id	sid	tid	token	lemma	pos
ejpam-5321	1	1	european	european	PROPN
ejpam-5321	1	2	journal	journal	PROPN
ejpam-5321	1	3	of	of	ADP
ejpam-5321	1	4	pure	pure	ADJ
ejpam-5321	1	5	and	and	CCONJ
ejpam-5321	1	6	applied	apply	VERB
ejpam-5321	1	7	mathematics	mathematic	NOUN
ejpam-5321	1	8	vol	vol	NOUN
ejpam-5321	1	9	.	.	PROPN
ejpam-5321	2	1	17	17	NUM
ejpam-5321	2	2	,	,	PUNCT
ejpam-5321	2	3	no	no	INTJ
ejpam-5321	2	4	.	.	NOUN
ejpam-5321	2	5	3	3	NUM
ejpam-5321	2	6	,	,	PUNCT
ejpam-5321	2	7	2024	2024	NUM
ejpam-5321	2	8	,	,	PUNCT
ejpam-5321	2	9	2142	2142	NUM
ejpam-5321	2	10	-	-	SYM
ejpam-5321	2	11	2154	2154	NUM
ejpam-5321	2	12	issn	issn	PROPN
ejpam-5321	2	13	1307	1307	NUM
ejpam-5321	2	14	-	-	SYM
ejpam-5321	2	15	5543	5543	NUM
ejpam-5321	2	16	–	–	PUNCT
ejpam-5321	2	17	ejpam.com	ejpam.com	X
ejpam-5321	2	18	published	publish	VERB
ejpam-5321	2	19	by	by	ADP
ejpam-5321	2	20	new	new	PROPN
ejpam-5321	2	21	york	york	PROPN
ejpam-5321	2	22	business	business	PROPN
ejpam-5321	2	23	global	global	PROPN
ejpam-5321	2	24	upper	upper	ADJ
ejpam-5321	2	25	and	and	CCONJ
ejpam-5321	2	26	lower	low	ADJ
ejpam-5321	2	27	slight	slight	ADJ
ejpam-5321	2	28	α(τ1	α(τ1	NOUN
ejpam-5321	2	29	,	,	PUNCT
ejpam-5321	2	30	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5321	2	31	chokchai	chokchai	ADJ
ejpam-5321	2	32	viriyapong1	viriyapong1	PROPN
ejpam-5321	2	33	,	,	PUNCT
ejpam-5321	2	34	supannee	supannee	PROPN
ejpam-5321	2	35	sompong2	sompong2	PROPN
ejpam-5321	2	36	,	,	PUNCT
ejpam-5321	2	37	chawalit	chawalit	VERB
ejpam-5321	2	38	boonpok1,∗	boonpok1,∗	NOUN
ejpam-5321	2	39	1	1	NUM
ejpam-5321	2	40	mathematics	mathematic	NOUN
ejpam-5321	2	41	and	and	CCONJ
ejpam-5321	2	42	applied	apply	VERB
ejpam-5321	2	43	mathematics	mathematics	PROPN
ejpam-5321	2	44	research	research	NOUN
ejpam-5321	2	45	unit	unit	NOUN
ejpam-5321	2	46	,	,	PUNCT
ejpam-5321	2	47	department	department	NOUN
ejpam-5321	2	48	of	of	ADP
ejpam-5321	2	49	mathematics	mathematic	NOUN
ejpam-5321	2	50	,	,	PUNCT
ejpam-5321	2	51	faculty	faculty	NOUN
ejpam-5321	2	52	of	of	ADP
ejpam-5321	2	53	science	science	NOUN
ejpam-5321	2	54	,	,	PUNCT
ejpam-5321	2	55	mahasarakham	mahasarakham	PROPN
ejpam-5321	2	56	university	university	PROPN
ejpam-5321	2	57	,	,	PUNCT
ejpam-5321	2	58	maha	maha	PROPN
ejpam-5321	2	59	sarakham	sarakham	PROPN
ejpam-5321	2	60	,	,	PUNCT
ejpam-5321	2	61	44150	44150	NUM
ejpam-5321	2	62	,	,	PUNCT
ejpam-5321	2	63	thailand	thailand	PROPN
ejpam-5321	2	64	2	2	NUM
ejpam-5321	2	65	department	department	NOUN
ejpam-5321	2	66	of	of	ADP
ejpam-5321	2	67	mathematics	mathematic	NOUN
ejpam-5321	2	68	and	and	CCONJ
ejpam-5321	2	69	statistics	statistic	NOUN
ejpam-5321	2	70	,	,	PUNCT
ejpam-5321	2	71	faculty	faculty	NOUN
ejpam-5321	2	72	of	of	ADP
ejpam-5321	2	73	science	science	NOUN
ejpam-5321	2	74	and	and	CCONJ
ejpam-5321	2	75	technology	technology	NOUN
ejpam-5321	2	76	,	,	PUNCT
ejpam-5321	2	77	sakon	sakon	PROPN
ejpam-5321	2	78	nakhon	nakhon	PROPN
ejpam-5321	2	79	rajbhat	rajbhat	PROPN
ejpam-5321	2	80	university	university	PROPN
ejpam-5321	2	81	,	,	PUNCT
ejpam-5321	2	82	sakon	sakon	PROPN
ejpam-5321	2	83	nakhon	nakhon	PROPN
ejpam-5321	2	84	,	,	PUNCT
ejpam-5321	2	85	47000	47000	NUM
ejpam-5321	2	86	,	,	PUNCT
ejpam-5321	2	87	thailand	thailand	PROPN
ejpam-5321	2	88	abstract	abstract	NOUN
ejpam-5321	2	89	.	.	PUNCT
ejpam-5321	3	1	this	this	DET
ejpam-5321	3	2	paper	paper	NOUN
ejpam-5321	3	3	deals	deal	NOUN
ejpam-5321	3	4	with	with	ADP
ejpam-5321	3	5	the	the	DET
ejpam-5321	3	6	notions	notion	NOUN
ejpam-5321	3	7	of	of	ADP
ejpam-5321	3	8	upper	upper	ADJ
ejpam-5321	3	9	and	and	CCONJ
ejpam-5321	3	10	lower	low	ADJ
ejpam-5321	3	11	slightly	slightly	ADJ
ejpam-5321	3	12	α(τ1	α(τ1	NOUN
ejpam-5321	3	13	,	,	PUNCT
ejpam-5321	3	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	3	15	multifunctions	multifunction	NOUN
ejpam-5321	3	16	.	.	PUNCT
ejpam-5321	4	1	furthermore	furthermore	ADV
ejpam-5321	4	2	,	,	PUNCT
ejpam-5321	4	3	several	several	ADJ
ejpam-5321	4	4	characterizations	characterization	NOUN
ejpam-5321	4	5	of	of	ADP
ejpam-5321	4	6	upper	upper	ADJ
ejpam-5321	4	7	and	and	CCONJ
ejpam-5321	4	8	lower	low	ADJ
ejpam-5321	4	9	slightly	slightly	ADJ
ejpam-5321	4	10	α(τ1	α(τ1	NOUN
ejpam-5321	4	11	,	,	PUNCT
ejpam-5321	4	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	4	13	multifunctions	multifunction	NOUN
ejpam-5321	4	14	are	be	AUX
ejpam-5321	4	15	discussed	discuss	VERB
ejpam-5321	4	16	.	.	PUNCT
ejpam-5321	5	1	2020	2020	NUM
ejpam-5321	5	2	mathematics	mathematic	NOUN
ejpam-5321	5	3	subject	subject	NOUN
ejpam-5321	5	4	classifications	classification	NOUN
ejpam-5321	5	5	:	:	PUNCT
ejpam-5321	5	6	54c08	54c08	NUM
ejpam-5321	5	7	,	,	PUNCT
ejpam-5321	5	8	54c60	54c60	NUM
ejpam-5321	5	9	,	,	PUNCT
ejpam-5321	5	10	54e55	54e55	NUM
ejpam-5321	5	11	key	key	ADJ
ejpam-5321	5	12	words	word	NOUN
ejpam-5321	5	13	and	and	CCONJ
ejpam-5321	5	14	phrases	phrase	NOUN
ejpam-5321	5	15	:	:	PUNCT
ejpam-5321	5	16	α(τ1	α(τ1	NOUN
ejpam-5321	5	17	,	,	PUNCT
ejpam-5321	5	18	τ2)-open	τ2)-open	ADJ
ejpam-5321	5	19	set	set	NOUN
ejpam-5321	5	20	,	,	PUNCT
ejpam-5321	5	21	upper	upper	ADJ
ejpam-5321	5	22	slightly	slightly	ADV
ejpam-5321	5	23	α(τ1	α(τ1	NOUN
ejpam-5321	5	24	,	,	PUNCT
ejpam-5321	5	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	5	26	multifunction	multifunction	NOUN
ejpam-5321	5	27	,	,	PUNCT
ejpam-5321	5	28	lower	low	ADJ
ejpam-5321	5	29	slightly	slightly	ADV
ejpam-5321	5	30	α(τ1	α(τ1	NOUN
ejpam-5321	5	31	,	,	PUNCT
ejpam-5321	5	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	5	33	multifunction	multifunction	NOUN
ejpam-5321	5	34	1	1	NUM
ejpam-5321	5	35	.	.	PUNCT
ejpam-5321	5	36	introduction	introduction	NOUN
ejpam-5321	5	37	in	in	ADP
ejpam-5321	5	38	1980	1980	NUM
ejpam-5321	5	39	,	,	PUNCT
ejpam-5321	5	40	jain	jain	PROPN
ejpam-5321	6	1	[	[	X
ejpam-5321	6	2	25	25	NUM
ejpam-5321	6	3	]	]	PUNCT
ejpam-5321	6	4	introduced	introduce	VERB
ejpam-5321	6	5	the	the	DET
ejpam-5321	6	6	notion	notion	NOUN
ejpam-5321	6	7	of	of	ADP
ejpam-5321	6	8	slightly	slightly	ADV
ejpam-5321	6	9	continuous	continuous	ADJ
ejpam-5321	6	10	functions	function	NOUN
ejpam-5321	6	11	.	.	PUNCT
ejpam-5321	7	1	nour	nour	PROPN
ejpam-5321	8	1	[	[	X
ejpam-5321	8	2	32	32	NUM
ejpam-5321	8	3	]	]	PUNCT
ejpam-5321	8	4	defined	define	VERB
ejpam-5321	8	5	slightly	slightly	ADV
ejpam-5321	8	6	semi	semi	ADJ
ejpam-5321	8	7	-	-	ADJ
ejpam-5321	8	8	continuous	continuous	ADJ
ejpam-5321	8	9	functions	function	NOUN
ejpam-5321	8	10	as	as	ADP
ejpam-5321	8	11	a	a	DET
ejpam-5321	8	12	weak	weak	ADJ
ejpam-5321	8	13	form	form	NOUN
ejpam-5321	8	14	of	of	ADP
ejpam-5321	8	15	slight	slight	ADJ
ejpam-5321	8	16	continuity	continuity	NOUN
ejpam-5321	8	17	and	and	CCONJ
ejpam-5321	8	18	investigated	investigate	VERB
ejpam-5321	8	19	some	some	DET
ejpam-5321	8	20	characterizations	characterization	NOUN
ejpam-5321	8	21	of	of	ADP
ejpam-5321	8	22	slightly	slightly	ADV
ejpam-5321	8	23	semi	semi	ADJ
ejpam-5321	8	24	-	-	ADJ
ejpam-5321	8	25	continuous	continuous	ADJ
ejpam-5321	8	26	functions	function	NOUN
ejpam-5321	8	27	.	.	PUNCT
ejpam-5321	9	1	noiri	noiri	PROPN
ejpam-5321	9	2	and	and	CCONJ
ejpam-5321	9	3	chae	chae	PROPN
ejpam-5321	9	4	[	[	X
ejpam-5321	9	5	30	30	NUM
ejpam-5321	9	6	]	]	PUNCT
ejpam-5321	9	7	have	have	AUX
ejpam-5321	9	8	further	far	ADV
ejpam-5321	9	9	investigated	investigate	VERB
ejpam-5321	9	10	slightly	slightly	ADV
ejpam-5321	9	11	semi	semi	ADJ
ejpam-5321	9	12	-	-	ADJ
ejpam-5321	9	13	continuous	continuous	ADJ
ejpam-5321	9	14	functions	function	NOUN
ejpam-5321	9	15	.	.	PUNCT
ejpam-5321	10	1	pal	pal	NOUN
ejpam-5321	10	2	and	and	CCONJ
ejpam-5321	10	3	bhattacharyya	bhattacharyya	ADJ
ejpam-5321	10	4	[	[	X
ejpam-5321	10	5	33	33	NUM
ejpam-5321	10	6	]	]	PUNCT
ejpam-5321	10	7	introduced	introduce	VERB
ejpam-5321	10	8	and	and	CCONJ
ejpam-5321	10	9	studied	study	VERB
ejpam-5321	10	10	the	the	DET
ejpam-5321	10	11	concept	concept	NOUN
ejpam-5321	10	12	of	of	ADP
ejpam-5321	10	13	faintly	faintly	ADV
ejpam-5321	10	14	precontinuous	precontinuous	ADJ
ejpam-5321	10	15	functions	function	NOUN
ejpam-5321	10	16	.	.	PUNCT
ejpam-5321	11	1	slight	slight	ADJ
ejpam-5321	11	2	continuity	continuity	NOUN
ejpam-5321	11	3	implies	imply	VERB
ejpam-5321	11	4	both	both	DET
ejpam-5321	11	5	slight	slight	ADJ
ejpam-5321	11	6	semi	semi	ADJ
ejpam-5321	11	7	-	-	ADJ
ejpam-5321	11	8	continuity	continuity	ADJ
ejpam-5321	11	9	and	and	CCONJ
ejpam-5321	11	10	faint	faint	ADJ
ejpam-5321	11	11	precontinuity	precontinuity	NOUN
ejpam-5321	11	12	.	.	PUNCT
ejpam-5321	12	1	noiri	noiri	PROPN
ejpam-5321	13	1	[	[	X
ejpam-5321	13	2	29	29	NUM
ejpam-5321	13	3	]	]	PUNCT
ejpam-5321	13	4	introduced	introduce	VERB
ejpam-5321	13	5	and	and	CCONJ
ejpam-5321	13	6	studied	study	VERB
ejpam-5321	13	7	the	the	DET
ejpam-5321	13	8	notion	notion	NOUN
ejpam-5321	13	9	of	of	ADP
ejpam-5321	13	10	slight	slight	ADJ
ejpam-5321	13	11	β	β	NOUN
ejpam-5321	13	12	-	-	NOUN
ejpam-5321	13	13	continuity	continuity	NOUN
ejpam-5321	13	14	which	which	PRON
ejpam-5321	13	15	is	be	AUX
ejpam-5321	13	16	implied	imply	VERB
ejpam-5321	13	17	by	by	ADP
ejpam-5321	13	18	both	both	DET
ejpam-5321	13	19	slight	slight	ADJ
ejpam-5321	13	20	semi	semi	ADJ
ejpam-5321	13	21	-	-	ADJ
ejpam-5321	13	22	continuity	continuity	ADJ
ejpam-5321	13	23	and	and	CCONJ
ejpam-5321	13	24	faint	faint	ADJ
ejpam-5321	13	25	precontinuity	precontinuity	NOUN
ejpam-5321	13	26	.	.	PUNCT
ejpam-5321	14	1	duangphui	duangphui	NOUN
ejpam-5321	14	2	et	et	PROPN
ejpam-5321	14	3	al	al	PROPN
ejpam-5321	14	4	.	.	PUNCT
ejpam-5321	15	1	[	[	X
ejpam-5321	15	2	21	21	NUM
ejpam-5321	15	3	]	]	PUNCT
ejpam-5321	15	4	introduced	introduce	VERB
ejpam-5321	15	5	and	and	CCONJ
ejpam-5321	15	6	investigated	investigate	VERB
ejpam-5321	15	7	the	the	DET
ejpam-5321	15	8	notion	notion	NOUN
ejpam-5321	15	9	of	of	ADP
ejpam-5321	15	10	almost	almost	ADV
ejpam-5321	15	11	(	(	PUNCT
ejpam-5321	15	12	µ	µ	NUM
ejpam-5321	15	13	,	,	PUNCT
ejpam-5321	15	14	µ′)(m	µ′)(m	VERB
ejpam-5321	15	15	,	,	PUNCT
ejpam-5321	15	16	n)-continuous	n)-continuous	ADJ
ejpam-5321	15	17	functions	function	NOUN
ejpam-5321	15	18	.	.	PUNCT
ejpam-5321	16	1	thongmoon	thongmoon	NOUN
ejpam-5321	16	2	and	and	CCONJ
ejpam-5321	16	3	boonpok	boonpok	VERB
ejpam-5321	16	4	[	[	X
ejpam-5321	16	5	42	42	NUM
ejpam-5321	16	6	]	]	PUNCT
ejpam-5321	16	7	introduced	introduce	VERB
ejpam-5321	16	8	and	and	CCONJ
ejpam-5321	16	9	studied	study	VERB
ejpam-5321	16	10	the	the	DET
ejpam-5321	16	11	notion	notion	NOUN
ejpam-5321	16	12	of	of	ADP
ejpam-5321	16	13	strongly	strongly	ADV
ejpam-5321	16	14	θ(λ	θ(λ	ADJ
ejpam-5321	16	15	,	,	PUNCT
ejpam-5321	16	16	p)-continuous	p)-continuous	ADJ
ejpam-5321	16	17	functions	function	NOUN
ejpam-5321	16	18	.	.	PUNCT
ejpam-5321	17	1	moreover	moreover	ADV
ejpam-5321	17	2	,	,	PUNCT
ejpam-5321	17	3	several	several	ADJ
ejpam-5321	17	4	characterizations	characterization	NOUN
ejpam-5321	17	5	of	of	ADP
ejpam-5321	17	6	almost	almost	ADV
ejpam-5321	17	7	(	(	PUNCT
ejpam-5321	17	8	λ	λ	PROPN
ejpam-5321	17	9	,	,	PUNCT
ejpam-5321	17	10	p)-continuous	p)-continuous	ADJ
ejpam-5321	17	11	functions	function	NOUN
ejpam-5321	17	12	,	,	PUNCT
ejpam-5321	17	13	almost	almost	ADV
ejpam-5321	17	14	strongly	strongly	ADV
ejpam-5321	17	15	θ(λ	θ(λ	VERB
ejpam-5321	17	16	,	,	PUNCT
ejpam-5321	17	17	p)-continuous	p)-continuous	ADJ
ejpam-5321	17	18	functions	function	NOUN
ejpam-5321	17	19	,	,	PUNCT
ejpam-5321	17	20	θ(λ	θ(λ	PROPN
ejpam-5321	17	21	,	,	PUNCT
ejpam-5321	17	22	p)-continuous	p)-continuous	ADJ
ejpam-5321	17	23	functions	function	NOUN
ejpam-5321	17	24	,	,	PUNCT
ejpam-5321	17	25	weakly	weakly	ADJ
ejpam-5321	17	26	(	(	PUNCT
ejpam-5321	17	27	λ	λ	PROPN
ejpam-5321	17	28	,	,	PUNCT
ejpam-5321	17	29	b)-continuous	b)-continuous	ADJ
ejpam-5321	17	30	functions	function	NOUN
ejpam-5321	17	31	,	,	PUNCT
ejpam-5321	17	32	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5321	17	33	functions	function	NOUN
ejpam-5321	17	34	,	,	PUNCT
ejpam-5321	17	35	⋆-continuous	⋆-continuous	ADJ
ejpam-5321	17	36	functions	function	NOUN
ejpam-5321	17	37	,	,	PUNCT
ejpam-5321	17	38	θ	θ	PROPN
ejpam-5321	17	39	-	-	ADJ
ejpam-5321	17	40	i	i	NOUN
ejpam-5321	17	41	-continuous	-continuous	ADJ
ejpam-5321	17	42	functions	function	NOUN
ejpam-5321	17	43	,	,	PUNCT
ejpam-5321	17	44	almost	almost	ADV
ejpam-5321	17	45	(	(	PUNCT
ejpam-5321	17	46	g	g	NOUN
ejpam-5321	17	47	,	,	PUNCT
ejpam-5321	17	48	m)-continuous	m)-continuous	ADJ
ejpam-5321	17	49	functions	function	NOUN
ejpam-5321	17	50	,	,	PUNCT
ejpam-5321	17	51	(	(	PUNCT
ejpam-5321	17	52	λ	λ	NOUN
ejpam-5321	17	53	,	,	PUNCT
ejpam-5321	17	54	sp)-continuous	sp)-continuous	ADJ
ejpam-5321	17	55	functions	function	NOUN
ejpam-5321	17	56	,	,	PUNCT
ejpam-5321	17	57	δp(λ	δp(λ	NOUN
ejpam-5321	17	58	,	,	PUNCT
ejpam-5321	17	59	s)-continuous	s)-continuous	ADJ
ejpam-5321	17	60	functions	function	NOUN
ejpam-5321	17	61	,	,	PUNCT
ejpam-5321	17	62	(	(	PUNCT
ejpam-5321	17	63	λ	λ	NOUN
ejpam-5321	17	64	,	,	PUNCT
ejpam-5321	17	65	p(⋆))-continuous	p(⋆))-continuous	ADJ
ejpam-5321	17	66	functions	function	NOUN
ejpam-5321	17	67	,	,	PUNCT
ejpam-5321	17	68	pairwise	pairwise	NOUN
ejpam-5321	17	69	almost	almost	ADV
ejpam-5321	17	70	m	m	VERB
ejpam-5321	17	71	-continuous	-continuous	ADJ
ejpam-5321	17	72	functions	function	NOUN
ejpam-5321	17	73	,	,	PUNCT
ejpam-5321	17	74	(	(	PUNCT
ejpam-5321	17	75	τ1	τ1	NOUN
ejpam-5321	17	76	,	,	PUNCT
ejpam-5321	17	77	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	17	78	functions	function	NOUN
ejpam-5321	17	79	,	,	PUNCT
ejpam-5321	17	80	almost	almost	ADV
ejpam-5321	17	81	(	(	PUNCT
ejpam-5321	17	82	τ1	τ1	NOUN
ejpam-5321	17	83	,	,	PUNCT
ejpam-5321	17	84	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	17	85	functions	function	NOUN
ejpam-5321	17	86	and	and	CCONJ
ejpam-5321	17	87	weakly	weakly	ADJ
ejpam-5321	17	88	(	(	PUNCT
ejpam-5321	17	89	τ1	τ1	NOUN
ejpam-5321	17	90	,	,	PUNCT
ejpam-5321	17	91	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	17	92	functions	function	NOUN
ejpam-5321	17	93	were	be	AUX
ejpam-5321	17	94	presented	present	VERB
ejpam-5321	17	95	in	in	ADP
ejpam-5321	17	96	∗corresponding	∗corresponde	VERB
ejpam-5321	17	97	author	author	NOUN
ejpam-5321	17	98	.	.	PUNCT
ejpam-5321	18	1	doi	doi	NOUN
ejpam-5321	18	2	:	:	PUNCT
ejpam-5321	18	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5321	https://doi.org/10.29020/nybg.ejpam.v17i3.5321	PRON
ejpam-5321	18	4	email	email	NOUN
ejpam-5321	18	5	addresses	address	NOUN
ejpam-5321	18	6	:	:	PUNCT
ejpam-5321	18	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	PROPN
ejpam-5321	18	8	(	(	PUNCT
ejpam-5321	18	9	c.	c.	PROPN
ejpam-5321	18	10	viriyapong	viriyapong	PROPN
ejpam-5321	18	11	)	)	PUNCT
ejpam-5321	18	12	,	,	PUNCT
ejpam-5321	18	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-5321	18	14	(	(	PUNCT
ejpam-5321	18	15	s.	s.	PROPN
ejpam-5321	18	16	sompong	sompong	PROPN
ejpam-5321	18	17	)	)	PUNCT
ejpam-5321	18	18	,	,	PUNCT
ejpam-5321	18	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5321	18	20	(	(	PUNCT
ejpam-5321	18	21	c.	c.	PROPN
ejpam-5321	18	22	boonpok	boonpok	PROPN
ejpam-5321	18	23	)	)	PUNCT
ejpam-5321	18	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5321	18	25	2142	2142	NUM
ejpam-5321	19	1	©	©	ADP
ejpam-5321	19	2	2024	2024	NUM
ejpam-5321	19	3	ejpam	ejpam	NOUN
ejpam-5321	19	4	all	all	DET
ejpam-5321	19	5	rights	right	NOUN
ejpam-5321	19	6	reserved	reserve	VERB
ejpam-5321	19	7	.	.	PUNCT
ejpam-5321	20	1	c.	c.	PROPN
ejpam-5321	20	2	viriyapong	viriyapong	PROPN
ejpam-5321	20	3	,	,	PUNCT
ejpam-5321	20	4	s.	s.	PROPN
ejpam-5321	20	5	sompong	sompong	PROPN
ejpam-5321	20	6	,	,	PUNCT
ejpam-5321	20	7	c.	c.	PROPN
ejpam-5321	20	8	boonpok	boonpok	PROPN
ejpam-5321	20	9	/	/	SYM
ejpam-5321	20	10	eur	eur	PROPN
ejpam-5321	20	11	.	.	PUNCT
ejpam-5321	21	1	j.	j.	PROPN
ejpam-5321	21	2	pure	pure	PROPN
ejpam-5321	21	3	appl	appl	PROPN
ejpam-5321	21	4	.	.	PROPN
ejpam-5321	21	5	math	math	PROPN
ejpam-5321	21	6	,	,	PUNCT
ejpam-5321	21	7	17	17	NUM
ejpam-5321	21	8	(	(	PUNCT
ejpam-5321	21	9	3	3	NUM
ejpam-5321	21	10	)	)	PUNCT
ejpam-5321	21	11	(	(	PUNCT
ejpam-5321	21	12	2024	2024	NUM
ejpam-5321	21	13	)	)	PUNCT
ejpam-5321	21	14	,	,	PUNCT
ejpam-5321	21	15	2142	2142	NUM
ejpam-5321	21	16	-	-	SYM
ejpam-5321	21	17	2154	2154	NUM
ejpam-5321	21	18	2143	2143	NUM
ejpam-5321	22	1	[	[	X
ejpam-5321	22	2	40	40	NUM
ejpam-5321	22	3	]	]	PUNCT
ejpam-5321	22	4	,	,	PUNCT
ejpam-5321	23	1	[	[	X
ejpam-5321	23	2	11	11	NUM
ejpam-5321	23	3	]	]	PUNCT
ejpam-5321	23	4	,	,	PUNCT
ejpam-5321	23	5	[	[	X
ejpam-5321	23	6	37	37	NUM
ejpam-5321	23	7	]	]	PUNCT
ejpam-5321	23	8	,	,	PUNCT
ejpam-5321	23	9	[	[	X
ejpam-5321	23	10	16	16	NUM
ejpam-5321	23	11	]	]	PUNCT
ejpam-5321	23	12	,	,	PUNCT
ejpam-5321	23	13	[	[	X
ejpam-5321	23	14	10	10	NUM
ejpam-5321	23	15	]	]	PUNCT
ejpam-5321	23	16	,	,	PUNCT
ejpam-5321	24	1	[	[	X
ejpam-5321	24	2	9	9	NUM
ejpam-5321	24	3	]	]	PUNCT
ejpam-5321	24	4	,	,	PUNCT
ejpam-5321	24	5	[	[	X
ejpam-5321	24	6	5	5	NUM
ejpam-5321	24	7	]	]	PUNCT
ejpam-5321	24	8	,	,	PUNCT
ejpam-5321	24	9	[	[	X
ejpam-5321	24	10	2	2	NUM
ejpam-5321	24	11	]	]	PUNCT
ejpam-5321	24	12	,	,	PUNCT
ejpam-5321	24	13	[	[	X
ejpam-5321	24	14	44	44	NUM
ejpam-5321	24	15	]	]	PUNCT
ejpam-5321	24	16	,	,	PUNCT
ejpam-5321	24	17	[	[	X
ejpam-5321	24	18	41	41	NUM
ejpam-5321	24	19	]	]	PUNCT
ejpam-5321	24	20	,	,	PUNCT
ejpam-5321	24	21	[	[	X
ejpam-5321	24	22	8	8	NUM
ejpam-5321	24	23	]	]	PUNCT
ejpam-5321	24	24	,	,	PUNCT
ejpam-5321	24	25	[	[	X
ejpam-5321	24	26	3	3	NUM
ejpam-5321	24	27	]	]	PUNCT
ejpam-5321	24	28	,	,	PUNCT
ejpam-5321	24	29	[	[	X
ejpam-5321	24	30	17	17	NUM
ejpam-5321	24	31	]	]	PUNCT
ejpam-5321	24	32	,	,	PUNCT
ejpam-5321	24	33	[	[	X
ejpam-5321	24	34	15	15	NUM
ejpam-5321	24	35	]	]	PUNCT
ejpam-5321	24	36	and	and	CCONJ
ejpam-5321	24	37	[	[	X
ejpam-5321	24	38	12	12	NUM
ejpam-5321	24	39	]	]	PUNCT
ejpam-5321	24	40	,	,	PUNCT
ejpam-5321	24	41	respectively	respectively	ADV
ejpam-5321	24	42	.	.	PUNCT
ejpam-5321	25	1	sangviset	sangviset	NOUN
ejpam-5321	25	2	et	et	PROPN
ejpam-5321	25	3	al	al	PROPN
ejpam-5321	25	4	.	.	PUNCT
ejpam-5321	26	1	[	[	X
ejpam-5321	26	2	39	39	NUM
ejpam-5321	26	3	]	]	PUNCT
ejpam-5321	26	4	introduced	introduce	VERB
ejpam-5321	26	5	the	the	DET
ejpam-5321	26	6	notion	notion	NOUN
ejpam-5321	26	7	of	of	ADP
ejpam-5321	26	8	slightly	slightly	ADV
ejpam-5321	26	9	(	(	PUNCT
ejpam-5321	26	10	m,µ)-continuous	m,µ)-continuous	ADJ
ejpam-5321	26	11	functions	function	NOUN
ejpam-5321	26	12	as	as	ADP
ejpam-5321	26	13	functions	function	NOUN
ejpam-5321	26	14	from	from	ADP
ejpam-5321	26	15	an	an	DET
ejpam-5321	26	16	m	m	NOUN
ejpam-5321	26	17	-	-	PUNCT
ejpam-5321	26	18	spaces	space	NOUN
ejpam-5321	26	19	into	into	ADP
ejpam-5321	26	20	a	a	DET
ejpam-5321	26	21	generalized	generalized	ADJ
ejpam-5321	26	22	topological	topological	ADJ
ejpam-5321	26	23	space	space	NOUN
ejpam-5321	26	24	and	and	CCONJ
ejpam-5321	26	25	investigated	investigate	VERB
ejpam-5321	26	26	several	several	ADJ
ejpam-5321	26	27	characterizations	characterization	NOUN
ejpam-5321	26	28	of	of	ADP
ejpam-5321	26	29	slightly	slightly	ADV
ejpam-5321	26	30	(	(	PUNCT
ejpam-5321	26	31	m,µ)-continuous	m,µ)-continuous	ADJ
ejpam-5321	26	32	functions	function	NOUN
ejpam-5321	26	33	.	.	PUNCT
ejpam-5321	27	1	in	in	ADP
ejpam-5321	27	2	2005	2005	NUM
ejpam-5321	27	3	,	,	PUNCT
ejpam-5321	27	4	ekici	ekici	NOUN
ejpam-5321	27	5	[	[	X
ejpam-5321	27	6	23	23	NUM
ejpam-5321	27	7	]	]	PUNCT
ejpam-5321	27	8	introduced	introduce	VERB
ejpam-5321	27	9	and	and	CCONJ
ejpam-5321	27	10	investigated	investigate	VERB
ejpam-5321	27	11	the	the	DET
ejpam-5321	27	12	notion	notion	NOUN
ejpam-5321	27	13	of	of	ADP
ejpam-5321	27	14	upper	upper	ADJ
ejpam-5321	27	15	(	(	PUNCT
ejpam-5321	27	16	lower	low	ADJ
ejpam-5321	27	17	)	)	PUNCT
ejpam-5321	27	18	slightly	slightly	ADV
ejpam-5321	27	19	α	α	X
ejpam-5321	27	20	-	-	ADJ
ejpam-5321	27	21	continuous	continuous	ADJ
ejpam-5321	27	22	multifunctions	multifunction	NOUN
ejpam-5321	27	23	as	as	ADP
ejpam-5321	27	24	a	a	DET
ejpam-5321	27	25	generalization	generalization	NOUN
ejpam-5321	27	26	of	of	ADP
ejpam-5321	27	27	upper	upper	ADJ
ejpam-5321	27	28	(	(	PUNCT
ejpam-5321	27	29	lower	low	ADJ
ejpam-5321	27	30	)	)	PUNCT
ejpam-5321	27	31	α	α	NUM
ejpam-5321	27	32	-	-	ADJ
ejpam-5321	27	33	continuous	continuous	ADJ
ejpam-5321	27	34	multifunctions	multifunction	NOUN
ejpam-5321	27	35	due	due	ADP
ejpam-5321	27	36	to	to	ADP
ejpam-5321	27	37	neubrunn	neubrunn	NOUN
ejpam-5321	27	38	[	[	X
ejpam-5321	27	39	28	28	NUM
ejpam-5321	27	40	]	]	PUNCT
ejpam-5321	27	41	.	.	PUNCT
ejpam-5321	28	1	popa	popa	NOUN
ejpam-5321	28	2	and	and	CCONJ
ejpam-5321	28	3	noiri	noiri	ADV
ejpam-5321	29	1	[	[	X
ejpam-5321	29	2	36	36	NUM
ejpam-5321	29	3	]	]	PUNCT
ejpam-5321	29	4	introduced	introduce	VERB
ejpam-5321	29	5	and	and	CCONJ
ejpam-5321	29	6	studied	study	VERB
ejpam-5321	29	7	the	the	DET
ejpam-5321	29	8	notion	notion	NOUN
ejpam-5321	29	9	of	of	ADP
ejpam-5321	29	10	upper	upper	ADJ
ejpam-5321	29	11	(	(	PUNCT
ejpam-5321	29	12	lower	low	ADJ
ejpam-5321	29	13	)	)	PUNCT
ejpam-5321	29	14	β	β	X
ejpam-5321	29	15	-	-	ADJ
ejpam-5321	29	16	continuous	continuous	ADJ
ejpam-5321	29	17	multifunctions	multifunction	NOUN
ejpam-5321	29	18	.	.	PUNCT
ejpam-5321	30	1	furthermore	furthermore	ADV
ejpam-5321	30	2	,	,	PUNCT
ejpam-5321	30	3	ekici	ekici	NOUN
ejpam-5321	30	4	[	[	X
ejpam-5321	30	5	22	22	NUM
ejpam-5321	30	6	]	]	PUNCT
ejpam-5321	30	7	introduced	introduce	VERB
ejpam-5321	30	8	and	and	CCONJ
ejpam-5321	30	9	studied	study	VERB
ejpam-5321	30	10	upper	upper	ADJ
ejpam-5321	30	11	(	(	PUNCT
ejpam-5321	30	12	lower	low	ADJ
ejpam-5321	30	13	)	)	PUNCT
ejpam-5321	30	14	slightly	slightly	ADV
ejpam-5321	30	15	β	β	X
ejpam-5321	30	16	-	-	ADJ
ejpam-5321	30	17	continuous	continuous	ADJ
ejpam-5321	30	18	multifunctions	multifunction	NOUN
ejpam-5321	30	19	as	as	ADP
ejpam-5321	30	20	a	a	DET
ejpam-5321	30	21	generalization	generalization	NOUN
ejpam-5321	30	22	of	of	ADP
ejpam-5321	30	23	upper	upper	ADJ
ejpam-5321	30	24	(	(	PUNCT
ejpam-5321	30	25	lower	low	ADJ
ejpam-5321	30	26	)	)	PUNCT
ejpam-5321	30	27	semicontinuous	semicontinuous	ADJ
ejpam-5321	30	28	multifunctions	multifunction	NOUN
ejpam-5321	30	29	,	,	PUNCT
ejpam-5321	30	30	upper	upper	ADJ
ejpam-5321	30	31	(	(	PUNCT
ejpam-5321	30	32	lower	low	ADJ
ejpam-5321	30	33	)	)	PUNCT
ejpam-5321	30	34	α	α	NUM
ejpam-5321	30	35	-	-	ADJ
ejpam-5321	30	36	continuous	continuous	ADJ
ejpam-5321	30	37	multifunctions	multifunction	NOUN
ejpam-5321	30	38	,	,	PUNCT
ejpam-5321	30	39	upper	upper	ADJ
ejpam-5321	30	40	(	(	PUNCT
ejpam-5321	30	41	lower	low	ADJ
ejpam-5321	30	42	)	)	PUNCT
ejpam-5321	30	43	precontinuous	precontinuous	ADJ
ejpam-5321	30	44	multifunctions	multifunction	NOUN
ejpam-5321	31	1	[	[	X
ejpam-5321	31	2	35	35	NUM
ejpam-5321	31	3	]	]	X
ejpam-5321	31	4	,	,	PUNCT
ejpam-5321	31	5	upper	upper	ADJ
ejpam-5321	31	6	(	(	PUNCT
ejpam-5321	31	7	lower	low	ADJ
ejpam-5321	31	8	)	)	PUNCT
ejpam-5321	31	9	quasi	quasi	ADJ
ejpam-5321	31	10	-	-	ADJ
ejpam-5321	31	11	continuous	continuous	ADJ
ejpam-5321	31	12	multifunctions	multifunction	NOUN
ejpam-5321	31	13	[	[	X
ejpam-5321	31	14	34	34	NUM
ejpam-5321	31	15	]	]	PUNCT
ejpam-5321	31	16	,	,	PUNCT
ejpam-5321	31	17	upper	upper	ADJ
ejpam-5321	31	18	(	(	PUNCT
ejpam-5321	31	19	lower	low	ADJ
ejpam-5321	31	20	)	)	PUNCT
ejpam-5321	31	21	γ	γ	ADJ
ejpam-5321	31	22	-	-	ADJ
ejpam-5321	31	23	continuous	continuous	ADJ
ejpam-5321	31	24	multifunctions	multifunction	NOUN
ejpam-5321	32	1	[	[	X
ejpam-5321	32	2	24	24	NUM
ejpam-5321	32	3	]	]	PUNCT
ejpam-5321	32	4	,	,	PUNCT
ejpam-5321	32	5	upper	upper	ADJ
ejpam-5321	32	6	(	(	PUNCT
ejpam-5321	32	7	lower	low	ADJ
ejpam-5321	32	8	)	)	PUNCT
ejpam-5321	32	9	β	β	X
ejpam-5321	32	10	-	-	ADJ
ejpam-5321	32	11	continuous	continuous	ADJ
ejpam-5321	32	12	multifunctions	multifunction	NOUN
ejpam-5321	32	13	and	and	CCONJ
ejpam-5321	32	14	slightly	slightly	ADV
ejpam-5321	32	15	β	β	ADJ
ejpam-5321	32	16	-	-	ADJ
ejpam-5321	32	17	continuous	continuous	ADJ
ejpam-5321	32	18	functions	function	NOUN
ejpam-5321	32	19	.	.	PUNCT
ejpam-5321	33	1	noiri	noiri	PROPN
ejpam-5321	33	2	and	and	CCONJ
ejpam-5321	33	3	popa	popa	NOUN
ejpam-5321	34	1	[	[	X
ejpam-5321	34	2	31	31	NUM
ejpam-5321	34	3	]	]	PUNCT
ejpam-5321	34	4	introduced	introduce	VERB
ejpam-5321	34	5	the	the	DET
ejpam-5321	34	6	notion	notion	NOUN
ejpam-5321	34	7	of	of	ADP
ejpam-5321	34	8	slightly	slightly	ADV
ejpam-5321	34	9	m	m	ADJ
ejpam-5321	34	10	-	-	ADJ
ejpam-5321	34	11	continuous	continuous	ADJ
ejpam-5321	34	12	multifunctions	multifunction	NOUN
ejpam-5321	34	13	and	and	CCONJ
ejpam-5321	34	14	established	establish	VERB
ejpam-5321	34	15	the	the	DET
ejpam-5321	34	16	relationships	relationship	NOUN
ejpam-5321	34	17	amongm	amongm	ADJ
ejpam-5321	34	18	-	-	PUNCT
ejpam-5321	34	19	continuity	continuity	NOUN
ejpam-5321	34	20	,	,	PUNCT
ejpam-5321	34	21	almost	almost	ADV
ejpam-5321	34	22	m	m	NOUN
ejpam-5321	34	23	-	-	NOUN
ejpam-5321	34	24	continuity	continuity	NOUN
ejpam-5321	34	25	,	,	PUNCT
ejpam-5321	34	26	weak	weak	ADJ
ejpam-5321	34	27	m	m	NOUN
ejpam-5321	34	28	-	-	NOUN
ejpam-5321	34	29	continuity	continuity	NOUN
ejpam-5321	34	30	and	and	CCONJ
ejpam-5321	34	31	slight	slight	ADJ
ejpam-5321	34	32	m	m	NOUN
ejpam-5321	34	33	-	-	NOUN
ejpam-5321	34	34	continuity	continuity	NOUN
ejpam-5321	34	35	for	for	ADP
ejpam-5321	34	36	multifunctions	multifunction	NOUN
ejpam-5321	34	37	.	.	PUNCT
ejpam-5321	35	1	laprom	laprom	ADP
ejpam-5321	35	2	et	et	PROPN
ejpam-5321	35	3	al	al	PROPN
ejpam-5321	35	4	.	.	PUNCT
ejpam-5321	36	1	[	[	X
ejpam-5321	36	2	27	27	NUM
ejpam-5321	36	3	]	]	PUNCT
ejpam-5321	36	4	introduced	introduce	VERB
ejpam-5321	36	5	and	and	CCONJ
ejpam-5321	36	6	investigated	investigate	VERB
ejpam-5321	36	7	the	the	DET
ejpam-5321	36	8	notion	notion	NOUN
ejpam-5321	36	9	of	of	ADP
ejpam-5321	36	10	β(τ1	β(τ1	NOUN
ejpam-5321	36	11	,	,	PUNCT
ejpam-5321	36	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	36	13	multifunctions	multifunction	NOUN
ejpam-5321	36	14	.	.	PUNCT
ejpam-5321	37	1	furthermore	furthermore	ADV
ejpam-5321	37	2	,	,	PUNCT
ejpam-5321	37	3	several	several	ADJ
ejpam-5321	37	4	characterizations	characterization	NOUN
ejpam-5321	37	5	of	of	ADP
ejpam-5321	37	6	(	(	PUNCT
ejpam-5321	37	7	τ1	τ1	NOUN
ejpam-5321	37	8	,	,	PUNCT
ejpam-5321	37	9	τ2)δ	τ2)δ	ADJ
ejpam-5321	37	10	-	-	PUNCT
ejpam-5321	37	11	semicontinuous	semicontinuous	ADJ
ejpam-5321	37	12	multifunctions	multifunction	NOUN
ejpam-5321	37	13	,	,	PUNCT
ejpam-5321	37	14	almost	almost	ADV
ejpam-5321	37	15	weakly	weakly	ADJ
ejpam-5321	37	16	⋆-continuous	⋆-continuous	ADJ
ejpam-5321	37	17	multifunctions	multifunction	NOUN
ejpam-5321	37	18	,	,	PUNCT
ejpam-5321	37	19	weakly	weakly	ADJ
ejpam-5321	37	20	⋆-continuous	⋆-continuous	ADJ
ejpam-5321	37	21	multifunctions	multifunction	NOUN
ejpam-5321	37	22	,	,	PUNCT
ejpam-5321	37	23	weakly	weakly	ADJ
ejpam-5321	37	24	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5321	37	25	multifunctions	multifunction	NOUN
ejpam-5321	37	26	,	,	PUNCT
ejpam-5321	37	27	ı⋆-continuous	ı⋆-continuous	ADJ
ejpam-5321	37	28	multifunctions	multifunction	NOUN
ejpam-5321	37	29	,	,	PUNCT
ejpam-5321	37	30	almost	almost	ADV
ejpam-5321	37	31	β(⋆)-continuous	β(⋆)-continuous	ADJ
ejpam-5321	37	32	multifunctions	multifunction	NOUN
ejpam-5321	37	33	,	,	PUNCT
ejpam-5321	37	34	almost	almost	ADV
ejpam-5321	37	35	weakly	weakly	ADJ
ejpam-5321	37	36	(	(	PUNCT
ejpam-5321	37	37	τ1	τ1	NOUN
ejpam-5321	37	38	,	,	PUNCT
ejpam-5321	37	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	37	40	multifunctions	multifunction	NOUN
ejpam-5321	37	41	,	,	PUNCT
ejpam-5321	37	42	almost	almost	ADV
ejpam-5321	37	43	(	(	PUNCT
ejpam-5321	37	44	τ1	τ1	NOUN
ejpam-5321	37	45	,	,	PUNCT
ejpam-5321	37	46	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	37	47	multifunctions	multifunction	NOUN
ejpam-5321	37	48	and	and	CCONJ
ejpam-5321	37	49	(	(	PUNCT
ejpam-5321	37	50	τ1	τ1	NOUN
ejpam-5321	37	51	,	,	PUNCT
ejpam-5321	37	52	τ2)α	τ2)α	ADJ
ejpam-5321	37	53	-	-	PUNCT
ejpam-5321	37	54	continuous	continuous	ADJ
ejpam-5321	37	55	multifunctions	multifunction	NOUN
ejpam-5321	37	56	were	be	AUX
ejpam-5321	37	57	investigated	investigate	VERB
ejpam-5321	37	58	in	in	ADP
ejpam-5321	37	59	[	[	X
ejpam-5321	37	60	6	6	NUM
ejpam-5321	37	61	]	]	PUNCT
ejpam-5321	37	62	,	,	PUNCT
ejpam-5321	37	63	[	[	X
ejpam-5321	37	64	18	18	NUM
ejpam-5321	37	65	]	]	PUNCT
ejpam-5321	37	66	,	,	PUNCT
ejpam-5321	37	67	[	[	X
ejpam-5321	37	68	4	4	NUM
ejpam-5321	37	69	]	]	PUNCT
ejpam-5321	37	70	,	,	PUNCT
ejpam-5321	37	71	[	[	X
ejpam-5321	37	72	14	14	NUM
ejpam-5321	37	73	]	]	PUNCT
ejpam-5321	37	74	,	,	PUNCT
ejpam-5321	37	75	[	[	X
ejpam-5321	37	76	13	13	NUM
ejpam-5321	37	77	]	]	PUNCT
ejpam-5321	37	78	,	,	PUNCT
ejpam-5321	37	79	[	[	X
ejpam-5321	37	80	7	7	NUM
ejpam-5321	37	81	]	]	PUNCT
ejpam-5321	37	82	,	,	PUNCT
ejpam-5321	37	83	[	[	X
ejpam-5321	37	84	19	19	NUM
ejpam-5321	37	85	]	]	PUNCT
ejpam-5321	37	86	,	,	PUNCT
ejpam-5321	37	87	[	[	X
ejpam-5321	37	88	26	26	NUM
ejpam-5321	37	89	]	]	PUNCT
ejpam-5321	37	90	and	and	CCONJ
ejpam-5321	37	91	[	[	X
ejpam-5321	37	92	43	43	NUM
ejpam-5321	37	93	]	]	X
ejpam-5321	37	94	,	,	PUNCT
ejpam-5321	37	95	respectively	respectively	ADV
ejpam-5321	37	96	.	.	PUNCT
ejpam-5321	38	1	pue	pue	NOUN
ejpam-5321	38	2	-	-	PUNCT
ejpam-5321	38	3	on	on	NOUN
ejpam-5321	38	4	et	et	PROPN
ejpam-5321	38	5	al	al	PROPN
ejpam-5321	38	6	.	.	PUNCT
ejpam-5321	39	1	[	[	X
ejpam-5321	39	2	38	38	NUM
ejpam-5321	39	3	]	]	PUNCT
ejpam-5321	39	4	introduce	introduce	NOUN
ejpam-5321	39	5	and	and	CCONJ
ejpam-5321	39	6	studied	study	VERB
ejpam-5321	39	7	the	the	DET
ejpam-5321	39	8	notions	notion	NOUN
ejpam-5321	39	9	of	of	ADP
ejpam-5321	39	10	upper	upper	ADJ
ejpam-5321	39	11	and	and	CCONJ
ejpam-5321	39	12	lower	low	ADJ
ejpam-5321	39	13	(	(	PUNCT
ejpam-5321	39	14	τ1	τ1	NOUN
ejpam-5321	39	15	,	,	PUNCT
ejpam-5321	39	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	39	17	multifunctions	multifunction	NOUN
ejpam-5321	39	18	.	.	PUNCT
ejpam-5321	40	1	in	in	ADP
ejpam-5321	40	2	this	this	DET
ejpam-5321	40	3	paper	paper	NOUN
ejpam-5321	40	4	,	,	PUNCT
ejpam-5321	40	5	we	we	PRON
ejpam-5321	40	6	introduce	introduce	VERB
ejpam-5321	40	7	the	the	DET
ejpam-5321	40	8	concepts	concept	NOUN
ejpam-5321	40	9	of	of	ADP
ejpam-5321	40	10	upper	upper	ADJ
ejpam-5321	40	11	and	and	CCONJ
ejpam-5321	40	12	lower	low	ADJ
