id	sid	tid	token	lemma	pos
ejpam-6565	1	1	european	european	PROPN
ejpam-6565	1	2	journal	journal	PROPN
ejpam-6565	1	3	of	of	ADP
ejpam-6565	1	4	pure	pure	ADJ
ejpam-6565	1	5	and	and	CCONJ
ejpam-6565	1	6	applied	applied	ADJ
ejpam-6565	1	7	mathematics	mathematic	NOUN
ejpam-6565	1	8	2025	2025	NUM
ejpam-6565	1	9	,	,	PUNCT
ejpam-6565	1	10	vol	vol	NOUN
ejpam-6565	1	11	.	.	PROPN
ejpam-6565	1	12	18	18	NUM
ejpam-6565	1	13	,	,	PUNCT
ejpam-6565	1	14	issue	issue	NOUN
ejpam-6565	1	15	3	3	NUM
ejpam-6565	1	16	,	,	PUNCT
ejpam-6565	1	17	article	article	NOUN
ejpam-6565	1	18	number	number	NOUN
ejpam-6565	1	19	6565	6565	NUM
ejpam-6565	1	20	issn	issn	PROPN
ejpam-6565	1	21	1307	1307	NUM
ejpam-6565	1	22	-	-	SYM
ejpam-6565	1	23	5543	5543	NUM
ejpam-6565	1	24	–	–	PUNCT
ejpam-6565	1	25	ejpam.com	ejpam.com	X
ejpam-6565	1	26	published	publish	VERB
ejpam-6565	1	27	by	by	ADP
ejpam-6565	1	28	new	new	PROPN
ejpam-6565	1	29	york	york	PROPN
ejpam-6565	1	30	business	business	PROPN
ejpam-6565	1	31	global	global	PROPN
ejpam-6565	1	32	upper	upper	ADJ
ejpam-6565	1	33	and	and	CCONJ
ejpam-6565	1	34	lower	low	ADJ
ejpam-6565	1	35	continuous	continuous	ADJ
ejpam-6565	1	36	multifunctions	multifunction	NOUN
ejpam-6565	1	37	defined	define	VERB
ejpam-6565	1	38	between	between	ADP
ejpam-6565	1	39	an	an	DET
ejpam-6565	1	40	ideal	ideal	ADJ
ejpam-6565	1	41	topological	topological	ADJ
ejpam-6565	1	42	space	space	NOUN
ejpam-6565	1	43	and	and	CCONJ
ejpam-6565	1	44	a	a	DET
ejpam-6565	1	45	bitopological	bitopological	ADJ
ejpam-6565	1	46	space	space	NOUN
ejpam-6565	1	47	jeeranunt	jeeranunt	NOUN
ejpam-6565	1	48	khampakdee1	khampakdee1	PROPN
ejpam-6565	1	49	,	,	PUNCT
ejpam-6565	1	50	areeyuth	areeyuth	NOUN
ejpam-6565	1	51	sama	sama	NOUN
ejpam-6565	1	52	-	-	PUNCT
ejpam-6565	1	53	ae2	ae2	PROPN
ejpam-6565	1	54	,	,	PUNCT
ejpam-6565	1	55	chawalit	chawalit	VERB
ejpam-6565	1	56	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6565	1	57	1	1	NUM
ejpam-6565	1	58	mathematics	mathematic	NOUN
ejpam-6565	1	59	and	and	CCONJ
ejpam-6565	1	60	applied	apply	VERB
ejpam-6565	1	61	mathematics	mathematics	PROPN
ejpam-6565	1	62	research	research	NOUN
ejpam-6565	1	63	unit	unit	NOUN
ejpam-6565	1	64	,	,	PUNCT
ejpam-6565	1	65	department	department	NOUN
ejpam-6565	1	66	of	of	ADP
ejpam-6565	1	67	mathematics	mathematic	NOUN
ejpam-6565	1	68	,	,	PUNCT
ejpam-6565	1	69	faculty	faculty	NOUN
ejpam-6565	1	70	of	of	ADP
ejpam-6565	1	71	science	science	NOUN
ejpam-6565	1	72	,	,	PUNCT
ejpam-6565	1	73	mahasarakham	mahasarakham	PROPN
ejpam-6565	1	74	university	university	PROPN
ejpam-6565	1	75	,	,	PUNCT
ejpam-6565	1	76	maha	maha	PROPN
ejpam-6565	1	77	sarakham	sarakham	PROPN
ejpam-6565	1	78	,	,	PUNCT
ejpam-6565	1	79	44150	44150	NUM
ejpam-6565	1	80	,	,	PUNCT
ejpam-6565	1	81	thailand	thailand	PROPN
ejpam-6565	1	82	2	2	NUM
ejpam-6565	1	83	department	department	NOUN
ejpam-6565	1	84	of	of	ADP
ejpam-6565	1	85	mathematics	mathematic	NOUN
ejpam-6565	1	86	and	and	CCONJ
ejpam-6565	1	87	computer	computer	NOUN
ejpam-6565	1	88	science	science	NOUN
ejpam-6565	1	89	,	,	PUNCT
ejpam-6565	1	90	faculty	faculty	NOUN
ejpam-6565	1	91	of	of	ADP
ejpam-6565	1	92	science	science	NOUN
ejpam-6565	1	93	and	and	CCONJ
ejpam-6565	1	94	technology	technology	NOUN
ejpam-6565	1	95	,	,	PUNCT
ejpam-6565	1	96	prince	prince	NOUN
ejpam-6565	1	97	of	of	ADP
ejpam-6565	1	98	songkla	songkla	PROPN
ejpam-6565	1	99	university	university	PROPN
ejpam-6565	1	100	,	,	PUNCT
ejpam-6565	1	101	pattani	pattani	NOUN
ejpam-6565	1	102	campus	campus	NOUN
ejpam-6565	1	103	,	,	PUNCT
ejpam-6565	1	104	pattani	pattani	NOUN
ejpam-6565	1	105	,	,	PUNCT
ejpam-6565	1	106	94000	94000	NUM
ejpam-6565	1	107	,	,	PUNCT
ejpam-6565	1	108	thailand	thailand	PROPN
ejpam-6565	1	109	abstract	abstract	PROPN
ejpam-6565	1	110	.	.	PUNCT
ejpam-6565	2	1	this	this	DET
ejpam-6565	2	2	paper	paper	NOUN
ejpam-6565	2	3	presents	present	VERB
ejpam-6565	2	4	new	new	ADJ
ejpam-6565	2	5	concepts	concept	NOUN
ejpam-6565	2	6	of	of	ADP
ejpam-6565	2	7	continuous	continuous	ADJ
ejpam-6565	2	8	multifunctions	multifunction	NOUN
ejpam-6565	2	9	defined	define	VERB
ejpam-6565	2	10	from	from	ADP
ejpam-6565	2	11	an	an	DET
ejpam-6565	2	12	ideal	ideal	ADJ
ejpam-6565	2	13	topological	topological	ADJ
ejpam-6565	2	14	space	space	NOUN
ejpam-6565	2	15	into	into	ADP
ejpam-6565	2	16	a	a	DET
ejpam-6565	2	17	bitopological	bitopological	ADJ
ejpam-6565	2	18	space	space	NOUN
ejpam-6565	2	19	,	,	PUNCT
ejpam-6565	2	20	called	call	VERB
ejpam-6565	2	21	upper	upper	ADJ
ejpam-6565	2	22	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	2	23	,	,	PUNCT
ejpam-6565	2	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	2	25	multifunctions	multifunction	NOUN
ejpam-6565	2	26	and	and	CCONJ
ejpam-6565	2	27	lower	low	ADJ
ejpam-6565	2	28	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	2	29	,	,	PUNCT
ejpam-6565	2	30	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	2	31	multifunctions	multifunction	NOUN
ejpam-6565	2	32	.	.	PUNCT
ejpam-6565	3	1	furthermore	furthermore	ADV
ejpam-6565	3	2	,	,	PUNCT
ejpam-6565	3	3	several	several	ADJ
ejpam-6565	3	4	characterizations	characterization	NOUN
ejpam-6565	3	5	and	and	CCONJ
ejpam-6565	3	6	some	some	DET
ejpam-6565	3	7	properties	property	NOUN
ejpam-6565	3	8	concerning	concern	VERB
ejpam-6565	3	9	upper	upper	ADJ
ejpam-6565	3	10	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	3	11	,	,	PUNCT
ejpam-6565	3	12	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	3	13	multifunctions	multifunction	NOUN
ejpam-6565	3	14	and	and	CCONJ
ejpam-6565	3	15	lower	low	ADJ
ejpam-6565	3	16	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	3	17	,	,	PUNCT
ejpam-6565	3	18	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	3	19	multifunctions	multifunction	NOUN
ejpam-6565	3	20	are	be	AUX
ejpam-6565	3	21	investigated	investigate	VERB
ejpam-6565	3	22	.	.	PUNCT
ejpam-6565	4	1	2020	2020	NUM
ejpam-6565	4	2	mathematics	mathematic	NOUN
ejpam-6565	4	3	subject	subject	NOUN
ejpam-6565	4	4	classifications	classification	NOUN
ejpam-6565	4	5	:	:	PUNCT
ejpam-6565	4	6	54c08	54c08	NUM
ejpam-6565	4	7	,	,	PUNCT
ejpam-6565	4	8	54c60	54c60	NUM
ejpam-6565	4	9	key	key	ADJ
ejpam-6565	4	10	words	word	NOUN
ejpam-6565	4	11	and	and	CCONJ
ejpam-6565	4	12	phrases	phrase	NOUN
ejpam-6565	4	13	:	:	PUNCT
ejpam-6565	4	14	upper	upper	ADJ
ejpam-6565	4	15	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	4	16	,	,	PUNCT
ejpam-6565	4	17	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	4	18	multifunction	multifunction	NOUN
ejpam-6565	4	19	,	,	PUNCT
ejpam-6565	4	20	lower	low	ADJ
ejpam-6565	4	21	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	4	22	,	,	PUNCT
ejpam-6565	4	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	4	24	multifunction	multifunction	NOUN
ejpam-6565	4	25	1	1	NUM
ejpam-6565	4	26	.	.	PUNCT
ejpam-6565	4	27	introduction	introduction	NOUN
ejpam-6565	4	28	the	the	DET
ejpam-6565	4	29	field	field	NOUN
ejpam-6565	4	30	of	of	ADP
ejpam-6565	4	31	the	the	DET
ejpam-6565	4	32	mathematical	mathematical	ADJ
ejpam-6565	4	33	science	science	NOUN
ejpam-6565	4	34	which	which	PRON
ejpam-6565	4	35	goes	go	VERB
ejpam-6565	4	36	under	under	ADP
ejpam-6565	4	37	the	the	DET
ejpam-6565	4	38	name	name	NOUN
ejpam-6565	4	39	of	of	ADP
ejpam-6565	4	40	topology	topology	NOUN
ejpam-6565	4	41	is	be	AUX
ejpam-6565	4	42	concerned	concern	VERB
ejpam-6565	4	43	with	with	ADP
ejpam-6565	4	44	all	all	DET
ejpam-6565	4	45	questions	question	NOUN
ejpam-6565	4	46	directly	directly	ADV
ejpam-6565	4	47	or	or	CCONJ
ejpam-6565	4	48	indirectly	indirectly	ADV
ejpam-6565	4	49	related	relate	VERB
ejpam-6565	4	50	to	to	ADP
ejpam-6565	4	51	continuity	continuity	NOUN
ejpam-6565	4	52	.	.	PUNCT
ejpam-6565	5	1	the	the	DET
ejpam-6565	5	2	concept	concept	NOUN
ejpam-6565	5	3	of	of	ADP
ejpam-6565	5	4	ideals	ideal	NOUN
ejpam-6565	5	5	in	in	ADP
ejpam-6565	5	6	topological	topological	ADJ
ejpam-6565	5	7	spaces	space	NOUN
ejpam-6565	5	8	has	have	AUX
ejpam-6565	5	9	been	be	AUX
ejpam-6565	5	10	introduced	introduce	VERB
ejpam-6565	5	11	and	and	CCONJ
ejpam-6565	5	12	studied	study	VERB
ejpam-6565	5	13	by	by	ADP
ejpam-6565	5	14	kuratowski	kuratowski	ADJ
ejpam-6565	5	15	[	[	X
ejpam-6565	5	16	1	1	NUM
ejpam-6565	5	17	]	]	PUNCT
ejpam-6565	5	18	and	and	CCONJ
ejpam-6565	5	19	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-6565	6	1	[	[	X
ejpam-6565	6	2	2	2	NUM
ejpam-6565	6	3	]	]	PUNCT
ejpam-6565	6	4	which	which	PRON
ejpam-6565	6	5	is	be	AUX
ejpam-6565	6	6	one	one	NUM
ejpam-6565	6	7	of	of	ADP
ejpam-6565	6	8	the	the	DET
ejpam-6565	6	9	important	important	ADJ
ejpam-6565	6	10	areas	area	NOUN
ejpam-6565	6	11	of	of	ADP
ejpam-6565	6	12	research	research	NOUN
ejpam-6565	6	13	in	in	ADP
ejpam-6565	6	14	the	the	DET
ejpam-6565	6	15	branch	branch	NOUN
ejpam-6565	6	16	of	of	ADP
ejpam-6565	6	17	mathematics	mathematic	NOUN
ejpam-6565	6	18	.	.	PUNCT
ejpam-6565	7	1	stronger	strong	ADJ
ejpam-6565	7	2	and	and	CCONJ
ejpam-6565	7	3	weaker	weak	ADJ
ejpam-6565	7	4	forms	form	NOUN
ejpam-6565	7	5	of	of	ADP
ejpam-6565	7	6	open	open	ADJ
ejpam-6565	7	7	sets	set	NOUN
ejpam-6565	7	8	in	in	ADP
ejpam-6565	7	9	ideal	ideal	ADJ
ejpam-6565	7	10	topological	topological	ADJ
ejpam-6565	7	11	spaces	space	NOUN
ejpam-6565	7	12	such	such	ADJ
ejpam-6565	7	13	as	as	ADP
ejpam-6565	7	14	semi	semi	ADJ
ejpam-6565	7	15	-	-	ADJ
ejpam-6565	7	16	i	i	ADJ
ejpam-6565	7	17	-open	-open	NOUN
ejpam-6565	7	18	sets	set	NOUN
ejpam-6565	7	19	,	,	PUNCT
ejpam-6565	7	20	pre	pre	ADJ
ejpam-6565	7	21	-	-	ADJ
ejpam-6565	7	22	i	i	ADJ
ejpam-6565	7	23	-open	-open	NOUN
ejpam-6565	7	24	sets	set	NOUN
ejpam-6565	7	25	,	,	PUNCT
ejpam-6565	7	26	α	α	X
ejpam-6565	7	27	-	-	PUNCT
ejpam-6565	7	28	i	i	PRON
ejpam-6565	7	29	-open	-open	NOUN
ejpam-6565	7	30	sets	set	NOUN
ejpam-6565	7	31	,	,	PUNCT
ejpam-6565	7	32	β	β	X
ejpam-6565	7	33	-	-	ADJ
ejpam-6565	7	34	i	i	PRON
ejpam-6565	7	35	-open	-open	NOUN
ejpam-6565	7	36	sets	set	NOUN
ejpam-6565	7	37	and	and	CCONJ
ejpam-6565	7	38	δ	δ	PROPN
ejpam-6565	7	39	-	-	PUNCT
ejpam-6565	7	40	i	i	PRON
ejpam-6565	7	41	-open	-open	NOUN
ejpam-6565	7	42	sets	set	NOUN
ejpam-6565	7	43	play	play	VERB
ejpam-6565	7	44	an	an	DET
ejpam-6565	7	45	important	important	ADJ
ejpam-6565	7	46	role	role	NOUN
ejpam-6565	7	47	in	in	ADP
ejpam-6565	7	48	the	the	DET
ejpam-6565	7	49	research	research	NOUN
ejpam-6565	7	50	of	of	ADP
ejpam-6565	7	51	generalizations	generalization	NOUN
ejpam-6565	7	52	of	of	ADP
ejpam-6565	7	53	continuity	continuity	NOUN
ejpam-6565	7	54	.	.	PUNCT
ejpam-6565	8	1	using	use	VERB
ejpam-6565	8	2	these	these	DET
ejpam-6565	8	3	notions	notion	NOUN
ejpam-6565	8	4	many	many	ADJ
ejpam-6565	8	5	authors	author	NOUN
ejpam-6565	8	6	introduced	introduce	VERB
ejpam-6565	8	7	and	and	CCONJ
ejpam-6565	8	8	studied	study	VERB
ejpam-6565	8	9	various	various	ADJ
ejpam-6565	8	10	types	type	NOUN
ejpam-6565	8	11	of	of	ADP
ejpam-6565	8	12	generalizations	generalization	NOUN
ejpam-6565	8	13	of	of	ADP
ejpam-6565	8	14	continuity	continuity	NOUN
ejpam-6565	8	15	for	for	ADP
ejpam-6565	8	16	functions	function	NOUN
ejpam-6565	8	17	and	and	CCONJ
ejpam-6565	8	18	multifunctions	multifunction	NOUN
ejpam-6565	8	19	.	.	PUNCT
ejpam-6565	9	1	hatir	hatir	PROPN
ejpam-6565	9	2	and	and	CCONJ
ejpam-6565	9	3	noiri	noiri	ADV
ejpam-6565	10	1	[	[	X
ejpam-6565	10	2	3	3	X
ejpam-6565	10	3	]	]	PUNCT
ejpam-6565	10	4	introduced	introduce	VERB
ejpam-6565	10	5	and	and	CCONJ
ejpam-6565	10	6	investigated	investigate	VERB
ejpam-6565	10	7	the	the	DET
ejpam-6565	10	8	notions	notion	NOUN
ejpam-6565	10	9	of	of	ADP
ejpam-6565	10	10	weakly	weakly	ADJ
ejpam-6565	10	11	pre	pre	ADJ
ejpam-6565	10	12	-	-	ADJ
ejpam-6565	10	13	i	i	PRON
ejpam-6565	10	14	-open	-open	NOUN
ejpam-6565	10	15	sets	set	NOUN
ejpam-6565	10	16	and	and	CCONJ
ejpam-6565	10	17	weakly	weakly	ADJ
ejpam-6565	10	18	pre	pre	ADJ
ejpam-6565	10	19	-	-	ADJ
ejpam-6565	10	20	i	i	ADJ
ejpam-6565	10	21	-continuous	-continuous	ADJ
ejpam-6565	10	22	functions	function	NOUN
ejpam-6565	10	23	.	.	PUNCT
ejpam-6565	11	1	moreover	moreover	ADV
ejpam-6565	11	2	,	,	PUNCT
ejpam-6565	11	3	hatir	hatir	PROPN
ejpam-6565	11	4	and	and	CCONJ
ejpam-6565	11	5	noiri	noiri	ADV
ejpam-6565	12	1	[	[	X
ejpam-6565	12	2	4	4	X
ejpam-6565	12	3	]	]	PUNCT
ejpam-6565	12	4	investigated	investigate	VERB
ejpam-6565	12	5	further	further	ADJ
ejpam-6565	12	6	properties	property	NOUN
ejpam-6565	12	7	of	of	ADP
ejpam-6565	12	8	semi	semi	ADJ
ejpam-6565	12	9	-	-	ADJ
ejpam-6565	12	10	i	i	PRON
ejpam-6565	12	11	-open	-open	NOUN
ejpam-6565	12	12	sets	set	NOUN
ejpam-6565	12	13	and	and	CCONJ
ejpam-6565	12	14	semi	semi	ADJ
ejpam-6565	12	15	-	-	ADJ
ejpam-6565	12	16	i	i	PRON
ejpam-6565	12	17	-continuous	-continuous	ADJ
ejpam-6565	12	18	∗corresponding	∗corresponding	NOUN
ejpam-6565	12	19	author	author	NOUN
ejpam-6565	12	20	.	.	PUNCT
ejpam-6565	13	1	doi	doi	NOUN
ejpam-6565	13	2	:	:	PUNCT
ejpam-6565	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6565	https://doi.org/10.29020/nybg.ejpam.v18i3.6565	ADP
ejpam-6565	13	4	email	email	NOUN
ejpam-6565	13	5	addresses	address	NOUN
ejpam-6565	13	6	:	:	PUNCT
ejpam-6565	13	7	jeeranunt.k@msu.ac.th	jeeranunt.k@msu.ac.th	INTJ
ejpam-6565	13	8	(	(	PUNCT
ejpam-6565	13	9	j.	j.	PROPN
ejpam-6565	13	10	khampakdee	khampakdee	PROPN
ejpam-6565	13	11	)	)	PUNCT
ejpam-6565	13	12	,	,	PUNCT
ejpam-6565	13	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-6565	13	14	(	(	PUNCT
ejpam-6565	13	15	a.	a.	PROPN
ejpam-6565	13	16	sama	sama	PROPN
ejpam-6565	13	17	-	-	PUNCT
ejpam-6565	13	18	ae	ae	PROPN
ejpam-6565	13	19	)	)	PUNCT
ejpam-6565	13	20	,	,	PUNCT
ejpam-6565	13	21	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6565	13	22	(	(	PUNCT
ejpam-6565	13	23	c.	c.	PROPN
ejpam-6565	13	24	boonpok	boonpok	PROPN
ejpam-6565	13	25	)	)	PUNCT
ejpam-6565	13	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6565	14	1	1	1	NUM
ejpam-6565	14	2	copyright	copyright	NOUN
ejpam-6565	14	3	:	:	PUNCT
ejpam-6565	14	4	©	©	PROPN
ejpam-6565	14	5	2025	2025	NUM
ejpam-6565	14	6	the	the	DET
ejpam-6565	14	7	author(s	author(s	NOUN
ejpam-6565	14	8	)	)	PUNCT
ejpam-6565	14	9	.	.	PUNCT
ejpam-6565	15	1	(	(	PUNCT
ejpam-6565	15	2	cc	cc	NOUN
ejpam-6565	15	3	by	by	ADP
ejpam-6565	15	4	-	-	PUNCT
ejpam-6565	15	5	nc	nc	PROPN
ejpam-6565	15	6	4.0	4.0	NUM
ejpam-6565	15	7	)	)	PUNCT
ejpam-6565	15	8	j.	j.	PROPN
ejpam-6565	15	9	khampakdee	khampakdee	PROPN
ejpam-6565	15	10	,	,	PUNCT
ejpam-6565	15	11	a.	a.	PROPN
ejpam-6565	15	12	sama	sama	PROPN
ejpam-6565	15	13	-	-	PUNCT
ejpam-6565	15	14	ae	ae	PROPN
ejpam-6565	15	15	,	,	PUNCT
ejpam-6565	15	16	c.	c.	PROPN
ejpam-6565	15	17	boonpok	boonpok	PROPN
ejpam-6565	15	18	/	/	SYM
ejpam-6565	15	19	eur	eur	PROPN
ejpam-6565	15	20	.	.	PUNCT
ejpam-6565	16	1	j.	j.	PROPN
ejpam-6565	16	2	pure	pure	PROPN
ejpam-6565	16	3	appl	appl	PROPN
ejpam-6565	16	4	.	.	PROPN
ejpam-6565	16	5	math	math	PROPN
ejpam-6565	16	6	,	,	PUNCT
ejpam-6565	16	7	18	18	NUM
ejpam-6565	16	8	(	(	PUNCT
ejpam-6565	16	9	3	3	NUM
ejpam-6565	16	10	)	)	PUNCT
ejpam-6565	16	11	(	(	PUNCT
ejpam-6565	16	12	2025	2025	NUM
ejpam-6565	16	13	)	)	PUNCT
ejpam-6565	16	14	,	,	PUNCT
ejpam-6565	16	15	6565	6565	NUM
ejpam-6565	16	16	2	2	NUM
ejpam-6565	16	17	of	of	ADP
ejpam-6565	16	18	9	9	NUM
ejpam-6565	16	19	functions	function	NOUN
ejpam-6565	16	20	.	.	PUNCT
ejpam-6565	17	1	on	on	ADP
ejpam-6565	17	2	the	the	DET
ejpam-6565	17	3	other	other	ADJ
ejpam-6565	17	4	hand	hand	NOUN
ejpam-6565	17	5	,	,	PUNCT
ejpam-6565	17	6	the	the	DET
ejpam-6565	17	7	present	present	ADJ
ejpam-6565	17	8	author	author	NOUN
ejpam-6565	17	9	[	[	X
ejpam-6565	17	10	5	5	NUM
ejpam-6565	17	11	]	]	PUNCT
ejpam-6565	17	12	introduced	introduce	VERB
ejpam-6565	17	13	new	new	ADJ
ejpam-6565	17	14	classes	class	NOUN
ejpam-6565	17	15	of	of	ADP
ejpam-6565	17	16	multifunctions	multifunction	NOUN
ejpam-6565	17	17	between	between	ADP
ejpam-6565	17	18	ideal	ideal	ADJ
ejpam-6565	17	19	topological	topological	ADJ
ejpam-6565	17	20	spaces	space	NOUN
ejpam-6565	17	21	,	,	PUNCT
ejpam-6565	17	22	namely	namely	ADV
ejpam-6565	17	23	upper	upper	ADJ
ejpam-6565	17	24	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	17	25	multifunctions	multifunction	NOUN
ejpam-6565	17	26	and	and	CCONJ
ejpam-6565	17	27	lower	low	ADJ
ejpam-6565	17	28	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	17	29	multifunctions	multifunction	NOUN
ejpam-6565	17	30	.	.	PUNCT
ejpam-6565	18	1	furthermore	furthermore	ADV
ejpam-6565	18	2	,	,	PUNCT
ejpam-6565	18	3	several	several	ADJ
ejpam-6565	18	4	characterizations	characterization	NOUN
ejpam-6565	18	5	of	of	ADP
ejpam-6565	18	6	upper	upper	ADJ
ejpam-6565	18	7	⋆continuous	⋆continuous	ADJ
ejpam-6565	18	8	multifunctions	multifunction	NOUN
ejpam-6565	18	9	,	,	PUNCT
ejpam-6565	18	10	lower	low	ADJ
ejpam-6565	18	11	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	18	12	multifunctions	multifunction	NOUN
ejpam-6565	18	13	,	,	PUNCT
ejpam-6565	18	14	upper	upper	ADJ
ejpam-6565	18	15	almost	almost	ADV
ejpam-6565	18	16	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	18	17	multifunctions	multifunction	NOUN
ejpam-6565	18	18	,	,	PUNCT
ejpam-6565	18	19	lower	low	ADJ
ejpam-6565	18	20	almost	almost	ADV
ejpam-6565	18	21	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	18	22	multifunctions	multifunction	NOUN
ejpam-6565	18	23	,	,	PUNCT
ejpam-6565	18	24	upper	upper	ADJ
ejpam-6565	18	25	weakly	weakly	ADJ
ejpam-6565	18	26	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	18	27	multifunctions	multifunction	NOUN
ejpam-6565	18	28	and	and	CCONJ
ejpam-6565	18	29	lower	low	ADJ
ejpam-6565	18	30	weakly	weakly	ADJ
ejpam-6565	18	31	⋆-continuous	⋆-continuous	ADJ
ejpam-6565	18	32	multifunctions	multifunction	NOUN
ejpam-6565	18	33	were	be	AUX
ejpam-6565	18	34	considered	consider	VERB
ejpam-6565	18	35	in	in	ADP
ejpam-6565	18	36	[	[	X
ejpam-6565	18	37	5	5	NUM
ejpam-6565	18	38	]	]	PUNCT
ejpam-6565	18	39	.	.	PUNCT
ejpam-6565	19	1	quite	quite	ADV
ejpam-6565	19	2	recently	recently	ADV
ejpam-6565	19	3	,	,	PUNCT
ejpam-6565	19	4	the	the	DET
ejpam-6565	19	5	present	present	ADJ
ejpam-6565	19	6	author	author	NOUN
ejpam-6565	19	7	[	[	X
ejpam-6565	19	8	6	6	NUM
ejpam-6565	19	9	]	]	PUNCT
ejpam-6565	19	10	introduced	introduce	VERB
ejpam-6565	19	11	and	and	CCONJ
ejpam-6565	19	12	investigated	investigate	VERB
ejpam-6565	19	13	the	the	DET
ejpam-6565	19	14	notions	notion	NOUN
ejpam-6565	19	15	of	of	ADP
ejpam-6565	19	16	pı	pı	ADJ
ejpam-6565	19	17	-	-	ADJ
ejpam-6565	19	18	continuous	continuous	ADJ
ejpam-6565	19	19	multifunctions	multifunction	NOUN
ejpam-6565	19	20	and	and	CCONJ
ejpam-6565	19	21	weakly	weakly	ADJ
ejpam-6565	19	22	pı	pı	ADJ
ejpam-6565	19	23	-	-	ADJ
ejpam-6565	19	24	continuous	continuous	ADJ
ejpam-6565	19	25	multifunctions	multifunction	NOUN
ejpam-6565	19	26	.	.	PUNCT
ejpam-6565	20	1	pue	pue	NOUN
ejpam-6565	20	2	-	-	PUNCT
ejpam-6565	20	3	on	on	NOUN
ejpam-6565	20	4	et	et	PROPN
ejpam-6565	20	5	al	al	PROPN
ejpam-6565	20	6	.	.	PUNCT
ejpam-6565	21	1	[	[	X
ejpam-6565	21	2	7	7	X
ejpam-6565	21	3	]	]	PUNCT
ejpam-6565	21	4	introduced	introduce	VERB
ejpam-6565	21	5	and	and	CCONJ
ejpam-6565	21	6	studied	study	VERB
ejpam-6565	21	7	the	the	DET
ejpam-6565	21	8	concepts	concept	NOUN
ejpam-6565	21	9	of	of	ADP
ejpam-6565	21	10	upper	upper	ADJ
ejpam-6565	21	11	(	(	PUNCT
ejpam-6565	21	12	τ1	τ1	NOUN
ejpam-6565	21	13	,	,	PUNCT
ejpam-6565	21	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	21	15	multifunctions	multifunction	NOUN
ejpam-6565	21	16	and	and	CCONJ
ejpam-6565	21	17	lower	low	ADJ
ejpam-6565	21	18	(	(	PUNCT
ejpam-6565	21	19	τ1	τ1	NOUN
ejpam-6565	21	20	,	,	PUNCT
ejpam-6565	21	21	τ2)continuous	τ2)continuous	ADJ
ejpam-6565	21	22	multifunctions	multifunction	NOUN
ejpam-6565	21	23	.	.	PUNCT
ejpam-6565	22	1	klanarong	klanarong	NOUN
ejpam-6565	22	2	et	et	PROPN
ejpam-6565	22	3	al	al	PROPN
ejpam-6565	22	4	.	.	PUNCT
ejpam-6565	23	1	[	[	X
ejpam-6565	23	2	8	8	NUM
ejpam-6565	23	3	]	]	PUNCT
ejpam-6565	23	4	investigated	investigate	VERB
ejpam-6565	23	5	several	several	ADJ
ejpam-6565	23	6	characterizations	characterization	NOUN
ejpam-6565	23	7	of	of	ADP
ejpam-6565	23	8	upper	upper	ADJ
ejpam-6565	23	9	(	(	PUNCT
ejpam-6565	23	10	τ1	τ1	NOUN
ejpam-6565	23	11	,	,	PUNCT
ejpam-6565	23	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	23	13	multifunctions	multifunction	NOUN
ejpam-6565	23	14	and	and	CCONJ
ejpam-6565	23	15	lower	low	ADJ
ejpam-6565	23	16	(	(	PUNCT
ejpam-6565	23	17	τ1	τ1	NOUN
ejpam-6565	23	18	,	,	PUNCT
ejpam-6565	23	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	23	20	multifunctions	multifunction	NOUN
ejpam-6565	23	21	by	by	ADP
ejpam-6565	23	22	utilizing	utilize	VERB
ejpam-6565	23	23	the	the	DET
ejpam-6565	23	24	notions	notion	NOUN
ejpam-6565	23	25	of	of	ADP
ejpam-6565	23	26	(	(	PUNCT
ejpam-6565	23	27	τ1	τ1	NOUN
ejpam-6565	23	28	,	,	PUNCT
ejpam-6565	23	29	τ2)θ	τ2)θ	ADJ
ejpam-6565	23	30	-	-	PUNCT
ejpam-6565	23	31	closed	close	VERB
ejpam-6565	23	32	sets	set	NOUN
ejpam-6565	23	33	and	and	CCONJ
ejpam-6565	23	34	(	(	PUNCT
ejpam-6565	23	35	τ1	τ1	NOUN
ejpam-6565	23	36	,	,	PUNCT
ejpam-6565	23	37	τ2)θ	τ2)θ	ADJ
ejpam-6565	23	38	-	-	PUNCT
ejpam-6565	23	39	open	open	ADJ
ejpam-6565	23	40	sets	set	NOUN
ejpam-6565	23	41	.	.	PUNCT
ejpam-6565	24	1	thongmoon	thongmoon	NOUN
ejpam-6565	24	2	et	et	PROPN
ejpam-6565	24	3	al	al	PROPN
ejpam-6565	24	4	.	.	PUNCT
ejpam-6565	25	1	[	[	X
ejpam-6565	25	2	9	9	NUM
ejpam-6565	25	3	]	]	PUNCT
ejpam-6565	25	4	studied	study	VERB
ejpam-6565	25	5	some	some	DET
ejpam-6565	25	6	characterizations	characterization	NOUN
ejpam-6565	25	7	of	of	ADP
ejpam-6565	25	8	upper	upper	ADJ
ejpam-6565	25	9	(	(	PUNCT
ejpam-6565	25	10	τ1	τ1	NOUN
