id	sid	tid	token	lemma	pos
ejpam-6039	1	1	european	european	PROPN
ejpam-6039	1	2	journal	journal	PROPN
ejpam-6039	1	3	of	of	ADP
ejpam-6039	1	4	pure	pure	ADJ
ejpam-6039	1	5	and	and	CCONJ
ejpam-6039	1	6	applied	applied	ADJ
ejpam-6039	1	7	mathematics	mathematic	NOUN
ejpam-6039	1	8	2025	2025	NUM
ejpam-6039	1	9	,	,	PUNCT
ejpam-6039	1	10	vol	vol	NOUN
ejpam-6039	1	11	.	.	PROPN
ejpam-6039	1	12	18	18	NUM
ejpam-6039	1	13	,	,	PUNCT
ejpam-6039	1	14	issue	issue	NOUN
ejpam-6039	1	15	2	2	NUM
ejpam-6039	1	16	,	,	PUNCT
ejpam-6039	1	17	article	article	NOUN
ejpam-6039	1	18	number	number	NOUN
ejpam-6039	1	19	6039	6039	NUM
ejpam-6039	1	20	issn	issn	VERB
ejpam-6039	1	21	1307	1307	NUM
ejpam-6039	1	22	-	-	SYM
ejpam-6039	1	23	5543	5543	NUM
ejpam-6039	1	24	–	–	PUNCT
ejpam-6039	1	25	ejpam.com	ejpam.com	X
ejpam-6039	1	26	published	publish	VERB
ejpam-6039	1	27	by	by	ADP
ejpam-6039	1	28	new	new	PROPN
ejpam-6039	1	29	york	york	PROPN
ejpam-6039	1	30	business	business	PROPN
ejpam-6039	1	31	global	global	PROPN
ejpam-6039	1	32	on	on	ADP
ejpam-6039	1	33	contra-(τ1	contra-(τ1	PROPN
ejpam-6039	1	34	,	,	PUNCT
ejpam-6039	1	35	τ2)p	τ2)p	ADJ
ejpam-6039	1	36	-	-	PUNCT
ejpam-6039	1	37	continuous	continuous	ADJ
ejpam-6039	1	38	functions	function	NOUN
ejpam-6039	1	39	monchaya	monchaya	NOUN
ejpam-6039	1	40	chiangpradit1	chiangpradit1	PROPN
ejpam-6039	1	41	,	,	PUNCT
ejpam-6039	1	42	supunnee	supunnee	PROPN
ejpam-6039	1	43	sompong2	sompong2	PROPN
ejpam-6039	1	44	,	,	PUNCT
ejpam-6039	1	45	chawalit	chawalit	VERB
ejpam-6039	1	46	boonpok1,∗	boonpok1,∗	NOUN
ejpam-6039	1	47	1	1	NUM
ejpam-6039	1	48	mathematics	mathematic	NOUN
ejpam-6039	1	49	and	and	CCONJ
ejpam-6039	1	50	applied	apply	VERB
ejpam-6039	1	51	mathematics	mathematics	PROPN
ejpam-6039	1	52	research	research	NOUN
ejpam-6039	1	53	unit	unit	NOUN
ejpam-6039	1	54	,	,	PUNCT
ejpam-6039	1	55	department	department	NOUN
ejpam-6039	1	56	of	of	ADP
ejpam-6039	1	57	mathematics	mathematic	NOUN
ejpam-6039	1	58	,	,	PUNCT
ejpam-6039	1	59	faculty	faculty	NOUN
ejpam-6039	1	60	of	of	ADP
ejpam-6039	1	61	science	science	NOUN
ejpam-6039	1	62	,	,	PUNCT
ejpam-6039	1	63	mahasarakham	mahasarakham	PROPN
ejpam-6039	1	64	university	university	PROPN
ejpam-6039	1	65	,	,	PUNCT
ejpam-6039	1	66	maha	maha	PROPN
ejpam-6039	1	67	sarakham	sarakham	PROPN
ejpam-6039	1	68	,	,	PUNCT
ejpam-6039	1	69	44150	44150	NUM
ejpam-6039	1	70	,	,	PUNCT
ejpam-6039	1	71	thailand	thailand	PROPN
ejpam-6039	1	72	2	2	NUM
ejpam-6039	1	73	department	department	NOUN
ejpam-6039	1	74	of	of	ADP
ejpam-6039	1	75	mathematics	mathematic	NOUN
ejpam-6039	1	76	and	and	CCONJ
ejpam-6039	1	77	statistics	statistic	NOUN
ejpam-6039	1	78	,	,	PUNCT
ejpam-6039	1	79	faculty	faculty	NOUN
ejpam-6039	1	80	of	of	ADP
ejpam-6039	1	81	science	science	NOUN
ejpam-6039	1	82	and	and	CCONJ
ejpam-6039	1	83	technology	technology	NOUN
ejpam-6039	1	84	,	,	PUNCT
ejpam-6039	1	85	sakon	sakon	PROPN
ejpam-6039	1	86	nakhon	nakhon	PROPN
ejpam-6039	1	87	rajbhat	rajbhat	PROPN
ejpam-6039	1	88	university	university	PROPN
ejpam-6039	1	89	,	,	PUNCT
ejpam-6039	1	90	sakon	sakon	PROPN
ejpam-6039	1	91	nakhon	nakhon	PROPN
ejpam-6039	1	92	,	,	PUNCT
ejpam-6039	1	93	47000	47000	NUM
ejpam-6039	1	94	,	,	PUNCT
ejpam-6039	1	95	thailand	thailand	PROPN
ejpam-6039	1	96	abstract	abstract	NOUN
ejpam-6039	1	97	.	.	PUNCT
ejpam-6039	2	1	this	this	DET
ejpam-6039	2	2	paper	paper	NOUN
ejpam-6039	2	3	presents	present	VERB
ejpam-6039	2	4	a	a	DET
ejpam-6039	2	5	new	new	ADJ
ejpam-6039	2	6	class	class	NOUN
ejpam-6039	2	7	of	of	ADP
ejpam-6039	2	8	functions	function	NOUN
ejpam-6039	2	9	called	call	VERB
ejpam-6039	2	10	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	2	11	,	,	PUNCT
ejpam-6039	2	12	τ2)p	τ2)p	ADJ
ejpam-6039	2	13	-	-	PUNCT
ejpam-6039	2	14	continuous	continuous	ADJ
ejpam-6039	2	15	functions	function	NOUN
ejpam-6039	2	16	.	.	PUNCT
ejpam-6039	3	1	furthermore	furthermore	ADV
ejpam-6039	3	2	,	,	PUNCT
ejpam-6039	3	3	several	several	ADJ
ejpam-6039	3	4	characterizations	characterization	NOUN
ejpam-6039	3	5	and	and	CCONJ
ejpam-6039	3	6	some	some	DET
ejpam-6039	3	7	properties	property	NOUN
ejpam-6039	3	8	concerning	concern	VERB
ejpam-6039	3	9	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	3	10	,	,	PUNCT
ejpam-6039	3	11	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-6039	3	12	functions	function	NOUN
ejpam-6039	3	13	are	be	AUX
ejpam-6039	3	14	considered	consider	VERB
ejpam-6039	3	15	.	.	PUNCT
ejpam-6039	4	1	2020	2020	NUM
ejpam-6039	4	2	mathematics	mathematic	NOUN
ejpam-6039	4	3	subject	subject	NOUN
ejpam-6039	4	4	classifications	classification	NOUN
ejpam-6039	4	5	:	:	PUNCT
ejpam-6039	4	6	54c08	54c08	NUM
ejpam-6039	4	7	,	,	PUNCT
ejpam-6039	4	8	54e55	54e55	NUM
ejpam-6039	4	9	key	key	ADJ
ejpam-6039	4	10	words	word	NOUN
ejpam-6039	4	11	and	and	CCONJ
ejpam-6039	4	12	phrases	phrase	NOUN
ejpam-6039	4	13	:	:	PUNCT
ejpam-6039	4	14	τ1τ2	τ1τ2	ADJ
ejpam-6039	4	15	-	-	ADJ
ejpam-6039	4	16	open	open	ADJ
ejpam-6039	4	17	set	set	NOUN
ejpam-6039	4	18	,	,	PUNCT
ejpam-6039	4	19	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	4	20	,	,	PUNCT
ejpam-6039	4	21	τ2)p	τ2)p	ADJ
ejpam-6039	4	22	-	-	ADJ
ejpam-6039	4	23	continuous	continuous	ADJ
ejpam-6039	4	24	function	function	NOUN
ejpam-6039	4	25	1	1	NUM
ejpam-6039	4	26	.	.	PUNCT
ejpam-6039	5	1	introduction	introduction	NOUN
ejpam-6039	5	2	the	the	DET
ejpam-6039	5	3	notions	notion	NOUN
ejpam-6039	5	4	of	of	ADP
ejpam-6039	5	5	contra	contra	PROPN
ejpam-6039	5	6	-	-	NOUN
ejpam-6039	5	7	continuity	continuity	NOUN
ejpam-6039	5	8	and	and	CCONJ
ejpam-6039	5	9	strong	strong	ADJ
ejpam-6039	5	10	s	s	NOUN
ejpam-6039	5	11	-	-	NOUN
ejpam-6039	5	12	closedness	closedness	ADJ
ejpam-6039	5	13	in	in	ADP
ejpam-6039	5	14	topological	topological	ADJ
ejpam-6039	5	15	spaces	space	NOUN
ejpam-6039	5	16	were	be	AUX
ejpam-6039	5	17	introduced	introduce	VERB
ejpam-6039	5	18	by	by	ADP
ejpam-6039	5	19	dontchev	dontchev	NOUN
ejpam-6039	5	20	[	[	X
ejpam-6039	5	21	1	1	NUM
ejpam-6039	5	22	]	]	PUNCT
ejpam-6039	5	23	.	.	PUNCT
ejpam-6039	6	1	furthermore	furthermore	ADV
ejpam-6039	6	2	,	,	PUNCT
ejpam-6039	6	3	dontchev	dontchev	ADJ
ejpam-6039	6	4	[	[	X
ejpam-6039	6	5	1	1	NUM
ejpam-6039	6	6	]	]	PUNCT
ejpam-6039	6	7	obtained	obtain	VERB
ejpam-6039	6	8	very	very	ADV
ejpam-6039	6	9	interesting	interesting	ADJ
ejpam-6039	6	10	and	and	CCONJ
ejpam-6039	6	11	important	important	ADJ
ejpam-6039	6	12	results	result	NOUN
ejpam-6039	6	13	concerning	concern	VERB
ejpam-6039	6	14	contra	contra	NOUN
ejpam-6039	6	15	-	-	ADJ
ejpam-6039	6	16	continuity	continuity	NOUN
ejpam-6039	6	17	,	,	PUNCT
ejpam-6039	6	18	compactness	compactness	NOUN
ejpam-6039	6	19	,	,	PUNCT
ejpam-6039	6	20	s	s	NOUN
ejpam-6039	6	21	-	-	PUNCT
ejpam-6039	6	22	closedness	closedness	ADJ
ejpam-6039	6	23	and	and	CCONJ
ejpam-6039	6	24	strong	strong	ADJ
ejpam-6039	6	25	sclosedness	sclosedness	NOUN
ejpam-6039	6	26	.	.	PUNCT
ejpam-6039	7	1	dontchev	dontchev	NOUN
ejpam-6039	7	2	and	and	CCONJ
ejpam-6039	7	3	noiri	noiri	ADV
ejpam-6039	8	1	[	[	X
ejpam-6039	8	2	2	2	NUM
ejpam-6039	8	3	]	]	PUNCT
ejpam-6039	8	4	introduced	introduce	VERB
ejpam-6039	8	5	and	and	CCONJ
ejpam-6039	8	6	studied	study	VERB
ejpam-6039	8	7	the	the	DET
ejpam-6039	8	8	concept	concept	NOUN
ejpam-6039	8	9	of	of	ADP
ejpam-6039	8	10	rc	rc	NOUN
ejpam-6039	8	11	-	-	NOUN
ejpam-6039	8	12	continuity	continuity	NOUN
ejpam-6039	8	13	between	between	ADP
ejpam-6039	8	14	topological	topological	ADJ
ejpam-6039	8	15	spaces	space	NOUN
ejpam-6039	8	16	which	which	PRON
ejpam-6039	8	17	is	be	AUX
ejpam-6039	8	18	weaker	weak	ADJ
ejpam-6039	8	19	than	than	ADP
ejpam-6039	8	20	contra	contra	NOUN
ejpam-6039	8	21	-	-	NOUN
ejpam-6039	8	22	continuity	continuity	NOUN
ejpam-6039	8	23	.	.	PUNCT
ejpam-6039	9	1	jafari	jafari	PROPN
ejpam-6039	9	2	and	and	CCONJ
ejpam-6039	9	3	noiri	noiri	ADV
ejpam-6039	10	1	[	[	X
ejpam-6039	10	2	3	3	X
ejpam-6039	10	3	]	]	PUNCT
ejpam-6039	10	4	introduced	introduce	VERB
ejpam-6039	10	5	and	and	CCONJ
ejpam-6039	10	6	investigated	investigate	VERB
ejpam-6039	10	7	a	a	DET
ejpam-6039	10	8	new	new	ADJ
ejpam-6039	10	9	class	class	NOUN
ejpam-6039	10	10	of	of	ADP
ejpam-6039	10	11	functions	function	NOUN
ejpam-6039	10	12	called	call	VERB
ejpam-6039	10	13	contra	contra	NOUN
ejpam-6039	10	14	-	-	PUNCT
ejpam-6039	10	15	super	super	ADJ
ejpam-6039	10	16	-	-	ADJ
ejpam-6039	10	17	continuous	continuous	ADJ
ejpam-6039	10	18	functions	function	NOUN
ejpam-6039	10	19	which	which	PRON
ejpam-6039	10	20	lies	lie	VERB
ejpam-6039	10	21	between	between	ADP
ejpam-6039	10	22	classes	class	NOUN
ejpam-6039	10	23	of	of	ADP
ejpam-6039	10	24	rc	rc	PROPN
ejpam-6039	10	25	-	-	ADJ
ejpam-6039	10	26	continuous	continuous	ADJ
ejpam-6039	10	27	functions	function	NOUN
ejpam-6039	10	28	and	and	CCONJ
ejpam-6039	10	29	contra	contra	ADJ
ejpam-6039	10	30	-	-	ADJ
ejpam-6039	10	31	continuous	continuous	ADJ
ejpam-6039	10	32	functions	function	NOUN
ejpam-6039	10	33	.	.	PUNCT
ejpam-6039	11	1	in	in	ADP
ejpam-6039	11	2	2002	2002	NUM
ejpam-6039	11	3	,	,	PUNCT
ejpam-6039	11	4	jafari	jafari	ADJ
ejpam-6039	11	5	and	and	CCONJ
ejpam-6039	11	6	noiri	noiri	ADV
ejpam-6039	11	7	[	[	X
ejpam-6039	11	8	4	4	X
ejpam-6039	11	9	]	]	PUNCT
ejpam-6039	11	10	introduced	introduce	VERB
ejpam-6039	11	11	a	a	DET
ejpam-6039	11	12	new	new	ADJ
ejpam-6039	11	13	class	class	NOUN
ejpam-6039	11	14	of	of	ADP
ejpam-6039	11	15	function	function	NOUN
ejpam-6039	11	16	called	call	VERB
ejpam-6039	11	17	contraprecontinuous	contraprecontinuous	ADJ
ejpam-6039	11	18	functions	function	NOUN
ejpam-6039	11	19	which	which	PRON
ejpam-6039	11	20	is	be	AUX
ejpam-6039	11	21	weaker	weak	ADJ
ejpam-6039	11	22	than	than	ADP
ejpam-6039	11	23	contra	contra	ADJ
ejpam-6039	11	24	-	-	ADJ
ejpam-6039	11	25	continuous	continuous	ADJ
ejpam-6039	11	26	functions	function	NOUN
ejpam-6039	11	27	and	and	CCONJ
ejpam-6039	11	28	studied	study	VERB
ejpam-6039	11	29	several	several	ADJ
ejpam-6039	11	30	basic	basic	ADJ
ejpam-6039	11	31	properties	property	NOUN
ejpam-6039	11	32	of	of	ADP
ejpam-6039	11	33	contra	contra	ADJ
ejpam-6039	11	34	-	-	ADJ
ejpam-6039	11	35	precontinuous	precontinuous	ADJ
ejpam-6039	11	36	functions	function	NOUN
ejpam-6039	11	37	.	.	PUNCT
ejpam-6039	12	1	moreover	moreover	ADV
ejpam-6039	12	2	,	,	PUNCT
ejpam-6039	12	3	the	the	DET
ejpam-6039	12	4	present	present	ADJ
ejpam-6039	12	5	authors	author	NOUN
ejpam-6039	12	6	[	[	X
ejpam-6039	12	7	4	4	NUM
ejpam-6039	12	8	]	]	PUNCT
ejpam-6039	12	9	defined	define	VERB
ejpam-6039	12	10	contra	contra	PROPN
ejpam-6039	12	11	-	-	ADJ
ejpam-6039	12	12	preclosed	preclose	VERB
ejpam-6039	12	13	graphs	graph	NOUN
ejpam-6039	12	14	and	and	CCONJ
ejpam-6039	12	15	investigated	investigate	VERB
ejpam-6039	12	16	relations	relation	NOUN
ejpam-6039	12	17	between	between	ADP
ejpam-6039	12	18	contra	contra	PROPN
ejpam-6039	12	19	-	-	NOUN
ejpam-6039	12	20	precontinuity	precontinuity	NOUN
ejpam-6039	12	21	and	and	CCONJ
ejpam-6039	12	22	contra	contra	PROPN
ejpam-6039	12	23	-	-	PUNCT
ejpam-6039	12	24	preclosed	preclose	VERB
ejpam-6039	12	25	graphs	graph	NOUN
ejpam-6039	12	26	.	.	PUNCT
ejpam-6039	13	1	in	in	ADP
ejpam-6039	13	2	2004	2004	NUM
ejpam-6039	13	3	,	,	PUNCT
ejpam-6039	13	4	ekici	ekici	NOUN
ejpam-6039	13	5	[	[	X
ejpam-6039	13	6	5	5	NUM
ejpam-6039	13	7	]	]	PUNCT
ejpam-6039	13	8	introduced	introduce	VERB
ejpam-6039	13	9	and	and	CCONJ
ejpam-6039	13	10	studied	study	VERB
ejpam-6039	13	11	a	a	DET
ejpam-6039	13	12	new	new	ADJ
ejpam-6039	13	13	class	class	NOUN
ejpam-6039	13	14	of	of	ADP
ejpam-6039	13	15	functions	function	NOUN
ejpam-6039	13	16	called	call	VERB
ejpam-6039	13	17	almost	almost	ADV
ejpam-6039	13	18	contra	contra	ADJ
ejpam-6039	13	19	-	-	ADJ
ejpam-6039	13	20	precontinuous	precontinuous	ADJ
ejpam-6039	13	21	functions	function	NOUN
ejpam-6039	13	22	which	which	PRON
ejpam-6039	13	23	generalize	generalize	VERB
ejpam-6039	13	24	classes	class	NOUN
ejpam-6039	13	25	of	of	ADP
ejpam-6039	13	26	regular	regular	ADJ
ejpam-6039	13	27	setconnected	setconnecte	VERB
ejpam-6039	13	28	functions	function	NOUN
ejpam-6039	13	29	[	[	X
ejpam-6039	13	30	6	6	NUM
ejpam-6039	13	31	]	]	PUNCT
ejpam-6039	13	32	,	,	PUNCT
ejpam-6039	13	33	contra	contra	ADJ
ejpam-6039	13	34	-	-	ADJ
ejpam-6039	13	35	precontinuous	precontinuous	ADJ
ejpam-6039	13	36	functions	function	NOUN
ejpam-6039	13	37	[	[	X
ejpam-6039	13	38	4	4	NUM
ejpam-6039	13	39	]	]	PUNCT
ejpam-6039	13	40	,	,	PUNCT
ejpam-6039	13	41	contra	contra	ADJ
ejpam-6039	13	42	-	-	ADJ
ejpam-6039	13	43	continuous	continuous	ADJ
ejpam-6039	13	44	functions	function	NOUN
ejpam-6039	13	45	[	[	X
ejpam-6039	13	46	1	1	NUM
ejpam-6039	13	47	]	]	PUNCT
ejpam-6039	13	48	,	,	PUNCT
ejpam-6039	13	49	almost	almost	ADV
ejpam-6039	13	50	s	s	NOUN
ejpam-6039	13	51	-	-	ADJ
ejpam-6039	13	52	continuous	continuous	ADJ
ejpam-6039	13	53	functions	function	NOUN
ejpam-6039	13	54	[	[	X
ejpam-6039	13	55	7	7	NUM
ejpam-6039	13	56	]	]	PUNCT
ejpam-6039	13	57	and	and	CCONJ
ejpam-6039	13	58	perfectly	perfectly	ADV
ejpam-6039	13	59	continuous	continuous	ADJ
ejpam-6039	13	60	functions	function	NOUN
ejpam-6039	13	61	[	[	X
ejpam-6039	13	62	8	8	NUM
ejpam-6039	13	63	]	]	PUNCT
ejpam-6039	13	64	.	.	PUNCT
ejpam-6039	14	1	in	in	ADP
ejpam-6039	14	2	2007	2007	NUM
ejpam-6039	14	3	,	,	PUNCT
ejpam-6039	14	4	alomari	alomari	PROPN
ejpam-6039	14	5	and	and	CCONJ
ejpam-6039	14	6	noorani	noorani	ADJ
ejpam-6039	15	1	[	[	X
ejpam-6039	15	2	9	9	NUM
ejpam-6039	15	3	]	]	PUNCT
ejpam-6039	15	4	introduced	introduce	VERB
ejpam-6039	15	5	the	the	DET
ejpam-6039	15	6	concept	concept	NOUN
ejpam-6039	15	7	of	of	ADP
ejpam-6039	15	8	almost	almost	ADV
ejpam-6039	15	9	contra	contra	PROPN
ejpam-6039	15	10	ω	ω	ADJ
ejpam-6039	15	11	-	-	ADJ
ejpam-6039	15	12	continuous	continuous	ADJ
ejpam-6039	15	13	functions	function	NOUN
ejpam-6039	15	14	via	via	ADP
ejpam-6039	15	15	the	the	DET
ejpam-6039	15	16	notion	notion	NOUN
ejpam-6039	15	17	of	of	ADP
ejpam-6039	15	18	ω	ω	VERB
ejpam-6039	15	19	-	-	ADJ
ejpam-6039	15	20	open	open	ADJ
ejpam-6039	15	21	sets	set	NOUN
ejpam-6039	15	22	and	and	CCONJ
ejpam-6039	15	23	investigated	investigate	VERB
ejpam-6039	15	24	several	several	ADJ
ejpam-6039	15	25	characterizations	characterization	NOUN
ejpam-6039	15	26	of	of	ADP
ejpam-6039	15	27	contra	contra	PROPN
ejpam-6039	15	28	ω	ω	PROPN
ejpam-6039	15	29	-	-	ADJ
ejpam-6039	15	30	continuous	continuous	ADJ
ejpam-6039	15	31	∗corresponding	∗corresponding	NOUN
ejpam-6039	15	32	author	author	NOUN
ejpam-6039	15	33	.	.	PUNCT
ejpam-6039	16	1	doi	doi	NOUN
ejpam-6039	16	2	:	:	PUNCT
ejpam-6039	16	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6039	https://doi.org/10.29020/nybg.ejpam.v18i2.6039	NOUN
ejpam-6039	16	4	email	email	NOUN
ejpam-6039	16	5	addresses	address	VERB
ejpam-6039	16	6	:	:	PUNCT
ejpam-6039	16	7	monchaya.c@msu.ac.th	monchaya.c@msu.ac.th	PROPN
ejpam-6039	16	8	(	(	PUNCT
ejpam-6039	16	9	m.	m.	NOUN
ejpam-6039	16	10	chiangpradit	chiangpradit	PROPN
ejpam-6039	16	11	)	)	PUNCT
ejpam-6039	16	12	,	,	PUNCT
ejpam-6039	16	13	s−sompong@snru.ac.th	s−sompong@snru.ac.th	PRON
ejpam-6039	16	14	(	(	PUNCT
ejpam-6039	16	15	s.	s.	PROPN
ejpam-6039	16	16	sompong	sompong	PROPN
ejpam-6039	16	17	)	)	PUNCT
ejpam-6039	16	18	,	,	PUNCT
ejpam-6039	16	19	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-6039	16	20	(	(	PUNCT
ejpam-6039	16	21	c.	c.	PROPN
ejpam-6039	16	22	boonpok	boonpok	PROPN
ejpam-6039	16	23	)	)	PUNCT
ejpam-6039	16	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6039	16	25	1	1	NUM
ejpam-6039	16	26	copyright	copyright	NOUN
ejpam-6039	16	27	:	:	PUNCT
ejpam-6039	17	1	©	©	PROPN
ejpam-6039	17	2	2025	2025	NUM
ejpam-6039	17	3	the	the	DET
ejpam-6039	17	4	author(s	author(s	NOUN
ejpam-6039	17	5	)	)	PUNCT
ejpam-6039	17	6	.	.	PUNCT
ejpam-6039	18	1	(	(	PUNCT
ejpam-6039	18	2	cc	cc	NOUN
ejpam-6039	18	3	by	by	ADP
ejpam-6039	18	4	-	-	PUNCT
ejpam-6039	18	5	nc	nc	PROPN
ejpam-6039	18	6	4.0	4.0	NUM
ejpam-6039	18	7	)	)	PUNCT
ejpam-6039	18	8	m.	m.	NOUN
ejpam-6039	18	9	chiangpradit	chiangpradit	NOUN
ejpam-6039	18	10	,	,	PUNCT
ejpam-6039	18	11	s.	s.	PROPN
ejpam-6039	18	12	sompong	sompong	PROPN
ejpam-6039	18	13	,	,	PUNCT
ejpam-6039	18	14	c.	c.	PROPN
ejpam-6039	18	15	boonpok	boonpok	PROPN
ejpam-6039	18	16	/	/	SYM
ejpam-6039	18	17	eur	eur	PROPN
ejpam-6039	18	18	.	.	PUNCT
ejpam-6039	19	1	j.	j.	PROPN
ejpam-6039	19	2	pure	pure	PROPN
ejpam-6039	19	3	appl	appl	PROPN
ejpam-6039	19	4	.	.	PROPN
ejpam-6039	19	5	math	math	PROPN
ejpam-6039	19	6	,	,	PUNCT
ejpam-6039	19	7	18	18	NUM
ejpam-6039	19	8	(	(	PUNCT
ejpam-6039	19	9	2	2	NUM
ejpam-6039	19	10	)	)	PUNCT
ejpam-6039	19	11	(	(	PUNCT
ejpam-6039	19	12	2025	2025	NUM
ejpam-6039	19	13	)	)	PUNCT
ejpam-6039	19	14	,	,	PUNCT
ejpam-6039	19	15	6039	6039	NUM
ejpam-6039	19	16	2	2	NUM
ejpam-6039	19	17	of	of	ADP
ejpam-6039	19	18	11	11	NUM
ejpam-6039	19	19	functions	function	NOUN
ejpam-6039	19	20	and	and	CCONJ
ejpam-6039	19	21	almost	almost	ADV
ejpam-6039	19	22	contra	contra	PROPN
ejpam-6039	19	23	ω	ω	ADJ
ejpam-6039	19	24	-	-	ADJ
ejpam-6039	19	25	continuous	continuous	ADJ
ejpam-6039	19	26	functions	function	NOUN
ejpam-6039	19	27	.	.	PUNCT
ejpam-6039	20	1	noiri	noiri	PROPN
ejpam-6039	20	2	and	and	CCONJ
ejpam-6039	20	3	popa	popa	NOUN
ejpam-6039	21	1	[	[	X
ejpam-6039	21	2	10	10	NUM
ejpam-6039	21	3	]	]	PUNCT
ejpam-6039	21	4	introduced	introduce	VERB
ejpam-6039	21	5	the	the	DET
ejpam-6039	21	6	notion	notion	NOUN
ejpam-6039	21	7	of	of	ADP
ejpam-6039	21	8	contra	contra	PROPN
ejpam-6039	21	9	m	m	PROPN
ejpam-6039	21	10	-	-	ADJ
ejpam-6039	21	11	continuous	continuous	ADJ
ejpam-6039	21	12	functions	function	NOUN
ejpam-6039	21	13	as	as	ADP
ejpam-6039	21	14	functions	function	NOUN
ejpam-6039	21	15	from	from	ADP
ejpam-6039	21	16	a	a	DET
ejpam-6039	21	17	set	set	NOUN
ejpam-6039	21	18	satisfying	satisfy	VERB
ejpam-6039	21	19	some	some	DET
ejpam-6039	21	20	minimal	minimal	ADJ
ejpam-6039	21	21	conditions	condition	NOUN
ejpam-6039	21	22	into	into	ADP
ejpam-6039	21	23	a	a	DET
ejpam-6039	21	24	topological	topological	ADJ
ejpam-6039	21	25	space	space	NOUN
ejpam-6039	21	26	and	and	CCONJ
ejpam-6039	21	27	investigated	investigate	VERB
ejpam-6039	21	28	some	some	DET
ejpam-6039	21	29	characterizations	characterization	NOUN
ejpam-6039	21	30	and	and	CCONJ
ejpam-6039	21	31	the	the	DET
ejpam-6039	21	32	relationships	relationship	NOUN
ejpam-6039	21	33	between	between	ADP
ejpam-6039	21	34	contra	contra	PROPN
ejpam-6039	21	35	m	m	PROPN
ejpam-6039	21	36	-	-	PUNCT
ejpam-6039	21	37	continuity	continuity	NOUN
ejpam-6039	21	38	and	and	CCONJ
ejpam-6039	21	39	other	other	ADJ
ejpam-6039	21	40	related	related	ADJ
ejpam-6039	21	41	generalized	generalized	ADJ
ejpam-6039	21	42	forms	form	NOUN
ejpam-6039	21	43	of	of	ADP
ejpam-6039	21	44	continuity	continuity	NOUN
ejpam-6039	21	45	.	.	PUNCT
ejpam-6039	22	1	it	it	PRON
ejpam-6039	22	2	turns	turn	VERB
ejpam-6039	22	3	out	out	ADP
ejpam-6039	22	4	that	that	SCONJ
ejpam-6039	22	5	the	the	DET
ejpam-6039	22	6	contra	contra	PROPN
ejpam-6039	22	7	m	m	PROPN
ejpam-6039	22	8	-	-	PUNCT
ejpam-6039	22	9	continuity	continuity	NOUN
ejpam-6039	22	10	is	be	AUX
ejpam-6039	22	11	a	a	DET
ejpam-6039	22	12	unified	unified	ADJ
ejpam-6039	22	13	form	form	NOUN
ejpam-6039	22	14	of	of	ADP
ejpam-6039	22	15	several	several	ADJ
ejpam-6039	22	16	modifications	modification	NOUN
ejpam-6039	22	17	of	of	ADP
ejpam-6039	22	18	weak	weak	ADJ
ejpam-6039	22	19	contra	contra	ADJ
ejpam-6039	22	20	-	-	NOUN
ejpam-6039	22	21	continuity	continuity	NOUN
ejpam-6039	22	22	due	due	ADP
ejpam-6039	22	23	to	to	ADP
ejpam-6039	22	24	baker	baker	PROPN
ejpam-6039	23	1	[	[	X
ejpam-6039	23	2	11	11	NUM
ejpam-6039	23	3	]	]	PUNCT
ejpam-6039	23	4	.	.	PUNCT
ejpam-6039	24	1	on	on	ADP
ejpam-6039	24	2	the	the	DET
ejpam-6039	24	3	other	other	ADJ
ejpam-6039	24	4	hand	hand	NOUN
ejpam-6039	24	5	,	,	PUNCT
ejpam-6039	24	6	the	the	DET
ejpam-6039	24	7	present	present	ADJ
ejpam-6039	24	8	authors	author	NOUN
ejpam-6039	24	9	introduced	introduce	VERB
ejpam-6039	24	10	and	and	CCONJ
ejpam-6039	24	11	studied	study	VERB
ejpam-6039	24	12	the	the	DET
ejpam-6039	24	13	notions	notion	NOUN
ejpam-6039	24	14	of	of	ADP
ejpam-6039	24	15	(	(	PUNCT
ejpam-6039	24	16	τ1	τ1	PROPN
ejpam-6039	24	17	,	,	PUNCT
ejpam-6039	24	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	19	functions	function	NOUN
ejpam-6039	24	20	[	[	X
ejpam-6039	24	21	12	12	NUM
ejpam-6039	24	22	]	]	PUNCT
ejpam-6039	24	23	,	,	PUNCT
ejpam-6039	24	24	almost	almost	ADV
ejpam-6039	24	25	(	(	PUNCT
ejpam-6039	24	26	τ1	τ1	NOUN
ejpam-6039	24	27	,	,	PUNCT
ejpam-6039	24	28	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	29	functions	function	NOUN
ejpam-6039	24	30	[	[	X
ejpam-6039	24	31	13	13	NUM
ejpam-6039	24	32	]	]	PUNCT
ejpam-6039	24	33	,	,	PUNCT
ejpam-6039	24	34	weakly	weakly	ADJ
ejpam-6039	24	35	(	(	PUNCT
ejpam-6039	24	36	τ1	τ1	NOUN
ejpam-6039	24	37	,	,	PUNCT
ejpam-6039	24	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	39	functions	function	NOUN
ejpam-6039	24	40	[	[	X
ejpam-6039	24	41	14	14	NUM
ejpam-6039	24	42	]	]	PUNCT
ejpam-6039	24	43	,	,	PUNCT
ejpam-6039	24	44	quasi	quasi	NOUN
ejpam-6039	24	45	θ(τ1	θ(τ1	NOUN
ejpam-6039	24	46	,	,	PUNCT
ejpam-6039	24	47	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	48	functions	function	NOUN
ejpam-6039	24	49	[	[	X
ejpam-6039	24	50	15	15	NUM
ejpam-6039	24	51	]	]	PUNCT
ejpam-6039	24	52	,	,	PUNCT
ejpam-6039	24	53	δ(τ1	δ(τ1	PROPN
ejpam-6039	24	54	,	,	PUNCT
ejpam-6039	24	55	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	56	functions	function	NOUN
ejpam-6039	24	57	[	[	X
ejpam-6039	24	58	16	16	NUM
ejpam-6039	24	59	]	]	PUNCT
ejpam-6039	24	60	,	,	PUNCT
ejpam-6039	24	61	almost	almost	ADV
ejpam-6039	24	62	quasi	quasi	NOUN
ejpam-6039	24	63	(	(	PUNCT
ejpam-6039	24	64	τ1	τ1	NOUN
ejpam-6039	24	65	,	,	PUNCT
ejpam-6039	24	66	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	67	functions	function	NOUN
ejpam-6039	24	68	[	[	X
ejpam-6039	24	69	17	17	NUM
ejpam-6039	24	70	]	]	PUNCT
ejpam-6039	24	71	,	,	PUNCT
ejpam-6039	24	72	weakly	weakly	ADJ
ejpam-6039	24	73	quasi	quasi	NOUN
ejpam-6039	24	74	(	(	PUNCT
ejpam-6039	24	75	τ1	τ1	PROPN
ejpam-6039	24	76	,	,	PUNCT
ejpam-6039	24	77	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	78	functions	function	NOUN
ejpam-6039	24	79	[	[	X
ejpam-6039	24	80	18	18	NUM
ejpam-6039	24	81	]	]	PUNCT
ejpam-6039	24	82	,	,	PUNCT
ejpam-6039	24	83	faintly	faintly	ADV
ejpam-6039	24	84	(	(	PUNCT
ejpam-6039	24	85	τ1	τ1	PROPN
ejpam-6039	24	86	,	,	PUNCT
ejpam-6039	24	87	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	88	functions	function	NOUN
ejpam-6039	24	89	[	[	X
ejpam-6039	24	90	19	19	NUM
ejpam-6039	24	91	]	]	PUNCT
ejpam-6039	24	92	and	and	CCONJ
ejpam-6039	24	93	almost	almost	ADV
ejpam-6039	24	94	nearly	nearly	ADV
ejpam-6039	24	95	(	(	PUNCT
ejpam-6039	24	96	τ1	τ1	NOUN
ejpam-6039	24	97	,	,	PUNCT
ejpam-6039	24	98	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	24	99	functions	function	NOUN
ejpam-6039	24	100	[	[	X
ejpam-6039	24	101	20	20	NUM
ejpam-6039	24	102	]	]	PUNCT
ejpam-6039	24	103	.	.	PUNCT
ejpam-6039	25	1	in	in	ADP
ejpam-6039	25	2	this	this	DET
ejpam-6039	25	3	paper	paper	NOUN
ejpam-6039	25	4	,	,	PUNCT
ejpam-6039	25	5	we	we	PRON
ejpam-6039	25	6	introduce	introduce	VERB
ejpam-6039	25	7	the	the	DET
ejpam-6039	25	8	concept	concept	NOUN
ejpam-6039	25	9	of	of	ADP
ejpam-6039	25	10	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	25	11	,	,	PUNCT
ejpam-6039	25	12	τ2)p	τ2)p	ADJ
ejpam-6039	25	13	-	-	PUNCT
ejpam-6039	25	14	continuous	continuous	ADJ
ejpam-6039	25	15	functions	function	NOUN
ejpam-6039	25	16	.	.	PUNCT
ejpam-6039	26	1	we	we	PRON
ejpam-6039	26	2	also	also	ADV
ejpam-6039	26	3	investigate	investigate	VERB
ejpam-6039	26	4	some	some	DET
ejpam-6039	26	5	characterizations	characterization	NOUN
ejpam-6039	26	6	of	of	ADP
ejpam-6039	26	7	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	26	8	,	,	PUNCT
ejpam-6039	26	9	τ2)p	τ2)p	ADJ
ejpam-6039	26	10	-	-	PUNCT
ejpam-6039	26	11	continuous	continuous	ADJ
ejpam-6039	26	12	functions	function	NOUN
ejpam-6039	26	13	.	.	PUNCT
ejpam-6039	27	1	2	2	X
ejpam-6039	27	2	.	.	NUM
ejpam-6039	27	3	preliminaries	preliminary	NOUN
ejpam-6039	27	4	throughout	throughout	ADP
ejpam-6039	27	5	the	the	DET
ejpam-6039	27	6	present	present	ADJ
ejpam-6039	27	7	paper	paper	NOUN
ejpam-6039	27	8	,	,	PUNCT
ejpam-6039	27	9	spaces	space	NOUN
ejpam-6039	27	10	(	(	PUNCT
ejpam-6039	27	11	x	x	NOUN
ejpam-6039	27	12	,	,	PUNCT
ejpam-6039	27	13	τ1	τ1	NOUN
ejpam-6039	27	14	,	,	PUNCT
ejpam-6039	27	15	τ2	τ2	NOUN
ejpam-6039	27	16	)	)	PUNCT
ejpam-6039	27	17	and	and	CCONJ
ejpam-6039	27	18	(	(	PUNCT
ejpam-6039	27	19	y	y	PROPN
ejpam-6039	27	20	,	,	PUNCT
ejpam-6039	27	21	σ1	σ1	PROPN
ejpam-6039	27	22	,	,	PUNCT
ejpam-6039	27	23	σ2	σ2	NOUN
ejpam-6039	27	24	)	)	PUNCT
ejpam-6039	27	25	(	(	PUNCT
ejpam-6039	27	26	or	or	CCONJ
ejpam-6039	27	27	simply	simply	ADV
ejpam-6039	27	28	x	x	X
ejpam-6039	27	29	and	and	CCONJ
ejpam-6039	27	30	y	y	PROPN
ejpam-6039	27	31	)	)	PUNCT
ejpam-6039	27	32	always	always	ADV
ejpam-6039	27	33	mean	mean	VERB
ejpam-6039	27	34	bitopological	bitopological	ADJ
ejpam-6039	27	35	spaces	space	NOUN
ejpam-6039	27	36	on	on	ADP
ejpam-6039	27	37	which	which	PRON
ejpam-6039	27	38	no	no	DET
ejpam-6039	27	39	separation	separation	NOUN
ejpam-6039	27	40	axioms	axiom	NOUN
ejpam-6039	27	41	are	be	AUX
ejpam-6039	27	42	assumed	assume	VERB
ejpam-6039	27	43	unless	unless	SCONJ
ejpam-6039	27	44	explicitly	explicitly	ADV
ejpam-6039	27	45	stated	state	VERB
ejpam-6039	27	46	.	.	PUNCT
ejpam-6039	28	1	let	let	VERB
ejpam-6039	28	2	a	a	DET
ejpam-6039	28	3	be	be	AUX
ejpam-6039	28	4	a	a	DET
ejpam-6039	28	5	subset	subset	NOUN
ejpam-6039	28	6	of	of	ADP
ejpam-6039	28	7	a	a	DET
ejpam-6039	28	8	bitopological	bitopological	ADJ
ejpam-6039	28	9	space	space	NOUN
ejpam-6039	28	10	(	(	PUNCT
ejpam-6039	28	11	x	x	NOUN
ejpam-6039	28	12	,	,	PUNCT
ejpam-6039	28	13	τ1	τ1	NOUN
ejpam-6039	28	14	,	,	PUNCT
ejpam-6039	28	15	τ2	τ2	NOUN
ejpam-6039	28	16	)	)	PUNCT
ejpam-6039	28	17	.	.	PUNCT
ejpam-6039	29	1	the	the	DET
ejpam-6039	29	2	closure	closure	NOUN
ejpam-6039	29	3	of	of	ADP
ejpam-6039	29	4	a	a	PRON
ejpam-6039	29	5	and	and	CCONJ
ejpam-6039	29	6	the	the	DET
ejpam-6039	29	7	interior	interior	NOUN
ejpam-6039	29	8	of	of	ADP
ejpam-6039	29	9	a	a	PRON
ejpam-6039	29	10	with	with	ADP
ejpam-6039	29	11	respect	respect	NOUN
ejpam-6039	29	12	to	to	ADP
ejpam-6039	29	13	τi	τi	PROPN
ejpam-6039	29	14	are	be	AUX
ejpam-6039	29	15	denoted	denote	VERB
ejpam-6039	29	16	by	by	ADP
ejpam-6039	29	17	τi	τi	NOUN
ejpam-6039	29	18	-	-	PUNCT
ejpam-6039	29	19	cl(a	cl(a	NUM
ejpam-6039	29	20	)	)	PUNCT
ejpam-6039	29	21	and	and	CCONJ
ejpam-6039	29	22	τi	τi	NOUN
ejpam-6039	29	23	-	-	PUNCT
ejpam-6039	29	24	int(a	int(a	NOUN
ejpam-6039	29	25	)	)	PUNCT
ejpam-6039	29	26	,	,	PUNCT
ejpam-6039	29	27	respectively	respectively	ADV
ejpam-6039	29	28	,	,	PUNCT
ejpam-6039	29	29	for	for	ADP
ejpam-6039	29	30	i	i	PROPN
ejpam-6039	29	31	=	=	SYM
ejpam-6039	29	32	1	1	NUM
ejpam-6039	29	33	,	,	PUNCT
ejpam-6039	29	34	2	2	NUM
ejpam-6039	29	35	.	.	X
ejpam-6039	29	36	a	a	DET
ejpam-6039	29	37	subset	subset	NOUN
ejpam-6039	29	38	a	a	PRON
ejpam-6039	29	39	of	of	ADP
ejpam-6039	29	40	a	a	DET
ejpam-6039	29	41	bitopological	bitopological	ADJ
ejpam-6039	29	42	space	space	NOUN
ejpam-6039	29	43	(	(	PUNCT
ejpam-6039	29	44	x	x	NOUN
ejpam-6039	29	45	,	,	PUNCT
ejpam-6039	29	46	τ1	τ1	NOUN
ejpam-6039	29	47	,	,	PUNCT
ejpam-6039	29	48	τ2	τ2	NOUN
ejpam-6039	29	49	)	)	PUNCT
ejpam-6039	29	50	is	be	AUX
ejpam-6039	29	51	called	call	VERB
ejpam-6039	29	52	τ1τ2	τ1τ2	VERB
ejpam-6039	29	53	-	-	ADJ
ejpam-6039	29	54	closed	closed	ADJ
ejpam-6039	29	55	[	[	X
ejpam-6039	29	56	21	21	NUM
ejpam-6039	29	57	]	]	X
ejpam-6039	29	58	if	if	SCONJ
ejpam-6039	29	59	a	a	DET
ejpam-6039	29	60	=	=	NOUN
ejpam-6039	29	61	τ1	τ1	NOUN
ejpam-6039	29	62	-	-	PUNCT
ejpam-6039	29	63	cl(τ2	cl(τ2	NOUN
ejpam-6039	29	64	-	-	PUNCT
ejpam-6039	29	65	cl(a	cl(a	NUM
ejpam-6039	29	66	)	)	PUNCT
ejpam-6039	29	67	)	)	PUNCT
ejpam-6039	29	68	.	.	PUNCT
ejpam-6039	30	1	the	the	DET
ejpam-6039	30	2	complement	complement	NOUN
ejpam-6039	30	3	of	of	ADP
ejpam-6039	30	4	a	a	DET
ejpam-6039	30	5	τ1τ2	τ1τ2	ADJ
ejpam-6039	30	6	-	-	ADJ
ejpam-6039	30	7	closed	closed	ADJ
ejpam-6039	30	8	set	set	NOUN
ejpam-6039	30	9	is	be	AUX
ejpam-6039	30	10	called	call	VERB
ejpam-6039	30	11	τ1τ2	τ1τ2	NOUN
ejpam-6039	30	12	-	-	ADJ
ejpam-6039	30	13	open	open	ADJ