ejpam-5321	40	13	slightly	slightly	ADJ
ejpam-5321	40	14	α(τ1	α(τ1	NOUN
ejpam-5321	40	15	,	,	PUNCT
ejpam-5321	40	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	40	17	multifunctions	multifunction	NOUN
ejpam-5321	40	18	.	.	PUNCT
ejpam-5321	41	1	we	we	PRON
ejpam-5321	41	2	also	also	ADV
ejpam-5321	41	3	investigate	investigate	VERB
ejpam-5321	41	4	several	several	ADJ
ejpam-5321	41	5	characterizations	characterization	NOUN
ejpam-5321	41	6	of	of	ADP
ejpam-5321	41	7	upper	upper	ADJ
ejpam-5321	41	8	and	and	CCONJ
ejpam-5321	41	9	lower	low	ADJ
ejpam-5321	41	10	slightly	slightly	ADJ
ejpam-5321	41	11	α(τ1	α(τ1	NOUN
ejpam-5321	41	12	,	,	PUNCT
ejpam-5321	41	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	41	14	multifunctions	multifunction	NOUN
ejpam-5321	41	15	.	.	PUNCT
ejpam-5321	42	1	2	2	X
ejpam-5321	42	2	.	.	X
ejpam-5321	42	3	preliminaries	preliminary	NOUN
ejpam-5321	42	4	throughout	throughout	ADP
ejpam-5321	42	5	the	the	DET
ejpam-5321	42	6	present	present	ADJ
ejpam-5321	42	7	paper	paper	NOUN
ejpam-5321	42	8	,	,	PUNCT
ejpam-5321	42	9	spaces	space	NOUN
ejpam-5321	42	10	(	(	PUNCT
ejpam-5321	42	11	x	x	NOUN
ejpam-5321	42	12	,	,	PUNCT
ejpam-5321	42	13	τ1	τ1	NOUN
ejpam-5321	42	14	,	,	PUNCT
ejpam-5321	42	15	τ2	τ2	NOUN
ejpam-5321	42	16	)	)	PUNCT
ejpam-5321	42	17	and	and	CCONJ
ejpam-5321	42	18	(	(	PUNCT
ejpam-5321	42	19	y	y	PROPN
ejpam-5321	42	20	,	,	PUNCT
ejpam-5321	42	21	σ1	σ1	PROPN
ejpam-5321	42	22	,	,	PUNCT
ejpam-5321	42	23	σ2	σ2	NOUN
ejpam-5321	42	24	)	)	PUNCT
ejpam-5321	42	25	(	(	PUNCT
ejpam-5321	42	26	or	or	CCONJ
ejpam-5321	42	27	simply	simply	ADV
ejpam-5321	42	28	x	x	X
ejpam-5321	42	29	and	and	CCONJ
ejpam-5321	42	30	y	y	PROPN
ejpam-5321	42	31	)	)	PUNCT
ejpam-5321	42	32	always	always	ADV
ejpam-5321	42	33	mean	mean	VERB
ejpam-5321	42	34	bitopological	bitopological	ADJ
ejpam-5321	42	35	spaces	space	NOUN
ejpam-5321	42	36	on	on	ADP
ejpam-5321	42	37	which	which	PRON
ejpam-5321	42	38	no	no	DET
ejpam-5321	42	39	separation	separation	NOUN
ejpam-5321	42	40	axioms	axiom	NOUN
ejpam-5321	42	41	are	be	AUX
ejpam-5321	42	42	assumed	assume	VERB
ejpam-5321	42	43	unless	unless	SCONJ
ejpam-5321	42	44	explicitly	explicitly	ADV
ejpam-5321	42	45	stated	state	VERB
ejpam-5321	42	46	.	.	PUNCT
ejpam-5321	43	1	let	let	VERB
ejpam-5321	43	2	a	a	DET
ejpam-5321	43	3	be	be	AUX
ejpam-5321	43	4	a	a	DET
ejpam-5321	43	5	subset	subset	NOUN
ejpam-5321	43	6	of	of	ADP
ejpam-5321	43	7	a	a	DET
ejpam-5321	43	8	bitopological	bitopological	ADJ
ejpam-5321	43	9	space	space	NOUN
ejpam-5321	43	10	(	(	PUNCT
ejpam-5321	43	11	x	x	NOUN
ejpam-5321	43	12	,	,	PUNCT
ejpam-5321	43	13	τ1	τ1	NOUN
ejpam-5321	43	14	,	,	PUNCT
ejpam-5321	43	15	τ2	τ2	NOUN
ejpam-5321	43	16	)	)	PUNCT
ejpam-5321	43	17	.	.	PUNCT
ejpam-5321	44	1	the	the	DET
ejpam-5321	44	2	closure	closure	NOUN
ejpam-5321	44	3	of	of	ADP
ejpam-5321	44	4	a	a	PRON
ejpam-5321	44	5	and	and	CCONJ
ejpam-5321	44	6	the	the	DET
ejpam-5321	44	7	interior	interior	NOUN
ejpam-5321	44	8	of	of	ADP
ejpam-5321	44	9	a	a	PRON
ejpam-5321	44	10	with	with	ADP
ejpam-5321	44	11	respect	respect	NOUN
ejpam-5321	44	12	to	to	ADP
ejpam-5321	44	13	τi	τi	PROPN
ejpam-5321	44	14	are	be	AUX
ejpam-5321	44	15	denoted	denote	VERB
ejpam-5321	44	16	by	by	ADP
ejpam-5321	44	17	τi	τi	NOUN
ejpam-5321	44	18	-	-	PUNCT
ejpam-5321	44	19	cl(a	cl(a	NUM
ejpam-5321	44	20	)	)	PUNCT
ejpam-5321	44	21	and	and	CCONJ
ejpam-5321	44	22	τi	τi	NOUN
ejpam-5321	44	23	-	-	PUNCT
ejpam-5321	44	24	int(a	int(a	NOUN
ejpam-5321	44	25	)	)	PUNCT
ejpam-5321	44	26	,	,	PUNCT
ejpam-5321	44	27	respectively	respectively	ADV
ejpam-5321	44	28	,	,	PUNCT
ejpam-5321	44	29	for	for	ADP
ejpam-5321	44	30	i	i	PROPN
ejpam-5321	44	31	=	=	SYM
ejpam-5321	44	32	1	1	NUM
ejpam-5321	44	33	,	,	PUNCT
ejpam-5321	44	34	2	2	NUM
ejpam-5321	44	35	.	.	X
ejpam-5321	44	36	a	a	DET
ejpam-5321	44	37	subset	subset	NOUN
ejpam-5321	44	38	a	a	PRON
ejpam-5321	44	39	of	of	ADP
ejpam-5321	44	40	a	a	DET
ejpam-5321	44	41	bitopological	bitopological	ADJ
ejpam-5321	44	42	space	space	NOUN
ejpam-5321	44	43	(	(	PUNCT
ejpam-5321	44	44	x	x	NOUN
ejpam-5321	44	45	,	,	PUNCT
ejpam-5321	44	46	τ1	τ1	NOUN
ejpam-5321	44	47	,	,	PUNCT
ejpam-5321	44	48	τ2	τ2	NOUN
ejpam-5321	44	49	)	)	PUNCT
ejpam-5321	44	50	is	be	AUX
ejpam-5321	44	51	called	call	VERB
ejpam-5321	44	52	τ1τ2	τ1τ2	VERB
ejpam-5321	44	53	-	-	ADJ
ejpam-5321	44	54	closed	closed	ADJ
ejpam-5321	44	55	[	[	X
ejpam-5321	44	56	20	20	NUM
ejpam-5321	44	57	]	]	PUNCT
ejpam-5321	44	58	if	if	SCONJ
ejpam-5321	44	59	a	a	DET
ejpam-5321	44	60	=	=	NOUN
ejpam-5321	44	61	τ1	τ1	NOUN
ejpam-5321	44	62	-	-	PUNCT
ejpam-5321	44	63	cl(τ2	cl(τ2	NOUN
ejpam-5321	44	64	-	-	PUNCT
ejpam-5321	44	65	cl(a	cl(a	NUM
ejpam-5321	44	66	)	)	PUNCT
ejpam-5321	44	67	)	)	PUNCT
ejpam-5321	44	68	.	.	PUNCT
ejpam-5321	45	1	the	the	DET
ejpam-5321	45	2	complement	complement	NOUN
ejpam-5321	45	3	of	of	ADP
ejpam-5321	45	4	a	a	DET
ejpam-5321	45	5	τ1τ2	τ1τ2	ADJ
ejpam-5321	45	6	-	-	ADJ
ejpam-5321	45	7	closed	closed	ADJ
ejpam-5321	45	8	set	set	NOUN
ejpam-5321	45	9	is	be	AUX
ejpam-5321	45	10	called	call	VERB
ejpam-5321	45	11	τ1τ2	τ1τ2	NOUN
ejpam-5321	45	12	-	-	ADJ
ejpam-5321	45	13	open	open	ADJ
ejpam-5321	45	14	.	.	PUNCT
ejpam-5321	46	1	let	let	VERB
ejpam-5321	46	2	a	a	DET
ejpam-5321	46	3	be	be	AUX
ejpam-5321	46	4	a	a	DET
ejpam-5321	46	5	subset	subset	NOUN
ejpam-5321	46	6	of	of	ADP
ejpam-5321	46	7	a	a	DET
ejpam-5321	46	8	bitopological	bitopological	ADJ
ejpam-5321	46	9	space	space	NOUN
ejpam-5321	46	10	(	(	PUNCT
ejpam-5321	46	11	x	x	NOUN
ejpam-5321	46	12	,	,	PUNCT
ejpam-5321	46	13	τ1	τ1	NOUN
ejpam-5321	46	14	,	,	PUNCT
ejpam-5321	46	15	τ2	τ2	NOUN
ejpam-5321	46	16	)	)	PUNCT
ejpam-5321	46	17	.	.	PUNCT
ejpam-5321	47	1	the	the	DET
ejpam-5321	47	2	intersection	intersection	NOUN
ejpam-5321	47	3	of	of	ADP
ejpam-5321	47	4	all	all	DET
ejpam-5321	47	5	τ1τ2	τ1τ2	ADJ
ejpam-5321	47	6	-	-	ADJ
ejpam-5321	47	7	closed	closed	ADJ
ejpam-5321	47	8	sets	set	NOUN
ejpam-5321	47	9	of	of	ADP
ejpam-5321	47	10	x	x	PUNCT
ejpam-5321	47	11	containing	contain	VERB
ejpam-5321	47	12	a	a	PRON
ejpam-5321	47	13	is	be	AUX
ejpam-5321	47	14	called	call	VERB
ejpam-5321	47	15	the	the	DET
ejpam-5321	47	16	τ1τ2	τ1τ2	NOUN
ejpam-5321	47	17	-	-	NOUN
ejpam-5321	47	18	closure	closure	NOUN
ejpam-5321	47	19	[	[	X
ejpam-5321	47	20	20	20	NUM
ejpam-5321	47	21	]	]	PUNCT
ejpam-5321	47	22	of	of	ADP
ejpam-5321	47	23	a	a	PRON
ejpam-5321	47	24	and	and	CCONJ
ejpam-5321	47	25	is	be	AUX
ejpam-5321	47	26	denoted	denote	VERB
ejpam-5321	47	27	by	by	ADP
ejpam-5321	47	28	τ1τ2	τ1τ2	NOUN
ejpam-5321	47	29	-	-	NUM
ejpam-5321	47	30	cl(a	cl(a	NUM
ejpam-5321	47	31	)	)	PUNCT
ejpam-5321	47	32	.	.	PUNCT
ejpam-5321	48	1	the	the	DET
ejpam-5321	48	2	union	union	NOUN
ejpam-5321	48	3	of	of	ADP
ejpam-5321	48	4	all	all	DET
ejpam-5321	48	5	τ1τ2	τ1τ2	ADJ
ejpam-5321	48	6	-	-	ADJ
ejpam-5321	48	7	open	open	ADJ
ejpam-5321	48	8	sets	set	NOUN
ejpam-5321	48	9	of	of	ADP
ejpam-5321	48	10	x	x	PUNCT
ejpam-5321	48	11	contained	contain	VERB
ejpam-5321	48	12	in	in	ADP
ejpam-5321	48	13	a	a	PRON
ejpam-5321	48	14	is	be	AUX
ejpam-5321	48	15	called	call	VERB
ejpam-5321	48	16	the	the	DET
ejpam-5321	48	17	τ1τ2	τ1τ2	NOUN
ejpam-5321	48	18	-	-	ADJ
ejpam-5321	48	19	interior	interior	ADJ
ejpam-5321	48	20	[	[	X
ejpam-5321	48	21	20	20	NUM
ejpam-5321	48	22	]	]	PUNCT
ejpam-5321	48	23	of	of	ADP
ejpam-5321	48	24	a	a	PRON
ejpam-5321	48	25	and	and	CCONJ
ejpam-5321	48	26	is	be	AUX
ejpam-5321	48	27	denoted	denote	VERB
ejpam-5321	48	28	by	by	ADP
ejpam-5321	48	29	τ1τ2	τ1τ2	NOUN
ejpam-5321	48	30	-	-	ADJ
ejpam-5321	48	31	int(a	int(a	NOUN
ejpam-5321	48	32	)	)	PUNCT
ejpam-5321	48	33	.	.	PUNCT
ejpam-5321	49	1	lemma	lemma	PROPN
ejpam-5321	49	2	1	1	NUM
ejpam-5321	49	3	.	.	PUNCT
ejpam-5321	50	1	[	[	X
ejpam-5321	50	2	20	20	NUM
ejpam-5321	50	3	]	]	PUNCT
ejpam-5321	50	4	let	let	VERB
ejpam-5321	50	5	a	a	PRON
ejpam-5321	50	6	and	and	CCONJ
ejpam-5321	50	7	b	b	NOUN
ejpam-5321	50	8	be	be	AUX
ejpam-5321	50	9	subsets	subset	NOUN
ejpam-5321	50	10	of	of	ADP
ejpam-5321	50	11	a	a	DET
ejpam-5321	50	12	bitopological	bitopological	ADJ
ejpam-5321	50	13	space	space	NOUN
ejpam-5321	50	14	(	(	PUNCT
ejpam-5321	50	15	x	x	NOUN
ejpam-5321	50	16	,	,	PUNCT
ejpam-5321	50	17	τ1	τ1	NOUN
ejpam-5321	50	18	,	,	PUNCT
ejpam-5321	50	19	τ2	τ2	NOUN
ejpam-5321	50	20	)	)	PUNCT
ejpam-5321	50	21	.	.	PUNCT
ejpam-5321	51	1	for	for	ADP
ejpam-5321	51	2	the	the	DET
ejpam-5321	51	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-5321	51	4	,	,	PUNCT
ejpam-5321	51	5	the	the	DET
ejpam-5321	51	6	following	follow	VERB
ejpam-5321	51	7	properties	property	NOUN
ejpam-5321	51	8	hold	hold	VERB
ejpam-5321	51	9	:	:	PUNCT
ejpam-5321	51	10	(	(	PUNCT
ejpam-5321	51	11	1	1	X
ejpam-5321	51	12	)	)	PUNCT
ejpam-5321	51	13	a	a	DET
ejpam-5321	51	14	⊆	⊆	NUM
ejpam-5321	51	15	τ1τ2	τ1τ2	NOUN
ejpam-5321	51	16	-	-	NUM
ejpam-5321	51	17	cl(a	cl(a	NUM
ejpam-5321	51	18	)	)	PUNCT
ejpam-5321	51	19	and	and	CCONJ
ejpam-5321	51	20	τ1τ2	τ1τ2	NOUN
ejpam-5321	51	21	-	-	ADJ
ejpam-5321	51	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5321	51	23	-	-	PUNCT
ejpam-5321	51	24	cl(a	cl(a	NUM
ejpam-5321	51	25	)	)	PUNCT
ejpam-5321	51	26	)	)	PUNCT
ejpam-5321	52	1	=	=	PUNCT
ejpam-5321	52	2	τ1τ2	τ1τ2	NOUN
ejpam-5321	52	3	-	-	NUM
ejpam-5321	52	4	cl(a	cl(a	NUM
ejpam-5321	52	5	)	)	PUNCT
ejpam-5321	52	6	.	.	PUNCT
ejpam-5321	53	1	c.	c.	PROPN
ejpam-5321	53	2	viriyapong	viriyapong	PROPN
ejpam-5321	53	3	,	,	PUNCT
ejpam-5321	53	4	s.	s.	PROPN
ejpam-5321	53	5	sompong	sompong	PROPN
ejpam-5321	53	6	,	,	PUNCT
ejpam-5321	53	7	c.	c.	PROPN
ejpam-5321	53	8	boonpok	boonpok	PROPN
ejpam-5321	53	9	/	/	SYM
ejpam-5321	53	10	eur	eur	PROPN
ejpam-5321	53	11	.	.	PUNCT
ejpam-5321	54	1	j.	j.	PROPN
ejpam-5321	54	2	pure	pure	PROPN
ejpam-5321	54	3	appl	appl	PROPN
ejpam-5321	54	4	.	.	PROPN
ejpam-5321	54	5	math	math	PROPN
ejpam-5321	54	6	,	,	PUNCT
ejpam-5321	54	7	17	17	NUM
ejpam-5321	54	8	(	(	PUNCT
ejpam-5321	54	9	3	3	NUM
ejpam-5321	54	10	)	)	PUNCT
ejpam-5321	54	11	(	(	PUNCT
ejpam-5321	54	12	2024	2024	NUM
ejpam-5321	54	13	)	)	PUNCT
ejpam-5321	54	14	,	,	PUNCT
ejpam-5321	54	15	2142	2142	NUM
ejpam-5321	54	16	-	-	SYM
ejpam-5321	54	17	2154	2154	NUM
ejpam-5321	54	18	2144	2144	NUM
ejpam-5321	54	19	(	(	PUNCT
ejpam-5321	54	20	2	2	NUM
ejpam-5321	54	21	)	)	PUNCT
ejpam-5321	55	1	if	if	SCONJ
ejpam-5321	55	2	a	a	DET
ejpam-5321	55	3	⊆	⊆	NUM
ejpam-5321	55	4	b	b	NOUN
ejpam-5321	55	5	,	,	PUNCT
ejpam-5321	55	6	then	then	ADV
ejpam-5321	55	7	τ1τ2	τ1τ2	NOUN
ejpam-5321	55	8	-	-	NUM
ejpam-5321	55	9	cl(a	cl(a	NUM
ejpam-5321	55	10	)	)	PUNCT
ejpam-5321	55	11	⊆	⊆	NUM
ejpam-5321	55	12	τ1τ2	τ1τ2	NOUN
ejpam-5321	55	13	-	-	NOUN
ejpam-5321	55	14	cl(b	cl(b	NOUN
ejpam-5321	55	15	)	)	PUNCT
ejpam-5321	55	16	.	.	PUNCT
ejpam-5321	56	1	(	(	PUNCT
ejpam-5321	56	2	3	3	X
ejpam-5321	56	3	)	)	PUNCT
ejpam-5321	56	4	τ1τ2	τ1τ2	NOUN
ejpam-5321	56	5	-	-	NUM
ejpam-5321	56	6	cl(a	cl(a	NUM
ejpam-5321	56	7	)	)	PUNCT
ejpam-5321	56	8	is	be	AUX
ejpam-5321	56	9	τ1τ2	τ1τ2	NOUN
ejpam-5321	56	10	-	-	ADJ
ejpam-5321	56	11	closed	closed	ADJ
ejpam-5321	56	12	.	.	PUNCT
ejpam-5321	57	1	(	(	PUNCT
ejpam-5321	57	2	4	4	X
ejpam-5321	57	3	)	)	PUNCT
ejpam-5321	57	4	a	a	PRON
ejpam-5321	57	5	is	be	AUX
ejpam-5321	57	6	τ1τ2	τ1τ2	NOUN
ejpam-5321	57	7	-	-	ADJ
ejpam-5321	57	8	closed	closed	ADJ
ejpam-5321	57	9	if	if	SCONJ
ejpam-5321	57	10	and	and	CCONJ
ejpam-5321	57	11	only	only	ADV
ejpam-5321	57	12	if	if	SCONJ
ejpam-5321	57	13	a	a	DET
ejpam-5321	57	14	=	=	PUNCT
ejpam-5321	57	15	τ1τ2	τ1τ2	NOUN
ejpam-5321	57	16	-	-	NUM
ejpam-5321	57	17	cl(a	cl(a	NUM
ejpam-5321	57	18	)	)	PUNCT
ejpam-5321	57	19	.	.	PUNCT
ejpam-5321	58	1	(	(	PUNCT
ejpam-5321	58	2	5	5	X
ejpam-5321	58	3	)	)	PUNCT
ejpam-5321	58	4	τ1τ2	τ1τ2	NOUN
ejpam-5321	58	5	-	-	NOUN
ejpam-5321	58	6	cl(x	cl(x	X
ejpam-5321	58	7	−a	−a	NOUN
ejpam-5321	58	8	)	)	PUNCT
ejpam-5321	59	1	=	=	PUNCT
ejpam-5321	59	2	x	x	X
ejpam-5321	60	1	−	−	ADP
ejpam-5321	60	2	τ1τ2	τ1τ2	NOUN
ejpam-5321	60	3	-	-	PUNCT
ejpam-5321	60	4	int(a	int(a	NOUN
ejpam-5321	60	5	)	)	PUNCT
ejpam-5321	60	6	.	.	PUNCT
ejpam-5321	61	1	a	a	DET
ejpam-5321	61	2	subset	subset	NOUN
ejpam-5321	61	3	a	a	PRON
ejpam-5321	61	4	of	of	ADP
ejpam-5321	61	5	a	a	DET
ejpam-5321	61	6	bitopological	bitopological	ADJ
ejpam-5321	61	7	space	space	NOUN
ejpam-5321	61	8	(	(	PUNCT
ejpam-5321	61	9	x	x	NOUN
ejpam-5321	61	10	,	,	PUNCT
ejpam-5321	61	11	τ1	τ1	NOUN
ejpam-5321	61	12	,	,	PUNCT
ejpam-5321	61	13	τ2	τ2	NOUN
ejpam-5321	61	14	)	)	PUNCT
ejpam-5321	61	15	is	be	AUX
ejpam-5321	61	16	said	say	VERB
ejpam-5321	61	17	to	to	PART
ejpam-5321	61	18	be	be	AUX
ejpam-5321	61	19	τ1τ2	τ1τ2	NOUN
ejpam-5321	61	20	-	-	ADJ
ejpam-5321	61	21	clopen	clopen	ADJ
ejpam-5321	62	1	[	[	X
ejpam-5321	62	2	20	20	NUM
ejpam-5321	62	3	]	]	X
ejpam-5321	62	4	if	if	SCONJ
ejpam-5321	62	5	a	a	PRON
ejpam-5321	62	6	is	be	AUX
ejpam-5321	62	7	both	both	PRON
ejpam-5321	62	8	τ1τ2	τ1τ2	ADJ
ejpam-5321	62	9	-	-	ADJ
ejpam-5321	62	10	open	open	ADJ
ejpam-5321	62	11	and	and	CCONJ
ejpam-5321	62	12	τ1τ2	τ1τ2	NOUN
ejpam-5321	62	13	-	-	ADJ
ejpam-5321	62	14	closed	closed	ADJ
ejpam-5321	62	15	.	.	PUNCT
ejpam-5321	63	1	a	a	DET
ejpam-5321	63	2	subset	subset	NOUN
ejpam-5321	63	3	a	a	PRON
ejpam-5321	63	4	of	of	ADP
ejpam-5321	63	5	a	a	DET
ejpam-5321	63	6	bitopological	bitopological	ADJ
ejpam-5321	63	7	space	space	NOUN
ejpam-5321	63	8	(	(	PUNCT
ejpam-5321	63	9	x	x	NOUN
ejpam-5321	63	10	,	,	PUNCT
ejpam-5321	63	11	τ1	τ1	NOUN
ejpam-5321	63	12	,	,	PUNCT
ejpam-5321	63	13	τ2	τ2	NOUN
ejpam-5321	63	14	)	)	PUNCT
ejpam-5321	63	15	is	be	AUX
ejpam-5321	63	16	said	say	VERB
ejpam-5321	63	17	to	to	PART
ejpam-5321	63	18	be	be	AUX
ejpam-5321	63	19	(	(	PUNCT
ejpam-5321	63	20	τ1	τ1	NOUN
ejpam-5321	63	21	,	,	PUNCT
ejpam-5321	63	22	τ2)r	τ2)r	NOUN
ejpam-5321	63	23	-	-	PUNCT
ejpam-5321	63	24	open	open	NOUN
ejpam-5321	64	1	[	[	X
ejpam-5321	64	2	43	43	NUM
ejpam-5321	64	3	]	]	PUNCT
ejpam-5321	64	4	(	(	PUNCT
ejpam-5321	64	5	resp	resp	NOUN
ejpam-5321	64	6	.	.	PUNCT
ejpam-5321	65	1	(	(	PUNCT
ejpam-5321	65	2	τ1	τ1	NOUN
ejpam-5321	65	3	,	,	PUNCT
ejpam-5321	65	4	τ2)s	τ2)s	NOUN
ejpam-5321	65	5	-	-	PUNCT
ejpam-5321	65	6	open	open	ADJ
ejpam-5321	65	7	[	[	X
ejpam-5321	65	8	6	6	NUM
ejpam-5321	65	9	]	]	PUNCT
ejpam-5321	65	10	,	,	PUNCT
ejpam-5321	65	11	(	(	PUNCT
ejpam-5321	65	12	τ1	τ1	NOUN
ejpam-5321	65	13	,	,	PUNCT
ejpam-5321	65	14	τ2)p	τ2)p	NOUN
ejpam-5321	65	15	-	-	ADJ
ejpam-5321	65	16	open	open	ADJ
ejpam-5321	65	17	[	[	X
ejpam-5321	65	18	6	6	NUM
ejpam-5321	65	19	]	]	PUNCT
ejpam-5321	65	20	,	,	PUNCT
ejpam-5321	65	21	(	(	PUNCT
ejpam-5321	65	22	τ1	τ1	NOUN
ejpam-5321	65	23	,	,	PUNCT
ejpam-5321	65	24	τ2)β	τ2)β	ADJ
ejpam-5321	65	25	-	-	PUNCT
ejpam-5321	65	26	open	open	NOUN
ejpam-5321	66	1	[	[	X
ejpam-5321	66	2	6	6	NUM
ejpam-5321	66	3	]	]	PUNCT
ejpam-5321	66	4	)	)	PUNCT
ejpam-5321	66	5	if	if	SCONJ
ejpam-5321	66	6	a	a	DET
ejpam-5321	66	7	=	=	PUNCT
ejpam-5321	66	8	τ1τ2	τ1τ2	NOUN
ejpam-5321	66	9	-	-	NOUN
ejpam-5321	66	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5321	66	11	-	-	PUNCT
ejpam-5321	66	12	cl(a	cl(a	NUM
ejpam-5321	66	13	)	)	PUNCT
ejpam-5321	66	14	)	)	PUNCT
ejpam-5321	66	15	(	(	PUNCT
ejpam-5321	66	16	resp	resp	NOUN
ejpam-5321	66	17	.	.	PUNCT
ejpam-5321	67	1	a	a	DET
ejpam-5321	67	2	⊆	⊆	NUM
ejpam-5321	67	3	τ1τ2	τ1τ2	NOUN
ejpam-5321	67	4	-	-	ADJ
ejpam-5321	67	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5321	67	6	-	-	PUNCT
ejpam-5321	67	7	int(a	int(a	NOUN
ejpam-5321	67	8	)	)	PUNCT
ejpam-5321	67	9	)	)	PUNCT
ejpam-5321	67	10	,	,	PUNCT
ejpam-5321	67	11	a	a	DET
ejpam-5321	67	12	⊆	⊆	NUM
ejpam-5321	67	13	τ1τ2	τ1τ2	NOUN
ejpam-5321	67	14	-	-	NOUN
ejpam-5321	67	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5321	67	16	-	-	PUNCT
ejpam-5321	67	17	cl(a	cl(a	NUM
ejpam-5321	67	18	)	)	PUNCT
ejpam-5321	67	19	)	)	PUNCT
ejpam-5321	67	20	,	,	PUNCT
ejpam-5321	67	21	a	a	DET
ejpam-5321	67	22	⊆	⊆	NUM
ejpam-5321	67	23	τ1τ2	τ1τ2	NOUN
ejpam-5321	67	24	-	-	PUNCT
ejpam-5321	67	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5321	67	26	-	-	PUNCT
ejpam-5321	67	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5321	67	28	-	-	PUNCT
ejpam-5321	67	29	cl(a	cl(a	NUM
ejpam-5321	67	30	)	)	PUNCT
ejpam-5321	67	31	)	)	PUNCT
ejpam-5321	67	32	)	)	PUNCT
ejpam-5321	67	33	)	)	PUNCT
ejpam-5321	67	34	.	.	PUNCT
ejpam-5321	68	1	the	the	DET
ejpam-5321	68	2	complement	complement	NOUN
ejpam-5321	68	3	of	of	ADP
ejpam-5321	68	4	a	a	DET
ejpam-5321	68	5	(	(	PUNCT
ejpam-5321	68	6	τ1	τ1	NOUN
ejpam-5321	68	7	,	,	PUNCT
ejpam-5321	68	8	τ2)r	τ2)r	NOUN
ejpam-5321	68	9	-	-	PUNCT
ejpam-5321	68	10	open	open	ADJ
ejpam-5321	68	11	(	(	PUNCT
ejpam-5321	68	12	resp	resp	NOUN
ejpam-5321	68	13	.	.	PUNCT
ejpam-5321	69	1	(	(	PUNCT
ejpam-5321	69	2	τ1	τ1	NOUN
ejpam-5321	69	3	,	,	PUNCT
ejpam-5321	69	4	τ2)sopen	τ2)sopen	ADJ
ejpam-5321	69	5	,	,	PUNCT
ejpam-5321	69	6	(	(	PUNCT
ejpam-5321	69	7	τ1	τ1	NOUN
ejpam-5321	69	8	,	,	PUNCT
ejpam-5321	69	9	τ2)p	τ2)p	NOUN
ejpam-5321	69	10	-	-	ADJ
ejpam-5321	69	11	open	open	ADJ
ejpam-5321	69	12	,	,	PUNCT
ejpam-5321	69	13	(	(	PUNCT
ejpam-5321	69	14	τ1	τ1	NOUN
ejpam-5321	69	15	,	,	PUNCT
ejpam-5321	69	16	τ2)β	τ2)β	ADJ
ejpam-5321	69	17	-	-	PUNCT
ejpam-5321	69	18	open	open	ADJ
ejpam-5321	69	19	)	)	PUNCT
ejpam-5321	69	20	set	set	NOUN
ejpam-5321	69	21	is	be	AUX
ejpam-5321	69	22	called	call	VERB
ejpam-5321	69	23	(	(	PUNCT
ejpam-5321	69	24	τ1	τ1	NOUN
ejpam-5321	69	25	,	,	PUNCT
ejpam-5321	69	26	τ2)r	τ2)r	NOUN
ejpam-5321	69	27	-	-	PUNCT
ejpam-5321	69	28	closed	closed	ADJ
ejpam-5321	69	29	(	(	PUNCT
ejpam-5321	69	30	resp	resp	NOUN
ejpam-5321	69	31	.	.	PUNCT
ejpam-5321	70	1	(	(	PUNCT
ejpam-5321	70	2	τ1	τ1	NOUN
ejpam-5321	70	3	,	,	PUNCT
ejpam-5321	70	4	τ2)s	τ2)s	NOUN
ejpam-5321	70	5	-	-	PUNCT
ejpam-5321	70	6	closed	closed	ADJ
ejpam-5321	70	7	,	,	PUNCT
ejpam-5321	70	8	(	(	PUNCT
ejpam-5321	70	9	τ1	τ1	NOUN
ejpam-5321	70	10	,	,	PUNCT
ejpam-5321	70	11	τ2)p	τ2)p	NOUN
ejpam-5321	70	12	-	-	PUNCT
ejpam-5321	70	13	closed	closed	ADJ
ejpam-5321	70	14	,	,	PUNCT
ejpam-5321	70	15	(	(	PUNCT
ejpam-5321	70	16	τ1	τ1	NOUN
ejpam-5321	70	17	,	,	PUNCT
ejpam-5321	70	18	τ2)β	τ2)β	ADJ
ejpam-5321	70	19	-	-	PUNCT
ejpam-5321	70	20	closed	closed	ADJ
ejpam-5321	70	21	)	)	PUNCT
ejpam-5321	70	22	.	.	PUNCT
ejpam-5321	71	1	a	a	DET
ejpam-5321	71	2	subset	subset	NOUN
ejpam-5321	71	3	a	a	PRON
ejpam-5321	71	4	of	of	ADP
ejpam-5321	71	5	a	a	DET
ejpam-5321	71	6	bitopological	bitopological	ADJ
ejpam-5321	71	7	space	space	NOUN
ejpam-5321	71	8	(	(	PUNCT
ejpam-5321	71	9	x	x	NOUN
ejpam-5321	71	10	,	,	PUNCT
ejpam-5321	71	11	τ1	τ1	NOUN
ejpam-5321	71	12	,	,	PUNCT
ejpam-5321	71	13	τ2	τ2	NOUN
ejpam-5321	71	14	)	)	PUNCT
ejpam-5321	71	15	is	be	AUX
ejpam-5321	71	16	said	say	VERB
ejpam-5321	71	17	to	to	PART
ejpam-5321	71	18	be	be	AUX
ejpam-5321	71	19	α(τ1	α(τ1	NOUN
ejpam-5321	71	20	,	,	PUNCT
ejpam-5321	71	21	τ2)-open	τ2)-open	ADJ
ejpam-5321	72	1	[	[	X
ejpam-5321	72	2	45	45	NUM
ejpam-5321	72	3	]	]	PUNCT
ejpam-5321	72	4	if	if	SCONJ
ejpam-5321	72	5	a	a	DET
ejpam-5321	72	6	⊆	⊆	NUM
ejpam-5321	72	7	τ1τ2	τ1τ2	NOUN
ejpam-5321	72	8	-	-	PUNCT
ejpam-5321	72	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5321	72	10	-	-	PUNCT
ejpam-5321	72	11	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5321	72	12	-	-	PUNCT
ejpam-5321	72	13	int(a	int(a	NOUN
ejpam-5321	72	14	)	)	PUNCT
ejpam-5321	72	15	)	)	PUNCT
ejpam-5321	72	16	)	)	PUNCT
ejpam-5321	72	17	.	.	PUNCT
ejpam-5321	73	1	the	the	DET
ejpam-5321	73	2	complement	complement	NOUN
ejpam-5321	73	3	of	of	ADP
ejpam-5321	73	4	an	an	DET
ejpam-5321	73	5	α(τ1	α(τ1	NOUN
ejpam-5321	73	6	,	,	PUNCT
ejpam-5321	73	7	τ2)-open	τ2)-open	ADJ
ejpam-5321	73	8	set	set	NOUN
ejpam-5321	73	9	is	be	AUX
ejpam-5321	73	10	said	say	VERB
ejpam-5321	73	11	to	to	PART
ejpam-5321	73	12	be	be	AUX
ejpam-5321	73	13	α(τ1	α(τ1	NOUN
ejpam-5321	73	14	,	,	PUNCT
ejpam-5321	73	15	τ2)-closed	τ2)-close	VERB
ejpam-5321	73	16	.	.	PUNCT
ejpam-5321	74	1	let	let	VERB
ejpam-5321	74	2	a	a	DET
ejpam-5321	74	3	be	be	AUX
ejpam-5321	74	4	a	a	DET
ejpam-5321	74	5	subset	subset	NOUN
ejpam-5321	74	6	of	of	ADP
ejpam-5321	74	7	a	a	DET
ejpam-5321	74	8	bitopological	bitopological	ADJ
ejpam-5321	74	9	space	space	NOUN
ejpam-5321	74	10	(	(	PUNCT
ejpam-5321	74	11	x	x	NOUN
ejpam-5321	74	12	,	,	PUNCT
ejpam-5321	74	13	τ1	τ1	NOUN
ejpam-5321	74	14	,	,	PUNCT
ejpam-5321	74	15	τ2	τ2	NOUN
ejpam-5321	74	16	)	)	PUNCT
ejpam-5321	74	17	.	.	PUNCT
ejpam-5321	75	1	the	the	DET
ejpam-5321	75	2	intersection	intersection	NOUN
ejpam-5321	75	3	of	of	ADP
ejpam-5321	75	4	all	all	DET
ejpam-5321	75	5	α(τ1	α(τ1	NOUN
ejpam-5321	75	6	,	,	PUNCT
ejpam-5321	75	7	τ2)-closed	τ2)-close	VERB
ejpam-5321	75	8	sets	set	NOUN
ejpam-5321	75	9	of	of	ADP
ejpam-5321	75	10	x	x	PUNCT
ejpam-5321	75	11	containing	contain	VERB
ejpam-5321	75	12	a	a	PRON
ejpam-5321	75	13	is	be	AUX
ejpam-5321	75	14	called	call	VERB
ejpam-5321	75	15	the	the	DET
ejpam-5321	75	16	α(τ1	α(τ1	NOUN
ejpam-5321	75	17	,	,	PUNCT
ejpam-5321	75	18	τ2)-closure	τ2)-closure	NOUN
ejpam-5321	75	19	of	of	ADP
ejpam-5321	75	20	a	a	PRON
ejpam-5321	75	21	and	and	CCONJ
ejpam-5321	75	22	is	be	AUX
ejpam-5321	75	23	denoted	denote	VERB
ejpam-5321	75	24	by	by	ADP
ejpam-5321	75	25	α(τ1	α(τ1	NOUN
ejpam-5321	75	26	,	,	PUNCT
ejpam-5321	75	27	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	75	28	)	)	PUNCT
ejpam-5321	75	29	.	.	PUNCT
ejpam-5321	76	1	the	the	DET
ejpam-5321	76	2	union	union	NOUN
ejpam-5321	76	3	of	of	ADP
ejpam-5321	76	4	all	all	DET
ejpam-5321	76	5	α(τ1	α(τ1	NOUN
ejpam-5321	76	6	,	,	PUNCT
ejpam-5321	76	7	τ2)open	τ2)open	ADJ
ejpam-5321	76	8	sets	set	NOUN
ejpam-5321	76	9	of	of	ADP
ejpam-5321	76	10	x	x	PUNCT
ejpam-5321	76	11	contained	contain	VERB
ejpam-5321	76	12	in	in	ADP
ejpam-5321	76	13	a	a	PRON
ejpam-5321	76	14	is	be	AUX
ejpam-5321	76	15	called	call	VERB
ejpam-5321	76	16	the	the	DET
ejpam-5321	76	17	α(τ1	α(τ1	NOUN
ejpam-5321	76	18	,	,	PUNCT
ejpam-5321	76	19	τ2)-interior	τ2)-interior	PRON
ejpam-5321	76	20	of	of	ADP
ejpam-5321	76	21	a	a	PRON
ejpam-5321	76	22	and	and	CCONJ
ejpam-5321	76	23	is	be	AUX
ejpam-5321	76	24	denoted	denote	VERB
ejpam-5321	76	25	by	by	ADP
ejpam-5321	76	26	α(τ1	α(τ1	NOUN
ejpam-5321	76	27	,	,	PUNCT
ejpam-5321	76	28	τ2)-int(a	τ2)-int(a	NOUN
ejpam-5321	76	29	)	)	PUNCT
ejpam-5321	76	30	.	.	PUNCT
ejpam-5321	77	1	lemma	lemma	PROPN
ejpam-5321	77	2	2	2	NUM
ejpam-5321	77	3	.	.	X
ejpam-5321	78	1	for	for	ADP
ejpam-5321	78	2	subsets	subset	NOUN
ejpam-5321	78	3	a	a	PRON
ejpam-5321	78	4	and	and	CCONJ
ejpam-5321	78	5	b	b	NOUN
ejpam-5321	78	6	of	of	ADP
ejpam-5321	78	7	a	a	DET
ejpam-5321	78	8	bitopological	bitopological	ADJ
ejpam-5321	78	9	space	space	NOUN
ejpam-5321	78	10	(	(	PUNCT
ejpam-5321	78	11	x	x	NOUN
ejpam-5321	78	12	,	,	PUNCT
ejpam-5321	78	13	τ1	τ1	NOUN
ejpam-5321	78	14	,	,	PUNCT
ejpam-5321	78	15	τ2	τ2	NOUN
ejpam-5321	78	16	)	)	PUNCT
ejpam-5321	78	17	,	,	PUNCT
ejpam-5321	78	18	the	the	DET
ejpam-5321	78	19	following	follow	VERB
ejpam-5321	78	20	properties	property	NOUN
ejpam-5321	78	21	hold	hold	VERB
ejpam-5321	78	22	:	:	PUNCT
ejpam-5321	78	23	(	(	PUNCT
ejpam-5321	78	24	1	1	X
ejpam-5321	78	25	)	)	PUNCT
ejpam-5321	78	26	a	a	DET
ejpam-5321	78	27	⊆	⊆	NUM
ejpam-5321	78	28	α(τ1	α(τ1	NOUN
ejpam-5321	78	29	,	,	PUNCT
ejpam-5321	78	30	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	78	31	)	)	PUNCT
ejpam-5321	78	32	and	and	CCONJ
ejpam-5321	78	33	α(τ1	α(τ1	NOUN
ejpam-5321	78	34	,	,	PUNCT
ejpam-5321	78	35	τ2)-cl(α(τ1	τ2)-cl(α(τ1	PROPN
ejpam-5321	78	36	,	,	PUNCT
ejpam-5321	78	37	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	78	38	)	)	PUNCT
ejpam-5321	78	39	)	)	PUNCT
ejpam-5321	79	1	=	=	SYM
ejpam-5321	79	2	α(τ1	α(τ1	NOUN
ejpam-5321	79	3	,	,	PUNCT
ejpam-5321	79	4	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	79	5	)	)	PUNCT
ejpam-5321	79	6	.	.	PUNCT
ejpam-5321	80	1	(	(	PUNCT
ejpam-5321	80	2	2	2	X
ejpam-5321	80	3	)	)	PUNCT
ejpam-5321	80	4	if	if	SCONJ
ejpam-5321	80	5	a	a	DET
ejpam-5321	80	6	⊆	⊆	NUM
ejpam-5321	80	7	b	b	NOUN
ejpam-5321	80	8	,	,	PUNCT
ejpam-5321	80	9	then	then	ADV
ejpam-5321	80	10	α(τ1	α(τ1	NOUN
ejpam-5321	80	11	,	,	PUNCT
ejpam-5321	80	12	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	80	13	)	)	PUNCT
ejpam-5321	80	14	⊆	⊆	NUM
ejpam-5321	80	15	α(τ1	α(τ1	NOUN
ejpam-5321	80	16	,	,	PUNCT
ejpam-5321	80	17	τ2)-cl(b	τ2)-cl(b	NOUN
ejpam-5321	80	18	)	)	PUNCT
ejpam-5321	80	19	.	.	PUNCT
ejpam-5321	81	1	(	(	PUNCT
ejpam-5321	81	2	3	3	X
ejpam-5321	81	3	)	)	PUNCT
ejpam-5321	81	4	α(τ1	α(τ1	NOUN
ejpam-5321	81	5	,	,	PUNCT
ejpam-5321	81	6	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	81	7	)	)	PUNCT
ejpam-5321	81	8	is	be	AUX
ejpam-5321	81	9	α(τ1	α(τ1	NOUN
ejpam-5321	81	10	,	,	PUNCT
ejpam-5321	81	11	τ2)-closed	τ2)-closed	ADJ
ejpam-5321	81	12	.	.	PUNCT
ejpam-5321	82	1	(	(	PUNCT
ejpam-5321	82	2	4	4	X
ejpam-5321	82	3	)	)	PUNCT
ejpam-5321	82	4	a	a	PRON
ejpam-5321	82	5	is	be	AUX
ejpam-5321	82	6	α(τ1	α(τ1	NOUN
ejpam-5321	82	7	,	,	PUNCT
ejpam-5321	82	8	τ2)-closed	τ2)-close	VERB
ejpam-5321	82	9	if	if	SCONJ
ejpam-5321	82	10	and	and	CCONJ
ejpam-5321	82	11	only	only	ADV
ejpam-5321	82	12	if	if	SCONJ
ejpam-5321	82	13	a	a	DET
ejpam-5321	82	14	=	=	NOUN
ejpam-5321	82	15	α(τ1	α(τ1	NOUN
ejpam-5321	82	16	,	,	PUNCT
ejpam-5321	82	17	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5321	82	18	)	)	PUNCT
ejpam-5321	82	19	.	.	PUNCT
ejpam-5321	83	1	(	(	PUNCT
ejpam-5321	83	2	5	5	NUM
ejpam-5321	83	3	)	)	PUNCT
ejpam-5321	83	4	α(τ1	α(τ1	NOUN
ejpam-5321	83	5	,	,	PUNCT
ejpam-5321	83	6	τ2)-cl(x	τ2)-cl(x	PROPN
ejpam-5321	83	7	−a	−a	NOUN
ejpam-5321	83	8	)	)	PUNCT
ejpam-5321	84	1	=	=	PUNCT
ejpam-5321	84	2	x	x	SYM
ejpam-5321	85	1	−	−	PROPN
ejpam-5321	85	2	α(τ1	α(τ1	NOUN
ejpam-5321	85	3	,	,	PUNCT
ejpam-5321	85	4	τ2)-int(a	τ2)-int(a	NOUN
ejpam-5321	85	5	)	)	PUNCT
ejpam-5321	85	6	.	.	PUNCT
ejpam-5321	86	1	by	by	ADP
ejpam-5321	86	2	a	a	DET
ejpam-5321	86	3	multifunction	multifunction	NOUN
ejpam-5321	86	4	f	f	NOUN
ejpam-5321	86	5	:	:	PUNCT
ejpam-5321	86	6	x	x	X
ejpam-5321	86	7	→	→	SYM
ejpam-5321	86	8	y	y	PROPN
ejpam-5321	86	9	,	,	PUNCT
ejpam-5321	86	10	we	we	PRON
ejpam-5321	86	11	mean	mean	VERB
ejpam-5321	86	12	a	a	DET
ejpam-5321	86	13	point	point	NOUN
ejpam-5321	86	14	-	-	PUNCT
ejpam-5321	86	15	to	to	ADP
ejpam-5321	86	16	-	-	PUNCT
ejpam-5321	86	17	set	set	VERB
ejpam-5321	86	18	correspondence	correspondence	NOUN
ejpam-5321	86	19	from	from	ADP
ejpam-5321	86	20	x	x	PUNCT
ejpam-5321	86	21	into	into	ADP
ejpam-5321	86	22	y	y	PROPN
ejpam-5321	86	23	,	,	PUNCT
ejpam-5321	86	24	and	and	CCONJ
ejpam-5321	86	25	we	we	PRON
ejpam-5321	86	26	always	always	ADV
ejpam-5321	86	27	assume	assume	VERB
ejpam-5321	86	28	that	that	SCONJ
ejpam-5321	86	29	f	f	PROPN
ejpam-5321	86	30	(	(	PUNCT
ejpam-5321	86	31	x	x	X
ejpam-5321	86	32	)	)	PUNCT
ejpam-5321	86	33	̸=	̸=	NOUN
ejpam-5321	86	34	∅	∅	NOUN
ejpam-5321	86	35	for	for	ADP
ejpam-5321	86	36	all	all	PRON
ejpam-5321	86	37	x	x	SYM
ejpam-5321	86	38	∈	∈	ADJ
ejpam-5321	86	39	x.	x.	NOUN
ejpam-5321	86	40	for	for	ADP
ejpam-5321	86	41	a	a	DET
ejpam-5321	86	42	multifunction	multifunction	NOUN
ejpam-5321	86	43	f	f	NOUN
ejpam-5321	87	1	:	:	PUNCT
ejpam-5321	87	2	x	x	X
ejpam-5321	87	3	→	→	SYM
ejpam-5321	87	4	y	y	PROPN
ejpam-5321	87	5	,	,	PUNCT
ejpam-5321	87	6	following	follow	VERB
ejpam-5321	87	7	[	[	X
ejpam-5321	87	8	1	1	X
ejpam-5321	87	9	]	]	PUNCT
ejpam-5321	87	10	we	we	PRON
ejpam-5321	87	11	shall	shall	AUX
ejpam-5321	87	12	denote	denote	VERB
ejpam-5321	87	13	the	the	DET
ejpam-5321	87	14	upper	upper	ADJ
ejpam-5321	87	15	and	and	CCONJ
ejpam-5321	87	16	lower	low	ADJ
ejpam-5321	87	17	inverse	inverse	NOUN
ejpam-5321	87	18	of	of	ADP
ejpam-5321	87	19	a	a	DET
ejpam-5321	87	20	set	set	NOUN
ejpam-5321	87	21	b	b	PROPN
ejpam-5321	87	22	of	of	ADP
ejpam-5321	87	23	y	y	PROPN
ejpam-5321	87	24	by	by	ADP
ejpam-5321	87	25	f+(b	f+(b	NOUN
ejpam-5321	87	26	)	)	PUNCT
ejpam-5321	87	27	and	and	CCONJ
ejpam-5321	87	28	f−(b	f−(b	NOUN
ejpam-5321	87	29	)	)	PUNCT
ejpam-5321	87	30	,	,	PUNCT
ejpam-5321	87	31	respectively	respectively	ADV
ejpam-5321	87	32	,	,	PUNCT
ejpam-5321	87	33	that	that	ADV
ejpam-5321	87	34	is	is	ADV
ejpam-5321	87	35	,	,	PUNCT
ejpam-5321	87	36	f+(b	f+(b	NOUN
ejpam-5321	87	37	)	)	PUNCT
ejpam-5321	87	38	=	=	PRON
ejpam-5321	88	1	{	{	PUNCT
ejpam-5321	88	2	x	x	PUNCT
ejpam-5321	88	3	∈	∈	PROPN
ejpam-5321	88	4	x	x	INTJ
ejpam-5321	89	1	|	|	NOUN
ejpam-5321	89	2	f	f	X
ejpam-5321	89	3	(	(	PUNCT
ejpam-5321	89	4	x	x	NOUN
ejpam-5321	89	5	)	)	PUNCT
ejpam-5321	89	6	⊆	⊆	NUM
ejpam-5321	89	7	b	b	NOUN
ejpam-5321	89	8	}	}	PUNCT
ejpam-5321	89	9	and	and	CCONJ
ejpam-5321	89	10	f−(b	f−(b	PROPN
ejpam-5321	89	11	)	)	PUNCT
ejpam-5321	89	12	=	=	PRON
ejpam-5321	90	1	{	{	PUNCT
ejpam-5321	90	2	x	x	PUNCT
ejpam-5321	90	3	∈	∈	PROPN
ejpam-5321	90	4	x	x	INTJ
ejpam-5321	91	1	|	|	NOUN
ejpam-5321	91	2	f	f	X
ejpam-5321	91	3	(	(	PUNCT
ejpam-5321	91	4	x	x	NOUN
ejpam-5321	91	5	)	)	PUNCT
ejpam-5321	91	6	∩b	∩b	NOUN
ejpam-5321	91	7	̸=	̸=	PROPN
ejpam-5321	91	8	∅	∅	NOUN
ejpam-5321	91	9	}	}	PUNCT
ejpam-5321	91	10	.	.	PUNCT
ejpam-5321	92	1	in	in	ADP
ejpam-5321	92	2	particular	particular	ADJ
ejpam-5321	92	3	,	,	PUNCT
ejpam-5321	92	4	f−(y	f−(y	NOUN
ejpam-5321	92	5	)	)	PUNCT
ejpam-5321	92	6	=	=	SYM
ejpam-5321	93	1	{	{	PUNCT
ejpam-5321	93	2	x	x	PUNCT
ejpam-5321	93	3	∈	∈	PROPN
ejpam-5321	93	4	x	x	INTJ
ejpam-5321	94	1	|	|	ADV
ejpam-5321	94	2	y	y	PROPN
ejpam-5321	94	3	∈	∈	PROPN
ejpam-5321	94	4	f	f	X
ejpam-5321	94	5	(	(	PUNCT
ejpam-5321	94	6	x	x	NOUN
ejpam-5321	94	7	)	)	PUNCT
ejpam-5321	94	8	}	}	PUNCT
ejpam-5321	94	9	for	for	ADP
ejpam-5321	94	10	each	each	DET
ejpam-5321	94	11	point	point	NOUN
ejpam-5321	94	12	y	y	PROPN
ejpam-5321	94	13	∈	∈	PROPN
ejpam-5321	94	14	y	y	PROPN
ejpam-5321	94	15	.	.	PUNCT
ejpam-5321	95	1	for	for	ADP
ejpam-5321	95	2	each	each	PRON
ejpam-5321	95	3	a	a	DET
ejpam-5321	95	4	⊆	⊆	NUM
ejpam-5321	95	5	x	x	SYM
ejpam-5321	95	6	,	,	PUNCT
ejpam-5321	95	7	f	f	PROPN
ejpam-5321	95	8	(	(	PUNCT
ejpam-5321	95	9	a	a	NOUN
ejpam-5321	95	10	)	)	PUNCT
ejpam-5321	95	11	=	=	SYM
ejpam-5321	95	12	∪x∈af	∪x∈af	NOUN
ejpam-5321	95	13	(	(	PUNCT
ejpam-5321	95	14	x	x	NOUN
ejpam-5321	95	15	)	)	PUNCT
ejpam-5321	95	16	.	.	PUNCT
ejpam-5321	96	1	c.	c.	PROPN
ejpam-5321	96	2	viriyapong	viriyapong	PROPN
ejpam-5321	96	3	,	,	PUNCT
ejpam-5321	96	4	s.	s.	PROPN
ejpam-5321	96	5	sompong	sompong	PROPN
ejpam-5321	96	6	,	,	PUNCT
ejpam-5321	96	7	c.	c.	PROPN
ejpam-5321	96	8	boonpok	boonpok	PROPN
ejpam-5321	96	9	/	/	SYM
ejpam-5321	96	10	eur	eur	PROPN