ejpam-6565	25	11	,	,	PUNCT
ejpam-6565	25	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	25	13	multifunctions	multifunction	NOUN
ejpam-6565	25	14	and	and	CCONJ
ejpam-6565	25	15	lower	low	ADJ
ejpam-6565	25	16	(	(	PUNCT
ejpam-6565	25	17	τ1	τ1	NOUN
ejpam-6565	25	18	,	,	PUNCT
ejpam-6565	25	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	25	20	multifunctions	multifunction	NOUN
ejpam-6565	25	21	by	by	ADP
ejpam-6565	25	22	using	use	VERB
ejpam-6565	25	23	τ1τ2	τ1τ2	ADJ
ejpam-6565	25	24	-	-	ADJ
ejpam-6565	25	25	δ	δ	ADJ
ejpam-6565	25	26	-	-	ADJ
ejpam-6565	25	27	open	open	ADJ
ejpam-6565	25	28	sets	set	NOUN
ejpam-6565	25	29	and	and	CCONJ
ejpam-6565	25	30	τ1τ2	τ1τ2	NOUN
ejpam-6565	25	31	-	-	ADJ
ejpam-6565	25	32	δ	δ	NOUN
ejpam-6565	25	33	-	-	PUNCT
ejpam-6565	25	34	closed	close	VERB
ejpam-6565	25	35	sets	set	NOUN
ejpam-6565	25	36	.	.	PUNCT
ejpam-6565	26	1	in	in	ADP
ejpam-6565	26	2	this	this	DET
ejpam-6565	26	3	paper	paper	NOUN
ejpam-6565	26	4	,	,	PUNCT
ejpam-6565	26	5	we	we	PRON
ejpam-6565	26	6	introduce	introduce	VERB
ejpam-6565	26	7	the	the	DET
ejpam-6565	26	8	concepts	concept	NOUN
ejpam-6565	26	9	of	of	ADP
ejpam-6565	26	10	upper	upper	ADJ
ejpam-6565	26	11	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	26	12	,	,	PUNCT
ejpam-6565	26	13	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	26	14	multifunctions	multifunction	NOUN
ejpam-6565	26	15	and	and	CCONJ
ejpam-6565	26	16	lower	low	ADJ
ejpam-6565	26	17	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	26	18	,	,	PUNCT
ejpam-6565	26	19	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	26	20	multifunctions	multifunction	NOUN
ejpam-6565	26	21	.	.	PUNCT
ejpam-6565	27	1	we	we	PRON
ejpam-6565	27	2	also	also	ADV
ejpam-6565	27	3	investigate	investigate	VERB
ejpam-6565	27	4	several	several	ADJ
ejpam-6565	27	5	characterizations	characterization	NOUN
ejpam-6565	27	6	of	of	ADP
ejpam-6565	27	7	upper	upper	ADJ
ejpam-6565	27	8	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	27	9	,	,	PUNCT
ejpam-6565	27	10	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	27	11	multifunctions	multifunction	NOUN
ejpam-6565	27	12	and	and	CCONJ
ejpam-6565	27	13	lower	low	ADJ
ejpam-6565	27	14	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	27	15	,	,	PUNCT
ejpam-6565	27	16	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	27	17	multifunctions	multifunction	NOUN
ejpam-6565	27	18	.	.	PUNCT
ejpam-6565	28	1	2	2	X
ejpam-6565	28	2	.	.	X
ejpam-6565	28	3	preliminaries	preliminary	NOUN
ejpam-6565	28	4	throughout	throughout	ADP
ejpam-6565	28	5	the	the	DET
ejpam-6565	28	6	present	present	ADJ
ejpam-6565	28	7	paper	paper	NOUN
ejpam-6565	28	8	,	,	PUNCT
ejpam-6565	28	9	spaces	space	NOUN
ejpam-6565	28	10	(	(	PUNCT
ejpam-6565	28	11	x	x	NOUN
ejpam-6565	28	12	,	,	PUNCT
ejpam-6565	28	13	τ1	τ1	NOUN
ejpam-6565	28	14	,	,	PUNCT
ejpam-6565	28	15	τ2	τ2	NOUN
ejpam-6565	28	16	)	)	PUNCT
ejpam-6565	28	17	and	and	CCONJ
ejpam-6565	28	18	(	(	PUNCT
ejpam-6565	28	19	y	y	PROPN
ejpam-6565	28	20	,	,	PUNCT
ejpam-6565	28	21	σ1	σ1	PROPN
ejpam-6565	28	22	,	,	PUNCT
ejpam-6565	28	23	σ2	σ2	NOUN
ejpam-6565	28	24	)	)	PUNCT
ejpam-6565	28	25	(	(	PUNCT
ejpam-6565	28	26	or	or	CCONJ
ejpam-6565	28	27	simply	simply	ADV
ejpam-6565	28	28	x	x	X
ejpam-6565	28	29	and	and	CCONJ
ejpam-6565	28	30	y	y	PROPN
ejpam-6565	28	31	)	)	PUNCT
ejpam-6565	28	32	always	always	ADV
ejpam-6565	28	33	mean	mean	VERB
ejpam-6565	28	34	bitopological	bitopological	ADJ
ejpam-6565	28	35	spaces	space	NOUN
ejpam-6565	28	36	on	on	ADP
ejpam-6565	28	37	which	which	PRON
ejpam-6565	28	38	no	no	DET
ejpam-6565	28	39	separation	separation	NOUN
ejpam-6565	28	40	axioms	axiom	NOUN
ejpam-6565	28	41	are	be	AUX
ejpam-6565	28	42	assumed	assume	VERB
ejpam-6565	28	43	unless	unless	SCONJ
ejpam-6565	28	44	explicitly	explicitly	ADV
ejpam-6565	28	45	stated	state	VERB
ejpam-6565	28	46	.	.	PUNCT
ejpam-6565	29	1	let	let	VERB
ejpam-6565	29	2	a	a	DET
ejpam-6565	29	3	be	be	AUX
ejpam-6565	29	4	a	a	DET
ejpam-6565	29	5	subset	subset	NOUN
ejpam-6565	29	6	of	of	ADP
ejpam-6565	29	7	a	a	DET
ejpam-6565	29	8	bitopological	bitopological	ADJ
ejpam-6565	29	9	space	space	NOUN
ejpam-6565	29	10	(	(	PUNCT
ejpam-6565	29	11	x	x	NOUN
ejpam-6565	29	12	,	,	PUNCT
ejpam-6565	29	13	τ1	τ1	NOUN
ejpam-6565	29	14	,	,	PUNCT
ejpam-6565	29	15	τ2	τ2	NOUN
ejpam-6565	29	16	)	)	PUNCT
ejpam-6565	29	17	.	.	PUNCT
ejpam-6565	30	1	the	the	DET
ejpam-6565	30	2	closure	closure	NOUN
ejpam-6565	30	3	of	of	ADP
ejpam-6565	30	4	a	a	PRON
ejpam-6565	30	5	and	and	CCONJ
ejpam-6565	30	6	the	the	DET
ejpam-6565	30	7	interior	interior	NOUN
ejpam-6565	30	8	of	of	ADP
ejpam-6565	30	9	a	a	PRON
ejpam-6565	30	10	with	with	ADP
ejpam-6565	30	11	respect	respect	NOUN
ejpam-6565	30	12	to	to	ADP
ejpam-6565	30	13	τi	τi	PROPN
ejpam-6565	30	14	are	be	AUX
ejpam-6565	30	15	denoted	denote	VERB
ejpam-6565	30	16	by	by	ADP
ejpam-6565	30	17	τi	τi	NOUN
ejpam-6565	30	18	-	-	PUNCT
ejpam-6565	30	19	cl(a	cl(a	NUM
ejpam-6565	30	20	)	)	PUNCT
ejpam-6565	30	21	and	and	CCONJ
ejpam-6565	30	22	τi	τi	NOUN
ejpam-6565	30	23	-	-	PUNCT
ejpam-6565	30	24	int(a	int(a	NOUN
ejpam-6565	30	25	)	)	PUNCT
ejpam-6565	30	26	,	,	PUNCT
ejpam-6565	30	27	respectively	respectively	ADV
ejpam-6565	30	28	,	,	PUNCT
ejpam-6565	30	29	for	for	ADP
ejpam-6565	30	30	i	i	PROPN
ejpam-6565	30	31	=	=	SYM
ejpam-6565	30	32	1	1	NUM
ejpam-6565	30	33	,	,	PUNCT
ejpam-6565	30	34	2	2	NUM
ejpam-6565	30	35	.	.	X
ejpam-6565	30	36	a	a	DET
ejpam-6565	30	37	subset	subset	NOUN
ejpam-6565	30	38	a	a	PRON
ejpam-6565	30	39	of	of	ADP
ejpam-6565	30	40	a	a	DET
ejpam-6565	30	41	bitopological	bitopological	ADJ
ejpam-6565	30	42	space	space	NOUN
ejpam-6565	30	43	(	(	PUNCT
ejpam-6565	30	44	x	x	NOUN
ejpam-6565	30	45	,	,	PUNCT
ejpam-6565	30	46	τ1	τ1	NOUN
ejpam-6565	30	47	,	,	PUNCT
ejpam-6565	30	48	τ2	τ2	NOUN
ejpam-6565	30	49	)	)	PUNCT
ejpam-6565	30	50	is	be	AUX
ejpam-6565	30	51	called	call	VERB
ejpam-6565	30	52	τ1τ2	τ1τ2	VERB
ejpam-6565	30	53	-	-	ADJ
ejpam-6565	30	54	closed	closed	ADJ
ejpam-6565	30	55	[	[	X
ejpam-6565	30	56	10	10	NUM
ejpam-6565	30	57	]	]	X
ejpam-6565	30	58	if	if	SCONJ
ejpam-6565	30	59	a	a	DET
ejpam-6565	30	60	=	=	NOUN
ejpam-6565	30	61	τ1	τ1	NOUN
ejpam-6565	30	62	-	-	PUNCT
ejpam-6565	30	63	cl(τ2	cl(τ2	NOUN
ejpam-6565	30	64	-	-	PUNCT
ejpam-6565	30	65	cl(a	cl(a	NUM
ejpam-6565	30	66	)	)	PUNCT
ejpam-6565	30	67	)	)	PUNCT
ejpam-6565	30	68	.	.	PUNCT
ejpam-6565	31	1	the	the	DET
ejpam-6565	31	2	complement	complement	NOUN
ejpam-6565	31	3	of	of	ADP
ejpam-6565	31	4	a	a	DET
ejpam-6565	31	5	τ1τ2	τ1τ2	ADJ
ejpam-6565	31	6	-	-	ADJ
ejpam-6565	31	7	closed	closed	ADJ
ejpam-6565	31	8	set	set	NOUN
ejpam-6565	31	9	is	be	AUX
ejpam-6565	31	10	called	call	VERB
ejpam-6565	31	11	τ1τ2	τ1τ2	NOUN
ejpam-6565	31	12	-	-	ADJ
ejpam-6565	31	13	open	open	ADJ
ejpam-6565	31	14	.	.	PUNCT
ejpam-6565	32	1	the	the	DET
ejpam-6565	32	2	intersection	intersection	NOUN
ejpam-6565	32	3	of	of	ADP
ejpam-6565	32	4	all	all	DET
ejpam-6565	32	5	τ1τ2	τ1τ2	ADJ
ejpam-6565	32	6	-	-	ADJ
ejpam-6565	32	7	closed	closed	ADJ
ejpam-6565	32	8	sets	set	NOUN
ejpam-6565	32	9	of	of	ADP
ejpam-6565	32	10	x	x	PUNCT
ejpam-6565	32	11	containing	contain	VERB
ejpam-6565	32	12	a	a	PRON
ejpam-6565	32	13	is	be	AUX
ejpam-6565	32	14	called	call	VERB
ejpam-6565	32	15	the	the	DET
ejpam-6565	32	16	τ1τ2	τ1τ2	NOUN
ejpam-6565	32	17	-	-	NOUN
ejpam-6565	32	18	closure	closure	NOUN
ejpam-6565	32	19	[	[	X
ejpam-6565	32	20	10	10	NUM
ejpam-6565	32	21	]	]	PUNCT
ejpam-6565	32	22	of	of	ADP
ejpam-6565	32	23	a	a	PRON
ejpam-6565	32	24	and	and	CCONJ
ejpam-6565	32	25	is	be	AUX
ejpam-6565	32	26	denoted	denote	VERB
ejpam-6565	32	27	by	by	ADP
ejpam-6565	32	28	τ1τ2	τ1τ2	NOUN
ejpam-6565	32	29	-	-	NUM
ejpam-6565	32	30	cl(a	cl(a	NUM
ejpam-6565	32	31	)	)	PUNCT
ejpam-6565	32	32	.	.	PUNCT
ejpam-6565	33	1	the	the	DET
ejpam-6565	33	2	union	union	NOUN
ejpam-6565	33	3	of	of	ADP
ejpam-6565	33	4	all	all	DET
ejpam-6565	33	5	τ1τ2	τ1τ2	ADJ
ejpam-6565	33	6	-	-	ADJ
ejpam-6565	33	7	open	open	ADJ
ejpam-6565	33	8	sets	set	NOUN
ejpam-6565	33	9	of	of	ADP
ejpam-6565	33	10	x	x	PUNCT
ejpam-6565	33	11	contained	contain	VERB
ejpam-6565	33	12	in	in	ADP
ejpam-6565	33	13	a	a	PRON
ejpam-6565	33	14	is	be	AUX
ejpam-6565	33	15	called	call	VERB
ejpam-6565	33	16	the	the	DET
ejpam-6565	33	17	τ1τ2	τ1τ2	NOUN
ejpam-6565	33	18	-	-	ADJ
ejpam-6565	33	19	interior	interior	ADJ
ejpam-6565	33	20	[	[	X
ejpam-6565	33	21	10	10	NUM
ejpam-6565	33	22	]	]	PUNCT
ejpam-6565	33	23	of	of	ADP
ejpam-6565	33	24	a	a	PRON
ejpam-6565	33	25	and	and	CCONJ
ejpam-6565	33	26	is	be	AUX
ejpam-6565	33	27	denoted	denote	VERB
ejpam-6565	33	28	by	by	ADP
ejpam-6565	33	29	τ1τ2	τ1τ2	NOUN
ejpam-6565	33	30	-	-	ADJ
ejpam-6565	33	31	int(a	int(a	NOUN
ejpam-6565	33	32	)	)	PUNCT
ejpam-6565	33	33	.	.	PUNCT
ejpam-6565	34	1	lemma	lemma	PROPN
ejpam-6565	34	2	1	1	NUM
ejpam-6565	34	3	.	.	PUNCT
ejpam-6565	35	1	[	[	X
ejpam-6565	35	2	10	10	NUM
ejpam-6565	35	3	]	]	PUNCT
ejpam-6565	35	4	let	let	VERB
ejpam-6565	35	5	a	a	PRON
ejpam-6565	35	6	and	and	CCONJ
ejpam-6565	35	7	b	b	NOUN
ejpam-6565	35	8	be	be	AUX
ejpam-6565	35	9	subsets	subset	NOUN
ejpam-6565	35	10	of	of	ADP
ejpam-6565	35	11	a	a	DET
ejpam-6565	35	12	bitopological	bitopological	ADJ
ejpam-6565	35	13	space	space	NOUN
ejpam-6565	35	14	(	(	PUNCT
ejpam-6565	35	15	x	x	NOUN
ejpam-6565	35	16	,	,	PUNCT
ejpam-6565	35	17	τ1	τ1	NOUN
ejpam-6565	35	18	,	,	PUNCT
ejpam-6565	35	19	τ2	τ2	NOUN
ejpam-6565	35	20	)	)	PUNCT
ejpam-6565	35	21	.	.	PUNCT
ejpam-6565	36	1	for	for	ADP
ejpam-6565	36	2	the	the	DET
ejpam-6565	36	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6565	36	4	,	,	PUNCT
ejpam-6565	36	5	the	the	DET
ejpam-6565	36	6	following	follow	VERB
ejpam-6565	36	7	properties	property	NOUN
ejpam-6565	36	8	hold	hold	VERB
ejpam-6565	36	9	:	:	PUNCT
ejpam-6565	36	10	(	(	PUNCT
ejpam-6565	36	11	1	1	X
ejpam-6565	36	12	)	)	PUNCT
ejpam-6565	36	13	a	a	DET
ejpam-6565	36	14	⊆	⊆	NUM
ejpam-6565	36	15	τ1τ2	τ1τ2	NOUN
ejpam-6565	36	16	-	-	NUM
ejpam-6565	36	17	cl(a	cl(a	NUM
ejpam-6565	36	18	)	)	PUNCT
ejpam-6565	36	19	and	and	CCONJ
ejpam-6565	36	20	τ1τ2	τ1τ2	NOUN
ejpam-6565	36	21	-	-	ADJ
ejpam-6565	36	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6565	36	23	-	-	PUNCT
ejpam-6565	36	24	cl(a	cl(a	NUM
ejpam-6565	36	25	)	)	PUNCT
ejpam-6565	36	26	)	)	PUNCT
ejpam-6565	37	1	=	=	PUNCT
ejpam-6565	37	2	τ1τ2	τ1τ2	NOUN
ejpam-6565	37	3	-	-	NUM
ejpam-6565	37	4	cl(a	cl(a	NUM
ejpam-6565	37	5	)	)	PUNCT
ejpam-6565	37	6	.	.	PUNCT
ejpam-6565	38	1	(	(	PUNCT
ejpam-6565	38	2	2	2	X
ejpam-6565	38	3	)	)	PUNCT
ejpam-6565	38	4	if	if	SCONJ
ejpam-6565	38	5	a	a	DET
ejpam-6565	38	6	⊆	⊆	NUM
ejpam-6565	38	7	b	b	NOUN
ejpam-6565	38	8	,	,	PUNCT
ejpam-6565	38	9	then	then	ADV
ejpam-6565	38	10	τ1τ2	τ1τ2	NOUN
ejpam-6565	38	11	-	-	NUM
ejpam-6565	38	12	cl(a	cl(a	NUM
ejpam-6565	38	13	)	)	PUNCT
ejpam-6565	38	14	⊆	⊆	NUM
ejpam-6565	38	15	τ1τ2	τ1τ2	NOUN
ejpam-6565	38	16	-	-	NOUN
ejpam-6565	38	17	cl(b	cl(b	NOUN
ejpam-6565	38	18	)	)	PUNCT
ejpam-6565	38	19	.	.	PUNCT
ejpam-6565	39	1	(	(	PUNCT
ejpam-6565	39	2	3	3	X
ejpam-6565	39	3	)	)	PUNCT
ejpam-6565	39	4	τ1τ2	τ1τ2	NOUN
ejpam-6565	39	5	-	-	NUM
ejpam-6565	39	6	cl(a	cl(a	NUM
ejpam-6565	39	7	)	)	PUNCT
ejpam-6565	39	8	is	be	AUX
ejpam-6565	39	9	τ1τ2	τ1τ2	NOUN
ejpam-6565	39	10	-	-	ADJ
ejpam-6565	39	11	closed	closed	ADJ
ejpam-6565	39	12	.	.	PUNCT
ejpam-6565	40	1	(	(	PUNCT
ejpam-6565	40	2	4	4	X
ejpam-6565	40	3	)	)	PUNCT
ejpam-6565	40	4	a	a	PRON
ejpam-6565	40	5	is	be	AUX
ejpam-6565	40	6	τ1τ2	τ1τ2	NOUN
ejpam-6565	40	7	-	-	ADJ
ejpam-6565	40	8	closed	closed	ADJ
ejpam-6565	40	9	if	if	SCONJ
ejpam-6565	40	10	and	and	CCONJ
ejpam-6565	40	11	only	only	ADV
ejpam-6565	40	12	if	if	SCONJ
ejpam-6565	40	13	a	a	DET
ejpam-6565	40	14	=	=	PUNCT
ejpam-6565	40	15	τ1τ2	τ1τ2	NOUN
ejpam-6565	40	16	-	-	NUM
ejpam-6565	40	17	cl(a	cl(a	NUM
ejpam-6565	40	18	)	)	PUNCT
ejpam-6565	40	19	.	.	PUNCT
ejpam-6565	41	1	(	(	PUNCT
ejpam-6565	41	2	5	5	X
ejpam-6565	41	3	)	)	PUNCT
ejpam-6565	41	4	τ1τ2	τ1τ2	NOUN
ejpam-6565	41	5	-	-	NOUN
ejpam-6565	41	6	cl(x	cl(x	X
ejpam-6565	41	7	−a	−a	NOUN
ejpam-6565	41	8	)	)	PUNCT
ejpam-6565	42	1	=	=	PUNCT
ejpam-6565	42	2	x	x	X
ejpam-6565	43	1	−	−	ADP
ejpam-6565	43	2	τ1τ2	τ1τ2	NOUN
ejpam-6565	43	3	-	-	PUNCT
ejpam-6565	43	4	int(a	int(a	NOUN
ejpam-6565	43	5	)	)	PUNCT
ejpam-6565	43	6	.	.	PUNCT
ejpam-6565	44	1	a	a	DET
ejpam-6565	44	2	subset	subset	NOUN
ejpam-6565	44	3	a	a	PRON
ejpam-6565	44	4	of	of	ADP
ejpam-6565	44	5	a	a	DET
ejpam-6565	44	6	bitopological	bitopological	ADJ
ejpam-6565	44	7	space	space	NOUN
ejpam-6565	44	8	(	(	PUNCT
ejpam-6565	44	9	x	x	NOUN
ejpam-6565	44	10	,	,	PUNCT
ejpam-6565	44	11	τ1	τ1	NOUN
ejpam-6565	44	12	,	,	PUNCT
ejpam-6565	44	13	τ2	τ2	NOUN
ejpam-6565	44	14	)	)	PUNCT
ejpam-6565	44	15	is	be	AUX
ejpam-6565	44	16	said	say	VERB
ejpam-6565	44	17	to	to	PART
ejpam-6565	44	18	be	be	AUX
ejpam-6565	44	19	(	(	PUNCT
ejpam-6565	44	20	τ1	τ1	NOUN
ejpam-6565	44	21	,	,	PUNCT
ejpam-6565	44	22	τ2)r	τ2)r	NOUN
ejpam-6565	44	23	-	-	PUNCT
ejpam-6565	44	24	open	open	NOUN
ejpam-6565	44	25	[	[	X
ejpam-6565	44	26	11	11	NUM
ejpam-6565	44	27	]	]	PUNCT
ejpam-6565	44	28	(	(	PUNCT
ejpam-6565	44	29	resp	resp	NOUN
ejpam-6565	44	30	.	.	PUNCT
ejpam-6565	45	1	(	(	PUNCT
ejpam-6565	45	2	τ1	τ1	NOUN
ejpam-6565	45	3	,	,	PUNCT
ejpam-6565	45	4	τ2)s	τ2)s	NOUN
ejpam-6565	45	5	-	-	PUNCT
ejpam-6565	45	6	open	open	ADJ
ejpam-6565	45	7	[	[	X
ejpam-6565	45	8	12	12	NUM
ejpam-6565	45	9	]	]	PUNCT
ejpam-6565	45	10	,	,	PUNCT
ejpam-6565	45	11	(	(	PUNCT
ejpam-6565	45	12	τ1	τ1	NOUN
ejpam-6565	45	13	,	,	PUNCT
ejpam-6565	45	14	τ2)p	τ2)p	NOUN
ejpam-6565	45	15	-	-	ADJ
ejpam-6565	45	16	open	open	ADJ
ejpam-6565	45	17	[	[	X
ejpam-6565	45	18	12	12	NUM
ejpam-6565	45	19	]	]	PUNCT
ejpam-6565	45	20	,	,	PUNCT
ejpam-6565	45	21	(	(	PUNCT
ejpam-6565	45	22	τ1	τ1	NOUN
ejpam-6565	45	23	,	,	PUNCT
ejpam-6565	45	24	τ2)β	τ2)β	ADJ
ejpam-6565	45	25	-	-	PUNCT
ejpam-6565	45	26	open	open	NOUN
ejpam-6565	46	1	[	[	X
ejpam-6565	46	2	12	12	NUM
ejpam-6565	46	3	]	]	PUNCT
ejpam-6565	46	4	)	)	PUNCT
ejpam-6565	46	5	if	if	SCONJ
ejpam-6565	46	6	a	a	DET
ejpam-6565	46	7	=	=	PUNCT
ejpam-6565	46	8	τ1τ2	τ1τ2	NOUN
ejpam-6565	46	9	-	-	NOUN
ejpam-6565	46	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6565	46	11	-	-	PUNCT
ejpam-6565	46	12	cl(a	cl(a	NUM
ejpam-6565	46	13	)	)	PUNCT
ejpam-6565	46	14	)	)	PUNCT
ejpam-6565	46	15	(	(	PUNCT
ejpam-6565	46	16	resp	resp	NOUN
ejpam-6565	46	17	.	.	PUNCT
ejpam-6565	47	1	a	a	DET
ejpam-6565	47	2	⊆	⊆	NUM
ejpam-6565	47	3	τ1τ2	τ1τ2	NOUN
ejpam-6565	47	4	-	-	ADJ
ejpam-6565	47	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6565	47	6	-	-	PUNCT
ejpam-6565	47	7	int(a	int(a	NOUN
ejpam-6565	47	8	)	)	PUNCT
ejpam-6565	47	9	)	)	PUNCT
ejpam-6565	47	10	,	,	PUNCT
ejpam-6565	47	11	a	a	DET
ejpam-6565	47	12	⊆	⊆	NUM
ejpam-6565	47	13	τ1τ2	τ1τ2	NOUN
ejpam-6565	47	14	-	-	NOUN
ejpam-6565	47	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6565	47	16	-	-	PUNCT
ejpam-6565	47	17	cl(a	cl(a	NUM
ejpam-6565	47	18	)	)	PUNCT
ejpam-6565	47	19	)	)	PUNCT
ejpam-6565	47	20	,	,	PUNCT
ejpam-6565	47	21	a	a	DET
ejpam-6565	47	22	⊆	⊆	NUM
ejpam-6565	47	23	τ1τ2	τ1τ2	NOUN
ejpam-6565	47	24	-	-	PUNCT
ejpam-6565	47	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6565	47	26	-	-	PUNCT
ejpam-6565	47	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6565	47	28	-	-	PUNCT
ejpam-6565	47	29	cl(a	cl(a	NUM
ejpam-6565	47	30	)	)	PUNCT
ejpam-6565	47	31	)	)	PUNCT
ejpam-6565	47	32	)	)	PUNCT
ejpam-6565	47	33	)	)	PUNCT
ejpam-6565	47	34	.	.	PUNCT
ejpam-6565	48	1	j.	j.	PROPN
ejpam-6565	48	2	khampakdee	khampakdee	PROPN
ejpam-6565	48	3	,	,	PUNCT
ejpam-6565	48	4	a.	a.	PROPN
ejpam-6565	48	5	sama	sama	PROPN
ejpam-6565	48	6	-	-	PUNCT
ejpam-6565	48	7	ae	ae	PROPN
ejpam-6565	48	8	,	,	PUNCT
ejpam-6565	48	9	c.	c.	PROPN
ejpam-6565	48	10	boonpok	boonpok	PROPN
ejpam-6565	48	11	/	/	SYM
ejpam-6565	48	12	eur	eur	PROPN
ejpam-6565	48	13	.	.	PUNCT
ejpam-6565	49	1	j.	j.	PROPN
ejpam-6565	49	2	pure	pure	PROPN
ejpam-6565	49	3	appl	appl	PROPN
ejpam-6565	49	4	.	.	PROPN
ejpam-6565	49	5	math	math	PROPN
ejpam-6565	49	6	,	,	PUNCT
ejpam-6565	49	7	18	18	NUM
ejpam-6565	49	8	(	(	PUNCT
ejpam-6565	49	9	3	3	NUM
ejpam-6565	49	10	)	)	PUNCT
ejpam-6565	49	11	(	(	PUNCT
ejpam-6565	49	12	2025	2025	NUM
ejpam-6565	49	13	)	)	PUNCT
ejpam-6565	49	14	,	,	PUNCT
ejpam-6565	49	15	6565	6565	NUM
ejpam-6565	49	16	3	3	NUM
ejpam-6565	49	17	of	of	ADP
ejpam-6565	49	18	9	9	NUM
ejpam-6565	49	19	the	the	DET
ejpam-6565	49	20	complement	complement	NOUN
ejpam-6565	49	21	of	of	ADP
ejpam-6565	49	22	a	a	DET
ejpam-6565	49	23	(	(	PUNCT
ejpam-6565	49	24	τ1	τ1	NOUN
ejpam-6565	49	25	,	,	PUNCT
ejpam-6565	49	26	τ2)r	τ2)r	NOUN
ejpam-6565	49	27	-	-	PUNCT
ejpam-6565	49	28	open	open	ADJ
ejpam-6565	49	29	(	(	PUNCT
ejpam-6565	49	30	resp	resp	NOUN
ejpam-6565	49	31	.	.	PUNCT
ejpam-6565	50	1	(	(	PUNCT
ejpam-6565	50	2	τ1	τ1	NOUN
ejpam-6565	50	3	,	,	PUNCT
ejpam-6565	50	4	τ2)s	τ2)s	NOUN
ejpam-6565	50	5	-	-	PUNCT
ejpam-6565	50	6	open	open	ADJ
ejpam-6565	50	7	,	,	PUNCT
ejpam-6565	50	8	(	(	PUNCT
ejpam-6565	50	9	τ1	τ1	NOUN
ejpam-6565	50	10	,	,	PUNCT
ejpam-6565	50	11	τ2)p	τ2)p	NOUN
ejpam-6565	50	12	-	-	ADJ
ejpam-6565	50	13	open	open	ADJ
ejpam-6565	50	14	,	,	PUNCT
ejpam-6565	50	15	(	(	PUNCT
ejpam-6565	50	16	τ1	τ1	NOUN
ejpam-6565	50	17	,	,	PUNCT
ejpam-6565	50	18	τ2)β	τ2)β	ADJ
ejpam-6565	50	19	-	-	PUNCT
ejpam-6565	50	20	open	open	ADJ
ejpam-6565	50	21	)	)	PUNCT
ejpam-6565	50	22	set	set	NOUN
ejpam-6565	50	23	is	be	AUX
ejpam-6565	50	24	said	say	VERB
ejpam-6565	50	25	to	to	PART
ejpam-6565	50	26	be	be	AUX
ejpam-6565	50	27	(	(	PUNCT
ejpam-6565	50	28	τ1	τ1	NOUN
ejpam-6565	50	29	,	,	PUNCT
ejpam-6565	50	30	τ2)r	τ2)r	NOUN
ejpam-6565	50	31	-	-	PUNCT
ejpam-6565	50	32	closed	closed	ADJ
ejpam-6565	50	33	(	(	PUNCT
ejpam-6565	50	34	resp	resp	NOUN
ejpam-6565	50	35	.	.	PUNCT
ejpam-6565	51	1	(	(	PUNCT
ejpam-6565	51	2	τ1	τ1	NOUN
ejpam-6565	51	3	,	,	PUNCT
ejpam-6565	51	4	τ2)s	τ2)s	NOUN
ejpam-6565	51	5	-	-	PUNCT
ejpam-6565	51	6	closed	closed	ADJ
ejpam-6565	51	7	,	,	PUNCT
ejpam-6565	51	8	(	(	PUNCT
ejpam-6565	51	9	τ1	τ1	NOUN
ejpam-6565	51	10	,	,	PUNCT
ejpam-6565	51	11	τ2)p	τ2)p	NOUN
ejpam-6565	51	12	-	-	PUNCT
ejpam-6565	51	13	closed	closed	ADJ
ejpam-6565	51	14	,	,	PUNCT
ejpam-6565	51	15	(	(	PUNCT
ejpam-6565	51	16	τ1	τ1	NOUN
ejpam-6565	51	17	,	,	PUNCT
ejpam-6565	51	18	τ2)β	τ2)β	ADJ
ejpam-6565	51	19	-	-	PUNCT
ejpam-6565	51	20	closed	closed	ADJ
ejpam-6565	51	21	)	)	PUNCT
ejpam-6565	51	22	.	.	PUNCT
ejpam-6565	52	1	a	a	DET
ejpam-6565	52	2	subset	subset	NOUN
ejpam-6565	52	3	a	a	PRON
ejpam-6565	52	4	of	of	ADP
ejpam-6565	52	5	a	a	DET
ejpam-6565	52	6	bitopological	bitopological	ADJ
ejpam-6565	52	7	space	space	NOUN
ejpam-6565	52	8	(	(	PUNCT
ejpam-6565	52	9	x	x	NOUN
ejpam-6565	52	10	,	,	PUNCT
ejpam-6565	52	11	τ1	τ1	NOUN
ejpam-6565	52	12	,	,	PUNCT
ejpam-6565	52	13	τ2	τ2	NOUN
ejpam-6565	52	14	)	)	PUNCT
ejpam-6565	52	15	is	be	AUX
ejpam-6565	52	16	said	say	VERB
ejpam-6565	52	17	to	to	PART
ejpam-6565	52	18	be	be	AUX
ejpam-6565	52	19	α(τ1	α(τ1	NOUN
ejpam-6565	52	20	,	,	PUNCT
ejpam-6565	52	21	τ2)-open	τ2)-open	ADJ
ejpam-6565	52	22	[	[	X
ejpam-6565	52	23	13	13	NUM
ejpam-6565	52	24	]	]	PUNCT
ejpam-6565	52	25	if	if	SCONJ
ejpam-6565	52	26	a	a	DET
ejpam-6565	52	27	⊆	⊆	NUM
ejpam-6565	52	28	τ1τ2	τ1τ2	NOUN
ejpam-6565	52	29	-	-	PUNCT
ejpam-6565	52	30	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6565	52	31	-	-	PUNCT
ejpam-6565	52	32	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6565	52	33	-	-	PUNCT
ejpam-6565	52	34	int(a	int(a	NOUN
ejpam-6565	52	35	)	)	PUNCT
ejpam-6565	52	36	)	)	PUNCT
ejpam-6565	52	37	)	)	PUNCT
ejpam-6565	52	38	.	.	PUNCT
ejpam-6565	53	1	the	the	DET
ejpam-6565	53	2	complement	complement	NOUN
ejpam-6565	53	3	of	of	ADP
ejpam-6565	53	4	an	an	DET
ejpam-6565	53	5	α(τ1	α(τ1	NOUN
ejpam-6565	53	6	,	,	PUNCT
ejpam-6565	53	7	τ2)-open	τ2)-open	ADJ
ejpam-6565	53	8	set	set	NOUN
ejpam-6565	53	9	is	be	AUX
ejpam-6565	53	10	said	say	VERB
ejpam-6565	53	11	to	to	PART
ejpam-6565	53	12	be	be	AUX
ejpam-6565	53	13	α(τ1	α(τ1	NOUN
ejpam-6565	53	14	,	,	PUNCT
ejpam-6565	53	15	τ2)-closed	τ2)-closed	ADJ
ejpam-6565	53	16	.	.	PUNCT
ejpam-6565	54	1	a	a	DET
ejpam-6565	54	2	subset	subset	NOUN
ejpam-6565	54	3	a	a	PRON
ejpam-6565	54	4	of	of	ADP
ejpam-6565	54	5	a	a	DET
ejpam-6565	54	6	bitopological	bitopological	ADJ
ejpam-6565	54	7	space	space	NOUN
ejpam-6565	54	8	(	(	PUNCT
ejpam-6565	54	9	x	x	NOUN
ejpam-6565	54	10	,	,	PUNCT
ejpam-6565	54	11	τ1	τ1	NOUN
ejpam-6565	54	12	,	,	PUNCT
ejpam-6565	54	13	τ2	τ2	NOUN
ejpam-6565	54	14	)	)	PUNCT
ejpam-6565	54	15	is	be	AUX
ejpam-6565	54	16	said	say	VERB
ejpam-6565	54	17	to	to	PART
ejpam-6565	54	18	be	be	AUX
ejpam-6565	54	19	τ1τ2	τ1τ2	NOUN
ejpam-6565	54	20	-	-	ADJ
ejpam-6565	54	21	δ	δ	NOUN
ejpam-6565	54	22	-	-	NOUN
ejpam-6565	54	23	open	open	ADJ
ejpam-6565	54	24	if	if	SCONJ
ejpam-6565	54	25	a	a	PRON
ejpam-6565	54	26	is	be	AUX
ejpam-6565	54	27	the	the	DET
ejpam-6565	54	28	union	union	NOUN
ejpam-6565	54	29	of	of	ADP
ejpam-6565	54	30	(	(	PUNCT
ejpam-6565	54	31	τ1	τ1	NOUN
ejpam-6565	54	32	,	,	PUNCT
ejpam-6565	54	33	τ2)r	τ2)r	ADJ
ejpam-6565	54	34	-	-	PUNCT
ejpam-6565	54	35	open	open	ADJ
ejpam-6565	54	36	sets	set	NOUN
ejpam-6565	54	37	of	of	ADP
ejpam-6565	54	38	x.	x.	NOUN
ejpam-6565	54	39	the	the	DET
ejpam-6565	54	40	complement	complement	NOUN
ejpam-6565	54	41	of	of	ADP
ejpam-6565	54	42	a	a	DET
ejpam-6565	54	43	τ1τ2	τ1τ2	ADJ
ejpam-6565	54	44	-	-	ADJ
ejpam-6565	54	45	δ	δ	NOUN
ejpam-6565	54	46	-	-	ADJ
ejpam-6565	54	47	open	open	ADJ
ejpam-6565	54	48	set	set	NOUN
ejpam-6565	54	49	is	be	AUX
ejpam-6565	54	50	called	call	VERB
ejpam-6565	54	51	τ1τ2	τ1τ2	NOUN
ejpam-6565	54	52	-	-	ADJ
ejpam-6565	54	53	δ	δ	NOUN
ejpam-6565	54	54	-	-	PUNCT
