id	sid	tid	token	lemma	pos
ejpam-6014	1	1	european	european	PROPN
ejpam-6014	1	2	journal	journal	PROPN
ejpam-6014	1	3	of	of	ADP
ejpam-6014	1	4	pure	pure	ADJ
ejpam-6014	1	5	and	and	CCONJ
ejpam-6014	1	6	applied	applied	ADJ
ejpam-6014	1	7	mathematics	mathematic	NOUN
ejpam-6014	1	8	2025	2025	NUM
ejpam-6014	1	9	,	,	PUNCT
ejpam-6014	1	10	vol	vol	NOUN
ejpam-6014	1	11	.	.	PROPN
ejpam-6014	1	12	18	18	NUM
ejpam-6014	1	13	,	,	PUNCT
ejpam-6014	1	14	issue	issue	NOUN
ejpam-6014	1	15	2	2	NUM
ejpam-6014	1	16	,	,	PUNCT
ejpam-6014	1	17	article	article	NOUN
ejpam-6014	1	18	number	number	NOUN
ejpam-6014	1	19	6014	6014	NUM
ejpam-6014	1	20	issn	issn	PROPN
ejpam-6014	1	21	1307	1307	NUM
ejpam-6014	1	22	-	-	SYM
ejpam-6014	1	23	5543	5543	NUM
ejpam-6014	1	24	–	–	PUNCT
ejpam-6014	1	25	ejpam.com	ejpam.com	X
ejpam-6014	1	26	published	publish	VERB
ejpam-6014	1	27	by	by	ADP
ejpam-6014	1	28	new	new	PROPN
ejpam-6014	1	29	york	york	PROPN
ejpam-6014	1	30	business	business	PROPN
ejpam-6014	1	31	global	global	PROPN
ejpam-6014	1	32	θ(τ1	θ(τ1	NOUN
ejpam-6014	1	33	,	,	PUNCT
ejpam-6014	1	34	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6014	1	35	for	for	ADP
ejpam-6014	1	36	functions	function	NOUN
ejpam-6014	1	37	montri	montri	PROPN
ejpam-6014	1	38	thongmoon1	thongmoon1	PROPN
ejpam-6014	1	39	,	,	PUNCT
ejpam-6014	1	40	supunnee	supunnee	PROPN
ejpam-6014	1	41	sompong2	sompong2	PROPN
ejpam-6014	1	42	,	,	PUNCT
ejpam-6014	1	43	chawalit	chawalit	VERB
ejpam-6014	1	44	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6014	1	45	1	1	NUM
ejpam-6014	1	46	mathematics	mathematic	NOUN
ejpam-6014	1	47	and	and	CCONJ
ejpam-6014	1	48	applied	apply	VERB
ejpam-6014	1	49	mathematics	mathematics	PROPN
ejpam-6014	1	50	research	research	NOUN
ejpam-6014	1	51	unit	unit	NOUN
ejpam-6014	1	52	,	,	PUNCT
ejpam-6014	1	53	department	department	NOUN
ejpam-6014	1	54	of	of	ADP
ejpam-6014	1	55	mathematics	mathematic	NOUN
ejpam-6014	1	56	,	,	PUNCT
ejpam-6014	1	57	faculty	faculty	NOUN
ejpam-6014	1	58	of	of	ADP
ejpam-6014	1	59	science	science	NOUN
ejpam-6014	1	60	,	,	PUNCT
ejpam-6014	1	61	mahasarakham	mahasarakham	PROPN
ejpam-6014	1	62	university	university	PROPN
ejpam-6014	1	63	,	,	PUNCT
ejpam-6014	1	64	maha	maha	PROPN
ejpam-6014	1	65	sarakham	sarakham	PROPN
ejpam-6014	1	66	,	,	PUNCT
ejpam-6014	1	67	44150	44150	NUM
ejpam-6014	1	68	,	,	PUNCT
ejpam-6014	1	69	thailand	thailand	PROPN
ejpam-6014	1	70	2	2	NUM
ejpam-6014	1	71	department	department	NOUN
ejpam-6014	1	72	of	of	ADP
ejpam-6014	1	73	mathematics	mathematic	NOUN
ejpam-6014	1	74	and	and	CCONJ
ejpam-6014	1	75	statistics	statistic	NOUN
ejpam-6014	1	76	,	,	PUNCT
ejpam-6014	1	77	faculty	faculty	NOUN
ejpam-6014	1	78	of	of	ADP
ejpam-6014	1	79	science	science	NOUN
ejpam-6014	1	80	and	and	CCONJ
ejpam-6014	1	81	technology	technology	NOUN
ejpam-6014	1	82	,	,	PUNCT
ejpam-6014	1	83	sakon	sakon	PROPN
ejpam-6014	1	84	nakhon	nakhon	PROPN
ejpam-6014	1	85	rajbhat	rajbhat	PROPN
ejpam-6014	1	86	university	university	PROPN
ejpam-6014	1	87	,	,	PUNCT
ejpam-6014	1	88	sakon	sakon	PROPN
ejpam-6014	1	89	nakhon	nakhon	PROPN
ejpam-6014	1	90	,	,	PUNCT
ejpam-6014	1	91	47000	47000	NUM
ejpam-6014	1	92	,	,	PUNCT
ejpam-6014	1	93	thailand	thailand	PROPN
ejpam-6014	1	94	abstract	abstract	NOUN
ejpam-6014	1	95	.	.	PUNCT
ejpam-6014	2	1	this	this	DET
ejpam-6014	2	2	paper	paper	NOUN
ejpam-6014	2	3	introduces	introduce	VERB
ejpam-6014	2	4	a	a	DET
ejpam-6014	2	5	new	new	ADJ
ejpam-6014	2	6	class	class	NOUN
ejpam-6014	2	7	of	of	ADP
ejpam-6014	2	8	functions	function	NOUN
ejpam-6014	2	9	between	between	ADP
ejpam-6014	2	10	bitopological	bitopological	ADJ
ejpam-6014	2	11	spaces	space	NOUN
ejpam-6014	2	12	,	,	PUNCT
ejpam-6014	2	13	namely	namely	ADV
ejpam-6014	2	14	θ(τ1	θ(τ1	NOUN
ejpam-6014	2	15	,	,	PUNCT
ejpam-6014	2	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	2	17	functions	function	NOUN
ejpam-6014	2	18	.	.	PUNCT
ejpam-6014	3	1	moreover	moreover	ADV
ejpam-6014	3	2	,	,	PUNCT
ejpam-6014	3	3	several	several	ADJ
ejpam-6014	3	4	characterizations	characterization	NOUN
ejpam-6014	3	5	and	and	CCONJ
ejpam-6014	3	6	some	some	DET
ejpam-6014	3	7	properties	property	NOUN
ejpam-6014	3	8	concerning	concern	VERB
ejpam-6014	3	9	θ(τ1	θ(τ1	NOUN
ejpam-6014	3	10	,	,	PUNCT
ejpam-6014	3	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	3	12	functions	function	NOUN
ejpam-6014	3	13	are	be	AUX
ejpam-6014	3	14	investigated	investigate	VERB
ejpam-6014	3	15	.	.	PUNCT
ejpam-6014	4	1	2020	2020	NUM
ejpam-6014	4	2	mathematics	mathematic	NOUN
ejpam-6014	4	3	subject	subject	NOUN
ejpam-6014	4	4	classifications	classification	NOUN
ejpam-6014	4	5	:	:	PUNCT
ejpam-6014	4	6	54c08	54c08	NUM
ejpam-6014	4	7	;	;	PUNCT
ejpam-6014	4	8	54e55	54e55	NUM
ejpam-6014	4	9	key	key	ADJ
ejpam-6014	4	10	words	word	NOUN
ejpam-6014	4	11	and	and	CCONJ
ejpam-6014	4	12	phrases	phrase	NOUN
ejpam-6014	4	13	:	:	PUNCT
ejpam-6014	4	14	(	(	PUNCT
ejpam-6014	4	15	τ1	τ1	NOUN
ejpam-6014	4	16	,	,	PUNCT
ejpam-6014	4	17	τ2)θ	τ2)θ	ADJ
ejpam-6014	4	18	-	-	PUNCT
ejpam-6014	4	19	open	open	ADJ
ejpam-6014	4	20	set	set	NOUN
ejpam-6014	4	21	,	,	PUNCT
ejpam-6014	4	22	θ(τ1	θ(τ1	NOUN
ejpam-6014	4	23	,	,	PUNCT
ejpam-6014	4	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	4	25	function	function	NOUN
ejpam-6014	4	26	1	1	NUM
ejpam-6014	4	27	.	.	PUNCT
ejpam-6014	5	1	introduction	introduction	NOUN
ejpam-6014	5	2	stronger	strong	ADJ
ejpam-6014	5	3	and	and	CCONJ
ejpam-6014	5	4	weaker	weak	ADJ
ejpam-6014	5	5	forms	form	NOUN
ejpam-6014	5	6	of	of	ADP
ejpam-6014	5	7	open	open	ADJ
ejpam-6014	5	8	sets	set	NOUN
ejpam-6014	5	9	in	in	ADP
ejpam-6014	5	10	topological	topological	ADJ
ejpam-6014	5	11	spaces	space	NOUN
ejpam-6014	5	12	such	such	ADJ
ejpam-6014	5	13	as	as	ADP
ejpam-6014	5	14	semi	semi	ADJ
ejpam-6014	5	15	-	-	ADJ
ejpam-6014	5	16	open	open	ADJ
ejpam-6014	5	17	sets	set	NOUN
ejpam-6014	5	18	,	,	PUNCT
ejpam-6014	5	19	preopen	preopen	ADJ
ejpam-6014	5	20	sets	set	NOUN
ejpam-6014	5	21	,	,	PUNCT
ejpam-6014	5	22	α	α	NOUN
ejpam-6014	5	23	-	-	ADJ
ejpam-6014	5	24	open	open	ADJ
ejpam-6014	5	25	sets	set	NOUN
ejpam-6014	5	26	,	,	PUNCT
ejpam-6014	5	27	β	β	ADJ
ejpam-6014	5	28	-	-	ADJ
ejpam-6014	5	29	open	open	ADJ
ejpam-6014	5	30	sets	set	NOUN
ejpam-6014	5	31	,	,	PUNCT
ejpam-6014	5	32	δ	δ	NOUN
ejpam-6014	5	33	-	-	ADJ
ejpam-6014	5	34	open	open	ADJ
ejpam-6014	5	35	sets	set	NOUN
ejpam-6014	5	36	and	and	CCONJ
ejpam-6014	5	37	θ	θ	ADJ
ejpam-6014	5	38	-	-	ADJ
ejpam-6014	5	39	open	open	ADJ
ejpam-6014	5	40	sets	set	NOUN
ejpam-6014	5	41	play	play	VERB
ejpam-6014	5	42	an	an	DET
ejpam-6014	5	43	important	important	ADJ
ejpam-6014	5	44	role	role	NOUN
ejpam-6014	5	45	in	in	ADP
ejpam-6014	5	46	the	the	DET
ejpam-6014	5	47	research	research	NOUN
ejpam-6014	5	48	of	of	ADP
ejpam-6014	5	49	generalizations	generalization	NOUN
ejpam-6014	5	50	of	of	ADP
ejpam-6014	5	51	continuity	continuity	NOUN
ejpam-6014	5	52	.	.	PUNCT
ejpam-6014	6	1	by	by	ADP
ejpam-6014	6	2	using	use	VERB
ejpam-6014	6	3	these	these	DET
ejpam-6014	6	4	sets	set	NOUN
ejpam-6014	6	5	many	many	ADJ
ejpam-6014	6	6	authors	author	NOUN
ejpam-6014	6	7	introduced	introduce	VERB
ejpam-6014	6	8	and	and	CCONJ
ejpam-6014	6	9	investigated	investigate	VERB
ejpam-6014	6	10	various	various	ADJ
ejpam-6014	6	11	types	type	NOUN
ejpam-6014	6	12	of	of	ADP
ejpam-6014	6	13	continuity	continuity	NOUN
ejpam-6014	6	14	.	.	PUNCT
ejpam-6014	7	1	the	the	DET
ejpam-6014	7	2	notions	notion	NOUN
ejpam-6014	7	3	of	of	ADP
ejpam-6014	7	4	(	(	PUNCT
ejpam-6014	7	5	λ	λ	PROPN
ejpam-6014	7	6	,	,	PUNCT
ejpam-6014	7	7	sp)-open	sp)-open	ADJ
ejpam-6014	7	8	sets	set	NOUN
ejpam-6014	7	9	,	,	PUNCT
ejpam-6014	7	10	s(λ	s(λ	PROPN
ejpam-6014	7	11	,	,	PUNCT
ejpam-6014	7	12	sp)-open	sp)-open	ADJ
ejpam-6014	7	13	sets	set	NOUN
ejpam-6014	7	14	,	,	PUNCT
ejpam-6014	7	15	p(λ	p(λ	NOUN
ejpam-6014	7	16	,	,	PUNCT
ejpam-6014	7	17	sp)-open	sp)-open	ADJ
ejpam-6014	7	18	sets	set	NOUN
ejpam-6014	7	19	,	,	PUNCT
ejpam-6014	7	20	α(λ	α(λ	PROPN
ejpam-6014	7	21	,	,	PUNCT
ejpam-6014	7	22	sp)-open	sp)-open	ADJ
ejpam-6014	7	23	sets	set	NOUN
ejpam-6014	7	24	and	and	CCONJ
ejpam-6014	7	25	β(λ	β(λ	NOUN
ejpam-6014	7	26	,	,	PUNCT
ejpam-6014	7	27	sp)-open	sp)-open	ADJ
ejpam-6014	7	28	sets	set	NOUN
ejpam-6014	7	29	were	be	AUX
ejpam-6014	7	30	studied	study	VERB
ejpam-6014	7	31	in	in	ADP
ejpam-6014	7	32	[	[	X
ejpam-6014	7	33	1	1	NUM
ejpam-6014	7	34	]	]	PUNCT
ejpam-6014	7	35	.	.	PUNCT
ejpam-6014	8	1	viriyapong	viriyapong	PROPN
ejpam-6014	8	2	and	and	CCONJ
ejpam-6014	8	3	boonpok	boonpok	VERB
ejpam-6014	8	4	[	[	X
ejpam-6014	8	5	2	2	NUM
ejpam-6014	8	6	]	]	PUNCT
ejpam-6014	8	7	investigated	investigate	VERB
ejpam-6014	8	8	several	several	ADJ
ejpam-6014	8	9	characterizations	characterization	NOUN
ejpam-6014	8	10	of	of	ADP
ejpam-6014	8	11	(	(	PUNCT
ejpam-6014	8	12	λ	λ	PROPN
ejpam-6014	8	13	,	,	PUNCT
ejpam-6014	8	14	sp)continuous	sp)continuous	ADJ
ejpam-6014	8	15	functions	function	NOUN
ejpam-6014	8	16	by	by	ADP
ejpam-6014	8	17	utilizing	utilize	VERB
ejpam-6014	8	18	the	the	DET
ejpam-6014	8	19	notions	notion	NOUN
ejpam-6014	8	20	of	of	ADP
ejpam-6014	8	21	(	(	PUNCT
ejpam-6014	8	22	λ	λ	PROPN
ejpam-6014	8	23	,	,	PUNCT
ejpam-6014	8	24	sp)-open	sp)-open	ADJ
ejpam-6014	8	25	sets	set	NOUN
ejpam-6014	8	26	and	and	CCONJ
ejpam-6014	8	27	(	(	PUNCT
ejpam-6014	8	28	λ	λ	PROPN
ejpam-6014	8	29	,	,	PUNCT
ejpam-6014	8	30	sp)-closed	sp)-close	VERB
ejpam-6014	8	31	sets	set	NOUN
ejpam-6014	8	32	.	.	PUNCT
ejpam-6014	9	1	dungthaisong	dungthaisong	NOUN
ejpam-6014	9	2	et	et	PROPN
ejpam-6014	9	3	al	al	PROPN
ejpam-6014	9	4	.	.	PUNCT
ejpam-6014	10	1	[	[	X
ejpam-6014	10	2	3	3	NUM
ejpam-6014	10	3	]	]	PUNCT
ejpam-6014	10	4	introduced	introduce	VERB
ejpam-6014	10	5	and	and	CCONJ
ejpam-6014	10	6	studied	study	VERB
ejpam-6014	10	7	the	the	DET
ejpam-6014	10	8	concept	concept	NOUN
ejpam-6014	10	9	of	of	ADP
ejpam-6014	10	10	g(m	g(m	ADJ
ejpam-6014	10	11	,	,	PUNCT
ejpam-6014	10	12	n)-continuous	n)-continuous	ADJ
ejpam-6014	10	13	functions	function	NOUN
ejpam-6014	10	14	.	.	PUNCT
ejpam-6014	11	1	duangphui	duangphui	NOUN
ejpam-6014	11	2	et	et	PROPN
ejpam-6014	11	3	al	al	PROPN
ejpam-6014	11	4	.	.	PUNCT
ejpam-6014	12	1	[	[	X
ejpam-6014	12	2	4	4	X
ejpam-6014	12	3	]	]	PUNCT
ejpam-6014	12	4	introduced	introduce	VERB
ejpam-6014	12	5	and	and	CCONJ
ejpam-6014	12	6	investigated	investigate	VERB
ejpam-6014	12	7	the	the	DET
ejpam-6014	12	8	notion	notion	NOUN
ejpam-6014	12	9	of	of	ADP
ejpam-6014	12	10	almost	almost	ADV
ejpam-6014	12	11	(	(	PUNCT
ejpam-6014	12	12	µ	µ	NUM
ejpam-6014	12	13	,	,	PUNCT
ejpam-6014	12	14	µ′)(m	µ′)(m	VERB
ejpam-6014	12	15	,	,	PUNCT
ejpam-6014	12	16	n)continuous	n)continuous	ADJ
ejpam-6014	12	17	functions	function	NOUN
ejpam-6014	12	18	.	.	PUNCT
ejpam-6014	13	1	moreover	moreover	ADV
ejpam-6014	13	2	,	,	PUNCT
ejpam-6014	13	3	some	some	DET
ejpam-6014	13	4	characterizations	characterization	NOUN
ejpam-6014	13	5	of	of	ADP
ejpam-6014	13	6	almost	almost	ADV
ejpam-6014	13	7	(	(	PUNCT
ejpam-6014	13	8	λ	λ	PROPN
ejpam-6014	13	9	,	,	PUNCT
ejpam-6014	13	10	p)-continuous	p)-continuous	ADJ
ejpam-6014	13	11	functions	function	NOUN
ejpam-6014	13	12	,	,	PUNCT
ejpam-6014	13	13	almost	almost	ADV
ejpam-6014	13	14	strongly	strongly	ADV
ejpam-6014	13	15	θ(λ	θ(λ	VERB
ejpam-6014	13	16	,	,	PUNCT
ejpam-6014	13	17	p)-continuous	p)-continuous	ADJ
ejpam-6014	13	18	functions	function	NOUN
ejpam-6014	13	19	,	,	PUNCT
ejpam-6014	13	20	weakly	weakly	ADJ
ejpam-6014	13	21	(	(	PUNCT
ejpam-6014	13	22	λ	λ	PROPN
ejpam-6014	13	23	,	,	PUNCT
ejpam-6014	13	24	b)-continuous	b)-continuous	ADJ
ejpam-6014	13	25	functions	function	NOUN
ejpam-6014	13	26	,	,	PUNCT
ejpam-6014	13	27	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-6014	13	28	functions	function	NOUN
ejpam-6014	13	29	,	,	PUNCT
ejpam-6014	13	30	(	(	PUNCT
ejpam-6014	13	31	λ	λ	NOUN
ejpam-6014	13	32	,	,	PUNCT
ejpam-6014	13	33	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-6014	13	34	functions	function	NOUN
ejpam-6014	13	35	,	,	PUNCT
ejpam-6014	13	36	⋆-continuous	⋆-continuous	ADJ
ejpam-6014	13	37	functions	function	NOUN
ejpam-6014	13	38	,	,	PUNCT
ejpam-6014	13	39	θ	θ	PROPN
ejpam-6014	13	40	-	-	ADJ
ejpam-6014	13	41	i	i	VERB
ejpam-6014	13	42	continuous	continuous	ADJ
ejpam-6014	13	43	functions	function	NOUN
ejpam-6014	13	44	,	,	PUNCT
ejpam-6014	13	45	almost	almost	ADV
ejpam-6014	13	46	(	(	PUNCT
ejpam-6014	13	47	g	g	NOUN
ejpam-6014	13	48	,	,	PUNCT
ejpam-6014	13	49	m)-continuous	m)-continuous	ADJ
ejpam-6014	13	50	functions	function	NOUN
ejpam-6014	13	51	and	and	CCONJ
ejpam-6014	13	52	pairwise	pairwise	NOUN
ejpam-6014	13	53	weaklym	weaklym	PROPN
ejpam-6014	13	54	-continuous	-continuous	ADJ
ejpam-6014	13	55	functions	function	NOUN
ejpam-6014	13	56	were	be	AUX
ejpam-6014	13	57	presented	present	VERB
ejpam-6014	13	58	in	in	ADP
ejpam-6014	13	59	[	[	X
ejpam-6014	13	60	5	5	NUM
ejpam-6014	13	61	]	]	PUNCT
ejpam-6014	13	62	,	,	PUNCT
ejpam-6014	13	63	[	[	X
ejpam-6014	13	64	6	6	NUM
ejpam-6014	13	65	]	]	PUNCT
ejpam-6014	13	66	,	,	PUNCT
ejpam-6014	13	67	[	[	X
ejpam-6014	13	68	7	7	NUM
ejpam-6014	13	69	]	]	PUNCT
ejpam-6014	13	70	,	,	PUNCT
ejpam-6014	13	71	[	[	X
ejpam-6014	13	72	8	8	NUM
ejpam-6014	13	73	]	]	PUNCT
ejpam-6014	13	74	,	,	PUNCT
ejpam-6014	14	1	[	[	X
ejpam-6014	14	2	9	9	NUM
ejpam-6014	14	3	]	]	PUNCT
ejpam-6014	14	4	,	,	PUNCT
ejpam-6014	14	5	[	[	X
ejpam-6014	14	6	10	10	NUM
ejpam-6014	14	7	]	]	PUNCT
ejpam-6014	14	8	,	,	PUNCT
ejpam-6014	15	1	[	[	X
ejpam-6014	15	2	11	11	NUM
ejpam-6014	15	3	]	]	PUNCT
ejpam-6014	15	4	,	,	PUNCT
ejpam-6014	15	5	[	[	X
ejpam-6014	15	6	12	12	NUM
ejpam-6014	15	7	]	]	PUNCT
ejpam-6014	15	8	and	and	CCONJ
ejpam-6014	16	1	[	[	X
ejpam-6014	16	2	13	13	NUM
ejpam-6014	16	3	]	]	PUNCT
ejpam-6014	16	4	,	,	PUNCT
ejpam-6014	16	5	respectively	respectively	ADV
ejpam-6014	16	6	.	.	PUNCT
ejpam-6014	17	1	the	the	DET
ejpam-6014	17	2	concept	concept	NOUN
ejpam-6014	17	3	of	of	ADP
ejpam-6014	17	4	θ	θ	ADJ
ejpam-6014	17	5	-	-	ADJ
ejpam-6014	17	6	continuous	continuous	ADJ
ejpam-6014	17	7	functions	function	NOUN
ejpam-6014	17	8	was	be	AUX
ejpam-6014	17	9	introduced	introduce	VERB
ejpam-6014	17	10	by	by	ADP
ejpam-6014	17	11	fomin	fomin	NOUN
ejpam-6014	18	1	[	[	X
ejpam-6014	18	2	14	14	NUM
ejpam-6014	18	3	]	]	PUNCT
ejpam-6014	18	4	.	.	PUNCT
ejpam-6014	19	1	noiri	noiri	PROPN
ejpam-6014	20	1	[	[	X
ejpam-6014	20	2	15	15	NUM
ejpam-6014	20	3	]	]	PUNCT
ejpam-6014	20	4	studied	study	VERB
ejpam-6014	20	5	some	some	DET
ejpam-6014	20	6	properties	property	NOUN
ejpam-6014	20	7	of	of	ADP
ejpam-6014	20	8	θ	θ	ADJ
ejpam-6014	20	9	-	-	ADJ
ejpam-6014	20	10	continuous	continuous	ADJ
ejpam-6014	20	11	functions	function	NOUN
ejpam-6014	20	12	.	.	PUNCT
ejpam-6014	21	1	furthermore	furthermore	ADV
ejpam-6014	21	2	,	,	PUNCT
ejpam-6014	21	3	the	the	DET
ejpam-6014	21	4	present	present	ADJ
ejpam-6014	21	5	author	author	NOUN
ejpam-6014	21	6	[	[	X
ejpam-6014	21	7	16	16	NUM
ejpam-6014	21	8	]	]	PUNCT
ejpam-6014	21	9	investigated	investigate	VERB
ejpam-6014	21	10	∗corresponding	∗corresponde	VERB
ejpam-6014	21	11	author	author	NOUN
ejpam-6014	21	12	.	.	PUNCT
ejpam-6014	22	1	doi	doi	NOUN
ejpam-6014	22	2	:	:	PUNCT
ejpam-6014	22	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6014	https://doi.org/10.29020/nybg.ejpam.v18i2.6014	NUM
ejpam-6014	22	4	email	email	NOUN
ejpam-6014	22	5	addresses	address	NOUN
ejpam-6014	22	6	:	:	PUNCT
ejpam-6014	22	7	montri.t@msu.ac.th	montri.t@msu.ac.th	PROPN
ejpam-6014	22	8	(	(	PUNCT
ejpam-6014	22	9	m.	m.	NOUN
ejpam-6014	22	10	thongmoon	thongmoon	PROPN
ejpam-6014	22	11	)	)	PUNCT
ejpam-6014	22	12	,	,	PUNCT
ejpam-6014	22	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-6014	22	14	(	(	PUNCT
ejpam-6014	22	15	s.	s.	PROPN
ejpam-6014	22	16	sompong	sompong	PROPN
ejpam-6014	22	17	)	)	PUNCT
ejpam-6014	22	18	,	,	PUNCT
ejpam-6014	22	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6014	22	20	(	(	PUNCT
ejpam-6014	22	21	c.	c.	PROPN
ejpam-6014	22	22	boonpok	boonpok	PROPN
ejpam-6014	22	23	)	)	PUNCT
ejpam-6014	22	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6014	23	1	1	1	NUM
ejpam-6014	23	2	copyright	copyright	NOUN
ejpam-6014	23	3	:	:	PUNCT
ejpam-6014	23	4	©	©	PROPN
ejpam-6014	23	5	2025	2025	NUM
ejpam-6014	23	6	the	the	DET
ejpam-6014	23	7	author(s	author(s	NOUN
ejpam-6014	23	8	)	)	PUNCT
ejpam-6014	23	9	.	.	PUNCT
ejpam-6014	24	1	(	(	PUNCT
ejpam-6014	24	2	cc	cc	NOUN
ejpam-6014	24	3	by	by	ADP
ejpam-6014	24	4	-	-	PUNCT
ejpam-6014	24	5	nc	nc	PROPN
ejpam-6014	24	6	4.0	4.0	NUM
ejpam-6014	24	7	)	)	PUNCT
ejpam-6014	24	8	m.	m.	NOUN
ejpam-6014	24	9	thongmoon	thongmoon	NOUN
ejpam-6014	24	10	,	,	PUNCT
ejpam-6014	24	11	s.	s.	PROPN
ejpam-6014	24	12	sompong	sompong	PROPN
ejpam-6014	24	13	,	,	PUNCT
ejpam-6014	24	14	c.	c.	PROPN
ejpam-6014	24	15	boonpok	boonpok	PROPN
ejpam-6014	24	16	/	/	SYM
ejpam-6014	24	17	eur	eur	PROPN
ejpam-6014	24	18	.	.	PUNCT
ejpam-6014	25	1	j.	j.	PROPN
ejpam-6014	25	2	pure	pure	PROPN
ejpam-6014	25	3	appl	appl	PROPN
ejpam-6014	25	4	.	.	PROPN
ejpam-6014	25	5	math	math	PROPN
ejpam-6014	25	6	,	,	PUNCT
ejpam-6014	25	7	18	18	NUM
ejpam-6014	25	8	(	(	PUNCT
ejpam-6014	25	9	2	2	NUM
ejpam-6014	25	10	)	)	PUNCT
ejpam-6014	25	11	(	(	PUNCT
ejpam-6014	25	12	2025	2025	NUM
ejpam-6014	25	13	)	)	PUNCT
ejpam-6014	25	14	,	,	PUNCT
ejpam-6014	25	15	6014	6014	NUM
ejpam-6014	25	16	2	2	NUM
ejpam-6014	25	17	of	of	ADP
ejpam-6014	25	18	13	13	NUM
ejpam-6014	25	19	several	several	ADJ
ejpam-6014	25	20	characterizations	characterization	NOUN
ejpam-6014	25	21	of	of	ADP
ejpam-6014	25	22	θ	θ	ADJ
ejpam-6014	25	23	-	-	ADJ
ejpam-6014	25	24	continuous	continuous	ADJ
ejpam-6014	25	25	functions	function	NOUN
ejpam-6014	25	26	.	.	PUNCT
ejpam-6014	26	1	arya	arya	PROPN
ejpam-6014	26	2	and	and	CCONJ
ejpam-6014	26	3	bhamini	bhamini	PROPN
ejpam-6014	27	1	[	[	X
ejpam-6014	27	2	17	17	NUM
ejpam-6014	27	3	]	]	PUNCT
ejpam-6014	27	4	introduced	introduce	VERB
ejpam-6014	27	5	the	the	DET
ejpam-6014	27	6	notion	notion	NOUN
ejpam-6014	27	7	of	of	ADP
ejpam-6014	27	8	θ	θ	NOUN
ejpam-6014	27	9	-	-	PUNCT
ejpam-6014	27	10	semi	semi	ADJ
ejpam-6014	27	11	-	-	ADJ
ejpam-6014	27	12	continuous	continuous	ADJ
ejpam-6014	27	13	functions	function	NOUN
ejpam-6014	27	14	.	.	PUNCT
ejpam-6014	28	1	jafari	jafari	PROPN
ejpam-6014	28	2	and	and	CCONJ
ejpam-6014	28	3	noiri	noiri	ADV
ejpam-6014	29	1	[	[	X
ejpam-6014	29	2	18	18	NUM
ejpam-6014	29	3	]	]	PUNCT
ejpam-6014	29	4	investigated	investigate	VERB
ejpam-6014	29	5	several	several	ADJ
ejpam-6014	29	6	characterizations	characterization	NOUN
ejpam-6014	29	7	of	of	ADP
ejpam-6014	29	8	θ	θ	NOUN
ejpam-6014	29	9	-	-	PUNCT
ejpam-6014	29	10	semi	semi	ADJ
ejpam-6014	29	11	-	-	ADJ
ejpam-6014	29	12	continuous	continuous	ADJ
ejpam-6014	29	13	functions	function	NOUN
ejpam-6014	29	14	.	.	PUNCT
ejpam-6014	30	1	noiri	noiri	PROPN
ejpam-6014	31	1	[	[	X
ejpam-6014	31	2	19	19	NUM
ejpam-6014	31	3	]	]	PUNCT
ejpam-6014	31	4	introduced	introduce	VERB
ejpam-6014	31	5	and	and	CCONJ
ejpam-6014	31	6	investigated	investigate	VERB
ejpam-6014	31	7	the	the	DET
ejpam-6014	31	8	concept	concept	NOUN
ejpam-6014	31	9	of	of	ADP
ejpam-6014	31	10	θ	θ	ADJ
ejpam-6014	31	11	-	-	ADJ
ejpam-6014	31	12	precontinuous	precontinuous	ADJ
ejpam-6014	31	13	functions	function	NOUN
ejpam-6014	31	14	.	.	PUNCT
ejpam-6014	32	1	baker	baker	NOUN
ejpam-6014	33	1	[	[	X
ejpam-6014	33	2	20	20	NUM
ejpam-6014	33	3	]	]	PUNCT
ejpam-6014	33	4	introduced	introduce	VERB
ejpam-6014	33	5	and	and	CCONJ
ejpam-6014	33	6	studied	study	VERB
ejpam-6014	33	7	the	the	DET
ejpam-6014	33	8	notion	notion	NOUN
ejpam-6014	33	9	of	of	ADP
ejpam-6014	33	10	weakly	weakly	ADJ
ejpam-6014	33	11	θprecontinuous	θprecontinuous	ADJ
ejpam-6014	33	12	functions	function	NOUN
ejpam-6014	33	13	.	.	PUNCT
ejpam-6014	34	1	noiri	noiri	PROPN
ejpam-6014	34	2	and	and	CCONJ
ejpam-6014	34	3	popa	popa	NOUN
ejpam-6014	34	4	[	[	X
ejpam-6014	34	5	21	21	NUM
ejpam-6014	34	6	]	]	PUNCT
ejpam-6014	34	7	introduced	introduce	VERB
ejpam-6014	34	8	the	the	DET
ejpam-6014	34	9	concept	concept	NOUN
ejpam-6014	34	10	of	of	ADP
ejpam-6014	34	11	θ	θ	PROPN
ejpam-6014	34	12	-	-	ADJ
ejpam-6014	34	13	m	m	NOUN
ejpam-6014	34	14	-continuous	-continuous	ADJ
ejpam-6014	34	15	functions	function	NOUN
ejpam-6014	34	16	as	as	ADP
ejpam-6014	34	17	functions	function	NOUN
ejpam-6014	34	18	from	from	ADP
ejpam-6014	34	19	a	a	DET
ejpam-6014	34	20	set	set	NOUN
ejpam-6014	34	21	satisfying	satisfy	VERB
ejpam-6014	34	22	some	some	DET
ejpam-6014	34	23	minimal	minimal	ADJ
ejpam-6014	34	24	conditions	condition	NOUN
ejpam-6014	34	25	into	into	ADP
ejpam-6014	34	26	a	a	DET
ejpam-6014	34	27	set	set	NOUN
ejpam-6014	34	28	satisfying	satisfy	VERB
ejpam-6014	34	29	some	some	DET
ejpam-6014	34	30	minimal	minimal	ADJ
ejpam-6014	34	31	conditions	condition	NOUN
ejpam-6014	34	32	and	and	CCONJ
ejpam-6014	34	33	investigated	investigate	VERB
ejpam-6014	34	34	some	some	DET
ejpam-6014	34	35	characterizations	characterization	NOUN
ejpam-6014	34	36	and	and	CCONJ
ejpam-6014	34	37	several	several	ADJ
ejpam-6014	34	38	properties	property	NOUN
ejpam-6014	34	39	of	of	ADP
ejpam-6014	34	40	θ	θ	PROPN
ejpam-6014	34	41	-	-	ADJ
ejpam-6014	34	42	m	m	NOUN
ejpam-6014	34	43	-continuous	-continuous	ADJ
ejpam-6014	34	44	functions	function	NOUN
ejpam-6014	34	45	.	.	PUNCT
ejpam-6014	35	1	in	in	ADP
ejpam-6014	35	2	particular	particular	ADJ
ejpam-6014	35	3	,	,	PUNCT
ejpam-6014	35	4	noiri	noiri	PROPN
ejpam-6014	35	5	and	and	CCONJ
ejpam-6014	35	6	popa	popa	NOUN
ejpam-6014	35	7	[	[	X
ejpam-6014	35	8	21	21	NUM
ejpam-6014	35	9	]	]	PUNCT
ejpam-6014	35	10	defined	define	VERB
ejpam-6014	35	11	and	and	CCONJ
ejpam-6014	35	12	studied	study	VERB
ejpam-6014	35	13	the	the	DET
ejpam-6014	35	14	notion	notion	NOUN
ejpam-6014	35	15	of	of	ADP
ejpam-6014	35	16	strongly	strongly	ADV
ejpam-6014	35	17	θ	θ	NOUN
ejpam-6014	35	18	-	-	PUNCT
ejpam-6014	35	19	m	m	PUNCT
ejpam-6014	35	20	-closed	-close	VERB
ejpam-6014	35	21	graphs	graph	NOUN
ejpam-6014	35	22	.	.	PUNCT
ejpam-6014	36	1	noiri	noiri	PROPN
ejpam-6014	36	2	and	and	CCONJ
ejpam-6014	36	3	popa	popa	NOUN
ejpam-6014	37	1	[	[	X
ejpam-6014	37	2	22	22	NUM
ejpam-6014	37	3	]	]	PUNCT
ejpam-6014	37	4	introduced	introduce	VERB
ejpam-6014	37	5	the	the	DET
ejpam-6014	37	6	concept	concept	NOUN
ejpam-6014	37	7	of	of	ADP
ejpam-6014	37	8	θm	θm	NOUN
ejpam-6014	37	9	-	-	PUNCT
ejpam-6014	37	10	continuous	continuous	ADJ
ejpam-6014	37	11	functions	function	NOUN
ejpam-6014	37	12	as	as	ADP
ejpam-6014	37	13	functions	function	NOUN
ejpam-6014	37	14	from	from	ADP
ejpam-6014	37	15	a	a	DET
ejpam-6014	37	16	set	set	NOUN
ejpam-6014	37	17	satisfying	satisfy	VERB
ejpam-6014	37	18	some	some	DET
ejpam-6014	37	19	minimal	minimal	ADJ
ejpam-6014	37	20	conditions	condition	NOUN
ejpam-6014	37	21	into	into	ADP
ejpam-6014	37	22	a	a	DET
ejpam-6014	37	23	topological	topological	ADJ
ejpam-6014	37	24	space	space	NOUN
ejpam-6014	37	25	and	and	CCONJ
ejpam-6014	37	26	obtained	obtain	VERB
ejpam-6014	37	27	several	several	ADJ
ejpam-6014	37	28	characterizations	characterization	NOUN
ejpam-6014	37	29	of	of	ADP
ejpam-6014	37	30	such	such	ADJ
ejpam-6014	37	31	functions	function	NOUN
ejpam-6014	37	32	.	.	PUNCT
ejpam-6014	38	1	long	long	ADJ
ejpam-6014	38	2	and	and	CCONJ
ejpam-6014	38	3	herrington	herrington	PROPN
ejpam-6014	39	1	[	[	X
ejpam-6014	39	2	23	23	NUM
ejpam-6014	39	3	]	]	PUNCT
ejpam-6014	39	4	investigated	investigate	VERB
ejpam-6014	39	5	some	some	DET
ejpam-6014	39	6	characterizations	characterization	NOUN
ejpam-6014	39	7	of	of	ADP
ejpam-6014	39	8	strongly	strongly	ADV
ejpam-6014	39	9	θ	θ	ADJ
ejpam-6014	39	10	-	-	ADJ
ejpam-6014	39	11	continuous	continuous	ADJ
ejpam-6014	39	12	functions	function	NOUN
ejpam-6014	39	13	.	.	PUNCT
ejpam-6014	40	1	jafari	jafari	PROPN
ejpam-6014	40	2	and	and	CCONJ
ejpam-6014	40	3	noiri	noiri	ADV
ejpam-6014	41	1	[	[	X
ejpam-6014	41	2	24	24	NUM
ejpam-6014	41	3	]	]	PUNCT
ejpam-6014	41	4	introduced	introduce	VERB
ejpam-6014	41	5	and	and	CCONJ
ejpam-6014	41	6	studied	study	VERB
ejpam-6014	41	7	the	the	DET
ejpam-6014	41	8	notion	notion	NOUN
ejpam-6014	41	9	of	of	ADP
ejpam-6014	41	10	strongly	strongly	ADV
ejpam-6014	41	11	θ	θ	NOUN
ejpam-6014	41	12	-	-	PUNCT
ejpam-6014	41	13	semi	semi	ADJ
ejpam-6014	41	14	-	-	ADJ
ejpam-6014	41	15	continuous	continuous	ADJ
ejpam-6014	41	16	functions	function	NOUN
ejpam-6014	41	17	.	.	PUNCT
ejpam-6014	42	1	noiri	noiri	PROPN
ejpam-6014	43	1	[	[	X
ejpam-6014	43	2	25	25	NUM
ejpam-6014	43	3	]	]	PUNCT
ejpam-6014	43	4	introduced	introduce	VERB
ejpam-6014	43	5	and	and	CCONJ
ejpam-6014	43	6	investigated	investigate	VERB
ejpam-6014	43	7	the	the	DET
ejpam-6014	43	8	concept	concept	NOUN
ejpam-6014	43	9	of	of	ADP
ejpam-6014	43	10	strongly	strongly	ADV
ejpam-6014	43	11	θ	θ	ADJ
ejpam-6014	43	12	-	-	ADJ
ejpam-6014	43	13	precontinuous	precontinuous	ADJ
ejpam-6014	43	14	functions	function	NOUN
ejpam-6014	43	15	.	.	PUNCT
ejpam-6014	44	1	pue	pue	NOUN
ejpam-6014	44	2	-	-	PUNCT
ejpam-6014	44	3	on	on	NOUN
ejpam-6014	44	4	and	and	CCONJ
ejpam-6014	44	5	boonpok	boonpok	VERB
ejpam-6014	44	6	[	[	X
ejpam-6014	44	7	26	26	NUM
ejpam-6014	44	8	]	]	PUNCT
ejpam-6014	44	9	introduced	introduce	VERB
ejpam-6014	44	10	and	and	CCONJ
ejpam-6014	44	11	studied	study	VERB
ejpam-6014	44	12	the	the	DET
ejpam-6014	44	13	concept	concept	NOUN
ejpam-6014	44	14	of	of	ADP
ejpam-6014	44	15	θ(λ	θ(λ	PROPN
ejpam-6014	44	16	,	,	PUNCT
ejpam-6014	44	17	p)-continuous	p)-continuous	ADJ
ejpam-6014	44	18	functions	function	NOUN
ejpam-6014	44	19	.	.	PUNCT
ejpam-6014	45	1	quite	quite	ADV
ejpam-6014	45	2	recently	recently	ADV
ejpam-6014	45	3	,	,	PUNCT
ejpam-6014	45	4	thongmoon	thongmoon	NOUN
ejpam-6014	45	5	and	and	CCONJ
ejpam-6014	45	6	boonpok	boonpok	VERB
ejpam-6014	46	1	[	[	X
ejpam-6014	46	2	27	27	NUM
ejpam-6014	46	3	]	]	PUNCT
ejpam-6014	46	4	introduced	introduce	VERB
ejpam-6014	46	5	and	and	CCONJ
ejpam-6014	46	6	investigated	investigate	VERB
ejpam-6014	46	7	the	the	DET
ejpam-6014	46	8	notion	notion	NOUN
ejpam-6014	46	9	of	of	ADP
ejpam-6014	46	10	strongly	strongly	ADV
ejpam-6014	46	11	θ(λ	θ(λ	ADJ
ejpam-6014	46	12	,	,	PUNCT
ejpam-6014	46	13	p)-continuous	p)-continuous	ADJ
ejpam-6014	46	14	functions	function	NOUN
ejpam-6014	46	15	.	.	PUNCT
ejpam-6014	47	1	on	on	ADP
ejpam-6014	47	2	the	the	DET
ejpam-6014	47	3	other	other	ADJ
ejpam-6014	47	4	hand	hand	NOUN
ejpam-6014	47	5	,	,	PUNCT
ejpam-6014	47	6	the	the	DET
ejpam-6014	47	7	present	present	ADJ
ejpam-6014	47	8	authors	author	NOUN
ejpam-6014	47	9	introduced	introduce	VERB
ejpam-6014	47	10	and	and	CCONJ
ejpam-6014	47	11	studied	study	VERB
ejpam-6014	47	12	the	the	DET
ejpam-6014	47	13	concepts	concept	NOUN
ejpam-6014	47	14	of	of	ADP
ejpam-6014	47	15	(	(	PUNCT
ejpam-6014	47	16	τ1	τ1	PROPN
ejpam-6014	47	17	,	,	PUNCT
ejpam-6014	47	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	47	19	functions	function	NOUN
ejpam-6014	47	20	[	[	X
ejpam-6014	47	21	28	28	NUM
ejpam-6014	47	22	]	]	PUNCT
ejpam-6014	47	23	,	,	PUNCT
ejpam-6014	47	24	almost	almost	ADV
ejpam-6014	47	25	(	(	PUNCT
ejpam-6014	47	26	τ1	τ1	NOUN
ejpam-6014	47	27	,	,	PUNCT
ejpam-6014	47	28	τ2)continuous	τ2)continuous	ADJ
ejpam-6014	47	29	functions	function	NOUN
ejpam-6014	47	30	[	[	X
ejpam-6014	47	31	29	29	NUM
ejpam-6014	47	32	]	]	PUNCT
ejpam-6014	47	33	,	,	PUNCT
ejpam-6014	47	34	weakly	weakly	ADJ
ejpam-6014	47	35	(	(	PUNCT
ejpam-6014	47	36	τ1	τ1	NOUN
ejpam-6014	47	37	,	,	PUNCT
ejpam-6014	47	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	47	39	functions	function	NOUN
ejpam-6014	47	40	[	[	X
ejpam-6014	47	41	30	30	NUM
ejpam-6014	47	42	]	]	PUNCT
ejpam-6014	47	43	and	and	CCONJ
ejpam-6014	47	44	quasi	quasi	PROPN
ejpam-6014	47	45	θ(τ1	θ(τ1	NOUN
ejpam-6014	47	46	,	,	PUNCT
ejpam-6014	47	47	τ2)continuous	τ2)continuous	ADJ
ejpam-6014	47	48	functions	function	NOUN
ejpam-6014	47	49	[	[	X
ejpam-6014	47	50	31	31	NUM
ejpam-6014	47	51	]	]	PUNCT
ejpam-6014	47	52	.	.	PUNCT
ejpam-6014	48	1	in	in	ADP
ejpam-6014	48	2	this	this	DET
ejpam-6014	48	3	paper	paper	NOUN
ejpam-6014	48	4	,	,	PUNCT
ejpam-6014	48	5	we	we	PRON
ejpam-6014	48	6	introduce	introduce	VERB
ejpam-6014	48	7	the	the	DET
ejpam-6014	48	8	concept	concept	NOUN
ejpam-6014	48	9	of	of	ADP
ejpam-6014	48	10	θ(τ1	θ(τ1	NOUN
ejpam-6014	48	11	,	,	PUNCT
ejpam-6014	48	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	48	13	functions	function	NOUN
ejpam-6014	48	14	.	.	PUNCT
ejpam-6014	49	1	we	we	PRON
ejpam-6014	49	2	also	also	ADV
ejpam-6014	49	3	investigate	investigate	VERB
ejpam-6014	49	4	several	several	ADJ
ejpam-6014	49	5	characterizations	characterization	NOUN
ejpam-6014	49	6	of	of	ADP
ejpam-6014	49	7	θ(τ1	θ(τ1	NOUN
ejpam-6014	49	8	,	,	PUNCT
ejpam-6014	49	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	49	10	functions	function	NOUN
ejpam-6014	49	11	.	.	PUNCT
ejpam-6014	50	1	2	2	X
ejpam-6014	50	2	.	.	NUM
ejpam-6014	50	3	preliminaries	preliminary	NOUN
ejpam-6014	50	4	throughout	throughout	ADP
ejpam-6014	50	5	the	the	DET
ejpam-6014	50	6	present	present	ADJ
ejpam-6014	50	7	paper	paper	NOUN
ejpam-6014	50	8	,	,	PUNCT
ejpam-6014	50	9	spaces	space	NOUN
ejpam-6014	50	10	(	(	PUNCT
ejpam-6014	50	11	x	x	NOUN
ejpam-6014	50	12	,	,	PUNCT
ejpam-6014	50	13	τ1	τ1	NOUN
ejpam-6014	50	14	,	,	PUNCT
ejpam-6014	50	15	τ2	τ2	NOUN
ejpam-6014	50	16	)	)	PUNCT
ejpam-6014	50	17	and	and	CCONJ
ejpam-6014	50	18	(	(	PUNCT
ejpam-6014	50	19	y	y	PROPN
ejpam-6014	50	20	,	,	PUNCT
ejpam-6014	50	21	σ1	σ1	PROPN
ejpam-6014	50	22	,	,	PUNCT
ejpam-6014	50	23	σ2	σ2	NOUN
ejpam-6014	50	24	)	)	PUNCT
ejpam-6014	50	25	(	(	PUNCT
ejpam-6014	50	26	or	or	CCONJ
ejpam-6014	50	27	simply	simply	ADV
ejpam-6014	50	28	x	x	X
ejpam-6014	50	29	and	and	CCONJ
ejpam-6014	50	30	y	y	PROPN
ejpam-6014	50	31	)	)	PUNCT
ejpam-6014	50	32	always	always	ADV
ejpam-6014	50	33	mean	mean	VERB
ejpam-6014	50	34	bitopological	bitopological	ADJ
ejpam-6014	50	35	spaces	space	NOUN
ejpam-6014	50	36	on	on	ADP
ejpam-6014	50	37	which	which	PRON
ejpam-6014	50	38	no	no	DET
ejpam-6014	50	39	separation	separation	NOUN
ejpam-6014	50	40	axioms	axiom	NOUN
ejpam-6014	50	41	are	be	AUX
ejpam-6014	50	42	assumed	assume	VERB
ejpam-6014	50	43	unless	unless	SCONJ
ejpam-6014	50	44	explicitly	explicitly	ADV
ejpam-6014	50	45	stated	state	VERB
ejpam-6014	50	46	.	.	PUNCT
ejpam-6014	51	1	let	let	VERB
ejpam-6014	51	2	a	a	DET
ejpam-6014	51	3	be	be	AUX
ejpam-6014	51	4	a	a	DET
ejpam-6014	51	5	subset	subset	NOUN
ejpam-6014	51	6	of	of	ADP
ejpam-6014	51	7	a	a	DET
ejpam-6014	51	8	bitopological	bitopological	ADJ
ejpam-6014	51	9	space	space	NOUN
ejpam-6014	51	10	(	(	PUNCT
ejpam-6014	51	11	x	x	NOUN
ejpam-6014	51	12	,	,	PUNCT
ejpam-6014	51	13	τ1	τ1	NOUN
ejpam-6014	51	14	,	,	PUNCT
ejpam-6014	51	15	τ2	τ2	NOUN
ejpam-6014	51	16	)	)	PUNCT
ejpam-6014	51	17	.	.	PUNCT
ejpam-6014	52	1	the	the	DET
ejpam-6014	52	2	closure	closure	NOUN
ejpam-6014	52	3	of	of	ADP
ejpam-6014	52	4	a	a	PRON
ejpam-6014	52	5	and	and	CCONJ
ejpam-6014	52	6	the	the	DET
ejpam-6014	52	7	interior	interior	NOUN
ejpam-6014	52	8	of	of	ADP
ejpam-6014	52	9	a	a	PRON
ejpam-6014	52	10	with	with	ADP
ejpam-6014	52	11	respect	respect	NOUN
ejpam-6014	52	12	to	to	ADP
ejpam-6014	52	13	τi	τi	PROPN
ejpam-6014	52	14	are	be	AUX
ejpam-6014	52	15	denoted	denote	VERB
ejpam-6014	52	16	by	by	ADP
ejpam-6014	52	17	τi	τi	NOUN
ejpam-6014	52	18	-	-	PUNCT
ejpam-6014	52	19	cl(a	cl(a	NUM
ejpam-6014	52	20	)	)	PUNCT
ejpam-6014	52	21	and	and	CCONJ
ejpam-6014	52	22	τi	τi	NOUN
ejpam-6014	52	23	-	-	PUNCT
ejpam-6014	52	24	int(a	int(a	NOUN
ejpam-6014	52	25	)	)	PUNCT
ejpam-6014	52	26	,	,	PUNCT
ejpam-6014	52	27	respectively	respectively	ADV
ejpam-6014	52	28	,	,	PUNCT
ejpam-6014	52	29	for	for	ADP
ejpam-6014	52	30	i	i	PROPN
ejpam-6014	52	31	=	=	SYM
ejpam-6014	52	32	1	1	NUM
ejpam-6014	52	33	,	,	PUNCT
ejpam-6014	52	34	2	2	NUM
ejpam-6014	52	35	.	.	X
ejpam-6014	52	36	a	a	DET
ejpam-6014	52	37	subset	subset	NOUN
ejpam-6014	52	38	a	a	PRON
ejpam-6014	52	39	of	of	ADP
ejpam-6014	52	40	a	a	DET
ejpam-6014	52	41	bitopological	bitopological	ADJ
ejpam-6014	52	42	space	space	NOUN
ejpam-6014	52	43	(	(	PUNCT
ejpam-6014	52	44	x	x	NOUN
ejpam-6014	52	45	,	,	PUNCT
ejpam-6014	52	46	τ1	τ1	NOUN
ejpam-6014	52	47	,	,	PUNCT
ejpam-6014	52	48	τ2	τ2	NOUN
ejpam-6014	52	49	)	)	PUNCT
ejpam-6014	52	50	is	be	AUX
ejpam-6014	52	51	called	call	VERB
ejpam-6014	52	52	τ1τ2	τ1τ2	VERB
ejpam-6014	52	53	-	-	ADJ
ejpam-6014	52	54	closed	closed	ADJ
ejpam-6014	52	55	[	[	X
ejpam-6014	52	56	32	32	NUM
ejpam-6014	52	57	]	]	PUNCT
ejpam-6014	52	58	if	if	SCONJ
ejpam-6014	52	59	a	a	DET
ejpam-6014	52	60	=	=	NOUN
ejpam-6014	52	61	τ1	τ1	NOUN
ejpam-6014	52	62	-	-	PUNCT
ejpam-6014	52	63	cl(τ2	cl(τ2	NOUN
ejpam-6014	52	64	-	-	PUNCT
ejpam-6014	52	65	cl(a	cl(a	NUM
ejpam-6014	52	66	)	)	PUNCT
ejpam-6014	52	67	)	)	PUNCT
ejpam-6014	52	68	.	.	PUNCT
ejpam-6014	53	1	the	the	DET
ejpam-6014	53	2	complement	complement	NOUN
ejpam-6014	53	3	of	of	ADP
ejpam-6014	53	4	a	a	DET
ejpam-6014	53	5	τ1τ2	τ1τ2	ADJ
ejpam-6014	53	6	-	-	ADJ
ejpam-6014	53	7	closed	closed	ADJ
ejpam-6014	53	8	set	set	NOUN
ejpam-6014	53	9	is	be	AUX
ejpam-6014	53	10	called	call	VERB
ejpam-6014	53	11	τ1τ2	τ1τ2	NOUN
ejpam-6014	53	12	-	-	ADJ
ejpam-6014	53	13	open	open	ADJ
ejpam-6014	53	14	.	.	PUNCT
ejpam-6014	54	1	the	the	DET
ejpam-6014	54	2	intersection	intersection	NOUN
