id	sid	tid	token	lemma	pos
ejpam-5335	1	1	european	european	PROPN
ejpam-5335	1	2	journal	journal	PROPN
ejpam-5335	1	3	of	of	ADP
ejpam-5335	1	4	pure	pure	ADJ
ejpam-5335	1	5	and	and	CCONJ
ejpam-5335	1	6	applied	apply	VERB
ejpam-5335	1	7	mathematics	mathematic	NOUN
ejpam-5335	1	8	vol	vol	NOUN
ejpam-5335	1	9	.	.	PROPN
ejpam-5335	2	1	17	17	NUM
ejpam-5335	2	2	,	,	PUNCT
ejpam-5335	2	3	no	no	INTJ
ejpam-5335	2	4	.	.	NOUN
ejpam-5335	2	5	4	4	NUM
ejpam-5335	2	6	,	,	PUNCT
ejpam-5335	2	7	2024	2024	NUM
ejpam-5335	2	8	,	,	PUNCT
ejpam-5335	2	9	3730	3730	NUM
ejpam-5335	2	10	-	-	SYM
ejpam-5335	2	11	3742	3742	NUM
ejpam-5335	2	12	issn	issn	PROPN
ejpam-5335	2	13	1307	1307	NUM
ejpam-5335	2	14	-	-	SYM
ejpam-5335	2	15	5543	5543	NUM
ejpam-5335	2	16	–	–	PUNCT
ejpam-5335	2	17	ejpam.com	ejpam.com	X
ejpam-5335	2	18	published	publish	VERB
ejpam-5335	2	19	by	by	ADP
ejpam-5335	2	20	new	new	PROPN
ejpam-5335	2	21	york	york	PROPN
ejpam-5335	2	22	business	business	PROPN
ejpam-5335	2	23	global	global	PROPN
ejpam-5335	2	24	δ(τ1	δ(τ1	PROPN
ejpam-5335	2	25	,	,	PUNCT
ejpam-5335	2	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	2	27	functions	function	NOUN
ejpam-5335	2	28	chatchadaporn	chatchadaporn	VERB
ejpam-5335	2	29	prachanpol1	prachanpol1	ADV
ejpam-5335	2	30	,	,	PUNCT
ejpam-5335	2	31	chawalit	chawalit	VERB
ejpam-5335	2	32	boonpok1	boonpok1	PROPN
ejpam-5335	2	33	,	,	PUNCT
ejpam-5335	2	34	chokchai	chokchai	ADJ
ejpam-5335	2	35	viriyapong1,∗	viriyapong1,∗	NOUN
ejpam-5335	2	36	1	1	NUM
ejpam-5335	2	37	mathematics	mathematic	NOUN
ejpam-5335	2	38	and	and	CCONJ
ejpam-5335	2	39	applied	apply	VERB
ejpam-5335	2	40	mathematics	mathematics	PROPN
ejpam-5335	2	41	research	research	NOUN
ejpam-5335	2	42	unit	unit	NOUN
ejpam-5335	2	43	,	,	PUNCT
ejpam-5335	2	44	department	department	NOUN
ejpam-5335	2	45	of	of	ADP
ejpam-5335	2	46	mathematics	mathematic	NOUN
ejpam-5335	2	47	,	,	PUNCT
ejpam-5335	2	48	faculty	faculty	NOUN
ejpam-5335	2	49	of	of	ADP
ejpam-5335	2	50	science	science	NOUN
ejpam-5335	2	51	,	,	PUNCT
ejpam-5335	2	52	mahasarakham	mahasarakham	PROPN
ejpam-5335	2	53	university	university	PROPN
ejpam-5335	2	54	,	,	PUNCT
ejpam-5335	2	55	maha	maha	PROPN
ejpam-5335	2	56	sarakham	sarakham	PROPN
ejpam-5335	2	57	,	,	PUNCT
ejpam-5335	2	58	44150	44150	NUM
ejpam-5335	2	59	,	,	PUNCT
ejpam-5335	2	60	thailand	thailand	PROPN
ejpam-5335	2	61	abstract	abstract	PROPN
ejpam-5335	2	62	.	.	PUNCT
ejpam-5335	3	1	this	this	DET
ejpam-5335	3	2	paper	paper	NOUN
ejpam-5335	3	3	introduces	introduce	VERB
ejpam-5335	3	4	a	a	DET
ejpam-5335	3	5	new	new	ADJ
ejpam-5335	3	6	class	class	NOUN
ejpam-5335	3	7	of	of	ADP
ejpam-5335	3	8	functions	function	NOUN
ejpam-5335	3	9	called	call	VERB
ejpam-5335	3	10	δ(τ1	δ(τ1	NOUN
ejpam-5335	3	11	,	,	PUNCT
ejpam-5335	3	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	3	13	functions	function	NOUN
ejpam-5335	3	14	.	.	PUNCT
ejpam-5335	4	1	several	several	ADJ
ejpam-5335	4	2	characterizations	characterization	NOUN
ejpam-5335	4	3	of	of	ADP
ejpam-5335	4	4	δ(τ1	δ(τ1	NOUN
ejpam-5335	4	5	,	,	PUNCT
ejpam-5335	4	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	4	7	functions	function	NOUN
ejpam-5335	4	8	are	be	AUX
ejpam-5335	4	9	investigated	investigate	VERB
ejpam-5335	4	10	.	.	PUNCT
ejpam-5335	5	1	the	the	DET
ejpam-5335	5	2	relationships	relationship	NOUN
ejpam-5335	5	3	between	between	ADP
ejpam-5335	5	4	δ(τ1	δ(τ1	NOUN
ejpam-5335	5	5	,	,	PUNCT
ejpam-5335	5	6	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	5	7	and	and	CCONJ
ejpam-5335	5	8	the	the	DET
ejpam-5335	5	9	other	other	ADJ
ejpam-5335	5	10	types	type	NOUN
ejpam-5335	5	11	of	of	ADP
ejpam-5335	5	12	δ(τ1	δ(τ1	NOUN
ejpam-5335	5	13	,	,	PUNCT
ejpam-5335	5	14	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	5	15	are	be	AUX
ejpam-5335	5	16	also	also	ADV
ejpam-5335	5	17	discussed	discuss	VERB
ejpam-5335	5	18	.	.	PUNCT
ejpam-5335	6	1	2020	2020	NUM
ejpam-5335	6	2	mathematics	mathematic	NOUN
ejpam-5335	6	3	subject	subject	NOUN
ejpam-5335	6	4	classifications	classification	NOUN
ejpam-5335	6	5	:	:	PUNCT
ejpam-5335	6	6	54c05	54c05	NUM
ejpam-5335	6	7	,	,	PUNCT
ejpam-5335	6	8	54c08	54c08	NUM
ejpam-5335	6	9	,	,	PUNCT
ejpam-5335	6	10	54e55	54e55	NUM
ejpam-5335	6	11	key	key	ADJ
ejpam-5335	6	12	words	word	NOUN
ejpam-5335	6	13	and	and	CCONJ
ejpam-5335	6	14	phrases	phrase	NOUN
ejpam-5335	6	15	:	:	PUNCT
ejpam-5335	6	16	δ(τ1	δ(τ1	NOUN
ejpam-5335	6	17	,	,	PUNCT
ejpam-5335	6	18	τ2)-open	τ2)-open	ADJ
ejpam-5335	6	19	set	set	NOUN
ejpam-5335	6	20	,	,	PUNCT
ejpam-5335	6	21	δ(τ1	δ(τ1	PROPN
ejpam-5335	6	22	,	,	PUNCT
ejpam-5335	6	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	6	24	function	function	NOUN
ejpam-5335	6	25	1	1	NUM
ejpam-5335	6	26	.	.	PUNCT
ejpam-5335	6	27	introduction	introduction	NOUN
ejpam-5335	6	28	the	the	DET
ejpam-5335	6	29	field	field	NOUN
ejpam-5335	6	30	of	of	ADP
ejpam-5335	6	31	the	the	DET
ejpam-5335	6	32	mathematical	mathematical	ADJ
ejpam-5335	6	33	science	science	NOUN
ejpam-5335	6	34	which	which	PRON
ejpam-5335	6	35	goes	go	VERB
ejpam-5335	6	36	under	under	ADP
ejpam-5335	6	37	the	the	DET
ejpam-5335	6	38	name	name	NOUN
ejpam-5335	6	39	of	of	ADP
ejpam-5335	6	40	topology	topology	NOUN
ejpam-5335	6	41	is	be	AUX
ejpam-5335	6	42	concerned	concern	VERB
ejpam-5335	6	43	with	with	ADP
ejpam-5335	6	44	all	all	DET
ejpam-5335	6	45	questions	question	NOUN
ejpam-5335	6	46	directly	directly	ADV
ejpam-5335	6	47	or	or	CCONJ
ejpam-5335	6	48	indirectly	indirectly	ADV
ejpam-5335	6	49	related	relate	VERB
ejpam-5335	6	50	to	to	ADP
ejpam-5335	6	51	continuity	continuity	NOUN
ejpam-5335	6	52	.	.	PUNCT
ejpam-5335	7	1	by	by	ADP
ejpam-5335	7	2	using	use	VERB
ejpam-5335	7	3	various	various	ADJ
ejpam-5335	7	4	forms	form	NOUN
ejpam-5335	7	5	of	of	ADP
ejpam-5335	7	6	open	open	ADJ
ejpam-5335	7	7	sets	set	NOUN
ejpam-5335	7	8	many	many	ADJ
ejpam-5335	7	9	authors	author	NOUN
ejpam-5335	7	10	introduced	introduce	VERB
ejpam-5335	7	11	and	and	CCONJ
ejpam-5335	7	12	studied	study	VERB
ejpam-5335	7	13	various	various	ADJ
ejpam-5335	7	14	types	type	NOUN
ejpam-5335	7	15	of	of	ADP
ejpam-5335	7	16	continuity	continuity	NOUN
ejpam-5335	7	17	.	.	PUNCT
ejpam-5335	8	1	in	in	ADP
ejpam-5335	8	2	1968	1968	NUM
ejpam-5335	8	3	,	,	PUNCT
ejpam-5335	8	4	veličko	veličko	PROPN
ejpam-5335	8	5	[	[	X
ejpam-5335	8	6	22	22	NUM
ejpam-5335	8	7	]	]	PUNCT
ejpam-5335	8	8	introduced	introduce	VERB
ejpam-5335	8	9	a	a	DET
ejpam-5335	8	10	new	new	ADJ
ejpam-5335	8	11	class	class	NOUN
ejpam-5335	8	12	of	of	ADP
ejpam-5335	8	13	open	open	ADJ
ejpam-5335	8	14	sets	set	NOUN
ejpam-5335	8	15	in	in	ADP
ejpam-5335	8	16	topological	topological	ADJ
ejpam-5335	8	17	spaces	space	NOUN
ejpam-5335	8	18	called	call	VERB
ejpam-5335	8	19	δ	δ	NOUN
ejpam-5335	8	20	-	-	ADJ
ejpam-5335	8	21	open	open	ADJ
ejpam-5335	8	22	sets	set	NOUN
ejpam-5335	8	23	and	and	CCONJ
ejpam-5335	8	24	investigated	investigate	VERB
ejpam-5335	8	25	some	some	DET
ejpam-5335	8	26	properties	property	NOUN
ejpam-5335	8	27	of	of	ADP
ejpam-5335	8	28	δ	δ	NOUN
ejpam-5335	8	29	-	-	PUNCT
ejpam-5335	8	30	closed	close	VERB
ejpam-5335	8	31	sets	set	NOUN
ejpam-5335	8	32	and	and	CCONJ
ejpam-5335	8	33	δ	δ	NOUN
ejpam-5335	8	34	-	-	PUNCT
ejpam-5335	8	35	open	open	ADJ
ejpam-5335	8	36	sets	set	NOUN
ejpam-5335	8	37	.	.	PUNCT
ejpam-5335	9	1	the	the	DET
ejpam-5335	9	2	class	class	NOUN
ejpam-5335	9	3	of	of	ADP
ejpam-5335	9	4	open	open	ADJ
ejpam-5335	9	5	sets	set	NOUN
ejpam-5335	9	6	including	include	VERB
ejpam-5335	9	7	the	the	DET
ejpam-5335	9	8	class	class	NOUN
ejpam-5335	9	9	of	of	ADP
ejpam-5335	9	10	δ	δ	PROPN
ejpam-5335	9	11	-	-	PUNCT
ejpam-5335	9	12	open	open	ADJ
ejpam-5335	9	13	sets	set	NOUN
ejpam-5335	9	14	.	.	PUNCT
ejpam-5335	10	1	in	in	ADP
ejpam-5335	10	2	1980	1980	NUM
ejpam-5335	10	3	,	,	PUNCT
ejpam-5335	10	4	noiri	noiri	ADV
ejpam-5335	10	5	[	[	X
ejpam-5335	10	6	17	17	NUM
ejpam-5335	10	7	]	]	PUNCT
ejpam-5335	10	8	introduced	introduce	VERB
ejpam-5335	10	9	and	and	CCONJ
ejpam-5335	10	10	studied	study	VERB
ejpam-5335	10	11	the	the	DET
ejpam-5335	10	12	notion	notion	NOUN
ejpam-5335	10	13	of	of	ADP
ejpam-5335	10	14	δ	δ	NOUN
ejpam-5335	10	15	-	-	ADJ
ejpam-5335	10	16	continuous	continuous	ADJ
ejpam-5335	10	17	functions	function	NOUN
ejpam-5335	10	18	.	.	PUNCT
ejpam-5335	11	1	munshi	munshi	PROPN
ejpam-5335	11	2	and	and	CCONJ
ejpam-5335	11	3	bassan	bassan	NOUN
ejpam-5335	11	4	[	[	X
ejpam-5335	11	5	16	16	NUM
ejpam-5335	11	6	]	]	PUNCT
ejpam-5335	11	7	defined	define	VERB
ejpam-5335	11	8	and	and	CCONJ
ejpam-5335	11	9	developed	develop	VERB
ejpam-5335	11	10	the	the	DET
ejpam-5335	11	11	concept	concept	NOUN
ejpam-5335	11	12	of	of	ADP
ejpam-5335	11	13	super	super	NOUN
ejpam-5335	11	14	-	-	NOUN
ejpam-5335	11	15	continuity	continuity	NOUN
ejpam-5335	11	16	.	.	PUNCT
ejpam-5335	12	1	the	the	DET
ejpam-5335	12	2	concept	concept	NOUN
ejpam-5335	12	3	has	have	AUX
ejpam-5335	12	4	been	be	AUX
ejpam-5335	12	5	investigated	investigate	VERB
ejpam-5335	12	6	further	far	ADV
ejpam-5335	12	7	by	by	ADP
ejpam-5335	12	8	reilly	reilly	ADJ
ejpam-5335	12	9	and	and	CCONJ
ejpam-5335	12	10	vamanamurthy	vamanamurthy	ADJ
ejpam-5335	12	11	[	[	X
ejpam-5335	12	12	21	21	NUM
ejpam-5335	12	13	]	]	PUNCT
ejpam-5335	12	14	where	where	SCONJ
ejpam-5335	12	15	super	super	NOUN
ejpam-5335	12	16	-	-	NOUN
ejpam-5335	12	17	continuity	continuity	NOUN
ejpam-5335	12	18	is	be	AUX
ejpam-5335	12	19	characterized	characterize	VERB
ejpam-5335	12	20	in	in	ADP
ejpam-5335	12	21	terms	term	NOUN
ejpam-5335	12	22	of	of	ADP
ejpam-5335	12	23	the	the	DET
ejpam-5335	12	24	semi	semi	ADJ
ejpam-5335	12	25	-	-	ADJ
ejpam-5335	12	26	regularization	regularization	ADJ
ejpam-5335	12	27	topology	topology	NOUN
ejpam-5335	12	28	.	.	PUNCT
ejpam-5335	13	1	super	super	ADJ
ejpam-5335	13	2	-	-	NOUN
ejpam-5335	13	3	continuity	continuity	NOUN
ejpam-5335	13	4	is	be	AUX
ejpam-5335	13	5	related	relate	VERB
ejpam-5335	13	6	to	to	ADP
ejpam-5335	13	7	the	the	DET
ejpam-5335	13	8	concepts	concept	NOUN
ejpam-5335	13	9	of	of	ADP
ejpam-5335	13	10	δ	δ	NOUN
ejpam-5335	13	11	-	-	PUNCT
ejpam-5335	13	12	continuity	continuity	NOUN
ejpam-5335	13	13	and	and	CCONJ
ejpam-5335	13	14	strong	strong	ADJ
ejpam-5335	13	15	θ	θ	NOUN
ejpam-5335	13	16	-	-	PUNCT
ejpam-5335	13	17	continuity	continuity	NOUN
ejpam-5335	13	18	developed	develop	VERB
ejpam-5335	13	19	by	by	ADP
ejpam-5335	13	20	noiri	noiri	PROPN
ejpam-5335	13	21	[	[	X
ejpam-5335	13	22	17	17	NUM
ejpam-5335	13	23	]	]	PUNCT
ejpam-5335	13	24	.	.	PUNCT
ejpam-5335	14	1	in	in	ADP
ejpam-5335	14	2	particular	particular	ADJ
ejpam-5335	14	3	,	,	PUNCT
ejpam-5335	14	4	super	super	ADJ
ejpam-5335	14	5	-	-	NOUN
ejpam-5335	14	6	continuity	continuity	NOUN
ejpam-5335	14	7	is	be	AUX
ejpam-5335	14	8	strictly	strictly	ADV
ejpam-5335	14	9	between	between	ADP
ejpam-5335	14	10	strong	strong	ADJ
ejpam-5335	14	11	θ	θ	NOUN
ejpam-5335	14	12	-	-	PUNCT
ejpam-5335	14	13	continuity	continuity	NOUN
ejpam-5335	14	14	and	and	CCONJ
ejpam-5335	14	15	δ	δ	NOUN
ejpam-5335	14	16	-	-	PUNCT
ejpam-5335	14	17	continuity	continuity	NOUN
ejpam-5335	14	18	and	and	CCONJ
ejpam-5335	14	19	strictly	strictly	ADV
ejpam-5335	14	20	between	between	ADP
ejpam-5335	14	21	complete	complete	ADJ
ejpam-5335	14	22	continuity	continuity	NOUN
ejpam-5335	14	23	[	[	X
ejpam-5335	14	24	1	1	NUM
ejpam-5335	14	25	]	]	PUNCT
ejpam-5335	14	26	and	and	CCONJ
ejpam-5335	14	27	δ	δ	NOUN
ejpam-5335	14	28	-	-	NOUN
ejpam-5335	14	29	continuity	continuity	NOUN
ejpam-5335	14	30	.	.	PUNCT
ejpam-5335	15	1	raychaudhuri	raychaudhuri	PROPN
ejpam-5335	15	2	and	and	CCONJ
ejpam-5335	15	3	mukherjee	mukherjee	NOUN
ejpam-5335	16	1	[	[	X
ejpam-5335	16	2	20	20	NUM
ejpam-5335	16	3	]	]	PUNCT
ejpam-5335	16	4	introduced	introduce	VERB
ejpam-5335	16	5	the	the	DET
ejpam-5335	16	6	concept	concept	NOUN
ejpam-5335	16	7	of	of	ADP
ejpam-5335	16	8	δ	δ	PROPN
ejpam-5335	16	9	-	-	PUNCT
ejpam-5335	16	10	preopen	preopen	ADJ
ejpam-5335	16	11	sets	set	NOUN
ejpam-5335	16	12	which	which	PRON
ejpam-5335	16	13	is	be	AUX
ejpam-5335	16	14	weaker	weak	ADJ
ejpam-5335	16	15	than	than	ADP
ejpam-5335	16	16	that	that	PRON
ejpam-5335	16	17	of	of	ADP
ejpam-5335	16	18	preopen	preopen	ADJ
ejpam-5335	16	19	sets	set	NOUN
ejpam-5335	16	20	and	and	CCONJ
ejpam-5335	16	21	used	use	VERB
ejpam-5335	16	22	this	this	DET
ejpam-5335	16	23	concept	concept	NOUN
ejpam-5335	16	24	to	to	PART
ejpam-5335	16	25	define	define	VERB
ejpam-5335	16	26	the	the	DET
ejpam-5335	16	27	notion	notion	NOUN
ejpam-5335	16	28	of	of	ADP
ejpam-5335	16	29	δ	δ	NOUN
ejpam-5335	16	30	-	-	PUNCT
ejpam-5335	16	31	almost	almost	ADV
ejpam-5335	16	32	continuous	continuous	ADJ
ejpam-5335	16	33	functions	function	NOUN
ejpam-5335	16	34	as	as	ADP
ejpam-5335	16	35	a	a	DET
ejpam-5335	16	36	generalization	generalization	NOUN
ejpam-5335	16	37	of	of	ADP
ejpam-5335	16	38	precontinuous	precontinuous	ADJ
ejpam-5335	16	39	functions	function	NOUN
ejpam-5335	16	40	due	due	ADP
ejpam-5335	16	41	to	to	ADP
ejpam-5335	16	42	mashhour	mashhour	PROPN
ejpam-5335	16	43	et	et	PROPN
ejpam-5335	16	44	al	al	PROPN
ejpam-5335	16	45	.	.	PUNCT
ejpam-5335	17	1	[	[	X
ejpam-5335	17	2	15	15	NUM
ejpam-5335	17	3	]	]	PUNCT
ejpam-5335	17	4	.	.	PUNCT
ejpam-5335	18	1	baker	baker	PROPN
ejpam-5335	19	1	[	[	X
ejpam-5335	19	2	2	2	NUM
ejpam-5335	19	3	]	]	PUNCT
ejpam-5335	19	4	introduced	introduce	VERB
ejpam-5335	19	5	and	and	CCONJ
ejpam-5335	19	6	investigated	investigate	VERB
ejpam-5335	19	7	the	the	DET
ejpam-5335	19	8	notion	notion	NOUN
ejpam-5335	19	9	of	of	ADP
ejpam-5335	19	10	weakly	weakly	ADJ
ejpam-5335	19	11	δ	δ	NOUN
ejpam-5335	19	12	-	-	ADJ
ejpam-5335	19	13	continuous	continuous	ADJ
ejpam-5335	19	14	functions	function	NOUN
ejpam-5335	19	15	.	.	PUNCT
ejpam-5335	20	1	the	the	DET
ejpam-5335	20	2	class	class	NOUN
ejpam-5335	20	3	of	of	ADP
ejpam-5335	20	4	weakly	weakly	ADJ
ejpam-5335	20	5	δ	δ	NOUN
ejpam-5335	20	6	-	-	ADJ
ejpam-5335	20	7	continuous	continuous	ADJ
ejpam-5335	20	8	functions	function	NOUN
ejpam-5335	20	9	is	be	AUX
ejpam-5335	20	10	a	a	DET
ejpam-5335	20	11	generalization	generalization	NOUN
ejpam-5335	20	12	of	of	ADP
ejpam-5335	20	13	δ	δ	PROPN
ejpam-5335	20	14	-	-	ADJ
ejpam-5335	20	15	continuous	continuous	ADJ
ejpam-5335	20	16	functions	function	NOUN
ejpam-5335	20	17	.	.	PUNCT
ejpam-5335	21	1	in	in	ADP
ejpam-5335	21	2	1997	1997	NUM
ejpam-5335	21	3	,	,	PUNCT
ejpam-5335	21	4	park	park	NOUN
ejpam-5335	21	5	et	et	PROPN
ejpam-5335	21	6	al	al	PROPN
ejpam-5335	21	7	.	.	PUNCT
ejpam-5335	22	1	[	[	X
ejpam-5335	22	2	19	19	NUM
ejpam-5335	22	3	]	]	PUNCT
ejpam-5335	22	4	introduced	introduce	VERB
ejpam-5335	22	5	the	the	DET
ejpam-5335	22	6	notion	notion	NOUN
ejpam-5335	22	7	of	of	ADP
ejpam-5335	22	8	δ	δ	PROPN
ejpam-5335	22	9	-	-	PUNCT
ejpam-5335	22	10	semiopen	semiopen	VERB
ejpam-5335	22	11	sets	set	NOUN
ejpam-5335	22	12	by	by	ADP
ejpam-5335	22	13	using	use	VERB
ejpam-5335	22	14	δ	δ	PROPN
ejpam-5335	22	15	-	-	ADJ
ejpam-5335	22	16	open	open	ADJ
ejpam-5335	22	17	sets	set	NOUN
ejpam-5335	22	18	due	due	ADP
ejpam-5335	22	19	to	to	ADP
ejpam-5335	22	20	valičko	valičko	PROPN
ejpam-5335	22	21	[	[	X
ejpam-5335	22	22	22	22	NUM
ejpam-5335	22	23	]	]	PUNCT
ejpam-5335	22	24	.	.	PUNCT
ejpam-5335	23	1	in	in	ADP
ejpam-5335	23	2	2005	2005	NUM
ejpam-5335	23	3	,	,	PUNCT
ejpam-5335	23	4	ekici	ekici	NOUN
ejpam-5335	23	5	and	and	CCONJ
ejpam-5335	23	6	navalagi	navalagi	ADJ
ejpam-5335	23	7	[	[	X
ejpam-5335	23	8	13	13	NUM
ejpam-5335	23	9	]	]	PUNCT
ejpam-5335	23	10	introduced	introduce	VERB
ejpam-5335	23	11	and	and	CCONJ
ejpam-5335	23	12	investigated	investigate	VERB
ejpam-5335	23	13	∗corresponding	∗corresponde	VERB
ejpam-5335	23	14	author	author	NOUN
ejpam-5335	23	15	.	.	PUNCT
ejpam-5335	24	1	doi	doi	NOUN
ejpam-5335	24	2	:	:	PUNCT
ejpam-5335	24	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5335	https://doi.org/10.29020/nybg.ejpam.v17i4.5335	NOUN
ejpam-5335	24	4	email	email	NOUN
ejpam-5335	24	5	addresses	address	NOUN
ejpam-5335	24	6	:	:	PUNCT
ejpam-5335	24	7	prachanpol.ch@gmail.com	prachanpol.ch@gmail.com	PROPN
ejpam-5335	24	8	(	(	PUNCT
ejpam-5335	24	9	c.	c.	PROPN
ejpam-5335	24	10	prachanpol	prachanpol	PROPN
ejpam-5335	24	11	)	)	PUNCT
ejpam-5335	24	12	,	,	PUNCT
ejpam-5335	25	1	chawalit.b@msu.ac.th	chawalit.b@msu.ac.th	PROPN
ejpam-5335	25	2	(	(	PUNCT
ejpam-5335	25	3	c.	c.	PROPN
ejpam-5335	25	4	boonpok	boonpok	PROPN
ejpam-5335	25	5	)	)	PUNCT
ejpam-5335	25	6	,	,	PUNCT
ejpam-5335	25	7	chokchai.v@msu.ac.th	chokchai.v@msu.ac.th	INTJ
ejpam-5335	25	8	(	(	PUNCT
ejpam-5335	25	9	c.	c.	PROPN
ejpam-5335	25	10	viriyapong	viriyapong	PROPN
ejpam-5335	25	11	)	)	PUNCT
ejpam-5335	25	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5335	25	13	3730	3730	NUM
ejpam-5335	26	1	copyright	copyright	NOUN
ejpam-5335	26	2	:	:	PUNCT
ejpam-5335	26	3	©	©	PROPN
ejpam-5335	26	4	2024	2024	NUM
ejpam-5335	26	5	the	the	DET
ejpam-5335	26	6	author(s	author(s	NOUN
ejpam-5335	26	7	)	)	PUNCT
ejpam-5335	26	8	.	.	PUNCT
ejpam-5335	27	1	(	(	PUNCT
ejpam-5335	27	2	cc	cc	NOUN
ejpam-5335	27	3	by	by	ADP
ejpam-5335	27	4	-	-	PUNCT
ejpam-5335	27	5	nc	nc	PROPN
ejpam-5335	27	6	4.0	4.0	NUM
ejpam-5335	27	7	)	)	PUNCT
ejpam-5335	27	8	c.	c.	NOUN
ejpam-5335	27	9	prachanpol	prachanpol	NOUN
ejpam-5335	27	10	,	,	PUNCT
ejpam-5335	27	11	c.	c.	PROPN
ejpam-5335	27	12	boonpok	boonpok	PROPN
ejpam-5335	27	13	,	,	PUNCT
ejpam-5335	27	14	c.	c.	PROPN
ejpam-5335	27	15	viriyapong	viriyapong	PROPN
ejpam-5335	27	16	/	/	SYM
ejpam-5335	27	17	eur	eur	PROPN
ejpam-5335	27	18	.	.	PUNCT
ejpam-5335	28	1	j.	j.	PROPN
ejpam-5335	28	2	pure	pure	PROPN
ejpam-5335	28	3	appl	appl	PROPN
ejpam-5335	28	4	.	.	PROPN
ejpam-5335	28	5	math	math	PROPN
ejpam-5335	28	6	,	,	PUNCT
ejpam-5335	28	7	17	17	NUM
ejpam-5335	28	8	(	(	PUNCT
ejpam-5335	28	9	4	4	NUM
ejpam-5335	28	10	)	)	PUNCT
ejpam-5335	28	11	(	(	PUNCT
ejpam-5335	28	12	2024	2024	NUM
ejpam-5335	28	13	)	)	PUNCT
ejpam-5335	28	14	,	,	PUNCT
ejpam-5335	28	15	3730	3730	NUM
ejpam-5335	28	16	-	-	SYM
ejpam-5335	28	17	3742	3742	NUM
ejpam-5335	28	18	3731	3731	NUM
ejpam-5335	28	19	the	the	DET
ejpam-5335	28	20	concept	concept	NOUN
ejpam-5335	28	21	of	of	ADP
ejpam-5335	28	22	δ	δ	NOUN
ejpam-5335	28	23	-	-	PUNCT
ejpam-5335	28	24	semicontinuous	semicontinuous	ADJ
ejpam-5335	28	25	functions	function	NOUN
ejpam-5335	28	26	.	.	PUNCT
ejpam-5335	29	1	the	the	DET
ejpam-5335	29	2	class	class	NOUN
ejpam-5335	29	3	of	of	ADP
ejpam-5335	29	4	δ	δ	PROPN
ejpam-5335	29	5	-	-	PUNCT
ejpam-5335	29	6	semicontinuous	semicontinuous	ADJ
ejpam-5335	29	7	functions	function	NOUN
ejpam-5335	29	8	is	be	AUX
ejpam-5335	29	9	a	a	DET
ejpam-5335	29	10	weaker	weak	ADJ
ejpam-5335	29	11	form	form	NOUN
ejpam-5335	29	12	of	of	ADP
ejpam-5335	29	13	the	the	DET
ejpam-5335	29	14	classes	class	NOUN
ejpam-5335	29	15	of	of	ADP
ejpam-5335	29	16	perfectly	perfectly	ADV
ejpam-5335	29	17	continuous	continuous	ADJ
ejpam-5335	29	18	functions	function	NOUN
ejpam-5335	29	19	[	[	X
ejpam-5335	29	20	18	18	NUM
ejpam-5335	29	21	]	]	PUNCT
ejpam-5335	29	22	,	,	PUNCT
ejpam-5335	29	23	strongly	strongly	ADV
ejpam-5335	29	24	θ	θ	ADJ
ejpam-5335	29	25	-	-	ADJ
ejpam-5335	29	26	continuous	continuous	ADJ
ejpam-5335	29	27	functions	function	NOUN
ejpam-5335	29	28	[	[	X
ejpam-5335	29	29	14	14	NUM
ejpam-5335	29	30	]	]	PUNCT
ejpam-5335	29	31	and	and	CCONJ
ejpam-5335	29	32	super	super	ADJ
ejpam-5335	29	33	-	-	ADJ
ejpam-5335	29	34	continuous	continuous	ADJ
ejpam-5335	29	35	functions	function	NOUN
ejpam-5335	29	36	.	.	PUNCT
ejpam-5335	30	1	ekici	ekici	NOUN
ejpam-5335	31	1	[	[	X
ejpam-5335	31	2	12	12	NUM
ejpam-5335	31	3	]	]	PUNCT
ejpam-5335	31	4	introduced	introduce	VERB
ejpam-5335	31	5	the	the	DET
ejpam-5335	31	6	notion	notion	NOUN
ejpam-5335	31	7	of	of	ADP
ejpam-5335	31	8	almost	almost	ADV
ejpam-5335	31	9	δsemicontinuous	δsemicontinuous	ADJ
ejpam-5335	31	10	functions	function	NOUN
ejpam-5335	31	11	which	which	PRON
ejpam-5335	31	12	generalize	generalize	VERB
ejpam-5335	31	13	r	r	NOUN
ejpam-5335	31	14	-	-	PUNCT
ejpam-5335	31	15	maps	map	NOUN
ejpam-5335	31	16	[	[	X
ejpam-5335	31	17	11	11	NUM
ejpam-5335	31	18	]	]	PUNCT
ejpam-5335	31	19	and	and	CCONJ
ejpam-5335	31	20	δ	δ	NOUN
ejpam-5335	31	21	-	-	ADJ
ejpam-5335	31	22	continuous	continuous	ADJ
ejpam-5335	31	23	functions	function	NOUN
ejpam-5335	31	24	.	.	PUNCT
ejpam-5335	32	1	yüksel	yüksel	PROPN
ejpam-5335	32	2	et	et	PROPN
ejpam-5335	32	3	al	al	PROPN
ejpam-5335	32	4	.	.	PUNCT
ejpam-5335	33	1	[	[	X
ejpam-5335	33	2	25	25	NUM
ejpam-5335	33	3	]	]	PUNCT
ejpam-5335	33	4	extended	extend	VERB
ejpam-5335	33	5	the	the	DET
ejpam-5335	33	6	concept	concept	NOUN
ejpam-5335	33	7	of	of	ADP
ejpam-5335	33	8	δ	δ	NOUN
ejpam-5335	33	9	-	-	ADJ
ejpam-5335	33	10	open	open	ADJ
ejpam-5335	33	11	sets	set	NOUN
ejpam-5335	33	12	to	to	PART
ejpam-5335	33	13	ideal	ideal	VERB
ejpam-5335	33	14	topological	topological	ADJ
ejpam-5335	33	15	spaces	space	NOUN
ejpam-5335	33	16	and	and	CCONJ
ejpam-5335	33	17	defined	define	VERB
ejpam-5335	33	18	δi	δi	ADJ
ejpam-5335	33	19	-	-	PUNCT
ejpam-5335	33	20	continuous	continuous	ADJ