ejpam-6039	30	14	.	.	PUNCT
ejpam-6039	31	1	the	the	DET
ejpam-6039	31	2	intersection	intersection	NOUN
ejpam-6039	31	3	of	of	ADP
ejpam-6039	31	4	all	all	DET
ejpam-6039	31	5	τ1τ2	τ1τ2	ADJ
ejpam-6039	31	6	-	-	ADJ
ejpam-6039	31	7	closed	closed	ADJ
ejpam-6039	31	8	sets	set	NOUN
ejpam-6039	31	9	of	of	ADP
ejpam-6039	31	10	x	x	PUNCT
ejpam-6039	31	11	containing	contain	VERB
ejpam-6039	31	12	a	a	PRON
ejpam-6039	31	13	is	be	AUX
ejpam-6039	31	14	called	call	VERB
ejpam-6039	31	15	the	the	DET
ejpam-6039	31	16	τ1τ2	τ1τ2	NOUN
ejpam-6039	31	17	-	-	NOUN
ejpam-6039	31	18	closure	closure	NOUN
ejpam-6039	31	19	[	[	X
ejpam-6039	31	20	21	21	NUM
ejpam-6039	31	21	]	]	PUNCT
ejpam-6039	31	22	of	of	ADP
ejpam-6039	31	23	a	a	PRON
ejpam-6039	31	24	and	and	CCONJ
ejpam-6039	31	25	is	be	AUX
ejpam-6039	31	26	denoted	denote	VERB
ejpam-6039	31	27	by	by	ADP
ejpam-6039	31	28	τ1τ2	τ1τ2	NOUN
ejpam-6039	31	29	-	-	NUM
ejpam-6039	31	30	cl(a	cl(a	NUM
ejpam-6039	31	31	)	)	PUNCT
ejpam-6039	31	32	.	.	PUNCT
ejpam-6039	32	1	the	the	DET
ejpam-6039	32	2	union	union	NOUN
ejpam-6039	32	3	of	of	ADP
ejpam-6039	32	4	all	all	DET
ejpam-6039	32	5	τ1τ2	τ1τ2	ADJ
ejpam-6039	32	6	-	-	ADJ
ejpam-6039	32	7	open	open	ADJ
ejpam-6039	32	8	sets	set	NOUN
ejpam-6039	32	9	of	of	ADP
ejpam-6039	32	10	x	x	PUNCT
ejpam-6039	32	11	contained	contain	VERB
ejpam-6039	32	12	in	in	ADP
ejpam-6039	32	13	a	a	PRON
ejpam-6039	32	14	is	be	AUX
ejpam-6039	32	15	called	call	VERB
ejpam-6039	32	16	the	the	DET
ejpam-6039	32	17	τ1τ2	τ1τ2	NOUN
ejpam-6039	32	18	-	-	ADJ
ejpam-6039	32	19	interior	interior	ADJ
ejpam-6039	32	20	[	[	X
ejpam-6039	32	21	21	21	NUM
ejpam-6039	32	22	]	]	PUNCT
ejpam-6039	32	23	of	of	ADP
ejpam-6039	32	24	a	a	PRON
ejpam-6039	32	25	and	and	CCONJ
ejpam-6039	32	26	is	be	AUX
ejpam-6039	32	27	denoted	denote	VERB
ejpam-6039	32	28	by	by	ADP
ejpam-6039	32	29	τ1τ2	τ1τ2	NOUN
ejpam-6039	32	30	-	-	ADJ
ejpam-6039	32	31	int(a	int(a	NOUN
ejpam-6039	32	32	)	)	PUNCT
ejpam-6039	32	33	.	.	PUNCT
ejpam-6039	33	1	lemma	lemma	PROPN
ejpam-6039	33	2	1	1	NUM
ejpam-6039	33	3	.	.	PUNCT
ejpam-6039	34	1	[	[	X
ejpam-6039	34	2	21	21	NUM
ejpam-6039	34	3	]	]	PUNCT
ejpam-6039	34	4	let	let	VERB
ejpam-6039	34	5	a	a	PRON
ejpam-6039	34	6	and	and	CCONJ
ejpam-6039	34	7	b	b	NOUN
ejpam-6039	34	8	be	be	AUX
ejpam-6039	34	9	subsets	subset	NOUN
ejpam-6039	34	10	of	of	ADP
ejpam-6039	34	11	a	a	DET
ejpam-6039	34	12	bitopological	bitopological	ADJ
ejpam-6039	34	13	space	space	NOUN
ejpam-6039	34	14	(	(	PUNCT
ejpam-6039	34	15	x	x	NOUN
ejpam-6039	34	16	,	,	PUNCT
ejpam-6039	34	17	τ1	τ1	NOUN
ejpam-6039	34	18	,	,	PUNCT
ejpam-6039	34	19	τ2	τ2	NOUN
ejpam-6039	34	20	)	)	PUNCT
ejpam-6039	34	21	.	.	PUNCT
ejpam-6039	35	1	for	for	ADP
ejpam-6039	35	2	the	the	DET
ejpam-6039	35	3	τ1τ2closure	τ1τ2closure	NOUN
ejpam-6039	35	4	,	,	PUNCT
ejpam-6039	35	5	the	the	DET
ejpam-6039	35	6	following	follow	VERB
ejpam-6039	35	7	properties	property	NOUN
ejpam-6039	35	8	hold	hold	VERB
ejpam-6039	35	9	:	:	PUNCT
ejpam-6039	35	10	(	(	PUNCT
ejpam-6039	35	11	1	1	X
ejpam-6039	35	12	)	)	PUNCT
ejpam-6039	35	13	a	a	DET
ejpam-6039	35	14	⊆	⊆	NUM
ejpam-6039	35	15	τ1τ2	τ1τ2	NOUN
ejpam-6039	35	16	-	-	NUM
ejpam-6039	35	17	cl(a	cl(a	NUM
ejpam-6039	35	18	)	)	PUNCT
ejpam-6039	35	19	and	and	CCONJ
ejpam-6039	35	20	τ1τ2	τ1τ2	NOUN
ejpam-6039	35	21	-	-	ADJ
ejpam-6039	35	22	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	35	23	-	-	PUNCT
ejpam-6039	35	24	cl(a	cl(a	NUM
ejpam-6039	35	25	)	)	PUNCT
ejpam-6039	35	26	)	)	PUNCT
ejpam-6039	36	1	=	=	PUNCT
ejpam-6039	36	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	36	3	-	-	NUM
ejpam-6039	36	4	cl(a	cl(a	NUM
ejpam-6039	36	5	)	)	PUNCT
ejpam-6039	36	6	.	.	PUNCT
ejpam-6039	37	1	(	(	PUNCT
ejpam-6039	37	2	2	2	X
ejpam-6039	37	3	)	)	PUNCT
ejpam-6039	37	4	if	if	SCONJ
ejpam-6039	37	5	a	a	DET
ejpam-6039	37	6	⊆	⊆	NUM
ejpam-6039	37	7	b	b	NOUN
ejpam-6039	37	8	,	,	PUNCT
ejpam-6039	37	9	then	then	ADV
ejpam-6039	37	10	τ1τ2	τ1τ2	NOUN
ejpam-6039	37	11	-	-	NUM
ejpam-6039	37	12	cl(a	cl(a	NUM
ejpam-6039	37	13	)	)	PUNCT
ejpam-6039	37	14	⊆	⊆	NUM
ejpam-6039	37	15	τ1τ2	τ1τ2	NOUN
ejpam-6039	37	16	-	-	NOUN
ejpam-6039	37	17	cl(b	cl(b	NOUN
ejpam-6039	37	18	)	)	PUNCT
ejpam-6039	37	19	.	.	PUNCT
ejpam-6039	38	1	(	(	PUNCT
ejpam-6039	38	2	3	3	X
ejpam-6039	38	3	)	)	PUNCT
ejpam-6039	38	4	τ1τ2	τ1τ2	NOUN
ejpam-6039	38	5	-	-	NUM
ejpam-6039	38	6	cl(a	cl(a	NUM
ejpam-6039	38	7	)	)	PUNCT
ejpam-6039	38	8	is	be	AUX
ejpam-6039	38	9	τ1τ2	τ1τ2	NOUN
ejpam-6039	38	10	-	-	ADJ
ejpam-6039	38	11	closed	closed	ADJ
ejpam-6039	38	12	.	.	PUNCT
ejpam-6039	39	1	(	(	PUNCT
ejpam-6039	39	2	4	4	X
ejpam-6039	39	3	)	)	PUNCT
ejpam-6039	39	4	a	a	PRON
ejpam-6039	39	5	is	be	AUX
ejpam-6039	39	6	τ1τ2	τ1τ2	NOUN
ejpam-6039	39	7	-	-	ADJ
ejpam-6039	39	8	closed	closed	ADJ
ejpam-6039	39	9	if	if	SCONJ
ejpam-6039	39	10	and	and	CCONJ
ejpam-6039	39	11	only	only	ADV
ejpam-6039	39	12	if	if	SCONJ
ejpam-6039	39	13	a	a	DET
ejpam-6039	39	14	=	=	PUNCT
ejpam-6039	39	15	τ1τ2	τ1τ2	NOUN
ejpam-6039	39	16	-	-	NUM
ejpam-6039	39	17	cl(a	cl(a	NUM
ejpam-6039	39	18	)	)	PUNCT
ejpam-6039	39	19	.	.	PUNCT
ejpam-6039	40	1	(	(	PUNCT
ejpam-6039	40	2	5	5	X
ejpam-6039	40	3	)	)	PUNCT
ejpam-6039	40	4	τ1τ2	τ1τ2	NOUN
ejpam-6039	40	5	-	-	NOUN
ejpam-6039	40	6	cl(x	cl(x	X
ejpam-6039	40	7	−a	−a	NOUN
ejpam-6039	40	8	)	)	PUNCT
ejpam-6039	41	1	=	=	PUNCT
ejpam-6039	41	2	x	x	X
ejpam-6039	42	1	−	−	ADP
ejpam-6039	42	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	42	3	-	-	PUNCT
ejpam-6039	42	4	int(a	int(a	NOUN
ejpam-6039	42	5	)	)	PUNCT
ejpam-6039	42	6	.	.	PUNCT
ejpam-6039	43	1	a	a	DET
ejpam-6039	43	2	subset	subset	NOUN
ejpam-6039	43	3	a	a	PRON
ejpam-6039	43	4	of	of	ADP
ejpam-6039	43	5	a	a	DET
ejpam-6039	43	6	bitopological	bitopological	ADJ
ejpam-6039	43	7	space	space	NOUN
ejpam-6039	43	8	(	(	PUNCT
ejpam-6039	43	9	x	x	NOUN
ejpam-6039	43	10	,	,	PUNCT
ejpam-6039	43	11	τ1	τ1	NOUN
ejpam-6039	43	12	,	,	PUNCT
ejpam-6039	43	13	τ2	τ2	NOUN
ejpam-6039	43	14	)	)	PUNCT
ejpam-6039	43	15	is	be	AUX
ejpam-6039	43	16	said	say	VERB
ejpam-6039	43	17	to	to	PART
ejpam-6039	43	18	be	be	AUX
ejpam-6039	43	19	(	(	PUNCT
ejpam-6039	43	20	τ1	τ1	NOUN
ejpam-6039	43	21	,	,	PUNCT
ejpam-6039	43	22	τ2)r	τ2)r	NOUN
ejpam-6039	43	23	-	-	PUNCT
ejpam-6039	43	24	open	open	NOUN
ejpam-6039	44	1	[	[	X
ejpam-6039	44	2	22	22	NUM
ejpam-6039	44	3	]	]	PUNCT
ejpam-6039	44	4	(	(	PUNCT
ejpam-6039	44	5	resp	resp	NOUN
ejpam-6039	44	6	.	.	PUNCT
ejpam-6039	45	1	(	(	PUNCT
ejpam-6039	45	2	τ1	τ1	NOUN
ejpam-6039	45	3	,	,	PUNCT
ejpam-6039	45	4	τ2)s	τ2)s	NOUN
ejpam-6039	45	5	-	-	PUNCT
ejpam-6039	45	6	open	open	ADJ
ejpam-6039	45	7	[	[	X
ejpam-6039	45	8	23	23	NUM
ejpam-6039	45	9	]	]	PUNCT
ejpam-6039	45	10	,	,	PUNCT
ejpam-6039	45	11	(	(	PUNCT
ejpam-6039	45	12	τ1	τ1	NOUN
ejpam-6039	45	13	,	,	PUNCT
ejpam-6039	45	14	τ2)p	τ2)p	NOUN
ejpam-6039	45	15	-	-	ADJ
ejpam-6039	45	16	open	open	ADJ
ejpam-6039	45	17	[	[	X
ejpam-6039	45	18	23	23	NUM
ejpam-6039	45	19	]	]	PUNCT
ejpam-6039	45	20	,	,	PUNCT
ejpam-6039	45	21	(	(	PUNCT
ejpam-6039	45	22	τ1	τ1	NOUN
ejpam-6039	45	23	,	,	PUNCT
ejpam-6039	45	24	τ2)β	τ2)β	ADJ
ejpam-6039	45	25	-	-	PUNCT
ejpam-6039	45	26	open	open	NOUN
ejpam-6039	46	1	[	[	X
ejpam-6039	46	2	23	23	NUM
ejpam-6039	46	3	]	]	SYM
ejpam-6039	46	4	)	)	PUNCT
ejpam-6039	46	5	if	if	SCONJ
ejpam-6039	46	6	a	a	DET
ejpam-6039	46	7	=	=	PUNCT
ejpam-6039	46	8	τ1τ2	τ1τ2	NOUN
ejpam-6039	46	9	-	-	NOUN
ejpam-6039	46	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	46	11	-	-	PUNCT
ejpam-6039	46	12	cl(a	cl(a	NUM
ejpam-6039	46	13	)	)	PUNCT
ejpam-6039	46	14	)	)	PUNCT
ejpam-6039	46	15	(	(	PUNCT
ejpam-6039	46	16	resp	resp	NOUN
ejpam-6039	46	17	.	.	PUNCT
ejpam-6039	47	1	a	a	DET
ejpam-6039	47	2	⊆	⊆	NUM
ejpam-6039	47	3	τ1τ2	τ1τ2	NOUN
ejpam-6039	47	4	-	-	ADJ
ejpam-6039	47	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	47	6	-	-	PUNCT
ejpam-6039	47	7	int(a	int(a	NOUN
ejpam-6039	47	8	)	)	PUNCT
ejpam-6039	47	9	)	)	PUNCT
ejpam-6039	47	10	,	,	PUNCT
ejpam-6039	47	11	a	a	DET
ejpam-6039	47	12	⊆	⊆	NUM
ejpam-6039	47	13	τ1τ2	τ1τ2	NOUN
ejpam-6039	47	14	-	-	NOUN
ejpam-6039	47	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	47	16	-	-	PUNCT
ejpam-6039	47	17	cl(a	cl(a	NUM
ejpam-6039	47	18	)	)	PUNCT
ejpam-6039	47	19	)	)	PUNCT
ejpam-6039	47	20	,	,	PUNCT
ejpam-6039	47	21	a	a	DET
ejpam-6039	47	22	⊆	⊆	NUM
ejpam-6039	47	23	τ1τ2	τ1τ2	NOUN
ejpam-6039	47	24	-	-	PUNCT
ejpam-6039	47	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	47	26	-	-	PUNCT
ejpam-6039	47	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	47	28	-	-	PUNCT
ejpam-6039	47	29	cl(a	cl(a	NUM
ejpam-6039	47	30	)	)	PUNCT
ejpam-6039	47	31	)	)	PUNCT
ejpam-6039	47	32	)	)	PUNCT
ejpam-6039	47	33	)	)	PUNCT
ejpam-6039	47	34	.	.	PUNCT
ejpam-6039	48	1	the	the	DET
ejpam-6039	48	2	complement	complement	NOUN
ejpam-6039	48	3	of	of	ADP
ejpam-6039	48	4	a	a	DET
ejpam-6039	48	5	(	(	PUNCT
ejpam-6039	48	6	τ1	τ1	NOUN
ejpam-6039	48	7	,	,	PUNCT
ejpam-6039	48	8	τ2)r	τ2)r	NOUN
ejpam-6039	48	9	-	-	PUNCT
ejpam-6039	48	10	open	open	ADJ
ejpam-6039	48	11	(	(	PUNCT
ejpam-6039	48	12	resp	resp	NOUN
ejpam-6039	48	13	.	.	PUNCT
ejpam-6039	49	1	(	(	PUNCT
ejpam-6039	49	2	τ1	τ1	NOUN
ejpam-6039	49	3	,	,	PUNCT
ejpam-6039	49	4	τ2)s	τ2)s	NOUN
ejpam-6039	49	5	-	-	PUNCT
ejpam-6039	49	6	open	open	ADJ
ejpam-6039	49	7	,	,	PUNCT
ejpam-6039	49	8	(	(	PUNCT
ejpam-6039	49	9	τ1	τ1	NOUN
ejpam-6039	49	10	,	,	PUNCT
ejpam-6039	49	11	τ2)p	τ2)p	NOUN
ejpam-6039	49	12	-	-	ADJ
ejpam-6039	49	13	open	open	ADJ
ejpam-6039	49	14	,	,	PUNCT
ejpam-6039	49	15	(	(	PUNCT
ejpam-6039	49	16	τ1	τ1	NOUN
ejpam-6039	49	17	,	,	PUNCT
ejpam-6039	49	18	τ2)β	τ2)β	ADJ
ejpam-6039	49	19	-	-	PUNCT
ejpam-6039	49	20	open	open	ADJ
ejpam-6039	49	21	)	)	PUNCT
ejpam-6039	49	22	set	set	NOUN
ejpam-6039	49	23	is	be	AUX
ejpam-6039	49	24	said	say	VERB
ejpam-6039	49	25	to	to	PART
ejpam-6039	49	26	be	be	AUX
ejpam-6039	49	27	(	(	PUNCT
ejpam-6039	49	28	τ1	τ1	NOUN
ejpam-6039	49	29	,	,	PUNCT
ejpam-6039	49	30	τ2)r	τ2)r	NOUN
ejpam-6039	49	31	-	-	PUNCT
ejpam-6039	49	32	closed	closed	ADJ
ejpam-6039	49	33	(	(	PUNCT
ejpam-6039	49	34	resp	resp	NOUN
ejpam-6039	49	35	.	.	PUNCT
ejpam-6039	50	1	(	(	PUNCT
ejpam-6039	50	2	τ1	τ1	NOUN
ejpam-6039	50	3	,	,	PUNCT
ejpam-6039	50	4	τ2)s	τ2)s	NOUN
ejpam-6039	50	5	-	-	PUNCT
ejpam-6039	50	6	closed	closed	ADJ
ejpam-6039	50	7	,	,	PUNCT
ejpam-6039	50	8	(	(	PUNCT
ejpam-6039	50	9	τ1	τ1	NOUN
ejpam-6039	50	10	,	,	PUNCT
ejpam-6039	50	11	τ2)p	τ2)p	NOUN
ejpam-6039	50	12	-	-	PUNCT
ejpam-6039	50	13	closed	closed	ADJ
ejpam-6039	50	14	,	,	PUNCT
ejpam-6039	50	15	(	(	PUNCT
ejpam-6039	50	16	τ1	τ1	NOUN
ejpam-6039	50	17	,	,	PUNCT
ejpam-6039	50	18	τ2)β	τ2)β	ADJ
ejpam-6039	50	19	-	-	PUNCT
ejpam-6039	50	20	closed	closed	ADJ
ejpam-6039	50	21	)	)	PUNCT
ejpam-6039	50	22	.	.	PUNCT
ejpam-6039	51	1	a	a	DET
ejpam-6039	51	2	subset	subset	NOUN
ejpam-6039	51	3	a	a	PRON
ejpam-6039	51	4	of	of	ADP
ejpam-6039	51	5	a	a	DET
ejpam-6039	51	6	bitopological	bitopological	ADJ
ejpam-6039	51	7	space	space	NOUN
ejpam-6039	51	8	(	(	PUNCT
ejpam-6039	51	9	x	x	NOUN
ejpam-6039	51	10	,	,	PUNCT
ejpam-6039	51	11	τ1	τ1	NOUN
ejpam-6039	51	12	,	,	PUNCT
ejpam-6039	51	13	τ2	τ2	NOUN
ejpam-6039	51	14	)	)	PUNCT
ejpam-6039	51	15	is	be	AUX
ejpam-6039	51	16	said	say	VERB
ejpam-6039	51	17	to	to	PART
ejpam-6039	51	18	be	be	AUX
ejpam-6039	51	19	α(τ1	α(τ1	NOUN
ejpam-6039	51	20	,	,	PUNCT
ejpam-6039	51	21	τ2)-open	τ2)-open	ADJ
ejpam-6039	51	22	[	[	X
ejpam-6039	51	23	24	24	NUM
ejpam-6039	51	24	]	]	X
ejpam-6039	51	25	if	if	SCONJ
ejpam-6039	51	26	a	a	DET
ejpam-6039	51	27	⊆	⊆	NUM
ejpam-6039	51	28	m.	m.	NOUN
ejpam-6039	51	29	chiangpradit	chiangpradit	NOUN
ejpam-6039	51	30	,	,	PUNCT
ejpam-6039	51	31	s.	s.	PROPN
ejpam-6039	51	32	sompong	sompong	PROPN
ejpam-6039	51	33	,	,	PUNCT
ejpam-6039	51	34	c.	c.	PROPN
ejpam-6039	51	35	boonpok	boonpok	PROPN
ejpam-6039	51	36	/	/	SYM
ejpam-6039	51	37	eur	eur	PROPN
ejpam-6039	51	38	.	.	PUNCT
ejpam-6039	52	1	j.	j.	PROPN
ejpam-6039	52	2	pure	pure	PROPN
ejpam-6039	52	3	appl	appl	PROPN
ejpam-6039	52	4	.	.	PROPN
ejpam-6039	52	5	math	math	PROPN
ejpam-6039	52	6	,	,	PUNCT
ejpam-6039	52	7	18	18	NUM
ejpam-6039	52	8	(	(	PUNCT
ejpam-6039	52	9	2	2	NUM
ejpam-6039	52	10	)	)	PUNCT
ejpam-6039	52	11	(	(	PUNCT
ejpam-6039	52	12	2025	2025	NUM
ejpam-6039	52	13	)	)	PUNCT
ejpam-6039	52	14	,	,	PUNCT
ejpam-6039	52	15	6039	6039	NUM
ejpam-6039	52	16	3	3	NUM
ejpam-6039	52	17	of	of	ADP
ejpam-6039	52	18	11	11	NUM
ejpam-6039	52	19	τ1τ2	τ1τ2	NOUN
ejpam-6039	52	20	-	-	NOUN
ejpam-6039	52	21	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	52	22	-	-	PUNCT
ejpam-6039	52	23	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	52	24	-	-	PUNCT
ejpam-6039	52	25	int(a	int(a	NOUN
ejpam-6039	52	26	)	)	PUNCT
ejpam-6039	52	27	)	)	PUNCT
ejpam-6039	52	28	)	)	PUNCT
ejpam-6039	52	29	.	.	PUNCT
ejpam-6039	53	1	the	the	DET
ejpam-6039	53	2	complement	complement	NOUN
ejpam-6039	53	3	of	of	ADP
ejpam-6039	53	4	an	an	DET
ejpam-6039	53	5	α(τ1	α(τ1	NOUN
ejpam-6039	53	6	,	,	PUNCT
ejpam-6039	53	7	τ2)-open	τ2)-open	ADJ
ejpam-6039	53	8	set	set	NOUN
ejpam-6039	53	9	is	be	AUX
ejpam-6039	53	10	said	say	VERB
ejpam-6039	53	11	to	to	PART
ejpam-6039	53	12	be	be	AUX
ejpam-6039	53	13	α(τ1	α(τ1	NOUN
ejpam-6039	53	14	,	,	PUNCT
ejpam-6039	53	15	τ2)-closed	τ2)-close	VERB
ejpam-6039	53	16	.	.	PUNCT
ejpam-6039	54	1	let	let	VERB
ejpam-6039	54	2	a	a	DET
ejpam-6039	54	3	be	be	AUX
ejpam-6039	54	4	a	a	DET
ejpam-6039	54	5	subset	subset	NOUN
ejpam-6039	54	6	of	of	ADP
ejpam-6039	54	7	a	a	DET
ejpam-6039	54	8	bitopological	bitopological	ADJ
ejpam-6039	54	9	space	space	NOUN
ejpam-6039	54	10	(	(	PUNCT
ejpam-6039	54	11	x	x	NOUN
ejpam-6039	54	12	,	,	PUNCT
ejpam-6039	54	13	τ1	τ1	NOUN
ejpam-6039	54	14	,	,	PUNCT
ejpam-6039	54	15	τ2	τ2	NOUN
ejpam-6039	54	16	)	)	PUNCT
ejpam-6039	54	17	.	.	PUNCT
ejpam-6039	55	1	the	the	DET
ejpam-6039	55	2	set	set	NOUN
ejpam-6039	55	3	∩{g	∩{g	INTJ
ejpam-6039	55	4	|	|	ADV
ejpam-6039	55	5	a	a	DET
ejpam-6039	55	6	⊆	⊆	NUM
ejpam-6039	55	7	g	g	NOUN
ejpam-6039	55	8	and	and	CCONJ
ejpam-6039	55	9	g	g	PROPN
ejpam-6039	55	10	is	be	AUX
ejpam-6039	55	11	τ1τ2	τ1τ2	VERB
ejpam-6039	55	12	-	-	ADJ
ejpam-6039	55	13	open	open	ADJ
ejpam-6039	55	14	}	}	PUNCT
ejpam-6039	55	15	is	be	AUX
ejpam-6039	55	16	called	call	VERB
ejpam-6039	55	17	the	the	DET
ejpam-6039	55	18	τ1τ2	τ1τ2	NOUN
ejpam-6039	55	19	-	-	NOUN
ejpam-6039	55	20	kernel	kernel	NOUN
ejpam-6039	56	1	[	[	X
ejpam-6039	56	2	21	21	NUM
ejpam-6039	56	3	]	]	PUNCT
ejpam-6039	56	4	of	of	ADP
ejpam-6039	56	5	a	a	PRON
ejpam-6039	56	6	and	and	CCONJ
ejpam-6039	56	7	is	be	AUX
ejpam-6039	56	8	denoted	denote	VERB
ejpam-6039	56	9	by	by	ADP
ejpam-6039	56	10	τ1τ2	τ1τ2	NOUN
ejpam-6039	56	11	-	-	ADJ
ejpam-6039	56	12	ker(a	ker(a	ADJ
ejpam-6039	56	13	)	)	PUNCT
ejpam-6039	56	14	.	.	PUNCT
ejpam-6039	57	1	lemma	lemma	PROPN
ejpam-6039	57	2	2	2	NUM
ejpam-6039	57	3	.	.	PUNCT
ejpam-6039	58	1	[	[	X
ejpam-6039	58	2	21	21	NUM
ejpam-6039	58	3	]	]	PUNCT
ejpam-6039	58	4	for	for	ADP
ejpam-6039	58	5	subsets	subset	NOUN
ejpam-6039	58	6	a	a	DET
ejpam-6039	58	7	,	,	PUNCT
ejpam-6039	58	8	b	b	NOUN
ejpam-6039	58	9	of	of	ADP
ejpam-6039	58	10	a	a	DET
ejpam-6039	58	11	bitopological	bitopological	ADJ
ejpam-6039	58	12	space	space	NOUN
ejpam-6039	58	13	(	(	PUNCT
ejpam-6039	58	14	x	x	NOUN
ejpam-6039	58	15	,	,	PUNCT
ejpam-6039	58	16	τ1	τ1	NOUN
ejpam-6039	58	17	,	,	PUNCT
ejpam-6039	58	18	τ2	τ2	NOUN
ejpam-6039	58	19	)	)	PUNCT
ejpam-6039	58	20	,	,	PUNCT
ejpam-6039	58	21	the	the	DET
ejpam-6039	58	22	following	follow	VERB
ejpam-6039	58	23	properties	property	NOUN
ejpam-6039	58	24	hold	hold	VERB
ejpam-6039	58	25	:	:	PUNCT
ejpam-6039	58	26	(	(	PUNCT
ejpam-6039	58	27	1	1	X
ejpam-6039	58	28	)	)	PUNCT
ejpam-6039	58	29	a	a	DET
ejpam-6039	58	30	⊆	⊆	NUM
ejpam-6039	58	31	τ1τ2	τ1τ2	NOUN
ejpam-6039	58	32	-	-	ADJ
ejpam-6039	58	33	ker(a	ker(a	ADJ
ejpam-6039	58	34	)	)	PUNCT
ejpam-6039	58	35	.	.	PUNCT
ejpam-6039	59	1	(	(	PUNCT
ejpam-6039	59	2	2	2	X
ejpam-6039	59	3	)	)	PUNCT
ejpam-6039	59	4	if	if	SCONJ
ejpam-6039	59	5	a	a	DET
ejpam-6039	59	6	⊆	⊆	NUM
ejpam-6039	59	7	b	b	NOUN
ejpam-6039	59	8	,	,	PUNCT
ejpam-6039	59	9	then	then	ADV
ejpam-6039	59	10	τ1τ2	τ1τ2	NOUN
ejpam-6039	59	11	-	-	ADJ
ejpam-6039	59	12	ker(a	ker(a	ADJ
ejpam-6039	59	13	)	)	PUNCT
ejpam-6039	59	14	⊆	⊆	NUM
ejpam-6039	59	15	τ1τ2	τ1τ2	PROPN
ejpam-6039	59	16	-	-	ADJ
ejpam-6039	59	17	ker(b	ker(b	PROPN
ejpam-6039	59	18	)	)	PUNCT
ejpam-6039	59	19	.	.	PUNCT
ejpam-6039	60	1	(	(	PUNCT
ejpam-6039	60	2	3	3	X
ejpam-6039	60	3	)	)	PUNCT
ejpam-6039	60	4	if	if	SCONJ
ejpam-6039	60	5	a	a	PRON
ejpam-6039	60	6	is	be	AUX
ejpam-6039	60	7	τ1τ2	τ1τ2	NOUN
ejpam-6039	60	8	-	-	ADJ
ejpam-6039	60	9	open	open	ADJ
ejpam-6039	60	10	,	,	PUNCT
ejpam-6039	60	11	then	then	ADV
ejpam-6039	60	12	τ1τ2	τ1τ2	NOUN
ejpam-6039	60	13	-	-	ADJ
ejpam-6039	60	14	ker(a	ker(a	ADJ
ejpam-6039	60	15	)	)	PUNCT
ejpam-6039	60	16	=	=	SYM
ejpam-6039	60	17	a.	a.	NOUN
ejpam-6039	60	18	(	(	PUNCT
ejpam-6039	60	19	4	4	NUM
ejpam-6039	60	20	)	)	PUNCT
ejpam-6039	60	21	x	x	SYM
ejpam-6039	60	22	∈	∈	PROPN
ejpam-6039	60	23	τ1τ2	τ1τ2	NOUN
ejpam-6039	60	24	-	-	ADJ
ejpam-6039	60	25	ker(a	ker(a	ADJ
ejpam-6039	60	26	)	)	PUNCT
ejpam-6039	60	27	if	if	SCONJ
ejpam-6039	60	28	and	and	CCONJ
ejpam-6039	60	29	only	only	ADV
ejpam-6039	60	30	if	if	SCONJ
ejpam-6039	60	31	a	a	DET
ejpam-6039	60	32	∩h	∩h	ADJ
ejpam-6039	60	33	̸=	̸=	PROPN
ejpam-6039	60	34	∅	∅	NOUN
ejpam-6039	60	35	for	for	ADP
ejpam-6039	60	36	every	every	DET
ejpam-6039	60	37	τ1τ2	τ1τ2	ADJ
ejpam-6039	60	38	-	-	ADJ
ejpam-6039	60	39	closed	closed	ADJ
ejpam-6039	60	40	set	set	ADJ
ejpam-6039	60	41	h	h	NOUN
ejpam-6039	60	42	containing	contain	VERB
ejpam-6039	60	43	x.	x.	NOUN
ejpam-6039	60	44	let	let	VERB
ejpam-6039	60	45	a	a	PRON
ejpam-6039	60	46	be	be	AUX
ejpam-6039	60	47	a	a	DET
ejpam-6039	60	48	subset	subset	NOUN
ejpam-6039	60	49	of	of	ADP
ejpam-6039	60	50	a	a	DET
ejpam-6039	60	51	bitopological	bitopological	ADJ
ejpam-6039	60	52	space	space	NOUN
ejpam-6039	60	53	(	(	PUNCT
ejpam-6039	60	54	x	x	NOUN
ejpam-6039	60	55	,	,	PUNCT
ejpam-6039	60	56	τ1	τ1	NOUN
ejpam-6039	60	57	,	,	PUNCT
ejpam-6039	60	58	τ2	τ2	NOUN
ejpam-6039	60	59	)	)	PUNCT
ejpam-6039	60	60	.	.	PUNCT
ejpam-6039	61	1	the	the	DET
ejpam-6039	61	2	intersection	intersection	NOUN
ejpam-6039	61	3	of	of	ADP
ejpam-6039	61	4	all	all	DET
ejpam-6039	61	5	(	(	PUNCT
ejpam-6039	61	6	τ1	τ1	NOUN
ejpam-6039	61	7	,	,	PUNCT
ejpam-6039	61	8	τ2)pclosed	τ2)pclose	VERB
ejpam-6039	61	9	sets	set	NOUN
ejpam-6039	61	10	of	of	ADP
ejpam-6039	61	11	x	x	PUNCT
ejpam-6039	61	12	containing	contain	VERB
ejpam-6039	61	13	a	a	PRON
ejpam-6039	61	14	is	be	AUX
ejpam-6039	61	15	called	call	VERB
ejpam-6039	61	16	the	the	DET
ejpam-6039	61	17	(	(	PUNCT
ejpam-6039	61	18	τ1	τ1	NOUN
ejpam-6039	61	19	,	,	PUNCT
ejpam-6039	61	20	τ2)p	τ2)p	NOUN
ejpam-6039	61	21	-	-	NOUN
ejpam-6039	61	22	closure	closure	NOUN
ejpam-6039	61	23	[	[	X
ejpam-6039	61	24	25	25	NUM
ejpam-6039	61	25	]	]	PUNCT
ejpam-6039	61	26	of	of	ADP
ejpam-6039	61	27	a	a	PRON
ejpam-6039	61	28	and	and	CCONJ
ejpam-6039	61	29	is	be	AUX
ejpam-6039	61	30	denoted	denote	VERB
ejpam-6039	61	31	by	by	ADP
ejpam-6039	61	32	(	(	PUNCT
ejpam-6039	61	33	τ1	τ1	PROPN
ejpam-6039	61	34	,	,	PUNCT
ejpam-6039	61	35	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6039	61	36	)	)	PUNCT
ejpam-6039	61	37	.	.	PUNCT
ejpam-6039	62	1	the	the	DET
ejpam-6039	62	2	union	union	NOUN
ejpam-6039	62	3	of	of	ADP
ejpam-6039	62	4	all	all	DET
ejpam-6039	62	5	(	(	PUNCT
ejpam-6039	62	6	τ1	τ1	NOUN
ejpam-6039	62	7	,	,	PUNCT
ejpam-6039	62	8	τ2)p	τ2)p	ADJ
ejpam-6039	62	9	-	-	PUNCT
ejpam-6039	62	10	open	open	ADJ
ejpam-6039	62	11	sets	set	NOUN
ejpam-6039	62	12	of	of	ADP
ejpam-6039	62	13	x	x	PUNCT
ejpam-6039	62	14	contained	contain	VERB
ejpam-6039	62	15	in	in	ADP
ejpam-6039	62	16	a	a	PRON
ejpam-6039	62	17	is	be	AUX
ejpam-6039	62	18	called	call	VERB
ejpam-6039	62	19	the	the	DET
ejpam-6039	62	20	(	(	PUNCT
ejpam-6039	62	21	τ1	τ1	NOUN
ejpam-6039	62	22	,	,	PUNCT
ejpam-6039	62	23	τ2)p	τ2)p	ADJ
ejpam-6039	62	24	-	-	NOUN
ejpam-6039	62	25	interior	interior	ADJ
ejpam-6039	62	26	[	[	X
ejpam-6039	62	27	25	25	NUM
ejpam-6039	62	28	]	]	PUNCT
ejpam-6039	62	29	of	of	ADP
ejpam-6039	62	30	a	a	PRON
ejpam-6039	62	31	and	and	CCONJ
ejpam-6039	62	32	is	be	AUX
ejpam-6039	62	33	denoted	denote	VERB
ejpam-6039	62	34	by	by	ADP
ejpam-6039	62	35	(	(	PUNCT
ejpam-6039	62	36	τ1	τ1	NOUN
ejpam-6039	62	37	,	,	PUNCT
ejpam-6039	62	38	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6039	62	39	)	)	PUNCT
ejpam-6039	62	40	.	.	PUNCT
ejpam-6039	63	1	lemma	lemma	PROPN
ejpam-6039	63	2	3	3	X
ejpam-6039	63	3	.	.	X
ejpam-6039	64	1	for	for	ADP
ejpam-6039	64	2	a	a	DET
ejpam-6039	64	3	subset	subset	NOUN
ejpam-6039	64	4	a	a	PRON
ejpam-6039	64	5	of	of	ADP
ejpam-6039	64	6	a	a	DET
ejpam-6039	64	7	bitopological	bitopological	ADJ
ejpam-6039	64	8	space	space	NOUN
ejpam-6039	64	9	(	(	PUNCT
ejpam-6039	64	10	x	x	NOUN
ejpam-6039	64	11	,	,	PUNCT
ejpam-6039	64	12	τ1	τ1	NOUN
ejpam-6039	64	13	,	,	PUNCT
ejpam-6039	64	14	τ2	τ2	NOUN
ejpam-6039	64	15	)	)	PUNCT
ejpam-6039	64	16	,	,	PUNCT
ejpam-6039	64	17	the	the	DET
ejpam-6039	64	18	following	follow	VERB
ejpam-6039	64	19	properties	property	NOUN
ejpam-6039	64	20	hold	hold	VERB
ejpam-6039	64	21	:	:	PUNCT
ejpam-6039	64	22	(	(	PUNCT
ejpam-6039	64	23	1	1	X
ejpam-6039	64	24	)	)	PUNCT
ejpam-6039	64	25	(	(	PUNCT
ejpam-6039	64	26	τ1	τ1	NOUN
ejpam-6039	64	27	,	,	PUNCT
ejpam-6039	64	28	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-6039	64	29	)	)	PUNCT
ejpam-6039	64	30	=	=	PUNCT
ejpam-6039	65	1	τ1τ2	τ1τ2	NOUN
ejpam-6039	65	2	-	-	ADJ
ejpam-6039	65	3	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	65	4	-	-	PUNCT
ejpam-6039	65	5	int(a	int(a	NOUN
ejpam-6039	65	6	)	)	PUNCT
ejpam-6039	65	7	)	)	PUNCT
ejpam-6039	65	8	∪a	∪a	X
ejpam-6039	66	1	[	[	X
ejpam-6039	66	2	25	25	NUM
ejpam-6039	66	3	]	]	PUNCT
ejpam-6039	66	4	;	;	PUNCT
ejpam-6039	66	5	(	(	PUNCT
ejpam-6039	66	6	2	2	X
ejpam-6039	66	7	)	)	PUNCT
ejpam-6039	66	8	(	(	PUNCT
ejpam-6039	66	9	τ1	τ1	NOUN
ejpam-6039	66	10	,	,	PUNCT
ejpam-6039	66	11	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6039	66	12	)	)	PUNCT
ejpam-6039	66	13	=	=	PUNCT
ejpam-6039	67	1	τ1τ2	τ1τ2	NOUN
ejpam-6039	67	2	-	-	NOUN
ejpam-6039	67	3	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	67	4	-	-	PUNCT
ejpam-6039	67	5	cl(a	cl(a	NUM
ejpam-6039	67	6	)	)	PUNCT
ejpam-6039	67	7	)	)	PUNCT
ejpam-6039	68	1	∩a	∩a	PROPN
ejpam-6039	69	1	[	[	X
ejpam-6039	69	2	26	26	NUM
ejpam-6039	69	3	]	]	PUNCT
ejpam-6039	69	4	.	.	PUNCT
ejpam-6039	70	1	3	3	X
ejpam-6039	70	2	.	.	X
ejpam-6039	70	3	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	70	4	,	,	PUNCT
ejpam-6039	70	5	τ2)p	τ2)p	ADJ
ejpam-6039	70	6	-	-	PUNCT
ejpam-6039	70	7	continuous	continuous	ADJ
ejpam-6039	70	8	functions	function	NOUN
ejpam-6039	70	9	in	in	ADP
ejpam-6039	70	10	this	this	DET
ejpam-6039	70	11	section	section	NOUN
ejpam-6039	70	12	,	,	PUNCT
ejpam-6039	70	13	we	we	PRON
ejpam-6039	70	14	introduce	introduce	VERB
ejpam-6039	70	15	the	the	DET
ejpam-6039	70	16	concept	concept	NOUN
ejpam-6039	70	17	of	of	ADP
ejpam-6039	70	18	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	70	19	,	,	PUNCT
ejpam-6039	70	20	τ2)p	τ2)p	ADJ
ejpam-6039	70	21	-	-	PUNCT
ejpam-6039	70	22	continuous	continuous	ADJ
ejpam-6039	70	23	functions	function	NOUN
ejpam-6039	70	24	.	.	PUNCT
ejpam-6039	71	1	moreover	moreover	ADV
ejpam-6039	71	2	,	,	PUNCT
ejpam-6039	71	3	some	some	DET
ejpam-6039	71	4	characterizations	characterization	NOUN
ejpam-6039	71	5	of	of	ADP
ejpam-6039	71	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	71	7	,	,	PUNCT
ejpam-6039	71	8	τ2)p	τ2)p	ADJ
ejpam-6039	71	9	-	-	PUNCT
ejpam-6039	71	10	continuous	continuous	ADJ
ejpam-6039	71	11	functions	function	NOUN
ejpam-6039	71	12	are	be	AUX
ejpam-6039	71	13	discussed	discuss	VERB
ejpam-6039	71	14	.	.	PUNCT
ejpam-6039	72	1	definition	definition	NOUN
ejpam-6039	72	2	1	1	NUM
ejpam-6039	72	3	.	.	PUNCT
ejpam-6039	73	1	a	a	DET
ejpam-6039	73	2	function	function	NOUN
ejpam-6039	73	3	f	f	NOUN
ejpam-6039	73	4	:	:	PUNCT
ejpam-6039	73	5	(	(	PUNCT
ejpam-6039	73	6	x	x	NOUN
ejpam-6039	73	7	,	,	PUNCT
ejpam-6039	73	8	τ1	τ1	NOUN
ejpam-6039	73	9	,	,	PUNCT
ejpam-6039	73	10	τ2	τ2	NOUN
ejpam-6039	73	11	)	)	PUNCT
ejpam-6039	73	12	→	→	SYM
ejpam-6039	73	13	(	(	PUNCT
ejpam-6039	73	14	y	y	PROPN
ejpam-6039	73	15	,	,	PUNCT
ejpam-6039	73	16	σ1	σ1	PROPN
ejpam-6039	73	17	,	,	PUNCT
ejpam-6039	73	18	σ2	σ2	PROPN
ejpam-6039	73	19	)	)	PUNCT
ejpam-6039	73	20	is	be	AUX
ejpam-6039	73	21	said	say	VERB
ejpam-6039	73	22	to	to	PART
ejpam-6039	73	23	be	be	AUX
ejpam-6039	73	24	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	73	25	,	,	PUNCT
ejpam-6039	73	26	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-6039	73	27	if	if	SCONJ
ejpam-6039	73	28	for	for	ADP
ejpam-6039	73	29	each	each	DET
ejpam-6039	73	30	x	x	SYM
ejpam-6039	73	31	∈	∈	PROPN
ejpam-6039	73	32	x	x	X
ejpam-6039	73	33	and	and	CCONJ
ejpam-6039	73	34	for	for	ADP
ejpam-6039	73	35	each	each	DET
ejpam-6039	73	36	σ1σ2	σ1σ2	NUM
ejpam-6039	73	37	-	-	PUNCT
ejpam-6039	73	38	closed	closed	ADJ
ejpam-6039	73	39	set	set	ADJ
ejpam-6039	73	40	f	f	PROPN
ejpam-6039	73	41	of	of	ADP
ejpam-6039	73	42	y	y	PROPN
ejpam-6039	73	43	containing	contain	VERB
ejpam-6039	73	44	f(x	f(x	PROPN
ejpam-6039	73	45	)	)	PUNCT
ejpam-6039	73	46	,	,	PUNCT
ejpam-6039	73	47	there	there	PRON
ejpam-6039	73	48	exists	exist	VERB
ejpam-6039	73	49	a	a	DET
ejpam-6039	73	50	(	(	PUNCT
ejpam-6039	73	51	τ1	τ1	NOUN
ejpam-6039	73	52	,	,	PUNCT
ejpam-6039	73	53	τ2)p	τ2)p	ADJ
ejpam-6039	73	54	-	-	PUNCT
ejpam-6039	73	55	open	open	ADJ
ejpam-6039	73	56	set	set	NOUN
ejpam-6039	73	57	u	u	NOUN
ejpam-6039	73	58	of	of	ADP
ejpam-6039	73	59	x	x	PUNCT
ejpam-6039	73	60	containing	contain	VERB
ejpam-6039	73	61	x	x	PUNCT
ejpam-6039	73	62	such	such	ADJ
ejpam-6039	73	63	that	that	DET
ejpam-6039	73	64	f(u	f(u	PROPN
ejpam-6039	73	65	)	)	PUNCT
ejpam-6039	73	66	⊆	⊆	NUM
ejpam-6039	73	67	f	f	PROPN
ejpam-6039	73	68	.	.	PUNCT
ejpam-6039	74	1	theorem	theorem	NOUN
ejpam-6039	74	2	1	1	NUM
ejpam-6039	74	3	.	.	X
ejpam-6039	74	4	for	for	ADP
ejpam-6039	74	5	a	a	DET
ejpam-6039	74	6	function	function	NOUN
ejpam-6039	74	7	f	f	NOUN
ejpam-6039	74	8	:	:	PUNCT
ejpam-6039	74	9	(	(	PUNCT
ejpam-6039	74	10	x	x	NOUN
ejpam-6039	74	11	,	,	PUNCT
ejpam-6039	74	12	τ1	τ1	NOUN
ejpam-6039	74	13	,	,	PUNCT
ejpam-6039	74	14	τ2	τ2	NOUN
ejpam-6039	74	15	)	)	PUNCT
ejpam-6039	74	16	→	→	SYM
ejpam-6039	74	17	(	(	PUNCT
ejpam-6039	74	18	y	y	PROPN
ejpam-6039	74	19	,	,	PUNCT
ejpam-6039	74	20	σ1	σ1	PROPN
ejpam-6039	74	21	,	,	PUNCT
ejpam-6039	74	22	σ2	σ2	NOUN
ejpam-6039	74	23	)	)	PUNCT
ejpam-6039	74	24	,	,	PUNCT
ejpam-6039	74	25	the	the	DET
ejpam-6039	74	26	following	follow	VERB
ejpam-6039	74	27	properties	property	NOUN
ejpam-6039	74	28	are	be	AUX
ejpam-6039	74	29	equivalent	equivalent	ADJ
ejpam-6039	74	30	:	:	PUNCT
ejpam-6039	74	31	(	(	PUNCT
ejpam-6039	74	32	1	1	X
ejpam-6039	74	33	)	)	PUNCT
ejpam-6039	74	34	f	f	PROPN
ejpam-6039	74	35	is	be	AUX
ejpam-6039	74	36	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	74	37	,	,	PUNCT
ejpam-6039	74	38	τ2)p	τ2)p	ADJ
ejpam-6039	74	39	-	-	NOUN
ejpam-6039	74	40	continuous	continuous	ADJ
ejpam-6039	74	41	;	;	PUNCT
ejpam-6039	74	42	(	(	PUNCT
ejpam-6039	74	43	2	2	X
ejpam-6039	74	44	)	)	PUNCT
ejpam-6039	74	45	f−1(f	f−1(f	NOUN
ejpam-6039	74	46	)	)	PUNCT
ejpam-6039	74	47	is	be	AUX
ejpam-6039	74	48	(	(	PUNCT
ejpam-6039	74	49	τ1	τ1	NOUN
ejpam-6039	74	50	,	,	PUNCT
ejpam-6039	74	51	τ2)p	τ2)p	NOUN
ejpam-6039	74	52	-	-	PUNCT
ejpam-6039	74	53	open	open	ADJ
ejpam-6039	74	54	in	in	ADP
ejpam-6039	74	55	x	x	PUNCT
ejpam-6039	74	56	for	for	ADP
ejpam-6039	74	57	every	every	DET
ejpam-6039	74	58	σ1σ2	σ1σ2	NUM
ejpam-6039	74	59	-	-	PUNCT
ejpam-6039	74	60	closed	closed	ADJ
ejpam-6039	74	61	set	set	ADJ
ejpam-6039	74	62	f	f	PROPN
ejpam-6039	74	63	of	of	ADP
ejpam-6039	74	64	y	y	PROPN
ejpam-6039	74	65	;	;	PUNCT
ejpam-6039	74	66	(	(	PUNCT
ejpam-6039	74	67	3	3	X
ejpam-6039	74	68	)	)	PUNCT
ejpam-6039	74	69	f−1(v	f−1(v	NOUN
ejpam-6039	74	70	)	)	PUNCT
ejpam-6039	75	1	is	be	AUX
ejpam-6039	75	2	(	(	PUNCT
ejpam-6039	75	3	τ1	τ1	NOUN
ejpam-6039	75	4	,	,	PUNCT
ejpam-6039	75	5	τ2)p	τ2)p	NOUN
ejpam-6039	75	6	-	-	PUNCT
ejpam-6039	75	7	closed	closed	ADJ
ejpam-6039	75	8	in	in	ADP
ejpam-6039	75	9	x	x	PUNCT
ejpam-6039	75	10	for	for	ADP
ejpam-6039	75	11	every	every	DET
ejpam-6039	75	12	σ1σ2	σ1σ2	NOUN
ejpam-6039	75	13	-	-	ADJ
ejpam-6039	75	14	open	open	ADJ
