	253	37	τ1	τ1	NOUN
ejpam-5721	253	38	,	,	PUNCT
ejpam-5721	253	39	τ2)-h	τ2)-h	PUNCT
ejpam-5721	253	40	-closed	-closed	ADJ
ejpam-5721	253	41	[	[	PUNCT
ejpam-5721	253	42	34	34	NUM
ejpam-5721	253	43	]	]	PUNCT
ejpam-5721	253	44	if	if	SCONJ
ejpam-5721	253	45	every	every	DET
ejpam-5721	253	46	τ1τ2	τ1τ2	ADJ
ejpam-5721	253	47	-	-	ADJ
ejpam-5721	253	48	open	open	ADJ
ejpam-5721	253	49	cover	cover	NOUN
ejpam-5721	253	50	{	{	PUNCT
ejpam-5721	253	51	uγ	uγ	ADV
ejpam-5721	253	52	|	|	ADV
ejpam-5721	253	53	γ	γ	X
ejpam-5721	253	54	∈	∈	PROPN
ejpam-5721	253	55	γ	γ	X
ejpam-5721	253	56	}	}	PUNCT
ejpam-5721	253	57	,	,	PUNCT
ejpam-5721	253	58	there	there	PRON
ejpam-5721	253	59	exists	exist	VERB
ejpam-5721	253	60	a	a	DET
ejpam-5721	253	61	finite	finite	NOUN
ejpam-5721	253	62	subset	subset	NOUN
ejpam-5721	253	63	γ0	γ0	NOUN
ejpam-5721	253	64	of	of	ADP
ejpam-5721	253	65	γ	γ	NOUN
ejpam-5721	254	1	such	such	ADJ
ejpam-5721	254	2	that	that	SCONJ
ejpam-5721	254	3	x	x	X
ejpam-5721	254	4	=	=	PUNCT
ejpam-5721	254	5	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-5721	254	6	-	-	NOUN
ejpam-5721	254	7	cl(uγ	cl(uγ	NOUN
ejpam-5721	254	8	)	)	PUNCT
ejpam-5721	254	9	|	|	ADV
ejpam-5721	254	10	γ	γ	PROPN
ejpam-5721	254	11	∈	∈	PROPN
ejpam-5721	254	12	γ0	γ0	PROPN
ejpam-5721	254	13	}	}	PUNCT
ejpam-5721	254	14	.	.	PUNCT
ejpam-5721	255	1	a	a	DET
ejpam-5721	255	2	subset	subset	NOUN
ejpam-5721	255	3	k	k	NOUN
ejpam-5721	255	4	of	of	ADP
ejpam-5721	255	5	a	a	DET
ejpam-5721	255	6	bitopological	bitopological	ADJ
ejpam-5721	255	7	space	space	NOUN
ejpam-5721	255	8	(	(	PUNCT
ejpam-5721	255	9	x	x	NOUN
ejpam-5721	255	10	,	,	PUNCT
ejpam-5721	255	11	τ1	τ1	NOUN
ejpam-5721	255	12	,	,	PUNCT
ejpam-5721	255	13	τ2	τ2	NOUN
ejpam-5721	255	14	)	)	PUNCT
ejpam-5721	255	15	is	be	AUX
ejpam-5721	255	16	said	say	VERB
ejpam-5721	255	17	to	to	PART
ejpam-5721	255	18	be	be	AUX
ejpam-5721	255	19	quasi	quasi	X
ejpam-5721	255	20	(	(	PUNCT
ejpam-5721	255	21	τ1	τ1	NOUN
ejpam-5721	255	22	,	,	PUNCT
ejpam-5721	255	23	τ2)-h	τ2)-h	NOUN
ejpam-5721	255	24	-closed	-close	VERB
ejpam-5721	255	25	relative	relative	ADJ
ejpam-5721	255	26	to	to	ADP
ejpam-5721	255	27	x	x	PRON
ejpam-5721	255	28	if	if	SCONJ
ejpam-5721	255	29	for	for	ADP
ejpam-5721	255	30	any	any	DET
ejpam-5721	255	31	cover	cover	NOUN
ejpam-5721	255	32	{	{	PUNCT
ejpam-5721	255	33	vγ	vγ	NOUN
ejpam-5721	255	34	|	|	ADV
ejpam-5721	255	35	γ	γ	PROPN
ejpam-5721	255	36	∈	∈	PROPN
ejpam-5721	255	37	γ	γ	X
ejpam-5721	255	38	}	}	PUNCT
ejpam-5721	255	39	by	by	ADP
ejpam-5721	255	40	τ1τ2	τ1τ2	ADJ
ejpam-5721	255	41	-	-	ADJ
ejpam-5721	255	42	open	open	ADJ
ejpam-5721	255	43	sets	set	NOUN
ejpam-5721	255	44	of	of	ADP
ejpam-5721	255	45	x	x	NOUN
ejpam-5721	255	46	,	,	PUNCT
ejpam-5721	255	47	there	there	PRON
ejpam-5721	255	48	exists	exist	VERB
ejpam-5721	255	49	a	a	DET
ejpam-5721	255	50	finite	finite	NOUN
ejpam-5721	255	51	subset	subset	NOUN
ejpam-5721	255	52	γ0	γ0	NOUN
ejpam-5721	255	53	of	of	ADP
ejpam-5721	255	54	γ	γ	PRON
ejpam-5721	255	55	such	such	ADJ
ejpam-5721	255	56	that	that	SCONJ
ejpam-5721	255	57	k	k	PROPN
ejpam-5721	255	58	⊆	⊆	NUM
ejpam-5721	255	59	∪{τ1τ2	∪{τ1τ2	ADJ
ejpam-5721	255	60	-	-	ADJ
ejpam-5721	255	61	cl(vγ	cl(vγ	ADJ
ejpam-5721	255	62	)	)	PUNCT
ejpam-5721	255	63	|	|	ADV
ejpam-5721	255	64	γ	γ	PROPN
ejpam-5721	255	65	∈	∈	PROPN
ejpam-5721	255	66	γ0	γ0	PROPN
ejpam-5721	255	67	}	}	PUNCT
ejpam-5721	255	68	.	.	PUNCT
ejpam-5721	256	1	a	a	DET
ejpam-5721	256	2	subset	subset	NOUN
ejpam-5721	256	3	a	a	PRON
ejpam-5721	256	4	of	of	ADP
ejpam-5721	256	5	a	a	DET
ejpam-5721	256	6	bitopological	bitopological	ADJ
ejpam-5721	256	7	space	space	NOUN
ejpam-5721	256	8	(	(	PUNCT
ejpam-5721	256	9	x	x	NOUN
ejpam-5721	256	10	,	,	PUNCT
ejpam-5721	256	11	τ1	τ1	NOUN
ejpam-5721	256	12	,	,	PUNCT
ejpam-5721	256	13	τ2	τ2	NOUN
ejpam-5721	256	14	)	)	PUNCT
ejpam-5721	256	15	is	be	AUX
ejpam-5721	256	16	said	say	VERB
ejpam-5721	256	17	to	to	PART
ejpam-5721	256	18	be	be	AUX
ejpam-5721	256	19	(	(	PUNCT
ejpam-5721	256	20	τ1	τ1	NOUN
ejpam-5721	256	21	,	,	PUNCT
ejpam-5721	256	22	τ2)p	τ2)p	ADJ
ejpam-5721	256	23	-	-	ADJ
ejpam-5721	256	24	compact	compact	ADJ
ejpam-5721	256	25	relative	relative	NOUN
ejpam-5721	256	26	to	to	ADP
ejpam-5721	256	27	x	x	PRON
ejpam-5721	256	28	if	if	SCONJ
ejpam-5721	256	29	every	every	DET
ejpam-5721	256	30	cover	cover	NOUN
ejpam-5721	256	31	of	of	ADP
ejpam-5721	256	32	a	a	DET
ejpam-5721	256	33	by	by	ADP
ejpam-5721	256	34	(	(	PUNCT
ejpam-5721	256	35	τ1	τ1	NOUN
ejpam-5721	256	36	,	,	PUNCT
ejpam-5721	256	37	τ2)p	τ2)p	ADJ
ejpam-5721	256	38	-	-	PUNCT
ejpam-5721	256	39	open	open	ADJ
ejpam-5721	256	40	sets	set	NOUN
ejpam-5721	256	41	of	of	ADP
ejpam-5721	256	42	x	x	PUNCT
ejpam-5721	256	43	has	have	VERB
ejpam-5721	256	44	a	a	DET
ejpam-5721	256	45	finite	finite	ADJ
ejpam-5721	256	46	subcover	subcover	PROPN
ejpam-5721	256	47	.	.	PUNCT
ejpam-5721	257	1	if	if	SCONJ
ejpam-5721	257	2	a	a	DET
ejpam-5721	257	3	=	=	SYM
ejpam-5721	257	4	x	x	NOUN
ejpam-5721	257	5	,	,	PUNCT
ejpam-5721	257	6	then	then	ADV
ejpam-5721	257	7	x	x	PUNCT
ejpam-5721	257	8	is	be	AUX
ejpam-5721	257	9	said	say	VERB
ejpam-5721	257	10	to	to	PART
ejpam-5721	257	11	be	be	AUX
ejpam-5721	257	12	(	(	PUNCT
ejpam-5721	257	13	τ1	τ1	NOUN
ejpam-5721	257	14	,	,	PUNCT
ejpam-5721	257	15	τ2)p	τ2)p	ADJ
ejpam-5721	257	16	-	-	ADJ
ejpam-5721	257	17	compact	compact	ADJ
ejpam-5721	257	18	.	.	PUNCT
ejpam-5721	258	1	theorem	theorem	VERB
ejpam-5721	258	2	8	8	NUM
ejpam-5721	258	3	.	.	PUNCT
ejpam-5721	259	1	if	if	SCONJ
ejpam-5721	259	2	f	f	PROPN
ejpam-5721	259	3	:	:	PUNCT
ejpam-5721	259	4	(	(	PUNCT
ejpam-5721	259	5	x	x	NOUN
ejpam-5721	259	6	,	,	PUNCT
ejpam-5721	259	7	τ1	τ1	NOUN
ejpam-5721	259	8	,	,	PUNCT
ejpam-5721	259	9	τ2	τ2	NOUN
ejpam-5721	259	10	)	)	PUNCT
ejpam-5721	259	11	→	→	SYM
ejpam-5721	259	12	(	(	PUNCT
ejpam-5721	259	13	y	y	PROPN
ejpam-5721	259	14	,	,	PUNCT
ejpam-5721	259	15	σ1	σ1	PROPN
ejpam-5721	259	16	,	,	PUNCT
ejpam-5721	259	17	σ2	σ2	PROPN
ejpam-5721	259	18	)	)	PUNCT
ejpam-5721	259	19	is	be	AUX
ejpam-5721	259	20	almost	almost	ADV
ejpam-5721	259	21	weakly	weakly	ADJ
ejpam-5721	259	22	(	(	PUNCT
ejpam-5721	259	23	τ1	τ1	NOUN
ejpam-5721	259	24	,	,	PUNCT
ejpam-5721	259	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	259	26	and	and	CCONJ
ejpam-5721	259	27	k	k	PROPN
ejpam-5721	259	28	is	be	AUX
ejpam-5721	259	29	(	(	PUNCT
ejpam-5721	259	30	τ1	τ1	NOUN
ejpam-5721	259	31	,	,	PUNCT
ejpam-5721	259	32	τ2)p	τ2)p	ADJ
ejpam-5721	259	33	-	-	ADJ
ejpam-5721	259	34	compact	compact	ADJ
ejpam-5721	259	35	relative	relative	NOUN
ejpam-5721	259	36	to	to	ADP
ejpam-5721	259	37	x	x	PRON
ejpam-5721	259	38	,	,	PUNCT
ejpam-5721	259	39	then	then	ADV
ejpam-5721	259	40	f(k	f(k	VERB
ejpam-5721	259	41	)	)	PUNCT
ejpam-5721	259	42	is	be	AUX
ejpam-5721	259	43	quasi	quasi	X
ejpam-5721	259	44	(	(	PUNCT
ejpam-5721	259	45	σ1	σ1	PROPN
ejpam-5721	259	46	,	,	PUNCT
ejpam-5721	259	47	σ2)-h	σ2)-h	PROPN
ejpam-5721	259	48	-closed	-closed	ADJ
ejpam-5721	259	49	relative	relative	ADJ
ejpam-5721	259	50	to	to	ADP
ejpam-5721	259	51	y	y	PROPN
ejpam-5721	259	52	.	.	PUNCT
ejpam-5721	260	1	proof	proof	NOUN
ejpam-5721	260	2	.	.	PUNCT
ejpam-5721	261	1	let	let	VERB
ejpam-5721	261	2	{	{	PUNCT
ejpam-5721	261	3	vγ	vγ	VERB
ejpam-5721	261	4	|	|	ADV
ejpam-5721	261	5	γ	γ	PROPN
ejpam-5721	261	6	∈	∈	PROPN
ejpam-5721	261	7	γ	γ	AUX
ejpam-5721	261	8	}	}	PUNCT
ejpam-5721	261	9	be	be	AUX
ejpam-5721	261	10	a	a	DET
ejpam-5721	261	11	cover	cover	NOUN
ejpam-5721	261	12	of	of	ADP
ejpam-5721	261	13	f(k	f(k	VERB
ejpam-5721	261	14	)	)	PUNCT
ejpam-5721	261	15	by	by	ADP
ejpam-5721	261	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	261	17	-	-	PUNCT
ejpam-5721	261	18	open	open	ADJ
ejpam-5721	261	19	sets	set	NOUN
ejpam-5721	261	20	in	in	ADP
ejpam-5721	261	21	y	y	PROPN
ejpam-5721	261	22	.	.	PUNCT
ejpam-5721	262	1	for	for	ADP
ejpam-5721	262	2	each	each	DET
ejpam-5721	262	3	k	k	PROPN
ejpam-5721	262	4	∈	∈	PROPN
ejpam-5721	262	5	k	k	NOUN
ejpam-5721	262	6	,	,	PUNCT
ejpam-5721	262	7	there	there	PRON
ejpam-5721	262	8	exists	exist	VERB
ejpam-5721	262	9	γ(k	γ(k	PROPN
ejpam-5721	262	10	)	)	PUNCT
ejpam-5721	263	1	∈	∈	PROPN
ejpam-5721	263	2	γ	γ	NOUN
ejpam-5721	263	3	such	such	ADJ
ejpam-5721	263	4	that	that	SCONJ
ejpam-5721	263	5	f(k	f(k	ADJ
ejpam-5721	263	6	)	)	PUNCT
ejpam-5721	263	7	∈	∈	PROPN
ejpam-5721	263	8	vγ(k	vγ(k	NOUN
ejpam-5721	263	9	)	)	PUNCT
ejpam-5721	263	10	.	.	PUNCT
ejpam-5721	264	1	since	since	SCONJ
ejpam-5721	264	2	f	f	PROPN
ejpam-5721	264	3	is	be	AUX
ejpam-5721	264	4	almost	almost	ADV
ejpam-5721	264	5	weakly	weakly	ADJ
ejpam-5721	264	6	(	(	PUNCT
ejpam-5721	264	7	τ1	τ1	NOUN
ejpam-5721	264	8	,	,	PUNCT
ejpam-5721	264	9	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	264	10	,	,	PUNCT
ejpam-5721	264	11	by	by	ADP
ejpam-5721	264	12	theorem	theorem	NOUN
ejpam-5721	264	13	1	1	NUM
ejpam-5721	264	14	there	there	ADV
ejpam-5721	264	15	exists	exist	VERB
ejpam-5721	264	16	a	a	DET
ejpam-5721	264	17	(	(	PUNCT
ejpam-5721	264	18	τ1	τ1	NOUN
ejpam-5721	264	19	,	,	PUNCT
ejpam-5721	264	20	τ2)p	τ2)p	ADJ
ejpam-5721	264	21	-	-	PUNCT
ejpam-5721	264	22	open	open	ADJ
ejpam-5721	264	23	set	set	ADJ
ejpam-5721	264	24	uk	uk	PROPN
ejpam-5721	264	25	of	of	ADP
ejpam-5721	264	26	x	x	PUNCT
ejpam-5721	264	27	containing	contain	VERB
ejpam-5721	264	28	k	k	PROPN
ejpam-5721	264	29	such	such	ADJ
ejpam-5721	264	30	that	that	SCONJ
ejpam-5721	264	31	f(uk	f(uk	PROPN
ejpam-5721	264	32	)	)	PUNCT