ejpam-6014	54	3	of	of	ADP
ejpam-6014	54	4	all	all	DET
ejpam-6014	54	5	τ1τ2	τ1τ2	ADJ
ejpam-6014	54	6	-	-	ADJ
ejpam-6014	54	7	closed	closed	ADJ
ejpam-6014	54	8	sets	set	NOUN
ejpam-6014	54	9	of	of	ADP
ejpam-6014	54	10	x	x	PUNCT
ejpam-6014	54	11	containing	contain	VERB
ejpam-6014	54	12	a	a	PRON
ejpam-6014	54	13	is	be	AUX
ejpam-6014	54	14	called	call	VERB
ejpam-6014	54	15	the	the	DET
ejpam-6014	54	16	τ1τ2	τ1τ2	NOUN
ejpam-6014	54	17	-	-	NOUN
ejpam-6014	54	18	closure	closure	NOUN
ejpam-6014	54	19	[	[	X
ejpam-6014	54	20	32	32	NUM
ejpam-6014	54	21	]	]	PUNCT
ejpam-6014	54	22	of	of	ADP
ejpam-6014	54	23	a	a	PRON
ejpam-6014	54	24	and	and	CCONJ
ejpam-6014	54	25	is	be	AUX
ejpam-6014	54	26	denoted	denote	VERB
ejpam-6014	54	27	by	by	ADP
ejpam-6014	54	28	τ1τ2	τ1τ2	NOUN
ejpam-6014	54	29	-	-	NUM
ejpam-6014	54	30	cl(a	cl(a	NUM
ejpam-6014	54	31	)	)	PUNCT
ejpam-6014	54	32	.	.	PUNCT
ejpam-6014	55	1	the	the	DET
ejpam-6014	55	2	union	union	NOUN
ejpam-6014	55	3	of	of	ADP
ejpam-6014	55	4	all	all	DET
ejpam-6014	55	5	τ1τ2	τ1τ2	ADJ
ejpam-6014	55	6	-	-	ADJ
ejpam-6014	55	7	open	open	ADJ
ejpam-6014	55	8	sets	set	NOUN
ejpam-6014	55	9	of	of	ADP
ejpam-6014	55	10	x	x	PUNCT
ejpam-6014	55	11	contained	contain	VERB
ejpam-6014	55	12	in	in	ADP
ejpam-6014	55	13	a	a	PRON
ejpam-6014	55	14	is	be	AUX
ejpam-6014	55	15	called	call	VERB
ejpam-6014	55	16	the	the	DET
ejpam-6014	55	17	τ1τ2	τ1τ2	NOUN
ejpam-6014	55	18	-	-	ADJ
ejpam-6014	55	19	interior	interior	ADJ
ejpam-6014	55	20	[	[	X
ejpam-6014	55	21	32	32	NUM
ejpam-6014	55	22	]	]	PUNCT
ejpam-6014	55	23	of	of	ADP
ejpam-6014	55	24	a	a	PRON
ejpam-6014	55	25	and	and	CCONJ
ejpam-6014	55	26	is	be	AUX
ejpam-6014	55	27	denoted	denote	VERB
ejpam-6014	55	28	by	by	ADP
ejpam-6014	55	29	τ1τ2	τ1τ2	NOUN
ejpam-6014	55	30	-	-	ADJ
ejpam-6014	55	31	int(a	int(a	NOUN
ejpam-6014	55	32	)	)	PUNCT
ejpam-6014	55	33	.	.	PUNCT
ejpam-6014	56	1	lemma	lemma	PROPN
ejpam-6014	56	2	1	1	NUM
ejpam-6014	56	3	.	.	PUNCT
ejpam-6014	57	1	[	[	X
ejpam-6014	57	2	32	32	NUM
ejpam-6014	57	3	]	]	PUNCT
ejpam-6014	57	4	let	let	VERB
ejpam-6014	57	5	a	a	PRON
ejpam-6014	57	6	and	and	CCONJ
ejpam-6014	57	7	b	b	NOUN
ejpam-6014	57	8	be	be	AUX
ejpam-6014	57	9	subsets	subset	NOUN
ejpam-6014	57	10	of	of	ADP
ejpam-6014	57	11	a	a	DET
ejpam-6014	57	12	bitopological	bitopological	ADJ
ejpam-6014	57	13	space	space	NOUN
ejpam-6014	57	14	(	(	PUNCT
ejpam-6014	57	15	x	x	NOUN
ejpam-6014	57	16	,	,	PUNCT
ejpam-6014	57	17	τ1	τ1	NOUN
ejpam-6014	57	18	,	,	PUNCT
ejpam-6014	57	19	τ2	τ2	NOUN
ejpam-6014	57	20	)	)	PUNCT
ejpam-6014	57	21	.	.	PUNCT
ejpam-6014	58	1	for	for	ADP
ejpam-6014	58	2	the	the	DET
ejpam-6014	58	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6014	58	4	,	,	PUNCT
ejpam-6014	58	5	the	the	DET
ejpam-6014	58	6	following	follow	VERB
ejpam-6014	58	7	properties	property	NOUN
ejpam-6014	58	8	hold	hold	VERB
ejpam-6014	58	9	:	:	PUNCT
ejpam-6014	58	10	(	(	PUNCT
ejpam-6014	58	11	1	1	X
ejpam-6014	58	12	)	)	PUNCT
ejpam-6014	58	13	a	a	DET
ejpam-6014	58	14	⊆	⊆	NUM
ejpam-6014	58	15	τ1τ2	τ1τ2	NOUN
ejpam-6014	58	16	-	-	NUM
ejpam-6014	58	17	cl(a	cl(a	NUM
ejpam-6014	58	18	)	)	PUNCT
ejpam-6014	58	19	and	and	CCONJ
ejpam-6014	58	20	τ1τ2	τ1τ2	NOUN
ejpam-6014	58	21	-	-	ADJ
ejpam-6014	58	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6014	58	23	-	-	PUNCT
ejpam-6014	58	24	cl(a	cl(a	NUM
ejpam-6014	58	25	)	)	PUNCT
ejpam-6014	58	26	)	)	PUNCT
ejpam-6014	59	1	=	=	PUNCT
ejpam-6014	59	2	τ1τ2	τ1τ2	NOUN
ejpam-6014	59	3	-	-	NUM
ejpam-6014	59	4	cl(a	cl(a	NUM
ejpam-6014	59	5	)	)	PUNCT
ejpam-6014	59	6	.	.	PUNCT
ejpam-6014	60	1	(	(	PUNCT
ejpam-6014	60	2	2	2	X
ejpam-6014	60	3	)	)	PUNCT
ejpam-6014	60	4	if	if	SCONJ
ejpam-6014	60	5	a	a	DET
ejpam-6014	60	6	⊆	⊆	NUM
ejpam-6014	60	7	b	b	NOUN
ejpam-6014	60	8	,	,	PUNCT
ejpam-6014	60	9	then	then	ADV
ejpam-6014	60	10	τ1τ2	τ1τ2	NOUN
ejpam-6014	60	11	-	-	NUM
ejpam-6014	60	12	cl(a	cl(a	NUM
ejpam-6014	60	13	)	)	PUNCT
ejpam-6014	60	14	⊆	⊆	NUM
ejpam-6014	60	15	τ1τ2	τ1τ2	NOUN
ejpam-6014	60	16	-	-	NOUN
ejpam-6014	60	17	cl(b	cl(b	NOUN
ejpam-6014	60	18	)	)	PUNCT
ejpam-6014	60	19	.	.	PUNCT
ejpam-6014	61	1	(	(	PUNCT
ejpam-6014	61	2	3	3	X
ejpam-6014	61	3	)	)	PUNCT
ejpam-6014	61	4	τ1τ2	τ1τ2	NOUN
ejpam-6014	61	5	-	-	NUM
ejpam-6014	61	6	cl(a	cl(a	NUM
ejpam-6014	61	7	)	)	PUNCT
ejpam-6014	61	8	is	be	AUX
ejpam-6014	61	9	τ1τ2	τ1τ2	NOUN
ejpam-6014	61	10	-	-	ADJ
ejpam-6014	61	11	closed	closed	ADJ
ejpam-6014	61	12	.	.	PUNCT
ejpam-6014	62	1	(	(	PUNCT
ejpam-6014	62	2	4	4	X
ejpam-6014	62	3	)	)	PUNCT
ejpam-6014	62	4	a	a	PRON
ejpam-6014	62	5	is	be	AUX
ejpam-6014	62	6	τ1τ2	τ1τ2	NOUN
ejpam-6014	62	7	-	-	ADJ
ejpam-6014	62	8	closed	closed	ADJ
ejpam-6014	62	9	if	if	SCONJ
ejpam-6014	62	10	and	and	CCONJ
ejpam-6014	62	11	only	only	ADV
ejpam-6014	62	12	if	if	SCONJ
ejpam-6014	62	13	a	a	DET
ejpam-6014	62	14	=	=	PUNCT
ejpam-6014	62	15	τ1τ2	τ1τ2	NOUN
ejpam-6014	62	16	-	-	NUM
ejpam-6014	62	17	cl(a	cl(a	NUM
ejpam-6014	62	18	)	)	PUNCT
ejpam-6014	62	19	.	.	PUNCT
ejpam-6014	63	1	m.	m.	NOUN
ejpam-6014	63	2	thongmoon	thongmoon	PROPN
ejpam-6014	63	3	,	,	PUNCT
ejpam-6014	63	4	s.	s.	PROPN
ejpam-6014	63	5	sompong	sompong	PROPN
ejpam-6014	63	6	,	,	PUNCT
ejpam-6014	63	7	c.	c.	PROPN
ejpam-6014	63	8	boonpok	boonpok	PROPN
ejpam-6014	63	9	/	/	SYM
ejpam-6014	63	10	eur	eur	PROPN
ejpam-6014	63	11	.	.	PUNCT
ejpam-6014	64	1	j.	j.	PROPN
ejpam-6014	64	2	pure	pure	PROPN
ejpam-6014	64	3	appl	appl	PROPN
ejpam-6014	64	4	.	.	PROPN
ejpam-6014	64	5	math	math	PROPN
ejpam-6014	64	6	,	,	PUNCT
ejpam-6014	64	7	18	18	NUM
ejpam-6014	64	8	(	(	PUNCT
ejpam-6014	64	9	2	2	NUM
ejpam-6014	64	10	)	)	PUNCT
ejpam-6014	64	11	(	(	PUNCT
ejpam-6014	64	12	2025	2025	NUM
ejpam-6014	64	13	)	)	PUNCT
ejpam-6014	64	14	,	,	PUNCT
ejpam-6014	64	15	6014	6014	NUM
ejpam-6014	64	16	3	3	NUM
ejpam-6014	64	17	of	of	ADP
ejpam-6014	64	18	13	13	NUM
ejpam-6014	64	19	(	(	PUNCT
ejpam-6014	64	20	5	5	NUM
ejpam-6014	64	21	)	)	PUNCT
ejpam-6014	64	22	τ1τ2	τ1τ2	NOUN
ejpam-6014	64	23	-	-	NOUN
ejpam-6014	64	24	cl(x	cl(x	X
ejpam-6014	64	25	−a	−a	NOUN
ejpam-6014	64	26	)	)	PUNCT
ejpam-6014	64	27	=	=	PUNCT
ejpam-6014	65	1	x	x	X
ejpam-6014	65	2	−	−	ADP
ejpam-6014	65	3	τ1τ2	τ1τ2	NOUN
ejpam-6014	65	4	-	-	PUNCT
ejpam-6014	65	5	int(a	int(a	NOUN
ejpam-6014	65	6	)	)	PUNCT
ejpam-6014	65	7	.	.	PUNCT
ejpam-6014	66	1	a	a	DET
ejpam-6014	66	2	subset	subset	NOUN
ejpam-6014	66	3	a	a	PRON
ejpam-6014	66	4	of	of	ADP
ejpam-6014	66	5	a	a	DET
ejpam-6014	66	6	bitopological	bitopological	ADJ
ejpam-6014	66	7	space	space	NOUN
ejpam-6014	66	8	(	(	PUNCT
ejpam-6014	66	9	x	x	NOUN
ejpam-6014	66	10	,	,	PUNCT
ejpam-6014	66	11	τ1	τ1	NOUN
ejpam-6014	66	12	,	,	PUNCT
ejpam-6014	66	13	τ2	τ2	NOUN
ejpam-6014	66	14	)	)	PUNCT
ejpam-6014	66	15	is	be	AUX
ejpam-6014	66	16	said	say	VERB
ejpam-6014	66	17	to	to	PART
ejpam-6014	66	18	be	be	AUX
ejpam-6014	66	19	(	(	PUNCT
ejpam-6014	66	20	τ1	τ1	NOUN
ejpam-6014	66	21	,	,	PUNCT
ejpam-6014	66	22	τ2)r	τ2)r	NOUN
ejpam-6014	66	23	-	-	PUNCT
ejpam-6014	66	24	open	open	ADJ
ejpam-6014	67	1	[	[	X
ejpam-6014	67	2	33	33	NUM
ejpam-6014	67	3	]	]	PUNCT
ejpam-6014	67	4	(	(	PUNCT
ejpam-6014	67	5	resp	resp	NOUN
ejpam-6014	67	6	.	.	PUNCT
ejpam-6014	68	1	(	(	PUNCT
ejpam-6014	68	2	τ1	τ1	NOUN
ejpam-6014	68	3	,	,	PUNCT
ejpam-6014	68	4	τ2)s	τ2)s	NOUN
ejpam-6014	68	5	-	-	PUNCT
ejpam-6014	68	6	open	open	ADJ
ejpam-6014	68	7	[	[	X
ejpam-6014	68	8	34	34	NUM
ejpam-6014	68	9	]	]	PUNCT
ejpam-6014	68	10	,	,	PUNCT
ejpam-6014	68	11	(	(	PUNCT
ejpam-6014	68	12	τ1	τ1	NOUN
ejpam-6014	68	13	,	,	PUNCT
ejpam-6014	68	14	τ2)p	τ2)p	NOUN
ejpam-6014	68	15	-	-	ADJ
ejpam-6014	68	16	open	open	ADJ
ejpam-6014	69	1	[	[	X
ejpam-6014	69	2	34	34	NUM
ejpam-6014	69	3	]	]	PUNCT
ejpam-6014	69	4	,	,	PUNCT
ejpam-6014	69	5	(	(	PUNCT
ejpam-6014	69	6	τ1	τ1	NOUN
ejpam-6014	69	7	,	,	PUNCT
ejpam-6014	69	8	τ2)β	τ2)β	ADJ
ejpam-6014	69	9	-	-	PUNCT
ejpam-6014	69	10	open	open	NOUN
ejpam-6014	70	1	[	[	X
ejpam-6014	70	2	34	34	NUM
ejpam-6014	70	3	]	]	SYM
ejpam-6014	70	4	)	)	PUNCT
ejpam-6014	70	5	if	if	SCONJ
ejpam-6014	70	6	a	a	DET
ejpam-6014	70	7	=	=	PUNCT
ejpam-6014	70	8	τ1τ2	τ1τ2	NOUN
ejpam-6014	70	9	-	-	NOUN
ejpam-6014	70	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6014	70	11	-	-	PUNCT
ejpam-6014	70	12	cl(a	cl(a	NUM
ejpam-6014	70	13	)	)	PUNCT
ejpam-6014	70	14	)	)	PUNCT
ejpam-6014	70	15	(	(	PUNCT
ejpam-6014	70	16	resp	resp	NOUN
ejpam-6014	70	17	.	.	PUNCT
ejpam-6014	71	1	a	a	DET
ejpam-6014	71	2	⊆	⊆	NUM
ejpam-6014	71	3	τ1τ2	τ1τ2	NOUN
ejpam-6014	71	4	-	-	ADJ
ejpam-6014	71	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6014	71	6	-	-	PUNCT
ejpam-6014	71	7	int(a	int(a	NOUN
ejpam-6014	71	8	)	)	PUNCT
ejpam-6014	71	9	)	)	PUNCT
ejpam-6014	71	10	,	,	PUNCT
ejpam-6014	71	11	a	a	DET
ejpam-6014	71	12	⊆	⊆	NUM
ejpam-6014	71	13	τ1τ2	τ1τ2	NOUN
ejpam-6014	71	14	-	-	NOUN
ejpam-6014	71	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6014	71	16	-	-	PUNCT
ejpam-6014	71	17	cl(a	cl(a	NUM
ejpam-6014	71	18	)	)	PUNCT
ejpam-6014	71	19	)	)	PUNCT
ejpam-6014	71	20	,	,	PUNCT
ejpam-6014	71	21	a	a	DET
ejpam-6014	71	22	⊆	⊆	NUM
ejpam-6014	71	23	τ1τ2	τ1τ2	NOUN
ejpam-6014	71	24	-	-	PUNCT
ejpam-6014	71	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6014	71	26	-	-	PUNCT
ejpam-6014	71	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6014	71	28	-	-	PUNCT
ejpam-6014	71	29	cl(a	cl(a	NUM
ejpam-6014	71	30	)	)	PUNCT
ejpam-6014	71	31	)	)	PUNCT
ejpam-6014	71	32	)	)	PUNCT
ejpam-6014	71	33	)	)	PUNCT
ejpam-6014	71	34	.	.	PUNCT
ejpam-6014	72	1	the	the	DET
ejpam-6014	72	2	complement	complement	NOUN
ejpam-6014	72	3	of	of	ADP
ejpam-6014	72	4	a	a	DET
ejpam-6014	72	5	(	(	PUNCT
ejpam-6014	72	6	τ1	τ1	NOUN
ejpam-6014	72	7	,	,	PUNCT
ejpam-6014	72	8	τ2)r	τ2)r	NOUN
ejpam-6014	72	9	-	-	PUNCT
ejpam-6014	72	10	open	open	ADJ
ejpam-6014	72	11	(	(	PUNCT
ejpam-6014	72	12	resp	resp	NOUN
ejpam-6014	72	13	.	.	PUNCT
ejpam-6014	73	1	(	(	PUNCT
ejpam-6014	73	2	τ1	τ1	NOUN
ejpam-6014	73	3	,	,	PUNCT
ejpam-6014	73	4	τ2)s	τ2)s	NOUN
ejpam-6014	73	5	-	-	PUNCT
ejpam-6014	73	6	open	open	ADJ
ejpam-6014	73	7	,	,	PUNCT
ejpam-6014	73	8	(	(	PUNCT
ejpam-6014	73	9	τ1	τ1	NOUN
ejpam-6014	73	10	,	,	PUNCT
ejpam-6014	73	11	τ2)p	τ2)p	NOUN
ejpam-6014	73	12	-	-	ADJ
ejpam-6014	73	13	open	open	ADJ
ejpam-6014	73	14	,	,	PUNCT
ejpam-6014	73	15	(	(	PUNCT
ejpam-6014	73	16	τ1	τ1	NOUN
ejpam-6014	73	17	,	,	PUNCT
ejpam-6014	73	18	τ2)β	τ2)β	ADJ
ejpam-6014	73	19	-	-	PUNCT
ejpam-6014	73	20	open	open	ADJ
ejpam-6014	73	21	)	)	PUNCT
ejpam-6014	73	22	set	set	NOUN
ejpam-6014	73	23	is	be	AUX
ejpam-6014	73	24	called	call	VERB
ejpam-6014	73	25	(	(	PUNCT
ejpam-6014	73	26	τ1	τ1	NOUN
ejpam-6014	73	27	,	,	PUNCT
ejpam-6014	73	28	τ2)r	τ2)r	NOUN
ejpam-6014	73	29	-	-	PUNCT
ejpam-6014	73	30	closed	closed	ADJ
ejpam-6014	73	31	(	(	PUNCT
ejpam-6014	73	32	resp	resp	NOUN
ejpam-6014	73	33	.	.	PUNCT
ejpam-6014	74	1	(	(	PUNCT
ejpam-6014	74	2	τ1	τ1	NOUN
ejpam-6014	74	3	,	,	PUNCT
ejpam-6014	74	4	τ2)s	τ2)s	NOUN
ejpam-6014	74	5	-	-	PUNCT
ejpam-6014	74	6	closed	closed	ADJ
ejpam-6014	74	7	,	,	PUNCT
ejpam-6014	74	8	(	(	PUNCT
ejpam-6014	74	9	τ1	τ1	NOUN
ejpam-6014	74	10	,	,	PUNCT
ejpam-6014	74	11	τ2)p	τ2)p	NOUN
ejpam-6014	74	12	-	-	PUNCT
ejpam-6014	74	13	closed	closed	ADJ
ejpam-6014	74	14	,	,	PUNCT
ejpam-6014	74	15	(	(	PUNCT
ejpam-6014	74	16	τ1	τ1	NOUN
ejpam-6014	74	17	,	,	PUNCT
ejpam-6014	74	18	τ2)β	τ2)β	ADJ
ejpam-6014	74	19	-	-	PUNCT
ejpam-6014	74	20	closed	closed	ADJ
ejpam-6014	74	21	)	)	PUNCT
ejpam-6014	74	22	.	.	PUNCT
ejpam-6014	75	1	a	a	DET
ejpam-6014	75	2	subset	subset	NOUN
ejpam-6014	75	3	a	a	PRON
ejpam-6014	75	4	of	of	ADP
ejpam-6014	75	5	a	a	DET
ejpam-6014	75	6	bitopological	bitopological	ADJ
ejpam-6014	75	7	space	space	NOUN
ejpam-6014	75	8	(	(	PUNCT
ejpam-6014	75	9	x	x	NOUN
ejpam-6014	75	10	,	,	PUNCT
ejpam-6014	75	11	τ1	τ1	NOUN
ejpam-6014	75	12	,	,	PUNCT
ejpam-6014	75	13	τ2	τ2	NOUN
ejpam-6014	75	14	)	)	PUNCT
ejpam-6014	75	15	is	be	AUX
ejpam-6014	75	16	said	say	VERB
ejpam-6014	75	17	to	to	PART
ejpam-6014	75	18	be	be	AUX
ejpam-6014	75	19	α(τ1	α(τ1	NOUN
ejpam-6014	75	20	,	,	PUNCT
ejpam-6014	75	21	τ2)-open	τ2)-open	ADJ
ejpam-6014	75	22	[	[	X
ejpam-6014	75	23	35	35	NUM
ejpam-6014	75	24	]	]	X
ejpam-6014	75	25	if	if	SCONJ
ejpam-6014	75	26	a	a	DET
ejpam-6014	75	27	⊆	⊆	NUM
ejpam-6014	75	28	τ1τ2	τ1τ2	NOUN
ejpam-6014	75	29	-	-	PUNCT
ejpam-6014	75	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6014	75	31	-	-	PUNCT
ejpam-6014	75	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6014	75	33	-	-	PUNCT
ejpam-6014	75	34	int(a	int(a	NOUN
ejpam-6014	75	35	)	)	PUNCT
ejpam-6014	75	36	)	)	PUNCT
ejpam-6014	75	37	)	)	PUNCT
ejpam-6014	75	38	.	.	PUNCT
ejpam-6014	76	1	the	the	DET
ejpam-6014	76	2	complement	complement	NOUN
ejpam-6014	76	3	of	of	ADP
ejpam-6014	76	4	an	an	DET
ejpam-6014	76	5	α(τ1	α(τ1	NOUN
ejpam-6014	76	6	,	,	PUNCT
ejpam-6014	76	7	τ2)-open	τ2)-open	ADJ
ejpam-6014	76	8	set	set	NOUN
ejpam-6014	76	9	is	be	AUX
ejpam-6014	76	10	said	say	VERB
ejpam-6014	76	11	to	to	PART
ejpam-6014	76	12	be	be	AUX
ejpam-6014	76	13	α(τ1	α(τ1	NOUN
ejpam-6014	76	14	,	,	PUNCT
ejpam-6014	76	15	τ2)-closed	τ2)-close	VERB
ejpam-6014	76	16	.	.	PUNCT
ejpam-6014	77	1	let	let	VERB
ejpam-6014	77	2	a	a	DET
ejpam-6014	77	3	be	be	AUX
ejpam-6014	77	4	a	a	DET
ejpam-6014	77	5	subset	subset	NOUN
ejpam-6014	77	6	of	of	ADP
ejpam-6014	77	7	a	a	DET
ejpam-6014	77	8	bitopological	bitopological	ADJ
ejpam-6014	77	9	space	space	NOUN
ejpam-6014	77	10	(	(	PUNCT
ejpam-6014	77	11	x	x	NOUN
ejpam-6014	77	12	,	,	PUNCT
ejpam-6014	77	13	τ1	τ1	NOUN
ejpam-6014	77	14	,	,	PUNCT
ejpam-6014	77	15	τ2	τ2	NOUN
ejpam-6014	77	16	)	)	PUNCT
ejpam-6014	77	17	.	.	PUNCT
ejpam-6014	78	1	a	a	DET
ejpam-6014	78	2	point	point	NOUN
ejpam-6014	78	3	x	x	X
ejpam-6014	78	4	∈	∈	NOUN
ejpam-6014	78	5	x	x	PUNCT
ejpam-6014	78	6	is	be	AUX
ejpam-6014	78	7	called	call	VERB
ejpam-6014	78	8	a	a	DET
ejpam-6014	78	9	(	(	PUNCT
ejpam-6014	78	10	τ1	τ1	NOUN
ejpam-6014	78	11	,	,	PUNCT
ejpam-6014	78	12	τ2)θ	τ2)θ	ADJ
ejpam-6014	78	13	-	-	PUNCT
ejpam-6014	78	14	cluster	cluster	NOUN
ejpam-6014	78	15	point	point	NOUN
ejpam-6014	78	16	[	[	X
ejpam-6014	78	17	33	33	NUM
ejpam-6014	78	18	]	]	PUNCT
ejpam-6014	78	19	of	of	ADP
ejpam-6014	78	20	a	a	DET
ejpam-6014	78	21	if	if	SCONJ
ejpam-6014	78	22	τ1τ2	τ1τ2	NOUN
ejpam-6014	78	23	-	-	NOUN
ejpam-6014	78	24	cl(u	cl(u	NOUN
ejpam-6014	78	25	)	)	PUNCT
ejpam-6014	78	26	∩	∩	NOUN
ejpam-6014	78	27	a	a	DET
ejpam-6014	78	28	̸=	̸=	PROPN
ejpam-6014	78	29	∅	∅	NOUN
ejpam-6014	78	30	for	for	ADP
ejpam-6014	78	31	every	every	DET
ejpam-6014	78	32	τ1τ2	τ1τ2	ADJ
ejpam-6014	78	33	-	-	ADJ
ejpam-6014	78	34	open	open	ADJ
ejpam-6014	78	35	set	set	NOUN
ejpam-6014	78	36	u	u	NOUN
ejpam-6014	78	37	containing	contain	VERB
ejpam-6014	78	38	x.	x.	NOUN
ejpam-6014	78	39	the	the	DET
ejpam-6014	78	40	set	set	NOUN
ejpam-6014	78	41	of	of	ADP
ejpam-6014	78	42	all	all	DET
ejpam-6014	78	43	(	(	PUNCT
ejpam-6014	78	44	τ1	τ1	NOUN
ejpam-6014	78	45	,	,	PUNCT
ejpam-6014	78	46	τ2)θ	τ2)θ	ADJ
ejpam-6014	78	47	-	-	PUNCT
ejpam-6014	78	48	cluster	cluster	NOUN
ejpam-6014	78	49	points	point	NOUN
ejpam-6014	78	50	of	of	ADP
ejpam-6014	78	51	a	a	PRON
ejpam-6014	78	52	is	be	AUX
ejpam-6014	78	53	called	call	VERB
ejpam-6014	78	54	the	the	DET
ejpam-6014	78	55	(	(	PUNCT
ejpam-6014	78	56	τ1	τ1	NOUN
ejpam-6014	78	57	,	,	PUNCT
ejpam-6014	78	58	τ2)θ	τ2)θ	ADJ
ejpam-6014	78	59	-	-	PUNCT
ejpam-6014	78	60	closure	closure	NOUN
ejpam-6014	78	61	[	[	X
ejpam-6014	78	62	33	33	NUM
ejpam-6014	78	63	]	]	PUNCT
ejpam-6014	78	64	of	of	ADP
ejpam-6014	78	65	a	a	PRON
ejpam-6014	78	66	and	and	CCONJ
ejpam-6014	78	67	is	be	AUX
ejpam-6014	78	68	denoted	denote	VERB
ejpam-6014	78	69	by	by	ADP
ejpam-6014	78	70	(	(	PUNCT
ejpam-6014	78	71	τ1	τ1	NOUN
ejpam-6014	78	72	,	,	PUNCT
ejpam-6014	78	73	τ2)θ	τ2)θ	NOUN
ejpam-6014	78	74	-	-	PUNCT
ejpam-6014	78	75	cl(a	cl(a	NUM
ejpam-6014	78	76	)	)	PUNCT
ejpam-6014	78	77	.	.	PUNCT
ejpam-6014	79	1	a	a	DET
ejpam-6014	79	2	subset	subset	NOUN
ejpam-6014	79	3	a	a	PRON
ejpam-6014	79	4	of	of	ADP
ejpam-6014	79	5	a	a	DET
ejpam-6014	79	6	bitopological	bitopological	ADJ
ejpam-6014	79	7	space	space	NOUN
ejpam-6014	79	8	(	(	PUNCT
ejpam-6014	79	9	x	x	NOUN
ejpam-6014	79	10	,	,	PUNCT
ejpam-6014	79	11	τ1	τ1	NOUN
ejpam-6014	79	12	,	,	PUNCT
ejpam-6014	79	13	τ2	τ2	NOUN
ejpam-6014	79	14	)	)	PUNCT
ejpam-6014	79	15	is	be	AUX
ejpam-6014	79	16	said	say	VERB
ejpam-6014	79	17	to	to	PART
ejpam-6014	79	18	be	be	AUX
ejpam-6014	79	19	(	(	PUNCT
ejpam-6014	79	20	τ1	τ1	NOUN
ejpam-6014	79	21	,	,	PUNCT
ejpam-6014	79	22	τ2)θ	τ2)θ	NOUN
ejpam-6014	79	23	-	-	PUNCT
ejpam-6014	79	24	closed	closed	ADJ
ejpam-6014	79	25	[	[	X
ejpam-6014	79	26	33	33	NUM
ejpam-6014	79	27	]	]	PUNCT
ejpam-6014	79	28	if	if	SCONJ
ejpam-6014	79	29	a	a	PRON
ejpam-6014	79	30	=	=	X
ejpam-6014	79	31	(	(	PUNCT
ejpam-6014	79	32	τ1	τ1	NOUN
ejpam-6014	79	33	,	,	PUNCT
ejpam-6014	79	34	τ2)θ	τ2)θ	NOUN
ejpam-6014	79	35	-	-	PUNCT
ejpam-6014	79	36	cl(a	cl(a	NUM
ejpam-6014	79	37	)	)	PUNCT
ejpam-6014	79	38	.	.	PUNCT
ejpam-6014	80	1	the	the	DET
ejpam-6014	80	2	complement	complement	NOUN
ejpam-6014	80	3	of	of	ADP
ejpam-6014	80	4	a	a	DET
ejpam-6014	80	5	(	(	PUNCT
ejpam-6014	80	6	τ1	τ1	NOUN
ejpam-6014	80	7	,	,	PUNCT
ejpam-6014	80	8	τ2)θ	τ2)θ	ADJ
ejpam-6014	80	9	-	-	PUNCT
ejpam-6014	80	10	closed	close	VERB
ejpam-6014	80	11	set	set	NOUN
ejpam-6014	80	12	is	be	AUX
ejpam-6014	80	13	said	say	VERB
ejpam-6014	80	14	to	to	PART
ejpam-6014	80	15	be	be	AUX
ejpam-6014	80	16	(	(	PUNCT
ejpam-6014	80	17	τ1	τ1	NOUN
ejpam-6014	80	18	,	,	PUNCT
ejpam-6014	80	19	τ2)θ	τ2)θ	NOUN
ejpam-6014	80	20	-	-	PUNCT
ejpam-6014	80	21	open	open	ADJ
ejpam-6014	80	22	.	.	PUNCT
ejpam-6014	81	1	the	the	DET
ejpam-6014	81	2	union	union	NOUN
ejpam-6014	81	3	of	of	ADP
ejpam-6014	81	4	all	all	DET
ejpam-6014	81	5	(	(	PUNCT
ejpam-6014	81	6	τ1	τ1	NOUN
ejpam-6014	81	7	,	,	PUNCT
ejpam-6014	81	8	τ2)θ	τ2)θ	ADJ
ejpam-6014	81	9	-	-	PUNCT
ejpam-6014	81	10	open	open	ADJ
ejpam-6014	81	11	sets	set	NOUN
ejpam-6014	81	12	contained	contain	VERB
ejpam-6014	81	13	in	in	ADP
ejpam-6014	81	14	a	a	PRON
ejpam-6014	81	15	is	be	AUX
ejpam-6014	81	16	called	call	VERB
ejpam-6014	81	17	the	the	DET
ejpam-6014	81	18	(	(	PUNCT
ejpam-6014	81	19	τ1	τ1	NOUN
ejpam-6014	81	20	,	,	PUNCT
ejpam-6014	81	21	τ2)θ	τ2)θ	ADJ
ejpam-6014	81	22	-	-	PUNCT
ejpam-6014	81	23	interior	interior	NOUN
ejpam-6014	81	24	[	[	X
ejpam-6014	81	25	33	33	NUM
ejpam-6014	81	26	]	]	PUNCT
ejpam-6014	81	27	of	of	ADP
ejpam-6014	81	28	a	a	PRON
ejpam-6014	81	29	and	and	CCONJ
ejpam-6014	81	30	is	be	AUX
ejpam-6014	81	31	denoted	denote	VERB
ejpam-6014	81	32	by	by	ADP
ejpam-6014	81	33	(	(	PUNCT
ejpam-6014	81	34	τ1	τ1	NOUN
ejpam-6014	81	35	,	,	PUNCT
ejpam-6014	81	36	τ2)θ	τ2)θ	NOUN
ejpam-6014	81	37	-	-	PUNCT
ejpam-6014	81	38	int(a	int(a	NOUN
ejpam-6014	81	39	)	)	PUNCT
ejpam-6014	81	40	.	.	PUNCT
ejpam-6014	82	1	lemma	lemma	PROPN
ejpam-6014	82	2	2	2	NUM
ejpam-6014	82	3	.	.	PUNCT
ejpam-6014	83	1	[	[	X
ejpam-6014	83	2	33	33	NUM
ejpam-6014	83	3	]	]	PUNCT
ejpam-6014	83	4	for	for	ADP
ejpam-6014	83	5	a	a	DET
ejpam-6014	83	6	subset	subset	NOUN
ejpam-6014	83	7	a	a	PRON
ejpam-6014	83	8	of	of	ADP
ejpam-6014	83	9	a	a	DET
ejpam-6014	83	10	bitopological	bitopological	ADJ
ejpam-6014	83	11	space	space	NOUN
ejpam-6014	83	12	(	(	PUNCT
ejpam-6014	83	13	x	x	NOUN
ejpam-6014	83	14	,	,	PUNCT
ejpam-6014	83	15	τ1	τ1	NOUN
ejpam-6014	83	16	,	,	PUNCT
ejpam-6014	83	17	τ2	τ2	NOUN
ejpam-6014	83	18	)	)	PUNCT
ejpam-6014	83	19	,	,	PUNCT
ejpam-6014	83	20	the	the	DET
ejpam-6014	83	21	following	follow	VERB
ejpam-6014	83	22	properties	property	NOUN
ejpam-6014	83	23	hold	hold	VERB
ejpam-6014	83	24	:	:	PUNCT
ejpam-6014	83	25	(	(	PUNCT
ejpam-6014	83	26	1	1	X
ejpam-6014	83	27	)	)	PUNCT
ejpam-6014	83	28	if	if	SCONJ
ejpam-6014	83	29	a	a	PRON
ejpam-6014	83	30	is	be	AUX
ejpam-6014	83	31	τ2τ2	τ2τ2	VERB
ejpam-6014	83	32	-	-	VERB
ejpam-6014	83	33	open	open	ADJ
ejpam-6014	83	34	in	in	ADP
ejpam-6014	83	35	x	x	NOUN
ejpam-6014	83	36	,	,	PUNCT
ejpam-6014	83	37	then	then	ADV
ejpam-6014	83	38	τ1τ2	τ1τ2	NOUN
ejpam-6014	83	39	-	-	NUM
ejpam-6014	83	40	cl(a	cl(a	NUM
ejpam-6014	83	41	)	)	PUNCT
ejpam-6014	83	42	=	=	PUNCT
ejpam-6014	83	43	(	(	PUNCT
ejpam-6014	83	44	τ1	τ1	NOUN
ejpam-6014	83	45	,	,	PUNCT
ejpam-6014	83	46	τ2)θ	τ2)θ	NOUN
ejpam-6014	83	47	-	-	PUNCT
ejpam-6014	83	48	cl(a	cl(a	NUM
ejpam-6014	83	49	)	)	PUNCT
ejpam-6014	83	50	.	.	PUNCT
ejpam-6014	84	1	(	(	PUNCT
ejpam-6014	84	2	2	2	X
ejpam-6014	84	3	)	)	PUNCT
ejpam-6014	84	4	(	(	PUNCT
ejpam-6014	84	5	τ1	τ1	NOUN
ejpam-6014	84	6	,	,	PUNCT
ejpam-6014	84	7	τ2)θ	τ2)θ	NOUN
ejpam-6014	84	8	-	-	PUNCT
ejpam-6014	84	9	cl(a	cl(a	NUM
ejpam-6014	84	10	)	)	PUNCT
ejpam-6014	84	11	is	be	AUX
ejpam-6014	84	12	τ1τ2	τ1τ2	NOUN
ejpam-6014	84	13	-	-	ADJ
ejpam-6014	84	14	closed	closed	ADJ
ejpam-6014	84	15	in	in	ADP
ejpam-6014	84	16	x.	x.	NOUN
ejpam-6014	84	17	3	3	NUM
ejpam-6014	84	18	.	.	NOUN
ejpam-6014	84	19	θ(τ1	θ(τ1	NOUN
ejpam-6014	84	20	,	,	PUNCT
ejpam-6014	84	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	84	22	functions	function	NOUN
ejpam-6014	84	23	in	in	ADP
ejpam-6014	84	24	this	this	DET
ejpam-6014	84	25	section	section	NOUN
ejpam-6014	84	26	,	,	PUNCT
ejpam-6014	84	27	we	we	PRON
ejpam-6014	84	28	introduce	introduce	VERB
ejpam-6014	84	29	the	the	DET
ejpam-6014	84	30	concept	concept	NOUN
ejpam-6014	84	31	of	of	ADP
ejpam-6014	84	32	θ(τ1	θ(τ1	NOUN
ejpam-6014	84	33	,	,	PUNCT
ejpam-6014	84	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	84	35	functions	function	NOUN
ejpam-6014	84	36	.	.	PUNCT
ejpam-6014	85	1	furthermore	furthermore	ADV
ejpam-6014	85	2	,	,	PUNCT
ejpam-6014	85	3	several	several	ADJ
ejpam-6014	85	4	characterizations	characterization	NOUN
ejpam-6014	85	5	of	of	ADP
ejpam-6014	85	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	85	7	,	,	PUNCT
ejpam-6014	85	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	85	9	functions	function	NOUN
ejpam-6014	85	10	are	be	AUX
ejpam-6014	85	11	discussed	discuss	VERB
ejpam-6014	85	12	.	.	PUNCT
ejpam-6014	86	1	definition	definition	NOUN
ejpam-6014	86	2	1	1	NUM
ejpam-6014	86	3	.	.	PUNCT
ejpam-6014	87	1	a	a	DET
ejpam-6014	87	2	function	function	NOUN
ejpam-6014	87	3	f	f	NOUN
ejpam-6014	87	4	:	:	PUNCT
ejpam-6014	87	5	(	(	PUNCT
ejpam-6014	87	6	x	x	NOUN
ejpam-6014	87	7	,	,	PUNCT
ejpam-6014	87	8	τ1	τ1	NOUN
ejpam-6014	87	9	,	,	PUNCT
ejpam-6014	87	10	τ2	τ2	NOUN
ejpam-6014	87	11	)	)	PUNCT
ejpam-6014	87	12	→	→	SYM
ejpam-6014	87	13	(	(	PUNCT
ejpam-6014	87	14	y	y	PROPN
ejpam-6014	87	15	,	,	PUNCT
ejpam-6014	87	16	σ1	σ1	PROPN
ejpam-6014	87	17	,	,	PUNCT
ejpam-6014	87	18	σ2	σ2	PROPN
ejpam-6014	87	19	)	)	PUNCT
ejpam-6014	87	20	is	be	AUX
ejpam-6014	87	21	said	say	VERB
ejpam-6014	87	22	to	to	PART
ejpam-6014	87	23	be	be	AUX
ejpam-6014	87	24	θ(τ1	θ(τ1	NOUN
ejpam-6014	87	25	,	,	PUNCT
ejpam-6014	87	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	87	27	at	at	ADP
ejpam-6014	87	28	a	a	DET
ejpam-6014	87	29	point	point	NOUN
ejpam-6014	87	30	x	x	SYM
ejpam-6014	87	31	∈	∈	NOUN
ejpam-6014	87	32	x	x	PUNCT
ejpam-6014	87	33	if	if	SCONJ
ejpam-6014	87	34	for	for	ADP
ejpam-6014	87	35	each	each	DET
ejpam-6014	87	36	σ1σ2	σ1σ2	VERB
ejpam-6014	87	37	-	-	ADJ
ejpam-6014	87	38	open	open	ADJ
ejpam-6014	87	39	set	set	NOUN
ejpam-6014	87	40	v	v	NOUN
ejpam-6014	87	41	of	of	ADP
ejpam-6014	87	42	y	y	NOUN
ejpam-6014	87	43	containing	contain	VERB
ejpam-6014	87	44	f(x	f(x	PROPN
ejpam-6014	87	45	)	)	PUNCT
ejpam-6014	87	46	,	,	PUNCT
ejpam-6014	87	47	there	there	PRON
ejpam-6014	87	48	exists	exist	VERB
ejpam-6014	87	49	a	a	DET
ejpam-6014	87	50	τ1τ2	τ1τ2	NOUN
ejpam-6014	87	51	-	-	ADJ
ejpam-6014	87	52	open	open	ADJ
ejpam-6014	87	53	set	set	ADJ
ejpam-6014	87	54	u	u	NOUN
ejpam-6014	87	55	of	of	ADP
ejpam-6014	87	56	x	x	PUNCT
ejpam-6014	87	57	containing	contain	VERB
ejpam-6014	87	58	x	x	PUNCT
ejpam-6014	87	59	such	such	ADJ
ejpam-6014	87	60	that	that	SCONJ
ejpam-6014	87	61	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	87	62	-	-	PUNCT
ejpam-6014	87	63	cl(u	cl(u	NOUN
ejpam-6014	87	64	)	)	PUNCT
ejpam-6014	87	65	)	)	PUNCT
ejpam-6014	88	1	⊆	⊆	X
ejpam-6014	88	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	88	3	-	-	NUM
ejpam-6014	88	4	cl(v	cl(v	NOUN
ejpam-6014	88	5	)	)	PUNCT
ejpam-6014	88	6	.	.	PUNCT
ejpam-6014	89	1	a	a	DET
ejpam-6014	89	2	function	function	NOUN
ejpam-6014	89	3	f	f	NOUN
ejpam-6014	89	4	:	:	PUNCT
ejpam-6014	89	5	(	(	PUNCT
ejpam-6014	89	6	x	x	NOUN
ejpam-6014	89	7	,	,	PUNCT
ejpam-6014	89	8	τ1	τ1	NOUN
ejpam-6014	89	9	,	,	PUNCT
ejpam-6014	89	10	τ2	τ2	NOUN
ejpam-6014	89	11	)	)	PUNCT
ejpam-6014	89	12	→	→	SYM
ejpam-6014	89	13	(	(	PUNCT
ejpam-6014	89	14	y	y	PROPN
ejpam-6014	89	15	,	,	PUNCT
ejpam-6014	89	16	σ1	σ1	PROPN
ejpam-6014	89	17	,	,	PUNCT
ejpam-6014	89	18	σ2	σ2	PROPN
ejpam-6014	89	19	)	)	PUNCT
ejpam-6014	89	20	is	be	AUX
ejpam-6014	89	21	said	say	VERB
ejpam-6014	89	22	to	to	PART
ejpam-6014	89	23	be	be	AUX
ejpam-6014	89	24	θ(τ1	θ(τ1	NOUN
ejpam-6014	89	25	,	,	PUNCT
ejpam-6014	89	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	89	27	if	if	SCONJ
ejpam-6014	89	28	f	f	PROPN
ejpam-6014	89	29	is	be	AUX
ejpam-6014	89	30	θ(τ1	θ(τ1	NOUN
ejpam-6014	89	31	,	,	PUNCT
ejpam-6014	89	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	89	33	at	at	ADP
ejpam-6014	89	34	each	each	DET
ejpam-6014	89	35	point	point	NOUN
ejpam-6014	89	36	x	x	PUNCT
ejpam-6014	89	37	of	of	ADP
ejpam-6014	89	38	x.	x.	PROPN
ejpam-6014	89	39	theorem	theorem	VERB
ejpam-6014	89	40	1	1	NUM
ejpam-6014	89	41	.	.	PUNCT
ejpam-6014	90	1	a	a	DET
ejpam-6014	90	2	function	function	NOUN
ejpam-6014	90	3	f	f	NOUN
ejpam-6014	90	4	:	:	PUNCT
ejpam-6014	90	5	(	(	PUNCT
ejpam-6014	90	6	x	x	NOUN
ejpam-6014	90	7	,	,	PUNCT
ejpam-6014	90	8	τ1	τ1	NOUN
ejpam-6014	90	9	,	,	PUNCT
ejpam-6014	90	10	τ2	τ2	NOUN
ejpam-6014	90	11	)	)	PUNCT
ejpam-6014	90	12	→	→	SYM
ejpam-6014	90	13	(	(	PUNCT
ejpam-6014	90	14	y	y	PROPN
ejpam-6014	90	15	,	,	PUNCT
ejpam-6014	90	16	σ1	σ1	PROPN
ejpam-6014	90	17	,	,	PUNCT
ejpam-6014	90	18	σ2	σ2	PROPN
ejpam-6014	90	19	)	)	PUNCT
ejpam-6014	90	20	is	be	AUX
ejpam-6014	90	21	θ(τ1	θ(τ1	NOUN
ejpam-6014	90	22	,	,	PUNCT
ejpam-6014	90	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	90	24	at	at	ADP
ejpam-6014	90	25	x	x	X
ejpam-6014	90	26	∈	∈	PROPN
ejpam-6014	90	27	x	x	SYM
ejpam-6014	90	28	if	if	SCONJ
ejpam-6014	91	1	and	and	CCONJ
ejpam-6014	91	2	only	only	ADV
ejpam-6014	91	3	if	if	SCONJ
ejpam-6014	91	4	for	for	ADP
ejpam-6014	91	5	each	each	DET
ejpam-6014	91	6	σ1σ2	σ1σ2	VERB
ejpam-6014	91	7	-	-	ADJ
ejpam-6014	91	8	open	open	ADJ
ejpam-6014	91	9	set	set	NOUN
ejpam-6014	91	10	v	v	NOUN
ejpam-6014	91	11	of	of	ADP
ejpam-6014	91	12	y	y	NOUN
ejpam-6014	91	13	containing	contain	VERB
ejpam-6014	91	14	f(x	f(x	PROPN
ejpam-6014	91	15	)	)	PUNCT
ejpam-6014	91	16	,	,	PUNCT
ejpam-6014	91	17	x	x	PUNCT
ejpam-6014	91	18	∈	∈	PROPN
ejpam-6014	91	19	(	(	PUNCT
ejpam-6014	91	20	τ1	τ1	NOUN
ejpam-6014	91	21	,	,	PUNCT
ejpam-6014	91	22	τ2)θ	τ2)θ	NOUN
ejpam-6014	91	23	-	-	PUNCT
ejpam-6014	91	24	int(f	int(f	VERB
ejpam-6014	91	25	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	91	26	-	-	PUNCT
ejpam-6014	91	27	cl(v	cl(v	NOUN
ejpam-6014	91	28	)	)	PUNCT
ejpam-6014	91	29	)	)	PUNCT
ejpam-6014	91	30	)	)	PUNCT
ejpam-6014	91	31	.	.	PUNCT
ejpam-6014	92	1	proof	proof	NOUN
ejpam-6014	92	2	.	.	PUNCT
ejpam-6014	93	1	let	let	VERB
ejpam-6014	93	2	x	x	PUNCT
ejpam-6014	93	3	∈	∈	PROPN
ejpam-6014	93	4	x	x	X
ejpam-6014	93	5	and	and	CCONJ
ejpam-6014	93	6	v	v	X
ejpam-6014	93	7	be	be	AUX
ejpam-6014	93	8	any	any	DET
ejpam-6014	93	9	σ1σ2	σ1σ2	NOUN
ejpam-6014	93	10	-	-	ADJ
ejpam-6014	93	11	open	open	ADJ
ejpam-6014	93	12	set	set	NOUN
ejpam-6014	93	13	of	of	ADP
ejpam-6014	93	14	y	y	PROPN
ejpam-6014	93	15	containing	contain	VERB
ejpam-6014	93	16	f(x	f(x	PROPN
ejpam-6014	93	17	)	)	PUNCT
ejpam-6014	93	18	.	.	PUNCT
ejpam-6014	94	1	since	since	SCONJ
ejpam-6014	94	2	f	f	PROPN
ejpam-6014	94	3	is	be	AUX
ejpam-6014	94	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	94	5	,	,	PUNCT
ejpam-6014	94	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	94	7	at	at	ADP
ejpam-6014	94	8	x	x	X
ejpam-6014	94	9	,	,	PUNCT
ejpam-6014	94	10	there	there	PRON
ejpam-6014	94	11	exists	exist	VERB
ejpam-6014	94	12	a	a	DET
ejpam-6014	94	13	τ1τ2	τ1τ2	NOUN
ejpam-6014	94	14	-	-	ADJ
ejpam-6014	94	15	open	open	ADJ
ejpam-6014	94	16	set	set	ADJ
ejpam-6014	94	17	u	u	NOUN
ejpam-6014	94	18	of	of	ADP
ejpam-6014	94	19	x	x	PUNCT
ejpam-6014	94	20	containing	contain	VERB
ejpam-6014	94	21	x	x	PUNCT
ejpam-6014	94	22	such	such	ADJ
ejpam-6014	94	23	that	that	SCONJ
ejpam-6014	94	24	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	94	25	-	-	PUNCT
ejpam-6014	94	26	cl(u	cl(u	NOUN
ejpam-6014	94	27	)	)	PUNCT
ejpam-6014	94	28	)	)	PUNCT
ejpam-6014	95	1	⊆	⊆	X
ejpam-6014	95	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	95	3	-	-	NUM
ejpam-6014	95	4	cl(v	cl(v	NOUN
ejpam-6014	95	5	)	)	PUNCT
ejpam-6014	95	6	.	.	PUNCT
ejpam-6014	96	1	then	then	ADV
ejpam-6014	96	2	,	,	PUNCT
ejpam-6014	96	3	we	we	PRON
ejpam-6014	96	4	have	have	VERB
ejpam-6014	96	5	x	x	X
ejpam-6014	96	6	∈	∈	PROPN
ejpam-6014	96	7	u	u	NOUN
ejpam-6014	96	8	⊆	⊆	NUM
ejpam-6014	96	9	τ1τ2	τ1τ2	NOUN
ejpam-6014	96	10	-	-	NOUN
ejpam-6014	96	11	cl(u	cl(u	NOUN
ejpam-6014	96	12	)	)	PUNCT
ejpam-6014	96	13	⊆	⊆	NUM
ejpam-6014	96	14	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	96	15	-	-	PUNCT
ejpam-6014	96	16	cl(v	cl(v	NOUN
ejpam-6014	96	17	)	)	PUNCT
ejpam-6014	96	18	)	)	PUNCT
ejpam-6014	97	1	and	and	CCONJ
ejpam-6014	97	2	hence	hence	ADV
ejpam-6014	97	3	x	x	X
ejpam-6014	97	4	∈	∈	PROPN
ejpam-6014	97	5	(	(	PUNCT
ejpam-6014	97	6	τ1	τ1	NOUN
ejpam-6014	97	7	,	,	PUNCT
ejpam-6014	97	8	τ2)θ	τ2)θ	NOUN
ejpam-6014	97	9	-	-	PUNCT
ejpam-6014	97	10	int(f	int(f	VERB
ejpam-6014	97	11	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	97	12	-	-	PUNCT
ejpam-6014	97	13	cl(v	cl(v	NOUN
ejpam-6014	97	14	)	)	PUNCT
ejpam-6014	97	15	)	)	PUNCT
ejpam-6014	97	16	)	)	PUNCT
ejpam-6014	97	17	.	.	PUNCT
ejpam-6014	98	1	conversely	conversely	ADV
ejpam-6014	98	2	,	,	PUNCT
ejpam-6014	98	3	let	let	VERB
ejpam-6014	98	4	v	v	PART
ejpam-6014	98	5	be	be	AUX
ejpam-6014	98	6	any	any	DET
ejpam-6014	98	7	σ1σ2	σ1σ2	NOUN
ejpam-6014	98	8	-	-	ADJ
ejpam-6014	98	9	open	open	ADJ
ejpam-6014	98	10	set	set	NOUN
ejpam-6014	98	11	of	of	ADP
ejpam-6014	98	12	y	y	PROPN
ejpam-6014	98	13	containing	contain	VERB
ejpam-6014	98	14	f(x	f(x	PROPN
ejpam-6014	98	15	)	)	PUNCT
ejpam-6014	98	16	.	.	PUNCT
ejpam-6014	99	1	then	then	ADV
ejpam-6014	99	2	,	,	PUNCT
ejpam-6014	99	3	by	by	ADP
ejpam-6014	99	4	the	the	DET
ejpam-6014	99	5	hypothesis	hypothesis	NOUN
ejpam-6014	99	6	we	we	PRON
ejpam-6014	99	7	have	have	VERB
ejpam-6014	99	8	x	x	X
ejpam-6014	99	9	∈	∈	PROPN
ejpam-6014	99	10	(	(	PUNCT
ejpam-6014	99	11	τ1	τ1	NOUN
ejpam-6014	99	12	,	,	PUNCT
ejpam-6014	99	13	τ2)θ	τ2)θ	NOUN
ejpam-6014	99	14	-	-	PUNCT
ejpam-6014	99	15	int(f	int(f	VERB
ejpam-6014	99	16	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	99	17	-	-	PUNCT
ejpam-6014	99	18	cl(v	cl(v	NOUN
ejpam-6014	99	19	)	)	PUNCT
ejpam-6014	99	20	)	)	PUNCT
ejpam-6014	99	21	)	)	PUNCT
ejpam-6014	99	22	.	.	PUNCT
ejpam-6014	100	1	there	there	PRON
ejpam-6014	100	2	exists	exist	VERB
ejpam-6014	100	3	a	a	DET
ejpam-6014	100	4	τ1τ2	τ1τ2	NOUN
ejpam-6014	100	5	-	-	ADJ
ejpam-6014	100	6	open	open	ADJ
ejpam-6014	100	7	set	set	ADJ
ejpam-6014	100	8	u	u	NOUN
ejpam-6014	100	9	of	of	ADP
ejpam-6014	100	10	x	x	SYM
ejpam-6014	100	11	such	such	ADJ
ejpam-6014	100	12	that	that	SCONJ
ejpam-6014	100	13	x	x	SYM
ejpam-6014	100	14	∈	∈	PROPN
ejpam-6014	100	15	u	u	NOUN
ejpam-6014	100	16	⊆	⊆	NUM
ejpam-6014	100	17	τ1τ2	τ1τ2	NOUN
ejpam-6014	100	18	-	-	NOUN
ejpam-6014	100	19	cl(u	cl(u	NOUN
ejpam-6014	100	20	)	)	PUNCT
ejpam-6014	100	21	⊆	⊆	NUM
ejpam-6014	100	22	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	100	23	-	-	PUNCT
ejpam-6014	100	24	cl(v	cl(v	NOUN
ejpam-6014	100	25	)	)	PUNCT
ejpam-6014	100	26	)	)	PUNCT
ejpam-6014	100	27	;	;	PUNCT
ejpam-6014	100	28	hence	hence	ADV
ejpam-6014	100	29	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	100	30	-	-	PUNCT
ejpam-6014	100	31	cl(u	cl(u	NOUN
ejpam-6014	100	32	)	)	PUNCT
ejpam-6014	100	33	)	)	PUNCT
ejpam-6014	101	1	⊆	⊆	X
ejpam-6014	101	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	101	3	-	-	NUM
ejpam-6014	101	4	cl(v	cl(v	NOUN
ejpam-6014	101	5	)	)	PUNCT
ejpam-6014	101	6	.	.	PUNCT
ejpam-6014	102	1	this	this	PRON
ejpam-6014	102	2	shows	show	VERB
ejpam-6014	102	3	that	that	SCONJ
ejpam-6014	102	4	f	f	PROPN
ejpam-6014	102	5	is	be	AUX
ejpam-6014	102	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	102	7	,	,	PUNCT
ejpam-6014	102	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	102	9	at	at	ADP
ejpam-6014	102	10	x.	x.	PROPN
ejpam-6014	102	11	m.	m.	PROPN
ejpam-6014	102	12	thongmoon	thongmoon	PROPN
ejpam-6014	102	13	,	,	PUNCT
ejpam-6014	102	14	s.	s.	PROPN
ejpam-6014	102	15	sompong	sompong	PROPN
ejpam-6014	102	16	,	,	PUNCT
ejpam-6014	102	17	c.	c.	PROPN
ejpam-6014	102	18	boonpok	boonpok	PROPN
ejpam-6014	102	19	/	/	SYM
ejpam-6014	102	20	eur	eur	PROPN
ejpam-6014	102	21	.	.	PUNCT
ejpam-6014	103	1	j.	j.	PROPN
ejpam-6014	103	2	pure	pure	PROPN
ejpam-6014	103	3	appl	appl	PROPN
ejpam-6014	103	4	.	.	PROPN
ejpam-6014	103	5	math	math	PROPN
ejpam-6014	103	6	,	,	PUNCT