ejpam-6039	75	15	set	set	NOUN
ejpam-6039	75	16	v	v	NOUN
ejpam-6039	75	17	of	of	ADP
ejpam-6039	75	18	y	y	PROPN
ejpam-6039	75	19	;	;	PUNCT
ejpam-6039	75	20	(	(	PUNCT
ejpam-6039	75	21	4	4	X
ejpam-6039	75	22	)	)	PUNCT
ejpam-6039	75	23	f((τ1	f((τ1	PROPN
ejpam-6039	75	24	,	,	PUNCT
ejpam-6039	75	25	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6039	75	26	)	)	PUNCT
ejpam-6039	75	27	)	)	PUNCT
ejpam-6039	76	1	⊆	⊆	X
ejpam-6039	76	2	σ1σ2	σ1σ2	NUM
ejpam-6039	76	3	-	-	PUNCT
ejpam-6039	76	4	ker(f(a	ker(f(a	NOUN
ejpam-6039	76	5	)	)	PUNCT
ejpam-6039	76	6	)	)	PUNCT
ejpam-6039	76	7	for	for	ADP
ejpam-6039	76	8	every	every	DET
ejpam-6039	76	9	subset	subset	NOUN
ejpam-6039	76	10	a	a	PRON
ejpam-6039	76	11	of	of	ADP
ejpam-6039	76	12	x	x	NOUN
ejpam-6039	76	13	;	;	PUNCT
ejpam-6039	76	14	m.	m.	NOUN
ejpam-6039	76	15	chiangpradit	chiangpradit	NOUN
ejpam-6039	76	16	,	,	PUNCT
ejpam-6039	76	17	s.	s.	PROPN
ejpam-6039	76	18	sompong	sompong	PROPN
ejpam-6039	76	19	,	,	PUNCT
ejpam-6039	76	20	c.	c.	PROPN
ejpam-6039	76	21	boonpok	boonpok	PROPN
ejpam-6039	76	22	/	/	SYM
ejpam-6039	76	23	eur	eur	PROPN
ejpam-6039	76	24	.	.	PUNCT
ejpam-6039	77	1	j.	j.	PROPN
ejpam-6039	77	2	pure	pure	PROPN
ejpam-6039	77	3	appl	appl	PROPN
ejpam-6039	77	4	.	.	PROPN
ejpam-6039	77	5	math	math	PROPN
ejpam-6039	77	6	,	,	PUNCT
ejpam-6039	77	7	18	18	NUM
ejpam-6039	77	8	(	(	PUNCT
ejpam-6039	77	9	2	2	NUM
ejpam-6039	77	10	)	)	PUNCT
ejpam-6039	77	11	(	(	PUNCT
ejpam-6039	77	12	2025	2025	NUM
ejpam-6039	77	13	)	)	PUNCT
ejpam-6039	77	14	,	,	PUNCT
ejpam-6039	77	15	6039	6039	NUM
ejpam-6039	77	16	4	4	NUM
ejpam-6039	77	17	of	of	ADP
ejpam-6039	77	18	11	11	NUM
ejpam-6039	77	19	(	(	PUNCT
ejpam-6039	77	20	5	5	NUM
ejpam-6039	77	21	)	)	PUNCT
ejpam-6039	77	22	(	(	PUNCT
ejpam-6039	77	23	τ1	τ1	NOUN
ejpam-6039	77	24	,	,	PUNCT
ejpam-6039	77	25	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-6039	77	26	−1(b	−1(b	NOUN
ejpam-6039	77	27	)	)	PUNCT
ejpam-6039	77	28	)	)	PUNCT
ejpam-6039	78	1	⊆	⊆	NUM
ejpam-6039	78	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	78	3	-	-	PUNCT
ejpam-6039	78	4	ker(b	ker(b	PROPN
ejpam-6039	78	5	)	)	PUNCT
ejpam-6039	78	6	)	)	PUNCT
ejpam-6039	78	7	for	for	ADP
ejpam-6039	78	8	every	every	DET
ejpam-6039	78	9	subset	subset	NOUN
ejpam-6039	78	10	b	b	PROPN
ejpam-6039	78	11	of	of	ADP
ejpam-6039	78	12	y	y	PROPN
ejpam-6039	78	13	.	.	PUNCT
ejpam-6039	79	1	proof	proof	NOUN
ejpam-6039	79	2	.	.	PUNCT
ejpam-6039	80	1	(	(	PUNCT
ejpam-6039	80	2	1	1	X
ejpam-6039	80	3	)	)	PUNCT
ejpam-6039	80	4	⇒	⇒	NOUN
ejpam-6039	80	5	(	(	PUNCT
ejpam-6039	80	6	2	2	NUM
ejpam-6039	80	7	):	):	PUNCT
ejpam-6039	80	8	let	let	VERB
ejpam-6039	80	9	f	f	PRON
ejpam-6039	80	10	be	be	AUX
ejpam-6039	80	11	any	any	DET
ejpam-6039	80	12	σ1σ2	σ1σ2	NUM
ejpam-6039	80	13	-	-	PUNCT
ejpam-6039	80	14	closed	closed	ADJ
ejpam-6039	80	15	set	set	NOUN
ejpam-6039	80	16	of	of	ADP
ejpam-6039	80	17	y	y	PROPN
ejpam-6039	80	18	and	and	CCONJ
ejpam-6039	80	19	x	x	PROPN
ejpam-6039	80	20	∈	∈	PROPN
ejpam-6039	80	21	f−1(f	f−1(f	PROPN
ejpam-6039	80	22	)	)	PUNCT
ejpam-6039	80	23	.	.	PUNCT
ejpam-6039	81	1	then	then	ADV
ejpam-6039	81	2	,	,	PUNCT
ejpam-6039	81	3	f(x	f(x	PROPN
ejpam-6039	81	4	)	)	PUNCT
ejpam-6039	81	5	∈	∈	PROPN
ejpam-6039	81	6	f	f	PROPN
ejpam-6039	81	7	.	.	PUNCT
ejpam-6039	82	1	since	since	SCONJ
ejpam-6039	82	2	f	f	PROPN
ejpam-6039	82	3	is	be	AUX
ejpam-6039	82	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	82	5	,	,	PUNCT
ejpam-6039	82	6	τ2)p	τ2)p	ADJ
ejpam-6039	82	7	-	-	ADJ
ejpam-6039	82	8	continuous	continuous	ADJ
ejpam-6039	82	9	,	,	PUNCT
ejpam-6039	82	10	there	there	PRON
ejpam-6039	82	11	exists	exist	VERB
ejpam-6039	82	12	a	a	DET
ejpam-6039	82	13	(	(	PUNCT
ejpam-6039	82	14	τ1	τ1	NOUN
ejpam-6039	82	15	,	,	PUNCT
ejpam-6039	82	16	τ2)p	τ2)p	ADJ
ejpam-6039	82	17	-	-	PUNCT
ejpam-6039	82	18	open	open	ADJ
ejpam-6039	82	19	set	set	NOUN
ejpam-6039	82	20	u	u	NOUN
ejpam-6039	82	21	of	of	ADP
ejpam-6039	82	22	x	x	PUNCT
ejpam-6039	82	23	containing	contain	VERB
ejpam-6039	82	24	x	x	PUNCT
ejpam-6039	82	25	such	such	ADJ
ejpam-6039	82	26	that	that	DET
ejpam-6039	82	27	f(u	f(u	PROPN
ejpam-6039	82	28	)	)	PUNCT
ejpam-6039	82	29	⊆	⊆	NUM
ejpam-6039	82	30	f	f	NOUN
ejpam-6039	82	31	.	.	PUNCT
ejpam-6039	83	1	thus	thus	ADV
ejpam-6039	83	2	,	,	PUNCT
ejpam-6039	83	3	u	u	PROPN
ejpam-6039	83	4	⊆	⊆	NUM
ejpam-6039	83	5	f−1(f	f−1(f	PROPN
ejpam-6039	83	6	)	)	PUNCT
ejpam-6039	83	7	and	and	CCONJ
ejpam-6039	83	8	hence	hence	ADV
ejpam-6039	83	9	x	x	PART
ejpam-6039	83	10	∈	∈	PROPN
ejpam-6039	83	11	u	u	NOUN
ejpam-6039	83	12	⊆	⊆	NUM
ejpam-6039	83	13	f−1(f	f−1(f	PROPN
ejpam-6039	83	14	)	)	PUNCT
ejpam-6039	83	15	.	.	PUNCT
ejpam-6039	84	1	therefore	therefore	ADV
ejpam-6039	84	2	,	,	PUNCT
ejpam-6039	84	3	x	x	X
ejpam-6039	84	4	∈	∈	PROPN
ejpam-6039	84	5	(	(	PUNCT
ejpam-6039	84	6	τ1	τ1	NOUN
ejpam-6039	84	7	,	,	PUNCT
ejpam-6039	84	8	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-6039	84	9	−1(f	−1(f	NUM
ejpam-6039	84	10	)	)	PUNCT
ejpam-6039	84	11	)	)	PUNCT
ejpam-6039	84	12	.	.	PUNCT
ejpam-6039	85	1	this	this	PRON
ejpam-6039	85	2	implies	imply	VERB
ejpam-6039	85	3	that	that	SCONJ
ejpam-6039	85	4	f−1(f	f−1(f	NOUN
ejpam-6039	85	5	)	)	PUNCT
ejpam-6039	86	1	⊆	⊆	NUM
ejpam-6039	86	2	(	(	PUNCT
ejpam-6039	86	3	τ1	τ1	NOUN
ejpam-6039	86	4	,	,	PUNCT
ejpam-6039	86	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-6039	86	6	−1(f	−1(f	NUM
ejpam-6039	86	7	)	)	PUNCT
ejpam-6039	86	8	)	)	PUNCT
ejpam-6039	86	9	.	.	PUNCT
ejpam-6039	87	1	thus	thus	ADV
ejpam-6039	87	2	,	,	PUNCT
ejpam-6039	87	3	f−1(f	f−1(f	PROPN
ejpam-6039	87	4	)	)	PUNCT
ejpam-6039	87	5	is	be	AUX
ejpam-6039	87	6	(	(	PUNCT
ejpam-6039	87	7	τ1	τ1	NOUN
ejpam-6039	87	8	,	,	PUNCT
ejpam-6039	87	9	τ2)p	τ2)p	NOUN
ejpam-6039	87	10	-	-	PUNCT
ejpam-6039	87	11	open	open	ADJ
ejpam-6039	87	12	in	in	ADP
ejpam-6039	87	13	x.	x.	NOUN
ejpam-6039	87	14	(	(	PUNCT
ejpam-6039	87	15	2	2	X
ejpam-6039	87	16	)	)	PUNCT
ejpam-6039	87	17	⇔	⇔	X
ejpam-6039	87	18	(	(	PUNCT
ejpam-6039	87	19	3	3	NUM
ejpam-6039	87	20	):	):	PUNCT
ejpam-6039	87	21	let	let	VERB
ejpam-6039	87	22	v	v	PART
ejpam-6039	87	23	be	be	AUX
ejpam-6039	87	24	any	any	DET
ejpam-6039	87	25	σ1σ2	σ1σ2	NOUN
ejpam-6039	87	26	-	-	ADJ
ejpam-6039	87	27	open	open	ADJ
ejpam-6039	87	28	set	set	NOUN
ejpam-6039	87	29	of	of	ADP
ejpam-6039	87	30	y	y	PROPN
ejpam-6039	87	31	.	.	PUNCT
ejpam-6039	88	1	then	then	ADV
ejpam-6039	88	2	,	,	PUNCT
ejpam-6039	88	3	y	y	PROPN
ejpam-6039	88	4	−	−	PROPN
ejpam-6039	88	5	v	v	NOUN
ejpam-6039	88	6	is	be	AUX
ejpam-6039	88	7	σ1σ2	σ1σ2	NOUN
ejpam-6039	88	8	-	-	ADJ
ejpam-6039	88	9	closed	closed	ADJ
ejpam-6039	88	10	in	in	ADP
ejpam-6039	88	11	y	y	PROPN
ejpam-6039	88	12	.	.	PUNCT
ejpam-6039	89	1	by	by	ADP
ejpam-6039	89	2	(	(	PUNCT
ejpam-6039	89	3	2	2	NUM
ejpam-6039	89	4	)	)	PUNCT
ejpam-6039	89	5	,	,	PUNCT
ejpam-6039	89	6	we	we	PRON
ejpam-6039	89	7	have	have	VERB
ejpam-6039	89	8	f−1(y	f−1(y	NOUN
ejpam-6039	89	9	−	−	PROPN
ejpam-6039	89	10	v	v	NOUN
ejpam-6039	89	11	)	)	PUNCT
ejpam-6039	89	12	=	=	PUNCT
ejpam-6039	89	13	x	x	X
ejpam-6039	89	14	−	−	PROPN
ejpam-6039	89	15	f−1(v	f−1(v	PROPN
ejpam-6039	89	16	)	)	PUNCT
ejpam-6039	89	17	is	be	AUX
ejpam-6039	89	18	(	(	PUNCT
ejpam-6039	89	19	τ1	τ1	NOUN
ejpam-6039	89	20	,	,	PUNCT
ejpam-6039	89	21	τ2)p	τ2)p	NOUN
ejpam-6039	89	22	-	-	PUNCT
ejpam-6039	89	23	open	open	ADJ
ejpam-6039	89	24	in	in	ADP
ejpam-6039	89	25	x	x	X
ejpam-6039	89	26	and	and	CCONJ
ejpam-6039	89	27	hence	hence	ADV
ejpam-6039	89	28	f−1(v	f−1(v	NOUN
ejpam-6039	89	29	)	)	PUNCT
ejpam-6039	90	1	is	be	AUX
ejpam-6039	90	2	(	(	PUNCT
ejpam-6039	90	3	τ1	τ1	NOUN
ejpam-6039	90	4	,	,	PUNCT
ejpam-6039	90	5	τ2)p	τ2)p	NOUN
ejpam-6039	90	6	-	-	PUNCT
ejpam-6039	90	7	closed	closed	ADJ
ejpam-6039	90	8	in	in	ADP
ejpam-6039	90	9	x.	x.	NOUN
ejpam-6039	90	10	the	the	DET
ejpam-6039	90	11	converse	converse	NOUN
ejpam-6039	90	12	can	can	AUX
ejpam-6039	90	13	be	be	AUX
ejpam-6039	90	14	shown	show	VERB
ejpam-6039	90	15	easily	easily	ADV
ejpam-6039	90	16	.	.	PUNCT
ejpam-6039	91	1	(	(	PUNCT
ejpam-6039	91	2	2	2	X
ejpam-6039	91	3	)	)	PUNCT
ejpam-6039	91	4	⇒	⇒	NOUN
ejpam-6039	91	5	(	(	PUNCT
ejpam-6039	91	6	4	4	NUM
ejpam-6039	91	7	):	):	PUNCT
ejpam-6039	91	8	let	let	VERB
ejpam-6039	91	9	a	a	DET
ejpam-6039	91	10	be	be	AUX
ejpam-6039	91	11	any	any	DET
ejpam-6039	91	12	subset	subset	NOUN
ejpam-6039	91	13	of	of	ADP
ejpam-6039	91	14	x.	x.	PROPN
ejpam-6039	91	15	suppose	suppose	VERB
ejpam-6039	91	16	that	that	SCONJ
ejpam-6039	91	17	y	y	PROPN
ejpam-6039	91	18	̸∈	̸∈	PROPN
ejpam-6039	91	19	σ1σ2	σ1σ2	NOUN
ejpam-6039	91	20	-	-	PUNCT
ejpam-6039	91	21	ker(f(a	ker(f(a	NOUN
ejpam-6039	91	22	)	)	PUNCT
ejpam-6039	91	23	)	)	PUNCT
ejpam-6039	91	24	.	.	PUNCT
ejpam-6039	92	1	then	then	ADV
ejpam-6039	92	2	by	by	ADP
ejpam-6039	92	3	lemma	lemma	PROPN
ejpam-6039	92	4	2	2	NUM
ejpam-6039	92	5	,	,	PUNCT
ejpam-6039	92	6	there	there	PRON
ejpam-6039	92	7	exists	exist	VERB
ejpam-6039	92	8	a	a	DET
ejpam-6039	92	9	σ1σ2	σ1σ2	NUM
ejpam-6039	92	10	-	-	PUNCT
ejpam-6039	92	11	closed	closed	ADJ
ejpam-6039	92	12	set	set	NOUN
ejpam-6039	92	13	k	k	PROPN
ejpam-6039	92	14	of	of	ADP
ejpam-6039	92	15	y	y	PROPN
ejpam-6039	92	16	containing	contain	VERB
ejpam-6039	92	17	y	y	PRON
ejpam-6039	92	18	such	such	ADJ
ejpam-6039	92	19	that	that	DET
ejpam-6039	92	20	f(a	f(a	NOUN
ejpam-6039	92	21	)	)	PUNCT
ejpam-6039	92	22	∩	∩	NOUN
ejpam-6039	93	1	k	k	NOUN
ejpam-6039	94	1	=	=	PUNCT
ejpam-6039	94	2	∅.	∅.	VERB
ejpam-6039	94	3	thus	thus	ADV
ejpam-6039	94	4	,	,	PUNCT
ejpam-6039	94	5	a	a	DET
ejpam-6039	94	6	∩	∩	ADJ
ejpam-6039	94	7	f−1(k	f−1(k	PROPN
ejpam-6039	94	8	)	)	PUNCT
ejpam-6039	94	9	=	=	SYM
ejpam-6039	94	10	∅	∅	NOUN
ejpam-6039	94	11	and	and	CCONJ
ejpam-6039	94	12	hence	hence	ADV
ejpam-6039	94	13	(	(	PUNCT
ejpam-6039	94	14	τ1	τ1	PROPN
ejpam-6039	94	15	,	,	PUNCT
ejpam-6039	94	16	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-6039	94	17	)	)	PUNCT
ejpam-6039	94	18	∩	∩	ADJ
ejpam-6039	94	19	f−1(k	f−1(k	PROPN
ejpam-6039	94	20	)	)	PUNCT
ejpam-6039	94	21	=	=	PUNCT
ejpam-6039	94	22	∅.	∅.	VERB
ejpam-6039	94	23	therefore	therefore	ADV
ejpam-6039	94	24	,	,	PUNCT
ejpam-6039	94	25	f((τ1	f((τ1	PROPN
ejpam-6039	94	26	,	,	PUNCT
ejpam-6039	94	27	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6039	94	28	)	)	PUNCT
ejpam-6039	94	29	)	)	PUNCT
ejpam-6039	94	30	∩k	∩k	NOUN
ejpam-6039	94	31	=	=	SYM
ejpam-6039	94	32	∅	∅	NOUN
ejpam-6039	94	33	and	and	CCONJ
ejpam-6039	94	34	y	y	PROPN
ejpam-6039	94	35	̸∈	̸∈	PROPN
ejpam-6039	94	36	f((τ1	f((τ1	PROPN
ejpam-6039	94	37	,	,	PUNCT
ejpam-6039	94	38	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6039	94	39	)	)	PUNCT
ejpam-6039	94	40	)	)	PUNCT
ejpam-6039	94	41	.	.	PUNCT
ejpam-6039	95	1	this	this	PRON
ejpam-6039	95	2	shows	show	VERB
ejpam-6039	95	3	that	that	SCONJ
ejpam-6039	95	4	f((τ1	f((τ1	PROPN
ejpam-6039	95	5	,	,	PUNCT
ejpam-6039	95	6	τ2)-pcl(a	τ2)-pcl(a	NOUN
ejpam-6039	95	7	)	)	PUNCT
ejpam-6039	95	8	)	)	PUNCT
ejpam-6039	96	1	⊆	⊆	X
ejpam-6039	96	2	σ1σ2	σ1σ2	NUM
ejpam-6039	96	3	-	-	PUNCT
ejpam-6039	96	4	ker(f(a	ker(f(a	NOUN
ejpam-6039	96	5	)	)	PUNCT
ejpam-6039	96	6	)	)	PUNCT
ejpam-6039	96	7	.	.	PUNCT
ejpam-6039	97	1	(	(	PUNCT
ejpam-6039	97	2	4	4	X
ejpam-6039	97	3	)	)	PUNCT
ejpam-6039	97	4	⇒	⇒	NOUN
ejpam-6039	97	5	(	(	PUNCT
ejpam-6039	97	6	5	5	NUM
ejpam-6039	97	7	):	):	PUNCT
ejpam-6039	97	8	let	let	VERB
ejpam-6039	97	9	b	b	X
ejpam-6039	97	10	be	be	AUX
ejpam-6039	97	11	any	any	DET
ejpam-6039	97	12	subset	subset	NOUN
ejpam-6039	97	13	of	of	ADP
ejpam-6039	97	14	y	y	PROPN
ejpam-6039	97	15	.	.	PUNCT
ejpam-6039	98	1	by	by	ADP
ejpam-6039	98	2	(	(	PUNCT
ejpam-6039	98	3	4	4	NUM
ejpam-6039	98	4	)	)	PUNCT
ejpam-6039	98	5	and	and	CCONJ
ejpam-6039	98	6	lemma	lemma	PROPN
ejpam-6039	98	7	2	2	NUM
ejpam-6039	98	8	,	,	PUNCT
ejpam-6039	98	9	we	we	PRON
ejpam-6039	98	10	have	have	AUX
ejpam-6039	98	11	f((τ1	f((τ1	VERB
ejpam-6039	98	12	,	,	PUNCT
ejpam-6039	98	13	τ2)-pcl(f	τ2)-pcl(f	NOUN
ejpam-6039	98	14	−1(b	−1(b	NOUN
ejpam-6039	98	15	)	)	PUNCT
ejpam-6039	98	16	)	)	PUNCT
ejpam-6039	98	17	)	)	PUNCT
ejpam-6039	99	1	⊆	⊆	X
ejpam-6039	99	2	σ1σ2	σ1σ2	NUM
ejpam-6039	99	3	-	-	PUNCT
ejpam-6039	99	4	ker(f(f	ker(f(f	NOUN
ejpam-6039	99	5	−1(b	−1(b	NOUN
ejpam-6039	99	6	)	)	PUNCT
ejpam-6039	99	7	)	)	PUNCT
ejpam-6039	99	8	)	)	PUNCT
ejpam-6039	100	1	⊆	⊆	X
ejpam-6039	100	2	σ1σ2	σ1σ2	X
ejpam-6039	100	3	-	-	PUNCT
ejpam-6039	100	4	ker(b	ker(b	NOUN
ejpam-6039	100	5	)	)	PUNCT
ejpam-6039	100	6	and	and	CCONJ
ejpam-6039	100	7	hence	hence	ADV
ejpam-6039	100	8	(	(	PUNCT
ejpam-6039	100	9	τ1	τ1	PROPN
ejpam-6039	100	10	,	,	PUNCT
ejpam-6039	100	11	τ2)-pcl(f	τ2)-pcl(f	NOUN
ejpam-6039	100	12	−1(b	−1(b	NOUN
ejpam-6039	100	13	)	)	PUNCT
ejpam-6039	100	14	)	)	PUNCT
ejpam-6039	101	1	⊆	⊆	NUM
ejpam-6039	101	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	101	3	-	-	PUNCT
ejpam-6039	101	4	ker(b	ker(b	PROPN
ejpam-6039	101	5	)	)	PUNCT
ejpam-6039	101	6	)	)	PUNCT
ejpam-6039	101	7	.	.	PUNCT
ejpam-6039	102	1	(	(	PUNCT
ejpam-6039	102	2	5	5	X
ejpam-6039	102	3	)	)	PUNCT
ejpam-6039	102	4	⇒	⇒	NOUN
ejpam-6039	102	5	(	(	PUNCT
ejpam-6039	102	6	3	3	NUM
ejpam-6039	102	7	):	):	PUNCT
ejpam-6039	102	8	let	let	VERB
ejpam-6039	102	9	v	v	PART
ejpam-6039	102	10	be	be	AUX
ejpam-6039	102	11	any	any	DET
ejpam-6039	102	12	σ1σ2	σ1σ2	NOUN
ejpam-6039	102	13	-	-	ADJ
ejpam-6039	102	14	open	open	ADJ
ejpam-6039	102	15	set	set	NOUN
ejpam-6039	102	16	of	of	ADP
ejpam-6039	102	17	y	y	PROPN
ejpam-6039	102	18	.	.	PUNCT
ejpam-6039	103	1	then	then	ADV
ejpam-6039	103	2	by	by	ADP
ejpam-6039	103	3	(	(	PUNCT
ejpam-6039	103	4	5	5	NUM
ejpam-6039	103	5	)	)	PUNCT
ejpam-6039	103	6	and	and	CCONJ
ejpam-6039	103	7	lemma	lemma	PROPN
ejpam-6039	103	8	2	2	NUM
ejpam-6039	103	9	,	,	PUNCT
ejpam-6039	103	10	we	we	PRON
ejpam-6039	103	11	have	have	VERB
ejpam-6039	103	12	(	(	PUNCT
ejpam-6039	103	13	τ1	τ1	NOUN
ejpam-6039	103	14	,	,	PUNCT
ejpam-6039	103	15	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-6039	103	16	−1(v	−1(v	PROPN
ejpam-6039	103	17	)	)	PUNCT
ejpam-6039	103	18	)	)	PUNCT
ejpam-6039	104	1	⊆	⊆	NUM
ejpam-6039	104	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	104	3	-	-	PUNCT
ejpam-6039	104	4	ker(v	ker(v	NOUN
ejpam-6039	104	5	)	)	PUNCT
ejpam-6039	104	6	)	)	PUNCT
ejpam-6039	105	1	=	=	SYM
ejpam-6039	105	2	f−1(v	f−1(v	PROPN
ejpam-6039	105	3	)	)	PUNCT
ejpam-6039	105	4	and	and	CCONJ
ejpam-6039	105	5	hence	hence	ADV
ejpam-6039	105	6	f−1(v	f−1(v	PROPN
ejpam-6039	105	7	)	)	PUNCT
ejpam-6039	105	8	is	be	AUX
ejpam-6039	105	9	(	(	PUNCT
ejpam-6039	105	10	τ1	τ1	NOUN
ejpam-6039	105	11	,	,	PUNCT
ejpam-6039	105	12	τ2)p	τ2)p	NOUN
ejpam-6039	105	13	-	-	PUNCT
ejpam-6039	105	14	closed	closed	ADJ
ejpam-6039	105	15	in	in	ADP
ejpam-6039	105	16	x.	x.	NOUN
ejpam-6039	105	17	(	(	PUNCT
ejpam-6039	105	18	2	2	NUM
ejpam-6039	105	19	)	)	PUNCT
ejpam-6039	105	20	⇒	⇒	NOUN
ejpam-6039	105	21	(	(	PUNCT
ejpam-6039	105	22	1	1	NUM
ejpam-6039	105	23	):	):	PUNCT
ejpam-6039	105	24	let	let	VERB
ejpam-6039	105	25	f	f	PRON
ejpam-6039	105	26	be	be	AUX
ejpam-6039	105	27	any	any	DET
ejpam-6039	105	28	σ1σ2	σ1σ2	NUM
ejpam-6039	105	29	-	-	PUNCT
ejpam-6039	105	30	closed	closed	ADJ
ejpam-6039	105	31	set	set	NOUN
ejpam-6039	105	32	of	of	ADP
ejpam-6039	105	33	y	y	PROPN
ejpam-6039	105	34	containing	contain	VERB
ejpam-6039	105	35	f(x	f(x	PROPN
ejpam-6039	105	36	)	)	PUNCT
ejpam-6039	105	37	.	.	PUNCT
ejpam-6039	106	1	by	by	ADP
ejpam-6039	106	2	(	(	PUNCT
ejpam-6039	106	3	2	2	NUM
ejpam-6039	106	4	)	)	PUNCT
ejpam-6039	106	5	,	,	PUNCT
ejpam-6039	106	6	f−1(f	f−1(f	PROPN
ejpam-6039	106	7	)	)	PUNCT
ejpam-6039	106	8	is	be	AUX
ejpam-6039	106	9	(	(	PUNCT
ejpam-6039	106	10	τ1	τ1	NOUN
ejpam-6039	106	11	,	,	PUNCT
ejpam-6039	106	12	τ2)p	τ2)p	NOUN
ejpam-6039	106	13	-	-	PUNCT
ejpam-6039	106	14	open	open	ADJ
ejpam-6039	106	15	in	in	ADP
ejpam-6039	106	16	x.	x.	NOUN
ejpam-6039	106	17	then	then	ADV
ejpam-6039	106	18	we	we	PRON
ejpam-6039	106	19	have	have	VERB
ejpam-6039	106	20	,	,	PUNCT
ejpam-6039	106	21	x	x	SYM
ejpam-6039	106	22	∈	∈	PROPN
ejpam-6039	106	23	(	(	PUNCT
ejpam-6039	106	24	τ1	τ1	NOUN
ejpam-6039	106	25	,	,	PUNCT
ejpam-6039	106	26	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-6039	106	27	−1(f	−1(f	NUM
ejpam-6039	106	28	)	)	PUNCT
ejpam-6039	106	29	)	)	PUNCT
ejpam-6039	106	30	and	and	CCONJ
ejpam-6039	106	31	therefore	therefore	ADV
ejpam-6039	106	32	there	there	PRON
ejpam-6039	106	33	exists	exist	VERB
ejpam-6039	106	34	a	a	DET
ejpam-6039	106	35	(	(	PUNCT
ejpam-6039	106	36	τ1	τ1	NOUN
ejpam-6039	106	37	,	,	PUNCT
ejpam-6039	106	38	τ2)p	τ2)p	ADJ
ejpam-6039	106	39	-	-	PUNCT
ejpam-6039	106	40	open	open	ADJ
ejpam-6039	106	41	set	set	NOUN
ejpam-6039	106	42	u	u	NOUN
ejpam-6039	106	43	of	of	ADP
ejpam-6039	106	44	x	x	SYM
ejpam-6039	106	45	such	such	ADJ
ejpam-6039	106	46	that	that	SCONJ
ejpam-6039	106	47	x	x	SYM
ejpam-6039	106	48	∈	∈	PROPN
ejpam-6039	106	49	u	u	NOUN
ejpam-6039	106	50	⊆	⊆	NUM
ejpam-6039	106	51	f−1(f	f−1(f	PROPN
ejpam-6039	106	52	)	)	PUNCT
ejpam-6039	106	53	;	;	PUNCT
ejpam-6039	106	54	hence	hence	ADV
ejpam-6039	106	55	f(u	f(u	PROPN
ejpam-6039	106	56	)	)	PUNCT
ejpam-6039	106	57	⊆	⊆	NUM
ejpam-6039	106	58	f	f	NOUN
ejpam-6039	106	59	.	.	PUNCT
ejpam-6039	107	1	this	this	PRON
ejpam-6039	107	2	shows	show	VERB
ejpam-6039	107	3	that	that	SCONJ
ejpam-6039	107	4	f	f	PROPN
ejpam-6039	107	5	is	be	AUX
ejpam-6039	107	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	107	7	,	,	PUNCT
ejpam-6039	107	8	τ2)p	τ2)p	ADJ
ejpam-6039	107	9	-	-	ADJ
ejpam-6039	107	10	continuous	continuous	ADJ
ejpam-6039	107	11	.	.	PUNCT
ejpam-6039	108	1	recall	recall	VERB
ejpam-6039	108	2	that	that	SCONJ
ejpam-6039	108	3	a	a	DET
ejpam-6039	108	4	bitopological	bitopological	ADJ
ejpam-6039	108	5	space	space	NOUN
ejpam-6039	108	6	(	(	PUNCT
ejpam-6039	108	7	x	x	NOUN
ejpam-6039	108	8	,	,	PUNCT
ejpam-6039	108	9	τ1	τ1	NOUN
ejpam-6039	108	10	,	,	PUNCT
ejpam-6039	108	11	τ2	τ2	NOUN
ejpam-6039	108	12	)	)	PUNCT
ejpam-6039	108	13	is	be	AUX
ejpam-6039	108	14	said	say	VERB
ejpam-6039	108	15	to	to	PART
ejpam-6039	108	16	be	be	AUX
ejpam-6039	108	17	(	(	PUNCT
ejpam-6039	108	18	τ1	τ1	NOUN
ejpam-6039	108	19	,	,	PUNCT
ejpam-6039	109	1	τ2)-regular	τ2)-regular	ADJ
ejpam-6039	109	2	[	[	X
ejpam-6039	109	3	27	27	NUM
ejpam-6039	109	4	]	]	X
ejpam-6039	109	5	if	if	SCONJ
ejpam-6039	109	6	for	for	ADP
ejpam-6039	109	7	each	each	DET
ejpam-6039	109	8	τ1τ2	τ1τ2	ADJ
ejpam-6039	109	9	-	-	ADJ
ejpam-6039	109	10	closed	closed	ADJ
ejpam-6039	109	11	set	set	VERB
ejpam-6039	109	12	f	f	NOUN
ejpam-6039	109	13	and	and	CCONJ
ejpam-6039	109	14	each	each	DET
ejpam-6039	109	15	point	point	NOUN
ejpam-6039	109	16	x	x	X
ejpam-6039	109	17	∈	∈	NOUN
ejpam-6039	109	18	x	x	X
ejpam-6039	109	19	−	−	PROPN
ejpam-6039	109	20	f	f	NOUN
ejpam-6039	109	21	,	,	PUNCT
ejpam-6039	109	22	there	there	PRON
ejpam-6039	109	23	exist	exist	VERB
ejpam-6039	109	24	disjoint	disjoint	ADJ
ejpam-6039	109	25	τ1τ2	τ1τ2	ADJ
ejpam-6039	109	26	-	-	ADJ
ejpam-6039	109	27	open	open	ADJ
ejpam-6039	109	28	sets	set	NOUN
ejpam-6039	109	29	u	u	NOUN
ejpam-6039	109	30	and	and	CCONJ
ejpam-6039	109	31	v	v	ADP
ejpam-6039	109	32	such	such	ADJ
ejpam-6039	109	33	that	that	SCONJ
ejpam-6039	109	34	x	x	SYM
ejpam-6039	109	35	∈	∈	PROPN
ejpam-6039	109	36	u	u	NOUN
ejpam-6039	109	37	and	and	CCONJ
ejpam-6039	109	38	f	f	PROPN
ejpam-6039	109	39	⊆	⊆	NUM
ejpam-6039	109	40	v	v	NOUN
ejpam-6039	109	41	.	.	PUNCT
ejpam-6039	110	1	definition	definition	NOUN
ejpam-6039	110	2	2	2	NUM
ejpam-6039	110	3	.	.	PUNCT
ejpam-6039	111	1	a	a	DET
ejpam-6039	111	2	function	function	NOUN
ejpam-6039	111	3	f	f	NOUN
ejpam-6039	111	4	:	:	PUNCT
ejpam-6039	111	5	(	(	PUNCT
ejpam-6039	111	6	x	x	NOUN
ejpam-6039	111	7	,	,	PUNCT
ejpam-6039	111	8	τ1	τ1	NOUN
ejpam-6039	111	9	,	,	PUNCT
ejpam-6039	111	10	τ2	τ2	NOUN
ejpam-6039	111	11	)	)	PUNCT
ejpam-6039	111	12	→	→	SYM
ejpam-6039	111	13	(	(	PUNCT
ejpam-6039	111	14	y	y	PROPN
ejpam-6039	111	15	,	,	PUNCT
ejpam-6039	111	16	σ1	σ1	PROPN
ejpam-6039	111	17	,	,	PUNCT
ejpam-6039	111	18	σ2	σ2	PROPN
ejpam-6039	111	19	)	)	PUNCT
ejpam-6039	111	20	is	be	AUX
ejpam-6039	111	21	said	say	VERB
ejpam-6039	111	22	to	to	PART
ejpam-6039	111	23	be	be	AUX
ejpam-6039	111	24	(	(	PUNCT
ejpam-6039	111	25	τ1	τ1	NOUN
ejpam-6039	111	26	,	,	PUNCT
ejpam-6039	111	27	τ2)p	τ2)p	ADJ
ejpam-6039	111	28	-	-	ADJ
ejpam-6039	111	29	continuous	continuous	ADJ
ejpam-6039	111	30	if	if	SCONJ
ejpam-6039	111	31	for	for	ADP
ejpam-6039	111	32	each	each	DET
ejpam-6039	111	33	x	x	SYM
ejpam-6039	111	34	∈	∈	PROPN
ejpam-6039	111	35	x	x	X
ejpam-6039	111	36	and	and	CCONJ
ejpam-6039	111	37	for	for	ADP
ejpam-6039	111	38	each	each	DET
ejpam-6039	111	39	σ1σ2	σ1σ2	VERB
ejpam-6039	111	40	-	-	ADJ
ejpam-6039	111	41	open	open	ADJ
ejpam-6039	111	42	set	set	NOUN
ejpam-6039	111	43	v	v	NOUN
ejpam-6039	111	44	of	of	ADP
ejpam-6039	111	45	y	y	NOUN
ejpam-6039	111	46	containing	contain	VERB
ejpam-6039	111	47	f(x	f(x	PROPN
ejpam-6039	111	48	)	)	PUNCT
ejpam-6039	111	49	,	,	PUNCT
ejpam-6039	111	50	there	there	PRON
ejpam-6039	111	51	exists	exist	VERB
ejpam-6039	111	52	a	a	DET
ejpam-6039	111	53	(	(	PUNCT
ejpam-6039	111	54	τ1	τ1	NOUN
ejpam-6039	111	55	,	,	PUNCT
ejpam-6039	111	56	τ2)p	τ2)p	ADJ
ejpam-6039	111	57	-	-	PUNCT
ejpam-6039	111	58	open	open	ADJ
ejpam-6039	111	59	set	set	NOUN
ejpam-6039	111	60	u	u	NOUN
ejpam-6039	111	61	of	of	ADP
ejpam-6039	111	62	x	x	PUNCT
ejpam-6039	111	63	containing	contain	VERB
ejpam-6039	111	64	x	x	PUNCT
ejpam-6039	111	65	such	such	ADJ
ejpam-6039	111	66	that	that	DET
ejpam-6039	111	67	f(u	f(u	PROPN
ejpam-6039	111	68	)	)	PUNCT
ejpam-6039	111	69	⊆	⊆	NUM
ejpam-6039	111	70	v	v	NOUN
ejpam-6039	111	71	.	.	PUNCT
ejpam-6039	112	1	theorem	theorem	NOUN
ejpam-6039	112	2	2	2	NUM
ejpam-6039	112	3	.	.	PUNCT
ejpam-6039	113	1	if	if	SCONJ
ejpam-6039	113	2	a	a	DET
ejpam-6039	113	3	function	function	NOUN
ejpam-6039	113	4	f	f	X
ejpam-6039	113	5	:	:	PUNCT
ejpam-6039	113	6	(	(	PUNCT
ejpam-6039	113	7	x	x	NOUN
ejpam-6039	113	8	,	,	PUNCT
ejpam-6039	113	9	τ1	τ1	NOUN
ejpam-6039	113	10	,	,	PUNCT
ejpam-6039	113	11	τ2	τ2	NOUN
ejpam-6039	113	12	)	)	PUNCT
ejpam-6039	113	13	→	→	SYM
ejpam-6039	113	14	(	(	PUNCT
ejpam-6039	113	15	y	y	PROPN
ejpam-6039	113	16	,	,	PUNCT
ejpam-6039	113	17	σ1	σ1	PROPN
ejpam-6039	113	18	,	,	PUNCT
ejpam-6039	113	19	σ2	σ2	PROPN
ejpam-6039	113	20	)	)	PUNCT
ejpam-6039	113	21	is	be	AUX
ejpam-6039	113	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	113	23	,	,	PUNCT
ejpam-6039	113	24	τ2)p	τ2)p	ADJ
ejpam-6039	113	25	-	-	ADJ
ejpam-6039	113	26	continuous	continuous	ADJ
ejpam-6039	113	27	and	and	CCONJ
ejpam-6039	113	28	(	(	PUNCT
ejpam-6039	113	29	y	y	PROPN
ejpam-6039	113	30	,	,	PUNCT
ejpam-6039	113	31	σ1	σ1	PROPN
ejpam-6039	113	32	,	,	PUNCT
ejpam-6039	113	33	σ2	σ2	PROPN
ejpam-6039	113	34	)	)	PUNCT
ejpam-6039	113	35	is	be	AUX
ejpam-6039	113	36	(	(	PUNCT
ejpam-6039	113	37	σ1	σ1	NOUN
ejpam-6039	113	38	,	,	PUNCT
ejpam-6039	113	39	σ2)-regular	σ2)-regular	ADJ
ejpam-6039	113	40	,	,	PUNCT
ejpam-6039	113	41	then	then	ADV
ejpam-6039	113	42	f	f	PROPN
ejpam-6039	113	43	is	be	AUX
ejpam-6039	113	44	(	(	PUNCT
ejpam-6039	113	45	τ1	τ1	NOUN
ejpam-6039	113	46	,	,	PUNCT
ejpam-6039	113	47	τ2)p	τ2)p	ADJ
ejpam-6039	113	48	-	-	ADJ
ejpam-6039	113	49	continuous	continuous	ADJ
ejpam-6039	113	50	.	.	PUNCT
ejpam-6039	114	1	proof	proof	NOUN
ejpam-6039	114	2	.	.	PUNCT
ejpam-6039	115	1	let	let	VERB
ejpam-6039	115	2	x	x	PUNCT
ejpam-6039	115	3	∈	∈	PROPN
ejpam-6039	115	4	x	x	X
ejpam-6039	115	5	and	and	CCONJ
ejpam-6039	115	6	v	v	X
ejpam-6039	115	7	be	be	AUX
ejpam-6039	115	8	any	any	DET
ejpam-6039	115	9	σ1σ2	σ1σ2	NOUN
ejpam-6039	115	10	-	-	ADJ
ejpam-6039	115	11	open	open	ADJ
ejpam-6039	115	12	set	set	NOUN
ejpam-6039	115	13	of	of	ADP
ejpam-6039	115	14	y	y	PROPN
ejpam-6039	115	15	containing	contain	VERB
ejpam-6039	115	16	f(x	f(x	PROPN
ejpam-6039	115	17	)	)	PUNCT
ejpam-6039	115	18	.	.	PUNCT
ejpam-6039	116	1	since	since	SCONJ
ejpam-6039	116	2	(	(	PUNCT
ejpam-6039	116	3	y	y	PROPN
ejpam-6039	116	4	,	,	PUNCT
ejpam-6039	116	5	σ1	σ1	PROPN
ejpam-6039	116	6	,	,	PUNCT
ejpam-6039	116	7	σ2	σ2	PROPN
ejpam-6039	116	8	)	)	PUNCT
ejpam-6039	116	9	is	be	AUX
ejpam-6039	116	10	(	(	PUNCT
ejpam-6039	116	11	σ1	σ1	NOUN
ejpam-6039	116	12	,	,	PUNCT
ejpam-6039	116	13	σ2)-regular	σ2)-regular	ADJ
ejpam-6039	116	14	,	,	PUNCT
ejpam-6039	116	15	there	there	PRON
ejpam-6039	116	16	exists	exist	VERB
ejpam-6039	116	17	a	a	DET
ejpam-6039	116	18	σ1σ2	σ1σ2	NUM
ejpam-6039	116	19	-	-	ADJ
ejpam-6039	116	20	open	open	ADJ
ejpam-6039	116	21	set	set	NOUN
ejpam-6039	116	22	w	w	PROPN
ejpam-6039	116	23	of	of	ADP
ejpam-6039	116	24	y	y	PROPN
ejpam-6039	116	25	containing	contain	VERB
ejpam-6039	116	26	f(x	f(x	PROPN
ejpam-6039	116	27	)	)	PUNCT
ejpam-6039	116	28	such	such	ADJ
ejpam-6039	116	29	that	that	SCONJ
ejpam-6039	116	30	σ1σ2	σ1σ2	NOUN
ejpam-6039	116	31	-	-	PUNCT
ejpam-6039	116	32	cl(w	cl(w	NOUN
ejpam-6039	116	33	)	)	PUNCT
ejpam-6039	116	34	⊆	⊆	PROPN
ejpam-6039	116	35	v.	v.	ADP
ejpam-6039	116	36	m.	m.	PROPN
ejpam-6039	116	37	chiangpradit	chiangpradit	PROPN
ejpam-6039	116	38	,	,	PUNCT
ejpam-6039	116	39	s.	s.	PROPN
ejpam-6039	116	40	sompong	sompong	PROPN
ejpam-6039	116	41	,	,	PUNCT
ejpam-6039	116	42	c.	c.	PROPN
ejpam-6039	116	43	boonpok	boonpok	PROPN
ejpam-6039	116	44	/	/	SYM
ejpam-6039	116	45	eur	eur	PROPN
ejpam-6039	116	46	.	.	PUNCT
ejpam-6039	117	1	j.	j.	PROPN
ejpam-6039	117	2	pure	pure	PROPN
ejpam-6039	117	3	appl	appl	PROPN
ejpam-6039	117	4	.	.	PROPN
ejpam-6039	117	5	math	math	PROPN
ejpam-6039	117	6	,	,	PUNCT
ejpam-6039	117	7	18	18	NUM
ejpam-6039	117	8	(	(	PUNCT
ejpam-6039	117	9	2	2	NUM
ejpam-6039	117	10	)	)	PUNCT
ejpam-6039	117	11	(	(	PUNCT
ejpam-6039	117	12	2025	2025	NUM
ejpam-6039	117	13	)	)	PUNCT
ejpam-6039	117	14	,	,	PUNCT
ejpam-6039	117	15	6039	6039	NUM
ejpam-6039	117	16	5	5	NUM
ejpam-6039	117	17	of	of	ADP
ejpam-6039	117	18	11	11	NUM
ejpam-6039	117	19	since	since	SCONJ
ejpam-6039	117	20	f	f	PROPN
ejpam-6039	117	21	is	be	AUX
ejpam-6039	117	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	117	23	,	,	PUNCT
ejpam-6039	117	24	τ2)p	τ2)p	ADJ
ejpam-6039	117	25	-	-	ADJ
ejpam-6039	117	26	continuous	continuous	ADJ
ejpam-6039	117	27	,	,	PUNCT
ejpam-6039	117	28	there	there	PRON
ejpam-6039	117	29	exists	exist	VERB
ejpam-6039	117	30	a	a	DET
ejpam-6039	117	31	(	(	PUNCT
ejpam-6039	117	32	τ1	τ1	NOUN
ejpam-6039	117	33	,	,	PUNCT
ejpam-6039	117	34	τ2)p	τ2)p	ADJ
ejpam-6039	117	35	-	-	PUNCT
ejpam-6039	117	36	open	open	ADJ
ejpam-6039	117	37	set	set	NOUN
ejpam-6039	117	38	u	u	NOUN
ejpam-6039	117	39	of	of	ADP
ejpam-6039	117	40	x	x	PUNCT
ejpam-6039	117	41	containing	contain	VERB
ejpam-6039	117	42	x	x	PUNCT
ejpam-6039	117	43	such	such	ADJ
ejpam-6039	117	44	that	that	DET
ejpam-6039	117	45	f(u	f(u	PROPN
ejpam-6039	117	46	)	)	PUNCT
ejpam-6039	118	1	⊆	⊆	NUM
ejpam-6039	118	2	σ1σ2	σ1σ2	NOUN
ejpam-6039	118	3	-	-	PUNCT
ejpam-6039	118	4	cl(w	cl(w	NOUN
ejpam-6039	118	5	)	)	PUNCT
ejpam-6039	118	6	.	.	PUNCT
ejpam-6039	119	1	thus	thus	ADV
ejpam-6039	119	2	,	,	PUNCT
ejpam-6039	119	3	f(u	f(u	PROPN
ejpam-6039	119	4	)	)	PUNCT
ejpam-6039	119	5	⊆	⊆	NUM
ejpam-6039	119	6	σ1σ2	σ1σ2	NOUN
ejpam-6039	119	7	-	-	PUNCT
ejpam-6039	119	8	cl(w	cl(w	NOUN
ejpam-6039	119	9	)	)	PUNCT
ejpam-6039	119	10	⊆	⊆	NUM
ejpam-6039	119	11	v	v	NOUN
ejpam-6039	119	12	and	and	CCONJ
ejpam-6039	119	13	hence	hence	ADV
ejpam-6039	119	14	f	f	PROPN
ejpam-6039	119	15	is	be	AUX
ejpam-6039	119	16	(	(	PUNCT
ejpam-6039	119	17	τ1	τ1	NOUN
ejpam-6039	119	18	,	,	PUNCT
ejpam-6039	119	19	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-6039	119	20	.	.	PUNCT
ejpam-6039	120	1	definition	definition	NOUN
ejpam-6039	120	2	3	3	NUM
ejpam-6039	120	3	.	.	PUNCT
ejpam-6039	121	1	a	a	DET
ejpam-6039	121	2	function	function	NOUN
ejpam-6039	121	3	f	f	NOUN
ejpam-6039	121	4	:	:	PUNCT
ejpam-6039	121	5	(	(	PUNCT
ejpam-6039	121	6	x	x	NOUN
ejpam-6039	121	7	,	,	PUNCT
ejpam-6039	121	8	τ1	τ1	NOUN
ejpam-6039	121	9	,	,	PUNCT
ejpam-6039	121	10	τ2	τ2	NOUN
ejpam-6039	121	11	)	)	PUNCT
ejpam-6039	121	12	→	→	SYM
ejpam-6039	121	13	(	(	PUNCT
ejpam-6039	121	14	y	y	PROPN
ejpam-6039	121	15	,	,	PUNCT
ejpam-6039	121	16	σ1	σ1	PROPN
ejpam-6039	121	17	,	,	PUNCT
ejpam-6039	121	18	σ2	σ2	PROPN
ejpam-6039	121	19	)	)	PUNCT
ejpam-6039	121	20	is	be	AUX
ejpam-6039	121	21	said	say	VERB
ejpam-6039	121	22	to	to	PART
ejpam-6039	121	23	be	be	AUX
ejpam-6039	121	24	almost	almost	ADV
ejpam-6039	121	25	(	(	PUNCT
ejpam-6039	121	26	τ1	τ1	NOUN
ejpam-6039	121	27	,	,	PUNCT