ejpam-5721	264	33	⊆	⊆	NUM
ejpam-5721	264	34	σ1σ2	σ1σ2	NUM
ejpam-5721	264	35	-	-	PUNCT
ejpam-5721	264	36	cl(vγ(k	cl(vγ(k	NOUN
ejpam-5721	264	37	)	)	PUNCT
ejpam-5721	264	38	)	)	PUNCT
ejpam-5721	264	39	.	.	PUNCT
ejpam-5721	265	1	since	since	SCONJ
ejpam-5721	265	2	{	{	PUNCT
ejpam-5721	265	3	uk	uk	PROPN
ejpam-5721	265	4	|	|	ADV
ejpam-5721	265	5	k	k	PROPN
ejpam-5721	265	6	∈	∈	PROPN
ejpam-5721	265	7	k	k	AUX
ejpam-5721	265	8	}	}	PUNCT
ejpam-5721	265	9	is	be	AUX
ejpam-5721	265	10	a	a	DET
ejpam-5721	265	11	cover	cover	NOUN
ejpam-5721	265	12	of	of	ADP
ejpam-5721	265	13	k	k	X
ejpam-5721	265	14	by	by	ADP
ejpam-5721	265	15	(	(	PUNCT
ejpam-5721	265	16	τ1	τ1	NOUN
ejpam-5721	265	17	,	,	PUNCT
ejpam-5721	265	18	τ2)p	τ2)p	ADJ
ejpam-5721	265	19	-	-	PUNCT
ejpam-5721	265	20	open	open	ADJ
ejpam-5721	265	21	sets	set	NOUN
ejpam-5721	265	22	in	in	ADP
ejpam-5721	265	23	x	x	NOUN
ejpam-5721	265	24	,	,	PUNCT
ejpam-5721	265	25	there	there	PRON
ejpam-5721	265	26	exists	exist	VERB
ejpam-5721	265	27	a	a	DET
ejpam-5721	265	28	finite	finite	NOUN
ejpam-5721	265	29	subset	subset	VERB
ejpam-5721	265	30	k0	k0	PROPN
ejpam-5721	265	31	of	of	ADP
ejpam-5721	265	32	k	k	PROPN
ejpam-5721	265	33	such	such	ADJ
ejpam-5721	265	34	that	that	SCONJ
ejpam-5721	265	35	k	k	PROPN
ejpam-5721	265	36	⊆	⊆	NUM
ejpam-5721	265	37	∪{uk	∪{uk	PROPN
ejpam-5721	265	38	|	|	NOUN
ejpam-5721	265	39	k	k	PROPN
ejpam-5721	265	40	∈	∈	PROPN
ejpam-5721	265	41	k0	k0	PROPN
ejpam-5721	265	42	}	}	PUNCT
ejpam-5721	265	43	.	.	PUNCT
ejpam-5721	266	1	thus	thus	ADV
ejpam-5721	266	2	,	,	PUNCT
ejpam-5721	266	3	f(k	f(k	VERB
ejpam-5721	266	4	)	)	PUNCT
ejpam-5721	266	5	⊆	⊆	NUM
ejpam-5721	266	6	∪{f(uk	∪{f(uk	NUM
ejpam-5721	266	7	)	)	PUNCT
ejpam-5721	266	8	|	|	ADV
ejpam-5721	266	9	k	k	PROPN
ejpam-5721	266	10	∈	∈	PROPN
ejpam-5721	266	11	k0	k0	PROPN
ejpam-5721	266	12	}	}	PUNCT
ejpam-5721	266	13	⊆	⊆	NUM
ejpam-5721	266	14	∪{σ1σ2	∪{σ1σ2	NOUN
ejpam-5721	266	15	-	-	PUNCT
ejpam-5721	266	16	cl(vγ(k	cl(vγ(k	NOUN
ejpam-5721	266	17	)	)	PUNCT
ejpam-5721	266	18	)	)	PUNCT
ejpam-5721	267	1	|	|	ADV
ejpam-5721	267	2	k	k	PROPN
ejpam-5721	267	3	∈	∈	PROPN
ejpam-5721	267	4	k0	k0	PROPN
ejpam-5721	267	5	}	}	PUNCT
ejpam-5721	267	6	.	.	PUNCT
ejpam-5721	268	1	this	this	PRON
ejpam-5721	268	2	shows	show	VERB
ejpam-5721	268	3	that	that	SCONJ
ejpam-5721	268	4	f(k	f(k	VERB
ejpam-5721	268	5	)	)	PUNCT
ejpam-5721	268	6	is	be	AUX
ejpam-5721	268	7	quasi	quasi	X
ejpam-5721	268	8	(	(	PUNCT
ejpam-5721	268	9	σ1	σ1	PROPN
ejpam-5721	268	10	,	,	PUNCT
ejpam-5721	268	11	σ2)-h	σ2)-h	PROPN
ejpam-5721	268	12	-closed	-closed	ADJ
ejpam-5721	268	13	relative	relative	ADJ
ejpam-5721	268	14	to	to	ADP
ejpam-5721	268	15	y	y	PROPN
ejpam-5721	268	16	.	.	PUNCT
ejpam-5721	269	1	corollary	corollary	ADJ
ejpam-5721	269	2	1	1	NUM
ejpam-5721	269	3	.	.	PUNCT
ejpam-5721	270	1	if	if	SCONJ
ejpam-5721	270	2	f	f	PROPN
ejpam-5721	270	3	:	:	PUNCT
ejpam-5721	270	4	(	(	PUNCT
ejpam-5721	270	5	x	x	NOUN
ejpam-5721	270	6	,	,	PUNCT
ejpam-5721	270	7	τ1	τ1	NOUN
ejpam-5721	270	8	,	,	PUNCT
ejpam-5721	270	9	τ2	τ2	NOUN
ejpam-5721	270	10	)	)	PUNCT
ejpam-5721	270	11	→	→	SYM
ejpam-5721	270	12	(	(	PUNCT
ejpam-5721	270	13	y	y	PROPN
ejpam-5721	270	14	,	,	PUNCT
ejpam-5721	270	15	σ1	σ1	PROPN
ejpam-5721	270	16	,	,	PUNCT
ejpam-5721	270	17	σ2	σ2	PROPN
ejpam-5721	270	18	)	)	PUNCT
ejpam-5721	270	19	is	be	AUX
ejpam-5721	270	20	an	an	DET
ejpam-5721	270	21	almost	almost	ADV
ejpam-5721	270	22	weakly	weakly	ADJ
ejpam-5721	270	23	(	(	PUNCT
ejpam-5721	270	24	τ1	τ1	NOUN
ejpam-5721	270	25	,	,	PUNCT
ejpam-5721	270	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	270	27	surjection	surjection	NOUN
ejpam-5721	270	28	and	and	CCONJ
ejpam-5721	270	29	(	(	PUNCT
ejpam-5721	270	30	x	x	NOUN
ejpam-5721	270	31	,	,	PUNCT
ejpam-5721	270	32	τ1	τ1	NOUN
ejpam-5721	270	33	,	,	PUNCT
ejpam-5721	270	34	τ2	τ2	NOUN
ejpam-5721	270	35	)	)	PUNCT
ejpam-5721	270	36	is	be	AUX
ejpam-5721	270	37	(	(	PUNCT
ejpam-5721	270	38	τ1	τ1	NOUN
ejpam-5721	270	39	,	,	PUNCT
ejpam-5721	270	40	τ2)p	τ2)p	ADJ
ejpam-5721	270	41	-	-	ADJ
ejpam-5721	270	42	compact	compact	ADJ
ejpam-5721	270	43	,	,	PUNCT
ejpam-5721	270	44	then	then	ADV
ejpam-5721	270	45	(	(	PUNCT
ejpam-5721	270	46	y	y	PROPN
ejpam-5721	270	47	,	,	PUNCT
ejpam-5721	270	48	σ1	σ1	PROPN
ejpam-5721	270	49	,	,	PUNCT
ejpam-5721	270	50	σ2	σ2	PROPN
ejpam-5721	270	51	)	)	PUNCT
ejpam-5721	270	52	is	be	AUX
ejpam-5721	270	53	quasi	quasi	X
ejpam-5721	270	54	(	(	PUNCT
ejpam-5721	270	55	σ1	σ1	PROPN
ejpam-5721	270	56	,	,	PUNCT
ejpam-5721	270	57	σ2)-h	σ2)-h	PROPN
ejpam-5721	270	58	-closed	-closed	ADJ
ejpam-5721	270	59	.	.	PUNCT
ejpam-5721	271	1	definition	definition	NOUN
ejpam-5721	271	2	9	9	NUM
ejpam-5721	271	3	.	.	PUNCT
ejpam-5721	272	1	[	[	X
ejpam-5721	272	2	14	14	NUM
ejpam-5721	272	3	]	]	PUNCT
ejpam-5721	272	4	a	a	DET
ejpam-5721	272	5	bitopological	bitopological	ADJ
ejpam-5721	272	6	space	space	NOUN
ejpam-5721	272	7	(	(	PUNCT
ejpam-5721	272	8	x	x	NOUN
ejpam-5721	272	9	,	,	PUNCT
ejpam-5721	272	10	τ1	τ1	NOUN
ejpam-5721	272	11	,	,	PUNCT
ejpam-5721	272	12	τ2	τ2	NOUN
ejpam-5721	272	13	)	)	PUNCT
ejpam-5721	272	14	is	be	AUX
ejpam-5721	272	15	said	say	VERB
ejpam-5721	272	16	to	to	PART
ejpam-5721	272	17	be	be	AUX
ejpam-5721	272	18	τ1τ2	τ1τ2	NOUN
ejpam-5721	272	19	-	-	ADJ
ejpam-5721	272	20	connected	connected	ADJ
ejpam-5721	272	21	if	if	SCONJ
ejpam-5721	272	22	x	x	PRON
ejpam-5721	272	23	can	can	AUX
ejpam-5721	272	24	not	not	PART
ejpam-5721	272	25	be	be	AUX
ejpam-5721	272	26	written	write	VERB
ejpam-5721	272	27	as	as	ADP
ejpam-5721	272	28	the	the	DET
ejpam-5721	272	29	union	union	NOUN
ejpam-5721	272	30	of	of	ADP
ejpam-5721	272	31	two	two	NUM
ejpam-5721	272	32	disjoint	disjoint	NOUN
ejpam-5721	272	33	nonempty	nonempty	ADJ
ejpam-5721	272	34	τ1τ2	τ1τ2	ADJ
ejpam-5721	272	35	-	-	ADJ
ejpam-5721	272	36	open	open	ADJ
ejpam-5721	272	37	sets	set	NOUN
ejpam-5721	272	38	.	.	PUNCT
ejpam-5721	273	1	definition	definition	NOUN
ejpam-5721	273	2	10	10	NUM
ejpam-5721	273	3	.	.	PUNCT
ejpam-5721	274	1	a	a	DET
ejpam-5721	274	2	bitopological	bitopological	ADJ
ejpam-5721	274	3	space	space	NOUN
ejpam-5721	274	4	(	(	PUNCT
ejpam-5721	274	5	x	x	NOUN
ejpam-5721	274	6	,	,	PUNCT
ejpam-5721	274	7	τ1	τ1	NOUN
ejpam-5721	274	8	,	,	PUNCT
ejpam-5721	274	9	τ2	τ2	NOUN
ejpam-5721	274	10	)	)	PUNCT
ejpam-5721	274	11	is	be	AUX
ejpam-5721	274	12	said	say	VERB
ejpam-5721	274	13	to	to	PART
ejpam-5721	274	14	be	be	AUX
ejpam-5721	274	15	(	(	PUNCT
ejpam-5721	274	16	τ1	τ1	NOUN
ejpam-5721	274	17	,	,	PUNCT
ejpam-5721	274	18	τ2)p	τ2)p	NOUN
ejpam-5721	274	19	-	-	PUNCT
ejpam-5721	274	20	connected	connected	ADJ
ejpam-5721	274	21	if	if	SCONJ
ejpam-5721	274	22	x	x	PRON
ejpam-5721	274	23	can	can	AUX
ejpam-5721	274	24	not	not	PART
ejpam-5721	274	25	be	be	AUX
ejpam-5721	274	26	written	write	VERB
ejpam-5721	274	27	as	as	ADP
ejpam-5721	274	28	the	the	DET
ejpam-5721	274	29	union	union	NOUN
ejpam-5721	274	30	of	of	ADP
ejpam-5721	274	31	two	two	NUM
ejpam-5721	274	32	disjoint	disjoint	NOUN
ejpam-5721	274	33	nonempty	nonempty	NOUN
ejpam-5721	274	34	(	(	PUNCT
ejpam-5721	274	35	τ1	τ1	NOUN
ejpam-5721	274	36	,	,	PUNCT
ejpam-5721	274	37	τ2)p	τ2)p	ADJ
ejpam-5721	274	38	-	-	PUNCT
ejpam-5721	274	39	open	open	ADJ
ejpam-5721	274	40	sets	set	NOUN
ejpam-5721	274	41	.	.	PUNCT
ejpam-5721	275	1	theorem	theorem	NOUN
ejpam-5721	275	2	9	9	NUM
ejpam-5721	275	3	.	.	PUNCT
ejpam-5721	276	1	if	if	SCONJ
ejpam-5721	276	2	f	f	PROPN
ejpam-5721	276	3	:	:	PUNCT
ejpam-5721	276	4	(	(	PUNCT
ejpam-5721	276	5	x	x	NOUN
ejpam-5721	276	6	,	,	PUNCT
ejpam-5721	276	7	τ1	τ1	NOUN
ejpam-5721	276	8	,	,	PUNCT
ejpam-5721	276	9	τ2	τ2	NOUN
ejpam-5721	276	10	)	)	PUNCT
ejpam-5721	276	11	→	→	SYM
ejpam-5721	276	12	(	(	PUNCT
ejpam-5721	276	13	y	y	PROPN
ejpam-5721	276	14	,	,	PUNCT
ejpam-5721	276	15	σ1	σ1	PROPN
ejpam-5721	276	16	,	,	PUNCT
ejpam-5721	276	17	σ2	σ2	PROPN
ejpam-5721	276	18	)	)	PUNCT
ejpam-5721	276	19	is	be	AUX
ejpam-5721	276	20	an	an	DET
ejpam-5721	276	21	almost	almost	ADV
ejpam-5721	276	22	weakly	weakly	ADJ
ejpam-5721	276	23	(	(	PUNCT
ejpam-5721	276	24	τ1	τ1	NOUN
ejpam-5721	276	25	,	,	PUNCT
ejpam-5721	276	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	276	27	surjection	surjection	NOUN
ejpam-5721	276	28	and	and	CCONJ
ejpam-5721	276	29	(	(	PUNCT
ejpam-5721	276	30	x	x	NOUN
ejpam-5721	276	31	,	,	PUNCT
ejpam-5721	276	32	τ1	τ1	NOUN
ejpam-5721	276	33	,	,	PUNCT
ejpam-5721	276	34	τ2	τ2	NOUN
ejpam-5721	276	35	)	)	PUNCT
ejpam-5721	276	36	is	be	AUX
ejpam-5721	276	37	(	(	PUNCT
ejpam-5721	276	38	τ1	τ1	NOUN
ejpam-5721	276	39	,	,	PUNCT
ejpam-5721	276	40	τ2)p	τ2)p	NOUN
ejpam-5721	276	41	-	-	PUNCT
ejpam-5721	276	42	connected	connect	VERB
ejpam-5721	276	43	,	,	PUNCT
ejpam-5721	276	44	then	then	ADV
ejpam-5721	276	45	(	(	PUNCT
ejpam-5721	276	46	y	y	PROPN
ejpam-5721	276	47	,	,	PUNCT
ejpam-5721	276	48	σ1	σ1	PROPN
ejpam-5721	276	49	,	,	PUNCT
ejpam-5721	276	50	σ2	σ2	PROPN
ejpam-5721	276	51	)	)	PUNCT
ejpam-5721	276	52	is	be	AUX
ejpam-5721	276	53	σ1σ2	σ1σ2	NOUN
ejpam-5721	276	54	-	-	PUNCT
ejpam-5721	276	55	connected	connected	ADJ
ejpam-5721	276	56	.	.	PUNCT
ejpam-5721	277	1	proof	proof	NOUN
ejpam-5721	277	2	.	.	PUNCT
ejpam-5721	278	1	suppose	suppose	VERB
ejpam-5721	278	2	that	that	SCONJ
ejpam-5721	278	3	(	(	PUNCT
ejpam-5721	278	4	y	y	PROPN
ejpam-5721	278	5	,	,	PUNCT
ejpam-5721	278	6	σ1	σ1	PROPN
ejpam-5721	278	7	,	,	PUNCT
ejpam-5721	278	8	σ2	σ2	PROPN
ejpam-5721	278	9	)	)	PUNCT