ejpam-5335	33	21	functions	function	NOUN
ejpam-5335	33	22	.	.	PUNCT
ejpam-5335	34	1	moreover	moreover	ADV
ejpam-5335	34	2	,	,	PUNCT
ejpam-5335	34	3	some	some	DET
ejpam-5335	34	4	characterizations	characterization	NOUN
ejpam-5335	34	5	of	of	ADP
ejpam-5335	34	6	θ	θ	PROPN
ejpam-5335	34	7	-	-	ADJ
ejpam-5335	34	8	i	i	PROPN
ejpam-5335	34	9	-continuous	-continuous	ADJ
ejpam-5335	34	10	functions	function	NOUN
ejpam-5335	34	11	,	,	PUNCT
ejpam-5335	34	12	⋆-continuous	⋆-continuous	ADJ
ejpam-5335	34	13	functions	function	NOUN
ejpam-5335	34	14	and	and	CCONJ
ejpam-5335	34	15	θ(⋆)-precontinuous	θ(⋆)-precontinuous	ADJ
ejpam-5335	34	16	functions	function	NOUN
ejpam-5335	34	17	were	be	AUX
ejpam-5335	34	18	presented	present	VERB
ejpam-5335	34	19	in	in	ADP
ejpam-5335	34	20	[	[	X
ejpam-5335	34	21	3	3	NUM
ejpam-5335	34	22	]	]	PUNCT
ejpam-5335	34	23	,	,	PUNCT
ejpam-5335	34	24	[	[	X
ejpam-5335	34	25	5	5	NUM
ejpam-5335	34	26	]	]	PUNCT
ejpam-5335	34	27	and	and	CCONJ
ejpam-5335	34	28	[	[	X
ejpam-5335	34	29	6	6	NUM
ejpam-5335	34	30	]	]	PUNCT
ejpam-5335	34	31	,	,	PUNCT
ejpam-5335	34	32	respectively	respectively	ADV
ejpam-5335	34	33	.	.	PUNCT
ejpam-5335	35	1	in	in	ADP
ejpam-5335	35	2	[	[	X
ejpam-5335	35	3	9	9	NUM
ejpam-5335	35	4	]	]	PUNCT
ejpam-5335	35	5	,	,	PUNCT
ejpam-5335	35	6	the	the	DET
ejpam-5335	35	7	present	present	ADJ
ejpam-5335	35	8	authors	author	NOUN
ejpam-5335	35	9	introduced	introduce	VERB
ejpam-5335	35	10	and	and	CCONJ
ejpam-5335	35	11	studied	study	VERB
ejpam-5335	35	12	the	the	DET
ejpam-5335	35	13	concept	concept	NOUN
ejpam-5335	35	14	of	of	ADP
ejpam-5335	35	15	(	(	PUNCT
ejpam-5335	35	16	τ1	τ1	NOUN
ejpam-5335	35	17	,	,	PUNCT
ejpam-5335	35	18	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	35	19	functions	function	NOUN
ejpam-5335	35	20	.	.	PUNCT
ejpam-5335	36	1	futhermore	futhermore	NOUN
ejpam-5335	36	2	,	,	PUNCT
ejpam-5335	36	3	several	several	ADJ
ejpam-5335	36	4	characterizations	characterization	NOUN
ejpam-5335	36	5	of	of	ADP
ejpam-5335	36	6	almost	almost	ADV
ejpam-5335	36	7	(	(	PUNCT
ejpam-5335	36	8	τ1	τ1	NOUN
ejpam-5335	36	9	,	,	PUNCT
ejpam-5335	36	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	36	11	functions	function	NOUN
ejpam-5335	36	12	and	and	CCONJ
ejpam-5335	36	13	weakly	weakly	ADJ
ejpam-5335	36	14	(	(	PUNCT
ejpam-5335	36	15	τ1	τ1	NOUN
ejpam-5335	36	16	,	,	PUNCT
ejpam-5335	36	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	36	18	functions	function	NOUN
ejpam-5335	36	19	were	be	AUX
ejpam-5335	36	20	established	establish	VERB
ejpam-5335	36	21	in	in	ADP
ejpam-5335	36	22	[	[	X
ejpam-5335	36	23	8	8	NUM
ejpam-5335	36	24	]	]	PUNCT
ejpam-5335	36	25	and	and	CCONJ
ejpam-5335	36	26	[	[	X
ejpam-5335	36	27	7	7	NUM
ejpam-5335	36	28	]	]	PUNCT
ejpam-5335	36	29	,	,	PUNCT
ejpam-5335	36	30	respectively	respectively	ADV
ejpam-5335	36	31	.	.	PUNCT
ejpam-5335	37	1	in	in	ADP
ejpam-5335	37	2	this	this	DET
ejpam-5335	37	3	paper	paper	NOUN
ejpam-5335	37	4	,	,	PUNCT
ejpam-5335	37	5	we	we	PRON
ejpam-5335	37	6	introduce	introduce	VERB
ejpam-5335	37	7	the	the	DET
ejpam-5335	37	8	notions	notion	NOUN
ejpam-5335	37	9	of	of	ADP
ejpam-5335	37	10	δ(τ1	δ(τ1	NOUN
ejpam-5335	37	11	,	,	PUNCT
ejpam-5335	37	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	37	13	functions	function	NOUN
ejpam-5335	37	14	,	,	PUNCT
ejpam-5335	37	15	almost	almost	ADV
ejpam-5335	37	16	δ(τ1	δ(τ1	NOUN
ejpam-5335	37	17	,	,	PUNCT
ejpam-5335	37	18	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	37	19	functions	function	NOUN
ejpam-5335	37	20	,	,	PUNCT
ejpam-5335	37	21	and	and	CCONJ
ejpam-5335	37	22	weakly	weakly	ADJ
ejpam-5335	37	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	37	24	,	,	PUNCT
ejpam-5335	37	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	37	26	functions	function	NOUN
ejpam-5335	37	27	.	.	PUNCT
ejpam-5335	38	1	we	we	PRON
ejpam-5335	38	2	also	also	ADV
ejpam-5335	38	3	investigate	investigate	VERB
ejpam-5335	38	4	several	several	ADJ
ejpam-5335	38	5	characterizations	characterization	NOUN
ejpam-5335	38	6	of	of	ADP
ejpam-5335	38	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	38	8	,	,	PUNCT
ejpam-5335	38	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	38	10	functions	function	NOUN
ejpam-5335	38	11	,	,	PUNCT
ejpam-5335	38	12	almost	almost	ADV
ejpam-5335	38	13	δ(τ1	δ(τ1	NOUN
ejpam-5335	38	14	,	,	PUNCT
ejpam-5335	38	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	38	16	functions	function	NOUN
ejpam-5335	38	17	,	,	PUNCT
ejpam-5335	38	18	and	and	CCONJ
ejpam-5335	38	19	weakly	weakly	ADJ
ejpam-5335	38	20	δ(τ1	δ(τ1	NOUN
ejpam-5335	38	21	,	,	PUNCT
ejpam-5335	38	22	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	38	23	functions	function	NOUN
ejpam-5335	38	24	.	.	PUNCT
ejpam-5335	39	1	finally	finally	ADV
ejpam-5335	39	2	,	,	PUNCT
ejpam-5335	39	3	the	the	DET
ejpam-5335	39	4	relationships	relationship	NOUN
ejpam-5335	39	5	among	among	ADP
ejpam-5335	39	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	39	7	,	,	PUNCT
ejpam-5335	39	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	39	9	functions	function	NOUN
ejpam-5335	39	10	,	,	PUNCT
ejpam-5335	39	11	almost	almost	ADV
ejpam-5335	39	12	δ(τ1	δ(τ1	NOUN
ejpam-5335	39	13	,	,	PUNCT
ejpam-5335	39	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	39	15	functions	function	NOUN
ejpam-5335	39	16	,	,	PUNCT
ejpam-5335	39	17	and	and	CCONJ
ejpam-5335	39	18	weakly	weakly	ADJ
ejpam-5335	39	19	δ(τ1	δ(τ1	NOUN
ejpam-5335	39	20	,	,	PUNCT
ejpam-5335	39	21	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	39	22	functions	function	NOUN
ejpam-5335	39	23	are	be	AUX
ejpam-5335	39	24	discussed	discuss	VERB
ejpam-5335	39	25	.	.	PUNCT
ejpam-5335	40	1	2	2	X
ejpam-5335	40	2	.	.	X
ejpam-5335	40	3	preliminaries	preliminary	NOUN
ejpam-5335	40	4	throughout	throughout	ADP
ejpam-5335	40	5	the	the	DET
ejpam-5335	40	6	present	present	ADJ
ejpam-5335	40	7	paper	paper	NOUN
ejpam-5335	40	8	,	,	PUNCT
ejpam-5335	40	9	spaces	space	NOUN
ejpam-5335	40	10	(	(	PUNCT
ejpam-5335	40	11	x	x	NOUN
ejpam-5335	40	12	,	,	PUNCT
ejpam-5335	40	13	τ1	τ1	NOUN
ejpam-5335	40	14	,	,	PUNCT
ejpam-5335	40	15	τ2	τ2	NOUN
ejpam-5335	40	16	)	)	PUNCT
ejpam-5335	40	17	and	and	CCONJ
ejpam-5335	40	18	(	(	PUNCT
ejpam-5335	40	19	y	y	PROPN
ejpam-5335	40	20	,	,	PUNCT
ejpam-5335	40	21	σ1	σ1	PROPN
ejpam-5335	40	22	,	,	PUNCT
ejpam-5335	40	23	σ2	σ2	NOUN
ejpam-5335	40	24	)	)	PUNCT
ejpam-5335	40	25	(	(	PUNCT
ejpam-5335	40	26	or	or	CCONJ
ejpam-5335	40	27	simply	simply	ADV
ejpam-5335	40	28	x	x	X
ejpam-5335	40	29	and	and	CCONJ
ejpam-5335	40	30	y	y	PROPN
ejpam-5335	40	31	)	)	PUNCT
ejpam-5335	40	32	always	always	ADV
ejpam-5335	40	33	mean	mean	VERB
ejpam-5335	40	34	bitopological	bitopological	ADJ
ejpam-5335	40	35	spaces	space	NOUN
ejpam-5335	40	36	on	on	ADP
ejpam-5335	40	37	which	which	PRON
ejpam-5335	40	38	no	no	DET
ejpam-5335	40	39	separation	separation	NOUN
ejpam-5335	40	40	axioms	axiom	NOUN
ejpam-5335	40	41	are	be	AUX
ejpam-5335	40	42	assumed	assume	VERB
ejpam-5335	40	43	unless	unless	SCONJ
ejpam-5335	40	44	explicitly	explicitly	ADV
ejpam-5335	40	45	stated	state	VERB
ejpam-5335	40	46	.	.	PUNCT
ejpam-5335	41	1	let	let	VERB
ejpam-5335	41	2	a	a	DET
ejpam-5335	41	3	be	be	AUX
ejpam-5335	41	4	a	a	DET
ejpam-5335	41	5	subset	subset	NOUN
ejpam-5335	41	6	of	of	ADP
ejpam-5335	41	7	a	a	DET
ejpam-5335	41	8	bitopological	bitopological	ADJ
ejpam-5335	41	9	space	space	NOUN
ejpam-5335	41	10	(	(	PUNCT
ejpam-5335	41	11	x	x	NOUN
ejpam-5335	41	12	,	,	PUNCT
ejpam-5335	41	13	τ1	τ1	NOUN
ejpam-5335	41	14	,	,	PUNCT
ejpam-5335	41	15	τ2	τ2	NOUN
ejpam-5335	41	16	)	)	PUNCT
ejpam-5335	41	17	.	.	PUNCT
ejpam-5335	42	1	the	the	DET
ejpam-5335	42	2	closure	closure	NOUN
ejpam-5335	42	3	of	of	ADP
ejpam-5335	42	4	a	a	PRON
ejpam-5335	42	5	and	and	CCONJ
ejpam-5335	42	6	the	the	DET
ejpam-5335	42	7	interior	interior	NOUN
ejpam-5335	42	8	of	of	ADP
ejpam-5335	42	9	a	a	PRON
ejpam-5335	42	10	with	with	ADP
ejpam-5335	42	11	respect	respect	NOUN
ejpam-5335	42	12	to	to	ADP
ejpam-5335	42	13	τi	τi	PROPN
ejpam-5335	42	14	are	be	AUX
ejpam-5335	42	15	denoted	denote	VERB
ejpam-5335	42	16	by	by	ADP
ejpam-5335	42	17	τi	τi	NOUN
ejpam-5335	42	18	-	-	PUNCT
ejpam-5335	42	19	cl(a	cl(a	NUM
ejpam-5335	42	20	)	)	PUNCT
ejpam-5335	42	21	and	and	CCONJ
ejpam-5335	42	22	τi	τi	NOUN
ejpam-5335	42	23	-	-	PUNCT
ejpam-5335	42	24	int(a	int(a	NOUN
ejpam-5335	42	25	)	)	PUNCT
ejpam-5335	42	26	,	,	PUNCT
ejpam-5335	42	27	respectively	respectively	ADV
ejpam-5335	42	28	,	,	PUNCT
ejpam-5335	42	29	for	for	ADP
ejpam-5335	42	30	i	i	PROPN
ejpam-5335	42	31	=	=	SYM
ejpam-5335	42	32	1	1	NUM
ejpam-5335	42	33	,	,	PUNCT
ejpam-5335	42	34	2	2	NUM
ejpam-5335	42	35	.	.	X
ejpam-5335	42	36	a	a	DET
ejpam-5335	42	37	subset	subset	NOUN
ejpam-5335	42	38	a	a	PRON
ejpam-5335	42	39	of	of	ADP
ejpam-5335	42	40	a	a	DET
ejpam-5335	42	41	bitopological	bitopological	ADJ
ejpam-5335	42	42	space	space	NOUN
ejpam-5335	42	43	(	(	PUNCT
ejpam-5335	42	44	x	x	NOUN
ejpam-5335	42	45	,	,	PUNCT
ejpam-5335	42	46	τ1	τ1	NOUN
ejpam-5335	42	47	,	,	PUNCT
ejpam-5335	42	48	τ2	τ2	NOUN
ejpam-5335	42	49	)	)	PUNCT
ejpam-5335	42	50	is	be	AUX
ejpam-5335	42	51	called	call	VERB
ejpam-5335	42	52	τ1τ2	τ1τ2	VERB
ejpam-5335	42	53	-	-	ADJ
ejpam-5335	42	54	closed	closed	ADJ
ejpam-5335	42	55	[	[	X
ejpam-5335	42	56	10	10	NUM
ejpam-5335	42	57	]	]	X
ejpam-5335	42	58	if	if	SCONJ
ejpam-5335	42	59	a	a	DET
ejpam-5335	42	60	=	=	NOUN
ejpam-5335	42	61	τ1	τ1	NOUN
ejpam-5335	42	62	-	-	PUNCT
ejpam-5335	42	63	cl(τ2	cl(τ2	NOUN
ejpam-5335	42	64	-	-	PUNCT
ejpam-5335	42	65	cl(a	cl(a	NUM
ejpam-5335	42	66	)	)	PUNCT
ejpam-5335	42	67	)	)	PUNCT
ejpam-5335	42	68	.	.	PUNCT
ejpam-5335	43	1	the	the	DET
ejpam-5335	43	2	complement	complement	NOUN
ejpam-5335	43	3	of	of	ADP
ejpam-5335	43	4	a	a	DET
ejpam-5335	43	5	τ1τ2	τ1τ2	ADJ
ejpam-5335	43	6	-	-	ADJ
ejpam-5335	43	7	closed	closed	ADJ
ejpam-5335	43	8	set	set	NOUN
ejpam-5335	43	9	is	be	AUX
ejpam-5335	43	10	called	call	VERB
ejpam-5335	43	11	τ1τ2	τ1τ2	NOUN
ejpam-5335	43	12	-	-	ADJ
ejpam-5335	43	13	open	open	ADJ
ejpam-5335	43	14	.	.	PUNCT
ejpam-5335	44	1	the	the	DET
ejpam-5335	44	2	intersection	intersection	NOUN
ejpam-5335	44	3	of	of	ADP
ejpam-5335	44	4	all	all	DET
ejpam-5335	44	5	τ1τ2	τ1τ2	ADJ
ejpam-5335	44	6	-	-	ADJ
ejpam-5335	44	7	closed	closed	ADJ
ejpam-5335	44	8	sets	set	NOUN
ejpam-5335	44	9	of	of	ADP
ejpam-5335	44	10	x	x	PUNCT
ejpam-5335	44	11	containing	contain	VERB
ejpam-5335	44	12	a	a	PRON
ejpam-5335	44	13	is	be	AUX
ejpam-5335	44	14	called	call	VERB
ejpam-5335	44	15	the	the	DET
ejpam-5335	44	16	τ1τ2	τ1τ2	NOUN
ejpam-5335	44	17	-	-	NOUN
ejpam-5335	44	18	closure	closure	NOUN
ejpam-5335	44	19	[	[	X
ejpam-5335	44	20	10	10	NUM
ejpam-5335	44	21	]	]	PUNCT
ejpam-5335	44	22	of	of	ADP
ejpam-5335	44	23	a	a	PRON
ejpam-5335	44	24	and	and	CCONJ
ejpam-5335	44	25	is	be	AUX
ejpam-5335	44	26	denoted	denote	VERB
ejpam-5335	44	27	by	by	ADP
ejpam-5335	44	28	τ1τ2	τ1τ2	NOUN
ejpam-5335	44	29	-	-	NUM
ejpam-5335	44	30	cl(a	cl(a	NUM
ejpam-5335	44	31	)	)	PUNCT
ejpam-5335	44	32	.	.	PUNCT
ejpam-5335	45	1	the	the	DET
ejpam-5335	45	2	union	union	NOUN
ejpam-5335	45	3	of	of	ADP
ejpam-5335	45	4	all	all	DET
ejpam-5335	45	5	τ1τ2	τ1τ2	ADJ
ejpam-5335	45	6	-	-	ADJ
ejpam-5335	45	7	open	open	ADJ
ejpam-5335	45	8	sets	set	NOUN
ejpam-5335	45	9	of	of	ADP
ejpam-5335	45	10	x	x	PUNCT
ejpam-5335	45	11	contained	contain	VERB
ejpam-5335	45	12	in	in	ADP
ejpam-5335	45	13	a	a	PRON
ejpam-5335	45	14	is	be	AUX
ejpam-5335	45	15	called	call	VERB
ejpam-5335	45	16	the	the	DET
ejpam-5335	45	17	τ1τ2	τ1τ2	NOUN
ejpam-5335	45	18	-	-	ADJ
ejpam-5335	45	19	interior	interior	ADJ
ejpam-5335	45	20	[	[	X
ejpam-5335	45	21	10	10	NUM
ejpam-5335	45	22	]	]	PUNCT
ejpam-5335	45	23	of	of	ADP
ejpam-5335	45	24	a	a	PRON
ejpam-5335	45	25	and	and	CCONJ
ejpam-5335	45	26	is	be	AUX
ejpam-5335	45	27	denoted	denote	VERB
ejpam-5335	45	28	by	by	ADP
ejpam-5335	45	29	τ1τ2	τ1τ2	NOUN
ejpam-5335	45	30	-	-	ADJ
ejpam-5335	45	31	int(a	int(a	NOUN
ejpam-5335	45	32	)	)	PUNCT
ejpam-5335	45	33	.	.	PUNCT
ejpam-5335	46	1	a	a	DET
ejpam-5335	46	2	subset	subset	NOUN
ejpam-5335	46	3	a	a	PRON
ejpam-5335	46	4	of	of	ADP
ejpam-5335	46	5	a	a	DET
ejpam-5335	46	6	bitopological	bitopological	ADJ
ejpam-5335	46	7	space	space	NOUN
ejpam-5335	46	8	(	(	PUNCT
ejpam-5335	46	9	x	x	NOUN
ejpam-5335	46	10	,	,	PUNCT
ejpam-5335	46	11	τ1	τ1	NOUN
ejpam-5335	46	12	,	,	PUNCT
ejpam-5335	46	13	τ2	τ2	NOUN
ejpam-5335	46	14	)	)	PUNCT
ejpam-5335	46	15	is	be	AUX
ejpam-5335	46	16	called	call	VERB
ejpam-5335	46	17	(	(	PUNCT
ejpam-5335	46	18	τ1	τ1	NOUN
ejpam-5335	46	19	,	,	PUNCT
ejpam-5335	46	20	τ2)r	τ2)r	NOUN
ejpam-5335	46	21	-	-	PUNCT
ejpam-5335	46	22	open	open	NOUN
ejpam-5335	46	23	[	[	X
ejpam-5335	46	24	23	23	NUM
ejpam-5335	46	25	]	]	PUNCT
ejpam-5335	46	26	(	(	PUNCT
ejpam-5335	46	27	resp	resp	NOUN
ejpam-5335	46	28	.	.	PUNCT
ejpam-5335	47	1	(	(	PUNCT
ejpam-5335	47	2	τ1	τ1	NOUN
ejpam-5335	47	3	,	,	PUNCT
ejpam-5335	47	4	τ2)s	τ2)s	NOUN
ejpam-5335	47	5	-	-	PUNCT
ejpam-5335	47	6	open	open	ADJ
ejpam-5335	47	7	[	[	X
ejpam-5335	47	8	4	4	NUM
ejpam-5335	47	9	]	]	PUNCT
ejpam-5335	47	10	,	,	PUNCT
ejpam-5335	47	11	(	(	PUNCT
ejpam-5335	47	12	τ1	τ1	NOUN
ejpam-5335	47	13	,	,	PUNCT
ejpam-5335	47	14	τ2)p	τ2)p	NOUN
ejpam-5335	47	15	-	-	ADJ
ejpam-5335	47	16	open	open	ADJ
ejpam-5335	47	17	[	[	X
ejpam-5335	47	18	4	4	NUM
ejpam-5335	47	19	]	]	PUNCT
ejpam-5335	47	20	,	,	PUNCT
ejpam-5335	47	21	(	(	PUNCT
ejpam-5335	47	22	τ1	τ1	NOUN
ejpam-5335	47	23	,	,	PUNCT
ejpam-5335	47	24	τ2)β	τ2)β	ADJ
ejpam-5335	47	25	-	-	PUNCT
ejpam-5335	47	26	open	open	NOUN
ejpam-5335	48	1	[	[	X
ejpam-5335	48	2	4	4	NUM
ejpam-5335	48	3	]	]	PUNCT
ejpam-5335	48	4	,	,	PUNCT
ejpam-5335	48	5	α(τ1	α(τ1	NOUN
ejpam-5335	48	6	,	,	PUNCT
ejpam-5335	48	7	τ2)-open	τ2)-open	ADJ
ejpam-5335	48	8	)	)	PUNCT
ejpam-5335	49	1	[	[	X
ejpam-5335	49	2	24	24	NUM
ejpam-5335	49	3	]	]	SYM
ejpam-5335	49	4	)	)	PUNCT
ejpam-5335	49	5	if	if	SCONJ
ejpam-5335	49	6	a	a	DET
ejpam-5335	49	7	=	=	PUNCT
ejpam-5335	49	8	τ1τ2	τ1τ2	NOUN
ejpam-5335	49	9	-	-	NOUN
ejpam-5335	49	10	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5335	49	11	-	-	PUNCT
ejpam-5335	49	12	cl(a	cl(a	NUM
ejpam-5335	49	13	)	)	PUNCT
ejpam-5335	49	14	)	)	PUNCT
ejpam-5335	49	15	(	(	PUNCT
ejpam-5335	49	16	resp	resp	NOUN
ejpam-5335	49	17	.	.	PUNCT
ejpam-5335	50	1	a	a	DET
ejpam-5335	50	2	⊆	⊆	NUM
ejpam-5335	50	3	τ1τ2	τ1τ2	NOUN
ejpam-5335	50	4	-	-	ADJ
ejpam-5335	50	5	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5335	50	6	-	-	PUNCT
ejpam-5335	50	7	int(a	int(a	NOUN
ejpam-5335	50	8	)	)	PUNCT
ejpam-5335	50	9	)	)	PUNCT
ejpam-5335	50	10	,	,	PUNCT
ejpam-5335	50	11	a	a	DET
ejpam-5335	50	12	⊆	⊆	NUM
ejpam-5335	50	13	τ1τ2	τ1τ2	NOUN
ejpam-5335	50	14	-	-	NOUN
ejpam-5335	50	15	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5335	50	16	-	-	PUNCT
ejpam-5335	50	17	cl(a	cl(a	NUM
ejpam-5335	50	18	)	)	PUNCT
ejpam-5335	50	19	)	)	PUNCT
ejpam-5335	50	20	,	,	PUNCT
ejpam-5335	50	21	a	a	DET
ejpam-5335	50	22	⊆	⊆	NUM
ejpam-5335	50	23	τ1τ2	τ1τ2	NOUN
ejpam-5335	50	24	-	-	PUNCT
ejpam-5335	50	25	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5335	50	26	-	-	PUNCT
ejpam-5335	50	27	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5335	50	28	-	-	PUNCT
ejpam-5335	50	29	cl(a	cl(a	NUM
ejpam-5335	50	30	)	)	PUNCT
ejpam-5335	50	31	)	)	PUNCT
ejpam-5335	50	32	)	)	PUNCT
ejpam-5335	50	33	,	,	PUNCT
ejpam-5335	50	34	a	a	DET
ejpam-5335	50	35	⊆	⊆	NUM
ejpam-5335	50	36	τ1τ2	τ1τ2	NOUN
ejpam-5335	50	37	-	-	PUNCT
ejpam-5335	50	38	int(τ1τ2	int(τ1τ2	NOUN
ejpam-5335	50	39	-	-	PUNCT
ejpam-5335	50	40	cl(τ1τ2	cl(τ1τ2	NOUN
ejpam-5335	50	41	-	-	PUNCT
ejpam-5335	50	42	int(a	int(a	NOUN
ejpam-5335	50	43	)	)	PUNCT
ejpam-5335	50	44	)	)	PUNCT
ejpam-5335	50	45	)	)	PUNCT
ejpam-5335	50	46	)	)	PUNCT
ejpam-5335	50	47	.	.	PUNCT
ejpam-5335	51	1	the	the	DET
ejpam-5335	51	2	complement	complement	NOUN
ejpam-5335	51	3	of	of	ADP
ejpam-5335	51	4	a	a	DET
ejpam-5335	51	5	(	(	PUNCT
ejpam-5335	51	6	τ1	τ1	NOUN
ejpam-5335	51	7	,	,	PUNCT
ejpam-5335	51	8	τ2)r	τ2)r	NOUN
ejpam-5335	51	9	-	-	PUNCT
ejpam-5335	51	10	open	open	ADJ
ejpam-5335	51	11	(	(	PUNCT
ejpam-5335	51	12	resp	resp	NOUN
ejpam-5335	51	13	.	.	PUNCT
ejpam-5335	52	1	(	(	PUNCT
ejpam-5335	52	2	τ1	τ1	NOUN
ejpam-5335	52	3	,	,	PUNCT
ejpam-5335	52	4	τ2)sopen	τ2)sopen	ADJ
ejpam-5335	52	5	,	,	PUNCT
ejpam-5335	52	6	(	(	PUNCT
ejpam-5335	52	7	τ1	τ1	NOUN
ejpam-5335	52	8	,	,	PUNCT
ejpam-5335	52	9	τ2)p	τ2)p	NOUN
ejpam-5335	52	10	-	-	ADJ
ejpam-5335	52	11	open	open	ADJ
ejpam-5335	52	12	,	,	PUNCT
ejpam-5335	52	13	(	(	PUNCT
ejpam-5335	52	14	τ1	τ1	NOUN
ejpam-5335	52	15	,	,	PUNCT
ejpam-5335	52	16	τ2)β	τ2)β	ADJ
ejpam-5335	52	17	-	-	PUNCT
ejpam-5335	52	18	open	open	ADJ
ejpam-5335	52	19	,	,	PUNCT
ejpam-5335	52	20	α(τ1	α(τ1	NOUN
ejpam-5335	52	21	,	,	PUNCT
ejpam-5335	52	22	τ2)-open	τ2)-open	ADJ
ejpam-5335	52	23	)	)	PUNCT
ejpam-5335	52	24	set	set	NOUN
ejpam-5335	52	25	is	be	AUX
ejpam-5335	52	26	called	call	VERB
ejpam-5335	52	27	(	(	PUNCT
ejpam-5335	52	28	τ1	τ1	NOUN
ejpam-5335	52	29	,	,	PUNCT
ejpam-5335	52	30	τ2)r	τ2)r	NOUN
ejpam-5335	52	31	-	-	PUNCT
ejpam-5335	52	32	closed	closed	ADJ
ejpam-5335	52	33	(	(	PUNCT
ejpam-5335	52	34	resp	resp	NOUN
ejpam-5335	52	35	.	.	PUNCT
ejpam-5335	53	1	(	(	PUNCT
ejpam-5335	53	2	τ1	τ1	NOUN
ejpam-5335	53	3	,	,	PUNCT
ejpam-5335	53	4	τ2)s	τ2)s	NOUN
ejpam-5335	53	5	-	-	PUNCT
ejpam-5335	53	6	closed	closed	ADJ
ejpam-5335	53	7	,	,	PUNCT
ejpam-5335	53	8	(	(	PUNCT
ejpam-5335	53	9	τ1	τ1	NOUN
ejpam-5335	53	10	,	,	PUNCT
ejpam-5335	53	11	τ2)p	τ2)p	NOUN
ejpam-5335	53	12	-	-	PUNCT
ejpam-5335	53	13	closed	closed	ADJ
ejpam-5335	53	14	,	,	PUNCT
ejpam-5335	53	15	(	(	PUNCT
ejpam-5335	53	16	τ1	τ1	NOUN
ejpam-5335	53	17	,	,	PUNCT
ejpam-5335	53	18	τ2)β	τ2)β	ADJ
ejpam-5335	53	19	-	-	PUNCT
ejpam-5335	53	20	closed	closed	ADJ
ejpam-5335	53	21	,	,	PUNCT
ejpam-5335	53	22	α(τ1	α(τ1	NOUN
ejpam-5335	53	23	,	,	PUNCT
ejpam-5335	53	24	τ2)-closed	τ2)-closed	ADJ
ejpam-5335	53	25	)	)	PUNCT
ejpam-5335	53	26	.	.	PUNCT
ejpam-5335	54	1	let	let	VERB
ejpam-5335	54	2	a	a	DET
ejpam-5335	54	3	be	be	AUX
ejpam-5335	54	4	a	a	DET
ejpam-5335	54	5	subset	subset	NOUN
ejpam-5335	54	6	of	of	ADP
ejpam-5335	54	7	a	a	DET
ejpam-5335	54	8	bitopological	bitopological	ADJ
ejpam-5335	54	9	space	space	NOUN
ejpam-5335	54	10	(	(	PUNCT
ejpam-5335	54	11	x	x	NOUN
ejpam-5335	54	12	,	,	PUNCT
ejpam-5335	54	13	τ1	τ1	NOUN
ejpam-5335	54	14	,	,	PUNCT
ejpam-5335	54	15	τ2	τ2	NOUN
ejpam-5335	54	16	)	)	PUNCT
ejpam-5335	54	17	.	.	PUNCT
ejpam-5335	55	1	a	a	DET
ejpam-5335	55	2	point	point	NOUN
ejpam-5335	55	3	x	x	PUNCT
ejpam-5335	55	4	of	of	ADP
ejpam-5335	55	5	x	x	PROPN
ejpam-5335	55	6	is	be	AUX
ejpam-5335	55	7	called	call	VERB
ejpam-5335	55	8	a	a	DET
ejpam-5335	55	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	55	10	,	,	PUNCT
ejpam-5335	55	11	τ2)-cluster	τ2)-cluster	VERB
ejpam-5335	55	12	point	point	NOUN
ejpam-5335	55	13	of	of	ADP
ejpam-5335	55	14	a	a	DET
ejpam-5335	55	15	if	if	SCONJ
ejpam-5335	55	16	v	v	ADP
ejpam-5335	55	17	∩	∩	NOUN
ejpam-5335	55	18	a	a	DET
ejpam-5335	55	19	̸=	̸=	PROPN
ejpam-5335	55	20	∅	∅	NOUN
ejpam-5335	55	21	for	for	ADP
ejpam-5335	55	22	every	every	DET
ejpam-5335	55	23	(	(	PUNCT
ejpam-5335	55	24	τ1	τ1	NOUN
ejpam-5335	55	25	,	,	PUNCT
ejpam-5335	55	26	τ2)r	τ2)r	ADV
ejpam-5335	55	27	-	-	PUNCT
ejpam-5335	55	28	open	open	NOUN
ejpam-5335	55	29	set	set	VERB
ejpam-5335	55	30	v	v	NOUN
ejpam-5335	55	31	containing	contain	VERB
ejpam-5335	55	32	x.	x.	NOUN
ejpam-5335	55	33	the	the	DET
ejpam-5335	55	34	set	set	NOUN
ejpam-5335	55	35	of	of	ADP
ejpam-5335	55	36	all	all	DET
ejpam-5335	55	37	δ(τ1	δ(τ1	NOUN
ejpam-5335	55	38	,	,	PUNCT
ejpam-5335	55	39	τ2)-cluster	τ2)-cluster	ADJ
ejpam-5335	55	40	points	point	NOUN
ejpam-5335	55	41	of	of	ADP
ejpam-5335	55	42	a	a	PRON
ejpam-5335	55	43	is	be	AUX
ejpam-5335	55	44	called	call	VERB
ejpam-5335	55	45	the	the	DET
ejpam-5335	55	46	δ(τ1	δ(τ1	NOUN
ejpam-5335	55	47	,	,	PUNCT
ejpam-5335	55	48	τ2)-closure	τ2)-closure	NOUN
ejpam-5335	55	49	of	of	ADP
ejpam-5335	55	50	a	a	PRON
ejpam-5335	55	51	and	and	CCONJ
ejpam-5335	55	52	is	be	AUX
ejpam-5335	55	53	denoted	denote	VERB
ejpam-5335	55	54	by	by	ADP
ejpam-5335	55	55	δ(τ1	δ(τ1	NOUN
ejpam-5335	55	56	,	,	PUNCT
ejpam-5335	55	57	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	55	58	)	)	PUNCT
ejpam-5335	55	59	.	.	PUNCT
ejpam-5335	56	1	a	a	DET
ejpam-5335	56	2	subset	subset	NOUN
ejpam-5335	56	3	a	a	PRON
ejpam-5335	56	4	of	of	ADP
ejpam-5335	56	5	a	a	DET
ejpam-5335	56	6	bitopological	bitopological	ADJ
ejpam-5335	56	7	space	space	NOUN
ejpam-5335	56	8	(	(	PUNCT
ejpam-5335	56	9	x	x	NOUN
ejpam-5335	56	10	,	,	PUNCT
ejpam-5335	56	11	τ1	τ1	NOUN
ejpam-5335	56	12	,	,	PUNCT
ejpam-5335	56	13	τ2	τ2	NOUN
ejpam-5335	56	14	)	)	PUNCT
ejpam-5335	56	15	is	be	AUX
ejpam-5335	56	16	called	call	VERB
ejpam-5335	56	17	δ(τ1	δ(τ1	NOUN
ejpam-5335	56	18	,	,	PUNCT
ejpam-5335	56	19	τ2)-closed	τ2)-close	VERB
ejpam-5335	56	20	if	if	SCONJ
ejpam-5335	56	21	a	a	DET
ejpam-5335	56	22	=	=	PUNCT