ejpam-6565	54	55	closed	closed	ADJ
ejpam-6565	54	56	.	.	PUNCT
ejpam-6565	55	1	the	the	DET
ejpam-6565	55	2	union	union	NOUN
ejpam-6565	55	3	of	of	ADP
ejpam-6565	55	4	all	all	DET
ejpam-6565	55	5	τ1τ2	τ1τ2	NOUN
ejpam-6565	55	6	-	-	ADJ
ejpam-6565	55	7	δ	δ	NOUN
ejpam-6565	55	8	-	-	ADJ
ejpam-6565	55	9	open	open	ADJ
ejpam-6565	55	10	sets	set	NOUN
ejpam-6565	55	11	of	of	ADP
ejpam-6565	55	12	x	x	PUNCT
ejpam-6565	55	13	contained	contain	VERB
ejpam-6565	55	14	in	in	ADP
ejpam-6565	55	15	a	a	PRON
ejpam-6565	55	16	is	be	AUX
ejpam-6565	55	17	called	call	VERB
ejpam-6565	55	18	the	the	DET
ejpam-6565	55	19	τ1τ2	τ1τ2	ADJ
ejpam-6565	55	20	-	-	ADJ
ejpam-6565	55	21	δ	δ	NOUN
ejpam-6565	55	22	-	-	NOUN
ejpam-6565	55	23	interior	interior	NOUN
ejpam-6565	55	24	of	of	ADP
ejpam-6565	55	25	a	a	PRON
ejpam-6565	55	26	and	and	CCONJ
ejpam-6565	55	27	is	be	AUX
ejpam-6565	55	28	denoted	denote	VERB
ejpam-6565	55	29	by	by	ADP
ejpam-6565	55	30	τ1τ2	τ1τ2	ADJ
ejpam-6565	55	31	-	-	ADJ
ejpam-6565	55	32	δ	δ	NOUN
ejpam-6565	55	33	-	-	PUNCT
ejpam-6565	55	34	int(a	int(a	PROPN
ejpam-6565	55	35	)	)	PUNCT
ejpam-6565	55	36	.	.	PUNCT
ejpam-6565	56	1	the	the	DET
ejpam-6565	56	2	intersection	intersection	NOUN
ejpam-6565	56	3	of	of	ADP
ejpam-6565	56	4	all	all	DET
ejpam-6565	56	5	τ1τ2	τ1τ2	NOUN
ejpam-6565	56	6	-	-	ADJ
ejpam-6565	56	7	δ	δ	NOUN
ejpam-6565	56	8	-	-	PUNCT
ejpam-6565	56	9	closed	close	VERB
ejpam-6565	56	10	sets	set	NOUN
ejpam-6565	56	11	of	of	ADP
ejpam-6565	56	12	x	x	PUNCT
ejpam-6565	56	13	containing	contain	VERB
ejpam-6565	56	14	a	a	PRON
ejpam-6565	56	15	is	be	AUX
ejpam-6565	56	16	called	call	VERB
ejpam-6565	56	17	the	the	DET
ejpam-6565	56	18	τ1τ2	τ1τ2	ADJ
ejpam-6565	56	19	-	-	ADJ
ejpam-6565	56	20	δ	δ	NOUN
ejpam-6565	56	21	-	-	NOUN
ejpam-6565	56	22	closure	closure	NOUN
ejpam-6565	56	23	of	of	ADP
ejpam-6565	56	24	a	a	PRON
ejpam-6565	56	25	and	and	CCONJ
ejpam-6565	56	26	is	be	AUX
ejpam-6565	56	27	denoted	denote	VERB
ejpam-6565	56	28	by	by	ADP
ejpam-6565	56	29	τ1τ2	τ1τ2	ADJ
ejpam-6565	56	30	-	-	ADJ
ejpam-6565	56	31	δ	δ	NOUN
ejpam-6565	56	32	-	-	PUNCT
ejpam-6565	56	33	cl(a	cl(a	NUM
ejpam-6565	56	34	)	)	PUNCT
ejpam-6565	57	1	[	[	X
ejpam-6565	57	2	14	14	NUM
ejpam-6565	57	3	]	]	PUNCT
ejpam-6565	57	4	.	.	PUNCT
ejpam-6565	58	1	for	for	ADP
ejpam-6565	58	2	a	a	DET
ejpam-6565	58	3	subset	subset	NOUN
ejpam-6565	58	4	a	a	PRON
ejpam-6565	58	5	of	of	ADP
ejpam-6565	58	6	a	a	DET
ejpam-6565	58	7	bitopological	bitopological	ADJ
ejpam-6565	58	8	space	space	NOUN
ejpam-6565	58	9	(	(	PUNCT
ejpam-6565	58	10	x	x	NOUN
ejpam-6565	58	11	,	,	PUNCT
ejpam-6565	58	12	τ1	τ1	NOUN
ejpam-6565	58	13	,	,	PUNCT
ejpam-6565	58	14	τ2	τ2	PROPN
ejpam-6565	58	15	)	)	PUNCT
ejpam-6565	58	16	,	,	PUNCT
ejpam-6565	58	17	a	a	DET
ejpam-6565	58	18	point	point	NOUN
ejpam-6565	58	19	x	x	X
ejpam-6565	58	20	∈	∈	NOUN
ejpam-6565	58	21	x	x	PUNCT
ejpam-6565	58	22	is	be	AUX
ejpam-6565	58	23	called	call	VERB
ejpam-6565	58	24	a	a	DET
ejpam-6565	58	25	(	(	PUNCT
ejpam-6565	58	26	τ1	τ1	NOUN
ejpam-6565	58	27	,	,	PUNCT
ejpam-6565	58	28	τ2)θcluster	τ2)θcluster	NOUN
ejpam-6565	58	29	point	point	NOUN
ejpam-6565	58	30	of	of	ADP
ejpam-6565	58	31	a	a	DET
ejpam-6565	58	32	if	if	SCONJ
ejpam-6565	58	33	τ1τ2	τ1τ2	NOUN
ejpam-6565	58	34	-	-	NOUN
ejpam-6565	58	35	cl(u	cl(u	NOUN
ejpam-6565	58	36	)	)	PUNCT
ejpam-6565	58	37	∩	∩	NOUN
ejpam-6565	58	38	a	a	DET
ejpam-6565	58	39	̸=	̸=	PROPN
ejpam-6565	58	40	∅	∅	NOUN
ejpam-6565	58	41	for	for	ADP
ejpam-6565	58	42	every	every	DET
ejpam-6565	58	43	τ1τ2	τ1τ2	ADJ
ejpam-6565	58	44	-	-	ADJ
ejpam-6565	58	45	open	open	ADJ
ejpam-6565	58	46	set	set	NOUN
ejpam-6565	58	47	u	u	NOUN
ejpam-6565	58	48	containing	contain	VERB
ejpam-6565	58	49	x.	x.	NOUN
ejpam-6565	58	50	the	the	DET
ejpam-6565	58	51	set	set	NOUN
ejpam-6565	58	52	of	of	ADP
ejpam-6565	58	53	all	all	DET
ejpam-6565	58	54	(	(	PUNCT
ejpam-6565	58	55	τ1	τ1	NOUN
ejpam-6565	58	56	,	,	PUNCT
ejpam-6565	58	57	τ2)θ	τ2)θ	ADJ
ejpam-6565	58	58	-	-	PUNCT
ejpam-6565	58	59	cluster	cluster	NOUN
ejpam-6565	58	60	points	point	NOUN
ejpam-6565	58	61	of	of	ADP
ejpam-6565	58	62	a	a	PRON
ejpam-6565	58	63	is	be	AUX
ejpam-6565	58	64	called	call	VERB
ejpam-6565	58	65	the	the	DET
ejpam-6565	58	66	(	(	PUNCT
ejpam-6565	58	67	τ1	τ1	NOUN
ejpam-6565	58	68	,	,	PUNCT
ejpam-6565	58	69	τ2)θ	τ2)θ	NOUN
ejpam-6565	58	70	-	-	PUNCT
ejpam-6565	58	71	closure	closure	NOUN
ejpam-6565	58	72	of	of	ADP
ejpam-6565	58	73	a	a	PRON
ejpam-6565	58	74	and	and	CCONJ
ejpam-6565	58	75	is	be	AUX
ejpam-6565	58	76	denoted	denote	VERB
ejpam-6565	58	77	by	by	ADP
ejpam-6565	58	78	(	(	PUNCT
ejpam-6565	58	79	τ1	τ1	NOUN
ejpam-6565	58	80	,	,	PUNCT
ejpam-6565	58	81	τ2)θ	τ2)θ	NOUN
ejpam-6565	58	82	-	-	PUNCT
ejpam-6565	58	83	cl(a	cl(a	NUM
ejpam-6565	58	84	)	)	PUNCT
ejpam-6565	58	85	.	.	PUNCT
ejpam-6565	59	1	a	a	DET
ejpam-6565	59	2	subset	subset	NOUN
ejpam-6565	59	3	a	a	PRON
ejpam-6565	59	4	of	of	ADP
ejpam-6565	59	5	a	a	DET
ejpam-6565	59	6	bitopological	bitopological	ADJ
ejpam-6565	59	7	space	space	NOUN
ejpam-6565	59	8	(	(	PUNCT
ejpam-6565	59	9	x	x	NOUN
ejpam-6565	59	10	,	,	PUNCT
ejpam-6565	59	11	τ1	τ1	NOUN
ejpam-6565	59	12	,	,	PUNCT
ejpam-6565	59	13	τ2	τ2	NOUN
ejpam-6565	59	14	)	)	PUNCT
ejpam-6565	59	15	is	be	AUX
ejpam-6565	59	16	said	say	VERB
ejpam-6565	59	17	to	to	PART
ejpam-6565	59	18	be	be	AUX
ejpam-6565	59	19	(	(	PUNCT
ejpam-6565	59	20	τ1	τ1	NOUN
ejpam-6565	59	21	,	,	PUNCT
ejpam-6565	59	22	τ2)θ	τ2)θ	NOUN
ejpam-6565	59	23	-	-	PUNCT
ejpam-6565	59	24	closed	closed	ADJ
ejpam-6565	59	25	if	if	SCONJ
ejpam-6565	59	26	(	(	PUNCT
ejpam-6565	59	27	τ1	τ1	NOUN
ejpam-6565	59	28	,	,	PUNCT
ejpam-6565	59	29	τ2)θ	τ2)θ	NOUN
ejpam-6565	59	30	-	-	PUNCT
ejpam-6565	59	31	cl(a	cl(a	NUM
ejpam-6565	59	32	)	)	PUNCT
ejpam-6565	59	33	=	=	PUNCT
ejpam-6565	59	34	a.	a.	NOUN
ejpam-6565	59	35	the	the	DET
ejpam-6565	59	36	complement	complement	NOUN
ejpam-6565	59	37	of	of	ADP
ejpam-6565	59	38	a	a	DET
ejpam-6565	59	39	(	(	PUNCT
ejpam-6565	59	40	τ1	τ1	NOUN
ejpam-6565	59	41	,	,	PUNCT
ejpam-6565	59	42	τ2)θ	τ2)θ	ADJ
ejpam-6565	59	43	-	-	PUNCT
ejpam-6565	59	44	closed	close	VERB
ejpam-6565	59	45	set	set	NOUN
ejpam-6565	59	46	is	be	AUX
ejpam-6565	59	47	said	say	VERB
ejpam-6565	59	48	to	to	PART
ejpam-6565	59	49	be	be	AUX
ejpam-6565	59	50	(	(	PUNCT
ejpam-6565	59	51	τ1	τ1	NOUN
ejpam-6565	59	52	,	,	PUNCT
ejpam-6565	59	53	τ2)θ	τ2)θ	NOUN
ejpam-6565	59	54	-	-	PUNCT
ejpam-6565	59	55	open	open	ADJ
ejpam-6565	59	56	.	.	PUNCT
ejpam-6565	60	1	the	the	DET
ejpam-6565	60	2	union	union	NOUN
ejpam-6565	60	3	of	of	ADP
ejpam-6565	60	4	all	all	DET
ejpam-6565	60	5	(	(	PUNCT
ejpam-6565	60	6	τ1	τ1	NOUN
ejpam-6565	60	7	,	,	PUNCT
ejpam-6565	60	8	τ2)θ	τ2)θ	ADJ
ejpam-6565	60	9	-	-	PUNCT
ejpam-6565	60	10	open	open	ADJ
ejpam-6565	60	11	sets	set	NOUN
ejpam-6565	60	12	of	of	ADP
ejpam-6565	60	13	x	x	PUNCT
ejpam-6565	60	14	contained	contain	VERB
ejpam-6565	60	15	in	in	ADP
ejpam-6565	60	16	a	a	PRON
ejpam-6565	60	17	is	be	AUX
ejpam-6565	60	18	called	call	VERB
ejpam-6565	60	19	the	the	DET
ejpam-6565	60	20	(	(	PUNCT
ejpam-6565	60	21	τ1	τ1	NOUN
ejpam-6565	60	22	,	,	PUNCT
ejpam-6565	60	23	τ2)θ	τ2)θ	ADJ
ejpam-6565	60	24	-	-	PUNCT
ejpam-6565	60	25	interior	interior	NOUN
ejpam-6565	60	26	of	of	ADP
ejpam-6565	60	27	a	a	PRON
ejpam-6565	60	28	and	and	CCONJ
ejpam-6565	60	29	is	be	AUX
ejpam-6565	60	30	denoted	denote	VERB
ejpam-6565	60	31	by	by	ADP
ejpam-6565	60	32	(	(	PUNCT
ejpam-6565	60	33	τ1	τ1	NOUN
ejpam-6565	60	34	,	,	PUNCT
ejpam-6565	60	35	τ2)θ	τ2)θ	NOUN
ejpam-6565	60	36	-	-	PUNCT
ejpam-6565	60	37	int(a	int(a	NOUN
ejpam-6565	60	38	)	)	PUNCT
ejpam-6565	61	1	[	[	X
ejpam-6565	61	2	11	11	NUM
ejpam-6565	61	3	]	]	PUNCT
ejpam-6565	61	4	.	.	PUNCT
ejpam-6565	62	1	an	an	DET
ejpam-6565	62	2	ideal	ideal	NOUN
ejpam-6565	62	3	i	i	PRON
ejpam-6565	62	4	on	on	ADP
ejpam-6565	62	5	a	a	DET
ejpam-6565	62	6	topological	topological	ADJ
ejpam-6565	62	7	space	space	NOUN
ejpam-6565	62	8	(	(	PUNCT
ejpam-6565	62	9	x	x	X
ejpam-6565	62	10	,	,	PUNCT
ejpam-6565	62	11	τ	τ	X
ejpam-6565	62	12	)	)	PUNCT
ejpam-6565	62	13	is	be	AUX
ejpam-6565	62	14	a	a	DET
ejpam-6565	62	15	nonempty	nonempty	ADJ
ejpam-6565	62	16	collection	collection	NOUN
ejpam-6565	62	17	of	of	ADP
ejpam-6565	62	18	subsets	subset	NOUN
ejpam-6565	62	19	of	of	ADP
ejpam-6565	62	20	x	x	PUNCT
ejpam-6565	62	21	satisfying	satisfy	VERB
ejpam-6565	62	22	the	the	DET
ejpam-6565	62	23	following	follow	VERB
ejpam-6565	62	24	properties	property	NOUN
ejpam-6565	62	25	:	:	PUNCT
ejpam-6565	62	26	(	(	PUNCT
ejpam-6565	62	27	1	1	X
ejpam-6565	62	28	)	)	PUNCT
ejpam-6565	62	29	a	a	DET
ejpam-6565	62	30	∈	∈	NOUN
ejpam-6565	62	31	i	i	PRON
ejpam-6565	62	32	and	and	CCONJ
ejpam-6565	62	33	b	b	X
ejpam-6565	62	34	⊆	⊆	NUM
ejpam-6565	62	35	a	a	DET
ejpam-6565	62	36	imply	imply	NOUN
ejpam-6565	62	37	b	b	X
ejpam-6565	62	38	∈	∈	PROPN
ejpam-6565	62	39	i	i	PRON
ejpam-6565	62	40	;	;	PUNCT
ejpam-6565	62	41	(	(	PUNCT
ejpam-6565	62	42	2	2	X
ejpam-6565	62	43	)	)	PUNCT
ejpam-6565	63	1	a	a	PRON
ejpam-6565	63	2	∈	∈	NOUN
ejpam-6565	64	1	i	i	PRON
ejpam-6565	64	2	and	and	CCONJ
ejpam-6565	64	3	b	b	X
ejpam-6565	64	4	∈	∈	NOUN
ejpam-6565	65	1	i	i	PRON
ejpam-6565	65	2	imply	imply	VERB
ejpam-6565	65	3	a	a	DET
ejpam-6565	65	4	∪	∪	X
ejpam-6565	65	5	b	b	NOUN
ejpam-6565	65	6	∈	∈	NOUN
ejpam-6565	66	1	i	i	PRON
ejpam-6565	66	2	.	.	PUNCT
ejpam-6565	67	1	a	a	DET
ejpam-6565	67	2	topological	topological	ADJ
ejpam-6565	67	3	space	space	NOUN
ejpam-6565	67	4	(	(	PUNCT
ejpam-6565	67	5	x	x	X
ejpam-6565	67	6	,	,	PUNCT
ejpam-6565	67	7	τ	τ	X
ejpam-6565	67	8	)	)	PUNCT
ejpam-6565	67	9	with	with	ADP
ejpam-6565	67	10	an	an	DET
ejpam-6565	67	11	ideal	ideal	ADJ
ejpam-6565	67	12	i	i	PRON
ejpam-6565	67	13	on	on	ADP
ejpam-6565	67	14	x	x	SYM
ejpam-6565	67	15	is	be	AUX
ejpam-6565	67	16	called	call	VERB
ejpam-6565	67	17	an	an	DET
ejpam-6565	67	18	ideal	ideal	ADJ
ejpam-6565	67	19	topological	topological	ADJ
ejpam-6565	67	20	space	space	NOUN
ejpam-6565	67	21	and	and	CCONJ
ejpam-6565	67	22	is	be	AUX
ejpam-6565	67	23	denoted	denote	VERB
ejpam-6565	67	24	by	by	ADP
ejpam-6565	67	25	(	(	PUNCT
ejpam-6565	67	26	x	x	X
ejpam-6565	67	27	,	,	PUNCT
ejpam-6565	67	28	τ	τ	PROPN
ejpam-6565	67	29	,	,	PUNCT
ejpam-6565	67	30	i	i	NOUN
ejpam-6565	67	31	)	)	PUNCT
ejpam-6565	67	32	.	.	PUNCT
ejpam-6565	68	1	for	for	ADP
ejpam-6565	68	2	an	an	DET
ejpam-6565	68	3	ideal	ideal	ADJ
ejpam-6565	68	4	topological	topological	ADJ
ejpam-6565	68	5	space	space	NOUN
ejpam-6565	68	6	(	(	PUNCT
ejpam-6565	68	7	x	x	X
ejpam-6565	68	8	,	,	PUNCT
ejpam-6565	68	9	τ	τ	PROPN
ejpam-6565	68	10	,	,	PUNCT
ejpam-6565	68	11	i	i	PROPN
ejpam-6565	68	12	)	)	PUNCT
ejpam-6565	68	13	and	and	CCONJ
ejpam-6565	68	14	a	a	DET
ejpam-6565	68	15	subset	subset	NOUN
ejpam-6565	68	16	a	a	PRON
ejpam-6565	68	17	of	of	ADP
ejpam-6565	68	18	x	x	PRON
ejpam-6565	68	19	,	,	PUNCT
ejpam-6565	68	20	a⋆(i	a⋆(i	PROPN
ejpam-6565	68	21	)	)	PUNCT
ejpam-6565	68	22	is	be	AUX
ejpam-6565	68	23	defined	define	VERB
ejpam-6565	68	24	as	as	SCONJ
ejpam-6565	68	25	follows	follow	VERB
ejpam-6565	68	26	:	:	PUNCT
ejpam-6565	68	27	a⋆(i	a⋆(i	NOUN
ejpam-6565	68	28	)	)	PUNCT
ejpam-6565	69	1	=	=	PUNCT
ejpam-6565	69	2	{	{	PUNCT
ejpam-6565	69	3	x	x	PUNCT
ejpam-6565	69	4	∈	∈	PROPN
ejpam-6565	69	5	x	x	X
ejpam-6565	69	6	:	:	PUNCT
ejpam-6565	69	7	u	u	X
ejpam-6565	69	8	∩a	∩a	PROPN
ejpam-6565	69	9	̸∈	̸∈	PROPN
ejpam-6565	69	10	i	i	PRON
ejpam-6565	69	11	for	for	ADP
ejpam-6565	69	12	every	every	DET
ejpam-6565	69	13	open	open	ADJ
ejpam-6565	69	14	neighbourhood	neighbourhood	NOUN
ejpam-6565	69	15	u	u	NOUN
ejpam-6565	69	16	of	of	ADP
ejpam-6565	69	17	x	x	NOUN
ejpam-6565	69	18	}	}	PUNCT
ejpam-6565	69	19	.	.	PUNCT
ejpam-6565	70	1	in	in	ADP
ejpam-6565	70	2	case	case	NOUN
ejpam-6565	70	3	there	there	PRON
ejpam-6565	70	4	is	be	VERB
ejpam-6565	70	5	no	no	DET
ejpam-6565	70	6	chance	chance	NOUN
ejpam-6565	70	7	for	for	ADP
ejpam-6565	70	8	confusion	confusion	NOUN
ejpam-6565	70	9	,	,	PUNCT
ejpam-6565	70	10	a⋆(i	a⋆(i	NOUN
ejpam-6565	70	11	)	)	PUNCT
ejpam-6565	70	12	is	be	AUX
ejpam-6565	70	13	simply	simply	ADV
ejpam-6565	70	14	written	write	VERB
ejpam-6565	70	15	as	as	ADP
ejpam-6565	70	16	a⋆.	a⋆.	NOUN
ejpam-6565	70	17	in	in	ADP
ejpam-6565	70	18	[	[	X
ejpam-6565	70	19	1	1	NUM
ejpam-6565	70	20	]	]	PUNCT
ejpam-6565	70	21	,	,	PUNCT
ejpam-6565	70	22	a⋆	a⋆	ADV
ejpam-6565	70	23	is	be	AUX
ejpam-6565	70	24	called	call	VERB
ejpam-6565	70	25	the	the	DET
ejpam-6565	70	26	local	local	ADJ
ejpam-6565	70	27	function	function	NOUN
ejpam-6565	70	28	of	of	ADP
ejpam-6565	70	29	a	a	PRON
ejpam-6565	70	30	with	with	ADP
ejpam-6565	70	31	respect	respect	NOUN
ejpam-6565	70	32	to	to	ADP
ejpam-6565	70	33	i	i	PRON
ejpam-6565	70	34	and	and	CCONJ
ejpam-6565	70	35	τ	τ	PROPN
ejpam-6565	70	36	and	and	CCONJ
ejpam-6565	70	37	cl⋆(a	cl⋆(a	PROPN
ejpam-6565	70	38	)	)	PUNCT
ejpam-6565	70	39	=	=	NOUN
ejpam-6565	70	40	a⋆∪a	a⋆∪a	NOUN
ejpam-6565	70	41	defines	define	VERB
ejpam-6565	70	42	a	a	DET
ejpam-6565	70	43	kuratowski	kuratowski	ADJ
ejpam-6565	70	44	closure	closure	NOUN
ejpam-6565	70	45	operator	operator	NOUN
ejpam-6565	70	46	for	for	ADP
ejpam-6565	70	47	a	a	DET
ejpam-6565	70	48	topology	topology	NOUN
ejpam-6565	70	49	τ⋆(i	τ⋆(i	NOUN
ejpam-6565	70	50	)	)	PUNCT
ejpam-6565	70	51	finer	fine	ADJ
ejpam-6565	70	52	than	than	ADP
ejpam-6565	70	53	τ	τ	PROPN
ejpam-6565	70	54	.	.	PUNCT
ejpam-6565	71	1	a	a	DET
ejpam-6565	71	2	subset	subset	NOUN
ejpam-6565	71	3	a	a	PRON
ejpam-6565	71	4	is	be	AUX
ejpam-6565	71	5	said	say	VERB
ejpam-6565	71	6	to	to	PART
ejpam-6565	71	7	be	be	AUX
ejpam-6565	71	8	⋆-closed	⋆-close	VERB
ejpam-6565	71	9	[	[	X
ejpam-6565	71	10	15	15	NUM
ejpam-6565	71	11	]	]	X
ejpam-6565	71	12	if	if	SCONJ
ejpam-6565	71	13	a⋆	a⋆	ADJ
ejpam-6565	71	14	⊆	⊆	NUM
ejpam-6565	71	15	a.	a.	NOUN
ejpam-6565	71	16	the	the	DET
ejpam-6565	71	17	interior	interior	NOUN
ejpam-6565	71	18	of	of	ADP
ejpam-6565	71	19	a	a	DET
ejpam-6565	71	20	subset	subset	NOUN
ejpam-6565	71	21	a	a	DET
ejpam-6565	71	22	in	in	ADP
ejpam-6565	71	23	(	(	PUNCT
ejpam-6565	71	24	x	x	X
ejpam-6565	71	25	,	,	PUNCT
ejpam-6565	71	26	τ⋆(i	τ⋆(i	NOUN
ejpam-6565	71	27	)	)	PUNCT
ejpam-6565	71	28	)	)	PUNCT
ejpam-6565	71	29	is	be	AUX
ejpam-6565	71	30	denoted	denote	VERB
ejpam-6565	71	31	by	by	ADP
ejpam-6565	71	32	int⋆(a	int⋆(a	NOUN
ejpam-6565	71	33	)	)	PUNCT
ejpam-6565	71	34	.	.	PUNCT
ejpam-6565	72	1	a	a	DET
ejpam-6565	72	2	subset	subset	NOUN
ejpam-6565	72	3	a	a	PRON
ejpam-6565	72	4	of	of	ADP
ejpam-6565	72	5	an	an	DET
ejpam-6565	72	6	ideal	ideal	ADJ
ejpam-6565	72	7	topological	topological	ADJ
ejpam-6565	72	8	space	space	NOUN
ejpam-6565	72	9	(	(	PUNCT
ejpam-6565	72	10	x	x	X
ejpam-6565	72	11	,	,	PUNCT
ejpam-6565	72	12	τ	τ	PROPN
ejpam-6565	72	13	,	,	PUNCT
ejpam-6565	72	14	i	i	PROPN
ejpam-6565	72	15	)	)	PUNCT
ejpam-6565	72	16	is	be	AUX
ejpam-6565	72	17	said	say	VERB
ejpam-6565	72	18	to	to	PART
ejpam-6565	72	19	be	be	AUX
ejpam-6565	72	20	semi⋆-i	semi⋆-i	PUNCT
ejpam-6565	72	21	-open	-open	VERB
ejpam-6565	72	22	[	[	PUNCT
ejpam-6565	72	23	16	16	NUM
ejpam-6565	72	24	]	]	PUNCT
ejpam-6565	72	25	(	(	PUNCT
ejpam-6565	72	26	resp	resp	NOUN
ejpam-6565	72	27	.	.	PUNCT
ejpam-6565	73	1	semi	semi	ADJ
ejpam-6565	73	2	-	-	VERB
ejpam-6565	73	3	i	i	PRON
ejpam-6565	73	4	-open	-open	NOUN
ejpam-6565	74	1	[	[	X
ejpam-6565	74	2	4	4	NUM
ejpam-6565	74	3	]	]	PUNCT
ejpam-6565	74	4	)	)	PUNCT
ejpam-6565	74	5	if	if	SCONJ
ejpam-6565	74	6	a	a	DET
ejpam-6565	74	7	⊆	⊆	NUM
ejpam-6565	74	8	cl(int⋆(a	cl(int⋆(a	NUM
ejpam-6565	74	9	)	)	PUNCT
ejpam-6565	74	10	)	)	PUNCT
ejpam-6565	74	11	(	(	PUNCT
ejpam-6565	74	12	resp	resp	NOUN
ejpam-6565	74	13	.	.	PUNCT
ejpam-6565	75	1	a	a	DET
ejpam-6565	75	2	⊆	⊆	NUM
ejpam-6565	75	3	cl⋆(int(a	cl⋆(int(a	PROPN
ejpam-6565	75	4	)	)	PUNCT
ejpam-6565	75	5	)	)	PUNCT
ejpam-6565	75	6	)	)	PUNCT
ejpam-6565	75	7	.	.	PUNCT
ejpam-6565	76	1	the	the	DET
ejpam-6565	76	2	complement	complement	NOUN
ejpam-6565	76	3	of	of	ADP
ejpam-6565	76	4	a	a	DET
ejpam-6565	76	5	semi⋆-i	semi⋆-i	PUNCT
ejpam-6565	76	6	-open	-open	ADJ
ejpam-6565	76	7	(	(	PUNCT
ejpam-6565	76	8	resp	resp	NOUN
ejpam-6565	76	9	.	.	PUNCT
ejpam-6565	77	1	semi	semi	ADJ
ejpam-6565	77	2	-	-	VERB
ejpam-6565	77	3	i	i	PRON
ejpam-6565	77	4	-open	-open	NOUN
ejpam-6565	77	5	)	)	PUNCT
ejpam-6565	78	1	set	set	NOUN
ejpam-6565	78	2	is	be	AUX
ejpam-6565	78	3	said	say	VERB
ejpam-6565	78	4	to	to	PART
ejpam-6565	78	5	be	be	AUX
ejpam-6565	78	6	semi⋆-i	semi⋆-i	PUNCT
ejpam-6565	78	7	-closed	-close	VERB
ejpam-6565	78	8	[	[	PUNCT
ejpam-6565	78	9	16	16	NUM
ejpam-6565	78	10	]	]	PUNCT
ejpam-6565	78	11	(	(	PUNCT
ejpam-6565	78	12	resp	resp	NOUN
ejpam-6565	78	13	.	.	PUNCT
ejpam-6565	79	1	semi	semi	ADJ
ejpam-6565	79	2	-	-	VERB
ejpam-6565	79	3	i	i	PRON
ejpam-6565	79	4	-closed	-close	VERB
ejpam-6565	80	1	[	[	X
ejpam-6565	80	2	4	4	NUM
ejpam-6565	80	3	]	]	NUM
ejpam-6565	80	4	)	)	PUNCT
ejpam-6565	80	5	.	.	PUNCT
ejpam-6565	81	1	by	by	ADP
ejpam-6565	81	2	a	a	DET
ejpam-6565	81	3	multifunction	multifunction	NOUN
ejpam-6565	81	4	f	f	NOUN
ejpam-6565	81	5	:	:	PUNCT
ejpam-6565	81	6	x	x	X
ejpam-6565	81	7	→	→	SYM
ejpam-6565	81	8	y	y	PROPN
ejpam-6565	81	9	,	,	PUNCT
ejpam-6565	81	10	we	we	PRON
ejpam-6565	81	11	mean	mean	VERB
ejpam-6565	81	12	a	a	DET
ejpam-6565	81	13	point	point	NOUN
ejpam-6565	81	14	-	-	PUNCT
ejpam-6565	81	15	to	to	ADP
ejpam-6565	81	16	-	-	PUNCT
ejpam-6565	81	17	set	set	VERB
ejpam-6565	81	18	correspondence	correspondence	NOUN
ejpam-6565	81	19	from	from	ADP
ejpam-6565	81	20	x	x	PUNCT
ejpam-6565	81	21	into	into	ADP
ejpam-6565	81	22	y	y	PROPN
ejpam-6565	81	23	,	,	PUNCT
ejpam-6565	81	24	and	and	CCONJ
ejpam-6565	81	25	we	we	PRON
ejpam-6565	81	26	always	always	ADV
ejpam-6565	81	27	assume	assume	VERB
ejpam-6565	81	28	that	that	SCONJ
ejpam-6565	81	29	f	f	PROPN
ejpam-6565	81	30	(	(	PUNCT
ejpam-6565	81	31	x	x	X
ejpam-6565	81	32	)	)	PUNCT
ejpam-6565	81	33	̸=	̸=	NOUN
ejpam-6565	81	34	∅	∅	NOUN
ejpam-6565	81	35	for	for	ADP
ejpam-6565	81	36	all	all	PRON
ejpam-6565	81	37	x	x	SYM
ejpam-6565	81	38	∈	∈	ADJ
ejpam-6565	81	39	x.	x.	NOUN
ejpam-6565	81	40	for	for	ADP
ejpam-6565	81	41	a	a	DET
ejpam-6565	81	42	multifunction	multifunction	NOUN
ejpam-6565	81	43	f	f	NOUN
ejpam-6565	81	44	:	:	PUNCT
ejpam-6565	81	45	x	x	X
ejpam-6565	81	46	→	→	SYM
ejpam-6565	81	47	y	y	PROPN
ejpam-6565	81	48	,	,	PUNCT
ejpam-6565	81	49	we	we	PRON
ejpam-6565	81	50	shall	shall	AUX
ejpam-6565	81	51	denote	denote	VERB
ejpam-6565	81	52	the	the	DET
ejpam-6565	81	53	upper	upper	ADJ
ejpam-6565	81	54	and	and	CCONJ
ejpam-6565	81	55	lower	low	ADJ
ejpam-6565	81	56	inverse	inverse	NOUN
ejpam-6565	81	57	of	of	ADP
ejpam-6565	81	58	a	a	DET
ejpam-6565	81	59	set	set	NOUN
ejpam-6565	81	60	b	b	PROPN
ejpam-6565	81	61	of	of	ADP
ejpam-6565	81	62	y	y	PROPN
ejpam-6565	81	63	by	by	ADP
ejpam-6565	81	64	f+(b	f+(b	NOUN
ejpam-6565	81	65	)	)	PUNCT
ejpam-6565	81	66	and	and	CCONJ
ejpam-6565	81	67	f−(b	f−(b	NOUN
ejpam-6565	81	68	)	)	PUNCT
ejpam-6565	81	69	,	,	PUNCT
ejpam-6565	81	70	respectively	respectively	ADV
ejpam-6565	81	71	,	,	PUNCT
ejpam-6565	81	72	that	that	ADV
ejpam-6565	81	73	is	is	ADV
ejpam-6565	81	74	,	,	PUNCT
ejpam-6565	81	75	f+(b	f+(b	NOUN
ejpam-6565	81	76	)	)	PUNCT
ejpam-6565	81	77	=	=	PRON
ejpam-6565	82	1	{	{	PUNCT
ejpam-6565	82	2	x	x	PUNCT
ejpam-6565	82	3	∈	∈	PROPN
ejpam-6565	82	4	x	x	INTJ
ejpam-6565	83	1	|	|	NOUN
ejpam-6565	83	2	f	f	X
ejpam-6565	83	3	(	(	PUNCT
ejpam-6565	83	4	x	x	NOUN
ejpam-6565	83	5	)	)	PUNCT
ejpam-6565	83	6	⊆	⊆	NUM
ejpam-6565	83	7	b	b	NOUN
ejpam-6565	83	8	}	}	PUNCT
ejpam-6565	83	9	and	and	CCONJ
ejpam-6565	83	10	f−(b	f−(b	PROPN
ejpam-6565	83	11	)	)	PUNCT
ejpam-6565	83	12	=	=	PRON
ejpam-6565	84	1	{	{	PUNCT
ejpam-6565	84	2	x	x	PUNCT
ejpam-6565	84	3	∈	∈	PROPN
ejpam-6565	84	4	x	x	INTJ
ejpam-6565	85	1	|	|	NOUN
ejpam-6565	85	2	f	f	X
ejpam-6565	85	3	(	(	PUNCT
ejpam-6565	85	4	x	x	NOUN
ejpam-6565	85	5	)	)	PUNCT
ejpam-6565	85	6	∩	∩	NOUN
ejpam-6565	85	7	b	b	PROPN
ejpam-6565	85	8	̸=	̸=	PROPN
ejpam-6565	85	9	∅	∅	NOUN
ejpam-6565	85	10	}	}	PUNCT
ejpam-6565	85	11	.	.	PUNCT
ejpam-6565	86	1	in	in	ADP
ejpam-6565	86	2	particular	particular	ADJ
ejpam-6565	86	3	,	,	PUNCT
ejpam-6565	86	4	f−(y	f−(y	NOUN
ejpam-6565	86	5	)	)	PUNCT
ejpam-6565	86	6	=	=	SYM
ejpam-6565	87	1	{	{	PUNCT
ejpam-6565	87	2	x	x	PUNCT
ejpam-6565	87	3	∈	∈	PROPN
ejpam-6565	87	4	x	x	INTJ
ejpam-6565	88	1	|	|	ADV
ejpam-6565	88	2	y	y	PROPN
ejpam-6565	88	3	∈	∈	PROPN
ejpam-6565	88	4	f	f	X
ejpam-6565	88	5	(	(	PUNCT
ejpam-6565	88	6	x	x	NOUN
ejpam-6565	88	7	)	)	PUNCT
ejpam-6565	88	8	}	}	PUNCT
ejpam-6565	88	9	for	for	ADP
ejpam-6565	88	10	each	each	DET
ejpam-6565	88	11	point	point	NOUN
ejpam-6565	88	12	y	y	PROPN
ejpam-6565	88	13	∈	∈	PROPN
ejpam-6565	88	14	y	y	PROPN
ejpam-6565	88	15	.	.	PUNCT
ejpam-6565	89	1	for	for	ADP
ejpam-6565	89	2	each	each	DET
ejpam-6565	89	3	a	a	DET
ejpam-6565	89	4	⊆	⊆	NUM
ejpam-6565	89	5	x	x	SYM
ejpam-6565	89	6	,	,	PUNCT
ejpam-6565	89	7	f	f	PROPN
ejpam-6565	89	8	(	(	PUNCT
ejpam-6565	89	9	a	a	NOUN
ejpam-6565	89	10	)	)	PUNCT
ejpam-6565	89	11	=	=	SYM
ejpam-6565	89	12	∪x∈af	∪x∈af	NOUN
ejpam-6565	89	13	(	(	PUNCT
ejpam-6565	89	14	x	x	NOUN
ejpam-6565	89	15	)	)	PUNCT
ejpam-6565	89	16	.	.	PUNCT
ejpam-6565	90	1	3	3	X
ejpam-6565	90	2	.	.	X
ejpam-6565	90	3	upper	upper	ADJ
ejpam-6565	90	4	and	and	CCONJ
ejpam-6565	90	5	lower	low	ADJ
ejpam-6565	90	6	τ	τ	X
ejpam-6565	90	7	⋆(σ1	⋆(σ1	NOUN
ejpam-6565	90	8	,	,	PUNCT
ejpam-6565	90	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	90	10	multifunctions	multifunction	NOUN
ejpam-6565	90	11	in	in	ADP
ejpam-6565	90	12	this	this	DET
ejpam-6565	90	13	section	section	NOUN
ejpam-6565	90	14	,	,	PUNCT
ejpam-6565	90	15	we	we	PRON
ejpam-6565	90	16	introduce	introduce	VERB
ejpam-6565	90	17	the	the	DET
ejpam-6565	90	18	concepts	concept	NOUN
ejpam-6565	90	19	of	of	ADP