ejpam-5721	278	10	is	be	AUX
ejpam-5721	278	11	not	not	PART
ejpam-5721	278	12	σ1σ2	σ1σ2	VERB
ejpam-5721	278	13	-	-	PUNCT
ejpam-5721	278	14	connected	connect	VERB
ejpam-5721	278	15	.	.	PUNCT
ejpam-5721	279	1	then	then	ADV
ejpam-5721	279	2	,	,	PUNCT
ejpam-5721	279	3	there	there	PRON
ejpam-5721	279	4	exist	exist	VERB
ejpam-5721	279	5	nonempty	nonempty	ADV
ejpam-5721	279	6	σ1σ2	σ1σ2	NOUN
ejpam-5721	279	7	-	-	ADJ
ejpam-5721	279	8	open	open	ADJ
ejpam-5721	279	9	sets	set	NOUN
ejpam-5721	279	10	u	u	NOUN
ejpam-5721	279	11	and	and	CCONJ
ejpam-5721	279	12	v	v	NOUN
ejpam-5721	279	13	of	of	ADP
ejpam-5721	279	14	y	y	PRON
ejpam-5721	279	15	such	such	ADJ
ejpam-5721	279	16	that	that	SCONJ
ejpam-5721	279	17	u	u	PROPN
ejpam-5721	279	18	∩	∩	NOUN
ejpam-5721	279	19	v	v	NOUN
ejpam-5721	279	20	=	=	NOUN
ejpam-5721	279	21	∅	∅	NOUN
ejpam-5721	279	22	and	and	CCONJ
ejpam-5721	279	23	u	u	NOUN
ejpam-5721	279	24	∪	∪	NOUN
ejpam-5721	279	25	v	v	ADP
ejpam-5721	279	26	=	=	SYM
ejpam-5721	279	27	y	y	PROPN
ejpam-5721	279	28	.	.	PUNCT
ejpam-5721	280	1	it	it	PRON
ejpam-5721	280	2	follows	follow	VERB
ejpam-5721	280	3	that	that	SCONJ
ejpam-5721	280	4	f−1(u	f−1(u	NOUN
ejpam-5721	280	5	)	)	PUNCT
ejpam-5721	280	6	∩	∩	ADJ
ejpam-5721	280	7	f−1(v	f−1(v	NOUN
ejpam-5721	280	8	)	)	PUNCT
ejpam-5721	281	1	=	=	NOUN
ejpam-5721	281	2	∅	∅	NOUN
ejpam-5721	281	3	and	and	CCONJ
ejpam-5721	281	4	f−1(u	f−1(u	NOUN
ejpam-5721	281	5	)	)	PUNCT
ejpam-5721	281	6	∪	∪	NOUN
ejpam-5721	281	7	f−1(v	f−1(v	NOUN
ejpam-5721	281	8	)	)	PUNCT
ejpam-5721	282	1	=	=	PUNCT
ejpam-5721	282	2	x.	x.	NOUN
ejpam-5721	282	3	since	since	SCONJ
ejpam-5721	282	4	f	f	PROPN
ejpam-5721	282	5	is	be	AUX
ejpam-5721	282	6	surjective	surjective	ADJ
ejpam-5721	282	7	and	and	CCONJ
ejpam-5721	282	8	u	u	NOUN
ejpam-5721	282	9	,	,	PUNCT
ejpam-5721	282	10	v	v	NUM
ejpam-5721	282	11	are	be	AUX
ejpam-5721	282	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	282	13	-	-	ADJ
ejpam-5721	282	14	closed	closed	ADJ
ejpam-5721	282	15	and	and	CCONJ
ejpam-5721	282	16	σ1σ2	σ1σ2	NOUN
ejpam-5721	282	17	-	-	ADJ
ejpam-5721	282	18	open	open	ADJ
ejpam-5721	282	19	,	,	PUNCT
ejpam-5721	282	20	by	by	ADP
ejpam-5721	282	21	theorem	theorem	NOUN
ejpam-5721	282	22	1	1	NUM
ejpam-5721	282	23	the	the	DET
ejpam-5721	282	24	inverse	inverse	NOUN
ejpam-5721	282	25	images	image	NOUN
ejpam-5721	282	26	of	of	ADP
ejpam-5721	282	27	u	u	NOUN
ejpam-5721	282	28	and	and	CCONJ
ejpam-5721	282	29	v	v	NOUN
ejpam-5721	282	30	are	be	AUX
ejpam-5721	282	31	nonempty	nonempty	ADJ
ejpam-5721	282	32	(	(	PUNCT
ejpam-5721	282	33	τ1	τ1	NOUN
ejpam-5721	282	34	,	,	PUNCT
ejpam-5721	282	35	τ2)p	τ2)p	ADJ
ejpam-5721	282	36	-	-	PUNCT
ejpam-5721	282	37	open	open	ADJ
ejpam-5721	282	38	sets	set	NOUN
ejpam-5721	282	39	in	in	ADP
ejpam-5721	282	40	x.	x.	NOUN
ejpam-5721	282	41	this	this	PRON
ejpam-5721	282	42	means	mean	VERB
ejpam-5721	282	43	that	that	SCONJ
ejpam-5721	282	44	(	(	PUNCT
ejpam-5721	282	45	x	x	NOUN
ejpam-5721	282	46	,	,	PUNCT
ejpam-5721	282	47	τ1	τ1	NOUN
ejpam-5721	282	48	,	,	PUNCT
ejpam-5721	282	49	τ2	τ2	NOUN
ejpam-5721	282	50	)	)	PUNCT
ejpam-5721	282	51	is	be	AUX
ejpam-5721	282	52	not	not	PART
ejpam-5721	282	53	(	(	PUNCT
ejpam-5721	282	54	τ1	τ1	NOUN
ejpam-5721	282	55	,	,	PUNCT
ejpam-5721	282	56	τ2)p	τ2)p	NOUN
ejpam-5721	282	57	-	-	PUNCT
ejpam-5721	282	58	connected	connect	VERB
ejpam-5721	282	59	.	.	PUNCT
ejpam-5721	283	1	this	this	PRON
ejpam-5721	283	2	is	be	AUX
ejpam-5721	283	3	a	a	DET
ejpam-5721	283	4	contradiction	contradiction	NOUN
ejpam-5721	283	5	.	.	PUNCT
ejpam-5721	284	1	it	it	PRON
ejpam-5721	284	2	follows	follow	VERB
ejpam-5721	284	3	that	that	SCONJ
ejpam-5721	284	4	(	(	PUNCT
ejpam-5721	284	5	y	y	PROPN
ejpam-5721	284	6	,	,	PUNCT
ejpam-5721	284	7	σ1	σ1	PROPN
ejpam-5721	284	8	,	,	PUNCT
ejpam-5721	284	9	σ2	σ2	PROPN
ejpam-5721	284	10	)	)	PUNCT
ejpam-5721	284	11	is	be	AUX
ejpam-5721	284	12	σ1σ2	σ1σ2	NOUN
ejpam-5721	284	13	-	-	PUNCT
ejpam-5721	284	14	connected	connect	VERB
ejpam-5721	284	15	.	.	PUNCT
ejpam-5721	285	1	corollary	corollary	ADJ
ejpam-5721	285	2	2	2	NUM
ejpam-5721	285	3	.	.	PUNCT
ejpam-5721	286	1	if	if	SCONJ
ejpam-5721	286	2	f	f	PROPN
ejpam-5721	286	3	:	:	PUNCT
ejpam-5721	286	4	(	(	PUNCT
ejpam-5721	286	5	x	x	NOUN
ejpam-5721	286	6	,	,	PUNCT
ejpam-5721	286	7	τ1	τ1	NOUN
ejpam-5721	286	8	,	,	PUNCT
ejpam-5721	286	9	τ2	τ2	NOUN
ejpam-5721	286	10	)	)	PUNCT
ejpam-5721	286	11	→	→	SYM
ejpam-5721	286	12	(	(	PUNCT
ejpam-5721	286	13	y	y	PROPN
ejpam-5721	286	14	,	,	PUNCT
ejpam-5721	286	15	σ1	σ1	PROPN
ejpam-5721	286	16	,	,	PUNCT
ejpam-5721	286	17	σ2	σ2	PROPN
ejpam-5721	286	18	)	)	PUNCT
ejpam-5721	286	19	is	be	AUX
ejpam-5721	286	20	a	a	DET
ejpam-5721	286	21	(	(	PUNCT
ejpam-5721	286	22	τ1	τ1	NOUN
ejpam-5721	286	23	,	,	PUNCT
ejpam-5721	286	24	τ2)p	τ2)p	ADJ
ejpam-5721	286	25	-	-	ADJ
ejpam-5721	286	26	continuous	continuous	ADJ
ejpam-5721	286	27	surjection	surjection	NOUN
ejpam-5721	286	28	and	and	CCONJ
ejpam-5721	286	29	(	(	PUNCT
ejpam-5721	286	30	x	x	NOUN
ejpam-5721	286	31	,	,	PUNCT
ejpam-5721	286	32	τ1	τ1	NOUN
ejpam-5721	286	33	,	,	PUNCT
ejpam-5721	286	34	τ2	τ2	NOUN
ejpam-5721	286	35	)	)	PUNCT
ejpam-5721	286	36	is	be	AUX
ejpam-5721	286	37	(	(	PUNCT
ejpam-5721	286	38	τ1	τ1	NOUN
ejpam-5721	286	39	,	,	PUNCT
ejpam-5721	286	40	τ2)p	τ2)p	NOUN
ejpam-5721	286	41	-	-	PUNCT
ejpam-5721	286	42	connected	connect	VERB
ejpam-5721	286	43	,	,	PUNCT
ejpam-5721	286	44	then	then	ADV
ejpam-5721	286	45	(	(	PUNCT
ejpam-5721	286	46	y	y	PROPN
ejpam-5721	286	47	,	,	PUNCT
ejpam-5721	286	48	σ1	σ1	PROPN
ejpam-5721	286	49	,	,	PUNCT
ejpam-5721	286	50	σ2	σ2	PROPN
ejpam-5721	286	51	)	)	PUNCT
ejpam-5721	286	52	is	be	AUX
ejpam-5721	286	53	σ1σ2	σ1σ2	NOUN
ejpam-5721	286	54	-	-	PUNCT
ejpam-5721	286	55	connected	connect	VERB
ejpam-5721	286	56	.	.	PUNCT
ejpam-5721	287	1	definition	definition	NOUN
ejpam-5721	287	2	11	11	NUM
ejpam-5721	287	3	.	.	PUNCT
ejpam-5721	288	1	a	a	DET
ejpam-5721	288	2	function	function	NOUN
ejpam-5721	288	3	f	f	NOUN
ejpam-5721	288	4	:	:	PUNCT
ejpam-5721	288	5	(	(	PUNCT
ejpam-5721	288	6	x	x	NOUN
ejpam-5721	288	7	,	,	PUNCT
ejpam-5721	288	8	τ1	τ1	NOUN
ejpam-5721	288	9	,	,	PUNCT
ejpam-5721	288	10	τ2	τ2	NOUN
ejpam-5721	288	11	)	)	PUNCT
ejpam-5721	288	12	→	→	SYM
ejpam-5721	288	13	(	(	PUNCT
ejpam-5721	288	14	y	y	PROPN
ejpam-5721	288	15	,	,	PUNCT
ejpam-5721	288	16	σ1	σ1	PROPN
ejpam-5721	288	17	,	,	PUNCT
ejpam-5721	288	18	σ2	σ2	PROPN
ejpam-5721	288	19	)	)	PUNCT
ejpam-5721	288	20	is	be	AUX
ejpam-5721	288	21	said	say	VERB
ejpam-5721	288	22	to	to	PART
ejpam-5721	288	23	be	be	AUX
ejpam-5721	288	24	(	(	PUNCT
ejpam-5721	288	25	τ1	τ1	NOUN
ejpam-5721	288	26	,	,	PUNCT
ejpam-5721	288	27	τ2)p	τ2)p	NOUN
ejpam-5721	288	28	-	-	PUNCT
ejpam-5721	288	29	irresolute	irresolute	ADJ
ejpam-5721	288	30	if	if	SCONJ
ejpam-5721	288	31	for	for	ADP
ejpam-5721	288	32	each	each	DET
ejpam-5721	288	33	x	x	SYM
ejpam-5721	288	34	∈	∈	PROPN
ejpam-5721	288	35	x	x	X
ejpam-5721	288	36	and	and	CCONJ
ejpam-5721	288	37	each	each	DET
ejpam-5721	288	38	(	(	PUNCT
ejpam-5721	288	39	σ1	σ1	PROPN
ejpam-5721	288	40	,	,	PUNCT
ejpam-5721	288	41	σ2)p	σ2)p	NOUN
ejpam-5721	288	42	-	-	PUNCT
ejpam-5721	288	43	open	open	NOUN
ejpam-5721	288	44	set	set	NOUN
ejpam-5721	288	45	v	v	NOUN
ejpam-5721	288	46	of	of	ADP
ejpam-5721	288	47	y	y	NOUN
ejpam-5721	288	48	containing	contain	VERB
ejpam-5721	288	49	f(x	f(x	PROPN
ejpam-5721	288	50	)	)	PUNCT
ejpam-5721	288	51	,	,	PUNCT
ejpam-5721	288	52	there	there	PRON
ejpam-5721	288	53	exists	exist	VERB
ejpam-5721	288	54	a	a	DET
ejpam-5721	288	55	(	(	PUNCT
ejpam-5721	288	56	τ1	τ1	NOUN
ejpam-5721	288	57	,	,	PUNCT
ejpam-5721	288	58	τ2)p	τ2)p	ADJ
ejpam-5721	288	59	-	-	PUNCT
ejpam-5721	288	60	open	open	ADJ
ejpam-5721	288	61	set	set	NOUN
ejpam-5721	288	62	u	u	NOUN
ejpam-5721	288	63	of	of	ADP
ejpam-5721	288	64	x	x	SYM
ejpam-5721	288	65	such	such	ADJ
ejpam-5721	288	66	that	that	DET
ejpam-5721	288	67	f(u	f(u	PROPN
ejpam-5721	288	68	)	)	PUNCT
ejpam-5721	288	69	⊆	⊆	NUM
ejpam-5721	288	70	v	v	NOUN
ejpam-5721	288	71	.	.	PUNCT
ejpam-5721	289	1	j.	j.	PROPN
ejpam-5721	289	2	khampakdee	khampakdee	PROPN
ejpam-5721	289	3	,	,	PUNCT
ejpam-5721	289	4	s.	s.	PROPN
ejpam-5721	289	5	sompong	sompong	PROPN
ejpam-5721	289	6	,	,	PUNCT
ejpam-5721	289	7	c.	c.	PROPN
ejpam-5721	289	8	boonpok	boonpok	PROPN
ejpam-5721	289	9	/	/	SYM
ejpam-5721	289	10	eur	eur	PROPN
ejpam-5721	289	11	.	.	PUNCT
ejpam-5721	290	1	j.	j.	PROPN
ejpam-5721	290	2	pure	pure	PROPN
ejpam-5721	290	3	appl	appl	PROPN
ejpam-5721	290	4	.	.	PROPN
ejpam-5721	290	5	math	math	PROPN
ejpam-5721	290	6	,	,	PUNCT
ejpam-5721	290	7	18	18	NUM
ejpam-5721	290	8	(	(	PUNCT
ejpam-5721	290	9	1	1	NUM
ejpam-5721	290	10	)	)	PUNCT
ejpam-5721	290	11	(	(	PUNCT
ejpam-5721	290	12	2025	2025	NUM
ejpam-5721	290	13	)	)	PUNCT
ejpam-5721	290	14	,	,	PUNCT
ejpam-5721	290	15	5721	5721	NUM
ejpam-5721	290	16	10	10	NUM
ejpam-5721	290	17	of	of	ADP
ejpam-5721	290	18	12	12	NUM
ejpam-5721	290	19	theorem	theorem	NOUN
ejpam-5721	290	20	10	10	NUM
ejpam-5721	290	21	.	.	PUNCT
ejpam-5721	291	1	if	if	SCONJ
ejpam-5721	291	2	f	f	PROPN
ejpam-5721	291	3	:	:	PUNCT
ejpam-5721	291	4	(	(	PUNCT
ejpam-5721	291	5	x	x	NOUN
ejpam-5721	291	6	,	,	PUNCT
ejpam-5721	291	7	τ1	τ1	NOUN
ejpam-5721	291	8	,	,	PUNCT
ejpam-5721	291	9	τ2	τ2	NOUN
ejpam-5721	291	10	)	)	PUNCT
ejpam-5721	291	11	→	→	SYM
ejpam-5721	291	12	(	(	PUNCT
ejpam-5721	291	13	y	y	PROPN
ejpam-5721	291	14	,	,	PUNCT
ejpam-5721	291	15	σ1	σ1	PROPN