ejpam-6014	103	7	18	18	NUM
ejpam-6014	103	8	(	(	PUNCT
ejpam-6014	103	9	2	2	NUM
ejpam-6014	103	10	)	)	PUNCT
ejpam-6014	103	11	(	(	PUNCT
ejpam-6014	103	12	2025	2025	NUM
ejpam-6014	103	13	)	)	PUNCT
ejpam-6014	103	14	,	,	PUNCT
ejpam-6014	103	15	6014	6014	NUM
ejpam-6014	103	16	4	4	NUM
ejpam-6014	103	17	of	of	ADP
ejpam-6014	103	18	13	13	NUM
ejpam-6014	103	19	theorem	theorem	NOUN
ejpam-6014	103	20	2	2	NUM
ejpam-6014	103	21	.	.	PUNCT
ejpam-6014	104	1	a	a	DET
ejpam-6014	104	2	function	function	NOUN
ejpam-6014	104	3	f	f	NOUN
ejpam-6014	104	4	:	:	PUNCT
ejpam-6014	104	5	(	(	PUNCT
ejpam-6014	104	6	x	x	NOUN
ejpam-6014	104	7	,	,	PUNCT
ejpam-6014	104	8	τ1	τ1	NOUN
ejpam-6014	104	9	,	,	PUNCT
ejpam-6014	104	10	τ2	τ2	NOUN
ejpam-6014	104	11	)	)	PUNCT
ejpam-6014	104	12	→	→	SYM
ejpam-6014	104	13	(	(	PUNCT
ejpam-6014	104	14	y	y	PROPN
ejpam-6014	104	15	,	,	PUNCT
ejpam-6014	104	16	σ1	σ1	PROPN
ejpam-6014	104	17	,	,	PUNCT
ejpam-6014	104	18	σ2	σ2	PROPN
ejpam-6014	104	19	)	)	PUNCT
ejpam-6014	104	20	is	be	AUX
ejpam-6014	104	21	θ(τ1	θ(τ1	NOUN
ejpam-6014	104	22	,	,	PUNCT
ejpam-6014	104	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	104	24	if	if	SCONJ
ejpam-6014	104	25	and	and	CCONJ
ejpam-6014	104	26	only	only	ADV
ejpam-6014	104	27	if	if	SCONJ
ejpam-6014	104	28	f−1(v	f−1(v	PROPN
ejpam-6014	104	29	)	)	PUNCT
ejpam-6014	105	1	⊆	⊆	NUM
ejpam-6014	105	2	(	(	PUNCT
ejpam-6014	105	3	τ1	τ1	NOUN
ejpam-6014	105	4	,	,	PUNCT
ejpam-6014	105	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	105	6	-	-	PUNCT
ejpam-6014	105	7	int(f	int(f	VERB
ejpam-6014	105	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	105	9	-	-	PUNCT
ejpam-6014	105	10	cl(v	cl(v	NOUN
ejpam-6014	105	11	)	)	PUNCT
ejpam-6014	105	12	)	)	PUNCT
ejpam-6014	105	13	)	)	PUNCT
ejpam-6014	105	14	for	for	ADP
ejpam-6014	105	15	every	every	DET
ejpam-6014	105	16	σ1σ2	σ1σ2	NOUN
ejpam-6014	105	17	-	-	ADJ
ejpam-6014	105	18	open	open	ADJ
ejpam-6014	105	19	set	set	NOUN
ejpam-6014	105	20	v	v	NOUN
ejpam-6014	105	21	of	of	ADP
ejpam-6014	105	22	y	y	PROPN
ejpam-6014	105	23	.	.	PUNCT
ejpam-6014	106	1	proof	proof	NOUN
ejpam-6014	106	2	.	.	PUNCT
ejpam-6014	107	1	let	let	VERB
ejpam-6014	107	2	v	v	PART
ejpam-6014	107	3	be	be	AUX
ejpam-6014	107	4	any	any	DET
ejpam-6014	107	5	σ1σ2	σ1σ2	NOUN
ejpam-6014	107	6	-	-	ADJ
ejpam-6014	107	7	open	open	ADJ
ejpam-6014	107	8	set	set	NOUN
ejpam-6014	107	9	of	of	ADP
ejpam-6014	107	10	y	y	PROPN
ejpam-6014	107	11	and	and	CCONJ
ejpam-6014	107	12	x	x	PROPN
ejpam-6014	107	13	∈	∈	PROPN
ejpam-6014	107	14	f−1(v	f−1(v	NOUN
ejpam-6014	107	15	)	)	PUNCT
ejpam-6014	107	16	.	.	PUNCT
ejpam-6014	108	1	then	then	ADV
ejpam-6014	108	2	,	,	PUNCT
ejpam-6014	108	3	f(x	f(x	PROPN
ejpam-6014	108	4	)	)	PUNCT
ejpam-6014	108	5	∈	∈	PROPN
ejpam-6014	108	6	v	v	NOUN
ejpam-6014	108	7	.	.	PUNCT
ejpam-6014	109	1	since	since	SCONJ
ejpam-6014	109	2	f	f	PROPN
ejpam-6014	109	3	is	be	AUX
ejpam-6014	109	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	109	5	,	,	PUNCT
ejpam-6014	109	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	109	7	at	at	ADP
ejpam-6014	109	8	x	x	X
ejpam-6014	109	9	,	,	PUNCT
ejpam-6014	109	10	by	by	ADP
ejpam-6014	109	11	theorem	theorem	NOUN
ejpam-6014	109	12	1	1	NUM
ejpam-6014	109	13	we	we	PRON
ejpam-6014	109	14	have	have	VERB
ejpam-6014	109	15	x	x	X
ejpam-6014	109	16	∈	∈	PROPN
ejpam-6014	109	17	(	(	PUNCT
ejpam-6014	109	18	τ1	τ1	NOUN
ejpam-6014	109	19	,	,	PUNCT
ejpam-6014	109	20	τ2)θ	τ2)θ	NOUN
ejpam-6014	109	21	-	-	PUNCT
ejpam-6014	109	22	int(f	int(f	VERB
ejpam-6014	109	23	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	109	24	-	-	PUNCT
ejpam-6014	109	25	cl(v	cl(v	NOUN
ejpam-6014	109	26	)	)	PUNCT
ejpam-6014	109	27	)	)	PUNCT
ejpam-6014	109	28	)	)	PUNCT
ejpam-6014	109	29	and	and	CCONJ
ejpam-6014	109	30	hence	hence	ADV
ejpam-6014	109	31	f−1(v	f−1(v	NOUN
ejpam-6014	109	32	)	)	PUNCT
ejpam-6014	110	1	⊆	⊆	NUM
ejpam-6014	110	2	(	(	PUNCT
ejpam-6014	110	3	τ1	τ1	NOUN
ejpam-6014	110	4	,	,	PUNCT
ejpam-6014	110	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	110	6	-	-	PUNCT
ejpam-6014	110	7	int(f	int(f	VERB
ejpam-6014	110	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	110	9	-	-	PUNCT
ejpam-6014	110	10	cl(v	cl(v	NOUN
ejpam-6014	110	11	)	)	PUNCT
ejpam-6014	110	12	)	)	PUNCT
ejpam-6014	110	13	)	)	PUNCT
ejpam-6014	110	14	.	.	PUNCT
ejpam-6014	111	1	conversely	conversely	ADV
ejpam-6014	111	2	,	,	PUNCT
ejpam-6014	111	3	let	let	VERB
ejpam-6014	111	4	x	x	X
ejpam-6014	111	5	∈	∈	PROPN
ejpam-6014	111	6	x	x	X
ejpam-6014	111	7	and	and	CCONJ
ejpam-6014	111	8	v	v	AUX
ejpam-6014	111	9	be	be	AUX
ejpam-6014	111	10	any	any	DET
ejpam-6014	111	11	σ1σ2	σ1σ2	NOUN
ejpam-6014	111	12	-	-	ADJ
ejpam-6014	111	13	open	open	ADJ
ejpam-6014	111	14	set	set	NOUN
ejpam-6014	111	15	of	of	ADP
ejpam-6014	111	16	y	y	PROPN
ejpam-6014	111	17	containing	contain	VERB
ejpam-6014	111	18	f(x	f(x	PROPN
ejpam-6014	111	19	)	)	PUNCT
ejpam-6014	111	20	.	.	PUNCT
ejpam-6014	112	1	then	then	ADV
ejpam-6014	112	2	,	,	PUNCT
ejpam-6014	112	3	we	we	PRON
ejpam-6014	112	4	have	have	VERB
ejpam-6014	112	5	x	x	X
ejpam-6014	112	6	∈	∈	PROPN
ejpam-6014	112	7	f−1(v	f−1(v	NOUN
ejpam-6014	112	8	)	)	PUNCT
ejpam-6014	113	1	⊆	⊆	NUM
ejpam-6014	113	2	(	(	PUNCT
ejpam-6014	113	3	τ1	τ1	NOUN
ejpam-6014	113	4	,	,	PUNCT
ejpam-6014	113	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	113	6	-	-	PUNCT
ejpam-6014	113	7	int(f	int(f	VERB
ejpam-6014	113	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	113	9	-	-	PUNCT
ejpam-6014	113	10	cl(v	cl(v	NOUN
ejpam-6014	113	11	)	)	PUNCT
ejpam-6014	113	12	)	)	PUNCT
ejpam-6014	113	13	)	)	PUNCT
ejpam-6014	113	14	and	and	CCONJ
ejpam-6014	113	15	hence	hence	ADV
ejpam-6014	113	16	x	x	X
ejpam-6014	113	17	∈	∈	PROPN
ejpam-6014	113	18	(	(	PUNCT
ejpam-6014	113	19	τ1	τ1	NOUN
ejpam-6014	113	20	,	,	PUNCT
ejpam-6014	113	21	τ2)θ	τ2)θ	NOUN
ejpam-6014	113	22	-	-	PUNCT
ejpam-6014	113	23	int(f	int(f	VERB
ejpam-6014	113	24	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	113	25	-	-	PUNCT
ejpam-6014	113	26	cl(v	cl(v	NOUN
ejpam-6014	113	27	)	)	PUNCT
ejpam-6014	113	28	)	)	PUNCT
ejpam-6014	113	29	)	)	PUNCT
ejpam-6014	113	30	.	.	PUNCT
ejpam-6014	114	1	by	by	ADP
ejpam-6014	114	2	theorem	theorem	NOUN
ejpam-6014	114	3	1	1	NUM
ejpam-6014	114	4	,	,	PUNCT
ejpam-6014	114	5	f	f	PROPN
ejpam-6014	114	6	is	be	AUX
ejpam-6014	114	7	θ(τ1	θ(τ1	NOUN
ejpam-6014	114	8	,	,	PUNCT
ejpam-6014	114	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	114	10	.	.	PUNCT
ejpam-6014	115	1	theorem	theorem	NOUN
ejpam-6014	115	2	3	3	NUM
ejpam-6014	115	3	.	.	X
ejpam-6014	115	4	for	for	ADP
ejpam-6014	115	5	a	a	DET
ejpam-6014	115	6	function	function	NOUN
ejpam-6014	115	7	(	(	PUNCT
ejpam-6014	115	8	x	x	NOUN
ejpam-6014	115	9	,	,	PUNCT
ejpam-6014	115	10	τ1	τ1	NOUN
ejpam-6014	115	11	,	,	PUNCT
ejpam-6014	115	12	τ2	τ2	NOUN
ejpam-6014	115	13	)	)	PUNCT
ejpam-6014	115	14	→	→	SYM
ejpam-6014	115	15	(	(	PUNCT
ejpam-6014	115	16	y	y	PROPN
ejpam-6014	115	17	,	,	PUNCT
ejpam-6014	115	18	σ1	σ1	PROPN
ejpam-6014	115	19	,	,	PUNCT
ejpam-6014	115	20	σ2	σ2	NOUN
ejpam-6014	115	21	)	)	PUNCT
ejpam-6014	115	22	,	,	PUNCT
ejpam-6014	115	23	the	the	DET
ejpam-6014	115	24	following	follow	VERB
ejpam-6014	115	25	properties	property	NOUN
ejpam-6014	115	26	are	be	AUX
ejpam-6014	115	27	equivalent	equivalent	ADJ
ejpam-6014	115	28	:	:	PUNCT
ejpam-6014	115	29	(	(	PUNCT
ejpam-6014	115	30	1	1	X
ejpam-6014	115	31	)	)	PUNCT
ejpam-6014	115	32	f	f	PROPN
ejpam-6014	115	33	is	be	AUX
ejpam-6014	115	34	θ(τ1	θ(τ1	NOUN
ejpam-6014	115	35	,	,	PUNCT
ejpam-6014	115	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	115	37	;	;	PUNCT
ejpam-6014	115	38	(	(	PUNCT
ejpam-6014	115	39	2	2	X
ejpam-6014	115	40	)	)	PUNCT
ejpam-6014	115	41	(	(	PUNCT
ejpam-6014	115	42	τ1	τ1	NOUN
ejpam-6014	115	43	,	,	PUNCT
ejpam-6014	115	44	τ2)θ	τ2)θ	PROPN
ejpam-6014	115	45	-	-	PUNCT
ejpam-6014	115	46	cl(f	cl(f	NOUN
ejpam-6014	115	47	−1(b	−1(b	NOUN
ejpam-6014	115	48	)	)	PUNCT
ejpam-6014	115	49	)	)	PUNCT
ejpam-6014	115	50	⊆	⊆	NUM
ejpam-6014	115	51	f−1(σ1	f−1(σ1	NOUN
ejpam-6014	115	52	,	,	PUNCT
ejpam-6014	115	53	σ2)θ	σ2)θ	ADJ
ejpam-6014	115	54	-	-	PUNCT
ejpam-6014	115	55	cl(b	cl(b	NOUN
ejpam-6014	115	56	)	)	PUNCT
ejpam-6014	115	57	)	)	PUNCT
ejpam-6014	115	58	for	for	ADP
ejpam-6014	115	59	every	every	DET
ejpam-6014	115	60	subset	subset	NOUN
ejpam-6014	115	61	b	b	PROPN
ejpam-6014	115	62	of	of	ADP
ejpam-6014	115	63	y	y	PROPN
ejpam-6014	115	64	;	;	PUNCT
ejpam-6014	115	65	(	(	PUNCT
ejpam-6014	115	66	3	3	X
ejpam-6014	115	67	)	)	PUNCT
ejpam-6014	115	68	(	(	PUNCT
ejpam-6014	115	69	τ1	τ1	NOUN
ejpam-6014	115	70	,	,	PUNCT
ejpam-6014	115	71	τ2)θ	τ2)θ	PROPN
ejpam-6014	115	72	-	-	PUNCT
ejpam-6014	115	73	cl(f	cl(f	PROPN
ejpam-6014	115	74	−1(v	−1(v	PROPN
ejpam-6014	115	75	)	)	PUNCT
ejpam-6014	115	76	)	)	PUNCT
ejpam-6014	116	1	⊆	⊆	NUM
ejpam-6014	116	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	116	3	-	-	PUNCT
ejpam-6014	116	4	cl(v	cl(v	NOUN
ejpam-6014	116	5	)	)	PUNCT
ejpam-6014	116	6	)	)	PUNCT
ejpam-6014	116	7	for	for	ADP
ejpam-6014	116	8	every	every	DET
ejpam-6014	116	9	σ1σ2	σ1σ2	NOUN
ejpam-6014	116	10	-	-	ADJ
ejpam-6014	116	11	open	open	ADJ
ejpam-6014	116	12	set	set	NOUN
ejpam-6014	116	13	v	v	NOUN
ejpam-6014	116	14	of	of	ADP
ejpam-6014	116	15	y	y	PROPN
ejpam-6014	116	16	;	;	PUNCT
ejpam-6014	116	17	(	(	PUNCT
ejpam-6014	116	18	4	4	X
ejpam-6014	116	19	)	)	PUNCT
ejpam-6014	116	20	f−1(v	f−1(v	NOUN
ejpam-6014	116	21	)	)	PUNCT
ejpam-6014	117	1	⊆	⊆	NUM
ejpam-6014	117	2	(	(	PUNCT
ejpam-6014	117	3	τ1	τ1	NOUN
ejpam-6014	117	4	,	,	PUNCT
ejpam-6014	117	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	117	6	-	-	PUNCT
ejpam-6014	117	7	int(f	int(f	VERB
ejpam-6014	117	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	117	9	-	-	PUNCT
ejpam-6014	117	10	cl(v	cl(v	NOUN
ejpam-6014	117	11	)	)	PUNCT
ejpam-6014	117	12	)	)	PUNCT
ejpam-6014	117	13	)	)	PUNCT
ejpam-6014	118	1	for	for	ADP
ejpam-6014	118	2	every	every	DET
ejpam-6014	118	3	σ1σ2	σ1σ2	NOUN
ejpam-6014	118	4	-	-	ADJ
ejpam-6014	118	5	open	open	ADJ
ejpam-6014	118	6	set	set	NOUN
ejpam-6014	118	7	v	v	NOUN
ejpam-6014	118	8	of	of	ADP
ejpam-6014	118	9	y	y	PROPN
ejpam-6014	118	10	;	;	PUNCT
ejpam-6014	118	11	(	(	PUNCT
ejpam-6014	118	12	5	5	X
ejpam-6014	118	13	)	)	PUNCT
ejpam-6014	118	14	f((τ1	f((τ1	PROPN
ejpam-6014	118	15	,	,	PUNCT
ejpam-6014	118	16	τ2)θ	τ2)θ	NOUN
ejpam-6014	118	17	-	-	PUNCT
ejpam-6014	118	18	cl(a	cl(a	NUM
ejpam-6014	118	19	)	)	PUNCT
ejpam-6014	118	20	)	)	PUNCT
ejpam-6014	118	21	⊆	⊆	NUM
ejpam-6014	118	22	(	(	PUNCT
ejpam-6014	118	23	σ1	σ1	PROPN
ejpam-6014	118	24	,	,	PUNCT
ejpam-6014	118	25	σ2)θ	σ2)θ	NOUN
ejpam-6014	118	26	-	-	PUNCT
ejpam-6014	118	27	cl(f(a	cl(f(a	NOUN
ejpam-6014	118	28	)	)	PUNCT
ejpam-6014	118	29	)	)	PUNCT
ejpam-6014	118	30	for	for	ADP
ejpam-6014	118	31	every	every	DET
ejpam-6014	118	32	subset	subset	NOUN
ejpam-6014	118	33	a	a	PRON
ejpam-6014	118	34	of	of	ADP
ejpam-6014	118	35	x	x	PRON
ejpam-6014	118	36	;	;	PUNCT
ejpam-6014	118	37	(	(	PUNCT
ejpam-6014	118	38	6	6	NUM
ejpam-6014	118	39	)	)	PUNCT
ejpam-6014	118	40	(	(	PUNCT
ejpam-6014	118	41	τ1	τ1	NOUN
ejpam-6014	118	42	,	,	PUNCT
ejpam-6014	118	43	τ2)θ	τ2)θ	PROPN
ejpam-6014	118	44	-	-	PUNCT
ejpam-6014	118	45	cl(f	cl(f	NOUN
ejpam-6014	118	46	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	118	47	-	-	PUNCT
ejpam-6014	118	48	int((σ1	int((σ1	NUM
ejpam-6014	118	49	,	,	PUNCT
ejpam-6014	118	50	σ2)θ	σ2)θ	ADJ
ejpam-6014	118	51	-	-	PUNCT
ejpam-6014	118	52	cl(b	cl(b	NOUN
ejpam-6014	118	53	)	)	PUNCT
ejpam-6014	118	54	)	)	PUNCT
ejpam-6014	118	55	)	)	PUNCT
ejpam-6014	118	56	)	)	PUNCT
ejpam-6014	118	57	⊆	⊆	NUM
ejpam-6014	118	58	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	118	59	,	,	PUNCT
ejpam-6014	118	60	σ2)θ	σ2)θ	NOUN
ejpam-6014	118	61	-	-	PUNCT
ejpam-6014	118	62	cl(b	cl(b	NOUN
ejpam-6014	118	63	)	)	PUNCT
ejpam-6014	118	64	)	)	PUNCT
ejpam-6014	118	65	for	for	ADP
ejpam-6014	118	66	every	every	DET
ejpam-6014	118	67	subset	subset	NOUN
ejpam-6014	118	68	b	b	PROPN
ejpam-6014	118	69	of	of	ADP
ejpam-6014	118	70	y	y	PROPN
ejpam-6014	118	71	;	;	PUNCT
ejpam-6014	118	72	(	(	PUNCT
ejpam-6014	118	73	7	7	X
ejpam-6014	118	74	)	)	PUNCT
ejpam-6014	118	75	(	(	PUNCT
ejpam-6014	118	76	τ1	τ1	NOUN
ejpam-6014	118	77	,	,	PUNCT
ejpam-6014	118	78	τ2)θ	τ2)θ	PROPN
ejpam-6014	118	79	-	-	PUNCT
ejpam-6014	118	80	cl(f	cl(f	NOUN
ejpam-6014	118	81	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	118	82	-	-	PUNCT
ejpam-6014	118	83	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	118	84	-	-	PUNCT
ejpam-6014	118	85	cl(v	cl(v	NOUN
ejpam-6014	118	86	)	)	PUNCT
ejpam-6014	118	87	)	)	PUNCT
ejpam-6014	118	88	)	)	PUNCT
ejpam-6014	118	89	)	)	PUNCT
ejpam-6014	118	90	⊆	⊆	NUM
ejpam-6014	118	91	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	118	92	-	-	PUNCT
ejpam-6014	118	93	cl(v	cl(v	NOUN
ejpam-6014	118	94	)	)	PUNCT
ejpam-6014	118	95	)	)	PUNCT
ejpam-6014	118	96	for	for	ADP
ejpam-6014	118	97	every	every	DET
ejpam-6014	118	98	σ1σ2	σ1σ2	NOUN
ejpam-6014	118	99	-	-	ADJ
ejpam-6014	118	100	open	open	ADJ
ejpam-6014	118	101	set	set	NOUN
ejpam-6014	118	102	v	v	NOUN
ejpam-6014	118	103	of	of	ADP
ejpam-6014	118	104	y	y	PROPN
ejpam-6014	118	105	;	;	PUNCT
ejpam-6014	118	106	(	(	PUNCT
ejpam-6014	118	107	8)	8)	NUM
ejpam-6014	118	108	(	(	PUNCT
ejpam-6014	118	109	τ1	τ1	NOUN
ejpam-6014	118	110	,	,	PUNCT
ejpam-6014	118	111	τ2)θ	τ2)θ	PROPN
ejpam-6014	118	112	-	-	PUNCT
ejpam-6014	118	113	cl(f	cl(f	NOUN
ejpam-6014	118	114	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	118	115	-	-	PUNCT
ejpam-6014	118	116	int(k	int(k	NUM
ejpam-6014	118	117	)	)	PUNCT
ejpam-6014	118	118	)	)	PUNCT
ejpam-6014	118	119	)	)	PUNCT
ejpam-6014	119	1	⊆	⊆	NUM
ejpam-6014	119	2	f−1(k	f−1(k	PROPN
ejpam-6014	119	3	)	)	PUNCT
ejpam-6014	119	4	for	for	ADP
ejpam-6014	119	5	every	every	DET
ejpam-6014	119	6	(	(	PUNCT
ejpam-6014	119	7	σ1	σ1	PROPN
ejpam-6014	119	8	,	,	PUNCT
ejpam-6014	119	9	σ2)r	σ2)r	NOUN
ejpam-6014	119	10	-	-	PUNCT
ejpam-6014	119	11	closed	close	VERB
ejpam-6014	119	12	set	set	ADJ
ejpam-6014	119	13	k	k	PROPN
ejpam-6014	119	14	of	of	ADP
ejpam-6014	119	15	y	y	PROPN
ejpam-6014	119	16	;	;	PUNCT
ejpam-6014	119	17	(	(	PUNCT
ejpam-6014	119	18	9	9	X
ejpam-6014	119	19	)	)	PUNCT
ejpam-6014	119	20	(	(	PUNCT
ejpam-6014	119	21	τ1	τ1	NOUN
ejpam-6014	119	22	,	,	PUNCT
ejpam-6014	119	23	τ2)θ	τ2)θ	PROPN
ejpam-6014	119	24	-	-	PUNCT
ejpam-6014	119	25	cl(f	cl(f	NOUN
ejpam-6014	119	26	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	119	27	-	-	PUNCT
ejpam-6014	119	28	int(k	int(k	NUM
ejpam-6014	119	29	)	)	PUNCT
ejpam-6014	119	30	)	)	PUNCT
ejpam-6014	119	31	)	)	PUNCT
ejpam-6014	120	1	⊆	⊆	NUM
ejpam-6014	120	2	f−1(k	f−1(k	PROPN
ejpam-6014	120	3	)	)	PUNCT
ejpam-6014	120	4	for	for	ADP
ejpam-6014	120	5	every	every	DET
ejpam-6014	120	6	σ1σ2	σ1σ2	NUM
ejpam-6014	120	7	-	-	PUNCT
ejpam-6014	120	8	closed	closed	ADJ
ejpam-6014	120	9	set	set	NOUN
ejpam-6014	120	10	k	k	PROPN
ejpam-6014	120	11	of	of	ADP
ejpam-6014	120	12	y	y	PROPN
ejpam-6014	120	13	.	.	PUNCT
ejpam-6014	121	1	proof	proof	NOUN
ejpam-6014	121	2	.	.	PUNCT
ejpam-6014	122	1	(	(	PUNCT
ejpam-6014	122	2	1	1	X
ejpam-6014	122	3	)	)	PUNCT
ejpam-6014	122	4	⇒	⇒	NOUN
ejpam-6014	122	5	(	(	PUNCT
ejpam-6014	122	6	2	2	NUM
ejpam-6014	122	7	):	):	PUNCT
ejpam-6014	122	8	let	let	VERB
ejpam-6014	122	9	b	b	X
ejpam-6014	122	10	be	be	AUX
ejpam-6014	122	11	any	any	DET
ejpam-6014	122	12	subset	subset	NOUN
ejpam-6014	122	13	of	of	ADP
ejpam-6014	122	14	y	y	PROPN
ejpam-6014	122	15	.	.	PUNCT
ejpam-6014	122	16	suppose	suppose	VERB
ejpam-6014	122	17	that	that	SCONJ
ejpam-6014	122	18	x	x	PROPN
ejpam-6014	122	19	̸∈	̸∈	PROPN
ejpam-6014	122	20	f−1(σ1	f−1(σ1	NOUN
ejpam-6014	122	21	,	,	PUNCT
ejpam-6014	122	22	σ2)θ	σ2)θ	ADJ
ejpam-6014	122	23	-	-	PUNCT
ejpam-6014	122	24	cl(b	cl(b	NOUN
ejpam-6014	122	25	)	)	PUNCT
ejpam-6014	122	26	)	)	PUNCT
ejpam-6014	122	27	.	.	PUNCT
ejpam-6014	123	1	then	then	ADV
ejpam-6014	123	2	,	,	PUNCT
ejpam-6014	123	3	we	we	PRON
ejpam-6014	123	4	have	have	VERB
ejpam-6014	123	5	x	x	X
ejpam-6014	123	6	∈	∈	PROPN
ejpam-6014	123	7	f−1(y	f−1(y	PROPN
ejpam-6014	123	8	−	−	PROPN
ejpam-6014	123	9	(	(	PUNCT
ejpam-6014	123	10	σ1	σ1	PROPN
ejpam-6014	123	11	,	,	PUNCT
ejpam-6014	123	12	σ2)θ	σ2)θ	NOUN
ejpam-6014	123	13	-	-	PUNCT
ejpam-6014	123	14	cl(b	cl(b	NOUN
ejpam-6014	123	15	)	)	PUNCT
ejpam-6014	123	16	)	)	PUNCT
ejpam-6014	124	1	=	=	SYM
ejpam-6014	124	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	124	3	,	,	PUNCT
ejpam-6014	124	4	σ2)θ	σ2)θ	ADJ
ejpam-6014	124	5	-	-	PUNCT
ejpam-6014	124	6	int(y	int(y	PROPN
ejpam-6014	124	7	−	−	PROPN
ejpam-6014	124	8	b	b	NOUN
ejpam-6014	124	9	)	)	PUNCT
ejpam-6014	124	10	)	)	PUNCT
ejpam-6014	124	11	.	.	PUNCT
ejpam-6014	125	1	therefore	therefore	ADV
ejpam-6014	125	2	,	,	PUNCT
ejpam-6014	125	3	f(x	f(x	PROPN
ejpam-6014	125	4	)	)	PUNCT
ejpam-6014	125	5	∈	∈	PROPN
ejpam-6014	125	6	(	(	PUNCT
ejpam-6014	125	7	σ1	σ1	PROPN
ejpam-6014	125	8	,	,	PUNCT
ejpam-6014	125	9	σ2)θ	σ2)θ	ADJ
ejpam-6014	125	10	-	-	PUNCT
ejpam-6014	125	11	int(y	int(y	ADJ
ejpam-6014	125	12	−b	−b	NOUN
ejpam-6014	125	13	)	)	PUNCT
ejpam-6014	125	14	.	.	PUNCT
ejpam-6014	126	1	there	there	PRON
ejpam-6014	126	2	exists	exist	VERB
ejpam-6014	126	3	a	a	DET
ejpam-6014	126	4	σ1σ2	σ1σ2	NUM
ejpam-6014	126	5	-	-	ADJ
ejpam-6014	126	6	open	open	ADJ
ejpam-6014	126	7	set	set	NOUN
ejpam-6014	126	8	v	v	NOUN
ejpam-6014	126	9	of	of	ADP
ejpam-6014	126	10	y	y	PRON
ejpam-6014	126	11	such	such	ADJ
ejpam-6014	126	12	that	that	SCONJ
ejpam-6014	126	13	f(x	f(x	PROPN
ejpam-6014	126	14	)	)	PUNCT
ejpam-6014	126	15	∈	∈	PROPN
ejpam-6014	126	16	v	v	ADP
ejpam-6014	126	17	⊆	⊆	NUM
ejpam-6014	126	18	σ1σ2	σ1σ2	NOUN
ejpam-6014	126	19	-	-	PUNCT
ejpam-6014	126	20	cl(v	cl(v	NOUN
ejpam-6014	126	21	)	)	PUNCT
ejpam-6014	126	22	⊆	⊆	NUM
ejpam-6014	126	23	y	y	NOUN
ejpam-6014	126	24	−b	−b	VERB
ejpam-6014	126	25	.	.	PUNCT
ejpam-6014	127	1	since	since	SCONJ
ejpam-6014	127	2	f	f	PROPN
ejpam-6014	127	3	is	be	AUX
ejpam-6014	127	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	127	5	,	,	PUNCT
ejpam-6014	127	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	127	7	,	,	PUNCT
ejpam-6014	127	8	there	there	PRON
ejpam-6014	127	9	exists	exist	VERB
ejpam-6014	127	10	a	a	DET
ejpam-6014	127	11	τ1τ2	τ1τ2	NOUN
ejpam-6014	127	12	-	-	ADJ
ejpam-6014	127	13	open	open	ADJ
ejpam-6014	127	14	set	set	ADJ
ejpam-6014	127	15	u	u	NOUN
ejpam-6014	127	16	of	of	ADP
ejpam-6014	127	17	x	x	PUNCT
ejpam-6014	127	18	containing	contain	VERB
ejpam-6014	127	19	x	x	PUNCT
ejpam-6014	127	20	such	such	ADJ
ejpam-6014	127	21	that	that	SCONJ
ejpam-6014	127	22	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	127	23	-	-	PUNCT
ejpam-6014	127	24	cl(u	cl(u	NOUN
ejpam-6014	127	25	)	)	PUNCT
ejpam-6014	127	26	)	)	PUNCT
ejpam-6014	128	1	⊆	⊆	X
ejpam-6014	128	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	128	3	-	-	NUM
ejpam-6014	128	4	cl(v	cl(v	NOUN
ejpam-6014	128	5	)	)	PUNCT
ejpam-6014	128	6	;	;	PUNCT
ejpam-6014	128	7	hence	hence	ADV
ejpam-6014	128	8	τ1τ2	τ1τ2	NOUN
ejpam-6014	128	9	-	-	NOUN
ejpam-6014	128	10	cl(u	cl(u	ADJ
ejpam-6014	128	11	)	)	PUNCT
ejpam-6014	128	12	⊆	⊆	NUM
ejpam-6014	128	13	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	128	14	-	-	PUNCT
ejpam-6014	128	15	cl(v	cl(v	NOUN
ejpam-6014	128	16	)	)	PUNCT
ejpam-6014	128	17	)	)	PUNCT
ejpam-6014	129	1	⊆	⊆	X
ejpam-6014	129	2	f−1(y	f−1(y	NOUN
ejpam-6014	129	3	−b	−b	NOUN
ejpam-6014	129	4	)	)	PUNCT
ejpam-6014	130	1	=	=	PUNCT
ejpam-6014	130	2	x	x	X
ejpam-6014	131	1	−	−	NOUN
ejpam-6014	131	2	f−1(b	f−1(b	PROPN
ejpam-6014	131	3	)	)	PUNCT
ejpam-6014	131	4	.	.	PUNCT
ejpam-6014	132	1	thus	thus	ADV
ejpam-6014	132	2	,	,	PUNCT
ejpam-6014	132	3	τ1τ2	τ1τ2	NOUN
ejpam-6014	132	4	-	-	NOUN
ejpam-6014	132	5	cl(u	cl(u	ADJ
ejpam-6014	132	6	)	)	PUNCT
ejpam-6014	132	7	∩	∩	ADJ
ejpam-6014	132	8	f−1(b	f−1(b	PROPN
ejpam-6014	132	9	)	)	PUNCT
ejpam-6014	133	1	=	=	NOUN
ejpam-6014	133	2	∅	∅	NOUN
ejpam-6014	134	1	and	and	CCONJ
ejpam-6014	134	2	so	so	ADV
ejpam-6014	134	3	x	x	PUNCT
ejpam-6014	134	4	̸∈	̸∈	PROPN
ejpam-6014	134	5	(	(	PUNCT
ejpam-6014	134	6	τ1	τ1	PROPN
ejpam-6014	134	7	,	,	PUNCT
ejpam-6014	134	8	τ2)θ	τ2)θ	PROPN
ejpam-6014	134	9	-	-	PUNCT
ejpam-6014	134	10	cl(f	cl(f	NOUN
ejpam-6014	134	11	−1(b	−1(b	NOUN
ejpam-6014	134	12	)	)	PUNCT
ejpam-6014	134	13	)	)	PUNCT
ejpam-6014	134	14	.	.	PUNCT
ejpam-6014	135	1	m.	m.	NOUN
ejpam-6014	135	2	thongmoon	thongmoon	PROPN
ejpam-6014	135	3	,	,	PUNCT
ejpam-6014	135	4	s.	s.	PROPN
ejpam-6014	135	5	sompong	sompong	PROPN
ejpam-6014	135	6	,	,	PUNCT
ejpam-6014	135	7	c.	c.	PROPN
ejpam-6014	135	8	boonpok	boonpok	PROPN
ejpam-6014	135	9	/	/	SYM
ejpam-6014	135	10	eur	eur	PROPN
ejpam-6014	135	11	.	.	PUNCT
ejpam-6014	136	1	j.	j.	PROPN
ejpam-6014	136	2	pure	pure	PROPN
ejpam-6014	136	3	appl	appl	PROPN
ejpam-6014	136	4	.	.	PROPN
ejpam-6014	136	5	math	math	PROPN
ejpam-6014	136	6	,	,	PUNCT
ejpam-6014	136	7	18	18	NUM
ejpam-6014	136	8	(	(	PUNCT
ejpam-6014	136	9	2	2	NUM
ejpam-6014	136	10	)	)	PUNCT
ejpam-6014	136	11	(	(	PUNCT
ejpam-6014	136	12	2025	2025	NUM
ejpam-6014	136	13	)	)	PUNCT
ejpam-6014	136	14	,	,	PUNCT
ejpam-6014	136	15	6014	6014	NUM
ejpam-6014	136	16	5	5	NUM
ejpam-6014	136	17	of	of	ADP
ejpam-6014	136	18	13	13	NUM
ejpam-6014	136	19	(	(	PUNCT
ejpam-6014	136	20	2	2	NUM
ejpam-6014	136	21	)	)	PUNCT
ejpam-6014	136	22	⇒	⇒	NOUN
ejpam-6014	136	23	(	(	PUNCT
ejpam-6014	136	24	3	3	NUM
ejpam-6014	136	25	):	):	PUNCT
ejpam-6014	136	26	this	this	PRON
ejpam-6014	136	27	is	be	AUX
ejpam-6014	136	28	obvious	obvious	ADJ
ejpam-6014	136	29	since	since	SCONJ
ejpam-6014	136	30	σ1σ2	σ1σ2	NOUN
ejpam-6014	136	31	-	-	NOUN
ejpam-6014	136	32	cl(v	cl(v	X
ejpam-6014	136	33	)	)	PUNCT
ejpam-6014	137	1	=	=	SYM
ejpam-6014	137	2	(	(	PUNCT
ejpam-6014	137	3	σ1	σ1	PROPN
ejpam-6014	137	4	,	,	PUNCT
ejpam-6014	137	5	σ2)θ	σ2)θ	NOUN
ejpam-6014	137	6	-	-	PUNCT
ejpam-6014	137	7	cl(v	cl(v	NOUN
ejpam-6014	137	8	)	)	PUNCT
ejpam-6014	137	9	for	for	ADP
ejpam-6014	137	10	every	every	DET
ejpam-6014	137	11	σ1σ2	σ1σ2	NOUN
ejpam-6014	137	12	-	-	ADJ
ejpam-6014	137	13	open	open	ADJ
ejpam-6014	137	14	set	set	NOUN
ejpam-6014	137	15	v	v	NOUN
ejpam-6014	137	16	of	of	ADP
ejpam-6014	137	17	y	y	PROPN
ejpam-6014	137	18	.	.	PUNCT
ejpam-6014	138	1	(	(	PUNCT
ejpam-6014	138	2	3	3	X
ejpam-6014	138	3	)	)	PUNCT
ejpam-6014	138	4	⇒	⇒	NOUN
ejpam-6014	138	5	(	(	PUNCT
ejpam-6014	138	6	4	4	NUM
ejpam-6014	138	7	):	):	PUNCT
ejpam-6014	138	8	let	let	VERB
ejpam-6014	138	9	v	v	PART
ejpam-6014	138	10	be	be	AUX
ejpam-6014	138	11	any	any	DET
ejpam-6014	138	12	σ1σ2	σ1σ2	NOUN
ejpam-6014	138	13	-	-	ADJ
ejpam-6014	138	14	open	open	ADJ
ejpam-6014	138	15	set	set	NOUN
ejpam-6014	138	16	of	of	ADP
ejpam-6014	138	17	y	y	PROPN
ejpam-6014	138	18	.	.	PUNCT
ejpam-6014	139	1	thus	thus	ADV
ejpam-6014	139	2	by	by	ADP
ejpam-6014	139	3	(	(	PUNCT
ejpam-6014	139	4	3	3	NUM
ejpam-6014	139	5	)	)	PUNCT
ejpam-6014	139	6	,	,	PUNCT
ejpam-6014	139	7	we	we	PRON
ejpam-6014	139	8	have	have	VERB
ejpam-6014	139	9	x	x	X
ejpam-6014	139	10	−	−	PROPN
ejpam-6014	139	11	(	(	PUNCT
ejpam-6014	139	12	τ1	τ1	NOUN
ejpam-6014	139	13	,	,	PUNCT
ejpam-6014	139	14	τ2)θ	τ2)θ	NOUN
ejpam-6014	139	15	-	-	PUNCT
ejpam-6014	139	16	int(f	int(f	VERB
ejpam-6014	139	17	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	139	18	-	-	PUNCT
ejpam-6014	139	19	cl(v	cl(v	NOUN
ejpam-6014	139	20	)	)	PUNCT
ejpam-6014	139	21	)	)	PUNCT
ejpam-6014	139	22	)	)	PUNCT
ejpam-6014	140	1	=	=	PRON
ejpam-6014	140	2	(	(	PUNCT
ejpam-6014	140	3	τ1	τ1	NOUN
ejpam-6014	140	4	,	,	PUNCT
ejpam-6014	140	5	τ2)θ	τ2)θ	ADJ
ejpam-6014	140	6	-	-	PUNCT
ejpam-6014	140	7	cl(x	cl(x	PUNCT
ejpam-6014	140	8	−	−	NOUN
ejpam-6014	140	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	140	10	-	-	PUNCT
ejpam-6014	140	11	cl(v	cl(v	NOUN
ejpam-6014	140	12	)	)	PUNCT
ejpam-6014	140	13	)	)	PUNCT
ejpam-6014	140	14	)	)	PUNCT
ejpam-6014	141	1	=	=	PRON
ejpam-6014	141	2	(	(	PUNCT
ejpam-6014	141	3	τ1	τ1	NOUN
ejpam-6014	141	4	,	,	PUNCT
ejpam-6014	141	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	141	6	-	-	PUNCT
ejpam-6014	141	7	cl(f	cl(f	NOUN
ejpam-6014	141	8	−1(y	−1(y	VERB
ejpam-6014	141	9	−	−	PUNCT
ejpam-6014	141	10	σ1σ2	σ1σ2	NOUN
ejpam-6014	141	11	-	-	NUM
ejpam-6014	141	12	cl(v	cl(v	NOUN
ejpam-6014	141	13	)	)	PUNCT
ejpam-6014	141	14	)	)	PUNCT
ejpam-6014	141	15	)	)	PUNCT
ejpam-6014	142	1	⊆	⊆	NUM
ejpam-6014	142	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	142	3	-	-	PUNCT
ejpam-6014	142	4	cl(y	cl(y	NOUN
ejpam-6014	142	5	−	−	NOUN
ejpam-6014	142	6	σ1σ2	σ1σ2	NOUN
ejpam-6014	142	7	-	-	NUM
ejpam-6014	142	8	cl(v	cl(v	NOUN
ejpam-6014	142	9	)	)	PUNCT
ejpam-6014	142	10	)	)	PUNCT
ejpam-6014	142	11	)	)	PUNCT
ejpam-6014	143	1	⊆	⊆	NUM
ejpam-6014	143	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	143	3	-	-	PUNCT
ejpam-6014	143	4	cl(y	cl(y	NOUN
ejpam-6014	143	5	−	−	PROPN
ejpam-6014	143	6	v	v	NOUN
ejpam-6014	143	7	)	)	PUNCT
ejpam-6014	143	8	)	)	PUNCT
ejpam-6014	144	1	=	=	SYM
ejpam-6014	144	2	f−1(y	f−1(y	PROPN
ejpam-6014	144	3	−	−	PROPN
ejpam-6014	144	4	v	v	NOUN
ejpam-6014	144	5	)	)	PUNCT
ejpam-6014	144	6	=	=	PUNCT
ejpam-6014	145	1	x	x	PUNCT
ejpam-6014	145	2	−	−	PROPN
ejpam-6014	145	3	f−1(v	f−1(v	PROPN
ejpam-6014	145	4	)	)	PUNCT
ejpam-6014	145	5	and	and	CCONJ
ejpam-6014	145	6	hence	hence	ADV
ejpam-6014	145	7	f−1(v	f−1(v	NOUN
ejpam-6014	145	8	)	)	PUNCT
ejpam-6014	146	1	⊆	⊆	NUM
ejpam-6014	146	2	(	(	PUNCT
ejpam-6014	146	3	τ1	τ1	NOUN
ejpam-6014	146	4	,	,	PUNCT
ejpam-6014	146	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	146	6	-	-	PUNCT
ejpam-6014	146	7	int(f	int(f	VERB
ejpam-6014	146	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	146	9	-	-	PUNCT
ejpam-6014	146	10	cl(v	cl(v	NOUN
ejpam-6014	146	11	)	)	PUNCT
ejpam-6014	146	12	)	)	PUNCT
ejpam-6014	146	13	)	)	PUNCT
ejpam-6014	146	14	.	.	PUNCT
ejpam-6014	147	1	(	(	PUNCT
ejpam-6014	147	2	4	4	X
ejpam-6014	147	3	)	)	PUNCT
ejpam-6014	147	4	⇒	⇒	NOUN
ejpam-6014	147	5	(	(	PUNCT
ejpam-6014	147	6	1	1	NUM
ejpam-6014	147	7	):	):	PUNCT
ejpam-6014	147	8	it	it	PRON
ejpam-6014	147	9	follows	follow	VERB
ejpam-6014	147	10	from	from	ADP
ejpam-6014	147	11	theorem	theorem	ADJ
ejpam-6014	147	12	2	2	NUM
ejpam-6014	147	13	.	.	PUNCT
ejpam-6014	147	14	(	(	PUNCT
ejpam-6014	147	15	2	2	X
ejpam-6014	147	16	)	)	PUNCT
ejpam-6014	147	17	⇒	⇒	NOUN
ejpam-6014	147	18	(	(	PUNCT
ejpam-6014	147	19	5	5	NUM
ejpam-6014	147	20	):	):	PUNCT
ejpam-6014	147	21	let	let	VERB
ejpam-6014	147	22	a	a	PRON
ejpam-6014	147	23	be	be	AUX
ejpam-6014	147	24	any	any	DET
ejpam-6014	147	25	subset	subset	NOUN
ejpam-6014	147	26	of	of	ADP
ejpam-6014	147	27	x.	x.	NOUN
ejpam-6014	147	28	by	by	ADP
ejpam-6014	147	29	(	(	PUNCT
ejpam-6014	147	30	2	2	NUM
ejpam-6014	147	31	)	)	PUNCT
ejpam-6014	147	32	,	,	PUNCT
ejpam-6014	147	33	we	we	PRON
ejpam-6014	147	34	have	have	VERB
ejpam-6014	147	35	(	(	PUNCT
ejpam-6014	147	36	τ1	τ1	NOUN
ejpam-6014	147	37	,	,	PUNCT
ejpam-6014	147	38	τ2)θ	τ2)θ	NOUN
ejpam-6014	147	39	-	-	PUNCT
ejpam-6014	147	40	cl(a	cl(a	NUM
ejpam-6014	147	41	)	)	PUNCT
ejpam-6014	147	42	⊆	⊆	NUM
ejpam-6014	147	43	(	(	PUNCT
ejpam-6014	147	44	τ1	τ1	NOUN
ejpam-6014	147	45	,	,	PUNCT
ejpam-6014	147	46	τ2)θ	τ2)θ	PROPN
ejpam-6014	147	47	-	-	PUNCT
ejpam-6014	147	48	cl(f	cl(f	NOUN
ejpam-6014	147	49	−1(f(a	−1(f(a	NOUN
ejpam-6014	147	50	)	)	PUNCT
ejpam-6014	147	51	)	)	PUNCT
ejpam-6014	147	52	)	)	PUNCT
ejpam-6014	148	1	⊆	⊆	NUM
ejpam-6014	148	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	148	3	,	,	PUNCT
ejpam-6014	148	4	σ2)θ	σ2)θ	ADJ
ejpam-6014	148	5	-	-	PUNCT
ejpam-6014	148	6	cl(f(a	cl(f(a	NOUN
ejpam-6014	148	7	)	)	PUNCT
ejpam-6014	148	8	)	)	PUNCT
ejpam-6014	148	9	)	)	PUNCT
ejpam-6014	148	10	.	.	PUNCT
ejpam-6014	149	1	thus	thus	ADV
ejpam-6014	149	2	,	,	PUNCT
ejpam-6014	149	3	f((τ1	f((τ1	PROPN
ejpam-6014	149	4	,	,	PUNCT
ejpam-6014	149	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	149	6	-	-	PUNCT
ejpam-6014	149	7	cl(a	cl(a	NUM
ejpam-6014	149	8	)	)	PUNCT
ejpam-6014	149	9	)	)	PUNCT
ejpam-6014	150	1	⊆	⊆	NUM
ejpam-6014	150	2	(	(	PUNCT
ejpam-6014	150	3	σ1	σ1	PROPN
ejpam-6014	150	4	,	,	PUNCT
ejpam-6014	150	5	σ2)θ	σ2)θ	NOUN
ejpam-6014	150	6	-	-	PUNCT
ejpam-6014	150	7	cl(f(a	cl(f(a	NOUN
ejpam-6014	150	8	)	)	PUNCT
ejpam-6014	150	9	)	)	PUNCT
ejpam-6014	150	10	.	.	PUNCT
ejpam-6014	151	1	(	(	PUNCT
ejpam-6014	151	2	5	5	X
ejpam-6014	151	3	)	)	PUNCT
ejpam-6014	151	4	⇒	⇒	NOUN
ejpam-6014	151	5	(	(	PUNCT
ejpam-6014	151	6	2	2	NUM
ejpam-6014	151	7	):	):	PUNCT
ejpam-6014	151	8	let	let	VERB
ejpam-6014	151	9	b	b	X
ejpam-6014	151	10	be	be	AUX
ejpam-6014	151	11	any	any	DET
ejpam-6014	151	12	subset	subset	NOUN
ejpam-6014	151	13	of	of	ADP
ejpam-6014	151	14	y	y	PROPN
ejpam-6014	151	15	.	.	PUNCT
ejpam-6014	152	1	then	then	ADV
ejpam-6014	152	2	by	by	ADP
ejpam-6014	152	3	(	(	PUNCT
ejpam-6014	152	4	5	5	NUM
ejpam-6014	152	5	)	)	PUNCT
ejpam-6014	152	6	,	,	PUNCT
ejpam-6014	152	7	we	we	PRON
ejpam-6014	152	8	have	have	AUX
ejpam-6014	152	9	f((τ1	f((τ1	NOUN
ejpam-6014	152	10	,	,	PUNCT
ejpam-6014	152	11	τ2)θ	τ2)θ	ADJ
ejpam-6014	152	12	-	-	PUNCT
ejpam-6014	152	13	cl(f	cl(f	NOUN
ejpam-6014	152	14	−1(b	−1(b	NOUN
ejpam-6014	152	15	)	)	PUNCT
ejpam-6014	152	16	)	)	PUNCT
ejpam-6014	152	17	)	)	PUNCT
ejpam-6014	153	1	⊆	⊆	X
ejpam-6014	153	2	(	(	PUNCT
ejpam-6014	153	3	σ1	σ1	PROPN
ejpam-6014	153	4	,	,	PUNCT
ejpam-6014	153	5	σ2)θ	σ2)θ	ADJ
ejpam-6014	153	6	-	-	PUNCT
ejpam-6014	153	7	cl(f(f	cl(f(f	ADJ
ejpam-6014	153	8	−1(b	−1(b	NOUN
ejpam-6014	153	9	)	)	PUNCT
ejpam-6014	153	10	)	)	PUNCT
ejpam-6014	153	11	)	)	PUNCT
ejpam-6014	154	1	⊆	⊆	X
ejpam-6014	154	2	(	(	PUNCT
ejpam-6014	154	3	σ1	σ1	PROPN
ejpam-6014	154	4	,	,	PUNCT
ejpam-6014	154	5	σ2)θ	σ2)θ	NOUN
ejpam-6014	154	6	-	-	PUNCT
ejpam-6014	154	7	cl(b	cl(b	NOUN
ejpam-6014	154	8	)	)	PUNCT
ejpam-6014	154	9	and	and	CCONJ
ejpam-6014	154	10	hence	hence	ADV
ejpam-6014	154	11	(	(	PUNCT
ejpam-6014	154	12	τ1	τ1	NOUN
ejpam-6014	154	13	,	,	PUNCT
ejpam-6014	154	14	τ2)θ	τ2)θ	PROPN
ejpam-6014	154	15	-	-	PUNCT
ejpam-6014	154	16	cl(f	cl(f	NOUN
ejpam-6014	154	17	−1(b	−1(b	NOUN
ejpam-6014	154	18	)	)	PUNCT
ejpam-6014	154	19	)	)	PUNCT
ejpam-6014	154	20	⊆	⊆	NUM
ejpam-6014	154	21	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	154	22	,	,	PUNCT
ejpam-6014	154	23	σ2)θ	σ2)θ	NOUN
ejpam-6014	154	24	-	-	PUNCT
ejpam-6014	154	25	cl(b	cl(b	NOUN
ejpam-6014	154	26	)	)	PUNCT
ejpam-6014	154	27	)	)	PUNCT
ejpam-6014	154	28	.	.	PUNCT
ejpam-6014	155	1	(	(	PUNCT
ejpam-6014	155	2	3	3	X
ejpam-6014	155	3	)	)	PUNCT
ejpam-6014	155	4	⇒	⇒	NOUN
ejpam-6014	155	5	(	(	PUNCT
ejpam-6014	155	6	6	6	NUM
ejpam-6014	155	7	):	):	PUNCT
ejpam-6014	155	8	let	let	VERB
ejpam-6014	155	9	b	b	X
ejpam-6014	155	10	be	be	AUX
ejpam-6014	155	11	any	any	DET
ejpam-6014	155	12	subset	subset	NOUN
ejpam-6014	155	13	of	of	ADP
ejpam-6014	155	14	y	y	PROPN
ejpam-6014	155	15	.	.	PUNCT
ejpam-6014	156	1	since	since	SCONJ
ejpam-6014	156	2	(	(	PUNCT
ejpam-6014	156	3	σ1	σ1	PROPN
ejpam-6014	156	4	,	,	PUNCT
ejpam-6014	156	5	σ2)θ	σ2)θ	NOUN
ejpam-6014	156	6	-	-	PUNCT
ejpam-6014	156	7	cl(b	cl(b	NOUN
ejpam-6014	156	8	)	)	PUNCT
ejpam-6014	156	9	is	be	AUX
ejpam-6014	156	10	σ1σ2	σ1σ2	NOUN
ejpam-6014	156	11	-	-	ADJ
ejpam-6014	156	12	closed	closed	ADJ
ejpam-6014	156	13	in	in	ADP
ejpam-6014	156	14	y	y	PROPN
ejpam-6014	156	15	,	,	PUNCT
ejpam-6014	156	16	by	by	ADP
ejpam-6014	156	17	(	(	PUNCT
ejpam-6014	156	18	3	3	X
ejpam-6014	156	19	)	)	PUNCT
ejpam-6014	156	20	we	we	PRON
ejpam-6014	156	21	have	have	VERB
ejpam-6014	156	22	(	(	PUNCT
ejpam-6014	156	23	τ1	τ1	NOUN
ejpam-6014	156	24	,	,	PUNCT
ejpam-6014	156	25	τ2)θ	τ2)θ	PROPN
ejpam-6014	156	26	-	-	PUNCT
ejpam-6014	156	27	cl(f	cl(f	NOUN
ejpam-6014	156	28	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	156	29	-	-	PUNCT
ejpam-6014	156	30	int((σ1	int((σ1	NUM
ejpam-6014	156	31	,	,	PUNCT
ejpam-6014	156	32	σ2)θ	σ2)θ	ADJ
ejpam-6014	156	33	-	-	PUNCT
ejpam-6014	156	34	cl(b	cl(b	NOUN
ejpam-6014	156	35	)	)	PUNCT
ejpam-6014	156	36	)	)	PUNCT
ejpam-6014	156	37	)	)	PUNCT
ejpam-6014	156	38	)	)	PUNCT
ejpam-6014	157	1	⊆	⊆	NUM
ejpam-6014	157	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	157	3	-	-	PUNCT
ejpam-6014	157	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6014	157	5	-	-	PUNCT
ejpam-6014	157	6	int((σ1	int((σ1	PROPN
ejpam-6014	157	7	,	,	PUNCT
ejpam-6014	157	8	σ2)θ	σ2)θ	ADJ
ejpam-6014	157	9	-	-	PUNCT
ejpam-6014	157	10	cl(b	cl(b	NOUN
ejpam-6014	157	11	)	)	PUNCT
ejpam-6014	157	12	)	)	PUNCT
ejpam-6014	157	13	)	)	PUNCT
ejpam-6014	157	14	)	)	PUNCT
ejpam-6014	158	1	⊆	⊆	NUM
ejpam-6014	158	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	158	3	,	,	PUNCT
ejpam-6014	158	4	σ2)θ	σ2)θ	NOUN
ejpam-6014	158	5	-	-	PUNCT
ejpam-6014	158	6	cl(b	cl(b	NOUN
ejpam-6014	158	7	)	)	PUNCT
ejpam-6014	158	8	)	)	PUNCT
ejpam-6014	158	9	.	.	PUNCT
ejpam-6014	159	1	(	(	PUNCT
ejpam-6014	159	2	6	6	X
ejpam-6014	159	3	)	)	PUNCT
ejpam-6014	159	4	⇒	⇒	NOUN
ejpam-6014	159	5	(	(	PUNCT
ejpam-6014	159	6	7	7	NUM
ejpam-6014	159	7	):	):	PUNCT
ejpam-6014	159	8	this	this	PRON
ejpam-6014	159	9	is	be	AUX
ejpam-6014	159	10	obvious	obvious	ADJ
ejpam-6014	159	11	since	since	SCONJ
ejpam-6014	159	12	σ1σ2	σ1σ2	NOUN
ejpam-6014	159	13	-	-	NOUN
ejpam-6014	159	14	cl(v	cl(v	X
ejpam-6014	159	15	)	)	PUNCT
ejpam-6014	160	1	=	=	SYM