ejpam-5321	96	11	.	.	PUNCT
ejpam-5321	97	1	j.	j.	PROPN
ejpam-5321	97	2	pure	pure	PROPN
ejpam-5321	97	3	appl	appl	PROPN
ejpam-5321	97	4	.	.	PROPN
ejpam-5321	97	5	math	math	PROPN
ejpam-5321	97	6	,	,	PUNCT
ejpam-5321	97	7	17	17	NUM
ejpam-5321	97	8	(	(	PUNCT
ejpam-5321	97	9	3	3	NUM
ejpam-5321	97	10	)	)	PUNCT
ejpam-5321	97	11	(	(	PUNCT
ejpam-5321	97	12	2024	2024	NUM
ejpam-5321	97	13	)	)	PUNCT
ejpam-5321	97	14	,	,	PUNCT
ejpam-5321	97	15	2142	2142	NUM
ejpam-5321	97	16	-	-	SYM
ejpam-5321	97	17	2154	2154	NUM
ejpam-5321	97	18	2145	2145	NUM
ejpam-5321	97	19	3	3	NUM
ejpam-5321	97	20	.	.	PUNCT
ejpam-5321	97	21	upper	upper	ADJ
ejpam-5321	97	22	and	and	CCONJ
ejpam-5321	97	23	lower	low	ADJ
ejpam-5321	97	24	slightly	slightly	ADJ
ejpam-5321	97	25	α(τ1	α(τ1	NOUN
ejpam-5321	97	26	,	,	PUNCT
ejpam-5321	97	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	97	28	multifunctions	multifunction	NOUN
ejpam-5321	97	29	in	in	ADP
ejpam-5321	97	30	this	this	DET
ejpam-5321	97	31	section	section	NOUN
ejpam-5321	97	32	,	,	PUNCT
ejpam-5321	97	33	we	we	PRON
ejpam-5321	97	34	introduce	introduce	VERB
ejpam-5321	97	35	the	the	DET
ejpam-5321	97	36	notions	notion	NOUN
ejpam-5321	97	37	of	of	ADP
ejpam-5321	97	38	upper	upper	ADJ
ejpam-5321	97	39	and	and	CCONJ
ejpam-5321	97	40	lower	low	ADJ
ejpam-5321	97	41	slightly	slightly	ADJ
ejpam-5321	97	42	α(τ1	α(τ1	NOUN
ejpam-5321	97	43	,	,	PUNCT
ejpam-5321	97	44	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	97	45	multifunctions	multifunction	NOUN
ejpam-5321	97	46	.	.	PUNCT
ejpam-5321	98	1	moreover	moreover	ADV
ejpam-5321	98	2	,	,	PUNCT
ejpam-5321	98	3	some	some	DET
ejpam-5321	98	4	characterizations	characterization	NOUN
ejpam-5321	98	5	of	of	ADP
ejpam-5321	98	6	upper	upper	ADJ
ejpam-5321	98	7	and	and	CCONJ
ejpam-5321	98	8	lower	low	ADJ
ejpam-5321	98	9	slightly	slightly	ADJ
ejpam-5321	98	10	α(τ1	α(τ1	NOUN
ejpam-5321	98	11	,	,	PUNCT
ejpam-5321	98	12	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	98	13	multifunctions	multifunction	NOUN
ejpam-5321	98	14	are	be	AUX
ejpam-5321	98	15	discussed	discuss	VERB
ejpam-5321	98	16	.	.	PUNCT
ejpam-5321	99	1	definition	definition	NOUN
ejpam-5321	99	2	1	1	NUM
ejpam-5321	99	3	.	.	PUNCT
ejpam-5321	100	1	a	a	DET
ejpam-5321	100	2	multifunction	multifunction	NOUN
ejpam-5321	100	3	f	f	NOUN
ejpam-5321	100	4	:	:	PUNCT
ejpam-5321	100	5	(	(	PUNCT
ejpam-5321	100	6	x	x	NOUN
ejpam-5321	100	7	,	,	PUNCT
ejpam-5321	100	8	τ1	τ1	NOUN
ejpam-5321	100	9	,	,	PUNCT
ejpam-5321	100	10	τ2	τ2	NOUN
ejpam-5321	100	11	)	)	PUNCT
ejpam-5321	100	12	→	→	SYM
ejpam-5321	100	13	(	(	PUNCT
ejpam-5321	100	14	y	y	PROPN
ejpam-5321	100	15	,	,	PUNCT
ejpam-5321	100	16	σ1	σ1	PROPN
ejpam-5321	100	17	,	,	PUNCT
ejpam-5321	100	18	σ2	σ2	PROPN
ejpam-5321	100	19	)	)	PUNCT
ejpam-5321	100	20	is	be	AUX
ejpam-5321	100	21	said	say	VERB
ejpam-5321	100	22	to	to	PART
ejpam-5321	100	23	be	be	AUX
ejpam-5321	100	24	upper	upper	ADJ
ejpam-5321	100	25	slightly	slightly	ADV
ejpam-5321	100	26	α(τ1	α(τ1	NOUN
ejpam-5321	100	27	,	,	PUNCT
ejpam-5321	100	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	100	29	at	at	ADP
ejpam-5321	100	30	a	a	DET
ejpam-5321	100	31	point	point	NOUN
ejpam-5321	100	32	x	x	SYM
ejpam-5321	100	33	∈	∈	NOUN
ejpam-5321	100	34	x	x	PUNCT
ejpam-5321	100	35	if	if	SCONJ
ejpam-5321	100	36	for	for	ADP
ejpam-5321	100	37	each	each	DET
ejpam-5321	100	38	σ1σ2	σ1σ2	NUM
ejpam-5321	100	39	-	-	PUNCT
ejpam-5321	100	40	clopen	clopen	ADJ
ejpam-5321	100	41	set	set	NOUN
ejpam-5321	100	42	v	v	NOUN
ejpam-5321	100	43	of	of	ADP
ejpam-5321	100	44	y	y	PROPN
ejpam-5321	100	45	containing	contain	VERB
ejpam-5321	100	46	f	f	PROPN
ejpam-5321	100	47	(	(	PUNCT
ejpam-5321	100	48	x	x	NOUN
ejpam-5321	100	49	)	)	PUNCT
ejpam-5321	100	50	,	,	PUNCT
ejpam-5321	100	51	there	there	PRON
ejpam-5321	100	52	exists	exist	VERB
ejpam-5321	100	53	an	an	DET
ejpam-5321	100	54	α(τ1	α(τ1	NOUN
ejpam-5321	100	55	,	,	PUNCT
ejpam-5321	100	56	τ2)-open	τ2)-open	ADJ
ejpam-5321	100	57	set	set	VERB
ejpam-5321	100	58	u	u	NOUN
ejpam-5321	100	59	of	of	ADP
ejpam-5321	100	60	x	x	PUNCT
ejpam-5321	100	61	containing	contain	VERB
ejpam-5321	100	62	x	x	PUNCT
ejpam-5321	100	63	such	such	ADJ
ejpam-5321	100	64	that	that	SCONJ
ejpam-5321	100	65	f	f	PROPN
ejpam-5321	100	66	(	(	PUNCT
ejpam-5321	100	67	u	u	NOUN
ejpam-5321	100	68	)	)	PUNCT
ejpam-5321	100	69	⊆	⊆	NUM
ejpam-5321	100	70	v	v	NOUN
ejpam-5321	100	71	.	.	PUNCT
ejpam-5321	101	1	a	a	DET
ejpam-5321	101	2	multifunction	multifunction	NOUN
ejpam-5321	101	3	f	f	NOUN
ejpam-5321	101	4	:	:	PUNCT
ejpam-5321	101	5	(	(	PUNCT
ejpam-5321	101	6	x	x	NOUN
ejpam-5321	101	7	,	,	PUNCT
ejpam-5321	101	8	τ1	τ1	NOUN
ejpam-5321	101	9	,	,	PUNCT
ejpam-5321	101	10	τ2	τ2	NOUN
ejpam-5321	101	11	)	)	PUNCT
ejpam-5321	101	12	→	→	SYM
ejpam-5321	101	13	(	(	PUNCT
ejpam-5321	101	14	y	y	PROPN
ejpam-5321	101	15	,	,	PUNCT
ejpam-5321	101	16	σ1	σ1	PROPN
ejpam-5321	101	17	,	,	PUNCT
ejpam-5321	101	18	σ2	σ2	PROPN
ejpam-5321	101	19	)	)	PUNCT
ejpam-5321	101	20	is	be	AUX
ejpam-5321	101	21	said	say	VERB
ejpam-5321	101	22	to	to	PART
ejpam-5321	101	23	be	be	AUX
ejpam-5321	101	24	upper	upper	ADJ
ejpam-5321	101	25	slightly	slightly	ADV
ejpam-5321	101	26	α(τ1	α(τ1	NOUN
ejpam-5321	101	27	,	,	PUNCT
ejpam-5321	101	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	101	29	if	if	SCONJ
ejpam-5321	101	30	f	f	PROPN
ejpam-5321	101	31	has	have	VERB
ejpam-5321	101	32	this	this	DET
ejpam-5321	101	33	property	property	NOUN
ejpam-5321	101	34	at	at	ADP
ejpam-5321	101	35	every	every	DET
ejpam-5321	101	36	point	point	NOUN
ejpam-5321	101	37	of	of	ADP
ejpam-5321	101	38	x.	x.	NOUN
ejpam-5321	101	39	recall	recall	VERB
ejpam-5321	101	40	that	that	SCONJ
ejpam-5321	101	41	a	a	DET
ejpam-5321	101	42	net	net	NOUN
ejpam-5321	101	43	(	(	PUNCT
ejpam-5321	101	44	xγ	xγ	PROPN
ejpam-5321	101	45	)	)	PUNCT
ejpam-5321	101	46	in	in	ADP
ejpam-5321	101	47	a	a	DET
ejpam-5321	101	48	topological	topological	ADJ
ejpam-5321	101	49	space	space	NOUN
ejpam-5321	101	50	(	(	PUNCT
ejpam-5321	101	51	x	x	X
ejpam-5321	101	52	,	,	PUNCT
ejpam-5321	101	53	τ	τ	X
ejpam-5321	101	54	)	)	PUNCT
ejpam-5321	101	55	is	be	AUX
ejpam-5321	101	56	said	say	VERB
ejpam-5321	101	57	to	to	PART
ejpam-5321	101	58	be	be	AUX
ejpam-5321	101	59	eventually	eventually	ADV
ejpam-5321	101	60	in	in	ADP
ejpam-5321	101	61	the	the	DET
ejpam-5321	101	62	set	set	NOUN
ejpam-5321	101	63	u	u	NOUN
ejpam-5321	101	64	⊆	⊆	NUM
ejpam-5321	101	65	x	x	SYM
ejpam-5321	101	66	if	if	SCONJ
ejpam-5321	101	67	there	there	PRON
ejpam-5321	101	68	exists	exist	VERB
ejpam-5321	101	69	an	an	DET
ejpam-5321	101	70	index	index	NOUN
ejpam-5321	101	71	γ0	γ0	NOUN
ejpam-5321	101	72	∈	∈	PROPN
ejpam-5321	101	73	∇	∇	X
ejpam-5321	101	74	such	such	ADJ
ejpam-5321	101	75	that	that	SCONJ
ejpam-5321	101	76	xγ	xγ	VERB
ejpam-5321	101	77	∈	∈	PROPN
ejpam-5321	101	78	u	u	NOUN
ejpam-5321	101	79	for	for	ADP
ejpam-5321	101	80	all	all	DET
ejpam-5321	101	81	γ	γ	PROPN
ejpam-5321	101	82	≥	≥	PROPN
ejpam-5321	101	83	γ0	γ0	PROPN
ejpam-5321	101	84	.	.	PUNCT
ejpam-5321	102	1	definition	definition	NOUN
ejpam-5321	102	2	2	2	NUM
ejpam-5321	102	3	.	.	PUNCT
ejpam-5321	103	1	a	a	DET
ejpam-5321	103	2	sequence	sequence	NOUN
ejpam-5321	103	3	(	(	PUNCT
ejpam-5321	103	4	xn	xn	X
ejpam-5321	103	5	)	)	PUNCT
ejpam-5321	103	6	is	be	AUX
ejpam-5321	103	7	called	call	VERB
ejpam-5321	103	8	α(τ1	α(τ1	NOUN
ejpam-5321	103	9	,	,	PUNCT
ejpam-5321	103	10	τ2)-converge	τ2)-converge	VERB
ejpam-5321	103	11	to	to	ADP
ejpam-5321	103	12	a	a	DET
ejpam-5321	103	13	point	point	NOUN
ejpam-5321	103	14	x	x	PUNCT
ejpam-5321	103	15	if	if	SCONJ
ejpam-5321	103	16	for	for	ADP
ejpam-5321	103	17	every	every	DET
ejpam-5321	103	18	α(τ1	α(τ1	NOUN
ejpam-5321	103	19	,	,	PUNCT
ejpam-5321	103	20	τ2)-open	τ2)-open	ADP
ejpam-5321	103	21	set	set	VERB
ejpam-5321	103	22	v	v	NOUN
ejpam-5321	103	23	containing	contain	VERB
ejpam-5321	103	24	x	x	X
ejpam-5321	103	25	,	,	PUNCT
ejpam-5321	103	26	there	there	PRON
ejpam-5321	103	27	exists	exist	VERB
ejpam-5321	103	28	an	an	DET
ejpam-5321	103	29	index	index	NOUN
ejpam-5321	103	30	n0	n0	NOUN
ejpam-5321	103	31	such	such	ADJ
ejpam-5321	103	32	that	that	PRON
ejpam-5321	103	33	for	for	ADP
ejpam-5321	103	34	n	n	PRON
ejpam-5321	103	35	≥	≥	NOUN
ejpam-5321	103	36	n0	n0	NUM
ejpam-5321	103	37	,	,	PUNCT
ejpam-5321	103	38	xn	xn	PROPN
ejpam-5321	103	39	∈	∈	PROPN
ejpam-5321	103	40	v	v	NOUN
ejpam-5321	103	41	.	.	PUNCT
ejpam-5321	104	1	theorem	theorem	NOUN
ejpam-5321	104	2	1	1	NUM
ejpam-5321	104	3	.	.	X
ejpam-5321	104	4	for	for	ADP
ejpam-5321	104	5	a	a	DET
ejpam-5321	104	6	multifunction	multifunction	NOUN
ejpam-5321	105	1	f	f	NOUN
ejpam-5321	105	2	:	:	PUNCT
ejpam-5321	105	3	(	(	PUNCT
ejpam-5321	105	4	x	x	NOUN
ejpam-5321	105	5	,	,	PUNCT
ejpam-5321	105	6	τ1	τ1	NOUN
ejpam-5321	105	7	,	,	PUNCT
ejpam-5321	105	8	τ2	τ2	NOUN
ejpam-5321	105	9	)	)	PUNCT
ejpam-5321	105	10	→	→	SYM
ejpam-5321	105	11	(	(	PUNCT
ejpam-5321	105	12	y	y	PROPN
ejpam-5321	105	13	,	,	PUNCT
ejpam-5321	105	14	σ1	σ1	PROPN
ejpam-5321	105	15	,	,	PUNCT
ejpam-5321	105	16	σ2	σ2	NOUN
ejpam-5321	105	17	)	)	PUNCT
ejpam-5321	105	18	,	,	PUNCT
ejpam-5321	105	19	the	the	DET
ejpam-5321	105	20	following	follow	VERB
ejpam-5321	105	21	properties	property	NOUN
ejpam-5321	105	22	are	be	AUX
ejpam-5321	105	23	equivalent	equivalent	ADJ
ejpam-5321	105	24	:	:	PUNCT
ejpam-5321	105	25	(	(	PUNCT
ejpam-5321	105	26	1	1	X
ejpam-5321	105	27	)	)	PUNCT
ejpam-5321	105	28	f	f	PROPN
ejpam-5321	105	29	is	be	AUX
ejpam-5321	105	30	upper	upper	ADJ
ejpam-5321	105	31	slightly	slightly	ADV
ejpam-5321	105	32	α(τ1	α(τ1	NOUN
ejpam-5321	105	33	,	,	PUNCT
ejpam-5321	105	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	105	35	;	;	PUNCT
ejpam-5321	105	36	(	(	PUNCT
ejpam-5321	105	37	2	2	X
ejpam-5321	105	38	)	)	PUNCT
ejpam-5321	105	39	for	for	ADP
ejpam-5321	105	40	each	each	DET
ejpam-5321	105	41	x	x	SYM
ejpam-5321	105	42	∈	∈	PROPN
ejpam-5321	105	43	x	x	X
ejpam-5321	105	44	and	and	CCONJ
ejpam-5321	105	45	for	for	ADP
ejpam-5321	105	46	each	each	DET
ejpam-5321	105	47	σ1σ2	σ1σ2	NUM
ejpam-5321	105	48	-	-	PUNCT
ejpam-5321	105	49	clopen	clopen	ADJ
ejpam-5321	105	50	set	set	NOUN
ejpam-5321	105	51	v	v	NOUN
ejpam-5321	105	52	of	of	ADP
ejpam-5321	105	53	y	y	PRON
ejpam-5321	105	54	such	such	ADJ
ejpam-5321	105	55	that	that	SCONJ
ejpam-5321	105	56	x	x	SYM
ejpam-5321	105	57	∈	∈	PROPN
ejpam-5321	105	58	f+(v	f+(v	NOUN
ejpam-5321	105	59	)	)	PUNCT
ejpam-5321	105	60	,	,	PUNCT
ejpam-5321	105	61	there	there	PRON
ejpam-5321	105	62	exists	exist	VERB
ejpam-5321	105	63	an	an	DET
ejpam-5321	105	64	α(τ1	α(τ1	NOUN
ejpam-5321	105	65	,	,	PUNCT
ejpam-5321	105	66	τ2)-open	τ2)-open	ADJ
ejpam-5321	105	67	set	set	VERB
ejpam-5321	105	68	u	u	NOUN
ejpam-5321	105	69	of	of	ADP
ejpam-5321	105	70	x	x	PUNCT
ejpam-5321	105	71	containing	contain	VERB
ejpam-5321	105	72	x	x	PUNCT
ejpam-5321	105	73	such	such	ADJ
ejpam-5321	105	74	that	that	SCONJ
ejpam-5321	105	75	u	u	NOUN
ejpam-5321	105	76	⊆	⊆	NUM
ejpam-5321	105	77	f+(v	f+(v	NOUN
ejpam-5321	105	78	)	)	PUNCT
ejpam-5321	105	79	;	;	PUNCT
ejpam-5321	105	80	(	(	PUNCT
ejpam-5321	105	81	3	3	X
ejpam-5321	105	82	)	)	PUNCT
ejpam-5321	105	83	for	for	ADP
ejpam-5321	105	84	each	each	DET
ejpam-5321	105	85	x	x	SYM
ejpam-5321	105	86	∈	∈	PROPN
ejpam-5321	105	87	x	x	X
ejpam-5321	105	88	and	and	CCONJ
ejpam-5321	105	89	for	for	ADP
ejpam-5321	105	90	each	each	DET
ejpam-5321	105	91	σ1σ2	σ1σ2	NUM
ejpam-5321	105	92	-	-	PUNCT
ejpam-5321	105	93	clopen	clopen	ADJ
ejpam-5321	105	94	set	set	NOUN
ejpam-5321	105	95	v	v	NOUN
ejpam-5321	105	96	of	of	ADP
ejpam-5321	105	97	y	y	PRON
ejpam-5321	105	98	such	such	ADJ
ejpam-5321	105	99	that	that	SCONJ
ejpam-5321	105	100	x	x	SYM
ejpam-5321	105	101	∈	∈	PROPN
ejpam-5321	105	102	f+(y	f+(y	NOUN
ejpam-5321	105	103	−v	−v	NOUN
ejpam-5321	105	104	)	)	PUNCT
ejpam-5321	105	105	,	,	PUNCT
ejpam-5321	105	106	there	there	PRON
ejpam-5321	105	107	exists	exist	VERB
ejpam-5321	105	108	an	an	DET
ejpam-5321	105	109	α(τ1	α(τ1	NOUN
ejpam-5321	105	110	,	,	PUNCT
ejpam-5321	105	111	τ2)-closed	τ2)-close	VERB
ejpam-5321	105	112	set	set	ADJ
ejpam-5321	105	113	h	h	NOUN
ejpam-5321	105	114	of	of	ADP
ejpam-5321	105	115	x	x	INTJ
ejpam-5321	105	116	such	such	ADJ
ejpam-5321	105	117	that	that	SCONJ
ejpam-5321	105	118	x	x	SYM
ejpam-5321	105	119	∈	∈	NOUN
ejpam-5321	105	120	x	x	SYM
ejpam-5321	105	121	−h	−h	ADJ
ejpam-5321	105	122	and	and	CCONJ
ejpam-5321	105	123	f−(v	f−(v	ADJ
ejpam-5321	105	124	)	)	PUNCT
ejpam-5321	105	125	⊆	⊆	NUM
ejpam-5321	105	126	h	h	NOUN
ejpam-5321	105	127	;	;	PUNCT
ejpam-5321	105	128	(	(	PUNCT
ejpam-5321	105	129	4	4	NUM
ejpam-5321	105	130	)	)	PUNCT
ejpam-5321	105	131	f+(v	f+(v	NOUN
ejpam-5321	105	132	)	)	PUNCT
ejpam-5321	105	133	is	be	AUX
ejpam-5321	105	134	α(τ1	α(τ1	NOUN
ejpam-5321	105	135	,	,	PUNCT
ejpam-5321	105	136	τ2)-open	τ2)-open	ADJ
ejpam-5321	105	137	in	in	ADP
ejpam-5321	105	138	x	x	PUNCT
ejpam-5321	105	139	for	for	ADP
ejpam-5321	105	140	every	every	DET
ejpam-5321	105	141	σ1σ2	σ1σ2	NUM
ejpam-5321	105	142	-	-	PUNCT
ejpam-5321	105	143	clopen	clopen	ADJ
ejpam-5321	105	144	set	set	NOUN
ejpam-5321	105	145	v	v	NOUN
ejpam-5321	105	146	of	of	ADP
ejpam-5321	105	147	y	y	PROPN
ejpam-5321	105	148	;	;	PUNCT
ejpam-5321	105	149	(	(	PUNCT
ejpam-5321	105	150	5	5	X
ejpam-5321	105	151	)	)	PUNCT
ejpam-5321	105	152	f−(v	f−(v	NOUN
ejpam-5321	105	153	)	)	PUNCT
ejpam-5321	105	154	is	be	AUX
ejpam-5321	105	155	α(τ1	α(τ1	NOUN
ejpam-5321	105	156	,	,	PUNCT
ejpam-5321	105	157	τ2)-closed	τ2)-close	VERB
ejpam-5321	105	158	in	in	ADP
ejpam-5321	105	159	x	x	PUNCT
ejpam-5321	105	160	for	for	ADP
ejpam-5321	105	161	every	every	DET
ejpam-5321	105	162	σ1σ2	σ1σ2	NUM
ejpam-5321	105	163	-	-	PUNCT
ejpam-5321	105	164	clopen	clopen	ADJ
ejpam-5321	105	165	set	set	NOUN
ejpam-5321	105	166	v	v	NOUN
ejpam-5321	105	167	of	of	ADP
ejpam-5321	105	168	y	y	PROPN
ejpam-5321	105	169	;	;	PUNCT
ejpam-5321	105	170	(	(	PUNCT
ejpam-5321	105	171	6	6	X
ejpam-5321	105	172	)	)	PUNCT
ejpam-5321	105	173	f−(y	f−(y	NOUN
ejpam-5321	105	174	−	−	NOUN
ejpam-5321	105	175	v	v	NOUN
ejpam-5321	105	176	)	)	PUNCT
ejpam-5321	105	177	is	be	AUX
ejpam-5321	105	178	α(τ1	α(τ1	NOUN
ejpam-5321	105	179	,	,	PUNCT
ejpam-5321	105	180	τ2)-closed	τ2)-close	VERB
ejpam-5321	105	181	in	in	ADP
ejpam-5321	105	182	x	x	PUNCT
ejpam-5321	105	183	for	for	ADP
ejpam-5321	105	184	every	every	DET
ejpam-5321	105	185	σ1σ2	σ1σ2	NUM
ejpam-5321	105	186	-	-	PUNCT
ejpam-5321	105	187	clopen	clopen	ADJ
ejpam-5321	105	188	set	set	NOUN
ejpam-5321	105	189	v	v	NOUN
ejpam-5321	105	190	of	of	ADP
ejpam-5321	105	191	y	y	PROPN
ejpam-5321	105	192	;	;	PUNCT
ejpam-5321	105	193	(	(	PUNCT
ejpam-5321	105	194	7	7	X
ejpam-5321	105	195	)	)	PUNCT
ejpam-5321	105	196	f+(y	f+(y	NOUN
ejpam-5321	105	197	−	−	PROPN
ejpam-5321	105	198	v	v	NOUN
ejpam-5321	105	199	)	)	PUNCT
ejpam-5321	105	200	is	be	AUX
ejpam-5321	105	201	α(τ1	α(τ1	NOUN
ejpam-5321	105	202	,	,	PUNCT
ejpam-5321	105	203	τ2)-open	τ2)-open	ADJ
ejpam-5321	105	204	in	in	ADP
ejpam-5321	105	205	x	x	PUNCT
ejpam-5321	105	206	for	for	ADP
ejpam-5321	105	207	every	every	DET
ejpam-5321	105	208	σ1σ2	σ1σ2	NUM
ejpam-5321	105	209	-	-	PUNCT
ejpam-5321	105	210	clopen	clopen	ADJ
ejpam-5321	105	211	set	set	NOUN
ejpam-5321	105	212	v	v	NOUN
ejpam-5321	105	213	of	of	ADP
ejpam-5321	105	214	y	y	PROPN
ejpam-5321	105	215	;	;	PUNCT
ejpam-5321	105	216	(	(	PUNCT
ejpam-5321	105	217	8)	8)	NUM
ejpam-5321	105	218	for	for	ADP
ejpam-5321	105	219	each	each	DET
ejpam-5321	105	220	x	x	SYM
ejpam-5321	105	221	∈	∈	PROPN
ejpam-5321	105	222	x	x	X
ejpam-5321	105	223	and	and	CCONJ
ejpam-5321	105	224	for	for	ADP
ejpam-5321	105	225	each	each	DET
ejpam-5321	105	226	net	net	NOUN
ejpam-5321	105	227	(	(	PUNCT
ejpam-5321	105	228	xγ	xγ	PROPN
ejpam-5321	105	229	)	)	PUNCT
ejpam-5321	105	230	which	which	PRON
ejpam-5321	105	231	α(τ1	α(τ1	VERB
ejpam-5321	105	232	,	,	PUNCT
ejpam-5321	105	233	τ2)-converges	τ2)-converge	NOUN
ejpam-5321	105	234	to	to	ADP
ejpam-5321	105	235	x	x	PUNCT
ejpam-5321	105	236	in	in	ADP
ejpam-5321	105	237	x	x	X
ejpam-5321	105	238	and	and	CCONJ
ejpam-5321	105	239	for	for	ADP
ejpam-5321	105	240	each	each	DET
ejpam-5321	105	241	σ1σ2	σ1σ2	NUM
ejpam-5321	105	242	-	-	PUNCT
ejpam-5321	105	243	clopen	clopen	ADJ
ejpam-5321	105	244	set	set	NOUN
ejpam-5321	105	245	v	v	NOUN
ejpam-5321	105	246	of	of	ADP
ejpam-5321	105	247	y	y	PRON
ejpam-5321	105	248	such	such	ADJ
ejpam-5321	105	249	that	that	SCONJ
ejpam-5321	105	250	x	x	SYM
ejpam-5321	105	251	∈	∈	PROPN
ejpam-5321	105	252	f+(v	f+(v	NOUN
ejpam-5321	105	253	)	)	PUNCT
ejpam-5321	105	254	,	,	PUNCT
ejpam-5321	105	255	the	the	DET
ejpam-5321	105	256	net	net	NOUN
ejpam-5321	105	257	(	(	PUNCT
ejpam-5321	105	258	xγ	xγ	PROPN
ejpam-5321	105	259	)	)	PUNCT
ejpam-5321	105	260	is	be	AUX
ejpam-5321	105	261	eventually	eventually	ADV
ejpam-5321	105	262	in	in	ADP
ejpam-5321	105	263	f+(v	f+(v	PROPN
ejpam-5321	105	264	)	)	PUNCT
ejpam-5321	105	265	.	.	PUNCT
ejpam-5321	106	1	proof	proof	NOUN
ejpam-5321	106	2	.	.	PUNCT
ejpam-5321	107	1	(	(	PUNCT
ejpam-5321	107	2	1	1	X
ejpam-5321	107	3	)	)	PUNCT
ejpam-5321	107	4	⇔	⇔	X
ejpam-5321	107	5	(	(	PUNCT
ejpam-5321	107	6	2	2	NUM
ejpam-5321	107	7	):	):	PUNCT
ejpam-5321	107	8	obvious	obvious	ADJ
ejpam-5321	107	9	.	.	PUNCT
ejpam-5321	108	1	(	(	PUNCT
ejpam-5321	108	2	2	2	X
ejpam-5321	108	3	)	)	PUNCT
ejpam-5321	108	4	⇔	⇔	X
ejpam-5321	108	5	(	(	PUNCT
ejpam-5321	108	6	3	3	NUM
ejpam-5321	108	7	):	):	PUNCT
ejpam-5321	108	8	let	let	VERB
ejpam-5321	108	9	x	x	PUNCT
ejpam-5321	108	10	∈	∈	PROPN
ejpam-5321	108	11	x	x	X
ejpam-5321	108	12	and	and	CCONJ
ejpam-5321	108	13	v	v	X
ejpam-5321	108	14	be	be	AUX
ejpam-5321	108	15	any	any	DET
ejpam-5321	108	16	σ1σ2	σ1σ2	NOUN
ejpam-5321	108	17	-	-	PUNCT
ejpam-5321	108	18	clopen	clopen	ADJ
ejpam-5321	108	19	set	set	NOUN
ejpam-5321	108	20	of	of	ADP
ejpam-5321	108	21	y	y	PRON
ejpam-5321	108	22	such	such	ADJ
ejpam-5321	108	23	that	that	SCONJ
ejpam-5321	108	24	x	x	SYM
ejpam-5321	108	25	∈	∈	PROPN
ejpam-5321	108	26	f+(y	f+(y	NOUN
ejpam-5321	108	27	−	−	PROPN
ejpam-5321	108	28	v	v	NOUN
ejpam-5321	108	29	)	)	PUNCT
ejpam-5321	108	30	.	.	PUNCT
ejpam-5321	109	1	by	by	ADP
ejpam-5321	109	2	(	(	PUNCT
ejpam-5321	109	3	2	2	NUM
ejpam-5321	109	4	)	)	PUNCT
ejpam-5321	109	5	,	,	PUNCT
ejpam-5321	109	6	there	there	PRON
ejpam-5321	109	7	exists	exist	VERB
ejpam-5321	109	8	an	an	DET
ejpam-5321	109	9	α(τ1	α(τ1	NOUN
ejpam-5321	109	10	,	,	PUNCT
ejpam-5321	109	11	τ2)-open	τ2)-open	ADJ
ejpam-5321	109	12	set	set	VERB
ejpam-5321	109	13	u	u	NOUN
ejpam-5321	109	14	of	of	ADP
ejpam-5321	109	15	x	x	PUNCT
ejpam-5321	109	16	containing	contain	VERB
ejpam-5321	109	17	x	x	PUNCT
ejpam-5321	109	18	such	such	ADJ
ejpam-5321	109	19	that	that	SCONJ
ejpam-5321	109	20	u	u	PROPN
ejpam-5321	109	21	⊆	⊆	NUM
ejpam-5321	109	22	f+(y	f+(y	PROPN
ejpam-5321	109	23	−	−	PROPN
ejpam-5321	109	24	v	v	NOUN
ejpam-5321	109	25	)	)	PUNCT
ejpam-5321	109	26	.	.	PUNCT
ejpam-5321	110	1	then	then	ADV
ejpam-5321	110	2	,	,	PUNCT
ejpam-5321	110	3	f−(v	f−(v	ADJ
ejpam-5321	110	4	)	)	PUNCT
ejpam-5321	111	1	⊆	⊆	NUM
ejpam-5321	111	2	x−u	x−u	X
ejpam-5321	111	3	.	.	PUNCT
ejpam-5321	112	1	put	put	VERB
ejpam-5321	112	2	h	h	NOUN
ejpam-5321	112	3	=	=	PUNCT
ejpam-5321	112	4	x−u	x−u	PROPN
ejpam-5321	112	5	.	.	PUNCT
ejpam-5321	113	1	then	then	ADV
ejpam-5321	113	2	,	,	PUNCT
ejpam-5321	113	3	h	h	PROPN
ejpam-5321	113	4	is	be	AUX
ejpam-5321	113	5	α(τ1	α(τ1	NOUN
ejpam-5321	113	6	,	,	PUNCT
ejpam-5321	113	7	τ2)-closed	τ2)-close	VERB
ejpam-5321	113	8	in	in	ADP
ejpam-5321	113	9	x	x	X
ejpam-5321	113	10	and	and	CCONJ
ejpam-5321	113	11	x	x	PROPN
ejpam-5321	113	12	∈	∈	PROPN
ejpam-5321	113	13	x−h	x−h	PROPN
ejpam-5321	113	14	.	.	PUNCT
ejpam-5321	114	1	the	the	DET
ejpam-5321	114	2	converse	converse	NOUN
ejpam-5321	114	3	is	be	AUX
ejpam-5321	114	4	similar	similar	ADJ
ejpam-5321	114	5	.	.	PUNCT
ejpam-5321	115	1	(	(	PUNCT
ejpam-5321	115	2	1	1	X
ejpam-5321	115	3	)	)	PUNCT
ejpam-5321	115	4	⇔	⇔	X
ejpam-5321	115	5	(	(	PUNCT
ejpam-5321	115	6	4	4	NUM
ejpam-5321	115	7	):	):	PUNCT
ejpam-5321	115	8	let	let	VERB
ejpam-5321	115	9	v	v	PART
ejpam-5321	115	10	be	be	AUX
ejpam-5321	115	11	any	any	DET
ejpam-5321	115	12	σ1σ2	σ1σ2	NOUN
ejpam-5321	115	13	-	-	PUNCT
ejpam-5321	115	14	clopen	clopen	ADJ
ejpam-5321	115	15	set	set	NOUN
ejpam-5321	115	16	of	of	ADP
ejpam-5321	115	17	y	y	PROPN
ejpam-5321	115	18	and	and	CCONJ
ejpam-5321	115	19	x	x	PROPN
ejpam-5321	115	20	∈	∈	PROPN
ejpam-5321	115	21	f+(v	f+(v	NOUN
ejpam-5321	115	22	)	)	PUNCT
ejpam-5321	115	23	.	.	PUNCT
ejpam-5321	116	1	by	by	ADP
ejpam-5321	116	2	(	(	PUNCT
ejpam-5321	116	3	1	1	NUM
ejpam-5321	116	4	)	)	PUNCT
ejpam-5321	116	5	,	,	PUNCT
ejpam-5321	116	6	there	there	PRON
ejpam-5321	116	7	exists	exist	VERB
ejpam-5321	116	8	an	an	DET
ejpam-5321	116	9	α(τ1	α(τ1	NOUN
ejpam-5321	116	10	,	,	PUNCT
ejpam-5321	116	11	τ2)-open	τ2)-open	ADJ
ejpam-5321	116	12	set	set	VERB
ejpam-5321	116	13	ux	ux	NOUN
ejpam-5321	116	14	of	of	ADP
ejpam-5321	116	15	x	x	SYM
ejpam-5321	116	16	containing	contain	VERB
ejpam-5321	116	17	x	x	PUNCT
ejpam-5321	116	18	such	such	ADJ
ejpam-5321	116	19	that	that	PRON
ejpam-5321	116	20	ux	ux	PROPN
ejpam-5321	116	21	⊆	⊆	NUM
ejpam-5321	116	22	f+(v	f+(v	NOUN
ejpam-5321	116	23	)	)	PUNCT
ejpam-5321	116	24	.	.	PUNCT
ejpam-5321	117	1	it	it	PRON
ejpam-5321	117	2	follows	follow	VERB
ejpam-5321	117	3	that	that	PRON
ejpam-5321	117	4	f+(v	f+(v	NOUN
ejpam-5321	117	5	)	)	PUNCT
ejpam-5321	118	1	=	=	SYM
ejpam-5321	118	2	∪x∈f+(v	∪x∈f+(v	X
ejpam-5321	118	3	)	)	PUNCT
ejpam-5321	118	4	ux	ux	NOUN
ejpam-5321	118	5	and	and	CCONJ
ejpam-5321	118	6	hence	hence	ADV
ejpam-5321	118	7	f+(v	f+(v	PROPN
ejpam-5321	118	8	)	)	PUNCT
ejpam-5321	118	9	is	be	AUX
ejpam-5321	118	10	α(τ1	α(τ1	NOUN
ejpam-5321	118	11	,	,	PUNCT
ejpam-5321	118	12	τ2)-open	τ2)-open	ADJ
ejpam-5321	118	13	in	in	ADP
ejpam-5321	118	14	x.	x.	NOUN
ejpam-5321	119	1	the	the	DET
ejpam-5321	119	2	converse	converse	NOUN
ejpam-5321	119	3	can	can	AUX
ejpam-5321	119	4	be	be	AUX
ejpam-5321	119	5	shown	show	VERB
ejpam-5321	119	6	easily	easily	ADV
ejpam-5321	119	7	.	.	PUNCT
ejpam-5321	120	1	c.	c.	PROPN
ejpam-5321	120	2	viriyapong	viriyapong	PROPN
ejpam-5321	120	3	,	,	PUNCT
ejpam-5321	120	4	s.	s.	PROPN
ejpam-5321	120	5	sompong	sompong	PROPN
ejpam-5321	120	6	,	,	PUNCT
ejpam-5321	120	7	c.	c.	PROPN
ejpam-5321	120	8	boonpok	boonpok	PROPN
ejpam-5321	120	9	/	/	SYM
ejpam-5321	120	10	eur	eur	PROPN
ejpam-5321	120	11	.	.	PUNCT
ejpam-5321	121	1	j.	j.	PROPN
ejpam-5321	121	2	pure	pure	PROPN
ejpam-5321	121	3	appl	appl	PROPN
ejpam-5321	121	4	.	.	PROPN
ejpam-5321	121	5	math	math	PROPN
ejpam-5321	121	6	,	,	PUNCT
ejpam-5321	121	7	17	17	NUM
ejpam-5321	121	8	(	(	PUNCT
ejpam-5321	121	9	3	3	NUM
ejpam-5321	121	10	)	)	PUNCT
ejpam-5321	121	11	(	(	PUNCT
ejpam-5321	121	12	2024	2024	NUM
ejpam-5321	121	13	)	)	PUNCT
ejpam-5321	121	14	,	,	PUNCT
ejpam-5321	121	15	2142	2142	NUM
ejpam-5321	121	16	-	-	SYM
ejpam-5321	121	17	2154	2154	NUM
ejpam-5321	121	18	2146	2146	NUM
ejpam-5321	121	19	(	(	PUNCT
ejpam-5321	121	20	4	4	NUM
ejpam-5321	121	21	)	)	PUNCT
ejpam-5321	121	22	⇒	⇒	NOUN
ejpam-5321	121	23	(	(	PUNCT
ejpam-5321	121	24	5	5	NUM
ejpam-5321	121	25	):	):	PUNCT
ejpam-5321	121	26	let	let	VERB
ejpam-5321	121	27	v	v	PART
ejpam-5321	121	28	be	be	AUX
ejpam-5321	121	29	any	any	DET
ejpam-5321	121	30	σ1σ2	σ1σ2	NOUN
ejpam-5321	121	31	-	-	PUNCT
ejpam-5321	121	32	clopen	clopen	ADJ
ejpam-5321	121	33	set	set	NOUN
ejpam-5321	121	34	of	of	ADP
ejpam-5321	121	35	y	y	PROPN
ejpam-5321	121	36	.	.	PUNCT
ejpam-5321	122	1	then	then	ADV
ejpam-5321	122	2	,	,	PUNCT
ejpam-5321	122	3	y	y	PROPN
ejpam-5321	122	4	−	−	PROPN
ejpam-5321	122	5	v	v	NOUN
ejpam-5321	122	6	is	be	AUX
ejpam-5321	122	7	σ1σ2	σ1σ2	NOUN
ejpam-5321	122	8	-	-	PUNCT
ejpam-5321	122	9	clopen	clopen	ADJ
ejpam-5321	122	10	in	in	ADP
ejpam-5321	122	11	y	y	PROPN
ejpam-5321	122	12	and	and	CCONJ
ejpam-5321	122	13	by	by	ADP
ejpam-5321	122	14	(	(	PUNCT
ejpam-5321	122	15	4	4	NUM
ejpam-5321	122	16	)	)	PUNCT
ejpam-5321	122	17	,	,	PUNCT
ejpam-5321	122	18	f+(y	f+(y	PROPN
ejpam-5321	122	19	−	−	PROPN
ejpam-5321	122	20	v	v	NOUN
ejpam-5321	122	21	)	)	PUNCT
ejpam-5321	122	22	=	=	PUNCT
ejpam-5321	123	1	x	x	SYM
ejpam-5321	123	2	−	−	PROPN
ejpam-5321	123	3	f−(v	f−(v	PROPN
ejpam-5321	123	4	)	)	PUNCT
ejpam-5321	123	5	is	be	AUX
ejpam-5321	123	6	α(τ1	α(τ1	NOUN
ejpam-5321	123	7	,	,	PUNCT
ejpam-5321	123	8	τ2)-open	τ2)-open	ADJ
ejpam-5321	123	9	in	in	ADP
ejpam-5321	123	10	x.	x.	PROPN
ejpam-5321	123	11	thus	thus	ADV
ejpam-5321	123	12	,	,	PUNCT
ejpam-5321	123	13	f−(v	f−(v	ADJ
ejpam-5321	123	14	)	)	PUNCT
ejpam-5321	123	15	is	be	AUX
ejpam-5321	123	16	α(τ1	α(τ1	NOUN
ejpam-5321	123	17	,	,	PUNCT
ejpam-5321	123	18	τ2)-closed	τ2)-close	VERB
ejpam-5321	123	19	in	in	ADP
ejpam-5321	123	20	x.	x.	NOUN
ejpam-5321	123	21	(	(	PUNCT
ejpam-5321	123	22	5	5	NUM
ejpam-5321	123	23	)	)	PUNCT
ejpam-5321	123	24	⇒	⇒	NOUN
ejpam-5321	123	25	(	(	PUNCT
ejpam-5321	123	26	4	4	NUM
ejpam-5321	123	27	):	):	PUNCT
ejpam-5321	123	28	it	it	PRON
ejpam-5321	123	29	is	be	AUX
ejpam-5321	123	30	similar	similar	ADJ
ejpam-5321	123	31	to	to	ADP
ejpam-5321	123	32	that	that	PRON
ejpam-5321	123	33	of	of	ADP
ejpam-5321	123	34	(	(	PUNCT
ejpam-5321	123	35	4	4	NUM
ejpam-5321	123	36	)	)	PUNCT
ejpam-5321	123	37	⇒	⇒	NOUN
ejpam-5321	123	38	(	(	PUNCT
ejpam-5321	123	39	5	5	NUM
ejpam-5321	123	40	)	)	PUNCT
ejpam-5321	123	41	.	.	PUNCT
ejpam-5321	124	1	(	(	PUNCT
ejpam-5321	124	2	4	4	X
ejpam-5321	124	3	)	)	PUNCT
ejpam-5321	124	4	⇔	⇔	X
ejpam-5321	124	5	(	(	PUNCT
ejpam-5321	124	6	6	6	NUM
ejpam-5321	124	7	)	)	PUNCT
ejpam-5321	124	8	and	and	CCONJ
ejpam-5321	124	9	(	(	PUNCT
ejpam-5321	124	10	5	5	X
ejpam-5321	124	11	)	)	PUNCT
ejpam-5321	124	12	⇔	⇔	X
ejpam-5321	124	13	(	(	PUNCT
ejpam-5321	124	14	7	7	NUM
ejpam-5321	124	15	):	):	PUNCT
ejpam-5321	124	16	it	it	PRON
ejpam-5321	124	17	follows	follow	VERB
ejpam-5321	124	18	from	from	ADP
ejpam-5321	124	19	the	the	DET
ejpam-5321	124	20	fact	fact	NOUN
ejpam-5321	124	21	that	that	SCONJ
ejpam-5321	124	22	f−(y	f−(y	NOUN
ejpam-5321	124	23	−b	−b	VERB
ejpam-5321	124	24	)	)	PUNCT
ejpam-5321	124	25	=	=	PUNCT
ejpam-5321	125	1	x	x	PUNCT
ejpam-5321	125	2	−	−	NOUN
ejpam-5321	125	3	f+(b	f+(b	PROPN
ejpam-5321	125	4	)	)	PUNCT
ejpam-5321	125	5	and	and	CCONJ
ejpam-5321	125	6	f+(y	f+(y	X
ejpam-5321	125	7	−b	−b	ADJ
ejpam-5321	125	8	)	)	PUNCT
ejpam-5321	126	1	=	=	PUNCT
ejpam-5321	126	2	x	x	X
ejpam-5321	127	1	−	−	PROPN
ejpam-5321	127	2	f−(b	f−(b	PROPN
ejpam-5321	127	3	)	)	PUNCT
ejpam-5321	127	4	for	for	ADP
ejpam-5321	127	5	every	every	DET
ejpam-5321	127	6	subset	subset	NOUN
ejpam-5321	127	7	b	b	PROPN
ejpam-5321	127	8	of	of	ADP
ejpam-5321	127	9	y	y	PROPN
ejpam-5321	127	10	.	.	PUNCT
ejpam-5321	128	1	(	(	PUNCT
ejpam-5321	128	2	1	1	X
ejpam-5321	128	3	)	)	PUNCT
ejpam-5321	128	4	⇒	⇒	NOUN
ejpam-5321	128	5	(	(	PUNCT
ejpam-5321	128	6	8)	8)	NUM
ejpam-5321	128	7	:	:	PUNCT
ejpam-5321	128	8	let	let	VERB
ejpam-5321	128	9	(	(	PUNCT
ejpam-5321	128	10	xγ	xγ	VERB
ejpam-5321	128	11	)	)	PUNCT
ejpam-5321	128	12	be	be	AUX
ejpam-5321	128	13	a	a	DET
ejpam-5321	128	14	net	net	NOUN
ejpam-5321	128	15	which	which	PRON
ejpam-5321	128	16	α(τ1	α(τ1	VERB
ejpam-5321	128	17	,	,	PUNCT
ejpam-5321	128	18	τ2)-converges	τ2)-converge	NOUN
ejpam-5321	128	19	to	to	ADP
ejpam-5321	128	20	x	x	PUNCT
ejpam-5321	128	21	in	in	ADP
ejpam-5321	128	22	x	x	PUNCT
ejpam-5321	128	23	and	and	CCONJ
ejpam-5321	128	24	let	let	VERB
ejpam-5321	128	25	v	v	PART
ejpam-5321	128	26	be	be	AUX
ejpam-5321	128	27	any	any	DET
ejpam-5321	128	28	σ1σ2clopen	σ1σ2clopen	ADJ
ejpam-5321	128	29	set	set	NOUN
ejpam-5321	128	30	of	of	ADP
ejpam-5321	128	31	y	y	PRON
ejpam-5321	128	32	such	such	ADJ
ejpam-5321	128	33	that	that	SCONJ
ejpam-5321	128	34	x	x	SYM
ejpam-5321	128	35	∈	∈	PROPN
ejpam-5321	128	36	f+(v	f+(v	NOUN
ejpam-5321	128	37	)	)	PUNCT
ejpam-5321	128	38	.	.	PUNCT
ejpam-5321	129	1	since	since	SCONJ
ejpam-5321	129	2	f	f	PROPN
ejpam-5321	129	3	is	be	AUX
ejpam-5321	129	4	an	an	DET
ejpam-5321	129	5	upper	upper	ADJ
ejpam-5321	129	6	slightly	slightly	ADJ
ejpam-5321	129	7	α(τ1	α(τ1	NOUN
ejpam-5321	129	8	,	,	PUNCT
ejpam-5321	129	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	129	10	multifunction	multifunction	NOUN
ejpam-5321	129	11	,	,	PUNCT
ejpam-5321	129	12	there	there	PRON
ejpam-5321	129	13	exists	exist	VERB
ejpam-5321	129	14	an	an	DET
ejpam-5321	129	15	α(τ1	α(τ1	NOUN
ejpam-5321	129	16	,	,	PUNCT
ejpam-5321	129	17	τ2)-open	τ2)-open	ADJ
ejpam-5321	129	18	set	set	VERB
ejpam-5321	129	19	u	u	PRON
ejpam-5321	129	20	ofx	ofx	NOUN
ejpam-5321	129	21	containing	contain	VERB
ejpam-5321	129	22	x	x	PUNCT
ejpam-5321	129	23	such	such	ADJ
ejpam-5321	129	24	that	that	SCONJ
ejpam-5321	129	25	u	u	NOUN
ejpam-5321	129	26	⊆	⊆	NUM
ejpam-5321	129	27	f+(v	f+(v	NOUN
ejpam-5321	129	28	)	)	PUNCT
ejpam-5321	129	29	.	.	PUNCT
ejpam-5321	130	1	since	since	SCONJ
ejpam-5321	130	2	(	(	PUNCT
ejpam-5321	130	3	xγ	xγ	NOUN
ejpam-5321	130	4	)	)	PUNCT
ejpam-5321	130	5	α(τ1	α(τ1	NOUN
ejpam-5321	130	6	,	,	PUNCT
ejpam-5321	130	7	τ2)-converges	τ2)-converge	NOUN
ejpam-5321	130	8	to	to	ADP
ejpam-5321	130	9	x	x	PRON
ejpam-5321	130	10	,	,	PUNCT
ejpam-5321	130	11	it	it	PRON
ejpam-5321	130	12	follows	follow	VERB
ejpam-5321	130	13	that	that	SCONJ
ejpam-5321	130	14	there	there	PRON
ejpam-5321	130	15	exists	exist	VERB
ejpam-5321	130	16	an	an	DET
ejpam-5321	130	17	index	index	NOUN
ejpam-5321	130	18	γ0	γ0	NOUN
ejpam-5321	130	19	∈	∈	PROPN
ejpam-5321	130	20	∇	∇	X
ejpam-5321	130	21	such	such	ADJ
ejpam-5321	130	22	that	that	SCONJ
ejpam-5321	130	23	xγ	xγ	VERB
ejpam-5321	130	24	∈	∈	PROPN
ejpam-5321	130	25	u	u	NOUN
ejpam-5321	130	26	for	for	ADP
ejpam-5321	130	27	all	all	DET
ejpam-5321	130	28	γ	γ	PROPN
ejpam-5321	130	29	≥	≥	PROPN
ejpam-5321	130	30	γ0	γ0	PROPN
ejpam-5321	130	31	.	.	PUNCT
ejpam-5321	131	1	therefore	therefore	ADV
ejpam-5321	131	2	,	,	PUNCT
ejpam-5321	131	3	xγ	xγ	PROPN
ejpam-5321	131	4	∈	∈	PROPN
ejpam-5321	131	5	u	u	NOUN
ejpam-5321	131	6	⊆	⊆	NUM
ejpam-5321	131	7	f+(v	f+(v	NOUN
ejpam-5321	131	8	)	)	PUNCT
ejpam-5321	131	9	for	for	ADP
ejpam-5321	131	10	all	all	DET
ejpam-5321	131	11	γ	γ	PROPN
ejpam-5321	131	12	≥	≥	PROPN
ejpam-5321	131	13	γ0	γ0	PROPN
ejpam-5321	131	14	.	.	PUNCT
ejpam-5321	132	1	thus	thus	ADV
ejpam-5321	132	2	,	,	PUNCT
ejpam-5321	132	3	the	the	DET
ejpam-5321	132	4	net	net	NOUN
ejpam-5321	132	5	(	(	PUNCT
ejpam-5321	132	6	xγ	xγ	PROPN
ejpam-5321	132	7	)	)	PUNCT
ejpam-5321	132	8	is	be	AUX