ejpam-6039	121	28	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-6039	121	29	if	if	SCONJ
ejpam-6039	121	30	for	for	ADP
ejpam-6039	121	31	each	each	DET
ejpam-6039	121	32	x	x	SYM
ejpam-6039	121	33	∈	∈	PROPN
ejpam-6039	121	34	x	x	X
ejpam-6039	121	35	and	and	CCONJ
ejpam-6039	121	36	for	for	ADP
ejpam-6039	121	37	each	each	DET
ejpam-6039	121	38	σ1σ2	σ1σ2	VERB
ejpam-6039	121	39	-	-	ADJ
ejpam-6039	121	40	open	open	ADJ
ejpam-6039	121	41	set	set	NOUN
ejpam-6039	121	42	v	v	NOUN
ejpam-6039	121	43	of	of	ADP
ejpam-6039	121	44	y	y	NOUN
ejpam-6039	121	45	containing	contain	VERB
ejpam-6039	121	46	f(x	f(x	PROPN
ejpam-6039	121	47	)	)	PUNCT
ejpam-6039	121	48	,	,	PUNCT
ejpam-6039	121	49	there	there	PRON
ejpam-6039	121	50	exists	exist	VERB
ejpam-6039	121	51	a	a	DET
ejpam-6039	121	52	(	(	PUNCT
ejpam-6039	121	53	τ1	τ1	NOUN
ejpam-6039	121	54	,	,	PUNCT
ejpam-6039	121	55	τ2)p	τ2)p	ADJ
ejpam-6039	121	56	-	-	PUNCT
ejpam-6039	121	57	open	open	ADJ
ejpam-6039	121	58	set	set	NOUN
ejpam-6039	121	59	u	u	NOUN
ejpam-6039	121	60	of	of	ADP
ejpam-6039	121	61	x	x	PUNCT
ejpam-6039	121	62	containing	contain	VERB
ejpam-6039	121	63	x	x	PUNCT
ejpam-6039	121	64	such	such	ADJ
ejpam-6039	121	65	that	that	DET
ejpam-6039	121	66	f(u	f(u	PROPN
ejpam-6039	121	67	)	)	PUNCT
ejpam-6039	121	68	⊆	⊆	NUM
ejpam-6039	121	69	σ1σ2	σ1σ2	X
ejpam-6039	121	70	-	-	PUNCT
ejpam-6039	121	71	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6039	121	72	-	-	PUNCT
ejpam-6039	121	73	cl(v	cl(v	NOUN
ejpam-6039	121	74	)	)	PUNCT
ejpam-6039	121	75	)	)	PUNCT
ejpam-6039	121	76	.	.	PUNCT
ejpam-6039	122	1	definition	definition	NOUN
ejpam-6039	122	2	4	4	NUM
ejpam-6039	122	3	.	.	PUNCT
ejpam-6039	123	1	a	a	DET
ejpam-6039	123	2	functions	function	NOUN
ejpam-6039	123	3	f	f	X
ejpam-6039	123	4	:	:	PUNCT
ejpam-6039	123	5	(	(	PUNCT
ejpam-6039	123	6	x	x	NOUN
ejpam-6039	123	7	,	,	PUNCT
ejpam-6039	123	8	τ1	τ1	NOUN
ejpam-6039	123	9	,	,	PUNCT
ejpam-6039	123	10	τ2	τ2	NOUN
ejpam-6039	123	11	)	)	PUNCT
ejpam-6039	123	12	→	→	SYM
ejpam-6039	123	13	(	(	PUNCT
ejpam-6039	123	14	y	y	PROPN
ejpam-6039	123	15	,	,	PUNCT
ejpam-6039	123	16	σ1	σ1	PROPN
ejpam-6039	123	17	,	,	PUNCT
ejpam-6039	123	18	σ2	σ2	PROPN
ejpam-6039	123	19	)	)	PUNCT
ejpam-6039	123	20	is	be	AUX
ejpam-6039	123	21	said	say	VERB
ejpam-6039	123	22	to	to	PART
ejpam-6039	123	23	be	be	AUX
ejpam-6039	123	24	(	(	PUNCT
ejpam-6039	123	25	τ1	τ1	NOUN
ejpam-6039	123	26	,	,	PUNCT
ejpam-6039	123	27	τ2)p	τ2)p	NOUN
ejpam-6039	123	28	-	-	PUNCT
ejpam-6039	123	29	open	open	ADJ
ejpam-6039	123	30	if	if	SCONJ
ejpam-6039	123	31	f(u	f(u	PROPN
ejpam-6039	123	32	)	)	PUNCT
ejpam-6039	123	33	is	be	AUX
ejpam-6039	123	34	(	(	PUNCT
ejpam-6039	123	35	σ1	σ1	PROPN
ejpam-6039	123	36	,	,	PUNCT
ejpam-6039	123	37	σ2)p	σ2)p	NOUN
ejpam-6039	123	38	-	-	PUNCT
ejpam-6039	123	39	open	open	ADJ
ejpam-6039	123	40	in	in	ADP
ejpam-6039	123	41	y	y	PROPN
ejpam-6039	123	42	for	for	ADP
ejpam-6039	123	43	every	every	DET
ejpam-6039	123	44	(	(	PUNCT
ejpam-6039	123	45	τ1	τ1	NOUN
ejpam-6039	123	46	,	,	PUNCT
ejpam-6039	123	47	τ2)p	τ2)p	ADJ
ejpam-6039	123	48	-	-	PUNCT
ejpam-6039	123	49	open	open	ADJ
ejpam-6039	123	50	set	set	NOUN
ejpam-6039	123	51	u	u	PROPN
ejpam-6039	123	52	of	of	ADP
ejpam-6039	123	53	x.	x.	PROPN
ejpam-6039	123	54	theorem	theorem	VERB
ejpam-6039	123	55	3	3	X
ejpam-6039	123	56	.	.	PUNCT
ejpam-6039	124	1	if	if	SCONJ
ejpam-6039	124	2	f	f	PROPN
ejpam-6039	124	3	:	:	PUNCT
ejpam-6039	124	4	(	(	PUNCT
ejpam-6039	124	5	x	x	NOUN
ejpam-6039	124	6	,	,	PUNCT
ejpam-6039	124	7	τ1	τ1	NOUN
ejpam-6039	124	8	,	,	PUNCT
ejpam-6039	124	9	τ2	τ2	NOUN
ejpam-6039	124	10	)	)	PUNCT
ejpam-6039	124	11	→	→	SYM
ejpam-6039	124	12	(	(	PUNCT
ejpam-6039	124	13	y	y	PROPN
ejpam-6039	124	14	,	,	PUNCT
ejpam-6039	124	15	σ1	σ1	PROPN
ejpam-6039	124	16	,	,	PUNCT
ejpam-6039	124	17	σ2	σ2	PROPN
ejpam-6039	124	18	)	)	PUNCT
ejpam-6039	124	19	is	be	AUX
ejpam-6039	124	20	a	a	DET
ejpam-6039	124	21	(	(	PUNCT
ejpam-6039	124	22	τ1	τ1	NOUN
ejpam-6039	124	23	,	,	PUNCT
ejpam-6039	124	24	τ2)p	τ2)p	ADJ
ejpam-6039	124	25	-	-	PUNCT
ejpam-6039	124	26	open	open	ADJ
ejpam-6039	124	27	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	124	28	,	,	PUNCT
ejpam-6039	124	29	τ2)p	τ2)p	ADJ
ejpam-6039	124	30	-	-	ADJ
ejpam-6039	124	31	continuous	continuous	ADJ
ejpam-6039	124	32	function	function	NOUN
ejpam-6039	124	33	,	,	PUNCT
ejpam-6039	124	34	then	then	ADV
ejpam-6039	124	35	f	f	PROPN
ejpam-6039	124	36	is	be	AUX
ejpam-6039	124	37	almost	almost	ADV
ejpam-6039	124	38	(	(	PUNCT
ejpam-6039	124	39	τ1	τ1	NOUN
ejpam-6039	124	40	,	,	PUNCT
ejpam-6039	124	41	τ2)p	τ2)p	ADJ
ejpam-6039	124	42	-	-	ADJ
ejpam-6039	124	43	continuous	continuous	ADJ
ejpam-6039	124	44	.	.	PUNCT
ejpam-6039	125	1	proof	proof	NOUN
ejpam-6039	125	2	.	.	PUNCT
ejpam-6039	126	1	let	let	VERB
ejpam-6039	126	2	x	x	PUNCT
ejpam-6039	126	3	∈	∈	PROPN
ejpam-6039	126	4	x	x	X
ejpam-6039	126	5	and	and	CCONJ
ejpam-6039	126	6	v	v	X
ejpam-6039	126	7	be	be	AUX
ejpam-6039	126	8	any	any	DET
ejpam-6039	126	9	σ1σ2	σ1σ2	NOUN
ejpam-6039	126	10	-	-	ADJ
ejpam-6039	126	11	open	open	ADJ
ejpam-6039	126	12	set	set	NOUN
ejpam-6039	126	13	of	of	ADP
ejpam-6039	126	14	y	y	PROPN
ejpam-6039	126	15	containing	contain	VERB
ejpam-6039	126	16	f(x	f(x	PROPN
ejpam-6039	126	17	)	)	PUNCT
ejpam-6039	126	18	.	.	PUNCT
ejpam-6039	127	1	since	since	SCONJ
ejpam-6039	127	2	f	f	PROPN
ejpam-6039	127	3	is	be	AUX
ejpam-6039	127	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	127	5	,	,	PUNCT
ejpam-6039	127	6	τ2)p	τ2)p	ADJ
ejpam-6039	127	7	-	-	ADJ
ejpam-6039	127	8	continuous	continuous	ADJ
ejpam-6039	127	9	,	,	PUNCT
ejpam-6039	127	10	there	there	PRON
ejpam-6039	127	11	exists	exist	VERB
ejpam-6039	127	12	a	a	DET
ejpam-6039	127	13	(	(	PUNCT
ejpam-6039	127	14	τ1	τ1	NOUN
ejpam-6039	127	15	,	,	PUNCT
ejpam-6039	127	16	τ2)p	τ2)p	ADJ
ejpam-6039	127	17	-	-	PUNCT
ejpam-6039	127	18	open	open	ADJ
ejpam-6039	127	19	set	set	NOUN
ejpam-6039	127	20	u	u	NOUN
ejpam-6039	127	21	of	of	ADP
ejpam-6039	127	22	x	x	PUNCT
ejpam-6039	127	23	containing	contain	VERB
ejpam-6039	127	24	x	x	PUNCT
ejpam-6039	127	25	such	such	ADJ
ejpam-6039	127	26	that	that	DET
ejpam-6039	127	27	f(u	f(u	PROPN
ejpam-6039	127	28	)	)	PUNCT
ejpam-6039	127	29	⊆	⊆	NUM
ejpam-6039	127	30	σ1σ2	σ1σ2	NOUN
ejpam-6039	127	31	-	-	NUM
ejpam-6039	127	32	cl(v	cl(v	NOUN
ejpam-6039	127	33	)	)	PUNCT
ejpam-6039	127	34	.	.	PUNCT
ejpam-6039	128	1	since	since	SCONJ
ejpam-6039	128	2	f	f	PROPN
ejpam-6039	128	3	is	be	AUX
ejpam-6039	128	4	(	(	PUNCT
ejpam-6039	128	5	τ1	τ1	NOUN
ejpam-6039	128	6	,	,	PUNCT
ejpam-6039	128	7	τ2)p	τ2)p	ADJ
ejpam-6039	128	8	-	-	ADJ
ejpam-6039	128	9	open	open	ADJ
ejpam-6039	128	10	,	,	PUNCT
ejpam-6039	128	11	f(u	f(u	PROPN
ejpam-6039	128	12	)	)	PUNCT
ejpam-6039	128	13	is	be	AUX
ejpam-6039	128	14	σ1σ2	σ1σ2	NOUN
ejpam-6039	128	15	-	-	ADJ
ejpam-6039	128	16	open	open	ADJ
ejpam-6039	128	17	in	in	ADP
ejpam-6039	128	18	y	y	PROPN
ejpam-6039	128	19	.	.	PUNCT
ejpam-6039	129	1	therefore	therefore	ADV
ejpam-6039	129	2	,	,	PUNCT
ejpam-6039	129	3	f(u	f(u	PROPN
ejpam-6039	129	4	)	)	PUNCT
ejpam-6039	129	5	⊆	⊆	NUM
ejpam-6039	129	6	σ1σ2	σ1σ2	X
ejpam-6039	129	7	-	-	PUNCT
ejpam-6039	129	8	int(σ1σ2	int(σ1σ2	VERB
ejpam-6039	129	9	-	-	PUNCT
ejpam-6039	129	10	cl(f(u	cl(f(u	NOUN
ejpam-6039	129	11	)	)	PUNCT
ejpam-6039	129	12	)	)	PUNCT
ejpam-6039	129	13	)	)	PUNCT
ejpam-6039	130	1	⊆	⊆	X
ejpam-6039	130	2	σ1σ2	σ1σ2	X
ejpam-6039	130	3	-	-	PUNCT
ejpam-6039	130	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-6039	130	5	-	-	PUNCT
ejpam-6039	130	6	cl(v	cl(v	NOUN
ejpam-6039	130	7	)	)	PUNCT
ejpam-6039	130	8	)	)	PUNCT
ejpam-6039	130	9	.	.	PUNCT
ejpam-6039	131	1	this	this	PRON
ejpam-6039	131	2	shows	show	VERB
ejpam-6039	131	3	that	that	SCONJ
ejpam-6039	131	4	f	f	PROPN
ejpam-6039	131	5	is	be	AUX
ejpam-6039	131	6	almost	almost	ADV
ejpam-6039	131	7	(	(	PUNCT
ejpam-6039	131	8	τ1	τ1	NOUN
ejpam-6039	131	9	,	,	PUNCT
ejpam-6039	131	10	τ2)p	τ2)p	ADJ
ejpam-6039	131	11	-	-	ADJ
ejpam-6039	131	12	continuous	continuous	ADJ
ejpam-6039	131	13	.	.	PUNCT
ejpam-6039	132	1	definition	definition	NOUN
ejpam-6039	132	2	5	5	NUM
ejpam-6039	132	3	.	.	PUNCT
ejpam-6039	133	1	[	[	X
ejpam-6039	133	2	26	26	NUM
ejpam-6039	133	3	]	]	PUNCT
ejpam-6039	133	4	a	a	DET
ejpam-6039	133	5	function	function	NOUN
ejpam-6039	133	6	f	f	NOUN
ejpam-6039	133	7	:	:	PUNCT
ejpam-6039	133	8	(	(	PUNCT
ejpam-6039	133	9	x	x	NOUN
ejpam-6039	133	10	,	,	PUNCT
ejpam-6039	133	11	τ1	τ1	NOUN
ejpam-6039	133	12	,	,	PUNCT
ejpam-6039	133	13	τ2	τ2	NOUN
ejpam-6039	133	14	)	)	PUNCT
ejpam-6039	133	15	→	→	SYM
ejpam-6039	133	16	(	(	PUNCT
ejpam-6039	133	17	y	y	PROPN
ejpam-6039	133	18	,	,	PUNCT
ejpam-6039	133	19	σ1	σ1	PROPN
ejpam-6039	133	20	,	,	PUNCT
ejpam-6039	133	21	σ2	σ2	PROPN
ejpam-6039	133	22	)	)	PUNCT
ejpam-6039	133	23	is	be	AUX
ejpam-6039	133	24	said	say	VERB
ejpam-6039	133	25	to	to	PART
ejpam-6039	133	26	be	be	AUX
ejpam-6039	133	27	almost	almost	ADV
ejpam-6039	133	28	weakly	weakly	ADJ
ejpam-6039	133	29	(	(	PUNCT
ejpam-6039	133	30	τ1	τ1	NOUN
ejpam-6039	133	31	,	,	PUNCT
ejpam-6039	133	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	133	33	if	if	SCONJ
ejpam-6039	133	34	for	for	ADP
ejpam-6039	133	35	each	each	DET
ejpam-6039	133	36	x	x	SYM
ejpam-6039	133	37	∈	∈	PROPN
ejpam-6039	133	38	x	x	X
ejpam-6039	133	39	and	and	CCONJ
ejpam-6039	133	40	each	each	DET
ejpam-6039	133	41	σ1σ2	σ1σ2	VERB
ejpam-6039	133	42	-	-	ADJ
ejpam-6039	133	43	open	open	ADJ
ejpam-6039	133	44	set	set	NOUN
ejpam-6039	133	45	v	v	NOUN
ejpam-6039	133	46	of	of	ADP
ejpam-6039	133	47	y	y	NOUN
ejpam-6039	133	48	containing	contain	VERB
ejpam-6039	133	49	f(x	f(x	PROPN
ejpam-6039	133	50	)	)	PUNCT
ejpam-6039	133	51	,	,	PUNCT
ejpam-6039	133	52	x	x	PUNCT
ejpam-6039	133	53	∈	∈	ADP
ejpam-6039	133	54	τ1τ2	τ1τ2	NOUN
ejpam-6039	133	55	-	-	NOUN
ejpam-6039	133	56	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	133	57	-	-	PUNCT
ejpam-6039	133	58	cl(f	cl(f	NOUN
ejpam-6039	133	59	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6039	133	60	-	-	PUNCT
ejpam-6039	133	61	cl(v	cl(v	NOUN
ejpam-6039	133	62	)	)	PUNCT
ejpam-6039	133	63	)	)	PUNCT
ejpam-6039	133	64	)	)	PUNCT
ejpam-6039	133	65	)	)	PUNCT
ejpam-6039	133	66	.	.	PUNCT
ejpam-6039	134	1	lemma	lemma	PROPN
ejpam-6039	134	2	4	4	NUM
ejpam-6039	134	3	.	.	PUNCT
ejpam-6039	135	1	[	[	X
ejpam-6039	135	2	26	26	NUM
ejpam-6039	135	3	]	]	PUNCT
ejpam-6039	135	4	for	for	ADP
ejpam-6039	135	5	a	a	DET
ejpam-6039	135	6	function	function	NOUN
ejpam-6039	135	7	f	f	NOUN
ejpam-6039	135	8	:	:	PUNCT
ejpam-6039	135	9	(	(	PUNCT
ejpam-6039	135	10	x	x	NOUN
ejpam-6039	135	11	,	,	PUNCT
ejpam-6039	135	12	τ1	τ1	NOUN
ejpam-6039	135	13	,	,	PUNCT
ejpam-6039	135	14	τ2	τ2	NOUN
ejpam-6039	135	15	)	)	PUNCT
ejpam-6039	135	16	→	→	SYM
ejpam-6039	135	17	(	(	PUNCT
ejpam-6039	135	18	y	y	PROPN
ejpam-6039	135	19	,	,	PUNCT
ejpam-6039	135	20	σ1	σ1	PROPN
ejpam-6039	135	21	,	,	PUNCT
ejpam-6039	135	22	σ2	σ2	NOUN
ejpam-6039	135	23	)	)	PUNCT
ejpam-6039	135	24	,	,	PUNCT
ejpam-6039	135	25	the	the	DET
ejpam-6039	135	26	following	follow	VERB
ejpam-6039	135	27	properties	property	NOUN
ejpam-6039	135	28	are	be	AUX
ejpam-6039	135	29	equivalent	equivalent	ADJ
ejpam-6039	135	30	:	:	PUNCT
ejpam-6039	135	31	(	(	PUNCT
ejpam-6039	135	32	1	1	X
ejpam-6039	135	33	)	)	PUNCT
ejpam-6039	135	34	f	f	NOUN
ejpam-6039	135	35	is	be	AUX
ejpam-6039	135	36	almost	almost	ADV
ejpam-6039	135	37	weakly	weakly	ADJ
ejpam-6039	135	38	(	(	PUNCT
ejpam-6039	135	39	τ1	τ1	NOUN
ejpam-6039	135	40	,	,	PUNCT
ejpam-6039	135	41	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	135	42	;	;	PUNCT
ejpam-6039	135	43	(	(	PUNCT
ejpam-6039	135	44	2	2	X
ejpam-6039	135	45	)	)	PUNCT
ejpam-6039	135	46	f−1(v	f−1(v	NOUN
ejpam-6039	135	47	)	)	PUNCT
ejpam-6039	136	1	⊆	⊆	X
ejpam-6039	136	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	136	3	-	-	NOUN
ejpam-6039	136	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	136	5	-	-	PUNCT
ejpam-6039	136	6	cl(f	cl(f	NOUN
ejpam-6039	136	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6039	136	8	-	-	PUNCT
ejpam-6039	136	9	cl(v	cl(v	NOUN
ejpam-6039	136	10	)	)	PUNCT
ejpam-6039	136	11	)	)	PUNCT
ejpam-6039	136	12	)	)	PUNCT
ejpam-6039	136	13	)	)	PUNCT
ejpam-6039	136	14	for	for	ADP
ejpam-6039	136	15	every	every	DET
ejpam-6039	136	16	σ1σ2	σ1σ2	NOUN
ejpam-6039	136	17	-	-	ADJ
ejpam-6039	136	18	open	open	ADJ
ejpam-6039	136	19	set	set	NOUN
ejpam-6039	136	20	v	v	NOUN
ejpam-6039	136	21	of	of	ADP
ejpam-6039	136	22	y	y	PROPN
ejpam-6039	136	23	;	;	PUNCT
ejpam-6039	136	24	(	(	PUNCT
ejpam-6039	136	25	3	3	X
ejpam-6039	136	26	)	)	PUNCT
ejpam-6039	136	27	τ1τ2	τ1τ2	NOUN
ejpam-6039	136	28	-	-	NOUN
ejpam-6039	136	29	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	136	30	-	-	PUNCT
ejpam-6039	136	31	int(f	int(f	PROPN
ejpam-6039	136	32	−1(v	−1(v	NOUN
ejpam-6039	136	33	)	)	PUNCT
ejpam-6039	136	34	)	)	PUNCT
ejpam-6039	136	35	)	)	PUNCT
ejpam-6039	137	1	⊆	⊆	NUM
ejpam-6039	137	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	137	3	-	-	PUNCT
ejpam-6039	137	4	cl(v	cl(v	NOUN
ejpam-6039	137	5	)	)	PUNCT
ejpam-6039	137	6	)	)	PUNCT
ejpam-6039	137	7	for	for	ADP
ejpam-6039	137	8	every	every	DET
ejpam-6039	137	9	σ1σ2	σ1σ2	NOUN
ejpam-6039	137	10	-	-	ADJ
ejpam-6039	137	11	open	open	ADJ
ejpam-6039	137	12	set	set	NOUN
ejpam-6039	137	13	v	v	NOUN
ejpam-6039	137	14	of	of	ADP
ejpam-6039	137	15	y	y	PROPN
ejpam-6039	137	16	;	;	PUNCT
ejpam-6039	137	17	(	(	PUNCT
ejpam-6039	137	18	4	4	X
ejpam-6039	137	19	)	)	PUNCT
ejpam-6039	137	20	(	(	PUNCT
ejpam-6039	137	21	τ1	τ1	NOUN
ejpam-6039	137	22	,	,	PUNCT
ejpam-6039	137	23	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-6039	137	24	−1(v	−1(v	PROPN
ejpam-6039	137	25	)	)	PUNCT
ejpam-6039	137	26	)	)	PUNCT
ejpam-6039	138	1	⊆	⊆	NUM
ejpam-6039	138	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	138	3	-	-	PUNCT
ejpam-6039	138	4	cl(v	cl(v	NOUN
ejpam-6039	138	5	)	)	PUNCT
ejpam-6039	138	6	)	)	PUNCT
ejpam-6039	138	7	for	for	ADP
ejpam-6039	138	8	every	every	DET
ejpam-6039	138	9	σ1σ2	σ1σ2	NOUN
ejpam-6039	138	10	-	-	ADJ
ejpam-6039	138	11	open	open	ADJ
ejpam-6039	138	12	set	set	NOUN
ejpam-6039	138	13	v	v	NOUN
ejpam-6039	138	14	of	of	ADP
ejpam-6039	138	15	y	y	PROPN
ejpam-6039	138	16	;	;	PUNCT
ejpam-6039	138	17	(	(	PUNCT
ejpam-6039	138	18	5	5	X
ejpam-6039	138	19	)	)	PUNCT
ejpam-6039	138	20	f−1(v	f−1(v	NOUN
ejpam-6039	138	21	)	)	PUNCT
ejpam-6039	139	1	⊆	⊆	NUM
ejpam-6039	139	2	(	(	PUNCT
ejpam-6039	139	3	τ1	τ1	NOUN
ejpam-6039	139	4	,	,	PUNCT
ejpam-6039	139	5	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-6039	139	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6039	139	7	-	-	PUNCT
ejpam-6039	139	8	cl(v	cl(v	NOUN
ejpam-6039	139	9	)	)	PUNCT
ejpam-6039	139	10	)	)	PUNCT
ejpam-6039	139	11	)	)	PUNCT
ejpam-6039	139	12	for	for	ADP
ejpam-6039	139	13	every	every	DET
ejpam-6039	139	14	σ1σ2	σ1σ2	NOUN
ejpam-6039	139	15	-	-	ADJ
ejpam-6039	139	16	open	open	ADJ
ejpam-6039	139	17	set	set	NOUN
ejpam-6039	139	18	v	v	NOUN
ejpam-6039	139	19	of	of	ADP
ejpam-6039	139	20	y	y	PROPN
ejpam-6039	139	21	;	;	PUNCT
ejpam-6039	139	22	(	(	PUNCT
ejpam-6039	139	23	6	6	NUM
ejpam-6039	139	24	)	)	PUNCT
ejpam-6039	139	25	for	for	ADP
ejpam-6039	139	26	each	each	DET
ejpam-6039	139	27	x	x	SYM
ejpam-6039	139	28	∈	∈	PROPN
ejpam-6039	139	29	x	x	X
ejpam-6039	139	30	and	and	CCONJ
ejpam-6039	139	31	each	each	DET
ejpam-6039	139	32	σ1σ2	σ1σ2	VERB
ejpam-6039	139	33	-	-	ADJ
ejpam-6039	139	34	open	open	ADJ
ejpam-6039	139	35	set	set	NOUN
ejpam-6039	139	36	v	v	NOUN
ejpam-6039	139	37	of	of	ADP
ejpam-6039	139	38	y	y	NOUN
ejpam-6039	139	39	containing	contain	VERB
ejpam-6039	139	40	f(x	f(x	PROPN
ejpam-6039	139	41	)	)	PUNCT
ejpam-6039	139	42	,	,	PUNCT
ejpam-6039	139	43	there	there	PRON
ejpam-6039	139	44	exists	exist	VERB
ejpam-6039	139	45	a	a	DET
ejpam-6039	139	46	(	(	PUNCT
ejpam-6039	139	47	τ1	τ1	NOUN
ejpam-6039	139	48	,	,	PUNCT
ejpam-6039	139	49	τ2)p	τ2)p	ADJ
ejpam-6039	139	50	-	-	PUNCT
ejpam-6039	139	51	open	open	ADJ
ejpam-6039	139	52	set	set	NOUN
ejpam-6039	139	53	u	u	NOUN
ejpam-6039	139	54	of	of	ADP
ejpam-6039	139	55	x	x	PUNCT
ejpam-6039	139	56	containing	contain	VERB
ejpam-6039	139	57	x	x	PUNCT
ejpam-6039	139	58	such	such	ADJ
ejpam-6039	139	59	that	that	DET
ejpam-6039	139	60	f(u	f(u	PROPN
ejpam-6039	139	61	)	)	PUNCT
ejpam-6039	139	62	⊆	⊆	NUM
ejpam-6039	139	63	σ1σ2	σ1σ2	NOUN
ejpam-6039	139	64	-	-	NUM
ejpam-6039	139	65	cl(v	cl(v	NOUN
ejpam-6039	139	66	)	)	PUNCT
ejpam-6039	139	67	.	.	PUNCT
ejpam-6039	140	1	theorem	theorem	NOUN
ejpam-6039	140	2	4	4	NUM
ejpam-6039	140	3	.	.	PUNCT
ejpam-6039	141	1	if	if	SCONJ
ejpam-6039	141	2	a	a	DET
ejpam-6039	141	3	function	function	NOUN
ejpam-6039	141	4	f	f	X
ejpam-6039	141	5	:	:	PUNCT
ejpam-6039	141	6	(	(	PUNCT
ejpam-6039	141	7	x	x	NOUN
ejpam-6039	141	8	,	,	PUNCT
ejpam-6039	141	9	τ1	τ1	NOUN
ejpam-6039	141	10	,	,	PUNCT
ejpam-6039	141	11	τ2	τ2	NOUN
ejpam-6039	141	12	)	)	PUNCT
ejpam-6039	141	13	→	→	SYM
ejpam-6039	141	14	(	(	PUNCT
ejpam-6039	141	15	y	y	PROPN
ejpam-6039	141	16	,	,	PUNCT
ejpam-6039	141	17	σ1	σ1	PROPN
ejpam-6039	141	18	,	,	PUNCT
ejpam-6039	141	19	σ2	σ2	PROPN
ejpam-6039	141	20	)	)	PUNCT
ejpam-6039	141	21	is	be	AUX
ejpam-6039	141	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	141	23	,	,	PUNCT
ejpam-6039	141	24	τ2)p	τ2)p	ADJ
ejpam-6039	141	25	-	-	ADJ
ejpam-6039	141	26	continuous	continuous	ADJ
ejpam-6039	141	27	,	,	PUNCT
ejpam-6039	141	28	then	then	ADV
ejpam-6039	141	29	f	f	PROPN
ejpam-6039	141	30	is	be	AUX
ejpam-6039	141	31	almost	almost	ADV
ejpam-6039	141	32	weakly	weakly	ADJ
ejpam-6039	141	33	(	(	PUNCT
ejpam-6039	141	34	τ1	τ1	NOUN
ejpam-6039	141	35	,	,	PUNCT
ejpam-6039	141	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	141	37	.	.	PUNCT
ejpam-6039	142	1	proof	proof	NOUN
ejpam-6039	142	2	.	.	PUNCT
ejpam-6039	143	1	let	let	VERB
ejpam-6039	143	2	v	v	PART
ejpam-6039	143	3	be	be	AUX
ejpam-6039	143	4	any	any	DET
ejpam-6039	143	5	σ1σ2	σ1σ2	NOUN
ejpam-6039	143	6	-	-	ADJ
ejpam-6039	143	7	open	open	ADJ
ejpam-6039	143	8	set	set	NOUN
ejpam-6039	143	9	of	of	ADP
ejpam-6039	143	10	y	y	PROPN
ejpam-6039	143	11	.	.	PUNCT
ejpam-6039	144	1	since	since	SCONJ
ejpam-6039	144	2	f	f	PROPN
ejpam-6039	144	3	is	be	AUX
ejpam-6039	144	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	144	5	,	,	PUNCT
ejpam-6039	144	6	τ2)p	τ2)p	ADJ
ejpam-6039	144	7	-	-	ADJ
ejpam-6039	144	8	continuous	continuous	ADJ
ejpam-6039	144	9	and	and	CCONJ
ejpam-6039	144	10	σ1σ2	σ1σ2	NOUN
ejpam-6039	144	11	-	-	NUM
ejpam-6039	144	12	cl(v	cl(v	NOUN
ejpam-6039	144	13	)	)	PUNCT
ejpam-6039	144	14	is	be	AUX
ejpam-6039	144	15	σ1σ2	σ1σ2	NOUN
ejpam-6039	144	16	-	-	ADJ
ejpam-6039	144	17	closed	closed	ADJ
ejpam-6039	144	18	in	in	ADP
ejpam-6039	144	19	y	y	PROPN
ejpam-6039	144	20	,	,	PUNCT
ejpam-6039	144	21	by	by	ADP
ejpam-6039	144	22	theorem	theorem	NOUN
ejpam-6039	144	23	1	1	NUM
ejpam-6039	144	24	we	we	PRON
ejpam-6039	144	25	have	have	AUX
ejpam-6039	144	26	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	144	27	-	-	PUNCT
ejpam-6039	144	28	cl(v	cl(v	NOUN
ejpam-6039	144	29	)	)	PUNCT
ejpam-6039	144	30	)	)	PUNCT
ejpam-6039	145	1	is	be	AUX
ejpam-6039	145	2	(	(	PUNCT
ejpam-6039	145	3	τ1	τ1	NOUN
ejpam-6039	145	4	,	,	PUNCT
ejpam-6039	145	5	τ2)p	τ2)p	NOUN
ejpam-6039	145	6	-	-	PUNCT
ejpam-6039	145	7	open	open	ADJ
ejpam-6039	145	8	in	in	ADP
ejpam-6039	145	9	x.	x.	PROPN
ejpam-6039	145	10	thus	thus	ADV
ejpam-6039	145	11	,	,	PUNCT
ejpam-6039	145	12	f−1(v	f−1(v	PROPN
ejpam-6039	145	13	)	)	PUNCT
ejpam-6039	146	1	⊆	⊆	NUM
ejpam-6039	146	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-6039	146	3	-	-	PUNCT
ejpam-6039	146	4	cl(v	cl(v	NOUN
ejpam-6039	146	5	)	)	PUNCT
ejpam-6039	146	6	)	)	PUNCT
ejpam-6039	147	1	⊆	⊆	X
ejpam-6039	147	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	147	3	-	-	NOUN
ejpam-6039	147	4	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	147	5	-	-	PUNCT
ejpam-6039	147	6	cl(f	cl(f	NOUN
ejpam-6039	147	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-6039	147	8	-	-	PUNCT
ejpam-6039	147	9	cl(v	cl(v	NOUN
ejpam-6039	147	10	)	)	PUNCT
ejpam-6039	147	11	)	)	PUNCT
ejpam-6039	147	12	)	)	PUNCT
ejpam-6039	147	13	.	.	PUNCT
ejpam-6039	148	1	by	by	ADP
ejpam-6039	148	2	lemma	lemma	PROPN
ejpam-6039	148	3	4(2	4(2	PROPN
ejpam-6039	148	4	)	)	PUNCT
ejpam-6039	148	5	,	,	PUNCT
ejpam-6039	148	6	f	f	PROPN
ejpam-6039	148	7	is	be	AUX
ejpam-6039	148	8	almost	almost	ADV
ejpam-6039	148	9	weakly	weakly	ADJ
ejpam-6039	148	10	(	(	PUNCT
ejpam-6039	148	11	τ1	τ1	NOUN
ejpam-6039	148	12	,	,	PUNCT
ejpam-6039	148	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	148	14	.	.	PUNCT
ejpam-6039	148	15	m.	m.	NOUN
ejpam-6039	148	16	chiangpradit	chiangpradit	PROPN
ejpam-6039	148	17	,	,	PUNCT
ejpam-6039	148	18	s.	s.	PROPN
ejpam-6039	148	19	sompong	sompong	PROPN
ejpam-6039	148	20	,	,	PUNCT
ejpam-6039	148	21	c.	c.	PROPN
ejpam-6039	148	22	boonpok	boonpok	PROPN
ejpam-6039	148	23	/	/	SYM
ejpam-6039	148	24	eur	eur	PROPN
ejpam-6039	148	25	.	.	PUNCT
ejpam-6039	149	1	j.	j.	PROPN
ejpam-6039	149	2	pure	pure	PROPN
ejpam-6039	149	3	appl	appl	PROPN
ejpam-6039	149	4	.	.	PROPN
ejpam-6039	149	5	math	math	PROPN
ejpam-6039	149	6	,	,	PUNCT
ejpam-6039	149	7	18	18	NUM
ejpam-6039	149	8	(	(	PUNCT
ejpam-6039	149	9	2	2	NUM
ejpam-6039	149	10	)	)	PUNCT
ejpam-6039	149	11	(	(	PUNCT
ejpam-6039	149	12	2025	2025	NUM
ejpam-6039	149	13	)	)	PUNCT
ejpam-6039	149	14	,	,	PUNCT
ejpam-6039	149	15	6039	6039	NUM
ejpam-6039	149	16	6	6	NUM
ejpam-6039	149	17	of	of	ADP
ejpam-6039	149	18	11	11	NUM
ejpam-6039	149	19	the	the	DET
ejpam-6039	149	20	(	(	PUNCT
ejpam-6039	149	21	τ1	τ1	NOUN
ejpam-6039	149	22	,	,	PUNCT
ejpam-6039	149	23	τ2)p	τ2)p	ADJ
ejpam-6039	149	24	-	-	PUNCT
ejpam-6039	149	25	frontier	frontier	NOUN
ejpam-6039	149	26	[	[	X
ejpam-6039	149	27	25	25	NUM
ejpam-6039	149	28	]	]	PUNCT
ejpam-6039	149	29	of	of	ADP
ejpam-6039	149	30	a	a	DET
ejpam-6039	149	31	subset	subset	NOUN
ejpam-6039	149	32	a	a	PRON
ejpam-6039	149	33	of	of	ADP
ejpam-6039	149	34	a	a	DET
ejpam-6039	149	35	bitopological	bitopological	ADJ
ejpam-6039	149	36	space	space	NOUN
ejpam-6039	149	37	(	(	PUNCT
ejpam-6039	149	38	x	x	NOUN
ejpam-6039	149	39	,	,	PUNCT
ejpam-6039	149	40	τ1	τ1	NOUN
ejpam-6039	149	41	,	,	PUNCT
ejpam-6039	149	42	τ2	τ2	PROPN
ejpam-6039	149	43	)	)	PUNCT
ejpam-6039	149	44	,	,	PUNCT
ejpam-6039	149	45	denoted	denote	VERB
ejpam-6039	149	46	by	by	ADP
ejpam-6039	149	47	(	(	PUNCT
ejpam-6039	149	48	τ1	τ1	NOUN
ejpam-6039	149	49	,	,	PUNCT
ejpam-6039	149	50	τ2)-pfr(a	τ2)-pfr(a	NOUN
ejpam-6039	149	51	)	)	PUNCT
ejpam-6039	149	52	,	,	PUNCT
ejpam-6039	149	53	is	be	AUX
ejpam-6039	149	54	defined	define	VERB
ejpam-6039	149	55	by	by	ADP
ejpam-6039	149	56	(	(	PUNCT
ejpam-6039	149	57	τ1	τ1	NOUN
ejpam-6039	149	58	,	,	PUNCT
ejpam-6039	149	59	τ2)-pfr(a	τ2)-pfr(a	ADJ
ejpam-6039	149	60	)	)	PUNCT
ejpam-6039	149	61	=	=	SYM
ejpam-6039	149	62	(	(	PUNCT
ejpam-6039	149	63	τ1	τ1	PROPN
ejpam-6039	149	64	,	,	PUNCT
ejpam-6039	149	65	τ2)-pcl(a	τ2)-pcl(a	ADJ
ejpam-6039	149	66	)	)	PUNCT
ejpam-6039	149	67	∩	∩	NOUN
ejpam-6039	149	68	(	(	PUNCT
ejpam-6039	149	69	τ1	τ1	PROPN
ejpam-6039	149	70	,	,	PUNCT
ejpam-6039	149	71	τ2)-pcl(x	τ2)-pcl(x	NOUN
ejpam-6039	149	72	−a	−a	NOUN
ejpam-6039	149	73	)	)	PUNCT
ejpam-6039	150	1	=	=	PUNCT
ejpam-6039	150	2	(	(	PUNCT
ejpam-6039	150	3	τ1	τ1	NOUN
ejpam-6039	150	4	,	,	PUNCT
ejpam-6039	150	5	τ2)-pcl(a)−	τ2)-pcl(a)−	NOUN
ejpam-6039	150	6	(	(	PUNCT
ejpam-6039	150	7	τ1	τ1	NOUN
ejpam-6039	150	8	,	,	PUNCT
ejpam-6039	150	9	τ2)-pint(a	τ2)-pint(a	PROPN
ejpam-6039	150	10	)	)	PUNCT
ejpam-6039	150	11	.	.	PUNCT
ejpam-6039	151	1	theorem	theorem	NOUN
ejpam-6039	151	2	5	5	NUM
ejpam-6039	151	3	.	.	PUNCT
ejpam-6039	152	1	the	the	DET
ejpam-6039	152	2	set	set	NOUN
ejpam-6039	152	3	of	of	ADP
ejpam-6039	152	4	all	all	DET
ejpam-6039	152	5	points	point	NOUN
ejpam-6039	152	6	x	x	PUNCT
ejpam-6039	152	7	of	of	ADP
ejpam-6039	152	8	x	x	SYM
ejpam-6039	152	9	at	at	ADP
ejpam-6039	152	10	which	which	PRON
ejpam-6039	152	11	a	a	DET
ejpam-6039	152	12	function	function	NOUN
ejpam-6039	152	13	f	f	NOUN
ejpam-6039	152	14	:	:	PUNCT
ejpam-6039	152	15	(	(	PUNCT
ejpam-6039	152	16	x	x	NOUN
ejpam-6039	152	17	,	,	PUNCT
ejpam-6039	152	18	τ1	τ1	NOUN
ejpam-6039	152	19	,	,	PUNCT
ejpam-6039	152	20	τ2	τ2	NOUN
ejpam-6039	152	21	)	)	PUNCT
ejpam-6039	152	22	→	→	SYM
ejpam-6039	152	23	(	(	PUNCT
ejpam-6039	152	24	y	y	PROPN
ejpam-6039	152	25	,	,	PUNCT
ejpam-6039	152	26	σ1	σ1	PROPN
ejpam-6039	152	27	,	,	PUNCT
ejpam-6039	152	28	σ2	σ2	PROPN
ejpam-6039	152	29	)	)	PUNCT
ejpam-6039	152	30	is	be	AUX
ejpam-6039	152	31	not	not	PART
ejpam-6039	152	32	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	152	33	,	,	PUNCT
ejpam-6039	152	34	τ2)p	τ2)p	ADJ
ejpam-6039	152	35	-	-	ADJ
ejpam-6039	152	36	continuous	continuous	ADJ
ejpam-6039	152	37	is	be	AUX
ejpam-6039	152	38	identical	identical	ADJ
ejpam-6039	152	39	with	with	ADP
ejpam-6039	152	40	the	the	DET
ejpam-6039	152	41	union	union	NOUN
ejpam-6039	152	42	of	of	ADP
ejpam-6039	152	43	the	the	DET
ejpam-6039	152	44	(	(	PUNCT
ejpam-6039	152	45	τ1	τ1	NOUN
ejpam-6039	152	46	,	,	PUNCT
ejpam-6039	152	47	τ2)p	τ2)p	ADJ
ejpam-6039	152	48	-	-	PUNCT
ejpam-6039	152	49	frontier	frontier	NOUN
ejpam-6039	152	50	of	of	ADP
ejpam-6039	152	51	the	the	DET
ejpam-6039	152	52	inverse	inverse	NOUN
ejpam-6039	152	53	images	image	NOUN
ejpam-6039	152	54	of	of	ADP
ejpam-6039	152	55	σ1σ2	σ1σ2	NOUN
ejpam-6039	152	56	-	-	PUNCT
ejpam-6039	152	57	closed	closed	ADJ
ejpam-6039	152	58	sets	set	NOUN
ejpam-6039	152	59	of	of	ADP
ejpam-6039	152	60	y	y	NOUN
ejpam-6039	152	61	containing	contain	VERB
ejpam-6039	152	62	f(x	f(x	PROPN
ejpam-6039	152	63	)	)	PUNCT
ejpam-6039	152	64	.	.	PUNCT
ejpam-6039	153	1	proof	proof	NOUN
ejpam-6039	153	2	.	.	PUNCT
ejpam-6039	154	1	suppose	suppose	VERB
ejpam-6039	154	2	that	that	SCONJ
ejpam-6039	154	3	f	f	PROPN
ejpam-6039	154	4	is	be	AUX
ejpam-6039	154	5	not	not	PART
ejpam-6039	154	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	154	7	,	,	PUNCT
ejpam-6039	154	8	τ2)p	τ2)p	ADJ
ejpam-6039	154	9	-	-	ADJ
ejpam-6039	154	10	continuous	continuous	ADJ
ejpam-6039	154	11	at	at	ADP
ejpam-6039	154	12	x	x	SYM
ejpam-6039	154	13	∈	∈	PROPN
ejpam-6039	154	14	x.	x.	NOUN
ejpam-6039	154	15	then	then	ADV
ejpam-6039	154	16	,	,	PUNCT
ejpam-6039	154	17	there	there	PRON
ejpam-6039	154	18	exists	exist	VERB
ejpam-6039	154	19	a	a	DET
ejpam-6039	154	20	σ1σ2	σ1σ2	NUM
ejpam-6039	154	21	-	-	PUNCT
ejpam-6039	154	22	closed	closed	ADJ
ejpam-6039	154	23	set	set	ADJ
ejpam-6039	154	24	f	f	PROPN
ejpam-6039	154	25	of	of	ADP
ejpam-6039	154	26	y	y	PROPN
ejpam-6039	154	27	containing	contain	VERB
ejpam-6039	154	28	f(x	f(x	PROPN
ejpam-6039	154	29	)	)	PUNCT
ejpam-6039	154	30	such	such	ADJ
ejpam-6039	154	31	that	that	DET
ejpam-6039	154	32	f(u	f(u	PROPN
ejpam-6039	154	33	)	)	PUNCT
ejpam-6039	154	34	∩	∩	NOUN
ejpam-6039	154	35	(	(	PUNCT
ejpam-6039	154	36	y	y	PROPN
ejpam-6039	154	37	−	−	PROPN
ejpam-6039	154	38	f	f	PROPN
ejpam-6039	154	39	)	)	PUNCT
ejpam-6039	154	40	̸=	̸=	PROPN
ejpam-6039	154	41	∅	∅	NOUN
ejpam-6039	154	42	for	for	ADP
ejpam-6039	154	43	every	every	DET
ejpam-6039	154	44	(	(	PUNCT
ejpam-6039	154	45	τ1	τ1	NOUN
ejpam-6039	154	46	,	,	PUNCT
ejpam-6039	154	47	τ2)p	τ2)p	ADJ
ejpam-6039	154	48	-	-	PUNCT
ejpam-6039	154	49	open	open	ADJ
ejpam-6039	154	50	set	set	NOUN
ejpam-6039	154	51	u	u	NOUN
ejpam-6039	154	52	of	of	ADP
ejpam-6039	154	53	x	x	SYM
ejpam-6039	154	54	containing	contain	VERB
ejpam-6039	154	55	x.	x.	NOUN
ejpam-6039	154	56	this	this	PRON
ejpam-6039	154	57	implies	imply	VERB
ejpam-6039	154	58	that	that	SCONJ
ejpam-6039	154	59	u	u	PROPN
ejpam-6039	154	60	∩	∩	NOUN
ejpam-6039	154	61	f−1(y	f−1(y	NOUN
ejpam-6039	154	62	−f	−f	PROPN
ejpam-6039	154	63	)	)	PUNCT
ejpam-6039	154	64	̸=	̸=	PROPN
ejpam-6039	154	65	∅.	∅.	VERB
ejpam-6039	154	66	therefore	therefore	ADV
ejpam-6039	154	67	,	,	PUNCT
ejpam-6039	154	68	x	x	SYM
ejpam-6039	154	69	∈	∈	PROPN
ejpam-6039	154	70	(	(	PUNCT
ejpam-6039	154	71	τ1	τ1	PROPN
ejpam-6039	154	72	,	,	PUNCT
ejpam-6039	154	73	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-6039	154	74	−1(y	−1(y	PUNCT
ejpam-6039	154	75	−	−	PROPN
ejpam-6039	154	76	f	f	PROPN
ejpam-6039	154	77	)	)	PUNCT
ejpam-6039	154	78	)	)	PUNCT
ejpam-6039	155	1	=	=	PRON
ejpam-6039	155	2	(	(	PUNCT
ejpam-6039	155	3	τ1	τ1	PROPN
ejpam-6039	155	4	,	,	PUNCT
ejpam-6039	155	5	τ2)-pcl(x	τ2)-pcl(x	PUNCT
ejpam-6039	155	6	−	−	PROPN
ejpam-6039	155	7	f−1(f	f−1(f	PROPN
ejpam-6039	155	8	)	)	PUNCT
ejpam-6039	155	9	)	)	PUNCT