ejpam-5721	291	16	,	,	PUNCT
ejpam-5721	291	17	σ2	σ2	PROPN
ejpam-5721	291	18	)	)	PUNCT
ejpam-5721	291	19	is	be	AUX
ejpam-5721	291	20	(	(	PUNCT
ejpam-5721	291	21	τ1	τ1	NOUN
ejpam-5721	291	22	,	,	PUNCT
ejpam-5721	291	23	τ2)p	τ2)p	NOUN
ejpam-5721	291	24	-	-	PUNCT
ejpam-5721	291	25	irresolute	irresolute	ADJ
ejpam-5721	291	26	and	and	CCONJ
ejpam-5721	291	27	g	g	NOUN
ejpam-5721	291	28	:	:	PUNCT
ejpam-5721	291	29	(	(	PUNCT
ejpam-5721	291	30	y	y	PROPN
ejpam-5721	291	31	,	,	PUNCT
ejpam-5721	291	32	σ1	σ1	PROPN
ejpam-5721	291	33	,	,	PUNCT
ejpam-5721	291	34	σ2	σ2	NOUN
ejpam-5721	291	35	)	)	PUNCT
ejpam-5721	291	36	→	→	SYM
ejpam-5721	291	37	(	(	PUNCT
ejpam-5721	291	38	z	z	NOUN
ejpam-5721	291	39	,	,	PUNCT
ejpam-5721	291	40	ρ1	ρ1	NOUN
ejpam-5721	291	41	,	,	PUNCT
ejpam-5721	291	42	ρ2	ρ2	NOUN
ejpam-5721	291	43	)	)	PUNCT
ejpam-5721	291	44	is	be	AUX
ejpam-5721	291	45	almost	almost	ADV
ejpam-5721	291	46	weakly	weakly	ADJ
ejpam-5721	291	47	(	(	PUNCT
ejpam-5721	291	48	σ1	σ1	NOUN
ejpam-5721	291	49	,	,	PUNCT
ejpam-5721	291	50	σ2)-continuous	σ2)-continuous	PROPN
ejpam-5721	291	51	,	,	PUNCT
ejpam-5721	291	52	then	then	ADV
ejpam-5721	291	53	the	the	DET
ejpam-5721	291	54	composition	composition	NOUN
ejpam-5721	291	55	g	g	PROPN
ejpam-5721	291	56	◦	◦	NOUN
ejpam-5721	291	57	f	f	X
ejpam-5721	291	58	:	:	PUNCT
ejpam-5721	291	59	(	(	PUNCT
ejpam-5721	291	60	x	x	NOUN
ejpam-5721	291	61	,	,	PUNCT
ejpam-5721	291	62	τ1	τ1	NOUN
ejpam-5721	291	63	,	,	PUNCT
ejpam-5721	291	64	τ2	τ2	NOUN
ejpam-5721	291	65	)	)	PUNCT
ejpam-5721	291	66	→	→	SYM
ejpam-5721	292	1	(	(	PUNCT
ejpam-5721	292	2	z	z	NOUN
ejpam-5721	292	3	,	,	PUNCT
ejpam-5721	292	4	ρ1	ρ1	NOUN
ejpam-5721	292	5	,	,	PUNCT
ejpam-5721	292	6	ρ2	ρ2	NOUN
ejpam-5721	292	7	)	)	PUNCT
ejpam-5721	292	8	is	be	AUX
ejpam-5721	292	9	almost	almost	ADV
ejpam-5721	292	10	weakly	weakly	ADJ
ejpam-5721	292	11	(	(	PUNCT
ejpam-5721	292	12	σ1	σ1	NOUN
ejpam-5721	292	13	,	,	PUNCT
ejpam-5721	292	14	σ2)-continuous	σ2)-continuous	ADJ
ejpam-5721	292	15	.	.	PUNCT
ejpam-5721	293	1	proof	proof	NOUN
ejpam-5721	293	2	.	.	PUNCT
ejpam-5721	294	1	let	let	VERB
ejpam-5721	294	2	x	x	PUNCT
ejpam-5721	294	3	∈	∈	PROPN
ejpam-5721	294	4	x	x	X
ejpam-5721	294	5	and	and	CCONJ
ejpam-5721	294	6	w	w	PROPN
ejpam-5721	294	7	be	be	AUX
ejpam-5721	294	8	any	any	DET
ejpam-5721	294	9	ρ1ρ2	ρ1ρ2	NOUN
ejpam-5721	294	10	-	-	PUNCT
ejpam-5721	294	11	open	open	ADJ
ejpam-5721	294	12	set	set	NOUN
ejpam-5721	294	13	of	of	ADP
ejpam-5721	294	14	z	z	NOUN
ejpam-5721	294	15	containing	contain	VERB
ejpam-5721	294	16	(	(	PUNCT
ejpam-5721	294	17	g	g	PROPN
ejpam-5721	294	18	◦	◦	NOUN
ejpam-5721	294	19	f)(x	f)(x	PROPN
ejpam-5721	294	20	)	)	PUNCT
ejpam-5721	294	21	.	.	PUNCT
ejpam-5721	295	1	since	since	SCONJ
ejpam-5721	295	2	g	g	PROPN
ejpam-5721	295	3	is	be	AUX
ejpam-5721	295	4	almost	almost	ADV
ejpam-5721	295	5	weakly	weakly	ADJ
ejpam-5721	295	6	(	(	PUNCT
ejpam-5721	295	7	σ1	σ1	NOUN
ejpam-5721	295	8	,	,	PUNCT
ejpam-5721	295	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-5721	295	10	,	,	PUNCT
ejpam-5721	295	11	there	there	PRON
ejpam-5721	295	12	exists	exist	VERB
ejpam-5721	295	13	a	a	DET
ejpam-5721	295	14	(	(	PUNCT
ejpam-5721	295	15	σ1	σ1	PROPN
ejpam-5721	295	16	,	,	PUNCT
ejpam-5721	295	17	σ2)p	σ2)p	NOUN
ejpam-5721	295	18	-	-	PUNCT
ejpam-5721	295	19	open	open	NOUN
ejpam-5721	295	20	set	set	NOUN
ejpam-5721	295	21	v	v	NOUN
ejpam-5721	295	22	of	of	ADP
ejpam-5721	295	23	y	y	NOUN
ejpam-5721	295	24	containing	contain	VERB
ejpam-5721	295	25	f(x	f(x	PROPN
ejpam-5721	295	26	)	)	PUNCT
ejpam-5721	295	27	such	such	ADJ
ejpam-5721	295	28	that	that	SCONJ
ejpam-5721	295	29	g(v	g(v	PROPN
ejpam-5721	295	30	)	)	PUNCT
ejpam-5721	295	31	⊆	⊆	X
ejpam-5721	295	32	ρ1ρ2	ρ1ρ2	NOUN
ejpam-5721	295	33	-	-	PUNCT
ejpam-5721	295	34	cl(w	cl(w	NOUN
ejpam-5721	295	35	)	)	PUNCT
ejpam-5721	295	36	.	.	PUNCT
ejpam-5721	296	1	since	since	SCONJ
ejpam-5721	296	2	f	f	PROPN
ejpam-5721	296	3	is	be	AUX
ejpam-5721	296	4	(	(	PUNCT
ejpam-5721	296	5	τ1	τ1	NOUN
ejpam-5721	296	6	,	,	PUNCT
ejpam-5721	296	7	τ2)p	τ2)p	NOUN
ejpam-5721	296	8	-	-	PUNCT
ejpam-5721	296	9	irresolute	irresolute	ADJ
ejpam-5721	296	10	,	,	PUNCT
ejpam-5721	296	11	there	there	PRON
ejpam-5721	296	12	exists	exist	VERB
ejpam-5721	296	13	a	a	DET
ejpam-5721	296	14	(	(	PUNCT
ejpam-5721	296	15	τ1	τ1	NOUN
ejpam-5721	296	16	,	,	PUNCT
ejpam-5721	296	17	τ2)p	τ2)p	ADJ
ejpam-5721	296	18	-	-	PUNCT
ejpam-5721	296	19	open	open	ADJ
ejpam-5721	296	20	set	set	NOUN
ejpam-5721	296	21	u	u	NOUN
ejpam-5721	296	22	of	of	ADP
ejpam-5721	296	23	x	x	PUNCT
ejpam-5721	296	24	containing	contain	VERB
ejpam-5721	296	25	x	x	PUNCT
ejpam-5721	296	26	such	such	ADJ
ejpam-5721	296	27	that	that	DET
ejpam-5721	296	28	f(u	f(u	PROPN
ejpam-5721	296	29	)	)	PUNCT
ejpam-5721	296	30	⊆	⊆	NUM
ejpam-5721	296	31	v	v	NOUN
ejpam-5721	296	32	.	.	PUNCT
ejpam-5721	297	1	thus	thus	ADV
ejpam-5721	297	2	,	,	PUNCT
ejpam-5721	297	3	(	(	PUNCT
ejpam-5721	297	4	g	g	NOUN
ejpam-5721	297	5	◦	◦	NOUN
ejpam-5721	297	6	f)(u	f)(u	ADJ
ejpam-5721	297	7	)	)	PUNCT
ejpam-5721	297	8	=	=	SYM
ejpam-5721	297	9	g(f(u	g(f(u	NOUN
ejpam-5721	297	10	)	)	PUNCT
ejpam-5721	297	11	)	)	PUNCT
ejpam-5721	298	1	⊆	⊆	X
ejpam-5721	298	2	ρ1ρ2	ρ1ρ2	NOUN
ejpam-5721	298	3	-	-	PUNCT
ejpam-5721	298	4	cl(w	cl(w	NOUN
ejpam-5721	298	5	)	)	PUNCT
ejpam-5721	298	6	.	.	PUNCT
ejpam-5721	299	1	this	this	PRON
ejpam-5721	299	2	shows	show	VERB
ejpam-5721	299	3	that	that	SCONJ
ejpam-5721	299	4	g	g	PROPN
ejpam-5721	299	5	◦	◦	NOUN
ejpam-5721	299	6	f	f	PROPN
ejpam-5721	299	7	is	be	AUX
ejpam-5721	299	8	almost	almost	ADV
ejpam-5721	299	9	weakly	weakly	ADJ
ejpam-5721	299	10	(	(	PUNCT
ejpam-5721	299	11	σ1	σ1	NOUN
ejpam-5721	299	12	,	,	PUNCT
ejpam-5721	299	13	σ2)-continuous	σ2)-continuous	PROPN
ejpam-5721	299	14	.	.	NOUN
ejpam-5721	299	15	4	4	NUM
ejpam-5721	299	16	.	.	X
ejpam-5721	299	17	conclusion	conclusion	NOUN
ejpam-5721	299	18	stronger	strong	ADJ
ejpam-5721	299	19	and	and	CCONJ
ejpam-5721	299	20	weaker	weak	ADJ
ejpam-5721	299	21	forms	form	NOUN
ejpam-5721	299	22	of	of	ADP
ejpam-5721	299	23	open	open	ADJ
ejpam-5721	299	24	sets	set	NOUN
ejpam-5721	299	25	in	in	ADP
ejpam-5721	299	26	topological	topological	ADJ
ejpam-5721	299	27	spaces	space	NOUN
ejpam-5721	299	28	such	such	ADJ
ejpam-5721	299	29	as	as	ADP
ejpam-5721	299	30	semi	semi	ADJ
ejpam-5721	299	31	-	-	ADJ
ejpam-5721	299	32	open	open	ADJ
ejpam-5721	299	33	sets	set	NOUN
ejpam-5721	299	34	,	,	PUNCT
ejpam-5721	299	35	preopen	preopen	ADJ
ejpam-5721	299	36	sets	set	NOUN
ejpam-5721	299	37	,	,	PUNCT
ejpam-5721	299	38	α	α	NOUN
ejpam-5721	299	39	-	-	ADJ
ejpam-5721	299	40	open	open	ADJ
ejpam-5721	299	41	sets	set	NOUN
ejpam-5721	299	42	,	,	PUNCT
ejpam-5721	299	43	β	β	ADJ
ejpam-5721	299	44	-	-	ADJ
ejpam-5721	299	45	open	open	ADJ
ejpam-5721	299	46	sets	set	NOUN
ejpam-5721	299	47	,	,	PUNCT
ejpam-5721	299	48	θ	θ	ADJ
ejpam-5721	299	49	-	-	ADJ
ejpam-5721	299	50	open	open	ADJ
ejpam-5721	299	51	sets	set	NOUN
ejpam-5721	299	52	and	and	CCONJ
ejpam-5721	299	53	δ	δ	NOUN
ejpam-5721	299	54	-	-	PUNCT
ejpam-5721	299	55	open	open	ADJ
ejpam-5721	299	56	sets	set	NOUN
ejpam-5721	299	57	play	play	VERB
ejpam-5721	299	58	an	an	DET
ejpam-5721	299	59	important	important	ADJ
ejpam-5721	299	60	role	role	NOUN
ejpam-5721	299	61	in	in	ADP
ejpam-5721	299	62	the	the	DET
ejpam-5721	299	63	researching	researching	NOUN
ejpam-5721	299	64	of	of	ADP
ejpam-5721	299	65	generalizations	generalization	NOUN
ejpam-5721	299	66	of	of	ADP
ejpam-5721	299	67	continuity	continuity	NOUN
ejpam-5721	299	68	.	.	PUNCT
ejpam-5721	300	1	using	use	VERB
ejpam-5721	300	2	different	different	ADJ
ejpam-5721	300	3	forms	form	NOUN
ejpam-5721	300	4	of	of	ADP
ejpam-5721	300	5	open	open	ADJ
ejpam-5721	300	6	sets	set	NOUN
ejpam-5721	300	7	,	,	PUNCT
ejpam-5721	300	8	many	many	ADJ
ejpam-5721	300	9	authors	author	NOUN
ejpam-5721	300	10	have	have	AUX
ejpam-5721	300	11	introduced	introduce	VERB
ejpam-5721	300	12	and	and	CCONJ
ejpam-5721	300	13	investigated	investigate	VERB
ejpam-5721	300	14	various	various	ADJ
ejpam-5721	300	15	types	type	NOUN
ejpam-5721	300	16	of	of	ADP
ejpam-5721	300	17	weak	weak	ADJ
ejpam-5721	300	18	forms	form	NOUN
ejpam-5721	300	19	of	of	ADP
ejpam-5721	300	20	continuity	continuity	NOUN
ejpam-5721	300	21	for	for	ADP
ejpam-5721	300	22	functions	function	NOUN
ejpam-5721	300	23	and	and	CCONJ
ejpam-5721	300	24	multifunctions	multifunction	NOUN
ejpam-5721	300	25	.	.	PUNCT
ejpam-5721	301	1	this	this	DET
ejpam-5721	301	2	work	work	NOUN
ejpam-5721	301	3	deals	deal	VERB
ejpam-5721	301	4	with	with	ADP
ejpam-5721	301	5	the	the	DET
ejpam-5721	301	6	concept	concept	NOUN
ejpam-5721	301	7	of	of	ADP
ejpam-5721	301	8	almost	almost	ADV
ejpam-5721	301	9	weakly	weakly	ADJ
ejpam-5721	301	10	(	(	PUNCT
ejpam-5721	301	11	τ1	τ1	NOUN
ejpam-5721	301	12	,	,	PUNCT
ejpam-5721	301	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	301	14	functions	function	NOUN
ejpam-5721	301	15	.	.	PUNCT
ejpam-5721	302	1	additionally	additionally	ADV
ejpam-5721	302	2	,	,	PUNCT
ejpam-5721	302	3	several	several	ADJ
ejpam-5721	302	4	characterizations	characterization	NOUN
ejpam-5721	302	5	and	and	CCONJ
ejpam-5721	302	6	some	some	DET
ejpam-5721	302	7	properties	property	NOUN
ejpam-5721	302	8	concerning	concern	VERB
ejpam-5721	302	9	almost	almost	ADV
ejpam-5721	302	10	weakly	weakly	ADJ
ejpam-5721	302	11	(	(	PUNCT
ejpam-5721	302	12	τ1	τ1	NOUN
ejpam-5721	302	13	,	,	PUNCT