ejpam-5321	132	9	eventually	eventually	ADV
ejpam-5321	132	10	in	in	ADP
ejpam-5321	132	11	f+(v	f+(v	PROPN
ejpam-5321	132	12	)	)	PUNCT
ejpam-5321	132	13	.	.	PUNCT
ejpam-5321	133	1	(	(	PUNCT
ejpam-5321	133	2	8)	8)	NUM
ejpam-5321	133	3	⇒	⇒	NOUN
ejpam-5321	133	4	(	(	PUNCT
ejpam-5321	133	5	1	1	NUM
ejpam-5321	133	6	):	):	PUNCT
ejpam-5321	133	7	suppose	suppose	VERB
ejpam-5321	133	8	that	that	SCONJ
ejpam-5321	133	9	f	f	PROPN
ejpam-5321	133	10	is	be	AUX
ejpam-5321	133	11	not	not	PART
ejpam-5321	133	12	upper	upper	ADJ
ejpam-5321	133	13	slightly	slightly	ADV
ejpam-5321	133	14	α(τ1	α(τ1	NOUN
ejpam-5321	133	15	,	,	PUNCT
ejpam-5321	133	16	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	133	17	.	.	PUNCT
ejpam-5321	134	1	there	there	PRON
ejpam-5321	134	2	exists	exist	VERB
ejpam-5321	134	3	a	a	DET
ejpam-5321	134	4	point	point	NOUN
ejpam-5321	134	5	x	x	PUNCT
ejpam-5321	134	6	and	and	CCONJ
ejpam-5321	134	7	a	a	DET
ejpam-5321	134	8	σ1σ2	σ1σ2	NUM
ejpam-5321	134	9	-	-	PUNCT
ejpam-5321	134	10	clopen	clopen	ADJ
ejpam-5321	134	11	set	set	NOUN
ejpam-5321	134	12	v	v	NOUN
ejpam-5321	134	13	of	of	ADP
ejpam-5321	134	14	y	y	PROPN
ejpam-5321	134	15	with	with	ADP
ejpam-5321	134	16	x	x	PROPN
ejpam-5321	134	17	∈	∈	PROPN
ejpam-5321	134	18	f+(v	f+(v	NOUN
ejpam-5321	134	19	)	)	PUNCT
ejpam-5321	134	20	such	such	ADJ
ejpam-5321	134	21	that	that	SCONJ
ejpam-5321	134	22	u	u	NOUN
ejpam-5321	134	23	̸⊆	̸⊆	NOUN
ejpam-5321	134	24	f+(v	f+(v	NOUN
ejpam-5321	134	25	)	)	PUNCT
ejpam-5321	134	26	for	for	ADP
ejpam-5321	134	27	each	each	DET
ejpam-5321	134	28	α(τ1	α(τ1	NOUN
ejpam-5321	134	29	,	,	PUNCT
ejpam-5321	134	30	τ2)-open	τ2)-open	ADJ
ejpam-5321	134	31	set	set	VERB
ejpam-5321	134	32	u	u	NOUN
ejpam-5321	134	33	of	of	ADP
ejpam-5321	134	34	x	x	SYM
ejpam-5321	134	35	containing	contain	VERB
ejpam-5321	134	36	x.	x.	NOUN
ejpam-5321	134	37	let	let	VERB
ejpam-5321	134	38	xu	xu	PROPN
ejpam-5321	134	39	∈	∈	PROPN
ejpam-5321	134	40	u	u	PROPN
ejpam-5321	134	41	and	and	CCONJ
ejpam-5321	134	42	xu	xu	PROPN
ejpam-5321	134	43	̸∈	̸∈	PROPN
ejpam-5321	134	44	f+(v	f+(v	PROPN
ejpam-5321	134	45	)	)	PUNCT
ejpam-5321	134	46	for	for	ADP
ejpam-5321	134	47	each	each	DET
ejpam-5321	134	48	α(τ1	α(τ1	NOUN
ejpam-5321	134	49	,	,	PUNCT
ejpam-5321	134	50	τ2)open	τ2)open	PROPN
ejpam-5321	134	51	set	set	VERB
ejpam-5321	134	52	u	u	NOUN
ejpam-5321	134	53	of	of	ADP
ejpam-5321	134	54	x	x	SYM
ejpam-5321	134	55	containing	contain	VERB
ejpam-5321	134	56	x.	x.	NOUN
ejpam-5321	134	57	then	then	ADV
ejpam-5321	134	58	,	,	PUNCT
ejpam-5321	134	59	for	for	ADP
ejpam-5321	134	60	the	the	DET
ejpam-5321	134	61	α(τ1	α(τ1	NOUN
ejpam-5321	134	62	,	,	PUNCT
ejpam-5321	134	63	τ2)-neighbourhood	τ2)-neighbourhood	NOUN
ejpam-5321	134	64	net	net	NOUN
ejpam-5321	134	65	(	(	PUNCT
ejpam-5321	134	66	xu	xu	PROPN
ejpam-5321	134	67	)	)	PUNCT
ejpam-5321	134	68	,	,	PUNCT
ejpam-5321	134	69	(	(	PUNCT
ejpam-5321	134	70	xu	xu	INTJ
ejpam-5321	134	71	)	)	PUNCT
ejpam-5321	134	72	α(τ1	α(τ1	NOUN
ejpam-5321	134	73	,	,	PUNCT
ejpam-5321	134	74	τ2)-converges	τ2)-converge	NOUN
ejpam-5321	134	75	to	to	ADP
ejpam-5321	134	76	x	x	PRON
ejpam-5321	134	77	,	,	PUNCT
ejpam-5321	134	78	but	but	CCONJ
ejpam-5321	134	79	(	(	PUNCT
ejpam-5321	134	80	xu	xu	INTJ
ejpam-5321	134	81	)	)	PUNCT
ejpam-5321	134	82	is	be	AUX
ejpam-5321	134	83	not	not	PART
ejpam-5321	134	84	eventually	eventually	ADV
ejpam-5321	134	85	in	in	ADP
ejpam-5321	134	86	f+(v	f+(v	PROPN
ejpam-5321	134	87	)	)	PUNCT
ejpam-5321	134	88	.	.	PUNCT
ejpam-5321	135	1	this	this	PRON
ejpam-5321	135	2	is	be	AUX
ejpam-5321	135	3	a	a	DET
ejpam-5321	135	4	contradiction	contradiction	NOUN
ejpam-5321	135	5	.	.	PUNCT
ejpam-5321	136	1	thus	thus	ADV
ejpam-5321	136	2	,	,	PUNCT
ejpam-5321	136	3	f	f	PROPN
ejpam-5321	136	4	is	be	AUX
ejpam-5321	136	5	upper	upper	ADJ
ejpam-5321	136	6	slightly	slightly	ADV
ejpam-5321	136	7	α(τ1	α(τ1	NOUN
ejpam-5321	136	8	,	,	PUNCT
ejpam-5321	136	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	136	10	.	.	PUNCT
ejpam-5321	137	1	definition	definition	NOUN
ejpam-5321	137	2	3	3	NUM
ejpam-5321	137	3	.	.	PUNCT
ejpam-5321	138	1	a	a	DET
ejpam-5321	138	2	multifunction	multifunction	NOUN
ejpam-5321	138	3	f	f	NOUN
ejpam-5321	138	4	:	:	PUNCT
ejpam-5321	138	5	(	(	PUNCT
ejpam-5321	138	6	x	x	NOUN
ejpam-5321	138	7	,	,	PUNCT
ejpam-5321	138	8	τ1	τ1	NOUN
ejpam-5321	138	9	,	,	PUNCT
ejpam-5321	138	10	τ2	τ2	NOUN
ejpam-5321	138	11	)	)	PUNCT
ejpam-5321	138	12	→	→	SYM
ejpam-5321	138	13	(	(	PUNCT
ejpam-5321	138	14	y	y	PROPN
ejpam-5321	138	15	,	,	PUNCT
ejpam-5321	138	16	σ1	σ1	PROPN
ejpam-5321	138	17	,	,	PUNCT
ejpam-5321	138	18	σ2	σ2	PROPN
ejpam-5321	138	19	)	)	PUNCT
ejpam-5321	138	20	is	be	AUX
ejpam-5321	138	21	called	call	VERB
ejpam-5321	138	22	lower	low	ADJ
ejpam-5321	138	23	slightly	slightly	ADJ
ejpam-5321	138	24	α(τ1	α(τ1	NOUN
ejpam-5321	138	25	,	,	PUNCT
ejpam-5321	138	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	138	27	at	at	ADP
ejpam-5321	138	28	a	a	DET
ejpam-5321	138	29	point	point	NOUN
ejpam-5321	138	30	x	x	SYM
ejpam-5321	138	31	∈	∈	NOUN
ejpam-5321	138	32	x	x	PUNCT
ejpam-5321	138	33	if	if	SCONJ
ejpam-5321	138	34	for	for	ADP
ejpam-5321	138	35	each	each	DET
ejpam-5321	138	36	σ1σ2	σ1σ2	NUM
ejpam-5321	138	37	-	-	PUNCT
ejpam-5321	138	38	clopen	clopen	ADJ
ejpam-5321	138	39	set	set	NOUN
ejpam-5321	138	40	v	v	NOUN
ejpam-5321	138	41	of	of	ADP
ejpam-5321	138	42	y	y	PRON
ejpam-5321	138	43	such	such	ADJ
ejpam-5321	138	44	that	that	SCONJ
ejpam-5321	138	45	f	f	PROPN
ejpam-5321	138	46	(	(	PUNCT
ejpam-5321	138	47	x	x	NOUN
ejpam-5321	138	48	)	)	PUNCT
ejpam-5321	138	49	∩	∩	NOUN
ejpam-5321	138	50	v	v	ADP
ejpam-5321	138	51	̸=	̸=	PROPN
ejpam-5321	138	52	∅	∅	NOUN
ejpam-5321	138	53	,	,	PUNCT
ejpam-5321	138	54	there	there	PRON
ejpam-5321	138	55	exists	exist	VERB
ejpam-5321	138	56	an	an	DET
ejpam-5321	138	57	α(τ1	α(τ1	NOUN
ejpam-5321	138	58	,	,	PUNCT
ejpam-5321	138	59	τ2)-open	τ2)-open	ADJ
ejpam-5321	138	60	set	set	VERB
ejpam-5321	138	61	u	u	NOUN
ejpam-5321	138	62	of	of	ADP
ejpam-5321	138	63	x	x	PUNCT
ejpam-5321	138	64	containing	contain	VERB
ejpam-5321	138	65	x	x	PUNCT
ejpam-5321	138	66	such	such	ADJ
ejpam-5321	138	67	that	that	SCONJ
ejpam-5321	138	68	f	f	PROPN
ejpam-5321	138	69	(	(	PUNCT
ejpam-5321	138	70	z	z	NOUN
ejpam-5321	138	71	)	)	PUNCT
ejpam-5321	138	72	∩	∩	NOUN
ejpam-5321	138	73	v	v	ADP
ejpam-5321	138	74	̸=	̸=	PROPN
ejpam-5321	138	75	∅	∅	NOUN
ejpam-5321	138	76	for	for	ADP
ejpam-5321	138	77	each	each	DET
ejpam-5321	138	78	z	z	NOUN
ejpam-5321	138	79	∈	∈	PROPN
ejpam-5321	138	80	u	u	NOUN
ejpam-5321	138	81	.	.	PUNCT
ejpam-5321	139	1	a	a	DET
ejpam-5321	139	2	multifunction	multifunction	NOUN
ejpam-5321	139	3	f	f	NOUN
ejpam-5321	139	4	:	:	PUNCT
ejpam-5321	139	5	(	(	PUNCT
ejpam-5321	139	6	x	x	NOUN
ejpam-5321	139	7	,	,	PUNCT
ejpam-5321	139	8	τ1	τ1	NOUN
ejpam-5321	139	9	,	,	PUNCT
ejpam-5321	139	10	τ2	τ2	NOUN
ejpam-5321	139	11	)	)	PUNCT
ejpam-5321	139	12	→	→	SYM
ejpam-5321	139	13	(	(	PUNCT
ejpam-5321	139	14	y	y	PROPN
ejpam-5321	139	15	,	,	PUNCT
ejpam-5321	139	16	σ1	σ1	PROPN
ejpam-5321	139	17	,	,	PUNCT
ejpam-5321	139	18	σ2	σ2	PROPN
ejpam-5321	139	19	)	)	PUNCT
ejpam-5321	139	20	is	be	AUX
ejpam-5321	139	21	called	call	VERB
ejpam-5321	139	22	lower	low	ADJ
ejpam-5321	139	23	slightly	slightly	ADJ
ejpam-5321	139	24	α(τ1	α(τ1	NOUN
ejpam-5321	139	25	,	,	PUNCT
ejpam-5321	139	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	139	27	if	if	SCONJ
ejpam-5321	139	28	f	f	PROPN
ejpam-5321	139	29	has	have	VERB
ejpam-5321	139	30	this	this	DET
ejpam-5321	139	31	property	property	NOUN
ejpam-5321	139	32	at	at	ADP
ejpam-5321	139	33	every	every	DET
ejpam-5321	139	34	point	point	NOUN
ejpam-5321	139	35	of	of	ADP
ejpam-5321	139	36	x.	x.	NOUN
ejpam-5321	139	37	theorem	theorem	VERB
ejpam-5321	139	38	2	2	NUM
ejpam-5321	139	39	.	.	X
ejpam-5321	139	40	for	for	ADP
ejpam-5321	139	41	a	a	DET
ejpam-5321	139	42	multifunction	multifunction	NOUN
ejpam-5321	139	43	f	f	NOUN
ejpam-5321	139	44	:	:	PUNCT
ejpam-5321	139	45	(	(	PUNCT
ejpam-5321	139	46	x	x	NOUN
ejpam-5321	139	47	,	,	PUNCT
ejpam-5321	139	48	τ1	τ1	NOUN
ejpam-5321	139	49	,	,	PUNCT
ejpam-5321	139	50	τ2	τ2	NOUN
ejpam-5321	139	51	)	)	PUNCT
ejpam-5321	139	52	→	→	SYM
ejpam-5321	139	53	(	(	PUNCT
ejpam-5321	139	54	y	y	PROPN
ejpam-5321	139	55	,	,	PUNCT
ejpam-5321	139	56	σ1	σ1	PROPN
ejpam-5321	139	57	,	,	PUNCT
ejpam-5321	139	58	σ2	σ2	NOUN
ejpam-5321	139	59	)	)	PUNCT
ejpam-5321	139	60	,	,	PUNCT
ejpam-5321	139	61	the	the	DET
ejpam-5321	139	62	following	follow	VERB
ejpam-5321	139	63	properties	property	NOUN
ejpam-5321	139	64	are	be	AUX
ejpam-5321	139	65	equivalent	equivalent	ADJ
ejpam-5321	139	66	:	:	PUNCT
ejpam-5321	139	67	(	(	PUNCT
ejpam-5321	139	68	1	1	X
ejpam-5321	139	69	)	)	PUNCT
ejpam-5321	139	70	f	f	PROPN
ejpam-5321	139	71	is	be	AUX
ejpam-5321	139	72	lower	low	ADJ
ejpam-5321	139	73	slightly	slightly	ADV
ejpam-5321	139	74	α(τ1	α(τ1	NOUN
ejpam-5321	139	75	,	,	PUNCT
ejpam-5321	139	76	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	139	77	;	;	PUNCT
ejpam-5321	139	78	(	(	PUNCT
ejpam-5321	139	79	2	2	X
ejpam-5321	139	80	)	)	PUNCT
ejpam-5321	139	81	for	for	ADP
ejpam-5321	139	82	each	each	DET
ejpam-5321	139	83	x	x	SYM
ejpam-5321	139	84	∈	∈	PROPN
ejpam-5321	139	85	x	x	X
ejpam-5321	139	86	and	and	CCONJ
ejpam-5321	139	87	for	for	ADP
ejpam-5321	139	88	each	each	DET
ejpam-5321	139	89	σ1σ2	σ1σ2	NUM
ejpam-5321	139	90	-	-	PUNCT
ejpam-5321	139	91	clopen	clopen	ADJ
ejpam-5321	139	92	set	set	NOUN
ejpam-5321	139	93	v	v	NOUN
ejpam-5321	139	94	of	of	ADP
ejpam-5321	139	95	y	y	PRON
ejpam-5321	139	96	such	such	ADJ
ejpam-5321	139	97	that	that	SCONJ
ejpam-5321	139	98	x	x	SYM
ejpam-5321	139	99	∈	∈	PROPN
ejpam-5321	139	100	f−(v	f−(v	NOUN
ejpam-5321	139	101	)	)	PUNCT
ejpam-5321	139	102	,	,	PUNCT
ejpam-5321	139	103	there	there	PRON
ejpam-5321	139	104	exists	exist	VERB
ejpam-5321	139	105	an	an	DET
ejpam-5321	139	106	α(τ1	α(τ1	NOUN
ejpam-5321	139	107	,	,	PUNCT
ejpam-5321	139	108	τ2)-open	τ2)-open	ADJ
ejpam-5321	139	109	set	set	VERB
ejpam-5321	139	110	u	u	NOUN
ejpam-5321	139	111	of	of	ADP
ejpam-5321	139	112	x	x	PUNCT
ejpam-5321	139	113	containing	contain	VERB
ejpam-5321	139	114	x	x	PUNCT
ejpam-5321	139	115	such	such	ADJ
ejpam-5321	139	116	that	that	SCONJ
ejpam-5321	139	117	u	u	NOUN
ejpam-5321	139	118	⊆	⊆	NUM
ejpam-5321	139	119	f−(v	f−(v	NOUN
ejpam-5321	139	120	)	)	PUNCT
ejpam-5321	139	121	;	;	PUNCT
ejpam-5321	139	122	(	(	PUNCT
ejpam-5321	139	123	3	3	X
ejpam-5321	139	124	)	)	PUNCT
ejpam-5321	139	125	for	for	ADP
ejpam-5321	139	126	each	each	DET
ejpam-5321	139	127	x	x	SYM
ejpam-5321	139	128	∈	∈	PROPN
ejpam-5321	139	129	x	x	X
ejpam-5321	139	130	and	and	CCONJ
ejpam-5321	139	131	for	for	ADP
ejpam-5321	139	132	each	each	DET
ejpam-5321	139	133	σ1σ2	σ1σ2	NUM
ejpam-5321	139	134	-	-	PUNCT
ejpam-5321	139	135	clopen	clopen	ADJ
ejpam-5321	139	136	set	set	NOUN
ejpam-5321	139	137	v	v	NOUN
ejpam-5321	139	138	of	of	ADP
ejpam-5321	139	139	y	y	PRON
ejpam-5321	139	140	such	such	ADJ
ejpam-5321	139	141	that	that	SCONJ
ejpam-5321	139	142	x	x	SYM
ejpam-5321	139	143	∈	∈	NOUN
ejpam-5321	139	144	f−(y	f−(y	NOUN
ejpam-5321	139	145	−v	−v	NOUN
ejpam-5321	139	146	)	)	PUNCT
ejpam-5321	139	147	,	,	PUNCT
ejpam-5321	139	148	there	there	PRON
ejpam-5321	139	149	exists	exist	VERB
ejpam-5321	139	150	an	an	DET
ejpam-5321	139	151	α(τ1	α(τ1	NOUN
ejpam-5321	139	152	,	,	PUNCT
ejpam-5321	139	153	τ2)-closed	τ2)-close	VERB
ejpam-5321	139	154	set	set	ADJ
ejpam-5321	139	155	h	h	NOUN
ejpam-5321	139	156	of	of	ADP
ejpam-5321	139	157	x	x	INTJ
ejpam-5321	139	158	such	such	ADJ
ejpam-5321	139	159	that	that	SCONJ
ejpam-5321	139	160	x	x	SYM
ejpam-5321	139	161	∈	∈	NOUN
ejpam-5321	139	162	x	x	SYM
ejpam-5321	139	163	−h	−h	ADJ
ejpam-5321	139	164	and	and	CCONJ
ejpam-5321	139	165	f+(v	f+(v	NUM
ejpam-5321	139	166	)	)	PUNCT
ejpam-5321	139	167	⊆	⊆	NUM
ejpam-5321	139	168	h	h	NOUN
ejpam-5321	139	169	;	;	PUNCT
ejpam-5321	139	170	(	(	PUNCT
ejpam-5321	139	171	4	4	X
ejpam-5321	139	172	)	)	PUNCT
ejpam-5321	139	173	f−(v	f−(v	NOUN
ejpam-5321	139	174	)	)	PUNCT
ejpam-5321	139	175	is	be	AUX
ejpam-5321	139	176	α(τ1	α(τ1	NOUN
ejpam-5321	139	177	,	,	PUNCT
ejpam-5321	139	178	τ2)-open	τ2)-open	ADJ
ejpam-5321	139	179	in	in	ADP
ejpam-5321	139	180	x	x	PUNCT
ejpam-5321	139	181	for	for	ADP
ejpam-5321	139	182	every	every	DET
ejpam-5321	139	183	σ1σ2	σ1σ2	NUM
ejpam-5321	139	184	-	-	PUNCT
ejpam-5321	139	185	clopen	clopen	ADJ
ejpam-5321	139	186	set	set	NOUN
ejpam-5321	139	187	v	v	NOUN
ejpam-5321	139	188	of	of	ADP
ejpam-5321	139	189	y	y	PROPN
ejpam-5321	139	190	;	;	PUNCT
ejpam-5321	139	191	(	(	PUNCT
ejpam-5321	139	192	5	5	NUM
ejpam-5321	139	193	)	)	PUNCT
ejpam-5321	139	194	f+(v	f+(v	NOUN
ejpam-5321	139	195	)	)	PUNCT
ejpam-5321	139	196	is	be	AUX
ejpam-5321	139	197	α(τ1	α(τ1	NOUN
ejpam-5321	139	198	,	,	PUNCT
ejpam-5321	139	199	τ2)-closed	τ2)-close	VERB
ejpam-5321	139	200	in	in	ADP
ejpam-5321	139	201	x	x	PUNCT
ejpam-5321	139	202	for	for	ADP
ejpam-5321	139	203	every	every	DET
ejpam-5321	139	204	σ1σ2	σ1σ2	NUM
ejpam-5321	139	205	-	-	PUNCT
ejpam-5321	139	206	clopen	clopen	ADJ
ejpam-5321	139	207	set	set	NOUN
ejpam-5321	139	208	v	v	NOUN
ejpam-5321	139	209	of	of	ADP
ejpam-5321	139	210	y	y	PROPN
ejpam-5321	139	211	;	;	PUNCT
ejpam-5321	139	212	(	(	PUNCT
ejpam-5321	139	213	6	6	X
ejpam-5321	139	214	)	)	PUNCT
ejpam-5321	139	215	f+(y	f+(y	NOUN
ejpam-5321	139	216	−	−	PROPN
ejpam-5321	139	217	v	v	NOUN
ejpam-5321	139	218	)	)	PUNCT
ejpam-5321	139	219	is	be	AUX
ejpam-5321	139	220	α(τ1	α(τ1	NOUN
ejpam-5321	139	221	,	,	PUNCT
ejpam-5321	139	222	τ2)-closed	τ2)-close	VERB
ejpam-5321	139	223	in	in	ADP
ejpam-5321	139	224	x	x	PUNCT
ejpam-5321	139	225	for	for	ADP
ejpam-5321	139	226	every	every	DET
ejpam-5321	139	227	σ1σ2	σ1σ2	NUM
ejpam-5321	139	228	-	-	PUNCT
ejpam-5321	139	229	clopen	clopen	ADJ
ejpam-5321	139	230	set	set	NOUN
ejpam-5321	139	231	v	v	NOUN
ejpam-5321	139	232	of	of	ADP
ejpam-5321	139	233	y	y	PROPN
ejpam-5321	139	234	;	;	PUNCT
ejpam-5321	139	235	(	(	PUNCT
ejpam-5321	139	236	7	7	X
ejpam-5321	139	237	)	)	PUNCT
ejpam-5321	139	238	f−(y	f−(y	NOUN
ejpam-5321	139	239	−	−	NOUN
ejpam-5321	139	240	v	v	NOUN
ejpam-5321	139	241	)	)	PUNCT
ejpam-5321	139	242	is	be	AUX
ejpam-5321	139	243	α(τ1	α(τ1	NOUN
ejpam-5321	139	244	,	,	PUNCT
ejpam-5321	139	245	τ2)-open	τ2)-open	ADJ
ejpam-5321	139	246	in	in	ADP
ejpam-5321	139	247	x	x	PUNCT
ejpam-5321	139	248	for	for	ADP
ejpam-5321	139	249	every	every	DET
ejpam-5321	139	250	σ1σ2	σ1σ2	NUM
ejpam-5321	139	251	-	-	PUNCT
ejpam-5321	139	252	clopen	clopen	ADJ
ejpam-5321	139	253	set	set	NOUN
ejpam-5321	139	254	v	v	NOUN
ejpam-5321	139	255	of	of	ADP
ejpam-5321	139	256	y	y	PROPN
ejpam-5321	139	257	;	;	PUNCT
ejpam-5321	139	258	(	(	PUNCT
ejpam-5321	139	259	8)	8)	NUM
ejpam-5321	139	260	for	for	ADP
ejpam-5321	139	261	each	each	DET
ejpam-5321	139	262	x	x	SYM
ejpam-5321	139	263	∈	∈	PROPN
ejpam-5321	139	264	x	x	X
ejpam-5321	139	265	and	and	CCONJ
ejpam-5321	139	266	for	for	ADP
ejpam-5321	139	267	each	each	DET
ejpam-5321	139	268	net	net	NOUN
ejpam-5321	139	269	(	(	PUNCT
ejpam-5321	139	270	xγ	xγ	PROPN
ejpam-5321	139	271	)	)	PUNCT
ejpam-5321	139	272	which	which	PRON
ejpam-5321	139	273	α(τ1	α(τ1	VERB
ejpam-5321	139	274	,	,	PUNCT
ejpam-5321	139	275	τ2)-converges	τ2)-converge	NOUN
ejpam-5321	139	276	to	to	ADP
ejpam-5321	139	277	x	x	PUNCT
ejpam-5321	139	278	in	in	ADP
ejpam-5321	139	279	x	x	X
ejpam-5321	139	280	and	and	CCONJ
ejpam-5321	139	281	for	for	ADP
ejpam-5321	139	282	each	each	DET
ejpam-5321	139	283	σ1σ2	σ1σ2	NUM
ejpam-5321	139	284	-	-	PUNCT
ejpam-5321	139	285	clopen	clopen	ADJ
ejpam-5321	139	286	set	set	NOUN
ejpam-5321	139	287	v	v	NOUN
ejpam-5321	139	288	of	of	ADP
ejpam-5321	139	289	y	y	PRON
ejpam-5321	139	290	such	such	ADJ
ejpam-5321	139	291	that	that	SCONJ
ejpam-5321	139	292	x	x	SYM
ejpam-5321	139	293	∈	∈	PROPN
ejpam-5321	139	294	f−(v	f−(v	NOUN
ejpam-5321	139	295	)	)	PUNCT
ejpam-5321	139	296	,	,	PUNCT
ejpam-5321	139	297	the	the	DET
ejpam-5321	139	298	net	net	NOUN
ejpam-5321	139	299	(	(	PUNCT
ejpam-5321	139	300	xγ	xγ	PROPN
ejpam-5321	139	301	)	)	PUNCT
ejpam-5321	139	302	is	be	AUX
ejpam-5321	139	303	eventually	eventually	ADV
ejpam-5321	139	304	in	in	ADP
ejpam-5321	139	305	f−(v	f−(v	NOUN
ejpam-5321	139	306	)	)	PUNCT
ejpam-5321	139	307	.	.	PUNCT
ejpam-5321	140	1	c.	c.	PROPN
ejpam-5321	140	2	viriyapong	viriyapong	PROPN
ejpam-5321	140	3	,	,	PUNCT
ejpam-5321	140	4	s.	s.	PROPN
ejpam-5321	140	5	sompong	sompong	PROPN
ejpam-5321	140	6	,	,	PUNCT
ejpam-5321	140	7	c.	c.	PROPN
ejpam-5321	140	8	boonpok	boonpok	PROPN
ejpam-5321	140	9	/	/	SYM
ejpam-5321	140	10	eur	eur	PROPN
ejpam-5321	140	11	.	.	PUNCT
ejpam-5321	141	1	j.	j.	PROPN
ejpam-5321	141	2	pure	pure	PROPN
ejpam-5321	141	3	appl	appl	PROPN
ejpam-5321	141	4	.	.	PROPN
ejpam-5321	141	5	math	math	PROPN
ejpam-5321	141	6	,	,	PUNCT
ejpam-5321	141	7	17	17	NUM
ejpam-5321	141	8	(	(	PUNCT
ejpam-5321	141	9	3	3	NUM
ejpam-5321	141	10	)	)	PUNCT
ejpam-5321	141	11	(	(	PUNCT
ejpam-5321	141	12	2024	2024	NUM
ejpam-5321	141	13	)	)	PUNCT
ejpam-5321	141	14	,	,	PUNCT
ejpam-5321	141	15	2142	2142	NUM
ejpam-5321	141	16	-	-	SYM
ejpam-5321	141	17	2154	2154	NUM
ejpam-5321	141	18	2147	2147	NUM
ejpam-5321	141	19	proof	proof	NOUN
ejpam-5321	141	20	.	.	PUNCT
ejpam-5321	142	1	the	the	DET
ejpam-5321	142	2	proof	proof	NOUN
ejpam-5321	142	3	is	be	AUX
ejpam-5321	142	4	similar	similar	ADJ
ejpam-5321	142	5	to	to	ADP
ejpam-5321	142	6	that	that	PRON
ejpam-5321	142	7	of	of	ADP
ejpam-5321	142	8	theorem	theorem	NOUN
ejpam-5321	142	9	1	1	NUM
ejpam-5321	142	10	.	.	PUNCT
ejpam-5321	142	11	definition	definition	NOUN
ejpam-5321	142	12	4	4	NUM
ejpam-5321	142	13	.	.	PUNCT
ejpam-5321	143	1	a	a	DET
ejpam-5321	143	2	function	function	NOUN
ejpam-5321	143	3	f	f	NOUN
ejpam-5321	143	4	:	:	PUNCT
ejpam-5321	143	5	(	(	PUNCT
ejpam-5321	143	6	x	x	NOUN
ejpam-5321	143	7	,	,	PUNCT
ejpam-5321	143	8	τ1	τ1	NOUN
ejpam-5321	143	9	,	,	PUNCT
ejpam-5321	143	10	τ2	τ2	NOUN
ejpam-5321	143	11	)	)	PUNCT
ejpam-5321	143	12	→	→	SYM
ejpam-5321	143	13	(	(	PUNCT
ejpam-5321	143	14	y	y	PROPN
ejpam-5321	143	15	,	,	PUNCT
ejpam-5321	143	16	σ1	σ1	PROPN
ejpam-5321	143	17	,	,	PUNCT
ejpam-5321	143	18	σ2	σ2	PROPN
ejpam-5321	143	19	)	)	PUNCT
ejpam-5321	143	20	is	be	AUX
ejpam-5321	143	21	said	say	VERB
ejpam-5321	143	22	to	to	PART
ejpam-5321	143	23	be	be	AUX
ejpam-5321	143	24	slightly	slightly	ADV
ejpam-5321	143	25	α(τ1	α(τ1	NOUN
ejpam-5321	143	26	,	,	PUNCT
ejpam-5321	143	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	143	28	if	if	SCONJ
ejpam-5321	143	29	for	for	ADP
ejpam-5321	143	30	each	each	DET
ejpam-5321	143	31	x	x	SYM
ejpam-5321	143	32	∈	∈	PROPN
ejpam-5321	143	33	x	x	X
ejpam-5321	143	34	and	and	CCONJ
ejpam-5321	143	35	each	each	DET
ejpam-5321	143	36	σ1σ2	σ1σ2	NUM
ejpam-5321	143	37	-	-	PUNCT
ejpam-5321	143	38	clopen	clopen	ADJ
ejpam-5321	143	39	set	set	NOUN
ejpam-5321	143	40	v	v	NOUN
ejpam-5321	143	41	of	of	ADP
ejpam-5321	143	42	y	y	NOUN
ejpam-5321	143	43	containing	contain	VERB
ejpam-5321	143	44	f(x	f(x	PROPN
ejpam-5321	143	45	)	)	PUNCT
ejpam-5321	143	46	,	,	PUNCT
ejpam-5321	143	47	there	there	PRON
ejpam-5321	143	48	exists	exist	VERB
ejpam-5321	143	49	an	an	DET
ejpam-5321	143	50	α(τ1	α(τ1	NOUN
ejpam-5321	143	51	,	,	PUNCT
ejpam-5321	143	52	τ2)-open	τ2)-open	ADJ
ejpam-5321	143	53	set	set	VERB
ejpam-5321	143	54	u	u	NOUN
ejpam-5321	143	55	of	of	ADP
ejpam-5321	143	56	x	x	PUNCT
ejpam-5321	143	57	containing	contain	VERB
ejpam-5321	143	58	x	x	PUNCT
ejpam-5321	143	59	such	such	ADJ
ejpam-5321	143	60	that	that	DET
ejpam-5321	143	61	f(u	f(u	PROPN
ejpam-5321	143	62	)	)	PUNCT
ejpam-5321	143	63	⊆	⊆	NUM
ejpam-5321	143	64	v	v	NOUN
ejpam-5321	143	65	.	.	PUNCT
ejpam-5321	144	1	corollary	corollary	ADJ
ejpam-5321	144	2	1	1	NUM
ejpam-5321	144	3	.	.	PUNCT
ejpam-5321	145	1	for	for	ADP
ejpam-5321	145	2	a	a	DET
ejpam-5321	145	3	function	function	NOUN
ejpam-5321	145	4	f	f	NOUN
ejpam-5321	145	5	:	:	PUNCT
ejpam-5321	145	6	(	(	PUNCT
ejpam-5321	145	7	x	x	NOUN
ejpam-5321	145	8	,	,	PUNCT
ejpam-5321	145	9	τ1	τ1	NOUN
ejpam-5321	145	10	,	,	PUNCT
ejpam-5321	145	11	τ2	τ2	NOUN
ejpam-5321	145	12	)	)	PUNCT
ejpam-5321	145	13	→	→	SYM
ejpam-5321	145	14	(	(	PUNCT
ejpam-5321	145	15	y	y	PROPN
ejpam-5321	145	16	,	,	PUNCT
ejpam-5321	145	17	σ1	σ1	PROPN
ejpam-5321	145	18	,	,	PUNCT
ejpam-5321	145	19	σ2	σ2	NOUN
ejpam-5321	145	20	)	)	PUNCT
ejpam-5321	145	21	,	,	PUNCT
ejpam-5321	145	22	the	the	DET
ejpam-5321	145	23	following	follow	VERB
ejpam-5321	145	24	properties	property	NOUN
ejpam-5321	145	25	are	be	AUX
ejpam-5321	145	26	equivalent	equivalent	ADJ
ejpam-5321	145	27	:	:	PUNCT
ejpam-5321	145	28	(	(	PUNCT
ejpam-5321	145	29	1	1	X
ejpam-5321	145	30	)	)	PUNCT
ejpam-5321	145	31	f	f	PROPN
ejpam-5321	145	32	is	be	AUX
ejpam-5321	145	33	slightly	slightly	ADV
ejpam-5321	145	34	α(τ1	α(τ1	NOUN
ejpam-5321	145	35	,	,	PUNCT
ejpam-5321	145	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	145	37	;	;	PUNCT
ejpam-5321	145	38	(	(	PUNCT
ejpam-5321	145	39	2	2	X
ejpam-5321	145	40	)	)	PUNCT
ejpam-5321	145	41	f−1(v	f−1(v	NOUN
ejpam-5321	145	42	)	)	PUNCT
ejpam-5321	145	43	is	be	AUX
ejpam-5321	145	44	α(τ1	α(τ1	NOUN
ejpam-5321	145	45	,	,	PUNCT
ejpam-5321	145	46	τ2)-open	τ2)-open	ADJ
ejpam-5321	145	47	in	in	ADP
ejpam-5321	145	48	x	x	PUNCT
ejpam-5321	145	49	for	for	ADP
ejpam-5321	145	50	each	each	DET
ejpam-5321	145	51	σ1σ2	σ1σ2	NUM
ejpam-5321	145	52	-	-	PUNCT
ejpam-5321	145	53	clopen	clopen	ADJ
ejpam-5321	145	54	set	set	NOUN
ejpam-5321	145	55	v	v	NOUN
ejpam-5321	145	56	of	of	ADP
ejpam-5321	145	57	y	y	PROPN
ejpam-5321	145	58	;	;	PUNCT
ejpam-5321	145	59	(	(	PUNCT
ejpam-5321	145	60	3	3	X
ejpam-5321	145	61	)	)	PUNCT
ejpam-5321	145	62	f−1(v	f−1(v	NOUN
ejpam-5321	145	63	)	)	PUNCT
ejpam-5321	145	64	is	be	AUX
ejpam-5321	145	65	α(τ1	α(τ1	NOUN
ejpam-5321	145	66	,	,	PUNCT
ejpam-5321	145	67	τ2)-closed	τ2)-close	VERB
ejpam-5321	145	68	in	in	ADP
ejpam-5321	145	69	x	x	PUNCT
ejpam-5321	145	70	for	for	ADP
ejpam-5321	145	71	each	each	DET
ejpam-5321	145	72	σ1σ2	σ1σ2	NUM
ejpam-5321	145	73	-	-	PUNCT
ejpam-5321	145	74	clopen	clopen	ADJ
ejpam-5321	145	75	set	set	NOUN
ejpam-5321	145	76	v	v	NOUN
ejpam-5321	145	77	of	of	ADP
ejpam-5321	145	78	y	y	PROPN
ejpam-5321	145	79	;	;	PUNCT
ejpam-5321	145	80	(	(	PUNCT
ejpam-5321	145	81	4	4	X
ejpam-5321	145	82	)	)	PUNCT
ejpam-5321	145	83	for	for	ADP
ejpam-5321	145	84	each	each	DET
ejpam-5321	145	85	x	x	SYM
ejpam-5321	145	86	∈	∈	PROPN
ejpam-5321	145	87	x	x	X
ejpam-5321	145	88	and	and	CCONJ
ejpam-5321	145	89	for	for	ADP
ejpam-5321	145	90	each	each	DET
ejpam-5321	145	91	σ1σ2	σ1σ2	NUM
ejpam-5321	145	92	-	-	PUNCT
ejpam-5321	145	93	clopen	clopen	ADJ
ejpam-5321	145	94	set	set	NOUN
ejpam-5321	145	95	v	v	NOUN
ejpam-5321	145	96	of	of	ADP
ejpam-5321	145	97	y	y	NOUN
ejpam-5321	145	98	containing	contain	VERB
ejpam-5321	145	99	f(x	f(x	PROPN
ejpam-5321	145	100	)	)	PUNCT
ejpam-5321	145	101	,	,	PUNCT
ejpam-5321	145	102	there	there	PRON
ejpam-5321	145	103	exists	exist	VERB
ejpam-5321	145	104	an	an	DET
ejpam-5321	145	105	α(τ1	α(τ1	NOUN
ejpam-5321	145	106	,	,	PUNCT
ejpam-5321	145	107	τ2)-open	τ2)-open	ADJ
ejpam-5321	145	108	set	set	VERB
ejpam-5321	145	109	u	u	NOUN
ejpam-5321	145	110	of	of	ADP
ejpam-5321	145	111	x	x	PUNCT
ejpam-5321	145	112	containing	contain	VERB
ejpam-5321	145	113	x	x	PUNCT
ejpam-5321	145	114	such	such	ADJ
ejpam-5321	145	115	that	that	DET
ejpam-5321	145	116	f(u	f(u	PROPN
ejpam-5321	145	117	)	)	PUNCT
ejpam-5321	145	118	⊆	⊆	NUM
ejpam-5321	145	119	v	v	NOUN
ejpam-5321	145	120	.	.	PUNCT
ejpam-5321	146	1	definition	definition	NOUN
ejpam-5321	146	2	5	5	NUM
ejpam-5321	146	3	.	.	PUNCT
ejpam-5321	147	1	a	a	DET
ejpam-5321	147	2	bitopological	bitopological	ADJ
ejpam-5321	147	3	space	space	NOUN
ejpam-5321	147	4	(	(	PUNCT
ejpam-5321	147	5	x	x	NOUN
ejpam-5321	147	6	,	,	PUNCT
ejpam-5321	147	7	τ1	τ1	NOUN
ejpam-5321	147	8	,	,	PUNCT
ejpam-5321	147	9	τ2	τ2	NOUN
ejpam-5321	147	10	)	)	PUNCT
ejpam-5321	147	11	is	be	AUX
ejpam-5321	147	12	said	say	VERB
ejpam-5321	147	13	to	to	PART
ejpam-5321	147	14	be	be	AUX
ejpam-5321	147	15	mildly	mildly	ADV
ejpam-5321	147	16	τ1τ2	τ1τ2	ADJ
ejpam-5321	147	17	-	-	ADJ
ejpam-5321	147	18	compact	compact	ADJ
ejpam-5321	147	19	if	if	SCONJ
ejpam-5321	147	20	every	every	DET
ejpam-5321	147	21	cover	cover	NOUN
ejpam-5321	147	22	of	of	ADP
ejpam-5321	147	23	x	x	PUNCT
ejpam-5321	147	24	by	by	ADP
ejpam-5321	147	25	τ1τ2	τ1τ2	ADJ
ejpam-5321	147	26	-	-	ADJ
ejpam-5321	147	27	clopen	clopen	ADJ
ejpam-5321	147	28	sets	set	NOUN
ejpam-5321	147	29	of	of	ADP
ejpam-5321	147	30	x	x	PUNCT
ejpam-5321	147	31	has	have	VERB
ejpam-5321	147	32	a	a	DET
ejpam-5321	147	33	finite	finite	ADJ
ejpam-5321	147	34	subcover	subcover	PROPN
ejpam-5321	147	35	.	.	PUNCT
ejpam-5321	148	1	definition	definition	NOUN
ejpam-5321	148	2	6	6	NUM
ejpam-5321	148	3	.	.	PUNCT
ejpam-5321	149	1	a	a	DET
ejpam-5321	149	2	bitopological	bitopological	ADJ
ejpam-5321	149	3	space	space	NOUN
ejpam-5321	149	4	(	(	PUNCT
ejpam-5321	149	5	x	x	NOUN
ejpam-5321	149	6	,	,	PUNCT
ejpam-5321	149	7	τ1	τ1	NOUN
ejpam-5321	149	8	,	,	PUNCT
ejpam-5321	149	9	τ2	τ2	NOUN
ejpam-5321	149	10	)	)	PUNCT
ejpam-5321	149	11	is	be	AUX
ejpam-5321	149	12	said	say	VERB
ejpam-5321	149	13	to	to	PART
ejpam-5321	149	14	be	be	AUX
ejpam-5321	149	15	α(τ1	α(τ1	NOUN
ejpam-5321	149	16	,	,	PUNCT
ejpam-5321	149	17	τ2)-compact	τ2)-compact	ADJ
ejpam-5321	149	18	if	if	SCONJ
ejpam-5321	149	19	if	if	SCONJ
ejpam-5321	149	20	every	every	DET
ejpam-5321	149	21	cover	cover	NOUN
ejpam-5321	149	22	of	of	ADP
ejpam-5321	149	23	x	x	PUNCT
ejpam-5321	149	24	by	by	ADP
ejpam-5321	149	25	α(τ1	α(τ1	NOUN
ejpam-5321	149	26	,	,	PUNCT
ejpam-5321	149	27	τ2)-open	τ2)-open	ADJ
ejpam-5321	149	28	sets	set	NOUN
ejpam-5321	149	29	of	of	ADP
ejpam-5321	149	30	x	x	PUNCT
ejpam-5321	149	31	has	have	VERB
ejpam-5321	149	32	a	a	DET
ejpam-5321	149	33	finite	finite	ADJ
ejpam-5321	149	34	subcover	subcover	PROPN
ejpam-5321	149	35	.	.	PUNCT
ejpam-5321	150	1	theorem	theorem	VERB
ejpam-5321	150	2	3	3	X
ejpam-5321	150	3	.	.	PUNCT
ejpam-5321	151	1	let	let	AUX
ejpam-5321	151	2	f	f	NOUN
ejpam-5321	151	3	:	:	PUNCT
ejpam-5321	151	4	(	(	PUNCT
ejpam-5321	151	5	x	x	NOUN
ejpam-5321	151	6	,	,	PUNCT
ejpam-5321	151	7	τ1	τ1	NOUN
ejpam-5321	151	8	,	,	PUNCT
ejpam-5321	151	9	τ2	τ2	NOUN
ejpam-5321	151	10	)	)	PUNCT
ejpam-5321	151	11	→	→	SYM
ejpam-5321	151	12	(	(	PUNCT
ejpam-5321	151	13	y	y	PROPN
ejpam-5321	151	14	,	,	PUNCT
ejpam-5321	151	15	σ1	σ1	PROPN
ejpam-5321	151	16	,	,	PUNCT
ejpam-5321	151	17	σ2	σ2	PROPN
ejpam-5321	151	18	)	)	PUNCT
ejpam-5321	151	19	be	be	VERB
ejpam-5321	151	20	an	an	DET
ejpam-5321	151	21	upper	upper	ADJ
ejpam-5321	151	22	slightly	slightly	ADJ
ejpam-5321	151	23	α(τ1	α(τ1	NOUN
ejpam-5321	151	24	,	,	PUNCT
ejpam-5321	151	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	151	26	surjective	surjective	ADJ
ejpam-5321	151	27	multifunction	multifunction	NOUN
ejpam-5321	151	28	such	such	ADJ
ejpam-5321	151	29	that	that	SCONJ
ejpam-5321	151	30	f	f	PROPN
ejpam-5321	151	31	(	(	PUNCT
ejpam-5321	151	32	x	x	X
ejpam-5321	151	33	)	)	PUNCT
ejpam-5321	151	34	is	be	AUX
ejpam-5321	151	35	mildly	mildly	ADV
ejpam-5321	151	36	σ1σ2	σ1σ2	ADJ
ejpam-5321	151	37	-	-	ADJ
ejpam-5321	151	38	compact	compact	ADJ
ejpam-5321	151	39	for	for	ADP
ejpam-5321	151	40	each	each	DET
ejpam-5321	151	41	x	x	SYM
ejpam-5321	151	42	∈	∈	PROPN
ejpam-5321	151	43	x.	x.	NOUN
ejpam-5321	152	1	if	if	SCONJ
ejpam-5321	152	2	(	(	PUNCT
ejpam-5321	152	3	x	x	NOUN
ejpam-5321	152	4	,	,	PUNCT
ejpam-5321	152	5	τ1	τ1	NOUN
ejpam-5321	152	6	,	,	PUNCT
ejpam-5321	152	7	τ2	τ2	NOUN
ejpam-5321	152	8	)	)	PUNCT
ejpam-5321	152	9	is	be	AUX
ejpam-5321	152	10	α(τ1	α(τ1	NOUN
ejpam-5321	152	11	,	,	PUNCT
ejpam-5321	152	12	τ2)-compact	τ2)-compact	PROPN
ejpam-5321	152	13	,	,	PUNCT
ejpam-5321	152	14	then	then	ADV
ejpam-5321	152	15	(	(	PUNCT
ejpam-5321	152	16	y	y	PROPN
ejpam-5321	152	17	,	,	PUNCT
ejpam-5321	152	18	σ1	σ1	PROPN
ejpam-5321	152	19	,	,	PUNCT
ejpam-5321	152	20	σ2	σ2	NOUN
ejpam-5321	152	21	)	)	PUNCT
ejpam-5321	152	22	is	be	AUX
ejpam-5321	152	23	mildly	mildly	ADV
ejpam-5321	152	24	σ1σ2	σ1σ2	ADJ
ejpam-5321	152	25	-	-	ADJ
ejpam-5321	152	26	compact	compact	ADJ
ejpam-5321	152	27	.	.	PUNCT
ejpam-5321	153	1	proof	proof	NOUN
ejpam-5321	153	2	.	.	PUNCT
ejpam-5321	154	1	let	let	VERB
ejpam-5321	154	2	{	{	PUNCT
ejpam-5321	154	3	vγ	vγ	VERB
ejpam-5321	154	4	|	|	ADV
ejpam-5321	154	5	γ	γ	PROPN
ejpam-5321	154	6	∈	∈	PROPN
ejpam-5321	154	7	γ	γ	AUX
ejpam-5321	154	8	}	}	PUNCT
ejpam-5321	154	9	be	be	VERB
ejpam-5321	154	10	any	any	DET
ejpam-5321	154	11	σ1σ2	σ1σ2	NOUN
ejpam-5321	154	12	-	-	PUNCT
ejpam-5321	154	13	clopen	clopen	ADJ
ejpam-5321	154	14	cover	cover	NOUN
ejpam-5321	154	15	of	of	ADP
ejpam-5321	154	16	y	y	PROPN
ejpam-5321	154	17	.	.	PUNCT
ejpam-5321	155	1	since	since	SCONJ
ejpam-5321	155	2	f	f	PROPN
ejpam-5321	155	3	(	(	PUNCT
ejpam-5321	155	4	x	x	X
ejpam-5321	155	5	)	)	PUNCT
ejpam-5321	155	6	is	be	AUX
ejpam-5321	155	7	mildly	mildly	ADV
ejpam-5321	155	8	σ1σ2compact	σ1σ2compact	ADJ
ejpam-5321	155	9	for	for	ADP
ejpam-5321	155	10	each	each	DET
ejpam-5321	155	11	x	x	SYM
ejpam-5321	155	12	∈	∈	PROPN
ejpam-5321	155	13	x	x	X
ejpam-5321	155	14	,	,	PUNCT
ejpam-5321	155	15	there	there	PRON
ejpam-5321	155	16	exists	exist	VERB
ejpam-5321	155	17	a	a	DET
ejpam-5321	155	18	finite	finite	NOUN
ejpam-5321	155	19	subset	subset	NOUN
ejpam-5321	155	20	γ(x	γ(x	NOUN
ejpam-5321	155	21	)	)	PUNCT
ejpam-5321	155	22	of	of	ADP
ejpam-5321	155	23	γ	γ	PRON
ejpam-5321	155	24	such	such	ADJ
ejpam-5321	155	25	that	that	SCONJ
ejpam-5321	155	26	f	f	PROPN
ejpam-5321	155	27	(	(	PUNCT
ejpam-5321	155	28	x	x	X
ejpam-5321	155	29	)	)	PUNCT
ejpam-5321	155	30	⊆	⊆	NUM
ejpam-5321	155	31	∪{vγ	∪{vγ	PROPN
ejpam-5321	155	32	|	|	ADV
ejpam-5321	155	33	γ	γ	X
ejpam-5321	155	34	∈	∈	PROPN
ejpam-5321	155	35	γ(x	γ(x	PROPN
ejpam-5321	155	36	)	)	PUNCT
ejpam-5321	155	37	}	}	PUNCT
ejpam-5321	155	38	.	.	PUNCT
ejpam-5321	156	1	put	put	VERB
ejpam-5321	156	2	v	v	NOUN
ejpam-5321	156	3	(	(	PUNCT
ejpam-5321	156	4	x	x	NOUN
ejpam-5321	156	5	)	)	PUNCT
ejpam-5321	156	6	=	=	SYM
ejpam-5321	156	7	∪{vγ	∪{vγ	PROPN
ejpam-5321	156	8	|	|	ADV
ejpam-5321	156	9	γ	γ	X
ejpam-5321	156	10	∈	∈	PROPN
ejpam-5321	156	11	γ(x	γ(x	PROPN
ejpam-5321	156	12	)	)	PUNCT
ejpam-5321	156	13	}	}	PUNCT
ejpam-5321	156	14	.	.	PUNCT
ejpam-5321	157	1	since	since	SCONJ
ejpam-5321	157	2	f	f	PROPN
ejpam-5321	157	3	is	be	AUX
ejpam-5321	157	4	upper	upper	ADJ