ejpam-6565	90	20	upper	upper	ADJ
ejpam-6565	90	21	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	90	22	,	,	PUNCT
ejpam-6565	90	23	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	90	24	multifunctions	multifunction	NOUN
ejpam-6565	90	25	and	and	CCONJ
ejpam-6565	90	26	lower	low	ADJ
ejpam-6565	90	27	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	90	28	,	,	PUNCT
ejpam-6565	90	29	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	90	30	multifunctions	multifunction	NOUN
ejpam-6565	90	31	.	.	PUNCT
ejpam-6565	91	1	furthermore	furthermore	ADV
ejpam-6565	91	2	,	,	PUNCT
ejpam-6565	91	3	several	several	ADJ
ejpam-6565	91	4	characterizations	characterization	NOUN
ejpam-6565	91	5	of	of	ADP
ejpam-6565	91	6	upper	upper	ADJ
ejpam-6565	91	7	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	91	8	,	,	PUNCT
ejpam-6565	91	9	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	91	10	multifunctions	multifunction	NOUN
ejpam-6565	91	11	and	and	CCONJ
ejpam-6565	91	12	lower	low	ADJ
ejpam-6565	91	13	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	91	14	,	,	PUNCT
ejpam-6565	91	15	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	91	16	multifunctions	multifunction	NOUN
ejpam-6565	91	17	are	be	AUX
ejpam-6565	91	18	discussed	discuss	VERB
ejpam-6565	91	19	.	.	PUNCT
ejpam-6565	92	1	j.	j.	PROPN
ejpam-6565	92	2	khampakdee	khampakdee	PROPN
ejpam-6565	92	3	,	,	PUNCT
ejpam-6565	92	4	a.	a.	PROPN
ejpam-6565	92	5	sama	sama	PROPN
ejpam-6565	92	6	-	-	PUNCT
ejpam-6565	92	7	ae	ae	PROPN
ejpam-6565	92	8	,	,	PUNCT
ejpam-6565	92	9	c.	c.	PROPN
ejpam-6565	92	10	boonpok	boonpok	PROPN
ejpam-6565	92	11	/	/	SYM
ejpam-6565	92	12	eur	eur	PROPN
ejpam-6565	92	13	.	.	PUNCT
ejpam-6565	93	1	j.	j.	PROPN
ejpam-6565	93	2	pure	pure	PROPN
ejpam-6565	93	3	appl	appl	PROPN
ejpam-6565	93	4	.	.	PROPN
ejpam-6565	93	5	math	math	PROPN
ejpam-6565	93	6	,	,	PUNCT
ejpam-6565	93	7	18	18	NUM
ejpam-6565	93	8	(	(	PUNCT
ejpam-6565	93	9	3	3	NUM
ejpam-6565	93	10	)	)	PUNCT
ejpam-6565	93	11	(	(	PUNCT
ejpam-6565	93	12	2025	2025	NUM
ejpam-6565	93	13	)	)	PUNCT
ejpam-6565	93	14	,	,	PUNCT
ejpam-6565	93	15	6565	6565	NUM
ejpam-6565	93	16	4	4	NUM
ejpam-6565	93	17	of	of	ADP
ejpam-6565	93	18	9	9	NUM
ejpam-6565	93	19	definition	definition	NOUN
ejpam-6565	93	20	1	1	NUM
ejpam-6565	93	21	.	.	PUNCT
ejpam-6565	94	1	a	a	DET
ejpam-6565	94	2	multifunction	multifunction	NOUN
ejpam-6565	94	3	f	f	NOUN
ejpam-6565	94	4	:	:	PUNCT
ejpam-6565	94	5	(	(	PUNCT
ejpam-6565	94	6	x	x	X
ejpam-6565	94	7	,	,	PUNCT
ejpam-6565	94	8	τ	τ	PROPN
ejpam-6565	94	9	,	,	PUNCT
ejpam-6565	94	10	i	i	NOUN
ejpam-6565	94	11	)	)	PUNCT
ejpam-6565	94	12	→	→	PUNCT
ejpam-6565	94	13	(	(	PUNCT
ejpam-6565	94	14	y	y	PROPN
ejpam-6565	94	15	,	,	PUNCT
ejpam-6565	94	16	σ1	σ1	PROPN
ejpam-6565	94	17	,	,	PUNCT
ejpam-6565	94	18	σ2	σ2	PROPN
ejpam-6565	94	19	)	)	PUNCT
ejpam-6565	94	20	is	be	AUX
ejpam-6565	94	21	called	call	VERB
ejpam-6565	94	22	upper	upper	ADJ
ejpam-6565	94	23	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	94	24	,	,	PUNCT
ejpam-6565	94	25	σ2)continuous	σ2)continuous	ADJ
ejpam-6565	94	26	at	at	ADP
ejpam-6565	94	27	a	a	DET
ejpam-6565	94	28	point	point	NOUN
ejpam-6565	94	29	x	x	SYM
ejpam-6565	94	30	∈	∈	NOUN
ejpam-6565	94	31	x	x	PUNCT
ejpam-6565	94	32	if	if	SCONJ
ejpam-6565	94	33	for	for	ADP
ejpam-6565	94	34	each	each	DET
ejpam-6565	94	35	σ1σ2	σ1σ2	VERB
ejpam-6565	94	36	-	-	ADJ
ejpam-6565	94	37	open	open	ADJ
ejpam-6565	94	38	set	set	NOUN
ejpam-6565	94	39	v	v	NOUN
ejpam-6565	94	40	of	of	ADP
ejpam-6565	94	41	y	y	PRON
ejpam-6565	94	42	such	such	ADJ
ejpam-6565	94	43	that	that	SCONJ
ejpam-6565	94	44	f	f	PROPN
ejpam-6565	94	45	(	(	PUNCT
ejpam-6565	94	46	x	x	X
ejpam-6565	94	47	)	)	PUNCT
ejpam-6565	94	48	⊆	⊆	NUM
ejpam-6565	94	49	v	v	NOUN
ejpam-6565	94	50	,	,	PUNCT
ejpam-6565	94	51	there	there	PRON
ejpam-6565	94	52	exists	exist	VERB
ejpam-6565	94	53	a	a	DET
ejpam-6565	94	54	⋆-open	⋆-open	ADJ
ejpam-6565	94	55	set	set	NOUN
ejpam-6565	94	56	u	u	NOUN
ejpam-6565	94	57	of	of	ADP
ejpam-6565	94	58	x	x	PUNCT
ejpam-6565	94	59	containing	contain	VERB
ejpam-6565	94	60	x	x	PUNCT
ejpam-6565	94	61	such	such	ADJ
ejpam-6565	94	62	that	that	SCONJ
ejpam-6565	94	63	f	f	PROPN
ejpam-6565	94	64	(	(	PUNCT
ejpam-6565	94	65	u	u	NOUN
ejpam-6565	94	66	)	)	PUNCT
ejpam-6565	94	67	⊆	⊆	NUM
ejpam-6565	94	68	v	v	NOUN
ejpam-6565	94	69	.	.	PUNCT
ejpam-6565	95	1	a	a	DET
ejpam-6565	95	2	multifunction	multifunction	NOUN
ejpam-6565	95	3	f	f	NOUN
ejpam-6565	95	4	:	:	PUNCT
ejpam-6565	95	5	(	(	PUNCT
ejpam-6565	95	6	x	x	X
ejpam-6565	95	7	,	,	PUNCT
ejpam-6565	95	8	τ	τ	PROPN
ejpam-6565	95	9	,	,	PUNCT
ejpam-6565	95	10	i	i	NOUN
ejpam-6565	95	11	)	)	PUNCT
ejpam-6565	95	12	→	→	PUNCT
ejpam-6565	95	13	(	(	PUNCT
ejpam-6565	95	14	y	y	PROPN
ejpam-6565	95	15	,	,	PUNCT
ejpam-6565	95	16	σ1	σ1	PROPN
ejpam-6565	95	17	,	,	PUNCT
ejpam-6565	95	18	σ2	σ2	PROPN
ejpam-6565	95	19	)	)	PUNCT
ejpam-6565	95	20	is	be	AUX
ejpam-6565	95	21	called	call	VERB
ejpam-6565	95	22	upper	upper	ADJ
ejpam-6565	95	23	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	95	24	,	,	PUNCT
ejpam-6565	95	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	95	26	if	if	SCONJ
ejpam-6565	95	27	f	f	PROPN
ejpam-6565	95	28	is	be	AUX
ejpam-6565	95	29	upper	upper	ADJ
ejpam-6565	95	30	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6565	95	31	,	,	PUNCT
ejpam-6565	95	32	σ2)continuous	σ2)continuous	ADJ
ejpam-6565	95	33	at	at	ADP
ejpam-6565	95	34	each	each	DET
ejpam-6565	95	35	point	point	NOUN
ejpam-6565	95	36	x	x	PUNCT
ejpam-6565	95	37	of	of	ADP
ejpam-6565	95	38	x.	x.	PROPN
ejpam-6565	95	39	theorem	theorem	VERB
ejpam-6565	95	40	1	1	NUM
ejpam-6565	95	41	.	.	X
ejpam-6565	95	42	for	for	ADP
ejpam-6565	95	43	a	a	DET
ejpam-6565	95	44	multifunction	multifunction	NOUN
ejpam-6565	95	45	f	f	NOUN
ejpam-6565	95	46	:	:	PUNCT
ejpam-6565	95	47	(	(	PUNCT
ejpam-6565	95	48	x	x	X
ejpam-6565	95	49	,	,	PUNCT
ejpam-6565	95	50	τ	τ	PROPN
ejpam-6565	95	51	,	,	PUNCT
ejpam-6565	95	52	i	i	NOUN
ejpam-6565	95	53	)	)	PUNCT
ejpam-6565	96	1	→	→	PUNCT
ejpam-6565	96	2	(	(	PUNCT
ejpam-6565	96	3	y	y	PROPN
ejpam-6565	96	4	,	,	PUNCT
ejpam-6565	96	5	σ1	σ1	PROPN
ejpam-6565	96	6	,	,	PUNCT
ejpam-6565	96	7	σ2	σ2	NOUN
ejpam-6565	96	8	)	)	PUNCT
ejpam-6565	96	9	,	,	PUNCT
ejpam-6565	96	10	the	the	DET
ejpam-6565	96	11	following	follow	VERB
ejpam-6565	96	12	properties	property	NOUN
ejpam-6565	96	13	are	be	AUX
ejpam-6565	96	14	equivalent	equivalent	ADJ
ejpam-6565	96	15	:	:	PUNCT
ejpam-6565	96	16	(	(	PUNCT
ejpam-6565	96	17	1	1	X
ejpam-6565	96	18	)	)	PUNCT
ejpam-6565	96	19	f	f	PROPN
ejpam-6565	96	20	is	be	AUX
ejpam-6565	96	21	upper	upper	ADJ
ejpam-6565	96	22	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	96	23	,	,	PUNCT
ejpam-6565	96	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	96	25	;	;	PUNCT
ejpam-6565	96	26	(	(	PUNCT
ejpam-6565	96	27	2	2	NUM
ejpam-6565	96	28	)	)	PUNCT
ejpam-6565	96	29	f+(v	f+(v	NOUN
ejpam-6565	96	30	)	)	PUNCT
ejpam-6565	96	31	is	be	AUX
ejpam-6565	96	32	⋆-open	⋆-open	ADJ
ejpam-6565	96	33	in	in	ADP
ejpam-6565	96	34	x	x	PUNCT
ejpam-6565	96	35	for	for	ADP
ejpam-6565	96	36	every	every	DET
ejpam-6565	96	37	σ1σ2	σ1σ2	NOUN
ejpam-6565	96	38	-	-	ADJ
ejpam-6565	96	39	open	open	ADJ
ejpam-6565	96	40	set	set	NOUN
ejpam-6565	96	41	v	v	NOUN
ejpam-6565	96	42	of	of	ADP
ejpam-6565	96	43	y	y	PROPN
ejpam-6565	96	44	;	;	PUNCT
ejpam-6565	96	45	(	(	PUNCT
ejpam-6565	96	46	3	3	X
ejpam-6565	96	47	)	)	PUNCT
ejpam-6565	96	48	f−(k	f−(k	PROPN
ejpam-6565	96	49	)	)	PUNCT
ejpam-6565	96	50	is	be	AUX
ejpam-6565	96	51	⋆-closed	⋆-close	VERB
ejpam-6565	96	52	in	in	ADP
ejpam-6565	96	53	x	x	PUNCT
ejpam-6565	96	54	for	for	ADP
ejpam-6565	96	55	every	every	DET
ejpam-6565	96	56	σ1σ2	σ1σ2	NUM
ejpam-6565	96	57	-	-	PUNCT
ejpam-6565	96	58	closed	closed	ADJ
ejpam-6565	96	59	set	set	NOUN
ejpam-6565	96	60	k	k	PROPN
ejpam-6565	96	61	of	of	ADP
ejpam-6565	96	62	y	y	PROPN
ejpam-6565	96	63	;	;	PUNCT
ejpam-6565	96	64	(	(	PUNCT
ejpam-6565	96	65	4	4	X
ejpam-6565	96	66	)	)	PUNCT
ejpam-6565	96	67	cl⋆(f−(b	cl⋆(f−(b	NUM
ejpam-6565	96	68	)	)	PUNCT
ejpam-6565	96	69	)	)	PUNCT
ejpam-6565	97	1	⊆	⊆	X
ejpam-6565	97	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6565	97	3	-	-	PUNCT
ejpam-6565	97	4	cl(b	cl(b	NOUN
ejpam-6565	97	5	)	)	PUNCT
ejpam-6565	97	6	)	)	PUNCT
ejpam-6565	98	1	for	for	ADP
ejpam-6565	98	2	every	every	DET
ejpam-6565	98	3	subset	subset	NOUN
ejpam-6565	98	4	b	b	PROPN
ejpam-6565	98	5	of	of	ADP
ejpam-6565	98	6	y	y	PROPN
ejpam-6565	98	7	;	;	PUNCT
ejpam-6565	98	8	(	(	PUNCT
ejpam-6565	98	9	5	5	X
ejpam-6565	98	10	)	)	PUNCT
ejpam-6565	98	11	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6565	98	12	-	-	PUNCT
ejpam-6565	98	13	int(b	int(b	NOUN
ejpam-6565	98	14	)	)	PUNCT
ejpam-6565	98	15	)	)	PUNCT
ejpam-6565	98	16	⊆	⊆	NUM
ejpam-6565	98	17	int⋆(f+(b	int⋆(f+(b	NOUN
ejpam-6565	98	18	)	)	PUNCT
ejpam-6565	98	19	)	)	PUNCT
ejpam-6565	98	20	for	for	ADP
ejpam-6565	98	21	every	every	DET
ejpam-6565	98	22	subset	subset	NOUN
ejpam-6565	98	23	b	b	PROPN
ejpam-6565	98	24	of	of	ADP
ejpam-6565	98	25	y	y	PROPN
ejpam-6565	98	26	.	.	PUNCT
ejpam-6565	99	1	proof	proof	NOUN
ejpam-6565	99	2	.	.	PUNCT
ejpam-6565	100	1	(	(	PUNCT
ejpam-6565	100	2	1	1	X
ejpam-6565	100	3	)	)	PUNCT
ejpam-6565	100	4	⇒	⇒	NOUN
ejpam-6565	100	5	(	(	PUNCT
ejpam-6565	100	6	2	2	NUM
ejpam-6565	100	7	):	):	PUNCT
ejpam-6565	100	8	let	let	VERB
ejpam-6565	100	9	v	v	PART
ejpam-6565	100	10	be	be	AUX
ejpam-6565	100	11	any	any	DET
ejpam-6565	100	12	σ1σ2	σ1σ2	NOUN
ejpam-6565	100	13	-	-	ADJ
ejpam-6565	100	14	open	open	ADJ
ejpam-6565	100	15	set	set	NOUN
ejpam-6565	100	16	of	of	ADP
ejpam-6565	100	17	y	y	PROPN
ejpam-6565	100	18	and	and	CCONJ
ejpam-6565	100	19	x	x	PROPN
ejpam-6565	100	20	∈	∈	PROPN
ejpam-6565	100	21	f+(v	f+(v	NOUN
ejpam-6565	100	22	)	)	PUNCT
ejpam-6565	100	23	.	.	PUNCT
ejpam-6565	101	1	then	then	ADV
ejpam-6565	101	2	,	,	PUNCT
ejpam-6565	101	3	f	f	PROPN
ejpam-6565	101	4	(	(	PUNCT
ejpam-6565	101	5	x	x	X
ejpam-6565	101	6	)	)	PUNCT
ejpam-6565	101	7	⊆	⊆	NUM
ejpam-6565	101	8	v	v	NOUN
ejpam-6565	101	9	and	and	CCONJ
ejpam-6565	101	10	by	by	ADP
ejpam-6565	101	11	(	(	PUNCT
ejpam-6565	101	12	1	1	NUM
ejpam-6565	101	13	)	)	PUNCT
ejpam-6565	101	14	,	,	PUNCT
ejpam-6565	101	15	there	there	PRON
ejpam-6565	101	16	exists	exist	VERB
ejpam-6565	101	17	a	a	DET
ejpam-6565	101	18	⋆-open	⋆-open	ADJ
ejpam-6565	101	19	set	set	NOUN
ejpam-6565	101	20	u	u	NOUN
ejpam-6565	101	21	of	of	ADP
ejpam-6565	101	22	x	x	PUNCT
ejpam-6565	101	23	containing	contain	VERB
ejpam-6565	101	24	x	x	PUNCT
ejpam-6565	101	25	such	such	ADJ
ejpam-6565	101	26	that	that	SCONJ
ejpam-6565	101	27	f	f	PROPN
ejpam-6565	101	28	(	(	PUNCT
ejpam-6565	101	29	u	u	NOUN
ejpam-6565	101	30	)	)	PUNCT
ejpam-6565	101	31	⊆	⊆	NUM
ejpam-6565	101	32	v	v	NOUN
ejpam-6565	101	33	.	.	PUNCT
ejpam-6565	102	1	thus	thus	ADV
ejpam-6565	102	2	,	,	PUNCT
ejpam-6565	102	3	x	x	PUNCT
ejpam-6565	102	4	∈	∈	PROPN
ejpam-6565	102	5	u	u	NOUN
ejpam-6565	102	6	⊆	⊆	NUM
ejpam-6565	102	7	f+(v	f+(v	NOUN
ejpam-6565	102	8	)	)	PUNCT
ejpam-6565	102	9	and	and	CCONJ
ejpam-6565	102	10	hence	hence	ADV
ejpam-6565	102	11	x	x	X
ejpam-6565	102	12	∈	∈	PROPN
ejpam-6565	102	13	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6565	102	14	)	)	PUNCT
ejpam-6565	102	15	)	)	PUNCT
ejpam-6565	102	16	.	.	PUNCT
ejpam-6565	103	1	therefore	therefore	ADV
ejpam-6565	103	2	,	,	PUNCT
ejpam-6565	103	3	f+(v	f+(v	PROPN
ejpam-6565	103	4	)	)	PUNCT
ejpam-6565	104	1	⊆	⊆	NUM
ejpam-6565	104	2	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6565	104	3	)	)	PUNCT
ejpam-6565	104	4	)	)	PUNCT
ejpam-6565	104	5	.	.	PUNCT
ejpam-6565	105	1	this	this	PRON
ejpam-6565	105	2	shows	show	VERB
ejpam-6565	105	3	that	that	SCONJ
ejpam-6565	105	4	f+(v	f+(v	PROPN
ejpam-6565	105	5	)	)	PUNCT
ejpam-6565	105	6	is	be	AUX
ejpam-6565	105	7	⋆-open	⋆-open	ADJ
ejpam-6565	105	8	in	in	ADP
ejpam-6565	105	9	x.	x.	PROPN
ejpam-6565	105	10	(	(	PUNCT
ejpam-6565	105	11	2	2	NUM
ejpam-6565	105	12	)	)	PUNCT
ejpam-6565	105	13	⇒	⇒	NOUN
ejpam-6565	105	14	(	(	PUNCT
ejpam-6565	105	15	3	3	NUM
ejpam-6565	105	16	):	):	PUNCT
ejpam-6565	105	17	this	this	PRON
ejpam-6565	105	18	follows	follow	VERB
ejpam-6565	105	19	from	from	ADP
ejpam-6565	105	20	the	the	DET
ejpam-6565	105	21	fact	fact	NOUN
ejpam-6565	105	22	that	that	SCONJ
ejpam-6565	105	23	f+(y	f+(y	PROPN
ejpam-6565	105	24	−b	−b	ADV
ejpam-6565	105	25	)	)	PUNCT
ejpam-6565	105	26	=	=	PUNCT
ejpam-6565	106	1	x	x	X
ejpam-6565	106	2	−	−	PROPN
ejpam-6565	106	3	f−(b	f−(b	PROPN
ejpam-6565	106	4	)	)	PUNCT
ejpam-6565	106	5	for	for	ADP
ejpam-6565	106	6	every	every	DET
ejpam-6565	106	7	subset	subset	NOUN
ejpam-6565	106	8	b	b	PROPN
ejpam-6565	106	9	of	of	ADP
ejpam-6565	106	10	y	y	PROPN
ejpam-6565	106	11	.	.	PUNCT
ejpam-6565	107	1	(	(	PUNCT
ejpam-6565	107	2	3	3	X
ejpam-6565	107	3	)	)	PUNCT
ejpam-6565	107	4	⇒	⇒	NOUN
ejpam-6565	107	5	(	(	PUNCT
ejpam-6565	107	6	4	4	NUM
ejpam-6565	107	7	):	):	PUNCT
ejpam-6565	107	8	let	let	VERB
ejpam-6565	107	9	b	b	X
ejpam-6565	107	10	be	be	AUX
ejpam-6565	107	11	any	any	DET
ejpam-6565	107	12	subset	subset	NOUN
ejpam-6565	107	13	of	of	ADP
ejpam-6565	107	14	y	y	PROPN
ejpam-6565	107	15	.	.	PUNCT
ejpam-6565	108	1	then	then	ADV
ejpam-6565	108	2	,	,	PUNCT
ejpam-6565	108	3	σ1σ2	σ1σ2	NOUN
ejpam-6565	108	4	-	-	NOUN
ejpam-6565	108	5	cl(b	cl(b	NOUN
ejpam-6565	108	6	)	)	PUNCT
ejpam-6565	108	7	is	be	AUX
ejpam-6565	108	8	σ1σ2	σ1σ2	NOUN
ejpam-6565	108	9	-	-	ADJ
ejpam-6565	108	10	closed	closed	ADJ
ejpam-6565	108	11	in	in	ADP
ejpam-6565	108	12	y	y	PROPN
ejpam-6565	108	13	and	and	CCONJ
ejpam-6565	108	14	by	by	ADP
ejpam-6565	108	15	(	(	PUNCT
ejpam-6565	108	16	3	3	NUM
ejpam-6565	108	17	)	)	PUNCT
ejpam-6565	108	18	,	,	PUNCT
ejpam-6565	108	19	cl⋆(f−(b	cl⋆(f−(b	NUM
ejpam-6565	108	20	)	)	PUNCT
ejpam-6565	108	21	)	)	PUNCT
ejpam-6565	109	1	⊆	⊆	NUM
ejpam-6565	109	2	cl⋆(f−(σ1σ2	cl⋆(f−(σ1σ2	VERB
ejpam-6565	109	3	-	-	PUNCT
ejpam-6565	109	4	cl(b	cl(b	NOUN
ejpam-6565	109	5	)	)	PUNCT
ejpam-6565	109	6	)	)	PUNCT
ejpam-6565	109	7	)	)	PUNCT
ejpam-6565	110	1	=	=	PUNCT
ejpam-6565	110	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6565	110	3	-	-	PUNCT
ejpam-6565	110	4	cl(b	cl(b	NOUN
ejpam-6565	110	5	)	)	PUNCT
ejpam-6565	110	6	)	)	PUNCT
ejpam-6565	110	7	.	.	PUNCT
ejpam-6565	111	1	(	(	PUNCT
ejpam-6565	111	2	4	4	X
ejpam-6565	111	3	)	)	PUNCT
ejpam-6565	111	4	⇒	⇒	NOUN
ejpam-6565	111	5	(	(	PUNCT
ejpam-6565	111	6	5	5	NUM
ejpam-6565	111	7	):	):	PUNCT
ejpam-6565	111	8	let	let	VERB
ejpam-6565	111	9	b	b	X
ejpam-6565	111	10	be	be	AUX
ejpam-6565	111	11	any	any	DET
ejpam-6565	111	12	subset	subset	NOUN
ejpam-6565	111	13	of	of	ADP
ejpam-6565	111	14	y	y	PROPN
ejpam-6565	111	15	.	.	PUNCT
ejpam-6565	112	1	thus	thus	ADV
ejpam-6565	112	2	by	by	ADP
ejpam-6565	112	3	(	(	PUNCT
ejpam-6565	112	4	4	4	NUM
ejpam-6565	112	5	)	)	PUNCT
ejpam-6565	112	6	,	,	PUNCT
ejpam-6565	112	7	we	we	PRON
ejpam-6565	112	8	have	have	VERB
ejpam-6565	112	9	x	x	X
ejpam-6565	112	10	−	−	PRON
ejpam-6565	112	11	int⋆(f+(b	int⋆(f+(b	NOUN
ejpam-6565	112	12	)	)	PUNCT
ejpam-6565	112	13	)	)	PUNCT
ejpam-6565	113	1	=	=	PUNCT
ejpam-6565	113	2	cl⋆(x	cl⋆(x	NOUN
ejpam-6565	113	3	−	−	NOUN
ejpam-6565	113	4	f+(b	f+(b	NOUN
ejpam-6565	113	5	)	)	PUNCT
ejpam-6565	113	6	)	)	PUNCT
ejpam-6565	114	1	=	=	NOUN
ejpam-6565	114	2	cl⋆(f−(y	cl⋆(f−(y	NUM
ejpam-6565	114	3	−	−	PROPN
ejpam-6565	114	4	b	b	X
ejpam-6565	114	5	)	)	PUNCT
ejpam-6565	114	6	)	)	PUNCT
ejpam-6565	115	1	⊆	⊆	X
ejpam-6565	115	2	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6565	115	3	-	-	PUNCT
ejpam-6565	115	4	cl(y	cl(y	NOUN
ejpam-6565	115	5	−	−	PROPN
ejpam-6565	115	6	b	b	NOUN
ejpam-6565	115	7	)	)	PUNCT
ejpam-6565	115	8	)	)	PUNCT
ejpam-6565	116	1	=	=	PUNCT
ejpam-6565	116	2	f−(y	f−(y	NOUN
ejpam-6565	117	1	−	−	ADP
ejpam-6565	117	2	σ1σ2	σ1σ2	NOUN
ejpam-6565	117	3	-	-	PUNCT
ejpam-6565	117	4	int(b	int(b	NOUN
ejpam-6565	117	5	)	)	PUNCT
ejpam-6565	117	6	)	)	PUNCT
ejpam-6565	118	1	=	=	PUNCT
ejpam-6565	118	2	x	x	X
ejpam-6565	119	1	−	−	ADP
ejpam-6565	119	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6565	119	3	-	-	PUNCT
ejpam-6565	119	4	int(b	int(b	NOUN
ejpam-6565	119	5	)	)	PUNCT
ejpam-6565	119	6	)	)	PUNCT
ejpam-6565	119	7	and	and	CCONJ
ejpam-6565	119	8	hence	hence	ADV
ejpam-6565	119	9	f+(σ1σ2	f+(σ1σ2	ADV
ejpam-6565	119	10	-	-	PUNCT
ejpam-6565	119	11	int(b	int(b	NOUN
ejpam-6565	119	12	)	)	PUNCT
ejpam-6565	119	13	)	)	PUNCT
ejpam-6565	119	14	⊆	⊆	NUM
ejpam-6565	119	15	int⋆(f+(b	int⋆(f+(b	NOUN
ejpam-6565	119	16	)	)	PUNCT
ejpam-6565	119	17	)	)	PUNCT
ejpam-6565	119	18	.	.	PUNCT
ejpam-6565	120	1	(	(	PUNCT
ejpam-6565	120	2	5	5	X
ejpam-6565	120	3	)	)	PUNCT
ejpam-6565	120	4	⇒	⇒	NOUN
ejpam-6565	120	5	(	(	PUNCT
ejpam-6565	120	6	1	1	NUM
ejpam-6565	120	7	):	):	PUNCT
ejpam-6565	120	8	let	let	VERB
ejpam-6565	120	9	x	x	PUNCT
ejpam-6565	120	10	∈	∈	PROPN
ejpam-6565	120	11	x	x	X
ejpam-6565	120	12	and	and	CCONJ
ejpam-6565	120	13	v	v	X
ejpam-6565	120	14	be	be	AUX
ejpam-6565	120	15	any	any	DET
ejpam-6565	120	16	σ1σ2	σ1σ2	NOUN
ejpam-6565	120	17	-	-	ADJ
ejpam-6565	120	18	open	open	ADJ
ejpam-6565	120	19	set	set	NOUN
ejpam-6565	120	20	of	of	ADP
ejpam-6565	120	21	y	y	PRON
ejpam-6565	120	22	such	such	ADJ
ejpam-6565	120	23	that	that	SCONJ
ejpam-6565	120	24	f	f	PROPN
ejpam-6565	120	25	(	(	PUNCT
ejpam-6565	120	26	x	x	X
ejpam-6565	120	27	)	)	PUNCT
ejpam-6565	120	28	⊆	⊆	NUM
ejpam-6565	120	29	v	v	NOUN
ejpam-6565	120	30	.	.	PUNCT
ejpam-6565	121	1	then	then	ADV
ejpam-6565	121	2	,	,	PUNCT
ejpam-6565	121	3	x	x	X
ejpam-6565	121	4	∈	∈	NOUN
ejpam-6565	121	5	f+(v	f+(v	NOUN
ejpam-6565	121	6	)	)	PUNCT
ejpam-6565	122	1	=	=	SYM
ejpam-6565	122	2	int⋆(f+(v	int⋆(f+(v	PROPN
ejpam-6565	122	3	)	)	PUNCT
ejpam-6565	122	4	)	)	PUNCT
ejpam-6565	122	5	.	.	PUNCT
ejpam-6565	123	1	there	there	PRON
ejpam-6565	123	2	exists	exist	VERB
ejpam-6565	123	3	a	a	DET
ejpam-6565	123	4	⋆-open	⋆-open	ADJ
ejpam-6565	123	5	set	set	NOUN
ejpam-6565	123	6	u	u	NOUN
ejpam-6565	123	7	of	of	ADP
ejpam-6565	123	8	x	x	PUNCT
ejpam-6565	123	9	containing	contain	VERB
ejpam-6565	123	10	x	x	PUNCT
ejpam-6565	123	11	such	such	ADJ
ejpam-6565	123	12	that	that	SCONJ
ejpam-6565	123	13	u	u	NOUN
ejpam-6565	123	14	⊆	⊆	NUM
ejpam-6565	123	15	f+(v	f+(v	NOUN
ejpam-6565	123	16	)	)	PUNCT
ejpam-6565	123	17	;	;	PUNCT
ejpam-6565	123	18	hence	hence	ADV
ejpam-6565	123	19	f	f	PROPN
ejpam-6565	123	20	(	(	PUNCT
ejpam-6565	123	21	u	u	NOUN
ejpam-6565	123	22	)	)	PUNCT
ejpam-6565	123	23	⊆	⊆	NUM
ejpam-6565	123	24	v	v	NOUN
ejpam-6565	123	25	.	.	PUNCT
ejpam-6565	124	1	this	this	PRON
ejpam-6565	124	2	shows	show	VERB
ejpam-6565	124	3	that	that	SCONJ
ejpam-6565	124	4	f	f	PROPN
ejpam-6565	124	5	is	be	AUX
ejpam-6565	124	6	upper	upper	ADJ
ejpam-6565	124	7	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	124	8	,	,	PUNCT
ejpam-6565	124	9	σ2)-continuous	σ2)-continuous	PROPN
ejpam-6565	124	10	.	.	NOUN
ejpam-6565	124	11	definition	definition	NOUN
ejpam-6565	124	12	2	2	NUM
ejpam-6565	124	13	.	.	PUNCT
ejpam-6565	124	14	a	a	DET
ejpam-6565	124	15	multifunction	multifunction	NOUN
ejpam-6565	125	1	f	f	NOUN
ejpam-6565	125	2	:	:	PUNCT
ejpam-6565	125	3	(	(	PUNCT
ejpam-6565	125	4	x	x	X
ejpam-6565	125	5	,	,	PUNCT
ejpam-6565	125	6	τ	τ	PROPN
ejpam-6565	125	7	,	,	PUNCT
ejpam-6565	125	8	i	i	NOUN
ejpam-6565	125	9	)	)	PUNCT
ejpam-6565	125	10	→	→	PUNCT
ejpam-6565	125	11	(	(	PUNCT
ejpam-6565	125	12	y	y	PROPN
ejpam-6565	125	13	,	,	PUNCT
ejpam-6565	125	14	σ1	σ1	PROPN
ejpam-6565	125	15	,	,	PUNCT
ejpam-6565	125	16	σ2	σ2	PROPN
ejpam-6565	125	17	)	)	PUNCT
ejpam-6565	125	18	is	be	AUX
ejpam-6565	125	19	called	call	VERB
ejpam-6565	125	20	lower	low	ADJ
ejpam-6565	125	21	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	125	22	,	,	PUNCT
ejpam-6565	125	23	σ2)continuous	σ2)continuous	ADJ
ejpam-6565	125	24	at	at	ADP
ejpam-6565	125	25	a	a	DET
ejpam-6565	125	26	point	point	NOUN
ejpam-6565	125	27	x	x	SYM
ejpam-6565	125	28	∈	∈	NOUN
ejpam-6565	125	29	x	x	PUNCT
ejpam-6565	125	30	if	if	SCONJ
ejpam-6565	125	31	for	for	ADP
ejpam-6565	125	32	each	each	DET
ejpam-6565	125	33	σ1σ2	σ1σ2	VERB
ejpam-6565	125	34	-	-	ADJ
ejpam-6565	125	35	open	open	ADJ
ejpam-6565	125	36	set	set	NOUN
ejpam-6565	125	37	v	v	NOUN
ejpam-6565	125	38	of	of	ADP
ejpam-6565	125	39	y	y	PRON
ejpam-6565	125	40	such	such	ADJ
ejpam-6565	125	41	that	that	SCONJ
ejpam-6565	125	42	f	f	PROPN
ejpam-6565	125	43	(	(	PUNCT
ejpam-6565	125	44	x	x	NOUN
ejpam-6565	125	45	)	)	PUNCT
ejpam-6565	125	46	∩	∩	NOUN
ejpam-6565	125	47	v	v	ADP
ejpam-6565	125	48	̸=	̸=	PROPN
ejpam-6565	125	49	∅	∅	NOUN
ejpam-6565	125	50	,	,	PUNCT
ejpam-6565	125	51	there	there	PRON
ejpam-6565	125	52	exists	exist	VERB
ejpam-6565	125	53	a	a	DET
ejpam-6565	125	54	⋆-open	⋆-open	ADJ
ejpam-6565	125	55	set	set	NOUN
ejpam-6565	125	56	u	u	NOUN
ejpam-6565	125	57	of	of	ADP
ejpam-6565	125	58	x	x	PUNCT
ejpam-6565	125	59	containing	contain	VERB
ejpam-6565	125	60	x	x	PUNCT
ejpam-6565	125	61	such	such	ADJ
ejpam-6565	125	62	that	that	SCONJ
ejpam-6565	125	63	f	f	PROPN
ejpam-6565	125	64	(	(	PUNCT
ejpam-6565	125	65	z	z	NOUN
ejpam-6565	125	66	)	)	PUNCT
ejpam-6565	125	67	∩	∩	NOUN
ejpam-6565	125	68	v	v	ADP
ejpam-6565	125	69	̸=	̸=	PROPN
ejpam-6565	125	70	∅	∅	NOUN
ejpam-6565	125	71	for	for	ADP
ejpam-6565	125	72	every	every	DET
ejpam-6565	125	73	z	z	NOUN
ejpam-6565	125	74	∈	∈	PROPN
ejpam-6565	125	75	u	u	NOUN
ejpam-6565	125	76	.	.	PUNCT
ejpam-6565	126	1	a	a	DET
ejpam-6565	126	2	multifunction	multifunction	NOUN
ejpam-6565	126	3	f	f	NOUN
ejpam-6565	126	4	:	:	PUNCT
ejpam-6565	126	5	(	(	PUNCT
ejpam-6565	126	6	x	x	X
ejpam-6565	126	7	,	,	PUNCT
ejpam-6565	126	8	τ	τ	PROPN
ejpam-6565	126	9	,	,	PUNCT
ejpam-6565	126	10	i	i	NOUN
ejpam-6565	126	11	)	)	PUNCT
ejpam-6565	126	12	→	→	PUNCT