ejpam-5335	56	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	56	24	,	,	PUNCT
ejpam-5335	56	25	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	56	26	)	)	PUNCT
ejpam-5335	56	27	.	.	PUNCT
ejpam-5335	57	1	the	the	DET
ejpam-5335	57	2	complement	complement	NOUN
ejpam-5335	57	3	of	of	ADP
ejpam-5335	57	4	a	a	DET
ejpam-5335	57	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	57	6	,	,	PUNCT
ejpam-5335	57	7	τ2)-closed	τ2)-close	VERB
ejpam-5335	57	8	set	set	NOUN
ejpam-5335	57	9	is	be	AUX
ejpam-5335	57	10	called	call	VERB
ejpam-5335	57	11	δ(τ1	δ(τ1	NOUN
ejpam-5335	57	12	,	,	PUNCT
ejpam-5335	57	13	τ2)-open	τ2)-open	ADJ
ejpam-5335	57	14	(	(	PUNCT
ejpam-5335	57	15	τ1τ2	τ1τ2	ADJ
ejpam-5335	57	16	-	-	ADJ
ejpam-5335	57	17	δ	δ	NOUN
ejpam-5335	57	18	-	-	NOUN
ejpam-5335	57	19	open	open	ADJ
ejpam-5335	57	20	[	[	X
ejpam-5335	57	21	8	8	NUM
ejpam-5335	57	22	]	]	NUM
ejpam-5335	57	23	)	)	PUNCT
ejpam-5335	57	24	.	.	PUNCT
ejpam-5335	58	1	the	the	DET
ejpam-5335	58	2	family	family	NOUN
ejpam-5335	58	3	of	of	ADP
ejpam-5335	58	4	all	all	DET
ejpam-5335	58	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	58	6	,	,	PUNCT
ejpam-5335	58	7	τ2)-open	τ2)-open	ADJ
ejpam-5335	58	8	(	(	PUNCT
ejpam-5335	58	9	resp	resp	NOUN
ejpam-5335	58	10	.	.	PUNCT
ejpam-5335	58	11	δ(τ1	δ(τ1	PROPN
ejpam-5335	58	12	,	,	PUNCT
ejpam-5335	58	13	τ2)-closed	τ2)-closed	ADJ
ejpam-5335	58	14	)	)	PUNCT
ejpam-5335	58	15	sets	set	NOUN
ejpam-5335	58	16	of	of	ADP
ejpam-5335	58	17	a	a	DET
ejpam-5335	58	18	bitopological	bitopological	ADJ
ejpam-5335	58	19	space	space	NOUN
ejpam-5335	58	20	(	(	PUNCT
ejpam-5335	58	21	x	x	NOUN
ejpam-5335	58	22	,	,	PUNCT
ejpam-5335	58	23	τ1	τ1	NOUN
ejpam-5335	58	24	,	,	PUNCT
ejpam-5335	58	25	τ2	τ2	NOUN
ejpam-5335	58	26	)	)	PUNCT
ejpam-5335	58	27	is	be	AUX
ejpam-5335	58	28	denoted	denote	VERB
ejpam-5335	58	29	by	by	ADP
ejpam-5335	58	30	δ(τ1	δ(τ1	NOUN
ejpam-5335	58	31	,	,	PUNCT
ejpam-5335	58	32	τ2)o(x	τ2)o(x	PROPN
ejpam-5335	58	33	)	)	PUNCT
ejpam-5335	58	34	(	(	PUNCT
ejpam-5335	59	1	resp	resp	NOUN
ejpam-5335	59	2	.	.	PUNCT
ejpam-5335	59	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	59	4	,	,	PUNCT
ejpam-5335	59	5	τ2)c(x	τ2)c(x	NOUN
ejpam-5335	59	6	)	)	PUNCT
ejpam-5335	59	7	)	)	PUNCT
ejpam-5335	59	8	.	.	PUNCT
ejpam-5335	60	1	the	the	DET
ejpam-5335	60	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	60	3	,	,	PUNCT
ejpam-5335	60	4	τ2)-interior	τ2)-interior	PRON
ejpam-5335	60	5	of	of	ADP
ejpam-5335	60	6	a	a	DET
ejpam-5335	60	7	denoted	denote	VERB
ejpam-5335	60	8	by	by	ADP
ejpam-5335	60	9	c.	c.	PROPN
ejpam-5335	60	10	prachanpol	prachanpol	PROPN
ejpam-5335	60	11	,	,	PUNCT
ejpam-5335	60	12	c.	c.	PROPN
ejpam-5335	60	13	boonpok	boonpok	PROPN
ejpam-5335	60	14	,	,	PUNCT
ejpam-5335	60	15	c.	c.	PROPN
ejpam-5335	60	16	viriyapong	viriyapong	PROPN
ejpam-5335	60	17	/	/	SYM
ejpam-5335	60	18	eur	eur	PROPN
ejpam-5335	60	19	.	.	PUNCT
ejpam-5335	61	1	j.	j.	PROPN
ejpam-5335	61	2	pure	pure	PROPN
ejpam-5335	61	3	appl	appl	PROPN
ejpam-5335	61	4	.	.	PROPN
ejpam-5335	61	5	math	math	PROPN
ejpam-5335	61	6	,	,	PUNCT
ejpam-5335	61	7	17	17	NUM
ejpam-5335	61	8	(	(	PUNCT
ejpam-5335	61	9	4	4	NUM
ejpam-5335	61	10	)	)	PUNCT
ejpam-5335	61	11	(	(	PUNCT
ejpam-5335	61	12	2024	2024	NUM
ejpam-5335	61	13	)	)	PUNCT
ejpam-5335	61	14	,	,	PUNCT
ejpam-5335	61	15	3730	3730	NUM
ejpam-5335	61	16	-	-	SYM
ejpam-5335	61	17	3742	3742	NUM
ejpam-5335	61	18	3732	3732	NUM
ejpam-5335	61	19	δ(τ1	δ(τ1	NOUN
ejpam-5335	61	20	,	,	PUNCT
ejpam-5335	61	21	τ2)-int(a	τ2)-int(a	NOUN
ejpam-5335	61	22	)	)	PUNCT
ejpam-5335	61	23	is	be	AUX
ejpam-5335	61	24	defined	define	VERB
ejpam-5335	61	25	as	as	SCONJ
ejpam-5335	61	26	follows	follow	VERB
ejpam-5335	61	27	:	:	PUNCT
ejpam-5335	61	28	δ(τ1	δ(τ1	NOUN
ejpam-5335	61	29	,	,	PUNCT
ejpam-5335	61	30	τ2)-int(a	τ2)-int(a	NOUN
ejpam-5335	61	31	)	)	PUNCT
ejpam-5335	61	32	=	=	PUNCT
ejpam-5335	62	1	∪{g	∪{g	PROPN
ejpam-5335	62	2	⊆	⊆	NUM
ejpam-5335	62	3	x	x	X
ejpam-5335	62	4	|	|	ADV
ejpam-5335	62	5	g	g	PROPN
ejpam-5335	62	6	∈	∈	PROPN
ejpam-5335	62	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	62	8	,	,	PUNCT
ejpam-5335	62	9	τ2)o(x	τ2)o(x	PROPN
ejpam-5335	62	10	)	)	PUNCT
ejpam-5335	62	11	and	and	CCONJ
ejpam-5335	62	12	g	g	PROPN
ejpam-5335	62	13	⊆	⊆	NUM
ejpam-5335	62	14	a	a	PRON
ejpam-5335	62	15	}	}	PUNCT
ejpam-5335	62	16	.	.	PUNCT
ejpam-5335	63	1	lemma	lemma	PROPN
ejpam-5335	63	2	1	1	NUM
ejpam-5335	63	3	.	.	PUNCT
ejpam-5335	64	1	for	for	ADP
ejpam-5335	64	2	a	a	DET
ejpam-5335	64	3	subset	subset	NOUN
ejpam-5335	64	4	a	a	PRON
ejpam-5335	64	5	of	of	ADP
ejpam-5335	64	6	a	a	DET
ejpam-5335	64	7	bitopological	bitopological	ADJ
ejpam-5335	64	8	space	space	NOUN
ejpam-5335	64	9	(	(	PUNCT
ejpam-5335	64	10	x	x	NOUN
ejpam-5335	64	11	,	,	PUNCT
ejpam-5335	64	12	τ1	τ1	NOUN
ejpam-5335	64	13	,	,	PUNCT
ejpam-5335	64	14	τ2	τ2	NOUN
ejpam-5335	64	15	)	)	PUNCT
ejpam-5335	64	16	,	,	PUNCT
ejpam-5335	64	17	x	x	PROPN
ejpam-5335	64	18	∈	∈	PROPN
ejpam-5335	64	19	δ(τ1	δ(τ1	PROPN
ejpam-5335	64	20	,	,	PUNCT
ejpam-5335	64	21	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	64	22	)	)	PUNCT
ejpam-5335	64	23	if	if	SCONJ
ejpam-5335	64	24	and	and	CCONJ
ejpam-5335	64	25	only	only	ADV
ejpam-5335	64	26	if	if	SCONJ
ejpam-5335	64	27	v	v	NUM
ejpam-5335	64	28	∩a	∩a	PROPN
ejpam-5335	64	29	̸=	̸=	PROPN
ejpam-5335	64	30	∅	∅	NOUN
ejpam-5335	64	31	for	for	ADP
ejpam-5335	64	32	every	every	DET
ejpam-5335	64	33	v	v	PROPN
ejpam-5335	64	34	∈	∈	PROPN
ejpam-5335	64	35	δ(τ1	δ(τ1	NOUN
ejpam-5335	64	36	,	,	PUNCT
ejpam-5335	64	37	τ2)o(x	τ2)o(x	PROPN
ejpam-5335	64	38	)	)	PUNCT
ejpam-5335	64	39	containing	contain	VERB
ejpam-5335	64	40	x.	x.	PROPN
ejpam-5335	64	41	lemma	lemma	PROPN
ejpam-5335	65	1	2	2	NUM
ejpam-5335	65	2	.	.	X
ejpam-5335	65	3	for	for	ADP
ejpam-5335	65	4	a	a	DET
ejpam-5335	65	5	subset	subset	NOUN
ejpam-5335	65	6	a	a	PRON
ejpam-5335	65	7	of	of	ADP
ejpam-5335	65	8	a	a	DET
ejpam-5335	65	9	bitopological	bitopological	ADJ
ejpam-5335	65	10	space	space	NOUN
ejpam-5335	65	11	(	(	PUNCT
ejpam-5335	65	12	x	x	NOUN
ejpam-5335	65	13	,	,	PUNCT
ejpam-5335	65	14	τ1	τ1	NOUN
ejpam-5335	65	15	,	,	PUNCT
ejpam-5335	65	16	τ2	τ2	NOUN
ejpam-5335	65	17	)	)	PUNCT
ejpam-5335	65	18	,	,	PUNCT
ejpam-5335	65	19	the	the	DET
ejpam-5335	65	20	following	follow	VERB
ejpam-5335	65	21	properties	property	NOUN
ejpam-5335	65	22	hold	hold	VERB
ejpam-5335	65	23	:	:	PUNCT
ejpam-5335	65	24	(	(	PUNCT
ejpam-5335	65	25	1	1	X
ejpam-5335	65	26	)	)	PUNCT
ejpam-5335	65	27	δ(τ1	δ(τ1	NOUN
ejpam-5335	65	28	,	,	PUNCT
ejpam-5335	65	29	τ2)-int(x	τ2)-int(x	NOUN
ejpam-5335	65	30	−a	−a	NOUN
ejpam-5335	65	31	)	)	PUNCT
ejpam-5335	66	1	=	=	PUNCT
ejpam-5335	67	1	x	x	SYM
ejpam-5335	67	2	−	−	PROPN
ejpam-5335	67	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	67	4	,	,	PUNCT
ejpam-5335	67	5	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	67	6	)	)	PUNCT
ejpam-5335	67	7	.	.	PUNCT
ejpam-5335	68	1	(	(	PUNCT
ejpam-5335	68	2	2	2	X
ejpam-5335	68	3	)	)	PUNCT
ejpam-5335	68	4	δ(τ1	δ(τ1	NOUN
ejpam-5335	68	5	,	,	PUNCT
ejpam-5335	68	6	τ2)-cl(x	τ2)-cl(x	PROPN
ejpam-5335	68	7	−a	−a	NOUN
ejpam-5335	68	8	)	)	PUNCT
ejpam-5335	69	1	=	=	PUNCT
ejpam-5335	69	2	x	x	SYM
ejpam-5335	70	1	−	−	PROPN
ejpam-5335	70	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	70	3	,	,	PUNCT
ejpam-5335	70	4	τ2)-int(a	τ2)-int(a	NOUN
ejpam-5335	70	5	)	)	PUNCT
ejpam-5335	70	6	.	.	PUNCT
ejpam-5335	71	1	3	3	X
ejpam-5335	71	2	.	.	X
ejpam-5335	71	3	on	on	ADP
ejpam-5335	71	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	71	5	,	,	PUNCT
ejpam-5335	71	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	71	7	functions	function	NOUN
ejpam-5335	71	8	in	in	ADP
ejpam-5335	71	9	this	this	DET
ejpam-5335	71	10	section	section	NOUN
ejpam-5335	71	11	,	,	PUNCT
ejpam-5335	71	12	we	we	PRON
ejpam-5335	71	13	introduce	introduce	VERB
ejpam-5335	71	14	the	the	DET
ejpam-5335	71	15	notion	notion	NOUN
ejpam-5335	71	16	of	of	ADP
ejpam-5335	71	17	δ(τ1	δ(τ1	PROPN
ejpam-5335	71	18	,	,	PUNCT
ejpam-5335	71	19	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	71	20	functions	function	NOUN
ejpam-5335	71	21	.	.	PUNCT
ejpam-5335	72	1	we	we	PRON
ejpam-5335	72	2	also	also	ADV
ejpam-5335	72	3	discuss	discuss	VERB
ejpam-5335	72	4	several	several	ADJ
ejpam-5335	72	5	characterizations	characterization	NOUN
ejpam-5335	72	6	of	of	ADP
ejpam-5335	72	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	72	8	,	,	PUNCT
ejpam-5335	72	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	72	10	functions	function	NOUN
ejpam-5335	72	11	.	.	PUNCT
ejpam-5335	73	1	definition	definition	NOUN
ejpam-5335	73	2	1	1	NUM
ejpam-5335	73	3	.	.	PUNCT
ejpam-5335	74	1	a	a	DET
ejpam-5335	74	2	function	function	NOUN
ejpam-5335	74	3	f	f	NOUN
ejpam-5335	74	4	:	:	PUNCT
ejpam-5335	74	5	(	(	PUNCT
ejpam-5335	74	6	x	x	NOUN
ejpam-5335	74	7	,	,	PUNCT
ejpam-5335	74	8	τ1	τ1	NOUN
ejpam-5335	74	9	,	,	PUNCT
ejpam-5335	74	10	τ2	τ2	NOUN
ejpam-5335	74	11	)	)	PUNCT
ejpam-5335	74	12	→	→	SYM
ejpam-5335	74	13	(	(	PUNCT
ejpam-5335	74	14	y	y	PROPN
ejpam-5335	74	15	,	,	PUNCT
ejpam-5335	74	16	σ1	σ1	PROPN
ejpam-5335	74	17	,	,	PUNCT
ejpam-5335	74	18	σ2	σ2	PROPN
ejpam-5335	74	19	)	)	PUNCT
ejpam-5335	74	20	is	be	AUX
ejpam-5335	74	21	called	call	VERB
ejpam-5335	74	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	74	23	,	,	PUNCT
ejpam-5335	74	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	74	25	at	at	ADP
ejpam-5335	74	26	x	x	X
ejpam-5335	74	27	∈	∈	PROPN
ejpam-5335	74	28	x	x	SYM
ejpam-5335	74	29	if	if	SCONJ
ejpam-5335	74	30	for	for	ADP
ejpam-5335	74	31	each	each	DET
ejpam-5335	74	32	σ1σ2	σ1σ2	VERB
ejpam-5335	74	33	-	-	ADJ
ejpam-5335	74	34	open	open	ADJ
ejpam-5335	74	35	set	set	NOUN
ejpam-5335	74	36	v	v	NOUN
ejpam-5335	74	37	of	of	ADP
ejpam-5335	74	38	y	y	NOUN
ejpam-5335	74	39	containing	contain	VERB
ejpam-5335	74	40	f(x	f(x	PROPN
ejpam-5335	74	41	)	)	PUNCT
ejpam-5335	74	42	,	,	PUNCT
ejpam-5335	74	43	there	there	PRON
ejpam-5335	74	44	exists	exist	VERB
ejpam-5335	74	45	a	a	DET
ejpam-5335	74	46	δ(τ1	δ(τ1	NOUN
ejpam-5335	74	47	,	,	PUNCT
ejpam-5335	74	48	τ2)-open	τ2)-open	ADJ
ejpam-5335	74	49	set	set	ADJ
ejpam-5335	74	50	u	u	NOUN
ejpam-5335	74	51	of	of	ADP
ejpam-5335	74	52	x	x	PUNCT
ejpam-5335	74	53	containing	contain	VERB
ejpam-5335	74	54	x	x	PUNCT
ejpam-5335	74	55	such	such	ADJ
ejpam-5335	74	56	that	that	DET
ejpam-5335	74	57	f(u	f(u	PROPN
ejpam-5335	74	58	)	)	PUNCT
ejpam-5335	74	59	⊆	⊆	NUM
ejpam-5335	74	60	v	v	NOUN
ejpam-5335	74	61	.	.	PUNCT
ejpam-5335	75	1	a	a	DET
ejpam-5335	75	2	function	function	NOUN
ejpam-5335	75	3	f	f	NOUN
ejpam-5335	75	4	:	:	PUNCT
ejpam-5335	75	5	(	(	PUNCT
ejpam-5335	75	6	x	x	NOUN
ejpam-5335	75	7	,	,	PUNCT
ejpam-5335	75	8	τ1	τ1	NOUN
ejpam-5335	75	9	,	,	PUNCT
ejpam-5335	75	10	τ2	τ2	NOUN
ejpam-5335	75	11	)	)	PUNCT
ejpam-5335	75	12	→	→	SYM
ejpam-5335	75	13	(	(	PUNCT
ejpam-5335	75	14	y	y	PROPN
ejpam-5335	75	15	,	,	PUNCT
ejpam-5335	75	16	σ1	σ1	PROPN
ejpam-5335	75	17	,	,	PUNCT
ejpam-5335	75	18	σ2	σ2	PROPN
ejpam-5335	75	19	)	)	PUNCT
ejpam-5335	75	20	is	be	AUX
ejpam-5335	75	21	said	say	VERB
ejpam-5335	75	22	to	to	PART
ejpam-5335	75	23	be	be	AUX
ejpam-5335	75	24	δ(τ1	δ(τ1	NOUN
ejpam-5335	75	25	,	,	PUNCT
ejpam-5335	75	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	75	27	if	if	SCONJ
ejpam-5335	75	28	f	f	PROPN
ejpam-5335	75	29	is	be	AUX
ejpam-5335	75	30	δ(τ1	δ(τ1	NOUN
ejpam-5335	75	31	,	,	PUNCT
ejpam-5335	75	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	75	33	at	at	ADP
ejpam-5335	75	34	each	each	DET
ejpam-5335	75	35	point	point	NOUN
ejpam-5335	75	36	of	of	ADP
ejpam-5335	75	37	x.	x.	PROPN
ejpam-5335	75	38	example	example	NOUN
ejpam-5335	76	1	1	1	X
ejpam-5335	76	2	.	.	PUNCT
ejpam-5335	77	1	let	let	VERB
ejpam-5335	77	2	x	x	PUNCT
ejpam-5335	77	3	=	=	PRON
ejpam-5335	77	4	{	{	PUNCT
ejpam-5335	77	5	a	a	PRON
ejpam-5335	77	6	,	,	PUNCT
ejpam-5335	77	7	b	b	NOUN
ejpam-5335	77	8	,	,	PUNCT
ejpam-5335	77	9	c	c	NOUN
ejpam-5335	77	10	}	}	PUNCT
ejpam-5335	77	11	with	with	ADP
ejpam-5335	77	12	topologies	topology	NOUN
ejpam-5335	77	13	τ1	τ1	NOUN
ejpam-5335	77	14	=	=	SYM
ejpam-5335	77	15	{	{	PUNCT
ejpam-5335	77	16	∅	∅	NOUN
ejpam-5335	77	17	,	,	PUNCT
ejpam-5335	77	18	{	{	PUNCT
ejpam-5335	77	19	a	a	X
ejpam-5335	77	20	}	}	PUNCT
ejpam-5335	77	21	,	,	PUNCT
ejpam-5335	77	22	{	{	PUNCT
ejpam-5335	77	23	b	b	NOUN
ejpam-5335	77	24	}	}	PUNCT
ejpam-5335	77	25	,	,	PUNCT
ejpam-5335	77	26	{	{	PUNCT
ejpam-5335	77	27	a	a	DET
ejpam-5335	77	28	,	,	PUNCT
ejpam-5335	77	29	b	b	NOUN
ejpam-5335	77	30	}	}	PUNCT
ejpam-5335	77	31	,	,	PUNCT
ejpam-5335	77	32	{	{	PUNCT
ejpam-5335	77	33	a	a	X
ejpam-5335	77	34	,	,	PUNCT
ejpam-5335	77	35	c	c	NOUN
ejpam-5335	77	36	}	}	PUNCT
ejpam-5335	77	37	,	,	PUNCT
ejpam-5335	77	38	x	x	NOUN
ejpam-5335	77	39	}	}	PUNCT
ejpam-5335	77	40	and	and	CCONJ
ejpam-5335	77	41	τ2	τ2	NOUN
ejpam-5335	77	42	=	=	SYM
ejpam-5335	77	43	{	{	PUNCT
ejpam-5335	77	44	∅	∅	NOUN
ejpam-5335	77	45	,	,	PUNCT
ejpam-5335	77	46	{	{	PUNCT
ejpam-5335	77	47	a	a	X
ejpam-5335	77	48	}	}	PUNCT
ejpam-5335	77	49	,	,	PUNCT
ejpam-5335	77	50	{	{	PUNCT
ejpam-5335	77	51	b	b	NOUN
ejpam-5335	77	52	}	}	PUNCT
ejpam-5335	77	53	,	,	PUNCT
ejpam-5335	77	54	{	{	PUNCT
ejpam-5335	77	55	a	a	DET
ejpam-5335	77	56	,	,	PUNCT
ejpam-5335	77	57	b	b	NOUN
ejpam-5335	77	58	}	}	PUNCT
ejpam-5335	77	59	,	,	PUNCT
ejpam-5335	77	60	x	x	NOUN
ejpam-5335	77	61	}	}	PUNCT
ejpam-5335	77	62	.	.	PUNCT
ejpam-5335	78	1	let	let	VERB
ejpam-5335	78	2	y	y	PROPN
ejpam-5335	78	3	=	=	PUNCT
ejpam-5335	78	4	{	{	PUNCT
ejpam-5335	78	5	1	1	NUM
ejpam-5335	78	6	,	,	PUNCT
ejpam-5335	78	7	2	2	NUM
ejpam-5335	78	8	,	,	PUNCT
ejpam-5335	78	9	3	3	NUM
ejpam-5335	78	10	}	}	PUNCT
ejpam-5335	78	11	with	with	ADP
ejpam-5335	78	12	topologies	topology	NOUN
ejpam-5335	78	13	σ1	σ1	NOUN
ejpam-5335	78	14	=	=	SYM
ejpam-5335	78	15	{	{	PUNCT
ejpam-5335	78	16	∅	∅	NOUN
ejpam-5335	78	17	,	,	PUNCT
ejpam-5335	78	18	{	{	PUNCT
ejpam-5335	78	19	1	1	NUM
ejpam-5335	78	20	}	}	PUNCT
ejpam-5335	78	21	,	,	PUNCT
ejpam-5335	78	22	{	{	PUNCT
ejpam-5335	78	23	3	3	NUM
ejpam-5335	78	24	}	}	PUNCT
ejpam-5335	78	25	,	,	PUNCT
ejpam-5335	78	26	{	{	PUNCT
ejpam-5335	78	27	1	1	NUM
ejpam-5335	78	28	,	,	PUNCT
ejpam-5335	78	29	2	2	NUM
ejpam-5335	78	30	}	}	PUNCT
ejpam-5335	78	31	,	,	PUNCT
ejpam-5335	78	32	{	{	PUNCT
ejpam-5335	78	33	1	1	NUM
ejpam-5335	78	34	,	,	PUNCT
ejpam-5335	78	35	3	3	NUM
ejpam-5335	78	36	}	}	PUNCT
ejpam-5335	78	37	,	,	PUNCT
ejpam-5335	78	38	y	y	PROPN
ejpam-5335	78	39	}	}	PUNCT
ejpam-5335	78	40	and	and	CCONJ
ejpam-5335	78	41	σ2	σ2	PROPN
ejpam-5335	78	42	=	=	SYM
ejpam-5335	78	43	{	{	PUNCT
ejpam-5335	78	44	∅	∅	NOUN
ejpam-5335	78	45	,	,	PUNCT
ejpam-5335	78	46	{	{	PUNCT
ejpam-5335	78	47	1	1	NUM
ejpam-5335	78	48	}	}	PUNCT
ejpam-5335	78	49	,	,	PUNCT
ejpam-5335	78	50	{	{	PUNCT
ejpam-5335	78	51	3	3	NUM
ejpam-5335	78	52	}	}	PUNCT
ejpam-5335	78	53	,	,	PUNCT
ejpam-5335	78	54	{	{	PUNCT
ejpam-5335	78	55	1	1	NUM
ejpam-5335	78	56	,	,	PUNCT
ejpam-5335	78	57	3	3	NUM
ejpam-5335	78	58	}	}	PUNCT
ejpam-5335	78	59	,	,	PUNCT
ejpam-5335	78	60	y	y	PROPN
ejpam-5335	78	61	}	}	PUNCT
ejpam-5335	78	62	.	.	PUNCT
ejpam-5335	79	1	define	define	VERB
ejpam-5335	79	2	a	a	DET
ejpam-5335	79	3	function	function	NOUN
ejpam-5335	79	4	f	f	NOUN
ejpam-5335	79	5	:	:	PUNCT
ejpam-5335	79	6	(	(	PUNCT
ejpam-5335	79	7	x	x	NOUN
ejpam-5335	79	8	,	,	PUNCT
ejpam-5335	79	9	τ1	τ1	NOUN
ejpam-5335	79	10	,	,	PUNCT
ejpam-5335	79	11	τ2	τ2	NOUN
ejpam-5335	79	12	)	)	PUNCT
ejpam-5335	79	13	→	→	SYM
ejpam-5335	79	14	(	(	PUNCT
ejpam-5335	79	15	y	y	PROPN
ejpam-5335	79	16	,	,	PUNCT
ejpam-5335	79	17	σ1	σ1	PROPN
ejpam-5335	79	18	,	,	PUNCT
ejpam-5335	79	19	σ2	σ2	PROPN
ejpam-5335	79	20	)	)	PUNCT
ejpam-5335	79	21	as	as	SCONJ
ejpam-5335	79	22	follows	follow	VERB
ejpam-5335	79	23	:	:	PUNCT
ejpam-5335	79	24	f(a	f(a	NOUN
ejpam-5335	79	25	)	)	PUNCT
ejpam-5335	79	26	=	=	SYM
ejpam-5335	79	27	f(c	f(c	PROPN
ejpam-5335	79	28	)	)	PUNCT
ejpam-5335	79	29	=	=	SYM
ejpam-5335	79	30	2	2	NUM
ejpam-5335	79	31	and	and	CCONJ
ejpam-5335	79	32	f(b	f(b	PROPN
ejpam-5335	79	33	)	)	PUNCT
ejpam-5335	79	34	=	=	SYM
ejpam-5335	80	1	3	3	X
ejpam-5335	80	2	.	.	PUNCT
ejpam-5335	80	3	then	then	ADV
ejpam-5335	80	4	,	,	PUNCT
ejpam-5335	80	5	f	f	PROPN
ejpam-5335	80	6	is	be	AUX
ejpam-5335	80	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	80	8	,	,	PUNCT
ejpam-5335	80	9	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5335	80	10	.	.	PUNCT
ejpam-5335	81	1	theorem	theorem	NOUN
ejpam-5335	81	2	1	1	NUM
ejpam-5335	81	3	.	.	X
ejpam-5335	81	4	for	for	ADP
ejpam-5335	81	5	a	a	DET
ejpam-5335	81	6	function	function	NOUN
ejpam-5335	81	7	f	f	NOUN
ejpam-5335	81	8	:	:	PUNCT
ejpam-5335	81	9	(	(	PUNCT
ejpam-5335	81	10	x	x	NOUN
ejpam-5335	81	11	,	,	PUNCT
ejpam-5335	81	12	τ1	τ1	NOUN
ejpam-5335	81	13	,	,	PUNCT
ejpam-5335	81	14	τ2	τ2	NOUN
ejpam-5335	81	15	)	)	PUNCT
ejpam-5335	81	16	→	→	SYM
ejpam-5335	81	17	(	(	PUNCT
ejpam-5335	81	18	y	y	PROPN
ejpam-5335	81	19	,	,	PUNCT
ejpam-5335	81	20	σ1	σ1	PROPN
ejpam-5335	81	21	,	,	PUNCT
ejpam-5335	81	22	σ2	σ2	NOUN
ejpam-5335	81	23	)	)	PUNCT
ejpam-5335	81	24	,	,	PUNCT
ejpam-5335	81	25	the	the	DET
ejpam-5335	81	26	following	follow	VERB
ejpam-5335	81	27	properties	property	NOUN
ejpam-5335	81	28	are	be	AUX
ejpam-5335	81	29	equivalent	equivalent	ADJ
ejpam-5335	81	30	:	:	PUNCT
ejpam-5335	81	31	(	(	PUNCT
ejpam-5335	81	32	1	1	X
ejpam-5335	81	33	)	)	PUNCT
ejpam-5335	81	34	f	f	PROPN
ejpam-5335	81	35	is	be	AUX
ejpam-5335	81	36	δ(τ1	δ(τ1	NOUN
ejpam-5335	81	37	,	,	PUNCT
ejpam-5335	81	38	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	81	39	at	at	ADP
ejpam-5335	81	40	x	x	PRON
ejpam-5335	81	41	;	;	PUNCT
ejpam-5335	81	42	(	(	PUNCT
ejpam-5335	81	43	2	2	X
ejpam-5335	81	44	)	)	PUNCT
ejpam-5335	81	45	x	x	SYM
ejpam-5335	81	46	∈	∈	PROPN
ejpam-5335	81	47	δ(τ1	δ(τ1	PROPN
ejpam-5335	81	48	,	,	PUNCT
ejpam-5335	81	49	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	81	50	−1(v	−1(v	PROPN
ejpam-5335	81	51	)	)	PUNCT
ejpam-5335	81	52	)	)	PUNCT
ejpam-5335	81	53	for	for	ADP
ejpam-5335	81	54	every	every	DET
ejpam-5335	81	55	σ1σ2	σ1σ2	NOUN
ejpam-5335	81	56	-	-	ADJ
ejpam-5335	81	57	open	open	ADJ
ejpam-5335	81	58	set	set	NOUN
ejpam-5335	81	59	v	v	NOUN
ejpam-5335	81	60	of	of	ADP
ejpam-5335	81	61	y	y	NOUN
ejpam-5335	81	62	containing	contain	VERB
ejpam-5335	81	63	f(x	f(x	PROPN
ejpam-5335	81	64	)	)	PUNCT
ejpam-5335	81	65	;	;	PUNCT
ejpam-5335	82	1	(	(	PUNCT
ejpam-5335	82	2	3	3	X
ejpam-5335	82	3	)	)	PUNCT
ejpam-5335	82	4	x	x	SYM
ejpam-5335	82	5	∈	∈	PROPN
ejpam-5335	82	6	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	82	7	-	-	PUNCT
ejpam-5335	82	8	cl(f(a	cl(f(a	NOUN
ejpam-5335	82	9	)	)	PUNCT
ejpam-5335	82	10	)	)	PUNCT
ejpam-5335	82	11	)	)	PUNCT
ejpam-5335	83	1	for	for	ADP
ejpam-5335	83	2	every	every	DET
ejpam-5335	83	3	a	a	DET
ejpam-5335	83	4	⊆	⊆	NUM
ejpam-5335	83	5	x	x	SYM
ejpam-5335	83	6	such	such	ADJ
ejpam-5335	83	7	that	that	SCONJ
ejpam-5335	83	8	x	x	PROPN
ejpam-5335	83	9	∈	∈	PROPN
ejpam-5335	83	10	δ(τ1	δ(τ1	NOUN
ejpam-5335	83	11	,	,	PUNCT
ejpam-5335	83	12	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	83	13	)	)	PUNCT
ejpam-5335	83	14	;	;	PUNCT
ejpam-5335	83	15	(	(	PUNCT
ejpam-5335	83	16	4	4	X
ejpam-5335	83	17	)	)	PUNCT
ejpam-5335	83	18	x	x	SYM
ejpam-5335	83	19	∈	∈	PROPN
ejpam-5335	83	20	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	83	21	-	-	PUNCT
ejpam-5335	83	22	cl(b	cl(b	NOUN
ejpam-5335	83	23	)	)	PUNCT
ejpam-5335	83	24	)	)	PUNCT
ejpam-5335	84	1	for	for	ADP
ejpam-5335	84	2	every	every	DET
ejpam-5335	84	3	b	b	PROPN
ejpam-5335	84	4	⊆	⊆	NUM
ejpam-5335	84	5	y	y	NUM
ejpam-5335	84	6	such	such	ADJ
ejpam-5335	84	7	that	that	SCONJ
ejpam-5335	84	8	x	x	PROPN
ejpam-5335	84	9	∈	∈	PROPN
ejpam-5335	84	10	δ(τ1	δ(τ1	PROPN
ejpam-5335	84	11	,	,	PUNCT
ejpam-5335	84	12	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	84	13	−1(b	−1(b	PROPN
ejpam-5335	84	14	)	)	PUNCT
ejpam-5335	84	15	)	)	PUNCT
ejpam-5335	84	16	;	;	PUNCT
ejpam-5335	84	17	(	(	PUNCT
ejpam-5335	84	18	5	5	X
ejpam-5335	84	19	)	)	PUNCT
ejpam-5335	84	20	x	x	SYM
ejpam-5335	84	21	∈	∈	PROPN
ejpam-5335	84	22	δ(τ1	δ(τ1	PROPN
ejpam-5335	84	23	,	,	PUNCT
ejpam-5335	84	24	τ2)-int(f	τ2)-int(f	X
ejpam-5335	84	25	−1(b	−1(b	NOUN
ejpam-5335	84	26	)	)	PUNCT
ejpam-5335	84	27	)	)	PUNCT
ejpam-5335	84	28	for	for	ADP
ejpam-5335	84	29	every	every	DET
ejpam-5335	84	30	b	b	PROPN
ejpam-5335	84	31	⊆	⊆	NUM
ejpam-5335	84	32	y	y	NUM
ejpam-5335	84	33	such	such	ADJ
ejpam-5335	84	34	that	that	SCONJ
ejpam-5335	84	35	x	x	SYM
ejpam-5335	84	36	∈	∈	PROPN
ejpam-5335	84	37	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	84	38	-	-	PUNCT
ejpam-5335	84	39	int(b	int(b	NOUN
ejpam-5335	84	40	)	)	PUNCT
ejpam-5335	84	41	)	)	PUNCT
ejpam-5335	84	42	;	;	PUNCT
ejpam-5335	84	43	(	(	PUNCT
ejpam-5335	84	44	6	6	X
ejpam-5335	84	45	)	)	PUNCT
ejpam-5335	84	46	x	x	SYM
ejpam-5335	84	47	∈	∈	PROPN
ejpam-5335	84	48	f−1(f	f−1(f	PROPN
ejpam-5335	84	49	)	)	PUNCT
ejpam-5335	84	50	for	for	ADP
ejpam-5335	84	51	every	every	DET
ejpam-5335	84	52	σ1σ2	σ1σ2	NUM