ejpam-5321	157	5	slightly	slightly	ADV
ejpam-5321	157	6	α(τ1	α(τ1	NOUN
ejpam-5321	157	7	,	,	PUNCT
ejpam-5321	157	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	157	9	,	,	PUNCT
ejpam-5321	157	10	there	there	PRON
ejpam-5321	157	11	exists	exist	VERB
ejpam-5321	157	12	an	an	DET
ejpam-5321	157	13	α(τ1	α(τ1	NOUN
ejpam-5321	157	14	,	,	PUNCT
ejpam-5321	157	15	τ2)-open	τ2)-open	ADJ
ejpam-5321	157	16	set	set	VERB
ejpam-5321	157	17	u(x	u(x	NOUN
ejpam-5321	157	18	)	)	PUNCT
ejpam-5321	157	19	of	of	ADP
ejpam-5321	157	20	x	x	SYM
ejpam-5321	157	21	containing	contain	VERB
ejpam-5321	157	22	x	x	PUNCT
ejpam-5321	157	23	such	such	ADJ
ejpam-5321	157	24	that	that	SCONJ
ejpam-5321	157	25	f	f	PROPN
ejpam-5321	157	26	(	(	PUNCT
ejpam-5321	157	27	u(x	u(x	PROPN
ejpam-5321	157	28	)	)	PUNCT
ejpam-5321	157	29	)	)	PUNCT
ejpam-5321	158	1	⊆	⊆	NUM
ejpam-5321	158	2	v	v	X
ejpam-5321	158	3	(	(	PUNCT
ejpam-5321	158	4	x	x	NOUN
ejpam-5321	158	5	)	)	PUNCT
ejpam-5321	158	6	.	.	PUNCT
ejpam-5321	159	1	then	then	ADV
ejpam-5321	159	2	,	,	PUNCT
ejpam-5321	159	3	the	the	DET
ejpam-5321	159	4	family	family	NOUN
ejpam-5321	159	5	{	{	PUNCT
ejpam-5321	159	6	u(x	u(x	PROPN
ejpam-5321	159	7	)	)	PUNCT
ejpam-5321	159	8	|	|	ADV
ejpam-5321	159	9	x	x	SYM
ejpam-5321	159	10	∈	∈	NOUN
ejpam-5321	159	11	x	x	X
ejpam-5321	159	12	}	}	PUNCT
ejpam-5321	159	13	is	be	AUX
ejpam-5321	159	14	an	an	DET
ejpam-5321	159	15	α(τ1	α(τ1	NOUN
ejpam-5321	159	16	,	,	PUNCT
ejpam-5321	159	17	τ2)-open	τ2)-open	ADJ
ejpam-5321	159	18	cover	cover	NOUN
ejpam-5321	159	19	of	of	ADP
ejpam-5321	159	20	x.	x.	NOUN
ejpam-5321	159	21	since	since	SCONJ
ejpam-5321	159	22	(	(	PUNCT
ejpam-5321	159	23	x	x	NOUN
ejpam-5321	159	24	,	,	PUNCT
ejpam-5321	159	25	τ1	τ1	NOUN
ejpam-5321	159	26	,	,	PUNCT
ejpam-5321	159	27	τ2	τ2	NOUN
ejpam-5321	159	28	)	)	PUNCT
ejpam-5321	159	29	is	be	AUX
ejpam-5321	159	30	α(τ1	α(τ1	NOUN
ejpam-5321	159	31	,	,	PUNCT
ejpam-5321	159	32	τ2)-compact	τ2)-compact	PROPN
ejpam-5321	159	33	,	,	PUNCT
ejpam-5321	159	34	there	there	PRON
ejpam-5321	159	35	exists	exist	VERB
ejpam-5321	159	36	a	a	DET
ejpam-5321	159	37	finite	finite	ADJ
ejpam-5321	159	38	number	number	NOUN
ejpam-5321	159	39	of	of	ADP
ejpam-5321	159	40	points	point	NOUN
ejpam-5321	159	41	,	,	PUNCT
ejpam-5321	159	42	say	say	INTJ
ejpam-5321	159	43	,	,	PUNCT
ejpam-5321	159	44	x1	x1	PROPN
ejpam-5321	159	45	,	,	PUNCT
ejpam-5321	159	46	x2	x2	PROPN
ejpam-5321	159	47	,	,	PUNCT
ejpam-5321	159	48	...	...	PUNCT
ejpam-5321	159	49	,	,	PUNCT
ejpam-5321	159	50	xn	xn	PROPN
ejpam-5321	159	51	inx	inx	VERB
ejpam-5321	159	52	such	such	ADJ
ejpam-5321	159	53	thatx	thatx	NOUN
ejpam-5321	159	54	=	=	SYM
ejpam-5321	159	55	∪{u(xi	∪{u(xi	X
ejpam-5321	159	56	)	)	PUNCT
ejpam-5321	160	1	|	|	ADV
ejpam-5321	160	2	1	1	NUM
ejpam-5321	160	3	≤	≤	NUM
ejpam-5321	160	4	i	i	PRON
ejpam-5321	160	5	≤	≤	NOUN
ejpam-5321	160	6	n	n	CCONJ
ejpam-5321	160	7	}	}	PUNCT
ejpam-5321	160	8	.	.	PUNCT
ejpam-5321	161	1	thus	thus	ADV
ejpam-5321	161	2	,	,	PUNCT
ejpam-5321	161	3	y	y	PROPN
ejpam-5321	161	4	=	=	SYM
ejpam-5321	161	5	f	f	PROPN
ejpam-5321	161	6	(	(	PUNCT
ejpam-5321	161	7	x	x	X
ejpam-5321	161	8	)	)	PUNCT
ejpam-5321	161	9	=	=	SYM
ejpam-5321	161	10	∪	∪	X
ejpam-5321	161	11	{	{	PUNCT
ejpam-5321	161	12	f	f	X
ejpam-5321	161	13	(	(	PUNCT
ejpam-5321	161	14	u(xi	u(xi	PROPN
ejpam-5321	161	15	)	)	PUNCT
ejpam-5321	161	16	)	)	PUNCT
ejpam-5321	162	1	|	|	ADV
ejpam-5321	162	2	1	1	NUM
ejpam-5321	162	3	≤	≤	NUM
ejpam-5321	162	4	i	i	PRON
ejpam-5321	162	5	≤	≤	NOUN
ejpam-5321	162	6	n	n	CCONJ
ejpam-5321	162	7	}	}	PUNCT
ejpam-5321	162	8	⊆	⊆	NUM
ejpam-5321	162	9	∪{v	∪{v	PROPN
ejpam-5321	162	10	(	(	PUNCT
ejpam-5321	162	11	xi	xi	NOUN
ejpam-5321	162	12	)	)	PUNCT
ejpam-5321	162	13	|	|	ADV
ejpam-5321	162	14	1	1	NUM
ejpam-5321	162	15	≤	≤	NUM
ejpam-5321	162	16	i	i	PRON
ejpam-5321	162	17	≤	≤	NOUN
ejpam-5321	162	18	n	n	CCONJ
ejpam-5321	162	19	}	}	PUNCT
ejpam-5321	162	20	⊆	⊆	NUM
ejpam-5321	162	21	∪{vγ	∪{vγ	PROPN
ejpam-5321	162	22	|	|	ADV
ejpam-5321	162	23	γ	γ	X
ejpam-5321	162	24	∈	∈	PROPN
ejpam-5321	162	25	γ(xi	γ(xi	PROPN
ejpam-5321	162	26	)	)	PUNCT
ejpam-5321	162	27	,	,	PUNCT
ejpam-5321	162	28	1	1	NUM
ejpam-5321	162	29	≤	≤	NUM
ejpam-5321	162	30	i	i	PRON
ejpam-5321	162	31	≤	≤	NOUN
ejpam-5321	162	32	n	n	CCONJ
ejpam-5321	162	33	}	}	PUNCT
ejpam-5321	162	34	.	.	PUNCT
ejpam-5321	163	1	this	this	PRON
ejpam-5321	163	2	shows	show	VERB
ejpam-5321	163	3	that	that	SCONJ
ejpam-5321	163	4	(	(	PUNCT
ejpam-5321	163	5	y	y	PROPN
ejpam-5321	163	6	,	,	PUNCT
ejpam-5321	163	7	σ1	σ1	PROPN
ejpam-5321	163	8	,	,	PUNCT
ejpam-5321	163	9	σ2	σ2	NOUN
ejpam-5321	163	10	)	)	PUNCT
ejpam-5321	163	11	is	be	AUX
ejpam-5321	163	12	mildly	mildly	ADV
ejpam-5321	163	13	σ1σ2	σ1σ2	ADJ
ejpam-5321	163	14	-	-	ADJ
ejpam-5321	163	15	compact	compact	ADJ
ejpam-5321	163	16	.	.	PUNCT
ejpam-5321	164	1	recall	recall	VERB
ejpam-5321	164	2	that	that	SCONJ
ejpam-5321	164	3	a	a	DET
ejpam-5321	164	4	bitopological	bitopological	ADJ
ejpam-5321	164	5	space	space	NOUN
ejpam-5321	164	6	(	(	PUNCT
ejpam-5321	164	7	x	x	NOUN
ejpam-5321	164	8	,	,	PUNCT
ejpam-5321	164	9	τ1	τ1	NOUN
ejpam-5321	164	10	,	,	PUNCT
ejpam-5321	164	11	τ2	τ2	NOUN
ejpam-5321	164	12	)	)	PUNCT
ejpam-5321	164	13	is	be	AUX
ejpam-5321	164	14	said	say	VERB
ejpam-5321	164	15	to	to	PART
ejpam-5321	164	16	be	be	AUX
ejpam-5321	164	17	τ1τ2	τ1τ2	NOUN
ejpam-5321	164	18	-	-	ADJ
ejpam-5321	164	19	connected	connected	ADJ
ejpam-5321	165	1	[	[	X
ejpam-5321	165	2	20	20	NUM
ejpam-5321	165	3	]	]	PUNCT
ejpam-5321	165	4	if	if	SCONJ
ejpam-5321	165	5	x	x	PRON
ejpam-5321	165	6	can	can	AUX
ejpam-5321	165	7	not	not	PART
ejpam-5321	165	8	be	be	AUX
ejpam-5321	165	9	written	write	VERB
ejpam-5321	165	10	as	as	ADP
ejpam-5321	165	11	the	the	DET
ejpam-5321	165	12	union	union	NOUN
ejpam-5321	165	13	of	of	ADP
ejpam-5321	165	14	two	two	NUM
ejpam-5321	165	15	disjoint	disjoint	NOUN
ejpam-5321	165	16	nonempty	nonempty	ADJ
ejpam-5321	165	17	τ1τ2	τ1τ2	ADJ
ejpam-5321	165	18	-	-	ADJ
ejpam-5321	165	19	open	open	ADJ
ejpam-5321	165	20	sets	set	NOUN
ejpam-5321	165	21	.	.	PUNCT
ejpam-5321	166	1	c.	c.	PROPN
ejpam-5321	166	2	viriyapong	viriyapong	PROPN
ejpam-5321	166	3	,	,	PUNCT
ejpam-5321	166	4	s.	s.	PROPN
ejpam-5321	166	5	sompong	sompong	PROPN
ejpam-5321	166	6	,	,	PUNCT
ejpam-5321	166	7	c.	c.	PROPN
ejpam-5321	166	8	boonpok	boonpok	PROPN
ejpam-5321	166	9	/	/	SYM
ejpam-5321	166	10	eur	eur	PROPN
ejpam-5321	166	11	.	.	PUNCT
ejpam-5321	167	1	j.	j.	PROPN
ejpam-5321	167	2	pure	pure	PROPN
ejpam-5321	167	3	appl	appl	PROPN
ejpam-5321	167	4	.	.	PROPN
ejpam-5321	167	5	math	math	PROPN
ejpam-5321	167	6	,	,	PUNCT
ejpam-5321	167	7	17	17	NUM
ejpam-5321	167	8	(	(	PUNCT
ejpam-5321	167	9	3	3	NUM
ejpam-5321	167	10	)	)	PUNCT
ejpam-5321	167	11	(	(	PUNCT
ejpam-5321	167	12	2024	2024	NUM
ejpam-5321	167	13	)	)	PUNCT
ejpam-5321	167	14	,	,	PUNCT
ejpam-5321	167	15	2142	2142	NUM
ejpam-5321	167	16	-	-	SYM
ejpam-5321	167	17	2154	2154	NUM
ejpam-5321	167	18	2148	2148	NUM
ejpam-5321	167	19	definition	definition	NOUN
ejpam-5321	167	20	7	7	NUM
ejpam-5321	167	21	.	.	PUNCT
ejpam-5321	168	1	a	a	DET
ejpam-5321	168	2	bitopological	bitopological	ADJ
ejpam-5321	168	3	space	space	NOUN
ejpam-5321	168	4	(	(	PUNCT
ejpam-5321	168	5	x	x	NOUN
ejpam-5321	168	6	,	,	PUNCT
ejpam-5321	168	7	τ1	τ1	NOUN
ejpam-5321	168	8	,	,	PUNCT
ejpam-5321	168	9	τ2	τ2	NOUN
ejpam-5321	168	10	)	)	PUNCT
ejpam-5321	168	11	is	be	AUX
ejpam-5321	168	12	said	say	VERB
ejpam-5321	168	13	to	to	PART
ejpam-5321	168	14	be	be	AUX
ejpam-5321	168	15	α(τ1	α(τ1	NOUN
ejpam-5321	168	16	,	,	PUNCT
ejpam-5321	168	17	τ2)-connected	τ2)-connecte	VERB
ejpam-5321	168	18	provided	provide	VERB
ejpam-5321	168	19	that	that	SCONJ
ejpam-5321	168	20	x	x	PRON
ejpam-5321	168	21	is	be	AUX
ejpam-5321	168	22	not	not	PART
ejpam-5321	168	23	the	the	DET
ejpam-5321	168	24	union	union	NOUN
ejpam-5321	168	25	of	of	ADP
ejpam-5321	168	26	two	two	NUM
ejpam-5321	168	27	disjoint	disjoint	NOUN
ejpam-5321	168	28	nonempty	nonempty	NOUN
ejpam-5321	168	29	α(τ1	α(τ1	NOUN
ejpam-5321	168	30	,	,	PUNCT
ejpam-5321	168	31	τ2)-open	τ2)-open	ADJ
ejpam-5321	168	32	sets	set	NOUN
ejpam-5321	168	33	.	.	PUNCT
ejpam-5321	169	1	definition	definition	NOUN
ejpam-5321	169	2	8	8	NUM
ejpam-5321	169	3	.	.	PUNCT
ejpam-5321	170	1	a	a	DET
ejpam-5321	170	2	multifunction	multifunction	NOUN
ejpam-5321	170	3	f	f	NOUN
ejpam-5321	170	4	:	:	PUNCT
ejpam-5321	170	5	(	(	PUNCT
ejpam-5321	170	6	x	x	NOUN
ejpam-5321	170	7	,	,	PUNCT
ejpam-5321	170	8	τ1	τ1	NOUN
ejpam-5321	170	9	,	,	PUNCT
ejpam-5321	170	10	τ2	τ2	NOUN
ejpam-5321	170	11	)	)	PUNCT
ejpam-5321	170	12	→	→	SYM
ejpam-5321	170	13	(	(	PUNCT
ejpam-5321	170	14	y	y	PROPN
ejpam-5321	170	15	,	,	PUNCT
ejpam-5321	170	16	σ1	σ1	PROPN
ejpam-5321	170	17	,	,	PUNCT
ejpam-5321	170	18	σ2	σ2	PROPN
ejpam-5321	170	19	)	)	PUNCT
ejpam-5321	170	20	is	be	AUX
ejpam-5321	170	21	called	call	VERB
ejpam-5321	170	22	punctually	punctually	ADV
ejpam-5321	170	23	τ1τ2connected	τ1τ2connecte	VERB
ejpam-5321	170	24	if	if	SCONJ
ejpam-5321	170	25	,	,	PUNCT
ejpam-5321	170	26	for	for	ADP
ejpam-5321	170	27	each	each	DET
ejpam-5321	170	28	x	x	SYM
ejpam-5321	170	29	∈	∈	PROPN
ejpam-5321	170	30	x	x	X
ejpam-5321	170	31	,	,	PUNCT
ejpam-5321	170	32	f	f	PROPN
ejpam-5321	170	33	(	(	PUNCT
ejpam-5321	170	34	x	x	X
ejpam-5321	170	35	)	)	PUNCT
ejpam-5321	170	36	is	be	AUX
ejpam-5321	170	37	σ1σ2	σ1σ2	NOUN
ejpam-5321	170	38	-	-	PUNCT
ejpam-5321	170	39	connected	connect	VERB
ejpam-5321	170	40	.	.	PUNCT
ejpam-5321	171	1	theorem	theorem	ADJ
ejpam-5321	171	2	4	4	NUM
ejpam-5321	171	3	.	.	PUNCT
ejpam-5321	172	1	let	let	VERB
ejpam-5321	172	2	f	f	NOUN
ejpam-5321	172	3	:	:	PUNCT
ejpam-5321	172	4	(	(	PUNCT
ejpam-5321	172	5	x	x	NOUN
ejpam-5321	172	6	,	,	PUNCT
ejpam-5321	172	7	τ1	τ1	NOUN
ejpam-5321	172	8	,	,	PUNCT
ejpam-5321	172	9	τ2	τ2	NOUN
ejpam-5321	172	10	)	)	PUNCT
ejpam-5321	172	11	→	→	SYM
ejpam-5321	172	12	(	(	PUNCT
ejpam-5321	172	13	y	y	PROPN
ejpam-5321	172	14	,	,	PUNCT
ejpam-5321	172	15	σ1	σ1	PROPN
ejpam-5321	172	16	,	,	PUNCT
ejpam-5321	172	17	σ2	σ2	PROPN
ejpam-5321	172	18	)	)	PUNCT
ejpam-5321	172	19	be	be	VERB
ejpam-5321	172	20	an	an	DET
ejpam-5321	172	21	upper	upper	ADJ
ejpam-5321	172	22	slightly	slightly	ADJ
ejpam-5321	172	23	α(τ1	α(τ1	NOUN
ejpam-5321	172	24	,	,	PUNCT
ejpam-5321	172	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	172	26	multifunction	multifunction	NOUN
ejpam-5321	172	27	such	such	ADJ
ejpam-5321	172	28	that	that	SCONJ
ejpam-5321	172	29	f	f	PROPN
ejpam-5321	172	30	is	be	AUX
ejpam-5321	172	31	punctually	punctually	ADV
ejpam-5321	172	32	τ1τ2	τ1τ2	ADV
ejpam-5321	172	33	-	-	VERB
ejpam-5321	172	34	connected	connected	ADJ
ejpam-5321	172	35	.	.	PUNCT
ejpam-5321	173	1	if	if	SCONJ
ejpam-5321	173	2	(	(	PUNCT
ejpam-5321	173	3	x	x	NOUN
ejpam-5321	173	4	,	,	PUNCT
ejpam-5321	173	5	τ1	τ1	NOUN
ejpam-5321	173	6	,	,	PUNCT
ejpam-5321	173	7	τ2	τ2	NOUN
ejpam-5321	173	8	)	)	PUNCT
ejpam-5321	173	9	is	be	AUX
ejpam-5321	173	10	α(τ1	α(τ1	NOUN
ejpam-5321	173	11	,	,	PUNCT
ejpam-5321	173	12	τ2)-connected	τ2)-connecte	VERB
ejpam-5321	173	13	,	,	PUNCT
ejpam-5321	173	14	then	then	ADV
ejpam-5321	173	15	(	(	PUNCT
ejpam-5321	173	16	y	y	PROPN
ejpam-5321	173	17	,	,	PUNCT
ejpam-5321	173	18	σ1	σ1	PROPN
ejpam-5321	173	19	,	,	PUNCT
ejpam-5321	173	20	σ2	σ2	PROPN
ejpam-5321	173	21	)	)	PUNCT
ejpam-5321	173	22	is	be	AUX
ejpam-5321	173	23	σ1σ2	σ1σ2	NOUN
ejpam-5321	173	24	-	-	PUNCT
ejpam-5321	173	25	connected	connected	ADJ
ejpam-5321	173	26	.	.	PUNCT
ejpam-5321	174	1	proof	proof	NOUN
ejpam-5321	174	2	.	.	PUNCT
ejpam-5321	175	1	suppose	suppose	VERB
ejpam-5321	175	2	that	that	SCONJ
ejpam-5321	175	3	(	(	PUNCT
ejpam-5321	175	4	y	y	PROPN
ejpam-5321	175	5	,	,	PUNCT
ejpam-5321	175	6	σ1	σ1	PROPN
ejpam-5321	175	7	,	,	PUNCT
ejpam-5321	175	8	σ2	σ2	PROPN
ejpam-5321	175	9	)	)	PUNCT
ejpam-5321	175	10	is	be	AUX
ejpam-5321	175	11	not	not	PART
ejpam-5321	175	12	σ1σ2	σ1σ2	VERB
ejpam-5321	175	13	-	-	PUNCT
ejpam-5321	175	14	connected	connect	VERB
ejpam-5321	175	15	.	.	PUNCT
ejpam-5321	176	1	then	then	ADV
ejpam-5321	176	2	,	,	PUNCT
ejpam-5321	176	3	there	there	PRON
ejpam-5321	176	4	exist	exist	VERB
ejpam-5321	176	5	nonempty	nonempty	ADV
ejpam-5321	176	6	σ1σ2	σ1σ2	NOUN
ejpam-5321	176	7	-	-	ADJ
ejpam-5321	176	8	open	open	ADJ
ejpam-5321	176	9	sets	set	NOUN
ejpam-5321	176	10	u	u	NOUN
ejpam-5321	176	11	and	and	CCONJ
ejpam-5321	176	12	v	v	NOUN
ejpam-5321	176	13	of	of	ADP
ejpam-5321	176	14	y	y	PRON
ejpam-5321	176	15	such	such	ADJ
ejpam-5321	176	16	that	that	DET
ejpam-5321	176	17	u∩v	u∩v	NOUN
ejpam-5321	176	18	=	=	NOUN
ejpam-5321	176	19	∅	∅	NOUN
ejpam-5321	176	20	and	and	CCONJ
ejpam-5321	176	21	u∪v	u∪v	NOUN
ejpam-5321	176	22	=	=	SYM
ejpam-5321	176	23	y	y	PROPN
ejpam-5321	176	24	.	.	PUNCT
ejpam-5321	177	1	since	since	SCONJ
ejpam-5321	177	2	f	f	PROPN
ejpam-5321	177	3	is	be	AUX
ejpam-5321	177	4	upper	upper	ADJ
ejpam-5321	177	5	slightly	slightly	ADV
ejpam-5321	177	6	α(τ1	α(τ1	NOUN
ejpam-5321	177	7	,	,	PUNCT
ejpam-5321	177	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	177	9	,	,	PUNCT
ejpam-5321	177	10	f	f	PROPN
ejpam-5321	177	11	+	+	ADJ
ejpam-5321	177	12	(	(	PUNCT
ejpam-5321	177	13	u	u	NOUN
ejpam-5321	177	14	)	)	PUNCT
ejpam-5321	177	15	and	and	CCONJ
ejpam-5321	177	16	f+(v	f+(v	PROPN
ejpam-5321	177	17	)	)	PUNCT
ejpam-5321	177	18	are	be	AUX
ejpam-5321	177	19	α(τ1	α(τ1	NOUN
ejpam-5321	177	20	,	,	PUNCT
ejpam-5321	177	21	τ2)-open	τ2)-open	ADJ
ejpam-5321	177	22	sets	set	NOUN
ejpam-5321	177	23	of	of	ADP
ejpam-5321	177	24	x.	x.	NOUN
ejpam-5321	177	25	in	in	ADP
ejpam-5321	177	26	view	view	NOUN
ejpam-5321	177	27	of	of	ADP
ejpam-5321	177	28	the	the	DET
ejpam-5321	177	29	fact	fact	NOUN
ejpam-5321	177	30	that	that	SCONJ
ejpam-5321	177	31	f+(u	f+(u	NUM
ejpam-5321	177	32	)	)	PUNCT
ejpam-5321	177	33	,	,	PUNCT
ejpam-5321	177	34	f+(v	f+(v	PROPN
ejpam-5321	177	35	)	)	PUNCT
ejpam-5321	177	36	are	be	AUX
ejpam-5321	177	37	disjoint	disjoint	ADJ
ejpam-5321	177	38	and	and	CCONJ
ejpam-5321	177	39	f	f	PROPN
ejpam-5321	177	40	is	be	AUX
ejpam-5321	177	41	punctually	punctually	ADV
ejpam-5321	177	42	τ1τ2	τ1τ2	ADV
ejpam-5321	177	43	-	-	ADJ
ejpam-5321	177	44	connected	connected	ADJ
ejpam-5321	177	45	,	,	PUNCT
ejpam-5321	177	46	x	x	SYM
ejpam-5321	177	47	=	=	X
ejpam-5321	177	48	f+(u)∪f+(v	f+(u)∪f+(v	X
ejpam-5321	177	49	)	)	PUNCT
ejpam-5321	177	50	is	be	AUX
ejpam-5321	177	51	a	a	DET
ejpam-5321	177	52	partition	partition	NOUN
ejpam-5321	177	53	of	of	ADP
ejpam-5321	177	54	x.	x.	NOUN
ejpam-5321	177	55	this	this	PRON
ejpam-5321	177	56	is	be	AUX
ejpam-5321	177	57	contrary	contrary	ADJ
ejpam-5321	177	58	to	to	ADP
ejpam-5321	177	59	the	the	DET
ejpam-5321	177	60	α(τ1	α(τ1	NOUN
ejpam-5321	177	61	,	,	PUNCT
ejpam-5321	177	62	τ2)-connectedness	τ2)-connectedness	NOUN
ejpam-5321	177	63	of	of	ADP
ejpam-5321	177	64	(	(	PUNCT
ejpam-5321	177	65	x	x	NOUN
ejpam-5321	177	66	,	,	PUNCT
ejpam-5321	177	67	τ1	τ1	NOUN
ejpam-5321	177	68	,	,	PUNCT
ejpam-5321	177	69	τ2	τ2	NOUN
ejpam-5321	177	70	)	)	PUNCT
ejpam-5321	177	71	.	.	PUNCT
ejpam-5321	178	1	this	this	PRON
ejpam-5321	178	2	shows	show	VERB
ejpam-5321	178	3	that	that	SCONJ
ejpam-5321	178	4	(	(	PUNCT
ejpam-5321	178	5	y	y	PROPN
ejpam-5321	178	6	,	,	PUNCT
ejpam-5321	178	7	σ1	σ1	PROPN
ejpam-5321	178	8	,	,	PUNCT
ejpam-5321	178	9	σ2	σ2	PROPN
ejpam-5321	178	10	)	)	PUNCT
ejpam-5321	178	11	is	be	AUX
ejpam-5321	178	12	σ1σ2	σ1σ2	NOUN
ejpam-5321	178	13	-	-	PUNCT
ejpam-5321	178	14	connected	connect	VERB
ejpam-5321	178	15	.	.	PUNCT
ejpam-5321	179	1	theorem	theorem	NOUN
ejpam-5321	179	2	5	5	NUM
ejpam-5321	179	3	.	.	PUNCT
ejpam-5321	180	1	let	let	VERB
ejpam-5321	180	2	f	f	NOUN
ejpam-5321	180	3	:	:	PUNCT
ejpam-5321	180	4	(	(	PUNCT
ejpam-5321	180	5	x	x	NOUN
ejpam-5321	180	6	,	,	PUNCT
ejpam-5321	180	7	τ1	τ1	NOUN
ejpam-5321	180	8	,	,	PUNCT
ejpam-5321	180	9	τ2	τ2	NOUN
ejpam-5321	180	10	)	)	PUNCT
ejpam-5321	180	11	→	→	SYM
ejpam-5321	180	12	(	(	PUNCT
ejpam-5321	180	13	y	y	PROPN
ejpam-5321	180	14	,	,	PUNCT
ejpam-5321	180	15	σ1	σ1	PROPN
ejpam-5321	180	16	,	,	PUNCT
ejpam-5321	180	17	σ2	σ2	PROPN
ejpam-5321	180	18	)	)	PUNCT
ejpam-5321	180	19	be	be	AUX
ejpam-5321	180	20	a	a	DET
ejpam-5321	180	21	lower	low	ADJ
ejpam-5321	180	22	slightly	slightly	ADJ
ejpam-5321	180	23	α(τ1	α(τ1	NOUN
ejpam-5321	180	24	,	,	PUNCT
ejpam-5321	180	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	180	26	multifunction	multifunction	NOUN
ejpam-5321	180	27	such	such	ADJ
ejpam-5321	180	28	that	that	SCONJ
ejpam-5321	180	29	f	f	PROPN
ejpam-5321	180	30	is	be	AUX
ejpam-5321	180	31	punctually	punctually	ADV
ejpam-5321	180	32	τ1τ2	τ1τ2	ADV
ejpam-5321	180	33	-	-	VERB
ejpam-5321	180	34	connected	connected	ADJ
ejpam-5321	180	35	.	.	PUNCT
ejpam-5321	181	1	if	if	SCONJ
ejpam-5321	181	2	(	(	PUNCT
ejpam-5321	181	3	x	x	NOUN
ejpam-5321	181	4	,	,	PUNCT
ejpam-5321	181	5	τ1	τ1	NOUN
ejpam-5321	181	6	,	,	PUNCT
ejpam-5321	181	7	τ2	τ2	NOUN
ejpam-5321	181	8	)	)	PUNCT
ejpam-5321	181	9	is	be	AUX
ejpam-5321	181	10	α(τ1	α(τ1	NOUN
ejpam-5321	181	11	,	,	PUNCT
ejpam-5321	181	12	τ2)-connected	τ2)-connecte	VERB
ejpam-5321	181	13	,	,	PUNCT
ejpam-5321	181	14	then	then	ADV
ejpam-5321	181	15	(	(	PUNCT
ejpam-5321	181	16	y	y	PROPN
ejpam-5321	181	17	,	,	PUNCT
ejpam-5321	181	18	σ1	σ1	PROPN
ejpam-5321	181	19	,	,	PUNCT
ejpam-5321	181	20	σ2	σ2	PROPN
ejpam-5321	181	21	)	)	PUNCT
ejpam-5321	181	22	is	be	AUX
ejpam-5321	181	23	σ1σ2	σ1σ2	NOUN
ejpam-5321	181	24	-	-	PUNCT
ejpam-5321	181	25	connected	connected	ADJ
ejpam-5321	181	26	.	.	PUNCT
ejpam-5321	182	1	proof	proof	NOUN
ejpam-5321	182	2	.	.	PUNCT
ejpam-5321	183	1	the	the	DET
ejpam-5321	183	2	proof	proof	NOUN
ejpam-5321	183	3	is	be	AUX
ejpam-5321	183	4	similar	similar	ADJ
ejpam-5321	183	5	to	to	ADP
ejpam-5321	183	6	that	that	PRON
ejpam-5321	183	7	of	of	ADP
ejpam-5321	183	8	theorem	theorem	ADJ
ejpam-5321	183	9	4	4	NUM
ejpam-5321	183	10	.	.	PUNCT
ejpam-5321	183	11	definition	definition	NOUN
ejpam-5321	183	12	9	9	NUM
ejpam-5321	183	13	.	.	PUNCT
ejpam-5321	184	1	a	a	DET
ejpam-5321	184	2	bitopological	bitopological	ADJ
ejpam-5321	184	3	space	space	NOUN
ejpam-5321	184	4	(	(	PUNCT
ejpam-5321	184	5	x	x	NOUN
ejpam-5321	184	6	,	,	PUNCT
ejpam-5321	184	7	τ1	τ1	NOUN
ejpam-5321	184	8	,	,	PUNCT
ejpam-5321	184	9	τ2	τ2	NOUN
ejpam-5321	184	10	)	)	PUNCT
ejpam-5321	184	11	is	be	AUX
ejpam-5321	184	12	called	call	VERB
ejpam-5321	184	13	strongly	strongly	ADV
ejpam-5321	184	14	(	(	PUNCT
ejpam-5321	184	15	τ1	τ1	NOUN
ejpam-5321	184	16	,	,	PUNCT
ejpam-5321	184	17	τ2)-normal	τ2)-normal	ADJ
ejpam-5321	184	18	if	if	SCONJ
ejpam-5321	184	19	,	,	PUNCT
ejpam-5321	184	20	for	for	ADP
ejpam-5321	184	21	any	any	DET
ejpam-5321	184	22	disjoint	disjoint	ADJ
ejpam-5321	184	23	τ1τ2	τ1τ2	ADJ
ejpam-5321	184	24	-	-	ADJ
ejpam-5321	184	25	closed	closed	ADJ
ejpam-5321	184	26	sets	set	NOUN
ejpam-5321	184	27	f	f	PROPN
ejpam-5321	184	28	and	and	CCONJ
ejpam-5321	184	29	k	k	PROPN
ejpam-5321	184	30	of	of	ADP
ejpam-5321	184	31	x	x	PRON
ejpam-5321	184	32	,	,	PUNCT
ejpam-5321	184	33	there	there	PRON
ejpam-5321	184	34	exist	exist	VERB
ejpam-5321	184	35	τ1τ2	τ1τ2	ADJ
ejpam-5321	184	36	-	-	ADJ
ejpam-5321	184	37	clopen	clopen	ADJ
ejpam-5321	184	38	sets	set	NOUN
ejpam-5321	184	39	u	u	NOUN
ejpam-5321	184	40	and	and	CCONJ
ejpam-5321	184	41	v	v	NOUN
ejpam-5321	184	42	of	of	ADP
ejpam-5321	184	43	x	x	PUNCT
ejpam-5321	184	44	such	such	ADJ
ejpam-5321	184	45	that	that	SCONJ
ejpam-5321	184	46	f	f	PROPN
ejpam-5321	184	47	⊆	⊆	NUM
ejpam-5321	184	48	u	u	NOUN
ejpam-5321	184	49	,	,	PUNCT
ejpam-5321	184	50	k	k	PROPN
ejpam-5321	184	51	⊆	⊆	NUM
ejpam-5321	184	52	v	v	NOUN
ejpam-5321	184	53	and	and	CCONJ
ejpam-5321	184	54	u	u	NOUN
ejpam-5321	184	55	∩	∩	NOUN
ejpam-5321	184	56	v	v	NOUN
ejpam-5321	184	57	=	=	PUNCT
ejpam-5321	184	58	∅.	∅.	NOUN
ejpam-5321	184	59	definition	definition	NOUN
ejpam-5321	184	60	10	10	NUM
ejpam-5321	184	61	.	.	PUNCT
ejpam-5321	185	1	a	a	DET
ejpam-5321	185	2	bitopological	bitopological	ADJ
ejpam-5321	185	3	space	space	NOUN
ejpam-5321	185	4	(	(	PUNCT
ejpam-5321	185	5	x	x	NOUN
ejpam-5321	185	6	,	,	PUNCT
ejpam-5321	185	7	τ1	τ1	NOUN
ejpam-5321	185	8	,	,	PUNCT
ejpam-5321	185	9	τ2	τ2	NOUN
ejpam-5321	185	10	)	)	PUNCT
ejpam-5321	185	11	is	be	AUX
ejpam-5321	185	12	called	call	VERB
ejpam-5321	185	13	α(τ1	α(τ1	NOUN
ejpam-5321	185	14	,	,	PUNCT
ejpam-5321	185	15	τ2)-hausdorff	τ2)-hausdorff	NOUN
ejpam-5321	185	16	if	if	SCONJ
ejpam-5321	185	17	,	,	PUNCT
ejpam-5321	185	18	for	for	ADP
ejpam-5321	185	19	each	each	DET
ejpam-5321	185	20	pair	pair	NOUN
ejpam-5321	185	21	of	of	ADP
ejpam-5321	185	22	distinct	distinct	ADJ
ejpam-5321	185	23	points	point	NOUN
ejpam-5321	185	24	x	x	PUNCT
ejpam-5321	185	25	and	and	CCONJ
ejpam-5321	185	26	y	y	PROPN
ejpam-5321	185	27	in	in	ADP
ejpam-5321	185	28	x	x	SYM
ejpam-5321	185	29	,	,	PUNCT
ejpam-5321	185	30	there	there	PRON
ejpam-5321	185	31	exist	exist	VERB
ejpam-5321	185	32	disjoint	disjoint	NOUN
ejpam-5321	185	33	α(τ1	α(τ1	NOUN
ejpam-5321	185	34	,	,	PUNCT
ejpam-5321	185	35	τ2)-open	τ2)-open	ADJ
ejpam-5321	185	36	sets	set	NOUN
ejpam-5321	185	37	u	u	NOUN
ejpam-5321	185	38	and	and	CCONJ
ejpam-5321	185	39	v	v	NOUN
ejpam-5321	185	40	of	of	ADP
ejpam-5321	185	41	x	x	PUNCT
ejpam-5321	185	42	such	such	ADJ
ejpam-5321	185	43	that	that	SCONJ
ejpam-5321	185	44	x	x	SYM
ejpam-5321	185	45	∈	∈	PROPN
ejpam-5321	185	46	u	u	NOUN
ejpam-5321	185	47	and	and	CCONJ
ejpam-5321	185	48	y	y	PROPN
ejpam-5321	185	49	∈	∈	PROPN
ejpam-5321	185	50	v	v	NOUN
ejpam-5321	185	51	.	.	PUNCT
ejpam-5321	186	1	definition	definition	NOUN
ejpam-5321	186	2	11	11	NUM
ejpam-5321	186	3	.	.	PUNCT
ejpam-5321	187	1	a	a	DET
ejpam-5321	187	2	multifunction	multifunction	NOUN
ejpam-5321	187	3	f	f	NOUN
ejpam-5321	187	4	:	:	PUNCT
ejpam-5321	187	5	(	(	PUNCT
ejpam-5321	187	6	x	x	NOUN
ejpam-5321	187	7	,	,	PUNCT
ejpam-5321	187	8	τ1	τ1	NOUN
ejpam-5321	187	9	,	,	PUNCT
ejpam-5321	187	10	τ2	τ2	NOUN
ejpam-5321	187	11	)	)	PUNCT
ejpam-5321	187	12	→	→	SYM
ejpam-5321	187	13	(	(	PUNCT
ejpam-5321	187	14	y	y	PROPN
ejpam-5321	187	15	,	,	PUNCT
ejpam-5321	187	16	σ1	σ1	PROPN
ejpam-5321	187	17	,	,	PUNCT
ejpam-5321	187	18	σ2	σ2	PROPN
ejpam-5321	187	19	)	)	PUNCT
ejpam-5321	187	20	is	be	AUX
ejpam-5321	187	21	called	call	VERB
ejpam-5321	187	22	punctually	punctually	ADV
ejpam-5321	187	23	(	(	PUNCT
ejpam-5321	187	24	τ1	τ1	NOUN
ejpam-5321	187	25	,	,	PUNCT
ejpam-5321	187	26	τ2)closed	τ2)close	VERB
ejpam-5321	187	27	if	if	SCONJ
ejpam-5321	187	28	,	,	PUNCT
ejpam-5321	187	29	for	for	ADP
ejpam-5321	187	30	each	each	DET
ejpam-5321	187	31	x	x	SYM
ejpam-5321	187	32	∈	∈	PROPN
ejpam-5321	187	33	x	x	X
ejpam-5321	187	34	,	,	PUNCT
ejpam-5321	187	35	f	f	PROPN
ejpam-5321	187	36	(	(	PUNCT
ejpam-5321	187	37	x	x	X
ejpam-5321	187	38	)	)	PUNCT
ejpam-5321	187	39	is	be	AUX
ejpam-5321	187	40	σ1σ2	σ1σ2	NOUN
ejpam-5321	187	41	-	-	PUNCT
ejpam-5321	187	42	closed	closed	ADJ
ejpam-5321	187	43	.	.	PUNCT
ejpam-5321	188	1	theorem	theorem	NOUN
ejpam-5321	188	2	6	6	NUM
ejpam-5321	188	3	.	.	PUNCT
ejpam-5321	189	1	let	let	VERB
ejpam-5321	189	2	f	f	NOUN
ejpam-5321	189	3	:	:	PUNCT
ejpam-5321	189	4	(	(	PUNCT
ejpam-5321	189	5	x	x	NOUN
ejpam-5321	189	6	,	,	PUNCT
ejpam-5321	189	7	τ1	τ1	NOUN
ejpam-5321	189	8	,	,	PUNCT
ejpam-5321	189	9	τ2	τ2	NOUN
ejpam-5321	189	10	)	)	PUNCT
ejpam-5321	189	11	→	→	SYM
ejpam-5321	189	12	(	(	PUNCT
ejpam-5321	189	13	y	y	PROPN
ejpam-5321	189	14	,	,	PUNCT
ejpam-5321	189	15	σ1	σ1	PROPN
ejpam-5321	189	16	,	,	PUNCT
ejpam-5321	189	17	σ2	σ2	PROPN
ejpam-5321	189	18	)	)	PUNCT
ejpam-5321	189	19	be	be	VERB
ejpam-5321	189	20	an	an	DET
ejpam-5321	189	21	upper	upper	ADJ
ejpam-5321	189	22	slightly	slightly	ADJ
ejpam-5321	189	23	α(τ1	α(τ1	NOUN
ejpam-5321	189	24	,	,	PUNCT
ejpam-5321	189	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	189	26	multifunction	multifunction	NOUN
ejpam-5321	189	27	and	and	CCONJ
ejpam-5321	189	28	punctually	punctually	ADJ
ejpam-5321	189	29	(	(	PUNCT
ejpam-5321	189	30	τ1	τ1	NOUN
ejpam-5321	189	31	,	,	PUNCT
ejpam-5321	189	32	τ2)-closed	τ2)-close	VERB
ejpam-5321	189	33	from	from	ADP
ejpam-5321	189	34	a	a	DET
ejpam-5321	189	35	bitopological	bitopological	ADJ
ejpam-5321	189	36	space	space	NOUN
ejpam-5321	189	37	(	(	PUNCT
ejpam-5321	189	38	x	x	NOUN
ejpam-5321	189	39	,	,	PUNCT
ejpam-5321	189	40	τ1	τ1	NOUN
ejpam-5321	189	41	,	,	PUNCT
ejpam-5321	189	42	τ2	τ2	NOUN
ejpam-5321	189	43	)	)	PUNCT
ejpam-5321	189	44	to	to	ADP
ejpam-5321	189	45	a	a	DET
ejpam-5321	189	46	strongly	strongly	ADV
ejpam-5321	189	47	(	(	PUNCT
ejpam-5321	189	48	σ1	σ1	PROPN
ejpam-5321	189	49	,	,	PUNCT
ejpam-5321	189	50	σ2)-normal	σ2)-normal	ADJ
ejpam-5321	189	51	bitopological	bitopological	ADJ
ejpam-5321	189	52	space	space	NOUN
ejpam-5321	189	53	(	(	PUNCT
ejpam-5321	189	54	y	y	PROPN
ejpam-5321	189	55	,	,	PUNCT
ejpam-5321	189	56	σ1	σ1	PROPN
ejpam-5321	189	57	,	,	PUNCT
ejpam-5321	189	58	σ2	σ2	NOUN
ejpam-5321	189	59	)	)	PUNCT
ejpam-5321	189	60	and	and	CCONJ
ejpam-5321	189	61	let	let	VERB
ejpam-5321	189	62	f	f	PROPN
ejpam-5321	189	63	(	(	PUNCT
ejpam-5321	189	64	x	x	NOUN
ejpam-5321	189	65	)	)	PUNCT
ejpam-5321	189	66	∩	∩	ADJ
ejpam-5321	189	67	f	f	PROPN
ejpam-5321	189	68	(	(	PUNCT
ejpam-5321	189	69	y	y	NOUN
ejpam-5321	189	70	)	)	PUNCT
ejpam-5321	189	71	=	=	NOUN
ejpam-5321	189	72	∅	∅	NOUN
ejpam-5321	189	73	for	for	ADP
ejpam-5321	189	74	each	each	DET
ejpam-5321	189	75	pair	pair	NOUN
ejpam-5321	189	76	of	of	ADP
ejpam-5321	189	77	distinct	distinct	ADJ
ejpam-5321	189	78	points	point	NOUN
ejpam-5321	189	79	x	x	X
ejpam-5321	189	80	,	,	PUNCT
ejpam-5321	189	81	y	y	PROPN
ejpam-5321	189	82	∈	∈	PROPN
ejpam-5321	189	83	x.	x.	NOUN
ejpam-5321	189	84	then	then	ADV
ejpam-5321	189	85	,	,	PUNCT
ejpam-5321	189	86	(	(	PUNCT
ejpam-5321	189	87	x	x	NOUN
ejpam-5321	189	88	,	,	PUNCT
ejpam-5321	189	89	τ1	τ1	NOUN
ejpam-5321	189	90	,	,	PUNCT
ejpam-5321	189	91	τ2	τ2	NOUN
ejpam-5321	189	92	)	)	PUNCT
ejpam-5321	189	93	is	be	AUX
ejpam-5321	189	94	an	an	DET
ejpam-5321	189	95	α(τ1	α(τ1	NOUN
ejpam-5321	189	96	,	,	PUNCT
ejpam-5321	189	97	τ2)-hausdorff	τ2)-hausdorff	ADJ
ejpam-5321	189	98	space	space	NOUN
ejpam-5321	189	99	.	.	PUNCT
ejpam-5321	190	1	proof	proof	NOUN
ejpam-5321	190	2	.	.	PUNCT
ejpam-5321	191	1	let	let	VERB
ejpam-5321	191	2	x	x	PRON
ejpam-5321	191	3	and	and	CCONJ
ejpam-5321	191	4	y	y	PROPN
ejpam-5321	191	5	be	be	AUX
ejpam-5321	191	6	any	any	DET
ejpam-5321	191	7	two	two	NUM
ejpam-5321	191	8	distinct	distinct	ADJ
ejpam-5321	191	9	points	point	NOUN
ejpam-5321	191	10	in	in	ADP
ejpam-5321	191	11	x.	x.	NOUN
ejpam-5321	191	12	then	then	ADV
ejpam-5321	191	13	,	,	PUNCT
ejpam-5321	191	14	we	we	PRON
ejpam-5321	191	15	have	have	VERB
ejpam-5321	191	16	f	f	PROPN
ejpam-5321	191	17	(	(	PUNCT
ejpam-5321	191	18	x	x	NOUN
ejpam-5321	191	19	)	)	PUNCT
ejpam-5321	191	20	∩	∩	ADJ
ejpam-5321	191	21	f	f	PROPN
ejpam-5321	191	22	(	(	PUNCT
ejpam-5321	191	23	y	y	NOUN
ejpam-5321	191	24	)	)	PUNCT
ejpam-5321	191	25	=	=	PUNCT
ejpam-5321	191	26	∅.	∅.	NOUN
ejpam-5321	191	27	since	since	SCONJ
ejpam-5321	191	28	(	(	PUNCT
ejpam-5321	191	29	y	y	PROPN
ejpam-5321	191	30	,	,	PUNCT
ejpam-5321	191	31	σ1	σ1	PROPN
ejpam-5321	191	32	,	,	PUNCT
ejpam-5321	191	33	σ2	σ2	PROPN
ejpam-5321	191	34	)	)	PUNCT
ejpam-5321	191	35	is	be	AUX
ejpam-5321	191	36	strongly	strongly	ADV
ejpam-5321	191	37	(	(	PUNCT
ejpam-5321	191	38	σ1	σ1	PROPN
ejpam-5321	191	39	,	,	PUNCT
ejpam-5321	191	40	σ2)-normal	σ2)-normal	PROPN
ejpam-5321	191	41	,	,	PUNCT
ejpam-5321	191	42	it	it	PRON
ejpam-5321	191	43	follows	follow	VERB
ejpam-5321	191	44	that	that	SCONJ
ejpam-5321	191	45	there	there	PRON
ejpam-5321	191	46	exist	exist	VERB
ejpam-5321	191	47	disjoint	disjoint	ADJ
ejpam-5321	191	48	σ1σ2	σ1σ2	ADJ
ejpam-5321	191	49	-	-	PUNCT
ejpam-5321	191	50	clopen	clopen	ADJ
ejpam-5321	191	51	sets	set	NOUN
ejpam-5321	191	52	u	u	NOUN
ejpam-5321	191	53	and	and	CCONJ
ejpam-5321	191	54	v	v	NOUN
ejpam-5321	191	55	of	of	ADP
ejpam-5321	191	56	y	y	PROPN
ejpam-5321	191	57	containing	contain	VERB
ejpam-5321	191	58	f	f	PROPN
ejpam-5321	191	59	(	(	PUNCT
ejpam-5321	191	60	x	x	NOUN
ejpam-5321	191	61	)	)	PUNCT
ejpam-5321	191	62	and	and	CCONJ
ejpam-5321	191	63	f	f	PROPN
ejpam-5321	191	64	(	(	PUNCT
ejpam-5321	191	65	y	y	NOUN
ejpam-5321	191	66	)	)	PUNCT
ejpam-5321	191	67	,	,	PUNCT
ejpam-5321	191	68	respectively	respectively	ADV
ejpam-5321	191	69	.	.	PUNCT
ejpam-5321	192	1	thus	thus	ADV
ejpam-5321	192	2	,	,	PUNCT
ejpam-5321	192	3	f+(u	f+(u	NUM
ejpam-5321	192	4	)	)	PUNCT
ejpam-5321	192	5	and	and	CCONJ
ejpam-5321	192	6	f+(v	f+(v	NUM
ejpam-5321	192	7	)	)	PUNCT
ejpam-5321	192	8	are	be	AUX
ejpam-5321	192	9	disjoint	disjoint	NOUN
ejpam-5321	192	10	α(τ1	α(τ1	NOUN
ejpam-5321	192	11	,	,	PUNCT
ejpam-5321	192	12	τ2)-open	τ2)-open	ADJ
ejpam-5321	192	13	sets	set	NOUN
ejpam-5321	192	14	of	of	ADP
ejpam-5321	192	15	x	x	PUNCT
ejpam-5321	192	16	containing	contain	VERB
ejpam-5321	192	17	x	x	PROPN
ejpam-5321	192	18	and	and	CCONJ
ejpam-5321	192	19	y	y	PROPN
ejpam-5321	192	20	,	,	PUNCT
ejpam-5321	192	21	respectively	respectively	ADV
ejpam-5321	192	22	.	.	PUNCT
ejpam-5321	193	1	this	this	PRON
ejpam-5321	193	2	shows	show	VERB
ejpam-5321	193	3	that	that	SCONJ
ejpam-5321	193	4	(	(	PUNCT
ejpam-5321	193	5	x	x	NOUN
ejpam-5321	193	6	,	,	PUNCT
ejpam-5321	193	7	τ1	τ1	NOUN
ejpam-5321	193	8	,	,	PUNCT
ejpam-5321	193	9	τ2	τ2	NOUN
ejpam-5321	193	10	)	)	PUNCT
ejpam-5321	193	11	is	be	AUX
ejpam-5321	193	12	an	an	DET
ejpam-5321	193	13	α(τ1	α(τ1	NOUN
ejpam-5321	193	14	,	,	PUNCT
ejpam-5321	193	15	τ2)-hausdorff	τ2)-hausdorff	ADJ
ejpam-5321	193	16	space	space	NOUN
ejpam-5321	193	17	.	.	PUNCT
ejpam-5321	194	1	c.	c.	PROPN
ejpam-5321	194	2	viriyapong	viriyapong	PROPN
ejpam-5321	194	3	,	,	PUNCT
ejpam-5321	194	4	s.	s.	PROPN
ejpam-5321	194	5	sompong	sompong	PROPN
ejpam-5321	194	6	,	,	PUNCT
ejpam-5321	194	7	c.	c.	PROPN
ejpam-5321	194	8	boonpok	boonpok	PROPN
ejpam-5321	194	9	/	/	SYM
ejpam-5321	194	10	eur	eur	PROPN
ejpam-5321	194	11	.	.	PUNCT
ejpam-5321	195	1	j.	j.	PROPN
ejpam-5321	195	2	pure	pure	PROPN
ejpam-5321	195	3	appl	appl	PROPN
ejpam-5321	195	4	.	.	PROPN
ejpam-5321	195	5	math	math	PROPN
ejpam-5321	195	6	,	,	PUNCT
ejpam-5321	195	7	17	17	NUM
ejpam-5321	195	8	(	(	PUNCT
ejpam-5321	195	9	3	3	NUM
ejpam-5321	195	10	)	)	PUNCT
ejpam-5321	195	11	(	(	PUNCT
ejpam-5321	195	12	2024	2024	NUM
ejpam-5321	195	13	)	)	PUNCT
ejpam-5321	195	14	,	,	PUNCT
ejpam-5321	195	15	2142	2142	NUM
ejpam-5321	195	16	-	-	SYM
ejpam-5321	195	17	2154	2154	NUM
ejpam-5321	195	18	2149	2149	NUM
ejpam-5321	195	19	4	4	NUM