ejpam-6014	160	2	(	(	PUNCT
ejpam-6014	160	3	σ1	σ1	PROPN
ejpam-6014	160	4	,	,	PUNCT
ejpam-6014	160	5	σ2)θ	σ2)θ	NOUN
ejpam-6014	160	6	-	-	PUNCT
ejpam-6014	160	7	cl(v	cl(v	NOUN
ejpam-6014	160	8	)	)	PUNCT
ejpam-6014	160	9	for	for	ADP
ejpam-6014	160	10	every	every	DET
ejpam-6014	160	11	σ1σ2	σ1σ2	NOUN
ejpam-6014	160	12	-	-	ADJ
ejpam-6014	160	13	open	open	ADJ
ejpam-6014	160	14	set	set	NOUN
ejpam-6014	160	15	v	v	NOUN
ejpam-6014	160	16	of	of	ADP
ejpam-6014	160	17	y	y	PROPN
ejpam-6014	160	18	.	.	PUNCT
ejpam-6014	161	1	(	(	PUNCT
ejpam-6014	161	2	7	7	X
ejpam-6014	161	3	)	)	PUNCT
ejpam-6014	161	4	⇒	⇒	NOUN
ejpam-6014	161	5	(	(	PUNCT
ejpam-6014	161	6	8)	8)	NUM
ejpam-6014	161	7	:	:	PUNCT
ejpam-6014	161	8	let	let	VERB
ejpam-6014	161	9	k	k	X
ejpam-6014	161	10	be	be	AUX
ejpam-6014	161	11	any	any	DET
ejpam-6014	161	12	(	(	PUNCT
ejpam-6014	161	13	σ1	σ1	NOUN
ejpam-6014	161	14	,	,	PUNCT
ejpam-6014	161	15	σ2)r	σ2)r	NOUN
ejpam-6014	161	16	-	-	PUNCT
ejpam-6014	161	17	closed	close	VERB
ejpam-6014	161	18	set	set	NOUN
ejpam-6014	161	19	of	of	ADP
ejpam-6014	161	20	y	y	PROPN
ejpam-6014	161	21	.	.	PUNCT
ejpam-6014	162	1	thus	thus	ADV
ejpam-6014	162	2	by	by	ADP
ejpam-6014	162	3	(	(	PUNCT
ejpam-6014	162	4	7	7	NUM
ejpam-6014	162	5	)	)	PUNCT
ejpam-6014	162	6	,	,	PUNCT
ejpam-6014	162	7	we	we	PRON
ejpam-6014	162	8	have	have	VERB
ejpam-6014	162	9	(	(	PUNCT
ejpam-6014	162	10	τ1	τ1	NOUN
ejpam-6014	162	11	,	,	PUNCT
ejpam-6014	162	12	τ2)θ	τ2)θ	PROPN
ejpam-6014	162	13	-	-	PUNCT
ejpam-6014	162	14	cl(f	cl(f	NOUN
ejpam-6014	162	15	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	162	16	-	-	PUNCT
ejpam-6014	162	17	int(k	int(k	NUM
ejpam-6014	162	18	)	)	PUNCT
ejpam-6014	162	19	)	)	PUNCT
ejpam-6014	162	20	)	)	PUNCT
ejpam-6014	163	1	=	=	PRON
ejpam-6014	163	2	(	(	PUNCT
ejpam-6014	163	3	τ1	τ1	NOUN
ejpam-6014	163	4	,	,	PUNCT
ejpam-6014	163	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	163	6	-	-	PUNCT
ejpam-6014	163	7	cl(f	cl(f	NOUN
ejpam-6014	163	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	163	9	-	-	PUNCT
ejpam-6014	163	10	int(σ1σ2	int(σ1σ2	ADV
ejpam-6014	163	11	-	-	PUNCT
ejpam-6014	163	12	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6014	163	13	-	-	PUNCT
ejpam-6014	163	14	int(k	int(k	NOUN
ejpam-6014	163	15	)	)	PUNCT
ejpam-6014	163	16	)	)	PUNCT
ejpam-6014	163	17	)	)	PUNCT
ejpam-6014	163	18	)	)	PUNCT
ejpam-6014	163	19	)	)	PUNCT
ejpam-6014	164	1	⊆	⊆	NUM
ejpam-6014	164	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	164	3	-	-	PUNCT
ejpam-6014	164	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6014	164	5	-	-	PUNCT
ejpam-6014	164	6	int(k	int(k	NOUN
ejpam-6014	164	7	)	)	PUNCT
ejpam-6014	164	8	)	)	PUNCT
ejpam-6014	164	9	)	)	PUNCT
ejpam-6014	165	1	=	=	PUNCT
ejpam-6014	165	2	f−1(k	f−1(k	PROPN
ejpam-6014	165	3	)	)	PUNCT
ejpam-6014	165	4	.	.	PUNCT
ejpam-6014	166	1	(	(	PUNCT
ejpam-6014	166	2	8)	8)	NUM
ejpam-6014	166	3	⇒	⇒	NOUN
ejpam-6014	166	4	(	(	PUNCT
ejpam-6014	166	5	9	9	NUM
ejpam-6014	166	6	):	):	PUNCT
ejpam-6014	166	7	let	let	VERB
ejpam-6014	166	8	k	k	PRON
ejpam-6014	166	9	be	be	AUX
ejpam-6014	166	10	any	any	DET
ejpam-6014	166	11	σ1σ2	σ1σ2	NUM
ejpam-6014	166	12	-	-	PUNCT
ejpam-6014	166	13	closed	closed	ADJ
ejpam-6014	166	14	set	set	NOUN
ejpam-6014	166	15	of	of	ADP
ejpam-6014	166	16	y	y	PROPN
ejpam-6014	166	17	.	.	PUNCT
ejpam-6014	167	1	since	since	SCONJ
ejpam-6014	167	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	167	3	-	-	PUNCT
ejpam-6014	167	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6014	167	5	-	-	PUNCT
ejpam-6014	167	6	int(k	int(k	NOUN
ejpam-6014	167	7	)	)	PUNCT
ejpam-6014	167	8	)	)	PUNCT
ejpam-6014	167	9	is	be	AUX
ejpam-6014	167	10	(	(	PUNCT
ejpam-6014	167	11	σ1	σ1	PROPN
ejpam-6014	167	12	,	,	PUNCT
ejpam-6014	167	13	σ2)rclosed	σ2)rclose	VERB
ejpam-6014	167	14	in	in	ADP
ejpam-6014	167	15	y	y	PROPN
ejpam-6014	167	16	,	,	PUNCT
ejpam-6014	167	17	by	by	ADP
ejpam-6014	167	18	(	(	PUNCT
ejpam-6014	167	19	8)	8)	NUM
ejpam-6014	167	20	we	we	PRON
ejpam-6014	167	21	have	have	VERB
ejpam-6014	167	22	(	(	PUNCT
ejpam-6014	167	23	τ1	τ1	NOUN
ejpam-6014	167	24	,	,	PUNCT
ejpam-6014	167	25	τ2)θ	τ2)θ	PROPN
ejpam-6014	167	26	-	-	PUNCT
ejpam-6014	167	27	cl(f	cl(f	NOUN
ejpam-6014	167	28	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	167	29	-	-	PUNCT
ejpam-6014	167	30	int(k	int(k	NUM
ejpam-6014	167	31	)	)	PUNCT
ejpam-6014	167	32	)	)	PUNCT
ejpam-6014	167	33	)	)	PUNCT
ejpam-6014	168	1	=	=	PRON
ejpam-6014	168	2	(	(	PUNCT
ejpam-6014	168	3	τ1	τ1	NOUN
ejpam-6014	168	4	,	,	PUNCT
ejpam-6014	168	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	168	6	-	-	PUNCT
ejpam-6014	168	7	cl(f	cl(f	NOUN
ejpam-6014	168	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	168	9	-	-	PUNCT
ejpam-6014	168	10	int(σ1σ2	int(σ1σ2	ADV
ejpam-6014	168	11	-	-	PUNCT
ejpam-6014	168	12	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-6014	168	13	-	-	PUNCT
ejpam-6014	168	14	int(k	int(k	NOUN
ejpam-6014	168	15	)	)	PUNCT
ejpam-6014	168	16	)	)	PUNCT
ejpam-6014	168	17	)	)	PUNCT
ejpam-6014	168	18	)	)	PUNCT
ejpam-6014	168	19	)	)	PUNCT
ejpam-6014	169	1	⊆	⊆	NUM
ejpam-6014	169	2	f−1(k	f−1(k	NOUN
ejpam-6014	169	3	)	)	PUNCT
ejpam-6014	169	4	.	.	PUNCT
ejpam-6014	170	1	m.	m.	NOUN
ejpam-6014	170	2	thongmoon	thongmoon	PROPN
ejpam-6014	170	3	,	,	PUNCT
ejpam-6014	170	4	s.	s.	PROPN
ejpam-6014	170	5	sompong	sompong	PROPN
ejpam-6014	170	6	,	,	PUNCT
ejpam-6014	170	7	c.	c.	PROPN
ejpam-6014	170	8	boonpok	boonpok	PROPN
ejpam-6014	170	9	/	/	SYM
ejpam-6014	170	10	eur	eur	PROPN
ejpam-6014	170	11	.	.	PUNCT
ejpam-6014	171	1	j.	j.	PROPN
ejpam-6014	171	2	pure	pure	PROPN
ejpam-6014	171	3	appl	appl	PROPN
ejpam-6014	171	4	.	.	PROPN
ejpam-6014	171	5	math	math	PROPN
ejpam-6014	171	6	,	,	PUNCT
ejpam-6014	171	7	18	18	NUM
ejpam-6014	171	8	(	(	PUNCT
ejpam-6014	171	9	2	2	NUM
ejpam-6014	171	10	)	)	PUNCT
ejpam-6014	171	11	(	(	PUNCT
ejpam-6014	171	12	2025	2025	NUM
ejpam-6014	171	13	)	)	PUNCT
ejpam-6014	171	14	,	,	PUNCT
ejpam-6014	171	15	6014	6014	NUM
ejpam-6014	171	16	6	6	NUM
ejpam-6014	171	17	of	of	ADP
ejpam-6014	171	18	13	13	NUM
ejpam-6014	171	19	(	(	PUNCT
ejpam-6014	171	20	9	9	NUM
ejpam-6014	171	21	)	)	PUNCT
ejpam-6014	171	22	⇒	⇒	NOUN
ejpam-6014	171	23	(	(	PUNCT
ejpam-6014	171	24	4	4	NUM
ejpam-6014	171	25	):	):	PUNCT
ejpam-6014	171	26	let	let	VERB
ejpam-6014	171	27	v	v	PART
ejpam-6014	171	28	be	be	AUX
ejpam-6014	171	29	any	any	DET
ejpam-6014	171	30	σ1σ2	σ1σ2	NOUN
ejpam-6014	171	31	-	-	ADJ
ejpam-6014	171	32	open	open	ADJ
ejpam-6014	171	33	set	set	NOUN
ejpam-6014	171	34	of	of	ADP
ejpam-6014	171	35	y	y	PROPN
ejpam-6014	171	36	.	.	PUNCT
ejpam-6014	172	1	then	then	ADV
ejpam-6014	172	2	,	,	PUNCT
ejpam-6014	172	3	y	y	PROPN
ejpam-6014	172	4	−	−	PROPN
ejpam-6014	172	5	v	v	NOUN
ejpam-6014	172	6	is	be	AUX
ejpam-6014	172	7	σ1σ2	σ1σ2	NOUN
ejpam-6014	172	8	-	-	ADJ
ejpam-6014	172	9	closed	closed	ADJ
ejpam-6014	172	10	in	in	ADP
ejpam-6014	172	11	y	y	PROPN
ejpam-6014	172	12	and	and	CCONJ
ejpam-6014	172	13	by	by	ADP
ejpam-6014	172	14	(	(	PUNCT
ejpam-6014	172	15	9	9	NUM
ejpam-6014	172	16	)	)	PUNCT
ejpam-6014	172	17	,	,	PUNCT
ejpam-6014	172	18	(	(	PUNCT
ejpam-6014	172	19	τ1	τ1	NOUN
ejpam-6014	172	20	,	,	PUNCT
ejpam-6014	172	21	τ2)θ	τ2)θ	PROPN
ejpam-6014	172	22	-	-	PUNCT
ejpam-6014	172	23	cl(f	cl(f	NOUN
ejpam-6014	172	24	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	172	25	-	-	PUNCT
ejpam-6014	172	26	int(y	int(y	PROPN
ejpam-6014	172	27	−	−	PROPN
ejpam-6014	172	28	v	v	NOUN
ejpam-6014	172	29	)	)	PUNCT
ejpam-6014	172	30	)	)	PUNCT
ejpam-6014	172	31	)	)	PUNCT
ejpam-6014	173	1	⊆	⊆	X
ejpam-6014	173	2	f−1(y	f−1(y	NOUN
ejpam-6014	173	3	−	−	PROPN
ejpam-6014	173	4	v	v	NOUN
ejpam-6014	173	5	)	)	PUNCT
ejpam-6014	173	6	=	=	PUNCT
ejpam-6014	173	7	x	x	PUNCT
ejpam-6014	173	8	−	−	PROPN
ejpam-6014	173	9	f−1(v	f−1(v	PROPN
ejpam-6014	173	10	)	)	PUNCT
ejpam-6014	173	11	.	.	PUNCT
ejpam-6014	174	1	moreover	moreover	ADV
ejpam-6014	174	2	,	,	PUNCT
ejpam-6014	174	3	we	we	PRON
ejpam-6014	174	4	have	have	VERB
ejpam-6014	174	5	(	(	PUNCT
ejpam-6014	174	6	τ1	τ1	NOUN
ejpam-6014	174	7	,	,	PUNCT
ejpam-6014	174	8	τ2)θ	τ2)θ	PROPN
ejpam-6014	174	9	-	-	PUNCT
ejpam-6014	174	10	cl(f	cl(f	NOUN
ejpam-6014	174	11	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	174	12	-	-	PUNCT
ejpam-6014	174	13	int(y	int(y	PROPN
ejpam-6014	174	14	−	−	PROPN
ejpam-6014	174	15	v	v	NOUN
ejpam-6014	174	16	)	)	PUNCT
ejpam-6014	174	17	)	)	PUNCT
ejpam-6014	174	18	)	)	PUNCT
ejpam-6014	175	1	=	=	PRON
ejpam-6014	175	2	(	(	PUNCT
ejpam-6014	175	3	τ1	τ1	NOUN
ejpam-6014	175	4	,	,	PUNCT
ejpam-6014	175	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	175	6	-	-	PUNCT
ejpam-6014	175	7	cl(f	cl(f	NOUN
ejpam-6014	175	8	−1(y	−1(y	VERB
ejpam-6014	175	9	−	−	PUNCT
ejpam-6014	175	10	σ1σ2	σ1σ2	NOUN
ejpam-6014	175	11	-	-	NUM
ejpam-6014	175	12	cl(v	cl(v	NOUN
ejpam-6014	175	13	)	)	PUNCT
ejpam-6014	175	14	)	)	PUNCT
ejpam-6014	175	15	)	)	PUNCT
ejpam-6014	176	1	=	=	PRON
ejpam-6014	176	2	(	(	PUNCT
ejpam-6014	176	3	τ1	τ1	NOUN
ejpam-6014	176	4	,	,	PUNCT
ejpam-6014	176	5	τ2)θ	τ2)θ	ADJ
ejpam-6014	176	6	-	-	PUNCT
ejpam-6014	176	7	cl(x	cl(x	PUNCT
ejpam-6014	176	8	−	−	NOUN
ejpam-6014	176	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	176	10	-	-	PUNCT
ejpam-6014	176	11	cl(v	cl(v	NOUN
ejpam-6014	176	12	)	)	PUNCT
ejpam-6014	176	13	)	)	PUNCT
ejpam-6014	176	14	)	)	PUNCT
ejpam-6014	177	1	=	=	PUNCT
ejpam-6014	177	2	x	x	X
ejpam-6014	177	3	−	−	PROPN
ejpam-6014	177	4	(	(	PUNCT
ejpam-6014	177	5	τ1	τ1	NOUN
ejpam-6014	177	6	,	,	PUNCT
ejpam-6014	177	7	τ2)θ	τ2)θ	NOUN
ejpam-6014	177	8	-	-	PUNCT
ejpam-6014	177	9	int(f	int(f	VERB
ejpam-6014	177	10	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	177	11	-	-	PUNCT
ejpam-6014	177	12	cl(v	cl(v	NOUN
ejpam-6014	177	13	)	)	PUNCT
ejpam-6014	177	14	)	)	PUNCT
ejpam-6014	177	15	)	)	PUNCT
ejpam-6014	177	16	.	.	PUNCT
ejpam-6014	178	1	thus	thus	ADV
ejpam-6014	178	2	,	,	PUNCT
ejpam-6014	178	3	f−1(v	f−1(v	PROPN
ejpam-6014	178	4	)	)	PUNCT
ejpam-6014	179	1	⊆	⊆	NUM
ejpam-6014	179	2	(	(	PUNCT
ejpam-6014	179	3	τ1	τ1	NOUN
ejpam-6014	179	4	,	,	PUNCT
ejpam-6014	179	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	179	6	-	-	PUNCT
ejpam-6014	179	7	int(f	int(f	VERB
ejpam-6014	179	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	179	9	-	-	PUNCT
ejpam-6014	179	10	cl(v	cl(v	NOUN
ejpam-6014	179	11	)	)	PUNCT
ejpam-6014	179	12	)	)	PUNCT
ejpam-6014	179	13	)	)	PUNCT
ejpam-6014	179	14	.	.	PUNCT
ejpam-6014	180	1	theorem	theorem	VERB
ejpam-6014	180	2	4	4	NUM
ejpam-6014	180	3	.	.	PUNCT
ejpam-6014	181	1	a	a	DET
ejpam-6014	181	2	function	function	NOUN
ejpam-6014	181	3	f	f	NOUN
ejpam-6014	181	4	:	:	PUNCT
ejpam-6014	181	5	(	(	PUNCT
ejpam-6014	181	6	x	x	NOUN
ejpam-6014	181	7	,	,	PUNCT
ejpam-6014	181	8	τ1	τ1	NOUN
ejpam-6014	181	9	,	,	PUNCT
ejpam-6014	181	10	τ2	τ2	NOUN
ejpam-6014	181	11	)	)	PUNCT
ejpam-6014	181	12	→	→	SYM
ejpam-6014	181	13	(	(	PUNCT
ejpam-6014	181	14	y	y	PROPN
ejpam-6014	181	15	,	,	PUNCT
ejpam-6014	181	16	σ1	σ1	PROPN
ejpam-6014	181	17	,	,	PUNCT
ejpam-6014	181	18	σ2	σ2	PROPN
ejpam-6014	181	19	)	)	PUNCT
ejpam-6014	181	20	is	be	AUX
ejpam-6014	181	21	θ(τ1	θ(τ1	NOUN
ejpam-6014	181	22	,	,	PUNCT
ejpam-6014	181	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	181	24	if	if	SCONJ
ejpam-6014	181	25	and	and	CCONJ
ejpam-6014	181	26	only	only	ADV
ejpam-6014	181	27	if	if	SCONJ
ejpam-6014	181	28	(	(	PUNCT
ejpam-6014	181	29	τ1	τ1	NOUN
ejpam-6014	181	30	,	,	PUNCT
ejpam-6014	181	31	τ2)θ	τ2)θ	PROPN
ejpam-6014	181	32	-	-	PUNCT
ejpam-6014	181	33	cl(f	cl(f	PROPN
ejpam-6014	181	34	−1(v	−1(v	PROPN
ejpam-6014	181	35	)	)	PUNCT
ejpam-6014	181	36	)	)	PUNCT
ejpam-6014	182	1	⊆	⊆	NUM
ejpam-6014	182	2	f−1((σ1	f−1((σ1	NOUN
ejpam-6014	182	3	,	,	PUNCT
ejpam-6014	182	4	σ2)θ	σ2)θ	NOUN
ejpam-6014	182	5	-	-	PUNCT
ejpam-6014	182	6	cl(v	cl(v	NOUN
ejpam-6014	182	7	)	)	PUNCT
ejpam-6014	182	8	)	)	PUNCT
ejpam-6014	182	9	for	for	ADP
ejpam-6014	182	10	every	every	DET
ejpam-6014	182	11	σ1σ2	σ1σ2	NOUN
ejpam-6014	182	12	-	-	ADJ
ejpam-6014	182	13	open	open	ADJ
ejpam-6014	182	14	set	set	NOUN
ejpam-6014	182	15	v	v	NOUN
ejpam-6014	182	16	of	of	ADP
ejpam-6014	182	17	y	y	PROPN
ejpam-6014	182	18	.	.	PUNCT
ejpam-6014	183	1	proof	proof	NOUN
ejpam-6014	183	2	.	.	PUNCT
ejpam-6014	184	1	this	this	PRON
ejpam-6014	184	2	is	be	AUX
ejpam-6014	184	3	an	an	DET
ejpam-6014	184	4	immediate	immediate	ADJ
ejpam-6014	184	5	consequence	consequence	NOUN
ejpam-6014	184	6	of	of	ADP
ejpam-6014	184	7	theorem	theorem	ADJ
ejpam-6014	184	8	3	3	NUM
ejpam-6014	184	9	.	.	PUNCT
ejpam-6014	184	10	theorem	theorem	NOUN
ejpam-6014	184	11	5	5	NUM
ejpam-6014	184	12	.	.	X
ejpam-6014	184	13	for	for	ADP
ejpam-6014	184	14	a	a	DET
ejpam-6014	184	15	function	function	NOUN
ejpam-6014	184	16	(	(	PUNCT
ejpam-6014	184	17	x	x	NOUN
ejpam-6014	184	18	,	,	PUNCT
ejpam-6014	184	19	τ1	τ1	NOUN
ejpam-6014	184	20	,	,	PUNCT
ejpam-6014	184	21	τ2	τ2	NOUN
ejpam-6014	184	22	)	)	PUNCT
ejpam-6014	184	23	→	→	SYM
ejpam-6014	184	24	(	(	PUNCT
ejpam-6014	184	25	y	y	PROPN
ejpam-6014	184	26	,	,	PUNCT
ejpam-6014	184	27	σ1	σ1	PROPN
ejpam-6014	184	28	,	,	PUNCT
ejpam-6014	184	29	σ2	σ2	NOUN
ejpam-6014	184	30	)	)	PUNCT
ejpam-6014	184	31	,	,	PUNCT
ejpam-6014	184	32	the	the	DET
ejpam-6014	184	33	following	follow	VERB
ejpam-6014	184	34	properties	property	NOUN
ejpam-6014	184	35	are	be	AUX
ejpam-6014	184	36	equivalent	equivalent	ADJ
ejpam-6014	184	37	:	:	PUNCT
ejpam-6014	184	38	(	(	PUNCT
ejpam-6014	184	39	1	1	X
ejpam-6014	184	40	)	)	PUNCT
ejpam-6014	184	41	f	f	PROPN
ejpam-6014	184	42	is	be	AUX
ejpam-6014	184	43	θ(τ1	θ(τ1	NOUN
ejpam-6014	184	44	,	,	PUNCT
ejpam-6014	184	45	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	184	46	;	;	PUNCT
ejpam-6014	184	47	(	(	PUNCT
ejpam-6014	184	48	2	2	X
ejpam-6014	184	49	)	)	PUNCT
ejpam-6014	184	50	(	(	PUNCT
ejpam-6014	184	51	τ1	τ1	NOUN
ejpam-6014	184	52	,	,	PUNCT
ejpam-6014	184	53	τ2)θ	τ2)θ	PROPN
ejpam-6014	184	54	-	-	PUNCT
ejpam-6014	184	55	cl(f	cl(f	NOUN
ejpam-6014	184	56	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	184	57	-	-	PUNCT
ejpam-6014	184	58	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	184	59	-	-	PUNCT
ejpam-6014	184	60	cl(v	cl(v	NOUN
ejpam-6014	184	61	)	)	PUNCT
ejpam-6014	184	62	)	)	PUNCT
ejpam-6014	184	63	)	)	PUNCT
ejpam-6014	184	64	)	)	PUNCT
ejpam-6014	185	1	⊆	⊆	NUM
ejpam-6014	185	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	185	3	-	-	PUNCT
ejpam-6014	185	4	cl(v	cl(v	NOUN
ejpam-6014	185	5	)	)	PUNCT
ejpam-6014	185	6	)	)	PUNCT
ejpam-6014	185	7	for	for	ADP
ejpam-6014	185	8	every	every	DET
ejpam-6014	185	9	(	(	PUNCT
ejpam-6014	185	10	σ1	σ1	PROPN
ejpam-6014	185	11	,	,	PUNCT
ejpam-6014	185	12	σ2)β	σ2)β	NOUN
ejpam-6014	185	13	-	-	PUNCT
ejpam-6014	185	14	open	open	NOUN
ejpam-6014	185	15	set	set	NOUN
ejpam-6014	185	16	v	v	NOUN
ejpam-6014	185	17	of	of	ADP
ejpam-6014	185	18	y	y	PROPN
ejpam-6014	185	19	;	;	PUNCT
ejpam-6014	185	20	(	(	PUNCT
ejpam-6014	185	21	3	3	X
ejpam-6014	185	22	)	)	PUNCT
ejpam-6014	185	23	(	(	PUNCT
ejpam-6014	185	24	τ1	τ1	NOUN
ejpam-6014	185	25	,	,	PUNCT
ejpam-6014	185	26	τ2)θ	τ2)θ	PROPN
ejpam-6014	185	27	-	-	PUNCT
ejpam-6014	185	28	cl(f	cl(f	NOUN
ejpam-6014	185	29	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	185	30	-	-	PUNCT
ejpam-6014	185	31	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	185	32	-	-	PUNCT
ejpam-6014	185	33	cl(v	cl(v	NOUN
ejpam-6014	185	34	)	)	PUNCT
ejpam-6014	185	35	)	)	PUNCT
ejpam-6014	185	36	)	)	PUNCT
ejpam-6014	185	37	)	)	PUNCT
ejpam-6014	186	1	⊆	⊆	NUM
ejpam-6014	186	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	186	3	-	-	PUNCT
ejpam-6014	186	4	cl(v	cl(v	NOUN
ejpam-6014	186	5	)	)	PUNCT
ejpam-6014	186	6	)	)	PUNCT
ejpam-6014	186	7	for	for	ADP
ejpam-6014	186	8	every	every	DET
ejpam-6014	186	9	(	(	PUNCT
ejpam-6014	186	10	σ1	σ1	PROPN
ejpam-6014	186	11	,	,	PUNCT
ejpam-6014	186	12	σ2)s	σ2)s	NOUN
ejpam-6014	186	13	-	-	PUNCT
ejpam-6014	186	14	open	open	NOUN
ejpam-6014	186	15	set	set	NOUN
ejpam-6014	186	16	v	v	NOUN
ejpam-6014	186	17	of	of	ADP
ejpam-6014	186	18	y	y	PROPN
ejpam-6014	186	19	.	.	PUNCT
ejpam-6014	187	1	proof	proof	NOUN
ejpam-6014	187	2	.	.	PUNCT
ejpam-6014	188	1	(	(	PUNCT
ejpam-6014	188	2	1	1	X
ejpam-6014	188	3	)	)	PUNCT
ejpam-6014	188	4	⇒	⇒	NOUN
ejpam-6014	188	5	(	(	PUNCT
ejpam-6014	188	6	2	2	NUM
ejpam-6014	188	7	):	):	PUNCT
ejpam-6014	188	8	let	let	VERB
ejpam-6014	188	9	v	v	PART
ejpam-6014	188	10	be	be	AUX
ejpam-6014	188	11	any	any	DET
ejpam-6014	188	12	(	(	PUNCT
ejpam-6014	188	13	σ1	σ1	PROPN
ejpam-6014	188	14	,	,	PUNCT
ejpam-6014	188	15	σ2)β	σ2)β	NOUN
ejpam-6014	188	16	-	-	PUNCT
ejpam-6014	188	17	open	open	ADJ
ejpam-6014	188	18	set	set	NOUN
ejpam-6014	188	19	of	of	ADP
ejpam-6014	188	20	y	y	PROPN
ejpam-6014	188	21	.	.	PUNCT
ejpam-6014	189	1	then	then	ADV
ejpam-6014	189	2	,	,	PUNCT
ejpam-6014	189	3	v	v	ADP
ejpam-6014	189	4	⊆	⊆	NUM
ejpam-6014	189	5	σ1σ2	σ1σ2	NOUN
ejpam-6014	189	6	-	-	PUNCT
ejpam-6014	189	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6014	189	8	-	-	PUNCT
ejpam-6014	189	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	189	10	-	-	PUNCT
ejpam-6014	189	11	cl(v	cl(v	NOUN
ejpam-6014	189	12	)	)	PUNCT
ejpam-6014	189	13	)	)	PUNCT
ejpam-6014	189	14	)	)	PUNCT
ejpam-6014	189	15	and	and	CCONJ
ejpam-6014	189	16	σ1σ2	σ1σ2	NOUN
ejpam-6014	189	17	-	-	NUM
ejpam-6014	189	18	cl(v	cl(v	X
ejpam-6014	189	19	)	)	PUNCT
ejpam-6014	190	1	=	=	SYM
ejpam-6014	190	2	σ1σ2	σ1σ2	X
ejpam-6014	190	3	-	-	PUNCT
ejpam-6014	190	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6014	190	5	-	-	PUNCT
ejpam-6014	190	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	190	7	-	-	PUNCT
ejpam-6014	190	8	cl(v	cl(v	NOUN
ejpam-6014	190	9	)	)	PUNCT
ejpam-6014	190	10	)	)	PUNCT
ejpam-6014	190	11	)	)	PUNCT
ejpam-6014	190	12	.	.	PUNCT
ejpam-6014	191	1	since	since	SCONJ
ejpam-6014	191	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	191	3	-	-	PUNCT
ejpam-6014	191	4	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-6014	191	5	-	-	PUNCT
ejpam-6014	191	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	191	7	-	-	PUNCT
ejpam-6014	191	8	cl(v	cl(v	NOUN
ejpam-6014	191	9	)	)	PUNCT
ejpam-6014	191	10	)	)	PUNCT
ejpam-6014	191	11	)	)	PUNCT
ejpam-6014	191	12	is	be	AUX
ejpam-6014	191	13	(	(	PUNCT
ejpam-6014	191	14	σ1	σ1	NOUN
ejpam-6014	191	15	,	,	PUNCT
ejpam-6014	191	16	σ2)r	σ2)r	NOUN
ejpam-6014	191	17	-	-	PUNCT
ejpam-6014	191	18	closed	closed	ADJ
ejpam-6014	191	19	in	in	ADP
ejpam-6014	191	20	y	y	PROPN
ejpam-6014	191	21	,	,	PUNCT
ejpam-6014	191	22	by	by	ADP
ejpam-6014	191	23	theorem	theorem	NOUN
ejpam-6014	191	24	3	3	NUM
ejpam-6014	191	25	we	we	PRON
ejpam-6014	191	26	have	have	VERB
ejpam-6014	191	27	(	(	PUNCT
ejpam-6014	192	1	τ1	τ1	NOUN
ejpam-6014	192	2	,	,	PUNCT
ejpam-6014	192	3	τ2)θ	τ2)θ	PROPN
ejpam-6014	192	4	-	-	PUNCT
ejpam-6014	192	5	cl(f	cl(f	NOUN
ejpam-6014	192	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	192	7	-	-	PUNCT
ejpam-6014	192	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	192	9	-	-	PUNCT
ejpam-6014	192	10	cl(v	cl(v	NOUN
ejpam-6014	192	11	)	)	PUNCT
ejpam-6014	192	12	)	)	PUNCT
ejpam-6014	192	13	)	)	PUNCT
ejpam-6014	192	14	)	)	PUNCT
ejpam-6014	193	1	⊆	⊆	NUM
ejpam-6014	193	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	193	3	-	-	PUNCT
ejpam-6014	193	4	cl(v	cl(v	NOUN
ejpam-6014	193	5	)	)	PUNCT
ejpam-6014	193	6	)	)	PUNCT
ejpam-6014	193	7	.	.	PUNCT
ejpam-6014	194	1	(	(	PUNCT
ejpam-6014	194	2	2	2	X
ejpam-6014	194	3	)	)	PUNCT
ejpam-6014	194	4	⇒	⇒	NOUN
ejpam-6014	194	5	(	(	PUNCT
ejpam-6014	194	6	3	3	NUM
ejpam-6014	194	7	):	):	PUNCT
ejpam-6014	194	8	the	the	DET
ejpam-6014	194	9	proof	proof	NOUN
ejpam-6014	194	10	is	be	AUX
ejpam-6014	194	11	obvious	obvious	ADJ
ejpam-6014	194	12	.	.	PUNCT
ejpam-6014	195	1	(	(	PUNCT
ejpam-6014	195	2	3	3	X
ejpam-6014	195	3	)	)	PUNCT
ejpam-6014	195	4	⇒	⇒	NOUN
ejpam-6014	195	5	(	(	PUNCT
ejpam-6014	195	6	1	1	NUM
ejpam-6014	195	7	):	):	PUNCT
ejpam-6014	195	8	let	let	VERB
ejpam-6014	195	9	v	v	PART
ejpam-6014	195	10	be	be	AUX
ejpam-6014	195	11	any	any	DET
ejpam-6014	195	12	(	(	PUNCT
ejpam-6014	195	13	σ1	σ1	PROPN
ejpam-6014	195	14	,	,	PUNCT
ejpam-6014	195	15	σ2)β	σ2)β	NOUN
ejpam-6014	195	16	-	-	PUNCT
ejpam-6014	195	17	open	open	ADJ
ejpam-6014	195	18	set	set	NOUN
ejpam-6014	195	19	of	of	ADP
ejpam-6014	195	20	y	y	PROPN
ejpam-6014	195	21	.	.	PUNCT
ejpam-6014	196	1	since	since	SCONJ
ejpam-6014	196	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	196	3	-	-	NOUN
ejpam-6014	196	4	cl(v	cl(v	NOUN
ejpam-6014	196	5	)	)	PUNCT
ejpam-6014	196	6	is	be	AUX
ejpam-6014	196	7	(	(	PUNCT
ejpam-6014	196	8	σ1	σ1	PROPN
ejpam-6014	196	9	,	,	PUNCT
ejpam-6014	196	10	σ2)s	σ2)s	NOUN
ejpam-6014	196	11	-	-	PUNCT
ejpam-6014	196	12	open	open	ADJ
ejpam-6014	196	13	in	in	ADP
ejpam-6014	196	14	y	y	PROPN
ejpam-6014	196	15	,	,	PUNCT
ejpam-6014	196	16	by	by	ADP
ejpam-6014	196	17	(	(	PUNCT
ejpam-6014	196	18	3	3	X
ejpam-6014	196	19	)	)	PUNCT
ejpam-6014	196	20	we	we	PRON
ejpam-6014	196	21	have	have	VERB
ejpam-6014	196	22	(	(	PUNCT
ejpam-6014	197	1	τ1	τ1	NOUN
ejpam-6014	197	2	,	,	PUNCT
ejpam-6014	197	3	τ2)θ	τ2)θ	PROPN
ejpam-6014	197	4	-	-	PUNCT
ejpam-6014	197	5	cl(f	cl(f	NOUN
ejpam-6014	197	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	197	7	-	-	PUNCT
ejpam-6014	197	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	197	9	-	-	PUNCT
ejpam-6014	197	10	cl(v	cl(v	NOUN
ejpam-6014	197	11	)	)	PUNCT
ejpam-6014	197	12	)	)	PUNCT
ejpam-6014	197	13	)	)	PUNCT
ejpam-6014	197	14	)	)	PUNCT
ejpam-6014	198	1	⊆	⊆	NUM
ejpam-6014	198	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	198	3	-	-	PUNCT
ejpam-6014	198	4	cl(v	cl(v	NOUN
ejpam-6014	198	5	)	)	PUNCT
ejpam-6014	198	6	)	)	PUNCT
ejpam-6014	198	7	.	.	PUNCT
ejpam-6014	199	1	by	by	ADP
ejpam-6014	199	2	theorem	theorem	NOUN
ejpam-6014	199	3	3	3	NUM
ejpam-6014	199	4	,	,	PUNCT
ejpam-6014	199	5	f	f	PROPN
ejpam-6014	199	6	is	be	AUX
ejpam-6014	199	7	θ(τ1	θ(τ1	NOUN
ejpam-6014	199	8	,	,	PUNCT
ejpam-6014	199	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	199	10	.	.	PUNCT
ejpam-6014	200	1	theorem	theorem	VERB
ejpam-6014	200	2	6	6	NUM
ejpam-6014	200	3	.	.	PUNCT
ejpam-6014	200	4	for	for	ADP
ejpam-6014	200	5	a	a	DET
ejpam-6014	200	6	function	function	NOUN
ejpam-6014	200	7	(	(	PUNCT
ejpam-6014	200	8	x	x	NOUN
ejpam-6014	200	9	,	,	PUNCT
ejpam-6014	200	10	τ1	τ1	NOUN
ejpam-6014	200	11	,	,	PUNCT
ejpam-6014	200	12	τ2	τ2	NOUN
ejpam-6014	200	13	)	)	PUNCT
ejpam-6014	200	14	→	→	SYM
ejpam-6014	200	15	(	(	PUNCT
ejpam-6014	200	16	y	y	PROPN
ejpam-6014	200	17	,	,	PUNCT
ejpam-6014	200	18	σ1	σ1	PROPN
ejpam-6014	200	19	,	,	PUNCT
ejpam-6014	200	20	σ2	σ2	NOUN
ejpam-6014	200	21	)	)	PUNCT
ejpam-6014	200	22	,	,	PUNCT
ejpam-6014	200	23	the	the	DET
ejpam-6014	200	24	following	follow	VERB
ejpam-6014	200	25	properties	property	NOUN
ejpam-6014	200	26	are	be	AUX
ejpam-6014	200	27	equivalent	equivalent	ADJ
ejpam-6014	200	28	:	:	PUNCT
ejpam-6014	200	29	(	(	PUNCT
ejpam-6014	200	30	1	1	X
ejpam-6014	200	31	)	)	PUNCT
ejpam-6014	200	32	f	f	PROPN
ejpam-6014	200	33	is	be	AUX
ejpam-6014	200	34	θ(τ1	θ(τ1	NOUN
ejpam-6014	200	35	,	,	PUNCT
ejpam-6014	200	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	200	37	;	;	PUNCT
ejpam-6014	200	38	(	(	PUNCT
ejpam-6014	200	39	2	2	X
ejpam-6014	200	40	)	)	PUNCT
ejpam-6014	200	41	(	(	PUNCT
ejpam-6014	200	42	τ1	τ1	NOUN
ejpam-6014	200	43	,	,	PUNCT
ejpam-6014	200	44	τ2)θ	τ2)θ	PROPN
ejpam-6014	200	45	-	-	PUNCT
ejpam-6014	200	46	cl(f	cl(f	NOUN
ejpam-6014	200	47	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	200	48	-	-	PUNCT
ejpam-6014	200	49	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	200	50	-	-	PUNCT
ejpam-6014	200	51	cl(v	cl(v	NOUN
ejpam-6014	200	52	)	)	PUNCT
ejpam-6014	200	53	)	)	PUNCT
ejpam-6014	200	54	)	)	PUNCT
ejpam-6014	200	55	)	)	PUNCT
ejpam-6014	201	1	⊆	⊆	NUM
ejpam-6014	201	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	201	3	-	-	PUNCT
ejpam-6014	201	4	cl(v	cl(v	NOUN
ejpam-6014	201	5	)	)	PUNCT
ejpam-6014	201	6	)	)	PUNCT
ejpam-6014	201	7	for	for	ADP
ejpam-6014	201	8	every	every	DET
ejpam-6014	201	9	(	(	PUNCT
ejpam-6014	201	10	σ1	σ1	PROPN
ejpam-6014	201	11	,	,	PUNCT
ejpam-6014	201	12	σ2)p	σ2)p	NOUN
ejpam-6014	201	13	-	-	PUNCT
ejpam-6014	201	14	open	open	NOUN
ejpam-6014	201	15	set	set	NOUN
ejpam-6014	201	16	v	v	NOUN
ejpam-6014	201	17	of	of	ADP
ejpam-6014	201	18	y	y	PROPN
ejpam-6014	201	19	;	;	PUNCT
ejpam-6014	201	20	m.	m.	NOUN
ejpam-6014	201	21	thongmoon	thongmoon	NOUN
ejpam-6014	201	22	,	,	PUNCT
ejpam-6014	201	23	s.	s.	PROPN
ejpam-6014	201	24	sompong	sompong	PROPN
ejpam-6014	201	25	,	,	PUNCT
ejpam-6014	201	26	c.	c.	PROPN
ejpam-6014	201	27	boonpok	boonpok	PROPN
ejpam-6014	201	28	/	/	SYM
ejpam-6014	201	29	eur	eur	PROPN
ejpam-6014	201	30	.	.	PUNCT
ejpam-6014	202	1	j.	j.	PROPN
ejpam-6014	202	2	pure	pure	PROPN
ejpam-6014	202	3	appl	appl	PROPN
ejpam-6014	202	4	.	.	PROPN
ejpam-6014	202	5	math	math	PROPN
ejpam-6014	202	6	,	,	PUNCT
ejpam-6014	202	7	18	18	NUM
ejpam-6014	202	8	(	(	PUNCT
ejpam-6014	202	9	2	2	NUM
ejpam-6014	202	10	)	)	PUNCT
ejpam-6014	202	11	(	(	PUNCT
ejpam-6014	202	12	2025	2025	NUM
ejpam-6014	202	13	)	)	PUNCT
ejpam-6014	202	14	,	,	PUNCT
ejpam-6014	202	15	6014	6014	NUM
ejpam-6014	202	16	7	7	NUM
ejpam-6014	202	17	of	of	ADP
ejpam-6014	202	18	13	13	NUM
ejpam-6014	202	19	(	(	PUNCT
ejpam-6014	202	20	3	3	NUM
ejpam-6014	202	21	)	)	PUNCT
ejpam-6014	202	22	(	(	PUNCT
ejpam-6014	202	23	τ1	τ1	NOUN
ejpam-6014	202	24	,	,	PUNCT
ejpam-6014	202	25	τ2)θ	τ2)θ	PROPN
ejpam-6014	202	26	-	-	PUNCT
ejpam-6014	202	27	cl(f	cl(f	PROPN
ejpam-6014	202	28	−1(v	−1(v	PROPN
ejpam-6014	202	29	)	)	PUNCT
ejpam-6014	202	30	)	)	PUNCT
ejpam-6014	203	1	⊆	⊆	NUM
ejpam-6014	203	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	203	3	-	-	PUNCT
ejpam-6014	203	4	cl(v	cl(v	NOUN
ejpam-6014	203	5	)	)	PUNCT
ejpam-6014	203	6	)	)	PUNCT
ejpam-6014	203	7	for	for	ADP
ejpam-6014	203	8	every	every	DET
ejpam-6014	203	9	(	(	PUNCT
ejpam-6014	203	10	σ1	σ1	PROPN
ejpam-6014	203	11	,	,	PUNCT
ejpam-6014	203	12	σ2)p	σ2)p	NOUN
ejpam-6014	203	13	-	-	PUNCT
ejpam-6014	203	14	open	open	NOUN
ejpam-6014	203	15	set	set	NOUN
ejpam-6014	203	16	v	v	NOUN
ejpam-6014	203	17	of	of	ADP
ejpam-6014	203	18	y	y	PROPN
ejpam-6014	203	19	;	;	PUNCT
ejpam-6014	203	20	(	(	PUNCT
ejpam-6014	203	21	4	4	X
ejpam-6014	203	22	)	)	PUNCT
ejpam-6014	203	23	f−1(v	f−1(v	NOUN
ejpam-6014	203	24	)	)	PUNCT
ejpam-6014	204	1	⊆	⊆	NUM
ejpam-6014	204	2	(	(	PUNCT
ejpam-6014	204	3	τ1	τ1	NOUN
ejpam-6014	204	4	,	,	PUNCT
ejpam-6014	204	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	204	6	-	-	PUNCT
ejpam-6014	204	7	int(f	int(f	VERB
ejpam-6014	204	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	204	9	-	-	PUNCT
ejpam-6014	204	10	cl(v	cl(v	NOUN
ejpam-6014	204	11	)	)	PUNCT
ejpam-6014	204	12	)	)	PUNCT
ejpam-6014	204	13	)	)	PUNCT
ejpam-6014	205	1	for	for	ADP
ejpam-6014	205	2	every	every	DET
ejpam-6014	205	3	(	(	PUNCT
ejpam-6014	205	4	σ1	σ1	PROPN
ejpam-6014	205	5	,	,	PUNCT
ejpam-6014	205	6	σ2)p	σ2)p	NOUN
ejpam-6014	205	7	-	-	PUNCT
ejpam-6014	205	8	open	open	NOUN
ejpam-6014	205	9	set	set	NOUN
ejpam-6014	205	10	v	v	NOUN
ejpam-6014	205	11	of	of	ADP
ejpam-6014	205	12	y	y	PROPN
ejpam-6014	205	13	.	.	PUNCT
ejpam-6014	206	1	proof	proof	NOUN
ejpam-6014	206	2	.	.	PUNCT
ejpam-6014	207	1	(	(	PUNCT
ejpam-6014	207	2	1	1	X
ejpam-6014	207	3	)	)	PUNCT
ejpam-6014	207	4	⇒	⇒	NOUN
ejpam-6014	207	5	(	(	PUNCT
ejpam-6014	207	6	2	2	NUM
ejpam-6014	207	7	):	):	PUNCT
ejpam-6014	207	8	it	it	PRON
ejpam-6014	207	9	follows	follow	VERB
ejpam-6014	207	10	from	from	ADP
ejpam-6014	207	11	theorem	theorem	ADJ
ejpam-6014	207	12	5	5	NUM
ejpam-6014	207	13	.	.	PUNCT
ejpam-6014	207	14	(	(	PUNCT
ejpam-6014	207	15	2	2	X
ejpam-6014	207	16	)	)	PUNCT
ejpam-6014	207	17	⇒	⇒	NOUN
ejpam-6014	207	18	(	(	PUNCT
ejpam-6014	207	19	3	3	NUM
ejpam-6014	207	20	):	):	PUNCT
ejpam-6014	207	21	let	let	VERB
ejpam-6014	207	22	v	v	PART
ejpam-6014	207	23	be	be	AUX
ejpam-6014	207	24	any	any	DET
ejpam-6014	207	25	(	(	PUNCT
ejpam-6014	207	26	σ1	σ1	PROPN
ejpam-6014	207	27	,	,	PUNCT
ejpam-6014	207	28	σ2)p	σ2)p	NOUN
ejpam-6014	207	29	-	-	PUNCT
ejpam-6014	207	30	open	open	ADJ
ejpam-6014	207	31	set	set	NOUN
ejpam-6014	207	32	of	of	ADP
ejpam-6014	207	33	y	y	PROPN
ejpam-6014	207	34	.	.	PUNCT
ejpam-6014	208	1	then	then	ADV
ejpam-6014	208	2	by	by	ADP
ejpam-6014	208	3	(	(	PUNCT
ejpam-6014	208	4	2	2	NUM
ejpam-6014	208	5	)	)	PUNCT
ejpam-6014	208	6	,	,	PUNCT
ejpam-6014	208	7	we	we	PRON
ejpam-6014	208	8	have	have	VERB
ejpam-6014	208	9	(	(	PUNCT
ejpam-6014	208	10	τ1	τ1	NOUN
ejpam-6014	208	11	,	,	PUNCT
ejpam-6014	208	12	τ2)θ	τ2)θ	PROPN
ejpam-6014	208	13	-	-	PUNCT
ejpam-6014	208	14	cl(f	cl(f	PROPN
ejpam-6014	208	15	−1(v	−1(v	PROPN
ejpam-6014	208	16	)	)	PUNCT
ejpam-6014	208	17	)	)	PUNCT
ejpam-6014	209	1	⊆	⊆	NUM
ejpam-6014	209	2	(	(	PUNCT
ejpam-6014	209	3	τ1	τ1	NOUN
ejpam-6014	209	4	,	,	PUNCT
ejpam-6014	209	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	209	6	-	-	PUNCT
ejpam-6014	209	7	cl(f	cl(f	NOUN
ejpam-6014	209	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	209	9	-	-	PUNCT
ejpam-6014	209	10	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	209	11	-	-	PUNCT
ejpam-6014	209	12	cl(v	cl(v	NOUN
ejpam-6014	209	13	)	)	PUNCT
ejpam-6014	209	14	)	)	PUNCT
ejpam-6014	209	15	)	)	PUNCT
ejpam-6014	209	16	)	)	PUNCT
ejpam-6014	209	17	⊆	⊆	NUM
ejpam-6014	209	18	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	209	19	-	-	PUNCT
ejpam-6014	209	20	cl(v	cl(v	NOUN
ejpam-6014	209	21	)	)	PUNCT
ejpam-6014	209	22	)	)	PUNCT
ejpam-6014	209	23	.	.	PUNCT
ejpam-6014	210	1	(	(	PUNCT
ejpam-6014	210	2	3	3	X
ejpam-6014	210	3	)	)	PUNCT
ejpam-6014	210	4	⇒	⇒	NOUN
ejpam-6014	210	5	(	(	PUNCT
ejpam-6014	210	6	4	4	NUM
ejpam-6014	210	7	):	):	PUNCT
ejpam-6014	210	8	let	let	VERB
ejpam-6014	210	9	v	v	PART
ejpam-6014	210	10	be	be	AUX
ejpam-6014	210	11	any	any	DET
ejpam-6014	210	12	(	(	PUNCT
ejpam-6014	210	13	σ1	σ1	PROPN
ejpam-6014	210	14	,	,	PUNCT
ejpam-6014	210	15	σ2)p	σ2)p	NOUN
ejpam-6014	210	16	-	-	PUNCT
ejpam-6014	210	17	open	open	ADJ
ejpam-6014	210	18	set	set	NOUN
ejpam-6014	210	19	of	of	ADP
ejpam-6014	210	20	y	y	PROPN
ejpam-6014	210	21	.	.	PUNCT
ejpam-6014	211	1	by	by	ADP
ejpam-6014	211	2	(	(	PUNCT
ejpam-6014	211	3	3	3	NUM
ejpam-6014	211	4	)	)	PUNCT
ejpam-6014	211	5	,	,	PUNCT
ejpam-6014	211	6	we	we	PRON
ejpam-6014	211	7	have	have	VERB
ejpam-6014	211	8	x	x	X
ejpam-6014	211	9	−	−	PROPN
ejpam-6014	211	10	(	(	PUNCT
ejpam-6014	211	11	τ1	τ1	NOUN
ejpam-6014	211	12	,	,	PUNCT
ejpam-6014	211	13	τ2)θ	τ2)θ	NOUN
ejpam-6014	211	14	-	-	PUNCT
ejpam-6014	211	15	int(f	int(f	VERB
ejpam-6014	211	16	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	211	17	-	-	PUNCT
ejpam-6014	211	18	cl(v	cl(v	NOUN
ejpam-6014	211	19	)	)	PUNCT
ejpam-6014	211	20	)	)	PUNCT
ejpam-6014	211	21	)	)	PUNCT
ejpam-6014	212	1	=	=	PRON
ejpam-6014	212	2	(	(	PUNCT
ejpam-6014	212	3	τ1	τ1	NOUN
ejpam-6014	212	4	,	,	PUNCT
ejpam-6014	212	5	τ2)θ	τ2)θ	ADJ
ejpam-6014	212	6	-	-	PUNCT
ejpam-6014	212	7	cl(x	cl(x	PUNCT
ejpam-6014	212	8	−	−	NOUN
ejpam-6014	212	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	212	10	-	-	PUNCT
ejpam-6014	212	11	cl(v	cl(v	NOUN
ejpam-6014	212	12	)	)	PUNCT
ejpam-6014	212	13	)	)	PUNCT
ejpam-6014	212	14	)	)	PUNCT
ejpam-6014	213	1	=	=	PRON
ejpam-6014	213	2	(	(	PUNCT
ejpam-6014	213	3	τ1	τ1	NOUN
ejpam-6014	213	4	,	,	PUNCT
ejpam-6014	213	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	213	6	-	-	PUNCT
ejpam-6014	213	7	cl(f	cl(f	NOUN
ejpam-6014	213	8	−1(y	−1(y	VERB
ejpam-6014	213	9	−	−	PUNCT
ejpam-6014	213	10	σ1σ2	σ1σ2	NOUN
ejpam-6014	213	11	-	-	NUM
ejpam-6014	213	12	cl(v	cl(v	NOUN
ejpam-6014	213	13	)	)	PUNCT
ejpam-6014	213	14	)	)	PUNCT
ejpam-6014	213	15	)	)	PUNCT
ejpam-6014	214	1	⊆	⊆	NUM
ejpam-6014	214	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	214	3	-	-	PUNCT
ejpam-6014	214	4	cl(y	cl(y	NOUN
ejpam-6014	214	5	−	−	NOUN
ejpam-6014	214	6	σ1σ2	σ1σ2	NOUN
ejpam-6014	214	7	-	-	NUM
ejpam-6014	214	8	cl(v	cl(v	NOUN
ejpam-6014	214	9	)	)	PUNCT
ejpam-6014	214	10	)	)	PUNCT
ejpam-6014	214	11	)	)	PUNCT
ejpam-6014	215	1	=	=	PUNCT
ejpam-6014	215	2	x	x	X
ejpam-6014	215	3	−	−	PRON
ejpam-6014	215	4	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-6014	215	5	-	-	PUNCT
ejpam-6014	215	6	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6014	215	7	-	-	PUNCT
ejpam-6014	215	8	cl(v	cl(v	NOUN
ejpam-6014	215	9	)	)	PUNCT
ejpam-6014	215	10	)	)	PUNCT
ejpam-6014	215	11	)	)	PUNCT
ejpam-6014	216	1	⊆	⊆	NUM
ejpam-6014	216	2	x	x	SYM
ejpam-6014	216	3	−	−	PROPN
ejpam-6014	216	4	f−1(v	f−1(v	PROPN
ejpam-6014	216	5	)	)	PUNCT
ejpam-6014	216	6	and	and	CCONJ
ejpam-6014	216	7	hence	hence	ADV
ejpam-6014	216	8	f−1(v	f−1(v	NOUN
ejpam-6014	216	9	)	)	PUNCT
ejpam-6014	217	1	⊆	⊆	NUM
ejpam-6014	217	2	(	(	PUNCT
ejpam-6014	217	3	τ1	τ1	NOUN
ejpam-6014	217	4	,	,	PUNCT
ejpam-6014	217	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	217	6	-	-	PUNCT
ejpam-6014	217	7	int(f	int(f	VERB
ejpam-6014	217	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	217	9	-	-	PUNCT
ejpam-6014	217	10	cl(v	cl(v	NOUN
ejpam-6014	217	11	)	)	PUNCT
ejpam-6014	217	12	)	)	PUNCT
ejpam-6014	217	13	)	)	PUNCT
ejpam-6014	217	14	.	.	PUNCT
ejpam-6014	218	1	(	(	PUNCT
ejpam-6014	218	2	4	4	X
ejpam-6014	218	3	)	)	PUNCT