ejpam-5335	84	53	-	-	PUNCT
ejpam-5335	84	54	closed	closed	ADJ
ejpam-5335	84	55	set	set	ADJ
ejpam-5335	84	56	f	f	PROPN
ejpam-5335	84	57	of	of	ADP
ejpam-5335	84	58	y	y	PRON
ejpam-5335	84	59	such	such	ADJ
ejpam-5335	84	60	that	that	SCONJ
ejpam-5335	84	61	x	x	PROPN
ejpam-5335	84	62	∈	∈	PROPN
ejpam-5335	84	63	δ(τ1	δ(τ1	PROPN
ejpam-5335	84	64	,	,	PUNCT
ejpam-5335	84	65	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	84	66	−1(f	−1(f	NUM
ejpam-5335	84	67	)	)	PUNCT
ejpam-5335	84	68	)	)	PUNCT
ejpam-5335	84	69	.	.	PUNCT
ejpam-5335	85	1	proof	proof	NOUN
ejpam-5335	85	2	.	.	PUNCT
ejpam-5335	86	1	(	(	PUNCT
ejpam-5335	86	2	1	1	X
ejpam-5335	86	3	)	)	PUNCT
ejpam-5335	86	4	⇒	⇒	NOUN
ejpam-5335	86	5	(	(	PUNCT
ejpam-5335	86	6	2	2	NUM
ejpam-5335	86	7	):	):	PUNCT
ejpam-5335	86	8	let	let	VERB
ejpam-5335	86	9	v	v	PART
ejpam-5335	86	10	be	be	AUX
ejpam-5335	86	11	any	any	DET
ejpam-5335	86	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	86	13	-	-	ADJ
ejpam-5335	86	14	open	open	ADJ
ejpam-5335	86	15	set	set	NOUN
ejpam-5335	86	16	of	of	ADP
ejpam-5335	86	17	y	y	PROPN
ejpam-5335	86	18	containing	contain	VERB
ejpam-5335	86	19	f(x	f(x	PROPN
ejpam-5335	86	20	)	)	PUNCT
ejpam-5335	86	21	.	.	PUNCT
ejpam-5335	87	1	by	by	ADP
ejpam-5335	87	2	(	(	PUNCT
ejpam-5335	87	3	1	1	NUM
ejpam-5335	87	4	)	)	PUNCT
ejpam-5335	87	5	,	,	PUNCT
ejpam-5335	87	6	there	there	PRON
ejpam-5335	87	7	exists	exist	VERB
ejpam-5335	87	8	a	a	DET
ejpam-5335	87	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	87	10	,	,	PUNCT
ejpam-5335	87	11	τ2)-open	τ2)-open	ADJ
ejpam-5335	87	12	set	set	ADJ
ejpam-5335	87	13	u	u	NOUN
ejpam-5335	87	14	of	of	ADP
ejpam-5335	87	15	x	x	PUNCT
ejpam-5335	87	16	containing	contain	VERB
ejpam-5335	87	17	x	x	PUNCT
ejpam-5335	87	18	such	such	ADJ
ejpam-5335	87	19	that	that	DET
ejpam-5335	87	20	f(u	f(u	PROPN
ejpam-5335	87	21	)	)	PUNCT
ejpam-5335	87	22	⊆	⊆	NUM
ejpam-5335	87	23	v	v	NOUN
ejpam-5335	87	24	.	.	PUNCT
ejpam-5335	88	1	hence	hence	ADV
ejpam-5335	88	2	,	,	PUNCT
ejpam-5335	88	3	x	x	PUNCT
ejpam-5335	88	4	∈	∈	PROPN
ejpam-5335	88	5	u	u	NOUN
ejpam-5335	88	6	⊆	⊆	NUM
ejpam-5335	88	7	f−1(v	f−1(v	NOUN
ejpam-5335	88	8	)	)	PUNCT
ejpam-5335	88	9	.	.	PUNCT
ejpam-5335	89	1	therefore	therefore	ADV
ejpam-5335	89	2	,	,	PUNCT
ejpam-5335	89	3	x	x	PROPN
ejpam-5335	89	4	∈	∈	PROPN
ejpam-5335	89	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	89	6	,	,	PUNCT
ejpam-5335	89	7	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	89	8	−1(v	−1(v	PROPN
ejpam-5335	89	9	)	)	PUNCT
ejpam-5335	89	10	)	)	PUNCT
ejpam-5335	89	11	.	.	PUNCT
ejpam-5335	90	1	(	(	PUNCT
ejpam-5335	90	2	2	2	X
ejpam-5335	90	3	)	)	PUNCT
ejpam-5335	90	4	⇒	⇒	NOUN
ejpam-5335	90	5	(	(	PUNCT
ejpam-5335	90	6	3	3	NUM
ejpam-5335	90	7	):	):	PUNCT
ejpam-5335	90	8	let	let	VERB
ejpam-5335	90	9	a	a	DET
ejpam-5335	90	10	⊆	⊆	NUM
ejpam-5335	90	11	x	x	SYM
ejpam-5335	90	12	such	such	ADJ
ejpam-5335	90	13	that	that	SCONJ
ejpam-5335	90	14	x	x	PROPN
ejpam-5335	90	15	∈	∈	PROPN
ejpam-5335	90	16	δ(τ1	δ(τ1	PROPN
ejpam-5335	90	17	,	,	PUNCT
ejpam-5335	90	18	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	90	19	)	)	PUNCT
ejpam-5335	90	20	and	and	CCONJ
ejpam-5335	90	21	v	v	AUX
ejpam-5335	90	22	be	be	AUX
ejpam-5335	90	23	any	any	DET
ejpam-5335	90	24	σ1σ2	σ1σ2	NOUN
ejpam-5335	90	25	-	-	ADJ
ejpam-5335	90	26	open	open	ADJ
ejpam-5335	90	27	set	set	NOUN
ejpam-5335	90	28	of	of	ADP
ejpam-5335	90	29	y	y	PROPN
ejpam-5335	90	30	containing	contain	VERB
ejpam-5335	90	31	f(x	f(x	PROPN
ejpam-5335	90	32	)	)	PUNCT
ejpam-5335	90	33	.	.	PUNCT
ejpam-5335	91	1	by	by	ADP
ejpam-5335	91	2	(	(	PUNCT
ejpam-5335	91	3	2	2	NUM
ejpam-5335	91	4	)	)	PUNCT
ejpam-5335	91	5	,	,	PUNCT
ejpam-5335	91	6	x	x	PROPN
ejpam-5335	91	7	∈	∈	PROPN
ejpam-5335	91	8	δ(τ1	δ(τ1	PROPN
ejpam-5335	91	9	,	,	PUNCT
ejpam-5335	91	10	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	91	11	−1(v	−1(v	PROPN
ejpam-5335	91	12	)	)	PUNCT
ejpam-5335	91	13	)	)	PUNCT
ejpam-5335	91	14	.	.	PUNCT
ejpam-5335	92	1	then	then	ADV
ejpam-5335	92	2	,	,	PUNCT
ejpam-5335	92	3	there	there	PRON
ejpam-5335	92	4	exists	exist	VERB
ejpam-5335	92	5	a	a	DET
ejpam-5335	92	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	92	7	,	,	PUNCT
ejpam-5335	92	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	92	9	c.	c.	PROPN
ejpam-5335	92	10	prachanpol	prachanpol	NOUN
ejpam-5335	92	11	,	,	PUNCT
ejpam-5335	92	12	c.	c.	PROPN
ejpam-5335	92	13	boonpok	boonpok	PROPN
ejpam-5335	92	14	,	,	PUNCT
ejpam-5335	92	15	c.	c.	PROPN
ejpam-5335	92	16	viriyapong	viriyapong	PROPN
ejpam-5335	92	17	/	/	SYM
ejpam-5335	92	18	eur	eur	PROPN
ejpam-5335	92	19	.	.	PUNCT
ejpam-5335	93	1	j.	j.	PROPN
ejpam-5335	93	2	pure	pure	PROPN
ejpam-5335	93	3	appl	appl	PROPN
ejpam-5335	93	4	.	.	PROPN
ejpam-5335	93	5	math	math	PROPN
ejpam-5335	93	6	,	,	PUNCT
ejpam-5335	93	7	17	17	NUM
ejpam-5335	93	8	(	(	PUNCT
ejpam-5335	93	9	4	4	NUM
ejpam-5335	93	10	)	)	PUNCT
ejpam-5335	93	11	(	(	PUNCT
ejpam-5335	93	12	2024	2024	NUM
ejpam-5335	93	13	)	)	PUNCT
ejpam-5335	93	14	,	,	PUNCT
ejpam-5335	93	15	3730	3730	NUM
ejpam-5335	93	16	-	-	SYM
ejpam-5335	93	17	3742	3742	NUM
ejpam-5335	93	18	3733	3733	NUM
ejpam-5335	93	19	set	set	VERB
ejpam-5335	93	20	u	u	PROPN
ejpam-5335	93	21	of	of	ADP
ejpam-5335	93	22	x	x	PUNCT
ejpam-5335	93	23	containing	contain	VERB
ejpam-5335	93	24	x	x	PUNCT
ejpam-5335	93	25	such	such	ADJ
ejpam-5335	93	26	that	that	SCONJ
ejpam-5335	93	27	u	u	PROPN
ejpam-5335	93	28	⊆	⊆	NUM
ejpam-5335	93	29	f−1(v	f−1(v	NOUN
ejpam-5335	93	30	)	)	PUNCT
ejpam-5335	93	31	.	.	PUNCT
ejpam-5335	94	1	by	by	ADP
ejpam-5335	94	2	lemma	lemma	PROPN
ejpam-5335	94	3	1	1	NUM
ejpam-5335	94	4	,	,	PUNCT
ejpam-5335	94	5	we	we	PRON
ejpam-5335	94	6	have	have	VERB
ejpam-5335	94	7	u	u	NOUN
ejpam-5335	94	8	∩	∩	NOUN
ejpam-5335	94	9	a	a	DET
ejpam-5335	94	10	̸=	̸=	PROPN
ejpam-5335	94	11	∅	∅	NOUN
ejpam-5335	94	12	and	and	CCONJ
ejpam-5335	94	13	hence	hence	ADV
ejpam-5335	94	14	∅	∅	NOUN
ejpam-5335	94	15	=	=	NOUN
ejpam-5335	94	16	̸	̸	ADV
ejpam-5335	94	17	f(u	f(u	ADJ
ejpam-5335	94	18	∩	∩	NOUN
ejpam-5335	94	19	a	a	X
ejpam-5335	94	20	)	)	PUNCT
ejpam-5335	94	21	⊆	⊆	NUM
ejpam-5335	94	22	f(u	f(u	PROPN
ejpam-5335	94	23	)	)	PUNCT
ejpam-5335	94	24	∩	∩	ADJ
ejpam-5335	94	25	f(a	f(a	NOUN
ejpam-5335	94	26	)	)	PUNCT
ejpam-5335	94	27	⊆	⊆	NUM
ejpam-5335	94	28	v	v	ADP
ejpam-5335	94	29	∩	∩	ADJ
ejpam-5335	94	30	f(a	f(a	NOUN
ejpam-5335	94	31	)	)	PUNCT
ejpam-5335	94	32	.	.	PUNCT
ejpam-5335	95	1	thus	thus	ADV
ejpam-5335	95	2	,	,	PUNCT
ejpam-5335	95	3	f(x	f(x	PROPN
ejpam-5335	95	4	)	)	PUNCT
ejpam-5335	95	5	∈	∈	PROPN
ejpam-5335	95	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	95	7	-	-	PUNCT
ejpam-5335	95	8	cl(f(a	cl(f(a	NOUN
ejpam-5335	95	9	)	)	PUNCT
ejpam-5335	95	10	)	)	PUNCT
ejpam-5335	96	1	and	and	CCONJ
ejpam-5335	96	2	so	so	ADV
ejpam-5335	96	3	x	x	SYM
ejpam-5335	96	4	∈	∈	PROPN
ejpam-5335	96	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	96	6	-	-	PUNCT
ejpam-5335	96	7	cl(f(a	cl(f(a	NOUN
ejpam-5335	96	8	)	)	PUNCT
ejpam-5335	96	9	)	)	PUNCT
ejpam-5335	96	10	)	)	PUNCT
ejpam-5335	96	11	.	.	PUNCT
ejpam-5335	97	1	(	(	PUNCT
ejpam-5335	97	2	3	3	X
ejpam-5335	97	3	)	)	PUNCT
ejpam-5335	97	4	⇒	⇒	NOUN
ejpam-5335	97	5	(	(	PUNCT
ejpam-5335	97	6	4	4	NUM
ejpam-5335	97	7	):	):	PUNCT
ejpam-5335	97	8	let	let	VERB
ejpam-5335	97	9	b	b	X
ejpam-5335	97	10	be	be	AUX
ejpam-5335	97	11	any	any	DET
ejpam-5335	97	12	subset	subset	NOUN
ejpam-5335	97	13	of	of	ADP
ejpam-5335	97	14	y	y	PROPN
ejpam-5335	97	15	and	and	CCONJ
ejpam-5335	97	16	x	x	PROPN
ejpam-5335	97	17	∈	∈	PROPN
ejpam-5335	97	18	δ(τ1	δ(τ1	PROPN
ejpam-5335	97	19	,	,	PUNCT
ejpam-5335	97	20	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	97	21	−1(b	−1(b	PROPN
ejpam-5335	97	22	)	)	PUNCT
ejpam-5335	97	23	)	)	PUNCT
ejpam-5335	97	24	.	.	PUNCT
ejpam-5335	98	1	then	then	ADV
ejpam-5335	98	2	by	by	ADP
ejpam-5335	98	3	(	(	PUNCT
ejpam-5335	98	4	3	3	NUM
ejpam-5335	98	5	)	)	PUNCT
ejpam-5335	98	6	,	,	PUNCT
ejpam-5335	98	7	we	we	PRON
ejpam-5335	98	8	have	have	VERB
ejpam-5335	98	9	x	x	PART
ejpam-5335	98	10	∈	∈	PROPN
ejpam-5335	98	11	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-5335	98	12	-	-	PUNCT
ejpam-5335	98	13	cl(f(f	cl(f(f	X
ejpam-5335	98	14	−1(b	−1(b	NOUN
ejpam-5335	98	15	)	)	PUNCT
ejpam-5335	98	16	)	)	PUNCT
ejpam-5335	98	17	)	)	PUNCT
ejpam-5335	98	18	)	)	PUNCT
ejpam-5335	99	1	⊆	⊆	NUM
ejpam-5335	99	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	99	3	-	-	PUNCT
ejpam-5335	99	4	cl(b	cl(b	NOUN
ejpam-5335	99	5	)	)	PUNCT
ejpam-5335	99	6	)	)	PUNCT
ejpam-5335	99	7	.	.	PUNCT
ejpam-5335	100	1	(	(	PUNCT
ejpam-5335	100	2	4	4	X
ejpam-5335	100	3	)	)	PUNCT
ejpam-5335	100	4	⇒	⇒	NOUN
ejpam-5335	100	5	(	(	PUNCT
ejpam-5335	100	6	5	5	NUM
ejpam-5335	100	7	):	):	PUNCT
ejpam-5335	100	8	let	let	VERB
ejpam-5335	100	9	b	b	X
ejpam-5335	100	10	be	be	AUX
ejpam-5335	100	11	any	any	DET
ejpam-5335	100	12	subset	subset	NOUN
ejpam-5335	100	13	of	of	ADP
ejpam-5335	100	14	y	y	PROPN
ejpam-5335	100	15	and	and	CCONJ
ejpam-5335	100	16	x	x	PROPN
ejpam-5335	100	17	∈	∈	PROPN
ejpam-5335	100	18	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	100	19	-	-	PUNCT
ejpam-5335	100	20	int(b	int(b	NOUN
ejpam-5335	100	21	)	)	PUNCT
ejpam-5335	100	22	)	)	PUNCT
ejpam-5335	100	23	.	.	PUNCT
ejpam-5335	101	1	suppose	suppose	VERB
ejpam-5335	101	2	that	that	SCONJ
ejpam-5335	101	3	x	x	PROPN
ejpam-5335	101	4	/∈	/∈	SYM
ejpam-5335	101	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	101	6	,	,	PUNCT
ejpam-5335	101	7	τ2)-int(f	τ2)-int(f	X
ejpam-5335	101	8	−1(b	−1(b	NOUN
ejpam-5335	101	9	)	)	PUNCT
ejpam-5335	101	10	)	)	PUNCT
ejpam-5335	101	11	.	.	PUNCT
ejpam-5335	102	1	then	then	ADV
ejpam-5335	102	2	,	,	PUNCT
ejpam-5335	102	3	x	x	PUNCT
ejpam-5335	102	4	∈	∈	PROPN
ejpam-5335	102	5	x	x	SYM
ejpam-5335	102	6	−	−	PROPN
ejpam-5335	102	7	δ(τ1	δ(τ1	PROPN
ejpam-5335	102	8	,	,	PUNCT
ejpam-5335	102	9	τ2)-int(f	τ2)-int(f	X
ejpam-5335	102	10	−1(b	−1(b	NOUN
ejpam-5335	102	11	)	)	PUNCT
ejpam-5335	102	12	)	)	PUNCT
ejpam-5335	102	13	.	.	PUNCT
ejpam-5335	103	1	since	since	SCONJ
ejpam-5335	103	2	x	x	PART
ejpam-5335	103	3	−	−	PROPN
ejpam-5335	103	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	103	5	,	,	PUNCT
ejpam-5335	103	6	τ2)-int(f	τ2)-int(f	X
ejpam-5335	103	7	−1(b	−1(b	NOUN
ejpam-5335	103	8	)	)	PUNCT
ejpam-5335	103	9	)	)	PUNCT
ejpam-5335	103	10	=	=	SYM
ejpam-5335	103	11	δ(τ1	δ(τ1	PROPN
ejpam-5335	103	12	,	,	PUNCT
ejpam-5335	103	13	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	103	14	−1(y	−1(y	PUNCT
ejpam-5335	103	15	−b	−b	ADJ
ejpam-5335	103	16	)	)	PUNCT
ejpam-5335	103	17	)	)	PUNCT
ejpam-5335	103	18	and	and	CCONJ
ejpam-5335	103	19	by	by	ADP
ejpam-5335	103	20	(	(	PUNCT
ejpam-5335	103	21	4	4	NUM
ejpam-5335	103	22	)	)	PUNCT
ejpam-5335	103	23	,	,	PUNCT
ejpam-5335	103	24	we	we	PRON
ejpam-5335	103	25	obtain	obtain	VERB
ejpam-5335	103	26	that	that	SCONJ
ejpam-5335	103	27	x	x	SYM
ejpam-5335	103	28	∈	∈	PROPN
ejpam-5335	103	29	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-5335	103	30	-	-	PUNCT
ejpam-5335	103	31	cl(y	cl(y	NOUN
ejpam-5335	103	32	−b	−b	NOUN
ejpam-5335	103	33	)	)	PUNCT
ejpam-5335	103	34	)	)	PUNCT
ejpam-5335	104	1	=	=	PUNCT
ejpam-5335	104	2	x	x	X
ejpam-5335	104	3	−	−	PRON
ejpam-5335	104	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	104	5	-	-	PUNCT
ejpam-5335	104	6	int(b	int(b	NOUN
ejpam-5335	104	7	)	)	PUNCT
ejpam-5335	104	8	)	)	PUNCT
ejpam-5335	104	9	and	and	CCONJ
ejpam-5335	104	10	hence	hence	ADV
ejpam-5335	104	11	x	x	PROPN
ejpam-5335	104	12	/∈	/∈	PUNCT
ejpam-5335	104	13	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	104	14	-	-	PUNCT
ejpam-5335	104	15	int(b	int(b	NOUN
ejpam-5335	104	16	)	)	PUNCT
ejpam-5335	104	17	)	)	PUNCT
ejpam-5335	104	18	,	,	PUNCT
ejpam-5335	104	19	which	which	PRON
ejpam-5335	104	20	is	be	AUX
ejpam-5335	104	21	a	a	DET
ejpam-5335	104	22	contradiction	contradiction	NOUN
ejpam-5335	104	23	that	that	PRON
ejpam-5335	104	24	x	x	SYM
ejpam-5335	104	25	∈	∈	PROPN
ejpam-5335	104	26	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	104	27	-	-	PUNCT
ejpam-5335	104	28	int(b	int(b	NOUN
ejpam-5335	104	29	)	)	PUNCT
ejpam-5335	104	30	)	)	PUNCT
ejpam-5335	104	31	.	.	PUNCT
ejpam-5335	105	1	therefore	therefore	ADV
ejpam-5335	105	2	,	,	PUNCT
ejpam-5335	105	3	x	x	PROPN
ejpam-5335	105	4	∈	∈	PROPN
ejpam-5335	105	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	105	6	,	,	PUNCT
ejpam-5335	105	7	τ2)-int(f	τ2)-int(f	X
ejpam-5335	105	8	−1(b	−1(b	NOUN
ejpam-5335	105	9	)	)	PUNCT
ejpam-5335	105	10	)	)	PUNCT
ejpam-5335	105	11	.	.	PUNCT
ejpam-5335	106	1	(	(	PUNCT
ejpam-5335	106	2	5	5	X
ejpam-5335	106	3	)	)	PUNCT
ejpam-5335	106	4	⇒	⇒	NOUN
ejpam-5335	106	5	(	(	PUNCT
ejpam-5335	106	6	6	6	NUM
ejpam-5335	106	7	):	):	PUNCT
ejpam-5335	106	8	let	let	VERB
ejpam-5335	106	9	f	f	PRON
ejpam-5335	106	10	be	be	AUX
ejpam-5335	106	11	any	any	DET
ejpam-5335	106	12	σ1σ2	σ1σ2	NUM
ejpam-5335	106	13	-	-	PUNCT
ejpam-5335	106	14	closed	closed	ADJ
ejpam-5335	106	15	set	set	NOUN
ejpam-5335	106	16	of	of	ADP
ejpam-5335	106	17	y	y	PROPN
ejpam-5335	106	18	and	and	CCONJ
ejpam-5335	106	19	x	x	PROPN
ejpam-5335	106	20	∈	∈	PROPN
ejpam-5335	106	21	δ(τ1	δ(τ1	PROPN
ejpam-5335	106	22	,	,	PUNCT
ejpam-5335	106	23	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	106	24	−1(f	−1(f	NUM
ejpam-5335	106	25	)	)	PUNCT
ejpam-5335	106	26	)	)	PUNCT
ejpam-5335	106	27	.	.	PUNCT
ejpam-5335	107	1	suppose	suppose	VERB
ejpam-5335	107	2	that	that	SCONJ
ejpam-5335	107	3	x	x	PROPN
ejpam-5335	107	4	/∈	/∈	PROPN
ejpam-5335	107	5	f−1(f	f−1(f	PROPN
ejpam-5335	107	6	)	)	PUNCT
ejpam-5335	107	7	.	.	PUNCT
ejpam-5335	108	1	since	since	SCONJ
ejpam-5335	108	2	y	y	PROPN
ejpam-5335	108	3	−	−	PROPN
ejpam-5335	108	4	f	f	PROPN
ejpam-5335	108	5	is	be	AUX
ejpam-5335	108	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	108	7	-	-	ADJ
ejpam-5335	108	8	open	open	ADJ
ejpam-5335	108	9	in	in	ADP
ejpam-5335	108	10	y	y	PROPN
ejpam-5335	108	11	,	,	PUNCT
ejpam-5335	108	12	x	x	PUNCT
ejpam-5335	108	13	∈	∈	NOUN
ejpam-5335	108	14	x	x	PUNCT
ejpam-5335	108	15	−	−	PROPN
ejpam-5335	108	16	f−1(f	f−1(f	PROPN
ejpam-5335	108	17	)	)	PUNCT
ejpam-5335	109	1	=	=	PUNCT
ejpam-5335	109	2	f−1(y	f−1(y	PROPN
ejpam-5335	110	1	−	−	PROPN
ejpam-5335	110	2	f	f	PROPN
ejpam-5335	110	3	)	)	PUNCT
ejpam-5335	110	4	=	=	SYM
ejpam-5335	110	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	110	6	-	-	PUNCT
ejpam-5335	110	7	int(y	int(y	PROPN
ejpam-5335	110	8	−	−	PROPN
ejpam-5335	110	9	f	f	PROPN
ejpam-5335	110	10	)	)	PUNCT
ejpam-5335	110	11	)	)	PUNCT
ejpam-5335	110	12	.	.	PUNCT
ejpam-5335	111	1	by	by	ADP
ejpam-5335	111	2	(	(	PUNCT
ejpam-5335	111	3	5	5	NUM
ejpam-5335	111	4	)	)	PUNCT
ejpam-5335	111	5	and	and	CCONJ
ejpam-5335	111	6	lemma	lemma	PROPN
ejpam-5335	111	7	2	2	NUM
ejpam-5335	111	8	(	(	PUNCT
ejpam-5335	111	9	1	1	NUM
ejpam-5335	111	10	)	)	PUNCT
ejpam-5335	111	11	,	,	PUNCT
ejpam-5335	111	12	we	we	PRON
ejpam-5335	111	13	have	have	VERB
ejpam-5335	111	14	x	x	PROPN
ejpam-5335	111	15	∈	∈	PROPN
ejpam-5335	111	16	δ(τ1	δ(τ1	NOUN
ejpam-5335	111	17	,	,	PUNCT
ejpam-5335	111	18	τ2)-int(f	τ2)-int(f	VERB
ejpam-5335	112	1	−1(y	−1(y	X
ejpam-5335	113	1	−	−	PROPN
ejpam-5335	113	2	f	f	PROPN
ejpam-5335	113	3	)	)	PUNCT
ejpam-5335	113	4	)	)	PUNCT
ejpam-5335	114	1	=	=	PUNCT
ejpam-5335	115	1	x	x	X
ejpam-5335	115	2	−	−	PROPN
ejpam-5335	115	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	115	4	,	,	PUNCT
ejpam-5335	115	5	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	115	6	−1(f	−1(f	NUM
ejpam-5335	115	7	)	)	PUNCT
ejpam-5335	115	8	)	)	PUNCT
ejpam-5335	115	9	and	and	CCONJ
ejpam-5335	115	10	hence	hence	ADV
ejpam-5335	115	11	x	x	X
ejpam-5335	115	12	̸∈	̸∈	PROPN
ejpam-5335	115	13	δ(τ1	δ(τ1	PROPN
ejpam-5335	115	14	,	,	PUNCT
ejpam-5335	115	15	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	115	16	−1(f	−1(f	NUM
ejpam-5335	115	17	)	)	PUNCT
ejpam-5335	115	18	)	)	PUNCT
ejpam-5335	115	19	.	.	PUNCT
ejpam-5335	116	1	this	this	PRON
ejpam-5335	116	2	is	be	AUX
ejpam-5335	116	3	a	a	DET
ejpam-5335	116	4	contradiction	contradiction	NOUN
ejpam-5335	116	5	.	.	PUNCT
ejpam-5335	117	1	therefore	therefore	ADV
ejpam-5335	117	2	,	,	PUNCT
ejpam-5335	117	3	x	x	PROPN
ejpam-5335	117	4	∈	∈	PROPN
ejpam-5335	117	5	f−1(f	f−1(f	PROPN
ejpam-5335	117	6	)	)	PUNCT
ejpam-5335	117	7	.	.	PUNCT
ejpam-5335	118	1	(	(	PUNCT
ejpam-5335	118	2	6	6	X
ejpam-5335	118	3	)	)	PUNCT
ejpam-5335	118	4	⇒	⇒	NOUN
ejpam-5335	118	5	(	(	PUNCT
ejpam-5335	118	6	2	2	NUM
ejpam-5335	118	7	):	):	PUNCT
ejpam-5335	118	8	let	let	VERB
ejpam-5335	118	9	v	v	PART
ejpam-5335	118	10	be	be	AUX
ejpam-5335	118	11	any	any	DET
ejpam-5335	118	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	118	13	-	-	ADJ
ejpam-5335	118	14	open	open	ADJ
ejpam-5335	118	15	set	set	NOUN
ejpam-5335	118	16	of	of	ADP
ejpam-5335	118	17	y	y	PROPN
ejpam-5335	118	18	containing	contain	VERB
ejpam-5335	118	19	f(x	f(x	PROPN
ejpam-5335	118	20	)	)	PUNCT
ejpam-5335	118	21	.	.	PUNCT
ejpam-5335	119	1	then	then	ADV
ejpam-5335	119	2	,	,	PUNCT
ejpam-5335	119	3	x	x	PROPN
ejpam-5335	119	4	∈	∈	PROPN
ejpam-5335	119	5	f−1(v	f−1(v	NOUN
ejpam-5335	119	6	)	)	PUNCT
ejpam-5335	119	7	.	.	PUNCT
ejpam-5335	119	8	suppose	suppose	VERB
ejpam-5335	119	9	that	that	SCONJ
ejpam-5335	119	10	x	x	PROPN
ejpam-5335	119	11	̸∈	̸∈	PROPN
ejpam-5335	119	12	δ(τ1	δ(τ1	PROPN
ejpam-5335	119	13	,	,	PUNCT
ejpam-5335	119	14	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	119	15	−1(v	−1(v	PROPN
ejpam-5335	119	16	)	)	PUNCT
ejpam-5335	119	17	)	)	PUNCT
ejpam-5335	119	18	.	.	PUNCT
ejpam-5335	120	1	by	by	ADP
ejpam-5335	120	2	lemma	lemma	PROPN
ejpam-5335	120	3	2	2	NUM
ejpam-5335	120	4	(	(	PUNCT
ejpam-5335	120	5	2	2	NUM
ejpam-5335	120	6	)	)	PUNCT
ejpam-5335	120	7	,	,	PUNCT
ejpam-5335	120	8	x	x	PUNCT
ejpam-5335	120	9	∈	∈	PROPN
ejpam-5335	120	10	x−δ(τ1	x−δ(τ1	PROPN
ejpam-5335	120	11	,	,	PUNCT
ejpam-5335	120	12	τ2)-int(f	τ2)-int(f	VERB
ejpam-5335	120	13	−1(v	−1(v	PROPN
ejpam-5335	120	14	)	)	PUNCT
ejpam-5335	120	15	)	)	PUNCT
ejpam-5335	121	1	=	=	SYM
ejpam-5335	121	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	121	3	,	,	PUNCT
ejpam-5335	121	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	121	5	−1(y	−1(y	ADP
ejpam-5335	121	6	−	−	PROPN
ejpam-5335	121	7	v	v	NOUN
ejpam-5335	121	8	)	)	PUNCT
ejpam-5335	121	9	)	)	PUNCT
ejpam-5335	121	10	.	.	PUNCT
ejpam-5335	122	1	since	since	SCONJ
ejpam-5335	122	2	y	y	PROPN
ejpam-5335	122	3	−	−	PROPN
ejpam-5335	122	4	v	v	NOUN
ejpam-5335	122	5	is	be	AUX
ejpam-5335	122	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	122	7	-	-	ADJ
ejpam-5335	122	8	closed	closed	ADJ
ejpam-5335	122	9	in	in	ADP
ejpam-5335	122	10	y	y	PROPN
ejpam-5335	122	11	and	and	CCONJ
ejpam-5335	122	12	by	by	ADP
ejpam-5335	122	13	(	(	PUNCT
ejpam-5335	122	14	6	6	NUM
ejpam-5335	122	15	)	)	PUNCT
ejpam-5335	122	16	,	,	PUNCT
ejpam-5335	122	17	we	we	PRON
ejpam-5335	122	18	have	have	VERB
ejpam-5335	122	19	x	x	X
ejpam-5335	122	20	∈	∈	PROPN
ejpam-5335	122	21	f−1(y	f−1(y	NOUN
ejpam-5335	122	22	−	−	PROPN
ejpam-5335	122	23	v	v	NOUN
ejpam-5335	122	24	)	)	PUNCT
ejpam-5335	122	25	=	=	PUNCT
ejpam-5335	122	26	x	x	PUNCT
ejpam-5335	122	27	−	−	PROPN
ejpam-5335	122	28	f−1(v	f−1(v	PROPN
ejpam-5335	122	29	)	)	PUNCT
ejpam-5335	122	30	.	.	PUNCT
ejpam-5335	123	1	this	this	PRON
ejpam-5335	123	2	implies	imply	VERB
ejpam-5335	123	3	that	that	SCONJ
ejpam-5335	123	4	x	x	SYM
ejpam-5335	123	5	/∈	/∈	PUNCT
ejpam-5335	123	6	f−1(v	f−1(v	PROPN
ejpam-5335	123	7	)	)	PUNCT
ejpam-5335	123	8	,	,	PUNCT
ejpam-5335	123	9	which	which	PRON
ejpam-5335	123	10	is	be	AUX
ejpam-5335	123	11	a	a	DET
ejpam-5335	123	12	contradiction	contradiction	NOUN
ejpam-5335	123	13	.	.	PUNCT
ejpam-5335	124	1	thus	thus	ADV
ejpam-5335	124	2	,	,	PUNCT
ejpam-5335	124	3	x	x	PROPN
ejpam-5335	124	4	∈	∈	PROPN
ejpam-5335	124	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	124	6	,	,	PUNCT
ejpam-5335	124	7	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	124	8	−1(v	−1(v	PROPN
ejpam-5335	124	9	)	)	PUNCT
ejpam-5335	124	10	)	)	PUNCT
ejpam-5335	124	11	.	.	PUNCT
ejpam-5335	125	1	(	(	PUNCT
ejpam-5335	125	2	2	2	X
ejpam-5335	125	3	)	)	PUNCT
ejpam-5335	125	4	⇒	⇒	NOUN
ejpam-5335	125	5	(	(	PUNCT
ejpam-5335	125	6	1	1	NUM
ejpam-5335	125	7	):	):	PUNCT
ejpam-5335	125	8	let	let	VERB
ejpam-5335	125	9	x	x	PUNCT
ejpam-5335	125	10	∈	∈	PROPN
ejpam-5335	125	11	x	x	X
ejpam-5335	125	12	and	and	CCONJ
ejpam-5335	125	13	v	v	X
ejpam-5335	125	14	be	be	AUX
ejpam-5335	125	15	any	any	DET
ejpam-5335	125	16	σ1σ2	σ1σ2	NOUN
ejpam-5335	125	17	-	-	ADJ
ejpam-5335	125	18	open	open	ADJ
ejpam-5335	125	19	set	set	NOUN
ejpam-5335	125	20	of	of	ADP
ejpam-5335	125	21	y	y	PROPN
ejpam-5335	125	22	containing	contain	VERB
ejpam-5335	125	23	f(x	f(x	PROPN
ejpam-5335	125	24	)	)	PUNCT
ejpam-5335	125	25	.	.	PUNCT
ejpam-5335	126	1	by	by	ADP
ejpam-5335	126	2	(	(	PUNCT
ejpam-5335	126	3	2	2	NUM
ejpam-5335	126	4	)	)	PUNCT
ejpam-5335	126	5	,	,	PUNCT
ejpam-5335	126	6	we	we	PRON
ejpam-5335	126	7	have	have	VERB
ejpam-5335	126	8	x	x	PROPN