ejpam-5721	302	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	302	15	functions	function	NOUN
ejpam-5721	302	16	are	be	AUX
ejpam-5721	302	17	obtained	obtain	VERB
ejpam-5721	302	18	.	.	PUNCT
ejpam-5721	303	1	the	the	DET
ejpam-5721	303	2	ideas	idea	NOUN
ejpam-5721	303	3	and	and	CCONJ
ejpam-5721	303	4	results	result	NOUN
ejpam-5721	303	5	of	of	ADP
ejpam-5721	303	6	this	this	DET
ejpam-5721	303	7	work	work	NOUN
ejpam-5721	303	8	may	may	AUX
ejpam-5721	303	9	motivate	motivate	VERB
ejpam-5721	303	10	further	further	ADJ
ejpam-5721	303	11	research	research	NOUN
ejpam-5721	303	12	.	.	PUNCT
ejpam-5721	304	1	acknowledgements	acknowledgement	NOUN
ejpam-5721	304	2	this	this	DET
ejpam-5721	304	3	research	research	NOUN
ejpam-5721	304	4	project	project	NOUN
ejpam-5721	304	5	was	be	AUX
ejpam-5721	304	6	financially	financially	ADV
ejpam-5721	304	7	supported	support	VERB
ejpam-5721	304	8	by	by	ADP
ejpam-5721	304	9	mahasarakham	mahasarakham	PROPN
ejpam-5721	304	10	university	university	PROPN
ejpam-5721	304	11	.	.	PUNCT
ejpam-5721	305	1	references	reference	NOUN
ejpam-5721	305	2	[	[	X
ejpam-5721	305	3	1	1	NUM
ejpam-5721	305	4	]	]	PUNCT
ejpam-5721	305	5	c.	c.	PROPN
ejpam-5721	305	6	boonpok	boonpok	PROPN
ejpam-5721	305	7	.	.	PUNCT
ejpam-5721	306	1	almost	almost	ADV
ejpam-5721	306	2	(	(	PUNCT
ejpam-5721	306	3	g	g	NOUN
ejpam-5721	306	4	,	,	PUNCT
ejpam-5721	306	5	m)-continuous	m)-continuous	ADJ
ejpam-5721	306	6	functions	function	NOUN
ejpam-5721	306	7	.	.	PUNCT
ejpam-5721	307	1	international	international	ADJ
ejpam-5721	307	2	journal	journal	PROPN
ejpam-5721	307	3	of	of	ADP
ejpam-5721	307	4	mathematical	mathematical	ADJ
ejpam-5721	307	5	analysis	analysis	NOUN
ejpam-5721	307	6	,	,	PUNCT
ejpam-5721	307	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5721	307	8	,	,	PUNCT
ejpam-5721	307	9	2010	2010	NUM
ejpam-5721	307	10	.	.	PUNCT
ejpam-5721	308	1	[	[	X
ejpam-5721	308	2	2	2	NUM
ejpam-5721	308	3	]	]	PUNCT
ejpam-5721	308	4	c.	c.	PROPN
ejpam-5721	308	5	boonpok	boonpok	PROPN
ejpam-5721	308	6	.	.	PUNCT
ejpam-5721	309	1	m	m	VERB
ejpam-5721	309	2	-continuous	-continuous	ADJ
ejpam-5721	309	3	functions	function	NOUN
ejpam-5721	309	4	in	in	ADP
ejpam-5721	309	5	biminimal	biminimal	NOUN
ejpam-5721	309	6	structure	structure	NOUN
ejpam-5721	309	7	spaces	space	NOUN
ejpam-5721	309	8	.	.	PUNCT
ejpam-5721	310	1	far	far	PROPN
ejpam-5721	310	2	east	east	PROPN
ejpam-5721	310	3	journal	journal	PROPN
ejpam-5721	310	4	of	of	ADP
ejpam-5721	310	5	mathematical	mathematical	ADJ
ejpam-5721	310	6	sciences	science	NOUN
ejpam-5721	310	7	,	,	PUNCT
ejpam-5721	310	8	43(1):41–58	43(1):41–58	NUM
ejpam-5721	310	9	,	,	PUNCT
ejpam-5721	310	10	2010	2010	NUM
ejpam-5721	310	11	.	.	PUNCT
ejpam-5721	311	1	[	[	X
ejpam-5721	311	2	3	3	X
ejpam-5721	311	3	]	]	PUNCT
ejpam-5721	311	4	c.	c.	PROPN
ejpam-5721	311	5	boonpok	boonpok	PROPN
ejpam-5721	311	6	.	.	PUNCT
ejpam-5721	312	1	on	on	ADP
ejpam-5721	312	2	characterizations	characterization	NOUN
ejpam-5721	312	3	of	of	ADP
ejpam-5721	312	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5721	312	5	ideal	ideal	ADJ
ejpam-5721	312	6	topological	topological	ADJ
ejpam-5721	312	7	spaces	space	NOUN
ejpam-5721	312	8	.	.	PUNCT
ejpam-5721	313	1	journal	journal	NOUN
ejpam-5721	313	2	of	of	ADP
ejpam-5721	313	3	mathematics	mathematic	NOUN
ejpam-5721	313	4	,	,	PUNCT
ejpam-5721	313	5	2020:9387601	2020:9387601	NUM
ejpam-5721	313	6	,	,	PUNCT
ejpam-5721	313	7	2020	2020	NUM
ejpam-5721	313	8	.	.	PUNCT
ejpam-5721	314	1	[	[	X
ejpam-5721	314	2	4	4	NUM
ejpam-5721	314	3	]	]	PUNCT
ejpam-5721	314	4	c.	c.	PROPN
ejpam-5721	314	5	boonpok	boonpok	PROPN
ejpam-5721	314	6	.	.	PUNCT
ejpam-5721	315	1	(	(	PUNCT
ejpam-5721	315	2	τ1	τ1	NOUN
ejpam-5721	315	3	,	,	PUNCT
ejpam-5721	315	4	τ2)δ	τ2)δ	ADJ
ejpam-5721	315	5	-	-	PUNCT
ejpam-5721	315	6	semicontinuous	semicontinuous	ADJ
ejpam-5721	315	7	multifunctions	multifunction	NOUN
ejpam-5721	315	8	.	.	PUNCT
ejpam-5721	316	1	heliyon	heliyon	NOUN
ejpam-5721	316	2	,	,	PUNCT
ejpam-5721	316	3	6	6	NUM
ejpam-5721	316	4	:	:	SYM
ejpam-5721	316	5	e05367	e05367	PROPN
ejpam-5721	316	6	,	,	PUNCT
ejpam-5721	316	7	2020	2020	NUM
ejpam-5721	316	8	.	.	PUNCT
ejpam-5721	317	1	[	[	X
ejpam-5721	317	2	5	5	X
ejpam-5721	317	3	]	]	PUNCT
ejpam-5721	317	4	c.	c.	PROPN
ejpam-5721	317	5	boonpok	boonpok	PROPN
ejpam-5721	317	6	.	.	PUNCT
ejpam-5721	318	1	on	on	ADP
ejpam-5721	318	2	some	some	DET
ejpam-5721	318	3	closed	closed	ADJ
ejpam-5721	318	4	sets	set	NOUN
ejpam-5721	318	5	and	and	CCONJ
ejpam-5721	318	6	low	low	ADJ
ejpam-5721	318	7	separation	separation	NOUN
ejpam-5721	318	8	axioms	axiom	NOUN
ejpam-5721	318	9	via	via	ADP
ejpam-5721	318	10	topological	topological	ADJ
ejpam-5721	318	11	ideals	ideal	NOUN
ejpam-5721	318	12	.	.	PUNCT
ejpam-5721	319	1	european	european	ADJ
ejpam-5721	319	2	journal	journal	PROPN
ejpam-5721	319	3	of	of	ADP
ejpam-5721	319	4	pure	pure	ADJ
ejpam-5721	319	5	and	and	CCONJ
ejpam-5721	319	6	applied	applied	ADJ
ejpam-5721	319	7	mathematics	mathematic	NOUN
ejpam-5721	319	8	,	,	PUNCT
ejpam-5721	319	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-5721	319	10	,	,	PUNCT
ejpam-5721	319	11	2022	2022	NUM
ejpam-5721	319	12	.	.	PUNCT
ejpam-5721	320	1	[	[	X
ejpam-5721	320	2	6	6	NUM
ejpam-5721	320	3	]	]	PUNCT
ejpam-5721	320	4	c.	c.	PROPN
ejpam-5721	320	5	boonpok	boonpok	PROPN
ejpam-5721	320	6	.	.	PUNCT
ejpam-5721	321	1	on	on	ADP
ejpam-5721	321	2	some	some	DET
ejpam-5721	321	3	spaces	space	NOUN
ejpam-5721	321	4	via	via	ADP
ejpam-5721	321	5	topological	topological	ADJ
ejpam-5721	321	6	ideals	ideal	NOUN
ejpam-5721	321	7	.	.	PUNCT
ejpam-5721	322	1	open	open	ADJ
ejpam-5721	322	2	mathematics	mathematic	NOUN
ejpam-5721	322	3	,	,	PUNCT
ejpam-5721	322	4	21:20230118	21:20230118	NUM
ejpam-5721	322	5	,	,	PUNCT
ejpam-5721	322	6	2023	2023	NUM
ejpam-5721	322	7	.	.	PUNCT
ejpam-5721	323	1	[	[	X
ejpam-5721	323	2	7	7	X
ejpam-5721	323	3	]	]	X
ejpam-5721	323	4	c.	c.	PROPN
ejpam-5721	323	5	boonpok	boonpok	PROPN
ejpam-5721	323	6	.	.	PUNCT
ejpam-5721	324	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5721	324	2	.	.	PUNCT
ejpam-5721	325	1	mathematica	mathematica	PROPN
ejpam-5721	325	2	,	,	PUNCT
ejpam-5721	325	3	65(1):31–42	65(1):31–42	NUM
ejpam-5721	325	4	,	,	PUNCT
ejpam-5721	325	5	2023	2023	NUM
ejpam-5721	325	6	.	.	PUNCT
ejpam-5721	326	1	[	[	X
ejpam-5721	326	2	8	8	NUM
ejpam-5721	326	3	]	]	X
ejpam-5721	326	4	c.	c.	PROPN
ejpam-5721	326	5	boonpok	boonpok	PROPN
ejpam-5721	326	6	and	and	CCONJ
ejpam-5721	326	7	j.	j.	PROPN
ejpam-5721	326	8	khampakdee	khampakdee	PROPN
ejpam-5721	326	9	.	.	PUNCT
ejpam-5721	327	1	(	(	PUNCT
ejpam-5721	327	2	λ	λ	NOUN
ejpam-5721	327	3	,	,	PUNCT
ejpam-5721	327	4	sp)-open	sp)-open	ADJ
ejpam-5721	327	5	sets	set	NOUN
ejpam-5721	327	6	in	in	ADP
ejpam-5721	327	7	topological	topological	ADJ
ejpam-5721	327	8	spaces	space	NOUN
ejpam-5721	327	9	.	.	PUNCT
ejpam-5721	328	1	european	european	ADJ
ejpam-5721	328	2	journal	journal	PROPN
ejpam-5721	328	3	of	of	ADP
ejpam-5721	328	4	pure	pure	ADJ
ejpam-5721	328	5	and	and	CCONJ
ejpam-5721	328	6	applied	applied	ADJ
ejpam-5721	328	7	mathematics	mathematic	NOUN
ejpam-5721	328	8	,	,	PUNCT
ejpam-5721	328	9	15(2):572–588	15(2):572–588	NUM
ejpam-5721	328	10	,	,	PUNCT
ejpam-5721	328	11	2022	2022	NUM
ejpam-5721	328	12	.	.	PUNCT
ejpam-5721	329	1	j.	j.	PROPN
ejpam-5721	329	2	khampakdee	khampakdee	PROPN
ejpam-5721	329	3	,	,	PUNCT
ejpam-5721	329	4	s.	s.	PROPN
ejpam-5721	329	5	sompong	sompong	PROPN
ejpam-5721	329	6	,	,	PUNCT
ejpam-5721	329	7	c.	c.	PROPN
ejpam-5721	329	8	boonpok	boonpok	PROPN
ejpam-5721	329	9	/	/	SYM
ejpam-5721	329	10	eur	eur	PROPN
ejpam-5721	329	11	.	.	PUNCT
ejpam-5721	330	1	j.	j.	PROPN
ejpam-5721	330	2	pure	pure	PROPN
ejpam-5721	330	3	appl	appl	PROPN
ejpam-5721	330	4	.	.	PROPN
ejpam-5721	330	5	math	math	PROPN
ejpam-5721	330	6	,	,	PUNCT
ejpam-5721	330	7	18	18	NUM
ejpam-5721	330	8	(	(	PUNCT
ejpam-5721	330	9	1	1	NUM
ejpam-5721	330	10	)	)	PUNCT
ejpam-5721	330	11	(	(	PUNCT
ejpam-5721	330	12	2025	2025	NUM
ejpam-5721	330	13	)	)	PUNCT
ejpam-5721	330	14	,	,	PUNCT
ejpam-5721	330	15	5721	5721	NUM
ejpam-5721	330	16	11	11	NUM
ejpam-5721	330	17	of	of	ADP
ejpam-5721	330	18	12	12	NUM
ejpam-5721	330	19	[	[	SYM
ejpam-5721	330	20	9	9	NUM
ejpam-5721	330	21	]	]	PUNCT
ejpam-5721	330	22	c.	c.	NOUN
ejpam-5721	330	23	boonpok	boonpok	PROPN
ejpam-5721	330	24	and	and	CCONJ
ejpam-5721	330	25	j.	j.	PROPN
ejpam-5721	330	26	khampakdee	khampakdee	PROPN
ejpam-5721	330	27	.	.	PUNCT
ejpam-5721	331	1	almost	almost	ADV
ejpam-5721	331	2	strong	strong	ADJ
ejpam-5721	331	3	θ(λ	θ(λ	PROPN
ejpam-5721	331	4	,	,	PUNCT
ejpam-5721	331	5	p)-continuity	p)-continuity	NOUN
ejpam-5721	331	6	for	for	ADP
ejpam-5721	331	7	functions	function	NOUN
ejpam-5721	331	8	.	.	PUNCT
ejpam-5721	332	1	european	european	ADJ
ejpam-5721	332	2	journal	journal	PROPN
ejpam-5721	332	3	of	of	ADP
ejpam-5721	332	4	pure	pure	ADJ
ejpam-5721	332	5	and	and	CCONJ
ejpam-5721	332	6	applied	applied	ADJ
ejpam-5721	332	7	mathematics	mathematic	NOUN
ejpam-5721	332	8	,	,	PUNCT
ejpam-5721	332	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5721	332	10	,	,	PUNCT
ejpam-5721	332	11	2024	2024	NUM
ejpam-5721	332	12	.	.	PUNCT
ejpam-5721	333	1	[	[	X
ejpam-5721	333	2	10	10	NUM
ejpam-5721	333	3	]	]	X
ejpam-5721	333	4	c.	c.	PROPN
ejpam-5721	333	5	boonpok	boonpok	PROPN
ejpam-5721	333	6	and	and	CCONJ
ejpam-5721	333	7	c.	c.	PROPN