ejpam-6039	155	10	.	.	PUNCT
ejpam-6039	156	1	on	on	ADP
ejpam-6039	156	2	the	the	DET
ejpam-6039	156	3	other	other	ADJ
ejpam-6039	156	4	hand	hand	NOUN
ejpam-6039	156	5	,	,	PUNCT
ejpam-6039	156	6	we	we	PRON
ejpam-6039	156	7	have	have	VERB
ejpam-6039	156	8	x	x	PROPN
ejpam-6039	156	9	∈	∈	PROPN
ejpam-6039	156	10	f−1(f	f−1(f	PROPN
ejpam-6039	156	11	)	)	PUNCT
ejpam-6039	157	1	⊆	⊆	NUM
ejpam-6039	157	2	(	(	PUNCT
ejpam-6039	157	3	τ1	τ1	NOUN
ejpam-6039	157	4	,	,	PUNCT
ejpam-6039	157	5	τ2)-pcl(f	τ2)-pcl(f	PROPN
ejpam-6039	157	6	−1(f	−1(f	PUNCT
ejpam-6039	157	7	)	)	PUNCT
ejpam-6039	157	8	)	)	PUNCT
ejpam-6039	157	9	and	and	CCONJ
ejpam-6039	157	10	hence	hence	ADV
ejpam-6039	157	11	x	x	X
ejpam-6039	157	12	∈	∈	PROPN
ejpam-6039	157	13	(	(	PUNCT
ejpam-6039	157	14	τ1	τ1	NOUN
ejpam-6039	157	15	,	,	PUNCT
ejpam-6039	157	16	τ2)-pfr(f	τ2)-pfr(f	NOUN
ejpam-6039	157	17	−1(f	−1(f	PUNCT
ejpam-6039	157	18	)	)	PUNCT
ejpam-6039	157	19	)	)	PUNCT
ejpam-6039	157	20	.	.	PUNCT
ejpam-6039	158	1	conversely	conversely	ADV
ejpam-6039	158	2	,	,	PUNCT
ejpam-6039	158	3	suppose	suppose	VERB
ejpam-6039	158	4	that	that	SCONJ
ejpam-6039	158	5	x	x	SYM
ejpam-6039	158	6	∈	∈	PROPN
ejpam-6039	158	7	(	(	PUNCT
ejpam-6039	158	8	τ1	τ1	NOUN
ejpam-6039	158	9	,	,	PUNCT
ejpam-6039	158	10	τ2)-pfr(f	τ2)-pfr(f	NOUN
ejpam-6039	158	11	−1(f	−1(f	PUNCT
ejpam-6039	158	12	)	)	PUNCT
ejpam-6039	158	13	)	)	PUNCT
ejpam-6039	158	14	for	for	ADP
ejpam-6039	158	15	some	some	DET
ejpam-6039	158	16	σ1σ2	σ1σ2	NUM
ejpam-6039	158	17	-	-	PUNCT
ejpam-6039	158	18	closed	closed	ADJ
ejpam-6039	158	19	set	set	ADJ
ejpam-6039	158	20	f	f	PROPN
ejpam-6039	158	21	of	of	ADP
ejpam-6039	158	22	y	y	PROPN
ejpam-6039	158	23	containing	contain	VERB
ejpam-6039	158	24	f(x	f(x	PROPN
ejpam-6039	158	25	)	)	PUNCT
ejpam-6039	158	26	.	.	PUNCT
ejpam-6039	159	1	now	now	ADV
ejpam-6039	159	2	,	,	PUNCT
ejpam-6039	159	3	we	we	PRON
ejpam-6039	159	4	assume	assume	VERB
ejpam-6039	159	5	that	that	SCONJ
ejpam-6039	159	6	f	f	PROPN
ejpam-6039	159	7	is	be	AUX
ejpam-6039	159	8	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	159	9	,	,	PUNCT
ejpam-6039	159	10	τ2)p	τ2)p	ADJ
ejpam-6039	159	11	-	-	ADJ
ejpam-6039	159	12	continuous	continuous	ADJ
ejpam-6039	159	13	at	at	ADP
ejpam-6039	159	14	x.	x.	NOUN
ejpam-6039	159	15	then	then	ADV
ejpam-6039	159	16	,	,	PUNCT
ejpam-6039	159	17	there	there	PRON
ejpam-6039	159	18	exists	exist	VERB
ejpam-6039	159	19	a	a	DET
ejpam-6039	159	20	(	(	PUNCT
ejpam-6039	159	21	τ1	τ1	NOUN
ejpam-6039	159	22	,	,	PUNCT
ejpam-6039	159	23	τ2)p	τ2)p	ADJ
ejpam-6039	159	24	-	-	PUNCT
ejpam-6039	159	25	open	open	ADJ
ejpam-6039	159	26	set	set	NOUN
ejpam-6039	159	27	u	u	NOUN
ejpam-6039	159	28	of	of	ADP
ejpam-6039	159	29	x	x	PUNCT
ejpam-6039	159	30	containing	contain	VERB
ejpam-6039	159	31	x	x	PUNCT
ejpam-6039	159	32	such	such	ADJ
ejpam-6039	159	33	that	that	DET
ejpam-6039	159	34	f(u	f(u	PROPN
ejpam-6039	159	35	)	)	PUNCT
ejpam-6039	159	36	⊆	⊆	NUM
ejpam-6039	159	37	f	f	NOUN
ejpam-6039	159	38	.	.	PUNCT
ejpam-6039	160	1	thus	thus	ADV
ejpam-6039	160	2	,	,	PUNCT
ejpam-6039	160	3	u	u	PROPN
ejpam-6039	160	4	⊆	⊆	NUM
ejpam-6039	160	5	f−1(f	f−1(f	PROPN
ejpam-6039	160	6	)	)	PUNCT
ejpam-6039	160	7	and	and	CCONJ
ejpam-6039	160	8	hence	hence	ADV
ejpam-6039	160	9	x	x	X
ejpam-6039	160	10	∈	∈	PROPN
ejpam-6039	160	11	(	(	PUNCT
ejpam-6039	160	12	τ1	τ1	NOUN
ejpam-6039	160	13	,	,	PUNCT
ejpam-6039	160	14	τ2)-pint(f	τ2)-pint(f	PROPN
ejpam-6039	160	15	−1(f	−1(f	NUM
ejpam-6039	160	16	)	)	PUNCT
ejpam-6039	160	17	)	)	PUNCT
ejpam-6039	161	1	⊆	⊆	NUM
ejpam-6039	161	2	x	x	SYM
ejpam-6039	161	3	−	−	PROPN
ejpam-6039	161	4	(	(	PUNCT
ejpam-6039	161	5	τ1	τ1	NOUN
ejpam-6039	161	6	,	,	PUNCT
ejpam-6039	161	7	τ2)-pfr(f	τ2)-pfr(f	NOUN
ejpam-6039	161	8	−1(f	−1(f	PUNCT
ejpam-6039	161	9	)	)	PUNCT
ejpam-6039	161	10	)	)	PUNCT
ejpam-6039	161	11	.	.	PUNCT
ejpam-6039	162	1	this	this	PRON
ejpam-6039	162	2	is	be	AUX
ejpam-6039	162	3	a	a	DET
ejpam-6039	162	4	contradiction	contradiction	NOUN
ejpam-6039	162	5	.	.	PUNCT
ejpam-6039	163	1	this	this	PRON
ejpam-6039	163	2	means	mean	VERB
ejpam-6039	163	3	that	that	SCONJ
ejpam-6039	163	4	f	f	PROPN
ejpam-6039	163	5	is	be	AUX
ejpam-6039	163	6	not	not	PART
ejpam-6039	163	7	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	163	8	,	,	PUNCT
ejpam-6039	163	9	τ2)p	τ2)p	ADJ
ejpam-6039	163	10	-	-	ADJ
ejpam-6039	163	11	continuous	continuous	ADJ
ejpam-6039	163	12	.	.	PUNCT
ejpam-6039	164	1	definition	definition	NOUN
ejpam-6039	164	2	6	6	NUM
ejpam-6039	164	3	.	.	PUNCT
ejpam-6039	165	1	a	a	DET
ejpam-6039	165	2	function	function	NOUN
ejpam-6039	165	3	f	f	NOUN
ejpam-6039	165	4	:	:	PUNCT
ejpam-6039	165	5	(	(	PUNCT
ejpam-6039	165	6	x	x	NOUN
ejpam-6039	165	7	,	,	PUNCT
ejpam-6039	165	8	τ1	τ1	NOUN
ejpam-6039	165	9	,	,	PUNCT
ejpam-6039	165	10	τ2	τ2	NOUN
ejpam-6039	165	11	)	)	PUNCT
ejpam-6039	165	12	→	→	SYM
ejpam-6039	165	13	(	(	PUNCT
ejpam-6039	165	14	y	y	PROPN
ejpam-6039	165	15	,	,	PUNCT
ejpam-6039	165	16	σ1	σ1	PROPN
ejpam-6039	165	17	,	,	PUNCT
ejpam-6039	165	18	σ2	σ2	PROPN
ejpam-6039	165	19	)	)	PUNCT
ejpam-6039	165	20	is	be	AUX
ejpam-6039	165	21	called	call	VERB
ejpam-6039	165	22	contra	contra	PROPN
ejpam-6039	165	23	-	-	PUNCT
ejpam-6039	165	24	α(τ1	α(τ1	NOUN
ejpam-6039	165	25	,	,	PUNCT
ejpam-6039	165	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	165	27	if	if	SCONJ
ejpam-6039	165	28	f−1(v	f−1(v	PROPN
ejpam-6039	165	29	)	)	PUNCT
ejpam-6039	165	30	is	be	AUX
ejpam-6039	165	31	α(τ1	α(τ1	NOUN
ejpam-6039	165	32	,	,	PUNCT
ejpam-6039	165	33	τ2)-closed	τ2)-close	VERB
ejpam-6039	165	34	in	in	ADP
ejpam-6039	165	35	x	x	PUNCT
ejpam-6039	165	36	for	for	ADP
ejpam-6039	165	37	each	each	DET
ejpam-6039	165	38	σ1σ2	σ1σ2	VERB
ejpam-6039	165	39	-	-	ADJ
ejpam-6039	165	40	open	open	ADJ
ejpam-6039	165	41	set	set	NOUN
ejpam-6039	165	42	v	v	NOUN
ejpam-6039	165	43	of	of	ADP
ejpam-6039	165	44	y	y	PROPN
ejpam-6039	165	45	.	.	PUNCT
ejpam-6039	166	1	definition	definition	NOUN
ejpam-6039	166	2	7	7	NUM
ejpam-6039	166	3	.	.	PUNCT
ejpam-6039	167	1	a	a	DET
ejpam-6039	167	2	function	function	NOUN
ejpam-6039	167	3	f	f	NOUN
ejpam-6039	167	4	:	:	PUNCT
ejpam-6039	167	5	(	(	PUNCT
ejpam-6039	167	6	x	x	NOUN
ejpam-6039	167	7	,	,	PUNCT
ejpam-6039	167	8	τ1	τ1	NOUN
ejpam-6039	167	9	,	,	PUNCT
ejpam-6039	167	10	τ2	τ2	NOUN
ejpam-6039	167	11	)	)	PUNCT
ejpam-6039	167	12	→	→	SYM
ejpam-6039	167	13	(	(	PUNCT
ejpam-6039	167	14	y	y	PROPN
ejpam-6039	167	15	,	,	PUNCT
ejpam-6039	167	16	σ1	σ1	PROPN
ejpam-6039	167	17	,	,	PUNCT
ejpam-6039	167	18	σ2	σ2	PROPN
ejpam-6039	167	19	)	)	PUNCT
ejpam-6039	167	20	is	be	AUX
ejpam-6039	167	21	called	call	VERB
ejpam-6039	167	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	167	23	,	,	PUNCT
ejpam-6039	167	24	τ2)s	τ2)s	NOUN
ejpam-6039	167	25	-	-	PUNCT
ejpam-6039	167	26	continuous	continuous	ADJ
ejpam-6039	167	27	if	if	SCONJ
ejpam-6039	167	28	f−1(v	f−1(v	PROPN
ejpam-6039	167	29	)	)	PUNCT
ejpam-6039	168	1	is	be	AUX
ejpam-6039	168	2	(	(	PUNCT
ejpam-6039	168	3	τ1	τ1	NOUN
ejpam-6039	168	4	,	,	PUNCT
ejpam-6039	168	5	τ2)s	τ2)s	NOUN
ejpam-6039	168	6	-	-	PUNCT
ejpam-6039	168	7	closed	close	VERB
ejpam-6039	168	8	in	in	ADP
ejpam-6039	168	9	x	x	PUNCT
ejpam-6039	168	10	for	for	ADP
ejpam-6039	168	11	each	each	DET
ejpam-6039	168	12	σ1σ2	σ1σ2	VERB
ejpam-6039	168	13	-	-	ADJ
ejpam-6039	168	14	open	open	ADJ
ejpam-6039	168	15	set	set	NOUN
ejpam-6039	168	16	v	v	NOUN
ejpam-6039	168	17	of	of	ADP
ejpam-6039	168	18	y	y	PROPN
ejpam-6039	168	19	.	.	PUNCT
ejpam-6039	169	1	lemma	lemma	PROPN
ejpam-6039	169	2	5	5	NUM
ejpam-6039	169	3	.	.	PUNCT
ejpam-6039	170	1	for	for	ADP
ejpam-6039	170	2	a	a	DET
ejpam-6039	170	3	subset	subset	NOUN
ejpam-6039	170	4	a	a	PRON
ejpam-6039	170	5	of	of	ADP
ejpam-6039	170	6	a	a	DET
ejpam-6039	170	7	bitopological	bitopological	ADJ
ejpam-6039	170	8	space	space	NOUN
ejpam-6039	170	9	(	(	PUNCT
ejpam-6039	170	10	x	x	NOUN
ejpam-6039	170	11	,	,	PUNCT
ejpam-6039	170	12	τ1	τ1	NOUN
ejpam-6039	170	13	,	,	PUNCT
ejpam-6039	170	14	τ2	τ2	NOUN
ejpam-6039	170	15	)	)	PUNCT
ejpam-6039	170	16	,	,	PUNCT
ejpam-6039	170	17	the	the	DET
ejpam-6039	170	18	following	follow	VERB
ejpam-6039	170	19	properties	property	NOUN
ejpam-6039	170	20	are	be	AUX
ejpam-6039	170	21	equivalent	equivalent	ADJ
ejpam-6039	170	22	:	:	PUNCT
ejpam-6039	170	23	(	(	PUNCT
ejpam-6039	170	24	1	1	X
ejpam-6039	170	25	)	)	PUNCT
ejpam-6039	170	26	a	a	PRON
ejpam-6039	170	27	is	be	AUX
ejpam-6039	170	28	α(τ1	α(τ1	NOUN
ejpam-6039	170	29	,	,	PUNCT
ejpam-6039	170	30	τ2)-open	τ2)-open	ADJ
ejpam-6039	170	31	;	;	PUNCT
ejpam-6039	170	32	(	(	PUNCT
ejpam-6039	170	33	2	2	X
ejpam-6039	170	34	)	)	PUNCT
ejpam-6039	170	35	a	a	PRON
ejpam-6039	170	36	is	is	NOUN
ejpam-6039	170	37	(	(	PUNCT
ejpam-6039	170	38	τ1	τ1	NOUN
ejpam-6039	170	39	,	,	PUNCT
ejpam-6039	170	40	τ2)p	τ2)p	NOUN
ejpam-6039	170	41	-	-	PUNCT
ejpam-6039	170	42	open	open	ADJ
ejpam-6039	170	43	and	and	CCONJ
ejpam-6039	170	44	(	(	PUNCT
ejpam-6039	170	45	τ1	τ1	NOUN
ejpam-6039	170	46	,	,	PUNCT
ejpam-6039	170	47	τ2)s	τ2)s	NOUN
ejpam-6039	170	48	-	-	PUNCT
ejpam-6039	170	49	open	open	ADJ
ejpam-6039	170	50	.	.	PUNCT
ejpam-6039	171	1	proof	proof	NOUN
ejpam-6039	171	2	.	.	PUNCT
ejpam-6039	172	1	(	(	PUNCT
ejpam-6039	172	2	1	1	X
ejpam-6039	172	3	)	)	PUNCT
ejpam-6039	172	4	⇒	⇒	NOUN
ejpam-6039	172	5	(	(	PUNCT
ejpam-6039	172	6	2	2	NUM
ejpam-6039	172	7	):	):	PUNCT
ejpam-6039	172	8	let	let	VERB
ejpam-6039	172	9	a	a	DET
ejpam-6039	172	10	be	be	AUX
ejpam-6039	172	11	α(τ1	α(τ1	NOUN
ejpam-6039	172	12	,	,	PUNCT
ejpam-6039	172	13	τ2)-open	τ2)-open	ADJ
ejpam-6039	172	14	.	.	PUNCT
ejpam-6039	173	1	then	then	ADV
ejpam-6039	173	2	,	,	PUNCT
ejpam-6039	173	3	a	a	DET
ejpam-6039	173	4	⊆	⊆	NUM
ejpam-6039	173	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	173	6	-	-	PUNCT
ejpam-6039	173	7	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	173	8	-	-	PUNCT
ejpam-6039	173	9	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	173	10	-	-	PUNCT
ejpam-6039	173	11	int(a	int(a	NOUN
ejpam-6039	173	12	)	)	PUNCT
ejpam-6039	173	13	)	)	PUNCT
ejpam-6039	173	14	)	)	PUNCT
ejpam-6039	173	15	.	.	PUNCT
ejpam-6039	174	1	therefore	therefore	ADV
ejpam-6039	174	2	,	,	PUNCT
ejpam-6039	174	3	a	a	DET
ejpam-6039	174	4	⊆	⊆	NUM
ejpam-6039	174	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	174	6	-	-	NOUN
ejpam-6039	174	7	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	174	8	-	-	PUNCT
ejpam-6039	174	9	cl(a	cl(a	NUM
ejpam-6039	174	10	)	)	PUNCT
ejpam-6039	174	11	)	)	PUNCT
ejpam-6039	174	12	and	and	CCONJ
ejpam-6039	174	13	a	a	DET
ejpam-6039	174	14	⊆	⊆	NUM
ejpam-6039	174	15	τ1τ2	τ1τ2	NOUN
ejpam-6039	174	16	-	-	ADJ
ejpam-6039	174	17	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	174	18	-	-	PUNCT
ejpam-6039	174	19	int(a	int(a	NOUN
ejpam-6039	174	20	)	)	PUNCT
ejpam-6039	174	21	)	)	PUNCT
ejpam-6039	174	22	.	.	PUNCT
ejpam-6039	175	1	this	this	PRON
ejpam-6039	175	2	shows	show	VERB
ejpam-6039	175	3	that	that	SCONJ
ejpam-6039	175	4	a	a	PRON
ejpam-6039	175	5	is	be	AUX
ejpam-6039	175	6	(	(	PUNCT
ejpam-6039	175	7	τ1	τ1	NOUN
ejpam-6039	175	8	,	,	PUNCT
ejpam-6039	175	9	τ2)p	τ2)p	NOUN
ejpam-6039	175	10	-	-	PUNCT
ejpam-6039	175	11	open	open	ADJ
ejpam-6039	175	12	and	and	CCONJ
ejpam-6039	175	13	(	(	PUNCT
ejpam-6039	175	14	τ1	τ1	NOUN
ejpam-6039	175	15	,	,	PUNCT
ejpam-6039	175	16	τ2)s	τ2)s	NOUN
ejpam-6039	175	17	-	-	PUNCT
ejpam-6039	175	18	open	open	ADJ
ejpam-6039	175	19	.	.	PUNCT
ejpam-6039	176	1	(	(	PUNCT
ejpam-6039	176	2	2	2	X
ejpam-6039	176	3	)	)	PUNCT
ejpam-6039	176	4	⇒	⇒	NOUN
ejpam-6039	176	5	(	(	PUNCT
ejpam-6039	176	6	1	1	NUM
ejpam-6039	176	7	):	):	PUNCT
ejpam-6039	176	8	let	let	VERB
ejpam-6039	176	9	a	a	DET
ejpam-6039	176	10	be	be	AUX
ejpam-6039	176	11	(	(	PUNCT
ejpam-6039	176	12	τ1	τ1	NOUN
ejpam-6039	176	13	,	,	PUNCT
ejpam-6039	176	14	τ2)p	τ2)p	NOUN
ejpam-6039	176	15	-	-	PUNCT
ejpam-6039	176	16	open	open	ADJ
ejpam-6039	176	17	and	and	CCONJ
ejpam-6039	176	18	(	(	PUNCT
ejpam-6039	176	19	τ1	τ1	NOUN
ejpam-6039	176	20	,	,	PUNCT
ejpam-6039	176	21	τ2)s	τ2)s	NOUN
ejpam-6039	176	22	-	-	PUNCT
ejpam-6039	176	23	open	open	ADJ
ejpam-6039	176	24	.	.	PUNCT
ejpam-6039	177	1	then	then	ADV
ejpam-6039	177	2	,	,	PUNCT
ejpam-6039	177	3	a	a	DET
ejpam-6039	177	4	⊆	⊆	NUM
ejpam-6039	177	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	177	6	-	-	NOUN
ejpam-6039	177	7	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	177	8	-	-	PUNCT
ejpam-6039	177	9	cl(a	cl(a	NUM
ejpam-6039	177	10	)	)	PUNCT
ejpam-6039	177	11	)	)	PUNCT
ejpam-6039	177	12	and	and	CCONJ
ejpam-6039	177	13	a	a	DET
ejpam-6039	177	14	⊆	⊆	NUM
ejpam-6039	177	15	τ1τ2	τ1τ2	NOUN
ejpam-6039	177	16	-	-	ADJ
ejpam-6039	177	17	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	177	18	-	-	PUNCT
ejpam-6039	177	19	int(a	int(a	NOUN
ejpam-6039	177	20	)	)	PUNCT
ejpam-6039	177	21	)	)	PUNCT
ejpam-6039	177	22	.	.	PUNCT
ejpam-6039	178	1	thus	thus	ADV
ejpam-6039	178	2	,	,	PUNCT
ejpam-6039	178	3	a	a	DET
ejpam-6039	178	4	⊆	⊆	NUM
ejpam-6039	178	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	178	6	-	-	NOUN
ejpam-6039	178	7	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	178	8	-	-	PUNCT
ejpam-6039	178	9	cl(a	cl(a	NUM
ejpam-6039	178	10	)	)	PUNCT
ejpam-6039	178	11	)	)	PUNCT
ejpam-6039	178	12	⊆	⊆	X
ejpam-6039	178	13	τ1τ2	τ1τ2	NOUN
ejpam-6039	178	14	-	-	PUNCT
ejpam-6039	178	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	178	16	-	-	PUNCT
ejpam-6039	178	17	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	178	18	-	-	PUNCT
ejpam-6039	178	19	int(a	int(a	NOUN
ejpam-6039	178	20	)	)	PUNCT
ejpam-6039	178	21	)	)	PUNCT
ejpam-6039	178	22	)	)	PUNCT
ejpam-6039	178	23	and	and	CCONJ
ejpam-6039	178	24	hence	hence	ADV
ejpam-6039	178	25	a	a	PRON
ejpam-6039	178	26	is	be	AUX
ejpam-6039	178	27	α(τ1	α(τ1	NOUN
ejpam-6039	178	28	,	,	PUNCT
ejpam-6039	178	29	τ2)-open	τ2)-open	PROPN
ejpam-6039	178	30	.	.	PUNCT
ejpam-6039	178	31	theorem	theorem	VERB
ejpam-6039	178	32	6	6	NUM
ejpam-6039	178	33	.	.	PUNCT
ejpam-6039	178	34	for	for	ADP
ejpam-6039	178	35	a	a	DET
ejpam-6039	178	36	f	f	NOUN
ejpam-6039	178	37	:	:	PUNCT
ejpam-6039	178	38	(	(	PUNCT
ejpam-6039	178	39	x	x	NOUN
ejpam-6039	178	40	,	,	PUNCT
ejpam-6039	178	41	τ1	τ1	NOUN
ejpam-6039	178	42	,	,	PUNCT
ejpam-6039	178	43	τ2	τ2	NOUN
ejpam-6039	178	44	)	)	PUNCT
ejpam-6039	178	45	→	→	SYM
ejpam-6039	178	46	(	(	PUNCT
ejpam-6039	178	47	y	y	PROPN
ejpam-6039	178	48	,	,	PUNCT
ejpam-6039	178	49	σ1	σ1	PROPN
ejpam-6039	178	50	,	,	PUNCT
ejpam-6039	178	51	σ2	σ2	NOUN
ejpam-6039	178	52	)	)	PUNCT
ejpam-6039	178	53	,	,	PUNCT
ejpam-6039	178	54	the	the	DET
ejpam-6039	178	55	following	follow	VERB
ejpam-6039	178	56	properties	property	NOUN
ejpam-6039	178	57	are	be	AUX
ejpam-6039	178	58	equivalent	equivalent	ADJ
ejpam-6039	178	59	:	:	PUNCT
ejpam-6039	178	60	(	(	PUNCT
ejpam-6039	178	61	1	1	X
ejpam-6039	178	62	)	)	PUNCT
ejpam-6039	178	63	f	f	PROPN
ejpam-6039	178	64	is	be	AUX
ejpam-6039	178	65	contra	contra	PROPN
ejpam-6039	178	66	-	-	PUNCT
ejpam-6039	178	67	α(τ1	α(τ1	NOUN
ejpam-6039	178	68	,	,	PUNCT
ejpam-6039	178	69	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	178	70	;	;	PUNCT
ejpam-6039	178	71	m.	m.	NOUN
ejpam-6039	178	72	chiangpradit	chiangpradit	NOUN
ejpam-6039	178	73	,	,	PUNCT
ejpam-6039	178	74	s.	s.	PROPN
ejpam-6039	178	75	sompong	sompong	PROPN
ejpam-6039	178	76	,	,	PUNCT
ejpam-6039	178	77	c.	c.	PROPN
ejpam-6039	178	78	boonpok	boonpok	PROPN
ejpam-6039	178	79	/	/	SYM
ejpam-6039	178	80	eur	eur	PROPN
ejpam-6039	178	81	.	.	PUNCT
ejpam-6039	179	1	j.	j.	PROPN
ejpam-6039	179	2	pure	pure	PROPN
ejpam-6039	179	3	appl	appl	PROPN
ejpam-6039	179	4	.	.	PROPN
ejpam-6039	179	5	math	math	PROPN
ejpam-6039	179	6	,	,	PUNCT
ejpam-6039	179	7	18	18	NUM
ejpam-6039	179	8	(	(	PUNCT
ejpam-6039	179	9	2	2	NUM
ejpam-6039	179	10	)	)	PUNCT
ejpam-6039	179	11	(	(	PUNCT
ejpam-6039	179	12	2025	2025	NUM
ejpam-6039	179	13	)	)	PUNCT
ejpam-6039	179	14	,	,	PUNCT
ejpam-6039	179	15	6039	6039	NUM
ejpam-6039	179	16	7	7	NUM
ejpam-6039	179	17	of	of	ADP
ejpam-6039	179	18	11	11	NUM
ejpam-6039	179	19	(	(	PUNCT
ejpam-6039	179	20	2	2	NUM
ejpam-6039	179	21	)	)	PUNCT
ejpam-6039	179	22	f	f	PROPN
ejpam-6039	179	23	is	be	AUX
ejpam-6039	179	24	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	179	25	,	,	PUNCT
ejpam-6039	179	26	τ2)p	τ2)p	ADJ
ejpam-6039	179	27	-	-	ADJ
ejpam-6039	179	28	continuous	continuous	ADJ
ejpam-6039	179	29	and	and	CCONJ
ejpam-6039	179	30	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	179	31	,	,	PUNCT
ejpam-6039	179	32	τ2)s	τ2)s	NOUN
ejpam-6039	179	33	-	-	PUNCT
ejpam-6039	179	34	continuous	continuous	ADJ
ejpam-6039	179	35	.	.	PUNCT
ejpam-6039	180	1	proof	proof	NOUN
ejpam-6039	180	2	.	.	PUNCT
ejpam-6039	181	1	this	this	PRON
ejpam-6039	181	2	is	be	AUX
ejpam-6039	181	3	an	an	DET
ejpam-6039	181	4	immediate	immediate	ADJ
ejpam-6039	181	5	consequence	consequence	NOUN
ejpam-6039	181	6	of	of	ADP
ejpam-6039	181	7	lemma	lemma	PROPN
ejpam-6039	181	8	5	5	NUM
ejpam-6039	181	9	.	.	PUNCT
ejpam-6039	181	10	definition	definition	NOUN
ejpam-6039	181	11	8	8	NUM
ejpam-6039	181	12	.	.	PUNCT
ejpam-6039	182	1	a	a	DET
ejpam-6039	182	2	function	function	NOUN
ejpam-6039	182	3	f	f	NOUN
ejpam-6039	182	4	:	:	PUNCT
ejpam-6039	182	5	(	(	PUNCT
ejpam-6039	182	6	x	x	NOUN
ejpam-6039	182	7	,	,	PUNCT
ejpam-6039	182	8	τ1	τ1	NOUN
ejpam-6039	182	9	,	,	PUNCT
ejpam-6039	182	10	τ2	τ2	NOUN
ejpam-6039	182	11	)	)	PUNCT
ejpam-6039	182	12	→	→	SYM
ejpam-6039	182	13	(	(	PUNCT
ejpam-6039	182	14	y	y	PROPN
ejpam-6039	182	15	,	,	PUNCT
ejpam-6039	182	16	σ1	σ1	PROPN
ejpam-6039	182	17	,	,	PUNCT
ejpam-6039	182	18	σ2	σ2	PROPN
ejpam-6039	182	19	)	)	PUNCT
ejpam-6039	182	20	is	be	AUX
ejpam-6039	182	21	said	say	VERB
ejpam-6039	182	22	to	to	PART
ejpam-6039	182	23	be	be	AUX
ejpam-6039	182	24	rc-(τ1	rc-(τ1	NOUN
ejpam-6039	182	25	,	,	PUNCT
ejpam-6039	182	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	182	27	if	if	SCONJ
ejpam-6039	182	28	f−1(v	f−1(v	PROPN
ejpam-6039	182	29	)	)	PUNCT
ejpam-6039	182	30	is	be	AUX
ejpam-6039	182	31	(	(	PUNCT
ejpam-6039	182	32	τ1	τ1	NOUN
ejpam-6039	182	33	,	,	PUNCT
ejpam-6039	182	34	τ2)r	τ2)r	NOUN
ejpam-6039	182	35	-	-	PUNCT
ejpam-6039	182	36	closed	close	VERB
ejpam-6039	182	37	in	in	ADP
ejpam-6039	182	38	x	x	PUNCT
ejpam-6039	182	39	for	for	ADP
ejpam-6039	182	40	each	each	DET
ejpam-6039	182	41	σ1σ2	σ1σ2	VERB
ejpam-6039	182	42	-	-	ADJ
ejpam-6039	182	43	open	open	ADJ
ejpam-6039	182	44	set	set	NOUN
ejpam-6039	182	45	v	v	NOUN
ejpam-6039	182	46	of	of	ADP
ejpam-6039	182	47	y	y	PROPN
ejpam-6039	182	48	.	.	PUNCT
ejpam-6039	183	1	definition	definition	NOUN
ejpam-6039	183	2	9	9	NUM
ejpam-6039	183	3	.	.	PUNCT
ejpam-6039	184	1	a	a	DET
ejpam-6039	184	2	function	function	NOUN
ejpam-6039	184	3	f	f	NOUN
ejpam-6039	184	4	:	:	PUNCT
ejpam-6039	184	5	(	(	PUNCT
ejpam-6039	184	6	x	x	NOUN
ejpam-6039	184	7	,	,	PUNCT
ejpam-6039	184	8	τ1	τ1	NOUN
ejpam-6039	184	9	,	,	PUNCT
ejpam-6039	184	10	τ2	τ2	NOUN
ejpam-6039	184	11	)	)	PUNCT
ejpam-6039	184	12	→	→	SYM
ejpam-6039	184	13	(	(	PUNCT
ejpam-6039	184	14	y	y	PROPN
ejpam-6039	184	15	,	,	PUNCT
ejpam-6039	184	16	σ1	σ1	PROPN
ejpam-6039	184	17	,	,	PUNCT
ejpam-6039	184	18	σ2	σ2	PROPN
ejpam-6039	184	19	)	)	PUNCT
ejpam-6039	184	20	is	be	AUX
ejpam-6039	184	21	said	say	VERB
ejpam-6039	184	22	to	to	PART
ejpam-6039	184	23	be	be	AUX
ejpam-6039	184	24	(	(	PUNCT
ejpam-6039	184	25	τ1	τ1	NOUN
ejpam-6039	184	26	,	,	PUNCT
ejpam-6039	184	27	τ2)s	τ2)s	NOUN
ejpam-6039	184	28	-	-	ADJ
ejpam-6039	184	29	continuous	continuous	ADJ
ejpam-6039	184	30	if	if	SCONJ
ejpam-6039	184	31	for	for	ADP
ejpam-6039	184	32	x	x	SYM
ejpam-6039	184	33	∈	∈	PROPN
ejpam-6039	184	34	x	x	X
ejpam-6039	184	35	and	and	CCONJ
ejpam-6039	184	36	each	each	DET
ejpam-6039	184	37	σ1σ2	σ1σ2	VERB
ejpam-6039	184	38	-	-	ADJ
ejpam-6039	184	39	open	open	ADJ
ejpam-6039	184	40	set	set	NOUN
ejpam-6039	184	41	v	v	NOUN
ejpam-6039	184	42	of	of	ADP
ejpam-6039	184	43	y	y	NOUN
ejpam-6039	184	44	containing	contain	VERB
ejpam-6039	184	45	f(x	f(x	PROPN
ejpam-6039	184	46	)	)	PUNCT
ejpam-6039	184	47	,	,	PUNCT
ejpam-6039	184	48	there	there	PRON
ejpam-6039	184	49	exists	exist	VERB
ejpam-6039	184	50	a	a	DET
ejpam-6039	184	51	(	(	PUNCT
ejpam-6039	184	52	τ1	τ1	NOUN
ejpam-6039	184	53	,	,	PUNCT
ejpam-6039	184	54	τ2)s	τ2)s	NOUN
ejpam-6039	184	55	-	-	PUNCT
ejpam-6039	184	56	open	open	ADJ
ejpam-6039	184	57	set	set	NOUN
ejpam-6039	184	58	u	u	NOUN
ejpam-6039	184	59	of	of	ADP
ejpam-6039	184	60	x	x	PUNCT
ejpam-6039	184	61	containing	contain	VERB
ejpam-6039	184	62	x	x	PUNCT
ejpam-6039	184	63	such	such	ADJ
ejpam-6039	184	64	that	that	DET
ejpam-6039	184	65	f(u	f(u	PROPN
ejpam-6039	184	66	)	)	PUNCT
ejpam-6039	184	67	⊆	⊆	NUM
ejpam-6039	184	68	v	v	NOUN
ejpam-6039	184	69	.	.	PUNCT
ejpam-6039	185	1	definition	definition	NOUN
ejpam-6039	185	2	10	10	NUM
ejpam-6039	185	3	.	.	PUNCT
ejpam-6039	186	1	a	a	DET
ejpam-6039	186	2	function	function	NOUN
ejpam-6039	186	3	f	f	NOUN
ejpam-6039	186	4	:	:	PUNCT
ejpam-6039	186	5	(	(	PUNCT
ejpam-6039	186	6	x	x	NOUN
ejpam-6039	186	7	,	,	PUNCT
ejpam-6039	186	8	τ1	τ1	NOUN
ejpam-6039	186	9	,	,	PUNCT
ejpam-6039	186	10	τ2	τ2	NOUN
ejpam-6039	186	11	)	)	PUNCT
ejpam-6039	186	12	→	→	SYM
ejpam-6039	186	13	(	(	PUNCT
ejpam-6039	186	14	y	y	PROPN
ejpam-6039	186	15	,	,	PUNCT
ejpam-6039	186	16	σ1	σ1	PROPN
ejpam-6039	186	17	,	,	PUNCT
ejpam-6039	186	18	σ2	σ2	PROPN
ejpam-6039	186	19	)	)	PUNCT
ejpam-6039	186	20	is	be	AUX
ejpam-6039	186	21	said	say	VERB
ejpam-6039	186	22	to	to	PART
ejpam-6039	186	23	be	be	AUX
ejpam-6039	186	24	(	(	PUNCT
ejpam-6039	186	25	τ1	τ1	NOUN
ejpam-6039	186	26	,	,	PUNCT
ejpam-6039	186	27	τ2)β	τ2)β	ADJ
ejpam-6039	186	28	-	-	PUNCT
ejpam-6039	186	29	continuous	continuous	ADJ
ejpam-6039	186	30	if	if	SCONJ
ejpam-6039	186	31	for	for	ADP
ejpam-6039	186	32	x	x	SYM
ejpam-6039	186	33	∈	∈	PROPN
ejpam-6039	186	34	x	x	X
ejpam-6039	186	35	and	and	CCONJ
ejpam-6039	186	36	each	each	DET
ejpam-6039	186	37	σ1σ2	σ1σ2	VERB
ejpam-6039	186	38	-	-	ADJ
ejpam-6039	186	39	open	open	ADJ
ejpam-6039	186	40	set	set	NOUN
ejpam-6039	186	41	v	v	NOUN
ejpam-6039	186	42	of	of	ADP
ejpam-6039	186	43	y	y	NOUN
ejpam-6039	186	44	containing	contain	VERB
ejpam-6039	186	45	f(x	f(x	PROPN
ejpam-6039	186	46	)	)	PUNCT
ejpam-6039	186	47	,	,	PUNCT
ejpam-6039	186	48	there	there	PRON
ejpam-6039	186	49	exists	exist	VERB
ejpam-6039	186	50	a	a	DET
ejpam-6039	186	51	(	(	PUNCT
ejpam-6039	186	52	τ1	τ1	NOUN
ejpam-6039	186	53	,	,	PUNCT
ejpam-6039	186	54	τ2)β	τ2)β	ADJ
ejpam-6039	186	55	-	-	PUNCT
ejpam-6039	186	56	open	open	ADJ
ejpam-6039	186	57	set	set	NOUN
ejpam-6039	186	58	u	u	NOUN
ejpam-6039	186	59	of	of	ADP
ejpam-6039	186	60	x	x	PUNCT
ejpam-6039	186	61	containing	contain	VERB
ejpam-6039	186	62	x	x	PUNCT
ejpam-6039	186	63	such	such	ADJ
ejpam-6039	186	64	that	that	DET
ejpam-6039	186	65	f(u	f(u	PROPN
ejpam-6039	186	66	)	)	PUNCT
ejpam-6039	186	67	⊆	⊆	NUM
ejpam-6039	186	68	v	v	NOUN
ejpam-6039	186	69	.	.	PUNCT
ejpam-6039	187	1	lemma	lemma	PROPN
ejpam-6039	187	2	6	6	NUM
ejpam-6039	187	3	.	.	PUNCT
ejpam-6039	188	1	for	for	ADP
ejpam-6039	188	2	a	a	DET
ejpam-6039	188	3	subset	subset	NOUN
ejpam-6039	188	4	a	a	PRON
ejpam-6039	188	5	of	of	ADP
ejpam-6039	188	6	a	a	DET
ejpam-6039	188	7	bitopological	bitopological	ADJ
ejpam-6039	188	8	space	space	NOUN
ejpam-6039	188	9	(	(	PUNCT
ejpam-6039	188	10	x	x	NOUN
ejpam-6039	188	11	,	,	PUNCT
ejpam-6039	188	12	τ1	τ1	NOUN
ejpam-6039	188	13	,	,	PUNCT
ejpam-6039	188	14	τ2	τ2	NOUN
ejpam-6039	188	15	)	)	PUNCT
ejpam-6039	188	16	,	,	PUNCT
ejpam-6039	188	17	the	the	DET
ejpam-6039	188	18	following	follow	VERB
ejpam-6039	188	19	properties	property	NOUN
ejpam-6039	188	20	are	be	AUX
ejpam-6039	188	21	equivalent	equivalent	ADJ
ejpam-6039	188	22	:	:	PUNCT
ejpam-6039	188	23	(	(	PUNCT
ejpam-6039	188	24	1	1	X
ejpam-6039	188	25	)	)	PUNCT
ejpam-6039	188	26	a	a	PRON
ejpam-6039	188	27	is	is	NOUN
ejpam-6039	188	28	(	(	PUNCT
ejpam-6039	188	29	τ1	τ1	NOUN
ejpam-6039	188	30	,	,	PUNCT
ejpam-6039	188	31	τ2)r	τ2)r	NOUN
ejpam-6039	188	32	-	-	PUNCT
ejpam-6039	188	33	closed	closed	ADJ
ejpam-6039	188	34	;	;	PUNCT
ejpam-6039	188	35	(	(	PUNCT
ejpam-6039	188	36	2	2	X
ejpam-6039	188	37	)	)	PUNCT
ejpam-6039	188	38	a	a	PRON
ejpam-6039	188	39	is	is	NOUN
ejpam-6039	188	40	(	(	PUNCT
ejpam-6039	188	41	τ1	τ1	NOUN
ejpam-6039	188	42	,	,	PUNCT
ejpam-6039	188	43	τ2)p	τ2)p	NOUN
ejpam-6039	188	44	-	-	PUNCT
ejpam-6039	188	45	closed	closed	ADJ
ejpam-6039	188	46	and	and	CCONJ
ejpam-6039	188	47	(	(	PUNCT
ejpam-6039	188	48	τ1	τ1	NOUN
ejpam-6039	188	49	,	,	PUNCT
ejpam-6039	188	50	τ2)s	τ2)s	NOUN
ejpam-6039	188	51	-	-	PUNCT
ejpam-6039	188	52	open	open	ADJ
ejpam-6039	188	53	;	;	PUNCT
ejpam-6039	188	54	(	(	PUNCT
ejpam-6039	188	55	3	3	X
ejpam-6039	188	56	)	)	PUNCT
ejpam-6039	188	57	a	a	PRON
ejpam-6039	188	58	is	be	AUX
ejpam-6039	188	59	α(τ1	α(τ1	NOUN
ejpam-6039	188	60	,	,	PUNCT
ejpam-6039	188	61	τ2)-closed	τ2)-closed	ADJ
ejpam-6039	188	62	and	and	CCONJ
ejpam-6039	188	63	(	(	PUNCT
ejpam-6039	188	64	τ1	τ1	NOUN
ejpam-6039	188	65	,	,	PUNCT
ejpam-6039	188	66	τ2)β	τ2)β	ADJ
ejpam-6039	188	67	-	-	PUNCT
ejpam-6039	188	68	open	open	ADJ
ejpam-6039	188	69	.	.	PUNCT
ejpam-6039	189	1	proof	proof	NOUN
ejpam-6039	189	2	.	.	PUNCT
ejpam-6039	190	1	(	(	PUNCT
ejpam-6039	190	2	1	1	X
ejpam-6039	190	3	)	)	PUNCT
ejpam-6039	190	4	⇒	⇒	NOUN
ejpam-6039	190	5	(	(	PUNCT
ejpam-6039	190	6	2	2	NUM
ejpam-6039	190	7	):	):	PUNCT
ejpam-6039	190	8	let	let	VERB
ejpam-6039	190	9	a	a	DET
ejpam-6039	190	10	be	be	AUX
ejpam-6039	190	11	(	(	PUNCT
ejpam-6039	190	12	τ1	τ1	NOUN
ejpam-6039	190	13	,	,	PUNCT
ejpam-6039	190	14	τ2)r	τ2)r	NOUN
ejpam-6039	190	15	-	-	PUNCT
ejpam-6039	190	16	closed	closed	ADJ
ejpam-6039	190	17	.	.	PUNCT
ejpam-6039	191	1	then	then	ADV
ejpam-6039	191	2	,	,	PUNCT
ejpam-6039	191	3	we	we	PRON
ejpam-6039	191	4	have	have	VERB
ejpam-6039	191	5	a	a	DET
ejpam-6039	191	6	=	=	ADJ
ejpam-6039	191	7	τ1τ2	τ1τ2	NOUN
ejpam-6039	191	8	-	-	ADJ
ejpam-6039	191	9	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	191	10	-	-	PUNCT
ejpam-6039	191	11	int(a	int(a	NOUN
ejpam-6039	191	12	)	)	PUNCT
ejpam-6039	191	13	)	)	PUNCT
ejpam-6039	191	14	.	.	PUNCT
ejpam-6039	192	1	thus	thus	ADV
ejpam-6039	192	2	,	,	PUNCT
ejpam-6039	192	3	a	a	PRON
ejpam-6039	192	4	is	is	NOUN
ejpam-6039	192	5	(	(	PUNCT
ejpam-6039	192	6	τ1	τ1	NOUN
ejpam-6039	192	7	,	,	PUNCT
ejpam-6039	192	8	τ2)p	τ2)p	NOUN
ejpam-6039	192	9	-	-	PUNCT
ejpam-6039	192	10	closed	closed	ADJ
ejpam-6039	192	11	and	and	CCONJ
ejpam-6039	192	12	(	(	PUNCT
ejpam-6039	192	13	τ1	τ1	NOUN
ejpam-6039	192	14	,	,	PUNCT
ejpam-6039	192	15	τ2)s	τ2)s	NOUN
ejpam-6039	192	16	-	-	PUNCT
ejpam-6039	192	17	open	open	ADJ
ejpam-6039	192	18	.	.	PUNCT
ejpam-6039	193	1	(	(	PUNCT
ejpam-6039	193	2	2	2	X
ejpam-6039	193	3	)	)	PUNCT
ejpam-6039	193	4	⇒	⇒	NOUN
ejpam-6039	193	5	(	(	PUNCT
ejpam-6039	193	6	3	3	NUM
ejpam-6039	193	7	):	):	PUNCT
ejpam-6039	193	8	let	let	VERB
ejpam-6039	193	9	a	a	DET
ejpam-6039	193	10	be	be	AUX
ejpam-6039	193	11	(	(	PUNCT
ejpam-6039	193	12	τ1	τ1	NOUN
ejpam-6039	193	13	,	,	PUNCT
ejpam-6039	193	14	τ2)p	τ2)p	NOUN
ejpam-6039	193	15	-	-	PUNCT
ejpam-6039	193	16	closed	closed	ADJ
ejpam-6039	193	17	and	and	CCONJ
ejpam-6039	193	18	(	(	PUNCT
ejpam-6039	193	19	τ1	τ1	NOUN
ejpam-6039	193	20	,	,	PUNCT
ejpam-6039	193	21	τ2)s	τ2)s	NOUN
ejpam-6039	193	22	-	-	PUNCT
ejpam-6039	193	23	open	open	ADJ
ejpam-6039	193	24	.	.	PUNCT
ejpam-6039	194	1	then	then	ADV
ejpam-6039	194	2	,	,	PUNCT
ejpam-6039	194	3	τ1τ2	τ1τ2	NOUN
ejpam-6039	194	4	-	-	ADJ
ejpam-6039	194	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	194	6	-	-	PUNCT
ejpam-6039	194	7	int(a	int(a	NOUN
ejpam-6039	194	8	)	)	PUNCT
ejpam-6039	194	9	)	)	PUNCT
ejpam-6039	195	1	⊆	⊆	NUM
ejpam-6039	195	2	a	a	PRON
ejpam-6039	195	3	and	and	CCONJ
ejpam-6039	195	4	a	a	DET
ejpam-6039	195	5	⊆	⊆	NUM
ejpam-6039	195	6	τ1τ2	τ1τ2	NOUN
ejpam-6039	195	7	-	-	ADJ
ejpam-6039	195	8	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	195	9	-	-	PUNCT
ejpam-6039	195	10	int(a	int(a	NOUN
ejpam-6039	195	11	)	)	PUNCT
ejpam-6039	195	12	)	)	PUNCT
ejpam-6039	195	13	.	.	PUNCT
ejpam-6039	196	1	thus	thus	ADV
ejpam-6039	196	2	,	,	PUNCT
ejpam-6039	196	3	τ1τ2	τ1τ2	NOUN
ejpam-6039	196	4	-	-	ADJ
ejpam-6039	196	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	196	6	-	-	PUNCT
ejpam-6039	196	7	int(a	int(a	NOUN
ejpam-6039	196	8	)	)	PUNCT
ejpam-6039	196	9	)	)	PUNCT
ejpam-6039	197	1	=	=	PUNCT
ejpam-6039	197	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	197	3	-	-	NUM
ejpam-6039	197	4	cl(a	cl(a	NUM