ejpam-5335	126	9	∈	∈	PROPN
ejpam-5335	126	10	δ(τ1	δ(τ1	NOUN
ejpam-5335	126	11	,	,	PUNCT
ejpam-5335	126	12	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	126	13	−1(v	−1(v	PROPN
ejpam-5335	126	14	)	)	PUNCT
ejpam-5335	126	15	)	)	PUNCT
ejpam-5335	126	16	.	.	PUNCT
ejpam-5335	127	1	then	then	ADV
ejpam-5335	127	2	,	,	PUNCT
ejpam-5335	127	3	there	there	PRON
ejpam-5335	127	4	exists	exist	VERB
ejpam-5335	127	5	a	a	DET
ejpam-5335	127	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	127	7	,	,	PUNCT
ejpam-5335	127	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	127	9	set	set	VERB
ejpam-5335	127	10	u	u	PRON
ejpam-5335	127	11	such	such	ADJ
ejpam-5335	127	12	that	that	SCONJ
ejpam-5335	127	13	x	x	SYM
ejpam-5335	127	14	∈	∈	PROPN
ejpam-5335	127	15	u	u	NOUN
ejpam-5335	127	16	⊆	⊆	NUM
ejpam-5335	127	17	f−1(v	f−1(v	NOUN
ejpam-5335	127	18	)	)	PUNCT
ejpam-5335	127	19	.	.	PUNCT
ejpam-5335	128	1	thus	thus	ADV
ejpam-5335	128	2	,	,	PUNCT
ejpam-5335	128	3	f(u	f(u	PROPN
ejpam-5335	128	4	)	)	PUNCT
ejpam-5335	128	5	⊆	⊆	NUM
ejpam-5335	128	6	v	v	NOUN
ejpam-5335	128	7	.	.	PUNCT
ejpam-5335	129	1	this	this	PRON
ejpam-5335	129	2	shows	show	VERB
ejpam-5335	129	3	that	that	SCONJ
ejpam-5335	129	4	f	f	PROPN
ejpam-5335	129	5	is	be	AUX
ejpam-5335	129	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	129	7	,	,	PUNCT
ejpam-5335	129	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	129	9	at	at	ADP
ejpam-5335	129	10	x.	x.	NOUN
ejpam-5335	129	11	theorem	theorem	NOUN
ejpam-5335	129	12	2	2	NUM
ejpam-5335	129	13	.	.	X
ejpam-5335	129	14	for	for	ADP
ejpam-5335	129	15	a	a	DET
ejpam-5335	129	16	function	function	NOUN
ejpam-5335	129	17	f	f	NOUN
ejpam-5335	129	18	:	:	PUNCT
ejpam-5335	129	19	(	(	PUNCT
ejpam-5335	129	20	x	x	NOUN
ejpam-5335	129	21	,	,	PUNCT
ejpam-5335	129	22	τ1	τ1	NOUN
ejpam-5335	129	23	,	,	PUNCT
ejpam-5335	129	24	τ2	τ2	NOUN
ejpam-5335	129	25	)	)	PUNCT
ejpam-5335	129	26	→	→	SYM
ejpam-5335	129	27	(	(	PUNCT
ejpam-5335	129	28	y	y	PROPN
ejpam-5335	129	29	,	,	PUNCT
ejpam-5335	129	30	σ1	σ1	PROPN
ejpam-5335	129	31	,	,	PUNCT
ejpam-5335	129	32	σ2	σ2	NOUN
ejpam-5335	129	33	)	)	PUNCT
ejpam-5335	129	34	,	,	PUNCT
ejpam-5335	129	35	the	the	DET
ejpam-5335	129	36	following	follow	VERB
ejpam-5335	129	37	properties	property	NOUN
ejpam-5335	129	38	are	be	AUX
ejpam-5335	129	39	equivalent	equivalent	ADJ
ejpam-5335	129	40	:	:	PUNCT
ejpam-5335	129	41	(	(	PUNCT
ejpam-5335	129	42	1	1	X
ejpam-5335	129	43	)	)	PUNCT
ejpam-5335	129	44	f	f	PROPN
ejpam-5335	129	45	is	be	AUX
ejpam-5335	129	46	δ(τ1	δ(τ1	NOUN
ejpam-5335	129	47	,	,	PUNCT
ejpam-5335	129	48	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	129	49	;	;	PUNCT
ejpam-5335	129	50	(	(	PUNCT
ejpam-5335	129	51	2	2	X
ejpam-5335	129	52	)	)	PUNCT
ejpam-5335	129	53	f−1(v	f−1(v	NOUN
ejpam-5335	129	54	)	)	PUNCT
ejpam-5335	129	55	is	be	AUX
ejpam-5335	129	56	δ(τ1	δ(τ1	NOUN
ejpam-5335	129	57	,	,	PUNCT
ejpam-5335	129	58	τ2)-open	τ2)-open	ADJ
ejpam-5335	129	59	in	in	ADP
ejpam-5335	129	60	x	x	PUNCT
ejpam-5335	129	61	for	for	ADP
ejpam-5335	129	62	every	every	DET
ejpam-5335	129	63	σ1σ2	σ1σ2	NOUN
ejpam-5335	129	64	-	-	ADJ
ejpam-5335	129	65	open	open	ADJ
ejpam-5335	129	66	set	set	NOUN
ejpam-5335	129	67	v	v	NOUN
ejpam-5335	129	68	of	of	ADP
ejpam-5335	129	69	y	y	PROPN
ejpam-5335	129	70	;	;	PUNCT
ejpam-5335	129	71	(	(	PUNCT
ejpam-5335	129	72	3	3	X
ejpam-5335	129	73	)	)	PUNCT
ejpam-5335	129	74	f(δ(τ1	f(δ(τ1	PROPN
ejpam-5335	129	75	,	,	PUNCT
ejpam-5335	129	76	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	129	77	)	)	PUNCT
ejpam-5335	129	78	)	)	PUNCT
ejpam-5335	130	1	⊆	⊆	X
ejpam-5335	130	2	σ1σ2	σ1σ2	NUM
ejpam-5335	130	3	-	-	PUNCT
ejpam-5335	130	4	cl(f(a	cl(f(a	NOUN
ejpam-5335	130	5	)	)	PUNCT
ejpam-5335	130	6	)	)	PUNCT
ejpam-5335	130	7	for	for	ADP
ejpam-5335	130	8	every	every	DET
ejpam-5335	130	9	subset	subset	NOUN
ejpam-5335	130	10	a	a	PRON
ejpam-5335	130	11	of	of	ADP
ejpam-5335	130	12	x	x	PRON
ejpam-5335	130	13	;	;	PUNCT
ejpam-5335	130	14	(	(	PUNCT
ejpam-5335	130	15	4	4	X
ejpam-5335	130	16	)	)	PUNCT
ejpam-5335	130	17	δ(τ1	δ(τ1	NOUN
ejpam-5335	130	18	,	,	PUNCT
ejpam-5335	130	19	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	130	20	−1(b	−1(b	NOUN
ejpam-5335	130	21	)	)	PUNCT
ejpam-5335	130	22	)	)	PUNCT
ejpam-5335	130	23	⊆	⊆	NUM
ejpam-5335	130	24	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	130	25	-	-	PUNCT
ejpam-5335	130	26	cl(b	cl(b	NOUN
ejpam-5335	130	27	)	)	PUNCT
ejpam-5335	130	28	)	)	PUNCT
ejpam-5335	130	29	for	for	ADP
ejpam-5335	130	30	every	every	DET
ejpam-5335	130	31	subset	subset	NOUN
ejpam-5335	130	32	b	b	PROPN
ejpam-5335	130	33	of	of	ADP
ejpam-5335	130	34	y	y	PROPN
ejpam-5335	130	35	;	;	PUNCT
ejpam-5335	130	36	(	(	PUNCT
ejpam-5335	130	37	5	5	X
ejpam-5335	130	38	)	)	PUNCT
ejpam-5335	130	39	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	130	40	-	-	PUNCT
ejpam-5335	130	41	int(b	int(b	NOUN
ejpam-5335	130	42	)	)	PUNCT
ejpam-5335	130	43	)	)	PUNCT
ejpam-5335	131	1	⊆	⊆	NUM
ejpam-5335	131	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	131	3	,	,	PUNCT
ejpam-5335	131	4	τ2)-int(f	τ2)-int(f	X
ejpam-5335	131	5	−1(b	−1(b	NOUN
ejpam-5335	131	6	)	)	PUNCT
ejpam-5335	131	7	)	)	PUNCT
ejpam-5335	131	8	for	for	ADP
ejpam-5335	131	9	every	every	DET
ejpam-5335	131	10	subset	subset	NOUN
ejpam-5335	131	11	b	b	PROPN
ejpam-5335	131	12	of	of	ADP
ejpam-5335	131	13	y	y	PROPN
ejpam-5335	131	14	;	;	PUNCT
ejpam-5335	131	15	(	(	PUNCT
ejpam-5335	131	16	6	6	X
ejpam-5335	131	17	)	)	PUNCT
ejpam-5335	131	18	f−1(f	f−1(f	NOUN
ejpam-5335	131	19	)	)	PUNCT
ejpam-5335	131	20	is	be	AUX
ejpam-5335	131	21	δ(τ1	δ(τ1	NOUN
ejpam-5335	131	22	,	,	PUNCT
ejpam-5335	131	23	τ2)-closed	τ2)-close	VERB
ejpam-5335	131	24	in	in	ADP
ejpam-5335	131	25	x	x	PUNCT
ejpam-5335	131	26	for	for	ADP
ejpam-5335	131	27	every	every	DET
ejpam-5335	131	28	σ1σ2	σ1σ2	NUM
ejpam-5335	131	29	-	-	PUNCT
ejpam-5335	131	30	closed	closed	ADJ
ejpam-5335	131	31	set	set	ADJ
ejpam-5335	131	32	f	f	PROPN
ejpam-5335	131	33	of	of	ADP
ejpam-5335	131	34	y	y	PROPN
ejpam-5335	131	35	.	.	PUNCT
ejpam-5335	132	1	c.	c.	PROPN
ejpam-5335	132	2	prachanpol	prachanpol	PROPN
ejpam-5335	132	3	,	,	PUNCT
ejpam-5335	132	4	c.	c.	PROPN
ejpam-5335	132	5	boonpok	boonpok	PROPN
ejpam-5335	132	6	,	,	PUNCT
ejpam-5335	132	7	c.	c.	PROPN
ejpam-5335	132	8	viriyapong	viriyapong	PROPN
ejpam-5335	132	9	/	/	SYM
ejpam-5335	132	10	eur	eur	PROPN
ejpam-5335	132	11	.	.	PUNCT
ejpam-5335	133	1	j.	j.	PROPN
ejpam-5335	133	2	pure	pure	PROPN
ejpam-5335	133	3	appl	appl	PROPN
ejpam-5335	133	4	.	.	PROPN
ejpam-5335	133	5	math	math	PROPN
ejpam-5335	133	6	,	,	PUNCT
ejpam-5335	133	7	17	17	NUM
ejpam-5335	133	8	(	(	PUNCT
ejpam-5335	133	9	4	4	NUM
ejpam-5335	133	10	)	)	PUNCT
ejpam-5335	133	11	(	(	PUNCT
ejpam-5335	133	12	2024	2024	NUM
ejpam-5335	133	13	)	)	PUNCT
ejpam-5335	133	14	,	,	PUNCT
ejpam-5335	133	15	3730	3730	NUM
ejpam-5335	133	16	-	-	SYM
ejpam-5335	133	17	3742	3742	NUM
ejpam-5335	133	18	3734	3734	NUM
ejpam-5335	133	19	proof	proof	NOUN
ejpam-5335	133	20	.	.	PUNCT
ejpam-5335	134	1	(	(	PUNCT
ejpam-5335	134	2	1	1	X
ejpam-5335	134	3	)	)	PUNCT
ejpam-5335	134	4	⇒	⇒	NOUN
ejpam-5335	134	5	(	(	PUNCT
ejpam-5335	134	6	2	2	NUM
ejpam-5335	134	7	):	):	PUNCT
ejpam-5335	134	8	let	let	VERB
ejpam-5335	134	9	v	v	PART
ejpam-5335	134	10	be	be	AUX
ejpam-5335	134	11	any	any	DET
ejpam-5335	134	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	134	13	-	-	ADJ
ejpam-5335	134	14	open	open	ADJ
ejpam-5335	134	15	of	of	ADP
ejpam-5335	134	16	y	y	PROPN
ejpam-5335	134	17	and	and	CCONJ
ejpam-5335	134	18	x	x	PROPN
ejpam-5335	134	19	∈	∈	PROPN
ejpam-5335	134	20	f−1(v	f−1(v	NOUN
ejpam-5335	134	21	)	)	PUNCT
ejpam-5335	134	22	.	.	PUNCT
ejpam-5335	135	1	by	by	ADP
ejpam-5335	135	2	(	(	PUNCT
ejpam-5335	135	3	1	1	NUM
ejpam-5335	135	4	)	)	PUNCT
ejpam-5335	135	5	,	,	PUNCT
ejpam-5335	135	6	there	there	PRON
ejpam-5335	135	7	exists	exist	VERB
ejpam-5335	135	8	a	a	DET
ejpam-5335	135	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	135	10	,	,	PUNCT
ejpam-5335	135	11	τ2)-open	τ2)-open	ADJ
ejpam-5335	135	12	set	set	ADJ
ejpam-5335	135	13	u	u	NOUN
ejpam-5335	135	14	of	of	ADP
ejpam-5335	135	15	x	x	PUNCT
ejpam-5335	135	16	containing	contain	VERB
ejpam-5335	135	17	x	x	PUNCT
ejpam-5335	135	18	such	such	ADJ
ejpam-5335	135	19	that	that	DET
ejpam-5335	135	20	f(u	f(u	PROPN
ejpam-5335	135	21	)	)	PUNCT
ejpam-5335	135	22	⊆	⊆	NUM
ejpam-5335	135	23	v	v	NOUN
ejpam-5335	135	24	.	.	PUNCT
ejpam-5335	136	1	thus	thus	ADV
ejpam-5335	136	2	,	,	PUNCT
ejpam-5335	136	3	x	x	PUNCT
ejpam-5335	136	4	∈	∈	PROPN
ejpam-5335	136	5	u	u	NOUN
ejpam-5335	136	6	⊆	⊆	NUM
ejpam-5335	136	7	f−1(v	f−1(v	NOUN
ejpam-5335	136	8	)	)	PUNCT
ejpam-5335	136	9	and	and	CCONJ
ejpam-5335	136	10	hence	hence	ADV
ejpam-5335	136	11	x	x	X
ejpam-5335	136	12	∈	∈	PROPN
ejpam-5335	136	13	δ(τ1	δ(τ1	PROPN
ejpam-5335	136	14	,	,	PUNCT
ejpam-5335	136	15	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	136	16	−1(v	−1(v	PROPN
ejpam-5335	136	17	)	)	PUNCT
ejpam-5335	136	18	)	)	PUNCT
ejpam-5335	136	19	.	.	PUNCT
ejpam-5335	137	1	this	this	PRON
ejpam-5335	137	2	implies	imply	VERB
ejpam-5335	137	3	that	that	DET
ejpam-5335	137	4	f−1(v	f−1(v	PROPN
ejpam-5335	137	5	)	)	PUNCT
ejpam-5335	138	1	⊆	⊆	NUM
ejpam-5335	138	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	138	3	,	,	PUNCT
ejpam-5335	138	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	138	5	−1(v	−1(v	PROPN
ejpam-5335	138	6	)	)	PUNCT
ejpam-5335	138	7	)	)	PUNCT
ejpam-5335	138	8	.	.	PUNCT
ejpam-5335	139	1	therefore	therefore	ADV
ejpam-5335	139	2	,	,	PUNCT
ejpam-5335	139	3	f−1(v	f−1(v	PROPN
ejpam-5335	139	4	)	)	PUNCT
ejpam-5335	139	5	is	be	AUX
ejpam-5335	139	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	139	7	,	,	PUNCT
ejpam-5335	139	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	139	9	in	in	ADP
ejpam-5335	139	10	x.	x.	NOUN
ejpam-5335	139	11	(	(	PUNCT
ejpam-5335	139	12	2	2	NUM
ejpam-5335	139	13	)	)	PUNCT
ejpam-5335	139	14	⇒	⇒	NOUN
ejpam-5335	139	15	(	(	PUNCT
ejpam-5335	139	16	3	3	NUM
ejpam-5335	139	17	):	):	PUNCT
ejpam-5335	139	18	let	let	VERB
ejpam-5335	139	19	a	a	PRON
ejpam-5335	139	20	be	be	AUX
ejpam-5335	139	21	any	any	DET
ejpam-5335	139	22	subset	subset	NOUN
ejpam-5335	139	23	of	of	ADP
ejpam-5335	139	24	x	x	PUNCT
ejpam-5335	139	25	and	and	CCONJ
ejpam-5335	139	26	let	let	VERB
ejpam-5335	139	27	x	x	X
ejpam-5335	139	28	∈	∈	PROPN
ejpam-5335	139	29	δ(τ1	δ(τ1	PROPN
ejpam-5335	139	30	,	,	PUNCT
ejpam-5335	139	31	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	139	32	)	)	PUNCT
ejpam-5335	139	33	.	.	PUNCT
ejpam-5335	140	1	then	then	ADV
ejpam-5335	140	2	,	,	PUNCT
ejpam-5335	140	3	we	we	PRON
ejpam-5335	140	4	have	have	VERB
ejpam-5335	140	5	f(x	f(x	PROPN
ejpam-5335	140	6	)	)	PUNCT
ejpam-5335	140	7	∈	∈	PROPN
ejpam-5335	140	8	f(δ(τ1	f(δ(τ1	PROPN
ejpam-5335	140	9	,	,	PUNCT
ejpam-5335	140	10	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	140	11	)	)	PUNCT
ejpam-5335	140	12	)	)	PUNCT
ejpam-5335	140	13	.	.	PUNCT
ejpam-5335	141	1	let	let	VERB
ejpam-5335	141	2	v	v	PART
ejpam-5335	141	3	be	be	AUX
ejpam-5335	141	4	any	any	DET
ejpam-5335	141	5	σ1σ2	σ1σ2	NOUN
ejpam-5335	141	6	-	-	ADJ
ejpam-5335	141	7	open	open	ADJ
ejpam-5335	141	8	set	set	NOUN
ejpam-5335	141	9	of	of	ADP
ejpam-5335	141	10	y	y	PROPN
ejpam-5335	141	11	containing	contain	VERB
ejpam-5335	141	12	f(x	f(x	PROPN
ejpam-5335	141	13	)	)	PUNCT
ejpam-5335	141	14	.	.	PUNCT
ejpam-5335	142	1	by	by	ADP
ejpam-5335	142	2	(	(	PUNCT
ejpam-5335	142	3	2	2	NUM
ejpam-5335	142	4	)	)	PUNCT
ejpam-5335	142	5	,	,	PUNCT
ejpam-5335	142	6	f−1(v	f−1(v	PROPN
ejpam-5335	142	7	)	)	PUNCT
ejpam-5335	142	8	is	be	AUX
ejpam-5335	142	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	142	10	,	,	PUNCT
ejpam-5335	142	11	τ2)-open	τ2)-open	ADJ
ejpam-5335	142	12	in	in	ADP
ejpam-5335	142	13	x.	x.	NOUN
ejpam-5335	142	14	therefore	therefore	ADV
ejpam-5335	142	15	,	,	PUNCT
ejpam-5335	142	16	x	x	PROPN
ejpam-5335	142	17	∈	∈	PROPN
ejpam-5335	142	18	f−1(v	f−1(v	NOUN
ejpam-5335	142	19	)	)	PUNCT
ejpam-5335	143	1	=	=	SYM
ejpam-5335	143	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	143	3	,	,	PUNCT
ejpam-5335	143	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	143	5	−1(v	−1(v	PROPN
ejpam-5335	143	6	)	)	PUNCT
ejpam-5335	143	7	)	)	PUNCT
ejpam-5335	143	8	.	.	PUNCT
ejpam-5335	144	1	then	then	ADV
ejpam-5335	144	2	,	,	PUNCT
ejpam-5335	144	3	there	there	PRON
ejpam-5335	144	4	exists	exist	VERB
ejpam-5335	144	5	a	a	DET
ejpam-5335	144	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	144	7	,	,	PUNCT
ejpam-5335	144	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	144	9	set	set	ADJ
ejpam-5335	144	10	u	u	NOUN
ejpam-5335	144	11	of	of	ADP
ejpam-5335	144	12	x	x	PUNCT
ejpam-5335	144	13	containing	contain	VERB
ejpam-5335	144	14	x	x	PUNCT
ejpam-5335	144	15	such	such	ADJ
ejpam-5335	144	16	that	that	SCONJ
ejpam-5335	144	17	u	u	PROPN
ejpam-5335	144	18	⊆	⊆	NUM
ejpam-5335	144	19	f−1(v	f−1(v	NOUN
ejpam-5335	144	20	)	)	PUNCT
ejpam-5335	144	21	.	.	PUNCT
ejpam-5335	145	1	since	since	SCONJ
ejpam-5335	145	2	x	x	PROPN
ejpam-5335	145	3	∈	∈	PROPN
ejpam-5335	145	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	145	5	,	,	PUNCT
ejpam-5335	145	6	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	145	7	)	)	PUNCT
ejpam-5335	145	8	,	,	PUNCT
ejpam-5335	145	9	then	then	ADV
ejpam-5335	145	10	u	u	NOUN
ejpam-5335	145	11	∩	∩	NOUN
ejpam-5335	145	12	a	a	DET
ejpam-5335	145	13	̸=	̸=	PROPN
ejpam-5335	145	14	∅.	∅.	PRON
ejpam-5335	145	15	hence	hence	ADV
ejpam-5335	145	16	,	,	PUNCT
ejpam-5335	145	17	∅	∅	NOUN
ejpam-5335	145	18	̸=	̸=	PROPN
ejpam-5335	145	19	f(u	f(u	PROPN
ejpam-5335	145	20	∩	∩	NOUN
ejpam-5335	145	21	a	a	X
ejpam-5335	145	22	)	)	PUNCT
ejpam-5335	145	23	⊆	⊆	NUM
ejpam-5335	145	24	f(u	f(u	PROPN
ejpam-5335	145	25	)	)	PUNCT
ejpam-5335	145	26	∩	∩	ADJ
ejpam-5335	145	27	f(a	f(a	NOUN
ejpam-5335	145	28	)	)	PUNCT
ejpam-5335	145	29	⊆	⊆	NUM
ejpam-5335	145	30	v	v	ADP
ejpam-5335	145	31	∩	∩	ADJ
ejpam-5335	145	32	f(a	f(a	NOUN
ejpam-5335	145	33	)	)	PUNCT
ejpam-5335	145	34	.	.	PUNCT
ejpam-5335	146	1	thus	thus	ADV
ejpam-5335	146	2	,	,	PUNCT
ejpam-5335	146	3	f(x	f(x	PROPN
ejpam-5335	146	4	)	)	PUNCT
ejpam-5335	146	5	∈	∈	PROPN
ejpam-5335	146	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	146	7	-	-	PUNCT
ejpam-5335	146	8	cl(f(a	cl(f(a	NOUN
ejpam-5335	146	9	)	)	PUNCT
ejpam-5335	146	10	)	)	PUNCT
ejpam-5335	146	11	and	and	CCONJ
ejpam-5335	146	12	so	so	ADV
ejpam-5335	146	13	f(δ(τ1	f(δ(τ1	PROPN
ejpam-5335	146	14	,	,	PUNCT
ejpam-5335	146	15	τ2)-cl(a	τ2)-cl(a	NUM
ejpam-5335	146	16	)	)	PUNCT
ejpam-5335	146	17	)	)	PUNCT
ejpam-5335	147	1	⊆	⊆	X
ejpam-5335	147	2	σ1σ2	σ1σ2	NUM
ejpam-5335	147	3	-	-	PUNCT
ejpam-5335	147	4	cl(f(a	cl(f(a	NOUN
ejpam-5335	147	5	)	)	PUNCT
ejpam-5335	147	6	)	)	PUNCT
ejpam-5335	147	7	.	.	PUNCT
ejpam-5335	148	1	(	(	PUNCT
ejpam-5335	148	2	3	3	X
ejpam-5335	148	3	)	)	PUNCT
ejpam-5335	148	4	⇒	⇒	NOUN
ejpam-5335	148	5	(	(	PUNCT
ejpam-5335	148	6	4	4	NUM
ejpam-5335	148	7	):	):	PUNCT
ejpam-5335	148	8	let	let	VERB
ejpam-5335	148	9	b	b	X
ejpam-5335	148	10	be	be	AUX
ejpam-5335	148	11	any	any	DET
ejpam-5335	148	12	subset	subset	NOUN
ejpam-5335	148	13	of	of	ADP
ejpam-5335	148	14	y	y	PROPN
ejpam-5335	148	15	.	.	PUNCT
ejpam-5335	149	1	then	then	ADV
ejpam-5335	149	2	by	by	ADP
ejpam-5335	149	3	(	(	PUNCT
ejpam-5335	149	4	3	3	NUM
ejpam-5335	149	5	)	)	PUNCT
ejpam-5335	149	6	,	,	PUNCT
ejpam-5335	149	7	f(δ(τ1	f(δ(τ1	PROPN
ejpam-5335	149	8	,	,	PUNCT
ejpam-5335	149	9	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	149	10	−1(b	−1(b	NOUN
ejpam-5335	149	11	)	)	PUNCT
ejpam-5335	149	12	)	)	PUNCT
ejpam-5335	149	13	)	)	PUNCT
ejpam-5335	150	1	⊆	⊆	X
ejpam-5335	150	2	σ1σ2	σ1σ2	NUM
ejpam-5335	150	3	-	-	PUNCT
ejpam-5335	150	4	cl(f(f	cl(f(f	ADJ
ejpam-5335	150	5	−1(b	−1(b	NOUN
ejpam-5335	150	6	)	)	PUNCT
ejpam-5335	150	7	)	)	PUNCT
ejpam-5335	150	8	)	)	PUNCT
ejpam-5335	150	9	⊆	⊆	X
ejpam-5335	150	10	σ1σ2	σ1σ2	NUM
ejpam-5335	150	11	-	-	PUNCT
ejpam-5335	150	12	cl(b	cl(b	NOUN
ejpam-5335	150	13	)	)	PUNCT
ejpam-5335	150	14	.	.	PUNCT
ejpam-5335	151	1	therefore	therefore	ADV
ejpam-5335	151	2	,	,	PUNCT
ejpam-5335	151	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	151	4	,	,	PUNCT
ejpam-5335	151	5	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	151	6	−1(b	−1(b	NOUN
ejpam-5335	151	7	)	)	PUNCT
ejpam-5335	151	8	)	)	PUNCT
ejpam-5335	151	9	⊆	⊆	NUM
ejpam-5335	151	10	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	151	11	-	-	PUNCT
ejpam-5335	151	12	cl(b	cl(b	NOUN
ejpam-5335	151	13	)	)	PUNCT
ejpam-5335	151	14	)	)	PUNCT
ejpam-5335	151	15	.	.	PUNCT
ejpam-5335	152	1	(	(	PUNCT
ejpam-5335	152	2	4	4	X
ejpam-5335	152	3	)	)	PUNCT
ejpam-5335	152	4	⇒	⇒	NOUN
ejpam-5335	152	5	(	(	PUNCT
ejpam-5335	152	6	5	5	NUM
ejpam-5335	152	7	):	):	PUNCT
ejpam-5335	152	8	let	let	VERB
ejpam-5335	152	9	b	b	X
ejpam-5335	152	10	be	be	AUX
ejpam-5335	152	11	any	any	DET
ejpam-5335	152	12	subset	subset	NOUN
ejpam-5335	152	13	of	of	ADP
ejpam-5335	152	14	y	y	PROPN
ejpam-5335	152	15	.	.	PUNCT
ejpam-5335	153	1	by	by	ADP
ejpam-5335	153	2	(	(	PUNCT
ejpam-5335	153	3	4	4	NUM
ejpam-5335	153	4	)	)	PUNCT
ejpam-5335	153	5	and	and	CCONJ
ejpam-5335	153	6	lemma	lemma	PROPN
ejpam-5335	153	7	2	2	NUM
ejpam-5335	153	8	(	(	PUNCT
ejpam-5335	153	9	2	2	NUM
ejpam-5335	153	10	)	)	PUNCT
ejpam-5335	153	11	,	,	PUNCT
ejpam-5335	153	12	we	we	PRON
ejpam-5335	153	13	have	have	VERB
ejpam-5335	153	14	x	x	PART
ejpam-5335	153	15	−	−	PROPN
ejpam-5335	153	16	δ(τ1	δ(τ1	PROPN
ejpam-5335	153	17	,	,	PUNCT
ejpam-5335	153	18	τ2)-int(f	τ2)-int(f	X
ejpam-5335	153	19	−1(b	−1(b	NOUN
ejpam-5335	153	20	)	)	PUNCT
ejpam-5335	153	21	)	)	PUNCT
ejpam-5335	154	1	=	=	SYM
ejpam-5335	154	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	154	3	,	,	PUNCT
ejpam-5335	154	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	154	5	−1(y	−1(y	PUNCT
ejpam-5335	154	6	−b	−b	ADJ
ejpam-5335	154	7	)	)	PUNCT
ejpam-5335	154	8	)	)	PUNCT
ejpam-5335	155	1	⊆	⊆	NUM
ejpam-5335	155	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	155	3	-	-	PUNCT
ejpam-5335	155	4	cl(y	cl(y	NOUN
ejpam-5335	155	5	−b	−b	NOUN
ejpam-5335	155	6	)	)	PUNCT
ejpam-5335	155	7	)	)	PUNCT
ejpam-5335	156	1	=	=	PUNCT
ejpam-5335	156	2	x	x	X
ejpam-5335	156	3	−	−	PRON
ejpam-5335	156	4	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	156	5	-	-	PUNCT
ejpam-5335	156	6	int(b	int(b	NOUN
ejpam-5335	156	7	)	)	PUNCT
ejpam-5335	156	8	)	)	PUNCT
ejpam-5335	156	9	and	and	CCONJ
ejpam-5335	156	10	hence	hence	ADV
ejpam-5335	156	11	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	156	12	-	-	PUNCT
ejpam-5335	156	13	int(b	int(b	NOUN
ejpam-5335	156	14	)	)	PUNCT
ejpam-5335	156	15	)	)	PUNCT
ejpam-5335	157	1	⊆	⊆	NUM
ejpam-5335	157	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	157	3	,	,	PUNCT
ejpam-5335	157	4	τ2)-int(f	τ2)-int(f	X
ejpam-5335	157	5	−1(b	−1(b	NOUN
ejpam-5335	157	6	)	)	PUNCT
ejpam-5335	157	7	)	)	PUNCT
ejpam-5335	157	8	.	.	PUNCT
ejpam-5335	158	1	(	(	PUNCT
ejpam-5335	158	2	5	5	X
ejpam-5335	158	3	)	)	PUNCT
ejpam-5335	158	4	⇒	⇒	NOUN
ejpam-5335	158	5	(	(	PUNCT
ejpam-5335	158	6	6	6	NUM
ejpam-5335	158	7	):	):	PUNCT
ejpam-5335	158	8	let	let	VERB
ejpam-5335	158	9	f	f	PRON
ejpam-5335	158	10	be	be	AUX
ejpam-5335	158	11	any	any	DET
ejpam-5335	158	12	σ1σ2	σ1σ2	NUM
ejpam-5335	158	13	-	-	PUNCT
ejpam-5335	158	14	closed	closed	ADJ
ejpam-5335	158	15	set	set	NOUN
ejpam-5335	158	16	of	of	ADP
ejpam-5335	158	17	y	y	PROPN
ejpam-5335	158	18	.	.	PUNCT
ejpam-5335	159	1	then	then	ADV
ejpam-5335	159	2	,	,	PUNCT
ejpam-5335	159	3	y	y	PROPN
ejpam-5335	159	4	−f	−f	PROPN
ejpam-5335	159	5	is	be	AUX
ejpam-5335	159	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	159	7	-	-	ADJ
ejpam-5335	159	8	open	open	ADJ
ejpam-5335	159	9	in	in	ADP
ejpam-5335	159	10	y	y	PROPN
ejpam-5335	159	11	.	.	PUNCT
ejpam-5335	160	1	by	by	ADP
ejpam-5335	160	2	(	(	PUNCT
ejpam-5335	160	3	5	5	NUM
ejpam-5335	160	4	)	)	PUNCT
ejpam-5335	160	5	and	and	CCONJ
ejpam-5335	160	6	lemma	lemma	PROPN
ejpam-5335	160	7	2	2	NUM
ejpam-5335	160	8	(	(	PUNCT
ejpam-5335	160	9	1	1	NUM
ejpam-5335	160	10	)	)	PUNCT
ejpam-5335	160	11	,	,	PUNCT
ejpam-5335	160	12	we	we	PRON
ejpam-5335	160	13	obtain	obtain	VERB
ejpam-5335	160	14	that	that	SCONJ
ejpam-5335	160	15	x	x	PUNCT
ejpam-5335	160	16	−	−	PROPN
ejpam-5335	160	17	f−1(f	f−1(f	NOUN
ejpam-5335	160	18	)	)	PUNCT
ejpam-5335	161	1	=	=	PUNCT
ejpam-5335	161	2	f−1(y	f−1(y	PROPN
ejpam-5335	162	1	−	−	PROPN
ejpam-5335	162	2	f	f	PROPN
ejpam-5335	162	3	)	)	PUNCT
ejpam-5335	162	4	=	=	SYM
ejpam-5335	162	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	162	6	-	-	PUNCT
ejpam-5335	162	7	int(y	int(y	PROPN
ejpam-5335	162	8	−	−	PROPN
ejpam-5335	162	9	f	f	NOUN
ejpam-5335	162	10	)	)	PUNCT
ejpam-5335	162	11	)	)	PUNCT
ejpam-5335	163	1	⊆	⊆	NUM
ejpam-5335	163	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	163	3	,	,	PUNCT
ejpam-5335	163	4	τ2)-int(f	τ2)-int(f	VERB
ejpam-5335	163	5	−1(y	−1(y	X
ejpam-5335	164	1	−	−	PROPN
ejpam-5335	164	2	f	f	PROPN