ejpam-6014	218	4	⇒	⇒	NOUN
ejpam-6014	218	5	(	(	PUNCT
ejpam-6014	218	6	1	1	NUM
ejpam-6014	218	7	):	):	PUNCT
ejpam-6014	218	8	let	let	VERB
ejpam-6014	218	9	v	v	PART
ejpam-6014	218	10	be	be	AUX
ejpam-6014	218	11	any	any	DET
ejpam-6014	218	12	σ1σ2	σ1σ2	NOUN
ejpam-6014	218	13	-	-	ADJ
ejpam-6014	218	14	open	open	ADJ
ejpam-6014	218	15	set	set	NOUN
ejpam-6014	218	16	of	of	ADP
ejpam-6014	218	17	y	y	PROPN
ejpam-6014	218	18	.	.	PUNCT
ejpam-6014	219	1	then	then	ADV
ejpam-6014	219	2	,	,	PUNCT
ejpam-6014	219	3	v	v	NOUN
ejpam-6014	219	4	is	be	AUX
ejpam-6014	219	5	(	(	PUNCT
ejpam-6014	219	6	σ1	σ1	PROPN
ejpam-6014	219	7	,	,	PUNCT
ejpam-6014	219	8	σ2)p	σ2)p	NOUN
ejpam-6014	219	9	-	-	PUNCT
ejpam-6014	219	10	open	open	ADJ
ejpam-6014	219	11	in	in	ADP
ejpam-6014	219	12	y	y	PROPN
ejpam-6014	219	13	,	,	PUNCT
ejpam-6014	219	14	by	by	ADP
ejpam-6014	219	15	(	(	PUNCT
ejpam-6014	219	16	4	4	X
ejpam-6014	219	17	)	)	PUNCT
ejpam-6014	219	18	we	we	PRON
ejpam-6014	219	19	have	have	AUX
ejpam-6014	219	20	f−1(v	f−1(v	NOUN
ejpam-6014	219	21	)	)	PUNCT
ejpam-6014	220	1	⊆	⊆	NUM
ejpam-6014	220	2	(	(	PUNCT
ejpam-6014	220	3	τ1	τ1	NOUN
ejpam-6014	220	4	,	,	PUNCT
ejpam-6014	220	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	220	6	-	-	PUNCT
ejpam-6014	220	7	int(f	int(f	VERB
ejpam-6014	220	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	220	9	-	-	PUNCT
ejpam-6014	220	10	cl(v	cl(v	NOUN
ejpam-6014	220	11	)	)	PUNCT
ejpam-6014	220	12	)	)	PUNCT
ejpam-6014	220	13	)	)	PUNCT
ejpam-6014	220	14	.	.	PUNCT
ejpam-6014	221	1	by	by	ADP
ejpam-6014	221	2	theorem	theorem	NOUN
ejpam-6014	221	3	3	3	NUM
ejpam-6014	221	4	,	,	PUNCT
ejpam-6014	221	5	f	f	PROPN
ejpam-6014	221	6	is	be	AUX
ejpam-6014	221	7	θ(τ1	θ(τ1	NOUN
ejpam-6014	221	8	,	,	PUNCT
ejpam-6014	221	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	221	10	.	.	PUNCT
ejpam-6014	222	1	definition	definition	NOUN
ejpam-6014	222	2	2	2	NUM
ejpam-6014	222	3	.	.	PUNCT
ejpam-6014	223	1	[	[	X
ejpam-6014	223	2	36	36	NUM
ejpam-6014	223	3	]	]	PUNCT
ejpam-6014	223	4	a	a	DET
ejpam-6014	223	5	bitopological	bitopological	ADJ
ejpam-6014	223	6	space	space	NOUN
ejpam-6014	223	7	(	(	PUNCT
ejpam-6014	223	8	x	x	NOUN
ejpam-6014	223	9	,	,	PUNCT
ejpam-6014	223	10	τ1	τ1	NOUN
ejpam-6014	223	11	,	,	PUNCT
ejpam-6014	223	12	τ2	τ2	NOUN
ejpam-6014	223	13	)	)	PUNCT
ejpam-6014	223	14	is	be	AUX
ejpam-6014	223	15	said	say	VERB
ejpam-6014	223	16	to	to	PART
ejpam-6014	223	17	be	be	AUX
ejpam-6014	223	18	(	(	PUNCT
ejpam-6014	223	19	τ1	τ1	NOUN
ejpam-6014	223	20	,	,	PUNCT
ejpam-6014	223	21	τ2)-regular	τ2)-regular	ADJ
ejpam-6014	223	22	if	if	SCONJ
ejpam-6014	223	23	for	for	ADP
ejpam-6014	223	24	each	each	DET
ejpam-6014	223	25	τ1τ2	τ1τ2	ADJ
ejpam-6014	223	26	-	-	ADJ
ejpam-6014	223	27	closed	closed	ADJ
ejpam-6014	223	28	set	set	VERB
ejpam-6014	223	29	f	f	NOUN
ejpam-6014	223	30	and	and	CCONJ
ejpam-6014	223	31	each	each	DET
ejpam-6014	223	32	x	x	PROPN
ejpam-6014	223	33	̸∈	̸∈	PROPN
ejpam-6014	223	34	f	f	PROPN
ejpam-6014	223	35	,	,	PUNCT
ejpam-6014	223	36	there	there	PRON
ejpam-6014	223	37	exist	exist	VERB
ejpam-6014	223	38	disjoint	disjoint	ADJ
ejpam-6014	223	39	τ1τ2	τ1τ2	ADJ
ejpam-6014	223	40	-	-	ADJ
ejpam-6014	223	41	open	open	ADJ
ejpam-6014	223	42	sets	set	NOUN
ejpam-6014	223	43	u	u	NOUN
ejpam-6014	223	44	and	and	CCONJ
ejpam-6014	223	45	v	v	ADP
ejpam-6014	223	46	such	such	ADJ
ejpam-6014	223	47	that	that	SCONJ
ejpam-6014	223	48	x	x	SYM
ejpam-6014	223	49	∈	∈	PROPN
ejpam-6014	223	50	u	u	NOUN
ejpam-6014	223	51	and	and	CCONJ
ejpam-6014	223	52	f	f	PROPN
ejpam-6014	223	53	⊆	⊆	NUM
ejpam-6014	223	54	v	v	NOUN
ejpam-6014	223	55	.	.	PUNCT
ejpam-6014	224	1	lemma	lemma	PROPN
ejpam-6014	224	2	3	3	X
ejpam-6014	224	3	.	.	PUNCT
ejpam-6014	225	1	[	[	X
ejpam-6014	225	2	37	37	NUM
ejpam-6014	225	3	]	]	PUNCT
ejpam-6014	225	4	a	a	DET
ejpam-6014	225	5	bitopological	bitopological	ADJ
ejpam-6014	225	6	space	space	NOUN
ejpam-6014	225	7	(	(	PUNCT
ejpam-6014	225	8	x	x	NOUN
ejpam-6014	225	9	,	,	PUNCT
ejpam-6014	225	10	τ1	τ1	NOUN
ejpam-6014	225	11	,	,	PUNCT
ejpam-6014	225	12	τ2	τ2	NOUN
ejpam-6014	225	13	)	)	PUNCT
ejpam-6014	225	14	is	be	AUX
ejpam-6014	225	15	(	(	PUNCT
ejpam-6014	225	16	τ1	τ1	NOUN
ejpam-6014	225	17	,	,	PUNCT
ejpam-6014	225	18	τ2)-regular	τ2)-regular	ADJ
ejpam-6014	225	19	if	if	SCONJ
ejpam-6014	225	20	and	and	CCONJ
ejpam-6014	225	21	only	only	ADV
ejpam-6014	225	22	if	if	SCONJ
ejpam-6014	225	23	for	for	ADP
ejpam-6014	225	24	each	each	DET
ejpam-6014	225	25	x	x	SYM
ejpam-6014	225	26	∈	∈	PROPN
ejpam-6014	225	27	x	x	X
ejpam-6014	225	28	and	and	CCONJ
ejpam-6014	225	29	each	each	DET
ejpam-6014	225	30	τ1τ2	τ1τ2	ADJ
ejpam-6014	225	31	-	-	ADJ
ejpam-6014	225	32	open	open	ADJ
ejpam-6014	225	33	set	set	NOUN
ejpam-6014	225	34	u	u	NOUN
ejpam-6014	225	35	containing	contain	VERB
ejpam-6014	225	36	x	x	PRON
ejpam-6014	225	37	,	,	PUNCT
ejpam-6014	225	38	there	there	PRON
ejpam-6014	225	39	exists	exist	VERB
ejpam-6014	225	40	a	a	DET
ejpam-6014	225	41	τ1τ2	τ1τ2	NOUN
ejpam-6014	225	42	-	-	ADJ
ejpam-6014	225	43	open	open	ADJ
ejpam-6014	225	44	set	set	VERB
ejpam-6014	225	45	v	v	ADP
ejpam-6014	225	46	such	such	ADJ
ejpam-6014	225	47	that	that	SCONJ
ejpam-6014	225	48	x	x	SYM
ejpam-6014	225	49	∈	∈	NOUN
ejpam-6014	225	50	v	v	ADP
ejpam-6014	225	51	⊆	⊆	NUM
ejpam-6014	225	52	τ1τ2	τ1τ2	NOUN
ejpam-6014	225	53	-	-	NOUN
ejpam-6014	225	54	cl(v	cl(v	X
ejpam-6014	225	55	)	)	PUNCT
ejpam-6014	225	56	⊆	⊆	NUM
ejpam-6014	225	57	u	u	NOUN
ejpam-6014	225	58	.	.	PUNCT
ejpam-6014	226	1	lemma	lemma	PROPN
ejpam-6014	226	2	4	4	NUM
ejpam-6014	226	3	.	.	PUNCT
ejpam-6014	227	1	[	[	X
ejpam-6014	227	2	37	37	NUM
ejpam-6014	227	3	]	]	PUNCT
ejpam-6014	227	4	let	let	VERB
ejpam-6014	227	5	(	(	PUNCT
ejpam-6014	227	6	x	x	NOUN
ejpam-6014	227	7	,	,	PUNCT
ejpam-6014	227	8	τ1	τ1	NOUN
ejpam-6014	227	9	,	,	PUNCT
ejpam-6014	227	10	τ2	τ2	PROPN
ejpam-6014	227	11	)	)	PUNCT
ejpam-6014	227	12	be	be	VERB
ejpam-6014	227	13	a	a	DET
ejpam-6014	227	14	(	(	PUNCT
ejpam-6014	227	15	τ1	τ1	NOUN
ejpam-6014	227	16	,	,	PUNCT
ejpam-6014	227	17	τ2)-regular	τ2)-regular	ADJ
ejpam-6014	227	18	space	space	NOUN
ejpam-6014	227	19	.	.	PUNCT
ejpam-6014	228	1	then	then	ADV
ejpam-6014	228	2	,	,	PUNCT
ejpam-6014	228	3	the	the	DET
ejpam-6014	228	4	following	follow	VERB
ejpam-6014	228	5	properties	property	NOUN
ejpam-6014	228	6	hold	hold	VERB
ejpam-6014	228	7	:	:	PUNCT
ejpam-6014	228	8	(	(	PUNCT
ejpam-6014	228	9	1	1	X
ejpam-6014	228	10	)	)	PUNCT
ejpam-6014	228	11	τ1τ2	τ1τ2	NOUN
ejpam-6014	228	12	-	-	NUM
ejpam-6014	228	13	cl(a	cl(a	NUM
ejpam-6014	228	14	)	)	PUNCT
ejpam-6014	228	15	=	=	PUNCT
ejpam-6014	228	16	(	(	PUNCT
ejpam-6014	228	17	τ1	τ1	NOUN
ejpam-6014	228	18	,	,	PUNCT
ejpam-6014	228	19	τ2)θ	τ2)θ	NOUN
ejpam-6014	228	20	-	-	PUNCT
ejpam-6014	228	21	cl(a	cl(a	NUM
ejpam-6014	228	22	)	)	PUNCT
ejpam-6014	228	23	for	for	ADP
ejpam-6014	228	24	every	every	DET
ejpam-6014	228	25	subset	subset	NOUN
ejpam-6014	228	26	a	a	PRON
ejpam-6014	228	27	of	of	ADP
ejpam-6014	228	28	x.	x.	NOUN
ejpam-6014	228	29	(	(	PUNCT
ejpam-6014	228	30	2	2	NUM
ejpam-6014	228	31	)	)	PUNCT
ejpam-6014	228	32	every	every	DET
ejpam-6014	228	33	τ1τ2	τ1τ2	NOUN
ejpam-6014	228	34	-	-	ADJ
ejpam-6014	228	35	open	open	ADJ
ejpam-6014	228	36	set	set	NOUN
ejpam-6014	228	37	is	be	AUX
ejpam-6014	228	38	(	(	PUNCT
ejpam-6014	228	39	τ1	τ1	NOUN
ejpam-6014	228	40	,	,	PUNCT
ejpam-6014	228	41	τ2)θ	τ2)θ	ADJ
ejpam-6014	228	42	-	-	PUNCT
ejpam-6014	228	43	open	open	ADJ
ejpam-6014	228	44	.	.	PUNCT
ejpam-6014	229	1	definition	definition	NOUN
ejpam-6014	229	2	3	3	NUM
ejpam-6014	229	3	.	.	PUNCT
ejpam-6014	230	1	[	[	X
ejpam-6014	230	2	28	28	NUM
ejpam-6014	230	3	]	]	X
ejpam-6014	230	4	a	a	DET
ejpam-6014	230	5	function	function	NOUN
ejpam-6014	230	6	f	f	NOUN
ejpam-6014	230	7	:	:	PUNCT
ejpam-6014	230	8	(	(	PUNCT
ejpam-6014	230	9	x	x	NOUN
ejpam-6014	230	10	,	,	PUNCT
ejpam-6014	230	11	τ1	τ1	NOUN
ejpam-6014	230	12	,	,	PUNCT
ejpam-6014	230	13	τ2	τ2	NOUN
ejpam-6014	230	14	)	)	PUNCT
ejpam-6014	230	15	→	→	SYM
ejpam-6014	230	16	(	(	PUNCT
ejpam-6014	230	17	y	y	PROPN
ejpam-6014	230	18	,	,	PUNCT
ejpam-6014	230	19	σ1	σ1	PROPN
ejpam-6014	230	20	,	,	PUNCT
ejpam-6014	230	21	σ2	σ2	PROPN
ejpam-6014	230	22	)	)	PUNCT
ejpam-6014	230	23	is	be	AUX
ejpam-6014	230	24	called	call	VERB
ejpam-6014	230	25	(	(	PUNCT
ejpam-6014	230	26	τ1	τ1	NOUN
ejpam-6014	230	27	,	,	PUNCT
ejpam-6014	230	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	230	29	at	at	ADP
ejpam-6014	230	30	a	a	DET
ejpam-6014	230	31	point	point	NOUN
ejpam-6014	230	32	x	x	SYM
ejpam-6014	230	33	∈	∈	NOUN
ejpam-6014	230	34	x	x	PUNCT
ejpam-6014	230	35	if	if	SCONJ
ejpam-6014	230	36	for	for	ADP
ejpam-6014	230	37	each	each	DET
ejpam-6014	230	38	σ1σ2	σ1σ2	VERB
ejpam-6014	230	39	-	-	ADJ
ejpam-6014	230	40	open	open	ADJ
ejpam-6014	230	41	set	set	NOUN
ejpam-6014	230	42	v	v	NOUN
ejpam-6014	230	43	of	of	ADP
ejpam-6014	230	44	y	y	NOUN
ejpam-6014	230	45	containing	contain	VERB
ejpam-6014	230	46	f(x	f(x	PROPN
ejpam-6014	230	47	)	)	PUNCT
ejpam-6014	230	48	,	,	PUNCT
ejpam-6014	230	49	there	there	PRON
ejpam-6014	230	50	exists	exist	VERB
ejpam-6014	230	51	a	a	DET
ejpam-6014	230	52	τ1τ2	τ1τ2	NOUN
ejpam-6014	230	53	-	-	ADJ
ejpam-6014	230	54	open	open	ADJ
ejpam-6014	230	55	set	set	ADJ
ejpam-6014	230	56	u	u	NOUN
ejpam-6014	230	57	of	of	ADP
ejpam-6014	230	58	x	x	PUNCT
ejpam-6014	230	59	containing	contain	VERB
ejpam-6014	230	60	x	x	PUNCT
ejpam-6014	230	61	such	such	ADJ
ejpam-6014	230	62	that	that	DET
ejpam-6014	230	63	f(u	f(u	PROPN
ejpam-6014	230	64	)	)	PUNCT
ejpam-6014	230	65	⊆	⊆	NUM
ejpam-6014	230	66	v	v	NOUN
ejpam-6014	230	67	.	.	PUNCT
ejpam-6014	231	1	a	a	DET
ejpam-6014	231	2	function	function	NOUN
ejpam-6014	231	3	f	f	NOUN
ejpam-6014	231	4	:	:	PUNCT
ejpam-6014	231	5	(	(	PUNCT
ejpam-6014	231	6	x	x	NOUN
ejpam-6014	231	7	,	,	PUNCT
ejpam-6014	231	8	τ1	τ1	NOUN
ejpam-6014	231	9	,	,	PUNCT
ejpam-6014	231	10	τ2	τ2	NOUN
ejpam-6014	231	11	)	)	PUNCT
ejpam-6014	231	12	→	→	SYM
ejpam-6014	231	13	(	(	PUNCT
ejpam-6014	231	14	y	y	PROPN
ejpam-6014	231	15	,	,	PUNCT
ejpam-6014	231	16	σ1	σ1	PROPN
ejpam-6014	231	17	,	,	PUNCT
ejpam-6014	231	18	σ2	σ2	PROPN
ejpam-6014	231	19	)	)	PUNCT
ejpam-6014	231	20	is	be	AUX
ejpam-6014	231	21	called	call	VERB
ejpam-6014	231	22	(	(	PUNCT
ejpam-6014	231	23	τ1	τ1	NOUN
ejpam-6014	231	24	,	,	PUNCT
ejpam-6014	231	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	231	26	if	if	SCONJ
ejpam-6014	231	27	f	f	PROPN
ejpam-6014	231	28	has	have	VERB
ejpam-6014	231	29	this	this	DET
ejpam-6014	231	30	property	property	NOUN
ejpam-6014	231	31	at	at	ADP
ejpam-6014	231	32	each	each	DET
ejpam-6014	231	33	point	point	NOUN
ejpam-6014	231	34	of	of	ADP
ejpam-6014	231	35	x.	x.	NOUN
ejpam-6014	231	36	definition	definition	NOUN
ejpam-6014	231	37	4	4	NUM
ejpam-6014	231	38	.	.	PUNCT
ejpam-6014	232	1	[	[	X
ejpam-6014	232	2	30	30	NUM
ejpam-6014	232	3	]	]	X
ejpam-6014	232	4	a	a	DET
ejpam-6014	232	5	function	function	NOUN
ejpam-6014	232	6	f	f	NOUN
ejpam-6014	232	7	:	:	PUNCT
ejpam-6014	232	8	(	(	PUNCT
ejpam-6014	232	9	x	x	NOUN
ejpam-6014	232	10	,	,	PUNCT
ejpam-6014	232	11	τ1	τ1	NOUN
ejpam-6014	232	12	,	,	PUNCT
ejpam-6014	232	13	τ2	τ2	NOUN
ejpam-6014	232	14	)	)	PUNCT
ejpam-6014	232	15	→	→	SYM
ejpam-6014	232	16	(	(	PUNCT
ejpam-6014	232	17	y	y	PROPN
ejpam-6014	232	18	,	,	PUNCT
ejpam-6014	232	19	σ1	σ1	PROPN
ejpam-6014	232	20	,	,	PUNCT
ejpam-6014	232	21	σ2	σ2	PROPN
ejpam-6014	232	22	)	)	PUNCT
ejpam-6014	232	23	is	be	AUX
ejpam-6014	232	24	said	say	VERB
ejpam-6014	232	25	to	to	PART
ejpam-6014	232	26	be	be	AUX
ejpam-6014	232	27	weakly	weakly	ADJ
ejpam-6014	232	28	(	(	PUNCT
ejpam-6014	232	29	τ1	τ1	NOUN
ejpam-6014	232	30	,	,	PUNCT
ejpam-6014	232	31	τ2)continuous	τ2)continuous	ADJ
ejpam-6014	232	32	at	at	ADP
ejpam-6014	232	33	a	a	DET
ejpam-6014	232	34	point	point	NOUN
ejpam-6014	232	35	x	x	SYM
ejpam-6014	232	36	∈	∈	NOUN
ejpam-6014	232	37	x	x	PUNCT
ejpam-6014	232	38	if	if	SCONJ
ejpam-6014	232	39	for	for	ADP
ejpam-6014	232	40	each	each	DET
ejpam-6014	232	41	τ1τ2	τ1τ2	ADJ
ejpam-6014	232	42	-	-	ADJ
ejpam-6014	232	43	open	open	ADJ
ejpam-6014	232	44	set	set	VERB
ejpam-6014	232	45	v	v	NOUN
ejpam-6014	232	46	of	of	ADP
ejpam-6014	232	47	y	y	NOUN
ejpam-6014	232	48	containing	contain	VERB
ejpam-6014	232	49	f(x	f(x	PROPN
ejpam-6014	232	50	)	)	PUNCT
ejpam-6014	232	51	,	,	PUNCT
ejpam-6014	232	52	there	there	PRON
ejpam-6014	232	53	exists	exist	VERB
ejpam-6014	232	54	a	a	DET
ejpam-6014	232	55	τ1τ2	τ1τ2	NOUN
ejpam-6014	232	56	-	-	ADJ
ejpam-6014	232	57	open	open	ADJ
ejpam-6014	232	58	set	set	ADJ
ejpam-6014	232	59	u	u	NOUN
ejpam-6014	232	60	of	of	ADP
ejpam-6014	232	61	x	x	PUNCT
ejpam-6014	232	62	containing	contain	VERB
ejpam-6014	232	63	x	x	PUNCT
ejpam-6014	232	64	such	such	ADJ
ejpam-6014	232	65	that	that	DET
ejpam-6014	232	66	f(u	f(u	PROPN
ejpam-6014	232	67	)	)	PUNCT
ejpam-6014	232	68	⊆	⊆	NUM
ejpam-6014	232	69	σ1σ2	σ1σ2	NOUN
ejpam-6014	232	70	-	-	NUM
ejpam-6014	232	71	cl(v	cl(v	NOUN
ejpam-6014	232	72	)	)	PUNCT
ejpam-6014	232	73	.	.	PUNCT
ejpam-6014	233	1	a	a	DET
ejpam-6014	233	2	function	function	NOUN
ejpam-6014	233	3	f	f	NOUN
ejpam-6014	233	4	:	:	PUNCT
ejpam-6014	233	5	(	(	PUNCT
ejpam-6014	233	6	x	x	NOUN
ejpam-6014	233	7	,	,	PUNCT
ejpam-6014	233	8	τ1	τ1	NOUN
ejpam-6014	233	9	,	,	PUNCT
ejpam-6014	233	10	τ2	τ2	NOUN
ejpam-6014	233	11	)	)	PUNCT
ejpam-6014	233	12	→	→	SYM
ejpam-6014	233	13	(	(	PUNCT
ejpam-6014	233	14	y	y	PROPN
ejpam-6014	233	15	,	,	PUNCT
ejpam-6014	233	16	σ1	σ1	PROPN
ejpam-6014	233	17	,	,	PUNCT
ejpam-6014	233	18	σ2	σ2	PROPN
ejpam-6014	233	19	)	)	PUNCT
ejpam-6014	233	20	is	be	AUX
ejpam-6014	233	21	said	say	VERB
ejpam-6014	233	22	to	to	PART
ejpam-6014	233	23	be	be	AUX
ejpam-6014	233	24	weakly	weakly	ADJ
ejpam-6014	233	25	(	(	PUNCT
ejpam-6014	233	26	τ1	τ1	NOUN
ejpam-6014	233	27	,	,	PUNCT
ejpam-6014	233	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	233	29	if	if	SCONJ
ejpam-6014	233	30	f	f	PROPN
ejpam-6014	233	31	has	have	VERB
ejpam-6014	233	32	this	this	DET
ejpam-6014	233	33	property	property	NOUN
ejpam-6014	233	34	at	at	ADP
ejpam-6014	233	35	each	each	DET
ejpam-6014	233	36	point	point	NOUN
ejpam-6014	233	37	of	of	ADP
ejpam-6014	233	38	x.	x.	PROPN
ejpam-6014	233	39	m.	m.	PROPN
ejpam-6014	233	40	thongmoon	thongmoon	PROPN
ejpam-6014	233	41	,	,	PUNCT
ejpam-6014	233	42	s.	s.	PROPN
ejpam-6014	233	43	sompong	sompong	PROPN
ejpam-6014	233	44	,	,	PUNCT
ejpam-6014	233	45	c.	c.	PROPN
ejpam-6014	233	46	boonpok	boonpok	PROPN
ejpam-6014	233	47	/	/	SYM
ejpam-6014	233	48	eur	eur	PROPN
ejpam-6014	233	49	.	.	PUNCT
ejpam-6014	234	1	j.	j.	PROPN
ejpam-6014	234	2	pure	pure	PROPN
ejpam-6014	234	3	appl	appl	PROPN
ejpam-6014	234	4	.	.	PROPN
ejpam-6014	234	5	math	math	PROPN
ejpam-6014	234	6	,	,	PUNCT
ejpam-6014	234	7	18	18	NUM
ejpam-6014	234	8	(	(	PUNCT
ejpam-6014	234	9	2	2	NUM
ejpam-6014	234	10	)	)	PUNCT
ejpam-6014	234	11	(	(	PUNCT
ejpam-6014	234	12	2025	2025	NUM
ejpam-6014	234	13	)	)	PUNCT
ejpam-6014	234	14	,	,	PUNCT
ejpam-6014	234	15	6014	6014	NUM
ejpam-6014	234	16	8	8	NUM
ejpam-6014	234	17	of	of	ADP
ejpam-6014	234	18	13	13	NUM
ejpam-6014	234	19	lemma	lemma	PROPN
ejpam-6014	234	20	5	5	NUM
ejpam-6014	234	21	.	.	PUNCT
ejpam-6014	235	1	[	[	X
ejpam-6014	235	2	28	28	NUM
ejpam-6014	235	3	]	]	PUNCT
ejpam-6014	235	4	for	for	ADP
ejpam-6014	235	5	a	a	DET
ejpam-6014	235	6	function	function	NOUN
ejpam-6014	235	7	(	(	PUNCT
ejpam-6014	235	8	x	x	NOUN
ejpam-6014	235	9	,	,	PUNCT
ejpam-6014	235	10	τ1	τ1	NOUN
ejpam-6014	235	11	,	,	PUNCT
ejpam-6014	235	12	τ2	τ2	NOUN
ejpam-6014	235	13	)	)	PUNCT
ejpam-6014	235	14	→	→	SYM
ejpam-6014	235	15	(	(	PUNCT
ejpam-6014	235	16	y	y	PROPN
ejpam-6014	235	17	,	,	PUNCT
ejpam-6014	235	18	σ1	σ1	PROPN
ejpam-6014	235	19	,	,	PUNCT
ejpam-6014	235	20	σ2	σ2	NOUN
ejpam-6014	235	21	)	)	PUNCT
ejpam-6014	235	22	,	,	PUNCT
ejpam-6014	235	23	the	the	DET
ejpam-6014	235	24	following	follow	VERB
ejpam-6014	235	25	properties	property	NOUN
ejpam-6014	235	26	are	be	AUX
ejpam-6014	235	27	equivalent	equivalent	ADJ
ejpam-6014	235	28	:	:	PUNCT
ejpam-6014	235	29	(	(	PUNCT
ejpam-6014	235	30	1	1	X
ejpam-6014	235	31	)	)	PUNCT
ejpam-6014	235	32	f	f	PROPN
ejpam-6014	235	33	is	be	AUX
ejpam-6014	235	34	(	(	PUNCT
ejpam-6014	235	35	τ1	τ1	NOUN
ejpam-6014	235	36	,	,	PUNCT
ejpam-6014	235	37	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	235	38	;	;	PUNCT
ejpam-6014	235	39	(	(	PUNCT
ejpam-6014	235	40	2	2	X
ejpam-6014	235	41	)	)	PUNCT
ejpam-6014	235	42	f−1(v	f−1(v	NOUN
ejpam-6014	235	43	)	)	PUNCT
ejpam-6014	235	44	is	be	AUX
ejpam-6014	235	45	τ1τ2	τ1τ2	NOUN
ejpam-6014	235	46	-	-	ADJ
ejpam-6014	235	47	open	open	ADJ
ejpam-6014	235	48	in	in	ADP
ejpam-6014	235	49	x	x	PUNCT
ejpam-6014	235	50	for	for	ADP
ejpam-6014	235	51	every	every	DET
ejpam-6014	235	52	σ1σ2	σ1σ2	NOUN
ejpam-6014	235	53	-	-	ADJ
ejpam-6014	235	54	open	open	ADJ
ejpam-6014	235	55	set	set	NOUN
ejpam-6014	235	56	v	v	NOUN
ejpam-6014	235	57	of	of	ADP
ejpam-6014	235	58	y	y	PROPN
ejpam-6014	235	59	;	;	PUNCT
ejpam-6014	235	60	(	(	PUNCT
ejpam-6014	235	61	3	3	X
ejpam-6014	235	62	)	)	PUNCT
ejpam-6014	235	63	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	235	64	-	-	PUNCT
ejpam-6014	235	65	cl(a	cl(a	NUM
ejpam-6014	235	66	)	)	PUNCT
ejpam-6014	235	67	)	)	PUNCT
ejpam-6014	236	1	⊆	⊆	X
ejpam-6014	236	2	σ1σ2	σ1σ2	NUM
ejpam-6014	236	3	-	-	PUNCT
ejpam-6014	236	4	cl(f(a	cl(f(a	NOUN
ejpam-6014	236	5	)	)	PUNCT
ejpam-6014	236	6	)	)	PUNCT
ejpam-6014	236	7	for	for	ADP
ejpam-6014	236	8	every	every	DET
ejpam-6014	236	9	subset	subset	NOUN
ejpam-6014	236	10	a	a	PRON
ejpam-6014	236	11	of	of	ADP
ejpam-6014	236	12	x	x	PRON
ejpam-6014	236	13	;	;	PUNCT
ejpam-6014	236	14	(	(	PUNCT
ejpam-6014	236	15	4	4	X
ejpam-6014	236	16	)	)	PUNCT
ejpam-6014	236	17	τ1τ2	τ1τ2	NOUN
ejpam-6014	236	18	-	-	NOUN
ejpam-6014	236	19	cl(f	cl(f	NOUN
ejpam-6014	236	20	−1(b	−1(b	NOUN
ejpam-6014	236	21	)	)	PUNCT
ejpam-6014	236	22	)	)	PUNCT
ejpam-6014	237	1	⊆	⊆	NUM
ejpam-6014	237	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	237	3	-	-	PUNCT
ejpam-6014	237	4	cl(b	cl(b	NOUN
ejpam-6014	237	5	)	)	PUNCT
ejpam-6014	237	6	)	)	PUNCT
ejpam-6014	237	7	for	for	ADP
ejpam-6014	237	8	every	every	DET
ejpam-6014	237	9	subset	subset	NOUN
ejpam-6014	237	10	b	b	PROPN
ejpam-6014	237	11	of	of	ADP
ejpam-6014	237	12	y	y	PROPN
ejpam-6014	237	13	;	;	PUNCT
ejpam-6014	237	14	(	(	PUNCT
ejpam-6014	237	15	5	5	X
ejpam-6014	237	16	)	)	PUNCT
ejpam-6014	237	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	237	18	-	-	PUNCT
ejpam-6014	237	19	int(b	int(b	NOUN
ejpam-6014	237	20	)	)	PUNCT
ejpam-6014	237	21	)	)	PUNCT
ejpam-6014	238	1	⊆	⊆	X
ejpam-6014	238	2	τ1τ2	τ1τ2	NOUN
ejpam-6014	238	3	-	-	NUM
ejpam-6014	238	4	int(f	int(f	NOUN
ejpam-6014	238	5	−1(b	−1(b	NOUN
ejpam-6014	238	6	)	)	PUNCT
ejpam-6014	238	7	)	)	PUNCT
ejpam-6014	238	8	for	for	ADP
ejpam-6014	238	9	every	every	DET
ejpam-6014	238	10	subset	subset	NOUN
ejpam-6014	238	11	b	b	PROPN
ejpam-6014	238	12	of	of	ADP
ejpam-6014	238	13	y	y	PROPN
ejpam-6014	238	14	;	;	PUNCT
ejpam-6014	238	15	(	(	PUNCT
ejpam-6014	238	16	6	6	X
ejpam-6014	238	17	)	)	PUNCT
ejpam-6014	238	18	f−1(k	f−1(k	PROPN
ejpam-6014	238	19	)	)	PUNCT
ejpam-6014	238	20	is	be	AUX
ejpam-6014	238	21	τ1τ2	τ1τ2	NOUN
ejpam-6014	238	22	-	-	ADJ
ejpam-6014	238	23	closed	closed	ADJ
ejpam-6014	238	24	in	in	ADP
ejpam-6014	238	25	x	x	PUNCT
ejpam-6014	238	26	for	for	ADP
ejpam-6014	238	27	every	every	DET
ejpam-6014	238	28	σ1σ2	σ1σ2	NUM
ejpam-6014	238	29	-	-	PUNCT
ejpam-6014	238	30	closed	closed	ADJ
ejpam-6014	238	31	set	set	NOUN
ejpam-6014	238	32	k	k	PROPN
ejpam-6014	238	33	of	of	ADP
ejpam-6014	238	34	y	y	PROPN
ejpam-6014	238	35	.	.	PUNCT
ejpam-6014	239	1	theorem	theorem	VERB
ejpam-6014	239	2	7	7	NUM
ejpam-6014	239	3	.	.	X
ejpam-6014	239	4	for	for	ADP
ejpam-6014	239	5	a	a	DET
ejpam-6014	239	6	function	function	NOUN
ejpam-6014	239	7	f	f	NOUN
ejpam-6014	239	8	:	:	PUNCT
ejpam-6014	239	9	(	(	PUNCT
ejpam-6014	239	10	x	x	NOUN
ejpam-6014	239	11	,	,	PUNCT
ejpam-6014	239	12	τ1	τ1	NOUN
ejpam-6014	239	13	,	,	PUNCT
ejpam-6014	239	14	τ2	τ2	NOUN
ejpam-6014	239	15	)	)	PUNCT
ejpam-6014	239	16	→	→	SYM
ejpam-6014	239	17	(	(	PUNCT
ejpam-6014	239	18	y	y	PROPN
ejpam-6014	239	19	,	,	PUNCT
ejpam-6014	239	20	σ1	σ1	PROPN
ejpam-6014	239	21	,	,	PUNCT
ejpam-6014	239	22	σ2	σ2	NOUN
ejpam-6014	239	23	)	)	PUNCT
ejpam-6014	239	24	,	,	PUNCT
ejpam-6014	239	25	where	where	SCONJ
ejpam-6014	239	26	(	(	PUNCT
ejpam-6014	239	27	y	y	PROPN
ejpam-6014	239	28	,	,	PUNCT
ejpam-6014	239	29	σ1	σ1	PROPN
ejpam-6014	239	30	,	,	PUNCT
ejpam-6014	239	31	σ2	σ2	PROPN
ejpam-6014	239	32	)	)	PUNCT
ejpam-6014	239	33	is	be	AUX
ejpam-6014	239	34	(	(	PUNCT
ejpam-6014	239	35	σ1	σ1	PROPN
ejpam-6014	239	36	,	,	PUNCT
ejpam-6014	239	37	σ2)regular	σ2)regular	PROPN
ejpam-6014	239	38	,	,	PUNCT
ejpam-6014	239	39	the	the	DET
ejpam-6014	239	40	following	follow	VERB
ejpam-6014	239	41	properties	property	NOUN
ejpam-6014	239	42	are	be	AUX
ejpam-6014	239	43	equivalent	equivalent	ADJ
ejpam-6014	239	44	:	:	PUNCT
ejpam-6014	239	45	(	(	PUNCT
ejpam-6014	239	46	1	1	X
ejpam-6014	239	47	)	)	PUNCT
ejpam-6014	239	48	f	f	PROPN
ejpam-6014	239	49	is	be	AUX
ejpam-6014	239	50	(	(	PUNCT
ejpam-6014	239	51	τ1	τ1	NOUN
ejpam-6014	239	52	,	,	PUNCT
ejpam-6014	239	53	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	239	54	;	;	PUNCT
ejpam-6014	239	55	(	(	PUNCT
ejpam-6014	239	56	2	2	X
ejpam-6014	239	57	)	)	PUNCT
ejpam-6014	239	58	f	f	PROPN
ejpam-6014	239	59	is	be	AUX
ejpam-6014	239	60	θ(τ1	θ(τ1	NOUN
ejpam-6014	239	61	,	,	PUNCT
ejpam-6014	239	62	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	239	63	;	;	PUNCT
ejpam-6014	239	64	(	(	PUNCT
ejpam-6014	239	65	3	3	X
ejpam-6014	239	66	)	)	PUNCT
ejpam-6014	239	67	f	f	PROPN
ejpam-6014	239	68	is	be	AUX
ejpam-6014	239	69	weakly	weakly	ADJ
ejpam-6014	239	70	(	(	PUNCT
ejpam-6014	239	71	τ1	τ1	NOUN
ejpam-6014	239	72	,	,	PUNCT
ejpam-6014	239	73	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	239	74	.	.	PUNCT
ejpam-6014	240	1	proof	proof	NOUN
ejpam-6014	240	2	.	.	PUNCT
ejpam-6014	241	1	(	(	PUNCT
ejpam-6014	241	2	1	1	X
ejpam-6014	241	3	)	)	PUNCT
ejpam-6014	241	4	⇒	⇒	NOUN
ejpam-6014	241	5	(	(	PUNCT
ejpam-6014	241	6	2	2	NUM
ejpam-6014	241	7	):	):	PUNCT
ejpam-6014	241	8	let	let	VERB
ejpam-6014	241	9	v	v	PART
ejpam-6014	241	10	be	be	AUX
ejpam-6014	241	11	any	any	DET
ejpam-6014	241	12	σ1σ2	σ1σ2	NOUN
ejpam-6014	241	13	-	-	ADJ
ejpam-6014	241	14	open	open	ADJ
ejpam-6014	241	15	set	set	NOUN
ejpam-6014	241	16	of	of	ADP
ejpam-6014	241	17	y	y	PROPN
ejpam-6014	241	18	containing	contain	VERB
ejpam-6014	241	19	f(x	f(x	PROPN
ejpam-6014	241	20	)	)	PUNCT
ejpam-6014	241	21	.	.	PUNCT
ejpam-6014	242	1	thus	thus	ADV
ejpam-6014	242	2	by	by	ADP
ejpam-6014	242	3	lemma	lemma	PROPN
ejpam-6014	242	4	5	5	NUM
ejpam-6014	242	5	,	,	PUNCT
ejpam-6014	242	6	f−1(v	f−1(v	PROPN
ejpam-6014	242	7	)	)	PUNCT
ejpam-6014	242	8	is	be	AUX
ejpam-6014	242	9	τ1τ2	τ1τ2	NOUN
ejpam-6014	242	10	-	-	ADJ
ejpam-6014	242	11	open	open	ADJ
ejpam-6014	242	12	in	in	ADP
ejpam-6014	242	13	x.	x.	NOUN
ejpam-6014	242	14	since	since	SCONJ
ejpam-6014	242	15	σ1σ2	σ1σ2	NOUN
ejpam-6014	242	16	-	-	NOUN
ejpam-6014	242	17	cl(v	cl(v	NOUN
ejpam-6014	242	18	)	)	PUNCT
ejpam-6014	242	19	is	be	AUX
ejpam-6014	242	20	σ1σ2	σ1σ2	NOUN
ejpam-6014	242	21	-	-	ADJ
ejpam-6014	242	22	closed	closed	ADJ
ejpam-6014	242	23	,	,	PUNCT
ejpam-6014	242	24	by	by	ADP
ejpam-6014	242	25	lemma	lemma	PROPN
ejpam-6014	242	26	5	5	NUM
ejpam-6014	242	27	we	we	PRON
ejpam-6014	242	28	have	have	AUX
ejpam-6014	242	29	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	242	30	-	-	PUNCT
ejpam-6014	242	31	cl(v	cl(v	NOUN
ejpam-6014	242	32	)	)	PUNCT
ejpam-6014	242	33	)	)	PUNCT
ejpam-6014	243	1	is	be	AUX
ejpam-6014	243	2	τ1τ2	τ1τ2	NOUN
ejpam-6014	243	3	-	-	ADJ
ejpam-6014	243	4	closed	closed	ADJ
ejpam-6014	243	5	.	.	PUNCT
ejpam-6014	244	1	put	put	VERB
ejpam-6014	244	2	u	u	NOUN
ejpam-6014	244	3	=	=	NOUN
ejpam-6014	244	4	f−1(v	f−1(v	PROPN
ejpam-6014	244	5	)	)	PUNCT
ejpam-6014	244	6	.	.	PUNCT
ejpam-6014	245	1	then	then	ADV
ejpam-6014	245	2	,	,	PUNCT
ejpam-6014	245	3	u	u	NOUN
ejpam-6014	245	4	is	be	AUX
ejpam-6014	245	5	a	a	DET
ejpam-6014	245	6	τ1τ2	τ1τ2	ADJ
ejpam-6014	245	7	-	-	ADJ
ejpam-6014	245	8	open	open	ADJ
ejpam-6014	245	9	set	set	ADJ
ejpam-6014	245	10	u	u	NOUN
ejpam-6014	245	11	of	of	ADP
ejpam-6014	245	12	x	x	SYM
ejpam-6014	245	13	such	such	ADJ
ejpam-6014	245	14	that	that	SCONJ
ejpam-6014	245	15	x	x	SYM
ejpam-6014	245	16	∈	∈	PROPN
ejpam-6014	245	17	u	u	NOUN
ejpam-6014	245	18	⊆	⊆	NUM
ejpam-6014	245	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	245	20	-	-	PUNCT
ejpam-6014	245	21	cl(v	cl(v	NOUN
ejpam-6014	245	22	)	)	PUNCT
ejpam-6014	245	23	)	)	PUNCT
ejpam-6014	246	1	=	=	PUNCT
ejpam-6014	246	2	τ1τ2	τ1τ2	NOUN
ejpam-6014	246	3	-	-	ADJ
ejpam-6014	246	4	cl(f	cl(f	NOUN
ejpam-6014	246	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	246	6	-	-	PUNCT
ejpam-6014	246	7	cl(v	cl(v	NOUN
ejpam-6014	246	8	)	)	PUNCT
ejpam-6014	246	9	)	)	PUNCT
ejpam-6014	246	10	)	)	PUNCT
ejpam-6014	246	11	.	.	PUNCT
ejpam-6014	247	1	this	this	PRON
ejpam-6014	247	2	implies	imply	VERB
ejpam-6014	247	3	that	that	SCONJ
ejpam-6014	247	4	τ1τ2	τ1τ2	NOUN
ejpam-6014	247	5	-	-	NOUN
ejpam-6014	247	6	cl(u	cl(u	ADJ
ejpam-6014	247	7	)	)	PUNCT
ejpam-6014	247	8	⊆	⊆	NUM
ejpam-6014	247	9	τ1τ2	τ1τ2	NOUN
ejpam-6014	247	10	-	-	ADJ
ejpam-6014	247	11	cl(f	cl(f	NOUN
ejpam-6014	247	12	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	247	13	-	-	PUNCT
ejpam-6014	247	14	cl(v	cl(v	NOUN
ejpam-6014	247	15	)	)	PUNCT
ejpam-6014	247	16	)	)	PUNCT
ejpam-6014	247	17	)	)	PUNCT
ejpam-6014	247	18	=	=	PRON
ejpam-6014	247	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	247	20	-	-	PUNCT
ejpam-6014	247	21	cl(v	cl(v	NOUN
ejpam-6014	247	22	)	)	PUNCT
ejpam-6014	247	23	)	)	PUNCT
ejpam-6014	247	24	.	.	PUNCT
ejpam-6014	248	1	thus	thus	ADV
ejpam-6014	248	2	,	,	PUNCT
ejpam-6014	248	3	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	248	4	-	-	PUNCT
ejpam-6014	248	5	cl(u	cl(u	NOUN
ejpam-6014	248	6	)	)	PUNCT
ejpam-6014	248	7	)	)	PUNCT
ejpam-6014	249	1	⊆	⊆	X
ejpam-6014	249	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	249	3	-	-	NUM
ejpam-6014	249	4	cl(v	cl(v	NOUN
ejpam-6014	249	5	)	)	PUNCT
ejpam-6014	249	6	.	.	PUNCT
ejpam-6014	250	1	this	this	PRON
ejpam-6014	250	2	shows	show	VERB
ejpam-6014	250	3	that	that	SCONJ
ejpam-6014	250	4	f	f	PROPN
ejpam-6014	250	5	is	be	AUX
ejpam-6014	250	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	250	7	,	,	PUNCT
ejpam-6014	250	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	250	9	.	.	PUNCT
ejpam-6014	251	1	(	(	PUNCT
ejpam-6014	251	2	2	2	X
ejpam-6014	251	3	)	)	PUNCT
ejpam-6014	251	4	⇒	⇒	NOUN
ejpam-6014	251	5	(	(	PUNCT
ejpam-6014	251	6	3	3	NUM
ejpam-6014	251	7	):	):	PUNCT
ejpam-6014	251	8	the	the	DET
ejpam-6014	251	9	proof	proof	NOUN
ejpam-6014	251	10	is	be	AUX
ejpam-6014	251	11	obvious	obvious	ADJ
ejpam-6014	251	12	.	.	PUNCT
ejpam-6014	252	1	(	(	PUNCT
ejpam-6014	252	2	3	3	X
ejpam-6014	252	3	)	)	PUNCT
ejpam-6014	252	4	⇒	⇒	NOUN
ejpam-6014	252	5	(	(	PUNCT
ejpam-6014	252	6	1	1	NUM
ejpam-6014	252	7	):	):	PUNCT
ejpam-6014	252	8	let	let	VERB
ejpam-6014	252	9	x	x	PUNCT
ejpam-6014	252	10	∈	∈	PROPN
ejpam-6014	252	11	x	x	X
ejpam-6014	252	12	and	and	CCONJ
ejpam-6014	252	13	v	v	X
ejpam-6014	252	14	be	be	AUX
ejpam-6014	252	15	any	any	DET
ejpam-6014	252	16	σ1σ2	σ1σ2	NOUN
ejpam-6014	252	17	-	-	ADJ
ejpam-6014	252	18	open	open	ADJ
ejpam-6014	252	19	set	set	NOUN
ejpam-6014	252	20	of	of	ADP
ejpam-6014	252	21	y	y	PROPN
ejpam-6014	252	22	containing	contain	VERB
ejpam-6014	252	23	f(x	f(x	PROPN
ejpam-6014	252	24	)	)	PUNCT
ejpam-6014	252	25	.	.	PUNCT
ejpam-6014	253	1	since	since	SCONJ
ejpam-6014	253	2	(	(	PUNCT
ejpam-6014	253	3	y	y	PROPN
ejpam-6014	253	4	,	,	PUNCT
ejpam-6014	253	5	σ1	σ1	PROPN
ejpam-6014	253	6	,	,	PUNCT
ejpam-6014	253	7	σ2	σ2	PROPN
ejpam-6014	253	8	)	)	PUNCT
ejpam-6014	253	9	is	be	AUX
ejpam-6014	253	10	(	(	PUNCT
ejpam-6014	253	11	σ1	σ1	NOUN
ejpam-6014	253	12	,	,	PUNCT
ejpam-6014	253	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6014	253	14	,	,	PUNCT
ejpam-6014	253	15	by	by	ADP
ejpam-6014	253	16	lemma	lemma	PROPN
ejpam-6014	253	17	3	3	NUM
ejpam-6014	253	18	there	there	ADV
ejpam-6014	253	19	exists	exist	VERB
ejpam-6014	253	20	a	a	DET
ejpam-6014	253	21	σ1σ2	σ1σ2	NUM
ejpam-6014	253	22	-	-	ADJ
ejpam-6014	253	23	open	open	ADJ
ejpam-6014	253	24	set	set	NOUN
ejpam-6014	253	25	w	w	PROPN
ejpam-6014	253	26	of	of	ADP
ejpam-6014	253	27	y	y	PRON
ejpam-6014	253	28	such	such	ADJ
ejpam-6014	253	29	that	that	SCONJ
ejpam-6014	253	30	f(x	f(x	PROPN
ejpam-6014	253	31	)	)	PUNCT
ejpam-6014	253	32	∈	∈	PROPN
ejpam-6014	253	33	w	w	ADP
ejpam-6014	253	34	⊆	⊆	NUM
ejpam-6014	253	35	σ1σ2	σ1σ2	NOUN
ejpam-6014	253	36	-	-	PUNCT
ejpam-6014	253	37	cl(w	cl(w	NOUN
ejpam-6014	253	38	)	)	PUNCT
ejpam-6014	253	39	⊆	⊆	NUM
ejpam-6014	253	40	v	v	NOUN
ejpam-6014	253	41	.	.	PUNCT
ejpam-6014	254	1	since	since	SCONJ
ejpam-6014	254	2	f	f	PROPN
ejpam-6014	254	3	is	be	AUX
ejpam-6014	254	4	weakly	weakly	ADJ
ejpam-6014	254	5	(	(	PUNCT
ejpam-6014	254	6	τ1	τ1	NOUN
ejpam-6014	254	7	,	,	PUNCT
ejpam-6014	254	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	254	9	,	,	PUNCT
ejpam-6014	254	10	there	there	PRON
ejpam-6014	254	11	exists	exist	VERB
ejpam-6014	254	12	a	a	DET
ejpam-6014	254	13	τ1τ2	τ1τ2	NOUN
ejpam-6014	254	14	-	-	ADJ
ejpam-6014	254	15	open	open	ADJ
ejpam-6014	254	16	set	set	ADJ
ejpam-6014	254	17	u	u	NOUN
ejpam-6014	254	18	of	of	ADP
ejpam-6014	254	19	x	x	PUNCT
ejpam-6014	254	20	containing	contain	VERB
ejpam-6014	254	21	x	x	PUNCT
ejpam-6014	254	22	such	such	ADJ
ejpam-6014	254	23	that	that	DET
ejpam-6014	254	24	f(u	f(u	PROPN
ejpam-6014	254	25	)	)	PUNCT
ejpam-6014	255	1	⊆	⊆	NUM
ejpam-6014	255	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	255	3	-	-	PUNCT
ejpam-6014	255	4	cl(w	cl(w	NOUN
ejpam-6014	255	5	)	)	PUNCT
ejpam-6014	255	6	⊆	⊆	NUM
ejpam-6014	255	7	v	v	NOUN
ejpam-6014	255	8	.	.	PUNCT
ejpam-6014	256	1	thus	thus	ADV
ejpam-6014	256	2	,	,	PUNCT
ejpam-6014	256	3	f	f	PROPN
ejpam-6014	256	4	is	be	AUX
ejpam-6014	256	5	(	(	PUNCT
ejpam-6014	256	6	τ1	τ1	NOUN
ejpam-6014	256	7	,	,	PUNCT
ejpam-6014	256	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	256	9	.	.	PUNCT
ejpam-6014	257	1	theorem	theorem	NOUN
ejpam-6014	257	2	8	8	NUM
ejpam-6014	257	3	.	.	PUNCT
ejpam-6014	258	1	let	let	VERB
ejpam-6014	258	2	(	(	PUNCT
ejpam-6014	258	3	x	x	NOUN
ejpam-6014	258	4	,	,	PUNCT
ejpam-6014	258	5	τ1	τ1	NOUN
ejpam-6014	258	6	,	,	PUNCT
ejpam-6014	258	7	τ2	τ2	PROPN
ejpam-6014	258	8	)	)	PUNCT
ejpam-6014	258	9	be	be	AUX
ejpam-6014	258	10	(	(	PUNCT
ejpam-6014	258	11	τ1	τ1	NOUN
ejpam-6014	258	12	,	,	PUNCT
ejpam-6014	258	13	τ2)-regular	τ2)-regular	PROPN
ejpam-6014	258	14	.	.	PUNCT
ejpam-6014	259	1	then	then	ADV
ejpam-6014	259	2	a	a	DET
ejpam-6014	259	3	function	function	NOUN
ejpam-6014	259	4	f	f	NOUN
ejpam-6014	259	5	:	:	PUNCT
ejpam-6014	259	6	(	(	PUNCT
ejpam-6014	259	7	x	x	NOUN
ejpam-6014	259	8	,	,	PUNCT
ejpam-6014	259	9	τ1	τ1	NOUN
ejpam-6014	259	10	,	,	PUNCT
ejpam-6014	259	11	τ2	τ2	NOUN
ejpam-6014	259	12	)	)	PUNCT
ejpam-6014	259	13	→	→	SYM
ejpam-6014	259	14	(	(	PUNCT
ejpam-6014	259	15	y	y	PROPN
ejpam-6014	259	16	,	,	PUNCT
ejpam-6014	259	17	σ1	σ1	PROPN
ejpam-6014	259	18	,	,	PUNCT
ejpam-6014	259	19	σ2	σ2	PROPN
ejpam-6014	259	20	)	)	PUNCT
ejpam-6014	259	21	is	be	AUX
ejpam-6014	259	22	θ(τ1	θ(τ1	NOUN
ejpam-6014	259	23	,	,	PUNCT
ejpam-6014	259	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	259	25	if	if	SCONJ
ejpam-6014	259	26	and	and	CCONJ
ejpam-6014	259	27	only	only	ADV
ejpam-6014	259	28	if	if	SCONJ
ejpam-6014	259	29	f	f	PROPN
ejpam-6014	259	30	is	be	AUX
ejpam-6014	259	31	weakly	weakly	ADJ
ejpam-6014	259	32	(	(	PUNCT
ejpam-6014	259	33	τ1	τ1	NOUN
ejpam-6014	259	34	,	,	PUNCT
ejpam-6014	259	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	259	36	.	.	PUNCT
ejpam-6014	260	1	proof	proof	NOUN
ejpam-6014	260	2	.	.	PUNCT
ejpam-6014	261	1	we	we	PRON
ejpam-6014	261	2	prove	prove	VERB
ejpam-6014	261	3	only	only	ADV
ejpam-6014	261	4	the	the	DET
ejpam-6014	261	5	sufficiency	sufficiency	NOUN
ejpam-6014	261	6	.	.	PUNCT
ejpam-6014	262	1	suppose	suppose	VERB
ejpam-6014	262	2	that	that	SCONJ
ejpam-6014	262	3	f	f	PROPN
ejpam-6014	262	4	is	be	AUX
ejpam-6014	262	5	weakly	weakly	ADJ
ejpam-6014	262	6	(	(	PUNCT
ejpam-6014	262	7	τ1	τ1	NOUN
ejpam-6014	262	8	,	,	PUNCT
ejpam-6014	262	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	262	10	.	.	PUNCT
ejpam-6014	263	1	let	let	VERB
ejpam-6014	263	2	x	x	PUNCT
ejpam-6014	263	3	∈	∈	PROPN
ejpam-6014	263	4	x	x	X
ejpam-6014	263	5	and	and	CCONJ
ejpam-6014	263	6	v	v	X
ejpam-6014	263	7	be	be	AUX
ejpam-6014	263	8	any	any	DET
ejpam-6014	263	9	σ1σ2	σ1σ2	NOUN
ejpam-6014	263	10	-	-	ADJ
ejpam-6014	263	11	open	open	ADJ
ejpam-6014	263	12	set	set	NOUN
ejpam-6014	263	13	of	of	ADP
ejpam-6014	263	14	y	y	PROPN
ejpam-6014	263	15	containing	contain	VERB
ejpam-6014	263	16	f(x	f(x	PROPN
ejpam-6014	263	17	)	)	PUNCT
ejpam-6014	263	18	.	.	PUNCT
ejpam-6014	264	1	then	then	ADV
ejpam-6014	264	2	,	,	PUNCT
ejpam-6014	264	3	there	there	PRON
ejpam-6014	264	4	exists	exist	VERB
ejpam-6014	264	5	a	a	DET
ejpam-6014	264	6	τ1τ2	τ1τ2	NOUN
ejpam-6014	264	7	-	-	ADJ