ejpam-6565	126	13	(	(	PUNCT
ejpam-6565	126	14	y	y	PROPN
ejpam-6565	126	15	,	,	PUNCT
ejpam-6565	126	16	σ1	σ1	PROPN
ejpam-6565	126	17	,	,	PUNCT
ejpam-6565	126	18	σ2	σ2	PROPN
ejpam-6565	126	19	)	)	PUNCT
ejpam-6565	126	20	is	be	AUX
ejpam-6565	126	21	called	call	VERB
ejpam-6565	126	22	lower	low	ADJ
ejpam-6565	126	23	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	126	24	,	,	PUNCT
ejpam-6565	126	25	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	126	26	if	if	SCONJ
ejpam-6565	126	27	f	f	PROPN
ejpam-6565	126	28	is	be	AUX
ejpam-6565	126	29	lower	low	ADJ
ejpam-6565	126	30	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	126	31	,	,	PUNCT
ejpam-6565	126	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	126	33	at	at	ADP
ejpam-6565	126	34	each	each	DET
ejpam-6565	126	35	point	point	NOUN
ejpam-6565	126	36	x	x	PUNCT
ejpam-6565	126	37	of	of	ADP
ejpam-6565	126	38	x.	x.	PROPN
ejpam-6565	126	39	theorem	theorem	VERB
ejpam-6565	126	40	2	2	NUM
ejpam-6565	126	41	.	.	X
ejpam-6565	126	42	for	for	ADP
ejpam-6565	126	43	a	a	DET
ejpam-6565	126	44	multifunction	multifunction	NOUN
ejpam-6565	126	45	f	f	NOUN
ejpam-6565	126	46	:	:	PUNCT
ejpam-6565	126	47	(	(	PUNCT
ejpam-6565	126	48	x	x	X
ejpam-6565	126	49	,	,	PUNCT
ejpam-6565	126	50	τ	τ	PROPN
ejpam-6565	126	51	,	,	PUNCT
ejpam-6565	126	52	i	i	NOUN
ejpam-6565	126	53	)	)	PUNCT
ejpam-6565	127	1	→	→	PUNCT
ejpam-6565	127	2	(	(	PUNCT
ejpam-6565	127	3	y	y	PROPN
ejpam-6565	127	4	,	,	PUNCT
ejpam-6565	127	5	σ1	σ1	PROPN
ejpam-6565	127	6	,	,	PUNCT
ejpam-6565	127	7	σ2	σ2	NOUN
ejpam-6565	127	8	)	)	PUNCT
ejpam-6565	127	9	,	,	PUNCT
ejpam-6565	127	10	the	the	DET
ejpam-6565	127	11	following	follow	VERB
ejpam-6565	127	12	properties	property	NOUN
ejpam-6565	127	13	are	be	AUX
ejpam-6565	127	14	equivalent	equivalent	ADJ
ejpam-6565	127	15	:	:	PUNCT
ejpam-6565	127	16	(	(	PUNCT
ejpam-6565	127	17	1	1	X
ejpam-6565	127	18	)	)	PUNCT
ejpam-6565	127	19	f	f	PROPN
ejpam-6565	127	20	is	be	AUX
ejpam-6565	127	21	lower	low	ADJ
ejpam-6565	127	22	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	127	23	,	,	PUNCT
ejpam-6565	127	24	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	127	25	;	;	PUNCT
ejpam-6565	127	26	(	(	PUNCT
ejpam-6565	127	27	2	2	X
ejpam-6565	127	28	)	)	PUNCT
ejpam-6565	127	29	f−(v	f−(v	NOUN
ejpam-6565	127	30	)	)	PUNCT
ejpam-6565	127	31	is	be	AUX
ejpam-6565	127	32	⋆-open	⋆-open	ADJ
ejpam-6565	127	33	in	in	ADP
ejpam-6565	127	34	x	x	PUNCT
ejpam-6565	127	35	for	for	ADP
ejpam-6565	127	36	every	every	DET
ejpam-6565	127	37	σ1σ2	σ1σ2	NOUN
ejpam-6565	127	38	-	-	ADJ
ejpam-6565	127	39	open	open	ADJ
ejpam-6565	127	40	set	set	NOUN
ejpam-6565	127	41	v	v	NOUN
ejpam-6565	127	42	of	of	ADP
ejpam-6565	127	43	y	y	PROPN
ejpam-6565	127	44	;	;	PUNCT
ejpam-6565	127	45	j.	j.	PROPN
ejpam-6565	127	46	khampakdee	khampakdee	PROPN
ejpam-6565	127	47	,	,	PUNCT
ejpam-6565	127	48	a.	a.	PROPN
ejpam-6565	127	49	sama	sama	PROPN
ejpam-6565	127	50	-	-	PUNCT
ejpam-6565	127	51	ae	ae	PROPN
ejpam-6565	127	52	,	,	PUNCT
ejpam-6565	127	53	c.	c.	PROPN
ejpam-6565	127	54	boonpok	boonpok	PROPN
ejpam-6565	127	55	/	/	SYM
ejpam-6565	127	56	eur	eur	PROPN
ejpam-6565	127	57	.	.	PUNCT
ejpam-6565	128	1	j.	j.	PROPN
ejpam-6565	128	2	pure	pure	PROPN
ejpam-6565	128	3	appl	appl	PROPN
ejpam-6565	128	4	.	.	PROPN
ejpam-6565	128	5	math	math	PROPN
ejpam-6565	128	6	,	,	PUNCT
ejpam-6565	128	7	18	18	NUM
ejpam-6565	128	8	(	(	PUNCT
ejpam-6565	128	9	3	3	NUM
ejpam-6565	128	10	)	)	PUNCT
ejpam-6565	128	11	(	(	PUNCT
ejpam-6565	128	12	2025	2025	NUM
ejpam-6565	128	13	)	)	PUNCT
ejpam-6565	128	14	,	,	PUNCT
ejpam-6565	128	15	6565	6565	NUM
ejpam-6565	128	16	5	5	NUM
ejpam-6565	128	17	of	of	ADP
ejpam-6565	128	18	9	9	NUM
ejpam-6565	128	19	(	(	PUNCT
ejpam-6565	128	20	3	3	NUM
ejpam-6565	128	21	)	)	PUNCT
ejpam-6565	128	22	f+(k	f+(k	NUM
ejpam-6565	128	23	)	)	PUNCT
ejpam-6565	128	24	is	be	AUX
ejpam-6565	128	25	⋆-closed	⋆-close	VERB
ejpam-6565	128	26	in	in	ADP
ejpam-6565	128	27	x	x	PUNCT
ejpam-6565	128	28	for	for	ADP
ejpam-6565	128	29	every	every	DET
ejpam-6565	128	30	σ1σ2	σ1σ2	NUM
ejpam-6565	128	31	-	-	PUNCT
ejpam-6565	128	32	closed	closed	ADJ
ejpam-6565	128	33	set	set	NOUN
ejpam-6565	128	34	k	k	PROPN
ejpam-6565	128	35	of	of	ADP
ejpam-6565	128	36	y	y	PROPN
ejpam-6565	128	37	;	;	PUNCT
ejpam-6565	128	38	(	(	PUNCT
ejpam-6565	128	39	4	4	X
ejpam-6565	128	40	)	)	PUNCT
ejpam-6565	128	41	cl⋆(f+(b	cl⋆(f+(b	NOUN
ejpam-6565	128	42	)	)	PUNCT
ejpam-6565	128	43	)	)	PUNCT
ejpam-6565	129	1	⊆	⊆	NUM
ejpam-6565	129	2	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6565	129	3	-	-	PUNCT
ejpam-6565	129	4	cl(b	cl(b	NOUN
ejpam-6565	129	5	)	)	PUNCT
ejpam-6565	129	6	)	)	PUNCT
ejpam-6565	129	7	for	for	ADP
ejpam-6565	129	8	every	every	DET
ejpam-6565	129	9	subset	subset	NOUN
ejpam-6565	129	10	b	b	PROPN
ejpam-6565	129	11	of	of	ADP
ejpam-6565	129	12	y	y	PROPN
ejpam-6565	129	13	;	;	PUNCT
ejpam-6565	129	14	(	(	PUNCT
ejpam-6565	129	15	5	5	X
ejpam-6565	129	16	)	)	PUNCT
ejpam-6565	129	17	f	f	NOUN
ejpam-6565	129	18	(	(	PUNCT
ejpam-6565	129	19	cl⋆(a	cl⋆(a	PROPN
ejpam-6565	129	20	)	)	PUNCT
ejpam-6565	129	21	)	)	PUNCT
ejpam-6565	129	22	⊆	⊆	X
ejpam-6565	129	23	σ1σ2	σ1σ2	X
ejpam-6565	129	24	-	-	NUM
ejpam-6565	129	25	cl(f	cl(f	NOUN
ejpam-6565	129	26	(	(	PUNCT
ejpam-6565	129	27	a	a	NOUN
ejpam-6565	129	28	)	)	PUNCT
ejpam-6565	129	29	)	)	PUNCT
ejpam-6565	129	30	for	for	ADP
ejpam-6565	129	31	every	every	DET
ejpam-6565	129	32	subset	subset	NOUN
ejpam-6565	129	33	a	a	PRON
ejpam-6565	129	34	of	of	ADP
ejpam-6565	129	35	x	x	PRON
ejpam-6565	129	36	;	;	PUNCT
ejpam-6565	129	37	(	(	PUNCT
ejpam-6565	129	38	6	6	X
ejpam-6565	129	39	)	)	PUNCT
ejpam-6565	129	40	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6565	129	41	-	-	PUNCT
ejpam-6565	129	42	int(b	int(b	NOUN
ejpam-6565	129	43	)	)	PUNCT
ejpam-6565	129	44	)	)	PUNCT
ejpam-6565	129	45	⊆	⊆	NUM
ejpam-6565	129	46	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6565	129	47	)	)	PUNCT
ejpam-6565	129	48	)	)	PUNCT
ejpam-6565	129	49	for	for	ADP
ejpam-6565	129	50	every	every	DET
ejpam-6565	129	51	subset	subset	NOUN
ejpam-6565	129	52	b	b	PROPN
ejpam-6565	129	53	of	of	ADP
ejpam-6565	129	54	y	y	PROPN
ejpam-6565	129	55	.	.	PUNCT
ejpam-6565	130	1	proof	proof	NOUN
ejpam-6565	130	2	.	.	PUNCT
ejpam-6565	131	1	we	we	PRON
ejpam-6565	131	2	prove	prove	VERB
ejpam-6565	131	3	only	only	ADV
ejpam-6565	131	4	the	the	DET
ejpam-6565	131	5	implications	implication	NOUN
ejpam-6565	131	6	(	(	PUNCT
ejpam-6565	131	7	4	4	X
ejpam-6565	131	8	)	)	PUNCT
ejpam-6565	131	9	⇒	⇒	NOUN
ejpam-6565	131	10	(	(	PUNCT
ejpam-6565	131	11	5	5	NUM
ejpam-6565	131	12	)	)	PUNCT
ejpam-6565	131	13	and	and	CCONJ
ejpam-6565	131	14	(	(	PUNCT
ejpam-6565	131	15	5	5	X
ejpam-6565	131	16	)	)	PUNCT
ejpam-6565	131	17	⇒	⇒	NOUN
ejpam-6565	131	18	(	(	PUNCT
ejpam-6565	131	19	6	6	X
ejpam-6565	131	20	)	)	PUNCT
ejpam-6565	131	21	being	be	AUX
ejpam-6565	131	22	the	the	DET
ejpam-6565	131	23	proofs	proof	NOUN
ejpam-6565	131	24	of	of	ADP
ejpam-6565	131	25	the	the	DET
ejpam-6565	131	26	other	other	ADJ
ejpam-6565	131	27	similar	similar	ADJ
ejpam-6565	131	28	to	to	ADP
ejpam-6565	131	29	those	those	PRON
ejpam-6565	131	30	of	of	ADP
ejpam-6565	131	31	theorem	theorem	NOUN
ejpam-6565	131	32	1	1	NUM
ejpam-6565	131	33	.	.	PUNCT
ejpam-6565	131	34	(	(	PUNCT
ejpam-6565	131	35	4	4	X
ejpam-6565	131	36	)	)	PUNCT
ejpam-6565	131	37	⇒	⇒	NOUN
ejpam-6565	131	38	(	(	PUNCT
ejpam-6565	131	39	5	5	NUM
ejpam-6565	131	40	):	):	PUNCT
ejpam-6565	131	41	let	let	VERB
ejpam-6565	131	42	a	a	PRON
ejpam-6565	131	43	be	be	AUX
ejpam-6565	131	44	any	any	DET
ejpam-6565	131	45	subset	subset	ADJ
ejpam-6565	131	46	ofx	ofx	NOUN
ejpam-6565	131	47	.	.	PUNCT
ejpam-6565	132	1	then	then	ADV
ejpam-6565	132	2	by	by	ADP
ejpam-6565	132	3	(	(	PUNCT
ejpam-6565	132	4	4	4	NUM
ejpam-6565	132	5	)	)	PUNCT
ejpam-6565	132	6	,	,	PUNCT
ejpam-6565	132	7	we	we	PRON
ejpam-6565	132	8	have	have	VERB
ejpam-6565	132	9	cl⋆(a	cl⋆(a	NOUN
ejpam-6565	132	10	)	)	PUNCT
ejpam-6565	132	11	⊆	⊆	NUM
ejpam-6565	132	12	cl⋆(f+(f	cl⋆(f+(f	NOUN
ejpam-6565	132	13	(	(	PUNCT
ejpam-6565	132	14	a	a	NOUN
ejpam-6565	132	15	)	)	PUNCT
ejpam-6565	132	16	)	)	PUNCT
ejpam-6565	132	17	)	)	PUNCT
ejpam-6565	133	1	⊆	⊆	NUM
ejpam-6565	133	2	f+(cl⋆(f	f+(cl⋆(f	X
ejpam-6565	133	3	(	(	PUNCT
ejpam-6565	133	4	a	a	NOUN
ejpam-6565	133	5	)	)	PUNCT
ejpam-6565	133	6	)	)	PUNCT
ejpam-6565	133	7	)	)	PUNCT
ejpam-6565	133	8	and	and	CCONJ
ejpam-6565	133	9	so	so	ADV
ejpam-6565	133	10	f	f	PROPN
ejpam-6565	133	11	(	(	PUNCT
ejpam-6565	133	12	cl⋆(a	cl⋆(a	PROPN
ejpam-6565	133	13	)	)	PUNCT
ejpam-6565	133	14	)	)	PUNCT
ejpam-6565	133	15	⊆	⊆	X
ejpam-6565	133	16	σ1σ2	σ1σ2	X
ejpam-6565	133	17	-	-	NUM
ejpam-6565	133	18	cl(f	cl(f	NOUN
ejpam-6565	133	19	(	(	PUNCT
ejpam-6565	133	20	a	a	NOUN
ejpam-6565	133	21	)	)	PUNCT
ejpam-6565	133	22	)	)	PUNCT
ejpam-6565	133	23	.	.	PUNCT
ejpam-6565	134	1	(	(	PUNCT
ejpam-6565	134	2	5	5	X
ejpam-6565	134	3	)	)	PUNCT
ejpam-6565	134	4	⇒	⇒	NOUN
ejpam-6565	134	5	(	(	PUNCT
ejpam-6565	134	6	6	6	NUM
ejpam-6565	134	7	):	):	PUNCT
ejpam-6565	134	8	let	let	VERB
ejpam-6565	134	9	b	b	X
ejpam-6565	134	10	be	be	AUX
ejpam-6565	134	11	any	any	DET
ejpam-6565	134	12	subset	subset	NOUN
ejpam-6565	134	13	of	of	ADP
ejpam-6565	134	14	y	y	PROPN
ejpam-6565	134	15	.	.	PUNCT
ejpam-6565	135	1	by	by	ADP
ejpam-6565	135	2	(	(	PUNCT
ejpam-6565	135	3	5	5	NUM
ejpam-6565	135	4	)	)	PUNCT
ejpam-6565	135	5	,	,	PUNCT
ejpam-6565	135	6	f	f	PROPN
ejpam-6565	135	7	(	(	PUNCT
ejpam-6565	135	8	cl⋆(f+(y	cl⋆(f+(y	VERB
ejpam-6565	135	9	−b	−b	NOUN
ejpam-6565	135	10	)	)	PUNCT
ejpam-6565	135	11	)	)	PUNCT
ejpam-6565	135	12	)	)	PUNCT
ejpam-6565	136	1	⊆	⊆	X
ejpam-6565	136	2	σ1σ2	σ1σ2	X
ejpam-6565	136	3	-	-	NUM
ejpam-6565	136	4	cl(f	cl(f	NOUN
ejpam-6565	136	5	(	(	PUNCT
ejpam-6565	136	6	f+(y	f+(y	PROPN
ejpam-6565	136	7	−b	−b	PROPN
ejpam-6565	136	8	)	)	PUNCT
ejpam-6565	136	9	)	)	PUNCT
ejpam-6565	136	10	)	)	PUNCT
ejpam-6565	137	1	⊆	⊆	X
ejpam-6565	137	2	σ1σ2	σ1σ2	NUM
ejpam-6565	137	3	-	-	PUNCT
ejpam-6565	137	4	cl(y	cl(y	NOUN
ejpam-6565	137	5	−b	−b	NOUN
ejpam-6565	137	6	)	)	PUNCT
ejpam-6565	138	1	=	=	SYM
ejpam-6565	138	2	y	y	PROPN
ejpam-6565	138	3	−	−	ADP
ejpam-6565	138	4	σ1σ2	σ1σ2	X
ejpam-6565	138	5	-	-	PUNCT
ejpam-6565	138	6	int(b	int(b	NOUN
ejpam-6565	138	7	)	)	PUNCT
ejpam-6565	138	8	.	.	PUNCT
ejpam-6565	139	1	since	since	SCONJ
ejpam-6565	139	2	f	f	PROPN
ejpam-6565	139	3	(	(	PUNCT
ejpam-6565	139	4	cl⋆(f+(y	cl⋆(f+(y	VERB
ejpam-6565	139	5	−b	−b	NOUN
ejpam-6565	139	6	)	)	PUNCT
ejpam-6565	139	7	)	)	PUNCT
ejpam-6565	139	8	)	)	PUNCT
ejpam-6565	140	1	=	=	SYM
ejpam-6565	140	2	f	f	X
ejpam-6565	140	3	(	(	PUNCT
ejpam-6565	140	4	cl⋆(x	cl⋆(x	NOUN
ejpam-6565	140	5	−	−	PROPN
ejpam-6565	140	6	f−(b	f−(b	PROPN
ejpam-6565	140	7	)	)	PUNCT
ejpam-6565	140	8	)	)	PUNCT
ejpam-6565	140	9	)	)	PUNCT
ejpam-6565	141	1	=	=	SYM
ejpam-6565	141	2	f	f	X
ejpam-6565	141	3	(	(	PUNCT
ejpam-6565	141	4	x	x	X
ejpam-6565	141	5	−	−	PROPN
ejpam-6565	141	6	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6565	141	7	)	)	PUNCT
ejpam-6565	141	8	)	)	PUNCT
ejpam-6565	141	9	)	)	PUNCT
ejpam-6565	141	10	,	,	PUNCT
ejpam-6565	141	11	we	we	PRON
ejpam-6565	141	12	have	have	VERB
ejpam-6565	141	13	x	x	X
ejpam-6565	141	14	−	−	NOUN
ejpam-6565	141	15	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6565	141	16	)	)	PUNCT
ejpam-6565	141	17	)	)	PUNCT
ejpam-6565	142	1	⊆	⊆	NUM
ejpam-6565	142	2	f+(y	f+(y	ADP
ejpam-6565	142	3	−	−	NUM
ejpam-6565	142	4	σ1σ2	σ1σ2	SYM
ejpam-6565	142	5	-	-	PUNCT
ejpam-6565	142	6	int(b	int(b	NOUN
ejpam-6565	142	7	)	)	PUNCT
ejpam-6565	142	8	)	)	PUNCT
ejpam-6565	143	1	=	=	PUNCT
ejpam-6565	143	2	x	x	X
ejpam-6565	143	3	−	−	NOUN
ejpam-6565	143	4	f−(σ1σ2	f−(σ1σ2	NOUN
ejpam-6565	143	5	-	-	PUNCT
ejpam-6565	143	6	int(b	int(b	NOUN
ejpam-6565	143	7	)	)	PUNCT
ejpam-6565	143	8	)	)	PUNCT
ejpam-6565	143	9	and	and	CCONJ
ejpam-6565	143	10	hence	hence	ADV
ejpam-6565	143	11	f−(σ1σ2	f−(σ1σ2	ADV
ejpam-6565	143	12	-	-	PUNCT
ejpam-6565	143	13	int(b	int(b	NOUN
ejpam-6565	143	14	)	)	PUNCT
ejpam-6565	143	15	)	)	PUNCT
ejpam-6565	144	1	⊆	⊆	NUM
ejpam-6565	144	2	int⋆(f−(b	int⋆(f−(b	NOUN
ejpam-6565	144	3	)	)	PUNCT
ejpam-6565	144	4	)	)	PUNCT
ejpam-6565	144	5	.	.	PUNCT
ejpam-6565	145	1	definition	definition	NOUN
ejpam-6565	145	2	3	3	NUM
ejpam-6565	145	3	.	.	PUNCT
ejpam-6565	146	1	a	a	DET
ejpam-6565	146	2	function	function	NOUN
ejpam-6565	146	3	f	f	NOUN
ejpam-6565	146	4	:	:	PUNCT
ejpam-6565	146	5	(	(	PUNCT
ejpam-6565	146	6	x	x	X
ejpam-6565	146	7	,	,	PUNCT
ejpam-6565	146	8	τ	τ	PROPN
ejpam-6565	146	9	,	,	PUNCT
ejpam-6565	146	10	i	i	NOUN
ejpam-6565	146	11	)	)	PUNCT
ejpam-6565	146	12	→	→	PUNCT
ejpam-6565	146	13	(	(	PUNCT
ejpam-6565	146	14	y	y	PROPN
ejpam-6565	146	15	,	,	PUNCT
ejpam-6565	146	16	σ1	σ1	PROPN
ejpam-6565	146	17	,	,	PUNCT
ejpam-6565	146	18	σ2	σ2	PROPN
ejpam-6565	146	19	)	)	PUNCT
ejpam-6565	146	20	is	be	AUX
ejpam-6565	146	21	said	say	VERB
ejpam-6565	146	22	to	to	PART
ejpam-6565	146	23	be	be	AUX
ejpam-6565	146	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	146	25	,	,	PUNCT
ejpam-6565	146	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	146	27	at	at	ADP
ejpam-6565	146	28	a	a	DET
ejpam-6565	146	29	point	point	NOUN
ejpam-6565	146	30	x	x	SYM
ejpam-6565	146	31	∈	∈	NOUN
ejpam-6565	146	32	x	x	PUNCT
ejpam-6565	146	33	if	if	SCONJ
ejpam-6565	146	34	for	for	ADP
ejpam-6565	146	35	each	each	DET
ejpam-6565	146	36	σ1σ2	σ1σ2	VERB
ejpam-6565	146	37	-	-	ADJ
ejpam-6565	146	38	open	open	ADJ
ejpam-6565	146	39	set	set	NOUN
ejpam-6565	146	40	v	v	NOUN
ejpam-6565	146	41	of	of	ADP
ejpam-6565	146	42	y	y	NOUN
ejpam-6565	146	43	containing	contain	VERB
ejpam-6565	146	44	f(x	f(x	PROPN
ejpam-6565	146	45	)	)	PUNCT
ejpam-6565	146	46	,	,	PUNCT
ejpam-6565	146	47	there	there	PRON
ejpam-6565	146	48	exists	exist	VERB
ejpam-6565	146	49	a	a	DET
ejpam-6565	146	50	⋆-open	⋆-open	ADJ
ejpam-6565	146	51	set	set	NOUN
ejpam-6565	146	52	u	u	NOUN
ejpam-6565	146	53	of	of	ADP
ejpam-6565	146	54	x	x	PUNCT
ejpam-6565	146	55	containing	contain	VERB
ejpam-6565	146	56	x	x	PUNCT
ejpam-6565	146	57	such	such	ADJ
ejpam-6565	146	58	that	that	DET
ejpam-6565	146	59	f(u	f(u	PROPN
ejpam-6565	146	60	)	)	PUNCT
ejpam-6565	146	61	⊆	⊆	NUM
ejpam-6565	146	62	v	v	NOUN
ejpam-6565	146	63	.	.	PUNCT
ejpam-6565	147	1	a	a	DET
ejpam-6565	147	2	function	function	NOUN
ejpam-6565	147	3	f	f	NOUN
ejpam-6565	147	4	:	:	PUNCT
ejpam-6565	147	5	(	(	PUNCT
ejpam-6565	147	6	x	x	X
ejpam-6565	147	7	,	,	PUNCT
ejpam-6565	147	8	τ	τ	PROPN
ejpam-6565	147	9	,	,	PUNCT
ejpam-6565	147	10	i	i	NOUN
ejpam-6565	147	11	)	)	PUNCT
ejpam-6565	147	12	→	→	PUNCT
ejpam-6565	147	13	(	(	PUNCT
ejpam-6565	147	14	y	y	PROPN
ejpam-6565	147	15	,	,	PUNCT
ejpam-6565	147	16	σ1	σ1	PROPN
ejpam-6565	147	17	,	,	PUNCT
ejpam-6565	147	18	σ2	σ2	PROPN
ejpam-6565	147	19	)	)	PUNCT
ejpam-6565	147	20	is	be	AUX
ejpam-6565	147	21	said	say	VERB
ejpam-6565	147	22	to	to	PART
ejpam-6565	147	23	be	be	AUX
ejpam-6565	147	24	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	147	25	,	,	PUNCT
ejpam-6565	147	26	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	147	27	if	if	SCONJ
ejpam-6565	147	28	f	f	PROPN
ejpam-6565	147	29	is	be	AUX
ejpam-6565	147	30	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	147	31	,	,	PUNCT
ejpam-6565	147	32	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	147	33	at	at	ADP
ejpam-6565	147	34	each	each	DET
ejpam-6565	147	35	point	point	NOUN
ejpam-6565	147	36	x	x	PUNCT
ejpam-6565	147	37	of	of	ADP
ejpam-6565	147	38	x.	x.	PROPN
ejpam-6565	147	39	corollary	corollary	NOUN
ejpam-6565	147	40	1	1	NUM
ejpam-6565	147	41	.	.	PUNCT
ejpam-6565	148	1	for	for	ADP
ejpam-6565	148	2	a	a	DET
ejpam-6565	148	3	function	function	NOUN
ejpam-6565	148	4	f	f	NOUN
ejpam-6565	148	5	:	:	PUNCT
ejpam-6565	148	6	(	(	PUNCT
ejpam-6565	148	7	x	x	X
ejpam-6565	148	8	,	,	PUNCT
ejpam-6565	148	9	τ	τ	PROPN
ejpam-6565	148	10	,	,	PUNCT
ejpam-6565	148	11	i	i	NOUN
ejpam-6565	148	12	)	)	PUNCT
ejpam-6565	148	13	→	→	PUNCT
ejpam-6565	148	14	(	(	PUNCT
ejpam-6565	148	15	y	y	PROPN
ejpam-6565	148	16	,	,	PUNCT
ejpam-6565	148	17	σ1	σ1	PROPN
ejpam-6565	148	18	,	,	PUNCT
ejpam-6565	148	19	σ2	σ2	NOUN
ejpam-6565	148	20	)	)	PUNCT
ejpam-6565	148	21	,	,	PUNCT
ejpam-6565	148	22	the	the	DET
ejpam-6565	148	23	following	follow	VERB
ejpam-6565	148	24	properties	property	NOUN
ejpam-6565	148	25	are	be	AUX
ejpam-6565	148	26	equivalent	equivalent	ADJ
ejpam-6565	148	27	:	:	PUNCT
ejpam-6565	148	28	(	(	PUNCT
ejpam-6565	148	29	1	1	X
ejpam-6565	148	30	)	)	PUNCT
ejpam-6565	148	31	f	f	PROPN
ejpam-6565	148	32	is	be	AUX
ejpam-6565	148	33	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	148	34	,	,	PUNCT
ejpam-6565	148	35	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	148	36	;	;	PUNCT
ejpam-6565	148	37	(	(	PUNCT
ejpam-6565	148	38	2	2	X
ejpam-6565	148	39	)	)	PUNCT
ejpam-6565	148	40	f−1(v	f−1(v	NOUN
ejpam-6565	148	41	)	)	PUNCT
ejpam-6565	148	42	is	be	AUX
ejpam-6565	148	43	⋆-open	⋆-open	ADJ
ejpam-6565	148	44	in	in	ADP
ejpam-6565	148	45	x	x	PUNCT
ejpam-6565	148	46	for	for	ADP
ejpam-6565	148	47	every	every	DET
ejpam-6565	148	48	σ1σ2	σ1σ2	NOUN
ejpam-6565	148	49	-	-	ADJ
ejpam-6565	148	50	open	open	ADJ
ejpam-6565	148	51	set	set	NOUN
ejpam-6565	148	52	v	v	NOUN
ejpam-6565	148	53	of	of	ADP
ejpam-6565	148	54	y	y	PROPN
ejpam-6565	148	55	;	;	PUNCT
ejpam-6565	149	1	(	(	PUNCT
ejpam-6565	149	2	3	3	X
ejpam-6565	149	3	)	)	PUNCT
ejpam-6565	149	4	f−1(k	f−1(k	PROPN
ejpam-6565	149	5	)	)	PUNCT
ejpam-6565	149	6	is	be	AUX
ejpam-6565	149	7	⋆-closed	⋆-close	VERB
ejpam-6565	149	8	in	in	ADP
ejpam-6565	149	9	x	x	PUNCT
ejpam-6565	149	10	for	for	ADP
ejpam-6565	149	11	every	every	DET
ejpam-6565	149	12	σ1σ2	σ1σ2	NUM
ejpam-6565	149	13	-	-	PUNCT
ejpam-6565	149	14	closed	closed	ADJ
ejpam-6565	149	15	set	set	NOUN
ejpam-6565	149	16	k	k	PROPN
ejpam-6565	149	17	of	of	ADP
ejpam-6565	149	18	y	y	PROPN
ejpam-6565	149	19	;	;	PUNCT
ejpam-6565	149	20	(	(	PUNCT
ejpam-6565	149	21	4	4	X
ejpam-6565	149	22	)	)	PUNCT
ejpam-6565	149	23	cl⋆(f−1(b	cl⋆(f−1(b	NOUN
ejpam-6565	149	24	)	)	PUNCT
ejpam-6565	149	25	)	)	PUNCT
ejpam-6565	150	1	⊆	⊆	NUM
ejpam-6565	150	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6565	150	3	-	-	PUNCT
ejpam-6565	150	4	cl(b	cl(b	NOUN
ejpam-6565	150	5	)	)	PUNCT
ejpam-6565	150	6	)	)	PUNCT
ejpam-6565	150	7	for	for	ADP
ejpam-6565	150	8	every	every	DET
ejpam-6565	150	9	subset	subset	NOUN
ejpam-6565	150	10	b	b	PROPN
ejpam-6565	150	11	of	of	ADP
ejpam-6565	150	12	y	y	PROPN
ejpam-6565	150	13	;	;	PUNCT
ejpam-6565	150	14	(	(	PUNCT
ejpam-6565	150	15	5	5	X
ejpam-6565	150	16	)	)	PUNCT
ejpam-6565	150	17	f(cl⋆(a	f(cl⋆(a	NOUN
ejpam-6565	150	18	)	)	PUNCT
ejpam-6565	150	19	)	)	PUNCT
ejpam-6565	150	20	⊆	⊆	X
ejpam-6565	150	21	σ1σ2	σ1σ2	NUM
ejpam-6565	150	22	-	-	PUNCT
ejpam-6565	150	23	cl(f(a	cl(f(a	NOUN
ejpam-6565	150	24	)	)	PUNCT
ejpam-6565	150	25	)	)	PUNCT
ejpam-6565	150	26	for	for	ADP
ejpam-6565	150	27	every	every	DET
ejpam-6565	150	28	subset	subset	NOUN
ejpam-6565	150	29	a	a	PRON
ejpam-6565	150	30	of	of	ADP
ejpam-6565	150	31	x	x	PRON
ejpam-6565	150	32	;	;	PUNCT
ejpam-6565	150	33	(	(	PUNCT
ejpam-6565	150	34	6	6	X
ejpam-6565	150	35	)	)	PUNCT
ejpam-6565	150	36	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6565	150	37	-	-	PUNCT
ejpam-6565	150	38	int(b	int(b	NOUN
ejpam-6565	150	39	)	)	PUNCT
ejpam-6565	150	40	)	)	PUNCT
ejpam-6565	150	41	⊆	⊆	NUM
ejpam-6565	150	42	int⋆(f−1(b	int⋆(f−1(b	NUM
ejpam-6565	150	43	)	)	PUNCT
ejpam-6565	150	44	)	)	PUNCT
ejpam-6565	150	45	for	for	ADP
ejpam-6565	150	46	every	every	DET
ejpam-6565	150	47	subset	subset	NOUN
ejpam-6565	150	48	b	b	PROPN
ejpam-6565	150	49	of	of	ADP
ejpam-6565	150	50	y	y	PROPN
ejpam-6565	150	51	.	.	PUNCT
ejpam-6565	151	1	definition	definition	NOUN
ejpam-6565	151	2	4	4	NUM
ejpam-6565	151	3	.	.	PUNCT
ejpam-6565	152	1	[	[	X
ejpam-6565	152	2	9	9	NUM
ejpam-6565	152	3	]	]	PUNCT
ejpam-6565	152	4	a	a	DET
ejpam-6565	152	5	bitopological	bitopological	ADJ
ejpam-6565	152	6	space	space	NOUN
ejpam-6565	152	7	(	(	PUNCT
ejpam-6565	152	8	x	x	NOUN
ejpam-6565	152	9	,	,	PUNCT
ejpam-6565	152	10	τ1	τ1	NOUN
ejpam-6565	152	11	,	,	PUNCT
ejpam-6565	152	12	τ2	τ2	NOUN
ejpam-6565	152	13	)	)	PUNCT
ejpam-6565	152	14	is	be	AUX
ejpam-6565	152	15	said	say	VERB
ejpam-6565	152	16	to	to	PART
ejpam-6565	152	17	be	be	AUX
ejpam-6565	152	18	(	(	PUNCT
ejpam-6565	152	19	τ1	τ1	NOUN
ejpam-6565	152	20	,	,	PUNCT
ejpam-6565	152	21	τ2)s	τ2)s	NOUN
ejpam-6565	152	22	-	-	PUNCT
ejpam-6565	152	23	regular	regular	ADJ
ejpam-6565	152	24	if	if	SCONJ
ejpam-6565	152	25	for	for	ADP
ejpam-6565	152	26	each	each	DET
ejpam-6565	152	27	(	(	PUNCT
ejpam-6565	152	28	τ1	τ1	NOUN
ejpam-6565	152	29	,	,	PUNCT
ejpam-6565	152	30	τ2)s	τ2)s	NOUN
ejpam-6565	152	31	-	-	PUNCT
ejpam-6565	152	32	closed	close	VERB
ejpam-6565	152	33	set	set	VERB
ejpam-6565	152	34	f	f	NOUN
ejpam-6565	152	35	and	and	CCONJ
ejpam-6565	152	36	each	each	DET
ejpam-6565	152	37	x	x	PROPN
ejpam-6565	152	38	̸∈	̸∈	PROPN
ejpam-6565	152	39	f	f	PROPN
ejpam-6565	152	40	,	,	PUNCT
ejpam-6565	152	41	there	there	PRON
ejpam-6565	152	42	exist	exist	VERB
ejpam-6565	152	43	disjoint	disjoint	NOUN
ejpam-6565	152	44	(	(	PUNCT
ejpam-6565	152	45	τ1	τ1	NOUN
ejpam-6565	152	46	,	,	PUNCT
ejpam-6565	152	47	τ2)s	τ2)s	NOUN
ejpam-6565	152	48	-	-	PUNCT
ejpam-6565	152	49	open	open	ADJ
ejpam-6565	152	50	sets	set	VERB
ejpam-6565	152	51	u	u	NOUN
ejpam-6565	152	52	and	and	CCONJ
ejpam-6565	152	53	v	v	ADP
ejpam-6565	152	54	such	such	ADJ
ejpam-6565	152	55	that	that	SCONJ
ejpam-6565	152	56	x	x	SYM
ejpam-6565	152	57	∈	∈	PROPN
ejpam-6565	152	58	u	u	NOUN
ejpam-6565	152	59	and	and	CCONJ
ejpam-6565	152	60	f	f	PROPN
ejpam-6565	152	61	⊆	⊆	NUM
ejpam-6565	152	62	v	v	NOUN
ejpam-6565	152	63	.	.	PUNCT