ejpam-5335	164	3	)	)	PUNCT
ejpam-5335	164	4	)	)	PUNCT
ejpam-5335	165	1	=	=	PUNCT
ejpam-5335	166	1	x	x	X
ejpam-5335	166	2	−	−	PROPN
ejpam-5335	166	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	166	4	,	,	PUNCT
ejpam-5335	166	5	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	166	6	−1(f	−1(f	NUM
ejpam-5335	166	7	)	)	PUNCT
ejpam-5335	166	8	)	)	PUNCT
ejpam-5335	166	9	.	.	PUNCT
ejpam-5335	167	1	therefore	therefore	ADV
ejpam-5335	167	2	,	,	PUNCT
ejpam-5335	167	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	167	4	,	,	PUNCT
ejpam-5335	167	5	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	167	6	−1(f	−1(f	NUM
ejpam-5335	167	7	)	)	PUNCT
ejpam-5335	167	8	)	)	PUNCT
ejpam-5335	167	9	⊆	⊆	NUM
ejpam-5335	167	10	f−1(f	f−1(f	NOUN
ejpam-5335	167	11	)	)	PUNCT
ejpam-5335	167	12	.	.	PUNCT
ejpam-5335	168	1	this	this	PRON
ejpam-5335	168	2	shows	show	VERB
ejpam-5335	168	3	that	that	SCONJ
ejpam-5335	168	4	f−1(f	f−1(f	PROPN
ejpam-5335	168	5	)	)	PUNCT
ejpam-5335	168	6	is	be	AUX
ejpam-5335	168	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	168	8	,	,	PUNCT
ejpam-5335	168	9	τ2)-closed	τ2)-close	VERB
ejpam-5335	168	10	in	in	ADP
ejpam-5335	168	11	x.	x.	NOUN
ejpam-5335	168	12	(	(	PUNCT
ejpam-5335	168	13	6	6	NUM
ejpam-5335	168	14	)	)	PUNCT
ejpam-5335	168	15	⇒	⇒	NOUN
ejpam-5335	168	16	(	(	PUNCT
ejpam-5335	168	17	2	2	NUM
ejpam-5335	168	18	):	):	PUNCT
ejpam-5335	168	19	let	let	VERB
ejpam-5335	168	20	v	v	PART
ejpam-5335	168	21	be	be	AUX
ejpam-5335	168	22	any	any	DET
ejpam-5335	168	23	σ1σ2	σ1σ2	NOUN
ejpam-5335	168	24	-	-	ADJ
ejpam-5335	168	25	open	open	ADJ
ejpam-5335	168	26	set	set	NOUN
ejpam-5335	168	27	of	of	ADP
ejpam-5335	168	28	y	y	PROPN
ejpam-5335	168	29	.	.	PUNCT
ejpam-5335	169	1	then	then	ADV
ejpam-5335	169	2	,	,	PUNCT
ejpam-5335	169	3	y	y	PROPN
ejpam-5335	169	4	−v	−v	NOUN
ejpam-5335	169	5	is	be	AUX
ejpam-5335	169	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	169	7	-	-	ADJ
ejpam-5335	169	8	closed	closed	ADJ
ejpam-5335	169	9	in	in	ADP
ejpam-5335	169	10	y	y	PROPN
ejpam-5335	169	11	.	.	PUNCT
ejpam-5335	170	1	by	by	ADP
ejpam-5335	170	2	(	(	PUNCT
ejpam-5335	170	3	6	6	NUM
ejpam-5335	170	4	)	)	PUNCT
ejpam-5335	170	5	,	,	PUNCT
ejpam-5335	170	6	we	we	PRON
ejpam-5335	170	7	have	have	VERB
ejpam-5335	170	8	x−f−1(v	x−f−1(v	PUNCT
ejpam-5335	170	9	)	)	PUNCT
ejpam-5335	171	1	=	=	SYM
ejpam-5335	171	2	f−1(y	f−1(y	PROPN
ejpam-5335	171	3	−v	−v	NOUN
ejpam-5335	171	4	)	)	PUNCT
ejpam-5335	171	5	is	be	AUX
ejpam-5335	171	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	171	7	,	,	PUNCT
ejpam-5335	171	8	τ2)-closed	τ2)-close	VERB
ejpam-5335	171	9	in	in	ADP
ejpam-5335	171	10	x.	x.	PROPN
ejpam-5335	171	11	thus	thus	ADV
ejpam-5335	171	12	,	,	PUNCT
ejpam-5335	171	13	f−1(v	f−1(v	PROPN
ejpam-5335	171	14	)	)	PUNCT
ejpam-5335	171	15	is	be	AUX
ejpam-5335	171	16	δ(τ1	δ(τ1	NOUN
ejpam-5335	171	17	,	,	PUNCT
ejpam-5335	171	18	τ2)-open	τ2)-open	ADJ
ejpam-5335	171	19	in	in	ADP
ejpam-5335	171	20	x.	x.	NOUN
ejpam-5335	171	21	(	(	PUNCT
ejpam-5335	171	22	2	2	NUM
ejpam-5335	171	23	)	)	PUNCT
ejpam-5335	171	24	⇒	⇒	NOUN
ejpam-5335	171	25	(	(	PUNCT
ejpam-5335	171	26	1	1	NUM
ejpam-5335	171	27	):	):	PUNCT
ejpam-5335	171	28	let	let	VERB
ejpam-5335	171	29	x	x	PUNCT
ejpam-5335	171	30	∈	∈	PROPN
ejpam-5335	171	31	x	x	X
ejpam-5335	171	32	and	and	CCONJ
ejpam-5335	171	33	v	v	X
ejpam-5335	171	34	be	be	AUX
ejpam-5335	171	35	any	any	DET
ejpam-5335	171	36	σ1σ2	σ1σ2	NOUN
ejpam-5335	171	37	-	-	ADJ
ejpam-5335	171	38	open	open	ADJ
ejpam-5335	171	39	set	set	NOUN
ejpam-5335	171	40	of	of	ADP
ejpam-5335	171	41	y	y	PROPN
ejpam-5335	171	42	containing	contain	VERB
ejpam-5335	171	43	f(x	f(x	PROPN
ejpam-5335	171	44	)	)	PUNCT
ejpam-5335	171	45	.	.	PUNCT
ejpam-5335	172	1	by	by	ADP
ejpam-5335	172	2	(	(	PUNCT
ejpam-5335	172	3	2	2	NUM
ejpam-5335	172	4	)	)	PUNCT
ejpam-5335	172	5	,	,	PUNCT
ejpam-5335	172	6	f−1(v	f−1(v	PROPN
ejpam-5335	172	7	)	)	PUNCT
ejpam-5335	172	8	is	be	AUX
ejpam-5335	172	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	172	10	,	,	PUNCT
ejpam-5335	172	11	τ2)-open	τ2)-open	ADJ
ejpam-5335	172	12	in	in	ADP
ejpam-5335	172	13	x.	x.	NOUN
ejpam-5335	172	14	hence	hence	ADV
ejpam-5335	172	15	,	,	PUNCT
ejpam-5335	172	16	x	x	PROPN
ejpam-5335	172	17	∈	∈	PROPN
ejpam-5335	172	18	f−1(v	f−1(v	NOUN
ejpam-5335	172	19	)	)	PUNCT
ejpam-5335	173	1	=	=	SYM
ejpam-5335	173	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	173	3	,	,	PUNCT
ejpam-5335	173	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	173	5	−1(v	−1(v	PROPN
ejpam-5335	173	6	)	)	PUNCT
ejpam-5335	173	7	)	)	PUNCT
ejpam-5335	173	8	.	.	PUNCT
ejpam-5335	174	1	then	then	ADV
ejpam-5335	174	2	,	,	PUNCT
ejpam-5335	174	3	there	there	PRON
ejpam-5335	174	4	exists	exist	VERB
ejpam-5335	174	5	a	a	DET
ejpam-5335	174	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	174	7	,	,	PUNCT
ejpam-5335	174	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	174	9	set	set	VERB
ejpam-5335	174	10	u	u	PRON
ejpam-5335	174	11	such	such	ADJ
ejpam-5335	174	12	that	that	SCONJ
ejpam-5335	174	13	x	x	SYM
ejpam-5335	174	14	∈	∈	PROPN
ejpam-5335	174	15	u	u	NOUN
ejpam-5335	174	16	⊆	⊆	NUM
ejpam-5335	174	17	f−1(v	f−1(v	NOUN
ejpam-5335	174	18	)	)	PUNCT
ejpam-5335	174	19	.	.	PUNCT
ejpam-5335	175	1	therefore	therefore	ADV
ejpam-5335	175	2	,	,	PUNCT
ejpam-5335	175	3	f(u	f(u	PROPN
ejpam-5335	175	4	)	)	PUNCT
ejpam-5335	175	5	⊆	⊆	NUM
ejpam-5335	175	6	v	v	NOUN
ejpam-5335	175	7	.	.	PUNCT
ejpam-5335	176	1	thus	thus	ADV
ejpam-5335	176	2	,	,	PUNCT
ejpam-5335	176	3	f	f	PROPN
ejpam-5335	176	4	is	be	AUX
ejpam-5335	176	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	176	6	,	,	PUNCT
ejpam-5335	176	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	176	8	at	at	ADP
ejpam-5335	176	9	x.	x.	NOUN
ejpam-5335	176	10	this	this	PRON
ejpam-5335	176	11	shows	show	VERB
ejpam-5335	176	12	that	that	SCONJ
ejpam-5335	176	13	f	f	PROPN
ejpam-5335	176	14	is	be	AUX
ejpam-5335	176	15	δ(τ1	δ(τ1	NOUN
ejpam-5335	176	16	,	,	PUNCT
ejpam-5335	176	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	176	18	.	.	NOUN
ejpam-5335	177	1	4	4	X
ejpam-5335	177	2	.	.	X
ejpam-5335	177	3	on	on	ADP
ejpam-5335	177	4	almost	almost	ADV
ejpam-5335	177	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	177	6	,	,	PUNCT
ejpam-5335	177	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	177	8	functions	function	NOUN
ejpam-5335	177	9	in	in	ADP
ejpam-5335	177	10	this	this	DET
ejpam-5335	177	11	section	section	NOUN
ejpam-5335	177	12	,	,	PUNCT
ejpam-5335	177	13	we	we	PRON
ejpam-5335	177	14	introduce	introduce	VERB
ejpam-5335	177	15	the	the	DET
ejpam-5335	177	16	notion	notion	NOUN
ejpam-5335	177	17	of	of	ADP
ejpam-5335	177	18	almost	almost	ADV
ejpam-5335	177	19	δ(τ1	δ(τ1	NOUN
ejpam-5335	177	20	,	,	PUNCT
ejpam-5335	177	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	177	22	functions	function	NOUN
ejpam-5335	177	23	and	and	CCONJ
ejpam-5335	177	24	investigate	investigate	VERB
ejpam-5335	177	25	some	some	DET
ejpam-5335	177	26	characterizations	characterization	NOUN
ejpam-5335	177	27	of	of	ADP
ejpam-5335	177	28	almost	almost	ADV
ejpam-5335	177	29	δ(τ1	δ(τ1	NOUN
ejpam-5335	177	30	,	,	PUNCT
ejpam-5335	177	31	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	177	32	functions	function	NOUN
ejpam-5335	177	33	.	.	PUNCT
ejpam-5335	178	1	moreover	moreover	ADV
ejpam-5335	178	2	,	,	PUNCT
ejpam-5335	178	3	the	the	DET
ejpam-5335	178	4	relationships	relationship	NOUN
ejpam-5335	178	5	between	between	ADP
ejpam-5335	178	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	178	7	,	,	PUNCT
ejpam-5335	178	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	178	9	functions	function	NOUN
ejpam-5335	178	10	and	and	CCONJ
ejpam-5335	178	11	almost	almost	ADV
ejpam-5335	178	12	δ(τ1	δ(τ1	NOUN
ejpam-5335	178	13	,	,	PUNCT
ejpam-5335	178	14	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	178	15	functions	function	NOUN
ejpam-5335	178	16	are	be	AUX
ejpam-5335	178	17	considered	consider	VERB
ejpam-5335	178	18	.	.	PUNCT
ejpam-5335	179	1	definition	definition	NOUN
ejpam-5335	179	2	2	2	NUM
ejpam-5335	179	3	.	.	PUNCT
ejpam-5335	180	1	a	a	DET
ejpam-5335	180	2	function	function	NOUN
ejpam-5335	180	3	f	f	NOUN
ejpam-5335	180	4	:	:	PUNCT
ejpam-5335	180	5	(	(	PUNCT
ejpam-5335	180	6	x	x	NOUN
ejpam-5335	180	7	,	,	PUNCT
ejpam-5335	180	8	τ1	τ1	NOUN
ejpam-5335	180	9	,	,	PUNCT
ejpam-5335	180	10	τ2	τ2	NOUN
ejpam-5335	180	11	)	)	PUNCT
ejpam-5335	180	12	→	→	SYM
ejpam-5335	180	13	(	(	PUNCT
ejpam-5335	180	14	y	y	PROPN
ejpam-5335	180	15	,	,	PUNCT
ejpam-5335	180	16	σ1	σ1	PROPN
ejpam-5335	180	17	,	,	PUNCT
ejpam-5335	180	18	σ2	σ2	PROPN
ejpam-5335	180	19	)	)	PUNCT
ejpam-5335	180	20	is	be	AUX
ejpam-5335	180	21	said	say	VERB
ejpam-5335	180	22	to	to	PART
ejpam-5335	180	23	be	be	AUX
ejpam-5335	180	24	almost	almost	ADV
ejpam-5335	180	25	δ(τ1	δ(τ1	NOUN
ejpam-5335	180	26	,	,	PUNCT
ejpam-5335	180	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	180	28	at	at	ADP
ejpam-5335	180	29	x	x	X
ejpam-5335	180	30	∈	∈	PROPN
ejpam-5335	180	31	x	x	PUNCT
ejpam-5335	180	32	if	if	SCONJ
ejpam-5335	180	33	for	for	ADP
ejpam-5335	180	34	each	each	DET
ejpam-5335	180	35	σ1σ2	σ1σ2	VERB
ejpam-5335	180	36	-	-	ADJ
ejpam-5335	180	37	open	open	ADJ
ejpam-5335	180	38	set	set	NOUN
ejpam-5335	180	39	v	v	NOUN
ejpam-5335	180	40	of	of	ADP
ejpam-5335	180	41	y	y	NOUN
ejpam-5335	180	42	containing	contain	VERB
ejpam-5335	180	43	f(x	f(x	PROPN
ejpam-5335	180	44	)	)	PUNCT
ejpam-5335	180	45	,	,	PUNCT
ejpam-5335	180	46	there	there	PRON
ejpam-5335	180	47	exists	exist	VERB
ejpam-5335	180	48	a	a	DET
ejpam-5335	180	49	δ(τ1	δ(τ1	NOUN
ejpam-5335	180	50	,	,	PUNCT
ejpam-5335	180	51	τ2)-open	τ2)-open	ADJ
ejpam-5335	180	52	set	set	ADJ
ejpam-5335	180	53	u	u	NOUN
ejpam-5335	180	54	of	of	ADP
ejpam-5335	180	55	x	x	PUNCT
ejpam-5335	180	56	containing	contain	VERB
ejpam-5335	180	57	x	x	PUNCT
ejpam-5335	180	58	such	such	ADJ
ejpam-5335	180	59	that	that	DET
ejpam-5335	180	60	f(u	f(u	PROPN
ejpam-5335	180	61	)	)	PUNCT
ejpam-5335	180	62	⊆	⊆	NUM
ejpam-5335	180	63	σ1σ2	σ1σ2	X
ejpam-5335	180	64	-	-	PUNCT
ejpam-5335	180	65	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	180	66	-	-	PUNCT
ejpam-5335	180	67	cl(v	cl(v	NOUN
ejpam-5335	180	68	)	)	PUNCT
ejpam-5335	180	69	)	)	PUNCT
ejpam-5335	180	70	.	.	PUNCT
ejpam-5335	181	1	a	a	DET
ejpam-5335	181	2	function	function	NOUN
ejpam-5335	181	3	f	f	NOUN
ejpam-5335	181	4	:	:	PUNCT
ejpam-5335	181	5	(	(	PUNCT
ejpam-5335	181	6	x	x	NOUN
ejpam-5335	181	7	,	,	PUNCT
ejpam-5335	181	8	τ1	τ1	NOUN
ejpam-5335	181	9	,	,	PUNCT
ejpam-5335	181	10	τ2	τ2	NOUN
ejpam-5335	181	11	)	)	PUNCT
ejpam-5335	181	12	→	→	SYM
ejpam-5335	181	13	(	(	PUNCT
ejpam-5335	181	14	y	y	PROPN
ejpam-5335	181	15	,	,	PUNCT
ejpam-5335	181	16	σ1	σ1	PROPN
ejpam-5335	181	17	,	,	PUNCT
ejpam-5335	181	18	σ2	σ2	PROPN
ejpam-5335	181	19	)	)	PUNCT
ejpam-5335	181	20	is	be	AUX
ejpam-5335	181	21	called	call	VERB
ejpam-5335	181	22	almost	almost	ADV
ejpam-5335	181	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	181	24	,	,	PUNCT
ejpam-5335	181	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	181	26	if	if	SCONJ
ejpam-5335	181	27	f	f	PROPN
ejpam-5335	181	28	is	be	AUX
ejpam-5335	181	29	almost	almost	ADV
ejpam-5335	181	30	δ(τ1	δ(τ1	NOUN
ejpam-5335	181	31	,	,	PUNCT
ejpam-5335	181	32	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	181	33	at	at	ADP
ejpam-5335	181	34	each	each	DET
ejpam-5335	181	35	point	point	NOUN
ejpam-5335	181	36	of	of	ADP
ejpam-5335	181	37	x.	x.	PROPN
ejpam-5335	181	38	c.	c.	PROPN
ejpam-5335	181	39	prachanpol	prachanpol	PROPN
ejpam-5335	181	40	,	,	PUNCT
ejpam-5335	181	41	c.	c.	PROPN
ejpam-5335	181	42	boonpok	boonpok	PROPN
ejpam-5335	181	43	,	,	PUNCT
ejpam-5335	181	44	c.	c.	PROPN
ejpam-5335	181	45	viriyapong	viriyapong	PROPN
ejpam-5335	181	46	/	/	SYM
ejpam-5335	181	47	eur	eur	PROPN
ejpam-5335	181	48	.	.	PUNCT
ejpam-5335	182	1	j.	j.	PROPN
ejpam-5335	182	2	pure	pure	PROPN
ejpam-5335	182	3	appl	appl	PROPN
ejpam-5335	182	4	.	.	PROPN
ejpam-5335	182	5	math	math	PROPN
ejpam-5335	182	6	,	,	PUNCT
ejpam-5335	182	7	17	17	NUM
ejpam-5335	182	8	(	(	PUNCT
ejpam-5335	182	9	4	4	NUM
ejpam-5335	182	10	)	)	PUNCT
ejpam-5335	182	11	(	(	PUNCT
ejpam-5335	182	12	2024	2024	NUM
ejpam-5335	182	13	)	)	PUNCT
ejpam-5335	182	14	,	,	PUNCT
ejpam-5335	182	15	3730	3730	NUM
ejpam-5335	182	16	-	-	SYM
ejpam-5335	182	17	3742	3742	NUM
ejpam-5335	182	18	3735	3735	NUM
ejpam-5335	182	19	remark	remark	NOUN
ejpam-5335	182	20	1	1	NUM
ejpam-5335	182	21	.	.	PUNCT
ejpam-5335	183	1	for	for	ADP
ejpam-5335	183	2	a	a	DET
ejpam-5335	183	3	function	function	NOUN
ejpam-5335	183	4	f	f	NOUN
ejpam-5335	183	5	:	:	PUNCT
ejpam-5335	183	6	(	(	PUNCT
ejpam-5335	183	7	x	x	NOUN
ejpam-5335	183	8	,	,	PUNCT
ejpam-5335	183	9	τ1	τ1	NOUN
ejpam-5335	183	10	,	,	PUNCT
ejpam-5335	183	11	τ2	τ2	NOUN
ejpam-5335	183	12	)	)	PUNCT
ejpam-5335	183	13	→	→	SYM
ejpam-5335	183	14	(	(	PUNCT
ejpam-5335	183	15	y	y	PROPN
ejpam-5335	183	16	,	,	PUNCT
ejpam-5335	183	17	σ1	σ1	PROPN
ejpam-5335	183	18	,	,	PUNCT
ejpam-5335	183	19	σ2	σ2	NOUN
ejpam-5335	183	20	)	)	PUNCT
ejpam-5335	183	21	,	,	PUNCT
ejpam-5335	183	22	the	the	DET
ejpam-5335	183	23	following	follow	VERB
ejpam-5335	183	24	implication	implication	NOUN
ejpam-5335	183	25	holds	hold	VERB
ejpam-5335	183	26	:	:	PUNCT
ejpam-5335	183	27	δ(τ1	δ(τ1	NOUN
ejpam-5335	183	28	,	,	PUNCT
ejpam-5335	183	29	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	183	30	⇒	⇒	NOUN
ejpam-5335	183	31	almost	almost	ADV
ejpam-5335	183	32	δ(τ1	δ(τ1	NOUN
ejpam-5335	183	33	,	,	PUNCT
ejpam-5335	183	34	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	183	35	.	.	PUNCT
ejpam-5335	184	1	the	the	DET
ejpam-5335	184	2	converse	converse	NOUN
ejpam-5335	184	3	of	of	ADP
ejpam-5335	184	4	the	the	DET
ejpam-5335	184	5	implication	implication	NOUN
ejpam-5335	184	6	is	be	AUX
ejpam-5335	184	7	not	not	PART
ejpam-5335	184	8	true	true	ADJ
ejpam-5335	184	9	in	in	ADP
ejpam-5335	184	10	general	general	ADJ
ejpam-5335	184	11	.	.	PUNCT
ejpam-5335	185	1	we	we	PRON
ejpam-5335	185	2	give	give	VERB
ejpam-5335	185	3	an	an	DET
ejpam-5335	185	4	example	example	NOUN
ejpam-5335	185	5	for	for	ADP
ejpam-5335	185	6	the	the	DET
ejpam-5335	185	7	implication	implication	NOUN
ejpam-5335	185	8	as	as	SCONJ
ejpam-5335	185	9	follows	follow	VERB
ejpam-5335	185	10	.	.	PUNCT
ejpam-5335	185	11	example	example	NOUN
ejpam-5335	186	1	2	2	NUM
ejpam-5335	186	2	.	.	PUNCT
ejpam-5335	186	3	let	let	VERB
ejpam-5335	186	4	x	x	PUNCT
ejpam-5335	186	5	=	=	PRON
ejpam-5335	186	6	{	{	PUNCT
ejpam-5335	186	7	1	1	NUM
ejpam-5335	186	8	,	,	PUNCT
ejpam-5335	186	9	2	2	NUM
ejpam-5335	186	10	,	,	PUNCT
ejpam-5335	186	11	3	3	NUM
ejpam-5335	186	12	,	,	PUNCT
ejpam-5335	186	13	4	4	NUM
ejpam-5335	186	14	}	}	PUNCT
ejpam-5335	186	15	with	with	ADP
ejpam-5335	186	16	topologies	topology	NOUN
ejpam-5335	186	17	τ1	τ1	NOUN
ejpam-5335	186	18	=	=	SYM
ejpam-5335	186	19	{	{	PUNCT
ejpam-5335	186	20	∅	∅	NOUN
ejpam-5335	186	21	,	,	PUNCT
ejpam-5335	186	22	{	{	PUNCT
ejpam-5335	186	23	1	1	NUM
ejpam-5335	186	24	}	}	PUNCT
ejpam-5335	186	25	,	,	PUNCT
ejpam-5335	186	26	{	{	PUNCT
ejpam-5335	186	27	2	2	NUM
ejpam-5335	186	28	}	}	PUNCT
ejpam-5335	186	29	,	,	PUNCT
ejpam-5335	186	30	{	{	PUNCT
ejpam-5335	186	31	1	1	NUM
ejpam-5335	186	32	,	,	PUNCT
ejpam-5335	186	33	2	2	NUM
ejpam-5335	186	34	}	}	PUNCT
ejpam-5335	186	35	,	,	PUNCT
ejpam-5335	186	36	x	x	NOUN
ejpam-5335	186	37	}	}	PUNCT
ejpam-5335	186	38	and	and	CCONJ
ejpam-5335	186	39	τ2	τ2	NOUN
ejpam-5335	186	40	=	=	SYM
ejpam-5335	186	41	{	{	PUNCT
ejpam-5335	186	42	∅	∅	NOUN
ejpam-5335	186	43	,	,	PUNCT
ejpam-5335	186	44	{	{	PUNCT
ejpam-5335	186	45	1	1	NUM
ejpam-5335	186	46	}	}	PUNCT
ejpam-5335	186	47	,	,	PUNCT
ejpam-5335	186	48	{	{	PUNCT
ejpam-5335	186	49	2	2	NUM
ejpam-5335	186	50	}	}	PUNCT
ejpam-5335	186	51	,	,	PUNCT
ejpam-5335	186	52	{	{	PUNCT
ejpam-5335	186	53	3	3	NUM
ejpam-5335	186	54	}	}	PUNCT
ejpam-5335	186	55	,	,	PUNCT
ejpam-5335	186	56	{	{	PUNCT
ejpam-5335	186	57	1	1	NUM
ejpam-5335	186	58	,	,	PUNCT
ejpam-5335	186	59	2	2	NUM
ejpam-5335	186	60	}	}	PUNCT
ejpam-5335	186	61	,	,	PUNCT
ejpam-5335	186	62	{	{	PUNCT
ejpam-5335	186	63	1	1	NUM
ejpam-5335	186	64	,	,	PUNCT
ejpam-5335	186	65	3	3	NUM
ejpam-5335	186	66	}	}	PUNCT
ejpam-5335	186	67	,	,	PUNCT
ejpam-5335	186	68	{	{	PUNCT
ejpam-5335	186	69	2	2	NUM
ejpam-5335	186	70	,	,	PUNCT
ejpam-5335	186	71	3	3	NUM
ejpam-5335	186	72	}	}	PUNCT
ejpam-5335	186	73	,	,	PUNCT
ejpam-5335	186	74	{	{	PUNCT
ejpam-5335	186	75	1	1	NUM
ejpam-5335	186	76	,	,	PUNCT
ejpam-5335	186	77	2	2	NUM
ejpam-5335	186	78	,	,	PUNCT
ejpam-5335	186	79	3	3	NUM
ejpam-5335	186	80	}	}	PUNCT
ejpam-5335	186	81	,	,	PUNCT
ejpam-5335	186	82	x	x	NOUN
ejpam-5335	186	83	}	}	PUNCT
ejpam-5335	186	84	.	.	PUNCT
ejpam-5335	187	1	let	let	VERB
ejpam-5335	187	2	y	y	PROPN
ejpam-5335	187	3	=	=	PUNCT
ejpam-5335	187	4	{	{	PUNCT
ejpam-5335	187	5	a	a	PRON
ejpam-5335	187	6	,	,	PUNCT
ejpam-5335	187	7	b	b	NOUN
ejpam-5335	187	8	,	,	PUNCT
ejpam-5335	187	9	c	c	NOUN
ejpam-5335	187	10	}	}	PUNCT
ejpam-5335	187	11	with	with	ADP
ejpam-5335	187	12	topologies	topology	NOUN
ejpam-5335	187	13	σ1	σ1	NOUN
ejpam-5335	187	14	=	=	SYM
ejpam-5335	187	15	{	{	PUNCT
ejpam-5335	187	16	∅	∅	NOUN
ejpam-5335	187	17	,	,	PUNCT
ejpam-5335	187	18	{	{	PUNCT
ejpam-5335	187	19	a	a	X
ejpam-5335	187	20	}	}	PUNCT
ejpam-5335	187	21	,	,	PUNCT
ejpam-5335	187	22	{	{	PUNCT
ejpam-5335	187	23	b	b	NOUN
ejpam-5335	187	24	}	}	PUNCT
ejpam-5335	187	25	,	,	PUNCT
ejpam-5335	187	26	{	{	PUNCT
ejpam-5335	187	27	a	a	DET
ejpam-5335	187	28	,	,	PUNCT
ejpam-5335	187	29	b	b	NOUN
ejpam-5335	187	30	}	}	PUNCT
ejpam-5335	187	31	,	,	PUNCT
ejpam-5335	187	32	y	y	PROPN
ejpam-5335	187	33	}	}	PUNCT
ejpam-5335	187	34	and	and	CCONJ
ejpam-5335	187	35	σ2	σ2	PROPN
ejpam-5335	187	36	=	=	SYM
ejpam-5335	187	37	{	{	PUNCT
ejpam-5335	187	38	∅	∅	NOUN
ejpam-5335	187	39	,	,	PUNCT
ejpam-5335	187	40	{	{	PUNCT
ejpam-5335	187	41	a	a	X
ejpam-5335	187	42	}	}	PUNCT
ejpam-5335	187	43	,	,	PUNCT
ejpam-5335	187	44	{	{	PUNCT
ejpam-5335	187	45	b	b	NOUN
ejpam-5335	187	46	}	}	PUNCT
ejpam-5335	187	47	,	,	PUNCT
ejpam-5335	187	48	{	{	PUNCT
ejpam-5335	187	49	a	a	DET
ejpam-5335	187	50	,	,	PUNCT
ejpam-5335	187	51	b	b	NOUN
ejpam-5335	187	52	}	}	PUNCT
ejpam-5335	187	53	,	,	PUNCT
ejpam-5335	187	54	{	{	PUNCT
ejpam-5335	187	55	a	a	X
ejpam-5335	187	56	,	,	PUNCT
ejpam-5335	187	57	c	c	NOUN
ejpam-5335	187	58	}	}	PUNCT
ejpam-5335	187	59	,	,	PUNCT
ejpam-5335	187	60	y	y	PROPN
ejpam-5335	187	61	}	}	PUNCT
ejpam-5335	187	62	.	.	PUNCT
ejpam-5335	188	1	a	a	DET
ejpam-5335	188	2	function	function	NOUN
ejpam-5335	188	3	f	f	NOUN
ejpam-5335	188	4	:	:	PUNCT
ejpam-5335	188	5	(	(	PUNCT
ejpam-5335	188	6	x	x	NOUN
ejpam-5335	188	7	,	,	PUNCT
ejpam-5335	188	8	τ1	τ1	NOUN
ejpam-5335	188	9	,	,	PUNCT
ejpam-5335	188	10	τ2	τ2	NOUN
ejpam-5335	188	11	)	)	PUNCT
ejpam-5335	188	12	→	→	SYM
ejpam-5335	188	13	(	(	PUNCT
ejpam-5335	188	14	y	y	PROPN
ejpam-5335	188	15	,	,	PUNCT
ejpam-5335	188	16	σ1	σ1	PROPN
ejpam-5335	188	17	,	,	PUNCT
ejpam-5335	188	18	σ2	σ2	PROPN
ejpam-5335	188	19	)	)	PUNCT
ejpam-5335	188	20	is	be	AUX
ejpam-5335	188	21	defined	define	VERB
ejpam-5335	188	22	as	as	SCONJ
ejpam-5335	188	23	follows	follow	VERB
ejpam-5335	188	24	:	:	PUNCT
ejpam-5335	188	25	f(1	f(1	X
ejpam-5335	188	26	)	)	PUNCT
ejpam-5335	188	27	=	=	SYM
ejpam-5335	189	1	a	a	PRON
ejpam-5335	189	2	,	,	PUNCT
ejpam-5335	189	3	f(2	f(2	PROPN
ejpam-5335	189	4	)	)	PUNCT
ejpam-5335	189	5	=	=	SYM
ejpam-5335	189	6	b	b	PROPN
ejpam-5335	189	7	and	and	CCONJ
ejpam-5335	189	8	f(3	f(3	PROPN
ejpam-5335	189	9	)	)	PUNCT
ejpam-5335	189	10	=	=	SYM
ejpam-5335	189	11	f(4	f(4	PROPN
ejpam-5335	189	12	)	)	PUNCT
ejpam-5335	189	13	=	=	SYM
ejpam-5335	190	1	c.	c.	NOUN
ejpam-5335	190	2	then	then	ADV
ejpam-5335	190	3	f	f	PROPN
ejpam-5335	190	4	is	be	AUX
ejpam-5335	190	5	almost	almost	ADV
ejpam-5335	190	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	190	7	,	,	PUNCT
ejpam-5335	190	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	190	9	,	,	PUNCT
ejpam-5335	190	10	but	but	CCONJ
ejpam-5335	190	11	f	f	PROPN
ejpam-5335	190	12	is	be	AUX
ejpam-5335	190	13	not	not	PART
ejpam-5335	190	14	δ(τ1	δ(τ1	NOUN
ejpam-5335	190	15	,	,	PUNCT
ejpam-5335	190	16	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5335	190	17	.	.	PUNCT
ejpam-5335	191	1	theorem	theorem	NOUN
ejpam-5335	191	2	3	3	NUM
ejpam-5335	191	3	.	.	X
ejpam-5335	191	4	for	for	ADP
ejpam-5335	191	5	a	a	DET
ejpam-5335	191	6	function	function	NOUN
ejpam-5335	191	7	f	f	NOUN
ejpam-5335	191	8	:	:	PUNCT
ejpam-5335	191	9	(	(	PUNCT
ejpam-5335	191	10	x	x	NOUN
ejpam-5335	191	11	,	,	PUNCT
ejpam-5335	191	12	τ1	τ1	NOUN
ejpam-5335	191	13	,	,	PUNCT
ejpam-5335	191	14	τ2	τ2	NOUN
ejpam-5335	191	15	)	)	PUNCT
ejpam-5335	191	16	→	→	SYM
ejpam-5335	191	17	(	(	PUNCT
ejpam-5335	191	18	y	y	PROPN
ejpam-5335	191	19	,	,	PUNCT
ejpam-5335	191	20	σ1	σ1	PROPN
ejpam-5335	191	21	,	,	PUNCT
ejpam-5335	191	22	σ2	σ2	NOUN
ejpam-5335	191	23	)	)	PUNCT
ejpam-5335	191	24	,	,	PUNCT