ejpam-5321	195	20	.	.	PUNCT
ejpam-5321	195	21	slight	slight	PROPN
ejpam-5321	195	22	α(τ1	α(τ1	NOUN
ejpam-5321	195	23	,	,	PUNCT
ejpam-5321	195	24	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5321	195	25	and	and	CCONJ
ejpam-5321	195	26	other	other	ADJ
ejpam-5321	195	27	forms	form	NOUN
ejpam-5321	195	28	of	of	ADP
ejpam-5321	195	29	α(τ1	α(τ1	NOUN
ejpam-5321	195	30	,	,	PUNCT
ejpam-5321	195	31	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5321	195	32	we	we	PRON
ejpam-5321	195	33	begin	begin	VERB
ejpam-5321	195	34	this	this	DET
ejpam-5321	195	35	section	section	NOUN
ejpam-5321	195	36	by	by	ADP
ejpam-5321	195	37	introducing	introduce	VERB
ejpam-5321	195	38	the	the	DET
ejpam-5321	195	39	concept	concept	NOUN
ejpam-5321	195	40	of	of	ADP
ejpam-5321	195	41	upper	upper	ADJ
ejpam-5321	195	42	α(τ1	α(τ1	NOUN
ejpam-5321	195	43	,	,	PUNCT
ejpam-5321	195	44	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	195	45	multifunctions	multifunction	NOUN
ejpam-5321	195	46	.	.	PUNCT
ejpam-5321	196	1	definition	definition	NOUN
ejpam-5321	196	2	12	12	NUM
ejpam-5321	196	3	.	.	PUNCT
ejpam-5321	197	1	a	a	DET
ejpam-5321	197	2	multifunction	multifunction	NOUN
ejpam-5321	197	3	f	f	NOUN
ejpam-5321	197	4	:	:	PUNCT
ejpam-5321	197	5	(	(	PUNCT
ejpam-5321	197	6	x	x	NOUN
ejpam-5321	197	7	,	,	PUNCT
ejpam-5321	197	8	τ1	τ1	NOUN
ejpam-5321	197	9	,	,	PUNCT
ejpam-5321	197	10	τ2	τ2	NOUN
ejpam-5321	197	11	)	)	PUNCT
ejpam-5321	197	12	→	→	SYM
ejpam-5321	197	13	(	(	PUNCT
ejpam-5321	197	14	y	y	PROPN
ejpam-5321	197	15	,	,	PUNCT
ejpam-5321	197	16	σ1	σ1	PROPN
ejpam-5321	197	17	,	,	PUNCT
ejpam-5321	197	18	σ2	σ2	PROPN
ejpam-5321	197	19	)	)	PUNCT
ejpam-5321	197	20	is	be	AUX
ejpam-5321	197	21	said	say	VERB
ejpam-5321	197	22	to	to	PART
ejpam-5321	197	23	be	be	AUX
ejpam-5321	197	24	upper	upper	ADJ
ejpam-5321	197	25	α(τ1	α(τ1	NOUN
ejpam-5321	197	26	,	,	PUNCT
ejpam-5321	197	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	197	28	at	at	ADP
ejpam-5321	197	29	a	a	DET
ejpam-5321	197	30	point	point	NOUN
ejpam-5321	197	31	x	x	SYM
ejpam-5321	197	32	∈	∈	NOUN
ejpam-5321	197	33	x	x	PUNCT
ejpam-5321	197	34	if	if	SCONJ
ejpam-5321	197	35	for	for	ADP
ejpam-5321	197	36	each	each	DET
ejpam-5321	197	37	σ1σ2	σ1σ2	VERB
ejpam-5321	197	38	-	-	ADJ
ejpam-5321	197	39	open	open	ADJ
ejpam-5321	197	40	set	set	NOUN
ejpam-5321	197	41	v	v	NOUN
ejpam-5321	197	42	of	of	ADP
ejpam-5321	197	43	y	y	PROPN
ejpam-5321	197	44	containing	contain	VERB
ejpam-5321	197	45	f	f	PROPN
ejpam-5321	197	46	(	(	PUNCT
ejpam-5321	197	47	x	x	NOUN
ejpam-5321	197	48	)	)	PUNCT
ejpam-5321	197	49	,	,	PUNCT
ejpam-5321	197	50	there	there	PRON
ejpam-5321	197	51	exists	exist	VERB
ejpam-5321	197	52	an	an	DET
ejpam-5321	197	53	α(τ1	α(τ1	NOUN
ejpam-5321	197	54	,	,	PUNCT
ejpam-5321	197	55	τ2)-open	τ2)-open	ADJ
ejpam-5321	197	56	set	set	VERB
ejpam-5321	197	57	u	u	NOUN
ejpam-5321	197	58	of	of	ADP
ejpam-5321	197	59	x	x	PUNCT
ejpam-5321	197	60	containing	contain	VERB
ejpam-5321	197	61	x	x	PUNCT
ejpam-5321	197	62	such	such	ADJ
ejpam-5321	197	63	that	that	SCONJ
ejpam-5321	197	64	f	f	PROPN
ejpam-5321	197	65	(	(	PUNCT
ejpam-5321	197	66	u	u	NOUN
ejpam-5321	197	67	)	)	PUNCT
ejpam-5321	197	68	⊆	⊆	NUM
ejpam-5321	197	69	v	v	NOUN
ejpam-5321	197	70	.	.	PUNCT
ejpam-5321	198	1	a	a	DET
ejpam-5321	198	2	multifunction	multifunction	NOUN
ejpam-5321	198	3	f	f	NOUN
ejpam-5321	198	4	:	:	PUNCT
ejpam-5321	198	5	(	(	PUNCT
ejpam-5321	198	6	x	x	NOUN
ejpam-5321	198	7	,	,	PUNCT
ejpam-5321	198	8	τ1	τ1	NOUN
ejpam-5321	198	9	,	,	PUNCT
ejpam-5321	198	10	τ2	τ2	NOUN
ejpam-5321	198	11	)	)	PUNCT
ejpam-5321	198	12	→	→	SYM
ejpam-5321	198	13	(	(	PUNCT
ejpam-5321	198	14	y	y	PROPN
ejpam-5321	198	15	,	,	PUNCT
ejpam-5321	198	16	σ1	σ1	PROPN
ejpam-5321	198	17	,	,	PUNCT
ejpam-5321	198	18	σ2	σ2	PROPN
ejpam-5321	198	19	)	)	PUNCT
ejpam-5321	198	20	is	be	AUX
ejpam-5321	198	21	said	say	VERB
ejpam-5321	198	22	to	to	PART
ejpam-5321	198	23	be	be	AUX
ejpam-5321	198	24	upper	upper	ADJ
ejpam-5321	198	25	α(τ1	α(τ1	NOUN
ejpam-5321	198	26	,	,	PUNCT
ejpam-5321	198	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	198	28	if	if	SCONJ
ejpam-5321	198	29	f	f	PROPN
ejpam-5321	198	30	has	have	AUX
ejpam-5321	198	31	this	this	DET
ejpam-5321	198	32	property	property	NOUN
ejpam-5321	198	33	at	at	ADP
ejpam-5321	198	34	each	each	DET
ejpam-5321	198	35	point	point	NOUN
ejpam-5321	198	36	of	of	ADP
ejpam-5321	198	37	x.	x.	NOUN
ejpam-5321	198	38	theorem	theorem	VERB
ejpam-5321	198	39	7	7	NUM
ejpam-5321	198	40	.	.	PUNCT
ejpam-5321	199	1	if	if	SCONJ
ejpam-5321	199	2	a	a	DET
ejpam-5321	199	3	multifunction	multifunction	NOUN
ejpam-5321	199	4	f	f	NOUN
ejpam-5321	199	5	:	:	PUNCT
ejpam-5321	199	6	(	(	PUNCT
ejpam-5321	199	7	x	x	NOUN
ejpam-5321	199	8	,	,	PUNCT
ejpam-5321	199	9	τ1	τ1	NOUN
ejpam-5321	199	10	,	,	PUNCT
ejpam-5321	199	11	τ2	τ2	NOUN
ejpam-5321	199	12	)	)	PUNCT
ejpam-5321	199	13	→	→	SYM
ejpam-5321	199	14	(	(	PUNCT
ejpam-5321	199	15	y	y	PROPN
ejpam-5321	199	16	,	,	PUNCT
ejpam-5321	199	17	σ1	σ1	PROPN
ejpam-5321	199	18	,	,	PUNCT
ejpam-5321	199	19	σ2	σ2	PROPN
ejpam-5321	199	20	)	)	PUNCT
ejpam-5321	199	21	is	be	AUX
ejpam-5321	199	22	upper	upper	ADJ
ejpam-5321	199	23	α(τ1	α(τ1	NOUN
ejpam-5321	199	24	,	,	PUNCT
ejpam-5321	199	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	199	26	,	,	PUNCT
ejpam-5321	199	27	then	then	ADV
ejpam-5321	199	28	f	f	PROPN
ejpam-5321	199	29	is	be	AUX
ejpam-5321	199	30	upper	upper	ADJ
ejpam-5321	199	31	slightly	slightly	ADV
ejpam-5321	199	32	α(τ1	α(τ1	NOUN
ejpam-5321	199	33	,	,	PUNCT
ejpam-5321	199	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	199	35	.	.	PUNCT
ejpam-5321	200	1	proof	proof	NOUN
ejpam-5321	200	2	.	.	PUNCT
ejpam-5321	201	1	let	let	VERB
ejpam-5321	201	2	x	x	PUNCT
ejpam-5321	201	3	∈	∈	PROPN
ejpam-5321	201	4	x	x	X
ejpam-5321	201	5	and	and	CCONJ
ejpam-5321	201	6	v	v	X
ejpam-5321	201	7	be	be	AUX
ejpam-5321	201	8	any	any	DET
ejpam-5321	201	9	σ1σ2	σ1σ2	NOUN
ejpam-5321	201	10	-	-	PUNCT
ejpam-5321	201	11	clopen	clopen	ADJ
ejpam-5321	201	12	set	set	NOUN
ejpam-5321	201	13	of	of	ADP
ejpam-5321	201	14	y	y	PROPN
ejpam-5321	201	15	containing	contain	VERB
ejpam-5321	201	16	f	f	PROPN
ejpam-5321	201	17	(	(	PUNCT
ejpam-5321	201	18	x	x	NOUN
ejpam-5321	201	19	)	)	PUNCT
ejpam-5321	201	20	.	.	PUNCT
ejpam-5321	202	1	since	since	SCONJ
ejpam-5321	202	2	f	f	PROPN
ejpam-5321	202	3	is	be	AUX
ejpam-5321	202	4	upper	upper	ADJ
ejpam-5321	202	5	α(τ1	α(τ1	NOUN
ejpam-5321	202	6	,	,	PUNCT
ejpam-5321	202	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	202	8	,	,	PUNCT
ejpam-5321	202	9	there	there	PRON
ejpam-5321	202	10	exists	exist	VERB
ejpam-5321	202	11	an	an	DET
ejpam-5321	202	12	α(τ1	α(τ1	NOUN
ejpam-5321	202	13	,	,	PUNCT
ejpam-5321	202	14	τ2)-open	τ2)-open	ADJ
ejpam-5321	202	15	set	set	NOUN
ejpam-5321	202	16	of	of	ADP
ejpam-5321	202	17	x	x	PUNCT
ejpam-5321	202	18	containing	contain	VERB
ejpam-5321	202	19	x	x	PUNCT
ejpam-5321	202	20	such	such	ADJ
ejpam-5321	202	21	that	that	SCONJ
ejpam-5321	202	22	f	f	PROPN
ejpam-5321	202	23	(	(	PUNCT
ejpam-5321	202	24	u	u	NOUN
ejpam-5321	202	25	)	)	PUNCT
ejpam-5321	202	26	⊆	⊆	NUM
ejpam-5321	202	27	v	v	NOUN
ejpam-5321	202	28	.	.	PUNCT
ejpam-5321	203	1	this	this	PRON
ejpam-5321	203	2	shows	show	VERB
ejpam-5321	203	3	that	that	SCONJ
ejpam-5321	203	4	f	f	PROPN
ejpam-5321	203	5	is	be	AUX
ejpam-5321	203	6	upper	upper	ADJ
ejpam-5321	203	7	slightly	slightly	ADV
ejpam-5321	203	8	α(τ1	α(τ1	NOUN
ejpam-5321	203	9	,	,	PUNCT
ejpam-5321	203	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	203	11	.	.	PUNCT
ejpam-5321	204	1	definition	definition	NOUN
ejpam-5321	204	2	13	13	NUM
ejpam-5321	204	3	.	.	PUNCT
ejpam-5321	205	1	a	a	DET
ejpam-5321	205	2	multifunction	multifunction	NOUN
ejpam-5321	205	3	f	f	NOUN
ejpam-5321	205	4	:	:	PUNCT
ejpam-5321	205	5	(	(	PUNCT
ejpam-5321	205	6	x	x	NOUN
ejpam-5321	205	7	,	,	PUNCT
ejpam-5321	205	8	τ1	τ1	NOUN
ejpam-5321	205	9	,	,	PUNCT
ejpam-5321	205	10	τ2	τ2	NOUN
ejpam-5321	205	11	)	)	PUNCT
ejpam-5321	205	12	→	→	SYM
ejpam-5321	205	13	(	(	PUNCT
ejpam-5321	205	14	y	y	PROPN
ejpam-5321	205	15	,	,	PUNCT
ejpam-5321	205	16	σ1	σ1	PROPN
ejpam-5321	205	17	,	,	PUNCT
ejpam-5321	205	18	σ2	σ2	PROPN
ejpam-5321	205	19	)	)	PUNCT
ejpam-5321	205	20	is	be	AUX
ejpam-5321	205	21	called	call	VERB
ejpam-5321	205	22	lower	low	ADJ
ejpam-5321	205	23	α(τ1	α(τ1	NOUN
ejpam-5321	205	24	,	,	PUNCT
ejpam-5321	205	25	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	205	26	at	at	ADP
ejpam-5321	205	27	a	a	DET
ejpam-5321	205	28	point	point	NOUN
ejpam-5321	205	29	x	x	SYM
ejpam-5321	205	30	∈	∈	NOUN
ejpam-5321	205	31	x	x	PUNCT
ejpam-5321	205	32	if	if	SCONJ
ejpam-5321	205	33	for	for	ADP
ejpam-5321	205	34	each	each	DET
ejpam-5321	205	35	σ1σ2	σ1σ2	VERB
ejpam-5321	205	36	-	-	ADJ
ejpam-5321	205	37	open	open	ADJ
ejpam-5321	205	38	set	set	NOUN
ejpam-5321	205	39	v	v	NOUN
ejpam-5321	205	40	of	of	ADP
ejpam-5321	205	41	y	y	PRON
ejpam-5321	205	42	such	such	ADJ
ejpam-5321	205	43	that	that	SCONJ
ejpam-5321	205	44	f	f	PROPN
ejpam-5321	205	45	(	(	PUNCT
ejpam-5321	205	46	x	x	NOUN
ejpam-5321	205	47	)	)	PUNCT
ejpam-5321	205	48	∩	∩	NOUN
ejpam-5321	205	49	v	v	ADP
ejpam-5321	205	50	̸=	̸=	PROPN
ejpam-5321	205	51	∅	∅	NOUN
ejpam-5321	205	52	,	,	PUNCT
ejpam-5321	205	53	there	there	PRON
ejpam-5321	205	54	exists	exist	VERB
ejpam-5321	205	55	an	an	DET
ejpam-5321	205	56	α(τ1	α(τ1	NOUN
ejpam-5321	205	57	,	,	PUNCT
ejpam-5321	205	58	τ2)-open	τ2)-open	ADJ
ejpam-5321	205	59	set	set	VERB
ejpam-5321	205	60	u	u	NOUN
ejpam-5321	205	61	of	of	ADP
ejpam-5321	205	62	x	x	PUNCT
ejpam-5321	205	63	containing	contain	VERB
ejpam-5321	205	64	x	x	PUNCT
ejpam-5321	205	65	such	such	ADJ
ejpam-5321	205	66	that	that	SCONJ
ejpam-5321	205	67	f	f	PROPN
ejpam-5321	205	68	(	(	PUNCT
ejpam-5321	205	69	z	z	NOUN
ejpam-5321	205	70	)	)	PUNCT
ejpam-5321	205	71	∩	∩	NOUN
ejpam-5321	205	72	v	v	ADP
ejpam-5321	205	73	̸=	̸=	PROPN
ejpam-5321	205	74	∅	∅	NOUN
ejpam-5321	205	75	for	for	ADP
ejpam-5321	205	76	every	every	DET
ejpam-5321	205	77	z	z	NOUN
ejpam-5321	205	78	∈	∈	PROPN
ejpam-5321	205	79	u	u	NOUN
ejpam-5321	205	80	.	.	PUNCT
ejpam-5321	206	1	a	a	DET
ejpam-5321	206	2	multifunction	multifunction	NOUN
ejpam-5321	206	3	f	f	NOUN
ejpam-5321	206	4	:	:	PUNCT
ejpam-5321	206	5	(	(	PUNCT
ejpam-5321	206	6	x	x	NOUN
ejpam-5321	206	7	,	,	PUNCT
ejpam-5321	206	8	τ1	τ1	NOUN
ejpam-5321	206	9	,	,	PUNCT
ejpam-5321	206	10	τ2	τ2	NOUN
ejpam-5321	206	11	)	)	PUNCT
ejpam-5321	206	12	→	→	SYM
ejpam-5321	206	13	(	(	PUNCT
ejpam-5321	206	14	y	y	PROPN
ejpam-5321	206	15	,	,	PUNCT
ejpam-5321	206	16	σ1	σ1	PROPN
ejpam-5321	206	17	,	,	PUNCT
ejpam-5321	206	18	σ2	σ2	PROPN
ejpam-5321	206	19	)	)	PUNCT
ejpam-5321	206	20	is	be	AUX
ejpam-5321	206	21	called	call	VERB
ejpam-5321	206	22	lower	low	ADJ
ejpam-5321	206	23	α(τ1	α(τ1	NOUN
ejpam-5321	206	24	,	,	PUNCT
ejpam-5321	206	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	206	26	if	if	SCONJ
ejpam-5321	206	27	f	f	PROPN
ejpam-5321	206	28	has	have	VERB
ejpam-5321	206	29	this	this	DET
ejpam-5321	206	30	property	property	NOUN
ejpam-5321	206	31	at	at	ADP
ejpam-5321	206	32	each	each	DET
ejpam-5321	206	33	point	point	NOUN
ejpam-5321	206	34	of	of	ADP
ejpam-5321	206	35	x.	x.	NOUN
ejpam-5321	206	36	theorem	theorem	VERB
ejpam-5321	206	37	8	8	NUM
ejpam-5321	206	38	.	.	PUNCT
ejpam-5321	207	1	if	if	SCONJ
ejpam-5321	207	2	a	a	DET
ejpam-5321	207	3	multifunction	multifunction	NOUN
ejpam-5321	207	4	f	f	NOUN
ejpam-5321	207	5	:	:	PUNCT
ejpam-5321	207	6	(	(	PUNCT
ejpam-5321	207	7	x	x	NOUN
ejpam-5321	207	8	,	,	PUNCT
ejpam-5321	207	9	τ1	τ1	NOUN
ejpam-5321	207	10	,	,	PUNCT
ejpam-5321	207	11	τ2	τ2	NOUN
ejpam-5321	207	12	)	)	PUNCT
ejpam-5321	207	13	→	→	SYM
ejpam-5321	207	14	(	(	PUNCT
ejpam-5321	207	15	y	y	PROPN
ejpam-5321	207	16	,	,	PUNCT
ejpam-5321	207	17	σ1	σ1	PROPN
ejpam-5321	207	18	,	,	PUNCT
ejpam-5321	207	19	σ2	σ2	NOUN
ejpam-5321	207	20	)	)	PUNCT
ejpam-5321	207	21	is	be	AUX
ejpam-5321	207	22	lower	low	ADJ
ejpam-5321	207	23	α(τ1	α(τ1	NOUN
ejpam-5321	207	24	,	,	PUNCT
ejpam-5321	207	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	207	26	,	,	PUNCT
ejpam-5321	207	27	then	then	ADV
ejpam-5321	207	28	f	f	PROPN
ejpam-5321	207	29	is	be	AUX
ejpam-5321	207	30	lower	low	ADJ
ejpam-5321	207	31	slightly	slightly	ADV
ejpam-5321	207	32	α(τ1	α(τ1	NOUN
ejpam-5321	207	33	,	,	PUNCT
ejpam-5321	207	34	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	207	35	.	.	PUNCT
ejpam-5321	208	1	proof	proof	NOUN
ejpam-5321	208	2	.	.	PUNCT
ejpam-5321	209	1	the	the	DET
ejpam-5321	209	2	proof	proof	NOUN
ejpam-5321	209	3	is	be	AUX
ejpam-5321	209	4	similar	similar	ADJ
ejpam-5321	209	5	to	to	ADP
ejpam-5321	209	6	that	that	PRON
ejpam-5321	209	7	of	of	ADP
ejpam-5321	209	8	theorem	theorem	ADJ
ejpam-5321	209	9	7	7	NUM
ejpam-5321	209	10	.	.	NOUN
ejpam-5321	209	11	recall	recall	VERB
ejpam-5321	209	12	that	that	SCONJ
ejpam-5321	209	13	a	a	DET
ejpam-5321	209	14	bitopological	bitopological	ADJ
ejpam-5321	209	15	space	space	NOUN
ejpam-5321	209	16	(	(	PUNCT
ejpam-5321	209	17	x	x	NOUN
ejpam-5321	209	18	,	,	PUNCT
ejpam-5321	209	19	τ1	τ1	NOUN
ejpam-5321	209	20	,	,	PUNCT
ejpam-5321	209	21	τ2	τ2	NOUN
ejpam-5321	209	22	)	)	PUNCT
ejpam-5321	209	23	is	be	AUX
ejpam-5321	209	24	said	say	VERB
ejpam-5321	209	25	to	to	PART
ejpam-5321	209	26	be	be	AUX
ejpam-5321	209	27	(	(	PUNCT
ejpam-5321	209	28	τ1	τ1	NOUN
ejpam-5321	209	29	,	,	PUNCT
ejpam-5321	209	30	τ2)-extremally	τ2)-extremally	ADV
ejpam-5321	209	31	disconnected	disconnect	VERB
ejpam-5321	209	32	[	[	X
ejpam-5321	209	33	45	45	NUM
ejpam-5321	209	34	]	]	PUNCT
ejpam-5321	209	35	if	if	SCONJ
ejpam-5321	209	36	the	the	DET
ejpam-5321	209	37	τ1τ2	τ1τ2	NOUN
ejpam-5321	209	38	-	-	NOUN
ejpam-5321	209	39	closure	closure	NOUN
ejpam-5321	209	40	of	of	ADP
ejpam-5321	209	41	every	every	DET
ejpam-5321	209	42	τ1τ2	τ1τ2	NOUN
ejpam-5321	209	43	-	-	ADJ
ejpam-5321	209	44	open	open	ADJ
ejpam-5321	209	45	set	set	ADJ
ejpam-5321	209	46	u	u	NOUN
ejpam-5321	209	47	of	of	ADP
ejpam-5321	209	48	x	x	SYM
ejpam-5321	209	49	is	be	AUX
ejpam-5321	209	50	τ1τ2	τ1τ2	VERB
ejpam-5321	209	51	-	-	ADJ
ejpam-5321	209	52	open	open	ADJ
ejpam-5321	209	53	.	.	PUNCT
ejpam-5321	210	1	lemma	lemma	PROPN
ejpam-5321	210	2	3	3	X
ejpam-5321	210	3	.	.	PUNCT
ejpam-5321	211	1	[	[	X
ejpam-5321	211	2	45	45	NUM
ejpam-5321	211	3	]	]	PUNCT
ejpam-5321	211	4	for	for	ADP
ejpam-5321	211	5	a	a	DET
ejpam-5321	211	6	bitopological	bitopological	ADJ
ejpam-5321	211	7	space	space	NOUN
ejpam-5321	211	8	(	(	PUNCT
ejpam-5321	211	9	x	x	NOUN
ejpam-5321	211	10	,	,	PUNCT
ejpam-5321	211	11	τ1	τ1	NOUN
ejpam-5321	211	12	,	,	PUNCT
ejpam-5321	211	13	τ2	τ2	NOUN
ejpam-5321	211	14	)	)	PUNCT
ejpam-5321	211	15	,	,	PUNCT
ejpam-5321	211	16	the	the	DET
ejpam-5321	211	17	following	follow	VERB
ejpam-5321	211	18	properties	property	NOUN
ejpam-5321	211	19	are	be	AUX
ejpam-5321	211	20	equivalent	equivalent	ADJ
ejpam-5321	211	21	:	:	PUNCT
ejpam-5321	211	22	(	(	PUNCT
ejpam-5321	211	23	1	1	X
ejpam-5321	211	24	)	)	PUNCT
ejpam-5321	211	25	(	(	PUNCT
ejpam-5321	211	26	x	x	NOUN
ejpam-5321	211	27	,	,	PUNCT
ejpam-5321	211	28	τ1	τ1	NOUN
ejpam-5321	211	29	,	,	PUNCT
ejpam-5321	211	30	τ2	τ2	NOUN
ejpam-5321	211	31	)	)	PUNCT
ejpam-5321	211	32	is	be	AUX
ejpam-5321	211	33	(	(	PUNCT
ejpam-5321	211	34	τ1	τ1	NOUN
ejpam-5321	211	35	,	,	PUNCT
ejpam-5321	211	36	τ2)-extremally	τ2)-extremally	ADV
ejpam-5321	211	37	disconnected	disconnect	VERB
ejpam-5321	211	38	.	.	PUNCT
ejpam-5321	212	1	(	(	PUNCT
ejpam-5321	212	2	2	2	X
ejpam-5321	212	3	)	)	PUNCT
ejpam-5321	212	4	every	every	DET
ejpam-5321	212	5	(	(	PUNCT
ejpam-5321	212	6	τ1	τ1	NOUN
ejpam-5321	212	7	,	,	PUNCT
ejpam-5321	212	8	τ2)r	τ2)r	ADJ
ejpam-5321	212	9	-	-	PUNCT
ejpam-5321	212	10	open	open	ADJ
ejpam-5321	212	11	set	set	NOUN
ejpam-5321	212	12	of	of	ADP
ejpam-5321	212	13	x	x	PUNCT
ejpam-5321	212	14	is	be	AUX
ejpam-5321	212	15	τ1τ2	τ1τ2	VERB
ejpam-5321	212	16	-	-	ADJ
ejpam-5321	212	17	closed	closed	ADJ
ejpam-5321	212	18	.	.	PUNCT
ejpam-5321	213	1	(	(	PUNCT
ejpam-5321	213	2	3	3	X
ejpam-5321	213	3	)	)	PUNCT
ejpam-5321	213	4	every	every	DET
ejpam-5321	213	5	(	(	PUNCT
ejpam-5321	213	6	τ1	τ1	NOUN
ejpam-5321	213	7	,	,	PUNCT
ejpam-5321	213	8	τ2)r	τ2)r	NOUN
ejpam-5321	213	9	-	-	PUNCT
ejpam-5321	213	10	closed	close	VERB
ejpam-5321	213	11	set	set	NOUN
ejpam-5321	213	12	of	of	ADP
ejpam-5321	213	13	x	x	PUNCT
ejpam-5321	213	14	is	be	AUX
ejpam-5321	213	15	τ1τ2	τ1τ2	VERB
ejpam-5321	213	16	-	-	ADJ
ejpam-5321	213	17	open	open	ADJ
ejpam-5321	213	18	.	.	PUNCT
ejpam-5321	214	1	definition	definition	NOUN
ejpam-5321	214	2	14	14	NUM
ejpam-5321	214	3	.	.	PUNCT
ejpam-5321	215	1	a	a	DET
ejpam-5321	215	2	multifunction	multifunction	NOUN
ejpam-5321	215	3	f	f	NOUN
ejpam-5321	215	4	:	:	PUNCT
ejpam-5321	215	5	(	(	PUNCT
ejpam-5321	215	6	x	x	NOUN
ejpam-5321	215	7	,	,	PUNCT
ejpam-5321	215	8	τ1	τ1	NOUN
ejpam-5321	215	9	,	,	PUNCT
ejpam-5321	215	10	τ2	τ2	NOUN
ejpam-5321	215	11	)	)	PUNCT
ejpam-5321	215	12	→	→	SYM
ejpam-5321	215	13	(	(	PUNCT
ejpam-5321	215	14	y	y	PROPN
ejpam-5321	215	15	,	,	PUNCT
ejpam-5321	215	16	σ1	σ1	PROPN
ejpam-5321	215	17	,	,	PUNCT
ejpam-5321	215	18	σ2	σ2	PROPN
ejpam-5321	215	19	)	)	PUNCT
ejpam-5321	215	20	is	be	AUX
ejpam-5321	215	21	said	say	VERB
ejpam-5321	215	22	to	to	PART
ejpam-5321	215	23	be	be	AUX
ejpam-5321	215	24	upper	upper	ADJ
ejpam-5321	215	25	almost	almost	ADV
ejpam-5321	215	26	α(τ1	α(τ1	NOUN
ejpam-5321	215	27	,	,	PUNCT
ejpam-5321	215	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	215	29	at	at	ADP
ejpam-5321	215	30	a	a	DET
ejpam-5321	215	31	point	point	NOUN
ejpam-5321	215	32	x	x	SYM
ejpam-5321	215	33	∈	∈	NOUN
ejpam-5321	215	34	x	x	PUNCT
ejpam-5321	215	35	if	if	SCONJ
ejpam-5321	215	36	for	for	ADP
ejpam-5321	215	37	each	each	DET
ejpam-5321	215	38	σ1σ2	σ1σ2	VERB
ejpam-5321	215	39	-	-	ADJ
ejpam-5321	215	40	open	open	ADJ
ejpam-5321	215	41	set	set	NOUN
ejpam-5321	215	42	v	v	NOUN
ejpam-5321	215	43	of	of	ADP
ejpam-5321	215	44	y	y	PROPN
ejpam-5321	215	45	containing	contain	VERB
ejpam-5321	215	46	f	f	PROPN
ejpam-5321	215	47	(	(	PUNCT
ejpam-5321	215	48	x	x	NOUN
ejpam-5321	215	49	)	)	PUNCT
ejpam-5321	215	50	,	,	PUNCT
ejpam-5321	215	51	there	there	PRON
ejpam-5321	215	52	exists	exist	VERB
ejpam-5321	215	53	an	an	DET
ejpam-5321	215	54	α(τ1	α(τ1	NOUN
ejpam-5321	215	55	,	,	PUNCT
ejpam-5321	215	56	τ2)-open	τ2)-open	ADJ
ejpam-5321	215	57	set	set	VERB
ejpam-5321	215	58	u	u	NOUN
ejpam-5321	215	59	of	of	ADP
ejpam-5321	215	60	x	x	PUNCT
ejpam-5321	215	61	containing	contain	VERB
ejpam-5321	215	62	x	x	PUNCT
ejpam-5321	215	63	such	such	ADJ
ejpam-5321	215	64	that	that	SCONJ
ejpam-5321	215	65	f	f	PROPN
ejpam-5321	215	66	(	(	PUNCT
ejpam-5321	215	67	u	u	NOUN
ejpam-5321	215	68	)	)	PUNCT
ejpam-5321	215	69	⊆	⊆	NUM
ejpam-5321	215	70	σ1σ2	σ1σ2	X
ejpam-5321	215	71	-	-	PUNCT
ejpam-5321	215	72	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5321	215	73	-	-	PUNCT
ejpam-5321	215	74	cl(v	cl(v	NOUN
ejpam-5321	215	75	)	)	PUNCT
ejpam-5321	215	76	)	)	PUNCT
ejpam-5321	215	77	.	.	PUNCT
ejpam-5321	216	1	a	a	DET
ejpam-5321	216	2	multifunction	multifunction	NOUN
ejpam-5321	216	3	f	f	NOUN
ejpam-5321	216	4	:	:	PUNCT
ejpam-5321	216	5	(	(	PUNCT
ejpam-5321	216	6	x	x	NOUN
ejpam-5321	216	7	,	,	PUNCT
ejpam-5321	216	8	τ1	τ1	NOUN
ejpam-5321	216	9	,	,	PUNCT
ejpam-5321	216	10	τ2	τ2	NOUN
ejpam-5321	216	11	)	)	PUNCT
ejpam-5321	216	12	→	→	SYM
ejpam-5321	216	13	(	(	PUNCT
ejpam-5321	216	14	y	y	PROPN
ejpam-5321	216	15	,	,	PUNCT
ejpam-5321	216	16	σ1	σ1	PROPN
ejpam-5321	216	17	,	,	PUNCT
ejpam-5321	216	18	σ2	σ2	PROPN
ejpam-5321	216	19	)	)	PUNCT
ejpam-5321	216	20	is	be	AUX
ejpam-5321	216	21	said	say	VERB
ejpam-5321	216	22	to	to	PART
ejpam-5321	216	23	be	be	AUX
ejpam-5321	216	24	upper	upper	ADJ
ejpam-5321	216	25	almost	almost	ADV
ejpam-5321	216	26	α(τ1	α(τ1	NOUN
ejpam-5321	216	27	,	,	PUNCT
ejpam-5321	216	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	216	29	if	if	SCONJ
ejpam-5321	216	30	f	f	PROPN
ejpam-5321	216	31	has	have	VERB
ejpam-5321	216	32	this	this	DET
ejpam-5321	216	33	property	property	NOUN
ejpam-5321	216	34	at	at	ADP
ejpam-5321	216	35	each	each	DET
ejpam-5321	216	36	point	point	NOUN
ejpam-5321	216	37	of	of	ADP
ejpam-5321	216	38	x.	x.	PROPN
ejpam-5321	216	39	c.	c.	PROPN
ejpam-5321	216	40	viriyapong	viriyapong	PROPN
ejpam-5321	216	41	,	,	PUNCT
ejpam-5321	216	42	s.	s.	PROPN
ejpam-5321	216	43	sompong	sompong	PROPN
ejpam-5321	216	44	,	,	PUNCT
ejpam-5321	216	45	c.	c.	PROPN
ejpam-5321	216	46	boonpok	boonpok	PROPN
ejpam-5321	216	47	/	/	SYM
ejpam-5321	216	48	eur	eur	PROPN
ejpam-5321	216	49	.	.	PUNCT
ejpam-5321	217	1	j.	j.	PROPN
ejpam-5321	217	2	pure	pure	PROPN
ejpam-5321	217	3	appl	appl	PROPN
ejpam-5321	217	4	.	.	PROPN
ejpam-5321	217	5	math	math	PROPN
ejpam-5321	217	6	,	,	PUNCT
ejpam-5321	217	7	17	17	NUM
ejpam-5321	217	8	(	(	PUNCT
ejpam-5321	217	9	3	3	NUM
ejpam-5321	217	10	)	)	PUNCT
ejpam-5321	217	11	(	(	PUNCT
ejpam-5321	217	12	2024	2024	NUM
ejpam-5321	217	13	)	)	PUNCT
ejpam-5321	217	14	,	,	PUNCT
ejpam-5321	217	15	2142	2142	NUM
ejpam-5321	217	16	-	-	SYM
ejpam-5321	217	17	2154	2154	NUM
ejpam-5321	217	18	2150	2150	NUM
ejpam-5321	217	19	lemma	lemma	PROPN
ejpam-5321	217	20	4	4	NUM
ejpam-5321	217	21	.	.	PUNCT
ejpam-5321	217	22	for	for	ADP
ejpam-5321	217	23	a	a	DET
ejpam-5321	217	24	multifunction	multifunction	NOUN
ejpam-5321	217	25	f	f	NOUN
ejpam-5321	217	26	:	:	PUNCT
ejpam-5321	217	27	(	(	PUNCT
ejpam-5321	217	28	x	x	NOUN
ejpam-5321	217	29	,	,	PUNCT
ejpam-5321	217	30	τ1	τ1	NOUN
ejpam-5321	217	31	,	,	PUNCT
ejpam-5321	217	32	τ2	τ2	NOUN
ejpam-5321	217	33	)	)	PUNCT
ejpam-5321	217	34	→	→	SYM
ejpam-5321	217	35	(	(	PUNCT
ejpam-5321	217	36	y	y	PROPN
ejpam-5321	217	37	,	,	PUNCT
ejpam-5321	217	38	σ1	σ1	PROPN
ejpam-5321	217	39	,	,	PUNCT
ejpam-5321	217	40	σ2	σ2	NOUN
ejpam-5321	217	41	)	)	PUNCT
ejpam-5321	217	42	,	,	PUNCT
ejpam-5321	217	43	the	the	DET
ejpam-5321	217	44	following	follow	VERB
ejpam-5321	217	45	properties	property	NOUN
ejpam-5321	217	46	are	be	AUX
ejpam-5321	217	47	equivalent	equivalent	ADJ
ejpam-5321	217	48	:	:	PUNCT
ejpam-5321	217	49	(	(	PUNCT
ejpam-5321	217	50	1	1	X
ejpam-5321	217	51	)	)	PUNCT
ejpam-5321	217	52	f	f	PROPN
ejpam-5321	217	53	is	be	AUX
ejpam-5321	217	54	upper	upper	ADJ
ejpam-5321	217	55	almost	almost	ADV
ejpam-5321	217	56	α(τ1	α(τ1	NOUN
ejpam-5321	217	57	,	,	PUNCT
ejpam-5321	217	58	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	217	59	;	;	PUNCT
ejpam-5321	217	60	(	(	PUNCT
ejpam-5321	217	61	2	2	X
ejpam-5321	217	62	)	)	PUNCT
ejpam-5321	217	63	for	for	ADP
ejpam-5321	217	64	each	each	DET
ejpam-5321	217	65	x	x	SYM
ejpam-5321	217	66	∈	∈	PROPN
ejpam-5321	217	67	x	x	X
ejpam-5321	217	68	and	and	CCONJ
ejpam-5321	217	69	each	each	DET
ejpam-5321	217	70	(	(	PUNCT
ejpam-5321	217	71	σ1	σ1	PROPN
ejpam-5321	217	72	,	,	PUNCT
ejpam-5321	217	73	σ2)r	σ2)r	NOUN
ejpam-5321	217	74	-	-	PUNCT
ejpam-5321	217	75	open	open	ADJ
ejpam-5321	217	76	set	set	VERB
ejpam-5321	217	77	v	v	NOUN
ejpam-5321	217	78	of	of	ADP
ejpam-5321	217	79	y	y	PROPN
ejpam-5321	217	80	containing	contain	VERB
ejpam-5321	217	81	f	f	PROPN
ejpam-5321	217	82	(	(	PUNCT
ejpam-5321	217	83	x	x	NOUN
ejpam-5321	217	84	)	)	PUNCT
ejpam-5321	217	85	,	,	PUNCT
ejpam-5321	217	86	there	there	PRON
ejpam-5321	217	87	exists	exist	VERB
ejpam-5321	217	88	an	an	DET
ejpam-5321	217	89	α(τ1	α(τ1	NOUN
ejpam-5321	217	90	,	,	PUNCT
ejpam-5321	217	91	τ2)-open	τ2)-open	ADJ
ejpam-5321	217	92	set	set	NOUN
ejpam-5321	217	93	of	of	ADP
ejpam-5321	217	94	x	x	PUNCT
ejpam-5321	217	95	containing	contain	VERB
ejpam-5321	217	96	x	x	PUNCT
ejpam-5321	217	97	such	such	ADJ
ejpam-5321	217	98	that	that	SCONJ
ejpam-5321	217	99	f	f	PROPN
ejpam-5321	217	100	(	(	PUNCT
ejpam-5321	217	101	u	u	NOUN
ejpam-5321	217	102	)	)	PUNCT
ejpam-5321	217	103	⊆	⊆	NUM
ejpam-5321	217	104	v	v	NOUN
ejpam-5321	217	105	.	.	PUNCT
ejpam-5321	218	1	theorem	theorem	NOUN
ejpam-5321	218	2	9	9	NUM
ejpam-5321	218	3	.	.	PUNCT
ejpam-5321	219	1	if	if	SCONJ
ejpam-5321	219	2	a	a	DET
ejpam-5321	219	3	multifunction	multifunction	NOUN
ejpam-5321	219	4	f	f	NOUN
ejpam-5321	219	5	:	:	PUNCT
ejpam-5321	219	6	(	(	PUNCT
ejpam-5321	219	7	x	x	NOUN
ejpam-5321	219	8	,	,	PUNCT
ejpam-5321	219	9	τ1	τ1	NOUN
ejpam-5321	219	10	,	,	PUNCT
ejpam-5321	219	11	τ2	τ2	NOUN
ejpam-5321	219	12	)	)	PUNCT
ejpam-5321	219	13	→	→	SYM
ejpam-5321	219	14	(	(	PUNCT
ejpam-5321	219	15	y	y	PROPN
ejpam-5321	219	16	,	,	PUNCT
ejpam-5321	219	17	σ1	σ1	PROPN
ejpam-5321	219	18	,	,	PUNCT
ejpam-5321	219	19	σ2	σ2	PROPN
ejpam-5321	219	20	)	)	PUNCT
ejpam-5321	219	21	is	be	AUX
ejpam-5321	219	22	upper	upper	ADJ
ejpam-5321	219	23	slightly	slightly	ADV
ejpam-5321	219	24	α(τ1	α(τ1	NOUN
ejpam-5321	219	25	,	,	PUNCT
ejpam-5321	219	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	219	27	and	and	CCONJ
ejpam-5321	219	28	(	(	PUNCT
ejpam-5321	219	29	y	y	PROPN
ejpam-5321	219	30	,	,	PUNCT
ejpam-5321	219	31	σ1	σ1	PROPN
ejpam-5321	219	32	,	,	PUNCT
ejpam-5321	219	33	σ2	σ2	PROPN
ejpam-5321	219	34	)	)	PUNCT
ejpam-5321	219	35	is	be	AUX
ejpam-5321	219	36	(	(	PUNCT
ejpam-5321	219	37	σ1	σ1	NOUN
ejpam-5321	219	38	,	,	PUNCT
ejpam-5321	219	39	σ2)-extremally	σ2)-extremally	ADV
ejpam-5321	219	40	disconnected	disconnect	VERB
ejpam-5321	219	41	,	,	PUNCT
ejpam-5321	219	42	then	then	ADV
ejpam-5321	219	43	f	f	PROPN
ejpam-5321	219	44	is	be	AUX
ejpam-5321	219	45	upper	upper	ADJ
ejpam-5321	219	46	almost	almost	ADV
ejpam-5321	219	47	α(τ1	α(τ1	NOUN
ejpam-5321	219	48	,	,	PUNCT
ejpam-5321	219	49	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	219	50	.	.	PUNCT
ejpam-5321	220	1	proof	proof	NOUN
ejpam-5321	220	2	.	.	PUNCT
ejpam-5321	221	1	let	let	VERB
ejpam-5321	221	2	x	x	PUNCT
ejpam-5321	221	3	∈	∈	PROPN
ejpam-5321	221	4	x	x	X
ejpam-5321	221	5	and	and	CCONJ
ejpam-5321	221	6	v	v	X
ejpam-5321	221	7	be	be	AUX
ejpam-5321	221	8	any	any	DET
ejpam-5321	221	9	(	(	PUNCT
ejpam-5321	221	10	σ1	σ1	NOUN
ejpam-5321	221	11	,	,	PUNCT
ejpam-5321	221	12	σ2)r	σ2)r	NOUN
ejpam-5321	221	13	-	-	PUNCT
ejpam-5321	221	14	open	open	ADJ
ejpam-5321	221	15	set	set	NOUN
ejpam-5321	221	16	of	of	ADP
ejpam-5321	221	17	y	y	PROPN
ejpam-5321	221	18	containing	contain	VERB
ejpam-5321	221	19	f	f	PROPN
ejpam-5321	221	20	(	(	PUNCT
ejpam-5321	221	21	x	x	NOUN
ejpam-5321	221	22	)	)	PUNCT
ejpam-5321	221	23	.	.	PUNCT
ejpam-5321	222	1	then	then	ADV
ejpam-5321	222	2	,	,	PUNCT
ejpam-5321	222	3	by	by	ADP
ejpam-5321	222	4	lemma	lemma	PROPN
ejpam-5321	222	5	3	3	NUM
ejpam-5321	222	6	we	we	PRON
ejpam-5321	222	7	have	have	VERB
ejpam-5321	222	8	v	v	NOUN
ejpam-5321	222	9	is	be	AUX
ejpam-5321	222	10	σ1σ2	σ1σ2	NOUN
ejpam-5321	222	11	-	-	PUNCT
ejpam-5321	222	12	clopen	clopen	ADJ
ejpam-5321	222	13	in	in	ADP
ejpam-5321	222	14	y	y	PROPN
ejpam-5321	222	15	.	.	PUNCT
ejpam-5321	223	1	since	since	SCONJ
ejpam-5321	223	2	f	f	PROPN
ejpam-5321	223	3	is	be	AUX
ejpam-5321	223	4	upper	upper	ADJ
ejpam-5321	223	5	slightly	slightly	ADV
ejpam-5321	223	6	α(τ1	α(τ1	NOUN
ejpam-5321	223	7	,	,	PUNCT
ejpam-5321	223	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	223	9	,	,	PUNCT
ejpam-5321	223	10	there	there	PRON
ejpam-5321	223	11	exists	exist	VERB
ejpam-5321	223	12	an	an	DET
ejpam-5321	223	13	α(τ1	α(τ1	NOUN
ejpam-5321	223	14	,	,	PUNCT
ejpam-5321	223	15	τ2)-open	τ2)-open	ADJ
ejpam-5321	223	16	set	set	NOUN
ejpam-5321	223	17	of	of	ADP
ejpam-5321	223	18	x	x	PUNCT
ejpam-5321	223	19	containing	contain	VERB
ejpam-5321	223	20	x	x	PUNCT
ejpam-5321	223	21	such	such	ADJ
ejpam-5321	223	22	that	that	SCONJ
ejpam-5321	223	23	f	f	PROPN
ejpam-5321	223	24	(	(	PUNCT
ejpam-5321	223	25	u	u	NOUN
ejpam-5321	223	26	)	)	PUNCT
ejpam-5321	223	27	⊆	⊆	NUM
ejpam-5321	223	28	v	v	NOUN
ejpam-5321	223	29	.	.	PUNCT
ejpam-5321	224	1	by	by	ADP
ejpam-5321	224	2	lemma	lemma	PROPN
ejpam-5321	224	3	4	4	NUM
ejpam-5321	224	4	,	,	PUNCT
ejpam-5321	224	5	f	f	PROPN
ejpam-5321	224	6	is	be	AUX
ejpam-5321	224	7	upper	upper	ADJ
ejpam-5321	224	8	almost	almost	ADV
ejpam-5321	224	9	α(τ1	α(τ1	NOUN
ejpam-5321	224	10	,	,	PUNCT
ejpam-5321	224	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	224	12	.	.	PUNCT
ejpam-5321	225	1	definition	definition	NOUN
ejpam-5321	225	2	15	15	NUM
ejpam-5321	225	3	.	.	PUNCT
ejpam-5321	226	1	a	a	DET
ejpam-5321	226	2	multifunction	multifunction	NOUN
ejpam-5321	226	3	f	f	NOUN
ejpam-5321	226	4	:	:	PUNCT
ejpam-5321	226	5	(	(	PUNCT
ejpam-5321	226	6	x	x	NOUN
ejpam-5321	226	7	,	,	PUNCT
ejpam-5321	226	8	τ1	τ1	NOUN
ejpam-5321	226	9	,	,	PUNCT
ejpam-5321	226	10	τ2	τ2	NOUN
ejpam-5321	226	11	)	)	PUNCT
ejpam-5321	226	12	→	→	SYM
ejpam-5321	226	13	(	(	PUNCT
ejpam-5321	226	14	y	y	PROPN
ejpam-5321	226	15	,	,	PUNCT
ejpam-5321	226	16	σ1	σ1	PROPN
ejpam-5321	226	17	,	,	PUNCT
ejpam-5321	226	18	σ2	σ2	PROPN
ejpam-5321	226	19	)	)	PUNCT
ejpam-5321	226	20	is	be	AUX
ejpam-5321	226	21	called	call	VERB
ejpam-5321	226	22	lower	low	ADJ
ejpam-5321	226	23	almost	almost	ADV
ejpam-5321	226	24	α(τ1	α(τ1	NOUN
ejpam-5321	226	25	,	,	PUNCT
ejpam-5321	226	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	226	27	at	at	ADP
ejpam-5321	226	28	a	a	DET
ejpam-5321	226	29	point	point	NOUN
ejpam-5321	226	30	x	x	SYM
ejpam-5321	226	31	∈	∈	NOUN
ejpam-5321	226	32	x	x	PUNCT
ejpam-5321	226	33	if	if	SCONJ
ejpam-5321	226	34	for	for	ADP
ejpam-5321	226	35	each	each	DET
ejpam-5321	226	36	σ1σ2	σ1σ2	VERB
ejpam-5321	226	37	-	-	ADJ
ejpam-5321	226	38	open	open	ADJ
ejpam-5321	226	39	set	set	NOUN
ejpam-5321	226	40	v	v	NOUN
ejpam-5321	226	41	of	of	ADP
ejpam-5321	226	42	y	y	PRON
ejpam-5321	226	43	such	such	ADJ
ejpam-5321	226	44	that	that	SCONJ
ejpam-5321	226	45	f	f	PROPN
ejpam-5321	226	46	(	(	PUNCT
ejpam-5321	226	47	x	x	NOUN
ejpam-5321	226	48	)	)	PUNCT
ejpam-5321	226	49	∩	∩	NOUN
ejpam-5321	226	50	v	v	ADP
ejpam-5321	226	51	̸=	̸=	PROPN
ejpam-5321	226	52	∅	∅	NOUN
ejpam-5321	226	53	,	,	PUNCT
ejpam-5321	226	54	there	there	PRON
ejpam-5321	226	55	exists	exist	VERB
ejpam-5321	226	56	an	an	DET
ejpam-5321	226	57	α(τ1	α(τ1	NOUN
ejpam-5321	226	58	,	,	PUNCT
ejpam-5321	226	59	τ2)-open	τ2)-open	ADJ
ejpam-5321	226	60	set	set	VERB
ejpam-5321	226	61	u	u	NOUN
ejpam-5321	226	62	of	of	ADP
ejpam-5321	226	63	x	x	PUNCT
ejpam-5321	226	64	containing	contain	VERB
ejpam-5321	226	65	x	x	PUNCT
ejpam-5321	226	66	such	such	ADJ
ejpam-5321	226	67	that	that	SCONJ
ejpam-5321	226	68	σ1σ2	σ1σ2	ADV
ejpam-5321	226	69	-	-	PUNCT
ejpam-5321	226	70	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5321	226	71	-	-	PUNCT
ejpam-5321	226	72	cl(v	cl(v	NOUN