ejpam-6014	264	8	open	open	ADJ
ejpam-6014	264	9	set	set	NOUN
ejpam-6014	264	10	w	w	PROPN
ejpam-6014	264	11	of	of	ADP
ejpam-6014	264	12	x	x	PUNCT
ejpam-6014	264	13	containing	contain	VERB
ejpam-6014	264	14	x	x	PUNCT
ejpam-6014	264	15	such	such	ADJ
ejpam-6014	264	16	that	that	SCONJ
ejpam-6014	264	17	f(w	f(w	PROPN
ejpam-6014	264	18	)	)	PUNCT
ejpam-6014	265	1	⊆	⊆	X
ejpam-6014	265	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	265	3	-	-	NUM
ejpam-6014	265	4	cl(v	cl(v	NOUN
ejpam-6014	265	5	)	)	PUNCT
ejpam-6014	265	6	.	.	PUNCT
ejpam-6014	266	1	since	since	SCONJ
ejpam-6014	266	2	(	(	PUNCT
ejpam-6014	266	3	x	x	NOUN
ejpam-6014	266	4	,	,	PUNCT
ejpam-6014	266	5	τ1	τ1	NOUN
ejpam-6014	266	6	,	,	PUNCT
ejpam-6014	266	7	τ2	τ2	NOUN
ejpam-6014	266	8	)	)	PUNCT
ejpam-6014	266	9	is	be	AUX
ejpam-6014	266	10	(	(	PUNCT
ejpam-6014	266	11	τ1	τ1	NOUN
ejpam-6014	266	12	,	,	PUNCT
ejpam-6014	266	13	τ2)-regular	τ2)-regular	ADJ
ejpam-6014	266	14	,	,	PUNCT
ejpam-6014	266	15	by	by	ADP
ejpam-6014	266	16	lemma	lemma	PROPN
ejpam-6014	266	17	3	3	NUM
ejpam-6014	266	18	there	there	ADV
ejpam-6014	266	19	exists	exist	VERB
ejpam-6014	266	20	a	a	DET
ejpam-6014	266	21	τ1τ2	τ1τ2	NOUN
ejpam-6014	266	22	-	-	ADJ
ejpam-6014	266	23	open	open	ADJ
ejpam-6014	266	24	set	set	ADJ
ejpam-6014	266	25	u	u	NOUN
ejpam-6014	266	26	of	of	ADP
ejpam-6014	266	27	x	x	SYM
ejpam-6014	266	28	such	such	ADJ
ejpam-6014	266	29	that	that	SCONJ
ejpam-6014	266	30	x	x	SYM
ejpam-6014	266	31	∈	∈	PROPN
ejpam-6014	266	32	u	u	NOUN
ejpam-6014	266	33	⊆	⊆	NUM
ejpam-6014	266	34	τ1τ2	τ1τ2	NOUN
ejpam-6014	266	35	-	-	NOUN
ejpam-6014	266	36	cl(u	cl(u	ADJ
ejpam-6014	266	37	)	)	PUNCT
ejpam-6014	266	38	⊆	⊆	NUM
ejpam-6014	266	39	w	w	NOUN
ejpam-6014	266	40	.	.	PUNCT
ejpam-6014	267	1	thus	thus	ADV
ejpam-6014	267	2	,	,	PUNCT
ejpam-6014	267	3	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	267	4	-	-	PUNCT
ejpam-6014	267	5	cl(u	cl(u	NOUN
ejpam-6014	267	6	)	)	PUNCT
ejpam-6014	267	7	)	)	PUNCT
ejpam-6014	268	1	⊆	⊆	X
ejpam-6014	268	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	268	3	-	-	NUM
ejpam-6014	268	4	cl(v	cl(v	NOUN
ejpam-6014	268	5	)	)	PUNCT
ejpam-6014	268	6	.	.	PUNCT
ejpam-6014	269	1	this	this	PRON
ejpam-6014	269	2	shows	show	VERB
ejpam-6014	269	3	that	that	SCONJ
ejpam-6014	269	4	f	f	PROPN
ejpam-6014	269	5	is	be	AUX
ejpam-6014	269	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	269	7	,	,	PUNCT
ejpam-6014	269	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	269	9	.	.	PUNCT
ejpam-6014	269	10	m.	m.	NOUN
ejpam-6014	269	11	thongmoon	thongmoon	PROPN
ejpam-6014	269	12	,	,	PUNCT
ejpam-6014	269	13	s.	s.	PROPN
ejpam-6014	269	14	sompong	sompong	PROPN
ejpam-6014	269	15	,	,	PUNCT
ejpam-6014	269	16	c.	c.	PROPN
ejpam-6014	269	17	boonpok	boonpok	PROPN
ejpam-6014	269	18	/	/	SYM
ejpam-6014	269	19	eur	eur	PROPN
ejpam-6014	269	20	.	.	PUNCT
ejpam-6014	270	1	j.	j.	PROPN
ejpam-6014	270	2	pure	pure	PROPN
ejpam-6014	270	3	appl	appl	PROPN
ejpam-6014	270	4	.	.	PROPN
ejpam-6014	270	5	math	math	PROPN
ejpam-6014	270	6	,	,	PUNCT
ejpam-6014	270	7	18	18	NUM
ejpam-6014	270	8	(	(	PUNCT
ejpam-6014	270	9	2	2	NUM
ejpam-6014	270	10	)	)	PUNCT
ejpam-6014	270	11	(	(	PUNCT
ejpam-6014	270	12	2025	2025	NUM
ejpam-6014	270	13	)	)	PUNCT
ejpam-6014	270	14	,	,	PUNCT
ejpam-6014	270	15	6014	6014	NUM
ejpam-6014	270	16	9	9	NUM
ejpam-6014	270	17	of	of	ADP
ejpam-6014	270	18	13	13	NUM
ejpam-6014	270	19	4	4	NUM
ejpam-6014	270	20	.	.	PUNCT
ejpam-6014	271	1	some	some	DET
ejpam-6014	271	2	results	result	NOUN
ejpam-6014	271	3	on	on	ADP
ejpam-6014	271	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	271	5	,	,	PUNCT
ejpam-6014	271	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6014	271	7	recall	recall	VERB
ejpam-6014	271	8	that	that	SCONJ
ejpam-6014	271	9	a	a	DET
ejpam-6014	271	10	bitopological	bitopological	ADJ
ejpam-6014	271	11	space	space	NOUN
ejpam-6014	271	12	(	(	PUNCT
ejpam-6014	271	13	x	x	NOUN
ejpam-6014	271	14	,	,	PUNCT
ejpam-6014	271	15	τ1	τ1	NOUN
ejpam-6014	271	16	,	,	PUNCT
ejpam-6014	271	17	τ2	τ2	NOUN
ejpam-6014	271	18	)	)	PUNCT
ejpam-6014	271	19	is	be	AUX
ejpam-6014	271	20	said	say	VERB
ejpam-6014	271	21	to	to	PART
ejpam-6014	271	22	be	be	AUX
ejpam-6014	271	23	(	(	PUNCT
ejpam-6014	271	24	τ1	τ1	NOUN
ejpam-6014	271	25	,	,	PUNCT
ejpam-6014	271	26	τ2)-t2	τ2)-t2	X
ejpam-6014	272	1	[	[	X
ejpam-6014	272	2	38	38	NUM
ejpam-6014	272	3	]	]	PUNCT
ejpam-6014	272	4	if	if	SCONJ
ejpam-6014	272	5	for	for	ADP
ejpam-6014	272	6	any	any	DET
ejpam-6014	272	7	pair	pair	NOUN
ejpam-6014	272	8	of	of	ADP
ejpam-6014	272	9	distinct	distinct	ADJ
ejpam-6014	272	10	points	point	NOUN
ejpam-6014	272	11	x	x	X
ejpam-6014	272	12	,	,	PUNCT
ejpam-6014	272	13	y	y	PROPN
ejpam-6014	272	14	in	in	ADP
ejpam-6014	272	15	x	x	SYM
ejpam-6014	272	16	,	,	PUNCT
ejpam-6014	272	17	there	there	PRON
ejpam-6014	272	18	exist	exist	VERB
ejpam-6014	272	19	disjoint	disjoint	ADJ
ejpam-6014	272	20	τ1τ2	τ1τ2	ADJ
ejpam-6014	272	21	-	-	ADJ
ejpam-6014	272	22	open	open	ADJ
ejpam-6014	272	23	sets	set	NOUN
ejpam-6014	272	24	u	u	NOUN
ejpam-6014	272	25	and	and	CCONJ
ejpam-6014	272	26	v	v	NOUN
ejpam-6014	272	27	of	of	ADP
ejpam-6014	272	28	x	x	PUNCT
ejpam-6014	272	29	containing	contain	VERB
ejpam-6014	272	30	x	x	PROPN
ejpam-6014	272	31	and	and	CCONJ
ejpam-6014	272	32	y	y	PROPN
ejpam-6014	272	33	,	,	PUNCT
ejpam-6014	272	34	respectively	respectively	ADV
ejpam-6014	272	35	.	.	PUNCT
ejpam-6014	273	1	definition	definition	NOUN
ejpam-6014	273	2	5	5	NUM
ejpam-6014	273	3	.	.	PUNCT
ejpam-6014	274	1	[	[	X
ejpam-6014	274	2	39	39	NUM
ejpam-6014	274	3	]	]	PUNCT
ejpam-6014	274	4	a	a	DET
ejpam-6014	274	5	bitopological	bitopological	ADJ
ejpam-6014	274	6	space	space	NOUN
ejpam-6014	274	7	(	(	PUNCT
ejpam-6014	274	8	x	x	NOUN
ejpam-6014	274	9	,	,	PUNCT
ejpam-6014	274	10	τ1	τ1	NOUN
ejpam-6014	274	11	,	,	PUNCT
ejpam-6014	274	12	τ2	τ2	NOUN
ejpam-6014	274	13	)	)	PUNCT
ejpam-6014	274	14	is	be	AUX
ejpam-6014	274	15	said	say	VERB
ejpam-6014	274	16	to	to	PART
ejpam-6014	274	17	be	be	AUX
ejpam-6014	274	18	τ1τ2	τ1τ2	NOUN
ejpam-6014	274	19	-	-	ADJ
ejpam-6014	274	20	urysohn	urysohn	ADJ
ejpam-6014	274	21	if	if	SCONJ
ejpam-6014	274	22	for	for	ADP
ejpam-6014	274	23	each	each	DET
ejpam-6014	274	24	pair	pair	NOUN
ejpam-6014	274	25	of	of	ADP
ejpam-6014	274	26	distinct	distinct	ADJ
ejpam-6014	274	27	points	point	NOUN
ejpam-6014	274	28	x	x	PUNCT
ejpam-6014	274	29	and	and	CCONJ
ejpam-6014	274	30	y	y	PROPN
ejpam-6014	274	31	in	in	ADP
ejpam-6014	274	32	x	x	SYM
ejpam-6014	274	33	,	,	PUNCT
ejpam-6014	274	34	there	there	PRON
ejpam-6014	274	35	exist	exist	VERB
ejpam-6014	274	36	τ1τ2	τ1τ2	ADJ
ejpam-6014	274	37	-	-	ADJ
ejpam-6014	274	38	open	open	ADJ
ejpam-6014	274	39	sets	set	NOUN
ejpam-6014	274	40	u	u	NOUN
ejpam-6014	274	41	and	and	CCONJ
ejpam-6014	274	42	v	v	ADP
ejpam-6014	274	43	such	such	ADJ
ejpam-6014	274	44	that	that	SCONJ
ejpam-6014	274	45	x	x	SYM
ejpam-6014	274	46	∈	∈	PROPN
ejpam-6014	274	47	u	u	NOUN
ejpam-6014	274	48	,	,	PUNCT
ejpam-6014	274	49	y	y	PROPN
ejpam-6014	274	50	∈	∈	PROPN
ejpam-6014	274	51	v	v	NOUN
ejpam-6014	274	52	and	and	CCONJ
ejpam-6014	274	53	τ1τ2	τ1τ2	NOUN
ejpam-6014	274	54	-	-	NOUN
ejpam-6014	274	55	cl(u	cl(u	NOUN
ejpam-6014	274	56	)	)	PUNCT
ejpam-6014	274	57	∩	∩	NOUN
ejpam-6014	274	58	τ1τ2	τ1τ2	NOUN
ejpam-6014	274	59	-	-	NOUN
ejpam-6014	274	60	cl(v	cl(v	X
ejpam-6014	274	61	)	)	PUNCT
ejpam-6014	274	62	=	=	PUNCT
ejpam-6014	274	63	∅.	∅.	NOUN
ejpam-6014	274	64	theorem	theorem	VERB
ejpam-6014	274	65	9	9	NUM
ejpam-6014	274	66	.	.	PUNCT
ejpam-6014	275	1	let	let	AUX
ejpam-6014	275	2	(	(	PUNCT
ejpam-6014	275	3	x	x	NOUN
ejpam-6014	275	4	,	,	PUNCT
ejpam-6014	275	5	τ1	τ1	NOUN
ejpam-6014	275	6	,	,	PUNCT
ejpam-6014	275	7	τ2	τ2	PROPN
ejpam-6014	275	8	)	)	PUNCT
ejpam-6014	275	9	be	be	VERB
ejpam-6014	275	10	a	a	DET
ejpam-6014	275	11	bitopological	bitopological	ADJ
ejpam-6014	275	12	space	space	NOUN
ejpam-6014	275	13	.	.	PUNCT
ejpam-6014	276	1	if	if	SCONJ
ejpam-6014	276	2	for	for	ADP
ejpam-6014	276	3	any	any	DET
ejpam-6014	276	4	distinct	distinct	ADJ
ejpam-6014	276	5	points	point	NOUN
ejpam-6014	276	6	x	x	PUNCT
ejpam-6014	276	7	and	and	CCONJ
ejpam-6014	276	8	x′	x′	PROPN
ejpam-6014	276	9	in	in	ADP
ejpam-6014	276	10	x	x	SYM
ejpam-6014	276	11	,	,	PUNCT
ejpam-6014	276	12	there	there	PRON
ejpam-6014	276	13	exists	exist	VERB
ejpam-6014	276	14	a	a	DET
ejpam-6014	276	15	function	function	NOUN
ejpam-6014	276	16	f	f	NOUN
ejpam-6014	276	17	:	:	PUNCT
ejpam-6014	276	18	(	(	PUNCT
ejpam-6014	276	19	x	x	NOUN
ejpam-6014	276	20	,	,	PUNCT
ejpam-6014	276	21	τ1	τ1	NOUN
ejpam-6014	276	22	,	,	PUNCT
ejpam-6014	276	23	τ2	τ2	NOUN
ejpam-6014	276	24	)	)	PUNCT
ejpam-6014	276	25	→	→	SYM
ejpam-6014	276	26	(	(	PUNCT
ejpam-6014	276	27	y	y	PROPN
ejpam-6014	276	28	,	,	PUNCT
ejpam-6014	276	29	σ1	σ1	PROPN
ejpam-6014	276	30	,	,	PUNCT
ejpam-6014	276	31	σ2	σ2	NOUN
ejpam-6014	276	32	)	)	PUNCT
ejpam-6014	276	33	such	such	ADJ
ejpam-6014	276	34	that	that	SCONJ
ejpam-6014	276	35	(	(	PUNCT
ejpam-6014	276	36	1	1	NUM
ejpam-6014	276	37	)	)	PUNCT
ejpam-6014	276	38	(	(	PUNCT
ejpam-6014	276	39	y	y	PROPN
ejpam-6014	276	40	,	,	PUNCT
ejpam-6014	276	41	σ1	σ1	PROPN
ejpam-6014	276	42	,	,	PUNCT
ejpam-6014	276	43	σ2	σ2	PROPN
ejpam-6014	276	44	)	)	PUNCT
ejpam-6014	276	45	is	be	AUX
ejpam-6014	276	46	σ1σ2	σ1σ2	NOUN
ejpam-6014	276	47	-	-	ADJ
ejpam-6014	276	48	urysohn	urysohn	ADJ
ejpam-6014	276	49	,	,	PUNCT
ejpam-6014	276	50	(	(	PUNCT
ejpam-6014	276	51	2	2	X
ejpam-6014	276	52	)	)	PUNCT
ejpam-6014	276	53	f(x	f(x	PROPN
ejpam-6014	276	54	)	)	PUNCT
ejpam-6014	276	55	̸=	̸=	PROPN
ejpam-6014	276	56	f(x′	f(x′	NUM
ejpam-6014	276	57	)	)	PUNCT
ejpam-6014	276	58	,	,	PUNCT
ejpam-6014	276	59	and	and	CCONJ
ejpam-6014	276	60	(	(	PUNCT
ejpam-6014	276	61	3	3	X
ejpam-6014	276	62	)	)	PUNCT
ejpam-6014	276	63	f	f	PROPN
ejpam-6014	276	64	is	be	AUX
ejpam-6014	276	65	θ(τ1	θ(τ1	NOUN
ejpam-6014	276	66	,	,	PUNCT
ejpam-6014	276	67	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	276	68	at	at	ADP
ejpam-6014	276	69	x	x	PROPN
ejpam-6014	276	70	and	and	CCONJ
ejpam-6014	276	71	x′	x′	NUM
ejpam-6014	276	72	,	,	PUNCT
ejpam-6014	276	73	then	then	ADV
ejpam-6014	276	74	(	(	PUNCT
ejpam-6014	276	75	x	x	NOUN
ejpam-6014	276	76	,	,	PUNCT
ejpam-6014	276	77	τ1	τ1	NOUN
ejpam-6014	276	78	,	,	PUNCT
ejpam-6014	276	79	τ2	τ2	NOUN
ejpam-6014	276	80	)	)	PUNCT
ejpam-6014	276	81	is	be	AUX
ejpam-6014	276	82	τ1τ2	τ1τ2	NOUN
ejpam-6014	276	83	-	-	ADJ
ejpam-6014	276	84	urysohn	urysohn	ADJ
ejpam-6014	276	85	.	.	PUNCT
ejpam-6014	277	1	proof	proof	NOUN
ejpam-6014	277	2	.	.	PUNCT
ejpam-6014	278	1	let	let	VERB
ejpam-6014	278	2	x	x	PRON
ejpam-6014	278	3	,	,	PUNCT
ejpam-6014	278	4	x′	x′	PROPN
ejpam-6014	278	5	be	be	AUX
ejpam-6014	278	6	any	any	DET
ejpam-6014	278	7	distinct	distinct	ADJ
ejpam-6014	278	8	points	point	NOUN
ejpam-6014	278	9	of	of	ADP
ejpam-6014	278	10	x.	x.	NOUN
ejpam-6014	278	11	then	then	ADV
ejpam-6014	278	12	,	,	PUNCT
ejpam-6014	278	13	by	by	ADP
ejpam-6014	278	14	the	the	DET
ejpam-6014	278	15	hypothesis	hypothesis	NOUN
ejpam-6014	278	16	there	there	PRON
ejpam-6014	278	17	exists	exist	VERB
ejpam-6014	278	18	a	a	DET
ejpam-6014	278	19	function	function	NOUN
ejpam-6014	278	20	f	f	NOUN
ejpam-6014	278	21	:	:	PUNCT
ejpam-6014	278	22	(	(	PUNCT
ejpam-6014	278	23	x	x	NOUN
ejpam-6014	278	24	,	,	PUNCT
ejpam-6014	278	25	τ1	τ1	NOUN
ejpam-6014	278	26	,	,	PUNCT
ejpam-6014	278	27	τ2	τ2	NOUN
ejpam-6014	278	28	)	)	PUNCT
ejpam-6014	278	29	→	→	SYM
ejpam-6014	278	30	(	(	PUNCT
ejpam-6014	278	31	y	y	PROPN
ejpam-6014	278	32	,	,	PUNCT
ejpam-6014	278	33	σ1	σ1	PROPN
ejpam-6014	278	34	,	,	PUNCT
ejpam-6014	278	35	σ2	σ2	PROPN
ejpam-6014	278	36	)	)	PUNCT
ejpam-6014	278	37	which	which	PRON
ejpam-6014	278	38	satisfies	satisfy	VERB
ejpam-6014	278	39	three	three	NUM
ejpam-6014	278	40	conditions	condition	NOUN
ejpam-6014	278	41	.	.	PUNCT
ejpam-6014	279	1	now	now	ADV
ejpam-6014	279	2	let	let	VERB
ejpam-6014	279	3	y	y	PROPN
ejpam-6014	279	4	=	=	PUNCT
ejpam-6014	279	5	f(x	f(x	PROPN
ejpam-6014	279	6	)	)	PUNCT
ejpam-6014	279	7	and	and	CCONJ
ejpam-6014	279	8	y′	y′	NOUN
ejpam-6014	279	9	=	=	SYM
ejpam-6014	279	10	f(x′	f(x′	PROPN
ejpam-6014	279	11	)	)	PUNCT
ejpam-6014	279	12	.	.	PUNCT
ejpam-6014	280	1	then	then	ADV
ejpam-6014	280	2	,	,	PUNCT
ejpam-6014	280	3	y	y	PROPN
ejpam-6014	280	4	̸=	̸=	PROPN
ejpam-6014	280	5	y′.	y′.	VERB
ejpam-6014	280	6	since	since	SCONJ
ejpam-6014	280	7	(	(	PUNCT
ejpam-6014	280	8	y	y	PROPN
ejpam-6014	280	9	,	,	PUNCT
ejpam-6014	280	10	σ1	σ1	PROPN
ejpam-6014	280	11	,	,	PUNCT
ejpam-6014	280	12	σ2	σ2	PROPN
ejpam-6014	280	13	)	)	PUNCT
ejpam-6014	280	14	is	be	AUX
ejpam-6014	280	15	σ1σ2	σ1σ2	NOUN
ejpam-6014	280	16	-	-	PUNCT
ejpam-6014	280	17	urysohn	urysohn	ADJ
ejpam-6014	280	18	,	,	PUNCT
ejpam-6014	280	19	there	there	PRON
ejpam-6014	280	20	exist	exist	VERB
ejpam-6014	280	21	σ1σ2	σ1σ2	NOUN
ejpam-6014	280	22	-	-	ADJ
ejpam-6014	280	23	open	open	ADJ
ejpam-6014	280	24	sets	set	NOUN
ejpam-6014	280	25	v	v	ADP
ejpam-6014	280	26	and	and	CCONJ
ejpam-6014	280	27	v	v	ADP
ejpam-6014	280	28	′	′	NUM
ejpam-6014	280	29	of	of	ADP
ejpam-6014	280	30	y	y	PROPN
ejpam-6014	280	31	containing	contain	VERB
ejpam-6014	280	32	y	y	PROPN
ejpam-6014	280	33	and	and	CCONJ
ejpam-6014	280	34	y′	y′	NUM
ejpam-6014	280	35	,	,	PUNCT
ejpam-6014	280	36	respectively	respectively	ADV
ejpam-6014	280	37	,	,	PUNCT
ejpam-6014	280	38	such	such	ADJ
ejpam-6014	280	39	that	that	SCONJ
ejpam-6014	280	40	σ1σ2	σ1σ2	NOUN
ejpam-6014	280	41	-	-	NUM
ejpam-6014	280	42	cl(v	cl(v	NOUN
ejpam-6014	280	43	)	)	PUNCT
ejpam-6014	280	44	∩	∩	NOUN
ejpam-6014	280	45	σ1σ2	σ1σ2	NOUN
ejpam-6014	280	46	-	-	PUNCT
ejpam-6014	280	47	cl(v	cl(v	PRON
ejpam-6014	280	48	′	′	NOUN
ejpam-6014	280	49	)	)	PUNCT
ejpam-6014	280	50	=	=	PUNCT
ejpam-6014	280	51	∅.	∅.	NOUN
ejpam-6014	280	52	since	since	SCONJ
ejpam-6014	280	53	f	f	PROPN
ejpam-6014	280	54	is	be	AUX
ejpam-6014	280	55	θ(τ1	θ(τ1	NOUN
ejpam-6014	280	56	,	,	PUNCT
ejpam-6014	280	57	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	280	58	at	at	ADP
ejpam-6014	280	59	x	x	PROPN
ejpam-6014	280	60	and	and	CCONJ
ejpam-6014	280	61	x′	x′	NUM
ejpam-6014	280	62	,	,	PUNCT
ejpam-6014	280	63	there	there	PRON
ejpam-6014	280	64	exist	exist	VERB
ejpam-6014	280	65	τ1τ2	τ1τ2	ADJ
ejpam-6014	280	66	-	-	ADJ
ejpam-6014	280	67	open	open	ADJ
ejpam-6014	280	68	sets	set	NOUN
ejpam-6014	280	69	u	u	NOUN
ejpam-6014	280	70	and	and	CCONJ
ejpam-6014	280	71	u	u	NOUN
ejpam-6014	280	72	′	′	NOUN
ejpam-6014	280	73	of	of	ADP
ejpam-6014	280	74	x	x	PUNCT
ejpam-6014	280	75	containing	contain	VERB
ejpam-6014	280	76	x	x	PROPN
ejpam-6014	280	77	and	and	CCONJ
ejpam-6014	280	78	x′	x′	NUM
ejpam-6014	280	79	,	,	PUNCT
ejpam-6014	280	80	respectively	respectively	ADV
ejpam-6014	280	81	,	,	PUNCT
ejpam-6014	280	82	such	such	ADJ
ejpam-6014	280	83	that	that	SCONJ
ejpam-6014	280	84	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	280	85	-	-	PUNCT
ejpam-6014	280	86	cl(u	cl(u	NOUN
ejpam-6014	280	87	)	)	PUNCT
ejpam-6014	280	88	)	)	PUNCT
ejpam-6014	281	1	⊆	⊆	X
ejpam-6014	281	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	281	3	-	-	NUM
ejpam-6014	281	4	cl(v	cl(v	NOUN
ejpam-6014	281	5	)	)	PUNCT
ejpam-6014	281	6	and	and	CCONJ
ejpam-6014	281	7	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	281	8	-	-	PUNCT
ejpam-6014	281	9	cl(u	cl(u	NOUN
ejpam-6014	281	10	′	′	NOUN
ejpam-6014	281	11	)	)	PUNCT
ejpam-6014	281	12	)	)	PUNCT
ejpam-6014	281	13	⊆	⊆	X
ejpam-6014	281	14	σ1σ2	σ1σ2	NUM
ejpam-6014	281	15	-	-	PUNCT
ejpam-6014	281	16	cl(v	cl(v	PRON
ejpam-6014	281	17	′	′	NOUN
ejpam-6014	281	18	)	)	PUNCT
ejpam-6014	281	19	.	.	PUNCT
ejpam-6014	282	1	this	this	PRON
ejpam-6014	282	2	implies	imply	VERB
ejpam-6014	282	3	that	that	SCONJ
ejpam-6014	282	4	τ1τ2	τ1τ2	NOUN
ejpam-6014	282	5	-	-	NOUN
ejpam-6014	282	6	cl(u	cl(u	NOUN
ejpam-6014	282	7	)	)	PUNCT
ejpam-6014	282	8	∩	∩	NOUN
ejpam-6014	282	9	τ1τ2	τ1τ2	NOUN
ejpam-6014	282	10	-	-	NOUN
ejpam-6014	282	11	cl(u	cl(u	NOUN
ejpam-6014	282	12	′	′	NOUN
ejpam-6014	282	13	)	)	PUNCT
ejpam-6014	282	14	=	=	NOUN
ejpam-6014	282	15	∅.	∅.	ADP
ejpam-6014	282	16	thus	thus	ADV
ejpam-6014	282	17	,	,	PUNCT
ejpam-6014	282	18	(	(	PUNCT
ejpam-6014	282	19	x	x	NOUN
ejpam-6014	282	20	,	,	PUNCT
ejpam-6014	282	21	τ1	τ1	NOUN
ejpam-6014	282	22	,	,	PUNCT
ejpam-6014	282	23	τ2	τ2	NOUN
ejpam-6014	282	24	)	)	PUNCT
ejpam-6014	282	25	is	be	AUX
ejpam-6014	282	26	τ1τ2	τ1τ2	NOUN
ejpam-6014	282	27	-	-	ADJ
ejpam-6014	282	28	urysohn	urysohn	ADJ
ejpam-6014	282	29	.	.	PUNCT
ejpam-6014	283	1	definition	definition	NOUN
ejpam-6014	283	2	6	6	NUM
ejpam-6014	283	3	.	.	PUNCT
ejpam-6014	284	1	a	a	DET
ejpam-6014	284	2	function	function	NOUN
ejpam-6014	284	3	f	f	NOUN
ejpam-6014	284	4	:	:	PUNCT
ejpam-6014	284	5	(	(	PUNCT
ejpam-6014	284	6	x	x	NOUN
ejpam-6014	284	7	,	,	PUNCT
ejpam-6014	284	8	τ1	τ1	NOUN
ejpam-6014	284	9	,	,	PUNCT
ejpam-6014	284	10	τ2	τ2	NOUN
ejpam-6014	284	11	)	)	PUNCT
ejpam-6014	284	12	→	→	SYM
ejpam-6014	284	13	(	(	PUNCT
ejpam-6014	284	14	y	y	PROPN
ejpam-6014	284	15	,	,	PUNCT
ejpam-6014	284	16	σ1	σ1	PROPN
ejpam-6014	284	17	,	,	PUNCT
ejpam-6014	284	18	σ2	σ2	PROPN
ejpam-6014	284	19	)	)	PUNCT
ejpam-6014	284	20	is	be	AUX
ejpam-6014	284	21	said	say	VERB
ejpam-6014	284	22	to	to	PART
ejpam-6014	284	23	have	have	VERB
ejpam-6014	284	24	a	a	DET
ejpam-6014	284	25	strong	strong	ADJ
ejpam-6014	284	26	θ(τ1	θ(τ1	NOUN
ejpam-6014	284	27	,	,	PUNCT
ejpam-6014	284	28	τ2)closed	τ2)close	VERB
ejpam-6014	284	29	graph	graph	NOUN
ejpam-6014	284	30	if	if	SCONJ
ejpam-6014	284	31	for	for	ADP
ejpam-6014	284	32	each	each	DET
ejpam-6014	284	33	(	(	PUNCT
ejpam-6014	284	34	x	x	NOUN
ejpam-6014	284	35	,	,	PUNCT
ejpam-6014	284	36	y	y	NOUN
ejpam-6014	284	37	)	)	PUNCT
ejpam-6014	284	38	∈	∈	PROPN
ejpam-6014	284	39	(	(	PUNCT
ejpam-6014	284	40	x	x	SYM
ejpam-6014	284	41	×	×	PROPN
ejpam-6014	284	42	y	y	PROPN
ejpam-6014	284	43	)	)	PUNCT
ejpam-6014	285	1	−	−	PROPN
ejpam-6014	285	2	g(f	g(f	NOUN
ejpam-6014	285	3	)	)	PUNCT
ejpam-6014	285	4	,	,	PUNCT
ejpam-6014	285	5	there	there	PRON
ejpam-6014	285	6	exist	exist	VERB
ejpam-6014	285	7	a	a	DET
ejpam-6014	285	8	τ1τ2	τ1τ2	NOUN
ejpam-6014	285	9	-	-	ADJ
ejpam-6014	285	10	open	open	ADJ
ejpam-6014	285	11	set	set	ADJ
ejpam-6014	285	12	u	u	NOUN
ejpam-6014	285	13	of	of	ADP
ejpam-6014	285	14	x	x	PUNCT
ejpam-6014	285	15	containing	contain	VERB
ejpam-6014	285	16	x	x	X
ejpam-6014	285	17	and	and	CCONJ
ejpam-6014	285	18	a	a	DET
ejpam-6014	285	19	σ1σ2	σ1σ2	NUM
ejpam-6014	285	20	-	-	ADJ
ejpam-6014	285	21	open	open	ADJ
ejpam-6014	285	22	set	set	NOUN
ejpam-6014	285	23	v	v	NOUN
ejpam-6014	285	24	of	of	ADP
ejpam-6014	285	25	y	y	PROPN
ejpam-6014	285	26	containing	contain	VERB
ejpam-6014	285	27	y	y	PRON
ejpam-6014	285	28	such	such	ADJ
ejpam-6014	285	29	that	that	SCONJ
ejpam-6014	285	30	[	[	X
ejpam-6014	285	31	τ1τ2	τ1τ2	NOUN
ejpam-6014	285	32	-	-	PUNCT
ejpam-6014	285	33	cl(u)×	cl(u)×	PRON
ejpam-6014	285	34	σ1σ2	σ1σ2	NOUN
ejpam-6014	285	35	-	-	NUM
ejpam-6014	285	36	cl(v	cl(v	NOUN
ejpam-6014	285	37	)	)	PUNCT
ejpam-6014	285	38	]	]	PUNCT
ejpam-6014	285	39	∩g(f	∩g(f	PROPN
ejpam-6014	285	40	)	)	PUNCT
ejpam-6014	285	41	=	=	PUNCT
ejpam-6014	285	42	∅.	∅.	PRON
ejpam-6014	285	43	lemma	lemma	PROPN
ejpam-6014	285	44	6	6	NUM
ejpam-6014	285	45	.	.	PUNCT
ejpam-6014	286	1	a	a	DET
ejpam-6014	286	2	function	function	NOUN
ejpam-6014	286	3	f	f	NOUN
ejpam-6014	286	4	:	:	PUNCT
ejpam-6014	286	5	(	(	PUNCT
ejpam-6014	286	6	x	x	NOUN
ejpam-6014	286	7	,	,	PUNCT
ejpam-6014	286	8	τ1	τ1	NOUN
ejpam-6014	286	9	,	,	PUNCT
ejpam-6014	286	10	τ2	τ2	NOUN
ejpam-6014	286	11	)	)	PUNCT
ejpam-6014	286	12	→	→	SYM
ejpam-6014	286	13	(	(	PUNCT
ejpam-6014	286	14	y	y	PROPN
ejpam-6014	286	15	,	,	PUNCT
ejpam-6014	286	16	σ1	σ1	PROPN
ejpam-6014	286	17	,	,	PUNCT
ejpam-6014	286	18	σ2	σ2	NOUN
ejpam-6014	286	19	)	)	PUNCT
ejpam-6014	286	20	has	have	VERB
ejpam-6014	286	21	a	a	DET
ejpam-6014	286	22	strong	strong	ADJ
ejpam-6014	286	23	θ(τ1	θ(τ1	NOUN
ejpam-6014	286	24	,	,	PUNCT
ejpam-6014	286	25	τ2)s	τ2)s	NOUN
ejpam-6014	286	26	-	-	PUNCT
ejpam-6014	286	27	closed	closed	ADJ
ejpam-6014	286	28	graph	graph	NOUN
ejpam-6014	286	29	if	if	SCONJ
ejpam-6014	286	30	and	and	CCONJ
ejpam-6014	286	31	only	only	ADV
ejpam-6014	286	32	if	if	SCONJ
ejpam-6014	286	33	for	for	ADP
ejpam-6014	286	34	each	each	DET
ejpam-6014	286	35	(	(	PUNCT
ejpam-6014	286	36	x	x	NOUN
ejpam-6014	286	37	,	,	PUNCT
ejpam-6014	286	38	y	y	NOUN
ejpam-6014	286	39	)	)	PUNCT
ejpam-6014	286	40	∈	∈	PROPN
ejpam-6014	286	41	(	(	PUNCT
ejpam-6014	286	42	x×y	x×y	PROPN
ejpam-6014	286	43	)	)	PUNCT
ejpam-6014	286	44	−g(f	−g(f	NOUN
ejpam-6014	286	45	)	)	PUNCT
ejpam-6014	286	46	,	,	PUNCT
ejpam-6014	286	47	there	there	PRON
ejpam-6014	286	48	exist	exist	VERB
ejpam-6014	286	49	a	a	DET
ejpam-6014	286	50	τ1τ2	τ1τ2	NOUN
ejpam-6014	286	51	-	-	ADJ
ejpam-6014	286	52	open	open	ADJ
ejpam-6014	286	53	set	set	ADJ
ejpam-6014	286	54	u	u	NOUN
ejpam-6014	286	55	of	of	ADP
ejpam-6014	286	56	x	x	PUNCT
ejpam-6014	286	57	containing	contain	VERB
ejpam-6014	286	58	x	x	X
ejpam-6014	286	59	and	and	CCONJ
ejpam-6014	286	60	a	a	DET
ejpam-6014	286	61	σ1σ2	σ1σ2	NUM
ejpam-6014	286	62	-	-	ADJ
ejpam-6014	286	63	open	open	ADJ
ejpam-6014	286	64	set	set	NOUN
ejpam-6014	286	65	v	v	NOUN
ejpam-6014	286	66	of	of	ADP
ejpam-6014	286	67	y	y	PROPN
ejpam-6014	286	68	containing	contain	VERB
ejpam-6014	286	69	y	y	PRON
ejpam-6014	286	70	such	such	ADJ
ejpam-6014	286	71	that	that	SCONJ
ejpam-6014	286	72	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	286	73	-	-	PUNCT
ejpam-6014	286	74	cl(u	cl(u	NOUN
ejpam-6014	286	75	)	)	PUNCT
ejpam-6014	286	76	)	)	PUNCT
ejpam-6014	287	1	∩	∩	NOUN
ejpam-6014	287	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	287	3	-	-	NUM
ejpam-6014	287	4	cl(v	cl(v	X
ejpam-6014	287	5	)	)	PUNCT
ejpam-6014	287	6	=	=	PUNCT
ejpam-6014	287	7	∅.	∅.	NOUN
ejpam-6014	287	8	theorem	theorem	VERB
ejpam-6014	287	9	10	10	NUM
ejpam-6014	287	10	.	.	PUNCT
ejpam-6014	288	1	if	if	SCONJ
ejpam-6014	288	2	f	f	PROPN
ejpam-6014	288	3	:	:	PUNCT
ejpam-6014	288	4	(	(	PUNCT
ejpam-6014	288	5	x	x	NOUN
ejpam-6014	288	6	,	,	PUNCT
ejpam-6014	288	7	τ1	τ1	NOUN
ejpam-6014	288	8	,	,	PUNCT
ejpam-6014	288	9	τ2	τ2	NOUN
ejpam-6014	288	10	)	)	PUNCT
ejpam-6014	288	11	→	→	SYM
ejpam-6014	288	12	(	(	PUNCT
ejpam-6014	288	13	y	y	PROPN
ejpam-6014	288	14	,	,	PUNCT
ejpam-6014	288	15	σ1	σ1	PROPN
ejpam-6014	288	16	,	,	PUNCT
ejpam-6014	288	17	σ2	σ2	PROPN
ejpam-6014	288	18	)	)	PUNCT
ejpam-6014	288	19	is	be	AUX
ejpam-6014	288	20	θ(τ1	θ(τ1	NOUN
ejpam-6014	288	21	,	,	PUNCT
ejpam-6014	288	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	288	23	and	and	CCONJ
ejpam-6014	288	24	(	(	PUNCT
ejpam-6014	288	25	y	y	PROPN
ejpam-6014	288	26	,	,	PUNCT
ejpam-6014	288	27	σ1	σ1	PROPN
ejpam-6014	288	28	,	,	PUNCT
ejpam-6014	288	29	σ2	σ2	PROPN
ejpam-6014	288	30	)	)	PUNCT
ejpam-6014	288	31	is	be	AUX
ejpam-6014	288	32	σ1σ2	σ1σ2	NOUN
ejpam-6014	288	33	-	-	PUNCT
ejpam-6014	288	34	urysohn	urysohn	ADJ
ejpam-6014	288	35	,	,	PUNCT
ejpam-6014	288	36	then	then	ADV
ejpam-6014	288	37	g(f	g(f	PROPN
ejpam-6014	288	38	)	)	PUNCT
ejpam-6014	289	1	is	be	AUX
ejpam-6014	289	2	strong	strong	ADJ
ejpam-6014	289	3	θ(τ1	θ(τ1	NOUN
ejpam-6014	289	4	,	,	PUNCT
ejpam-6014	289	5	τ2)-closed	τ2)-closed	ADJ
ejpam-6014	289	6	.	.	PUNCT
ejpam-6014	290	1	proof	proof	NOUN
ejpam-6014	290	2	.	.	PUNCT
ejpam-6014	291	1	suppose	suppose	VERB
ejpam-6014	291	2	that	that	SCONJ
ejpam-6014	291	3	(	(	PUNCT
ejpam-6014	291	4	x	x	X
ejpam-6014	291	5	,	,	PUNCT
ejpam-6014	291	6	y	y	NOUN
ejpam-6014	291	7	)	)	PUNCT
ejpam-6014	291	8	∈	∈	PROPN
ejpam-6014	291	9	(	(	PUNCT
ejpam-6014	291	10	x×y	x×y	PROPN
ejpam-6014	291	11	)	)	PUNCT
ejpam-6014	291	12	−g(f	−g(f	NOUN
ejpam-6014	291	13	)	)	PUNCT
ejpam-6014	291	14	.	.	PUNCT
ejpam-6014	292	1	then	then	ADV
ejpam-6014	292	2	,	,	PUNCT
ejpam-6014	292	3	y	y	PROPN
ejpam-6014	292	4	̸=	̸=	PROPN
ejpam-6014	292	5	f(x	f(x	PROPN
ejpam-6014	292	6	)	)	PUNCT
ejpam-6014	292	7	.	.	PUNCT
ejpam-6014	293	1	since	since	SCONJ
ejpam-6014	293	2	(	(	PUNCT
ejpam-6014	293	3	y	y	PROPN
ejpam-6014	293	4	,	,	PUNCT
ejpam-6014	293	5	σ1	σ1	PROPN
ejpam-6014	293	6	,	,	PUNCT
ejpam-6014	293	7	σ2	σ2	PROPN
ejpam-6014	293	8	)	)	PUNCT
ejpam-6014	293	9	is	be	AUX
ejpam-6014	293	10	σ1σ2urysohn	σ1σ2urysohn	NUM
ejpam-6014	293	11	,	,	PUNCT
ejpam-6014	293	12	there	there	PRON
ejpam-6014	293	13	exist	exist	VERB
ejpam-6014	293	14	σ1σ2	σ1σ2	NOUN
ejpam-6014	293	15	-	-	ADJ
ejpam-6014	293	16	open	open	ADJ
ejpam-6014	293	17	sets	set	NOUN
ejpam-6014	293	18	v	v	ADP
ejpam-6014	293	19	and	and	CCONJ
ejpam-6014	293	20	w	w	PROPN
ejpam-6014	293	21	of	of	ADP
ejpam-6014	293	22	y	y	PROPN
ejpam-6014	293	23	containing	contain	VERB
ejpam-6014	293	24	y	y	PROPN
ejpam-6014	293	25	and	and	CCONJ
ejpam-6014	293	26	f(x	f(x	PROPN
ejpam-6014	293	27	)	)	PUNCT
ejpam-6014	293	28	,	,	PUNCT
ejpam-6014	293	29	respectively	respectively	ADV
ejpam-6014	293	30	,	,	PUNCT
ejpam-6014	293	31	such	such	ADJ
ejpam-6014	293	32	that	that	SCONJ
ejpam-6014	293	33	σ1σ2	σ1σ2	NOUN
ejpam-6014	293	34	-	-	PUNCT
ejpam-6014	293	35	cl(v	cl(v	NOUN
ejpam-6014	293	36	)	)	PUNCT
ejpam-6014	293	37	∩	∩	NOUN
ejpam-6014	293	38	σ1σ2	σ1σ2	NOUN
ejpam-6014	293	39	-	-	NUM
ejpam-6014	293	40	cl(w	cl(w	NOUN
ejpam-6014	293	41	)	)	PUNCT
ejpam-6014	294	1	=	=	PUNCT
ejpam-6014	294	2	∅.	∅.	NOUN
ejpam-6014	294	3	since	since	SCONJ
ejpam-6014	294	4	f	f	PROPN
ejpam-6014	294	5	is	be	AUX
ejpam-6014	294	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	294	7	,	,	PUNCT
ejpam-6014	294	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	294	9	,	,	PUNCT
ejpam-6014	294	10	there	there	PRON
ejpam-6014	294	11	exists	exist	VERB
ejpam-6014	294	12	a	a	DET
ejpam-6014	294	13	τ1τ2	τ1τ2	NOUN
ejpam-6014	294	14	-	-	ADJ
ejpam-6014	294	15	open	open	ADJ
ejpam-6014	294	16	set	set	ADJ
ejpam-6014	294	17	u	u	NOUN
ejpam-6014	294	18	of	of	ADP
ejpam-6014	294	19	x	x	PUNCT
ejpam-6014	294	20	containing	contain	VERB
ejpam-6014	294	21	x	x	PUNCT
ejpam-6014	294	22	such	such	ADJ
ejpam-6014	294	23	that	that	SCONJ
ejpam-6014	294	24	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	294	25	-	-	PUNCT
ejpam-6014	294	26	cl(u	cl(u	NOUN
ejpam-6014	294	27	)	)	PUNCT
ejpam-6014	294	28	)	)	PUNCT
ejpam-6014	295	1	⊆	⊆	X
ejpam-6014	295	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	295	3	-	-	PUNCT
ejpam-6014	295	4	cl(w	cl(w	NOUN
ejpam-6014	295	5	)	)	PUNCT
ejpam-6014	295	6	.	.	PUNCT
ejpam-6014	296	1	this	this	PRON
ejpam-6014	296	2	implies	imply	VERB
ejpam-6014	296	3	that	that	SCONJ
ejpam-6014	296	4	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	296	5	-	-	PUNCT
ejpam-6014	296	6	cl(u	cl(u	NOUN
ejpam-6014	296	7	)	)	PUNCT
ejpam-6014	296	8	)	)	PUNCT
ejpam-6014	297	1	∩	∩	NOUN
ejpam-6014	297	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	297	3	-	-	NUM
ejpam-6014	297	4	cl(v	cl(v	X
ejpam-6014	297	5	)	)	PUNCT
ejpam-6014	297	6	=	=	NOUN
ejpam-6014	297	7	∅	∅	NOUN
ejpam-6014	297	8	and	and	CCONJ
ejpam-6014	297	9	by	by	ADP
ejpam-6014	297	10	lemma	lemma	PROPN
ejpam-6014	297	11	6	6	NUM
ejpam-6014	297	12	,	,	PUNCT
ejpam-6014	297	13	g(f	g(f	PROPN
ejpam-6014	297	14	)	)	PUNCT
ejpam-6014	297	15	is	be	AUX
ejpam-6014	297	16	strong	strong	ADJ
ejpam-6014	297	17	θ(τ1	θ(τ1	NOUN
ejpam-6014	297	18	,	,	PUNCT
ejpam-6014	297	19	τ2)-closed	τ2)-closed	PROPN
ejpam-6014	297	20	.	.	PUNCT
ejpam-6014	297	21	m.	m.	NOUN
ejpam-6014	297	22	thongmoon	thongmoon	PROPN
ejpam-6014	297	23	,	,	PUNCT
ejpam-6014	297	24	s.	s.	PROPN
ejpam-6014	297	25	sompong	sompong	PROPN
ejpam-6014	297	26	,	,	PUNCT
ejpam-6014	297	27	c.	c.	PROPN
ejpam-6014	297	28	boonpok	boonpok	PROPN
ejpam-6014	297	29	/	/	SYM
ejpam-6014	297	30	eur	eur	PROPN
ejpam-6014	297	31	.	.	PUNCT
ejpam-6014	298	1	j.	j.	PROPN
ejpam-6014	298	2	pure	pure	PROPN
ejpam-6014	298	3	appl	appl	PROPN
ejpam-6014	298	4	.	.	PROPN
ejpam-6014	298	5	math	math	PROPN
ejpam-6014	298	6	,	,	PUNCT
ejpam-6014	298	7	18	18	NUM
ejpam-6014	298	8	(	(	PUNCT
ejpam-6014	298	9	2	2	NUM
ejpam-6014	298	10	)	)	PUNCT
ejpam-6014	298	11	(	(	PUNCT
ejpam-6014	298	12	2025	2025	NUM
ejpam-6014	298	13	)	)	PUNCT
ejpam-6014	298	14	,	,	PUNCT
ejpam-6014	298	15	6014	6014	NUM
ejpam-6014	298	16	10	10	NUM
ejpam-6014	298	17	of	of	ADP
ejpam-6014	298	18	13	13	NUM
ejpam-6014	298	19	theorem	theorem	NOUN
ejpam-6014	298	20	11	11	NUM
ejpam-6014	298	21	.	.	PUNCT
ejpam-6014	299	1	if	if	SCONJ
ejpam-6014	299	2	f	f	PROPN
ejpam-6014	299	3	:	:	PUNCT
ejpam-6014	299	4	(	(	PUNCT
ejpam-6014	299	5	x	x	NOUN
ejpam-6014	299	6	,	,	PUNCT
ejpam-6014	299	7	τ1	τ1	NOUN
ejpam-6014	299	8	,	,	PUNCT
ejpam-6014	299	9	τ2	τ2	NOUN
ejpam-6014	299	10	)	)	PUNCT
ejpam-6014	299	11	→	→	SYM
ejpam-6014	299	12	(	(	PUNCT
ejpam-6014	299	13	y	y	PROPN
ejpam-6014	299	14	,	,	PUNCT
ejpam-6014	299	15	σ1	σ1	PROPN
ejpam-6014	299	16	,	,	PUNCT
ejpam-6014	299	17	σ2	σ2	PROPN
ejpam-6014	299	18	)	)	PUNCT
ejpam-6014	299	19	is	be	AUX
ejpam-6014	299	20	an	an	DET
ejpam-6014	299	21	injective	injective	ADJ
ejpam-6014	299	22	θ(τ1	θ(τ1	NOUN
ejpam-6014	299	23	,	,	PUNCT
ejpam-6014	299	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	299	25	function	function	NOUN
ejpam-6014	299	26	with	with	ADP
ejpam-6014	299	27	a	a	DET
ejpam-6014	299	28	strong	strong	ADJ
ejpam-6014	299	29	θ(τ1	θ(τ1	NOUN
ejpam-6014	299	30	,	,	PUNCT
ejpam-6014	299	31	τ2)-closed	τ2)-close	VERB
ejpam-6014	299	32	graph	graph	NOUN
ejpam-6014	299	33	,	,	PUNCT
ejpam-6014	299	34	then	then	ADV
ejpam-6014	299	35	(	(	PUNCT
ejpam-6014	299	36	x	x	NOUN
ejpam-6014	299	37	,	,	PUNCT
ejpam-6014	299	38	τ1	τ1	NOUN
ejpam-6014	299	39	,	,	PUNCT
ejpam-6014	299	40	τ2	τ2	NOUN
ejpam-6014	299	41	)	)	PUNCT
ejpam-6014	299	42	is	be	AUX
ejpam-6014	299	43	σ1σ2	σ1σ2	NOUN
ejpam-6014	299	44	-	-	ADJ
ejpam-6014	299	45	urysohn	urysohn	ADJ
ejpam-6014	299	46	.	.	PUNCT
ejpam-6014	300	1	proof	proof	NOUN
ejpam-6014	300	2	.	.	PUNCT
ejpam-6014	301	1	let	let	VERB
ejpam-6014	301	2	x	x	PRON
ejpam-6014	301	3	and	and	CCONJ
ejpam-6014	301	4	y	y	PROPN
ejpam-6014	301	5	be	be	AUX
ejpam-6014	301	6	any	any	DET
ejpam-6014	301	7	distinct	distinct	ADJ
ejpam-6014	301	8	points	point	NOUN
ejpam-6014	301	9	of	of	ADP
ejpam-6014	301	10	x.	x.	NOUN
ejpam-6014	301	11	since	since	SCONJ
ejpam-6014	301	12	f	f	PROPN
ejpam-6014	301	13	is	be	AUX
ejpam-6014	301	14	injective	injective	ADJ
ejpam-6014	301	15	,	,	PUNCT
ejpam-6014	301	16	f(x	f(x	PROPN
ejpam-6014	301	17	)	)	PUNCT
ejpam-6014	301	18	̸=	̸=	PROPN
ejpam-6014	301	19	f(y	f(y	NOUN
ejpam-6014	301	20	)	)	PUNCT
ejpam-6014	301	21	.	.	PUNCT
ejpam-6014	302	1	then	then	ADV
ejpam-6014	302	2	,	,	PUNCT
ejpam-6014	302	3	we	we	PRON
ejpam-6014	302	4	have	have	VERB
ejpam-6014	302	5	(	(	PUNCT
ejpam-6014	302	6	x	x	NOUN
ejpam-6014	302	7	,	,	PUNCT
ejpam-6014	302	8	f(y	f(y	NOUN
ejpam-6014	302	9	)	)	PUNCT
ejpam-6014	302	10	)	)	PUNCT
ejpam-6014	303	1	∈	∈	PROPN
ejpam-6014	303	2	(	(	PUNCT
ejpam-6014	303	3	x	x	SYM
ejpam-6014	303	4	×	×	PROPN
ejpam-6014	303	5	y	y	PROPN
ejpam-6014	303	6	)	)	PUNCT
ejpam-6014	303	7	−	−	PROPN
ejpam-6014	303	8	g(f	g(f	NOUN
ejpam-6014	303	9	)	)	PUNCT
ejpam-6014	303	10	.	.	PUNCT
ejpam-6014	304	1	since	since	SCONJ
ejpam-6014	304	2	g(f	g(f	PROPN
ejpam-6014	304	3	)	)	PUNCT
ejpam-6014	304	4	is	be	AUX
ejpam-6014	304	5	strong	strong	ADJ
ejpam-6014	304	6	θ(τ1	θ(τ1	NOUN
ejpam-6014	304	7	,	,	PUNCT
ejpam-6014	304	8	τ2)-closed	τ2)-close	VERB
ejpam-6014	304	9	,	,	PUNCT
ejpam-6014	304	10	by	by	ADP
ejpam-6014	304	11	lemma	lemma	PROPN
ejpam-6014	304	12	6	6	NUM
ejpam-6014	304	13	there	there	ADV
ejpam-6014	304	14	exist	exist	VERB
ejpam-6014	304	15	a	a	DET
ejpam-6014	304	16	τ1τ2	τ1τ2	NOUN
ejpam-6014	304	17	-	-	ADJ
ejpam-6014	304	18	open	open	ADJ
ejpam-6014	304	19	set	set	ADJ
ejpam-6014	304	20	u	u	NOUN
ejpam-6014	304	21	of	of	ADP
ejpam-6014	304	22	x	x	PUNCT
ejpam-6014	304	23	containing	contain	VERB
ejpam-6014	304	24	x	x	X
ejpam-6014	304	25	and	and	CCONJ
ejpam-6014	304	26	a	a	DET
ejpam-6014	304	27	σ1σ2	σ1σ2	NUM
ejpam-6014	304	28	-	-	ADJ
ejpam-6014	304	29	open	open	ADJ
ejpam-6014	304	30	set	set	NOUN
ejpam-6014	304	31	v	v	NOUN
ejpam-6014	304	32	of	of	ADP
ejpam-6014	304	33	y	y	PROPN
ejpam-6014	304	34	containing	contain	VERB
ejpam-6014	304	35	f(y	f(y	NOUN
ejpam-6014	304	36	)	)	PUNCT
ejpam-6014	304	37	such	such	ADJ
ejpam-6014	304	38	that	that	SCONJ
ejpam-6014	304	39	such	such	ADJ
ejpam-6014	304	40	that	that	DET
ejpam-6014	304	41	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	304	42	-	-	PUNCT
ejpam-6014	304	43	cl(u	cl(u	NOUN
ejpam-6014	304	44	)	)	PUNCT
ejpam-6014	304	45	)	)	PUNCT
ejpam-6014	305	1	∩	∩	NOUN
ejpam-6014	305	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	305	3	-	-	NUM
ejpam-6014	305	4	cl(v	cl(v	X
ejpam-6014	305	5	)	)	PUNCT
ejpam-6014	305	6	=	=	PUNCT
ejpam-6014	305	7	∅.	∅.	NOUN
ejpam-6014	305	8	since	since	SCONJ
ejpam-6014	305	9	f	f	PROPN
ejpam-6014	305	10	is	be	AUX
ejpam-6014	305	11	θ(τ1	θ(τ1	NOUN
ejpam-6014	305	12	,	,	PUNCT
ejpam-6014	305	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	305	14	,	,	PUNCT
ejpam-6014	305	15	there	there	PRON
ejpam-6014	305	16	exists	exist	VERB
ejpam-6014	305	17	a	a	DET
ejpam-6014	305	18	τ1τ2	τ1τ2	NOUN
ejpam-6014	305	19	-	-	ADJ
ejpam-6014	305	20	open	open	ADJ
ejpam-6014	305	21	set	set	NOUN
ejpam-6014	305	22	w	w	PROPN
ejpam-6014	305	23	of	of	ADP
ejpam-6014	305	24	x	x	SYM