ejpam-6039	197	5	)	)	PUNCT
ejpam-6039	197	6	and	and	CCONJ
ejpam-6039	197	7	hence	hence	ADV
ejpam-6039	197	8	τ1τ2	τ1τ2	ADJ
ejpam-6039	197	9	-	-	ADJ
ejpam-6039	197	10	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	197	11	-	-	PUNCT
ejpam-6039	197	12	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	197	13	-	-	PUNCT
ejpam-6039	197	14	cl(a	cl(a	NUM
ejpam-6039	197	15	)	)	PUNCT
ejpam-6039	197	16	)	)	PUNCT
ejpam-6039	197	17	)	)	PUNCT
ejpam-6039	198	1	=	=	PUNCT
ejpam-6039	198	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	198	3	-	-	ADJ
ejpam-6039	198	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	198	5	-	-	PUNCT
ejpam-6039	198	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	198	7	-	-	PUNCT
ejpam-6039	198	8	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	198	9	-	-	PUNCT
ejpam-6039	198	10	int(a	int(a	NOUN
ejpam-6039	198	11	)	)	PUNCT
ejpam-6039	198	12	)	)	PUNCT
ejpam-6039	198	13	)	)	PUNCT
ejpam-6039	198	14	)	)	PUNCT
ejpam-6039	199	1	=	=	PUNCT
ejpam-6039	199	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	199	3	-	-	ADJ
ejpam-6039	199	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	199	5	-	-	PUNCT
ejpam-6039	199	6	int(a	int(a	NOUN
ejpam-6039	199	7	)	)	PUNCT
ejpam-6039	199	8	)	)	PUNCT
ejpam-6039	200	1	⊆	⊆	NUM
ejpam-6039	200	2	a.	a.	NOUN
ejpam-6039	200	3	this	this	PRON
ejpam-6039	200	4	shows	show	VERB
ejpam-6039	200	5	that	that	SCONJ
ejpam-6039	200	6	a	a	PRON
ejpam-6039	200	7	is	be	AUX
ejpam-6039	200	8	α(τ1	α(τ1	NOUN
ejpam-6039	200	9	,	,	PUNCT
ejpam-6039	200	10	τ2)-closed	τ2)-closed	ADJ
ejpam-6039	200	11	.	.	PUNCT
ejpam-6039	201	1	it	it	PRON
ejpam-6039	201	2	is	be	AUX
ejpam-6039	201	3	obvious	obvious	ADJ
ejpam-6039	201	4	that	that	SCONJ
ejpam-6039	201	5	a	a	PRON
ejpam-6039	201	6	is	be	AUX
ejpam-6039	201	7	(	(	PUNCT
ejpam-6039	201	8	τ1	τ1	NOUN
ejpam-6039	201	9	,	,	PUNCT
ejpam-6039	201	10	τ2)β	τ2)β	ADJ
ejpam-6039	201	11	-	-	PUNCT
ejpam-6039	201	12	open	open	ADJ
ejpam-6039	201	13	.	.	PUNCT
ejpam-6039	202	1	(	(	PUNCT
ejpam-6039	202	2	3	3	X
ejpam-6039	202	3	)	)	PUNCT
ejpam-6039	202	4	⇒	⇒	NOUN
ejpam-6039	202	5	(	(	PUNCT
ejpam-6039	202	6	1	1	NUM
ejpam-6039	202	7	):	):	PUNCT
ejpam-6039	202	8	let	let	VERB
ejpam-6039	202	9	a	a	DET
ejpam-6039	202	10	be	be	AUX
ejpam-6039	202	11	α(τ1	α(τ1	NOUN
ejpam-6039	202	12	,	,	PUNCT
ejpam-6039	202	13	τ2)-closed	τ2)-closed	ADJ
ejpam-6039	202	14	and	and	CCONJ
ejpam-6039	202	15	(	(	PUNCT
ejpam-6039	202	16	τ1	τ1	NOUN
ejpam-6039	202	17	,	,	PUNCT
ejpam-6039	202	18	τ2)β	τ2)β	ADJ
ejpam-6039	202	19	-	-	PUNCT
ejpam-6039	202	20	open	open	ADJ
ejpam-6039	202	21	.	.	PUNCT
ejpam-6039	203	1	then	then	ADV
ejpam-6039	203	2	,	,	PUNCT
ejpam-6039	203	3	we	we	PRON
ejpam-6039	203	4	have	have	VERB
ejpam-6039	203	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	203	6	-	-	ADJ
ejpam-6039	203	7	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	203	8	-	-	PUNCT
ejpam-6039	203	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	203	10	-	-	PUNCT
ejpam-6039	203	11	cl(a	cl(a	NUM
ejpam-6039	203	12	)	)	PUNCT
ejpam-6039	203	13	)	)	PUNCT
ejpam-6039	203	14	)	)	PUNCT
ejpam-6039	204	1	⊆	⊆	NUM
ejpam-6039	204	2	a	a	PRON
ejpam-6039	204	3	and	and	CCONJ
ejpam-6039	204	4	a	a	DET
ejpam-6039	204	5	⊆	⊆	NUM
ejpam-6039	204	6	τ1τ2	τ1τ2	NOUN
ejpam-6039	204	7	-	-	PUNCT
ejpam-6039	204	8	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	204	9	-	-	PUNCT
ejpam-6039	204	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	204	11	-	-	PUNCT
ejpam-6039	204	12	cl(a	cl(a	NUM
ejpam-6039	204	13	)	)	PUNCT
ejpam-6039	204	14	)	)	PUNCT
ejpam-6039	204	15	)	)	PUNCT
ejpam-6039	204	16	.	.	PUNCT
ejpam-6039	205	1	thus	thus	ADV
ejpam-6039	205	2	,	,	PUNCT
ejpam-6039	205	3	a	a	DET
ejpam-6039	205	4	=	=	ADJ
ejpam-6039	205	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	205	6	-	-	ADJ
ejpam-6039	205	7	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	205	8	-	-	PUNCT
ejpam-6039	205	9	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	205	10	-	-	PUNCT
ejpam-6039	205	11	cl(a	cl(a	NUM
ejpam-6039	205	12	)	)	PUNCT
ejpam-6039	205	13	)	)	PUNCT
ejpam-6039	205	14	)	)	PUNCT
ejpam-6039	205	15	and	and	CCONJ
ejpam-6039	205	16	hence	hence	ADV
ejpam-6039	205	17	τ1τ2	τ1τ2	NOUN
ejpam-6039	205	18	-	-	ADJ
ejpam-6039	205	19	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	205	20	-	-	PUNCT
ejpam-6039	205	21	int(a	int(a	NOUN
ejpam-6039	205	22	)	)	PUNCT
ejpam-6039	205	23	)	)	PUNCT
ejpam-6039	206	1	=	=	PUNCT
ejpam-6039	206	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	206	3	-	-	ADJ
ejpam-6039	206	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	206	5	-	-	PUNCT
ejpam-6039	206	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	206	7	-	-	PUNCT
ejpam-6039	206	8	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	206	9	-	-	PUNCT
ejpam-6039	206	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	206	11	-	-	PUNCT
ejpam-6039	206	12	cl(a	cl(a	NUM
ejpam-6039	206	13	)	)	PUNCT
ejpam-6039	206	14	)	)	PUNCT
ejpam-6039	206	15	)	)	PUNCT
ejpam-6039	206	16	)	)	PUNCT
ejpam-6039	206	17	)	)	PUNCT
ejpam-6039	207	1	=	=	PUNCT
ejpam-6039	207	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	207	3	-	-	ADJ
ejpam-6039	207	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	207	5	-	-	PUNCT
ejpam-6039	207	6	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	207	7	-	-	PUNCT
ejpam-6039	207	8	cl(a	cl(a	NUM
ejpam-6039	207	9	)	)	PUNCT
ejpam-6039	207	10	)	)	PUNCT
ejpam-6039	207	11	)	)	PUNCT
ejpam-6039	208	1	=	=	PUNCT
ejpam-6039	208	2	a.	a.	NOUN
ejpam-6039	208	3	therefore	therefore	ADV
ejpam-6039	208	4	,	,	PUNCT
ejpam-6039	208	5	a	a	PRON
ejpam-6039	208	6	is	is	NOUN
ejpam-6039	208	7	(	(	PUNCT
ejpam-6039	208	8	τ1	τ1	NOUN
ejpam-6039	208	9	,	,	PUNCT
ejpam-6039	208	10	τ2)r	τ2)r	NOUN
ejpam-6039	208	11	-	-	PUNCT
ejpam-6039	208	12	closed	closed	ADJ
ejpam-6039	208	13	.	.	PUNCT
ejpam-6039	209	1	as	as	ADP
ejpam-6039	209	2	a	a	DET
ejpam-6039	209	3	consequence	consequence	NOUN
ejpam-6039	209	4	of	of	ADP
ejpam-6039	209	5	lemma	lemma	PROPN
ejpam-6039	209	6	6	6	NUM
ejpam-6039	209	7	,	,	PUNCT
ejpam-6039	209	8	we	we	PRON
ejpam-6039	209	9	have	have	VERB
ejpam-6039	209	10	the	the	DET
ejpam-6039	209	11	following	follow	VERB
ejpam-6039	209	12	result	result	NOUN
ejpam-6039	209	13	:	:	PUNCT
ejpam-6039	209	14	theorem	theorem	VERB
ejpam-6039	209	15	7	7	NUM
ejpam-6039	209	16	.	.	X
ejpam-6039	209	17	for	for	ADP
ejpam-6039	209	18	a	a	DET
ejpam-6039	209	19	f	f	NOUN
ejpam-6039	209	20	:	:	PUNCT
ejpam-6039	209	21	(	(	PUNCT
ejpam-6039	209	22	x	x	NOUN
ejpam-6039	209	23	,	,	PUNCT
ejpam-6039	209	24	τ1	τ1	NOUN
ejpam-6039	209	25	,	,	PUNCT
ejpam-6039	209	26	τ2	τ2	NOUN
ejpam-6039	209	27	)	)	PUNCT
ejpam-6039	209	28	→	→	SYM
ejpam-6039	209	29	(	(	PUNCT
ejpam-6039	209	30	y	y	PROPN
ejpam-6039	209	31	,	,	PUNCT
ejpam-6039	209	32	σ1	σ1	PROPN
ejpam-6039	209	33	,	,	PUNCT
ejpam-6039	209	34	σ2	σ2	NOUN
ejpam-6039	209	35	)	)	PUNCT
ejpam-6039	209	36	,	,	PUNCT
ejpam-6039	209	37	the	the	DET
ejpam-6039	209	38	following	follow	VERB
ejpam-6039	209	39	properties	property	NOUN
ejpam-6039	209	40	are	be	AUX
ejpam-6039	209	41	equivalent	equivalent	ADJ
ejpam-6039	209	42	:	:	PUNCT
ejpam-6039	209	43	m.	m.	NOUN
ejpam-6039	209	44	chiangpradit	chiangpradit	NOUN
ejpam-6039	209	45	,	,	PUNCT
ejpam-6039	209	46	s.	s.	PROPN
ejpam-6039	209	47	sompong	sompong	PROPN
ejpam-6039	209	48	,	,	PUNCT
ejpam-6039	209	49	c.	c.	PROPN
ejpam-6039	209	50	boonpok	boonpok	PROPN
ejpam-6039	209	51	/	/	SYM
ejpam-6039	209	52	eur	eur	PROPN
ejpam-6039	209	53	.	.	PUNCT
ejpam-6039	210	1	j.	j.	PROPN
ejpam-6039	210	2	pure	pure	PROPN
ejpam-6039	210	3	appl	appl	PROPN
ejpam-6039	210	4	.	.	PROPN
ejpam-6039	210	5	math	math	PROPN
ejpam-6039	210	6	,	,	PUNCT
ejpam-6039	210	7	18	18	NUM
ejpam-6039	210	8	(	(	PUNCT
ejpam-6039	210	9	2	2	NUM
ejpam-6039	210	10	)	)	PUNCT
ejpam-6039	210	11	(	(	PUNCT
ejpam-6039	210	12	2025	2025	NUM
ejpam-6039	210	13	)	)	PUNCT
ejpam-6039	210	14	,	,	PUNCT
ejpam-6039	210	15	6039	6039	NUM
ejpam-6039	210	16	8	8	NUM
ejpam-6039	210	17	of	of	ADP
ejpam-6039	210	18	11	11	NUM
ejpam-6039	210	19	(	(	PUNCT
ejpam-6039	210	20	1	1	NUM
ejpam-6039	210	21	)	)	PUNCT
ejpam-6039	210	22	f	f	PROPN
ejpam-6039	210	23	is	be	AUX
ejpam-6039	210	24	rc-(τ1	rc-(τ1	NOUN
ejpam-6039	210	25	,	,	PUNCT
ejpam-6039	210	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	210	27	;	;	PUNCT
ejpam-6039	210	28	(	(	PUNCT
ejpam-6039	210	29	2	2	X
ejpam-6039	210	30	)	)	PUNCT
ejpam-6039	210	31	f	f	PROPN
ejpam-6039	210	32	is	be	AUX
ejpam-6039	210	33	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	210	34	,	,	PUNCT
ejpam-6039	210	35	τ2)p	τ2)p	ADJ
ejpam-6039	210	36	-	-	ADJ
ejpam-6039	210	37	continuous	continuous	ADJ
ejpam-6039	210	38	and	and	CCONJ
ejpam-6039	210	39	(	(	PUNCT
ejpam-6039	210	40	τ1	τ1	NOUN
ejpam-6039	210	41	,	,	PUNCT
ejpam-6039	210	42	τ2)s	τ2)s	NOUN
ejpam-6039	210	43	-	-	PUNCT
ejpam-6039	210	44	continuous	continuous	ADJ
ejpam-6039	210	45	;	;	PUNCT
ejpam-6039	210	46	(	(	PUNCT
ejpam-6039	210	47	3	3	X
ejpam-6039	210	48	)	)	PUNCT
ejpam-6039	210	49	f	f	PROPN
ejpam-6039	210	50	is	be	AUX
ejpam-6039	210	51	contra	contra	PROPN
ejpam-6039	210	52	-	-	PUNCT
ejpam-6039	210	53	α(τ1	α(τ1	NOUN
ejpam-6039	210	54	,	,	PUNCT
ejpam-6039	210	55	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	210	56	and	and	CCONJ
ejpam-6039	210	57	(	(	PUNCT
ejpam-6039	210	58	τ1	τ1	NOUN
ejpam-6039	210	59	,	,	PUNCT
ejpam-6039	210	60	τ2)β	τ2)β	ADJ
ejpam-6039	210	61	-	-	PUNCT
ejpam-6039	210	62	continuous	continuous	ADJ
ejpam-6039	210	63	.	.	PUNCT
ejpam-6039	211	1	definition	definition	NOUN
ejpam-6039	211	2	11	11	NUM
ejpam-6039	211	3	.	.	PUNCT
ejpam-6039	212	1	[	[	X
ejpam-6039	212	2	28	28	NUM
ejpam-6039	212	3	]	]	X
ejpam-6039	212	4	a	a	DET
ejpam-6039	212	5	bitopological	bitopological	ADJ
ejpam-6039	212	6	space	space	NOUN
ejpam-6039	212	7	(	(	PUNCT
ejpam-6039	212	8	x	x	NOUN
ejpam-6039	212	9	,	,	PUNCT
ejpam-6039	212	10	τ1	τ1	NOUN
ejpam-6039	212	11	,	,	PUNCT
ejpam-6039	212	12	τ2	τ2	NOUN
ejpam-6039	212	13	)	)	PUNCT
ejpam-6039	212	14	is	be	AUX
ejpam-6039	212	15	said	say	VERB
ejpam-6039	212	16	to	to	PART
ejpam-6039	212	17	be	be	AUX
ejpam-6039	212	18	τ1τ2	τ1τ2	NOUN
ejpam-6039	212	19	-	-	ADJ
ejpam-6039	212	20	urysohn	urysohn	ADJ
ejpam-6039	212	21	if	if	SCONJ
ejpam-6039	212	22	for	for	ADP
ejpam-6039	212	23	each	each	DET
ejpam-6039	212	24	pair	pair	NOUN
ejpam-6039	212	25	of	of	ADP
ejpam-6039	212	26	distinct	distinct	ADJ
ejpam-6039	212	27	points	point	NOUN
ejpam-6039	212	28	x	x	PUNCT
ejpam-6039	212	29	and	and	CCONJ
ejpam-6039	212	30	y	y	PROPN
ejpam-6039	212	31	in	in	ADP
ejpam-6039	212	32	x	x	SYM
ejpam-6039	212	33	,	,	PUNCT
ejpam-6039	212	34	there	there	PRON
ejpam-6039	212	35	exist	exist	VERB
ejpam-6039	212	36	τ1τ2	τ1τ2	ADJ
ejpam-6039	212	37	-	-	ADJ
ejpam-6039	212	38	open	open	ADJ
ejpam-6039	212	39	sets	set	NOUN
ejpam-6039	212	40	u	u	NOUN
ejpam-6039	212	41	and	and	CCONJ
ejpam-6039	212	42	v	v	ADP
ejpam-6039	212	43	such	such	ADJ
ejpam-6039	212	44	that	that	SCONJ
ejpam-6039	212	45	x	x	SYM
ejpam-6039	212	46	∈	∈	PROPN
ejpam-6039	212	47	u	u	NOUN
ejpam-6039	212	48	,	,	PUNCT
ejpam-6039	212	49	y	y	PROPN
ejpam-6039	212	50	∈	∈	PROPN
ejpam-6039	212	51	v	v	NOUN
ejpam-6039	212	52	and	and	CCONJ
ejpam-6039	212	53	τ1τ2	τ1τ2	NOUN
ejpam-6039	212	54	-	-	NOUN
ejpam-6039	212	55	cl(u	cl(u	NOUN
ejpam-6039	212	56	)	)	PUNCT
ejpam-6039	212	57	∩	∩	NOUN
ejpam-6039	212	58	τ1τ2	τ1τ2	NOUN
ejpam-6039	212	59	-	-	NOUN
ejpam-6039	212	60	cl(v	cl(v	X
ejpam-6039	212	61	)	)	PUNCT
ejpam-6039	212	62	=	=	PUNCT
ejpam-6039	212	63	∅.	∅.	PRON
ejpam-6039	212	64	definition	definition	NOUN
ejpam-6039	212	65	12	12	NUM
ejpam-6039	212	66	.	.	PUNCT
ejpam-6039	213	1	a	a	DET
ejpam-6039	213	2	function	function	NOUN
ejpam-6039	213	3	f	f	NOUN
ejpam-6039	213	4	:	:	PUNCT
ejpam-6039	213	5	(	(	PUNCT
ejpam-6039	213	6	x	x	NOUN
ejpam-6039	213	7	,	,	PUNCT
ejpam-6039	213	8	τ1	τ1	NOUN
ejpam-6039	213	9	,	,	PUNCT
ejpam-6039	213	10	τ2	τ2	NOUN
ejpam-6039	213	11	)	)	PUNCT
ejpam-6039	213	12	→	→	SYM
ejpam-6039	213	13	(	(	PUNCT
ejpam-6039	213	14	y	y	PROPN
ejpam-6039	213	15	,	,	PUNCT
ejpam-6039	213	16	σ1	σ1	PROPN
ejpam-6039	213	17	,	,	PUNCT
ejpam-6039	213	18	σ2	σ2	PROPN
ejpam-6039	213	19	)	)	PUNCT
ejpam-6039	213	20	is	be	AUX
ejpam-6039	213	21	said	say	VERB
ejpam-6039	213	22	to	to	PART
ejpam-6039	213	23	have	have	VERB
ejpam-6039	213	24	a	a	DET
ejpam-6039	213	25	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	213	26	,	,	PUNCT
ejpam-6039	213	27	τ2)pclosed	τ2)pclose	VERB
ejpam-6039	213	28	graph	graph	NOUN
ejpam-6039	213	29	if	if	SCONJ
ejpam-6039	213	30	for	for	ADP
ejpam-6039	213	31	each	each	DET
ejpam-6039	213	32	(	(	PUNCT
ejpam-6039	213	33	x	x	NOUN
ejpam-6039	213	34	,	,	PUNCT
ejpam-6039	213	35	y	y	NOUN
ejpam-6039	213	36	)	)	PUNCT
ejpam-6039	213	37	∈	∈	PROPN
ejpam-6039	213	38	(	(	PUNCT
ejpam-6039	213	39	x	x	SYM
ejpam-6039	213	40	×	×	PROPN
ejpam-6039	213	41	y	y	PROPN
ejpam-6039	213	42	)	)	PUNCT
ejpam-6039	213	43	−g(f	−g(f	PROPN
ejpam-6039	213	44	)	)	PUNCT
ejpam-6039	213	45	,	,	PUNCT
ejpam-6039	213	46	there	there	PRON
ejpam-6039	213	47	exists	exist	VERB
ejpam-6039	213	48	a	a	DET
ejpam-6039	213	49	(	(	PUNCT
ejpam-6039	213	50	τ1	τ1	NOUN
ejpam-6039	213	51	,	,	PUNCT
ejpam-6039	213	52	τ2)p	τ2)p	ADJ
ejpam-6039	213	53	-	-	PUNCT
ejpam-6039	213	54	open	open	ADJ
ejpam-6039	213	55	set	set	NOUN
ejpam-6039	213	56	u	u	NOUN
ejpam-6039	213	57	of	of	ADP
ejpam-6039	213	58	x	x	PUNCT
ejpam-6039	213	59	containing	contain	VERB
ejpam-6039	213	60	x	x	X
ejpam-6039	213	61	and	and	CCONJ
ejpam-6039	213	62	a	a	DET
ejpam-6039	213	63	σ1σ2	σ1σ2	NUM
ejpam-6039	213	64	-	-	PUNCT
ejpam-6039	213	65	closed	closed	ADJ
ejpam-6039	213	66	set	set	NOUN
ejpam-6039	213	67	k	k	PROPN
ejpam-6039	213	68	of	of	ADP
ejpam-6039	213	69	y	y	PROPN
ejpam-6039	213	70	containing	contain	VERB
ejpam-6039	213	71	y	y	PRON
ejpam-6039	213	72	such	such	ADJ
ejpam-6039	213	73	that	that	SCONJ
ejpam-6039	213	74	(	(	PUNCT
ejpam-6039	213	75	u	u	NOUN
ejpam-6039	213	76	×k	×k	NOUN
ejpam-6039	213	77	)	)	PUNCT
ejpam-6039	213	78	∩g(f	∩g(f	PROPN
ejpam-6039	213	79	)	)	PUNCT
ejpam-6039	213	80	=	=	PUNCT
ejpam-6039	214	1	∅.	∅.	PRON
ejpam-6039	214	2	lemma	lemma	PROPN
ejpam-6039	214	3	7	7	NUM
ejpam-6039	214	4	.	.	PUNCT
ejpam-6039	215	1	a	a	DET
ejpam-6039	215	2	function	function	NOUN
ejpam-6039	215	3	f	f	NOUN
ejpam-6039	215	4	:	:	PUNCT
ejpam-6039	215	5	(	(	PUNCT
ejpam-6039	215	6	x	x	NOUN
ejpam-6039	215	7	,	,	PUNCT
ejpam-6039	215	8	τ1	τ1	NOUN
ejpam-6039	215	9	,	,	PUNCT
ejpam-6039	215	10	τ2	τ2	NOUN
ejpam-6039	215	11	)	)	PUNCT
ejpam-6039	215	12	→	→	SYM
ejpam-6039	215	13	(	(	PUNCT
ejpam-6039	215	14	y	y	PROPN
ejpam-6039	215	15	,	,	PUNCT
ejpam-6039	215	16	σ1	σ1	PROPN
ejpam-6039	215	17	,	,	PUNCT
ejpam-6039	215	18	σ2	σ2	NOUN
ejpam-6039	215	19	)	)	PUNCT
ejpam-6039	215	20	has	have	VERB
ejpam-6039	215	21	a	a	DET
ejpam-6039	215	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	215	23	,	,	PUNCT
ejpam-6039	215	24	τ2)p	τ2)p	ADJ
ejpam-6039	215	25	-	-	PUNCT
ejpam-6039	215	26	closed	closed	ADJ
ejpam-6039	215	27	graph	graph	NOUN
ejpam-6039	215	28	if	if	SCONJ
ejpam-6039	215	29	and	and	CCONJ
ejpam-6039	215	30	only	only	ADV
ejpam-6039	215	31	if	if	SCONJ
ejpam-6039	215	32	for	for	ADP
ejpam-6039	215	33	each	each	DET
ejpam-6039	215	34	(	(	PUNCT
ejpam-6039	215	35	x	x	NOUN
ejpam-6039	215	36	,	,	PUNCT
ejpam-6039	215	37	y	y	NOUN
ejpam-6039	215	38	)	)	PUNCT
ejpam-6039	215	39	∈	∈	PROPN
ejpam-6039	215	40	(	(	PUNCT
ejpam-6039	215	41	x	x	SYM
ejpam-6039	215	42	×	×	PROPN
ejpam-6039	215	43	y	y	PROPN
ejpam-6039	215	44	)	)	PUNCT
ejpam-6039	215	45	−	−	PROPN
ejpam-6039	215	46	g(f	g(f	NOUN
ejpam-6039	215	47	)	)	PUNCT
ejpam-6039	215	48	,	,	PUNCT
ejpam-6039	215	49	there	there	PRON
ejpam-6039	215	50	exists	exist	VERB
ejpam-6039	215	51	a	a	DET
ejpam-6039	215	52	(	(	PUNCT
ejpam-6039	215	53	τ1	τ1	NOUN
ejpam-6039	215	54	,	,	PUNCT
ejpam-6039	215	55	τ2)p	τ2)p	ADJ
ejpam-6039	215	56	-	-	PUNCT
ejpam-6039	215	57	open	open	ADJ
ejpam-6039	215	58	set	set	NOUN
ejpam-6039	215	59	u	u	NOUN
ejpam-6039	215	60	of	of	ADP
ejpam-6039	215	61	x	x	PUNCT
ejpam-6039	215	62	containing	contain	VERB
ejpam-6039	215	63	x	x	X
ejpam-6039	215	64	and	and	CCONJ
ejpam-6039	215	65	a	a	DET
ejpam-6039	215	66	σ1σ2	σ1σ2	NUM
ejpam-6039	215	67	-	-	PUNCT
ejpam-6039	215	68	closed	closed	ADJ
ejpam-6039	215	69	set	set	NOUN
ejpam-6039	215	70	k	k	PROPN
ejpam-6039	215	71	of	of	ADP
ejpam-6039	215	72	y	y	PROPN
ejpam-6039	215	73	containing	contain	VERB
ejpam-6039	215	74	y	y	PRON
ejpam-6039	215	75	such	such	ADJ
ejpam-6039	215	76	that	that	DET
ejpam-6039	215	77	f(u	f(u	PROPN
ejpam-6039	215	78	)	)	PUNCT
ejpam-6039	215	79	∩k	∩k	NOUN
ejpam-6039	215	80	=	=	PRON
ejpam-6039	215	81	∅.	∅.	NOUN
ejpam-6039	215	82	theorem	theorem	VERB
ejpam-6039	215	83	8	8	NUM
ejpam-6039	215	84	.	.	PUNCT
ejpam-6039	216	1	if	if	SCONJ
ejpam-6039	216	2	a	a	DET
ejpam-6039	216	3	function	function	NOUN
ejpam-6039	216	4	f	f	X
ejpam-6039	216	5	:	:	PUNCT
ejpam-6039	216	6	(	(	PUNCT
ejpam-6039	216	7	x	x	NOUN
ejpam-6039	216	8	,	,	PUNCT
ejpam-6039	216	9	τ1	τ1	NOUN
ejpam-6039	216	10	,	,	PUNCT
ejpam-6039	216	11	τ2	τ2	NOUN
ejpam-6039	216	12	)	)	PUNCT
ejpam-6039	216	13	→	→	SYM
ejpam-6039	216	14	(	(	PUNCT
ejpam-6039	216	15	y	y	PROPN
ejpam-6039	216	16	,	,	PUNCT
ejpam-6039	216	17	σ1	σ1	PROPN
ejpam-6039	216	18	,	,	PUNCT
ejpam-6039	216	19	σ2	σ2	PROPN
ejpam-6039	216	20	)	)	PUNCT
ejpam-6039	216	21	is	be	AUX
ejpam-6039	216	22	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	216	23	,	,	PUNCT
ejpam-6039	216	24	τ2)p	τ2)p	ADJ
ejpam-6039	216	25	-	-	ADJ
ejpam-6039	216	26	continuous	continuous	ADJ
ejpam-6039	216	27	and	and	CCONJ
ejpam-6039	216	28	(	(	PUNCT
ejpam-6039	216	29	y	y	PROPN
ejpam-6039	216	30	,	,	PUNCT
ejpam-6039	216	31	σ1	σ1	PROPN
ejpam-6039	216	32	,	,	PUNCT
ejpam-6039	216	33	σ2	σ2	PROPN
ejpam-6039	216	34	)	)	PUNCT
ejpam-6039	216	35	is	be	AUX
ejpam-6039	216	36	σ1σ2	σ1σ2	NOUN
ejpam-6039	216	37	-	-	PUNCT
ejpam-6039	216	38	urysohn	urysohn	ADJ
ejpam-6039	216	39	,	,	PUNCT
ejpam-6039	216	40	then	then	ADV
ejpam-6039	216	41	g(f	g(f	PROPN
ejpam-6039	216	42	)	)	PUNCT
ejpam-6039	216	43	is	be	AUX
ejpam-6039	216	44	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	216	45	,	,	PUNCT
ejpam-6039	216	46	τ2)p	τ2)p	NOUN
ejpam-6039	216	47	-	-	PUNCT
ejpam-6039	216	48	closed	closed	ADJ
ejpam-6039	216	49	.	.	PUNCT
ejpam-6039	217	1	proof	proof	NOUN
ejpam-6039	217	2	.	.	PUNCT
ejpam-6039	218	1	let	let	VERB
ejpam-6039	218	2	(	(	PUNCT
ejpam-6039	218	3	x	x	NOUN
ejpam-6039	218	4	,	,	PUNCT
ejpam-6039	218	5	y	y	NOUN
ejpam-6039	218	6	)	)	PUNCT
ejpam-6039	218	7	∈	∈	PROPN
ejpam-6039	218	8	(	(	PUNCT
ejpam-6039	218	9	x	x	SYM
ejpam-6039	218	10	×	×	PROPN
ejpam-6039	218	11	y	y	PROPN
ejpam-6039	218	12	)	)	PUNCT
ejpam-6039	219	1	−	−	PROPN
ejpam-6039	219	2	g(f	g(f	NOUN
ejpam-6039	219	3	)	)	PUNCT
ejpam-6039	219	4	.	.	PUNCT
ejpam-6039	220	1	then	then	ADV
ejpam-6039	220	2	,	,	PUNCT
ejpam-6039	220	3	y	y	PROPN
ejpam-6039	220	4	̸=	̸=	PROPN
ejpam-6039	220	5	f(x	f(x	PROPN
ejpam-6039	220	6	)	)	PUNCT
ejpam-6039	220	7	.	.	PUNCT
ejpam-6039	221	1	since	since	SCONJ
ejpam-6039	221	2	(	(	PUNCT
ejpam-6039	221	3	y	y	PROPN
ejpam-6039	221	4	,	,	PUNCT
ejpam-6039	221	5	σ1	σ1	PROPN
ejpam-6039	221	6	,	,	PUNCT
ejpam-6039	221	7	σ2	σ2	PROPN
ejpam-6039	221	8	)	)	PUNCT
ejpam-6039	221	9	is	be	AUX
ejpam-6039	221	10	σ1σ2urysohn	σ1σ2urysohn	NUM
ejpam-6039	221	11	,	,	PUNCT
ejpam-6039	221	12	there	there	PRON
ejpam-6039	221	13	exist	exist	VERB
ejpam-6039	221	14	σ1σ2	σ1σ2	NOUN
ejpam-6039	221	15	-	-	ADJ
ejpam-6039	221	16	open	open	ADJ
ejpam-6039	221	17	sets	set	NOUN
ejpam-6039	221	18	v	v	ADP
ejpam-6039	221	19	and	and	CCONJ
ejpam-6039	221	20	w	w	PROPN
ejpam-6039	221	21	of	of	ADP
ejpam-6039	221	22	y	y	PROPN
ejpam-6039	221	23	containing	contain	VERB
ejpam-6039	221	24	y	y	PROPN
ejpam-6039	221	25	and	and	CCONJ
ejpam-6039	221	26	f(x	f(x	PROPN
ejpam-6039	221	27	)	)	PUNCT
ejpam-6039	221	28	,	,	PUNCT
ejpam-6039	221	29	respectively	respectively	ADV
ejpam-6039	221	30	,	,	PUNCT
ejpam-6039	221	31	such	such	ADJ
ejpam-6039	221	32	that	that	SCONJ
ejpam-6039	221	33	σ1σ2	σ1σ2	NOUN
ejpam-6039	221	34	-	-	PUNCT
ejpam-6039	221	35	cl(v	cl(v	NOUN
ejpam-6039	221	36	)	)	PUNCT
ejpam-6039	221	37	∩	∩	NOUN
ejpam-6039	221	38	σ1σ2	σ1σ2	NOUN
ejpam-6039	221	39	-	-	NUM
ejpam-6039	221	40	cl(w	cl(w	NOUN
ejpam-6039	221	41	)	)	PUNCT
ejpam-6039	222	1	=	=	PUNCT
ejpam-6039	222	2	∅.	∅.	NOUN
ejpam-6039	222	3	since	since	SCONJ
ejpam-6039	222	4	f	f	PROPN
ejpam-6039	222	5	is	be	AUX
ejpam-6039	222	6	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	222	7	,	,	PUNCT
ejpam-6039	222	8	τ2)p	τ2)p	ADJ
ejpam-6039	222	9	-	-	ADJ
ejpam-6039	222	10	continuous	continuous	ADJ
ejpam-6039	222	11	,	,	PUNCT
ejpam-6039	222	12	there	there	PRON
ejpam-6039	222	13	exists	exist	VERB
ejpam-6039	222	14	a	a	DET
ejpam-6039	222	15	(	(	PUNCT
ejpam-6039	222	16	τ1	τ1	NOUN
ejpam-6039	222	17	,	,	PUNCT
ejpam-6039	222	18	τ2)p	τ2)p	ADJ
ejpam-6039	222	19	-	-	PUNCT
ejpam-6039	222	20	open	open	ADJ
ejpam-6039	222	21	set	set	NOUN
ejpam-6039	222	22	u	u	NOUN
ejpam-6039	222	23	of	of	ADP
ejpam-6039	222	24	x	x	PUNCT
ejpam-6039	222	25	containing	contain	VERB
ejpam-6039	222	26	x	x	PUNCT
ejpam-6039	222	27	such	such	ADJ
ejpam-6039	222	28	that	that	DET
ejpam-6039	222	29	f(u	f(u	PROPN
ejpam-6039	222	30	)	)	PUNCT
ejpam-6039	222	31	⊆	⊆	NUM
ejpam-6039	222	32	σ1σ2	σ1σ2	NOUN
ejpam-6039	222	33	-	-	PUNCT
ejpam-6039	222	34	cl(w	cl(w	NOUN
ejpam-6039	222	35	)	)	PUNCT
ejpam-6039	222	36	.	.	PUNCT
ejpam-6039	223	1	thus	thus	ADV
ejpam-6039	223	2	,	,	PUNCT
ejpam-6039	223	3	f(u	f(u	PROPN
ejpam-6039	223	4	)	)	PUNCT
ejpam-6039	223	5	∩	∩	NOUN
ejpam-6039	223	6	σ1σ2	σ1σ2	NOUN
ejpam-6039	223	7	-	-	NUM
ejpam-6039	223	8	cl(v	cl(v	X
ejpam-6039	223	9	)	)	PUNCT
ejpam-6039	223	10	=	=	NOUN
ejpam-6039	223	11	∅	∅	NOUN
ejpam-6039	223	12	and	and	CCONJ
ejpam-6039	223	13	hence	hence	ADV
ejpam-6039	223	14	by	by	ADP
ejpam-6039	223	15	lemma	lemma	PROPN
ejpam-6039	223	16	7	7	NUM
ejpam-6039	223	17	,	,	PUNCT
ejpam-6039	223	18	g(f	g(f	PROPN
ejpam-6039	223	19	)	)	PUNCT
ejpam-6039	223	20	is	be	AUX
ejpam-6039	223	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	223	22	,	,	PUNCT
ejpam-6039	223	23	τ2)p	τ2)p	NOUN
ejpam-6039	223	24	-	-	PUNCT
ejpam-6039	223	25	closed	closed	ADJ
ejpam-6039	223	26	.	.	PUNCT
ejpam-6039	224	1	recall	recall	VERB
ejpam-6039	224	2	that	that	SCONJ
ejpam-6039	224	3	a	a	DET
ejpam-6039	224	4	bitopological	bitopological	ADJ
ejpam-6039	224	5	space	space	NOUN
ejpam-6039	224	6	(	(	PUNCT
ejpam-6039	224	7	x	x	NOUN
ejpam-6039	224	8	,	,	PUNCT
ejpam-6039	224	9	τ1	τ1	NOUN
ejpam-6039	224	10	,	,	PUNCT
ejpam-6039	224	11	τ2	τ2	NOUN
ejpam-6039	224	12	)	)	PUNCT
ejpam-6039	224	13	is	be	AUX
ejpam-6039	224	14	said	say	VERB
ejpam-6039	224	15	to	to	PART
ejpam-6039	224	16	be	be	AUX
ejpam-6039	224	17	τ1τ2	τ1τ2	NOUN
ejpam-6039	224	18	-	-	ADJ
ejpam-6039	224	19	connected	connected	ADJ
ejpam-6039	225	1	[	[	X
ejpam-6039	225	2	21	21	NUM
ejpam-6039	225	3	]	]	X
ejpam-6039	225	4	if	if	SCONJ
ejpam-6039	225	5	x	x	PRON
ejpam-6039	225	6	can	can	AUX
ejpam-6039	225	7	not	not	PART
ejpam-6039	225	8	be	be	AUX
ejpam-6039	225	9	written	write	VERB
ejpam-6039	225	10	as	as	ADP
ejpam-6039	225	11	the	the	DET
ejpam-6039	225	12	union	union	NOUN
ejpam-6039	225	13	of	of	ADP
ejpam-6039	225	14	two	two	NUM
ejpam-6039	225	15	nonempty	nonempty	ADV
ejpam-6039	225	16	disjoint	disjoint	NOUN
ejpam-6039	225	17	τ1τ2	τ1τ2	ADJ
ejpam-6039	225	18	-	-	ADJ
ejpam-6039	225	19	open	open	ADJ
ejpam-6039	225	20	sets	set	NOUN
ejpam-6039	225	21	.	.	PUNCT
ejpam-6039	226	1	definition	definition	NOUN
ejpam-6039	226	2	13	13	NUM
ejpam-6039	226	3	.	.	PUNCT
ejpam-6039	227	1	a	a	DET
ejpam-6039	227	2	bitopological	bitopological	ADJ
ejpam-6039	227	3	space	space	NOUN
ejpam-6039	227	4	(	(	PUNCT
ejpam-6039	227	5	x	x	NOUN
ejpam-6039	227	6	,	,	PUNCT
ejpam-6039	227	7	τ1	τ1	NOUN
ejpam-6039	227	8	,	,	PUNCT
ejpam-6039	227	9	τ2	τ2	NOUN
ejpam-6039	227	10	)	)	PUNCT
ejpam-6039	227	11	is	be	AUX
ejpam-6039	227	12	said	say	VERB
ejpam-6039	227	13	to	to	PART
ejpam-6039	227	14	be	be	AUX
ejpam-6039	227	15	(	(	PUNCT
ejpam-6039	227	16	τ1	τ1	NOUN
ejpam-6039	227	17	,	,	PUNCT
ejpam-6039	227	18	τ2)p	τ2)p	NOUN
ejpam-6039	227	19	-	-	PUNCT
ejpam-6039	227	20	connected	connected	ADJ
ejpam-6039	227	21	if	if	SCONJ
ejpam-6039	227	22	x	x	PRON
ejpam-6039	227	23	can	can	AUX
ejpam-6039	227	24	not	not	PART
ejpam-6039	227	25	be	be	AUX
ejpam-6039	227	26	written	write	VERB
ejpam-6039	227	27	as	as	ADP
ejpam-6039	227	28	the	the	DET
ejpam-6039	227	29	union	union	NOUN
ejpam-6039	227	30	of	of	ADP
ejpam-6039	227	31	two	two	NUM
ejpam-6039	227	32	nonempty	nonempty	ADJ
ejpam-6039	227	33	disjoint	disjoint	NOUN
ejpam-6039	227	34	(	(	PUNCT
ejpam-6039	227	35	τ1	τ1	NOUN
ejpam-6039	227	36	,	,	PUNCT
ejpam-6039	227	37	τ2)p	τ2)p	ADJ
ejpam-6039	227	38	-	-	PUNCT
ejpam-6039	227	39	open	open	ADJ
ejpam-6039	227	40	sets	set	NOUN
ejpam-6039	227	41	.	.	PUNCT
ejpam-6039	228	1	theorem	theorem	NOUN
ejpam-6039	228	2	9	9	NUM
ejpam-6039	228	3	.	.	PUNCT
ejpam-6039	229	1	if	if	SCONJ
ejpam-6039	229	2	f	f	PROPN
ejpam-6039	229	3	:	:	PUNCT
ejpam-6039	229	4	(	(	PUNCT
ejpam-6039	229	5	x	x	NOUN
ejpam-6039	229	6	,	,	PUNCT
ejpam-6039	229	7	τ1	τ1	NOUN
ejpam-6039	229	8	,	,	PUNCT
ejpam-6039	229	9	τ2	τ2	NOUN
ejpam-6039	229	10	)	)	PUNCT
ejpam-6039	229	11	→	→	SYM
ejpam-6039	229	12	(	(	PUNCT
ejpam-6039	229	13	y	y	PROPN
ejpam-6039	229	14	,	,	PUNCT
ejpam-6039	229	15	σ1	σ1	PROPN
ejpam-6039	229	16	,	,	PUNCT
ejpam-6039	229	17	σ2	σ2	PROPN
ejpam-6039	229	18	)	)	PUNCT
ejpam-6039	229	19	is	be	AUX
ejpam-6039	229	20	a	a	DET
ejpam-6039	229	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	229	22	,	,	PUNCT
ejpam-6039	229	23	τ2)p	τ2)p	ADJ
ejpam-6039	229	24	-	-	ADJ
ejpam-6039	229	25	continuous	continuous	ADJ
ejpam-6039	229	26	surjection	surjection	NOUN
ejpam-6039	229	27	and	and	CCONJ
ejpam-6039	229	28	(	(	PUNCT
ejpam-6039	229	29	x	x	NOUN
ejpam-6039	229	30	,	,	PUNCT
ejpam-6039	229	31	τ1	τ1	NOUN
ejpam-6039	229	32	,	,	PUNCT
ejpam-6039	229	33	τ2	τ2	NOUN
ejpam-6039	229	34	)	)	PUNCT
ejpam-6039	229	35	is	be	AUX
ejpam-6039	229	36	(	(	PUNCT
ejpam-6039	229	37	τ1	τ1	NOUN
ejpam-6039	229	38	,	,	PUNCT
ejpam-6039	229	39	τ2)p	τ2)p	NOUN
ejpam-6039	229	40	-	-	PUNCT
ejpam-6039	229	41	connected	connect	VERB
ejpam-6039	229	42	,	,	PUNCT
ejpam-6039	229	43	then	then	ADV
ejpam-6039	229	44	(	(	PUNCT
ejpam-6039	229	45	y	y	PROPN
ejpam-6039	229	46	,	,	PUNCT
ejpam-6039	229	47	σ1	σ1	PROPN
ejpam-6039	229	48	,	,	PUNCT
ejpam-6039	229	49	σ2	σ2	PROPN
ejpam-6039	229	50	)	)	PUNCT
ejpam-6039	229	51	is	be	AUX
ejpam-6039	229	52	σ1σ2	σ1σ2	NOUN
ejpam-6039	229	53	-	-	PUNCT
ejpam-6039	229	54	connected	connected	ADJ
ejpam-6039	229	55	.	.	PUNCT
ejpam-6039	230	1	proof	proof	NOUN
ejpam-6039	230	2	.	.	PUNCT
ejpam-6039	231	1	suppose	suppose	VERB
ejpam-6039	231	2	that	that	SCONJ
ejpam-6039	231	3	(	(	PUNCT
ejpam-6039	231	4	y	y	PROPN
ejpam-6039	231	5	,	,	PUNCT
ejpam-6039	231	6	σ1	σ1	PROPN
ejpam-6039	231	7	,	,	PUNCT
ejpam-6039	231	8	σ2	σ2	PROPN
ejpam-6039	231	9	)	)	PUNCT
ejpam-6039	231	10	is	be	AUX
ejpam-6039	231	11	not	not	PART
ejpam-6039	231	12	σ1σ2	σ1σ2	VERB
ejpam-6039	231	13	-	-	PUNCT
ejpam-6039	231	14	connected	connect	VERB
ejpam-6039	231	15	.	.	PUNCT
ejpam-6039	232	1	then	then	ADV
ejpam-6039	232	2	,	,	PUNCT
ejpam-6039	232	3	there	there	PRON
ejpam-6039	232	4	exist	exist	VERB
ejpam-6039	232	5	nonempty	nonempty	ADV
ejpam-6039	232	6	σ1σ2	σ1σ2	NOUN
ejpam-6039	232	7	-	-	ADJ
ejpam-6039	232	8	open	open	ADJ
ejpam-6039	232	9	sets	set	NOUN
ejpam-6039	232	10	v	v	ADP
ejpam-6039	232	11	and	and	CCONJ
ejpam-6039	232	12	w	w	ADP
ejpam-6039	232	13	such	such	ADJ
ejpam-6039	232	14	that	that	SCONJ
ejpam-6039	232	15	y	y	PROPN
ejpam-6039	232	16	=	=	PUNCT
ejpam-6039	232	17	v	v	PROPN
ejpam-6039	232	18	∪	∪	NOUN
ejpam-6039	232	19	w	w	PROPN
ejpam-6039	232	20	.	.	PUNCT
ejpam-6039	233	1	therefore	therefore	ADV
ejpam-6039	233	2	,	,	PUNCT
ejpam-6039	233	3	v	v	NOUN
ejpam-6039	233	4	and	and	CCONJ
ejpam-6039	233	5	w	w	NOUN
ejpam-6039	233	6	are	be	AUX
ejpam-6039	233	7	σ1σ2	σ1σ2	NOUN
ejpam-6039	233	8	-	-	PUNCT
ejpam-6039	233	9	clopen	clopen	ADJ
ejpam-6039	233	10	in	in	ADP
ejpam-6039	233	11	y	y	PROPN
ejpam-6039	233	12	.	.	PUNCT
ejpam-6039	234	1	since	since	SCONJ
ejpam-6039	234	2	f	f	PROPN
ejpam-6039	234	3	is	be	AUX
ejpam-6039	234	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	234	5	,	,	PUNCT
ejpam-6039	234	6	τ2)p	τ2)p	ADJ
ejpam-6039	234	7	-	-	ADJ