ejpam-5721	333	8	klanarong	klanarong	PROPN
ejpam-5721	333	9	.	.	PUNCT
ejpam-5721	334	1	on	on	ADP
ejpam-5721	334	2	weakly	weakly	ADJ
ejpam-5721	334	3	(	(	PUNCT
ejpam-5721	334	4	τ1	τ1	NOUN
ejpam-5721	334	5	,	,	PUNCT
ejpam-5721	334	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	334	7	functions	function	NOUN
ejpam-5721	334	8	.	.	PUNCT
ejpam-5721	335	1	european	european	ADJ
ejpam-5721	335	2	journal	journal	PROPN
ejpam-5721	335	3	of	of	ADP
ejpam-5721	335	4	pure	pure	ADJ
ejpam-5721	335	5	and	and	CCONJ
ejpam-5721	335	6	applied	applied	ADJ
ejpam-5721	335	7	mathematics	mathematic	NOUN
ejpam-5721	335	8	,	,	PUNCT
ejpam-5721	335	9	17(1):416–425	17(1):416–425	NUM
ejpam-5721	335	10	,	,	PUNCT
ejpam-5721	335	11	2024	2024	NUM
ejpam-5721	335	12	.	.	PUNCT
ejpam-5721	336	1	[	[	X
ejpam-5721	336	2	11	11	NUM
ejpam-5721	336	3	]	]	X
ejpam-5721	336	4	c.	c.	PROPN
ejpam-5721	336	5	boonpok	boonpok	PROPN
ejpam-5721	336	6	and	and	CCONJ
ejpam-5721	336	7	p.	p.	NOUN
ejpam-5721	336	8	pue	pue	NOUN
ejpam-5721	336	9	-	-	PUNCT
ejpam-5721	336	10	on	on	ADP
ejpam-5721	336	11	.	.	PUNCT
ejpam-5721	337	1	characterizations	characterization	NOUN
ejpam-5721	337	2	of	of	ADP
ejpam-5721	337	3	almost	almost	ADV
ejpam-5721	337	4	(	(	PUNCT
ejpam-5721	337	5	τ1	τ1	NOUN
ejpam-5721	337	6	,	,	PUNCT
ejpam-5721	337	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	337	8	functions	function	NOUN
ejpam-5721	337	9	.	.	PUNCT
ejpam-5721	338	1	international	international	ADJ
ejpam-5721	338	2	journal	journal	NOUN
ejpam-5721	338	3	of	of	ADP
ejpam-5721	338	4	analysis	analysis	NOUN
ejpam-5721	338	5	and	and	CCONJ
ejpam-5721	338	6	applications	application	NOUN
ejpam-5721	338	7	,	,	PUNCT
ejpam-5721	338	8	22:33	22:33	NUM
ejpam-5721	338	9	,	,	PUNCT
ejpam-5721	338	10	2024	2024	NUM
ejpam-5721	338	11	.	.	PUNCT
ejpam-5721	339	1	[	[	X
ejpam-5721	339	2	12	12	NUM
ejpam-5721	339	3	]	]	X
ejpam-5721	339	4	c.	c.	PROPN
ejpam-5721	339	5	boonpok	boonpok	PROPN
ejpam-5721	339	6	and	and	CCONJ
ejpam-5721	339	7	n.	n.	PROPN
ejpam-5721	339	8	srisarakham	srisarakham	PROPN
ejpam-5721	339	9	.	.	PUNCT
ejpam-5721	340	1	weak	weak	ADJ
ejpam-5721	340	2	forms	form	NOUN
ejpam-5721	340	3	of	of	ADP
ejpam-5721	340	4	(	(	PUNCT
ejpam-5721	340	5	λ	λ	PROPN
ejpam-5721	340	6	,	,	PUNCT
ejpam-5721	340	7	b)-open	b)-open	VERB
ejpam-5721	340	8	sets	set	NOUN
ejpam-5721	340	9	and	and	CCONJ
ejpam-5721	340	10	weak	weak	ADJ
ejpam-5721	340	11	(	(	PUNCT
ejpam-5721	340	12	λ	λ	NOUN
ejpam-5721	340	13	,	,	PUNCT
ejpam-5721	340	14	b)continuity	b)continuity	NOUN
ejpam-5721	340	15	.	.	PUNCT
ejpam-5721	341	1	european	european	PROPN
ejpam-5721	341	2	journal	journal	PROPN
ejpam-5721	341	3	of	of	ADP
ejpam-5721	341	4	pure	pure	ADJ
ejpam-5721	341	5	and	and	CCONJ
ejpam-5721	341	6	applied	applied	ADJ
ejpam-5721	341	7	mathematics	mathematic	NOUN
ejpam-5721	341	8	,	,	PUNCT
ejpam-5721	341	9	16(1):29–43	16(1):29–43	NUM
ejpam-5721	341	10	,	,	PUNCT
ejpam-5721	341	11	2023	2023	NUM
ejpam-5721	341	12	.	.	PUNCT
ejpam-5721	342	1	[	[	X
ejpam-5721	342	2	13	13	NUM
ejpam-5721	342	3	]	]	PUNCT
ejpam-5721	342	4	c.	c.	PROPN
ejpam-5721	342	5	boonpok	boonpok	PROPN
ejpam-5721	342	6	and	and	CCONJ
ejpam-5721	342	7	n.	n.	PROPN
ejpam-5721	342	8	srisarakham	srisarakham	PROPN
ejpam-5721	342	9	.	.	PUNCT
ejpam-5721	343	1	(	(	PUNCT
ejpam-5721	343	2	τ1	τ1	NOUN
ejpam-5721	343	3	,	,	PUNCT
ejpam-5721	343	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5721	343	5	for	for	ADP
ejpam-5721	343	6	functions	function	NOUN
ejpam-5721	343	7	.	.	PUNCT
ejpam-5721	344	1	asia	asia	PROPN
ejpam-5721	344	2	pacific	pacific	PROPN
ejpam-5721	344	3	journal	journal	PROPN
ejpam-5721	344	4	of	of	ADP
ejpam-5721	344	5	mathematics	mathematic	NOUN
ejpam-5721	344	6	,	,	PUNCT
ejpam-5721	344	7	11:21	11:21	NUM
ejpam-5721	344	8	,	,	PUNCT
ejpam-5721	344	9	2024	2024	NUM
ejpam-5721	344	10	.	.	PUNCT
ejpam-5721	345	1	[	[	X
ejpam-5721	345	2	14	14	NUM
ejpam-5721	345	3	]	]	X
ejpam-5721	345	4	c.	c.	PROPN
ejpam-5721	345	5	boonpok	boonpok	PROPN
ejpam-5721	345	6	,	,	PUNCT
ejpam-5721	345	7	c.	c.	PROPN
ejpam-5721	345	8	viriyapong	viriyapong	PROPN
ejpam-5721	345	9	,	,	PUNCT
ejpam-5721	345	10	and	and	CCONJ
ejpam-5721	345	11	m.	m.	NOUN
ejpam-5721	345	12	thongmoon	thongmoon	NOUN
ejpam-5721	345	13	.	.	PUNCT
ejpam-5721	346	1	on	on	ADP
ejpam-5721	346	2	upper	upper	ADJ
ejpam-5721	346	3	and	and	CCONJ
ejpam-5721	346	4	lower	low	ADJ
ejpam-5721	346	5	(	(	PUNCT
ejpam-5721	346	6	τ1	τ1	NOUN
ejpam-5721	346	7	,	,	PUNCT
ejpam-5721	346	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5721	346	9	multifunctions	multifunction	NOUN
ejpam-5721	346	10	.	.	PUNCT
ejpam-5721	347	1	journal	journal	PROPN
ejpam-5721	347	2	of	of	ADP
ejpam-5721	347	3	mathematics	mathematics	PROPN
ejpam-5721	347	4	and	and	CCONJ
ejpam-5721	347	5	computer	computer	NOUN
ejpam-5721	347	6	science	science	NOUN
ejpam-5721	347	7	,	,	PUNCT
ejpam-5721	347	8	18:282–293	18:282–293	NUM
ejpam-5721	347	9	,	,	PUNCT
ejpam-5721	347	10	2018	2018	NUM
ejpam-5721	347	11	.	.	PUNCT
ejpam-5721	348	1	[	[	X
ejpam-5721	348	2	15	15	NUM
ejpam-5721	348	3	]	]	X
ejpam-5721	348	4	m.	m.	NOUN
ejpam-5721	348	5	chiangpradit	chiangpradit	NOUN
ejpam-5721	348	6	,	,	PUNCT
ejpam-5721	348	7	s.	s.	PROPN
ejpam-5721	348	8	sompong	sompong	PROPN
ejpam-5721	348	9	,	,	PUNCT
ejpam-5721	348	10	and	and	CCONJ
ejpam-5721	348	11	c.	c.	PROPN
ejpam-5721	348	12	boonpok	boonpok	PROPN
ejpam-5721	348	13	.	.	PUNCT
ejpam-5721	349	1	on	on	ADP
ejpam-5721	349	2	characterizations	characterization	NOUN
ejpam-5721	349	3	of	of	ADP
ejpam-5721	349	4	(	(	PUNCT
ejpam-5721	349	5	τ1	τ1	NOUN
ejpam-5721	349	6	,	,	PUNCT
ejpam-5721	349	7	τ2)regular	τ2)regular	ADJ
ejpam-5721	349	8	spaces	space	NOUN
ejpam-5721	349	9	.	.	PUNCT
ejpam-5721	350	1	international	international	ADJ
ejpam-5721	350	2	journal	journal	PROPN
ejpam-5721	350	3	of	of	ADP
ejpam-5721	350	4	mathematics	mathematic	NOUN
ejpam-5721	350	5	and	and	CCONJ
ejpam-5721	350	6	computer	computer	NOUN
ejpam-5721	350	7	science	science	NOUN
ejpam-5721	350	8	,	,	PUNCT
ejpam-5721	350	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-5721	350	10	,	,	PUNCT
ejpam-5721	350	11	2024	2024	NUM
ejpam-5721	350	12	.	.	PUNCT
ejpam-5721	351	1	[	[	X
ejpam-5721	351	2	16	16	NUM
ejpam-5721	351	3	]	]	PUNCT
ejpam-5721	351	4	m.	m.	NOUN
ejpam-5721	351	5	chiangpradit	chiangpradit	NOUN
ejpam-5721	351	6	,	,	PUNCT
ejpam-5721	351	7	s.	s.	PROPN
ejpam-5721	351	8	sompong	sompong	PROPN
ejpam-5721	351	9	,	,	PUNCT
ejpam-5721	351	10	and	and	CCONJ
ejpam-5721	351	11	c.	c.	PROPN
ejpam-5721	351	12	boonpok	boonpok	PROPN
ejpam-5721	351	13	.	.	PUNCT
ejpam-5721	352	1	weakly	weakly	ADJ
ejpam-5721	352	2	quasi	quasi	NOUN
ejpam-5721	352	3	(	(	PUNCT
ejpam-5721	352	4	τ1	τ1	PROPN
ejpam-5721	352	5	,	,	PUNCT
ejpam-5721	352	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	352	7	functions	function	NOUN
ejpam-5721	352	8	.	.	PUNCT
ejpam-5721	353	1	international	international	ADJ
ejpam-5721	353	2	journal	journal	NOUN
ejpam-5721	353	3	of	of	ADP
ejpam-5721	353	4	analysis	analysis	NOUN
ejpam-5721	353	5	and	and	CCONJ
ejpam-5721	353	6	applications	application	NOUN
ejpam-5721	353	7	,	,	PUNCT
ejpam-5721	353	8	22:125	22:125	NUM
ejpam-5721	353	9	,	,	PUNCT
ejpam-5721	353	10	2024	2024	NUM
ejpam-5721	353	11	.	.	PUNCT
ejpam-5721	354	1	[	[	X
ejpam-5721	354	2	17	17	NUM
ejpam-5721	354	3	]	]	X
ejpam-5721	354	4	n.	n.	NOUN
ejpam-5721	354	5	chutiman	chutiman	NOUN
ejpam-5721	354	6	,	,	PUNCT
ejpam-5721	354	7	s.	s.	PROPN
ejpam-5721	354	8	sompong	sompong	PROPN
ejpam-5721	354	9	,	,	PUNCT
ejpam-5721	354	10	and	and	CCONJ
ejpam-5721	354	11	c.	c.	PROPN
ejpam-5721	354	12	boonpok	boonpok	PROPN
ejpam-5721	354	13	.	.	PUNCT
ejpam-5721	355	1	on	on	ADP
ejpam-5721	355	2	some	some	DET
ejpam-5721	355	3	separation	separation	NOUN
ejpam-5721	355	4	axioms	axiom	NOUN
ejpam-5721	355	5	in	in	ADP
ejpam-5721	355	6	bitopological	bitopological	ADJ
ejpam-5721	355	7	spaces	space	NOUN
ejpam-5721	355	8	.	.	PUNCT
ejpam-5721	356	1	asia	asia	PROPN
ejpam-5721	356	2	pacific	pacific	PROPN
ejpam-5721	356	3	journal	journal	PROPN
ejpam-5721	356	4	of	of	ADP
ejpam-5721	356	5	mathematics	mathematic	NOUN
ejpam-5721	356	6	,	,	PUNCT
ejpam-5721	356	7	11:41	11:41	NUM
ejpam-5721	356	8	,	,	PUNCT
ejpam-5721	356	9	2024	2024	NUM
ejpam-5721	356	10	.	.	PUNCT
ejpam-5721	357	1	[	[	X
ejpam-5721	357	2	18	18	NUM
ejpam-5721	357	3	]	]	PUNCT
ejpam-5721	357	4	t.	t.	PROPN
ejpam-5721	357	5	duangphui	duangphui	PROPN
ejpam-5721	357	6	,	,	PUNCT
ejpam-5721	357	7	c.	c.	PROPN
ejpam-5721	357	8	boonpok	boonpok	PROPN
ejpam-5721	357	9	,	,	PUNCT
ejpam-5721	357	10	and	and	CCONJ
ejpam-5721	357	11	c.	c.	PROPN
ejpam-5721	357	12	viriyapong	viriyapong	PROPN
ejpam-5721	357	13	.	.	PUNCT
ejpam-5721	358	1	continuous	continuous	ADJ
ejpam-5721	358	2	functions	function	NOUN
ejpam-5721	358	3	on	on	ADP
ejpam-5721	358	4	bigeneralized	bigeneralize	VERB
ejpam-5721	358	5	topological	topological	ADJ
ejpam-5721	358	6	spaces	space	NOUN
ejpam-5721	358	7	.	.	PUNCT
ejpam-5721	359	1	international	international	ADJ
ejpam-5721	359	2	journal	journal	PROPN
ejpam-5721	359	3	of	of	ADP
ejpam-5721	359	4	mathematical	mathematical	ADJ
ejpam-5721	359	5	analysis	analysis	NOUN
ejpam-5721	359	6	,	,	PUNCT
ejpam-5721	359	7	5(24):1165	5(24):1165	NUM
ejpam-5721	359	8	–	–	PUNCT
ejpam-5721	359	9	1174	1174	NUM
ejpam-5721	359	10	,	,	PUNCT
ejpam-5721	359	11	2011	2011	NUM
ejpam-5721	359	12	.	.	PUNCT
ejpam-5721	360	1	[	[	X