ejpam-5335	191	25	the	the	DET
ejpam-5335	191	26	following	follow	VERB
ejpam-5335	191	27	properties	property	NOUN
ejpam-5335	191	28	are	be	AUX
ejpam-5335	191	29	equivalent	equivalent	ADJ
ejpam-5335	191	30	:	:	PUNCT
ejpam-5335	191	31	(	(	PUNCT
ejpam-5335	191	32	1	1	X
ejpam-5335	191	33	)	)	PUNCT
ejpam-5335	191	34	f	f	NOUN
ejpam-5335	191	35	is	be	AUX
ejpam-5335	191	36	almost	almost	ADV
ejpam-5335	191	37	δ(τ1	δ(τ1	NOUN
ejpam-5335	191	38	,	,	PUNCT
ejpam-5335	191	39	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	191	40	at	at	ADP
ejpam-5335	191	41	x	x	PRON
ejpam-5335	191	42	;	;	PUNCT
ejpam-5335	191	43	(	(	PUNCT
ejpam-5335	191	44	2	2	X
ejpam-5335	191	45	)	)	PUNCT
ejpam-5335	191	46	x	x	SYM
ejpam-5335	191	47	∈	∈	PROPN
ejpam-5335	191	48	δ(τ1	δ(τ1	PROPN
ejpam-5335	191	49	,	,	PUNCT
ejpam-5335	191	50	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	191	51	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	191	52	-	-	PUNCT
ejpam-5335	191	53	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	191	54	-	-	PUNCT
ejpam-5335	191	55	cl(v	cl(v	NOUN
ejpam-5335	191	56	)	)	PUNCT
ejpam-5335	191	57	)	)	PUNCT
ejpam-5335	191	58	)	)	PUNCT
ejpam-5335	191	59	)	)	PUNCT
ejpam-5335	192	1	for	for	ADP
ejpam-5335	192	2	every	every	DET
ejpam-5335	192	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	192	4	-	-	ADJ
ejpam-5335	192	5	open	open	ADJ
ejpam-5335	192	6	set	set	NOUN
ejpam-5335	192	7	v	v	NOUN
ejpam-5335	192	8	of	of	ADP
ejpam-5335	192	9	y	y	NOUN
ejpam-5335	192	10	containing	contain	VERB
ejpam-5335	192	11	f(x	f(x	PROPN
ejpam-5335	192	12	)	)	PUNCT
ejpam-5335	192	13	;	;	PUNCT
ejpam-5335	192	14	(	(	PUNCT
ejpam-5335	192	15	3	3	X
ejpam-5335	192	16	)	)	PUNCT
ejpam-5335	192	17	x	x	SYM
ejpam-5335	192	18	∈	∈	PROPN
ejpam-5335	192	19	δ(τ1	δ(τ1	PROPN
ejpam-5335	192	20	,	,	PUNCT
ejpam-5335	192	21	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	192	22	−1(v	−1(v	PROPN
ejpam-5335	192	23	)	)	PUNCT
ejpam-5335	192	24	)	)	PUNCT
ejpam-5335	192	25	for	for	ADP
ejpam-5335	192	26	every	every	DET
ejpam-5335	192	27	(	(	PUNCT
ejpam-5335	192	28	σ1	σ1	PROPN
ejpam-5335	192	29	,	,	PUNCT
ejpam-5335	192	30	σ2)r	σ2)r	NOUN
ejpam-5335	192	31	-	-	PUNCT
ejpam-5335	192	32	open	open	ADJ
ejpam-5335	192	33	set	set	VERB
ejpam-5335	192	34	v	v	NOUN
ejpam-5335	192	35	of	of	ADP
ejpam-5335	192	36	y	y	NOUN
ejpam-5335	192	37	containing	contain	VERB
ejpam-5335	192	38	f(x	f(x	PROPN
ejpam-5335	192	39	)	)	PUNCT
ejpam-5335	192	40	;	;	PUNCT
ejpam-5335	192	41	(	(	PUNCT
ejpam-5335	192	42	4	4	X
ejpam-5335	192	43	)	)	PUNCT
ejpam-5335	192	44	for	for	ADP
ejpam-5335	192	45	each	each	DET
ejpam-5335	192	46	x	x	SYM
ejpam-5335	192	47	∈	∈	PROPN
ejpam-5335	192	48	x	x	X
ejpam-5335	192	49	and	and	CCONJ
ejpam-5335	192	50	each	each	DET
ejpam-5335	192	51	(	(	PUNCT
ejpam-5335	192	52	σ1	σ1	PROPN
ejpam-5335	192	53	,	,	PUNCT
ejpam-5335	192	54	σ2)r	σ2)r	NOUN
ejpam-5335	192	55	-	-	PUNCT
ejpam-5335	192	56	open	open	ADJ
ejpam-5335	192	57	set	set	VERB
ejpam-5335	192	58	v	v	NOUN
ejpam-5335	192	59	of	of	ADP
ejpam-5335	192	60	y	y	NOUN
ejpam-5335	192	61	containing	contain	VERB
ejpam-5335	192	62	f(x	f(x	PROPN
ejpam-5335	192	63	)	)	PUNCT
ejpam-5335	192	64	,	,	PUNCT
ejpam-5335	192	65	there	there	PRON
ejpam-5335	192	66	exists	exist	VERB
ejpam-5335	192	67	a	a	DET
ejpam-5335	192	68	δ(τ1	δ(τ1	NOUN
ejpam-5335	192	69	,	,	PUNCT
ejpam-5335	192	70	τ2)-open	τ2)-open	ADJ
ejpam-5335	192	71	set	set	ADJ
ejpam-5335	192	72	u	u	NOUN
ejpam-5335	192	73	of	of	ADP
ejpam-5335	192	74	x	x	PUNCT
ejpam-5335	192	75	containing	contain	VERB
ejpam-5335	192	76	x	x	PUNCT
ejpam-5335	192	77	such	such	ADJ
ejpam-5335	192	78	that	that	DET
ejpam-5335	192	79	f(u	f(u	PROPN
ejpam-5335	192	80	)	)	PUNCT
ejpam-5335	192	81	⊆	⊆	NUM
ejpam-5335	192	82	v	v	NOUN
ejpam-5335	192	83	.	.	PUNCT
ejpam-5335	193	1	proof	proof	NOUN
ejpam-5335	193	2	.	.	PUNCT
ejpam-5335	194	1	(	(	PUNCT
ejpam-5335	194	2	1	1	X
ejpam-5335	194	3	)	)	PUNCT
ejpam-5335	194	4	⇒	⇒	NOUN
ejpam-5335	194	5	(	(	PUNCT
ejpam-5335	194	6	2	2	NUM
ejpam-5335	194	7	):	):	PUNCT
ejpam-5335	194	8	let	let	VERB
ejpam-5335	194	9	v	v	PART
ejpam-5335	194	10	be	be	AUX
ejpam-5335	194	11	any	any	DET
ejpam-5335	194	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	194	13	-	-	ADJ
ejpam-5335	194	14	open	open	ADJ
ejpam-5335	194	15	set	set	NOUN
ejpam-5335	194	16	of	of	ADP
ejpam-5335	194	17	y	y	PROPN
ejpam-5335	194	18	containing	contain	VERB
ejpam-5335	194	19	f(x	f(x	PROPN
ejpam-5335	194	20	)	)	PUNCT
ejpam-5335	194	21	.	.	PUNCT
ejpam-5335	195	1	since	since	SCONJ
ejpam-5335	195	2	f	f	PROPN
ejpam-5335	195	3	is	be	AUX
ejpam-5335	195	4	almost	almost	ADV
ejpam-5335	195	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	195	6	,	,	PUNCT
ejpam-5335	195	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	195	8	at	at	ADP
ejpam-5335	195	9	x	x	SYM
ejpam-5335	195	10	∈	∈	ADJ
ejpam-5335	195	11	x.	x.	NOUN
ejpam-5335	195	12	there	there	PRON
ejpam-5335	195	13	exists	exist	VERB
ejpam-5335	195	14	a	a	DET
ejpam-5335	195	15	δ(τ1	δ(τ1	NOUN
ejpam-5335	195	16	,	,	PUNCT
ejpam-5335	195	17	τ2)-open	τ2)-open	ADJ
ejpam-5335	195	18	set	set	ADJ
ejpam-5335	195	19	u	u	NOUN
ejpam-5335	195	20	of	of	ADP
ejpam-5335	195	21	x	x	PUNCT
ejpam-5335	195	22	containing	contain	VERB
ejpam-5335	195	23	x	x	PUNCT
ejpam-5335	195	24	such	such	ADJ
ejpam-5335	195	25	that	that	DET
ejpam-5335	195	26	f(u	f(u	PROPN
ejpam-5335	195	27	)	)	PUNCT
ejpam-5335	195	28	⊆	⊆	NUM
ejpam-5335	195	29	σ1σ2	σ1σ2	X
ejpam-5335	195	30	-	-	PUNCT
ejpam-5335	195	31	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	195	32	-	-	PUNCT
ejpam-5335	195	33	cl(v	cl(v	NOUN
ejpam-5335	195	34	)	)	PUNCT
ejpam-5335	195	35	)	)	PUNCT
ejpam-5335	195	36	.	.	PUNCT
ejpam-5335	196	1	thus	thus	ADV
ejpam-5335	196	2	,	,	PUNCT
ejpam-5335	196	3	x	x	PUNCT
ejpam-5335	196	4	∈	∈	PROPN
ejpam-5335	196	5	u	u	NOUN
ejpam-5335	196	6	⊆	⊆	NUM
ejpam-5335	196	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	196	8	-	-	PUNCT
ejpam-5335	196	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	196	10	-	-	PUNCT
ejpam-5335	196	11	cl(v	cl(v	NOUN
ejpam-5335	196	12	)	)	PUNCT
ejpam-5335	196	13	)	)	PUNCT
ejpam-5335	196	14	)	)	PUNCT
ejpam-5335	196	15	.	.	PUNCT
ejpam-5335	197	1	therefore	therefore	ADV
ejpam-5335	197	2	,	,	PUNCT
ejpam-5335	197	3	x	x	PROPN
ejpam-5335	197	4	∈	∈	PROPN
ejpam-5335	197	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	197	6	,	,	PUNCT
ejpam-5335	197	7	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	197	8	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	197	9	-	-	PUNCT
ejpam-5335	197	10	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	197	11	-	-	PUNCT
ejpam-5335	197	12	cl(v	cl(v	NOUN
ejpam-5335	197	13	)	)	PUNCT
ejpam-5335	197	14	)	)	PUNCT
ejpam-5335	197	15	)	)	PUNCT
ejpam-5335	197	16	)	)	PUNCT
ejpam-5335	197	17	.	.	PUNCT
ejpam-5335	198	1	(	(	PUNCT
ejpam-5335	198	2	2	2	X
ejpam-5335	198	3	)	)	PUNCT
ejpam-5335	198	4	⇒	⇒	NOUN
ejpam-5335	198	5	(	(	PUNCT
ejpam-5335	198	6	3	3	NUM
ejpam-5335	198	7	):	):	PUNCT
ejpam-5335	198	8	let	let	VERB
ejpam-5335	198	9	v	v	PART
ejpam-5335	198	10	be	be	AUX
ejpam-5335	198	11	any	any	DET
ejpam-5335	198	12	(	(	PUNCT
ejpam-5335	198	13	σ1	σ1	NOUN
ejpam-5335	198	14	,	,	PUNCT
ejpam-5335	198	15	σ2)r	σ2)r	NOUN
ejpam-5335	198	16	-	-	PUNCT
ejpam-5335	198	17	open	open	ADJ
ejpam-5335	198	18	set	set	NOUN
ejpam-5335	198	19	of	of	ADP
ejpam-5335	198	20	y	y	PROPN
ejpam-5335	198	21	containing	contain	VERB
ejpam-5335	198	22	f(x	f(x	PROPN
ejpam-5335	198	23	)	)	PUNCT
ejpam-5335	198	24	.	.	PUNCT
ejpam-5335	199	1	then	then	ADV
ejpam-5335	199	2	,	,	PUNCT
ejpam-5335	199	3	v	v	NOUN
ejpam-5335	199	4	is	be	AUX
ejpam-5335	199	5	σ1σ2	σ1σ2	NOUN
ejpam-5335	199	6	-	-	ADJ
ejpam-5335	199	7	open	open	ADJ
ejpam-5335	199	8	in	in	ADP
ejpam-5335	199	9	y	y	PROPN
ejpam-5335	199	10	.	.	PUNCT
ejpam-5335	200	1	by	by	ADP
ejpam-5335	200	2	(	(	PUNCT
ejpam-5335	200	3	2	2	NUM
ejpam-5335	200	4	)	)	PUNCT
ejpam-5335	200	5	,	,	PUNCT
ejpam-5335	200	6	x	x	PROPN
ejpam-5335	200	7	∈	∈	PROPN
ejpam-5335	200	8	δ(τ1	δ(τ1	PROPN
ejpam-5335	200	9	,	,	PUNCT
ejpam-5335	200	10	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	200	11	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	200	12	-	-	PUNCT
ejpam-5335	200	13	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	200	14	-	-	PUNCT
ejpam-5335	200	15	cl(v	cl(v	NOUN
ejpam-5335	200	16	)	)	PUNCT
ejpam-5335	200	17	)	)	PUNCT
ejpam-5335	200	18	)	)	PUNCT
ejpam-5335	200	19	)	)	PUNCT
ejpam-5335	201	1	=	=	SYM
ejpam-5335	201	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	201	3	,	,	PUNCT
ejpam-5335	201	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	201	5	−1(v	−1(v	PROPN
ejpam-5335	201	6	)	)	PUNCT
ejpam-5335	201	7	)	)	PUNCT
ejpam-5335	201	8	.	.	PUNCT
ejpam-5335	202	1	(	(	PUNCT
ejpam-5335	202	2	3	3	X
ejpam-5335	202	3	)	)	PUNCT
ejpam-5335	202	4	⇒	⇒	NOUN
ejpam-5335	202	5	(	(	PUNCT
ejpam-5335	202	6	4	4	NUM
ejpam-5335	202	7	):	):	PUNCT
ejpam-5335	202	8	let	let	VERB
ejpam-5335	202	9	v	v	PART
ejpam-5335	202	10	be	be	AUX
ejpam-5335	202	11	any	any	DET
ejpam-5335	202	12	(	(	PUNCT
ejpam-5335	202	13	σ1	σ1	NOUN
ejpam-5335	202	14	,	,	PUNCT
ejpam-5335	202	15	σ2)r	σ2)r	NOUN
ejpam-5335	202	16	-	-	PUNCT
ejpam-5335	202	17	open	open	ADJ
ejpam-5335	202	18	set	set	NOUN
ejpam-5335	202	19	of	of	ADP
ejpam-5335	202	20	y	y	PROPN
ejpam-5335	202	21	containing	contain	VERB
ejpam-5335	202	22	f(x	f(x	PROPN
ejpam-5335	202	23	)	)	PUNCT
ejpam-5335	202	24	.	.	PUNCT
ejpam-5335	203	1	thus	thus	ADV
ejpam-5335	203	2	by	by	ADP
ejpam-5335	203	3	(	(	PUNCT
ejpam-5335	203	4	3	3	NUM
ejpam-5335	203	5	)	)	PUNCT
ejpam-5335	203	6	,	,	PUNCT
ejpam-5335	203	7	we	we	PRON
ejpam-5335	203	8	have	have	VERB
ejpam-5335	203	9	x	x	PROPN
ejpam-5335	203	10	∈	∈	PROPN
ejpam-5335	203	11	δ(τ1	δ(τ1	NOUN
ejpam-5335	203	12	,	,	PUNCT
ejpam-5335	203	13	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	203	14	−1(v	−1(v	PROPN
ejpam-5335	203	15	)	)	PUNCT
ejpam-5335	203	16	)	)	PUNCT
ejpam-5335	203	17	.	.	PUNCT
ejpam-5335	204	1	then	then	ADV
ejpam-5335	204	2	,	,	PUNCT
ejpam-5335	204	3	there	there	PRON
ejpam-5335	204	4	exists	exist	VERB
ejpam-5335	204	5	a	a	DET
ejpam-5335	204	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	204	7	,	,	PUNCT
ejpam-5335	204	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	204	9	set	set	ADJ
ejpam-5335	204	10	u	u	NOUN
ejpam-5335	204	11	of	of	ADP
ejpam-5335	204	12	x	x	SYM
ejpam-5335	204	13	such	such	ADJ
ejpam-5335	204	14	that	that	SCONJ
ejpam-5335	204	15	x	x	SYM
ejpam-5335	204	16	∈	∈	PROPN
ejpam-5335	204	17	u	u	NOUN
ejpam-5335	204	18	⊆	⊆	NUM
ejpam-5335	204	19	f−1(v	f−1(v	NOUN
ejpam-5335	204	20	)	)	PUNCT
ejpam-5335	204	21	.	.	PUNCT
ejpam-5335	205	1	therefore	therefore	ADV
ejpam-5335	205	2	,	,	PUNCT
ejpam-5335	205	3	f(u	f(u	PROPN
ejpam-5335	205	4	)	)	PUNCT
ejpam-5335	205	5	⊆	⊆	NUM
ejpam-5335	205	6	v	v	NOUN
ejpam-5335	205	7	.	.	PUNCT
ejpam-5335	206	1	(	(	PUNCT
ejpam-5335	206	2	4	4	X
ejpam-5335	206	3	)	)	PUNCT
ejpam-5335	206	4	⇒	⇒	NOUN
ejpam-5335	206	5	(	(	PUNCT
ejpam-5335	206	6	1	1	NUM
ejpam-5335	206	7	):	):	PUNCT
ejpam-5335	206	8	let	let	VERB
ejpam-5335	206	9	x	x	PUNCT
ejpam-5335	206	10	∈	∈	PROPN
ejpam-5335	206	11	x	x	X
ejpam-5335	206	12	and	and	CCONJ
ejpam-5335	206	13	v	v	X
ejpam-5335	206	14	be	be	AUX
ejpam-5335	206	15	any	any	DET
ejpam-5335	206	16	σ1σ2	σ1σ2	NOUN
ejpam-5335	206	17	-	-	ADJ
ejpam-5335	206	18	open	open	ADJ
ejpam-5335	206	19	set	set	NOUN
ejpam-5335	206	20	of	of	ADP
ejpam-5335	206	21	y	y	PROPN
ejpam-5335	206	22	containing	contain	VERB
ejpam-5335	206	23	f(x	f(x	PROPN
ejpam-5335	206	24	)	)	PUNCT
ejpam-5335	206	25	.	.	PUNCT
ejpam-5335	207	1	since	since	SCONJ
ejpam-5335	207	2	σ1σ2	σ1σ2	NOUN
ejpam-5335	207	3	-	-	PUNCT
ejpam-5335	207	4	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	207	5	-	-	PUNCT
ejpam-5335	207	6	cl(v	cl(v	NOUN
ejpam-5335	207	7	)	)	PUNCT
ejpam-5335	207	8	)	)	PUNCT
ejpam-5335	207	9	is	be	AUX
ejpam-5335	207	10	a	a	DET
ejpam-5335	207	11	(	(	PUNCT
ejpam-5335	207	12	σ1	σ1	NOUN
ejpam-5335	207	13	,	,	PUNCT
ejpam-5335	207	14	σ2)r	σ2)r	NOUN
ejpam-5335	207	15	-	-	PUNCT
ejpam-5335	207	16	open	open	ADJ
ejpam-5335	207	17	set	set	NOUN
ejpam-5335	207	18	and	and	CCONJ
ejpam-5335	207	19	by	by	ADP
ejpam-5335	207	20	(	(	PUNCT
ejpam-5335	207	21	4	4	NUM
ejpam-5335	207	22	)	)	PUNCT
ejpam-5335	207	23	,	,	PUNCT
ejpam-5335	207	24	there	there	PRON
ejpam-5335	207	25	exists	exist	VERB
ejpam-5335	207	26	a	a	DET
ejpam-5335	207	27	δ(τ1	δ(τ1	NOUN
ejpam-5335	207	28	,	,	PUNCT
ejpam-5335	207	29	τ2)-open	τ2)-open	ADJ
ejpam-5335	207	30	set	set	ADJ
ejpam-5335	207	31	u	u	NOUN
ejpam-5335	207	32	of	of	ADP
ejpam-5335	207	33	x	x	PUNCT
ejpam-5335	207	34	containing	contain	VERB
ejpam-5335	207	35	x	x	PUNCT
ejpam-5335	207	36	such	such	ADJ
ejpam-5335	207	37	that	that	DET
ejpam-5335	207	38	f(u	f(u	PROPN
ejpam-5335	207	39	)	)	PUNCT
ejpam-5335	207	40	⊆	⊆	NUM
ejpam-5335	207	41	σ1σ2	σ1σ2	X
ejpam-5335	207	42	-	-	PUNCT
ejpam-5335	207	43	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	207	44	-	-	PUNCT
ejpam-5335	207	45	cl(v	cl(v	NOUN
ejpam-5335	207	46	)	)	PUNCT
ejpam-5335	207	47	)	)	PUNCT
ejpam-5335	207	48	.	.	PUNCT
ejpam-5335	208	1	consequently	consequently	ADV
ejpam-5335	208	2	,	,	PUNCT
ejpam-5335	208	3	f	f	PROPN
ejpam-5335	208	4	is	be	AUX
ejpam-5335	208	5	almost	almost	ADV
ejpam-5335	208	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	208	7	,	,	PUNCT
ejpam-5335	208	8	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	208	9	at	at	ADP
ejpam-5335	208	10	x.	x.	NOUN
ejpam-5335	208	11	theorem	theorem	VERB
ejpam-5335	208	12	4	4	NUM
ejpam-5335	208	13	.	.	PUNCT
ejpam-5335	208	14	for	for	ADP
ejpam-5335	208	15	a	a	DET
ejpam-5335	208	16	function	function	NOUN
ejpam-5335	208	17	f	f	NOUN
ejpam-5335	208	18	:	:	PUNCT
ejpam-5335	208	19	(	(	PUNCT
ejpam-5335	208	20	x	x	NOUN
ejpam-5335	208	21	,	,	PUNCT
ejpam-5335	208	22	τ1	τ1	NOUN
ejpam-5335	208	23	,	,	PUNCT
ejpam-5335	208	24	τ2	τ2	NOUN
ejpam-5335	208	25	)	)	PUNCT
ejpam-5335	208	26	→	→	SYM
ejpam-5335	208	27	(	(	PUNCT
ejpam-5335	208	28	y	y	PROPN
ejpam-5335	208	29	,	,	PUNCT
ejpam-5335	208	30	σ1	σ1	PROPN
ejpam-5335	208	31	,	,	PUNCT
ejpam-5335	208	32	σ2	σ2	NOUN
ejpam-5335	208	33	)	)	PUNCT
ejpam-5335	208	34	,	,	PUNCT
ejpam-5335	208	35	the	the	DET
ejpam-5335	208	36	following	follow	VERB
ejpam-5335	208	37	properties	property	NOUN
ejpam-5335	208	38	are	be	AUX
ejpam-5335	208	39	equivalent	equivalent	ADJ
ejpam-5335	208	40	:	:	PUNCT
ejpam-5335	208	41	c.	c.	NOUN
ejpam-5335	208	42	prachanpol	prachanpol	NOUN
ejpam-5335	208	43	,	,	PUNCT
ejpam-5335	208	44	c.	c.	PROPN
ejpam-5335	208	45	boonpok	boonpok	PROPN
ejpam-5335	208	46	,	,	PUNCT
ejpam-5335	208	47	c.	c.	PROPN
ejpam-5335	208	48	viriyapong	viriyapong	PROPN
ejpam-5335	208	49	/	/	SYM
ejpam-5335	208	50	eur	eur	PROPN
ejpam-5335	208	51	.	.	PUNCT
ejpam-5335	209	1	j.	j.	PROPN
ejpam-5335	209	2	pure	pure	PROPN
ejpam-5335	209	3	appl	appl	PROPN
ejpam-5335	209	4	.	.	PROPN
ejpam-5335	209	5	math	math	PROPN
ejpam-5335	209	6	,	,	PUNCT
ejpam-5335	209	7	17	17	NUM
ejpam-5335	209	8	(	(	PUNCT
ejpam-5335	209	9	4	4	NUM
ejpam-5335	209	10	)	)	PUNCT
ejpam-5335	209	11	(	(	PUNCT
ejpam-5335	209	12	2024	2024	NUM
ejpam-5335	209	13	)	)	PUNCT
ejpam-5335	209	14	,	,	PUNCT
ejpam-5335	209	15	3730	3730	NUM
ejpam-5335	209	16	-	-	SYM
ejpam-5335	209	17	3742	3742	NUM
ejpam-5335	209	18	3736	3736	NUM
ejpam-5335	209	19	(	(	PUNCT
ejpam-5335	209	20	1	1	X
ejpam-5335	209	21	)	)	PUNCT
ejpam-5335	209	22	f	f	NOUN
ejpam-5335	209	23	is	be	AUX
ejpam-5335	209	24	almost	almost	ADV
ejpam-5335	209	25	δ(τ1	δ(τ1	NOUN
ejpam-5335	209	26	,	,	PUNCT
ejpam-5335	209	27	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	209	28	;	;	PUNCT
ejpam-5335	209	29	(	(	PUNCT
ejpam-5335	209	30	2	2	X
ejpam-5335	209	31	)	)	PUNCT
ejpam-5335	209	32	f−1(v	f−1(v	NOUN
ejpam-5335	209	33	)	)	PUNCT
ejpam-5335	210	1	⊆	⊆	NUM
ejpam-5335	210	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	210	3	,	,	PUNCT
ejpam-5335	210	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	210	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	210	6	-	-	PUNCT
ejpam-5335	210	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	210	8	-	-	PUNCT
ejpam-5335	210	9	cl(v	cl(v	NOUN
ejpam-5335	210	10	)	)	PUNCT
ejpam-5335	210	11	)	)	PUNCT
ejpam-5335	210	12	)	)	PUNCT
ejpam-5335	210	13	)	)	PUNCT
ejpam-5335	210	14	for	for	ADP
ejpam-5335	210	15	every	every	DET
ejpam-5335	210	16	σ1σ2	σ1σ2	NOUN
ejpam-5335	210	17	-	-	ADJ
ejpam-5335	210	18	open	open	ADJ
ejpam-5335	210	19	set	set	NOUN
ejpam-5335	210	20	v	v	NOUN
ejpam-5335	210	21	of	of	ADP
ejpam-5335	210	22	y	y	PROPN
ejpam-5335	210	23	;	;	PUNCT
ejpam-5335	210	24	(	(	PUNCT
ejpam-5335	210	25	3	3	X
ejpam-5335	210	26	)	)	PUNCT
ejpam-5335	210	27	δ(τ1	δ(τ1	NOUN
ejpam-5335	210	28	,	,	PUNCT
ejpam-5335	210	29	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	210	30	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	210	31	-	-	PUNCT
ejpam-5335	210	32	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	210	33	-	-	PUNCT
ejpam-5335	210	34	int(f	int(f	PROPN
ejpam-5335	210	35	)	)	PUNCT
ejpam-5335	210	36	)	)	PUNCT
ejpam-5335	210	37	)	)	PUNCT
ejpam-5335	210	38	)	)	PUNCT
ejpam-5335	211	1	⊆	⊆	NUM
ejpam-5335	211	2	f−1(f	f−1(f	PROPN
ejpam-5335	211	3	)	)	PUNCT
ejpam-5335	211	4	for	for	ADP
ejpam-5335	211	5	every	every	DET
ejpam-5335	211	6	σ1σ2	σ1σ2	NUM
ejpam-5335	211	7	-	-	PUNCT
ejpam-5335	211	8	closed	closed	ADJ
ejpam-5335	211	9	set	set	ADJ
ejpam-5335	211	10	f	f	PROPN
ejpam-5335	211	11	of	of	ADP
ejpam-5335	211	12	y	y	PROPN
ejpam-5335	211	13	;	;	PUNCT
ejpam-5335	211	14	(	(	PUNCT
ejpam-5335	211	15	4	4	X
ejpam-5335	211	16	)	)	PUNCT
ejpam-5335	211	17	δ(τ1	δ(τ1	NOUN
ejpam-5335	211	18	,	,	PUNCT
ejpam-5335	211	19	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	211	20	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	211	21	-	-	PUNCT
ejpam-5335	211	22	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	211	23	-	-	PUNCT
ejpam-5335	211	24	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	211	25	-	-	PUNCT
ejpam-5335	211	26	cl(b	cl(b	NOUN
ejpam-5335	211	27	)	)	PUNCT
ejpam-5335	211	28	)	)	PUNCT
ejpam-5335	211	29	)	)	PUNCT
ejpam-5335	211	30	)	)	PUNCT
ejpam-5335	211	31	)	)	PUNCT
ejpam-5335	212	1	⊆	⊆	NUM
ejpam-5335	212	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	212	3	-	-	PUNCT
ejpam-5335	212	4	cl(b	cl(b	NOUN
ejpam-5335	212	5	)	)	PUNCT
ejpam-5335	212	6	)	)	PUNCT
ejpam-5335	212	7	for	for	ADP
ejpam-5335	212	8	every	every	DET
ejpam-5335	212	9	subset	subset	NOUN
ejpam-5335	212	10	b	b	PROPN
ejpam-5335	212	11	of	of	ADP
ejpam-5335	212	12	y	y	PROPN
ejpam-5335	212	13	;	;	PUNCT
ejpam-5335	212	14	(	(	PUNCT
ejpam-5335	212	15	5	5	X
ejpam-5335	212	16	)	)	PUNCT
ejpam-5335	212	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	212	18	-	-	PUNCT
ejpam-5335	212	19	int(b	int(b	NOUN
ejpam-5335	212	20	)	)	PUNCT
ejpam-5335	212	21	)	)	PUNCT
ejpam-5335	212	22	⊆	⊆	NUM
ejpam-5335	212	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	212	24	,	,	PUNCT
ejpam-5335	212	25	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	212	26	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	212	27	-	-	PUNCT
ejpam-5335	212	28	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	212	29	-	-	PUNCT
ejpam-5335	212	30	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	212	31	-	-	PUNCT
ejpam-5335	212	32	int(b	int(b	NOUN
ejpam-5335	212	33	)	)	PUNCT
ejpam-5335	212	34	)	)	PUNCT
ejpam-5335	212	35	)	)	PUNCT
ejpam-5335	212	36	)	)	PUNCT
ejpam-5335	212	37	)	)	PUNCT
ejpam-5335	212	38	for	for	ADP
ejpam-5335	212	39	every	every	DET
ejpam-5335	212	40	subset	subset	NOUN
ejpam-5335	212	41	b	b	PROPN
ejpam-5335	212	42	of	of	ADP
ejpam-5335	212	43	y	y	PROPN
ejpam-5335	212	44	;	;	PUNCT
ejpam-5335	212	45	(	(	PUNCT
ejpam-5335	212	46	6	6	X
ejpam-5335	212	47	)	)	PUNCT
ejpam-5335	212	48	f−1(v	f−1(v	NOUN
ejpam-5335	212	49	)	)	PUNCT
ejpam-5335	212	50	is	be	AUX
ejpam-5335	212	51	δ(τ1	δ(τ1	NOUN
ejpam-5335	212	52	,	,	PUNCT
ejpam-5335	212	53	τ2)-open	τ2)-open	ADJ
ejpam-5335	212	54	in	in	ADP
ejpam-5335	212	55	x	x	PUNCT
ejpam-5335	212	56	for	for	ADP
ejpam-5335	212	57	every	every	DET
ejpam-5335	212	58	(	(	PUNCT
ejpam-5335	212	59	σ1	σ1	PROPN
ejpam-5335	212	60	,	,	PUNCT
ejpam-5335	212	61	σ2)r	σ2)r	NOUN
ejpam-5335	212	62	-	-	PUNCT
ejpam-5335	212	63	open	open	ADJ
ejpam-5335	212	64	set	set	VERB
ejpam-5335	212	65	v	v	NOUN
ejpam-5335	212	66	of	of	ADP
ejpam-5335	212	67	y	y	PROPN
ejpam-5335	212	68	;	;	PUNCT
ejpam-5335	212	69	(	(	PUNCT
ejpam-5335	212	70	7	7	X
ejpam-5335	212	71	)	)	PUNCT
ejpam-5335	212	72	f−1(f	f−1(f	NOUN
ejpam-5335	212	73	)	)	PUNCT
ejpam-5335	212	74	is	be	AUX
ejpam-5335	212	75	δ(τ1	δ(τ1	NOUN
ejpam-5335	212	76	,	,	PUNCT
ejpam-5335	212	77	τ2)-closed	τ2)-close	VERB
ejpam-5335	212	78	in	in	ADP
ejpam-5335	212	79	x	x	PUNCT
ejpam-5335	212	80	for	for	ADP
ejpam-5335	212	81	every	every	DET
ejpam-5335	212	82	(	(	PUNCT
ejpam-5335	212	83	σ1	σ1	PROPN
ejpam-5335	212	84	,	,	PUNCT
ejpam-5335	212	85	σ2)r	σ2)r	NOUN