ejpam-5321	226	73	)	)	PUNCT
ejpam-5321	226	74	)	)	PUNCT
ejpam-5321	227	1	∩	∩	PROPN
ejpam-5321	227	2	f	f	X
ejpam-5321	227	3	(	(	PUNCT
ejpam-5321	227	4	z	z	NOUN
ejpam-5321	227	5	)	)	PUNCT
ejpam-5321	227	6	̸=	̸=	NOUN
ejpam-5321	227	7	∅	∅	NOUN
ejpam-5321	227	8	for	for	ADP
ejpam-5321	227	9	every	every	DET
ejpam-5321	227	10	z	z	NOUN
ejpam-5321	227	11	∈	∈	PROPN
ejpam-5321	227	12	u	u	NOUN
ejpam-5321	227	13	.	.	PUNCT
ejpam-5321	228	1	a	a	DET
ejpam-5321	228	2	multifunction	multifunction	NOUN
ejpam-5321	228	3	f	f	NOUN
ejpam-5321	228	4	:	:	PUNCT
ejpam-5321	228	5	(	(	PUNCT
ejpam-5321	228	6	x	x	NOUN
ejpam-5321	228	7	,	,	PUNCT
ejpam-5321	228	8	τ1	τ1	NOUN
ejpam-5321	228	9	,	,	PUNCT
ejpam-5321	228	10	τ2	τ2	NOUN
ejpam-5321	228	11	)	)	PUNCT
ejpam-5321	228	12	→	→	SYM
ejpam-5321	228	13	(	(	PUNCT
ejpam-5321	228	14	y	y	PROPN
ejpam-5321	228	15	,	,	PUNCT
ejpam-5321	228	16	σ1	σ1	PROPN
ejpam-5321	228	17	,	,	PUNCT
ejpam-5321	228	18	σ2	σ2	PROPN
ejpam-5321	228	19	)	)	PUNCT
ejpam-5321	228	20	is	be	AUX
ejpam-5321	228	21	called	call	VERB
ejpam-5321	228	22	lower	low	ADJ
ejpam-5321	228	23	almost	almost	ADV
ejpam-5321	228	24	α(τ1	α(τ1	NOUN
ejpam-5321	228	25	,	,	PUNCT
ejpam-5321	228	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	228	27	if	if	SCONJ
ejpam-5321	228	28	f	f	PROPN
ejpam-5321	228	29	has	have	VERB
ejpam-5321	228	30	this	this	DET
ejpam-5321	228	31	property	property	NOUN
ejpam-5321	228	32	at	at	ADP
ejpam-5321	228	33	each	each	DET
ejpam-5321	228	34	point	point	NOUN
ejpam-5321	228	35	of	of	ADP
ejpam-5321	228	36	x.	x.	PROPN
ejpam-5321	228	37	lemma	lemma	PROPN
ejpam-5321	228	38	5	5	NUM
ejpam-5321	228	39	.	.	PUNCT
ejpam-5321	228	40	for	for	ADP
ejpam-5321	228	41	a	a	DET
ejpam-5321	228	42	multifunction	multifunction	NOUN
ejpam-5321	228	43	f	f	NOUN
ejpam-5321	228	44	:	:	PUNCT
ejpam-5321	228	45	(	(	PUNCT
ejpam-5321	228	46	x	x	NOUN
ejpam-5321	228	47	,	,	PUNCT
ejpam-5321	228	48	τ1	τ1	NOUN
ejpam-5321	228	49	,	,	PUNCT
ejpam-5321	228	50	τ2	τ2	NOUN
ejpam-5321	228	51	)	)	PUNCT
ejpam-5321	228	52	→	→	SYM
ejpam-5321	228	53	(	(	PUNCT
ejpam-5321	228	54	y	y	PROPN
ejpam-5321	228	55	,	,	PUNCT
ejpam-5321	228	56	σ1	σ1	PROPN
ejpam-5321	228	57	,	,	PUNCT
ejpam-5321	228	58	σ2	σ2	NOUN
ejpam-5321	228	59	)	)	PUNCT
ejpam-5321	228	60	,	,	PUNCT
ejpam-5321	228	61	the	the	DET
ejpam-5321	228	62	following	follow	VERB
ejpam-5321	228	63	properties	property	NOUN
ejpam-5321	228	64	are	be	AUX
ejpam-5321	228	65	equivalent	equivalent	ADJ
ejpam-5321	228	66	:	:	PUNCT
ejpam-5321	228	67	(	(	PUNCT
ejpam-5321	228	68	1	1	X
ejpam-5321	228	69	)	)	PUNCT
ejpam-5321	228	70	f	f	PROPN
ejpam-5321	228	71	is	be	AUX
ejpam-5321	228	72	lower	low	ADJ
ejpam-5321	228	73	almost	almost	ADV
ejpam-5321	228	74	α(τ1	α(τ1	NOUN
ejpam-5321	228	75	,	,	PUNCT
ejpam-5321	228	76	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	228	77	;	;	PUNCT
ejpam-5321	228	78	(	(	PUNCT
ejpam-5321	228	79	2	2	X
ejpam-5321	228	80	)	)	PUNCT
ejpam-5321	228	81	for	for	ADP
ejpam-5321	228	82	each	each	DET
ejpam-5321	228	83	x	x	SYM
ejpam-5321	228	84	∈	∈	PROPN
ejpam-5321	228	85	x	x	X
ejpam-5321	228	86	and	and	CCONJ
ejpam-5321	228	87	each	each	DET
ejpam-5321	228	88	(	(	PUNCT
ejpam-5321	228	89	σ1	σ1	PROPN
ejpam-5321	228	90	,	,	PUNCT
ejpam-5321	228	91	σ2)r	σ2)r	NOUN
ejpam-5321	228	92	-	-	PUNCT
ejpam-5321	228	93	open	open	ADJ
ejpam-5321	228	94	set	set	VERB
ejpam-5321	228	95	v	v	NOUN
ejpam-5321	228	96	of	of	ADP
ejpam-5321	228	97	y	y	PRON
ejpam-5321	228	98	such	such	ADJ
ejpam-5321	228	99	that	that	SCONJ
ejpam-5321	228	100	f	f	PROPN
ejpam-5321	228	101	(	(	PUNCT
ejpam-5321	228	102	x	x	NOUN
ejpam-5321	228	103	)	)	PUNCT
ejpam-5321	228	104	∩	∩	NOUN
ejpam-5321	228	105	v	v	ADP
ejpam-5321	228	106	̸=	̸=	PROPN
ejpam-5321	228	107	∅	∅	NOUN
ejpam-5321	228	108	,	,	PUNCT
ejpam-5321	228	109	there	there	PRON
ejpam-5321	228	110	exists	exist	VERB
ejpam-5321	228	111	an	an	DET
ejpam-5321	228	112	α(τ1	α(τ1	NOUN
ejpam-5321	228	113	,	,	PUNCT
ejpam-5321	228	114	τ2)-open	τ2)-open	ADJ
ejpam-5321	228	115	set	set	NOUN
ejpam-5321	228	116	of	of	ADP
ejpam-5321	228	117	x	x	PUNCT
ejpam-5321	228	118	containing	contain	VERB
ejpam-5321	228	119	x	x	PUNCT
ejpam-5321	228	120	such	such	ADJ
ejpam-5321	228	121	that	that	SCONJ
ejpam-5321	228	122	u	u	NOUN
ejpam-5321	228	123	⊆	⊆	NUM
ejpam-5321	228	124	f−(v	f−(v	NOUN
ejpam-5321	228	125	)	)	PUNCT
ejpam-5321	228	126	.	.	PUNCT
ejpam-5321	229	1	theorem	theorem	VERB
ejpam-5321	229	2	10	10	NUM
ejpam-5321	229	3	.	.	PUNCT
ejpam-5321	230	1	if	if	SCONJ
ejpam-5321	230	2	a	a	DET
ejpam-5321	230	3	multifunction	multifunction	NOUN
ejpam-5321	230	4	f	f	NOUN
ejpam-5321	230	5	:	:	PUNCT
ejpam-5321	230	6	(	(	PUNCT
ejpam-5321	230	7	x	x	NOUN
ejpam-5321	230	8	,	,	PUNCT
ejpam-5321	230	9	τ1	τ1	NOUN
ejpam-5321	230	10	,	,	PUNCT
ejpam-5321	230	11	τ2	τ2	NOUN
ejpam-5321	230	12	)	)	PUNCT
ejpam-5321	230	13	→	→	SYM
ejpam-5321	230	14	(	(	PUNCT
ejpam-5321	230	15	y	y	PROPN
ejpam-5321	230	16	,	,	PUNCT
ejpam-5321	230	17	σ1	σ1	PROPN
ejpam-5321	230	18	,	,	PUNCT
ejpam-5321	230	19	σ2	σ2	NOUN
ejpam-5321	230	20	)	)	PUNCT
ejpam-5321	230	21	is	be	AUX
ejpam-5321	230	22	lower	low	ADJ
ejpam-5321	230	23	slightly	slightly	ADV
ejpam-5321	230	24	α(τ1	α(τ1	NOUN
ejpam-5321	230	25	,	,	PUNCT
ejpam-5321	230	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	230	27	and	and	CCONJ
ejpam-5321	230	28	(	(	PUNCT
ejpam-5321	230	29	y	y	PROPN
ejpam-5321	230	30	,	,	PUNCT
ejpam-5321	230	31	σ1	σ1	PROPN
ejpam-5321	230	32	,	,	PUNCT
ejpam-5321	230	33	σ2	σ2	PROPN
ejpam-5321	230	34	)	)	PUNCT
ejpam-5321	230	35	is	be	AUX
ejpam-5321	230	36	(	(	PUNCT
ejpam-5321	230	37	σ1	σ1	NOUN
ejpam-5321	230	38	,	,	PUNCT
ejpam-5321	230	39	σ2)-extremally	σ2)-extremally	ADV
ejpam-5321	230	40	disconnected	disconnect	VERB
ejpam-5321	230	41	,	,	PUNCT
ejpam-5321	230	42	then	then	ADV
ejpam-5321	230	43	f	f	PROPN
ejpam-5321	230	44	is	be	AUX
ejpam-5321	230	45	lower	low	ADJ
ejpam-5321	230	46	almost	almost	ADV
ejpam-5321	230	47	α(τ1	α(τ1	NOUN
ejpam-5321	230	48	,	,	PUNCT
ejpam-5321	230	49	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	230	50	.	.	PUNCT
ejpam-5321	231	1	proof	proof	NOUN
ejpam-5321	231	2	.	.	PUNCT
ejpam-5321	232	1	by	by	ADP
ejpam-5321	232	2	utilizing	utilize	VERB
ejpam-5321	232	3	lemma	lemma	PROPN
ejpam-5321	232	4	5	5	NUM
ejpam-5321	232	5	,	,	PUNCT
ejpam-5321	232	6	this	this	PRON
ejpam-5321	232	7	can	can	AUX
ejpam-5321	232	8	be	be	AUX
ejpam-5321	232	9	proved	prove	VERB
ejpam-5321	232	10	similarly	similarly	ADV
ejpam-5321	232	11	to	to	ADP
ejpam-5321	232	12	that	that	PRON
ejpam-5321	232	13	of	of	ADP
ejpam-5321	232	14	theorem	theorem	ADJ
ejpam-5321	232	15	9	9	NUM
ejpam-5321	232	16	.	.	PUNCT
ejpam-5321	232	17	definition	definition	NOUN
ejpam-5321	232	18	16	16	NUM
ejpam-5321	232	19	.	.	PUNCT
ejpam-5321	233	1	a	a	DET
ejpam-5321	233	2	multifunction	multifunction	NOUN
ejpam-5321	233	3	f	f	NOUN
ejpam-5321	233	4	:	:	PUNCT
ejpam-5321	233	5	(	(	PUNCT
ejpam-5321	233	6	x	x	NOUN
ejpam-5321	233	7	,	,	PUNCT
ejpam-5321	233	8	τ1	τ1	NOUN
ejpam-5321	233	9	,	,	PUNCT
ejpam-5321	233	10	τ2	τ2	NOUN
ejpam-5321	233	11	)	)	PUNCT
ejpam-5321	233	12	→	→	SYM
ejpam-5321	233	13	(	(	PUNCT
ejpam-5321	233	14	y	y	PROPN
ejpam-5321	233	15	,	,	PUNCT
ejpam-5321	233	16	σ1	σ1	PROPN
ejpam-5321	233	17	,	,	PUNCT
ejpam-5321	233	18	σ2	σ2	PROPN
ejpam-5321	233	19	)	)	PUNCT
ejpam-5321	233	20	is	be	AUX
ejpam-5321	233	21	said	say	VERB
ejpam-5321	233	22	to	to	PART
ejpam-5321	233	23	be	be	AUX
ejpam-5321	233	24	upper	upper	ADJ
ejpam-5321	233	25	weakly	weakly	ADJ
ejpam-5321	233	26	α(τ1	α(τ1	NOUN
ejpam-5321	233	27	,	,	PUNCT
ejpam-5321	233	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	233	29	at	at	ADP
ejpam-5321	233	30	a	a	DET
ejpam-5321	233	31	point	point	NOUN
ejpam-5321	233	32	x	x	SYM
ejpam-5321	233	33	∈	∈	NOUN
ejpam-5321	233	34	x	x	PUNCT
ejpam-5321	233	35	if	if	SCONJ
ejpam-5321	233	36	for	for	ADP
ejpam-5321	233	37	each	each	DET
ejpam-5321	233	38	σ1σ2	σ1σ2	VERB
ejpam-5321	233	39	-	-	ADJ
ejpam-5321	233	40	open	open	ADJ
ejpam-5321	233	41	set	set	NOUN
ejpam-5321	233	42	v	v	NOUN
ejpam-5321	233	43	of	of	ADP
ejpam-5321	233	44	y	y	PROPN
ejpam-5321	233	45	containing	contain	VERB
ejpam-5321	233	46	f	f	PROPN
ejpam-5321	233	47	(	(	PUNCT
ejpam-5321	233	48	x	x	NOUN
ejpam-5321	233	49	)	)	PUNCT
ejpam-5321	233	50	,	,	PUNCT
ejpam-5321	233	51	there	there	PRON
ejpam-5321	233	52	exists	exist	VERB
ejpam-5321	233	53	an	an	DET
ejpam-5321	233	54	α(τ1	α(τ1	NOUN
ejpam-5321	233	55	,	,	PUNCT
ejpam-5321	233	56	τ2)-open	τ2)-open	ADJ
ejpam-5321	233	57	set	set	VERB
ejpam-5321	233	58	u	u	NOUN
ejpam-5321	233	59	of	of	ADP
ejpam-5321	233	60	x	x	PUNCT
ejpam-5321	233	61	containing	contain	VERB
ejpam-5321	233	62	x	x	PUNCT
ejpam-5321	233	63	such	such	ADJ
ejpam-5321	233	64	that	that	SCONJ
ejpam-5321	233	65	f	f	PROPN
ejpam-5321	233	66	(	(	PUNCT
ejpam-5321	233	67	u	u	NOUN
ejpam-5321	233	68	)	)	PUNCT
ejpam-5321	233	69	⊆	⊆	NUM
ejpam-5321	233	70	σ1σ2	σ1σ2	NOUN
ejpam-5321	233	71	-	-	NUM
ejpam-5321	233	72	cl(v	cl(v	NOUN
ejpam-5321	233	73	)	)	PUNCT
ejpam-5321	233	74	.	.	PUNCT
ejpam-5321	234	1	a	a	DET
ejpam-5321	234	2	multifunction	multifunction	NOUN
ejpam-5321	234	3	f	f	NOUN
ejpam-5321	234	4	:	:	PUNCT
ejpam-5321	234	5	(	(	PUNCT
ejpam-5321	234	6	x	x	NOUN
ejpam-5321	234	7	,	,	PUNCT
ejpam-5321	234	8	τ1	τ1	NOUN
ejpam-5321	234	9	,	,	PUNCT
ejpam-5321	234	10	τ2	τ2	NOUN
ejpam-5321	234	11	)	)	PUNCT
ejpam-5321	234	12	→	→	SYM
ejpam-5321	234	13	(	(	PUNCT
ejpam-5321	234	14	y	y	PROPN
ejpam-5321	234	15	,	,	PUNCT
ejpam-5321	234	16	σ1	σ1	PROPN
ejpam-5321	234	17	,	,	PUNCT
ejpam-5321	234	18	σ2	σ2	PROPN
ejpam-5321	234	19	)	)	PUNCT
ejpam-5321	234	20	is	be	AUX
ejpam-5321	234	21	said	say	VERB
ejpam-5321	234	22	to	to	PART
ejpam-5321	234	23	be	be	AUX
ejpam-5321	234	24	upper	upper	ADJ
ejpam-5321	234	25	weakly	weakly	ADJ
ejpam-5321	234	26	α(τ1	α(τ1	NOUN
ejpam-5321	234	27	,	,	PUNCT
ejpam-5321	234	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	234	29	if	if	SCONJ
ejpam-5321	234	30	f	f	PROPN
ejpam-5321	234	31	has	have	VERB
ejpam-5321	234	32	this	this	DET
ejpam-5321	234	33	property	property	NOUN
ejpam-5321	234	34	at	at	ADP
ejpam-5321	234	35	each	each	DET
ejpam-5321	234	36	point	point	NOUN
ejpam-5321	234	37	of	of	ADP
ejpam-5321	234	38	x.	x.	NOUN
ejpam-5321	234	39	references	reference	NOUN
ejpam-5321	234	40	2151	2151	NUM
ejpam-5321	234	41	theorem	theorem	VERB
ejpam-5321	234	42	11	11	NUM
ejpam-5321	234	43	.	.	PUNCT
ejpam-5321	235	1	if	if	SCONJ
ejpam-5321	235	2	a	a	DET
ejpam-5321	235	3	multifunction	multifunction	NOUN
ejpam-5321	235	4	f	f	NOUN
ejpam-5321	235	5	:	:	PUNCT
ejpam-5321	235	6	(	(	PUNCT
ejpam-5321	235	7	x	x	NOUN
ejpam-5321	235	8	,	,	PUNCT
ejpam-5321	235	9	τ1	τ1	NOUN
ejpam-5321	235	10	,	,	PUNCT
ejpam-5321	235	11	τ2	τ2	NOUN
ejpam-5321	235	12	)	)	PUNCT
ejpam-5321	235	13	→	→	SYM
ejpam-5321	235	14	(	(	PUNCT
ejpam-5321	235	15	y	y	PROPN
ejpam-5321	235	16	,	,	PUNCT
ejpam-5321	235	17	σ1	σ1	PROPN
ejpam-5321	235	18	,	,	PUNCT
ejpam-5321	235	19	σ2	σ2	PROPN
ejpam-5321	235	20	)	)	PUNCT
ejpam-5321	235	21	is	be	AUX
ejpam-5321	235	22	upper	upper	ADJ
ejpam-5321	235	23	weakly	weakly	ADJ
ejpam-5321	235	24	α(τ1	α(τ1	NOUN
ejpam-5321	235	25	,	,	PUNCT
ejpam-5321	235	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	235	27	,	,	PUNCT
ejpam-5321	235	28	then	then	ADV
ejpam-5321	235	29	f	f	PROPN
ejpam-5321	235	30	is	be	AUX
ejpam-5321	235	31	upper	upper	ADJ
ejpam-5321	235	32	slightly	slightly	ADV
ejpam-5321	235	33	α(τ1	α(τ1	NOUN
ejpam-5321	235	34	,	,	PUNCT
ejpam-5321	235	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	235	36	.	.	PUNCT
ejpam-5321	236	1	proof	proof	NOUN
ejpam-5321	236	2	.	.	PUNCT
ejpam-5321	237	1	let	let	VERB
ejpam-5321	237	2	x	x	PUNCT
ejpam-5321	237	3	∈	∈	PROPN
ejpam-5321	237	4	x	x	X
ejpam-5321	237	5	and	and	CCONJ
ejpam-5321	237	6	v	v	X
ejpam-5321	237	7	be	be	AUX
ejpam-5321	237	8	any	any	DET
ejpam-5321	237	9	σ1σ2	σ1σ2	NOUN
ejpam-5321	237	10	-	-	PUNCT
ejpam-5321	237	11	clopen	clopen	ADJ
ejpam-5321	237	12	set	set	NOUN
ejpam-5321	237	13	of	of	ADP
ejpam-5321	237	14	y	y	PROPN
ejpam-5321	237	15	containing	contain	VERB
ejpam-5321	237	16	f	f	PROPN
ejpam-5321	237	17	(	(	PUNCT
ejpam-5321	237	18	x	x	NOUN
ejpam-5321	237	19	)	)	PUNCT
ejpam-5321	237	20	.	.	PUNCT
ejpam-5321	238	1	since	since	SCONJ
ejpam-5321	238	2	f	f	PROPN
ejpam-5321	238	3	is	be	AUX
ejpam-5321	238	4	upper	upper	ADJ
ejpam-5321	238	5	weakly	weakly	ADJ
ejpam-5321	238	6	α(τ1	α(τ1	NOUN
ejpam-5321	238	7	,	,	PUNCT
ejpam-5321	238	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	238	9	,	,	PUNCT
ejpam-5321	238	10	there	there	PRON
ejpam-5321	238	11	exists	exist	VERB
ejpam-5321	238	12	an	an	DET
ejpam-5321	238	13	α(τ1	α(τ1	NOUN
ejpam-5321	238	14	,	,	PUNCT
ejpam-5321	238	15	τ2)-open	τ2)-open	ADJ
ejpam-5321	238	16	set	set	NOUN
ejpam-5321	238	17	of	of	ADP
ejpam-5321	238	18	x	x	PUNCT
ejpam-5321	238	19	containing	contain	VERB
ejpam-5321	238	20	x	x	PUNCT
ejpam-5321	238	21	such	such	ADJ
ejpam-5321	238	22	that	that	SCONJ
ejpam-5321	238	23	f	f	PROPN
ejpam-5321	238	24	(	(	PUNCT
ejpam-5321	238	25	u	u	NOUN
ejpam-5321	238	26	)	)	PUNCT
ejpam-5321	238	27	⊆	⊆	NUM
ejpam-5321	238	28	σ1σ2	σ1σ2	NOUN
ejpam-5321	238	29	-	-	NUM
ejpam-5321	238	30	cl(v	cl(v	X
ejpam-5321	238	31	)	)	PUNCT
ejpam-5321	238	32	=	=	SYM
ejpam-5321	238	33	v	v	NOUN
ejpam-5321	238	34	.	.	PUNCT
ejpam-5321	239	1	this	this	PRON
ejpam-5321	239	2	shows	show	VERB
ejpam-5321	239	3	that	that	SCONJ
ejpam-5321	239	4	f	f	PROPN
ejpam-5321	239	5	is	be	AUX
ejpam-5321	239	6	upper	upper	ADJ
ejpam-5321	239	7	slightly	slightly	ADV
ejpam-5321	239	8	α(τ1	α(τ1	NOUN
ejpam-5321	239	9	,	,	PUNCT
ejpam-5321	239	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	239	11	.	.	PUNCT
ejpam-5321	240	1	definition	definition	NOUN
ejpam-5321	240	2	17	17	NUM
ejpam-5321	240	3	.	.	PUNCT
ejpam-5321	241	1	a	a	DET
ejpam-5321	241	2	multifunction	multifunction	NOUN
ejpam-5321	241	3	f	f	NOUN
ejpam-5321	241	4	:	:	PUNCT
ejpam-5321	241	5	(	(	PUNCT
ejpam-5321	241	6	x	x	NOUN
ejpam-5321	241	7	,	,	PUNCT
ejpam-5321	241	8	τ1	τ1	NOUN
ejpam-5321	241	9	,	,	PUNCT
ejpam-5321	241	10	τ2	τ2	NOUN
ejpam-5321	241	11	)	)	PUNCT
ejpam-5321	241	12	→	→	SYM
ejpam-5321	241	13	(	(	PUNCT
ejpam-5321	241	14	y	y	PROPN
ejpam-5321	241	15	,	,	PUNCT
ejpam-5321	241	16	σ1	σ1	PROPN
ejpam-5321	241	17	,	,	PUNCT
ejpam-5321	241	18	σ2	σ2	PROPN
ejpam-5321	241	19	)	)	PUNCT
ejpam-5321	241	20	is	be	AUX
ejpam-5321	241	21	called	call	VERB
ejpam-5321	241	22	lower	low	ADJ
ejpam-5321	241	23	weakly	weakly	ADJ
ejpam-5321	241	24	α(τ1	α(τ1	NOUN
ejpam-5321	241	25	,	,	PUNCT
ejpam-5321	241	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	241	27	at	at	ADP
ejpam-5321	241	28	a	a	DET
ejpam-5321	241	29	point	point	NOUN
ejpam-5321	241	30	x	x	SYM
ejpam-5321	241	31	∈	∈	NOUN
ejpam-5321	241	32	x	x	PUNCT
ejpam-5321	241	33	if	if	SCONJ
ejpam-5321	241	34	for	for	ADP
ejpam-5321	241	35	each	each	DET
ejpam-5321	241	36	σ1σ2	σ1σ2	VERB
ejpam-5321	241	37	-	-	ADJ
ejpam-5321	241	38	open	open	ADJ
ejpam-5321	241	39	set	set	NOUN
ejpam-5321	241	40	v	v	NOUN
ejpam-5321	241	41	of	of	ADP
ejpam-5321	241	42	y	y	PRON
ejpam-5321	241	43	such	such	ADJ
ejpam-5321	241	44	that	that	SCONJ
ejpam-5321	241	45	f	f	PROPN
ejpam-5321	241	46	(	(	PUNCT
ejpam-5321	241	47	x	x	NOUN
ejpam-5321	241	48	)	)	PUNCT
ejpam-5321	241	49	∩	∩	NOUN
ejpam-5321	241	50	v	v	ADP
ejpam-5321	241	51	̸=	̸=	PROPN
ejpam-5321	241	52	∅	∅	NOUN
ejpam-5321	241	53	,	,	PUNCT
ejpam-5321	241	54	there	there	PRON
ejpam-5321	241	55	exists	exist	VERB
ejpam-5321	241	56	an	an	DET
ejpam-5321	241	57	α(τ1	α(τ1	NOUN
ejpam-5321	241	58	,	,	PUNCT
ejpam-5321	241	59	τ2)-open	τ2)-open	ADJ
ejpam-5321	241	60	set	set	VERB
ejpam-5321	241	61	u	u	NOUN
ejpam-5321	241	62	of	of	ADP
ejpam-5321	241	63	x	x	PUNCT
ejpam-5321	241	64	containing	contain	VERB
ejpam-5321	241	65	x	x	PUNCT
ejpam-5321	241	66	such	such	ADJ
ejpam-5321	241	67	that	that	SCONJ
ejpam-5321	241	68	σ1σ2	σ1σ2	NOUN
ejpam-5321	241	69	-	-	PUNCT
ejpam-5321	241	70	cl(v	cl(v	NOUN
ejpam-5321	241	71	)	)	PUNCT
ejpam-5321	241	72	∩	∩	PROPN
ejpam-5321	241	73	f	f	X
ejpam-5321	241	74	(	(	PUNCT
ejpam-5321	241	75	z	z	NOUN
ejpam-5321	241	76	)	)	PUNCT
ejpam-5321	241	77	̸=	̸=	NOUN
ejpam-5321	241	78	∅	∅	NOUN
ejpam-5321	241	79	for	for	ADP
ejpam-5321	241	80	every	every	DET
ejpam-5321	241	81	z	z	NOUN
ejpam-5321	241	82	∈	∈	PROPN
ejpam-5321	241	83	u	u	NOUN
ejpam-5321	241	84	.	.	PUNCT
ejpam-5321	242	1	a	a	DET
ejpam-5321	242	2	multifunction	multifunction	NOUN
ejpam-5321	242	3	f	f	NOUN
ejpam-5321	242	4	:	:	PUNCT
ejpam-5321	242	5	(	(	PUNCT
ejpam-5321	242	6	x	x	NOUN
ejpam-5321	242	7	,	,	PUNCT
ejpam-5321	242	8	τ1	τ1	NOUN
ejpam-5321	242	9	,	,	PUNCT
ejpam-5321	242	10	τ2	τ2	NOUN
ejpam-5321	242	11	)	)	PUNCT
ejpam-5321	242	12	→	→	SYM
ejpam-5321	242	13	(	(	PUNCT
ejpam-5321	242	14	y	y	PROPN
ejpam-5321	242	15	,	,	PUNCT
ejpam-5321	242	16	σ1	σ1	PROPN
ejpam-5321	242	17	,	,	PUNCT
ejpam-5321	242	18	σ2	σ2	PROPN
ejpam-5321	242	19	)	)	PUNCT
ejpam-5321	242	20	is	be	AUX
ejpam-5321	242	21	called	call	VERB
ejpam-5321	242	22	lower	low	ADJ
ejpam-5321	242	23	weakly	weakly	ADJ
ejpam-5321	242	24	α(τ1	α(τ1	NOUN
ejpam-5321	242	25	,	,	PUNCT
ejpam-5321	242	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	242	27	if	if	SCONJ
ejpam-5321	242	28	f	f	PROPN
ejpam-5321	242	29	has	have	VERB
ejpam-5321	242	30	this	this	DET
ejpam-5321	242	31	property	property	NOUN
ejpam-5321	242	32	at	at	ADP
ejpam-5321	242	33	each	each	DET
ejpam-5321	242	34	point	point	NOUN
ejpam-5321	242	35	of	of	ADP
ejpam-5321	242	36	x.	x.	NOUN
ejpam-5321	242	37	theorem	theorem	VERB
ejpam-5321	242	38	12	12	NUM
ejpam-5321	242	39	.	.	PUNCT
ejpam-5321	243	1	if	if	SCONJ
ejpam-5321	243	2	a	a	DET
ejpam-5321	243	3	multifunction	multifunction	NOUN
ejpam-5321	243	4	f	f	NOUN
ejpam-5321	243	5	:	:	PUNCT
ejpam-5321	243	6	(	(	PUNCT
ejpam-5321	243	7	x	x	NOUN
ejpam-5321	243	8	,	,	PUNCT
ejpam-5321	243	9	τ1	τ1	NOUN
ejpam-5321	243	10	,	,	PUNCT
ejpam-5321	243	11	τ2	τ2	NOUN
ejpam-5321	243	12	)	)	PUNCT
ejpam-5321	243	13	→	→	SYM
ejpam-5321	243	14	(	(	PUNCT
ejpam-5321	243	15	y	y	PROPN
ejpam-5321	243	16	,	,	PUNCT
ejpam-5321	243	17	σ1	σ1	PROPN
ejpam-5321	243	18	,	,	PUNCT
ejpam-5321	243	19	σ2	σ2	NOUN
ejpam-5321	243	20	)	)	PUNCT
ejpam-5321	243	21	is	be	AUX
ejpam-5321	243	22	lower	low	ADJ
ejpam-5321	243	23	weakly	weakly	ADJ
ejpam-5321	243	24	α(τ1	α(τ1	NOUN
ejpam-5321	243	25	,	,	PUNCT
ejpam-5321	243	26	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	243	27	,	,	PUNCT
ejpam-5321	243	28	then	then	ADV
ejpam-5321	243	29	f	f	PROPN
ejpam-5321	243	30	is	be	AUX
ejpam-5321	243	31	lower	low	ADJ
ejpam-5321	243	32	slightly	slightly	ADV
ejpam-5321	243	33	α(τ1	α(τ1	NOUN
ejpam-5321	243	34	,	,	PUNCT
ejpam-5321	243	35	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	243	36	.	.	PUNCT
ejpam-5321	244	1	proof	proof	NOUN
ejpam-5321	244	2	.	.	PUNCT
ejpam-5321	245	1	the	the	DET
ejpam-5321	245	2	proof	proof	NOUN
ejpam-5321	245	3	is	be	AUX
ejpam-5321	245	4	similar	similar	ADJ
ejpam-5321	245	5	to	to	ADP
ejpam-5321	245	6	that	that	PRON
ejpam-5321	245	7	of	of	ADP
ejpam-5321	245	8	theorem	theorem	NOUN
ejpam-5321	245	9	11	11	NUM
ejpam-5321	245	10	.	.	PUNCT
ejpam-5321	246	1	acknowledgements	acknowledgement	NOUN
ejpam-5321	246	2	this	this	DET
ejpam-5321	246	3	research	research	NOUN
ejpam-5321	246	4	project	project	NOUN
ejpam-5321	246	5	was	be	AUX
ejpam-5321	246	6	financially	financially	ADV
ejpam-5321	246	7	supported	support	VERB
ejpam-5321	246	8	by	by	ADP
ejpam-5321	246	9	mahasarakham	mahasarakham	PROPN
ejpam-5321	246	10	university	university	PROPN
ejpam-5321	246	11	.	.	PUNCT
ejpam-5321	247	1	references	reference	NOUN
ejpam-5321	247	2	[	[	X
ejpam-5321	247	3	1	1	NUM
ejpam-5321	247	4	]	]	PUNCT
ejpam-5321	247	5	c.	c.	PROPN
ejpam-5321	247	6	berge	berge	PROPN
ejpam-5321	247	7	.	.	PUNCT
ejpam-5321	248	1	espaces	espace	VERB
ejpam-5321	248	2	topologiques	topologique	NOUN
ejpam-5321	248	3	fonctions	fonction	NOUN
ejpam-5321	248	4	multivoques	multivoque	NOUN
ejpam-5321	248	5	.	.	PUNCT
ejpam-5321	249	1	dunod	dunod	PROPN
ejpam-5321	249	2	,	,	PUNCT
ejpam-5321	249	3	paris	paris	PROPN
ejpam-5321	249	4	,	,	PUNCT
ejpam-5321	249	5	1959	1959	NUM
ejpam-5321	249	6	.	.	PUNCT
ejpam-5321	250	1	[	[	X
ejpam-5321	250	2	2	2	NUM
ejpam-5321	250	3	]	]	PUNCT
ejpam-5321	250	4	c.	c.	PROPN
ejpam-5321	250	5	boonpok	boonpok	PROPN
ejpam-5321	250	6	.	.	PUNCT
ejpam-5321	251	1	almost	almost	ADV
ejpam-5321	251	2	(	(	PUNCT
ejpam-5321	251	3	g	g	NOUN
ejpam-5321	251	4	,	,	PUNCT
ejpam-5321	251	5	m)-continuous	m)-continuous	ADJ
ejpam-5321	251	6	functions	function	NOUN
ejpam-5321	251	7	.	.	PUNCT
ejpam-5321	252	1	international	international	ADJ
ejpam-5321	252	2	journal	journal	PROPN
ejpam-5321	252	3	of	of	ADP
ejpam-5321	252	4	mathematical	mathematical	ADJ
ejpam-5321	252	5	analysis	analysis	NOUN
ejpam-5321	252	6	,	,	PUNCT
ejpam-5321	252	7	4(40):1957–1964	4(40):1957–1964	NUM
ejpam-5321	252	8	,	,	PUNCT
ejpam-5321	252	9	2010	2010	NUM
ejpam-5321	252	10	.	.	PUNCT
ejpam-5321	253	1	[	[	X
ejpam-5321	253	2	3	3	X
ejpam-5321	253	3	]	]	PUNCT
ejpam-5321	253	4	c.	c.	PROPN
ejpam-5321	253	5	boonpok	boonpok	PROPN
ejpam-5321	253	6	.	.	PUNCT
ejpam-5321	254	1	m	m	VERB
ejpam-5321	254	2	-continuous	-continuous	ADJ
ejpam-5321	254	3	functions	function	NOUN
ejpam-5321	254	4	in	in	ADP
ejpam-5321	254	5	biminimal	biminimal	NOUN
ejpam-5321	254	6	structure	structure	NOUN
ejpam-5321	254	7	spaces	space	NOUN
ejpam-5321	254	8	.	.	PUNCT
ejpam-5321	255	1	far	far	PROPN
ejpam-5321	255	2	east	east	PROPN
ejpam-5321	255	3	journal	journal	PROPN
ejpam-5321	255	4	of	of	ADP
ejpam-5321	255	5	mathematical	mathematical	ADJ
ejpam-5321	255	6	sciences	science	NOUN
ejpam-5321	255	7	,	,	PUNCT
ejpam-5321	255	8	43(1):41–58	43(1):41–58	NUM
ejpam-5321	255	9	,	,	PUNCT
ejpam-5321	255	10	2010	2010	NUM
ejpam-5321	255	11	.	.	PUNCT
ejpam-5321	256	1	[	[	X
ejpam-5321	256	2	4	4	NUM
ejpam-5321	256	3	]	]	PUNCT
ejpam-5321	256	4	c.	c.	PROPN
ejpam-5321	256	5	boonpok	boonpok	PROPN
ejpam-5321	256	6	.	.	PUNCT
ejpam-5321	257	1	on	on	ADP
ejpam-5321	257	2	continuous	continuous	ADJ
ejpam-5321	257	3	multifunctions	multifunction	NOUN
ejpam-5321	257	4	in	in	ADP
ejpam-5321	257	5	ideal	ideal	ADJ
ejpam-5321	257	6	topological	topological	ADJ
ejpam-5321	257	7	spaces	space	NOUN
ejpam-5321	257	8	.	.	PUNCT
ejpam-5321	258	1	lobachevskii	lobachevskii	PROPN
ejpam-5321	258	2	journal	journal	PROPN
ejpam-5321	258	3	of	of	ADP
ejpam-5321	258	4	mathematics	mathematic	NOUN
ejpam-5321	258	5	,	,	PUNCT
ejpam-5321	258	6	40(1):24–35	40(1):24–35	NUM
ejpam-5321	258	7	,	,	PUNCT
ejpam-5321	258	8	2019	2019	NUM
ejpam-5321	258	9	.	.	PUNCT
ejpam-5321	259	1	[	[	X
ejpam-5321	259	2	5	5	X
ejpam-5321	259	3	]	]	PUNCT
ejpam-5321	259	4	c.	c.	PROPN
ejpam-5321	259	5	boonpok	boonpok	PROPN
ejpam-5321	259	6	.	.	PUNCT
ejpam-5321	260	1	on	on	ADP
ejpam-5321	260	2	characterizations	characterization	NOUN
ejpam-5321	260	3	of	of	ADP
ejpam-5321	260	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5321	260	5	ideal	ideal	ADJ
ejpam-5321	260	6	topological	topological	ADJ
ejpam-5321	260	7	spaces	space	NOUN
ejpam-5321	260	8	.	.	PUNCT
ejpam-5321	261	1	journal	journal	NOUN
ejpam-5321	261	2	of	of	ADP
ejpam-5321	261	3	mathematics	mathematic	NOUN
ejpam-5321	261	4	,	,	PUNCT
ejpam-5321	261	5	2020:9387601	2020:9387601	NUM
ejpam-5321	261	6	,	,	PUNCT
ejpam-5321	261	7	2020	2020	NUM
ejpam-5321	261	8	.	.	PUNCT
ejpam-5321	262	1	[	[	X
ejpam-5321	262	2	6	6	NUM
ejpam-5321	262	3	]	]	PUNCT
ejpam-5321	262	4	c.	c.	PROPN
ejpam-5321	262	5	boonpok	boonpok	PROPN
ejpam-5321	262	6	.	.	PUNCT
ejpam-5321	263	1	(	(	PUNCT
ejpam-5321	263	2	τ1	τ1	NOUN
ejpam-5321	263	3	,	,	PUNCT
ejpam-5321	263	4	τ2)δ	τ2)δ	ADJ
ejpam-5321	263	5	-	-	PUNCT
ejpam-5321	263	6	semicontinuous	semicontinuous	ADJ
ejpam-5321	263	7	multifunctions	multifunction	NOUN
ejpam-5321	263	8	.	.	PUNCT
ejpam-5321	264	1	heliyon	heliyon	NOUN
ejpam-5321	264	2	,	,	PUNCT
ejpam-5321	264	3	6	6	NUM
ejpam-5321	264	4	:	:	SYM
ejpam-5321	264	5	e05367	e05367	PROPN
ejpam-5321	264	6	,	,	PUNCT
ejpam-5321	264	7	2020	2020	NUM
ejpam-5321	264	8	.	.	PUNCT
ejpam-5321	265	1	[	[	X
ejpam-5321	265	2	7	7	X
ejpam-5321	265	3	]	]	X
ejpam-5321	265	4	c.	c.	PROPN
ejpam-5321	265	5	boonpok	boonpok	PROPN
ejpam-5321	265	6	.	.	PUNCT
ejpam-5321	266	1	upper	upper	ADJ
ejpam-5321	266	2	and	and	CCONJ
ejpam-5321	266	3	lower	low	ADJ
ejpam-5321	266	4	β(⋆)-continuity	β(⋆)-continuity	NOUN
ejpam-5321	266	5	.	.	PUNCT
ejpam-5321	266	6	heliyon	heliyon	NOUN
ejpam-5321	266	7	,	,	PUNCT
ejpam-5321	266	8	7	7	NUM
ejpam-5321	266	9	:	:	PUNCT
ejpam-5321	266	10	e05986	e05986	PROPN
ejpam-5321	266	11	,	,	PUNCT
ejpam-5321	266	12	2021	2021	NUM
ejpam-5321	266	13	.	.	PUNCT
ejpam-5321	267	1	[	[	X
ejpam-5321	267	2	8	8	NUM
ejpam-5321	267	3	]	]	X
ejpam-5321	267	4	c.	c.	PROPN
ejpam-5321	267	5	boonpok	boonpok	PROPN
ejpam-5321	267	6	.	.	PUNCT
ejpam-5321	268	1	on	on	ADP
ejpam-5321	268	2	some	some	DET
ejpam-5321	268	3	closed	closed	ADJ
ejpam-5321	268	4	sets	set	NOUN
ejpam-5321	268	5	and	and	CCONJ
ejpam-5321	268	6	low	low	ADJ
ejpam-5321	268	7	separation	separation	NOUN
ejpam-5321	268	8	axioms	axiom	NOUN
ejpam-5321	268	9	via	via	ADP
ejpam-5321	268	10	topological	topological	ADJ
ejpam-5321	268	11	ideals	ideal	NOUN
ejpam-5321	268	12	.	.	PUNCT
ejpam-5321	269	1	european	european	ADJ
ejpam-5321	269	2	journal	journal	PROPN
ejpam-5321	269	3	of	of	ADP
ejpam-5321	269	4	pure	pure	ADJ
ejpam-5321	269	5	and	and	CCONJ
ejpam-5321	269	6	applied	applied	ADJ
ejpam-5321	269	7	mathematics	mathematic	NOUN
ejpam-5321	269	8	,	,	PUNCT
ejpam-5321	269	9	15(3):300–309	15(3):300–309	NUM
ejpam-5321	269	10	,	,	PUNCT
ejpam-5321	269	11	2022	2022	NUM
ejpam-5321	269	12	.	.	PUNCT
ejpam-5321	270	1	references	reference	NOUN
ejpam-5321	270	2	2152	2152	NUM
ejpam-5321	270	3	[	[	X
ejpam-5321	270	4	9	9	NUM
ejpam-5321	270	5	]	]	PUNCT
ejpam-5321	270	6	c.	c.	PROPN
ejpam-5321	270	7	boonpok	boonpok	PROPN
ejpam-5321	270	8	.	.	PUNCT
ejpam-5321	271	1	on	on	ADP
ejpam-5321	271	2	some	some	DET
ejpam-5321	271	3	spaces	space	NOUN
ejpam-5321	271	4	via	via	ADP
ejpam-5321	271	5	topological	topological	ADJ
ejpam-5321	271	6	ideals	ideal	NOUN
ejpam-5321	271	7	.	.	PUNCT
ejpam-5321	272	1	open	open	ADJ
ejpam-5321	272	2	mathematics	mathematic	NOUN
ejpam-5321	272	3	,	,	PUNCT
ejpam-5321	272	4	21:20230118	21:20230118	NUM
ejpam-5321	272	5	,	,	PUNCT
ejpam-5321	272	6	2023	2023	NUM
ejpam-5321	272	7	.	.	PUNCT
ejpam-5321	273	1	[	[	X
ejpam-5321	273	2	10	10	NUM
ejpam-5321	273	3	]	]	X
ejpam-5321	273	4	c.	c.	PROPN
ejpam-5321	273	5	boonpok	boonpok	PROPN
ejpam-5321	273	6	.	.	PUNCT
ejpam-5321	274	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5321	274	2	.	.	PUNCT
ejpam-5321	275	1	mathematica	mathematica	PROPN
ejpam-5321	275	2	,	,	PUNCT
ejpam-5321	275	3	65(1):31–42	65(1):31–42	NUM
ejpam-5321	275	4	,	,	PUNCT
ejpam-5321	275	5	2023	2023	NUM
ejpam-5321	275	6	.	.	PUNCT
ejpam-5321	276	1	[	[	X
ejpam-5321	276	2	11	11	NUM
ejpam-5321	276	3	]	]	X
ejpam-5321	276	4	c.	c.	PROPN
ejpam-5321	276	5	boonpok	boonpok	PROPN
ejpam-5321	276	6	and	and	CCONJ
ejpam-5321	276	7	j.	j.	PROPN
ejpam-5321	276	8	khampakdee	khampakdee	PROPN
ejpam-5321	276	9	.	.	PUNCT
ejpam-5321	277	1	almost	almost	ADV
ejpam-5321	277	2	strong	strong	ADJ
ejpam-5321	277	3	θ(λ	θ(λ	PROPN
ejpam-5321	277	4	,	,	PUNCT
ejpam-5321	277	5	p)-continuity	p)-continuity	NOUN
ejpam-5321	277	6	for	for	ADP
ejpam-5321	277	7	functions	function	NOUN
ejpam-5321	277	8	.	.	PUNCT
ejpam-5321	278	1	european	european	ADJ
ejpam-5321	278	2	journal	journal	PROPN
ejpam-5321	278	3	of	of	ADP
ejpam-5321	278	4	pure	pure	ADJ
ejpam-5321	278	5	and	and	CCONJ
ejpam-5321	278	6	applied	applied	ADJ
ejpam-5321	278	7	mathematics	mathematic	NOUN
ejpam-5321	278	8	,	,	PUNCT
ejpam-5321	278	9	17(1):300–309	17(1):300–309	PROPN
ejpam-5321	278	10	,	,	PUNCT
ejpam-5321	278	11	2024	2024	NUM
ejpam-5321	278	12	.	.	PUNCT
ejpam-5321	279	1	[	[	X
ejpam-5321	279	2	12	12	NUM
ejpam-5321	279	3	]	]	X
ejpam-5321	279	4	c.	c.	PROPN
ejpam-5321	279	5	boonpok	boonpok	PROPN
ejpam-5321	279	6	and	and	CCONJ
ejpam-5321	279	7	c.	c.	PROPN
ejpam-5321	279	8	klanarong	klanarong	PROPN
ejpam-5321	279	9	.	.	PUNCT
ejpam-5321	280	1	on	on	ADP
ejpam-5321	280	2	weakly	weakly	ADJ
ejpam-5321	280	3	(	(	PUNCT
ejpam-5321	280	4	τ1	τ1	NOUN
ejpam-5321	280	5	,	,	PUNCT
ejpam-5321	280	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	280	7	functions	function	NOUN
ejpam-5321	280	8	.	.	PUNCT
ejpam-5321	281	1	european	european	ADJ
ejpam-5321	281	2	journal	journal	PROPN
ejpam-5321	281	3	of	of	ADP
ejpam-5321	281	4	pure	pure	ADJ
ejpam-5321	281	5	and	and	CCONJ
ejpam-5321	281	6	applied	applied	ADJ
ejpam-5321	281	7	mathematics	mathematic	NOUN
ejpam-5321	281	8	,	,	PUNCT
ejpam-5321	281	9	17(1):416–425	17(1):416–425	NUM
ejpam-5321	281	10	,	,	PUNCT
ejpam-5321	281	11	2024	2024	NUM
ejpam-5321	281	12	.	.	PUNCT
ejpam-5321	282	1	[	[	X
ejpam-5321	282	2	13	13	NUM
ejpam-5321	282	3	]	]	PUNCT
ejpam-5321	282	4	c.	c.	PROPN
ejpam-5321	282	5	boonpok	boonpok	PROPN
ejpam-5321	282	6	and	and	CCONJ
ejpam-5321	282	7	p.	p.	NOUN
ejpam-5321	282	8	pue	pue	NOUN
ejpam-5321	282	9	-	-	PUNCT
ejpam-5321	282	10	on	on	ADP
ejpam-5321	282	11	.	.	PUNCT
ejpam-5321	283	1	continuity	continuity	NOUN
ejpam-5321	283	2	for	for	ADP
ejpam-5321	283	3	multifunctions	multifunction	NOUN
ejpam-5321	283	4	in	in	ADP
ejpam-5321	283	5	ideal	ideal	ADJ
ejpam-5321	283	6	topological	topological	ADJ
ejpam-5321	283	7	spaces	space	NOUN
ejpam-5321	283	8	.	.	PUNCT
ejpam-5321	284	1	wseas	wseas	VERB
ejpam-5321	284	2	transactions	transaction	NOUN
ejpam-5321	284	3	on	on	ADP
ejpam-5321	284	4	mathematics	mathematic	NOUN
ejpam-5321	284	5	,	,	PUNCT
ejpam-5321	284	6	19:624–631	19:624–631	NUM
ejpam-5321	284	7	,	,	PUNCT
ejpam-5321	284	8	2020	2020	NUM
ejpam-5321	284	9	.	.	PUNCT
ejpam-5321	285	1	[	[	X
ejpam-5321	285	2	14	14	NUM
ejpam-5321	285	3	]	]	X
ejpam-5321	285	4	c.	c.	PROPN
ejpam-5321	285	5	boonpok	boonpok	PROPN
ejpam-5321	285	6	and	and	CCONJ
ejpam-5321	285	7	p.	p.	NOUN
ejpam-5321	285	8	pue	pue	NOUN
ejpam-5321	285	9	-	-	PUNCT
ejpam-5321	285	10	on	on	ADP
ejpam-5321	285	11	.	.	PUNCT
ejpam-5321	286	1	upper	upper	ADJ
ejpam-5321	286	2	and	and	CCONJ
ejpam-5321	286	3	lower	low	ADJ
ejpam-5321	286	4	weakly	weakly	ADJ
ejpam-5321	286	5	α-⋆-continuous	α-⋆-continuous	ADJ
ejpam-5321	286	6	multifunctions	multifunction	NOUN
ejpam-5321	286	7	.	.	PUNCT
ejpam-5321	287	1	international	international	ADJ
ejpam-5321	287	2	journal	journal	NOUN