ejpam-6565	153	1	lemma	lemma	PROPN
ejpam-6565	153	2	2	2	NUM
ejpam-6565	153	3	.	.	PUNCT
ejpam-6565	154	1	[	[	X
ejpam-6565	154	2	9	9	NUM
ejpam-6565	154	3	]	]	PUNCT
ejpam-6565	154	4	a	a	DET
ejpam-6565	154	5	bitopological	bitopological	ADJ
ejpam-6565	154	6	space	space	NOUN
ejpam-6565	154	7	(	(	PUNCT
ejpam-6565	154	8	x	x	NOUN
ejpam-6565	154	9	,	,	PUNCT
ejpam-6565	154	10	τ1	τ1	NOUN
ejpam-6565	154	11	,	,	PUNCT
ejpam-6565	154	12	τ2	τ2	NOUN
ejpam-6565	154	13	)	)	PUNCT
ejpam-6565	154	14	is	be	AUX
ejpam-6565	154	15	(	(	PUNCT
ejpam-6565	154	16	τ1	τ1	NOUN
ejpam-6565	154	17	,	,	PUNCT
ejpam-6565	154	18	τ2)s	τ2)s	NOUN
ejpam-6565	154	19	-	-	PUNCT
ejpam-6565	154	20	regular	regular	ADJ
ejpam-6565	154	21	if	if	SCONJ
ejpam-6565	154	22	and	and	CCONJ
ejpam-6565	154	23	only	only	ADV
ejpam-6565	154	24	if	if	SCONJ
ejpam-6565	154	25	for	for	ADP
ejpam-6565	154	26	each	each	DET
ejpam-6565	154	27	x	x	SYM
ejpam-6565	154	28	∈	∈	PROPN
ejpam-6565	154	29	x	x	X
ejpam-6565	154	30	and	and	CCONJ
ejpam-6565	154	31	each	each	DET
ejpam-6565	154	32	(	(	PUNCT
ejpam-6565	154	33	τ1	τ1	NOUN
ejpam-6565	154	34	,	,	PUNCT
ejpam-6565	154	35	τ2)s	τ2)s	NOUN
ejpam-6565	154	36	-	-	PUNCT
ejpam-6565	154	37	open	open	ADJ
ejpam-6565	154	38	set	set	NOUN
ejpam-6565	154	39	u	u	NOUN
ejpam-6565	154	40	containing	contain	VERB
ejpam-6565	154	41	x	x	PRON
ejpam-6565	154	42	,	,	PUNCT
ejpam-6565	154	43	there	there	PRON
ejpam-6565	154	44	exists	exist	VERB
ejpam-6565	154	45	a	a	DET
ejpam-6565	154	46	(	(	PUNCT
ejpam-6565	154	47	τ1	τ1	NOUN
ejpam-6565	154	48	,	,	PUNCT
ejpam-6565	154	49	τ2)s	τ2)s	NOUN
ejpam-6565	154	50	-	-	PUNCT
ejpam-6565	154	51	open	open	NOUN
ejpam-6565	154	52	set	set	VERB
ejpam-6565	154	53	v	v	ADP
ejpam-6565	155	1	such	such	ADJ
ejpam-6565	155	2	that	that	SCONJ
ejpam-6565	155	3	x	x	SYM
ejpam-6565	155	4	∈	∈	NOUN
ejpam-6565	155	5	v	v	ADP
ejpam-6565	155	6	⊆	⊆	NUM
ejpam-6565	155	7	(	(	PUNCT
ejpam-6565	155	8	τ1	τ1	NOUN
ejpam-6565	155	9	,	,	PUNCT
ejpam-6565	155	10	τ2)-scl(v	τ2)-scl(v	NOUN
ejpam-6565	155	11	)	)	PUNCT
ejpam-6565	155	12	⊆	⊆	NUM
ejpam-6565	155	13	u	u	NOUN
ejpam-6565	155	14	.	.	PUNCT
ejpam-6565	155	15	j.	j.	PROPN
ejpam-6565	155	16	khampakdee	khampakdee	PROPN
ejpam-6565	155	17	,	,	PUNCT
ejpam-6565	155	18	a.	a.	PROPN
ejpam-6565	155	19	sama	sama	PROPN
ejpam-6565	155	20	-	-	PUNCT
ejpam-6565	155	21	ae	ae	PROPN
ejpam-6565	155	22	,	,	PUNCT
ejpam-6565	155	23	c.	c.	PROPN
ejpam-6565	155	24	boonpok	boonpok	PROPN
ejpam-6565	155	25	/	/	SYM
ejpam-6565	155	26	eur	eur	PROPN
ejpam-6565	155	27	.	.	PUNCT
ejpam-6565	156	1	j.	j.	PROPN
ejpam-6565	156	2	pure	pure	PROPN
ejpam-6565	156	3	appl	appl	PROPN
ejpam-6565	156	4	.	.	PROPN
ejpam-6565	156	5	math	math	PROPN
ejpam-6565	156	6	,	,	PUNCT
ejpam-6565	156	7	18	18	NUM
ejpam-6565	156	8	(	(	PUNCT
ejpam-6565	156	9	3	3	NUM
ejpam-6565	156	10	)	)	PUNCT
ejpam-6565	156	11	(	(	PUNCT
ejpam-6565	156	12	2025	2025	NUM
ejpam-6565	156	13	)	)	PUNCT
ejpam-6565	156	14	,	,	PUNCT
ejpam-6565	156	15	6565	6565	NUM
ejpam-6565	156	16	6	6	NUM
ejpam-6565	156	17	of	of	ADP
ejpam-6565	156	18	9	9	NUM
ejpam-6565	156	19	lemma	lemma	PROPN
ejpam-6565	156	20	3	3	NUM
ejpam-6565	156	21	.	.	PUNCT
ejpam-6565	157	1	[	[	X
ejpam-6565	157	2	9	9	NUM
ejpam-6565	157	3	]	]	X
ejpam-6565	157	4	let	let	VERB
ejpam-6565	157	5	(	(	PUNCT
ejpam-6565	157	6	x	x	NOUN
ejpam-6565	157	7	,	,	PUNCT
ejpam-6565	157	8	τ1	τ1	NOUN
ejpam-6565	157	9	,	,	PUNCT
ejpam-6565	157	10	τ2	τ2	PROPN
ejpam-6565	157	11	)	)	PUNCT
ejpam-6565	157	12	be	be	VERB
ejpam-6565	157	13	a	a	DET
ejpam-6565	157	14	(	(	PUNCT
ejpam-6565	157	15	τ1	τ1	NOUN
ejpam-6565	157	16	,	,	PUNCT
ejpam-6565	157	17	τ2)s	τ2)s	NOUN
ejpam-6565	157	18	-	-	PUNCT
ejpam-6565	157	19	regular	regular	ADJ
ejpam-6565	157	20	space	space	NOUN
ejpam-6565	157	21	.	.	PUNCT
ejpam-6565	158	1	then	then	ADV
ejpam-6565	158	2	,	,	PUNCT
ejpam-6565	158	3	the	the	DET
ejpam-6565	158	4	following	follow	VERB
ejpam-6565	158	5	properties	property	NOUN
ejpam-6565	158	6	hold	hold	VERB
ejpam-6565	158	7	:	:	PUNCT
ejpam-6565	158	8	(	(	PUNCT
ejpam-6565	158	9	1	1	X
ejpam-6565	158	10	)	)	PUNCT
ejpam-6565	158	11	τ1τ2	τ1τ2	NOUN
ejpam-6565	158	12	-	-	NUM
ejpam-6565	158	13	cl(a	cl(a	NUM
ejpam-6565	158	14	)	)	PUNCT
ejpam-6565	158	15	=	=	PUNCT
ejpam-6565	159	1	τ1τ2	τ1τ2	PROPN
ejpam-6565	159	2	-	-	ADJ
ejpam-6565	159	3	δ	δ	NOUN
ejpam-6565	159	4	-	-	PUNCT
ejpam-6565	159	5	cl(a	cl(a	NUM
ejpam-6565	159	6	)	)	PUNCT
ejpam-6565	159	7	for	for	ADP
ejpam-6565	159	8	every	every	DET
ejpam-6565	159	9	subset	subset	NOUN
ejpam-6565	159	10	a	a	PRON
ejpam-6565	159	11	of	of	ADP
ejpam-6565	159	12	x.	x.	NOUN
ejpam-6565	159	13	(	(	PUNCT
ejpam-6565	159	14	2	2	NUM
ejpam-6565	159	15	)	)	PUNCT
ejpam-6565	159	16	every	every	DET
ejpam-6565	159	17	τ1τ2	τ1τ2	NOUN
ejpam-6565	159	18	-	-	ADJ
ejpam-6565	159	19	open	open	ADJ
ejpam-6565	159	20	set	set	NOUN
ejpam-6565	159	21	is	be	AUX
ejpam-6565	159	22	τ1τ2	τ1τ2	ADJ
ejpam-6565	159	23	-	-	ADJ
ejpam-6565	159	24	δ	δ	NOUN
ejpam-6565	159	25	-	-	ADJ
ejpam-6565	159	26	open	open	ADJ
ejpam-6565	159	27	.	.	PUNCT
ejpam-6565	160	1	theorem	theorem	NOUN
ejpam-6565	160	2	3	3	NUM
ejpam-6565	160	3	.	.	X
ejpam-6565	160	4	for	for	ADP
ejpam-6565	160	5	a	a	DET
ejpam-6565	160	6	multifunction	multifunction	NOUN
ejpam-6565	161	1	f	f	NOUN
ejpam-6565	161	2	:	:	PUNCT
ejpam-6565	161	3	(	(	PUNCT
ejpam-6565	161	4	x	x	X
ejpam-6565	161	5	,	,	PUNCT
ejpam-6565	161	6	τ	τ	PROPN
ejpam-6565	161	7	,	,	PUNCT
ejpam-6565	161	8	i	i	NOUN
ejpam-6565	161	9	)	)	PUNCT
ejpam-6565	161	10	→	→	PUNCT
ejpam-6565	161	11	(	(	PUNCT
ejpam-6565	161	12	y	y	PROPN
ejpam-6565	161	13	,	,	PUNCT
ejpam-6565	161	14	σ1	σ1	PROPN
ejpam-6565	161	15	,	,	PUNCT
ejpam-6565	161	16	σ2	σ2	NOUN
ejpam-6565	161	17	)	)	PUNCT
ejpam-6565	161	18	,	,	PUNCT
ejpam-6565	161	19	where	where	SCONJ
ejpam-6565	161	20	(	(	PUNCT
ejpam-6565	161	21	y	y	PROPN
ejpam-6565	161	22	,	,	PUNCT
ejpam-6565	161	23	σ1	σ1	PROPN
ejpam-6565	161	24	,	,	PUNCT
ejpam-6565	161	25	σ2	σ2	PROPN
ejpam-6565	161	26	)	)	PUNCT
ejpam-6565	161	27	is	be	AUX
ejpam-6565	161	28	(	(	PUNCT
ejpam-6565	161	29	σ1	σ1	PROPN
ejpam-6565	161	30	,	,	PUNCT
ejpam-6565	161	31	σ2)sregular	σ2)sregular	PROPN
ejpam-6565	161	32	,	,	PUNCT
ejpam-6565	161	33	the	the	DET
ejpam-6565	161	34	following	follow	VERB
ejpam-6565	161	35	properties	property	NOUN
ejpam-6565	161	36	are	be	AUX
ejpam-6565	161	37	equivalent	equivalent	ADJ
ejpam-6565	161	38	:	:	PUNCT
ejpam-6565	161	39	(	(	PUNCT
ejpam-6565	161	40	1	1	X
ejpam-6565	161	41	)	)	PUNCT
ejpam-6565	161	42	f	f	PROPN
ejpam-6565	161	43	is	be	AUX
ejpam-6565	161	44	upper	upper	ADJ
ejpam-6565	161	45	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	161	46	,	,	PUNCT
ejpam-6565	161	47	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	161	48	;	;	PUNCT
ejpam-6565	161	49	(	(	PUNCT
ejpam-6565	161	50	2	2	X
ejpam-6565	161	51	)	)	PUNCT
ejpam-6565	161	52	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6565	161	53	-	-	PUNCT
ejpam-6565	161	54	δ	δ	NOUN
ejpam-6565	161	55	-	-	NOUN
ejpam-6565	161	56	cl(b	cl(b	NOUN
ejpam-6565	161	57	)	)	PUNCT
ejpam-6565	161	58	)	)	PUNCT
ejpam-6565	161	59	is	be	AUX
ejpam-6565	161	60	⋆-closed	⋆-close	VERB
ejpam-6565	161	61	in	in	ADP
ejpam-6565	161	62	x	x	PUNCT
ejpam-6565	161	63	for	for	ADP
ejpam-6565	161	64	every	every	DET
ejpam-6565	161	65	subset	subset	NOUN
ejpam-6565	161	66	b	b	PROPN
ejpam-6565	161	67	of	of	ADP
ejpam-6565	161	68	y	y	PROPN
ejpam-6565	161	69	;	;	PUNCT
ejpam-6565	161	70	(	(	PUNCT
ejpam-6565	161	71	3	3	X
ejpam-6565	161	72	)	)	PUNCT
ejpam-6565	161	73	f−(k	f−(k	PROPN
ejpam-6565	161	74	)	)	PUNCT
ejpam-6565	161	75	is	be	AUX
ejpam-6565	161	76	⋆-closed	⋆-close	VERB
ejpam-6565	161	77	in	in	ADP
ejpam-6565	161	78	x	x	PUNCT
ejpam-6565	161	79	for	for	ADP
ejpam-6565	161	80	every	every	DET
ejpam-6565	161	81	σ1σ2	σ1σ2	NUM
ejpam-6565	161	82	-	-	PUNCT
ejpam-6565	161	83	δ	δ	NOUN
ejpam-6565	161	84	-	-	PUNCT
ejpam-6565	161	85	closed	close	VERB
ejpam-6565	161	86	set	set	ADJ
ejpam-6565	161	87	k	k	PROPN
ejpam-6565	161	88	of	of	ADP
ejpam-6565	161	89	y	y	PROPN
ejpam-6565	161	90	;	;	PUNCT
ejpam-6565	161	91	(	(	PUNCT
ejpam-6565	161	92	4	4	X
ejpam-6565	161	93	)	)	PUNCT
ejpam-6565	161	94	f+(v	f+(v	NOUN
ejpam-6565	161	95	)	)	PUNCT
ejpam-6565	162	1	is	be	AUX
ejpam-6565	162	2	⋆-open	⋆-open	ADJ
ejpam-6565	162	3	in	in	ADP
ejpam-6565	162	4	x	x	PUNCT
ejpam-6565	162	5	for	for	SCONJ
ejpam-6565	162	6	every	every	DET
ejpam-6565	162	7	σ1σ2	σ1σ2	NUM
ejpam-6565	162	8	-	-	PUNCT
ejpam-6565	162	9	δ	δ	NOUN
ejpam-6565	162	10	-	-	ADJ
ejpam-6565	162	11	open	open	ADJ
ejpam-6565	162	12	set	set	VERB
ejpam-6565	162	13	v	v	NOUN
ejpam-6565	162	14	of	of	ADP
ejpam-6565	162	15	y	y	PROPN
ejpam-6565	162	16	.	.	PUNCT
ejpam-6565	163	1	proof	proof	NOUN
ejpam-6565	163	2	.	.	PUNCT
ejpam-6565	164	1	(	(	PUNCT
ejpam-6565	164	2	1	1	X
ejpam-6565	164	3	)	)	PUNCT
ejpam-6565	164	4	⇒	⇒	NOUN
ejpam-6565	164	5	(	(	PUNCT
ejpam-6565	164	6	2	2	NUM
ejpam-6565	164	7	):	):	PUNCT
ejpam-6565	164	8	let	let	VERB
ejpam-6565	164	9	b	b	X
ejpam-6565	164	10	be	be	AUX
ejpam-6565	164	11	any	any	DET
ejpam-6565	164	12	subset	subset	NOUN
ejpam-6565	164	13	of	of	ADP
ejpam-6565	164	14	y	y	PROPN
ejpam-6565	164	15	.	.	PUNCT
ejpam-6565	165	1	by	by	ADP
ejpam-6565	165	2	lemma	lemma	PROPN
ejpam-6565	165	3	3	3	NUM
ejpam-6565	165	4	,	,	PUNCT
ejpam-6565	165	5	σ1σ2	σ1σ2	NOUN
ejpam-6565	165	6	-	-	PUNCT
ejpam-6565	165	7	δ	δ	NOUN
ejpam-6565	165	8	-	-	NOUN
ejpam-6565	165	9	cl(b	cl(b	NOUN
ejpam-6565	165	10	)	)	PUNCT
ejpam-6565	165	11	is	be	AUX
ejpam-6565	165	12	σ1σ2	σ1σ2	NOUN
ejpam-6565	165	13	-	-	ADJ
ejpam-6565	165	14	closed	closed	ADJ
ejpam-6565	165	15	in	in	ADP
ejpam-6565	165	16	y	y	PROPN
ejpam-6565	165	17	.	.	PUNCT
ejpam-6565	166	1	since	since	SCONJ
ejpam-6565	166	2	f	f	PROPN
ejpam-6565	166	3	is	be	AUX
ejpam-6565	166	4	upper	upper	ADJ
ejpam-6565	166	5	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	166	6	,	,	PUNCT
ejpam-6565	166	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	166	8	,	,	PUNCT
ejpam-6565	166	9	by	by	ADP
ejpam-6565	166	10	theorem	theorem	VERB
ejpam-6565	166	11	1	1	NUM
ejpam-6565	166	12	f−(σ1σ2	f−(σ1σ2	ADJ
ejpam-6565	166	13	-	-	PUNCT
ejpam-6565	166	14	δ	δ	NOUN
ejpam-6565	166	15	-	-	NOUN
ejpam-6565	166	16	cl(b	cl(b	NOUN
ejpam-6565	166	17	)	)	PUNCT
ejpam-6565	166	18	)	)	PUNCT
ejpam-6565	166	19	is	be	AUX
ejpam-6565	166	20	⋆-closed	⋆-close	VERB
ejpam-6565	166	21	in	in	ADP
ejpam-6565	166	22	x.	x.	PROPN
ejpam-6565	166	23	(	(	PUNCT
ejpam-6565	166	24	2	2	NUM
ejpam-6565	166	25	)	)	PUNCT
ejpam-6565	166	26	⇒	⇒	NOUN
ejpam-6565	166	27	(	(	PUNCT
ejpam-6565	166	28	3	3	NUM
ejpam-6565	166	29	):	):	PUNCT
ejpam-6565	166	30	let	let	VERB
ejpam-6565	166	31	k	k	PRON
ejpam-6565	166	32	be	be	AUX
ejpam-6565	166	33	any	any	DET
ejpam-6565	166	34	σ1σ2	σ1σ2	NUM
ejpam-6565	166	35	-	-	PUNCT
ejpam-6565	166	36	δ	δ	NOUN
ejpam-6565	166	37	-	-	PUNCT
ejpam-6565	166	38	closed	closed	ADJ
ejpam-6565	166	39	set	set	NOUN
ejpam-6565	166	40	of	of	ADP
ejpam-6565	166	41	y	y	PROPN
ejpam-6565	166	42	.	.	PUNCT
ejpam-6565	167	1	then	then	ADV
ejpam-6565	167	2	,	,	PUNCT
ejpam-6565	167	3	σ1σ2	σ1σ2	PROPN
ejpam-6565	167	4	-	-	PUNCT
ejpam-6565	167	5	δ	δ	NOUN
ejpam-6565	167	6	-	-	NOUN
ejpam-6565	167	7	cl(k	cl(k	NOUN
ejpam-6565	167	8	)	)	PUNCT
ejpam-6565	168	1	=	=	SYM
ejpam-6565	169	1	k	k	PROPN
ejpam-6565	169	2	and	and	CCONJ
ejpam-6565	169	3	by	by	ADP
ejpam-6565	169	4	(	(	PUNCT
ejpam-6565	169	5	2	2	NUM
ejpam-6565	169	6	)	)	PUNCT
ejpam-6565	169	7	,	,	PUNCT
ejpam-6565	169	8	we	we	PRON
ejpam-6565	169	9	have	have	VERB
ejpam-6565	169	10	f−(k	f−(k	PROPN
ejpam-6565	169	11	)	)	PUNCT
ejpam-6565	169	12	is	be	AUX
ejpam-6565	169	13	⋆-closed	⋆-close	VERB
ejpam-6565	169	14	in	in	ADP
ejpam-6565	169	15	x.	x.	PROPN
ejpam-6565	169	16	(	(	PUNCT
ejpam-6565	169	17	3	3	NUM
ejpam-6565	169	18	)	)	PUNCT
ejpam-6565	169	19	⇒	⇒	NOUN
ejpam-6565	169	20	(	(	PUNCT
ejpam-6565	169	21	4	4	NUM
ejpam-6565	169	22	):	):	PUNCT
ejpam-6565	169	23	this	this	PRON
ejpam-6565	169	24	follows	follow	VERB
ejpam-6565	169	25	from	from	ADP
ejpam-6565	169	26	the	the	DET
ejpam-6565	169	27	fact	fact	NOUN
ejpam-6565	169	28	that	that	SCONJ
ejpam-6565	169	29	f+(y	f+(y	PROPN
ejpam-6565	169	30	−b	−b	ADV
ejpam-6565	169	31	)	)	PUNCT
ejpam-6565	169	32	=	=	PUNCT
ejpam-6565	170	1	x	x	SYM
ejpam-6565	170	2	−f−(b	−f−(b	PROPN
ejpam-6565	170	3	)	)	PUNCT
ejpam-6565	170	4	for	for	ADP
ejpam-6565	170	5	any	any	DET
ejpam-6565	170	6	subset	subset	NOUN
ejpam-6565	170	7	b	b	PROPN
ejpam-6565	170	8	of	of	ADP
ejpam-6565	170	9	y	y	PROPN
ejpam-6565	170	10	.	.	PUNCT
ejpam-6565	171	1	(	(	PUNCT
ejpam-6565	171	2	4	4	X
ejpam-6565	171	3	)	)	PUNCT
ejpam-6565	171	4	⇒	⇒	NOUN
ejpam-6565	171	5	(	(	PUNCT
ejpam-6565	171	6	1	1	NUM
ejpam-6565	171	7	):	):	PUNCT
ejpam-6565	171	8	let	let	VERB
ejpam-6565	171	9	v	v	PART
ejpam-6565	171	10	be	be	AUX
ejpam-6565	171	11	any	any	DET
ejpam-6565	171	12	σ1σ2	σ1σ2	NOUN
ejpam-6565	171	13	-	-	ADJ
ejpam-6565	171	14	open	open	ADJ
ejpam-6565	171	15	set	set	NOUN
ejpam-6565	171	16	of	of	ADP
ejpam-6565	171	17	y	y	PROPN
ejpam-6565	171	18	.	.	PUNCT
ejpam-6565	172	1	since	since	SCONJ
ejpam-6565	172	2	(	(	PUNCT
ejpam-6565	172	3	y	y	PROPN
ejpam-6565	172	4	,	,	PUNCT
ejpam-6565	172	5	σ1	σ1	PROPN
ejpam-6565	172	6	,	,	PUNCT
ejpam-6565	172	7	σ2	σ2	PROPN
ejpam-6565	172	8	)	)	PUNCT
ejpam-6565	172	9	is	be	AUX
ejpam-6565	172	10	(	(	PUNCT
ejpam-6565	172	11	σ1	σ1	PROPN
ejpam-6565	172	12	,	,	PUNCT
ejpam-6565	172	13	σ2)s	σ2)s	NOUN
ejpam-6565	172	14	-	-	PUNCT
ejpam-6565	172	15	regular	regular	ADJ
ejpam-6565	172	16	,	,	PUNCT
ejpam-6565	172	17	we	we	PRON
ejpam-6565	172	18	have	have	VERB
ejpam-6565	172	19	v	v	NOUN
ejpam-6565	172	20	is	be	AUX
ejpam-6565	173	1	σ1σ2	σ1σ2	NOUN
ejpam-6565	173	2	-	-	PUNCT
ejpam-6565	173	3	δ	δ	NOUN
ejpam-6565	173	4	-	-	NOUN
ejpam-6565	173	5	open	open	ADJ
ejpam-6565	173	6	in	in	ADP
ejpam-6565	173	7	y	y	PROPN
ejpam-6565	173	8	and	and	CCONJ
ejpam-6565	173	9	by	by	ADP
ejpam-6565	173	10	(	(	PUNCT
ejpam-6565	173	11	4	4	NUM
ejpam-6565	173	12	)	)	PUNCT
ejpam-6565	173	13	,	,	PUNCT
ejpam-6565	173	14	f+(v	f+(v	PROPN
ejpam-6565	173	15	)	)	PUNCT
ejpam-6565	173	16	is	be	AUX
ejpam-6565	173	17	⋆-open	⋆-open	ADJ
ejpam-6565	173	18	in	in	ADP
ejpam-6565	173	19	x.	x.	PROPN
ejpam-6565	173	20	thus	thus	ADV
ejpam-6565	173	21	,	,	PUNCT
ejpam-6565	173	22	f	f	PROPN
ejpam-6565	173	23	is	be	AUX
ejpam-6565	173	24	upper	upper	ADJ
ejpam-6565	173	25	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6565	173	26	,	,	PUNCT
ejpam-6565	173	27	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	173	28	by	by	ADP
ejpam-6565	173	29	theorem	theorem	NOUN
ejpam-6565	173	30	1	1	NUM
ejpam-6565	173	31	.	.	PUNCT
ejpam-6565	173	32	theorem	theorem	NOUN
ejpam-6565	173	33	4	4	NUM
ejpam-6565	173	34	.	.	X
ejpam-6565	173	35	for	for	ADP
ejpam-6565	173	36	a	a	DET
ejpam-6565	173	37	multifunction	multifunction	NOUN
ejpam-6565	174	1	f	f	NOUN
ejpam-6565	174	2	:	:	PUNCT
ejpam-6565	174	3	(	(	PUNCT
ejpam-6565	174	4	x	x	X
ejpam-6565	174	5	,	,	PUNCT
ejpam-6565	174	6	τ	τ	PROPN
ejpam-6565	174	7	,	,	PUNCT
ejpam-6565	174	8	i	i	NOUN
ejpam-6565	174	9	)	)	PUNCT
ejpam-6565	174	10	→	→	PUNCT
ejpam-6565	174	11	(	(	PUNCT
ejpam-6565	174	12	y	y	PROPN
ejpam-6565	174	13	,	,	PUNCT
ejpam-6565	174	14	σ1	σ1	PROPN
ejpam-6565	174	15	,	,	PUNCT
ejpam-6565	174	16	σ2	σ2	NOUN
ejpam-6565	174	17	)	)	PUNCT
ejpam-6565	174	18	,	,	PUNCT
ejpam-6565	174	19	where	where	SCONJ
ejpam-6565	174	20	(	(	PUNCT
ejpam-6565	174	21	y	y	PROPN
ejpam-6565	174	22	,	,	PUNCT
ejpam-6565	174	23	σ1	σ1	PROPN
ejpam-6565	174	24	,	,	PUNCT
ejpam-6565	174	25	σ2	σ2	PROPN
ejpam-6565	174	26	)	)	PUNCT
ejpam-6565	174	27	is	be	AUX
ejpam-6565	174	28	(	(	PUNCT
ejpam-6565	174	29	σ1	σ1	PROPN
ejpam-6565	174	30	,	,	PUNCT
ejpam-6565	174	31	σ2)sregular	σ2)sregular	PROPN
ejpam-6565	174	32	,	,	PUNCT
ejpam-6565	174	33	the	the	DET
ejpam-6565	174	34	following	follow	VERB
ejpam-6565	174	35	properties	property	NOUN
ejpam-6565	174	36	are	be	AUX
ejpam-6565	174	37	equivalent	equivalent	ADJ
ejpam-6565	174	38	:	:	PUNCT
ejpam-6565	174	39	(	(	PUNCT
ejpam-6565	174	40	1	1	X
ejpam-6565	174	41	)	)	PUNCT
ejpam-6565	174	42	f	f	PROPN
ejpam-6565	174	43	is	be	AUX
ejpam-6565	174	44	lower	low	ADJ
ejpam-6565	174	45	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	174	46	,	,	PUNCT
ejpam-6565	174	47	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	174	48	;	;	PUNCT
ejpam-6565	174	49	(	(	PUNCT
ejpam-6565	174	50	2	2	X
ejpam-6565	174	51	)	)	PUNCT
ejpam-6565	174	52	f+(σ1σ2	f+(σ1σ2	NOUN
ejpam-6565	174	53	-	-	PUNCT
ejpam-6565	174	54	δ	δ	NOUN
ejpam-6565	174	55	-	-	NOUN
ejpam-6565	174	56	cl(b	cl(b	NOUN
ejpam-6565	174	57	)	)	PUNCT
ejpam-6565	174	58	)	)	PUNCT
ejpam-6565	174	59	is	be	AUX
ejpam-6565	174	60	⋆-closed	⋆-close	VERB
ejpam-6565	174	61	in	in	ADP
ejpam-6565	174	62	x	x	PUNCT
ejpam-6565	174	63	for	for	ADP
ejpam-6565	174	64	every	every	DET
ejpam-6565	174	65	subset	subset	NOUN
ejpam-6565	174	66	b	b	PROPN
ejpam-6565	174	67	of	of	ADP
ejpam-6565	174	68	y	y	PROPN
ejpam-6565	174	69	;	;	PUNCT
ejpam-6565	174	70	(	(	PUNCT
ejpam-6565	174	71	3	3	X
ejpam-6565	174	72	)	)	PUNCT
ejpam-6565	174	73	f+(k	f+(k	NUM
ejpam-6565	174	74	)	)	PUNCT
ejpam-6565	174	75	is	be	AUX
ejpam-6565	174	76	⋆-closed	⋆-close	VERB
ejpam-6565	174	77	in	in	ADP
ejpam-6565	174	78	x	x	PUNCT
ejpam-6565	174	79	for	for	ADP
ejpam-6565	174	80	every	every	DET
ejpam-6565	174	81	σ1σ2	σ1σ2	NUM
ejpam-6565	174	82	-	-	PUNCT
ejpam-6565	174	83	δ	δ	NOUN
ejpam-6565	174	84	-	-	PUNCT
ejpam-6565	174	85	closed	close	VERB
ejpam-6565	174	86	set	set	ADJ
ejpam-6565	174	87	k	k	PROPN
ejpam-6565	174	88	of	of	ADP
ejpam-6565	174	89	y	y	PROPN
ejpam-6565	174	90	;	;	PUNCT
ejpam-6565	174	91	(	(	PUNCT
ejpam-6565	174	92	4	4	X
ejpam-6565	174	93	)	)	PUNCT
ejpam-6565	174	94	f−(v	f−(v	NOUN
ejpam-6565	174	95	)	)	PUNCT
ejpam-6565	174	96	is	be	AUX
ejpam-6565	174	97	⋆-open	⋆-open	ADJ
ejpam-6565	174	98	in	in	ADP
ejpam-6565	174	99	x	x	PUNCT
ejpam-6565	174	100	for	for	SCONJ
ejpam-6565	174	101	every	every	DET
ejpam-6565	174	102	σ1σ2	σ1σ2	NUM
ejpam-6565	174	103	-	-	PUNCT
ejpam-6565	174	104	δ	δ	NOUN
ejpam-6565	174	105	-	-	ADJ
ejpam-6565	174	106	open	open	ADJ
ejpam-6565	174	107	set	set	VERB
ejpam-6565	174	108	v	v	NOUN
ejpam-6565	174	109	of	of	ADP
ejpam-6565	174	110	y	y	PROPN
ejpam-6565	174	111	.	.	PUNCT
ejpam-6565	175	1	proof	proof	NOUN
ejpam-6565	175	2	.	.	PUNCT
ejpam-6565	176	1	the	the	DET
ejpam-6565	176	2	proof	proof	NOUN
ejpam-6565	176	3	is	be	AUX
ejpam-6565	176	4	similar	similar	ADJ
ejpam-6565	176	5	to	to	ADP
ejpam-6565	176	6	that	that	PRON
ejpam-6565	176	7	of	of	ADP
ejpam-6565	176	8	theorem	theorem	ADJ
ejpam-6565	176	9	3	3	NUM
ejpam-6565	176	10	.	.	PUNCT
ejpam-6565	176	11	corollary	corollary	ADJ
ejpam-6565	176	12	2	2	NUM
ejpam-6565	176	13	.	.	PUNCT
ejpam-6565	176	14	for	for	ADP
ejpam-6565	176	15	a	a	DET
ejpam-6565	176	16	function	function	NOUN
ejpam-6565	176	17	f	f	NOUN
ejpam-6565	176	18	:	:	PUNCT
ejpam-6565	176	19	(	(	PUNCT
ejpam-6565	176	20	x	x	X
ejpam-6565	176	21	,	,	PUNCT
ejpam-6565	176	22	τ	τ	PROPN
ejpam-6565	176	23	,	,	PUNCT
ejpam-6565	176	24	i	i	NOUN
ejpam-6565	176	25	)	)	PUNCT
ejpam-6565	176	26	→	→	PUNCT
ejpam-6565	176	27	(	(	PUNCT
ejpam-6565	176	28	y	y	PROPN
ejpam-6565	176	29	,	,	PUNCT
ejpam-6565	176	30	σ1	σ1	PROPN
ejpam-6565	176	31	,	,	PUNCT
ejpam-6565	176	32	σ2	σ2	NOUN
ejpam-6565	176	33	)	)	PUNCT
ejpam-6565	176	34	,	,	PUNCT
ejpam-6565	176	35	where	where	SCONJ
ejpam-6565	176	36	(	(	PUNCT
ejpam-6565	176	37	y	y	PROPN
ejpam-6565	176	38	,	,	PUNCT
ejpam-6565	176	39	σ1	σ1	PROPN
ejpam-6565	176	40	,	,	PUNCT
ejpam-6565	176	41	σ2	σ2	PROPN
ejpam-6565	176	42	)	)	PUNCT
ejpam-6565	176	43	is	be	AUX
ejpam-6565	176	44	(	(	PUNCT
ejpam-6565	176	45	σ1	σ1	PROPN
ejpam-6565	176	46	,	,	PUNCT
ejpam-6565	176	47	σ2)sregular	σ2)sregular	PROPN
ejpam-6565	176	48	,	,	PUNCT
ejpam-6565	176	49	the	the	DET
ejpam-6565	176	50	following	follow	VERB
ejpam-6565	176	51	properties	property	NOUN
ejpam-6565	176	52	are	be	AUX
ejpam-6565	176	53	equivalent	equivalent	ADJ
ejpam-6565	176	54	:	:	PUNCT
ejpam-6565	176	55	(	(	PUNCT
ejpam-6565	176	56	1	1	X
ejpam-6565	176	57	)	)	PUNCT
ejpam-6565	176	58	f	f	PROPN
ejpam-6565	176	59	is	be	AUX
ejpam-6565	176	60	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	176	61	,	,	PUNCT
ejpam-6565	176	62	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	176	63	;	;	PUNCT
ejpam-6565	176	64	(	(	PUNCT
ejpam-6565	176	65	2	2	X
ejpam-6565	176	66	)	)	PUNCT
ejpam-6565	176	67	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6565	176	68	-	-	PUNCT
ejpam-6565	176	69	δ	δ	NOUN
ejpam-6565	176	70	-	-	NOUN
ejpam-6565	176	71	cl(b	cl(b	NOUN
ejpam-6565	176	72	)	)	PUNCT
ejpam-6565	176	73	)	)	PUNCT
ejpam-6565	176	74	is	be	AUX
ejpam-6565	176	75	⋆-closed	⋆-close	VERB
ejpam-6565	176	76	in	in	ADP
ejpam-6565	176	77	x	x	PUNCT
ejpam-6565	176	78	for	for	ADP
ejpam-6565	176	79	every	every	DET
ejpam-6565	176	80	subset	subset	NOUN
ejpam-6565	176	81	b	b	PROPN
ejpam-6565	176	82	of	of	ADP
ejpam-6565	176	83	y	y	PROPN
ejpam-6565	176	84	;	;	PUNCT
ejpam-6565	176	85	j.	j.	PROPN
ejpam-6565	176	86	khampakdee	khampakdee	PROPN
ejpam-6565	176	87	,	,	PUNCT
ejpam-6565	176	88	a.	a.	PROPN
ejpam-6565	176	89	sama	sama	PROPN
ejpam-6565	176	90	-	-	PUNCT
ejpam-6565	176	91	ae	ae	PROPN
ejpam-6565	176	92	,	,	PUNCT
ejpam-6565	176	93	c.	c.	PROPN
ejpam-6565	176	94	boonpok	boonpok	PROPN
ejpam-6565	176	95	/	/	SYM
ejpam-6565	176	96	eur	eur	PROPN
ejpam-6565	176	97	.	.	PUNCT
ejpam-6565	177	1	j.	j.	PROPN
ejpam-6565	177	2	pure	pure	PROPN
ejpam-6565	177	3	appl	appl	PROPN
ejpam-6565	177	4	.	.	PROPN
ejpam-6565	177	5	math	math	PROPN
ejpam-6565	177	6	,	,	PUNCT
ejpam-6565	177	7	18	18	NUM
ejpam-6565	177	8	(	(	PUNCT
ejpam-6565	177	9	3	3	NUM
ejpam-6565	177	10	)	)	PUNCT