ejpam-5721	360	2	19	19	NUM
ejpam-5721	360	3	]	]	PUNCT
ejpam-5721	360	4	t.	t.	NOUN
ejpam-5721	360	5	dungthaisong	dungthaisong	PROPN
ejpam-5721	360	6	,	,	PUNCT
ejpam-5721	360	7	c.	c.	PROPN
ejpam-5721	360	8	boonpok	boonpok	PROPN
ejpam-5721	360	9	,	,	PUNCT
ejpam-5721	360	10	and	and	CCONJ
ejpam-5721	360	11	c.	c.	PROPN
ejpam-5721	360	12	viriyapong	viriyapong	PROPN
ejpam-5721	360	13	.	.	PUNCT
ejpam-5721	361	1	generalized	generalize	VERB
ejpam-5721	361	2	closed	close	VERB
ejpam-5721	361	3	sets	set	NOUN
ejpam-5721	361	4	in	in	ADP
ejpam-5721	361	5	bigeneralized	bigeneralize	VERB
ejpam-5721	361	6	topological	topological	ADJ
ejpam-5721	361	7	spaces	space	NOUN
ejpam-5721	361	8	.	.	PUNCT
ejpam-5721	362	1	international	international	ADJ
ejpam-5721	362	2	journal	journal	PROPN
ejpam-5721	362	3	of	of	ADP
ejpam-5721	362	4	mathematical	mathematical	ADJ
ejpam-5721	362	5	analysis	analysis	NOUN
ejpam-5721	362	6	,	,	PUNCT
ejpam-5721	362	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-5721	362	8	,	,	PUNCT
ejpam-5721	362	9	2011	2011	NUM
ejpam-5721	362	10	.	.	PUNCT
ejpam-5721	363	1	[	[	X
ejpam-5721	363	2	20	20	NUM
ejpam-5721	363	3	]	]	PUNCT
ejpam-5721	363	4	m.	m.	NOUN
ejpam-5721	363	5	e.	e.	PROPN
ejpam-5721	363	6	abd	abd	PROPN
ejpam-5721	364	1	el	el	PROPN
ejpam-5721	364	2	-	-	PROPN
ejpam-5721	364	3	monsef	monsef	PROPN
ejpam-5721	364	4	,	,	PUNCT
ejpam-5721	364	5	s.	s.	PROPN
ejpam-5721	364	6	n.	n.	PROPN
ejpam-5721	364	7	el	el	PROPN
ejpam-5721	364	8	-	-	PROPN
ejpam-5721	364	9	deeb	deeb	PROPN
ejpam-5721	364	10	,	,	PUNCT
ejpam-5721	364	11	and	and	CCONJ
ejpam-5721	364	12	r.	r.	PROPN
ejpam-5721	364	13	a.	a.	PROPN
ejpam-5721	364	14	mahmoud	mahmoud	PROPN
ejpam-5721	364	15	.	.	PUNCT
ejpam-5721	365	1	β	β	X
ejpam-5721	365	2	-	-	ADJ
ejpam-5721	365	3	open	open	ADJ
ejpam-5721	365	4	sets	set	NOUN
ejpam-5721	365	5	and	and	CCONJ
ejpam-5721	365	6	βcontinuous	βcontinuous	ADJ
ejpam-5721	365	7	mappings	mapping	NOUN
ejpam-5721	365	8	.	.	PUNCT
ejpam-5721	366	1	bulletin	bulletin	NOUN
ejpam-5721	366	2	of	of	ADP
ejpam-5721	366	3	the	the	DET
ejpam-5721	366	4	faculty	faculty	NOUN
ejpam-5721	366	5	of	of	ADP
ejpam-5721	366	6	science	science	NOUN
ejpam-5721	366	7	,	,	PUNCT
ejpam-5721	366	8	assiut	assiut	NOUN
ejpam-5721	366	9	university	university	NOUN
ejpam-5721	366	10	,	,	PUNCT
ejpam-5721	366	11	12:77–90	12:77–90	NUM
ejpam-5721	366	12	,	,	PUNCT
ejpam-5721	366	13	1983	1983	NUM
ejpam-5721	366	14	.	.	PUNCT
ejpam-5721	367	1	[	[	X
ejpam-5721	367	2	21	21	NUM
ejpam-5721	367	3	]	]	PUNCT
ejpam-5721	367	4	t.	t.	PROPN
ejpam-5721	367	5	husain	husain	PROPN
ejpam-5721	367	6	.	.	PUNCT
ejpam-5721	368	1	almost	almost	ADV
ejpam-5721	368	2	continuous	continuous	ADJ
ejpam-5721	368	3	mappings	mapping	NOUN
ejpam-5721	368	4	.	.	PUNCT
ejpam-5721	369	1	prace	prace	PROPN
ejpam-5721	369	2	matematyczne	matematyczne	PROPN
ejpam-5721	369	3	,	,	PUNCT
ejpam-5721	369	4	10(1):1–7	10(1):1–7	NUM
ejpam-5721	369	5	,	,	PUNCT
ejpam-5721	369	6	1966	1966	NUM
ejpam-5721	369	7	.	.	PUNCT
ejpam-5721	370	1	[	[	X
ejpam-5721	370	2	22	22	NUM
ejpam-5721	370	3	]	]	X
ejpam-5721	371	1	d.	d.	PROPN
ejpam-5721	371	2	s.	s.	PROPN
ejpam-5721	371	3	janković.	janković.	PROPN
ejpam-5721	371	4	θ	θ	PROPN
ejpam-5721	371	5	-	-	ADJ
ejpam-5721	371	6	regular	regular	ADJ
ejpam-5721	371	7	spaces	space	NOUN
ejpam-5721	371	8	.	.	PUNCT
ejpam-5721	372	1	international	international	ADJ
ejpam-5721	372	2	journal	journal	PROPN
ejpam-5721	372	3	of	of	ADP
ejpam-5721	372	4	mathematics	mathematics	PROPN
ejpam-5721	372	5	and	and	CCONJ
ejpam-5721	372	6	mathematical	mathematical	ADJ
ejpam-5721	372	7	sciences	science	NOUN
ejpam-5721	372	8	,	,	PUNCT
ejpam-5721	372	9	8(3):615–619	8(3):615–619	NUM
ejpam-5721	372	10	,	,	PUNCT
ejpam-5721	372	11	1985	1985	NUM
ejpam-5721	372	12	.	.	PUNCT
ejpam-5721	373	1	[	[	X
ejpam-5721	373	2	23	23	NUM
ejpam-5721	373	3	]	]	X
ejpam-5721	373	4	b.	b.	PROPN
ejpam-5721	373	5	kong	kong	PROPN
ejpam-5721	373	6	-	-	PUNCT
ejpam-5721	373	7	ied	ied	PROPN
ejpam-5721	373	8	,	,	PUNCT
ejpam-5721	373	9	s.	s.	PROPN
ejpam-5721	373	10	sompong	sompong	PROPN
ejpam-5721	373	11	,	,	PUNCT
ejpam-5721	373	12	and	and	CCONJ
ejpam-5721	373	13	c.	c.	PROPN
ejpam-5721	373	14	boonpok	boonpok	PROPN
ejpam-5721	373	15	.	.	PUNCT
ejpam-5721	374	1	almost	almost	ADV
ejpam-5721	374	2	quasi	quasi	X
ejpam-5721	374	3	(	(	PUNCT
ejpam-5721	374	4	τ1	τ1	NOUN
ejpam-5721	374	5	,	,	PUNCT
ejpam-5721	374	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	374	7	functions	function	NOUN
ejpam-5721	374	8	.	.	PUNCT
ejpam-5721	375	1	asia	asia	PROPN
ejpam-5721	375	2	pacific	pacific	PROPN
ejpam-5721	375	3	journal	journal	PROPN
ejpam-5721	375	4	of	of	ADP
ejpam-5721	375	5	mathematics	mathematic	NOUN
ejpam-5721	375	6	,	,	PUNCT
ejpam-5721	375	7	11:64	11:64	NUM
ejpam-5721	375	8	,	,	PUNCT
ejpam-5721	375	9	2024	2024	NUM
ejpam-5721	375	10	.	.	PUNCT
ejpam-5721	376	1	[	[	X
ejpam-5721	376	2	24	24	NUM
ejpam-5721	376	3	]	]	X
ejpam-5721	376	4	n.	n.	PROPN
ejpam-5721	376	5	levine	levine	PROPN
ejpam-5721	376	6	.	.	PUNCT
ejpam-5721	377	1	a	a	DET
ejpam-5721	377	2	decomposition	decomposition	NOUN
ejpam-5721	377	3	of	of	ADP
ejpam-5721	377	4	continuity	continuity	NOUN
ejpam-5721	377	5	in	in	ADP
ejpam-5721	377	6	topological	topological	ADJ
ejpam-5721	377	7	spaces	space	NOUN
ejpam-5721	377	8	.	.	PUNCT
ejpam-5721	378	1	the	the	DET
ejpam-5721	378	2	american	american	PROPN
ejpam-5721	378	3	mathematical	mathematical	PROPN
ejpam-5721	378	4	monthly	monthly	ADV
ejpam-5721	378	5	,	,	PUNCT
ejpam-5721	378	6	60:44–46	60:44–46	NUM
ejpam-5721	378	7	,	,	PUNCT
ejpam-5721	378	8	1961	1961	NUM
ejpam-5721	378	9	.	.	PUNCT
ejpam-5721	379	1	[	[	X
ejpam-5721	379	2	25	25	NUM
ejpam-5721	379	3	]	]	X
ejpam-5721	379	4	n.	n.	PROPN
ejpam-5721	379	5	levine	levine	PROPN
ejpam-5721	379	6	.	.	PUNCT
ejpam-5721	380	1	semi	semi	ADJ
ejpam-5721	380	2	-	-	ADJ
ejpam-5721	380	3	open	open	ADJ
ejpam-5721	380	4	sets	set	NOUN
ejpam-5721	380	5	and	and	CCONJ
ejpam-5721	380	6	semi	semi	ADJ
ejpam-5721	380	7	-	-	NOUN
ejpam-5721	380	8	continuity	continuity	NOUN
ejpam-5721	380	9	in	in	ADP
ejpam-5721	380	10	topological	topological	ADJ
ejpam-5721	380	11	spaces	space	NOUN
ejpam-5721	380	12	.	.	PUNCT
ejpam-5721	381	1	the	the	DET
ejpam-5721	381	2	american	american	PROPN
ejpam-5721	381	3	mathematical	mathematical	PROPN
ejpam-5721	381	4	monthly	monthly	ADV
ejpam-5721	381	5	,	,	PUNCT
ejpam-5721	381	6	70:36–41	70:36–41	NUM
ejpam-5721	381	7	,	,	PUNCT
ejpam-5721	381	8	1963	1963	NUM
ejpam-5721	381	9	.	.	PUNCT
ejpam-5721	382	1	[	[	X
ejpam-5721	382	2	26	26	NUM
ejpam-5721	382	3	]	]	PUNCT
ejpam-5721	382	4	a.	a.	NOUN
ejpam-5721	382	5	s.	s.	PROPN
ejpam-5721	382	6	mashhour	mashhour	PROPN
ejpam-5721	382	7	,	,	PUNCT
ejpam-5721	382	8	m.	m.	PROPN
ejpam-5721	382	9	e.	e.	PROPN
ejpam-5721	382	10	abd	abd	PROPN
ejpam-5721	382	11	el	el	PROPN
ejpam-5721	382	12	-	-	PROPN
ejpam-5721	382	13	monsef	monsef	ADJ
ejpam-5721	382	14	,	,	PUNCT
ejpam-5721	382	15	and	and	CCONJ
ejpam-5721	382	16	s.	s.	PROPN
ejpam-5721	382	17	n.	n.	PROPN
ejpam-5721	382	18	el	el	PROPN
ejpam-5721	382	19	-	-	PROPN
ejpam-5721	382	20	deeb	deeb	PROPN
ejpam-5721	382	21	.	.	PUNCT
ejpam-5721	383	1	on	on	ADP
ejpam-5721	383	2	precontinuous	precontinuous	ADJ
ejpam-5721	383	3	and	and	CCONJ
ejpam-5721	383	4	weak	weak	ADJ
ejpam-5721	383	5	precontinuous	precontinuous	ADJ
ejpam-5721	383	6	mappings	mapping	NOUN
ejpam-5721	383	7	.	.	PUNCT
ejpam-5721	384	1	proceedings	proceeding	NOUN
ejpam-5721	384	2	of	of	ADP
ejpam-5721	384	3	the	the	DET
ejpam-5721	384	4	mathematical	mathematical	ADJ
ejpam-5721	384	5	and	and	CCONJ
ejpam-5721	384	6	physical	physical	ADJ
ejpam-5721	384	7	society	society	NOUN
ejpam-5721	384	8	of	of	ADP
ejpam-5721	384	9	egypt	egypt	PROPN
ejpam-5721	384	10	,	,	PUNCT
ejpam-5721	384	11	53:47–53	53:47–53	NUM
ejpam-5721	384	12	,	,	PUNCT
ejpam-5721	384	13	1982	1982	NUM
ejpam-5721	384	14	.	.	PUNCT
ejpam-5721	385	1	[	[	X
ejpam-5721	385	2	27	27	NUM
ejpam-5721	385	3	]	]	X
ejpam-5721	385	4	o.	o.	NOUN
ejpam-5721	385	5	nj̊astad	nj̊astad	NOUN
ejpam-5721	385	6	.	.	PUNCT
ejpam-5721	386	1	on	on	ADP
ejpam-5721	386	2	some	some	DET
ejpam-5721	386	3	classes	class	NOUN
ejpam-5721	386	4	of	of	ADP
ejpam-5721	386	5	nearly	nearly	ADV
ejpam-5721	386	6	open	open	ADJ
ejpam-5721	386	7	sets	set	NOUN
ejpam-5721	386	8	.	.	PUNCT
ejpam-5721	387	1	pacific	pacific	PROPN
ejpam-5721	387	2	journal	journal	PROPN
ejpam-5721	387	3	of	of	ADP
ejpam-5721	387	4	mathematics	mathematic	NOUN
ejpam-5721	387	5	,	,	PUNCT
ejpam-5721	387	6	15:961–970	15:961–970	PROPN
ejpam-5721	387	7	,	,	PUNCT
ejpam-5721	387	8	1965	1965	NUM
ejpam-5721	387	9	.	.	PUNCT
ejpam-5721	388	1	j.	j.	PROPN
ejpam-5721	388	2	khampakdee	khampakdee	PROPN
ejpam-5721	388	3	,	,	PUNCT
ejpam-5721	388	4	s.	s.	PROPN
ejpam-5721	388	5	sompong	sompong	PROPN
ejpam-5721	388	6	,	,	PUNCT
ejpam-5721	388	7	c.	c.	PROPN
ejpam-5721	388	8	boonpok	boonpok	PROPN
ejpam-5721	388	9	/	/	SYM
ejpam-5721	388	10	eur	eur	PROPN
ejpam-5721	388	11	.	.	PUNCT
ejpam-5721	389	1	j.	j.	PROPN
ejpam-5721	389	2	pure	pure	PROPN
ejpam-5721	389	3	appl	appl	PROPN
ejpam-5721	389	4	.	.	PROPN
ejpam-5721	389	5	math	math	PROPN
ejpam-5721	389	6	,	,	PUNCT
ejpam-5721	389	7	18	18	NUM
ejpam-5721	389	8	(	(	PUNCT
ejpam-5721	389	9	1	1	NUM
ejpam-5721	389	10	)	)	PUNCT
ejpam-5721	389	11	(	(	PUNCT
ejpam-5721	389	12	2025	2025	NUM
ejpam-5721	389	13	)	)	PUNCT
ejpam-5721	389	14	,	,	PUNCT
ejpam-5721	389	15	5721	5721	NUM