ejpam-6014	305	25	containing	contain	VERB
ejpam-6014	305	26	y	y	PRON
ejpam-6014	305	27	such	such	ADJ
ejpam-6014	305	28	that	that	SCONJ
ejpam-6014	305	29	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	305	30	-	-	PUNCT
ejpam-6014	305	31	cl(w	cl(w	NOUN
ejpam-6014	305	32	)	)	PUNCT
ejpam-6014	305	33	)	)	PUNCT
ejpam-6014	306	1	⊆	⊆	X
ejpam-6014	306	2	σ1σ2	σ1σ2	NOUN
ejpam-6014	306	3	-	-	NUM
ejpam-6014	306	4	cl(v	cl(v	NOUN
ejpam-6014	306	5	)	)	PUNCT
ejpam-6014	306	6	.	.	PUNCT
ejpam-6014	307	1	thus	thus	ADV
ejpam-6014	307	2	,	,	PUNCT
ejpam-6014	307	3	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	307	4	-	-	PUNCT
ejpam-6014	307	5	cl(u	cl(u	NOUN
ejpam-6014	307	6	)	)	PUNCT
ejpam-6014	307	7	)	)	PUNCT
ejpam-6014	307	8	∩	∩	NOUN
ejpam-6014	307	9	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	307	10	-	-	PUNCT
ejpam-6014	307	11	cl(w	cl(w	NOUN
ejpam-6014	307	12	)	)	PUNCT
ejpam-6014	307	13	)	)	PUNCT
ejpam-6014	308	1	=	=	NOUN
ejpam-6014	308	2	∅	∅	NOUN
ejpam-6014	308	3	and	and	CCONJ
ejpam-6014	308	4	hence	hence	ADV
ejpam-6014	308	5	τ1τ2	τ1τ2	NOUN
ejpam-6014	308	6	-	-	NOUN
ejpam-6014	308	7	cl(u	cl(u	NOUN
ejpam-6014	308	8	)	)	PUNCT
ejpam-6014	308	9	∩	∩	NOUN
ejpam-6014	308	10	τ1τ2	τ1τ2	NOUN
ejpam-6014	308	11	-	-	NOUN
ejpam-6014	308	12	cl(w	cl(w	NOUN
ejpam-6014	308	13	)	)	PUNCT
ejpam-6014	309	1	=	=	PUNCT
ejpam-6014	309	2	∅.	∅.	ADP
ejpam-6014	309	3	this	this	PRON
ejpam-6014	309	4	shows	show	VERB
ejpam-6014	309	5	that	that	SCONJ
ejpam-6014	309	6	(	(	PUNCT
ejpam-6014	309	7	x	x	NOUN
ejpam-6014	309	8	,	,	PUNCT
ejpam-6014	309	9	τ1	τ1	NOUN
ejpam-6014	309	10	,	,	PUNCT
ejpam-6014	309	11	τ2	τ2	NOUN
ejpam-6014	309	12	)	)	PUNCT
ejpam-6014	309	13	is	be	AUX
ejpam-6014	309	14	σ1σ2	σ1σ2	NOUN
ejpam-6014	309	15	-	-	ADJ
ejpam-6014	309	16	urysohn	urysohn	ADJ
ejpam-6014	309	17	.	.	PUNCT
ejpam-6014	310	1	recall	recall	VERB
ejpam-6014	310	2	that	that	SCONJ
ejpam-6014	310	3	a	a	DET
ejpam-6014	310	4	bitopological	bitopological	ADJ
ejpam-6014	310	5	space	space	NOUN
ejpam-6014	310	6	(	(	PUNCT
ejpam-6014	310	7	x	x	NOUN
ejpam-6014	310	8	,	,	PUNCT
ejpam-6014	310	9	τ1	τ1	NOUN
ejpam-6014	310	10	,	,	PUNCT
ejpam-6014	310	11	τ2	τ2	NOUN
ejpam-6014	310	12	)	)	PUNCT
ejpam-6014	310	13	is	be	AUX
ejpam-6014	310	14	said	say	VERB
ejpam-6014	310	15	to	to	PART
ejpam-6014	310	16	be	be	AUX
ejpam-6014	310	17	quasi	quasi	X
ejpam-6014	310	18	(	(	PUNCT
ejpam-6014	310	19	τ1	τ1	NOUN
ejpam-6014	310	20	,	,	PUNCT
ejpam-6014	311	1	τ2)-h	τ2)-h	PUNCT
ejpam-6014	311	2	-closed	-closed	ADJ
ejpam-6014	311	3	[	[	X
ejpam-6014	311	4	40	40	NUM
ejpam-6014	311	5	]	]	PUNCT
ejpam-6014	311	6	if	if	SCONJ
ejpam-6014	311	7	every	every	DET
ejpam-6014	311	8	τ1τ2	τ1τ2	ADJ
ejpam-6014	311	9	-	-	ADJ
ejpam-6014	311	10	open	open	ADJ
ejpam-6014	311	11	cover	cover	NOUN
ejpam-6014	311	12	{	{	PUNCT
ejpam-6014	311	13	uγ	uγ	ADV
ejpam-6014	311	14	|	|	ADV
ejpam-6014	311	15	γ	γ	X
ejpam-6014	311	16	∈	∈	PROPN
ejpam-6014	311	17	γ	γ	X
ejpam-6014	311	18	}	}	PUNCT
ejpam-6014	311	19	,	,	PUNCT
ejpam-6014	311	20	there	there	PRON
ejpam-6014	311	21	exists	exist	VERB
ejpam-6014	311	22	a	a	DET
ejpam-6014	311	23	finite	finite	NOUN
ejpam-6014	311	24	subset	subset	NOUN
ejpam-6014	311	25	γ0	γ0	NOUN
ejpam-6014	311	26	of	of	ADP
ejpam-6014	311	27	γ	γ	NOUN
ejpam-6014	312	1	such	such	ADJ
ejpam-6014	312	2	that	that	SCONJ
ejpam-6014	312	3	x	x	X
ejpam-6014	312	4	=	=	PUNCT
ejpam-6014	312	5	∪{τ1τ2	∪{τ1τ2	NOUN
ejpam-6014	312	6	-	-	NOUN
ejpam-6014	312	7	cl(uγ	cl(uγ	NOUN
ejpam-6014	312	8	)	)	PUNCT
ejpam-6014	312	9	|	|	ADV
ejpam-6014	312	10	γ	γ	PROPN
ejpam-6014	312	11	∈	∈	PROPN
ejpam-6014	312	12	γ0	γ0	PROPN
ejpam-6014	312	13	}	}	PUNCT
ejpam-6014	312	14	.	.	PUNCT
ejpam-6014	313	1	a	a	DET
ejpam-6014	313	2	subset	subset	NOUN
ejpam-6014	313	3	k	k	NOUN
ejpam-6014	313	4	of	of	ADP
ejpam-6014	313	5	a	a	DET
ejpam-6014	313	6	bitopological	bitopological	ADJ
ejpam-6014	313	7	space	space	NOUN
ejpam-6014	313	8	(	(	PUNCT
ejpam-6014	313	9	x	x	NOUN
ejpam-6014	313	10	,	,	PUNCT
ejpam-6014	313	11	τ1	τ1	NOUN
ejpam-6014	313	12	,	,	PUNCT
ejpam-6014	313	13	τ2	τ2	NOUN
ejpam-6014	313	14	)	)	PUNCT
ejpam-6014	313	15	is	be	AUX
ejpam-6014	313	16	said	say	VERB
ejpam-6014	313	17	to	to	PART
ejpam-6014	313	18	be	be	AUX
ejpam-6014	313	19	quasi	quasi	X
ejpam-6014	313	20	(	(	PUNCT
ejpam-6014	313	21	τ1	τ1	NOUN
ejpam-6014	313	22	,	,	PUNCT
ejpam-6014	313	23	τ2)-h	τ2)-h	NOUN
ejpam-6014	313	24	-closed	-close	VERB
ejpam-6014	313	25	relative	relative	ADJ
ejpam-6014	313	26	to	to	ADP
ejpam-6014	313	27	x	x	PRON
ejpam-6014	313	28	if	if	SCONJ
ejpam-6014	313	29	for	for	ADP
ejpam-6014	313	30	any	any	DET
ejpam-6014	313	31	cover	cover	NOUN
ejpam-6014	313	32	{	{	PUNCT
ejpam-6014	313	33	vγ	vγ	NOUN
ejpam-6014	313	34	|	|	ADV
ejpam-6014	313	35	γ	γ	PROPN
ejpam-6014	313	36	∈	∈	PROPN
ejpam-6014	313	37	γ	γ	X
ejpam-6014	313	38	}	}	PUNCT
ejpam-6014	313	39	by	by	ADP
ejpam-6014	313	40	τ1τ2	τ1τ2	ADJ
ejpam-6014	313	41	-	-	ADJ
ejpam-6014	313	42	open	open	ADJ
ejpam-6014	313	43	sets	set	NOUN
ejpam-6014	313	44	of	of	ADP
ejpam-6014	313	45	x	x	NOUN
ejpam-6014	313	46	,	,	PUNCT
ejpam-6014	313	47	there	there	PRON
ejpam-6014	313	48	exists	exist	VERB
ejpam-6014	313	49	a	a	DET
ejpam-6014	313	50	finite	finite	NOUN
ejpam-6014	313	51	subset	subset	NOUN
ejpam-6014	313	52	γ0	γ0	NOUN
ejpam-6014	313	53	of	of	ADP
ejpam-6014	313	54	γ	γ	PRON
ejpam-6014	313	55	such	such	ADJ
ejpam-6014	313	56	that	that	SCONJ
ejpam-6014	313	57	k	k	PROPN
ejpam-6014	313	58	⊆	⊆	NUM
ejpam-6014	313	59	∪{τ1τ2	∪{τ1τ2	ADJ
ejpam-6014	313	60	-	-	ADJ
ejpam-6014	313	61	cl(vγ	cl(vγ	ADJ
ejpam-6014	313	62	)	)	PUNCT
ejpam-6014	313	63	|	|	ADV
ejpam-6014	313	64	γ	γ	PROPN
ejpam-6014	313	65	∈	∈	PROPN
ejpam-6014	313	66	γ0	γ0	PROPN
ejpam-6014	313	67	}	}	PUNCT
ejpam-6014	313	68	.	.	PUNCT
ejpam-6014	314	1	theorem	theorem	NOUN
ejpam-6014	314	2	12	12	NUM
ejpam-6014	314	3	.	.	PUNCT
ejpam-6014	315	1	if	if	SCONJ
ejpam-6014	315	2	f	f	PROPN
ejpam-6014	315	3	:	:	PUNCT
ejpam-6014	315	4	(	(	PUNCT
ejpam-6014	315	5	x	x	NOUN
ejpam-6014	315	6	,	,	PUNCT
ejpam-6014	315	7	τ1	τ1	NOUN
ejpam-6014	315	8	,	,	PUNCT
ejpam-6014	315	9	τ2	τ2	NOUN
ejpam-6014	315	10	)	)	PUNCT
ejpam-6014	315	11	→	→	SYM
ejpam-6014	315	12	(	(	PUNCT
ejpam-6014	315	13	y	y	PROPN
ejpam-6014	315	14	,	,	PUNCT
ejpam-6014	315	15	σ1	σ1	PROPN
ejpam-6014	315	16	,	,	PUNCT
ejpam-6014	315	17	σ2	σ2	PROPN
ejpam-6014	315	18	)	)	PUNCT
ejpam-6014	315	19	is	be	AUX
ejpam-6014	315	20	θ(τ1	θ(τ1	NOUN
ejpam-6014	315	21	,	,	PUNCT
ejpam-6014	315	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	315	23	and	and	CCONJ
ejpam-6014	315	24	k	k	PROPN
ejpam-6014	315	25	is	be	AUX
ejpam-6014	315	26	quasi	quasi	NOUN
ejpam-6014	315	27	(	(	PUNCT
ejpam-6014	315	28	τ1	τ1	NOUN
ejpam-6014	315	29	,	,	PUNCT
ejpam-6014	315	30	τ2)h	τ2)h	NOUN
ejpam-6014	315	31	-closed	-close	VERB
ejpam-6014	315	32	relative	relative	ADJ
ejpam-6014	315	33	to	to	ADP
ejpam-6014	315	34	x	x	PRON
ejpam-6014	315	35	,	,	PUNCT
ejpam-6014	315	36	then	then	ADV
ejpam-6014	315	37	f(k	f(k	VERB
ejpam-6014	315	38	)	)	PUNCT
ejpam-6014	315	39	is	be	AUX
ejpam-6014	315	40	quasi	quasi	X
ejpam-6014	315	41	(	(	PUNCT
ejpam-6014	315	42	σ1	σ1	PROPN
ejpam-6014	315	43	,	,	PUNCT
ejpam-6014	315	44	σ2)-h	σ2)-h	PROPN
ejpam-6014	315	45	-closed	-closed	ADJ
ejpam-6014	315	46	relative	relative	ADJ
ejpam-6014	315	47	to	to	ADP
ejpam-6014	315	48	y	y	PROPN
ejpam-6014	315	49	.	.	PUNCT
ejpam-6014	316	1	proof	proof	NOUN
ejpam-6014	316	2	.	.	PUNCT
ejpam-6014	317	1	let	let	VERB
ejpam-6014	317	2	{	{	PUNCT
ejpam-6014	317	3	vγ	vγ	VERB
ejpam-6014	317	4	|	|	ADV
ejpam-6014	317	5	γ	γ	PROPN
ejpam-6014	317	6	∈	∈	PROPN
ejpam-6014	317	7	γ	γ	AUX
ejpam-6014	317	8	}	}	PUNCT
ejpam-6014	317	9	be	be	AUX
ejpam-6014	317	10	a	a	DET
ejpam-6014	317	11	cover	cover	NOUN
ejpam-6014	317	12	of	of	ADP
ejpam-6014	317	13	f(k	f(k	VERB
ejpam-6014	317	14	)	)	PUNCT
ejpam-6014	317	15	by	by	ADP
ejpam-6014	317	16	σ1σ2	σ1σ2	NOUN
ejpam-6014	317	17	-	-	PUNCT
ejpam-6014	317	18	open	open	ADJ
ejpam-6014	317	19	sets	set	NOUN
ejpam-6014	317	20	of	of	ADP
ejpam-6014	317	21	y	y	PROPN
ejpam-6014	317	22	.	.	PUNCT
ejpam-6014	318	1	for	for	ADP
ejpam-6014	318	2	each	each	DET
ejpam-6014	318	3	k	k	PROPN
ejpam-6014	318	4	∈	∈	PROPN
ejpam-6014	318	5	k	k	NOUN
ejpam-6014	318	6	,	,	PUNCT
ejpam-6014	318	7	there	there	PRON
ejpam-6014	318	8	exists	exist	VERB
ejpam-6014	318	9	γ(k	γ(k	PROPN
ejpam-6014	318	10	)	)	PUNCT
ejpam-6014	319	1	∈	∈	PROPN
ejpam-6014	319	2	γ	γ	NOUN
ejpam-6014	319	3	such	such	ADJ
ejpam-6014	319	4	that	that	SCONJ
ejpam-6014	319	5	f(k	f(k	ADJ
ejpam-6014	319	6	)	)	PUNCT
ejpam-6014	319	7	∈	∈	PROPN
ejpam-6014	319	8	vγ(k	vγ(k	NOUN
ejpam-6014	319	9	)	)	PUNCT
ejpam-6014	319	10	.	.	PUNCT
ejpam-6014	320	1	since	since	SCONJ
ejpam-6014	320	2	f	f	PROPN
ejpam-6014	320	3	is	be	AUX
ejpam-6014	320	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	320	5	,	,	PUNCT
ejpam-6014	320	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	320	7	,	,	PUNCT
ejpam-6014	320	8	there	there	PRON
ejpam-6014	320	9	exists	exist	VERB
ejpam-6014	320	10	a	a	DET
ejpam-6014	320	11	τ1τ2	τ1τ2	ADJ
ejpam-6014	320	12	-	-	ADJ
ejpam-6014	320	13	open	open	ADJ
ejpam-6014	320	14	set	set	ADJ
ejpam-6014	320	15	uk	uk	PROPN
ejpam-6014	320	16	of	of	ADP
ejpam-6014	320	17	x	x	PUNCT
ejpam-6014	320	18	containing	contain	VERB
ejpam-6014	320	19	k	k	PROPN
ejpam-6014	320	20	such	such	ADJ
ejpam-6014	320	21	that	that	SCONJ
ejpam-6014	320	22	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	320	23	-	-	PUNCT
ejpam-6014	320	24	cl(uk	cl(uk	PROPN
ejpam-6014	320	25	)	)	PUNCT
ejpam-6014	320	26	)	)	PUNCT
ejpam-6014	321	1	⊆	⊆	X
ejpam-6014	321	2	σ1σ2	σ1σ2	NUM
ejpam-6014	321	3	-	-	PUNCT
ejpam-6014	321	4	cl(vγ(k	cl(vγ(k	NOUN
ejpam-6014	321	5	)	)	PUNCT
ejpam-6014	321	6	)	)	PUNCT
ejpam-6014	321	7	.	.	PUNCT
ejpam-6014	322	1	since	since	SCONJ
ejpam-6014	322	2	{	{	PUNCT
ejpam-6014	322	3	uk	uk	PROPN
ejpam-6014	322	4	|	|	ADV
ejpam-6014	322	5	k	k	PROPN
ejpam-6014	322	6	∈	∈	PROPN
ejpam-6014	322	7	k	k	AUX
ejpam-6014	322	8	}	}	PUNCT
ejpam-6014	322	9	is	be	AUX
ejpam-6014	322	10	a	a	DET
ejpam-6014	322	11	cover	cover	NOUN
ejpam-6014	322	12	of	of	ADP
ejpam-6014	322	13	k	k	X
ejpam-6014	322	14	by	by	ADP
ejpam-6014	322	15	τ1τ2	τ1τ2	ADJ
ejpam-6014	322	16	-	-	ADJ
ejpam-6014	322	17	open	open	ADJ
ejpam-6014	322	18	sets	set	NOUN
ejpam-6014	322	19	in	in	ADP
ejpam-6014	322	20	x	x	NOUN
ejpam-6014	322	21	,	,	PUNCT
ejpam-6014	322	22	there	there	PRON
ejpam-6014	322	23	exists	exist	VERB
ejpam-6014	322	24	a	a	DET
ejpam-6014	322	25	finite	finite	NOUN
ejpam-6014	322	26	subset	subset	VERB
ejpam-6014	322	27	k0	k0	PROPN
ejpam-6014	322	28	of	of	ADP
ejpam-6014	322	29	k	k	PROPN
ejpam-6014	322	30	such	such	ADJ
ejpam-6014	322	31	that	that	SCONJ
ejpam-6014	322	32	k	k	PROPN
ejpam-6014	322	33	⊆	⊆	NUM
ejpam-6014	322	34	∪{τ1τ2	∪{τ1τ2	PROPN
ejpam-6014	322	35	-	-	PUNCT
ejpam-6014	322	36	cl(uk	cl(uk	VERB
ejpam-6014	322	37	)	)	PUNCT
ejpam-6014	323	1	|	|	ADV
ejpam-6014	323	2	k	k	PROPN
ejpam-6014	323	3	∈	∈	PROPN
ejpam-6014	323	4	k0	k0	PROPN
ejpam-6014	323	5	}	}	PUNCT
ejpam-6014	323	6	.	.	PUNCT
ejpam-6014	324	1	thus	thus	ADV
ejpam-6014	324	2	,	,	PUNCT
ejpam-6014	324	3	f(k	f(k	VERB
ejpam-6014	324	4	)	)	PUNCT
ejpam-6014	324	5	⊆	⊆	NUM
ejpam-6014	324	6	∪{f(τ1τ2	∪{f(τ1τ2	PROPN
ejpam-6014	324	7	-	-	PUNCT
ejpam-6014	324	8	cl(uk	cl(uk	PROPN
ejpam-6014	324	9	)	)	PUNCT
ejpam-6014	324	10	)	)	PUNCT
ejpam-6014	325	1	|	|	ADV
ejpam-6014	325	2	k	k	PROPN
ejpam-6014	325	3	∈	∈	PROPN
ejpam-6014	325	4	k0	k0	PROPN
ejpam-6014	325	5	}	}	PUNCT
ejpam-6014	325	6	⊆	⊆	NUM
ejpam-6014	325	7	∪{σ1σ2	∪{σ1σ2	NOUN
ejpam-6014	325	8	-	-	PUNCT
ejpam-6014	325	9	cl(vγ(k	cl(vγ(k	NOUN
ejpam-6014	325	10	)	)	PUNCT
ejpam-6014	325	11	)	)	PUNCT
ejpam-6014	326	1	|	|	ADV
ejpam-6014	326	2	k	k	PROPN
ejpam-6014	326	3	∈	∈	PROPN
ejpam-6014	326	4	k0	k0	PROPN
ejpam-6014	326	5	}	}	PUNCT
ejpam-6014	326	6	.	.	PUNCT
ejpam-6014	327	1	this	this	PRON
ejpam-6014	327	2	shows	show	VERB
ejpam-6014	327	3	that	that	SCONJ
ejpam-6014	327	4	f(k	f(k	VERB
ejpam-6014	327	5	)	)	PUNCT
ejpam-6014	327	6	is	be	AUX
ejpam-6014	327	7	quasi	quasi	X
ejpam-6014	327	8	(	(	PUNCT
ejpam-6014	327	9	σ1	σ1	PROPN
ejpam-6014	327	10	,	,	PUNCT
ejpam-6014	327	11	σ2)-h	σ2)-h	PROPN
ejpam-6014	327	12	-closed	-closed	ADJ
ejpam-6014	327	13	relative	relative	ADJ
ejpam-6014	327	14	to	to	ADP
ejpam-6014	327	15	y	y	PROPN
ejpam-6014	327	16	.	.	PUNCT
ejpam-6014	328	1	corollary	corollary	ADJ
ejpam-6014	328	2	1	1	NUM
ejpam-6014	328	3	.	.	PUNCT
ejpam-6014	329	1	if	if	SCONJ
ejpam-6014	329	2	f	f	PROPN
ejpam-6014	329	3	:	:	PUNCT
ejpam-6014	329	4	(	(	PUNCT
ejpam-6014	329	5	x	x	NOUN
ejpam-6014	329	6	,	,	PUNCT
ejpam-6014	329	7	τ1	τ1	NOUN
ejpam-6014	329	8	,	,	PUNCT
ejpam-6014	329	9	τ2	τ2	NOUN
ejpam-6014	329	10	)	)	PUNCT
ejpam-6014	329	11	→	→	SYM
ejpam-6014	329	12	(	(	PUNCT
ejpam-6014	329	13	y	y	PROPN
ejpam-6014	329	14	,	,	PUNCT
ejpam-6014	329	15	σ1	σ1	PROPN
ejpam-6014	329	16	,	,	PUNCT
ejpam-6014	329	17	σ2	σ2	PROPN
ejpam-6014	329	18	)	)	PUNCT
ejpam-6014	329	19	is	be	AUX
ejpam-6014	329	20	a	a	DET
ejpam-6014	329	21	θ(τ1	θ(τ1	NOUN
ejpam-6014	329	22	,	,	PUNCT
ejpam-6014	329	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	329	24	surjection	surjection	NOUN
ejpam-6014	329	25	and	and	CCONJ
ejpam-6014	329	26	(	(	PUNCT
ejpam-6014	329	27	x	x	NOUN
ejpam-6014	329	28	,	,	PUNCT
ejpam-6014	329	29	τ1	τ1	NOUN
ejpam-6014	329	30	,	,	PUNCT
ejpam-6014	329	31	τ2	τ2	NOUN
ejpam-6014	329	32	)	)	PUNCT
ejpam-6014	329	33	is	be	AUX
ejpam-6014	329	34	quasi	quasi	NOUN
ejpam-6014	329	35	(	(	PUNCT
ejpam-6014	329	36	τ1	τ1	NOUN
ejpam-6014	329	37	,	,	PUNCT
ejpam-6014	329	38	τ2)-h	τ2)-h	NOUN
ejpam-6014	329	39	-closed	-closed	ADJ
ejpam-6014	329	40	,	,	PUNCT
ejpam-6014	329	41	then	then	ADV
ejpam-6014	329	42	(	(	PUNCT
ejpam-6014	329	43	y	y	PROPN
ejpam-6014	329	44	,	,	PUNCT
ejpam-6014	329	45	σ1	σ1	PROPN
ejpam-6014	329	46	,	,	PUNCT
ejpam-6014	329	47	σ2	σ2	PROPN
ejpam-6014	329	48	)	)	PUNCT
ejpam-6014	329	49	is	be	AUX
ejpam-6014	329	50	quasi	quasi	X
ejpam-6014	329	51	(	(	PUNCT
ejpam-6014	329	52	σ1	σ1	PROPN
ejpam-6014	329	53	,	,	PUNCT
ejpam-6014	329	54	σ2)-h	σ2)-h	PROPN
ejpam-6014	329	55	-closed	-closed	ADJ
ejpam-6014	329	56	.	.	PUNCT
ejpam-6014	330	1	definition	definition	NOUN
ejpam-6014	330	2	7	7	NUM
ejpam-6014	330	3	.	.	PUNCT
ejpam-6014	331	1	let	let	VERB
ejpam-6014	331	2	a	a	DET
ejpam-6014	331	3	be	be	AUX
ejpam-6014	331	4	a	a	DET
ejpam-6014	331	5	subset	subset	NOUN
ejpam-6014	331	6	of	of	ADP
ejpam-6014	331	7	a	a	DET
ejpam-6014	331	8	bitopological	bitopological	ADJ
ejpam-6014	331	9	space	space	NOUN
ejpam-6014	331	10	(	(	PUNCT
ejpam-6014	331	11	x	x	NOUN
ejpam-6014	331	12	,	,	PUNCT
ejpam-6014	331	13	τ1	τ1	NOUN
ejpam-6014	331	14	,	,	PUNCT
ejpam-6014	331	15	τ2	τ2	NOUN
ejpam-6014	331	16	)	)	PUNCT
ejpam-6014	331	17	.	.	PUNCT
ejpam-6014	332	1	the	the	DET
ejpam-6014	332	2	(	(	PUNCT
ejpam-6014	332	3	τ1	τ1	NOUN
ejpam-6014	332	4	,	,	PUNCT
ejpam-6014	332	5	τ2)θ	τ2)θ	ADJ
ejpam-6014	332	6	-	-	PUNCT
ejpam-6014	332	7	frontier	frontier	NOUN
ejpam-6014	332	8	of	of	ADP
ejpam-6014	332	9	a	a	DET
ejpam-6014	332	10	,	,	PUNCT
ejpam-6014	332	11	(	(	PUNCT
ejpam-6014	332	12	τ1	τ1	NOUN
ejpam-6014	332	13	,	,	PUNCT
ejpam-6014	332	14	τ2)θ	τ2)θ	NOUN
ejpam-6014	332	15	-	-	PUNCT
ejpam-6014	332	16	fr(a	fr(a	NUM
ejpam-6014	332	17	)	)	PUNCT
ejpam-6014	332	18	,	,	PUNCT
ejpam-6014	332	19	is	be	AUX
ejpam-6014	332	20	defined	define	VERB
ejpam-6014	332	21	by	by	ADP
ejpam-6014	332	22	(	(	PUNCT
ejpam-6014	332	23	τ1	τ1	NOUN
ejpam-6014	332	24	,	,	PUNCT
ejpam-6014	332	25	τ2)θ	τ2)θ	NOUN
ejpam-6014	332	26	-	-	PUNCT
ejpam-6014	332	27	fr(a	fr(a	NUM
ejpam-6014	332	28	)	)	PUNCT
ejpam-6014	333	1	=	=	SYM
ejpam-6014	333	2	(	(	PUNCT
ejpam-6014	333	3	τ1	τ1	NOUN
ejpam-6014	333	4	,	,	PUNCT
ejpam-6014	333	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	333	6	-	-	PUNCT
ejpam-6014	333	7	cl(a	cl(a	NUM
ejpam-6014	333	8	)	)	PUNCT
ejpam-6014	333	9	∩	∩	NOUN
ejpam-6014	333	10	(	(	PUNCT
ejpam-6014	333	11	τ1	τ1	NOUN
ejpam-6014	333	12	,	,	PUNCT
ejpam-6014	333	13	τ2)θ	τ2)θ	ADJ
ejpam-6014	333	14	-	-	PUNCT
ejpam-6014	333	15	cl(x	cl(x	NOUN
ejpam-6014	333	16	−a	−a	NOUN
ejpam-6014	333	17	)	)	PUNCT
ejpam-6014	333	18	.	.	PUNCT
ejpam-6014	334	1	theorem	theorem	VERB
ejpam-6014	334	2	13	13	NUM
ejpam-6014	334	3	.	.	PUNCT
ejpam-6014	335	1	the	the	DET
ejpam-6014	335	2	set	set	NOUN
ejpam-6014	335	3	of	of	ADP
ejpam-6014	335	4	all	all	DET
ejpam-6014	335	5	points	point	NOUN
ejpam-6014	335	6	x	x	X
ejpam-6014	335	7	∈	∈	NOUN
ejpam-6014	335	8	x	x	PUNCT
ejpam-6014	335	9	at	at	ADP
ejpam-6014	335	10	which	which	PRON
ejpam-6014	335	11	a	a	DET
ejpam-6014	335	12	function	function	NOUN
ejpam-6014	335	13	f	f	NOUN
ejpam-6014	335	14	:	:	PUNCT
ejpam-6014	335	15	(	(	PUNCT
ejpam-6014	335	16	x	x	NOUN
ejpam-6014	335	17	,	,	PUNCT
ejpam-6014	335	18	τ1	τ1	NOUN
ejpam-6014	335	19	,	,	PUNCT
ejpam-6014	335	20	τ2	τ2	NOUN
ejpam-6014	335	21	)	)	PUNCT
ejpam-6014	335	22	→	→	SYM
ejpam-6014	335	23	(	(	PUNCT
ejpam-6014	335	24	y	y	PROPN
ejpam-6014	335	25	,	,	PUNCT
ejpam-6014	335	26	σ1	σ1	PROPN
ejpam-6014	335	27	,	,	PUNCT
ejpam-6014	335	28	σ2	σ2	PROPN
ejpam-6014	335	29	)	)	PUNCT
ejpam-6014	335	30	is	be	AUX
ejpam-6014	335	31	not	not	PART
ejpam-6014	335	32	θ(τ1	θ(τ1	ADJ
ejpam-6014	335	33	,	,	PUNCT
ejpam-6014	335	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	335	35	is	be	AUX
ejpam-6014	335	36	identical	identical	ADJ
ejpam-6014	335	37	with	with	ADP
ejpam-6014	335	38	the	the	DET
ejpam-6014	335	39	union	union	NOUN
ejpam-6014	335	40	of	of	ADP
ejpam-6014	335	41	the	the	DET
ejpam-6014	335	42	(	(	PUNCT
ejpam-6014	335	43	τ1	τ1	NOUN
ejpam-6014	335	44	,	,	PUNCT
ejpam-6014	335	45	τ2)θ	τ2)θ	ADJ
ejpam-6014	335	46	-	-	PUNCT
ejpam-6014	335	47	frontier	frontier	NOUN
ejpam-6014	335	48	of	of	ADP
ejpam-6014	335	49	the	the	DET
ejpam-6014	335	50	inverse	inverse	NOUN
ejpam-6014	335	51	images	image	NOUN
ejpam-6014	335	52	of	of	ADP
ejpam-6014	335	53	the	the	DET
ejpam-6014	335	54	σ1σ2	σ1σ2	NOUN
ejpam-6014	335	55	-	-	NOUN
ejpam-6014	335	56	closure	closure	NOUN
ejpam-6014	335	57	of	of	ADP
ejpam-6014	335	58	σ1σ2	σ1σ2	NOUN
ejpam-6014	335	59	-	-	PUNCT
ejpam-6014	335	60	open	open	ADJ
ejpam-6014	335	61	sets	set	NOUN
ejpam-6014	335	62	containing	contain	VERB
ejpam-6014	335	63	f(x	f(x	PROPN
ejpam-6014	335	64	)	)	PUNCT
ejpam-6014	335	65	.	.	PUNCT
ejpam-6014	336	1	proof	proof	NOUN
ejpam-6014	336	2	.	.	PUNCT
ejpam-6014	337	1	suppose	suppose	VERB
ejpam-6014	337	2	that	that	SCONJ
ejpam-6014	337	3	f	f	PROPN
ejpam-6014	337	4	is	be	AUX
ejpam-6014	337	5	not	not	PART
ejpam-6014	337	6	θ(τ1	θ(τ1	ADJ
ejpam-6014	337	7	,	,	PUNCT
ejpam-6014	337	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	337	9	.	.	PUNCT
ejpam-6014	338	1	then	then	ADV
ejpam-6014	338	2	,	,	PUNCT
ejpam-6014	338	3	there	there	PRON
ejpam-6014	338	4	exists	exist	VERB
ejpam-6014	338	5	a	a	DET
ejpam-6014	338	6	σ1σ2	σ1σ2	NUM
ejpam-6014	338	7	-	-	ADJ
ejpam-6014	338	8	open	open	ADJ
ejpam-6014	338	9	set	set	NOUN
ejpam-6014	338	10	v	v	NOUN
ejpam-6014	338	11	of	of	ADP
ejpam-6014	338	12	y	y	NOUN
ejpam-6014	338	13	containing	contain	VERB
ejpam-6014	338	14	f(x	f(x	PROPN
ejpam-6014	338	15	)	)	PUNCT
ejpam-6014	338	16	such	such	ADJ
ejpam-6014	338	17	that	that	SCONJ
ejpam-6014	338	18	f(τ1τ2	f(τ1τ2	NOUN
ejpam-6014	338	19	-	-	PUNCT
ejpam-6014	338	20	cl(u	cl(u	NOUN
ejpam-6014	338	21	)	)	PUNCT
ejpam-6014	338	22	)	)	PUNCT
ejpam-6014	338	23	is	be	AUX
ejpam-6014	338	24	not	not	PART
ejpam-6014	338	25	contained	contain	VERB
ejpam-6014	338	26	in	in	ADP
ejpam-6014	338	27	σ1σ2	σ1σ2	NOUN
ejpam-6014	338	28	-	-	NOUN
ejpam-6014	338	29	cl(v	cl(v	NOUN
ejpam-6014	338	30	)	)	PUNCT
ejpam-6014	338	31	for	for	ADP
ejpam-6014	338	32	every	every	DET
ejpam-6014	338	33	τ1τ2	τ1τ2	ADJ
ejpam-6014	338	34	-	-	ADJ
ejpam-6014	338	35	open	open	ADJ
ejpam-6014	338	36	set	set	ADJ
ejpam-6014	338	37	u	u	NOUN
ejpam-6014	338	38	of	of	ADP
ejpam-6014	338	39	x	x	SYM
ejpam-6014	338	40	containing	contain	VERB
ejpam-6014	338	41	x.	x.	NOUN
ejpam-6014	338	42	then	then	ADV
ejpam-6014	338	43	,	,	PUNCT
ejpam-6014	338	44	τ1τ2	τ1τ2	NOUN
ejpam-6014	338	45	-	-	NOUN
ejpam-6014	338	46	cl(u	cl(u	ADJ
ejpam-6014	338	47	)	)	PUNCT
ejpam-6014	338	48	∩	∩	NOUN
ejpam-6014	338	49	(	(	PUNCT
ejpam-6014	338	50	x	x	SYM
ejpam-6014	338	51	−	−	PRON
ejpam-6014	338	52	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	338	53	-	-	PUNCT
ejpam-6014	338	54	cl(v	cl(v	NOUN
ejpam-6014	338	55	)	)	PUNCT
ejpam-6014	338	56	)	)	PUNCT
ejpam-6014	338	57	)	)	PUNCT
ejpam-6014	339	1	̸=	̸=	NOUN
ejpam-6014	339	2	∅	∅	NOUN
ejpam-6014	339	3	for	for	ADP
ejpam-6014	339	4	every	every	DET
ejpam-6014	339	5	τ1τ2	τ1τ2	ADJ
ejpam-6014	339	6	-	-	ADJ
ejpam-6014	339	7	open	open	ADJ
ejpam-6014	339	8	set	set	ADJ
ejpam-6014	339	9	u	u	NOUN
ejpam-6014	339	10	of	of	ADP
ejpam-6014	339	11	x	x	SYM
ejpam-6014	339	12	containing	contain	VERB
ejpam-6014	339	13	x.	x.	NOUN
ejpam-6014	339	14	thus	thus	ADV
ejpam-6014	339	15	,	,	PUNCT
ejpam-6014	339	16	x	x	SYM
ejpam-6014	339	17	∈	∈	PROPN
ejpam-6014	339	18	(	(	PUNCT
ejpam-6014	339	19	τ1	τ1	NOUN
ejpam-6014	339	20	,	,	PUNCT
ejpam-6014	339	21	τ2)θ	τ2)θ	ADJ
ejpam-6014	339	22	-	-	PUNCT
ejpam-6014	339	23	cl(x	cl(x	PUNCT
ejpam-6014	339	24	−	−	NOUN
ejpam-6014	339	25	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6014	339	26	-	-	PUNCT
ejpam-6014	339	27	cl(v	cl(v	NOUN
ejpam-6014	339	28	)	)	PUNCT
ejpam-6014	339	29	)	)	PUNCT
ejpam-6014	339	30	)	)	PUNCT
ejpam-6014	339	31	.	.	PUNCT
ejpam-6014	340	1	on	on	ADP
ejpam-6014	340	2	the	the	DET
ejpam-6014	340	3	other	other	ADJ
ejpam-6014	340	4	hand	hand	NOUN
ejpam-6014	340	5	,	,	PUNCT
ejpam-6014	340	6	we	we	PRON
ejpam-6014	340	7	have	have	VERB
ejpam-6014	340	8	x	x	X
ejpam-6014	340	9	∈	∈	PROPN
ejpam-6014	340	10	f−1(v	f−1(v	NOUN
ejpam-6014	340	11	)	)	PUNCT
ejpam-6014	341	1	⊆	⊆	NUM
ejpam-6014	341	2	(	(	PUNCT
ejpam-6014	341	3	τ1	τ1	NOUN
ejpam-6014	341	4	,	,	PUNCT
ejpam-6014	341	5	τ2)θ	τ2)θ	PROPN
ejpam-6014	341	6	-	-	PUNCT
ejpam-6014	341	7	cl(f	cl(f	NOUN
ejpam-6014	341	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	341	9	-	-	PUNCT
ejpam-6014	341	10	cl(v	cl(v	NOUN
ejpam-6014	341	11	)	)	PUNCT
ejpam-6014	341	12	)	)	PUNCT
ejpam-6014	341	13	)	)	PUNCT
ejpam-6014	341	14	and	and	CCONJ
ejpam-6014	341	15	hence	hence	ADV
ejpam-6014	341	16	x	x	X
ejpam-6014	341	17	∈	∈	PROPN
ejpam-6014	341	18	(	(	PUNCT
ejpam-6014	341	19	τ1	τ1	NOUN
ejpam-6014	341	20	,	,	PUNCT
ejpam-6014	341	21	τ2)θ	τ2)θ	ADJ
ejpam-6014	341	22	-	-	PUNCT
ejpam-6014	341	23	fr(f	fr(f	NOUN
ejpam-6014	341	24	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	341	25	-	-	PUNCT
ejpam-6014	341	26	cl(v	cl(v	NOUN
ejpam-6014	341	27	)	)	PUNCT
ejpam-6014	341	28	)	)	PUNCT
ejpam-6014	341	29	)	)	PUNCT
ejpam-6014	341	30	.	.	PUNCT
ejpam-6014	341	31	m.	m.	NOUN
ejpam-6014	341	32	thongmoon	thongmoon	PROPN
ejpam-6014	341	33	,	,	PUNCT
ejpam-6014	341	34	s.	s.	PROPN
ejpam-6014	341	35	sompong	sompong	PROPN
ejpam-6014	341	36	,	,	PUNCT
ejpam-6014	341	37	c.	c.	PROPN
ejpam-6014	341	38	boonpok	boonpok	PROPN
ejpam-6014	341	39	/	/	SYM
ejpam-6014	341	40	eur	eur	PROPN
ejpam-6014	341	41	.	.	PUNCT
ejpam-6014	342	1	j.	j.	PROPN
ejpam-6014	342	2	pure	pure	PROPN
ejpam-6014	342	3	appl	appl	PROPN
ejpam-6014	342	4	.	.	PROPN
ejpam-6014	342	5	math	math	PROPN
ejpam-6014	342	6	,	,	PUNCT
ejpam-6014	342	7	18	18	NUM
ejpam-6014	342	8	(	(	PUNCT
ejpam-6014	342	9	2	2	NUM
ejpam-6014	342	10	)	)	PUNCT
ejpam-6014	342	11	(	(	PUNCT
ejpam-6014	342	12	2025	2025	NUM
ejpam-6014	342	13	)	)	PUNCT
ejpam-6014	342	14	,	,	PUNCT
ejpam-6014	342	15	6014	6014	NUM
ejpam-6014	342	16	11	11	NUM
ejpam-6014	342	17	of	of	ADP
ejpam-6014	342	18	13	13	NUM
ejpam-6014	342	19	conversely	conversely	ADV
ejpam-6014	342	20	,	,	PUNCT
ejpam-6014	342	21	suppose	suppose	VERB
ejpam-6014	342	22	that	that	SCONJ
ejpam-6014	342	23	f	f	PROPN
ejpam-6014	342	24	is	be	AUX
ejpam-6014	342	25	θ(τ1	θ(τ1	NOUN
ejpam-6014	342	26	,	,	PUNCT
ejpam-6014	342	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	342	28	at	at	ADP
ejpam-6014	342	29	x	x	SYM
ejpam-6014	342	30	∈	∈	PROPN
ejpam-6014	342	31	x.	x.	NOUN
ejpam-6014	342	32	let	let	VERB
ejpam-6014	342	33	v	v	PART
ejpam-6014	342	34	be	be	AUX
ejpam-6014	342	35	any	any	DET
ejpam-6014	342	36	σ1σ2	σ1σ2	NOUN
ejpam-6014	342	37	-	-	ADJ
ejpam-6014	342	38	open	open	ADJ
ejpam-6014	342	39	set	set	NOUN
ejpam-6014	342	40	of	of	ADP
ejpam-6014	342	41	y	y	PROPN
ejpam-6014	342	42	containing	contain	VERB
ejpam-6014	342	43	f(x	f(x	PROPN
ejpam-6014	342	44	)	)	PUNCT
ejpam-6014	342	45	.	.	PUNCT
ejpam-6014	343	1	then	then	ADV
ejpam-6014	343	2	by	by	ADP
ejpam-6014	343	3	theorem	theorem	NOUN
ejpam-6014	343	4	2	2	NUM
ejpam-6014	343	5	we	we	PRON
ejpam-6014	343	6	have	have	VERB
ejpam-6014	343	7	x	x	X
ejpam-6014	343	8	∈	∈	PROPN
ejpam-6014	343	9	f−1(v	f−1(v	NOUN
ejpam-6014	343	10	)	)	PUNCT
ejpam-6014	344	1	⊆	⊆	NUM
ejpam-6014	344	2	(	(	PUNCT
ejpam-6014	344	3	τ1	τ1	NOUN
ejpam-6014	344	4	,	,	PUNCT
ejpam-6014	344	5	τ2)θ	τ2)θ	NOUN
ejpam-6014	344	6	-	-	PUNCT
ejpam-6014	344	7	int(f	int(f	VERB
ejpam-6014	344	8	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	344	9	-	-	PUNCT
ejpam-6014	344	10	cl(v	cl(v	NOUN
ejpam-6014	344	11	)	)	PUNCT
ejpam-6014	344	12	)	)	PUNCT
ejpam-6014	344	13	)	)	PUNCT
ejpam-6014	344	14	.	.	PUNCT
ejpam-6014	345	1	thus	thus	ADV
ejpam-6014	345	2	,	,	PUNCT
ejpam-6014	345	3	x	x	PROPN
ejpam-6014	345	4	̸∈	̸∈	PROPN
ejpam-6014	345	5	(	(	PUNCT
ejpam-6014	345	6	τ1	τ1	PROPN
ejpam-6014	345	7	,	,	PUNCT
ejpam-6014	345	8	τ2)θ	τ2)θ	ADJ
ejpam-6014	345	9	-	-	PUNCT
ejpam-6014	345	10	fr(f	fr(f	NOUN
ejpam-6014	345	11	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6014	345	12	-	-	PUNCT
ejpam-6014	345	13	cl(v	cl(v	NOUN
ejpam-6014	345	14	)	)	PUNCT
ejpam-6014	345	15	)	)	PUNCT
ejpam-6014	345	16	)	)	PUNCT
ejpam-6014	346	1	for	for	ADP
ejpam-6014	346	2	every	every	DET
ejpam-6014	346	3	σ1σ2	σ1σ2	NOUN
ejpam-6014	346	4	-	-	ADJ
ejpam-6014	346	5	open	open	ADJ
ejpam-6014	346	6	set	set	NOUN
ejpam-6014	346	7	v	v	NOUN
ejpam-6014	346	8	of	of	ADP
ejpam-6014	346	9	y	y	NOUN
ejpam-6014	346	10	containing	contain	VERB
ejpam-6014	346	11	f(x	f(x	PROPN
ejpam-6014	346	12	)	)	PUNCT
ejpam-6014	346	13	.	.	PUNCT
ejpam-6014	347	1	this	this	PRON
ejpam-6014	347	2	completes	complete	VERB
ejpam-6014	347	3	the	the	DET
ejpam-6014	347	4	proof	proof	NOUN
ejpam-6014	347	5	.	.	PUNCT
ejpam-6014	348	1	acknowledgements	acknowledgement	NOUN
ejpam-6014	348	2	this	this	DET
ejpam-6014	348	3	research	research	NOUN
ejpam-6014	348	4	project	project	NOUN
ejpam-6014	348	5	was	be	AUX
ejpam-6014	348	6	financially	financially	ADV
ejpam-6014	348	7	supported	support	VERB
ejpam-6014	348	8	by	by	ADP
ejpam-6014	348	9	mahasarakham	mahasarakham	PROPN
ejpam-6014	348	10	university	university	PROPN
ejpam-6014	348	11	.	.	PUNCT
ejpam-6014	349	1	references	reference	NOUN
ejpam-6014	349	2	[	[	X
ejpam-6014	349	3	1	1	NUM
ejpam-6014	349	4	]	]	PUNCT
ejpam-6014	349	5	c.	c.	PROPN
ejpam-6014	349	6	boonpok	boonpok	PROPN
ejpam-6014	349	7	and	and	CCONJ
ejpam-6014	349	8	j.	j.	PROPN
ejpam-6014	349	9	khampakdee	khampakdee	PROPN
ejpam-6014	349	10	.	.	PUNCT
ejpam-6014	350	1	(	(	PUNCT
ejpam-6014	350	2	λ	λ	NOUN
ejpam-6014	350	3	,	,	PUNCT
ejpam-6014	350	4	sp)-open	sp)-open	ADJ
ejpam-6014	350	5	sets	set	NOUN
ejpam-6014	350	6	in	in	ADP
ejpam-6014	350	7	topological	topological	ADJ
ejpam-6014	350	8	spaces	space	NOUN
ejpam-6014	350	9	.	.	PUNCT
ejpam-6014	351	1	european	european	ADJ
ejpam-6014	351	2	journal	journal	PROPN
ejpam-6014	351	3	of	of	ADP
ejpam-6014	351	4	pure	pure	ADJ
ejpam-6014	351	5	and	and	CCONJ
ejpam-6014	351	6	applied	applied	ADJ
ejpam-6014	351	7	mathematics	mathematic	NOUN
ejpam-6014	351	8	,	,	PUNCT
ejpam-6014	351	9	15(2):572–588	15(2):572–588	NUM
ejpam-6014	351	10	,	,	PUNCT
ejpam-6014	351	11	2022	2022	NUM
ejpam-6014	351	12	.	.	PUNCT
ejpam-6014	352	1	[	[	X
ejpam-6014	352	2	2	2	NUM
ejpam-6014	352	3	]	]	PUNCT
ejpam-6014	352	4	c.	c.	PROPN
ejpam-6014	352	5	viriyapong	viriyapong	PROPN
ejpam-6014	352	6	and	and	CCONJ
ejpam-6014	352	7	c.	c.	PROPN
ejpam-6014	352	8	boonpok	boonpok	PROPN
ejpam-6014	352	9	.	.	PUNCT
ejpam-6014	353	1	(	(	PUNCT
ejpam-6014	353	2	λ	λ	X
ejpam-6014	353	3	,	,	PUNCT
ejpam-6014	353	4	sp)-continuous	sp)-continuous	ADJ
ejpam-6014	353	5	functions	function	NOUN
ejpam-6014	353	6	.	.	PUNCT
ejpam-6014	354	1	wseas	wseas	VERB
ejpam-6014	354	2	transactions	transaction	NOUN
ejpam-6014	354	3	on	on	ADP
ejpam-6014	354	4	mathematics	mathematic	NOUN
ejpam-6014	354	5	,	,	PUNCT
ejpam-6014	354	6	21:380–385	21:380–385	NUM
ejpam-6014	354	7	,	,	PUNCT
ejpam-6014	354	8	2022	2022	NUM
ejpam-6014	354	9	.	.	PUNCT
ejpam-6014	355	1	[	[	X
ejpam-6014	355	2	3	3	X
ejpam-6014	355	3	]	]	PUNCT
ejpam-6014	355	4	t.	t.	NOUN
ejpam-6014	355	5	dungthaisong	dungthaisong	PROPN
ejpam-6014	355	6	,	,	PUNCT
ejpam-6014	355	7	c.	c.	PROPN
ejpam-6014	355	8	boonpok	boonpok	PROPN
ejpam-6014	355	9	,	,	PUNCT
ejpam-6014	355	10	and	and	CCONJ
ejpam-6014	355	11	c.	c.	PROPN
ejpam-6014	355	12	viriyapong	viriyapong	PROPN
ejpam-6014	355	13	.	.	PUNCT
ejpam-6014	356	1	generalized	generalize	VERB
ejpam-6014	356	2	closed	close	VERB
ejpam-6014	356	3	sets	set	NOUN
ejpam-6014	356	4	in	in	ADP
ejpam-6014	356	5	bigeneralized	bigeneralize	VERB
ejpam-6014	356	6	topological	topological	ADJ
ejpam-6014	356	7	spaces	space	NOUN
ejpam-6014	356	8	.	.	PUNCT
ejpam-6014	357	1	international	international	ADJ
ejpam-6014	357	2	journal	journal	PROPN
ejpam-6014	357	3	of	of	ADP
ejpam-6014	357	4	mathematical	mathematical	ADJ
ejpam-6014	357	5	analysis	analysis	NOUN
ejpam-6014	357	6	,	,	PUNCT
ejpam-6014	357	7	5(24):1175–1184	5(24):1175–1184	NUM
ejpam-6014	357	8	,	,	PUNCT
ejpam-6014	357	9	2011	2011	NUM
ejpam-6014	357	10	.	.	PUNCT
ejpam-6014	358	1	[	[	X
ejpam-6014	358	2	4	4	X
ejpam-6014	358	3	]	]	PUNCT
ejpam-6014	358	4	t.	t.	PROPN
ejpam-6014	358	5	duangphui	duangphui	PROPN
ejpam-6014	358	6	,	,	PUNCT
ejpam-6014	358	7	c.	c.	PROPN
ejpam-6014	358	8	boonpok	boonpok	PROPN
ejpam-6014	358	9	,	,	PUNCT
ejpam-6014	358	10	and	and	CCONJ
ejpam-6014	358	11	c.	c.	PROPN
ejpam-6014	358	12	viriyapong	viriyapong	PROPN
ejpam-6014	358	13	.	.	PUNCT
ejpam-6014	359	1	continuous	continuous	ADJ
ejpam-6014	359	2	functions	function	NOUN
ejpam-6014	359	3	on	on	ADP
ejpam-6014	359	4	bigeneralized	bigeneralize	VERB
ejpam-6014	359	5	topological	topological	ADJ
ejpam-6014	359	6	spaces	space	NOUN
ejpam-6014	359	7	.	.	PUNCT
ejpam-6014	360	1	international	international	ADJ
ejpam-6014	360	2	journal	journal	PROPN
ejpam-6014	360	3	of	of	ADP
ejpam-6014	360	4	mathematical	mathematical	ADJ
ejpam-6014	360	5	analysis	analysis	NOUN
ejpam-6014	360	6	,	,	PUNCT
ejpam-6014	360	7	5(24):1165	5(24):1165	NUM
ejpam-6014	360	8	–	–	PUNCT
ejpam-6014	360	9	1174	1174	NUM
ejpam-6014	360	10	,	,	PUNCT
ejpam-6014	360	11	2011	2011	NUM
ejpam-6014	360	12	.	.	PUNCT
ejpam-6014	361	1	[	[	X
ejpam-6014	361	2	5	5	NUM
ejpam-6014	361	3	]	]	X
ejpam-6014	361	4	n.	n.	NOUN
ejpam-6014	361	5	srisarakham	srisarakham	PROPN
ejpam-6014	361	6	and	and	CCONJ
ejpam-6014	361	7	c.	c.	PROPN
ejpam-6014	361	8	boonpok	boonpok	PROPN
ejpam-6014	361	9	.	.	PUNCT
ejpam-6014	362	1	almost	almost	ADV
ejpam-6014	362	2	(	(	PUNCT
ejpam-6014	362	3	λ	λ	NOUN
ejpam-6014	362	4	,	,	PUNCT
ejpam-6014	362	5	p)-continuous	p)-continuous	ADJ
ejpam-6014	362	6	functions	function	NOUN
ejpam-6014	362	7	.	.	PUNCT
ejpam-6014	363	1	international	international	ADJ
ejpam-6014	363	2	journal	journal	PROPN
ejpam-6014	363	3	of	of	ADP
ejpam-6014	363	4	mathematics	mathematic	NOUN
ejpam-6014	363	5	and	and	CCONJ
ejpam-6014	363	6	computer	computer	NOUN
ejpam-6014	363	7	science	science	NOUN
ejpam-6014	363	8	,	,	PUNCT
ejpam-6014	363	9	18(2):255–259	18(2):255–259	NUM
ejpam-6014	363	10	,	,	PUNCT
ejpam-6014	363	11	2023	2023	NUM
ejpam-6014	363	12	.	.	PUNCT
ejpam-6014	364	1	[	[	X
ejpam-6014	364	2	6	6	NUM
ejpam-6014	364	3	]	]	PUNCT
ejpam-6014	364	4	c.	c.	PROPN
ejpam-6014	364	5	boonpok	boonpok	PROPN
ejpam-6014	364	6	and	and	CCONJ
ejpam-6014	364	7	j.	j.	PROPN
ejpam-6014	364	8	khampakdee	khampakdee	PROPN
ejpam-6014	364	9	.	.	PUNCT
ejpam-6014	365	1	almost	almost	ADV
ejpam-6014	365	2	strong	strong	ADJ
ejpam-6014	365	3	θ(λ	θ(λ	PROPN
ejpam-6014	365	4	,	,	PUNCT
ejpam-6014	365	5	p)-continuity	p)-continuity	NOUN
ejpam-6014	365	6	for	for	ADP
ejpam-6014	365	7	functions	function	NOUN
ejpam-6014	365	8	.	.	PUNCT
ejpam-6014	366	1	european	european	ADJ
ejpam-6014	366	2	journal	journal	PROPN
ejpam-6014	366	3	of	of	ADP
ejpam-6014	366	4	pure	pure	ADJ
ejpam-6014	366	5	and	and	CCONJ
ejpam-6014	366	6	applied	applied	ADJ
ejpam-6014	366	7	mathematics	mathematic	NOUN
ejpam-6014	366	8	,	,	PUNCT
ejpam-6014	366	9	17(1):300–309	17(1):300–309	PROPN
ejpam-6014	366	10	,	,	PUNCT
ejpam-6014	366	11	2024	2024	NUM
ejpam-6014	366	12	.	.	PUNCT
ejpam-6014	367	1	[	[	X
ejpam-6014	367	2	7	7	X
ejpam-6014	367	3	]	]	X
ejpam-6014	367	4	c.	c.	PROPN
ejpam-6014	367	5	boonpok	boonpok	PROPN
ejpam-6014	367	6	and	and	CCONJ
ejpam-6014	367	7	n.	n.	PROPN
ejpam-6014	367	8	srisarakham	srisarakham	PROPN
ejpam-6014	367	9	.	.	PUNCT
ejpam-6014	368	1	weak	weak	ADJ
ejpam-6014	368	2	forms	form	NOUN
ejpam-6014	368	3	of	of	ADP
ejpam-6014	368	4	(	(	PUNCT
ejpam-6014	368	5	λ	λ	PROPN
ejpam-6014	368	6	,	,	PUNCT
ejpam-6014	368	7	b)-open	b)-open	VERB
ejpam-6014	368	8	sets	set	NOUN