ejpam-5321	287	3	of	of	ADP
ejpam-5321	287	4	analysis	analysis	NOUN
ejpam-5321	287	5	and	and	CCONJ
ejpam-5321	287	6	applications	application	NOUN
ejpam-5321	287	7	,	,	PUNCT
ejpam-5321	287	8	21:90	21:90	NUM
ejpam-5321	287	9	,	,	PUNCT
ejpam-5321	287	10	2023	2023	NUM
ejpam-5321	287	11	.	.	PUNCT
ejpam-5321	288	1	[	[	X
ejpam-5321	288	2	15	15	NUM
ejpam-5321	288	3	]	]	X
ejpam-5321	288	4	c.	c.	PROPN
ejpam-5321	288	5	boonpok	boonpok	PROPN
ejpam-5321	288	6	and	and	CCONJ
ejpam-5321	288	7	p.	p.	NOUN
ejpam-5321	288	8	pue	pue	NOUN
ejpam-5321	288	9	-	-	PUNCT
ejpam-5321	288	10	on	on	ADP
ejpam-5321	288	11	.	.	PUNCT
ejpam-5321	289	1	characterizations	characterization	NOUN
ejpam-5321	289	2	of	of	ADP
ejpam-5321	289	3	almost	almost	ADV
ejpam-5321	289	4	(	(	PUNCT
ejpam-5321	289	5	τ1	τ1	NOUN
ejpam-5321	289	6	,	,	PUNCT
ejpam-5321	289	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	289	8	functions	function	NOUN
ejpam-5321	289	9	.	.	PUNCT
ejpam-5321	290	1	international	international	ADJ
ejpam-5321	290	2	journal	journal	NOUN
ejpam-5321	290	3	of	of	ADP
ejpam-5321	290	4	analysis	analysis	NOUN
ejpam-5321	290	5	and	and	CCONJ
ejpam-5321	290	6	applications	application	NOUN
ejpam-5321	290	7	,	,	PUNCT
ejpam-5321	290	8	22:33	22:33	NUM
ejpam-5321	290	9	,	,	PUNCT
ejpam-5321	290	10	2024	2024	NUM
ejpam-5321	290	11	.	.	PUNCT
ejpam-5321	291	1	[	[	X
ejpam-5321	291	2	16	16	NUM
ejpam-5321	291	3	]	]	X
ejpam-5321	291	4	c.	c.	PROPN
ejpam-5321	291	5	boonpok	boonpok	PROPN
ejpam-5321	291	6	and	and	CCONJ
ejpam-5321	291	7	n.	n.	PROPN
ejpam-5321	291	8	srisarakham	srisarakham	PROPN
ejpam-5321	291	9	.	.	PUNCT
ejpam-5321	292	1	weak	weak	ADJ
ejpam-5321	292	2	forms	form	NOUN
ejpam-5321	292	3	of	of	ADP
ejpam-5321	292	4	(	(	PUNCT
ejpam-5321	292	5	λ	λ	PROPN
ejpam-5321	292	6	,	,	PUNCT
ejpam-5321	292	7	b)-open	b)-open	VERB
ejpam-5321	292	8	sets	set	NOUN
ejpam-5321	292	9	and	and	CCONJ
ejpam-5321	292	10	weak	weak	ADJ
ejpam-5321	292	11	(	(	PUNCT
ejpam-5321	292	12	λ	λ	NOUN
ejpam-5321	292	13	,	,	PUNCT
ejpam-5321	292	14	b)continuity	b)continuity	NOUN
ejpam-5321	292	15	.	.	PUNCT
ejpam-5321	293	1	european	european	PROPN
ejpam-5321	293	2	journal	journal	PROPN
ejpam-5321	293	3	of	of	ADP
ejpam-5321	293	4	pure	pure	ADJ
ejpam-5321	293	5	and	and	CCONJ
ejpam-5321	293	6	applied	applied	ADJ
ejpam-5321	293	7	mathematics	mathematic	NOUN
ejpam-5321	293	8	,	,	PUNCT
ejpam-5321	293	9	16(1):29–43	16(1):29–43	NUM
ejpam-5321	293	10	,	,	PUNCT
ejpam-5321	293	11	2023	2023	NUM
ejpam-5321	293	12	.	.	PUNCT
ejpam-5321	294	1	[	[	X
ejpam-5321	294	2	17	17	NUM
ejpam-5321	294	3	]	]	X
ejpam-5321	294	4	c.	c.	PROPN
ejpam-5321	294	5	boonpok	boonpok	PROPN
ejpam-5321	294	6	and	and	CCONJ
ejpam-5321	294	7	n.	n.	PROPN
ejpam-5321	294	8	srisarakham	srisarakham	PROPN
ejpam-5321	294	9	.	.	PUNCT
ejpam-5321	295	1	(	(	PUNCT
ejpam-5321	295	2	τ1	τ1	NOUN
ejpam-5321	295	3	,	,	PUNCT
ejpam-5321	295	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5321	295	5	for	for	ADP
ejpam-5321	295	6	functions	function	NOUN
ejpam-5321	295	7	.	.	PUNCT
ejpam-5321	296	1	asia	asia	PROPN
ejpam-5321	296	2	pacific	pacific	PROPN
ejpam-5321	296	3	journal	journal	PROPN
ejpam-5321	296	4	of	of	ADP
ejpam-5321	296	5	mathematics	mathematic	NOUN
ejpam-5321	296	6	,	,	PUNCT
ejpam-5321	296	7	11:21	11:21	NUM
ejpam-5321	296	8	,	,	PUNCT
ejpam-5321	296	9	2024	2024	NUM
ejpam-5321	296	10	.	.	PUNCT
ejpam-5321	297	1	[	[	X
ejpam-5321	297	2	18	18	NUM
ejpam-5321	297	3	]	]	PUNCT
ejpam-5321	297	4	c.	c.	PROPN
ejpam-5321	297	5	boonpok	boonpok	PROPN
ejpam-5321	297	6	and	and	CCONJ
ejpam-5321	297	7	c.	c.	PROPN
ejpam-5321	297	8	viriyapong	viriyapong	PROPN
ejpam-5321	297	9	.	.	PUNCT
ejpam-5321	298	1	almost	almost	ADV
ejpam-5321	298	2	weak	weak	ADJ
ejpam-5321	298	3	continuity	continuity	NOUN
ejpam-5321	298	4	for	for	ADP
ejpam-5321	298	5	multifunctions	multifunction	NOUN
ejpam-5321	298	6	in	in	ADP
ejpam-5321	298	7	ideal	ideal	ADJ
ejpam-5321	298	8	topological	topological	ADJ
ejpam-5321	298	9	spaces	space	NOUN
ejpam-5321	298	10	.	.	PUNCT
ejpam-5321	299	1	wseas	wseas	VERB
ejpam-5321	299	2	transactions	transaction	NOUN
ejpam-5321	299	3	on	on	ADP
ejpam-5321	299	4	mathematics	mathematic	NOUN
ejpam-5321	299	5	,	,	PUNCT
ejpam-5321	299	6	19:367–372	19:367–372	PROPN
ejpam-5321	299	7	,	,	PUNCT
ejpam-5321	299	8	2020	2020	NUM
ejpam-5321	299	9	.	.	PUNCT
ejpam-5321	300	1	[	[	X
ejpam-5321	300	2	19	19	NUM
ejpam-5321	300	3	]	]	X
ejpam-5321	300	4	c.	c.	PROPN
ejpam-5321	300	5	boonpok	boonpok	PROPN
ejpam-5321	300	6	and	and	CCONJ
ejpam-5321	300	7	c.	c.	PROPN
ejpam-5321	300	8	viriyapong	viriyapong	PROPN
ejpam-5321	300	9	.	.	PUNCT
ejpam-5321	301	1	upper	upper	ADJ
ejpam-5321	301	2	and	and	CCONJ
ejpam-5321	301	3	lower	low	ADJ
ejpam-5321	301	4	almost	almost	ADV
ejpam-5321	301	5	weak	weak	ADJ
ejpam-5321	301	6	(	(	PUNCT
ejpam-5321	301	7	τ1	τ1	NOUN
ejpam-5321	301	8	,	,	PUNCT
ejpam-5321	301	9	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5321	301	10	.	.	PUNCT
ejpam-5321	302	1	european	european	PROPN
ejpam-5321	302	2	journal	journal	PROPN
ejpam-5321	302	3	of	of	ADP
ejpam-5321	302	4	pure	pure	ADJ
ejpam-5321	302	5	and	and	CCONJ
ejpam-5321	302	6	applied	applied	ADJ
ejpam-5321	302	7	mathematics	mathematic	NOUN
ejpam-5321	302	8	,	,	PUNCT
ejpam-5321	302	9	14(1):1212–1225	14(1):1212–1225	NUM
ejpam-5321	302	10	,	,	PUNCT
ejpam-5321	302	11	2021	2021	NUM
ejpam-5321	302	12	.	.	PUNCT
ejpam-5321	303	1	[	[	X
ejpam-5321	303	2	20	20	NUM
ejpam-5321	303	3	]	]	PUNCT
ejpam-5321	303	4	c.	c.	PROPN
ejpam-5321	303	5	boonpok	boonpok	PROPN
ejpam-5321	303	6	,	,	PUNCT
ejpam-5321	303	7	c.	c.	PROPN
ejpam-5321	303	8	viriyapong	viriyapong	PROPN
ejpam-5321	303	9	,	,	PUNCT
ejpam-5321	303	10	and	and	CCONJ
ejpam-5321	303	11	m.	m.	NOUN
ejpam-5321	303	12	thongmoon	thongmoon	NOUN
ejpam-5321	303	13	.	.	PUNCT
ejpam-5321	304	1	on	on	ADP
ejpam-5321	304	2	upper	upper	ADJ
ejpam-5321	304	3	and	and	CCONJ
ejpam-5321	304	4	lower	low	ADJ
ejpam-5321	304	5	(	(	PUNCT
ejpam-5321	304	6	τ1	τ1	NOUN
ejpam-5321	304	7	,	,	PUNCT
ejpam-5321	304	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5321	304	9	multifunctions	multifunction	NOUN
ejpam-5321	304	10	.	.	PUNCT
ejpam-5321	305	1	journal	journal	PROPN
ejpam-5321	305	2	of	of	ADP
ejpam-5321	305	3	mathematics	mathematics	PROPN
ejpam-5321	305	4	and	and	CCONJ
ejpam-5321	305	5	computer	computer	NOUN
ejpam-5321	305	6	science	science	NOUN
ejpam-5321	305	7	,	,	PUNCT
ejpam-5321	305	8	18:282–293	18:282–293	NUM
ejpam-5321	305	9	,	,	PUNCT
ejpam-5321	305	10	2018	2018	NUM
ejpam-5321	305	11	.	.	PUNCT
ejpam-5321	306	1	[	[	X
ejpam-5321	306	2	21	21	NUM
ejpam-5321	306	3	]	]	X
ejpam-5321	306	4	t.	t.	PROPN
ejpam-5321	306	5	duangphui	duangphui	PROPN
ejpam-5321	306	6	,	,	PUNCT
ejpam-5321	306	7	c.	c.	PROPN
ejpam-5321	306	8	boonpok	boonpok	PROPN
ejpam-5321	306	9	,	,	PUNCT
ejpam-5321	306	10	and	and	CCONJ
ejpam-5321	306	11	c.	c.	PROPN
ejpam-5321	306	12	viriyapong	viriyapong	PROPN
ejpam-5321	306	13	.	.	PUNCT
ejpam-5321	307	1	continuous	continuous	ADJ
ejpam-5321	307	2	functions	function	NOUN
ejpam-5321	307	3	on	on	ADP
ejpam-5321	307	4	bigeneralized	bigeneralize	VERB
ejpam-5321	307	5	topological	topological	ADJ
ejpam-5321	307	6	spaces	space	NOUN
ejpam-5321	307	7	.	.	PUNCT
ejpam-5321	308	1	international	international	ADJ
ejpam-5321	308	2	journal	journal	PROPN
ejpam-5321	308	3	of	of	ADP
ejpam-5321	308	4	mathematical	mathematical	ADJ
ejpam-5321	308	5	analysis	analysis	NOUN
ejpam-5321	308	6	,	,	PUNCT
ejpam-5321	308	7	5(24):1165	5(24):1165	NUM
ejpam-5321	308	8	–	–	PUNCT
ejpam-5321	308	9	1174	1174	NUM
ejpam-5321	308	10	,	,	PUNCT
ejpam-5321	308	11	2011	2011	NUM
ejpam-5321	308	12	.	.	PUNCT
ejpam-5321	309	1	[	[	X
ejpam-5321	309	2	22	22	NUM
ejpam-5321	309	3	]	]	X
ejpam-5321	309	4	e.	e.	PROPN
ejpam-5321	309	5	ekici	ekici	PROPN
ejpam-5321	309	6	.	.	PUNCT
ejpam-5321	310	1	slightly	slightly	ADV
ejpam-5321	310	2	β	β	X
ejpam-5321	310	3	-	-	ADJ
ejpam-5321	310	4	continuous	continuous	ADJ
ejpam-5321	310	5	multifunctions	multifunction	NOUN
ejpam-5321	310	6	.	.	PUNCT
ejpam-5321	311	1	demonstratio	demonstratio	PROPN
ejpam-5321	311	2	mathematica	mathematica	PROPN
ejpam-5321	311	3	,	,	PUNCT
ejpam-5321	311	4	38(2):469–484	38(2):469–484	PROPN
ejpam-5321	311	5	,	,	PUNCT
ejpam-5321	311	6	2005	2005	NUM
ejpam-5321	311	7	.	.	PUNCT
ejpam-5321	312	1	[	[	X
ejpam-5321	312	2	23	23	NUM
ejpam-5321	312	3	]	]	X
ejpam-5321	312	4	e.	e.	PROPN
ejpam-5321	312	5	ekici	ekici	PROPN
ejpam-5321	312	6	.	.	PUNCT
ejpam-5321	313	1	upper	upper	ADJ
ejpam-5321	313	2	and	and	CCONJ
ejpam-5321	313	3	lower	low	ADJ
ejpam-5321	313	4	slightly	slightly	ADV
ejpam-5321	313	5	α	α	NUM
ejpam-5321	313	6	-	-	ADJ
ejpam-5321	313	7	continuous	continuous	ADJ
ejpam-5321	313	8	multifunctions	multifunction	NOUN
ejpam-5321	313	9	.	.	PUNCT
ejpam-5321	314	1	miskolc	miskolc	ADJ
ejpam-5321	314	2	mathematical	mathematical	ADJ
ejpam-5321	314	3	notes	note	NOUN
ejpam-5321	314	4	,	,	PUNCT
ejpam-5321	314	5	6(1):31–41	6(1):31–41	NUM
ejpam-5321	314	6	,	,	PUNCT
ejpam-5321	314	7	2005	2005	NUM
ejpam-5321	314	8	.	.	PUNCT
ejpam-5321	315	1	[	[	X
ejpam-5321	315	2	24	24	NUM
ejpam-5321	315	3	]	]	PUNCT
ejpam-5321	315	4	m.	m.	NOUN
ejpam-5321	315	5	e.	e.	PROPN
ejpam-5321	315	6	abd	abd	PROPN
ejpam-5321	316	1	el	el	PROPN
ejpam-5321	316	2	-	-	PROPN
ejpam-5321	316	3	monsef	monsef	PROPN
ejpam-5321	316	4	and	and	CCONJ
ejpam-5321	316	5	a.	a.	NOUN
ejpam-5321	316	6	a.	a.	NOUN
ejpam-5321	316	7	nasef	nasef	PROPN
ejpam-5321	316	8	.	.	PUNCT
ejpam-5321	317	1	on	on	ADP
ejpam-5321	317	2	multifunctions	multifunction	NOUN
ejpam-5321	317	3	.	.	PUNCT
ejpam-5321	318	1	chaos	chaos	NOUN
ejpam-5321	318	2	,	,	PUNCT
ejpam-5321	318	3	solitons	soliton	NOUN
ejpam-5321	318	4	&	&	CCONJ
ejpam-5321	318	5	fractals	fractal	NOUN
ejpam-5321	318	6	,	,	PUNCT
ejpam-5321	318	7	12:2387–2394	12:2387–2394	NUM
ejpam-5321	318	8	,	,	PUNCT
ejpam-5321	318	9	2001	2001	NUM
ejpam-5321	318	10	.	.	PUNCT
ejpam-5321	319	1	references	reference	NOUN
ejpam-5321	319	2	2153	2153	NUM
ejpam-5321	319	3	[	[	X
ejpam-5321	319	4	25	25	NUM
ejpam-5321	319	5	]	]	PUNCT
ejpam-5321	319	6	r.	r.	PROPN
ejpam-5321	319	7	c.	c.	PROPN
ejpam-5321	319	8	jain	jain	PROPN
ejpam-5321	319	9	.	.	PUNCT
ejpam-5321	320	1	the	the	DET
ejpam-5321	320	2	role	role	NOUN
ejpam-5321	320	3	of	of	ADP
ejpam-5321	320	4	regularly	regularly	ADV
ejpam-5321	320	5	open	open	ADJ
ejpam-5321	320	6	sets	set	NOUN
ejpam-5321	320	7	in	in	ADP
ejpam-5321	320	8	general	general	ADJ
ejpam-5321	320	9	topology	topology	NOUN
ejpam-5321	320	10	.	.	PUNCT
ejpam-5321	321	1	ph.d	ph.d	PROPN
ejpam-5321	321	2	.	.	PUNCT
ejpam-5321	322	1	thesis	thesis	PROPN
ejpam-5321	322	2	,	,	PUNCT
ejpam-5321	322	3	meerut	meerut	PROPN
ejpam-5321	322	4	university	university	PROPN
ejpam-5321	322	5	,	,	PUNCT
ejpam-5321	322	6	meerut	meerut	PROPN
ejpam-5321	322	7	,	,	PUNCT
ejpam-5321	322	8	1980	1980	NUM
ejpam-5321	322	9	.	.	PUNCT
ejpam-5321	323	1	[	[	X
ejpam-5321	323	2	26	26	NUM
ejpam-5321	323	3	]	]	X
ejpam-5321	323	4	c.	c.	PROPN
ejpam-5321	323	5	klanarong	klanarong	PROPN
ejpam-5321	323	6	,	,	PUNCT
ejpam-5321	323	7	s.	s.	PROPN
ejpam-5321	323	8	sompong	sompong	PROPN
ejpam-5321	323	9	,	,	PUNCT
ejpam-5321	323	10	and	and	CCONJ
ejpam-5321	323	11	c.	c.	PROPN
ejpam-5321	323	12	boonpok	boonpok	PROPN
ejpam-5321	323	13	.	.	PUNCT
ejpam-5321	324	1	upper	upper	ADJ
ejpam-5321	324	2	and	and	CCONJ
ejpam-5321	324	3	lower	low	ADJ
ejpam-5321	324	4	almost	almost	ADV
ejpam-5321	324	5	(	(	PUNCT
ejpam-5321	324	6	τ1	τ1	NOUN
ejpam-5321	324	7	,	,	PUNCT
ejpam-5321	324	8	τ2)continuous	τ2)continuous	ADJ
ejpam-5321	324	9	multifunctions	multifunction	NOUN
ejpam-5321	324	10	.	.	PUNCT
ejpam-5321	325	1	european	european	ADJ
ejpam-5321	325	2	journal	journal	PROPN
ejpam-5321	325	3	of	of	ADP
ejpam-5321	325	4	pure	pure	ADJ
ejpam-5321	325	5	and	and	CCONJ
ejpam-5321	325	6	applied	applied	ADJ
ejpam-5321	325	7	mathematics	mathematic	NOUN
ejpam-5321	325	8	,	,	PUNCT
ejpam-5321	325	9	17(2):1244–1253	17(2):1244–1253	NUM
ejpam-5321	325	10	,	,	PUNCT
ejpam-5321	325	11	2024	2024	NUM
ejpam-5321	325	12	.	.	PUNCT
ejpam-5321	326	1	[	[	X
ejpam-5321	326	2	27	27	NUM
ejpam-5321	326	3	]	]	PUNCT
ejpam-5321	326	4	k.	k.	PROPN
ejpam-5321	326	5	laprom	laprom	PROPN
ejpam-5321	326	6	,	,	PUNCT
ejpam-5321	326	7	c.	c.	PROPN
ejpam-5321	326	8	boonpok	boonpok	PROPN
ejpam-5321	326	9	,	,	PUNCT
ejpam-5321	326	10	and	and	CCONJ
ejpam-5321	326	11	c.	c.	PROPN
ejpam-5321	326	12	viriyapong	viriyapong	PROPN
ejpam-5321	326	13	.	.	PUNCT
ejpam-5321	327	1	β(τ1	β(τ1	PROPN
ejpam-5321	327	2	,	,	PUNCT
ejpam-5321	327	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	327	4	multifunctions	multifunction	NOUN
ejpam-5321	327	5	on	on	ADP
ejpam-5321	327	6	bitopological	bitopological	ADJ
ejpam-5321	327	7	spaces	space	NOUN
ejpam-5321	327	8	.	.	PUNCT
ejpam-5321	328	1	journal	journal	NOUN
ejpam-5321	328	2	of	of	ADP
ejpam-5321	328	3	mathematics	mathematic	NOUN
ejpam-5321	328	4	,	,	PUNCT
ejpam-5321	328	5	2020:4020971	2020:4020971	NUM
ejpam-5321	328	6	,	,	PUNCT
ejpam-5321	328	7	2020	2020	NUM
ejpam-5321	328	8	.	.	PUNCT
ejpam-5321	329	1	[	[	X
ejpam-5321	329	2	28	28	NUM
ejpam-5321	329	3	]	]	X
ejpam-5321	329	4	t.	t.	NOUN
ejpam-5321	329	5	neubrunn	neubrunn	PROPN
ejpam-5321	329	6	.	.	PUNCT
ejpam-5321	330	1	strongly	strongly	ADV
ejpam-5321	330	2	quasi	quasi	ADJ
ejpam-5321	330	3	-	-	ADJ
ejpam-5321	330	4	continuous	continuous	ADJ
ejpam-5321	330	5	multivalued	multivalued	ADJ
ejpam-5321	330	6	mappings	mapping	NOUN
ejpam-5321	330	7	.	.	PUNCT
ejpam-5321	331	1	general	general	ADJ
ejpam-5321	331	2	topology	topology	NOUN
ejpam-5321	331	3	and	and	CCONJ
ejpam-5321	331	4	its	its	PRON
ejpam-5321	331	5	relations	relation	NOUN
ejpam-5321	331	6	to	to	ADP
ejpam-5321	331	7	modern	modern	ADJ
ejpam-5321	331	8	analysis	analysis	NOUN
ejpam-5321	331	9	and	and	CCONJ
ejpam-5321	331	10	algebra	algebra	NOUN
ejpam-5321	331	11	vi	vi	PROPN
ejpam-5321	331	12	(	(	PUNCT
ejpam-5321	331	13	prague	prague	NOUN
ejpam-5321	331	14	1986	1986	NUM
ejpam-5321	331	15	)	)	PUNCT
ejpam-5321	331	16	,	,	PUNCT
ejpam-5321	331	17	heldermann	heldermann	NOUN
ejpam-5321	331	18	,	,	PUNCT
ejpam-5321	331	19	berlin	berlin	PROPN
ejpam-5321	331	20	,	,	PUNCT
ejpam-5321	331	21	pages	page	NOUN
ejpam-5321	331	22	351–359	351–359	NUM
ejpam-5321	331	23	,	,	PUNCT
ejpam-5321	331	24	1988	1988	NUM
ejpam-5321	331	25	.	.	PUNCT
ejpam-5321	332	1	[	[	X
ejpam-5321	332	2	29	29	NUM
ejpam-5321	332	3	]	]	PUNCT
ejpam-5321	332	4	t.	t.	PROPN
ejpam-5321	332	5	noiri	noiri	PROPN
ejpam-5321	332	6	.	.	PUNCT
ejpam-5321	333	1	slightly	slightly	ADV
ejpam-5321	333	2	β	β	X
ejpam-5321	333	3	-	-	ADJ
ejpam-5321	333	4	continuous	continuous	ADJ
ejpam-5321	333	5	functions	function	NOUN
ejpam-5321	333	6	.	.	PUNCT
ejpam-5321	334	1	international	international	ADJ
ejpam-5321	334	2	journal	journal	PROPN
ejpam-5321	334	3	of	of	ADP
ejpam-5321	334	4	mathematics	mathematics	PROPN
ejpam-5321	334	5	and	and	CCONJ
ejpam-5321	334	6	mathematical	mathematical	ADJ
ejpam-5321	334	7	sciences	science	NOUN
ejpam-5321	334	8	,	,	PUNCT
ejpam-5321	334	9	28(8):469–478	28(8):469–478	PROPN
ejpam-5321	334	10	,	,	PUNCT
ejpam-5321	334	11	2001	2001	NUM
ejpam-5321	334	12	.	.	PUNCT
ejpam-5321	335	1	[	[	X
ejpam-5321	335	2	30	30	NUM
ejpam-5321	335	3	]	]	PUNCT
ejpam-5321	335	4	t.	t.	PROPN
ejpam-5321	335	5	noiri	noiri	PROPN
ejpam-5321	335	6	and	and	CCONJ
ejpam-5321	335	7	g.	g.	PROPN
ejpam-5321	335	8	i.	i.	PROPN
ejpam-5321	335	9	chae	chae	PROPN
ejpam-5321	335	10	.	.	PUNCT
ejpam-5321	336	1	a	a	DET
ejpam-5321	336	2	note	note	NOUN
ejpam-5321	336	3	on	on	ADP
ejpam-5321	336	4	slightly	slightly	ADV
ejpam-5321	336	5	semi	semi	ADJ
ejpam-5321	336	6	-	-	ADJ
ejpam-5321	336	7	continuous	continuous	ADJ
ejpam-5321	336	8	functions	function	NOUN
ejpam-5321	336	9	.	.	PUNCT
ejpam-5321	337	1	bulletin	bulletin	NOUN
ejpam-5321	337	2	of	of	ADP
ejpam-5321	337	3	the	the	DET
ejpam-5321	337	4	calcutta	calcutta	NOUN
ejpam-5321	337	5	mathematical	mathematical	ADJ
ejpam-5321	337	6	society	society	NOUN
ejpam-5321	337	7	,	,	PUNCT
ejpam-5321	337	8	92(2):87–92	92(2):87–92	NUM
ejpam-5321	337	9	,	,	PUNCT
ejpam-5321	337	10	2000	2000	NUM
ejpam-5321	337	11	.	.	PUNCT
ejpam-5321	338	1	[	[	X
ejpam-5321	338	2	31	31	NUM
ejpam-5321	338	3	]	]	PUNCT
ejpam-5321	338	4	t.	t.	PROPN
ejpam-5321	338	5	noiri	noiri	PROPN
ejpam-5321	338	6	and	and	CCONJ
ejpam-5321	338	7	v.	v.	ADP
ejpam-5321	338	8	popa	popa	NOUN
ejpam-5321	338	9	.	.	PUNCT
ejpam-5321	339	1	slightly	slightly	ADV
ejpam-5321	339	2	m	m	ADJ
ejpam-5321	339	3	-	-	ADJ
ejpam-5321	339	4	continuous	continuous	ADJ
ejpam-5321	339	5	multifunctions	multifunction	NOUN
ejpam-5321	339	6	.	.	PUNCT
ejpam-5321	340	1	bulletin	bulletin	NOUN
ejpam-5321	340	2	of	of	ADP
ejpam-5321	340	3	the	the	DET
ejpam-5321	340	4	institute	institute	PROPN
ejpam-5321	340	5	of	of	ADP
ejpam-5321	340	6	mathematics	mathematics	PROPN
ejpam-5321	340	7	academia	academia	PROPN
ejpam-5321	340	8	sinica	sinica	PROPN
ejpam-5321	340	9	(	(	PUNCT
ejpam-5321	340	10	new	new	ADJ
ejpam-5321	340	11	series	series	NOUN
ejpam-5321	340	12	)	)	PUNCT
ejpam-5321	340	13	,	,	PUNCT
ejpam-5321	340	14	1(4):485–501	1(4):485–501	NUM
ejpam-5321	340	15	,	,	PUNCT
ejpam-5321	340	16	2006	2006	NUM
ejpam-5321	340	17	.	.	PUNCT
ejpam-5321	341	1	[	[	X
ejpam-5321	341	2	32	32	NUM
ejpam-5321	341	3	]	]	PUNCT
ejpam-5321	341	4	t.	t.	PROPN
ejpam-5321	341	5	m.	m.	PROPN
ejpam-5321	341	6	nour	nour	PROPN
ejpam-5321	341	7	.	.	PUNCT
ejpam-5321	342	1	slightly	slightly	ADV
ejpam-5321	342	2	semi	semi	ADJ
ejpam-5321	342	3	-	-	ADJ
ejpam-5321	342	4	continuous	continuous	ADJ
ejpam-5321	342	5	functions	function	NOUN
ejpam-5321	342	6	.	.	PUNCT
ejpam-5321	343	1	bulletin	bulletin	NOUN
ejpam-5321	343	2	of	of	ADP
ejpam-5321	343	3	the	the	DET
ejpam-5321	343	4	calcutta	calcutta	PROPN
ejpam-5321	343	5	mathematical	mathematical	ADJ
ejpam-5321	343	6	society	society	NOUN
ejpam-5321	343	7	,	,	PUNCT
ejpam-5321	343	8	87(2):187–190	87(2):187–190	PROPN
ejpam-5321	343	9	,	,	PUNCT
ejpam-5321	343	10	1995	1995	NUM
ejpam-5321	343	11	.	.	PUNCT
ejpam-5321	344	1	[	[	X
ejpam-5321	344	2	33	33	NUM
ejpam-5321	344	3	]	]	PUNCT
ejpam-5321	344	4	m.	m.	NOUN
ejpam-5321	344	5	c.	c.	PROPN
ejpam-5321	344	6	pal	pal	PROPN
ejpam-5321	344	7	and	and	CCONJ
ejpam-5321	344	8	p.	p.	NOUN
ejpam-5321	344	9	bhattacharyya	bhattacharyya	ADJ
ejpam-5321	344	10	.	.	PUNCT
ejpam-5321	345	1	faint	faint	ADJ
ejpam-5321	345	2	precontinuous	precontinuous	ADJ
ejpam-5321	345	3	functions	function	NOUN
ejpam-5321	345	4	.	.	PUNCT
ejpam-5321	346	1	soochow	soochow	PROPN
ejpam-5321	346	2	journal	journal	PROPN
ejpam-5321	346	3	of	of	ADP
ejpam-5321	346	4	mathematics	mathematic	NOUN
ejpam-5321	346	5	,	,	PUNCT
ejpam-5321	346	6	21(3):273–289	21(3):273–289	PROPN
ejpam-5321	346	7	,	,	PUNCT
ejpam-5321	346	8	1995	1995	NUM
ejpam-5321	346	9	.	.	PUNCT
ejpam-5321	347	1	[	[	X
ejpam-5321	347	2	34	34	NUM
ejpam-5321	347	3	]	]	X
ejpam-5321	347	4	v.	v.	CCONJ
ejpam-5321	347	5	popa	popa	NOUN
ejpam-5321	347	6	.	.	PUNCT
ejpam-5321	348	1	sur	sur	VERB
ejpam-5321	348	2	certain	certain	ADJ
ejpam-5321	348	3	forms	form	NOUN
ejpam-5321	348	4	faibles	faible	NOUN
ejpam-5321	348	5	de	de	X
ejpam-5321	348	6	continuite	continuite	NOUN
ejpam-5321	348	7	pour	pour	PROPN
ejpam-5321	348	8	les	les	PROPN
ejpam-5321	348	9	multifunctions	multifunction	NOUN
ejpam-5321	348	10	.	.	PUNCT
ejpam-5321	349	1	revue	revue	PROPN
ejpam-5321	349	2	roumanie	roumanie	PROPN
ejpam-5321	349	3	de	de	PROPN
ejpam-5321	349	4	mathematiques	mathematiques	PROPN
ejpam-5321	349	5	pures	pure	NOUN
ejpam-5321	349	6	et	et	PROPN
ejpam-5321	349	7	appliquees	appliquee	NOUN
ejpam-5321	349	8	,	,	PUNCT
ejpam-5321	349	9	30:539–546	30:539–546	NUM
ejpam-5321	349	10	,	,	PUNCT
ejpam-5321	349	11	1985	1985	NUM
ejpam-5321	349	12	.	.	PUNCT
ejpam-5321	350	1	[	[	X
ejpam-5321	350	2	35	35	NUM
ejpam-5321	350	3	]	]	X
ejpam-5321	350	4	v.	v.	CCONJ
ejpam-5321	350	5	popa	popa	NOUN
ejpam-5321	350	6	.	.	PUNCT
ejpam-5321	351	1	some	some	DET
ejpam-5321	351	2	properties	property	NOUN
ejpam-5321	351	3	of	of	ADP
ejpam-5321	351	4	h	h	NOUN
ejpam-5321	351	5	-	-	PUNCT
ejpam-5321	351	6	almost	almost	ADV
ejpam-5321	351	7	continuous	continuous	ADJ
ejpam-5321	351	8	multifunctions	multifunction	NOUN
ejpam-5321	351	9	.	.	PUNCT
ejpam-5321	352	1	problemy	problemy	PROPN
ejpam-5321	352	2	matematyczne	matematyczne	PROPN
ejpam-5321	352	3	,	,	PUNCT
ejpam-5321	352	4	10:9–26	10:9–26	NUM
ejpam-5321	352	5	,	,	PUNCT
ejpam-5321	352	6	1988	1988	NUM
ejpam-5321	352	7	.	.	PUNCT
ejpam-5321	353	1	[	[	X
ejpam-5321	353	2	36	36	NUM
ejpam-5321	353	3	]	]	X
ejpam-5321	353	4	v.	v.	CCONJ
ejpam-5321	353	5	popa	popa	NOUN
ejpam-5321	353	6	and	and	CCONJ
ejpam-5321	353	7	t.	t.	PROPN
ejpam-5321	353	8	noiri	noiri	PROPN
ejpam-5321	353	9	.	.	PUNCT
ejpam-5321	354	1	on	on	ADP
ejpam-5321	354	2	upper	upper	ADJ
ejpam-5321	354	3	and	and	CCONJ
ejpam-5321	354	4	lower	low	ADJ
ejpam-5321	354	5	β	β	ADJ
ejpam-5321	354	6	-	-	ADJ
ejpam-5321	354	7	continuous	continuous	ADJ
ejpam-5321	354	8	multifunctions	multifunction	NOUN
ejpam-5321	354	9	.	.	PUNCT
ejpam-5321	355	1	real	real	ADJ
ejpam-5321	355	2	analysis	analysis	NOUN
ejpam-5321	355	3	exchange	exchange	NOUN
ejpam-5321	355	4	,	,	PUNCT
ejpam-5321	355	5	22:362–376	22:362–376	PROPN
ejpam-5321	355	6	,	,	PUNCT
ejpam-5321	355	7	1996/97	1996/97	NUM
ejpam-5321	355	8	.	.	PUNCT
ejpam-5321	356	1	[	[	X
ejpam-5321	356	2	37	37	NUM
ejpam-5321	356	3	]	]	X
ejpam-5321	356	4	p.	p.	NOUN
ejpam-5321	356	5	pue	pue	NOUN
ejpam-5321	356	6	-	-	PUNCT
ejpam-5321	356	7	on	on	ADP
ejpam-5321	356	8	and	and	CCONJ
ejpam-5321	356	9	c.	c.	PROPN
ejpam-5321	356	10	boonpok	boonpok	PROPN
ejpam-5321	356	11	.	.	PUNCT
ejpam-5321	357	1	θ(λ	θ(λ	PROPN
ejpam-5321	357	2	,	,	PUNCT
ejpam-5321	357	3	p)-continuity	p)-continuity	NOUN
ejpam-5321	357	4	for	for	ADP
ejpam-5321	357	5	functions	function	NOUN
ejpam-5321	357	6	.	.	PUNCT
ejpam-5321	358	1	international	international	ADJ
ejpam-5321	358	2	journal	journal	NOUN
ejpam-5321	358	3	of	of	ADP
ejpam-5321	358	4	mathematics	mathematic	NOUN
ejpam-5321	358	5	and	and	CCONJ
ejpam-5321	358	6	computer	computer	NOUN
ejpam-5321	358	7	science	science	NOUN
ejpam-5321	358	8	,	,	PUNCT
ejpam-5321	358	9	19(2):491–495	19(2):491–495	NUM
ejpam-5321	358	10	,	,	PUNCT
ejpam-5321	358	11	2024	2024	NUM
ejpam-5321	358	12	.	.	PUNCT
ejpam-5321	359	1	[	[	X
ejpam-5321	359	2	38	38	NUM
ejpam-5321	359	3	]	]	PUNCT
ejpam-5321	359	4	p.	p.	NOUN
ejpam-5321	359	5	pue	pue	NOUN
ejpam-5321	359	6	-	-	PUNCT
ejpam-5321	359	7	on	on	ADP
ejpam-5321	359	8	,	,	PUNCT
ejpam-5321	359	9	s.	s.	PROPN
ejpam-5321	359	10	sompong	sompong	PROPN
ejpam-5321	359	11	,	,	PUNCT
ejpam-5321	359	12	and	and	CCONJ
ejpam-5321	359	13	c.	c.	PROPN
ejpam-5321	359	14	boonpok	boonpok	PROPN
ejpam-5321	359	15	.	.	PUNCT
ejpam-5321	360	1	upper	upper	ADJ
ejpam-5321	360	2	and	and	CCONJ
ejpam-5321	360	3	lower	low	ADJ
ejpam-5321	360	4	(	(	PUNCT
ejpam-5321	360	5	τ1	τ1	NOUN
ejpam-5321	360	6	,	,	PUNCT
ejpam-5321	360	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5321	360	8	multifunctions	multifunction	NOUN
ejpam-5321	360	9	.	.	PUNCT
ejpam-5321	361	1	international	international	ADJ
ejpam-5321	361	2	journal	journal	PROPN
ejpam-5321	361	3	of	of	ADP
ejpam-5321	361	4	mathematics	mathematic	NOUN
ejpam-5321	361	5	and	and	CCONJ
ejpam-5321	361	6	computer	computer	NOUN
ejpam-5321	361	7	science	science	NOUN
ejpam-5321	361	8	,	,	PUNCT
ejpam-5321	361	9	19(4):1305	19(4):1305	NUM
ejpam-5321	361	10	–	–	PUNCT
ejpam-5321	361	11	1310	1310	NUM
ejpam-5321	361	12	,	,	PUNCT
ejpam-5321	361	13	2024	2024	NUM
ejpam-5321	361	14	.	.	PUNCT
ejpam-5321	362	1	[	[	X
ejpam-5321	362	2	39	39	NUM
ejpam-5321	362	3	]	]	PUNCT
ejpam-5321	362	4	p.	p.	NOUN
ejpam-5321	362	5	sangviset	sangviset	PROPN
ejpam-5321	362	6	,	,	PUNCT
ejpam-5321	362	7	c.	c.	PROPN
ejpam-5321	362	8	boonpok	boonpok	PROPN
ejpam-5321	362	9	,	,	PUNCT
ejpam-5321	362	10	and	and	CCONJ
ejpam-5321	362	11	c.	c.	PROPN
ejpam-5321	362	12	viriyapong	viriyapong	PROPN
ejpam-5321	362	13	.	.	PUNCT
ejpam-5321	363	1	slightly	slightly	ADV
ejpam-5321	363	2	(	(	PUNCT
ejpam-5321	363	3	m,µ)-continuous	m,µ)-continuous	ADJ
ejpam-5321	363	4	functions	function	NOUN
ejpam-5321	363	5	.	.	PUNCT
ejpam-5321	364	1	far	far	PROPN
ejpam-5321	364	2	east	east	PROPN
ejpam-5321	364	3	journal	journal	PROPN
ejpam-5321	364	4	of	of	ADP
ejpam-5321	364	5	mathematical	mathematical	ADJ
ejpam-5321	364	6	sciences	sciences	PROPN
ejpam-5321	364	7	,	,	PUNCT
ejpam-5321	364	8	85(2):165–176	85(2):165–176	NOUN
ejpam-5321	364	9	,	,	PUNCT
ejpam-5321	364	10	2014	2014	NUM
ejpam-5321	364	11	.	.	PUNCT
ejpam-5321	365	1	references	reference	NOUN
ejpam-5321	365	2	2154	2154	NUM
ejpam-5321	365	3	[	[	X
ejpam-5321	365	4	40	40	NUM
ejpam-5321	365	5	]	]	X
ejpam-5321	365	6	n.	n.	PROPN
ejpam-5321	365	7	srisarakham	srisarakham	PROPN
ejpam-5321	365	8	and	and	CCONJ
ejpam-5321	365	9	c.	c.	PROPN
ejpam-5321	365	10	boonpok	boonpok	PROPN
ejpam-5321	365	11	.	.	PUNCT
ejpam-5321	366	1	almost	almost	ADV
ejpam-5321	366	2	(	(	PUNCT
ejpam-5321	366	3	λ	λ	NOUN
ejpam-5321	366	4	,	,	PUNCT
ejpam-5321	366	5	p)-continuous	p)-continuous	ADJ
ejpam-5321	366	6	functions	function	NOUN
ejpam-5321	366	7	.	.	PUNCT
ejpam-5321	367	1	international	international	ADJ
ejpam-5321	367	2	journal	journal	PROPN
ejpam-5321	367	3	of	of	ADP
ejpam-5321	367	4	mathematics	mathematic	NOUN
ejpam-5321	367	5	and	and	CCONJ
ejpam-5321	367	6	computer	computer	NOUN
ejpam-5321	367	7	science	science	NOUN
ejpam-5321	367	8	,	,	PUNCT
ejpam-5321	367	9	18(2):255–259	18(2):255–259	NUM
ejpam-5321	367	10	,	,	PUNCT
ejpam-5321	367	11	2023	2023	NUM
ejpam-5321	367	12	.	.	PUNCT
ejpam-5321	368	1	[	[	X
ejpam-5321	368	2	41	41	NUM
ejpam-5321	368	3	]	]	X
ejpam-5321	368	4	n.	n.	NOUN
ejpam-5321	368	5	srisarakham	srisarakham	PROPN
ejpam-5321	368	6	and	and	CCONJ
ejpam-5321	368	7	c.	c.	PROPN
ejpam-5321	368	8	boonpok	boonpok	PROPN
ejpam-5321	368	9	.	.	PUNCT
ejpam-5321	369	1	on	on	ADP
ejpam-5321	369	2	characterizations	characterization	NOUN
ejpam-5321	369	3	of	of	ADP
ejpam-5321	369	4	δp(λ	δp(λ	NOUN
ejpam-5321	369	5	,	,	PUNCT
ejpam-5321	369	6	s)-d1	s)-d1	NOUN
ejpam-5321	369	7	spaces	space	NOUN
ejpam-5321	369	8	.	.	PUNCT
ejpam-5321	370	1	international	international	ADJ
ejpam-5321	370	2	journal	journal	PROPN
ejpam-5321	370	3	of	of	ADP
ejpam-5321	370	4	mathematics	mathematic	NOUN
ejpam-5321	370	5	and	and	CCONJ
ejpam-5321	370	6	computer	computer	NOUN
ejpam-5321	370	7	science	science	NOUN
ejpam-5321	370	8	,	,	PUNCT
ejpam-5321	370	9	18(4):743–747	18(4):743–747	PROPN
ejpam-5321	370	10	,	,	PUNCT
ejpam-5321	370	11	2023	2023	NUM
ejpam-5321	370	12	.	.	PUNCT
ejpam-5321	371	1	[	[	X
ejpam-5321	371	2	42	42	NUM
ejpam-5321	371	3	]	]	PUNCT
ejpam-5321	371	4	m.	m.	NOUN
ejpam-5321	371	5	thongmoon	thongmoon	NOUN
ejpam-5321	371	6	and	and	CCONJ
ejpam-5321	371	7	c.	c.	PROPN
ejpam-5321	371	8	boonpok	boonpok	PROPN
ejpam-5321	371	9	.	.	PUNCT
ejpam-5321	372	1	strongly	strongly	ADV
ejpam-5321	372	2	θ(λ	θ(λ	PROPN
ejpam-5321	372	3	,	,	PUNCT
ejpam-5321	372	4	p)-continuous	p)-continuous	ADJ
ejpam-5321	372	5	functions	function	NOUN
ejpam-5321	372	6	.	.	PUNCT
ejpam-5321	373	1	international	international	ADJ
ejpam-5321	373	2	journal	journal	PROPN
ejpam-5321	373	3	of	of	ADP
ejpam-5321	373	4	mathematics	mathematic	NOUN
ejpam-5321	373	5	and	and	CCONJ
ejpam-5321	373	6	computer	computer	NOUN
ejpam-5321	373	7	science	science	NOUN
ejpam-5321	373	8	,	,	PUNCT
ejpam-5321	373	9	19(2):475–479	19(2):475–479	PROPN
ejpam-5321	373	10	,	,	PUNCT
ejpam-5321	373	11	2024	2024	NUM
ejpam-5321	373	12	.	.	PUNCT
ejpam-5321	374	1	[	[	X
ejpam-5321	374	2	43	43	NUM
ejpam-5321	374	3	]	]	X
ejpam-5321	374	4	c.	c.	PROPN
ejpam-5321	374	5	viriyapong	viriyapong	PROPN
ejpam-5321	374	6	and	and	CCONJ
ejpam-5321	374	7	c.	c.	PROPN
ejpam-5321	374	8	boonpok	boonpok	PROPN
ejpam-5321	374	9	.	.	PUNCT
ejpam-5321	375	1	(	(	PUNCT
ejpam-5321	375	2	τ1	τ1	NOUN
ejpam-5321	375	3	,	,	PUNCT
ejpam-5321	375	4	τ2)α	τ2)α	NOUN
ejpam-5321	375	5	-	-	PUNCT
ejpam-5321	375	6	continuity	continuity	NOUN
ejpam-5321	375	7	for	for	ADP
ejpam-5321	375	8	multifunctions	multifunction	NOUN
ejpam-5321	375	9	.	.	PUNCT
ejpam-5321	376	1	journal	journal	PROPN
ejpam-5321	376	2	of	of	ADP
ejpam-5321	376	3	mathematics	mathematic	NOUN
ejpam-5321	376	4	,	,	PUNCT
ejpam-5321	376	5	2020:6285763	2020:6285763	NUM
ejpam-5321	376	6	,	,	PUNCT
ejpam-5321	376	7	2020	2020	NUM
ejpam-5321	376	8	.	.	PUNCT
ejpam-5321	377	1	[	[	X
ejpam-5321	377	2	44	44	NUM
ejpam-5321	377	3	]	]	PUNCT
ejpam-5321	377	4	c.	c.	PROPN
ejpam-5321	377	5	viriyapong	viriyapong	PROPN
ejpam-5321	377	6	and	and	CCONJ
ejpam-5321	377	7	c.	c.	PROPN
ejpam-5321	377	8	boonpok	boonpok	PROPN
ejpam-5321	377	9	.	.	PUNCT
ejpam-5321	378	1	(	(	PUNCT
ejpam-5321	378	2	λ	λ	X
ejpam-5321	378	3	,	,	PUNCT
ejpam-5321	378	4	sp)-continuous	sp)-continuous	ADJ
ejpam-5321	378	5	functions	function	NOUN
ejpam-5321	378	6	.	.	PUNCT
ejpam-5321	379	1	wseas	wseas	VERB
ejpam-5321	379	2	transactions	transaction	NOUN
ejpam-5321	379	3	on	on	ADP
ejpam-5321	379	4	mathematics	mathematic	NOUN
ejpam-5321	379	5	,	,	PUNCT
ejpam-5321	379	6	21:380–385	21:380–385	NUM
ejpam-5321	379	7	,	,	PUNCT
ejpam-5321	379	8	2022	2022	NUM
ejpam-5321	379	9	.	.	PUNCT
ejpam-5321	380	1	[	[	X
ejpam-5321	380	2	45	45	NUM
ejpam-5321	380	3	]	]	X
ejpam-5321	380	4	n.	n.	PROPN
ejpam-5321	380	5	viriyapong	viriyapong	PROPN
ejpam-5321	380	6	,	,	PUNCT
ejpam-5321	380	7	s.	s.	PROPN
ejpam-5321	380	8	sompong	sompong	PROPN
ejpam-5321	380	9	,	,	PUNCT
ejpam-5321	380	10	and	and	CCONJ
ejpam-5321	380	11	c.	c.	PROPN
ejpam-5321	380	12	boonpok	boonpok	PROPN
ejpam-5321	380	13	.	.	PUNCT
ejpam-5321	381	1	(	(	PUNCT
ejpam-5321	381	2	τ1	τ1	NOUN
ejpam-5321	381	3	,	,	PUNCT
ejpam-5321	381	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5321	381	5	disconnectedness	disconnectedness	NOUN
ejpam-5321	381	6	in	in	ADP
ejpam-5321	381	7	bitopological	bitopological	ADJ
ejpam-5321	381	8	spaces	space	NOUN
ejpam-5321	381	9	.	.	PUNCT
ejpam-5321	382	1	international	international	ADJ
ejpam-5321	382	2	journal	journal	PROPN
ejpam-5321	382	3	of	of	ADP
ejpam-5321	382	4	mathematics	mathematic	NOUN
ejpam-5321	382	5	and	and	CCONJ
ejpam-5321	382	6	computer	computer	NOUN
ejpam-5321	382	7	science	science	NOUN
ejpam-5321	382	8	,	,	PUNCT
ejpam-5321	382	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5321	382	10	,	,	PUNCT
ejpam-5321	382	11	2024	2024	NUM
ejpam-5321	382	12	.	.	PUNCT