ejpam-6565	177	11	(	(	PUNCT
ejpam-6565	177	12	2025	2025	NUM
ejpam-6565	177	13	)	)	PUNCT
ejpam-6565	177	14	,	,	PUNCT
ejpam-6565	177	15	6565	6565	NUM
ejpam-6565	177	16	7	7	NUM
ejpam-6565	177	17	of	of	ADP
ejpam-6565	177	18	9	9	NUM
ejpam-6565	177	19	(	(	PUNCT
ejpam-6565	177	20	3	3	NUM
ejpam-6565	177	21	)	)	PUNCT
ejpam-6565	177	22	f−1(k	f−1(k	PROPN
ejpam-6565	177	23	)	)	PUNCT
ejpam-6565	177	24	is	be	AUX
ejpam-6565	177	25	⋆-closed	⋆-close	VERB
ejpam-6565	177	26	in	in	ADP
ejpam-6565	177	27	x	x	PUNCT
ejpam-6565	177	28	for	for	ADP
ejpam-6565	177	29	every	every	DET
ejpam-6565	177	30	σ1σ2	σ1σ2	NUM
ejpam-6565	177	31	-	-	PUNCT
ejpam-6565	177	32	δ	δ	NOUN
ejpam-6565	177	33	-	-	PUNCT
ejpam-6565	177	34	closed	close	VERB
ejpam-6565	177	35	set	set	ADJ
ejpam-6565	177	36	k	k	PROPN
ejpam-6565	177	37	of	of	ADP
ejpam-6565	177	38	y	y	PROPN
ejpam-6565	177	39	;	;	PUNCT
ejpam-6565	177	40	(	(	PUNCT
ejpam-6565	177	41	4	4	X
ejpam-6565	177	42	)	)	PUNCT
ejpam-6565	177	43	f−1(v	f−1(v	NOUN
ejpam-6565	177	44	)	)	PUNCT
ejpam-6565	177	45	is	be	AUX
ejpam-6565	177	46	⋆-open	⋆-open	ADJ
ejpam-6565	177	47	in	in	ADP
ejpam-6565	177	48	x	x	PUNCT
ejpam-6565	177	49	for	for	SCONJ
ejpam-6565	177	50	every	every	DET
ejpam-6565	177	51	σ1σ2	σ1σ2	NUM
ejpam-6565	177	52	-	-	PUNCT
ejpam-6565	177	53	δ	δ	NOUN
ejpam-6565	177	54	-	-	ADJ
ejpam-6565	177	55	open	open	ADJ
ejpam-6565	177	56	set	set	VERB
ejpam-6565	177	57	v	v	NOUN
ejpam-6565	177	58	of	of	ADP
ejpam-6565	177	59	y	y	PROPN
ejpam-6565	177	60	.	.	PUNCT
ejpam-6565	178	1	definition	definition	NOUN
ejpam-6565	178	2	5	5	NUM
ejpam-6565	178	3	.	.	PUNCT
ejpam-6565	179	1	[	[	X
ejpam-6565	179	2	17	17	NUM
ejpam-6565	179	3	]	]	PUNCT
ejpam-6565	179	4	a	a	DET
ejpam-6565	179	5	bitopological	bitopological	ADJ
ejpam-6565	179	6	space	space	NOUN
ejpam-6565	179	7	(	(	PUNCT
ejpam-6565	179	8	x	x	NOUN
ejpam-6565	179	9	,	,	PUNCT
ejpam-6565	179	10	τ1	τ1	NOUN
ejpam-6565	179	11	,	,	PUNCT
ejpam-6565	179	12	τ2	τ2	NOUN
ejpam-6565	179	13	)	)	PUNCT
ejpam-6565	179	14	is	be	AUX
ejpam-6565	179	15	said	say	VERB
ejpam-6565	179	16	to	to	PART
ejpam-6565	179	17	be	be	AUX
ejpam-6565	179	18	(	(	PUNCT
ejpam-6565	179	19	τ1	τ1	NOUN
ejpam-6565	179	20	,	,	PUNCT
ejpam-6565	179	21	τ2)-regular	τ2)-regular	ADJ
ejpam-6565	179	22	if	if	SCONJ
ejpam-6565	179	23	for	for	ADP
ejpam-6565	179	24	each	each	DET
ejpam-6565	179	25	τ1τ2	τ1τ2	ADJ
ejpam-6565	179	26	-	-	ADJ
ejpam-6565	179	27	closed	closed	ADJ
ejpam-6565	179	28	set	set	VERB
ejpam-6565	179	29	f	f	NOUN
ejpam-6565	179	30	and	and	CCONJ
ejpam-6565	179	31	each	each	DET
ejpam-6565	179	32	x	x	PROPN
ejpam-6565	179	33	̸∈	̸∈	PROPN
ejpam-6565	179	34	f	f	PROPN
ejpam-6565	179	35	,	,	PUNCT
ejpam-6565	179	36	there	there	PRON
ejpam-6565	179	37	exist	exist	VERB
ejpam-6565	179	38	disjoint	disjoint	ADJ
ejpam-6565	179	39	τ1τ2	τ1τ2	ADJ
ejpam-6565	179	40	-	-	ADJ
ejpam-6565	179	41	open	open	ADJ
ejpam-6565	179	42	sets	set	NOUN
ejpam-6565	179	43	u	u	NOUN
ejpam-6565	179	44	and	and	CCONJ
ejpam-6565	179	45	v	v	ADP
ejpam-6565	179	46	such	such	ADJ
ejpam-6565	179	47	that	that	SCONJ
ejpam-6565	179	48	x	x	SYM
ejpam-6565	179	49	∈	∈	PROPN
ejpam-6565	179	50	u	u	NOUN
ejpam-6565	179	51	and	and	CCONJ
ejpam-6565	179	52	f	f	PROPN
ejpam-6565	179	53	⊆	⊆	NUM
ejpam-6565	179	54	v	v	NOUN
ejpam-6565	179	55	.	.	PUNCT
ejpam-6565	180	1	lemma	lemma	PROPN
ejpam-6565	180	2	4	4	NUM
ejpam-6565	180	3	.	.	PUNCT
ejpam-6565	181	1	[	[	X
ejpam-6565	181	2	17	17	NUM
ejpam-6565	181	3	]	]	PUNCT
ejpam-6565	181	4	a	a	DET
ejpam-6565	181	5	bitopological	bitopological	ADJ
ejpam-6565	181	6	space	space	NOUN
ejpam-6565	181	7	(	(	PUNCT
ejpam-6565	181	8	x	x	NOUN
ejpam-6565	181	9	,	,	PUNCT
ejpam-6565	181	10	τ1	τ1	NOUN
ejpam-6565	181	11	,	,	PUNCT
ejpam-6565	181	12	τ2	τ2	NOUN
ejpam-6565	181	13	)	)	PUNCT
ejpam-6565	181	14	is	be	AUX
ejpam-6565	181	15	(	(	PUNCT
ejpam-6565	181	16	τ1	τ1	NOUN
ejpam-6565	181	17	,	,	PUNCT
ejpam-6565	181	18	τ2)-regular	τ2)-regular	ADJ
ejpam-6565	181	19	if	if	SCONJ
ejpam-6565	181	20	and	and	CCONJ
ejpam-6565	181	21	only	only	ADV
ejpam-6565	181	22	if	if	SCONJ
ejpam-6565	181	23	for	for	ADP
ejpam-6565	181	24	each	each	DET
ejpam-6565	181	25	x	x	SYM
ejpam-6565	181	26	∈	∈	PROPN
ejpam-6565	181	27	x	x	X
ejpam-6565	181	28	and	and	CCONJ
ejpam-6565	181	29	each	each	DET
ejpam-6565	181	30	τ1τ2	τ1τ2	ADJ
ejpam-6565	181	31	-	-	ADJ
ejpam-6565	181	32	open	open	ADJ
ejpam-6565	181	33	set	set	NOUN
ejpam-6565	181	34	u	u	NOUN
ejpam-6565	181	35	containing	contain	VERB
ejpam-6565	181	36	x	x	PRON
ejpam-6565	181	37	,	,	PUNCT
ejpam-6565	181	38	there	there	PRON
ejpam-6565	181	39	exists	exist	VERB
ejpam-6565	181	40	a	a	DET
ejpam-6565	181	41	τ1τ2	τ1τ2	NOUN
ejpam-6565	181	42	-	-	ADJ
ejpam-6565	181	43	open	open	ADJ
ejpam-6565	181	44	set	set	VERB
ejpam-6565	181	45	v	v	ADP
ejpam-6565	181	46	such	such	ADJ
ejpam-6565	181	47	that	that	SCONJ
ejpam-6565	181	48	x	x	SYM
ejpam-6565	181	49	∈	∈	NOUN
ejpam-6565	181	50	v	v	ADP
ejpam-6565	181	51	⊆	⊆	NUM
ejpam-6565	181	52	τ1τ2	τ1τ2	NOUN
ejpam-6565	181	53	-	-	NOUN
ejpam-6565	181	54	cl(v	cl(v	X
ejpam-6565	181	55	)	)	PUNCT
ejpam-6565	181	56	⊆	⊆	NUM
ejpam-6565	181	57	u	u	NOUN
ejpam-6565	181	58	.	.	PUNCT
ejpam-6565	182	1	lemma	lemma	PROPN
ejpam-6565	182	2	5	5	NUM
ejpam-6565	182	3	.	.	PUNCT
ejpam-6565	183	1	[	[	X
ejpam-6565	183	2	8	8	NUM
ejpam-6565	183	3	]	]	X
ejpam-6565	183	4	let	let	VERB
ejpam-6565	183	5	(	(	PUNCT
ejpam-6565	183	6	x	x	NOUN
ejpam-6565	183	7	,	,	PUNCT
ejpam-6565	183	8	τ1	τ1	NOUN
ejpam-6565	183	9	,	,	PUNCT
ejpam-6565	183	10	τ2	τ2	PROPN
ejpam-6565	183	11	)	)	PUNCT
ejpam-6565	183	12	be	be	VERB
ejpam-6565	183	13	a	a	DET
ejpam-6565	183	14	(	(	PUNCT
ejpam-6565	183	15	τ1	τ1	NOUN
ejpam-6565	183	16	,	,	PUNCT
ejpam-6565	183	17	τ2)-regular	τ2)-regular	ADJ
ejpam-6565	183	18	space	space	NOUN
ejpam-6565	183	19	.	.	PUNCT
ejpam-6565	184	1	then	then	ADV
ejpam-6565	184	2	,	,	PUNCT
ejpam-6565	184	3	the	the	DET
ejpam-6565	184	4	following	follow	VERB
ejpam-6565	184	5	properties	property	NOUN
ejpam-6565	184	6	hold	hold	VERB
ejpam-6565	184	7	:	:	PUNCT
ejpam-6565	184	8	(	(	PUNCT
ejpam-6565	184	9	1	1	X
ejpam-6565	184	10	)	)	PUNCT
ejpam-6565	184	11	τ1τ2	τ1τ2	NOUN
ejpam-6565	184	12	-	-	NUM
ejpam-6565	184	13	cl(a	cl(a	NUM
ejpam-6565	184	14	)	)	PUNCT
ejpam-6565	184	15	=	=	PUNCT
ejpam-6565	184	16	(	(	PUNCT
ejpam-6565	184	17	τ1	τ1	NOUN
ejpam-6565	184	18	,	,	PUNCT
ejpam-6565	184	19	τ2)θ	τ2)θ	NOUN
ejpam-6565	184	20	-	-	PUNCT
ejpam-6565	184	21	cl(a	cl(a	NUM
ejpam-6565	184	22	)	)	PUNCT
ejpam-6565	184	23	for	for	ADP
ejpam-6565	184	24	every	every	DET
ejpam-6565	184	25	subset	subset	NOUN
ejpam-6565	184	26	a	a	PRON
ejpam-6565	184	27	of	of	ADP
ejpam-6565	184	28	x.	x.	NOUN
ejpam-6565	184	29	(	(	PUNCT
ejpam-6565	184	30	2	2	NUM
ejpam-6565	184	31	)	)	PUNCT
ejpam-6565	184	32	every	every	DET
ejpam-6565	184	33	τ1τ2	τ1τ2	NOUN
ejpam-6565	184	34	-	-	ADJ
ejpam-6565	184	35	open	open	ADJ
ejpam-6565	184	36	set	set	NOUN
ejpam-6565	184	37	is	be	AUX
ejpam-6565	184	38	(	(	PUNCT
ejpam-6565	184	39	τ1	τ1	NOUN
ejpam-6565	184	40	,	,	PUNCT
ejpam-6565	184	41	τ2)θ	τ2)θ	ADJ
ejpam-6565	184	42	-	-	PUNCT
ejpam-6565	184	43	open	open	ADJ
ejpam-6565	184	44	.	.	PUNCT
ejpam-6565	185	1	theorem	theorem	ADJ
ejpam-6565	185	2	5	5	NUM
ejpam-6565	185	3	.	.	X
ejpam-6565	185	4	for	for	ADP
ejpam-6565	185	5	a	a	DET
ejpam-6565	185	6	multifunction	multifunction	NOUN
ejpam-6565	186	1	f	f	NOUN
ejpam-6565	186	2	:	:	PUNCT
ejpam-6565	186	3	(	(	PUNCT
ejpam-6565	186	4	x	x	X
ejpam-6565	186	5	,	,	PUNCT
ejpam-6565	186	6	τ	τ	PROPN
ejpam-6565	186	7	,	,	PUNCT
ejpam-6565	186	8	i	i	NOUN
ejpam-6565	186	9	)	)	PUNCT
ejpam-6565	186	10	→	→	PUNCT
ejpam-6565	186	11	(	(	PUNCT
ejpam-6565	186	12	y	y	PROPN
ejpam-6565	186	13	,	,	PUNCT
ejpam-6565	186	14	σ1	σ1	PROPN
ejpam-6565	186	15	,	,	PUNCT
ejpam-6565	186	16	σ2	σ2	NOUN
ejpam-6565	186	17	)	)	PUNCT
ejpam-6565	186	18	,	,	PUNCT
ejpam-6565	186	19	where	where	SCONJ
ejpam-6565	186	20	(	(	PUNCT
ejpam-6565	186	21	y	y	PROPN
ejpam-6565	186	22	,	,	PUNCT
ejpam-6565	186	23	σ1	σ1	PROPN
ejpam-6565	186	24	,	,	PUNCT
ejpam-6565	186	25	σ2	σ2	PROPN
ejpam-6565	186	26	)	)	PUNCT
ejpam-6565	186	27	is	be	AUX
ejpam-6565	186	28	(	(	PUNCT
ejpam-6565	186	29	σ1	σ1	PROPN
ejpam-6565	186	30	,	,	PUNCT
ejpam-6565	186	31	σ2)regular	σ2)regular	PROPN
ejpam-6565	186	32	,	,	PUNCT
ejpam-6565	186	33	the	the	DET
ejpam-6565	186	34	following	follow	VERB
ejpam-6565	186	35	properties	property	NOUN
ejpam-6565	186	36	are	be	AUX
ejpam-6565	186	37	equivalent	equivalent	ADJ
ejpam-6565	186	38	:	:	PUNCT
ejpam-6565	186	39	(	(	PUNCT
ejpam-6565	186	40	1	1	X
ejpam-6565	186	41	)	)	PUNCT
ejpam-6565	186	42	f	f	PROPN
ejpam-6565	186	43	is	be	AUX
ejpam-6565	186	44	upper	upper	ADJ
ejpam-6565	186	45	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	186	46	,	,	PUNCT
ejpam-6565	186	47	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	186	48	;	;	PUNCT
ejpam-6565	186	49	(	(	PUNCT
ejpam-6565	186	50	2	2	X
ejpam-6565	186	51	)	)	PUNCT
ejpam-6565	186	52	f−((σ1	f−((σ1	NOUN
ejpam-6565	186	53	,	,	PUNCT
ejpam-6565	186	54	σ2)θ	σ2)θ	ADJ
ejpam-6565	186	55	-	-	PUNCT
ejpam-6565	186	56	cl(b	cl(b	NOUN
ejpam-6565	186	57	)	)	PUNCT
ejpam-6565	186	58	)	)	PUNCT
ejpam-6565	186	59	is	be	AUX
ejpam-6565	186	60	⋆-closed	⋆-close	VERB
ejpam-6565	186	61	in	in	ADP
ejpam-6565	186	62	x	x	PUNCT
ejpam-6565	186	63	for	for	ADP
ejpam-6565	186	64	every	every	DET
ejpam-6565	186	65	subset	subset	NOUN
ejpam-6565	186	66	b	b	PROPN
ejpam-6565	186	67	of	of	ADP
ejpam-6565	186	68	y	y	PROPN
ejpam-6565	186	69	;	;	PUNCT
ejpam-6565	186	70	(	(	PUNCT
ejpam-6565	186	71	3	3	X
ejpam-6565	186	72	)	)	PUNCT
ejpam-6565	186	73	f−(k	f−(k	PROPN
ejpam-6565	186	74	)	)	PUNCT
ejpam-6565	186	75	is	be	AUX
ejpam-6565	186	76	⋆-closed	⋆-close	VERB
ejpam-6565	186	77	in	in	ADP
ejpam-6565	186	78	x	x	PUNCT
ejpam-6565	186	79	for	for	ADP
ejpam-6565	186	80	every	every	DET
ejpam-6565	186	81	(	(	PUNCT
ejpam-6565	186	82	σ1	σ1	PROPN
ejpam-6565	186	83	,	,	PUNCT
ejpam-6565	186	84	σ2)θ	σ2)θ	NOUN
ejpam-6565	186	85	-	-	PUNCT
ejpam-6565	186	86	closed	close	VERB
ejpam-6565	186	87	set	set	NOUN
ejpam-6565	186	88	k	k	PROPN
ejpam-6565	186	89	of	of	ADP
ejpam-6565	186	90	y	y	PROPN
ejpam-6565	186	91	;	;	PUNCT
ejpam-6565	186	92	(	(	PUNCT
ejpam-6565	186	93	4	4	X
ejpam-6565	186	94	)	)	PUNCT
ejpam-6565	186	95	f+(v	f+(v	NOUN
ejpam-6565	186	96	)	)	PUNCT
ejpam-6565	186	97	is	be	AUX
ejpam-6565	186	98	⋆-open	⋆-open	ADJ
ejpam-6565	186	99	in	in	ADP
ejpam-6565	186	100	x	x	PUNCT
ejpam-6565	186	101	for	for	ADP
ejpam-6565	186	102	every	every	DET
ejpam-6565	186	103	(	(	PUNCT
ejpam-6565	186	104	σ1	σ1	PROPN
ejpam-6565	186	105	,	,	PUNCT
ejpam-6565	186	106	σ2)θ	σ2)θ	NOUN
ejpam-6565	186	107	-	-	PUNCT
ejpam-6565	186	108	open	open	ADJ
ejpam-6565	186	109	set	set	NOUN
ejpam-6565	186	110	v	v	NOUN
ejpam-6565	186	111	of	of	ADP
ejpam-6565	186	112	y	y	PROPN
ejpam-6565	186	113	.	.	PUNCT
ejpam-6565	187	1	proof	proof	NOUN
ejpam-6565	187	2	.	.	PUNCT
ejpam-6565	188	1	(	(	PUNCT
ejpam-6565	188	2	1	1	X
ejpam-6565	188	3	)	)	PUNCT
ejpam-6565	188	4	⇒	⇒	NOUN
ejpam-6565	188	5	(	(	PUNCT
ejpam-6565	188	6	2	2	NUM
ejpam-6565	188	7	):	):	PUNCT
ejpam-6565	188	8	let	let	VERB
ejpam-6565	188	9	b	b	X
ejpam-6565	188	10	be	be	AUX
ejpam-6565	188	11	any	any	DET
ejpam-6565	188	12	subset	subset	NOUN
ejpam-6565	188	13	of	of	ADP
ejpam-6565	188	14	y	y	PROPN
ejpam-6565	188	15	.	.	PUNCT
ejpam-6565	189	1	by	by	ADP
ejpam-6565	189	2	lemma	lemma	PROPN
ejpam-6565	189	3	5	5	NUM
ejpam-6565	189	4	,	,	PUNCT
ejpam-6565	189	5	(	(	PUNCT
ejpam-6565	189	6	σ1	σ1	PROPN
ejpam-6565	189	7	,	,	PUNCT
ejpam-6565	189	8	σ2)θ	σ2)θ	NOUN
ejpam-6565	189	9	-	-	PUNCT
ejpam-6565	189	10	cl(b	cl(b	NOUN
ejpam-6565	189	11	)	)	PUNCT
ejpam-6565	189	12	is	be	AUX
ejpam-6565	189	13	σ1σ2closed	σ1σ2close	VERB
ejpam-6565	189	14	in	in	ADP
ejpam-6565	189	15	y	y	PROPN
ejpam-6565	189	16	.	.	PUNCT
ejpam-6565	190	1	since	since	SCONJ
ejpam-6565	190	2	f	f	PROPN
ejpam-6565	190	3	is	be	AUX
ejpam-6565	190	4	upper	upper	ADJ
ejpam-6565	190	5	τ⋆(σ1	τ⋆(σ1	NOUN
ejpam-6565	190	6	,	,	PUNCT
ejpam-6565	190	7	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	190	8	,	,	PUNCT
ejpam-6565	190	9	by	by	ADP
ejpam-6565	190	10	theorem	theorem	NOUN
ejpam-6565	190	11	1	1	NUM
ejpam-6565	190	12	f−((σ1	f−((σ1	NOUN
ejpam-6565	190	13	,	,	PUNCT
ejpam-6565	190	14	σ2)θ	σ2)θ	ADJ
ejpam-6565	190	15	-	-	PUNCT
ejpam-6565	190	16	cl(b	cl(b	NOUN
ejpam-6565	190	17	)	)	PUNCT
ejpam-6565	190	18	)	)	PUNCT
ejpam-6565	190	19	is	be	AUX
ejpam-6565	190	20	⋆-closed	⋆-close	VERB
ejpam-6565	190	21	in	in	ADP
ejpam-6565	190	22	x.	x.	PROPN
ejpam-6565	190	23	(	(	PUNCT
ejpam-6565	190	24	2	2	NUM
ejpam-6565	190	25	)	)	PUNCT
ejpam-6565	190	26	⇒	⇒	NOUN
ejpam-6565	190	27	(	(	PUNCT
ejpam-6565	190	28	3	3	NUM
ejpam-6565	190	29	):	):	PUNCT
ejpam-6565	190	30	let	let	VERB
ejpam-6565	190	31	k	k	PRON
ejpam-6565	190	32	be	be	AUX
ejpam-6565	190	33	any	any	DET
ejpam-6565	190	34	(	(	PUNCT
ejpam-6565	190	35	σ1	σ1	PROPN
ejpam-6565	190	36	,	,	PUNCT
ejpam-6565	190	37	σ2)θ	σ2)θ	NOUN
ejpam-6565	190	38	-	-	PUNCT
ejpam-6565	190	39	closed	close	VERB
ejpam-6565	190	40	set	set	NOUN
ejpam-6565	190	41	of	of	ADP
ejpam-6565	190	42	y	y	PROPN
ejpam-6565	190	43	.	.	PUNCT
ejpam-6565	191	1	then	then	ADV
ejpam-6565	191	2	,	,	PUNCT
ejpam-6565	191	3	(	(	PUNCT
ejpam-6565	191	4	σ1	σ1	PROPN
ejpam-6565	191	5	,	,	PUNCT
ejpam-6565	191	6	σ2)θ	σ2)θ	NOUN
ejpam-6565	191	7	-	-	PUNCT
ejpam-6565	191	8	cl(k	cl(k	NOUN
ejpam-6565	191	9	)	)	PUNCT
ejpam-6565	192	1	=	=	SYM
ejpam-6565	193	1	k	k	PROPN
ejpam-6565	193	2	and	and	CCONJ
ejpam-6565	193	3	by	by	ADP
ejpam-6565	193	4	(	(	PUNCT
ejpam-6565	193	5	2	2	NUM
ejpam-6565	193	6	)	)	PUNCT
ejpam-6565	193	7	,	,	PUNCT
ejpam-6565	193	8	we	we	PRON
ejpam-6565	193	9	have	have	VERB
ejpam-6565	193	10	f−(k	f−(k	PROPN
ejpam-6565	193	11	)	)	PUNCT
ejpam-6565	193	12	is	be	AUX
ejpam-6565	193	13	⋆-closed	⋆-close	VERB
ejpam-6565	193	14	in	in	ADP
ejpam-6565	193	15	x.	x.	PROPN
ejpam-6565	193	16	(	(	PUNCT
ejpam-6565	193	17	3	3	NUM
ejpam-6565	193	18	)	)	PUNCT
ejpam-6565	193	19	⇒	⇒	NOUN
ejpam-6565	193	20	(	(	PUNCT
ejpam-6565	193	21	4	4	NUM
ejpam-6565	193	22	):	):	PUNCT
ejpam-6565	193	23	this	this	PRON
ejpam-6565	193	24	follows	follow	VERB
ejpam-6565	193	25	from	from	ADP
ejpam-6565	193	26	the	the	DET
ejpam-6565	193	27	fact	fact	NOUN
ejpam-6565	193	28	that	that	SCONJ
ejpam-6565	193	29	f+(y	f+(y	PROPN
ejpam-6565	193	30	−b	−b	ADV
ejpam-6565	193	31	)	)	PUNCT
ejpam-6565	194	1	=	=	PUNCT
ejpam-6565	194	2	x	x	X
ejpam-6565	195	1	−	−	PROPN
ejpam-6565	195	2	f−(b	f−(b	PROPN
ejpam-6565	195	3	)	)	PUNCT
ejpam-6565	195	4	for	for	ADP
ejpam-6565	195	5	every	every	DET
ejpam-6565	195	6	subset	subset	NOUN
ejpam-6565	195	7	b	b	PROPN
ejpam-6565	195	8	of	of	ADP
ejpam-6565	195	9	y	y	PROPN
ejpam-6565	195	10	.	.	PUNCT
ejpam-6565	196	1	(	(	PUNCT
ejpam-6565	196	2	4	4	X
ejpam-6565	196	3	)	)	PUNCT
ejpam-6565	196	4	⇒	⇒	NOUN
ejpam-6565	196	5	(	(	PUNCT
ejpam-6565	196	6	1	1	NUM
ejpam-6565	196	7	):	):	PUNCT
ejpam-6565	196	8	let	let	VERB
ejpam-6565	196	9	v	v	PART
ejpam-6565	196	10	be	be	AUX
ejpam-6565	196	11	any	any	DET
ejpam-6565	196	12	σ1σ2	σ1σ2	NOUN
ejpam-6565	196	13	-	-	ADJ
ejpam-6565	196	14	open	open	ADJ
ejpam-6565	196	15	set	set	NOUN
ejpam-6565	196	16	of	of	ADP
ejpam-6565	196	17	y	y	PROPN
ejpam-6565	196	18	.	.	PUNCT
ejpam-6565	197	1	since	since	SCONJ
ejpam-6565	197	2	(	(	PUNCT
ejpam-6565	197	3	y	y	PROPN
ejpam-6565	197	4	,	,	PUNCT
ejpam-6565	197	5	σ1	σ1	PROPN
ejpam-6565	197	6	,	,	PUNCT
ejpam-6565	197	7	σ2	σ2	PROPN
ejpam-6565	197	8	)	)	PUNCT
ejpam-6565	197	9	is	be	AUX
ejpam-6565	197	10	(	(	PUNCT
ejpam-6565	197	11	σ1	σ1	NOUN
ejpam-6565	197	12	,	,	PUNCT
ejpam-6565	197	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6565	197	14	,	,	PUNCT
ejpam-6565	197	15	we	we	PRON
ejpam-6565	197	16	have	have	VERB
ejpam-6565	197	17	v	v	NOUN
ejpam-6565	197	18	is	be	AUX
ejpam-6565	197	19	(	(	PUNCT
ejpam-6565	197	20	σ1	σ1	PROPN
ejpam-6565	197	21	,	,	PUNCT
ejpam-6565	197	22	σ2)θ	σ2)θ	NOUN
ejpam-6565	197	23	-	-	PUNCT
ejpam-6565	197	24	open	open	ADJ
ejpam-6565	197	25	in	in	ADP
ejpam-6565	197	26	y	y	PROPN
ejpam-6565	197	27	and	and	CCONJ
ejpam-6565	197	28	by	by	ADP
ejpam-6565	197	29	(	(	PUNCT
ejpam-6565	197	30	4	4	NUM
ejpam-6565	197	31	)	)	PUNCT
ejpam-6565	197	32	,	,	PUNCT
ejpam-6565	197	33	f+(v	f+(v	PROPN
ejpam-6565	197	34	)	)	PUNCT
ejpam-6565	197	35	is	be	AUX
ejpam-6565	197	36	⋆-open	⋆-open	ADJ
ejpam-6565	197	37	in	in	ADP
ejpam-6565	197	38	x.	x.	PROPN
ejpam-6565	197	39	thus	thus	ADV
ejpam-6565	197	40	,	,	PUNCT
ejpam-6565	197	41	f	f	PROPN
ejpam-6565	197	42	is	be	AUX
ejpam-6565	197	43	upper	upper	ADJ
ejpam-6565	197	44	τ⋆(σ1	τ⋆(σ1	ADV
ejpam-6565	197	45	,	,	PUNCT
ejpam-6565	197	46	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	197	47	by	by	ADP
ejpam-6565	197	48	theorem	theorem	NOUN
ejpam-6565	197	49	1	1	NUM
ejpam-6565	197	50	.	.	PUNCT
ejpam-6565	197	51	theorem	theorem	VERB
ejpam-6565	197	52	6	6	NUM
ejpam-6565	197	53	.	.	PUNCT
ejpam-6565	197	54	for	for	ADP
ejpam-6565	197	55	a	a	DET
ejpam-6565	197	56	multifunction	multifunction	NOUN
ejpam-6565	197	57	f	f	NOUN
ejpam-6565	197	58	:	:	PUNCT
ejpam-6565	197	59	(	(	PUNCT
ejpam-6565	197	60	x	x	X
ejpam-6565	197	61	,	,	PUNCT
ejpam-6565	197	62	τ	τ	PROPN
ejpam-6565	197	63	,	,	PUNCT
ejpam-6565	197	64	i	i	NOUN
ejpam-6565	197	65	)	)	PUNCT
ejpam-6565	197	66	→	→	PUNCT
ejpam-6565	197	67	(	(	PUNCT
ejpam-6565	197	68	y	y	PROPN
ejpam-6565	197	69	,	,	PUNCT
ejpam-6565	197	70	σ1	σ1	PROPN
ejpam-6565	197	71	,	,	PUNCT
ejpam-6565	197	72	σ2	σ2	NOUN
ejpam-6565	197	73	)	)	PUNCT
ejpam-6565	197	74	,	,	PUNCT
ejpam-6565	197	75	where	where	SCONJ
ejpam-6565	197	76	(	(	PUNCT
ejpam-6565	197	77	y	y	PROPN
ejpam-6565	197	78	,	,	PUNCT
ejpam-6565	197	79	σ1	σ1	PROPN
ejpam-6565	197	80	,	,	PUNCT
ejpam-6565	197	81	σ2	σ2	PROPN
ejpam-6565	197	82	)	)	PUNCT
ejpam-6565	197	83	is	be	AUX
ejpam-6565	197	84	(	(	PUNCT
ejpam-6565	197	85	σ1	σ1	PROPN
ejpam-6565	197	86	,	,	PUNCT
ejpam-6565	197	87	σ2)regular	σ2)regular	PROPN
ejpam-6565	197	88	,	,	PUNCT
ejpam-6565	197	89	the	the	DET
ejpam-6565	197	90	following	follow	VERB
ejpam-6565	197	91	properties	property	NOUN
ejpam-6565	197	92	are	be	AUX
ejpam-6565	197	93	equivalent	equivalent	ADJ
ejpam-6565	197	94	:	:	PUNCT
ejpam-6565	197	95	(	(	PUNCT
ejpam-6565	197	96	1	1	X
ejpam-6565	197	97	)	)	PUNCT
ejpam-6565	197	98	f	f	PROPN
ejpam-6565	197	99	is	be	AUX
ejpam-6565	197	100	lower	low	ADJ
ejpam-6565	197	101	τ⋆(σ1	τ⋆(σ1	ADP
ejpam-6565	197	102	,	,	PUNCT
ejpam-6565	197	103	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	197	104	;	;	PUNCT
ejpam-6565	197	105	(	(	PUNCT
ejpam-6565	197	106	2	2	X
ejpam-6565	197	107	)	)	PUNCT
ejpam-6565	197	108	f+((σ1	f+((σ1	NOUN
ejpam-6565	197	109	,	,	PUNCT
ejpam-6565	197	110	σ2)θ	σ2)θ	ADJ
ejpam-6565	197	111	-	-	PUNCT
ejpam-6565	197	112	cl(b	cl(b	NOUN
ejpam-6565	197	113	)	)	PUNCT
ejpam-6565	197	114	)	)	PUNCT
ejpam-6565	197	115	is	be	AUX
ejpam-6565	197	116	⋆-closed	⋆-close	VERB
ejpam-6565	197	117	in	in	ADP
ejpam-6565	197	118	x	x	PUNCT
ejpam-6565	197	119	for	for	ADP
ejpam-6565	197	120	every	every	DET
ejpam-6565	197	121	subset	subset	NOUN
ejpam-6565	197	122	b	b	PROPN
ejpam-6565	197	123	of	of	ADP
ejpam-6565	197	124	y	y	PROPN
ejpam-6565	197	125	;	;	PUNCT
ejpam-6565	197	126	(	(	PUNCT
ejpam-6565	197	127	3	3	X
ejpam-6565	197	128	)	)	PUNCT
ejpam-6565	197	129	f+(k	f+(k	NUM
ejpam-6565	197	130	)	)	PUNCT
ejpam-6565	197	131	is	be	AUX
ejpam-6565	197	132	⋆-closed	⋆-close	VERB
ejpam-6565	197	133	in	in	ADP
ejpam-6565	197	134	x	x	PUNCT
ejpam-6565	197	135	for	for	ADP
ejpam-6565	197	136	every	every	DET
ejpam-6565	197	137	(	(	PUNCT
ejpam-6565	197	138	σ1	σ1	PROPN
ejpam-6565	197	139	,	,	PUNCT
ejpam-6565	197	140	σ2)θ	σ2)θ	NOUN
ejpam-6565	197	141	-	-	PUNCT
ejpam-6565	197	142	closed	close	VERB
ejpam-6565	197	143	set	set	NOUN
ejpam-6565	197	144	k	k	PROPN
ejpam-6565	197	145	of	of	ADP
ejpam-6565	197	146	y	y	PROPN
ejpam-6565	197	147	;	;	PUNCT
ejpam-6565	197	148	j.	j.	PROPN
ejpam-6565	197	149	khampakdee	khampakdee	PROPN
ejpam-6565	197	150	,	,	PUNCT
ejpam-6565	197	151	a.	a.	PROPN
ejpam-6565	197	152	sama	sama	PROPN
ejpam-6565	197	153	-	-	PUNCT
ejpam-6565	197	154	ae	ae	PROPN
ejpam-6565	197	155	,	,	PUNCT
ejpam-6565	197	156	c.	c.	PROPN
ejpam-6565	197	157	boonpok	boonpok	PROPN
ejpam-6565	197	158	/	/	SYM
ejpam-6565	197	159	eur	eur	PROPN
ejpam-6565	197	160	.	.	PUNCT
ejpam-6565	198	1	j.	j.	PROPN
ejpam-6565	198	2	pure	pure	PROPN
ejpam-6565	198	3	appl	appl	PROPN
ejpam-6565	198	4	.	.	PROPN
ejpam-6565	198	5	math	math	PROPN
ejpam-6565	198	6	,	,	PUNCT
ejpam-6565	198	7	18	18	NUM
ejpam-6565	198	8	(	(	PUNCT
ejpam-6565	198	9	3	3	NUM
ejpam-6565	198	10	)	)	PUNCT
ejpam-6565	198	11	(	(	PUNCT
ejpam-6565	198	12	2025	2025	NUM
ejpam-6565	198	13	)	)	PUNCT
ejpam-6565	198	14	,	,	PUNCT
ejpam-6565	198	15	6565	6565	NUM
ejpam-6565	198	16	8	8	NUM
ejpam-6565	198	17	of	of	ADP
ejpam-6565	198	18	9	9	NUM
ejpam-6565	198	19	(	(	PUNCT
ejpam-6565	198	20	4	4	NUM
ejpam-6565	198	21	)	)	PUNCT
ejpam-6565	198	22	f−(v	f−(v	NOUN
ejpam-6565	198	23	)	)	PUNCT
ejpam-6565	198	24	is	be	AUX
ejpam-6565	198	25	⋆-open	⋆-open	ADJ
ejpam-6565	198	26	in	in	ADP
ejpam-6565	198	27	x	x	PUNCT
ejpam-6565	198	28	for	for	ADP
ejpam-6565	198	29	every	every	DET
ejpam-6565	198	30	(	(	PUNCT
ejpam-6565	198	31	σ1	σ1	PROPN
ejpam-6565	198	32	,	,	PUNCT
ejpam-6565	198	33	σ2)θ	σ2)θ	NOUN
ejpam-6565	198	34	-	-	PUNCT
ejpam-6565	198	35	open	open	ADJ
ejpam-6565	198	36	set	set	NOUN
ejpam-6565	198	37	v	v	NOUN
ejpam-6565	198	38	of	of	ADP
ejpam-6565	198	39	y	y	PROPN
ejpam-6565	198	40	.	.	PUNCT
ejpam-6565	199	1	proof	proof	NOUN
ejpam-6565	199	2	.	.	PUNCT
ejpam-6565	200	1	the	the	DET
ejpam-6565	200	2	proof	proof	NOUN
ejpam-6565	200	3	is	be	AUX
ejpam-6565	200	4	similar	similar	ADJ
ejpam-6565	200	5	to	to	ADP