ejpam-6014	368	9	and	and	CCONJ
ejpam-6014	368	10	weak	weak	ADJ
ejpam-6014	368	11	(	(	PUNCT
ejpam-6014	368	12	λ	λ	NOUN
ejpam-6014	368	13	,	,	PUNCT
ejpam-6014	368	14	b)continuity	b)continuity	NOUN
ejpam-6014	368	15	.	.	PUNCT
ejpam-6014	369	1	european	european	PROPN
ejpam-6014	369	2	journal	journal	PROPN
ejpam-6014	369	3	of	of	ADP
ejpam-6014	369	4	pure	pure	ADJ
ejpam-6014	369	5	and	and	CCONJ
ejpam-6014	369	6	applied	applied	ADJ
ejpam-6014	369	7	mathematics	mathematic	NOUN
ejpam-6014	369	8	,	,	PUNCT
ejpam-6014	369	9	16(1):29–43	16(1):29–43	NUM
ejpam-6014	369	10	,	,	PUNCT
ejpam-6014	369	11	2023	2023	NUM
ejpam-6014	369	12	.	.	PUNCT
ejpam-6014	370	1	[	[	X
ejpam-6014	370	2	8	8	NUM
ejpam-6014	370	3	]	]	X
ejpam-6014	370	4	c.	c.	PROPN
ejpam-6014	370	5	boonpok	boonpok	PROPN
ejpam-6014	370	6	.	.	PUNCT
ejpam-6014	371	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-6014	371	2	.	.	PUNCT
ejpam-6014	372	1	mathematica	mathematica	PROPN
ejpam-6014	372	2	,	,	PUNCT
ejpam-6014	372	3	65(1):31–42	65(1):31–42	NUM
ejpam-6014	372	4	,	,	PUNCT
ejpam-6014	372	5	2023	2023	NUM
ejpam-6014	372	6	.	.	PUNCT
ejpam-6014	373	1	[	[	X
ejpam-6014	373	2	9	9	NUM
ejpam-6014	373	3	]	]	PUNCT
ejpam-6014	373	4	c.	c.	PROPN
ejpam-6014	373	5	boonpok	boonpok	PROPN
ejpam-6014	373	6	.	.	PUNCT
ejpam-6014	374	1	on	on	ADP
ejpam-6014	374	2	some	some	DET
ejpam-6014	374	3	closed	closed	ADJ
ejpam-6014	374	4	sets	set	NOUN
ejpam-6014	374	5	and	and	CCONJ
ejpam-6014	374	6	low	low	ADJ
ejpam-6014	374	7	separation	separation	NOUN
ejpam-6014	374	8	axioms	axiom	NOUN
ejpam-6014	374	9	via	via	ADP
ejpam-6014	374	10	topological	topological	ADJ
ejpam-6014	374	11	ideals	ideal	NOUN
ejpam-6014	374	12	.	.	PUNCT
ejpam-6014	375	1	european	european	ADJ
ejpam-6014	375	2	journal	journal	PROPN
ejpam-6014	375	3	of	of	ADP
ejpam-6014	375	4	pure	pure	ADJ
ejpam-6014	375	5	and	and	CCONJ
ejpam-6014	375	6	applied	applied	ADJ
ejpam-6014	375	7	mathematics	mathematic	NOUN
ejpam-6014	375	8	,	,	PUNCT
ejpam-6014	375	9	15(3):1023–1046	15(3):1023–1046	NUM
ejpam-6014	375	10	,	,	PUNCT
ejpam-6014	375	11	2022	2022	NUM
ejpam-6014	375	12	.	.	PUNCT
ejpam-6014	376	1	[	[	X
ejpam-6014	376	2	10	10	NUM
ejpam-6014	376	3	]	]	X
ejpam-6014	376	4	c.	c.	PROPN
ejpam-6014	376	5	boonpok	boonpok	PROPN
ejpam-6014	376	6	.	.	PUNCT
ejpam-6014	377	1	on	on	ADP
ejpam-6014	377	2	some	some	DET
ejpam-6014	377	3	spaces	space	NOUN
ejpam-6014	377	4	via	via	ADP
ejpam-6014	377	5	topological	topological	ADJ
ejpam-6014	377	6	ideals	ideal	NOUN
ejpam-6014	377	7	.	.	PUNCT
ejpam-6014	378	1	open	open	ADJ
ejpam-6014	378	2	mathematics	mathematic	NOUN
ejpam-6014	378	3	,	,	PUNCT
ejpam-6014	378	4	21:20230118	21:20230118	NUM
ejpam-6014	378	5	,	,	PUNCT
ejpam-6014	378	6	2023	2023	NUM
ejpam-6014	378	7	.	.	PUNCT
ejpam-6014	379	1	[	[	X
ejpam-6014	379	2	11	11	NUM
ejpam-6014	379	3	]	]	PUNCT
ejpam-6014	379	4	c.	c.	PROPN
ejpam-6014	379	5	boonpok	boonpok	PROPN
ejpam-6014	379	6	.	.	PUNCT
ejpam-6014	380	1	on	on	ADP
ejpam-6014	380	2	characterizations	characterization	NOUN
ejpam-6014	380	3	of	of	ADP
ejpam-6014	380	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-6014	380	5	ideal	ideal	ADJ
ejpam-6014	380	6	topological	topological	ADJ
ejpam-6014	380	7	spaces	space	NOUN
ejpam-6014	380	8	.	.	PUNCT
ejpam-6014	381	1	journal	journal	NOUN
ejpam-6014	381	2	of	of	ADP
ejpam-6014	381	3	mathematics	mathematic	NOUN
ejpam-6014	381	4	,	,	PUNCT
ejpam-6014	381	5	2020:9387601	2020:9387601	NUM
ejpam-6014	381	6	,	,	PUNCT
ejpam-6014	381	7	2020	2020	NUM
ejpam-6014	381	8	.	.	PUNCT
ejpam-6014	382	1	[	[	X
ejpam-6014	382	2	12	12	NUM
ejpam-6014	382	3	]	]	PUNCT
ejpam-6014	382	4	c.	c.	PROPN
ejpam-6014	382	5	boonpok	boonpok	PROPN
ejpam-6014	382	6	.	.	PUNCT
ejpam-6014	383	1	almost	almost	ADV
ejpam-6014	383	2	(	(	PUNCT
ejpam-6014	383	3	g	g	NOUN
ejpam-6014	383	4	,	,	PUNCT
ejpam-6014	383	5	m)-continuous	m)-continuous	ADJ
ejpam-6014	383	6	functions	function	NOUN
ejpam-6014	383	7	.	.	PUNCT
ejpam-6014	384	1	international	international	ADJ
ejpam-6014	384	2	journal	journal	PROPN
ejpam-6014	384	3	of	of	ADP
ejpam-6014	384	4	mathematical	mathematical	ADJ
ejpam-6014	384	5	analysis	analysis	NOUN
ejpam-6014	384	6	,	,	PUNCT
ejpam-6014	384	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-6014	384	8	,	,	PUNCT
ejpam-6014	384	9	2010	2010	NUM
ejpam-6014	384	10	.	.	PUNCT
ejpam-6014	385	1	[	[	X
ejpam-6014	385	2	13	13	NUM
ejpam-6014	385	3	]	]	PUNCT
ejpam-6014	385	4	c.	c.	PROPN
ejpam-6014	385	5	boonpok	boonpok	PROPN
ejpam-6014	385	6	.	.	PUNCT
ejpam-6014	386	1	m	m	VERB
ejpam-6014	386	2	-continuous	-continuous	ADJ
ejpam-6014	386	3	functions	function	NOUN
ejpam-6014	386	4	in	in	ADP
ejpam-6014	386	5	biminimal	biminimal	NOUN
ejpam-6014	386	6	structure	structure	NOUN
ejpam-6014	386	7	spaces	space	NOUN
ejpam-6014	386	8	.	.	PUNCT
ejpam-6014	387	1	far	far	PROPN
ejpam-6014	387	2	east	east	PROPN
ejpam-6014	387	3	journal	journal	PROPN
ejpam-6014	387	4	of	of	ADP
ejpam-6014	387	5	mathematical	mathematical	ADJ
ejpam-6014	387	6	sciences	science	NOUN
ejpam-6014	387	7	,	,	PUNCT
ejpam-6014	387	8	43(1):41–58	43(1):41–58	NUM
ejpam-6014	387	9	,	,	PUNCT
ejpam-6014	387	10	2010	2010	NUM
ejpam-6014	387	11	.	.	PUNCT
ejpam-6014	388	1	[	[	X
ejpam-6014	388	2	14	14	NUM
ejpam-6014	388	3	]	]	X
ejpam-6014	388	4	s.	s.	PROPN
ejpam-6014	388	5	fomin	fomin	PROPN
ejpam-6014	388	6	.	.	PUNCT
ejpam-6014	389	1	extensions	extension	NOUN
ejpam-6014	389	2	of	of	ADP
ejpam-6014	389	3	topological	topological	ADJ
ejpam-6014	389	4	spaces	space	NOUN
ejpam-6014	389	5	.	.	PUNCT
ejpam-6014	390	1	doklady	doklady	PROPN
ejpam-6014	390	2	akademii	akademii	NOUN
ejpam-6014	390	3	nauk	nauk	NOUN
ejpam-6014	390	4	sssr	sssr	NOUN
ejpam-6014	390	5	,	,	PUNCT
ejpam-6014	390	6	32:114	32:114	NUM
ejpam-6014	390	7	–	–	PUNCT
ejpam-6014	390	8	116	116	NUM
ejpam-6014	390	9	,	,	PUNCT
ejpam-6014	390	10	1941	1941	NUM
ejpam-6014	390	11	.	.	PUNCT
ejpam-6014	391	1	m.	m.	NOUN
ejpam-6014	391	2	thongmoon	thongmoon	PROPN
ejpam-6014	391	3	,	,	PUNCT
ejpam-6014	391	4	s.	s.	PROPN
ejpam-6014	391	5	sompong	sompong	PROPN
ejpam-6014	391	6	,	,	PUNCT
ejpam-6014	391	7	c.	c.	PROPN
ejpam-6014	391	8	boonpok	boonpok	PROPN
ejpam-6014	391	9	/	/	SYM
ejpam-6014	391	10	eur	eur	PROPN
ejpam-6014	391	11	.	.	PUNCT
ejpam-6014	392	1	j.	j.	PROPN
ejpam-6014	392	2	pure	pure	PROPN
ejpam-6014	392	3	appl	appl	PROPN
ejpam-6014	392	4	.	.	PROPN
ejpam-6014	392	5	math	math	PROPN
ejpam-6014	392	6	,	,	PUNCT
ejpam-6014	392	7	18	18	NUM
ejpam-6014	392	8	(	(	PUNCT
ejpam-6014	392	9	2	2	NUM
ejpam-6014	392	10	)	)	PUNCT
ejpam-6014	392	11	(	(	PUNCT
ejpam-6014	392	12	2025	2025	NUM
ejpam-6014	392	13	)	)	PUNCT
ejpam-6014	392	14	,	,	PUNCT
ejpam-6014	392	15	6014	6014	NUM
ejpam-6014	392	16	12	12	NUM
ejpam-6014	392	17	of	of	ADP
ejpam-6014	392	18	13	13	NUM
ejpam-6014	392	19	[	[	SYM
ejpam-6014	392	20	15	15	NUM
ejpam-6014	392	21	]	]	PUNCT
ejpam-6014	392	22	t.	t.	PROPN
ejpam-6014	392	23	noiri	noiri	PROPN
ejpam-6014	392	24	.	.	PUNCT
ejpam-6014	393	1	properties	property	NOUN
ejpam-6014	393	2	of	of	ADP
ejpam-6014	393	3	θ	θ	ADJ
ejpam-6014	393	4	-	-	ADJ
ejpam-6014	393	5	continuous	continuous	ADJ
ejpam-6014	393	6	functions	function	NOUN
ejpam-6014	393	7	.	.	PUNCT
ejpam-6014	394	1	atti	atti	PROPN
ejpam-6014	394	2	della	della	PROPN
ejpam-6014	394	3	accademia	accademia	PROPN
ejpam-6014	394	4	nazionale	nazionale	PROPN
ejpam-6014	394	5	dei	dei	PROPN
ejpam-6014	394	6	lincei	lincei	NOUN
ejpam-6014	394	7	,	,	PUNCT
ejpam-6014	394	8	classe	classe	PROPN
ejpam-6014	394	9	di	di	PROPN
ejpam-6014	394	10	scienze	scienze	PROPN
ejpam-6014	394	11	fisiche	fisiche	PROPN
ejpam-6014	394	12	,	,	PUNCT
ejpam-6014	394	13	matematiche	matematiche	PROPN
ejpam-6014	394	14	e	e	X
ejpam-6014	394	15	naturali	naturali	X
ejpam-6014	394	16	.	.	PUNCT
ejpam-6014	395	1	rendiconti	rendiconti	PROPN
ejpam-6014	395	2	,	,	PUNCT
ejpam-6014	395	3	series	series	NOUN
ejpam-6014	395	4	(	(	PUNCT
ejpam-6014	395	5	8)	8)	NUM
ejpam-6014	395	6	,	,	PUNCT
ejpam-6014	395	7	58:887–891	58:887–891	NUM
ejpam-6014	395	8	,	,	PUNCT
ejpam-6014	395	9	1975	1975	NUM
ejpam-6014	395	10	.	.	PUNCT
ejpam-6014	396	1	[	[	X
ejpam-6014	396	2	16	16	NUM
ejpam-6014	396	3	]	]	PUNCT
ejpam-6014	396	4	v.	v.	CCONJ
ejpam-6014	396	5	popa	popa	NOUN
ejpam-6014	396	6	.	.	PUNCT
ejpam-6014	397	1	characterizations	characterization	NOUN
ejpam-6014	397	2	of	of	ADP
ejpam-6014	397	3	θ	θ	ADJ
ejpam-6014	397	4	-	-	ADJ
ejpam-6014	397	5	continuous	continuous	ADJ
ejpam-6014	397	6	functions	function	NOUN
ejpam-6014	397	7	.	.	PUNCT
ejpam-6014	398	1	studii	studii	PROPN
ejpam-6014	398	2	şi	şi	PROPN
ejpam-6014	398	3	cercetări	cercetări	PROPN
ejpam-6014	398	4	ştiinţifice	ştiinţifice	PROPN
ejpam-6014	398	5	.	.	PUNCT
ejpam-6014	399	1	seria	seria	PROPN
ejpam-6014	399	2	matematică	matematică	PROPN
ejpam-6014	399	3	,	,	PUNCT
ejpam-6014	399	4	32:113–119	32:113–119	PROPN
ejpam-6014	399	5	,	,	PUNCT
ejpam-6014	399	6	1980	1980	NUM
ejpam-6014	399	7	.	.	PUNCT
ejpam-6014	400	1	[	[	X
ejpam-6014	400	2	17	17	NUM
ejpam-6014	400	3	]	]	PUNCT
ejpam-6014	400	4	s.	s.	PROPN
ejpam-6014	400	5	p.	p.	PROPN
ejpam-6014	400	6	arya	arya	PROPN
ejpam-6014	400	7	and	and	CCONJ
ejpam-6014	400	8	m.	m.	PROPN
ejpam-6014	400	9	p.	p.	PROPN
ejpam-6014	400	10	bhamini	bhamini	PROPN
ejpam-6014	400	11	.	.	PUNCT
ejpam-6014	401	1	some	some	DET
ejpam-6014	401	2	weaker	weak	ADJ
ejpam-6014	401	3	forms	form	NOUN
ejpam-6014	401	4	of	of	ADP
ejpam-6014	401	5	semi	semi	ADJ
ejpam-6014	401	6	-	-	ADJ
ejpam-6014	401	7	continuous	continuous	ADJ
ejpam-6014	401	8	functions	function	NOUN
ejpam-6014	401	9	.	.	PUNCT
ejpam-6014	402	1	ganita	ganita	NOUN
ejpam-6014	402	2	,	,	PUNCT
ejpam-6014	402	3	33:124–134	33:124–134	NUM
ejpam-6014	402	4	,	,	PUNCT
ejpam-6014	402	5	1982	1982	NUM
ejpam-6014	402	6	.	.	PUNCT
ejpam-6014	403	1	[	[	X
ejpam-6014	403	2	18	18	NUM
ejpam-6014	403	3	]	]	X
ejpam-6014	403	4	s.	s.	PROPN
ejpam-6014	403	5	jafari	jafari	PROPN
ejpam-6014	403	6	and	and	CCONJ
ejpam-6014	403	7	t.	t.	PROPN
ejpam-6014	403	8	noiri	noiri	PROPN
ejpam-6014	403	9	.	.	PUNCT
ejpam-6014	404	1	properties	property	NOUN
ejpam-6014	404	2	of	of	ADP
ejpam-6014	404	3	θ	θ	NOUN
ejpam-6014	404	4	-	-	PUNCT
ejpam-6014	404	5	semi	semi	ADJ
ejpam-6014	404	6	-	-	ADJ
ejpam-6014	404	7	continuous	continuous	ADJ
ejpam-6014	404	8	functions	function	NOUN
ejpam-6014	404	9	.	.	PUNCT
ejpam-6014	405	1	journal	journal	PROPN
ejpam-6014	405	2	of	of	ADP
ejpam-6014	405	3	institute	institute	PROPN
ejpam-6014	405	4	of	of	ADP
ejpam-6014	405	5	mathematics	mathematics	PROPN
ejpam-6014	405	6	and	and	CCONJ
ejpam-6014	405	7	computer	computer	NOUN
ejpam-6014	405	8	sciences	science	NOUN
ejpam-6014	405	9	,	,	PUNCT
ejpam-6014	405	10	mathematics	mathematic	NOUN
ejpam-6014	405	11	series	series	NOUN
ejpam-6014	405	12	,	,	PUNCT
ejpam-6014	405	13	13:123–128	13:123–128	NUM
ejpam-6014	405	14	,	,	PUNCT
ejpam-6014	405	15	2000	2000	NUM
ejpam-6014	405	16	.	.	PUNCT
ejpam-6014	406	1	[	[	X
ejpam-6014	406	2	19	19	NUM
ejpam-6014	406	3	]	]	PUNCT
ejpam-6014	406	4	t.	t.	PROPN
ejpam-6014	406	5	noiri	noiri	PROPN
ejpam-6014	406	6	.	.	PUNCT
ejpam-6014	407	1	on	on	ADP
ejpam-6014	407	2	θ	θ	ADJ
ejpam-6014	407	3	-	-	ADJ
ejpam-6014	407	4	precontinuous	precontinuous	ADJ
ejpam-6014	407	5	functions	function	NOUN
ejpam-6014	407	6	.	.	PUNCT
ejpam-6014	408	1	international	international	ADJ
ejpam-6014	408	2	journal	journal	PROPN
ejpam-6014	408	3	of	of	ADP
ejpam-6014	408	4	mathematics	mathematics	PROPN
ejpam-6014	408	5	and	and	CCONJ
ejpam-6014	408	6	mathematical	mathematical	ADJ
ejpam-6014	408	7	sciences	science	NOUN
ejpam-6014	408	8	,	,	PUNCT
ejpam-6014	408	9	28:285–292	28:285–292	NUM
ejpam-6014	408	10	,	,	PUNCT
ejpam-6014	408	11	2001	2001	NUM
ejpam-6014	408	12	.	.	PUNCT
ejpam-6014	409	1	[	[	X
ejpam-6014	409	2	20	20	NUM
ejpam-6014	409	3	]	]	PUNCT
ejpam-6014	409	4	c.	c.	PROPN
ejpam-6014	409	5	w.	w.	PROPN
ejpam-6014	409	6	baker	baker	PROPN
ejpam-6014	409	7	.	.	PUNCT
ejpam-6014	410	1	weakly	weakly	ADJ
ejpam-6014	410	2	θ	θ	ADJ
ejpam-6014	410	3	-	-	ADJ
ejpam-6014	410	4	precontinuous	precontinuous	ADJ
ejpam-6014	410	5	functions	function	NOUN
ejpam-6014	410	6	.	.	PUNCT
ejpam-6014	411	1	acta	acta	PROPN
ejpam-6014	411	2	mathematica	mathematica	PROPN
ejpam-6014	411	3	hungarica	hungarica	PROPN
ejpam-6014	411	4	,	,	PUNCT
ejpam-6014	411	5	100:343–351	100:343–351	NUM
ejpam-6014	411	6	,	,	PUNCT
ejpam-6014	411	7	2003	2003	NUM
ejpam-6014	411	8	.	.	PUNCT
ejpam-6014	412	1	[	[	X
ejpam-6014	412	2	21	21	NUM
ejpam-6014	412	3	]	]	X
ejpam-6014	412	4	t.	t.	PROPN
ejpam-6014	412	5	noiri	noiri	PROPN
ejpam-6014	412	6	and	and	CCONJ
ejpam-6014	412	7	v.	v.	ADP
ejpam-6014	412	8	popa	popa	NOUN
ejpam-6014	412	9	.	.	PUNCT
ejpam-6014	413	1	a	a	DET
ejpam-6014	413	2	unified	unified	ADJ
ejpam-6014	413	3	theory	theory	NOUN
ejpam-6014	413	4	of	of	ADP
ejpam-6014	413	5	θ	θ	NOUN
ejpam-6014	413	6	-	-	NOUN
ejpam-6014	413	7	continuity	continuity	NOUN
ejpam-6014	413	8	for	for	ADP
ejpam-6014	413	9	functions	function	NOUN
ejpam-6014	413	10	.	.	PUNCT
ejpam-6014	414	1	rendiconti	rendiconti	ADJ
ejpam-6014	414	2	del	del	PROPN
ejpam-6014	414	3	circolo	circolo	PROPN
ejpam-6014	414	4	matematico	matematico	NOUN
ejpam-6014	414	5	di	di	PROPN
ejpam-6014	414	6	palermo	palermo	PROPN
ejpam-6014	414	7	series	series	PROPN
ejpam-6014	414	8	2	2	NUM
ejpam-6014	414	9	,	,	PUNCT
ejpam-6014	414	10	52:163–188	52:163–188	NUM
ejpam-6014	414	11	,	,	PUNCT
ejpam-6014	414	12	2003	2003	NUM
ejpam-6014	414	13	.	.	PUNCT
ejpam-6014	415	1	[	[	X
ejpam-6014	415	2	22	22	NUM
ejpam-6014	415	3	]	]	PUNCT
ejpam-6014	415	4	t.	t.	PROPN
ejpam-6014	415	5	noiri	noiri	PROPN
ejpam-6014	415	6	and	and	CCONJ
ejpam-6014	415	7	v.	v.	ADP
ejpam-6014	415	8	popa	popa	NOUN
ejpam-6014	415	9	.	.	PUNCT
ejpam-6014	416	1	on	on	ADP
ejpam-6014	416	2	θ	θ	PROPN
ejpam-6014	416	3	-	-	PUNCT
ejpam-6014	416	4	m	m	NOUN
ejpam-6014	416	5	-	-	PUNCT
ejpam-6014	416	6	continuous	continuous	ADJ
ejpam-6014	416	7	functions	function	NOUN
ejpam-6014	416	8	.	.	PUNCT
ejpam-6014	417	1	libertas	libertas	PROPN
ejpam-6014	417	2	mathematica	mathematica	PROPN
ejpam-6014	417	3	,	,	PUNCT
ejpam-6014	417	4	26:1–13	26:1–13	NUM
ejpam-6014	417	5	,	,	PUNCT
ejpam-6014	417	6	2006	2006	NUM
ejpam-6014	417	7	.	.	PUNCT
ejpam-6014	418	1	[	[	X
ejpam-6014	418	2	23	23	NUM
ejpam-6014	418	3	]	]	PUNCT
ejpam-6014	419	1	p.	p.	PROPN
ejpam-6014	419	2	e.	e.	PROPN
ejpam-6014	420	1	long	long	PROPN
ejpam-6014	420	2	and	and	CCONJ
ejpam-6014	420	3	l.	l.	PROPN
ejpam-6014	420	4	l.	l.	PROPN
ejpam-6014	420	5	herrington	herrington	PROPN
ejpam-6014	420	6	.	.	PUNCT
ejpam-6014	421	1	strongly	strongly	ADV
ejpam-6014	421	2	θ	θ	ADJ
ejpam-6014	421	3	-	-	ADJ
ejpam-6014	421	4	continuous	continuous	ADJ
ejpam-6014	421	5	functions	function	NOUN
ejpam-6014	421	6	.	.	PUNCT
ejpam-6014	422	1	journal	journal	NOUN
ejpam-6014	422	2	of	of	ADP
ejpam-6014	422	3	the	the	DET
ejpam-6014	422	4	korean	korean	PROPN
ejpam-6014	422	5	mathematical	mathematical	ADJ
ejpam-6014	422	6	society	society	NOUN
ejpam-6014	422	7	,	,	PUNCT
ejpam-6014	422	8	18:21–28	18:21–28	NUM
ejpam-6014	422	9	,	,	PUNCT
ejpam-6014	422	10	1981	1981	NUM
ejpam-6014	422	11	.	.	PUNCT
ejpam-6014	423	1	[	[	X
ejpam-6014	423	2	24	24	NUM
ejpam-6014	423	3	]	]	X
ejpam-6014	423	4	s.	s.	PROPN
ejpam-6014	423	5	jafari	jafari	PROPN
ejpam-6014	423	6	and	and	CCONJ
ejpam-6014	423	7	t.	t.	PROPN
ejpam-6014	423	8	noiri	noiri	PROPN
ejpam-6014	423	9	.	.	PUNCT
ejpam-6014	424	1	strongly	strongly	ADV
ejpam-6014	424	2	θ	θ	VERB
ejpam-6014	424	3	-	-	PUNCT
ejpam-6014	424	4	semi	semi	ADJ
ejpam-6014	424	5	-	-	ADJ
ejpam-6014	424	6	continuous	continuous	ADJ
ejpam-6014	424	7	functions	function	NOUN
ejpam-6014	424	8	.	.	PUNCT
ejpam-6014	425	1	indian	indian	ADJ
ejpam-6014	425	2	journal	journal	PROPN
ejpam-6014	425	3	of	of	ADP
ejpam-6014	425	4	pure	pure	ADJ
ejpam-6014	425	5	and	and	CCONJ
ejpam-6014	425	6	applied	applied	ADJ
ejpam-6014	425	7	mathematics	mathematic	NOUN
ejpam-6014	425	8	,	,	PUNCT
ejpam-6014	425	9	29:1195–1201	29:1195–1201	NUM
ejpam-6014	425	10	,	,	PUNCT
ejpam-6014	425	11	1998	1998	NUM
ejpam-6014	425	12	.	.	PUNCT
ejpam-6014	426	1	[	[	X
ejpam-6014	426	2	25	25	NUM
ejpam-6014	426	3	]	]	PUNCT
ejpam-6014	426	4	t.	t.	PROPN
ejpam-6014	426	5	noiri	noiri	PROPN
ejpam-6014	426	6	.	.	PUNCT
ejpam-6014	427	1	strongly	strongly	ADV
ejpam-6014	427	2	θ	θ	ADJ
ejpam-6014	427	3	-	-	ADJ
ejpam-6014	427	4	precontinuous	precontinuous	ADJ
ejpam-6014	427	5	functions	function	NOUN
ejpam-6014	427	6	.	.	PUNCT
ejpam-6014	428	1	acta	acta	PROPN
ejpam-6014	428	2	mathematica	mathematica	PROPN
ejpam-6014	428	3	hungarica	hungarica	PROPN
ejpam-6014	428	4	,	,	PUNCT
ejpam-6014	428	5	90:307	90:307	NUM
ejpam-6014	428	6	–	–	PUNCT
ejpam-6014	428	7	316	316	NUM
ejpam-6014	428	8	,	,	PUNCT
ejpam-6014	428	9	2001	2001	NUM
ejpam-6014	428	10	.	.	PUNCT
ejpam-6014	429	1	[	[	X
ejpam-6014	429	2	26	26	NUM
ejpam-6014	429	3	]	]	X
ejpam-6014	429	4	p.	p.	NOUN
ejpam-6014	429	5	pue	pue	NOUN
ejpam-6014	429	6	-	-	PUNCT
ejpam-6014	429	7	on	on	ADP
ejpam-6014	429	8	and	and	CCONJ
ejpam-6014	429	9	c.	c.	PROPN
ejpam-6014	429	10	boonpok	boonpok	PROPN
ejpam-6014	429	11	.	.	PUNCT
ejpam-6014	430	1	θ(λ	θ(λ	PROPN
ejpam-6014	430	2	,	,	PUNCT
ejpam-6014	430	3	p)-continuity	p)-continuity	NOUN
ejpam-6014	430	4	for	for	ADP
ejpam-6014	430	5	functions	function	NOUN
ejpam-6014	430	6	.	.	PUNCT
ejpam-6014	431	1	international	international	ADJ
ejpam-6014	431	2	journal	journal	NOUN
ejpam-6014	431	3	of	of	ADP
ejpam-6014	431	4	mathematics	mathematic	NOUN
ejpam-6014	431	5	and	and	CCONJ
ejpam-6014	431	6	computer	computer	NOUN
ejpam-6014	431	7	science	science	NOUN
ejpam-6014	431	8	,	,	PUNCT
ejpam-6014	431	9	19(2):491–495	19(2):491–495	NUM
ejpam-6014	431	10	,	,	PUNCT
ejpam-6014	431	11	2024	2024	NUM
ejpam-6014	431	12	.	.	PUNCT
ejpam-6014	432	1	[	[	X
ejpam-6014	432	2	27	27	NUM
ejpam-6014	432	3	]	]	PUNCT
ejpam-6014	432	4	m.	m.	NOUN
ejpam-6014	432	5	thongmoon	thongmoon	NOUN
ejpam-6014	432	6	and	and	CCONJ
ejpam-6014	432	7	c.	c.	PROPN
ejpam-6014	432	8	boonpok	boonpok	PROPN
ejpam-6014	432	9	.	.	PUNCT
ejpam-6014	433	1	strongly	strongly	ADV
ejpam-6014	433	2	θ(λ	θ(λ	PROPN
ejpam-6014	433	3	,	,	PUNCT
ejpam-6014	433	4	p)-continuous	p)-continuous	ADJ
ejpam-6014	433	5	functions	function	NOUN
ejpam-6014	433	6	.	.	PUNCT
ejpam-6014	434	1	international	international	ADJ
ejpam-6014	434	2	journal	journal	PROPN
ejpam-6014	434	3	of	of	ADP
ejpam-6014	434	4	mathematics	mathematic	NOUN
ejpam-6014	434	5	and	and	CCONJ
ejpam-6014	434	6	computer	computer	NOUN
ejpam-6014	434	7	science	science	NOUN
ejpam-6014	434	8	,	,	PUNCT
ejpam-6014	434	9	19(2):475–479	19(2):475–479	PROPN
ejpam-6014	434	10	,	,	PUNCT
ejpam-6014	434	11	2024	2024	NUM
ejpam-6014	434	12	.	.	PUNCT
ejpam-6014	435	1	[	[	X
ejpam-6014	435	2	28	28	NUM
ejpam-6014	435	3	]	]	X
ejpam-6014	435	4	c.	c.	PROPN
ejpam-6014	435	5	boonpok	boonpok	PROPN
ejpam-6014	435	6	and	and	CCONJ
ejpam-6014	435	7	n.	n.	PROPN
ejpam-6014	435	8	srisarakham	srisarakham	PROPN
ejpam-6014	435	9	.	.	PUNCT
ejpam-6014	436	1	(	(	PUNCT
ejpam-6014	436	2	τ1	τ1	NOUN
ejpam-6014	436	3	,	,	PUNCT
ejpam-6014	436	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6014	436	5	for	for	ADP
ejpam-6014	436	6	functions	function	NOUN
ejpam-6014	436	7	.	.	PUNCT
ejpam-6014	437	1	asia	asia	PROPN
ejpam-6014	437	2	pacific	pacific	PROPN
ejpam-6014	437	3	journal	journal	PROPN
ejpam-6014	437	4	of	of	ADP
ejpam-6014	437	5	mathematics	mathematic	NOUN
ejpam-6014	437	6	,	,	PUNCT
ejpam-6014	437	7	11:21	11:21	NUM
ejpam-6014	437	8	,	,	PUNCT
ejpam-6014	437	9	2024	2024	NUM
ejpam-6014	437	10	.	.	PUNCT
ejpam-6014	438	1	[	[	X
ejpam-6014	438	2	29	29	NUM
ejpam-6014	438	3	]	]	X
ejpam-6014	438	4	c.	c.	PROPN
ejpam-6014	438	5	boonpok	boonpok	PROPN
ejpam-6014	438	6	and	and	CCONJ
ejpam-6014	438	7	p.	p.	NOUN
ejpam-6014	438	8	pue	pue	NOUN
ejpam-6014	438	9	-	-	PUNCT
ejpam-6014	438	10	on	on	ADP
ejpam-6014	438	11	.	.	PUNCT
ejpam-6014	439	1	characterizations	characterization	NOUN
ejpam-6014	439	2	of	of	ADP
ejpam-6014	439	3	almost	almost	ADV
ejpam-6014	439	4	(	(	PUNCT
ejpam-6014	439	5	τ1	τ1	NOUN
ejpam-6014	439	6	,	,	PUNCT
ejpam-6014	439	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	439	8	functions	function	NOUN
ejpam-6014	439	9	.	.	PUNCT
ejpam-6014	440	1	international	international	ADJ
ejpam-6014	440	2	journal	journal	NOUN
ejpam-6014	440	3	of	of	ADP
ejpam-6014	440	4	analysis	analysis	NOUN
ejpam-6014	440	5	and	and	CCONJ
ejpam-6014	440	6	applications	application	NOUN
ejpam-6014	440	7	,	,	PUNCT
ejpam-6014	440	8	22:33	22:33	NUM
ejpam-6014	440	9	,	,	PUNCT
ejpam-6014	440	10	2024	2024	NUM
ejpam-6014	440	11	.	.	PUNCT
ejpam-6014	441	1	[	[	X
ejpam-6014	441	2	30	30	NUM
ejpam-6014	441	3	]	]	X
ejpam-6014	441	4	c.	c.	PROPN
ejpam-6014	441	5	boonpok	boonpok	PROPN
ejpam-6014	441	6	and	and	CCONJ
ejpam-6014	441	7	c.	c.	PROPN
ejpam-6014	441	8	klanarong	klanarong	PROPN
ejpam-6014	441	9	.	.	PUNCT
ejpam-6014	442	1	on	on	ADP
ejpam-6014	442	2	weakly	weakly	ADJ
ejpam-6014	442	3	(	(	PUNCT
ejpam-6014	442	4	τ1	τ1	NOUN
ejpam-6014	442	5	,	,	PUNCT
ejpam-6014	442	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	442	7	functions	function	NOUN
ejpam-6014	442	8	.	.	PUNCT
ejpam-6014	443	1	european	european	ADJ
ejpam-6014	443	2	journal	journal	PROPN
ejpam-6014	443	3	of	of	ADP
ejpam-6014	443	4	pure	pure	ADJ
ejpam-6014	443	5	and	and	CCONJ
ejpam-6014	443	6	applied	applied	ADJ
ejpam-6014	443	7	mathematics	mathematic	NOUN
ejpam-6014	443	8	,	,	PUNCT
ejpam-6014	443	9	17(1):416–425	17(1):416–425	NUM
ejpam-6014	443	10	,	,	PUNCT
ejpam-6014	443	11	2024	2024	NUM
ejpam-6014	443	12	.	.	PUNCT
ejpam-6014	444	1	[	[	X
ejpam-6014	444	2	31	31	NUM
ejpam-6014	444	3	]	]	X
ejpam-6014	444	4	n.	n.	PROPN
ejpam-6014	444	5	srisarakham	srisarakham	PROPN
ejpam-6014	444	6	,	,	PUNCT
ejpam-6014	444	7	s.	s.	PROPN
ejpam-6014	444	8	sompong	sompong	PROPN
ejpam-6014	444	9	,	,	PUNCT
ejpam-6014	444	10	and	and	CCONJ
ejpam-6014	444	11	c.	c.	PROPN
ejpam-6014	444	12	boonpok	boonpok	PROPN
ejpam-6014	444	13	.	.	PUNCT
ejpam-6014	445	1	quasi	quasi	PROPN
ejpam-6014	445	2	θ(τ1	θ(τ1	PROPN
ejpam-6014	445	3	,	,	PUNCT
ejpam-6014	445	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6014	445	5	functions	function	NOUN
ejpam-6014	445	6	.	.	PUNCT
ejpam-6014	446	1	european	european	ADJ
ejpam-6014	446	2	journal	journal	PROPN
ejpam-6014	446	3	of	of	ADP
ejpam-6014	446	4	pure	pure	ADJ
ejpam-6014	446	5	and	and	CCONJ
ejpam-6014	446	6	applied	applied	ADJ
ejpam-6014	446	7	mathematics	mathematic	NOUN
ejpam-6014	446	8	,	,	PUNCT
ejpam-6014	446	9	18(1):5722	18(1):5722	NUM
ejpam-6014	446	10	,	,	PUNCT
ejpam-6014	446	11	2025	2025	NUM
ejpam-6014	446	12	.	.	PUNCT
ejpam-6014	447	1	[	[	X
ejpam-6014	447	2	32	32	NUM
ejpam-6014	447	3	]	]	PUNCT
ejpam-6014	447	4	c.	c.	PROPN
ejpam-6014	447	5	boonpok	boonpok	PROPN
ejpam-6014	447	6	,	,	PUNCT
ejpam-6014	447	7	c.	c.	PROPN
ejpam-6014	447	8	viriyapong	viriyapong	PROPN
ejpam-6014	447	9	,	,	PUNCT
ejpam-6014	447	10	and	and	CCONJ
ejpam-6014	447	11	m.	m.	NOUN
ejpam-6014	447	12	thongmoon	thongmoon	NOUN
ejpam-6014	447	13	.	.	PUNCT
ejpam-6014	448	1	on	on	ADP
ejpam-6014	448	2	upper	upper	ADJ
ejpam-6014	448	3	and	and	CCONJ
ejpam-6014	448	4	lower	low	ADJ
ejpam-6014	448	5	(	(	PUNCT
ejpam-6014	448	6	τ1	τ1	NOUN
ejpam-6014	448	7	,	,	PUNCT
ejpam-6014	448	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6014	448	9	multifunctions	multifunction	NOUN
ejpam-6014	448	10	.	.	PUNCT
ejpam-6014	449	1	journal	journal	PROPN
ejpam-6014	449	2	of	of	ADP
ejpam-6014	449	3	mathematics	mathematics	PROPN
ejpam-6014	449	4	and	and	CCONJ
ejpam-6014	449	5	computer	computer	NOUN
ejpam-6014	449	6	science	science	NOUN
ejpam-6014	449	7	,	,	PUNCT
ejpam-6014	449	8	18:282–293	18:282–293	NUM
ejpam-6014	449	9	,	,	PUNCT
ejpam-6014	449	10	2018	2018	NUM
ejpam-6014	449	11	.	.	PUNCT
ejpam-6014	450	1	[	[	X
ejpam-6014	450	2	33	33	NUM
ejpam-6014	450	3	]	]	X
ejpam-6014	450	4	c.	c.	PROPN
ejpam-6014	450	5	viriyapong	viriyapong	PROPN
ejpam-6014	450	6	and	and	CCONJ
ejpam-6014	450	7	c.	c.	PROPN
ejpam-6014	450	8	boonpok	boonpok	PROPN
ejpam-6014	450	9	.	.	PUNCT
ejpam-6014	451	1	(	(	PUNCT
ejpam-6014	451	2	τ1	τ1	NOUN
ejpam-6014	451	3	,	,	PUNCT
ejpam-6014	451	4	τ2)α	τ2)α	NOUN
ejpam-6014	451	5	-	-	PUNCT
ejpam-6014	451	6	continuity	continuity	NOUN
ejpam-6014	451	7	for	for	ADP
ejpam-6014	451	8	multifunctions	multifunction	NOUN
ejpam-6014	451	9	.	.	PUNCT
ejpam-6014	452	1	journal	journal	PROPN
ejpam-6014	452	2	of	of	ADP
ejpam-6014	452	3	mathematics	mathematic	NOUN
ejpam-6014	452	4	,	,	PUNCT
ejpam-6014	452	5	2020:6285763	2020:6285763	NUM
ejpam-6014	452	6	,	,	PUNCT
ejpam-6014	452	7	2020	2020	NUM
ejpam-6014	452	8	.	.	PUNCT
ejpam-6014	453	1	[	[	X
ejpam-6014	453	2	34	34	NUM
ejpam-6014	453	3	]	]	PUNCT
ejpam-6014	453	4	c.	c.	PROPN
ejpam-6014	453	5	boonpok	boonpok	PROPN
ejpam-6014	453	6	.	.	PUNCT
ejpam-6014	454	1	(	(	PUNCT
ejpam-6014	454	2	τ1	τ1	NOUN
ejpam-6014	454	3	,	,	PUNCT
ejpam-6014	454	4	τ2)δ	τ2)δ	ADJ
ejpam-6014	454	5	-	-	PUNCT
ejpam-6014	454	6	semicontinuous	semicontinuous	ADJ
ejpam-6014	454	7	multifunctions	multifunction	NOUN
ejpam-6014	454	8	.	.	PUNCT
ejpam-6014	455	1	heliyon	heliyon	NOUN
ejpam-6014	455	2	,	,	PUNCT
ejpam-6014	455	3	6	6	NUM
ejpam-6014	455	4	:	:	SYM
ejpam-6014	455	5	e05367	e05367	PROPN
ejpam-6014	455	6	,	,	PUNCT
ejpam-6014	455	7	2020	2020	NUM
ejpam-6014	455	8	.	.	PUNCT
ejpam-6014	456	1	[	[	X
ejpam-6014	456	2	35	35	NUM
ejpam-6014	456	3	]	]	X
ejpam-6014	456	4	n.	n.	NOUN
ejpam-6014	456	5	viriyapong	viriyapong	PROPN
ejpam-6014	456	6	,	,	PUNCT
ejpam-6014	456	7	s.	s.	PROPN
ejpam-6014	456	8	sompong	sompong	PROPN
ejpam-6014	456	9	,	,	PUNCT
ejpam-6014	456	10	and	and	CCONJ
ejpam-6014	456	11	c.	c.	PROPN
ejpam-6014	456	12	boonpok	boonpok	PROPN
ejpam-6014	456	13	.	.	PUNCT
ejpam-6014	457	1	(	(	PUNCT
ejpam-6014	457	2	τ1	τ1	NOUN
ejpam-6014	457	3	,	,	PUNCT
ejpam-6014	457	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6014	457	5	disconnectedness	disconnectedness	NOUN
ejpam-6014	457	6	in	in	ADP
ejpam-6014	457	7	bitopological	bitopological	ADJ
ejpam-6014	457	8	spaces	space	NOUN
ejpam-6014	457	9	.	.	PUNCT
ejpam-6014	458	1	international	international	ADJ
ejpam-6014	458	2	journal	journal	PROPN
ejpam-6014	458	3	of	of	ADP
ejpam-6014	458	4	mathematics	mathematic	NOUN
ejpam-6014	458	5	and	and	CCONJ
ejpam-6014	458	6	computer	computer	NOUN
ejpam-6014	458	7	science	science	NOUN
ejpam-6014	458	8	,	,	PUNCT
ejpam-6014	458	9	m.	m.	NOUN
ejpam-6014	458	10	thongmoon	thongmoon	NOUN
ejpam-6014	458	11	,	,	PUNCT
ejpam-6014	458	12	s.	s.	PROPN
ejpam-6014	458	13	sompong	sompong	PROPN
ejpam-6014	458	14	,	,	PUNCT
ejpam-6014	458	15	c.	c.	PROPN
ejpam-6014	458	16	boonpok	boonpok	PROPN
ejpam-6014	458	17	/	/	SYM
ejpam-6014	458	18	eur	eur	PROPN
ejpam-6014	458	19	.	.	PUNCT
ejpam-6014	459	1	j.	j.	PROPN
ejpam-6014	459	2	pure	pure	PROPN
ejpam-6014	459	3	appl	appl	PROPN
ejpam-6014	459	4	.	.	PROPN
ejpam-6014	459	5	math	math	PROPN
ejpam-6014	459	6	,	,	PUNCT
ejpam-6014	459	7	18	18	NUM
ejpam-6014	459	8	(	(	PUNCT
ejpam-6014	459	9	2	2	NUM
ejpam-6014	459	10	)	)	PUNCT
ejpam-6014	459	11	(	(	PUNCT
ejpam-6014	459	12	2025	2025	NUM
ejpam-6014	459	13	)	)	PUNCT
ejpam-6014	459	14	,	,	PUNCT
ejpam-6014	459	15	6014	6014	NUM
ejpam-6014	459	16	13	13	NUM
ejpam-6014	459	17	of	of	ADP
ejpam-6014	459	18	13	13	NUM
ejpam-6014	459	19	19(3):855–860	19(3):855–860	NOUN
ejpam-6014	459	20	,	,	PUNCT
ejpam-6014	459	21	2024	2024	NUM
ejpam-6014	459	22	.	.	PUNCT
ejpam-6014	460	1	[	[	X
ejpam-6014	460	2	36	36	NUM
ejpam-6014	460	3	]	]	PUNCT
ejpam-6014	460	4	m.	m.	NOUN
ejpam-6014	460	5	chiangpradit	chiangpradit	NOUN
ejpam-6014	460	6	,	,	PUNCT
ejpam-6014	460	7	s.	s.	PROPN
ejpam-6014	460	8	sompong	sompong	PROPN
ejpam-6014	460	9	,	,	PUNCT
ejpam-6014	460	10	and	and	CCONJ
ejpam-6014	460	11	c.	c.	PROPN
ejpam-6014	460	12	boonpok	boonpok	PROPN
ejpam-6014	460	13	.	.	PUNCT
ejpam-6014	461	1	on	on	ADP
ejpam-6014	461	2	characterizations	characterization	NOUN
ejpam-6014	461	3	of	of	ADP
ejpam-6014	461	4	(	(	PUNCT
ejpam-6014	461	5	τ1	τ1	NOUN
ejpam-6014	461	6	,	,	PUNCT
ejpam-6014	461	7	τ2)regular	τ2)regular	ADJ
ejpam-6014	461	8	spaces	space	NOUN
ejpam-6014	461	9	.	.	PUNCT
ejpam-6014	462	1	international	international	ADJ
ejpam-6014	462	2	journal	journal	PROPN
ejpam-6014	462	3	of	of	ADP
ejpam-6014	462	4	mathematics	mathematic	NOUN
ejpam-6014	462	5	and	and	CCONJ
ejpam-6014	462	6	computer	computer	NOUN
ejpam-6014	462	7	science	science	NOUN
ejpam-6014	462	8	,	,	PUNCT
ejpam-6014	462	9	19(4):1229–1334	19(4):1229–1334	NUM
ejpam-6014	462	10	,	,	PUNCT
ejpam-6014	462	11	2024	2024	NUM
ejpam-6014	462	12	.	.	PUNCT
ejpam-6014	463	1	[	[	X
ejpam-6014	463	2	37	37	NUM
ejpam-6014	463	3	]	]	X
ejpam-6014	463	4	c.	c.	PROPN
ejpam-6014	463	5	klanarong	klanarong	PROPN
ejpam-6014	463	6	,	,	PUNCT
ejpam-6014	463	7	s.	s.	PROPN
ejpam-6014	463	8	sompong	sompong	PROPN
ejpam-6014	463	9	,	,	PUNCT
ejpam-6014	463	10	and	and	CCONJ
ejpam-6014	463	11	c.	c.	PROPN
ejpam-6014	463	12	boonpok	boonpok	PROPN
ejpam-6014	463	13	.	.	PUNCT
ejpam-6014	464	1	(	(	PUNCT
ejpam-6014	464	2	τ1	τ1	NOUN
ejpam-6014	464	3	,	,	PUNCT
ejpam-6014	464	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6014	464	5	and	and	CCONJ
ejpam-6014	464	6	(	(	PUNCT
ejpam-6014	464	7	τ1	τ1	NOUN
ejpam-6014	464	8	,	,	PUNCT
ejpam-6014	464	9	τ2)θ	τ2)θ	ADJ
ejpam-6014	464	10	-	-	PUNCT
ejpam-6014	464	11	closed	close	VERB
ejpam-6014	464	12	sets	set	NOUN
ejpam-6014	464	13	.	.	PUNCT
ejpam-6014	465	1	international	international	ADJ
ejpam-6014	465	2	journal	journal	NOUN
ejpam-6014	465	3	of	of	ADP
ejpam-6014	465	4	mathematics	mathematic	NOUN
ejpam-6014	465	5	and	and	CCONJ
ejpam-6014	465	6	computer	computer	NOUN
ejpam-6014	465	7	science	science	NOUN
ejpam-6014	465	8	,	,	PUNCT
ejpam-6014	465	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-6014	465	10	,	,	PUNCT
ejpam-6014	465	11	2024	2024	NUM
ejpam-6014	465	12	.	.	PUNCT
ejpam-6014	466	1	[	[	X
ejpam-6014	466	2	38	38	NUM
ejpam-6014	466	3	]	]	PUNCT
ejpam-6014	466	4	n.	n.	NOUN
ejpam-6014	466	5	chutiman	chutiman	NOUN
ejpam-6014	466	6	,	,	PUNCT
ejpam-6014	466	7	s.	s.	PROPN
ejpam-6014	466	8	sompong	sompong	PROPN
ejpam-6014	466	9	,	,	PUNCT
ejpam-6014	466	10	and	and	CCONJ
ejpam-6014	466	11	c.	c.	PROPN
ejpam-6014	466	12	boonpok	boonpok	PROPN
ejpam-6014	466	13	.	.	PUNCT
ejpam-6014	467	1	on	on	ADP
ejpam-6014	467	2	some	some	DET
ejpam-6014	467	3	separation	separation	NOUN
ejpam-6014	467	4	axioms	axiom	NOUN
ejpam-6014	467	5	in	in	ADP
ejpam-6014	467	6	bitopological	bitopological	ADJ
ejpam-6014	467	7	spaces	space	NOUN
ejpam-6014	467	8	.	.	PUNCT
ejpam-6014	468	1	asia	asia	PROPN
ejpam-6014	468	2	pacific	pacific	PROPN
ejpam-6014	468	3	journal	journal	PROPN
ejpam-6014	468	4	of	of	ADP
ejpam-6014	468	5	mathematics	mathematic	NOUN
ejpam-6014	468	6	,	,	PUNCT
ejpam-6014	468	7	11:41	11:41	NUM
ejpam-6014	468	8	,	,	PUNCT
ejpam-6014	468	9	2024	2024	NUM
ejpam-6014	468	10	.	.	PUNCT
ejpam-6014	469	1	[	[	X
ejpam-6014	469	2	39	39	NUM
ejpam-6014	469	3	]	]	PUNCT
ejpam-6014	469	4	p.	p.	NOUN
ejpam-6014	469	5	pue	pue	NOUN
ejpam-6014	469	6	-	-	PUNCT
ejpam-6014	469	7	on	on	ADP
ejpam-6014	469	8	,	,	PUNCT
ejpam-6014	469	9	a.	a.	PROPN
ejpam-6014	469	10	sama	sama	PROPN
ejpam-6014	469	11	-	-	PUNCT
ejpam-6014	469	12	ae	ae	PROPN
ejpam-6014	469	13	,	,	PUNCT
ejpam-6014	469	14	and	and	CCONJ
ejpam-6014	469	15	c.	c.	PROPN
ejpam-6014	469	16	boonpok	boonpok	PROPN
ejpam-6014	469	17	.	.	PUNCT
ejpam-6014	470	1	characterizations	characterization	NOUN
ejpam-6014	470	2	of	of	ADP
ejpam-6014	470	3	quasi	quasi	NOUN
ejpam-6014	470	4	θ(τ1	θ(τ1	NOUN
ejpam-6014	470	5	,	,	PUNCT
ejpam-6014	470	6	τ2)continuous	τ2)continuous	ADJ
ejpam-6014	470	7	multifunctions	multifunction	NOUN
ejpam-6014	470	8	.	.	PUNCT
ejpam-6014	471	1	international	international	ADJ
ejpam-6014	471	2	journal	journal	NOUN
ejpam-6014	471	3	of	of	ADP
ejpam-6014	471	4	analysis	analysis	NOUN
ejpam-6014	471	5	and	and	CCONJ
ejpam-6014	471	6	applications	application	NOUN
ejpam-6014	471	7	,	,	PUNCT
ejpam-6014	471	8	23:59	23:59	NUM
ejpam-6014	471	9	,	,	PUNCT
ejpam-6014	471	10	2025	2025	NUM
ejpam-6014	471	11	.	.	PUNCT
ejpam-6014	472	1	[	[	X
ejpam-6014	472	2	40	40	NUM
ejpam-6014	472	3	]	]	PUNCT
ejpam-6014	472	4	m.	m.	NOUN
ejpam-6014	472	5	thongmoon	thongmoon	NOUN
ejpam-6014	472	6	,	,	PUNCT
ejpam-6014	472	7	s.	s.	PROPN
ejpam-6014	472	8	sompong	sompong	PROPN
ejpam-6014	472	9	,	,	PUNCT
ejpam-6014	472	10	and	and	CCONJ
ejpam-6014	472	11	c.	c.	PROPN
ejpam-6014	472	12	boonpok	boonpok	PROPN
ejpam-6014	472	13	.	.	PUNCT
ejpam-6014	473	1	upper	upper	ADJ
ejpam-6014	473	2	and	and	CCONJ
ejpam-6014	473	3	lower	low	ADJ
ejpam-6014	473	4	weak	weak	ADJ
ejpam-6014	473	5	(	(	PUNCT
ejpam-6014	473	6	τ1	τ1	NOUN
ejpam-6014	473	7	,	,	PUNCT
ejpam-6014	473	8	τ2)continuity	τ2)continuity	PROPN
ejpam-6014	473	9	.	.	PUNCT
ejpam-6014	474	1	european	european	PROPN
ejpam-6014	474	2	journal	journal	PROPN
ejpam-6014	474	3	of	of	ADP
ejpam-6014	474	4	pure	pure	ADJ
ejpam-6014	474	5	and	and	CCONJ
ejpam-6014	474	6	applied	applied	ADJ
ejpam-6014	474	7	mathematics	mathematic	NOUN
ejpam-6014	474	8	,	,	PUNCT
ejpam-6014	474	9	17(3):1705–1716	17(3):1705–1716	NUM
ejpam-6014	474	10	,	,	PUNCT
ejpam-6014	474	11	2024	2024	NUM
ejpam-6014	474	12	.	.	PUNCT