ejpam-5721	389	16	12	12	NUM
ejpam-5721	389	17	of	of	ADP
ejpam-5721	389	18	12	12	NUM
ejpam-5721	389	19	[	[	SYM
ejpam-5721	389	20	28	28	NUM
ejpam-5721	389	21	]	]	PUNCT
ejpam-5721	389	22	t.	t.	PROPN
ejpam-5721	389	23	noiri	noiri	PROPN
ejpam-5721	389	24	.	.	PUNCT
ejpam-5721	390	1	properties	property	NOUN
ejpam-5721	390	2	of	of	ADP
ejpam-5721	390	3	some	some	DET
ejpam-5721	390	4	weak	weak	ADJ
ejpam-5721	390	5	forms	form	NOUN
ejpam-5721	390	6	of	of	ADP
ejpam-5721	390	7	continuity	continuity	NOUN
ejpam-5721	390	8	.	.	PUNCT
ejpam-5721	391	1	international	international	ADJ
ejpam-5721	391	2	journal	journal	PROPN
ejpam-5721	391	3	of	of	ADP
ejpam-5721	391	4	mathematics	mathematics	PROPN
ejpam-5721	391	5	and	and	CCONJ
ejpam-5721	391	6	mathematical	mathematical	ADJ
ejpam-5721	391	7	sciences	science	NOUN
ejpam-5721	391	8	,	,	PUNCT
ejpam-5721	391	9	10(1):97–111	10(1):97–111	NUM
ejpam-5721	391	10	,	,	PUNCT
ejpam-5721	391	11	1987	1987	NUM
ejpam-5721	391	12	.	.	PUNCT
ejpam-5721	392	1	[	[	X
ejpam-5721	392	2	29	29	NUM
ejpam-5721	392	3	]	]	PUNCT
ejpam-5721	392	4	v.	v.	CCONJ
ejpam-5721	392	5	popa	popa	NOUN
ejpam-5721	392	6	and	and	CCONJ
ejpam-5721	392	7	t.	t.	PROPN
ejpam-5721	392	8	noiri	noiri	PROPN
ejpam-5721	392	9	.	.	PUNCT
ejpam-5721	393	1	almost	almost	ADV
ejpam-5721	393	2	weakly	weakly	ADJ
ejpam-5721	393	3	continuous	continuous	ADJ
ejpam-5721	393	4	functions	function	NOUN
ejpam-5721	393	5	.	.	PUNCT
ejpam-5721	394	1	demonstratio	demonstratio	PROPN
ejpam-5721	394	2	mathematica	mathematica	PROPN
ejpam-5721	394	3	,	,	PUNCT
ejpam-5721	394	4	25:241–251	25:241–251	PROPN
ejpam-5721	394	5	,	,	PUNCT
ejpam-5721	394	6	1992	1992	NUM
ejpam-5721	394	7	.	.	PUNCT
ejpam-5721	395	1	[	[	X
ejpam-5721	395	2	30	30	NUM
ejpam-5721	395	3	]	]	X
ejpam-5721	395	4	c.	c.	NOUN
ejpam-5721	395	5	prachanpol	prachanpol	NOUN
ejpam-5721	395	6	,	,	PUNCT
ejpam-5721	395	7	c.	c.	PROPN
ejpam-5721	395	8	boonpok	boonpok	PROPN
ejpam-5721	395	9	,	,	PUNCT
ejpam-5721	395	10	and	and	CCONJ
ejpam-5721	395	11	c.	c.	PROPN
ejpam-5721	395	12	viriyapong	viriyapong	PROPN
ejpam-5721	395	13	.	.	PUNCT
ejpam-5721	396	1	δ(τ1	δ(τ1	PROPN
ejpam-5721	396	2	,	,	PUNCT
ejpam-5721	396	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5721	396	4	functions	function	NOUN
ejpam-5721	396	5	.	.	PUNCT
ejpam-5721	397	1	european	european	ADJ
ejpam-5721	397	2	journal	journal	PROPN
ejpam-5721	397	3	of	of	ADP
ejpam-5721	397	4	pure	pure	ADJ
ejpam-5721	397	5	and	and	CCONJ
ejpam-5721	397	6	applied	applied	ADJ
ejpam-5721	397	7	mathematics	mathematic	NOUN
ejpam-5721	397	8	,	,	PUNCT
ejpam-5721	397	9	17(4):3730–3742	17(4):3730–3742	NUM
ejpam-5721	397	10	,	,	PUNCT
ejpam-5721	397	11	2024	2024	NUM
ejpam-5721	397	12	.	.	PUNCT
ejpam-5721	398	1	[	[	X
ejpam-5721	398	2	31	31	NUM
ejpam-5721	398	3	]	]	PUNCT
ejpam-5721	398	4	p.	p.	NOUN
ejpam-5721	398	5	pue	pue	NOUN
ejpam-5721	398	6	-	-	PUNCT
ejpam-5721	398	7	on	on	ADP
ejpam-5721	398	8	and	and	CCONJ
ejpam-5721	398	9	c.	c.	PROPN
ejpam-5721	398	10	boonpok	boonpok	PROPN
ejpam-5721	398	11	.	.	PUNCT
ejpam-5721	399	1	θ(λ	θ(λ	PROPN
ejpam-5721	399	2	,	,	PUNCT
ejpam-5721	399	3	p)-continuity	p)-continuity	NOUN
ejpam-5721	399	4	for	for	ADP
ejpam-5721	399	5	functions	function	NOUN
ejpam-5721	399	6	.	.	PUNCT
ejpam-5721	400	1	international	international	ADJ
ejpam-5721	400	2	journal	journal	NOUN
ejpam-5721	400	3	of	of	ADP
ejpam-5721	400	4	mathematics	mathematic	NOUN
ejpam-5721	400	5	and	and	CCONJ
ejpam-5721	400	6	computer	computer	NOUN
ejpam-5721	400	7	science	science	NOUN
ejpam-5721	400	8	,	,	PUNCT
ejpam-5721	400	9	19(2):491–495	19(2):491–495	NUM
ejpam-5721	400	10	,	,	PUNCT
ejpam-5721	400	11	2024	2024	NUM
ejpam-5721	400	12	.	.	PUNCT
ejpam-5721	401	1	[	[	X
ejpam-5721	401	2	32	32	NUM
ejpam-5721	401	3	]	]	PUNCT
ejpam-5721	401	4	p.	p.	NOUN
ejpam-5721	401	5	pue	pue	NOUN
ejpam-5721	401	6	-	-	PUNCT
ejpam-5721	401	7	on	on	ADP
ejpam-5721	401	8	,	,	PUNCT
ejpam-5721	401	9	a.	a.	PROPN
ejpam-5721	401	10	sama	sama	PROPN
ejpam-5721	401	11	-	-	PUNCT
ejpam-5721	401	12	ae	ae	PROPN
ejpam-5721	401	13	,	,	PUNCT
ejpam-5721	401	14	and	and	CCONJ
ejpam-5721	401	15	c.	c.	PROPN
ejpam-5721	401	16	boonpok	boonpok	PROPN
ejpam-5721	401	17	.	.	PUNCT
ejpam-5721	402	1	characterizations	characterization	NOUN
ejpam-5721	402	2	of	of	ADP
ejpam-5721	402	3	quasi	quasi	NOUN
ejpam-5721	402	4	θ(τ1	θ(τ1	NOUN
ejpam-5721	402	5	,	,	PUNCT
ejpam-5721	402	6	τ2)continuous	τ2)continuous	ADJ
ejpam-5721	402	7	multifunctions	multifunction	NOUN
ejpam-5721	402	8	.	.	PUNCT
ejpam-5721	403	1	(	(	PUNCT
ejpam-5721	403	2	submitted	submit	VERB
ejpam-5721	403	3	)	)	PUNCT
ejpam-5721	403	4	.	.	PUNCT
ejpam-5721	404	1	[	[	X
ejpam-5721	404	2	33	33	NUM
ejpam-5721	404	3	]	]	X
ejpam-5721	404	4	n.	n.	PROPN
ejpam-5721	404	5	srisarakham	srisarakham	PROPN
ejpam-5721	404	6	and	and	CCONJ
ejpam-5721	404	7	c.	c.	PROPN
ejpam-5721	404	8	boonpok	boonpok	PROPN
ejpam-5721	404	9	.	.	PUNCT
ejpam-5721	405	1	almost	almost	ADV
ejpam-5721	405	2	(	(	PUNCT
ejpam-5721	405	3	λ	λ	NOUN
ejpam-5721	405	4	,	,	PUNCT
ejpam-5721	405	5	p)-continuous	p)-continuous	ADJ
ejpam-5721	405	6	functions	function	NOUN
ejpam-5721	405	7	.	.	PUNCT
ejpam-5721	406	1	international	international	ADJ
ejpam-5721	406	2	journal	journal	PROPN
ejpam-5721	406	3	of	of	ADP
ejpam-5721	406	4	mathematics	mathematic	NOUN
ejpam-5721	406	5	and	and	CCONJ
ejpam-5721	406	6	computer	computer	NOUN
ejpam-5721	406	7	science	science	NOUN
ejpam-5721	406	8	,	,	PUNCT
ejpam-5721	406	9	18(2):255–259	18(2):255–259	NUM
ejpam-5721	406	10	,	,	PUNCT
ejpam-5721	406	11	2023	2023	NUM
ejpam-5721	406	12	.	.	PUNCT
ejpam-5721	407	1	[	[	X
ejpam-5721	407	2	34	34	NUM
ejpam-5721	407	3	]	]	PUNCT
ejpam-5721	407	4	m.	m.	NOUN
ejpam-5721	407	5	thongmoon	thongmoon	NOUN
ejpam-5721	407	6	and	and	CCONJ
ejpam-5721	407	7	c.	c.	PROPN
ejpam-5721	407	8	boonpok	boonpok	PROPN
ejpam-5721	407	9	.	.	PUNCT
ejpam-5721	408	1	strongly	strongly	ADV
ejpam-5721	408	2	θ(λ	θ(λ	PROPN
ejpam-5721	408	3	,	,	PUNCT
ejpam-5721	408	4	p)-continuous	p)-continuous	ADJ
ejpam-5721	408	5	functions	function	NOUN
ejpam-5721	408	6	.	.	PUNCT
ejpam-5721	409	1	international	international	ADJ
ejpam-5721	409	2	journal	journal	PROPN
ejpam-5721	409	3	of	of	ADP
ejpam-5721	409	4	mathematics	mathematic	NOUN
ejpam-5721	409	5	and	and	CCONJ
ejpam-5721	409	6	computer	computer	NOUN
ejpam-5721	409	7	science	science	NOUN
ejpam-5721	409	8	,	,	PUNCT
ejpam-5721	409	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5721	409	10	,	,	PUNCT
ejpam-5721	409	11	2024	2024	NUM
ejpam-5721	409	12	.	.	PUNCT
ejpam-5721	410	1	[	[	X
ejpam-5721	410	2	35	35	NUM
ejpam-5721	410	3	]	]	X
ejpam-5721	410	4	c.	c.	PROPN
ejpam-5721	410	5	viriyapong	viriyapong	PROPN
ejpam-5721	410	6	and	and	CCONJ
ejpam-5721	410	7	c.	c.	PROPN
ejpam-5721	410	8	boonpok	boonpok	PROPN
ejpam-5721	410	9	.	.	PUNCT
ejpam-5721	411	1	(	(	PUNCT
ejpam-5721	411	2	τ1	τ1	NOUN
ejpam-5721	411	3	,	,	PUNCT
ejpam-5721	411	4	τ2)α	τ2)α	NOUN
ejpam-5721	411	5	-	-	PUNCT
ejpam-5721	411	6	continuity	continuity	NOUN
ejpam-5721	411	7	for	for	ADP
ejpam-5721	411	8	multifunctions	multifunction	NOUN
ejpam-5721	411	9	.	.	PUNCT
ejpam-5721	412	1	journal	journal	PROPN
ejpam-5721	412	2	of	of	ADP
ejpam-5721	412	3	mathematics	mathematic	NOUN
ejpam-5721	412	4	,	,	PUNCT
ejpam-5721	412	5	2020:6285763	2020:6285763	NUM
ejpam-5721	412	6	,	,	PUNCT
ejpam-5721	412	7	2020	2020	NUM
ejpam-5721	412	8	.	.	PUNCT
ejpam-5721	413	1	[	[	X
ejpam-5721	413	2	36	36	NUM
ejpam-5721	413	3	]	]	X
ejpam-5721	413	4	c.	c.	PROPN
ejpam-5721	413	5	viriyapong	viriyapong	PROPN
ejpam-5721	413	6	and	and	CCONJ
ejpam-5721	413	7	c.	c.	PROPN
ejpam-5721	413	8	boonpok	boonpok	PROPN
ejpam-5721	413	9	.	.	PUNCT
ejpam-5721	414	1	(	(	PUNCT
ejpam-5721	414	2	λ	λ	X
ejpam-5721	414	3	,	,	PUNCT
ejpam-5721	414	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5721	414	5	functions	function	NOUN
ejpam-5721	414	6	.	.	PUNCT
ejpam-5721	415	1	wseas	wseas	VERB
ejpam-5721	415	2	transactions	transaction	NOUN
ejpam-5721	415	3	on	on	ADP
ejpam-5721	415	4	mathematics	mathematic	NOUN
ejpam-5721	415	5	,	,	PUNCT
ejpam-5721	415	6	21:380–385	21:380–385	NUM
ejpam-5721	415	7	,	,	PUNCT
ejpam-5721	415	8	2022	2022	NUM
ejpam-5721	415	9	.	.	PUNCT
ejpam-5721	416	1	[	[	X
ejpam-5721	416	2	37	37	NUM
ejpam-5721	416	3	]	]	X
ejpam-5721	416	4	n.	n.	PROPN
ejpam-5721	416	5	viriyapong	viriyapong	PROPN
ejpam-5721	416	6	,	,	PUNCT
ejpam-5721	416	7	s.	s.	PROPN
ejpam-5721	416	8	sompong	sompong	PROPN
ejpam-5721	416	9	,	,	PUNCT
ejpam-5721	416	10	and	and	CCONJ
ejpam-5721	416	11	c.	c.	PROPN
ejpam-5721	416	12	boonpok	boonpok	PROPN
ejpam-5721	416	13	.	.	PUNCT
ejpam-5721	417	1	(	(	PUNCT
ejpam-5721	417	2	τ1	τ1	NOUN
ejpam-5721	417	3	,	,	PUNCT
ejpam-5721	417	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5721	417	5	disconnectedness	disconnectedness	NOUN
ejpam-5721	417	6	in	in	ADP
ejpam-5721	417	7	bitopological	bitopological	ADJ
ejpam-5721	417	8	spaces	space	NOUN
ejpam-5721	417	9	.	.	PUNCT
ejpam-5721	418	1	international	international	ADJ
ejpam-5721	418	2	journal	journal	PROPN
ejpam-5721	418	3	of	of	ADP
ejpam-5721	418	4	mathematics	mathematic	NOUN
ejpam-5721	418	5	and	and	CCONJ
ejpam-5721	418	6	computer	computer	NOUN
ejpam-5721	418	7	science	science	NOUN
ejpam-5721	418	8	,	,	PUNCT
ejpam-5721	418	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5721	418	10	,	,	PUNCT
ejpam-5721	418	11	2024	2024	NUM
ejpam-5721	418	12	.	.	PUNCT