ejpam-6565	200	6	that	that	PRON
ejpam-6565	200	7	of	of	ADP
ejpam-6565	200	8	theorem	theorem	ADJ
ejpam-6565	200	9	5	5	NUM
ejpam-6565	200	10	.	.	PUNCT
ejpam-6565	200	11	corollary	corollary	ADJ
ejpam-6565	200	12	3	3	NUM
ejpam-6565	200	13	.	.	PUNCT
ejpam-6565	201	1	for	for	ADP
ejpam-6565	201	2	a	a	DET
ejpam-6565	201	3	function	function	NOUN
ejpam-6565	201	4	f	f	NOUN
ejpam-6565	201	5	:	:	PUNCT
ejpam-6565	201	6	(	(	PUNCT
ejpam-6565	201	7	x	x	X
ejpam-6565	201	8	,	,	PUNCT
ejpam-6565	201	9	τ	τ	PROPN
ejpam-6565	201	10	,	,	PUNCT
ejpam-6565	201	11	i	i	NOUN
ejpam-6565	201	12	)	)	PUNCT
ejpam-6565	201	13	→	→	PUNCT
ejpam-6565	201	14	(	(	PUNCT
ejpam-6565	201	15	y	y	PROPN
ejpam-6565	201	16	,	,	PUNCT
ejpam-6565	201	17	σ1	σ1	PROPN
ejpam-6565	201	18	,	,	PUNCT
ejpam-6565	201	19	σ2	σ2	NOUN
ejpam-6565	201	20	)	)	PUNCT
ejpam-6565	201	21	,	,	PUNCT
ejpam-6565	201	22	where	where	SCONJ
ejpam-6565	201	23	(	(	PUNCT
ejpam-6565	201	24	y	y	PROPN
ejpam-6565	201	25	,	,	PUNCT
ejpam-6565	201	26	σ1	σ1	PROPN
ejpam-6565	201	27	,	,	PUNCT
ejpam-6565	201	28	σ2	σ2	PROPN
ejpam-6565	201	29	)	)	PUNCT
ejpam-6565	201	30	is	be	AUX
ejpam-6565	201	31	(	(	PUNCT
ejpam-6565	201	32	σ1	σ1	PROPN
ejpam-6565	201	33	,	,	PUNCT
ejpam-6565	201	34	σ2)regular	σ2)regular	PROPN
ejpam-6565	201	35	,	,	PUNCT
ejpam-6565	201	36	the	the	DET
ejpam-6565	201	37	following	follow	VERB
ejpam-6565	201	38	properties	property	NOUN
ejpam-6565	201	39	are	be	AUX
ejpam-6565	201	40	equivalent	equivalent	ADJ
ejpam-6565	201	41	:	:	PUNCT
ejpam-6565	201	42	(	(	PUNCT
ejpam-6565	201	43	1	1	X
ejpam-6565	201	44	)	)	PUNCT
ejpam-6565	201	45	f	f	PROPN
ejpam-6565	201	46	is	be	AUX
ejpam-6565	201	47	τ⋆(σ1	τ⋆(σ1	NUM
ejpam-6565	201	48	,	,	PUNCT
ejpam-6565	201	49	σ2)-continuous	σ2)-continuous	ADJ
ejpam-6565	201	50	;	;	PUNCT
ejpam-6565	201	51	(	(	PUNCT
ejpam-6565	201	52	2	2	X
ejpam-6565	201	53	)	)	PUNCT
ejpam-6565	201	54	f−1((σ1	f−1((σ1	NOUN
ejpam-6565	201	55	,	,	PUNCT
ejpam-6565	201	56	σ2)θ	σ2)θ	NOUN
ejpam-6565	201	57	-	-	PUNCT
ejpam-6565	201	58	cl(b	cl(b	NOUN
ejpam-6565	201	59	)	)	PUNCT
ejpam-6565	201	60	)	)	PUNCT
ejpam-6565	201	61	is	be	AUX
ejpam-6565	201	62	⋆-closed	⋆-close	VERB
ejpam-6565	201	63	in	in	ADP
ejpam-6565	201	64	x	x	PUNCT
ejpam-6565	201	65	for	for	ADP
ejpam-6565	201	66	every	every	DET
ejpam-6565	201	67	subset	subset	NOUN
ejpam-6565	201	68	b	b	PROPN
ejpam-6565	201	69	of	of	ADP
ejpam-6565	201	70	y	y	PROPN
ejpam-6565	201	71	;	;	PUNCT
ejpam-6565	201	72	(	(	PUNCT
ejpam-6565	201	73	3	3	X
ejpam-6565	201	74	)	)	PUNCT
ejpam-6565	201	75	f−1(k	f−1(k	PROPN
ejpam-6565	201	76	)	)	PUNCT
ejpam-6565	201	77	is	be	AUX
ejpam-6565	201	78	⋆-closed	⋆-close	VERB
ejpam-6565	201	79	in	in	ADP
ejpam-6565	201	80	x	x	PUNCT
ejpam-6565	201	81	for	for	SCONJ
ejpam-6565	201	82	every	every	DET
ejpam-6565	201	83	(	(	PUNCT
ejpam-6565	201	84	σ1	σ1	PROPN
ejpam-6565	201	85	,	,	PUNCT
ejpam-6565	201	86	σ2)θ	σ2)θ	NOUN
ejpam-6565	201	87	-	-	PUNCT
ejpam-6565	201	88	closed	close	VERB
ejpam-6565	201	89	set	set	NOUN
ejpam-6565	201	90	k	k	PROPN
ejpam-6565	201	91	of	of	ADP
ejpam-6565	201	92	y	y	PROPN
ejpam-6565	201	93	;	;	PUNCT
ejpam-6565	201	94	(	(	PUNCT
ejpam-6565	201	95	4	4	X
ejpam-6565	201	96	)	)	PUNCT
ejpam-6565	201	97	f−1(v	f−1(v	NOUN
ejpam-6565	201	98	)	)	PUNCT
ejpam-6565	201	99	is	be	AUX
ejpam-6565	201	100	⋆-open	⋆-open	ADJ
ejpam-6565	201	101	in	in	ADP
ejpam-6565	201	102	x	x	PUNCT
ejpam-6565	201	103	for	for	ADP
ejpam-6565	201	104	every	every	DET
ejpam-6565	201	105	(	(	PUNCT
ejpam-6565	201	106	σ1	σ1	PROPN
ejpam-6565	201	107	,	,	PUNCT
ejpam-6565	201	108	σ2)θ	σ2)θ	NOUN
ejpam-6565	201	109	-	-	PUNCT
ejpam-6565	201	110	open	open	ADJ
ejpam-6565	201	111	set	set	NOUN
ejpam-6565	201	112	v	v	NOUN
ejpam-6565	201	113	of	of	ADP
ejpam-6565	201	114	y	y	PROPN
ejpam-6565	201	115	.	.	PUNCT
ejpam-6565	202	1	acknowledgements	acknowledgement	NOUN
ejpam-6565	202	2	this	this	DET
ejpam-6565	202	3	research	research	NOUN
ejpam-6565	202	4	project	project	NOUN
ejpam-6565	202	5	was	be	AUX
ejpam-6565	202	6	financially	financially	ADV
ejpam-6565	202	7	supported	support	VERB
ejpam-6565	202	8	by	by	ADP
ejpam-6565	202	9	mahasarakham	mahasarakham	PROPN
ejpam-6565	202	10	university	university	PROPN
ejpam-6565	202	11	.	.	PUNCT
ejpam-6565	203	1	references	reference	NOUN
ejpam-6565	203	2	[	[	X
ejpam-6565	203	3	1	1	NUM
ejpam-6565	203	4	]	]	PUNCT
ejpam-6565	203	5	k.	k.	PROPN
ejpam-6565	203	6	kuratowski	kuratowski	PROPN
ejpam-6565	203	7	.	.	PUNCT
ejpam-6565	204	1	topology	topology	PROPN
ejpam-6565	204	2	,	,	PUNCT
ejpam-6565	204	3	vol	vol	NOUN
ejpam-6565	204	4	.	.	PUNCT
ejpam-6565	204	5	i.	i.	PROPN
ejpam-6565	204	6	academic	academic	PROPN
ejpam-6565	204	7	press	press	PROPN
ejpam-6565	204	8	,	,	PUNCT
ejpam-6565	204	9	new	new	PROPN
ejpam-6565	204	10	york	york	PROPN
ejpam-6565	204	11	,	,	PUNCT
ejpam-6565	204	12	1966	1966	NUM
ejpam-6565	204	13	.	.	PUNCT
ejpam-6565	205	1	[	[	X
ejpam-6565	205	2	2	2	NUM
ejpam-6565	205	3	]	]	X
ejpam-6565	205	4	r.	r.	NOUN
ejpam-6565	205	5	vaidyanathaswamy	vaidyanathaswamy	PROPN
ejpam-6565	205	6	.	.	PUNCT
ejpam-6565	206	1	the	the	DET
ejpam-6565	206	2	localisation	localisation	NOUN
ejpam-6565	206	3	theory	theory	NOUN
ejpam-6565	206	4	in	in	ADP
ejpam-6565	206	5	set	set	NOUN
ejpam-6565	206	6	-	-	PUNCT
ejpam-6565	206	7	topology	topology	NOUN
ejpam-6565	206	8	.	.	PUNCT
ejpam-6565	207	1	proceedings	proceeding	NOUN
ejpam-6565	207	2	of	of	ADP
ejpam-6565	207	3	the	the	DET
ejpam-6565	207	4	indian	indian	PROPN
ejpam-6565	207	5	academy	academy	PROPN
ejpam-6565	207	6	of	of	ADP
ejpam-6565	207	7	sciences	science	NOUN
ejpam-6565	207	8	-	-	PUNCT
ejpam-6565	207	9	section	section	NOUN
ejpam-6565	207	10	a	a	PRON
ejpam-6565	207	11	,	,	PUNCT
ejpam-6565	207	12	20:51–61	20:51–61	NUM
ejpam-6565	207	13	,	,	PUNCT
ejpam-6565	207	14	1944	1944	NUM
ejpam-6565	207	15	.	.	PUNCT
ejpam-6565	208	1	[	[	X
ejpam-6565	208	2	3	3	X
ejpam-6565	208	3	]	]	X
ejpam-6565	208	4	e.	e.	PROPN
ejpam-6565	208	5	hatir	hatir	PROPN
ejpam-6565	208	6	and	and	CCONJ
ejpam-6565	208	7	t.	t.	PROPN
ejpam-6565	208	8	noiri	noiri	PROPN
ejpam-6565	208	9	.	.	PUNCT
ejpam-6565	209	1	weakly	weakly	ADJ
ejpam-6565	209	2	pre	pre	ADJ
ejpam-6565	209	3	-	-	ADJ
ejpam-6565	209	4	i	i	PRON
ejpam-6565	209	5	-	-	PUNCT
ejpam-6565	209	6	open	open	ADJ
ejpam-6565	209	7	sets	set	NOUN
ejpam-6565	209	8	and	and	CCONJ
ejpam-6565	209	9	decomposition	decomposition	NOUN
ejpam-6565	209	10	of	of	ADP
ejpam-6565	209	11	continuity	continuity	NOUN
ejpam-6565	209	12	.	.	PUNCT
ejpam-6565	210	1	acta	acta	PROPN
ejpam-6565	210	2	mathematica	mathematica	PROPN
ejpam-6565	210	3	hungarica	hungarica	PROPN
ejpam-6565	210	4	,	,	PUNCT
ejpam-6565	210	5	106(3):227–238	106(3):227–238	NUM
ejpam-6565	210	6	,	,	PUNCT
ejpam-6565	210	7	2005	2005	NUM
ejpam-6565	210	8	.	.	PUNCT
ejpam-6565	211	1	[	[	X
ejpam-6565	211	2	4	4	X
ejpam-6565	211	3	]	]	X
ejpam-6565	211	4	e.	e.	PROPN
ejpam-6565	211	5	hatir	hatir	PROPN
ejpam-6565	211	6	and	and	CCONJ
ejpam-6565	211	7	t.	t.	PROPN
ejpam-6565	211	8	noiri	noiri	PROPN
ejpam-6565	211	9	.	.	PUNCT
ejpam-6565	212	1	on	on	ADP
ejpam-6565	212	2	decompositions	decomposition	NOUN
ejpam-6565	212	3	of	of	ADP
ejpam-6565	212	4	continuity	continuity	NOUN
ejpam-6565	212	5	via	via	ADP
ejpam-6565	212	6	idealization	idealization	NOUN
ejpam-6565	212	7	.	.	PUNCT
ejpam-6565	213	1	acta	acta	PROPN
ejpam-6565	213	2	mathematica	mathematica	PROPN
ejpam-6565	213	3	hungarica	hungarica	PROPN
ejpam-6565	213	4	,	,	PUNCT
ejpam-6565	213	5	96:341–349	96:341–349	PROPN
ejpam-6565	213	6	,	,	PUNCT
ejpam-6565	213	7	2002	2002	NUM
ejpam-6565	213	8	.	.	PUNCT
ejpam-6565	214	1	[	[	X
ejpam-6565	214	2	5	5	X
ejpam-6565	214	3	]	]	PUNCT
ejpam-6565	214	4	c.	c.	PROPN
ejpam-6565	214	5	boonpok	boonpok	PROPN
ejpam-6565	214	6	.	.	PUNCT
ejpam-6565	215	1	on	on	ADP
ejpam-6565	215	2	continuous	continuous	ADJ
ejpam-6565	215	3	multifunctions	multifunction	NOUN
ejpam-6565	215	4	in	in	ADP
ejpam-6565	215	5	ideal	ideal	ADJ
ejpam-6565	215	6	topological	topological	ADJ
ejpam-6565	215	7	spaces	space	NOUN
ejpam-6565	215	8	.	.	PUNCT
ejpam-6565	216	1	lobachevskii	lobachevskii	PROPN
ejpam-6565	216	2	journal	journal	PROPN
ejpam-6565	216	3	of	of	ADP
ejpam-6565	216	4	mathematics	mathematic	NOUN
ejpam-6565	216	5	,	,	PUNCT
ejpam-6565	216	6	40(1):24–35	40(1):24–35	NUM
ejpam-6565	216	7	,	,	PUNCT
ejpam-6565	216	8	2019	2019	NUM
ejpam-6565	216	9	.	.	PUNCT
ejpam-6565	217	1	[	[	X
ejpam-6565	217	2	6	6	NUM
ejpam-6565	217	3	]	]	PUNCT
ejpam-6565	217	4	c.	c.	PROPN
ejpam-6565	217	5	boonpok	boonpok	PROPN
ejpam-6565	217	6	.	.	PUNCT
ejpam-6565	218	1	pı	pı	NOUN
ejpam-6565	218	2	-	-	NOUN
ejpam-6565	218	3	continuity	continuity	NOUN
ejpam-6565	218	4	and	and	CCONJ
ejpam-6565	218	5	weak	weak	ADJ
ejpam-6565	218	6	pı	pı	NOUN
ejpam-6565	218	7	-	-	NOUN
ejpam-6565	218	8	continuity	continuity	NOUN
ejpam-6565	218	9	.	.	PUNCT
ejpam-6565	219	1	carpathian	carpathian	ADJ
ejpam-6565	219	2	mathematical	mathematical	ADJ
ejpam-6565	219	3	publications	publication	NOUN
ejpam-6565	219	4	,	,	PUNCT
ejpam-6565	219	5	17(1):171–186	17(1):171–186	PROPN
ejpam-6565	219	6	,	,	PUNCT
ejpam-6565	219	7	2025	2025	NUM
ejpam-6565	219	8	.	.	PUNCT
ejpam-6565	220	1	[	[	X
ejpam-6565	220	2	7	7	X
ejpam-6565	220	3	]	]	X
ejpam-6565	220	4	p.	p.	NOUN
ejpam-6565	220	5	pue	pue	NOUN
ejpam-6565	220	6	-	-	PUNCT
ejpam-6565	220	7	on	on	ADP
ejpam-6565	220	8	,	,	PUNCT
ejpam-6565	220	9	s.	s.	PROPN
ejpam-6565	220	10	sompong	sompong	PROPN
ejpam-6565	220	11	,	,	PUNCT
ejpam-6565	220	12	and	and	CCONJ
ejpam-6565	220	13	c.	c.	PROPN
ejpam-6565	220	14	boonpok	boonpok	PROPN
ejpam-6565	220	15	.	.	PUNCT
ejpam-6565	221	1	upper	upper	ADJ
ejpam-6565	221	2	and	and	CCONJ
ejpam-6565	221	3	lower	low	ADJ
ejpam-6565	221	4	(	(	PUNCT
ejpam-6565	221	5	τ1	τ1	NOUN
ejpam-6565	221	6	,	,	PUNCT
ejpam-6565	221	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	221	8	multifunctions	multifunction	NOUN
ejpam-6565	221	9	.	.	PUNCT
ejpam-6565	222	1	international	international	ADJ
ejpam-6565	222	2	journal	journal	PROPN
ejpam-6565	222	3	of	of	ADP
ejpam-6565	222	4	mathematics	mathematic	NOUN
ejpam-6565	222	5	and	and	CCONJ
ejpam-6565	222	6	computer	computer	NOUN
ejpam-6565	222	7	science	science	NOUN
ejpam-6565	222	8	,	,	PUNCT
ejpam-6565	222	9	19(4):1305	19(4):1305	NUM
ejpam-6565	222	10	–	–	PUNCT
ejpam-6565	222	11	1310	1310	NUM
ejpam-6565	222	12	,	,	PUNCT
ejpam-6565	222	13	2024	2024	NUM
ejpam-6565	222	14	.	.	PUNCT
ejpam-6565	223	1	[	[	X
ejpam-6565	223	2	8	8	NUM
ejpam-6565	223	3	]	]	X
ejpam-6565	223	4	c.	c.	PROPN
ejpam-6565	223	5	klanarong	klanarong	PROPN
ejpam-6565	223	6	,	,	PUNCT
ejpam-6565	223	7	s.	s.	PROPN
ejpam-6565	223	8	sompong	sompong	PROPN
ejpam-6565	223	9	,	,	PUNCT
ejpam-6565	223	10	and	and	CCONJ
ejpam-6565	223	11	c.	c.	PROPN
ejpam-6565	223	12	boonpok	boonpok	PROPN
ejpam-6565	223	13	.	.	PUNCT
ejpam-6565	224	1	(	(	PUNCT
ejpam-6565	224	2	τ1	τ1	NOUN
ejpam-6565	224	3	,	,	PUNCT
ejpam-6565	224	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6565	224	5	and	and	CCONJ
ejpam-6565	224	6	(	(	PUNCT
ejpam-6565	224	7	τ1	τ1	NOUN
ejpam-6565	224	8	,	,	PUNCT
ejpam-6565	224	9	τ2)θ	τ2)θ	ADJ
ejpam-6565	224	10	-	-	PUNCT
ejpam-6565	224	11	closed	close	VERB
ejpam-6565	224	12	sets	set	NOUN
ejpam-6565	224	13	.	.	PUNCT
ejpam-6565	225	1	international	international	ADJ
ejpam-6565	225	2	journal	journal	NOUN
ejpam-6565	225	3	of	of	ADP
ejpam-6565	225	4	mathematics	mathematic	NOUN
ejpam-6565	225	5	and	and	CCONJ
ejpam-6565	225	6	computer	computer	NOUN
ejpam-6565	225	7	science	science	NOUN
ejpam-6565	225	8	,	,	PUNCT
ejpam-6565	225	9	19(4):1299–1304	19(4):1299–1304	NUM
ejpam-6565	225	10	,	,	PUNCT
ejpam-6565	225	11	2024	2024	NUM
ejpam-6565	225	12	.	.	PUNCT
ejpam-6565	226	1	[	[	X
ejpam-6565	226	2	9	9	NUM
ejpam-6565	226	3	]	]	PUNCT
ejpam-6565	226	4	m.	m.	NOUN
ejpam-6565	226	5	thongmoon	thongmoon	NOUN
ejpam-6565	226	6	,	,	PUNCT
ejpam-6565	226	7	s.	s.	PROPN
ejpam-6565	226	8	sompong	sompong	PROPN
ejpam-6565	226	9	,	,	PUNCT
ejpam-6565	226	10	and	and	CCONJ
ejpam-6565	226	11	c.	c.	PROPN
ejpam-6565	226	12	boonpok	boonpok	PROPN
ejpam-6565	226	13	.	.	PUNCT
ejpam-6565	227	1	(	(	PUNCT
ejpam-6565	227	2	τ1	τ1	NOUN
ejpam-6565	227	3	,	,	PUNCT
ejpam-6565	227	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	227	5	multifunctions	multifunction	NOUN
ejpam-6565	227	6	and	and	CCONJ
ejpam-6565	227	7	τ1τ2	τ1τ2	NOUN
ejpam-6565	227	8	-	-	ADJ
ejpam-6565	227	9	δ	δ	NOUN
ejpam-6565	227	10	-	-	ADJ
ejpam-6565	227	11	open	open	ADJ
ejpam-6565	227	12	sets	set	NOUN
ejpam-6565	227	13	.	.	PUNCT
ejpam-6565	228	1	international	international	ADJ
ejpam-6565	228	2	journal	journal	NOUN
ejpam-6565	228	3	of	of	ADP
ejpam-6565	228	4	mathematics	mathematic	NOUN
ejpam-6565	228	5	and	and	CCONJ
ejpam-6565	228	6	computer	computer	NOUN
ejpam-6565	228	7	science	science	NOUN
ejpam-6565	228	8	,	,	PUNCT
ejpam-6565	228	9	19(4):1369–1375	19(4):1369–1375	NUM
ejpam-6565	228	10	,	,	PUNCT
ejpam-6565	228	11	2024	2024	NUM
ejpam-6565	228	12	.	.	PUNCT
ejpam-6565	229	1	[	[	X
ejpam-6565	229	2	10	10	NUM
ejpam-6565	229	3	]	]	X
ejpam-6565	229	4	c.	c.	PROPN
ejpam-6565	229	5	boonpok	boonpok	PROPN
ejpam-6565	229	6	,	,	PUNCT
ejpam-6565	229	7	c.	c.	PROPN
ejpam-6565	229	8	viriyapong	viriyapong	PROPN
ejpam-6565	229	9	,	,	PUNCT
ejpam-6565	229	10	and	and	CCONJ
ejpam-6565	229	11	m.	m.	NOUN
ejpam-6565	229	12	thongmoon	thongmoon	NOUN
ejpam-6565	229	13	.	.	PUNCT
ejpam-6565	230	1	on	on	ADP
ejpam-6565	230	2	upper	upper	ADJ
ejpam-6565	230	3	and	and	CCONJ
ejpam-6565	230	4	lower	low	ADJ
ejpam-6565	230	5	(	(	PUNCT
ejpam-6565	230	6	τ1	τ1	NOUN
ejpam-6565	230	7	,	,	PUNCT
ejpam-6565	230	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6565	230	9	multifunctions	multifunction	NOUN
ejpam-6565	230	10	.	.	PUNCT
ejpam-6565	231	1	journal	journal	PROPN
ejpam-6565	231	2	of	of	ADP
ejpam-6565	231	3	mathematics	mathematics	PROPN
ejpam-6565	231	4	and	and	CCONJ
ejpam-6565	231	5	computer	computer	NOUN
ejpam-6565	231	6	science	science	NOUN
ejpam-6565	231	7	,	,	PUNCT
ejpam-6565	231	8	18:282–293	18:282–293	NUM
ejpam-6565	231	9	,	,	PUNCT
ejpam-6565	231	10	2018	2018	NUM
ejpam-6565	231	11	.	.	PUNCT
ejpam-6565	232	1	j.	j.	PROPN
ejpam-6565	232	2	khampakdee	khampakdee	PROPN
ejpam-6565	232	3	,	,	PUNCT
ejpam-6565	232	4	a.	a.	PROPN
ejpam-6565	232	5	sama	sama	PROPN
ejpam-6565	232	6	-	-	PUNCT
ejpam-6565	232	7	ae	ae	PROPN
ejpam-6565	232	8	,	,	PUNCT
ejpam-6565	232	9	c.	c.	PROPN
ejpam-6565	232	10	boonpok	boonpok	PROPN
ejpam-6565	232	11	/	/	SYM
ejpam-6565	232	12	eur	eur	PROPN
ejpam-6565	232	13	.	.	PUNCT
ejpam-6565	233	1	j.	j.	PROPN
ejpam-6565	233	2	pure	pure	PROPN
ejpam-6565	233	3	appl	appl	PROPN
ejpam-6565	233	4	.	.	PROPN
ejpam-6565	233	5	math	math	PROPN
ejpam-6565	233	6	,	,	PUNCT
ejpam-6565	233	7	18	18	NUM
ejpam-6565	233	8	(	(	PUNCT
ejpam-6565	233	9	3	3	NUM
ejpam-6565	233	10	)	)	PUNCT
ejpam-6565	233	11	(	(	PUNCT
ejpam-6565	233	12	2025	2025	NUM
ejpam-6565	233	13	)	)	PUNCT
ejpam-6565	233	14	,	,	PUNCT
ejpam-6565	233	15	6565	6565	NUM
ejpam-6565	233	16	9	9	NUM
ejpam-6565	233	17	of	of	ADP
ejpam-6565	233	18	9	9	NUM
ejpam-6565	233	19	[	[	SYM
ejpam-6565	233	20	11	11	NUM
ejpam-6565	233	21	]	]	X
ejpam-6565	233	22	c.	c.	PROPN
ejpam-6565	233	23	viriyapong	viriyapong	PROPN
ejpam-6565	233	24	and	and	CCONJ
ejpam-6565	233	25	c.	c.	PROPN
ejpam-6565	233	26	boonpok	boonpok	PROPN
ejpam-6565	233	27	.	.	PUNCT
ejpam-6565	234	1	(	(	PUNCT
ejpam-6565	234	2	τ1	τ1	NOUN
ejpam-6565	234	3	,	,	PUNCT
ejpam-6565	234	4	τ2)α	τ2)α	NOUN
ejpam-6565	234	5	-	-	PUNCT
ejpam-6565	234	6	continuity	continuity	NOUN
ejpam-6565	234	7	for	for	ADP
ejpam-6565	234	8	multifunctions	multifunction	NOUN
ejpam-6565	234	9	.	.	PUNCT
ejpam-6565	235	1	journal	journal	PROPN
ejpam-6565	235	2	of	of	ADP
ejpam-6565	235	3	mathematics	mathematic	NOUN
ejpam-6565	235	4	,	,	PUNCT
ejpam-6565	235	5	2020:6285763	2020:6285763	NUM
ejpam-6565	235	6	,	,	PUNCT
ejpam-6565	235	7	2020	2020	NUM
ejpam-6565	235	8	.	.	PUNCT
ejpam-6565	236	1	[	[	X
ejpam-6565	236	2	12	12	NUM
ejpam-6565	236	3	]	]	PUNCT
ejpam-6565	236	4	c.	c.	PROPN
ejpam-6565	236	5	boonpok	boonpok	PROPN
ejpam-6565	236	6	.	.	PUNCT
ejpam-6565	237	1	(	(	PUNCT
ejpam-6565	237	2	τ1	τ1	NOUN
ejpam-6565	237	3	,	,	PUNCT
ejpam-6565	237	4	τ2)δ	τ2)δ	ADJ
ejpam-6565	237	5	-	-	PUNCT
ejpam-6565	237	6	semicontinuous	semicontinuous	ADJ
ejpam-6565	237	7	multifunctions	multifunction	NOUN
ejpam-6565	237	8	.	.	PUNCT
ejpam-6565	238	1	heliyon	heliyon	NOUN
ejpam-6565	238	2	,	,	PUNCT
ejpam-6565	238	3	6	6	NUM
ejpam-6565	238	4	:	:	SYM
ejpam-6565	238	5	e05367	e05367	PROPN
ejpam-6565	238	6	,	,	PUNCT
ejpam-6565	238	7	2020	2020	NUM
ejpam-6565	238	8	.	.	PUNCT
ejpam-6565	239	1	[	[	X
ejpam-6565	239	2	13	13	NUM
ejpam-6565	239	3	]	]	X
ejpam-6565	239	4	n.	n.	PROPN
ejpam-6565	239	5	viriyapong	viriyapong	PROPN
ejpam-6565	239	6	,	,	PUNCT
ejpam-6565	239	7	s.	s.	PROPN
ejpam-6565	239	8	sompong	sompong	PROPN
ejpam-6565	239	9	,	,	PUNCT
ejpam-6565	239	10	and	and	CCONJ
ejpam-6565	239	11	c.	c.	PROPN
ejpam-6565	239	12	boonpok	boonpok	PROPN
ejpam-6565	239	13	.	.	PUNCT
ejpam-6565	240	1	(	(	PUNCT
ejpam-6565	240	2	τ1	τ1	NOUN
ejpam-6565	240	3	,	,	PUNCT
ejpam-6565	240	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6565	240	5	disconnectedness	disconnectedness	NOUN
ejpam-6565	240	6	in	in	ADP
ejpam-6565	240	7	bitopological	bitopological	ADJ
ejpam-6565	240	8	spaces	space	NOUN
ejpam-6565	240	9	.	.	PUNCT
ejpam-6565	241	1	international	international	ADJ
ejpam-6565	241	2	journal	journal	PROPN
ejpam-6565	241	3	of	of	ADP
ejpam-6565	241	4	mathematics	mathematic	NOUN
ejpam-6565	241	5	and	and	CCONJ
ejpam-6565	241	6	computer	computer	NOUN
ejpam-6565	241	7	science	science	NOUN
ejpam-6565	241	8	,	,	PUNCT
ejpam-6565	241	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6565	241	10	,	,	PUNCT
ejpam-6565	241	11	2024	2024	NUM
ejpam-6565	241	12	.	.	PUNCT
ejpam-6565	242	1	[	[	X
ejpam-6565	242	2	14	14	NUM
ejpam-6565	242	3	]	]	X
ejpam-6565	242	4	c.	c.	PROPN
ejpam-6565	242	5	boonpok	boonpok	PROPN
ejpam-6565	242	6	and	and	CCONJ
ejpam-6565	242	7	p.	p.	NOUN
ejpam-6565	242	8	pue	pue	NOUN
ejpam-6565	242	9	-	-	PUNCT
ejpam-6565	242	10	on	on	ADP
ejpam-6565	242	11	.	.	PUNCT
ejpam-6565	243	1	characterizations	characterization	NOUN
ejpam-6565	243	2	of	of	ADP
ejpam-6565	243	3	almost	almost	ADV
ejpam-6565	243	4	(	(	PUNCT
ejpam-6565	243	5	τ1	τ1	NOUN
ejpam-6565	243	6	,	,	PUNCT
ejpam-6565	243	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6565	243	8	multifunctions	multifunction	NOUN
ejpam-6565	243	9	.	.	PUNCT
ejpam-6565	244	1	international	international	ADJ
ejpam-6565	244	2	journal	journal	NOUN
ejpam-6565	244	3	of	of	ADP
ejpam-6565	244	4	analysis	analysis	NOUN
ejpam-6565	244	5	and	and	CCONJ
ejpam-6565	244	6	applications	application	NOUN
ejpam-6565	244	7	,	,	PUNCT
ejpam-6565	244	8	22:33	22:33	NUM
ejpam-6565	244	9	,	,	PUNCT
ejpam-6565	244	10	2024	2024	NUM
ejpam-6565	244	11	.	.	PUNCT
ejpam-6565	245	1	[	[	X
ejpam-6565	245	2	15	15	NUM
ejpam-6565	245	3	]	]	X
ejpam-6565	245	4	d.	d.	PROPN
ejpam-6565	245	5	janković	janković	PROPN
ejpam-6565	245	6	and	and	CCONJ
ejpam-6565	245	7	t.	t.	PROPN
ejpam-6565	245	8	r.	r.	PROPN
ejpam-6565	245	9	hamlett	hamlett	PROPN
ejpam-6565	245	10	.	.	PUNCT
ejpam-6565	246	1	new	new	ADJ
ejpam-6565	246	2	topologies	topology	NOUN
ejpam-6565	246	3	from	from	ADP
ejpam-6565	246	4	old	old	ADJ
ejpam-6565	246	5	via	via	ADP
ejpam-6565	246	6	ideals	ideal	NOUN
ejpam-6565	246	7	.	.	PUNCT
ejpam-6565	247	1	the	the	DET
ejpam-6565	247	2	american	american	PROPN
ejpam-6565	247	3	mathematical	mathematical	PROPN
ejpam-6565	247	4	monthly	monthly	ADV
ejpam-6565	247	5	,	,	PUNCT
ejpam-6565	247	6	97:295–310	97:295–310	PROPN
ejpam-6565	247	7	,	,	PUNCT
ejpam-6565	247	8	1990	1990	NUM
ejpam-6565	247	9	.	.	PUNCT
ejpam-6565	248	1	[	[	X
ejpam-6565	248	2	16	16	NUM
ejpam-6565	248	3	]	]	X
ejpam-6565	248	4	e.	e.	PROPN
ejpam-6565	248	5	ekici	ekici	PROPN
ejpam-6565	248	6	and	and	CCONJ
ejpam-6565	248	7	t.	t.	PROPN
ejpam-6565	248	8	noiri	noiri	PROPN
ejpam-6565	248	9	.	.	PUNCT
ejpam-6565	249	1	⋆-extremally	⋆-extremally	ADV
ejpam-6565	249	2	disconnected	disconnect	VERB
ejpam-6565	249	3	ideal	ideal	ADJ
ejpam-6565	249	4	topological	topological	ADJ
ejpam-6565	249	5	spaces	space	NOUN
ejpam-6565	249	6	.	.	PUNCT
ejpam-6565	250	1	acta	acta	PROPN
ejpam-6565	250	2	mathematica	mathematica	PROPN
ejpam-6565	250	3	hungarica	hungarica	PROPN
ejpam-6565	250	4	,	,	PUNCT
ejpam-6565	250	5	122:81–90	122:81–90	NUM
ejpam-6565	250	6	,	,	PUNCT
ejpam-6565	250	7	2009	2009	NUM
ejpam-6565	250	8	.	.	PUNCT
ejpam-6565	251	1	[	[	X
ejpam-6565	251	2	17	17	NUM
ejpam-6565	251	3	]	]	PUNCT
ejpam-6565	251	4	m.	m.	NOUN
ejpam-6565	251	5	chiangpradit	chiangpradit	NOUN
ejpam-6565	251	6	,	,	PUNCT
ejpam-6565	251	7	s.	s.	PROPN
ejpam-6565	251	8	sompong	sompong	PROPN
ejpam-6565	251	9	,	,	PUNCT
ejpam-6565	251	10	and	and	CCONJ
ejpam-6565	251	11	c.	c.	PROPN
ejpam-6565	251	12	boonpok	boonpok	PROPN
ejpam-6565	251	13	.	.	PUNCT
ejpam-6565	252	1	on	on	ADP
ejpam-6565	252	2	characterizations	characterization	NOUN
ejpam-6565	252	3	of	of	ADP
ejpam-6565	252	4	(	(	PUNCT
ejpam-6565	252	5	τ1	τ1	NOUN
ejpam-6565	252	6	,	,	PUNCT
ejpam-6565	252	7	τ2)regular	τ2)regular	ADJ
ejpam-6565	252	8	spaces	space	NOUN
ejpam-6565	252	9	.	.	PUNCT
ejpam-6565	253	1	international	international	ADJ
ejpam-6565	253	2	journal	journal	PROPN
ejpam-6565	253	3	of	of	ADP
ejpam-6565	253	4	mathematics	mathematic	NOUN
ejpam-6565	253	5	and	and	CCONJ
ejpam-6565	253	6	computer	computer	NOUN
ejpam-6565	253	7	science	science	NOUN
ejpam-6565	253	8	,	,	PUNCT
ejpam-6565	253	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-6565	253	10	,	,	PUNCT
ejpam-6565	253	11	2024	2024	NUM
ejpam-6565	253	12	.	.	PUNCT