ejpam-6039	234	8	continuous	continuous	ADJ
ejpam-6039	234	9	,	,	PUNCT
ejpam-6039	234	10	f	f	PROPN
ejpam-6039	234	11	−1(v	−1(v	PROPN
ejpam-6039	234	12	)	)	PUNCT
ejpam-6039	234	13	and	and	CCONJ
ejpam-6039	234	14	f−1(w	f−1(w	ADV
ejpam-6039	234	15	)	)	PUNCT
ejpam-6039	234	16	are	be	AUX
ejpam-6039	234	17	(	(	PUNCT
ejpam-6039	234	18	τ1	τ1	NOUN
ejpam-6039	234	19	,	,	PUNCT
ejpam-6039	234	20	τ2)p	τ2)p	NOUN
ejpam-6039	234	21	-	-	PUNCT
ejpam-6039	234	22	open	open	ADJ
ejpam-6039	234	23	in	in	ADP
ejpam-6039	234	24	x.	x.	NOUN
ejpam-6039	234	25	moreover	moreover	ADV
ejpam-6039	234	26	,	,	PUNCT
ejpam-6039	234	27	f−1(v	f−1(v	PROPN
ejpam-6039	234	28	)	)	PUNCT
ejpam-6039	234	29	and	and	CCONJ
ejpam-6039	234	30	f−1(w	f−1(w	ADV
ejpam-6039	234	31	)	)	PUNCT
ejpam-6039	234	32	are	be	AUX
ejpam-6039	234	33	nonempty	nonempty	X
ejpam-6039	234	34	disjoint	disjoint	NOUN
ejpam-6039	234	35	and	and	CCONJ
ejpam-6039	234	36	x	x	X
ejpam-6039	234	37	=	=	SYM
ejpam-6039	234	38	f−1(v	f−1(v	PROPN
ejpam-6039	234	39	)	)	PUNCT
ejpam-6039	234	40	∪	∪	ADP
ejpam-6039	234	41	f−1(w	f−1(w	PROPN
ejpam-6039	234	42	)	)	PUNCT
ejpam-6039	234	43	.	.	PUNCT
ejpam-6039	235	1	this	this	PRON
ejpam-6039	235	2	shows	show	VERB
ejpam-6039	235	3	that	that	SCONJ
ejpam-6039	235	4	(	(	PUNCT
ejpam-6039	235	5	x	x	NOUN
ejpam-6039	235	6	,	,	PUNCT
ejpam-6039	235	7	τ1	τ1	NOUN
ejpam-6039	235	8	,	,	PUNCT
ejpam-6039	235	9	τ2	τ2	NOUN
ejpam-6039	235	10	)	)	PUNCT
ejpam-6039	235	11	is	be	AUX
ejpam-6039	235	12	not	not	PART
ejpam-6039	235	13	(	(	PUNCT
ejpam-6039	235	14	τ1	τ1	NOUN
ejpam-6039	235	15	,	,	PUNCT
ejpam-6039	235	16	τ2)p	τ2)p	NOUN
ejpam-6039	235	17	-	-	PUNCT
ejpam-6039	235	18	connected	connect	VERB
ejpam-6039	235	19	.	.	PUNCT
ejpam-6039	236	1	this	this	PRON
ejpam-6039	236	2	is	be	AUX
ejpam-6039	236	3	a	a	DET
ejpam-6039	236	4	contradiction	contradiction	NOUN
ejpam-6039	236	5	.	.	PUNCT
ejpam-6039	237	1	this	this	PRON
ejpam-6039	237	2	means	mean	VERB
ejpam-6039	237	3	that	that	SCONJ
ejpam-6039	237	4	(	(	PUNCT
ejpam-6039	237	5	y	y	PROPN
ejpam-6039	237	6	,	,	PUNCT
ejpam-6039	237	7	σ1	σ1	PROPN
ejpam-6039	237	8	,	,	PUNCT
ejpam-6039	237	9	σ2	σ2	PROPN
ejpam-6039	237	10	)	)	PUNCT
ejpam-6039	237	11	is	be	AUX
ejpam-6039	237	12	σ1σ2	σ1σ2	NOUN
ejpam-6039	237	13	-	-	PUNCT
ejpam-6039	237	14	connected	connect	VERB
ejpam-6039	237	15	.	.	PUNCT
ejpam-6039	238	1	definition	definition	NOUN
ejpam-6039	238	2	14	14	NUM
ejpam-6039	238	3	.	.	PUNCT
ejpam-6039	239	1	a	a	DET
ejpam-6039	239	2	function	function	NOUN
ejpam-6039	239	3	f	f	NOUN
ejpam-6039	239	4	:	:	PUNCT
ejpam-6039	239	5	(	(	PUNCT
ejpam-6039	239	6	x	x	NOUN
ejpam-6039	239	7	,	,	PUNCT
ejpam-6039	239	8	τ1	τ1	NOUN
ejpam-6039	239	9	,	,	PUNCT
ejpam-6039	239	10	τ2	τ2	NOUN
ejpam-6039	239	11	)	)	PUNCT
ejpam-6039	239	12	→	→	SYM
ejpam-6039	239	13	(	(	PUNCT
ejpam-6039	239	14	y	y	PROPN
ejpam-6039	239	15	,	,	PUNCT
ejpam-6039	239	16	σ1	σ1	PROPN
ejpam-6039	239	17	,	,	PUNCT
ejpam-6039	239	18	σ2	σ2	PROPN
ejpam-6039	239	19	)	)	PUNCT
ejpam-6039	239	20	is	be	AUX
ejpam-6039	239	21	called	call	VERB
ejpam-6039	239	22	perfectly	perfectly	ADV
ejpam-6039	239	23	(	(	PUNCT
ejpam-6039	239	24	τ1	τ1	NOUN
ejpam-6039	239	25	,	,	PUNCT
ejpam-6039	239	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	239	27	if	if	SCONJ
ejpam-6039	239	28	f−1(v	f−1(v	PROPN
ejpam-6039	239	29	)	)	PUNCT
ejpam-6039	239	30	is	be	AUX
ejpam-6039	239	31	τ1τ2	τ1τ2	NOUN
ejpam-6039	239	32	-	-	ADJ
ejpam-6039	239	33	clopen	clopen	ADJ
ejpam-6039	239	34	in	in	ADP
ejpam-6039	239	35	x	x	PUNCT
ejpam-6039	239	36	for	for	ADP
ejpam-6039	239	37	each	each	DET
ejpam-6039	239	38	σ1σ2	σ1σ2	VERB
ejpam-6039	239	39	-	-	ADJ
ejpam-6039	239	40	open	open	ADJ
ejpam-6039	239	41	set	set	NOUN
ejpam-6039	239	42	v	v	NOUN
ejpam-6039	239	43	of	of	ADP
ejpam-6039	239	44	y	y	PROPN
ejpam-6039	239	45	.	.	PUNCT
ejpam-6039	240	1	definition	definition	NOUN
ejpam-6039	240	2	15	15	NUM
ejpam-6039	240	3	.	.	PUNCT
ejpam-6039	241	1	a	a	DET
ejpam-6039	241	2	function	function	NOUN
ejpam-6039	241	3	f	f	NOUN
ejpam-6039	241	4	:	:	PUNCT
ejpam-6039	241	5	(	(	PUNCT
ejpam-6039	241	6	x	x	NOUN
ejpam-6039	241	7	,	,	PUNCT
ejpam-6039	241	8	τ1	τ1	NOUN
ejpam-6039	241	9	,	,	PUNCT
ejpam-6039	241	10	τ2	τ2	NOUN
ejpam-6039	241	11	)	)	PUNCT
ejpam-6039	241	12	→	→	SYM
ejpam-6039	241	13	(	(	PUNCT
ejpam-6039	241	14	y	y	PROPN
ejpam-6039	241	15	,	,	PUNCT
ejpam-6039	241	16	σ1	σ1	PROPN
ejpam-6039	241	17	,	,	PUNCT
ejpam-6039	241	18	σ2	σ2	PROPN
ejpam-6039	241	19	)	)	PUNCT
ejpam-6039	241	20	is	be	AUX
ejpam-6039	241	21	called	call	VERB
ejpam-6039	241	22	α(τ1	α(τ1	NOUN
ejpam-6039	241	23	,	,	PUNCT
ejpam-6039	241	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	241	25	if	if	SCONJ
ejpam-6039	241	26	f−1(v	f−1(v	PROPN
ejpam-6039	241	27	)	)	PUNCT
ejpam-6039	241	28	is	be	AUX
ejpam-6039	241	29	α(τ1	α(τ1	NOUN
ejpam-6039	241	30	,	,	PUNCT
ejpam-6039	241	31	τ2)-open	τ2)-open	ADJ
ejpam-6039	241	32	in	in	ADP
ejpam-6039	241	33	x	x	PUNCT
ejpam-6039	241	34	for	for	ADP
ejpam-6039	241	35	each	each	DET
ejpam-6039	241	36	σ1σ2	σ1σ2	VERB
ejpam-6039	241	37	-	-	ADJ
ejpam-6039	241	38	open	open	ADJ
ejpam-6039	241	39	set	set	NOUN
ejpam-6039	241	40	v	v	NOUN
ejpam-6039	241	41	of	of	ADP
ejpam-6039	241	42	y	y	PROPN
ejpam-6039	241	43	.	.	PUNCT
ejpam-6039	242	1	m.	m.	NOUN
ejpam-6039	242	2	chiangpradit	chiangpradit	PROPN
ejpam-6039	242	3	,	,	PUNCT
ejpam-6039	242	4	s.	s.	PROPN
ejpam-6039	242	5	sompong	sompong	PROPN
ejpam-6039	242	6	,	,	PUNCT
ejpam-6039	242	7	c.	c.	PROPN
ejpam-6039	242	8	boonpok	boonpok	PROPN
ejpam-6039	242	9	/	/	SYM
ejpam-6039	242	10	eur	eur	PROPN
ejpam-6039	242	11	.	.	PUNCT
ejpam-6039	243	1	j.	j.	PROPN
ejpam-6039	243	2	pure	pure	PROPN
ejpam-6039	243	3	appl	appl	PROPN
ejpam-6039	243	4	.	.	PROPN
ejpam-6039	243	5	math	math	PROPN
ejpam-6039	243	6	,	,	PUNCT
ejpam-6039	243	7	18	18	NUM
ejpam-6039	243	8	(	(	PUNCT
ejpam-6039	243	9	2	2	NUM
ejpam-6039	243	10	)	)	PUNCT
ejpam-6039	243	11	(	(	PUNCT
ejpam-6039	243	12	2025	2025	NUM
ejpam-6039	243	13	)	)	PUNCT
ejpam-6039	243	14	,	,	PUNCT
ejpam-6039	243	15	6039	6039	NUM
ejpam-6039	243	16	9	9	NUM
ejpam-6039	243	17	of	of	ADP
ejpam-6039	243	18	11	11	NUM
ejpam-6039	243	19	theorem	theorem	VERB
ejpam-6039	243	20	10	10	NUM
ejpam-6039	243	21	.	.	PUNCT
ejpam-6039	244	1	a	a	DET
ejpam-6039	244	2	function	function	NOUN
ejpam-6039	244	3	f	f	NOUN
ejpam-6039	244	4	:	:	PUNCT
ejpam-6039	244	5	(	(	PUNCT
ejpam-6039	244	6	x	x	NOUN
ejpam-6039	244	7	,	,	PUNCT
ejpam-6039	244	8	τ1	τ1	NOUN
ejpam-6039	244	9	,	,	PUNCT
ejpam-6039	244	10	τ2	τ2	NOUN
ejpam-6039	244	11	)	)	PUNCT
ejpam-6039	244	12	→	→	SYM
ejpam-6039	244	13	(	(	PUNCT
ejpam-6039	244	14	y	y	PROPN
ejpam-6039	244	15	,	,	PUNCT
ejpam-6039	244	16	σ1	σ1	PROPN
ejpam-6039	244	17	,	,	PUNCT
ejpam-6039	244	18	σ2	σ2	PROPN
ejpam-6039	244	19	)	)	PUNCT
ejpam-6039	244	20	is	be	AUX
ejpam-6039	244	21	perfectly	perfectly	ADV
ejpam-6039	244	22	(	(	PUNCT
ejpam-6039	244	23	τ1	τ1	NOUN
ejpam-6039	244	24	,	,	PUNCT
ejpam-6039	244	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	244	26	if	if	SCONJ
ejpam-6039	244	27	and	and	CCONJ
ejpam-6039	244	28	only	only	ADV
ejpam-6039	244	29	if	if	SCONJ
ejpam-6039	244	30	f	f	PROPN
ejpam-6039	244	31	is	be	AUX
ejpam-6039	244	32	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	244	33	,	,	PUNCT
ejpam-6039	244	34	τ2)p	τ2)p	ADJ
ejpam-6039	244	35	-	-	ADJ
ejpam-6039	244	36	continuous	continuous	ADJ
ejpam-6039	244	37	and	and	CCONJ
ejpam-6039	244	38	α(τ1	α(τ1	NOUN
ejpam-6039	244	39	,	,	PUNCT
ejpam-6039	244	40	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	244	41	.	.	PUNCT
ejpam-6039	245	1	proof	proof	NOUN
ejpam-6039	245	2	.	.	PUNCT
ejpam-6039	246	1	this	this	PRON
ejpam-6039	246	2	is	be	AUX
ejpam-6039	246	3	obvious	obvious	ADJ
ejpam-6039	246	4	.	.	PUNCT
ejpam-6039	247	1	conversely	conversely	ADV
ejpam-6039	247	2	,	,	PUNCT
ejpam-6039	247	3	let	let	VERB
ejpam-6039	247	4	v	v	PART
ejpam-6039	247	5	be	be	AUX
ejpam-6039	247	6	any	any	DET
ejpam-6039	247	7	σ1σ2	σ1σ2	NOUN
ejpam-6039	247	8	-	-	ADJ
ejpam-6039	247	9	open	open	ADJ
ejpam-6039	247	10	set	set	NOUN
ejpam-6039	247	11	of	of	ADP
ejpam-6039	247	12	y	y	PROPN
ejpam-6039	247	13	.	.	PUNCT
ejpam-6039	248	1	since	since	SCONJ
ejpam-6039	248	2	f	f	PROPN
ejpam-6039	248	3	is	be	AUX
ejpam-6039	248	4	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	248	5	,	,	PUNCT
ejpam-6039	248	6	τ2)p	τ2)p	ADJ
ejpam-6039	248	7	-	-	ADJ
ejpam-6039	248	8	continuous	continuous	ADJ
ejpam-6039	248	9	and	and	CCONJ
ejpam-6039	248	10	α(τ1	α(τ1	NOUN
ejpam-6039	248	11	,	,	PUNCT
ejpam-6039	248	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	248	13	,	,	PUNCT
ejpam-6039	248	14	f	f	PROPN
ejpam-6039	248	15	−1(v	−1(v	PROPN
ejpam-6039	248	16	)	)	PUNCT
ejpam-6039	248	17	is	be	AUX
ejpam-6039	248	18	(	(	PUNCT
ejpam-6039	248	19	τ1	τ1	NOUN
ejpam-6039	248	20	,	,	PUNCT
ejpam-6039	248	21	τ2)p	τ2)p	NOUN
ejpam-6039	248	22	-	-	PUNCT
ejpam-6039	248	23	closed	closed	ADJ
ejpam-6039	248	24	and	and	CCONJ
ejpam-6039	248	25	α(τ1	α(τ1	NOUN
ejpam-6039	248	26	,	,	PUNCT
ejpam-6039	248	27	τ2)-open	τ2)-open	ADJ
ejpam-6039	248	28	in	in	ADP
ejpam-6039	248	29	x.	x.	NOUN
ejpam-6039	248	30	therefore	therefore	ADV
ejpam-6039	248	31	,	,	PUNCT
ejpam-6039	248	32	we	we	PRON
ejpam-6039	248	33	have	have	VERB
ejpam-6039	248	34	τ1τ2	τ1τ2	NOUN
ejpam-6039	248	35	-	-	NOUN
ejpam-6039	248	36	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	248	37	-	-	PUNCT
ejpam-6039	248	38	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	248	39	-	-	PUNCT
ejpam-6039	248	40	int(f	int(f	PROPN
ejpam-6039	248	41	−1(v	−1(v	NOUN
ejpam-6039	248	42	)	)	PUNCT
ejpam-6039	248	43	)	)	PUNCT
ejpam-6039	248	44	)	)	PUNCT
ejpam-6039	248	45	)	)	PUNCT
ejpam-6039	249	1	⊆	⊆	X
ejpam-6039	249	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	249	3	-	-	NUM
ejpam-6039	249	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	249	5	-	-	PUNCT
ejpam-6039	249	6	int(f	int(f	PROPN
ejpam-6039	249	7	−1(v	−1(v	NOUN
ejpam-6039	249	8	)	)	PUNCT
ejpam-6039	249	9	)	)	PUNCT
ejpam-6039	249	10	)	)	PUNCT
ejpam-6039	250	1	⊆	⊆	NUM
ejpam-6039	250	2	f−1(v	f−1(v	NOUN
ejpam-6039	250	3	)	)	PUNCT
ejpam-6039	250	4	⊆	⊆	X
ejpam-6039	250	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	250	6	-	-	PUNCT
ejpam-6039	250	7	int(τ1τ2	int(τ1τ2	NOUN
ejpam-6039	250	8	-	-	PUNCT
ejpam-6039	250	9	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	250	10	-	-	PUNCT
ejpam-6039	250	11	int(f	int(f	PROPN
ejpam-6039	250	12	−1(v	−1(v	NOUN
ejpam-6039	250	13	)	)	PUNCT
ejpam-6039	250	14	)	)	PUNCT
ejpam-6039	250	15	)	)	PUNCT
ejpam-6039	250	16	)	)	PUNCT
ejpam-6039	251	1	⊆	⊆	X
ejpam-6039	251	2	τ1τ2	τ1τ2	NOUN
ejpam-6039	251	3	-	-	NUM
ejpam-6039	251	4	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-6039	251	5	-	-	PUNCT
ejpam-6039	251	6	int(f	int(f	PROPN
ejpam-6039	251	7	−1(v	−1(v	NOUN
ejpam-6039	251	8	)	)	PUNCT
ejpam-6039	251	9	)	)	PUNCT
ejpam-6039	251	10	)	)	PUNCT
ejpam-6039	251	11	.	.	PUNCT
ejpam-6039	252	1	this	this	PRON
ejpam-6039	252	2	implies	imply	VERB
ejpam-6039	252	3	that	that	DET
ejpam-6039	252	4	f−1(v	f−1(v	PROPN
ejpam-6039	252	5	)	)	PUNCT
ejpam-6039	252	6	is	be	AUX
ejpam-6039	252	7	τ1τ2	τ1τ2	NOUN
ejpam-6039	252	8	-	-	ADJ
ejpam-6039	252	9	clopen	clopen	ADJ
ejpam-6039	252	10	in	in	ADP
ejpam-6039	252	11	x.	x.	PROPN
ejpam-6039	252	12	thus	thus	ADV
ejpam-6039	252	13	,	,	PUNCT
ejpam-6039	252	14	f	f	PROPN
ejpam-6039	252	15	is	be	AUX
ejpam-6039	252	16	perfectly	perfectly	ADV
ejpam-6039	252	17	(	(	PUNCT
ejpam-6039	252	18	τ1	τ1	NOUN
ejpam-6039	252	19	,	,	PUNCT
ejpam-6039	252	20	τ2)-continuous	τ2)-continuous	PROPN
ejpam-6039	252	21	.	.	PUNCT
ejpam-6039	253	1	recall	recall	VERB
ejpam-6039	253	2	that	that	SCONJ
ejpam-6039	253	3	a	a	DET
ejpam-6039	253	4	bitopological	bitopological	ADJ
ejpam-6039	253	5	space	space	NOUN
ejpam-6039	253	6	(	(	PUNCT
ejpam-6039	253	7	x	x	NOUN
ejpam-6039	253	8	,	,	PUNCT
ejpam-6039	253	9	τ1	τ1	NOUN
ejpam-6039	253	10	,	,	PUNCT
ejpam-6039	253	11	τ2	τ2	NOUN
ejpam-6039	253	12	)	)	PUNCT
ejpam-6039	253	13	is	be	AUX
ejpam-6039	253	14	said	say	VERB
ejpam-6039	253	15	to	to	PART
ejpam-6039	253	16	be	be	AUX
ejpam-6039	253	17	τ1τ2	τ1τ2	NOUN
ejpam-6039	253	18	-	-	ADJ
ejpam-6039	253	19	compact	compact	ADJ
ejpam-6039	253	20	[	[	X
ejpam-6039	253	21	21	21	NUM
ejpam-6039	253	22	]	]	X
ejpam-6039	253	23	if	if	SCONJ
ejpam-6039	253	24	every	every	DET
ejpam-6039	253	25	cover	cover	NOUN
ejpam-6039	253	26	of	of	ADP
ejpam-6039	253	27	x	x	PUNCT
ejpam-6039	253	28	by	by	ADP
ejpam-6039	253	29	τ1τ2	τ1τ2	ADJ
ejpam-6039	253	30	-	-	ADJ
ejpam-6039	253	31	open	open	ADJ
ejpam-6039	253	32	sets	set	NOUN
ejpam-6039	253	33	has	have	VERB
ejpam-6039	253	34	a	a	DET
ejpam-6039	253	35	finite	finite	ADJ
ejpam-6039	253	36	subcover	subcover	PROPN
ejpam-6039	253	37	.	.	PUNCT
ejpam-6039	254	1	definition	definition	NOUN
ejpam-6039	254	2	16	16	NUM
ejpam-6039	254	3	.	.	PUNCT
ejpam-6039	255	1	a	a	DET
ejpam-6039	255	2	bitopological	bitopological	ADJ
ejpam-6039	255	3	space	space	NOUN
ejpam-6039	255	4	(	(	PUNCT
ejpam-6039	255	5	x	x	NOUN
ejpam-6039	255	6	,	,	PUNCT
ejpam-6039	255	7	τ1	τ1	NOUN
ejpam-6039	255	8	,	,	PUNCT
ejpam-6039	255	9	τ2	τ2	NOUN
ejpam-6039	255	10	)	)	PUNCT
ejpam-6039	255	11	is	be	AUX
ejpam-6039	255	12	said	say	VERB
ejpam-6039	255	13	to	to	PART
ejpam-6039	255	14	be	be	AUX
ejpam-6039	255	15	mildly	mildly	ADV
ejpam-6039	255	16	τ1τ2	τ1τ2	ADJ
ejpam-6039	255	17	-	-	ADJ
ejpam-6039	255	18	compact	compact	ADJ
ejpam-6039	255	19	if	if	SCONJ
ejpam-6039	255	20	every	every	DET
ejpam-6039	255	21	τ1τ2	τ1τ2	ADJ
ejpam-6039	255	22	-	-	ADJ
ejpam-6039	255	23	clopen	clopen	ADJ
ejpam-6039	255	24	cover	cover	NOUN
ejpam-6039	255	25	of	of	ADP
ejpam-6039	255	26	x	x	PUNCT
ejpam-6039	255	27	has	have	VERB
ejpam-6039	255	28	a	a	DET
ejpam-6039	255	29	finite	finite	ADJ
ejpam-6039	255	30	subcover	subcover	PROPN
ejpam-6039	255	31	.	.	PUNCT
ejpam-6039	256	1	theorem	theorem	VERB
ejpam-6039	256	2	11	11	NUM
ejpam-6039	256	3	.	.	PUNCT
ejpam-6039	257	1	if	if	SCONJ
ejpam-6039	257	2	f	f	PROPN
ejpam-6039	257	3	:	:	PUNCT
ejpam-6039	257	4	(	(	PUNCT
ejpam-6039	257	5	x	x	NOUN
ejpam-6039	257	6	,	,	PUNCT
ejpam-6039	257	7	τ1	τ1	NOUN
ejpam-6039	257	8	,	,	PUNCT
ejpam-6039	257	9	τ2	τ2	NOUN
ejpam-6039	257	10	)	)	PUNCT
ejpam-6039	257	11	→	→	SYM
ejpam-6039	257	12	(	(	PUNCT
ejpam-6039	257	13	y	y	PROPN
ejpam-6039	257	14	,	,	PUNCT
ejpam-6039	257	15	σ1	σ1	PROPN
ejpam-6039	257	16	,	,	PUNCT
ejpam-6039	257	17	σ2	σ2	PROPN
ejpam-6039	257	18	)	)	PUNCT
ejpam-6039	257	19	is	be	AUX
ejpam-6039	257	20	a	a	DET
ejpam-6039	257	21	perfectly	perfectly	ADV
ejpam-6039	257	22	(	(	PUNCT
ejpam-6039	257	23	τ1	τ1	NOUN
ejpam-6039	257	24	,	,	PUNCT
ejpam-6039	257	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	257	26	surjection	surjection	NOUN
ejpam-6039	257	27	and	and	CCONJ
ejpam-6039	257	28	(	(	PUNCT
ejpam-6039	257	29	x	x	NOUN
ejpam-6039	257	30	,	,	PUNCT
ejpam-6039	257	31	τ1	τ1	NOUN
ejpam-6039	257	32	,	,	PUNCT
ejpam-6039	257	33	τ2	τ2	NOUN
ejpam-6039	257	34	)	)	PUNCT
ejpam-6039	257	35	is	be	AUX
ejpam-6039	257	36	mildly	mildly	ADV
ejpam-6039	257	37	τ1τ2	τ1τ2	ADJ
ejpam-6039	257	38	-	-	ADJ
ejpam-6039	257	39	compact	compact	ADJ
ejpam-6039	257	40	,	,	PUNCT
ejpam-6039	257	41	then	then	ADV
ejpam-6039	257	42	(	(	PUNCT
ejpam-6039	257	43	y	y	PROPN
ejpam-6039	257	44	,	,	PUNCT
ejpam-6039	257	45	σ1	σ1	PROPN
ejpam-6039	257	46	,	,	PUNCT
ejpam-6039	257	47	σ2	σ2	PROPN
ejpam-6039	257	48	)	)	PUNCT
ejpam-6039	257	49	is	be	AUX
ejpam-6039	257	50	σ1σ2	σ1σ2	NOUN
ejpam-6039	257	51	-	-	ADJ
ejpam-6039	257	52	compact	compact	ADJ
ejpam-6039	257	53	.	.	PUNCT
ejpam-6039	258	1	proof	proof	NOUN
ejpam-6039	258	2	.	.	PUNCT
ejpam-6039	259	1	let	let	VERB
ejpam-6039	259	2	{	{	PUNCT
ejpam-6039	259	3	vγ	vγ	VERB
ejpam-6039	259	4	|	|	ADV
ejpam-6039	259	5	γ	γ	X
ejpam-6039	259	6	∈	∈	PROPN
ejpam-6039	259	7	∇	∇	X
ejpam-6039	259	8	}	}	PUNCT
ejpam-6039	259	9	be	be	VERB
ejpam-6039	259	10	any	any	DET
ejpam-6039	259	11	σ1σ2	σ1σ2	NOUN
ejpam-6039	259	12	-	-	PUNCT
ejpam-6039	259	13	open	open	ADJ
ejpam-6039	259	14	cover	cover	NOUN
ejpam-6039	259	15	of	of	ADP
ejpam-6039	259	16	y	y	PROPN
ejpam-6039	259	17	.	.	PUNCT
ejpam-6039	260	1	since	since	SCONJ
ejpam-6039	260	2	f	f	PROPN
ejpam-6039	260	3	is	be	AUX
ejpam-6039	260	4	perfectly	perfectly	ADV
ejpam-6039	260	5	(	(	PUNCT
ejpam-6039	260	6	τ1	τ1	NOUN
ejpam-6039	260	7	,	,	PUNCT
ejpam-6039	260	8	τ2)continuous	τ2)continuous	ADJ
ejpam-6039	260	9	,	,	PUNCT
ejpam-6039	260	10	we	we	PRON
ejpam-6039	260	11	have	have	VERB
ejpam-6039	260	12	{	{	PUNCT
ejpam-6039	260	13	f−1(vγ	f−1(vγ	PROPN
ejpam-6039	260	14	)	)	PUNCT
ejpam-6039	260	15	|	|	ADV
ejpam-6039	260	16	γ	γ	PROPN
ejpam-6039	260	17	∈	∈	PROPN
ejpam-6039	260	18	∇	∇	X
ejpam-6039	260	19	}	}	PUNCT
ejpam-6039	260	20	is	be	AUX
ejpam-6039	260	21	a	a	DET
ejpam-6039	260	22	τ1τ2	τ1τ2	ADJ
ejpam-6039	260	23	-	-	ADJ
ejpam-6039	260	24	clopen	clopen	ADJ
ejpam-6039	260	25	cover	cover	NOUN
ejpam-6039	260	26	of	of	ADP
ejpam-6039	260	27	x.	x.	NOUN
ejpam-6039	260	28	since	since	SCONJ
ejpam-6039	260	29	(	(	PUNCT
ejpam-6039	260	30	x	x	NOUN
ejpam-6039	260	31	,	,	PUNCT
ejpam-6039	260	32	τ1	τ1	NOUN
ejpam-6039	260	33	,	,	PUNCT
ejpam-6039	260	34	τ2	τ2	NOUN
ejpam-6039	260	35	)	)	PUNCT
ejpam-6039	260	36	is	be	AUX
ejpam-6039	260	37	mildly	mildly	ADV
ejpam-6039	260	38	τ1τ2	τ1τ2	ADJ
ejpam-6039	260	39	-	-	ADJ
ejpam-6039	260	40	compact	compact	ADJ
ejpam-6039	260	41	,	,	PUNCT
ejpam-6039	260	42	there	there	PRON
ejpam-6039	260	43	exists	exist	VERB
ejpam-6039	260	44	a	a	DET
ejpam-6039	260	45	finite	finite	NOUN
ejpam-6039	260	46	subset	subset	NOUN
ejpam-6039	260	47	∇0	∇0	NUM
ejpam-6039	260	48	of	of	ADP
ejpam-6039	260	49	∇	∇	NOUN
ejpam-6039	260	50	such	such	ADJ
ejpam-6039	260	51	that	that	SCONJ
ejpam-6039	260	52	x	x	SYM
ejpam-6039	260	53	=	=	SYM
ejpam-6039	260	54	∪{f−1(vγ	∪{f−1(vγ	PROPN
ejpam-6039	260	55	)	)	PUNCT
ejpam-6039	260	56	|	|	ADV
ejpam-6039	260	57	γ	γ	X
ejpam-6039	260	58	∈	∈	NOUN
ejpam-6039	260	59	∇0	∇0	NOUN
ejpam-6039	260	60	}	}	PUNCT
ejpam-6039	260	61	.	.	PUNCT
ejpam-6039	261	1	since	since	SCONJ
ejpam-6039	261	2	f	f	PROPN
ejpam-6039	261	3	is	be	AUX
ejpam-6039	261	4	surjective	surjective	ADJ
ejpam-6039	261	5	,	,	PUNCT
ejpam-6039	261	6	y	y	PROPN
ejpam-6039	261	7	=	=	SYM
ejpam-6039	261	8	∪{vγ	∪{vγ	PROPN
ejpam-6039	261	9	|	|	ADV
ejpam-6039	261	10	γ	γ	X
ejpam-6039	261	11	∈	∈	NOUN
ejpam-6039	261	12	∇0	∇0	VERB
ejpam-6039	261	13	}	}	PUNCT
ejpam-6039	261	14	and	and	CCONJ
ejpam-6039	261	15	(	(	PUNCT
ejpam-6039	261	16	y	y	PROPN
ejpam-6039	261	17	,	,	PUNCT
ejpam-6039	261	18	σ1	σ1	PROPN
ejpam-6039	261	19	,	,	PUNCT
ejpam-6039	261	20	σ2	σ2	PROPN
ejpam-6039	261	21	)	)	PUNCT
ejpam-6039	261	22	is	be	AUX
ejpam-6039	261	23	σ1σ2	σ1σ2	NOUN
ejpam-6039	261	24	-	-	ADJ
ejpam-6039	261	25	compact	compact	ADJ
ejpam-6039	261	26	.	.	PUNCT
ejpam-6039	262	1	definition	definition	NOUN
ejpam-6039	262	2	17	17	NUM
ejpam-6039	262	3	.	.	PUNCT
ejpam-6039	263	1	a	a	DET
ejpam-6039	263	2	bitopological	bitopological	ADJ
ejpam-6039	263	3	space	space	NOUN
ejpam-6039	263	4	(	(	PUNCT
ejpam-6039	263	5	x	x	NOUN
ejpam-6039	263	6	,	,	PUNCT
ejpam-6039	263	7	τ1	τ1	NOUN
ejpam-6039	263	8	,	,	PUNCT
ejpam-6039	263	9	τ2	τ2	NOUN
ejpam-6039	263	10	)	)	PUNCT
ejpam-6039	263	11	is	be	AUX
ejpam-6039	263	12	said	say	VERB
ejpam-6039	263	13	to	to	PART
ejpam-6039	263	14	be	be	AUX
ejpam-6039	263	15	:	:	PUNCT
ejpam-6039	263	16	(	(	PUNCT
ejpam-6039	263	17	1	1	X
ejpam-6039	263	18	)	)	PUNCT
ejpam-6039	263	19	(	(	PUNCT
ejpam-6039	263	20	τ1	τ1	NOUN
ejpam-6039	263	21	,	,	PUNCT
ejpam-6039	263	22	τ2)p	τ2)p	NOUN
ejpam-6039	263	23	-	-	PUNCT
ejpam-6039	263	24	irreducible	irreducible	ADJ
ejpam-6039	263	25	if	if	SCONJ
ejpam-6039	263	26	every	every	DET
ejpam-6039	263	27	pair	pair	NOUN
ejpam-6039	263	28	of	of	ADP
ejpam-6039	263	29	nonempty	nonempty	ADJ
ejpam-6039	263	30	(	(	PUNCT
ejpam-6039	263	31	τ1	τ1	NOUN
ejpam-6039	263	32	,	,	PUNCT
ejpam-6039	263	33	τ2)p	τ2)p	ADJ
ejpam-6039	263	34	-	-	PUNCT
ejpam-6039	263	35	closed	closed	ADJ
ejpam-6039	263	36	sets	set	NOUN
ejpam-6039	263	37	of	of	ADP
ejpam-6039	263	38	x	x	PUNCT
ejpam-6039	263	39	has	have	VERB
ejpam-6039	263	40	a	a	DET
ejpam-6039	263	41	nonempty	nonempty	ADJ
ejpam-6039	263	42	intersection	intersection	NOUN
ejpam-6039	263	43	;	;	PUNCT
ejpam-6039	263	44	(	(	PUNCT
ejpam-6039	263	45	2	2	X
ejpam-6039	263	46	)	)	PUNCT
ejpam-6039	263	47	τ1τ2	τ1τ2	NOUN
ejpam-6039	263	48	-	-	ADJ
ejpam-6039	263	49	hyperconnected	hyperconnecte	VERB
ejpam-6039	263	50	if	if	SCONJ
ejpam-6039	263	51	τ1τ2	τ1τ2	NOUN
ejpam-6039	263	52	-	-	NOUN
ejpam-6039	263	53	cl(v	cl(v	X
ejpam-6039	263	54	)	)	PUNCT
ejpam-6039	263	55	=	=	PUNCT
ejpam-6039	264	1	x	x	PUNCT
ejpam-6039	264	2	for	for	ADP
ejpam-6039	264	3	every	every	DET
ejpam-6039	264	4	nonempty	nonempty	ADJ
ejpam-6039	264	5	τ1τ2	τ1τ2	NOUN
ejpam-6039	264	6	-	-	ADJ
ejpam-6039	264	7	open	open	ADJ
ejpam-6039	264	8	set	set	NOUN
ejpam-6039	264	9	v	v	NOUN
ejpam-6039	264	10	of	of	ADP
ejpam-6039	264	11	x.	x.	NOUN
ejpam-6039	264	12	theorem	theorem	VERB
ejpam-6039	264	13	12	12	NUM
ejpam-6039	264	14	.	.	PUNCT
ejpam-6039	265	1	if	if	SCONJ
ejpam-6039	265	2	f	f	PROPN
ejpam-6039	265	3	:	:	PUNCT
ejpam-6039	265	4	(	(	PUNCT
ejpam-6039	265	5	x	x	NOUN
ejpam-6039	265	6	,	,	PUNCT
ejpam-6039	265	7	τ1	τ1	NOUN
ejpam-6039	265	8	,	,	PUNCT
ejpam-6039	265	9	τ2	τ2	NOUN
ejpam-6039	265	10	)	)	PUNCT
ejpam-6039	265	11	→	→	SYM
ejpam-6039	265	12	(	(	PUNCT
ejpam-6039	265	13	y	y	PROPN
ejpam-6039	265	14	,	,	PUNCT
ejpam-6039	265	15	σ1	σ1	PROPN
ejpam-6039	265	16	,	,	PUNCT
ejpam-6039	265	17	σ2	σ2	PROPN
ejpam-6039	265	18	)	)	PUNCT
ejpam-6039	265	19	is	be	AUX
ejpam-6039	265	20	a	a	DET
ejpam-6039	265	21	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	265	22	,	,	PUNCT
ejpam-6039	265	23	τ2)p	τ2)p	ADJ
ejpam-6039	265	24	-	-	ADJ
ejpam-6039	265	25	continuous	continuous	ADJ
ejpam-6039	265	26	surjection	surjection	NOUN
ejpam-6039	265	27	and	and	CCONJ
ejpam-6039	265	28	(	(	PUNCT
ejpam-6039	265	29	x	x	NOUN
ejpam-6039	265	30	,	,	PUNCT
ejpam-6039	265	31	τ1	τ1	NOUN
ejpam-6039	265	32	,	,	PUNCT
ejpam-6039	265	33	τ2	τ2	NOUN
ejpam-6039	265	34	)	)	PUNCT
ejpam-6039	265	35	is	be	AUX
ejpam-6039	265	36	(	(	PUNCT
ejpam-6039	265	37	τ1	τ1	NOUN
ejpam-6039	265	38	,	,	PUNCT
ejpam-6039	265	39	τ2)p	τ2)p	ADJ
ejpam-6039	265	40	-	-	PUNCT
ejpam-6039	265	41	irreducible	irreducible	ADJ
ejpam-6039	265	42	,	,	PUNCT
ejpam-6039	265	43	then	then	ADV
ejpam-6039	265	44	(	(	PUNCT
ejpam-6039	265	45	y	y	PROPN
ejpam-6039	265	46	,	,	PUNCT
ejpam-6039	265	47	σ1	σ1	PROPN
ejpam-6039	265	48	,	,	PUNCT
ejpam-6039	265	49	σ2	σ2	PROPN
ejpam-6039	265	50	)	)	PUNCT
ejpam-6039	265	51	is	be	AUX
ejpam-6039	265	52	σ1σ2	σ1σ2	NOUN
ejpam-6039	265	53	-	-	PUNCT
ejpam-6039	265	54	hyperconnected	hyperconnected	ADJ
ejpam-6039	265	55	.	.	PUNCT
ejpam-6039	266	1	proof	proof	NOUN
ejpam-6039	266	2	.	.	PUNCT
ejpam-6039	267	1	suppose	suppose	VERB
ejpam-6039	267	2	that	that	SCONJ
ejpam-6039	267	3	(	(	PUNCT
ejpam-6039	267	4	y	y	PROPN
ejpam-6039	267	5	,	,	PUNCT
ejpam-6039	267	6	σ1	σ1	PROPN
ejpam-6039	267	7	,	,	PUNCT
ejpam-6039	267	8	σ2	σ2	PROPN
ejpam-6039	267	9	)	)	PUNCT
ejpam-6039	267	10	is	be	AUX
ejpam-6039	267	11	not	not	PART
ejpam-6039	267	12	σ1σ2	σ1σ2	NOUN
ejpam-6039	267	13	-	-	PUNCT
ejpam-6039	267	14	hyperconnected	hyperconnected	ADJ
ejpam-6039	267	15	.	.	PUNCT
ejpam-6039	268	1	then	then	ADV
ejpam-6039	268	2	,	,	PUNCT
ejpam-6039	268	3	σ1σ2	σ1σ2	NOUN
ejpam-6039	268	4	-	-	NUM
ejpam-6039	268	5	cl(v	cl(v	NOUN
ejpam-6039	268	6	)	)	PUNCT
ejpam-6039	268	7	̸=	̸=	PROPN
ejpam-6039	268	8	y	y	NOUN
ejpam-6039	268	9	for	for	ADP
ejpam-6039	268	10	some	some	DET
ejpam-6039	268	11	nonempty	nonempty	ADJ
ejpam-6039	268	12	σ1σ2	σ1σ2	NOUN
ejpam-6039	268	13	-	-	ADJ
ejpam-6039	268	14	open	open	ADJ
ejpam-6039	268	15	set	set	NOUN
ejpam-6039	268	16	v	v	NOUN
ejpam-6039	268	17	of	of	ADP
ejpam-6039	268	18	y	y	PROPN
ejpam-6039	268	19	.	.	PUNCT
ejpam-6039	269	1	therefore	therefore	ADV
ejpam-6039	269	2	,	,	PUNCT
ejpam-6039	269	3	there	there	PRON
ejpam-6039	269	4	exists	exist	VERB
ejpam-6039	269	5	a	a	DET
ejpam-6039	269	6	point	point	NOUN
ejpam-6039	269	7	y	y	PROPN
ejpam-6039	269	8	̸∈	̸∈	PROPN
ejpam-6039	269	9	σ1σ2	σ1σ2	NOUN
ejpam-6039	269	10	-	-	NUM
ejpam-6039	269	11	cl(v	cl(v	NOUN
ejpam-6039	269	12	)	)	PUNCT
ejpam-6039	269	13	and	and	CCONJ
ejpam-6039	269	14	v	v	ADP
ejpam-6039	269	15	∩w	∩w	ADJ
ejpam-6039	269	16	=	=	NOUN
ejpam-6039	269	17	∅	∅	NOUN
ejpam-6039	269	18	for	for	ADP
ejpam-6039	269	19	some	some	DET
ejpam-6039	269	20	σ1σ2	σ1σ2	NOUN
ejpam-6039	269	21	-	-	ADJ
ejpam-6039	269	22	open	open	ADJ
ejpam-6039	269	23	set	set	NOUN
ejpam-6039	269	24	w	w	PROPN
ejpam-6039	269	25	of	of	ADP
ejpam-6039	269	26	y	y	PROPN
ejpam-6039	269	27	containing	contain	VERB
ejpam-6039	269	28	y.	y.	NOUN
ejpam-6039	269	29	since	since	SCONJ
ejpam-6039	269	30	f	f	PROPN
ejpam-6039	269	31	is	be	AUX
ejpam-6039	269	32	contra-(τ1	contra-(τ1	NOUN
ejpam-6039	269	33	,	,	PUNCT
ejpam-6039	269	34	τ2)pcontinuous	τ2)pcontinuous	ADJ
ejpam-6039	269	35	,	,	PUNCT
ejpam-6039	269	36	f−1(v	f−1(v	PROPN
ejpam-6039	269	37	)	)	PUNCT
ejpam-6039	269	38	and	and	CCONJ
ejpam-6039	269	39	f−1(w	f−1(w	PROPN
ejpam-6039	269	40	)	)	PUNCT
ejpam-6039	269	41	are	be	AUX
ejpam-6039	269	42	disjoint	disjoint	X
ejpam-6039	269	43	nonempty	nonempty	ADV
ejpam-6039	269	44	(	(	PUNCT
ejpam-6039	269	45	τ1	τ1	NOUN
ejpam-6039	269	46	,	,	PUNCT
ejpam-6039	269	47	τ2)p	τ2)p	ADJ
ejpam-6039	269	48	-	-	PUNCT
ejpam-6039	269	49	closed	closed	ADJ
ejpam-6039	269	50	sets	set	NOUN
ejpam-6039	269	51	of	of	ADP
ejpam-6039	269	52	x.	x.	NOUN
ejpam-6039	269	53	thus	thus	ADV
ejpam-6039	269	54	,	,	PUNCT
ejpam-6039	269	55	(	(	PUNCT
ejpam-6039	269	56	x	x	NOUN
ejpam-6039	269	57	,	,	PUNCT
ejpam-6039	269	58	τ1	τ1	NOUN
ejpam-6039	269	59	,	,	PUNCT
ejpam-6039	269	60	τ2	τ2	NOUN
ejpam-6039	269	61	)	)	PUNCT
ejpam-6039	269	62	is	be	AUX
ejpam-6039	269	63	not	not	PART
ejpam-6039	269	64	(	(	PUNCT
ejpam-6039	269	65	τ1	τ1	NOUN
ejpam-6039	269	66	,	,	PUNCT
ejpam-6039	269	67	τ2)p	τ2)p	ADJ
ejpam-6039	269	68	-	-	PUNCT
ejpam-6039	269	69	irreducible	irreducible	ADJ
ejpam-6039	269	70	.	.	PUNCT
ejpam-6039	270	1	acknowledgements	acknowledgement	NOUN
ejpam-6039	270	2	this	this	DET
ejpam-6039	270	3	research	research	NOUN
ejpam-6039	270	4	project	project	NOUN
ejpam-6039	270	5	was	be	AUX
ejpam-6039	270	6	financially	financially	ADV
ejpam-6039	270	7	supported	support	VERB
ejpam-6039	270	8	by	by	ADP
ejpam-6039	270	9	mahasarakham	mahasarakham	PROPN
ejpam-6039	270	10	university	university	PROPN
ejpam-6039	270	11	.	.	PUNCT
ejpam-6039	271	1	m.	m.	NOUN
ejpam-6039	271	2	chiangpradit	chiangpradit	PROPN
ejpam-6039	271	3	,	,	PUNCT
ejpam-6039	271	4	s.	s.	PROPN
ejpam-6039	271	5	sompong	sompong	PROPN
ejpam-6039	271	6	,	,	PUNCT
ejpam-6039	271	7	c.	c.	PROPN
ejpam-6039	271	8	boonpok	boonpok	PROPN
ejpam-6039	271	9	/	/	SYM
ejpam-6039	271	10	eur	eur	PROPN
ejpam-6039	271	11	.	.	PUNCT
ejpam-6039	272	1	j.	j.	PROPN
ejpam-6039	272	2	pure	pure	PROPN
ejpam-6039	272	3	appl	appl	PROPN
ejpam-6039	272	4	.	.	PROPN
ejpam-6039	272	5	math	math	PROPN
ejpam-6039	272	6	,	,	PUNCT
ejpam-6039	272	7	18	18	NUM
ejpam-6039	272	8	(	(	PUNCT
ejpam-6039	272	9	2	2	NUM
ejpam-6039	272	10	)	)	PUNCT
ejpam-6039	272	11	(	(	PUNCT
ejpam-6039	272	12	2025	2025	NUM
ejpam-6039	272	13	)	)	PUNCT
ejpam-6039	272	14	,	,	PUNCT
ejpam-6039	272	15	6039	6039	NUM
ejpam-6039	272	16	10	10	NUM
ejpam-6039	272	17	of	of	ADP
ejpam-6039	272	18	11	11	NUM
ejpam-6039	272	19	references	reference	NOUN
ejpam-6039	272	20	[	[	X
ejpam-6039	272	21	1	1	NUM
ejpam-6039	272	22	]	]	PUNCT
ejpam-6039	272	23	j.	j.	PROPN
ejpam-6039	272	24	dontchev	dontchev	PROPN
ejpam-6039	272	25	.	.	PUNCT
ejpam-6039	273	1	contra	contra	ADJ
ejpam-6039	273	2	-	-	ADJ
ejpam-6039	273	3	continuous	continuous	ADJ
ejpam-6039	273	4	functions	function	NOUN
ejpam-6039	273	5	and	and	CCONJ
ejpam-6039	273	6	strongly	strongly	ADV
ejpam-6039	273	7	s	s	NOUN
ejpam-6039	273	8	-	-	PUNCT
ejpam-6039	273	9	closed	closed	ADJ
ejpam-6039	273	10	spaces	space	NOUN
ejpam-6039	273	11	.	.	PUNCT
ejpam-6039	274	1	international	international	ADJ
ejpam-6039	274	2	journal	journal	PROPN
ejpam-6039	274	3	of	of	ADP
ejpam-6039	274	4	mathematics	mathematics	PROPN
ejpam-6039	274	5	and	and	CCONJ
ejpam-6039	274	6	mathematical	mathematical	ADJ
ejpam-6039	274	7	sciences	science	NOUN
ejpam-6039	274	8	,	,	PUNCT
ejpam-6039	274	9	19:303–310	19:303–310	PROPN
ejpam-6039	274	10	,	,	PUNCT
ejpam-6039	274	11	1966	1966	NUM
ejpam-6039	274	12	.	.	PUNCT
ejpam-6039	275	1	[	[	X
ejpam-6039	275	2	2	2	X
ejpam-6039	275	3	]	]	PUNCT
ejpam-6039	275	4	j.	j.	PROPN
ejpam-6039	275	5	dontchev	dontchev	PROPN
ejpam-6039	275	6	and	and	CCONJ
ejpam-6039	275	7	t.	t.	PROPN
ejpam-6039	275	8	noiri	noiri	PROPN
ejpam-6039	275	9	.	.	PUNCT
ejpam-6039	276	1	contra	contra	ADJ
ejpam-6039	276	2	-	-	ADJ
ejpam-6039	276	3	semicontinuous	semicontinuous	ADJ
ejpam-6039	276	4	functions	function	NOUN
ejpam-6039	276	5	.	.	PUNCT
ejpam-6039	277	1	mathematica	mathematica	PROPN
ejpam-6039	277	2	pannonica	pannonica	PROPN
ejpam-6039	277	3	,	,	PUNCT
ejpam-6039	277	4	10:159–168	10:159–168	NOUN
ejpam-6039	277	5	,	,	PUNCT
ejpam-6039	277	6	1999	1999	NUM
ejpam-6039	277	7	.	.	PUNCT
ejpam-6039	278	1	[	[	X
ejpam-6039	278	2	3	3	X
ejpam-6039	278	3	]	]	X
ejpam-6039	278	4	s.	s.	PROPN
ejpam-6039	278	5	jafari	jafari	PROPN
ejpam-6039	278	6	and	and	CCONJ
ejpam-6039	278	7	t.	t.	PROPN
ejpam-6039	278	8	noiri	noiri	PROPN
ejpam-6039	278	9	.	.	PUNCT
ejpam-6039	279	1	contra	contra	PROPN
ejpam-6039	279	2	-	-	ADJ
ejpam-6039	279	3	super	super	ADJ
ejpam-6039	279	4	-	-	ADJ
ejpam-6039	279	5	continuous	continuous	ADJ
ejpam-6039	279	6	functions	function	NOUN
ejpam-6039	279	7	.	.	PUNCT
ejpam-6039	280	1	annales	annales	PROPN
ejpam-6039	280	2	universitatis	universitatis	PROPN
ejpam-6039	280	3	scientiarum	scientiarum	PROPN
ejpam-6039	280	4	budapestinensis	budapestinensis	PROPN
ejpam-6039	280	5	de	de	PROPN
ejpam-6039	280	6	rolando	rolando	PROPN
ejpam-6039	280	7	eötvös	eötvös	PROPN
ejpam-6039	280	8	,	,	PUNCT
ejpam-6039	280	9	sectio	sectio	PROPN
ejpam-6039	280	10	mathematica	mathematica	PROPN
ejpam-6039	280	11	,	,	PUNCT
ejpam-6039	280	12	42:27–34	42:27–34	PROPN
ejpam-6039	280	13	,	,	PUNCT
ejpam-6039	280	14	1999	1999	NUM
ejpam-6039	280	15	.	.	PUNCT
ejpam-6039	281	1	[	[	X
ejpam-6039	281	2	4	4	X
ejpam-6039	281	3	]	]	PUNCT
ejpam-6039	281	4	s.	s.	PROPN
ejpam-6039	281	5	jafari	jafari	PROPN
ejpam-6039	281	6	and	and	CCONJ
ejpam-6039	281	7	t.	t.	PROPN
ejpam-6039	281	8	noiri	noiri	PROPN
ejpam-6039	281	9	.	.	PUNCT
ejpam-6039	282	1	on	on	ADP
ejpam-6039	282	2	contra	contra	ADJ
ejpam-6039	282	3	-	-	ADJ
ejpam-6039	282	4	precontinuous	precontinuous	ADJ
ejpam-6039	282	5	functions	function	NOUN
ejpam-6039	282	6	.	.	PUNCT
ejpam-6039	283	1	bulletin	bulletin	NOUN
ejpam-6039	283	2	of	of	ADP
ejpam-6039	283	3	the	the	DET
ejpam-6039	283	4	malaysian	malaysian	PROPN
ejpam-6039	283	5	mathematical	mathematical	PROPN
ejpam-6039	283	6	sciences	sciences	PROPN
ejpam-6039	283	7	society	society	NOUN
ejpam-6039	283	8	,	,	PUNCT
ejpam-6039	283	9	25:115–128	25:115–128	PROPN
ejpam-6039	283	10	,	,	PUNCT
ejpam-6039	283	11	2002	2002	NUM
ejpam-6039	283	12	.	.	PUNCT
ejpam-6039	284	1	[	[	X
ejpam-6039	284	2	5	5	X
ejpam-6039	284	3	]	]	PUNCT
ejpam-6039	284	4	e.	e.	PROPN
ejpam-6039	284	5	ekici	ekici	PROPN
ejpam-6039	284	6	.	.	PUNCT
ejpam-6039	285	1	almost	almost	ADV
ejpam-6039	285	2	contra	contra	ADJ
ejpam-6039	285	3	-	-	ADJ
ejpam-6039	285	4	precontinuous	precontinuous	ADJ
ejpam-6039	285	5	functions	function	NOUN
ejpam-6039	285	6	.	.	PUNCT
ejpam-6039	286	1	bulletin	bulletin	NOUN
ejpam-6039	286	2	of	of	ADP
ejpam-6039	286	3	the	the	DET
ejpam-6039	286	4	malaysian	malaysian	PROPN
ejpam-6039	286	5	mathematical	mathematical	PROPN
ejpam-6039	286	6	sciences	sciences	PROPN
ejpam-6039	286	7	society	society	NOUN
ejpam-6039	286	8	,	,	PUNCT
ejpam-6039	286	9	27:53–65	27:53–65	NUM
ejpam-6039	286	10	,	,	PUNCT
ejpam-6039	286	11	2004	2004	NUM
ejpam-6039	286	12	.	.	PUNCT
ejpam-6039	287	1	[	[	X
ejpam-6039	287	2	6	6	NUM
ejpam-6039	287	3	]	]	PUNCT
ejpam-6039	287	4	j.	j.	PROPN
ejpam-6039	287	5	dontchev	dontchev	PROPN
ejpam-6039	287	6	,	,	PUNCT
ejpam-6039	287	7	m.	m.	NOUN
ejpam-6039	287	8	ganster	ganster	NOUN
ejpam-6039	287	9	,	,	PUNCT
ejpam-6039	287	10	and	and	CCONJ
ejpam-6039	287	11	i.	i.	PROPN
ejpam-6039	287	12	reilly	reilly	PROPN
ejpam-6039	287	13	.	.	PUNCT
ejpam-6039	288	1	more	more	ADV
ejpam-6039	288	2	on	on	ADP
ejpam-6039	288	3	almost	almost	ADV
ejpam-6039	288	4	s	s	NOUN
ejpam-6039	288	5	-	-	NOUN
ejpam-6039	288	6	continuity	continuity	NOUN
ejpam-6039	288	7	.	.	PUNCT
ejpam-6039	289	1	indian	indian	ADJ
ejpam-6039	289	2	journal	journal	PROPN
ejpam-6039	289	3	of	of	ADP
ejpam-6039	289	4	mathematics	mathematics	PROPN
ejpam-6039	289	5	,	,	PUNCT
ejpam-6039	289	6	41:139–146	41:139–146	PROPN
ejpam-6039	289	7	,	,	PUNCT
ejpam-6039	289	8	1999	1999	NUM
ejpam-6039	289	9	.	.	PUNCT
ejpam-6039	290	1	[	[	X
ejpam-6039	290	2	7	7	X
ejpam-6039	290	3	]	]	X
ejpam-6039	290	4	t.	t.	PROPN
ejpam-6039	290	5	noiri	noiri	PROPN
ejpam-6039	290	6	,	,	PUNCT
ejpam-6039	290	7	b.	b.	PROPN
ejpam-6039	290	8	ahmad	ahmad	PROPN
ejpam-6039	290	9	,	,	PUNCT
ejpam-6039	290	10	and	and	CCONJ
ejpam-6039	290	11	m.	m.	PROPN
ejpam-6039	290	12	khan	khan	PROPN
ejpam-6039	290	13	.	.	PUNCT
ejpam-6039	291	1	almost	almost	ADV
ejpam-6039	291	2	s	s	NOUN
ejpam-6039	291	3	-	-	PUNCT
ejpam-6039	291	4	continuous	continuous	ADJ
ejpam-6039	291	5	functions	function	NOUN
ejpam-6039	291	6	.	.	PUNCT
ejpam-6039	292	1	kyungpook	kyungpook	PROPN
ejpam-6039	292	2	mathematical	mathematical	PROPN
ejpam-6039	292	3	journal	journal	PROPN
ejpam-6039	292	4	,	,	PUNCT
ejpam-6039	292	5	35:311–322	35:311–322	PROPN
ejpam-6039	292	6	,	,	PUNCT
ejpam-6039	292	7	1995	1995	NUM
ejpam-6039	292	8	.	.	PUNCT
ejpam-6039	293	1	[	[	X
ejpam-6039	293	2	8	8	X
ejpam-6039	293	3	]	]	PUNCT
ejpam-6039	293	4	t.	t.	PROPN
ejpam-6039	293	5	noiri	noiri	PROPN
ejpam-6039	293	6	.	.	PUNCT
ejpam-6039	294	1	super	super	ADJ
ejpam-6039	294	2	-	-	NOUN
ejpam-6039	294	3	continuity	continuity	NOUN
ejpam-6039	294	4	and	and	CCONJ
ejpam-6039	294	5	some	some	DET
ejpam-6039	294	6	strong	strong	ADJ
ejpam-6039	294	7	forms	form	NOUN
ejpam-6039	294	8	of	of	ADP
ejpam-6039	294	9	continuity	continuity	NOUN
ejpam-6039	294	10	.	.	PUNCT
ejpam-6039	295	1	indian	indian	ADJ
ejpam-6039	295	2	journal	journal	PROPN
ejpam-6039	295	3	of	of	ADP
ejpam-6039	295	4	pure	pure	ADJ
ejpam-6039	295	5	and	and	CCONJ
ejpam-6039	295	6	applied	applied	ADJ
ejpam-6039	295	7	mathematics	mathematic	NOUN
ejpam-6039	295	8	,	,	PUNCT
ejpam-6039	295	9	15:241–250	15:241–250	NUM
ejpam-6039	295	10	,	,	PUNCT
ejpam-6039	295	11	1984	1984	NUM
ejpam-6039	295	12	.	.	PUNCT
ejpam-6039	296	1	[	[	X
ejpam-6039	296	2	9	9	NUM
ejpam-6039	296	3	]	]	PUNCT
ejpam-6039	296	4	a.	a.	NOUN
ejpam-6039	296	5	al	al	PROPN
ejpam-6039	296	6	-	-	PUNCT
ejpam-6039	296	7	omari	omari	PROPN
ejpam-6039	296	8	and	and	CCONJ
ejpam-6039	296	9	m.	m.	PROPN
ejpam-6039	296	10	s.	s.	PROPN
ejpam-6039	296	11	m.	m.	PROPN
ejpam-6039	296	12	noorani	noorani	PROPN
ejpam-6039	296	13	.	.	PUNCT
ejpam-6039	297	1	contra	contra	PROPN
ejpam-6039	297	2	ω	ω	PROPN
ejpam-6039	297	3	-	-	ADJ
ejpam-6039	297	4	continuous	continuous	ADJ
ejpam-6039	297	5	and	and	CCONJ
ejpam-6039	297	6	almost	almost	ADV
ejpam-6039	297	7	contra	contra	PROPN
ejpam-6039	297	8	ωcontinuous	ωcontinuous	ADJ
ejpam-6039	297	9	functions	function	NOUN
ejpam-6039	297	10	.	.	PUNCT
ejpam-6039	298	1	international	international	ADJ
ejpam-6039	298	2	journal	journal	PROPN
ejpam-6039	298	3	of	of	ADP
ejpam-6039	298	4	mathematics	mathematics	PROPN
ejpam-6039	298	5	and	and	CCONJ
ejpam-6039	298	6	mathematical	mathematical	ADJ
ejpam-6039	298	7	sciences	science	NOUN
ejpam-6039	298	8	,	,	PUNCT
ejpam-6039	298	9	2007:40469	2007:40469	NUM
ejpam-6039	298	10	,	,	PUNCT
ejpam-6039	298	11	2007	2007	NUM
ejpam-6039	298	12	.	.	PUNCT
ejpam-6039	299	1	[	[	X
ejpam-6039	299	2	10	10	NUM
ejpam-6039	299	3	]	]	PUNCT
ejpam-6039	299	4	t.	t.	PROPN
ejpam-6039	299	5	noiri	noiri	PROPN
ejpam-6039	299	6	and	and	CCONJ
ejpam-6039	299	7	v.	v.	ADP
ejpam-6039	299	8	popa	popa	NOUN
ejpam-6039	299	9	.	.	PUNCT
ejpam-6039	300	1	a	a	DET
ejpam-6039	300	2	unified	unified	ADJ
ejpam-6039	300	3	theory	theory	NOUN
ejpam-6039	300	4	of	of	ADP
ejpam-6039	300	5	contra	contra	PROPN
ejpam-6039	300	6	-	-	NOUN
ejpam-6039	300	7	continuity	continuity	NOUN
ejpam-6039	300	8	for	for	ADP
ejpam-6039	300	9	functions	function	NOUN
ejpam-6039	300	10	.	.	PUNCT
ejpam-6039	301	1	annales	annales	PROPN
ejpam-6039	301	2	universitatis	universitatis	PROPN
ejpam-6039	301	3	scientiarum	scientiarum	PROPN
ejpam-6039	301	4	budapestinensis	budapestinensis	PROPN
ejpam-6039	301	5	de	de	PROPN
ejpam-6039	301	6	rolando	rolando	PROPN
ejpam-6039	301	7	eötvös	eötvös	PROPN
ejpam-6039	301	8	,	,	PUNCT
ejpam-6039	301	9	sectio	sectio	PROPN
ejpam-6039	301	10	mathematica	mathematica	PROPN
ejpam-6039	301	11	,	,	PUNCT
ejpam-6039	301	12	44:115–137	44:115–137	PROPN
ejpam-6039	301	13	,	,	PUNCT
ejpam-6039	301	14	2002	2002	NUM
ejpam-6039	301	15	.	.	PUNCT
ejpam-6039	302	1	[	[	X
ejpam-6039	302	2	11	11	NUM
ejpam-6039	302	3	]	]	X
ejpam-6039	302	4	c.	c.	PROPN
ejpam-6039	302	5	w.	w.	PROPN
ejpam-6039	302	6	baker	baker	PROPN
ejpam-6039	302	7	.	.	PUNCT
ejpam-6039	303	1	weakly	weakly	ADJ
ejpam-6039	303	2	contra	contra	ADJ
ejpam-6039	303	3	-	-	ADJ
ejpam-6039	303	4	continuous	continuous	ADJ
ejpam-6039	303	5	functions	function	NOUN
ejpam-6039	303	6	.	.	PUNCT
ejpam-6039	304	1	international	international	ADJ
ejpam-6039	304	2	journal	journal	NOUN
ejpam-6039	304	3	of	of	ADP
ejpam-6039	304	4	pure	pure	ADJ
ejpam-6039	304	5	and	and	CCONJ
ejpam-6039	304	6	applied	applied	ADJ
ejpam-6039	304	7	mathematics	mathematic	NOUN
ejpam-6039	304	8	,	,	PUNCT
ejpam-6039	304	9	40:265–271	40:265–271	PROPN
ejpam-6039	304	10	,	,	PUNCT
ejpam-6039	304	11	2007	2007	NUM
ejpam-6039	304	12	.	.	PUNCT
ejpam-6039	305	1	[	[	X
ejpam-6039	305	2	12	12	NUM
ejpam-6039	305	3	]	]	X
ejpam-6039	305	4	c.	c.	PROPN
ejpam-6039	305	5	boonpok	boonpok	PROPN
ejpam-6039	305	6	and	and	CCONJ
ejpam-6039	305	7	n.	n.	PROPN
ejpam-6039	305	8	srisarakham	srisarakham	PROPN
ejpam-6039	305	9	.	.	PUNCT
ejpam-6039	306	1	(	(	PUNCT
ejpam-6039	306	2	τ1	τ1	NOUN
ejpam-6039	306	3	,	,	PUNCT
ejpam-6039	306	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-6039	306	5	for	for	ADP
ejpam-6039	306	6	functions	function	NOUN
ejpam-6039	306	7	.	.	PUNCT
ejpam-6039	307	1	asia	asia	PROPN
ejpam-6039	307	2	pacific	pacific	PROPN
ejpam-6039	307	3	journal	journal	PROPN
ejpam-6039	307	4	of	of	ADP
ejpam-6039	307	5	mathematics	mathematic	NOUN
ejpam-6039	307	6	,	,	PUNCT
ejpam-6039	307	7	11:21	11:21	NUM
ejpam-6039	307	8	,	,	PUNCT
ejpam-6039	307	9	2024	2024	NUM
ejpam-6039	307	10	.	.	PUNCT
ejpam-6039	308	1	[	[	X
ejpam-6039	308	2	13	13	NUM
ejpam-6039	308	3	]	]	PUNCT
ejpam-6039	308	4	c.	c.	PROPN
ejpam-6039	308	5	boonpok	boonpok	PROPN
ejpam-6039	308	6	and	and	CCONJ
ejpam-6039	308	7	p.	p.	NOUN
ejpam-6039	308	8	pue	pue	NOUN
ejpam-6039	308	9	-	-	PUNCT
ejpam-6039	308	10	on	on	ADP
ejpam-6039	308	11	.	.	PUNCT
ejpam-6039	309	1	characterizations	characterization	NOUN
ejpam-6039	309	2	of	of	ADP
ejpam-6039	309	3	almost	almost	ADV
ejpam-6039	309	4	(	(	PUNCT
ejpam-6039	309	5	τ1	τ1	NOUN
ejpam-6039	309	6	,	,	PUNCT
ejpam-6039	309	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	309	8	functions	function	NOUN
ejpam-6039	309	9	.	.	PUNCT
ejpam-6039	310	1	international	international	ADJ
ejpam-6039	310	2	journal	journal	NOUN
ejpam-6039	310	3	of	of	ADP
ejpam-6039	310	4	analysis	analysis	NOUN
ejpam-6039	310	5	and	and	CCONJ
ejpam-6039	310	6	applications	application	NOUN
ejpam-6039	310	7	,	,	PUNCT
ejpam-6039	310	8	22:33	22:33	NUM
ejpam-6039	310	9	,	,	PUNCT
ejpam-6039	310	10	2024	2024	NUM
ejpam-6039	310	11	.	.	PUNCT
ejpam-6039	311	1	[	[	X
ejpam-6039	311	2	14	14	NUM
ejpam-6039	311	3	]	]	X
ejpam-6039	311	4	c.	c.	PROPN
ejpam-6039	311	5	boonpok	boonpok	PROPN
ejpam-6039	311	6	and	and	CCONJ
ejpam-6039	311	7	c.	c.	PROPN
ejpam-6039	311	8	klanarong	klanarong	PROPN
ejpam-6039	311	9	.	.	PUNCT
ejpam-6039	312	1	on	on	ADP
ejpam-6039	312	2	weakly	weakly	ADJ
ejpam-6039	312	3	(	(	PUNCT
ejpam-6039	312	4	τ1	τ1	NOUN
ejpam-6039	312	5	,	,	PUNCT
ejpam-6039	312	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	312	7	functions	function	NOUN
ejpam-6039	312	8	.	.	PUNCT
ejpam-6039	313	1	european	european	ADJ
ejpam-6039	313	2	journal	journal	PROPN
ejpam-6039	313	3	of	of	ADP
ejpam-6039	313	4	pure	pure	ADJ
ejpam-6039	313	5	and	and	CCONJ
ejpam-6039	313	6	applied	applied	ADJ
ejpam-6039	313	7	mathematics	mathematic	NOUN
ejpam-6039	313	8	,	,	PUNCT
ejpam-6039	313	9	17(1):416–425	17(1):416–425	NUM
ejpam-6039	313	10	,	,	PUNCT
ejpam-6039	313	11	2024	2024	NUM
ejpam-6039	313	12	.	.	PUNCT
ejpam-6039	314	1	[	[	X
ejpam-6039	314	2	15	15	NUM
ejpam-6039	314	3	]	]	X
ejpam-6039	314	4	n.	n.	NOUN
ejpam-6039	314	5	srisarakham	srisarakham	PROPN
ejpam-6039	314	6	,	,	PUNCT
ejpam-6039	314	7	s.	s.	PROPN
ejpam-6039	314	8	sompong	sompong	PROPN
ejpam-6039	314	9	,	,	PUNCT
ejpam-6039	314	10	and	and	CCONJ
ejpam-6039	314	11	c.	c.	PROPN
ejpam-6039	314	12	boonpok	boonpok	PROPN
ejpam-6039	314	13	.	.	PUNCT
ejpam-6039	315	1	quasi	quasi	PROPN
ejpam-6039	315	2	θ(τ1	θ(τ1	PROPN
ejpam-6039	315	3	,	,	PUNCT
ejpam-6039	315	4	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	315	5	functions	function	NOUN
ejpam-6039	315	6	.	.	PUNCT
ejpam-6039	316	1	european	european	ADJ
ejpam-6039	316	2	journal	journal	PROPN
ejpam-6039	316	3	of	of	ADP
ejpam-6039	316	4	pure	pure	ADJ
ejpam-6039	316	5	and	and	CCONJ
ejpam-6039	316	6	applied	applied	ADJ
ejpam-6039	316	7	mathematics	mathematic	NOUN
ejpam-6039	316	8	,	,	PUNCT
ejpam-6039	316	9	18(1):5722	18(1):5722	NUM
ejpam-6039	316	10	,	,	PUNCT
ejpam-6039	316	11	2025	2025	NUM
ejpam-6039	316	12	.	.	PUNCT
ejpam-6039	317	1	[	[	X
ejpam-6039	317	2	16	16	NUM
ejpam-6039	317	3	]	]	X
ejpam-6039	317	4	c.	c.	PROPN
ejpam-6039	317	5	prachanpol	prachanpol	NOUN
ejpam-6039	317	6	,	,	PUNCT
ejpam-6039	317	7	c.	c.	PROPN
ejpam-6039	317	8	boonpok	boonpok	PROPN
ejpam-6039	317	9	,	,	PUNCT
ejpam-6039	317	10	and	and	CCONJ
ejpam-6039	317	11	c.	c.	PROPN
ejpam-6039	317	12	viriyapong	viriyapong	PROPN
ejpam-6039	317	13	.	.	PUNCT
ejpam-6039	318	1	δ(τ1	δ(τ1	PROPN
ejpam-6039	318	2	,	,	PUNCT
ejpam-6039	318	3	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	318	4	functions	function	NOUN
ejpam-6039	318	5	.	.	PUNCT
ejpam-6039	319	1	european	european	ADJ
ejpam-6039	319	2	journal	journal	PROPN
ejpam-6039	319	3	of	of	ADP
ejpam-6039	319	4	pure	pure	ADJ
ejpam-6039	319	5	and	and	CCONJ
ejpam-6039	319	6	applied	applied	ADJ
ejpam-6039	319	7	mathematics	mathematic	NOUN
ejpam-6039	319	8	,	,	PUNCT
ejpam-6039	319	9	17(4):3730–3742	17(4):3730–3742	NUM
ejpam-6039	319	10	,	,	PUNCT
ejpam-6039	319	11	2024	2024	NUM
ejpam-6039	319	12	.	.	PUNCT
ejpam-6039	320	1	[	[	X
ejpam-6039	320	2	17	17	NUM
ejpam-6039	320	3	]	]	X
ejpam-6039	320	4	b.	b.	PROPN
ejpam-6039	320	5	kong	kong	PROPN
ejpam-6039	320	6	-	-	PUNCT
ejpam-6039	320	7	ied	ied	PROPN
ejpam-6039	320	8	,	,	PUNCT
ejpam-6039	320	9	s.	s.	PROPN
ejpam-6039	320	10	sompong	sompong	PROPN
ejpam-6039	320	11	,	,	PUNCT
ejpam-6039	320	12	and	and	CCONJ
ejpam-6039	320	13	c.	c.	PROPN
ejpam-6039	320	14	boonpok	boonpok	PROPN
ejpam-6039	320	15	.	.	PUNCT
ejpam-6039	321	1	almost	almost	ADV
ejpam-6039	321	2	quasi	quasi	X
ejpam-6039	321	3	(	(	PUNCT
ejpam-6039	321	4	τ1	τ1	NOUN
ejpam-6039	321	5	,	,	PUNCT
ejpam-6039	321	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	321	7	functions	function	NOUN
ejpam-6039	321	8	.	.	PUNCT
ejpam-6039	322	1	asia	asia	PROPN
ejpam-6039	322	2	pacific	pacific	PROPN
ejpam-6039	322	3	journal	journal	PROPN
ejpam-6039	322	4	of	of	ADP
ejpam-6039	322	5	mathematics	mathematic	NOUN
ejpam-6039	322	6	,	,	PUNCT
ejpam-6039	322	7	11:64	11:64	NUM
ejpam-6039	322	8	,	,	PUNCT
ejpam-6039	322	9	2024	2024	NUM
ejpam-6039	322	10	.	.	PUNCT
ejpam-6039	323	1	[	[	X
ejpam-6039	323	2	18	18	NUM
ejpam-6039	323	3	]	]	PUNCT
ejpam-6039	323	4	m.	m.	NOUN
ejpam-6039	323	5	chiangpradit	chiangpradit	NOUN
ejpam-6039	323	6	,	,	PUNCT
ejpam-6039	323	7	s.	s.	PROPN
ejpam-6039	323	8	sompong	sompong	PROPN
ejpam-6039	323	9	,	,	PUNCT
ejpam-6039	323	10	and	and	CCONJ
ejpam-6039	323	11	c.	c.	PROPN
ejpam-6039	323	12	boonpok	boonpok	PROPN
ejpam-6039	323	13	.	.	PUNCT
ejpam-6039	324	1	weakly	weakly	ADJ
ejpam-6039	324	2	quasi	quasi	NOUN
ejpam-6039	324	3	(	(	PUNCT
ejpam-6039	324	4	τ1	τ1	PROPN
ejpam-6039	324	5	,	,	PUNCT
ejpam-6039	324	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	324	7	functions	function	NOUN
ejpam-6039	324	8	.	.	PUNCT
ejpam-6039	325	1	international	international	ADJ
ejpam-6039	325	2	journal	journal	NOUN
ejpam-6039	325	3	of	of	ADP
ejpam-6039	325	4	analysis	analysis	NOUN
ejpam-6039	325	5	and	and	CCONJ
ejpam-6039	325	6	applications	application	NOUN
ejpam-6039	325	7	,	,	PUNCT
ejpam-6039	325	8	22:125	22:125	NUM
ejpam-6039	325	9	,	,	PUNCT
ejpam-6039	325	10	2024	2024	NUM
ejpam-6039	325	11	.	.	PUNCT
ejpam-6039	326	1	[	[	X
ejpam-6039	326	2	19	19	NUM
ejpam-6039	326	3	]	]	X
ejpam-6039	326	4	n.	n.	NOUN
ejpam-6039	326	5	srisarakham	srisarakham	PROPN
ejpam-6039	326	6	,	,	PUNCT
ejpam-6039	326	7	a.	a.	PROPN
ejpam-6039	326	8	sama	sama	PROPN
ejpam-6039	326	9	-	-	PUNCT
ejpam-6039	326	10	ae	ae	PROPN
ejpam-6039	326	11	,	,	PUNCT
ejpam-6039	326	12	and	and	CCONJ
ejpam-6039	326	13	c.	c.	PROPN
ejpam-6039	326	14	boonpok	boonpok	PROPN
ejpam-6039	326	15	.	.	PUNCT
ejpam-6039	327	1	characterizations	characterization	NOUN
ejpam-6039	327	2	of	of	ADP
ejpam-6039	327	3	faintly	faintly	ADV
ejpam-6039	327	4	(	(	PUNCT
ejpam-6039	327	5	τ1	τ1	PROPN
ejpam-6039	327	6	,	,	PUNCT
ejpam-6039	327	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	327	8	functions	function	NOUN
ejpam-6039	327	9	.	.	PUNCT
ejpam-6039	328	1	european	european	ADJ
ejpam-6039	328	2	journal	journal	PROPN
ejpam-6039	328	3	of	of	ADP
ejpam-6039	328	4	pure	pure	ADJ
ejpam-6039	328	5	and	and	CCONJ
ejpam-6039	328	6	applied	applied	ADJ
ejpam-6039	328	7	mathematics	mathematic	NOUN
ejpam-6039	328	8	,	,	PUNCT
ejpam-6039	328	9	17(4):2753–2762	17(4):2753–2762	NUM
ejpam-6039	328	10	,	,	PUNCT
ejpam-6039	328	11	2024	2024	NUM
ejpam-6039	328	12	.	.	PUNCT
ejpam-6039	328	13	m.	m.	NOUN
ejpam-6039	328	14	chiangpradit	chiangpradit	PROPN
ejpam-6039	328	15	,	,	PUNCT
ejpam-6039	328	16	s.	s.	PROPN
ejpam-6039	328	17	sompong	sompong	PROPN
ejpam-6039	328	18	,	,	PUNCT
ejpam-6039	328	19	c.	c.	PROPN
ejpam-6039	328	20	boonpok	boonpok	PROPN
ejpam-6039	328	21	/	/	SYM
ejpam-6039	328	22	eur	eur	PROPN
ejpam-6039	328	23	.	.	PUNCT
ejpam-6039	329	1	j.	j.	PROPN
ejpam-6039	329	2	pure	pure	PROPN
ejpam-6039	329	3	appl	appl	PROPN
ejpam-6039	329	4	.	.	PROPN
ejpam-6039	329	5	math	math	PROPN
ejpam-6039	329	6	,	,	PUNCT
ejpam-6039	329	7	18	18	NUM
ejpam-6039	329	8	(	(	PUNCT
ejpam-6039	329	9	2	2	NUM
ejpam-6039	329	10	)	)	PUNCT
ejpam-6039	329	11	(	(	PUNCT
ejpam-6039	329	12	2025	2025	NUM
ejpam-6039	329	13	)	)	PUNCT
ejpam-6039	329	14	,	,	PUNCT
ejpam-6039	329	15	6039	6039	NUM
ejpam-6039	329	16	11	11	NUM
ejpam-6039	329	17	of	of	ADP
ejpam-6039	329	18	11	11	NUM
ejpam-6039	329	19	[	[	SYM
ejpam-6039	329	20	20	20	NUM
ejpam-6039	329	21	]	]	PUNCT
ejpam-6039	329	22	b.	b.	PROPN
ejpam-6039	329	23	kong	kong	PROPN
ejpam-6039	329	24	-	-	PUNCT
ejpam-6039	329	25	ied	ied	PROPN
ejpam-6039	329	26	,	,	PUNCT
ejpam-6039	329	27	a.	a.	PROPN
ejpam-6039	329	28	sama	sama	PROPN
ejpam-6039	329	29	-	-	PUNCT
ejpam-6039	329	30	ae	ae	PROPN
ejpam-6039	329	31	,	,	PUNCT
ejpam-6039	329	32	and	and	CCONJ
ejpam-6039	329	33	c.	c.	PROPN
ejpam-6039	329	34	boonpok	boonpok	PROPN
ejpam-6039	329	35	.	.	PUNCT
ejpam-6039	330	1	almost	almost	ADV
ejpam-6039	330	2	nearly	nearly	ADV
ejpam-6039	330	3	(	(	PUNCT
ejpam-6039	330	4	τ1	τ1	NOUN
ejpam-6039	330	5	,	,	PUNCT
ejpam-6039	330	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	330	7	functions	function	NOUN
ejpam-6039	330	8	.	.	PUNCT
ejpam-6039	331	1	international	international	ADJ
ejpam-6039	331	2	journal	journal	NOUN
ejpam-6039	331	3	of	of	ADP
ejpam-6039	331	4	analysis	analysis	NOUN
ejpam-6039	331	5	and	and	CCONJ
ejpam-6039	331	6	applications	application	NOUN
ejpam-6039	331	7	,	,	PUNCT
ejpam-6039	331	8	23:14	23:14	NUM
ejpam-6039	331	9	,	,	PUNCT
ejpam-6039	331	10	2025	2025	NUM
ejpam-6039	331	11	.	.	PUNCT
ejpam-6039	332	1	[	[	X
ejpam-6039	332	2	21	21	NUM
ejpam-6039	332	3	]	]	X
ejpam-6039	332	4	c.	c.	PROPN
ejpam-6039	332	5	boonpok	boonpok	PROPN
ejpam-6039	332	6	,	,	PUNCT
ejpam-6039	332	7	c.	c.	PROPN
ejpam-6039	332	8	viriyapong	viriyapong	PROPN
ejpam-6039	332	9	,	,	PUNCT
ejpam-6039	332	10	and	and	CCONJ
ejpam-6039	332	11	m.	m.	NOUN
ejpam-6039	332	12	thongmoon	thongmoon	NOUN
ejpam-6039	332	13	.	.	PUNCT
ejpam-6039	333	1	on	on	ADP
ejpam-6039	333	2	upper	upper	ADJ
ejpam-6039	333	3	and	and	CCONJ
ejpam-6039	333	4	lower	low	ADJ
ejpam-6039	333	5	(	(	PUNCT
ejpam-6039	333	6	τ1	τ1	NOUN
ejpam-6039	333	7	,	,	PUNCT
ejpam-6039	333	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-6039	333	9	multifunctions	multifunction	NOUN
ejpam-6039	333	10	.	.	PUNCT
ejpam-6039	334	1	journal	journal	PROPN
ejpam-6039	334	2	of	of	ADP
ejpam-6039	334	3	mathematics	mathematics	PROPN
ejpam-6039	334	4	and	and	CCONJ
ejpam-6039	334	5	computer	computer	NOUN
ejpam-6039	334	6	science	science	NOUN
ejpam-6039	334	7	,	,	PUNCT
ejpam-6039	334	8	18:282–293	18:282–293	NUM
ejpam-6039	334	9	,	,	PUNCT
ejpam-6039	334	10	2018	2018	NUM
ejpam-6039	334	11	.	.	PUNCT
ejpam-6039	335	1	[	[	X
ejpam-6039	335	2	22	22	NUM
ejpam-6039	335	3	]	]	X
ejpam-6039	335	4	c.	c.	PROPN
ejpam-6039	335	5	viriyapong	viriyapong	PROPN
ejpam-6039	335	6	and	and	CCONJ
ejpam-6039	335	7	c.	c.	PROPN
ejpam-6039	335	8	boonpok	boonpok	PROPN
ejpam-6039	335	9	.	.	PUNCT
ejpam-6039	336	1	(	(	PUNCT
ejpam-6039	336	2	τ1	τ1	NOUN
ejpam-6039	336	3	,	,	PUNCT
ejpam-6039	336	4	τ2)α	τ2)α	NOUN
ejpam-6039	336	5	-	-	PUNCT
ejpam-6039	336	6	continuity	continuity	NOUN
ejpam-6039	336	7	for	for	ADP
ejpam-6039	336	8	multifunctions	multifunction	NOUN
ejpam-6039	336	9	.	.	PUNCT
ejpam-6039	337	1	journal	journal	PROPN
ejpam-6039	337	2	of	of	ADP
ejpam-6039	337	3	mathematics	mathematic	NOUN
ejpam-6039	337	4	,	,	PUNCT
ejpam-6039	337	5	2020:6285763	2020:6285763	NUM
ejpam-6039	337	6	,	,	PUNCT
ejpam-6039	337	7	2020	2020	NUM
ejpam-6039	337	8	.	.	PUNCT
ejpam-6039	338	1	[	[	X
ejpam-6039	338	2	23	23	NUM
ejpam-6039	338	3	]	]	X
ejpam-6039	338	4	c.	c.	PROPN
ejpam-6039	338	5	boonpok	boonpok	PROPN
ejpam-6039	338	6	.	.	PUNCT
ejpam-6039	339	1	(	(	PUNCT
ejpam-6039	339	2	τ1	τ1	NOUN
ejpam-6039	339	3	,	,	PUNCT
ejpam-6039	339	4	τ2)δ	τ2)δ	ADJ
ejpam-6039	339	5	-	-	PUNCT
ejpam-6039	339	6	semicontinuous	semicontinuous	ADJ
ejpam-6039	339	7	multifunctions	multifunction	NOUN
ejpam-6039	339	8	.	.	PUNCT
ejpam-6039	340	1	heliyon	heliyon	NOUN
ejpam-6039	340	2	,	,	PUNCT
ejpam-6039	340	3	6	6	NUM
ejpam-6039	340	4	:	:	SYM
ejpam-6039	340	5	e05367	e05367	PROPN
ejpam-6039	340	6	,	,	PUNCT
ejpam-6039	340	7	2020	2020	NUM
ejpam-6039	340	8	.	.	PUNCT
ejpam-6039	341	1	[	[	X
ejpam-6039	341	2	24	24	NUM
ejpam-6039	341	3	]	]	X
ejpam-6039	341	4	n.	n.	PROPN
ejpam-6039	341	5	viriyapong	viriyapong	PROPN
ejpam-6039	341	6	,	,	PUNCT
ejpam-6039	341	7	s.	s.	PROPN
ejpam-6039	341	8	sompong	sompong	PROPN
ejpam-6039	341	9	,	,	PUNCT
ejpam-6039	341	10	and	and	CCONJ
ejpam-6039	341	11	c.	c.	PROPN
ejpam-6039	341	12	boonpok	boonpok	PROPN
ejpam-6039	341	13	.	.	PUNCT
ejpam-6039	342	1	(	(	PUNCT
ejpam-6039	342	2	τ1	τ1	NOUN
ejpam-6039	342	3	,	,	PUNCT
ejpam-6039	342	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-6039	342	5	disconnectedness	disconnectedness	NOUN
ejpam-6039	342	6	in	in	ADP
ejpam-6039	342	7	bitopological	bitopological	ADJ
ejpam-6039	342	8	spaces	space	NOUN
ejpam-6039	342	9	.	.	PUNCT
ejpam-6039	343	1	international	international	ADJ
ejpam-6039	343	2	journal	journal	PROPN
ejpam-6039	343	3	of	of	ADP
ejpam-6039	343	4	mathematics	mathematic	NOUN
ejpam-6039	343	5	and	and	CCONJ
ejpam-6039	343	6	computer	computer	NOUN
ejpam-6039	343	7	science	science	NOUN
ejpam-6039	343	8	,	,	PUNCT
ejpam-6039	343	9	19(3):855–860	19(3):855–860	PROPN
ejpam-6039	343	10	,	,	PUNCT
ejpam-6039	343	11	2024	2024	NUM
ejpam-6039	343	12	.	.	PUNCT
ejpam-6039	344	1	[	[	X
ejpam-6039	344	2	25	25	NUM
ejpam-6039	344	3	]	]	X
ejpam-6039	344	4	n.	n.	PROPN
ejpam-6039	344	5	viriyapong	viriyapong	PROPN
ejpam-6039	344	6	,	,	PUNCT
ejpam-6039	344	7	s.	s.	PROPN
ejpam-6039	344	8	sompong	sompong	PROPN
ejpam-6039	344	9	,	,	PUNCT
ejpam-6039	344	10	and	and	CCONJ
ejpam-6039	344	11	c.	c.	PROPN
ejpam-6039	344	12	boonpok	boonpok	PROPN
ejpam-6039	344	13	.	.	PUNCT
ejpam-6039	345	1	upper	upper	ADJ
ejpam-6039	345	2	and	and	CCONJ
ejpam-6039	345	3	lower	low	ADJ
ejpam-6039	345	4	s-(τ1	s-(τ1	NOUN
ejpam-6039	345	5	,	,	PUNCT
ejpam-6039	345	6	τ2)p	τ2)p	ADJ
ejpam-6039	345	7	-	-	PUNCT
ejpam-6039	345	8	continuous	continuous	ADJ
ejpam-6039	345	9	multifunctions	multifunction	NOUN
ejpam-6039	345	10	.	.	PUNCT
ejpam-6039	346	1	european	european	ADJ
ejpam-6039	346	2	journal	journal	PROPN
ejpam-6039	346	3	of	of	ADP
ejpam-6039	346	4	pure	pure	ADJ
ejpam-6039	346	5	and	and	CCONJ
ejpam-6039	346	6	applied	applied	ADJ
ejpam-6039	346	7	mathematics	mathematic	NOUN
ejpam-6039	346	8	,	,	PUNCT
ejpam-6039	346	9	17(3):2210–2220	17(3):2210–2220	NUM
ejpam-6039	346	10	,	,	PUNCT
ejpam-6039	346	11	2024	2024	NUM
ejpam-6039	346	12	.	.	PUNCT
ejpam-6039	347	1	[	[	X
ejpam-6039	347	2	26	26	NUM
ejpam-6039	347	3	]	]	X
ejpam-6039	347	4	j.	j.	PROPN
ejpam-6039	347	5	khampakdee	khampakdee	PROPN
ejpam-6039	347	6	,	,	PUNCT
ejpam-6039	347	7	s.	s.	PROPN
ejpam-6039	347	8	sompong	sompong	PROPN
ejpam-6039	347	9	,	,	PUNCT
ejpam-6039	347	10	and	and	CCONJ
ejpam-6039	347	11	c.	c.	PROPN
ejpam-6039	347	12	boonpok	boonpok	PROPN
ejpam-6039	347	13	.	.	PUNCT
ejpam-6039	348	1	almost	almost	ADV
ejpam-6039	348	2	weakly	weakly	ADJ
ejpam-6039	348	3	(	(	PUNCT
ejpam-6039	348	4	τ1	τ1	NOUN
ejpam-6039	348	5	,	,	PUNCT
ejpam-6039	348	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-6039	348	7	functions	function	NOUN
ejpam-6039	348	8	.	.	PUNCT
ejpam-6039	349	1	european	european	ADJ
ejpam-6039	349	2	journal	journal	PROPN
ejpam-6039	349	3	of	of	ADP
ejpam-6039	349	4	pure	pure	ADJ
ejpam-6039	349	5	and	and	CCONJ
ejpam-6039	349	6	applied	applied	ADJ
ejpam-6039	349	7	mathematics	mathematic	NOUN
ejpam-6039	349	8	,	,	PUNCT
ejpam-6039	349	9	18(1):5721	18(1):5721	NUM
ejpam-6039	349	10	,	,	PUNCT
ejpam-6039	349	11	2025	2025	NUM
ejpam-6039	349	12	.	.	PUNCT
ejpam-6039	350	1	[	[	X
ejpam-6039	350	2	27	27	NUM
ejpam-6039	350	3	]	]	PUNCT
ejpam-6039	350	4	m.	m.	NOUN
ejpam-6039	350	5	chiangpradit	chiangpradit	NOUN
ejpam-6039	350	6	,	,	PUNCT
ejpam-6039	350	7	s.	s.	PROPN
ejpam-6039	350	8	sompong	sompong	PROPN
ejpam-6039	350	9	,	,	PUNCT
ejpam-6039	350	10	and	and	CCONJ
ejpam-6039	350	11	c.	c.	PROPN
ejpam-6039	350	12	boonpok	boonpok	PROPN
ejpam-6039	350	13	.	.	PUNCT
ejpam-6039	351	1	on	on	ADP
ejpam-6039	351	2	characterizations	characterization	NOUN
ejpam-6039	351	3	of	of	ADP
ejpam-6039	351	4	(	(	PUNCT
ejpam-6039	351	5	τ1	τ1	NOUN
ejpam-6039	351	6	,	,	PUNCT
ejpam-6039	351	7	τ2)regular	τ2)regular	ADJ
ejpam-6039	351	8	spaces	space	NOUN
ejpam-6039	351	9	.	.	PUNCT
ejpam-6039	352	1	international	international	ADJ
ejpam-6039	352	2	journal	journal	PROPN
ejpam-6039	352	3	of	of	ADP
ejpam-6039	352	4	mathematics	mathematic	NOUN
ejpam-6039	352	5	and	and	CCONJ
ejpam-6039	352	6	computer	computer	NOUN
ejpam-6039	352	7	science	science	NOUN
ejpam-6039	352	8	,	,	PUNCT
ejpam-6039	352	9	19(4):1329–1334	19(4):1329–1334	NUM
ejpam-6039	352	10	,	,	PUNCT
ejpam-6039	352	11	2024	2024	NUM
ejpam-6039	352	12	.	.	PUNCT
ejpam-6039	353	1	[	[	X
ejpam-6039	353	2	28	28	NUM
ejpam-6039	353	3	]	]	X
ejpam-6039	353	4	p.	p.	NOUN
ejpam-6039	353	5	pue	pue	NOUN
ejpam-6039	353	6	-	-	PUNCT
ejpam-6039	353	7	on	on	ADP
ejpam-6039	353	8	,	,	PUNCT
ejpam-6039	353	9	a.	a.	PROPN
ejpam-6039	353	10	sama	sama	PROPN
ejpam-6039	353	11	-	-	PUNCT
ejpam-6039	353	12	ae	ae	PROPN
ejpam-6039	353	13	,	,	PUNCT
ejpam-6039	353	14	and	and	CCONJ
ejpam-6039	353	15	c.	c.	PROPN
ejpam-6039	353	16	boonpok	boonpok	PROPN
ejpam-6039	353	17	.	.	PUNCT
ejpam-6039	354	1	characterizations	characterization	NOUN
ejpam-6039	354	2	of	of	ADP
ejpam-6039	354	3	quasi	quasi	NOUN
ejpam-6039	354	4	θ(τ1	θ(τ1	NOUN
ejpam-6039	354	5	,	,	PUNCT
ejpam-6039	354	6	τ2)continuous	τ2)continuous	ADJ
ejpam-6039	354	7	multifunctions	multifunction	NOUN
ejpam-6039	354	8	.	.	PUNCT
ejpam-6039	355	1	international	international	ADJ
ejpam-6039	355	2	journal	journal	NOUN
ejpam-6039	355	3	of	of	ADP
ejpam-6039	355	4	analysis	analysis	NOUN
ejpam-6039	355	5	and	and	CCONJ
ejpam-6039	355	6	applications	application	NOUN
ejpam-6039	355	7	,	,	PUNCT
ejpam-6039	355	8	23:59	23:59	NUM
ejpam-6039	355	9	,	,	PUNCT
ejpam-6039	355	10	2025	2025	NUM
ejpam-6039	355	11	.	.	PUNCT