ejpam-5335	212	86	-	-	PUNCT
ejpam-5335	212	87	closed	close	VERB
ejpam-5335	212	88	set	set	ADJ
ejpam-5335	212	89	f	f	PROPN
ejpam-5335	212	90	of	of	ADP
ejpam-5335	212	91	y	y	PROPN
ejpam-5335	212	92	.	.	PUNCT
ejpam-5335	213	1	proof	proof	NOUN
ejpam-5335	213	2	.	.	PUNCT
ejpam-5335	214	1	(	(	PUNCT
ejpam-5335	214	2	1	1	X
ejpam-5335	214	3	)	)	PUNCT
ejpam-5335	214	4	⇒	⇒	NOUN
ejpam-5335	214	5	(	(	PUNCT
ejpam-5335	214	6	2	2	NUM
ejpam-5335	214	7	):	):	PUNCT
ejpam-5335	214	8	let	let	VERB
ejpam-5335	214	9	v	v	PART
ejpam-5335	214	10	be	be	AUX
ejpam-5335	214	11	any	any	DET
ejpam-5335	214	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	214	13	-	-	ADJ
ejpam-5335	214	14	open	open	ADJ
ejpam-5335	214	15	set	set	NOUN
ejpam-5335	214	16	of	of	ADP
ejpam-5335	214	17	y	y	PROPN
ejpam-5335	214	18	and	and	CCONJ
ejpam-5335	214	19	x	x	PROPN
ejpam-5335	214	20	∈	∈	PROPN
ejpam-5335	214	21	f−1(v	f−1(v	NOUN
ejpam-5335	214	22	)	)	PUNCT
ejpam-5335	214	23	.	.	PUNCT
ejpam-5335	215	1	since	since	SCONJ
ejpam-5335	215	2	f	f	PROPN
ejpam-5335	215	3	is	be	AUX
ejpam-5335	215	4	almost	almost	ADV
ejpam-5335	215	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	215	6	,	,	PUNCT
ejpam-5335	215	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	215	8	,	,	PUNCT
ejpam-5335	215	9	there	there	PRON
ejpam-5335	215	10	exists	exist	VERB
ejpam-5335	215	11	a	a	DET
ejpam-5335	215	12	δ(τ1	δ(τ1	NOUN
ejpam-5335	215	13	,	,	PUNCT
ejpam-5335	215	14	τ2)-open	τ2)-open	ADJ
ejpam-5335	215	15	set	set	ADJ
ejpam-5335	215	16	u	u	NOUN
ejpam-5335	215	17	of	of	ADP
ejpam-5335	215	18	x	x	PUNCT
ejpam-5335	215	19	containing	contain	VERB
ejpam-5335	215	20	x	x	PUNCT
ejpam-5335	215	21	such	such	ADJ
ejpam-5335	215	22	that	that	DET
ejpam-5335	215	23	f(u	f(u	PROPN
ejpam-5335	215	24	)	)	PUNCT
ejpam-5335	215	25	⊆	⊆	NUM
ejpam-5335	215	26	σ1σ2	σ1σ2	X
ejpam-5335	215	27	-	-	PUNCT
ejpam-5335	215	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	215	29	-	-	PUNCT
ejpam-5335	215	30	cl(v	cl(v	NOUN
ejpam-5335	215	31	)	)	PUNCT
ejpam-5335	215	32	)	)	PUNCT
ejpam-5335	215	33	.	.	PUNCT
ejpam-5335	216	1	then	then	ADV
ejpam-5335	216	2	,	,	PUNCT
ejpam-5335	216	3	we	we	PRON
ejpam-5335	216	4	have	have	VERB
ejpam-5335	216	5	x	x	X
ejpam-5335	216	6	∈	∈	PROPN
ejpam-5335	216	7	u	u	NOUN
ejpam-5335	216	8	⊆	⊆	NUM
ejpam-5335	216	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	216	10	-	-	PUNCT
ejpam-5335	216	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	216	12	-	-	PUNCT
ejpam-5335	216	13	cl(v	cl(v	NOUN
ejpam-5335	216	14	)	)	PUNCT
ejpam-5335	216	15	)	)	PUNCT
ejpam-5335	216	16	)	)	PUNCT
ejpam-5335	217	1	and	and	CCONJ
ejpam-5335	217	2	hence	hence	ADV
ejpam-5335	217	3	x	x	X
ejpam-5335	217	4	∈	∈	PROPN
ejpam-5335	217	5	δ(τ1	δ(τ1	PROPN
ejpam-5335	217	6	,	,	PUNCT
ejpam-5335	217	7	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	217	8	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	217	9	-	-	PUNCT
ejpam-5335	217	10	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	217	11	-	-	PUNCT
ejpam-5335	217	12	cl(v	cl(v	NOUN
ejpam-5335	217	13	)	)	PUNCT
ejpam-5335	217	14	)	)	PUNCT
ejpam-5335	217	15	)	)	PUNCT
ejpam-5335	217	16	)	)	PUNCT
ejpam-5335	217	17	.	.	PUNCT
ejpam-5335	218	1	this	this	PRON
ejpam-5335	218	2	implies	imply	VERB
ejpam-5335	218	3	that	that	DET
ejpam-5335	218	4	f−1(v	f−1(v	PROPN
ejpam-5335	218	5	)	)	PUNCT
ejpam-5335	219	1	⊆	⊆	NUM
ejpam-5335	219	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	219	3	,	,	PUNCT
ejpam-5335	219	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	219	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	219	6	-	-	PUNCT
ejpam-5335	219	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	219	8	-	-	PUNCT
ejpam-5335	219	9	cl(v	cl(v	NOUN
ejpam-5335	219	10	)	)	PUNCT
ejpam-5335	219	11	)	)	PUNCT
ejpam-5335	219	12	)	)	PUNCT
ejpam-5335	219	13	)	)	PUNCT
ejpam-5335	219	14	.	.	PUNCT
ejpam-5335	220	1	(	(	PUNCT
ejpam-5335	220	2	2	2	X
ejpam-5335	220	3	)	)	PUNCT
ejpam-5335	220	4	⇒	⇒	NOUN
ejpam-5335	220	5	(	(	PUNCT
ejpam-5335	220	6	3	3	NUM
ejpam-5335	220	7	):	):	PUNCT
ejpam-5335	220	8	let	let	VERB
ejpam-5335	220	9	f	f	PRON
ejpam-5335	220	10	be	be	AUX
ejpam-5335	220	11	any	any	DET
ejpam-5335	220	12	σ1σ2	σ1σ2	NUM
ejpam-5335	220	13	-	-	PUNCT
ejpam-5335	220	14	closed	closed	ADJ
ejpam-5335	220	15	set	set	NOUN
ejpam-5335	220	16	of	of	ADP
ejpam-5335	220	17	y	y	PROPN
ejpam-5335	220	18	.	.	PUNCT
ejpam-5335	221	1	then	then	ADV
ejpam-5335	221	2	,	,	PUNCT
ejpam-5335	221	3	y	y	PROPN
ejpam-5335	221	4	−f	−f	PROPN
ejpam-5335	221	5	is	be	AUX
ejpam-5335	221	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	221	7	-	-	ADJ
ejpam-5335	221	8	open	open	ADJ
ejpam-5335	221	9	in	in	ADP
ejpam-5335	221	10	y	y	PROPN
ejpam-5335	221	11	.	.	PUNCT
ejpam-5335	222	1	thus	thus	ADV
ejpam-5335	222	2	by	by	ADP
ejpam-5335	222	3	(	(	PUNCT
ejpam-5335	222	4	2	2	NUM
ejpam-5335	222	5	)	)	PUNCT
ejpam-5335	222	6	,	,	PUNCT
ejpam-5335	222	7	we	we	PRON
ejpam-5335	222	8	have	have	VERB
ejpam-5335	222	9	x−	x−	PROPN
ejpam-5335	222	10	f−1(f	f−1(f	PROPN
ejpam-5335	222	11	)	)	PUNCT
ejpam-5335	223	1	=	=	PUNCT
ejpam-5335	223	2	f−1(y	f−1(y	PROPN
ejpam-5335	223	3	−f	−f	PROPN
ejpam-5335	223	4	)	)	PUNCT
ejpam-5335	223	5	⊆	⊆	NUM
ejpam-5335	223	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	223	7	,	,	PUNCT
ejpam-5335	223	8	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	223	9	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	223	10	-	-	PUNCT
ejpam-5335	223	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	223	12	-	-	PUNCT
ejpam-5335	223	13	cl(y	cl(y	NOUN
ejpam-5335	223	14	−f	−f	NOUN
ejpam-5335	223	15	)	)	PUNCT
ejpam-5335	223	16	)	)	PUNCT
ejpam-5335	223	17	)	)	PUNCT
ejpam-5335	223	18	)	)	PUNCT
ejpam-5335	224	1	=	=	SYM
ejpam-5335	224	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	224	3	,	,	PUNCT
ejpam-5335	224	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	224	5	−1(y−σ1σ2	−1(y−σ1σ2	NOUN
ejpam-5335	224	6	-	-	PUNCT
ejpam-5335	224	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	224	8	-	-	PUNCT
ejpam-5335	224	9	int(f	int(f	PROPN
ejpam-5335	224	10	)	)	PUNCT
ejpam-5335	224	11	)	)	PUNCT
ejpam-5335	224	12	)	)	PUNCT
ejpam-5335	224	13	)	)	PUNCT
ejpam-5335	225	1	=	=	SYM
ejpam-5335	225	2	x−δ(τ1	x−δ(τ1	PROPN
ejpam-5335	225	3	,	,	PUNCT
ejpam-5335	225	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	225	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	225	6	-	-	PUNCT
ejpam-5335	225	7	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	225	8	-	-	PUNCT
ejpam-5335	225	9	int(f	int(f	PROPN
ejpam-5335	225	10	)	)	PUNCT
ejpam-5335	225	11	)	)	PUNCT
ejpam-5335	225	12	)	)	PUNCT
ejpam-5335	225	13	)	)	PUNCT
ejpam-5335	225	14	and	and	CCONJ
ejpam-5335	225	15	hence	hence	ADV
ejpam-5335	225	16	δ(τ1	δ(τ1	PROPN
ejpam-5335	225	17	,	,	PUNCT
ejpam-5335	225	18	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	225	19	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	225	20	-	-	PUNCT
ejpam-5335	225	21	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	225	22	-	-	PUNCT
ejpam-5335	225	23	int(f	int(f	PROPN
ejpam-5335	225	24	)	)	PUNCT
ejpam-5335	225	25	)	)	PUNCT
ejpam-5335	225	26	)	)	PUNCT
ejpam-5335	225	27	)	)	PUNCT
ejpam-5335	226	1	⊆	⊆	NUM
ejpam-5335	226	2	f−1(f	f−1(f	NOUN
ejpam-5335	226	3	)	)	PUNCT
ejpam-5335	226	4	.	.	PUNCT
ejpam-5335	227	1	(	(	PUNCT
ejpam-5335	227	2	3	3	X
ejpam-5335	227	3	)	)	PUNCT
ejpam-5335	227	4	⇒	⇒	NOUN
ejpam-5335	227	5	(	(	PUNCT
ejpam-5335	227	6	4	4	NUM
ejpam-5335	227	7	):	):	PUNCT
ejpam-5335	227	8	let	let	VERB
ejpam-5335	227	9	b	b	X
ejpam-5335	227	10	be	be	AUX
ejpam-5335	227	11	any	any	DET
ejpam-5335	227	12	subset	subset	NOUN
ejpam-5335	227	13	of	of	ADP
ejpam-5335	227	14	y	y	PROPN
ejpam-5335	227	15	.	.	PUNCT
ejpam-5335	228	1	then	then	ADV
ejpam-5335	228	2	,	,	PUNCT
ejpam-5335	228	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	228	4	-	-	NOUN
ejpam-5335	228	5	cl(b	cl(b	NOUN
ejpam-5335	228	6	)	)	PUNCT
ejpam-5335	228	7	is	be	AUX
ejpam-5335	228	8	σ1σ2	σ1σ2	NOUN
ejpam-5335	228	9	-	-	ADJ
ejpam-5335	228	10	closed	closed	ADJ
ejpam-5335	228	11	in	in	ADP
ejpam-5335	228	12	y	y	PROPN
ejpam-5335	228	13	and	and	CCONJ
ejpam-5335	228	14	by	by	ADP
ejpam-5335	228	15	(	(	PUNCT
ejpam-5335	228	16	3	3	NUM
ejpam-5335	228	17	)	)	PUNCT
ejpam-5335	228	18	,	,	PUNCT
ejpam-5335	228	19	δ(τ1	δ(τ1	PROPN
ejpam-5335	228	20	,	,	PUNCT
ejpam-5335	228	21	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	228	22	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	228	23	-	-	PUNCT
ejpam-5335	228	24	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	228	25	-	-	PUNCT
ejpam-5335	228	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	228	27	-	-	PUNCT
ejpam-5335	228	28	cl(b	cl(b	NOUN
ejpam-5335	228	29	)	)	PUNCT
ejpam-5335	228	30	)	)	PUNCT
ejpam-5335	228	31	)	)	PUNCT
ejpam-5335	228	32	)	)	PUNCT
ejpam-5335	228	33	)	)	PUNCT
ejpam-5335	229	1	⊆	⊆	NUM
ejpam-5335	229	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	229	3	-	-	PUNCT
ejpam-5335	229	4	cl(b	cl(b	NOUN
ejpam-5335	229	5	)	)	PUNCT
ejpam-5335	229	6	)	)	PUNCT
ejpam-5335	229	7	.	.	PUNCT
ejpam-5335	230	1	(	(	PUNCT
ejpam-5335	230	2	4	4	X
ejpam-5335	230	3	)	)	PUNCT
ejpam-5335	230	4	⇒	⇒	NOUN
ejpam-5335	230	5	(	(	PUNCT
ejpam-5335	230	6	5	5	NUM
ejpam-5335	230	7	):	):	PUNCT
ejpam-5335	230	8	let	let	VERB
ejpam-5335	230	9	b	b	X
ejpam-5335	230	10	be	be	AUX
ejpam-5335	230	11	any	any	DET
ejpam-5335	230	12	subset	subset	NOUN
ejpam-5335	230	13	of	of	ADP
ejpam-5335	230	14	y	y	PROPN
ejpam-5335	230	15	.	.	PUNCT
ejpam-5335	231	1	then	then	ADV
ejpam-5335	231	2	by	by	ADP
ejpam-5335	231	3	(	(	PUNCT
ejpam-5335	231	4	4	4	NUM
ejpam-5335	231	5	)	)	PUNCT
ejpam-5335	231	6	,	,	PUNCT
ejpam-5335	231	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	231	8	-	-	PUNCT
ejpam-5335	231	9	int(b	int(b	NOUN
ejpam-5335	231	10	)	)	PUNCT
ejpam-5335	231	11	)	)	PUNCT
ejpam-5335	232	1	=	=	PUNCT
ejpam-5335	233	1	x	x	X
ejpam-5335	233	2	−	−	PRON
ejpam-5335	233	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	233	4	-	-	PUNCT
ejpam-5335	233	5	cl(y	cl(y	NOUN
ejpam-5335	233	6	−b	−b	NOUN
ejpam-5335	233	7	)	)	PUNCT
ejpam-5335	233	8	)	)	PUNCT
ejpam-5335	234	1	⊆	⊆	NUM
ejpam-5335	234	2	x	x	SYM
ejpam-5335	234	3	−	−	NOUN
ejpam-5335	234	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	234	5	,	,	PUNCT
ejpam-5335	234	6	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	234	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	234	8	-	-	PUNCT
ejpam-5335	234	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	234	10	-	-	PUNCT
ejpam-5335	234	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	234	12	-	-	PUNCT
ejpam-5335	234	13	cl(y	cl(y	NOUN
ejpam-5335	234	14	−b	−b	NOUN
ejpam-5335	234	15	)	)	PUNCT
ejpam-5335	234	16	)	)	PUNCT
ejpam-5335	234	17	)	)	PUNCT
ejpam-5335	234	18	)	)	PUNCT
ejpam-5335	234	19	)	)	PUNCT
ejpam-5335	235	1	=	=	SYM
ejpam-5335	235	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	235	3	,	,	PUNCT
ejpam-5335	235	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	235	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	235	6	-	-	PUNCT
ejpam-5335	235	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	235	8	-	-	PUNCT
ejpam-5335	235	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	235	10	-	-	PUNCT
ejpam-5335	235	11	int(b	int(b	NOUN
ejpam-5335	235	12	)	)	PUNCT
ejpam-5335	235	13	)	)	PUNCT
ejpam-5335	235	14	)	)	PUNCT
ejpam-5335	235	15	)	)	PUNCT
ejpam-5335	235	16	)	)	PUNCT
ejpam-5335	235	17	.	.	PUNCT
ejpam-5335	236	1	therefore	therefore	ADV
ejpam-5335	236	2	,	,	PUNCT
ejpam-5335	236	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	236	4	-	-	PUNCT
ejpam-5335	236	5	int(b	int(b	NOUN
ejpam-5335	236	6	)	)	PUNCT
ejpam-5335	236	7	)	)	PUNCT
ejpam-5335	237	1	⊆	⊆	NUM
ejpam-5335	237	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	237	3	,	,	PUNCT
ejpam-5335	237	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	237	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	237	6	-	-	PUNCT
ejpam-5335	237	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	237	8	-	-	PUNCT
ejpam-5335	237	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	237	10	-	-	PUNCT
ejpam-5335	237	11	int(b	int(b	NOUN
ejpam-5335	237	12	)	)	PUNCT
ejpam-5335	237	13	)	)	PUNCT
ejpam-5335	237	14	)	)	PUNCT
ejpam-5335	237	15	)	)	PUNCT
ejpam-5335	237	16	)	)	PUNCT
ejpam-5335	237	17	.	.	PUNCT
ejpam-5335	238	1	(	(	PUNCT
ejpam-5335	238	2	5	5	X
ejpam-5335	238	3	)	)	PUNCT
ejpam-5335	238	4	⇒	⇒	NOUN
ejpam-5335	238	5	(	(	PUNCT
ejpam-5335	238	6	6	6	NUM
ejpam-5335	238	7	):	):	PUNCT
ejpam-5335	238	8	let	let	VERB
ejpam-5335	238	9	v	v	PART
ejpam-5335	238	10	be	be	AUX
ejpam-5335	238	11	any	any	DET
ejpam-5335	238	12	(	(	PUNCT
ejpam-5335	238	13	σ1	σ1	NOUN
ejpam-5335	238	14	,	,	PUNCT
ejpam-5335	238	15	σ2)r	σ2)r	NOUN
ejpam-5335	238	16	-	-	PUNCT
ejpam-5335	238	17	open	open	ADJ
ejpam-5335	238	18	set	set	NOUN
ejpam-5335	238	19	of	of	ADP
ejpam-5335	238	20	y	y	PROPN
ejpam-5335	238	21	.	.	PUNCT
ejpam-5335	239	1	then	then	ADV
ejpam-5335	239	2	,	,	PUNCT
ejpam-5335	239	3	v	v	NOUN
ejpam-5335	239	4	is	be	AUX
ejpam-5335	239	5	a	a	DET
ejpam-5335	239	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	239	7	-	-	ADJ
ejpam-5335	239	8	open	open	ADJ
ejpam-5335	239	9	set	set	NOUN
ejpam-5335	239	10	of	of	ADP
ejpam-5335	239	11	y	y	PROPN
ejpam-5335	239	12	and	and	CCONJ
ejpam-5335	239	13	v	v	NOUN
ejpam-5335	239	14	=	=	SYM
ejpam-5335	239	15	σ1σ2	σ1σ2	NOUN
ejpam-5335	239	16	-	-	PUNCT
ejpam-5335	239	17	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	239	18	-	-	PUNCT
ejpam-5335	239	19	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	239	20	-	-	PUNCT
ejpam-5335	239	21	int(v	int(v	NOUN
ejpam-5335	239	22	)	)	PUNCT
ejpam-5335	239	23	)	)	PUNCT
ejpam-5335	239	24	)	)	PUNCT
ejpam-5335	239	25	.	.	PUNCT
ejpam-5335	240	1	by	by	ADP
ejpam-5335	240	2	(	(	PUNCT
ejpam-5335	240	3	5	5	NUM
ejpam-5335	240	4	)	)	PUNCT
ejpam-5335	240	5	,	,	PUNCT
ejpam-5335	240	6	f−1(v	f−1(v	NOUN
ejpam-5335	240	7	)	)	PUNCT
ejpam-5335	240	8	=	=	SYM
ejpam-5335	240	9	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	240	10	-	-	PUNCT
ejpam-5335	240	11	int(v	int(v	NOUN
ejpam-5335	240	12	)	)	PUNCT
ejpam-5335	240	13	)	)	PUNCT
ejpam-5335	241	1	⊆	⊆	NUM
ejpam-5335	241	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	241	3	,	,	PUNCT
ejpam-5335	241	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	241	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	241	6	-	-	PUNCT
ejpam-5335	241	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	241	8	-	-	PUNCT
ejpam-5335	241	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	241	10	-	-	PUNCT
ejpam-5335	241	11	int(v	int(v	NOUN
ejpam-5335	241	12	)	)	PUNCT
ejpam-5335	241	13	)	)	PUNCT
ejpam-5335	241	14	)	)	PUNCT
ejpam-5335	241	15	)	)	PUNCT
ejpam-5335	241	16	)	)	PUNCT
ejpam-5335	242	1	=	=	SYM
ejpam-5335	242	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	242	3	,	,	PUNCT
ejpam-5335	242	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	242	5	−1(v	−1(v	PROPN
ejpam-5335	242	6	)	)	PUNCT
ejpam-5335	242	7	)	)	PUNCT
ejpam-5335	242	8	and	and	CCONJ
ejpam-5335	242	9	so	so	ADV
ejpam-5335	242	10	f−1(v	f−1(v	PROPN
ejpam-5335	242	11	)	)	PUNCT
ejpam-5335	242	12	is	be	AUX
ejpam-5335	242	13	δ(τ1	δ(τ1	NOUN
ejpam-5335	242	14	,	,	PUNCT
ejpam-5335	242	15	τ2)-open	τ2)-open	ADJ
ejpam-5335	242	16	in	in	ADP
ejpam-5335	242	17	x.	x.	PROPN
ejpam-5335	242	18	(	(	PUNCT
ejpam-5335	242	19	6	6	NUM
ejpam-5335	242	20	)	)	PUNCT
ejpam-5335	242	21	⇒	⇒	NOUN
ejpam-5335	242	22	(	(	PUNCT
ejpam-5335	242	23	7	7	NUM
ejpam-5335	242	24	):	):	PUNCT
ejpam-5335	242	25	the	the	DET
ejpam-5335	242	26	proof	proof	NOUN
ejpam-5335	242	27	is	be	AUX
ejpam-5335	242	28	obvious	obvious	ADJ
ejpam-5335	242	29	.	.	PUNCT
ejpam-5335	243	1	(	(	PUNCT
ejpam-5335	243	2	7	7	X
ejpam-5335	243	3	)	)	PUNCT
ejpam-5335	243	4	⇒	⇒	NOUN
ejpam-5335	243	5	(	(	PUNCT
ejpam-5335	243	6	1	1	NUM
ejpam-5335	243	7	):	):	PUNCT
ejpam-5335	243	8	let	let	VERB
ejpam-5335	243	9	x	x	PUNCT
ejpam-5335	243	10	∈	∈	PROPN
ejpam-5335	243	11	x	x	X
ejpam-5335	243	12	and	and	CCONJ
ejpam-5335	243	13	v	v	AUX
ejpam-5335	243	14	be	be	AUX
ejpam-5335	243	15	any	any	DET
ejpam-5335	243	16	(	(	PUNCT
ejpam-5335	243	17	σ1	σ1	NOUN
ejpam-5335	243	18	,	,	PUNCT
ejpam-5335	243	19	σ2)r	σ2)r	NOUN
ejpam-5335	243	20	-	-	PUNCT
ejpam-5335	243	21	open	open	ADJ
ejpam-5335	243	22	set	set	NOUN
ejpam-5335	243	23	of	of	ADP
ejpam-5335	243	24	y	y	PROPN
ejpam-5335	243	25	containing	contain	VERB
ejpam-5335	243	26	f(x	f(x	PROPN
ejpam-5335	243	27	)	)	PUNCT
ejpam-5335	243	28	.	.	PUNCT
ejpam-5335	244	1	then	then	ADV
ejpam-5335	244	2	,	,	PUNCT
ejpam-5335	244	3	y	y	PROPN
ejpam-5335	244	4	−	−	PROPN
ejpam-5335	244	5	v	v	NOUN
ejpam-5335	244	6	is	be	AUX
ejpam-5335	244	7	(	(	PUNCT
ejpam-5335	244	8	σ1	σ1	NOUN
ejpam-5335	244	9	,	,	PUNCT
ejpam-5335	244	10	σ2)r	σ2)r	NOUN
ejpam-5335	244	11	-	-	PUNCT
ejpam-5335	244	12	closed	closed	ADJ
ejpam-5335	244	13	in	in	ADP
ejpam-5335	244	14	y	y	PROPN
ejpam-5335	244	15	.	.	PUNCT
ejpam-5335	245	1	thus	thus	ADV
ejpam-5335	245	2	by	by	ADP
ejpam-5335	245	3	(	(	PUNCT
ejpam-5335	245	4	7	7	NUM
ejpam-5335	245	5	)	)	PUNCT
ejpam-5335	245	6	,	,	PUNCT
ejpam-5335	245	7	we	we	PRON
ejpam-5335	245	8	have	have	VERB
ejpam-5335	245	9	x	x	X
ejpam-5335	245	10	−	−	PROPN
ejpam-5335	245	11	f−1(v	f−1(v	NOUN
ejpam-5335	245	12	)	)	PUNCT
ejpam-5335	246	1	=	=	PUNCT
ejpam-5335	246	2	f−1(y	f−1(y	PROPN
ejpam-5335	246	3	−	−	PROPN
ejpam-5335	246	4	v	v	NOUN
ejpam-5335	246	5	)	)	PUNCT
ejpam-5335	246	6	=	=	SYM
ejpam-5335	246	7	δ(τ1	δ(τ1	PROPN
ejpam-5335	246	8	,	,	PUNCT
ejpam-5335	246	9	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	246	10	−1(y	−1(y	VERB
ejpam-5335	246	11	−v	−v	NOUN
ejpam-5335	246	12	)	)	PUNCT
ejpam-5335	246	13	)	)	PUNCT
ejpam-5335	247	1	=	=	SYM
ejpam-5335	247	2	x−δ(τ1	x−δ(τ1	PROPN
ejpam-5335	247	3	,	,	PUNCT
ejpam-5335	247	4	τ2)-int(f	τ2)-int(f	VERB
ejpam-5335	247	5	−1(v	−1(v	PROPN
ejpam-5335	247	6	)	)	PUNCT
ejpam-5335	247	7	)	)	PUNCT
ejpam-5335	247	8	and	and	CCONJ
ejpam-5335	247	9	hence	hence	ADV
ejpam-5335	247	10	x	x	X
ejpam-5335	247	11	∈	∈	PROPN
ejpam-5335	247	12	δ(τ1	δ(τ1	PROPN
ejpam-5335	247	13	,	,	PUNCT
ejpam-5335	247	14	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	247	15	−1(v	−1(v	PROPN
ejpam-5335	247	16	)	)	PUNCT
ejpam-5335	247	17	)	)	PUNCT
ejpam-5335	247	18	.	.	PUNCT
ejpam-5335	248	1	then	then	ADV
ejpam-5335	248	2	,	,	PUNCT
ejpam-5335	248	3	there	there	PRON
ejpam-5335	248	4	exists	exist	VERB
ejpam-5335	248	5	a	a	DET
ejpam-5335	248	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	248	7	,	,	PUNCT
ejpam-5335	248	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	248	9	set	set	ADJ
ejpam-5335	248	10	u	u	NOUN
ejpam-5335	248	11	of	of	ADP
ejpam-5335	248	12	x	x	PUNCT
ejpam-5335	248	13	containing	contain	VERB
ejpam-5335	248	14	x	x	PUNCT
ejpam-5335	248	15	such	such	ADJ
ejpam-5335	248	16	that	that	SCONJ
ejpam-5335	248	17	u	u	PROPN
ejpam-5335	248	18	⊆	⊆	NUM
ejpam-5335	248	19	f−1(v	f−1(v	NOUN
ejpam-5335	248	20	)	)	PUNCT
ejpam-5335	248	21	.	.	PUNCT
ejpam-5335	249	1	thus	thus	ADV
ejpam-5335	249	2	,	,	PUNCT
ejpam-5335	249	3	f(u	f(u	PROPN
ejpam-5335	249	4	)	)	PUNCT
ejpam-5335	249	5	⊆	⊆	NUM
ejpam-5335	249	6	v	v	NOUN
ejpam-5335	249	7	.	.	PUNCT
ejpam-5335	250	1	by	by	ADP
ejpam-5335	250	2	theorem	theorem	NOUN
ejpam-5335	250	3	3	3	NUM
ejpam-5335	250	4	(	(	PUNCT
ejpam-5335	250	5	4	4	NUM
ejpam-5335	250	6	)	)	PUNCT
ejpam-5335	250	7	,	,	PUNCT
ejpam-5335	250	8	f	f	PROPN
ejpam-5335	250	9	is	be	AUX
ejpam-5335	250	10	almost	almost	ADV
ejpam-5335	250	11	δ(τ1	δ(τ1	NOUN
ejpam-5335	250	12	,	,	PUNCT
ejpam-5335	250	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	250	14	at	at	ADP
ejpam-5335	250	15	x.	x.	NOUN
ejpam-5335	250	16	this	this	PRON
ejpam-5335	250	17	shows	show	VERB
ejpam-5335	250	18	that	that	SCONJ
ejpam-5335	250	19	f	f	PROPN
ejpam-5335	250	20	is	be	AUX
ejpam-5335	250	21	almost	almost	ADV
ejpam-5335	250	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	250	23	,	,	PUNCT
ejpam-5335	250	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	250	25	.	.	PUNCT
ejpam-5335	251	1	c.	c.	NOUN
ejpam-5335	251	2	prachanpol	prachanpol	PROPN
ejpam-5335	251	3	,	,	PUNCT
ejpam-5335	251	4	c.	c.	PROPN
ejpam-5335	251	5	boonpok	boonpok	PROPN
ejpam-5335	251	6	,	,	PUNCT
ejpam-5335	251	7	c.	c.	PROPN
ejpam-5335	251	8	viriyapong	viriyapong	PROPN
ejpam-5335	251	9	/	/	SYM
ejpam-5335	251	10	eur	eur	PROPN
ejpam-5335	251	11	.	.	PUNCT
ejpam-5335	252	1	j.	j.	PROPN
ejpam-5335	252	2	pure	pure	PROPN
ejpam-5335	252	3	appl	appl	PROPN
ejpam-5335	252	4	.	.	PROPN
ejpam-5335	252	5	math	math	PROPN
ejpam-5335	252	6	,	,	PUNCT
ejpam-5335	252	7	17	17	NUM
ejpam-5335	252	8	(	(	PUNCT
ejpam-5335	252	9	4	4	NUM
ejpam-5335	252	10	)	)	PUNCT
ejpam-5335	252	11	(	(	PUNCT
ejpam-5335	252	12	2024	2024	NUM
ejpam-5335	252	13	)	)	PUNCT
ejpam-5335	252	14	,	,	PUNCT
ejpam-5335	252	15	3730	3730	NUM
ejpam-5335	252	16	-	-	SYM
ejpam-5335	252	17	3742	3742	NUM
ejpam-5335	252	18	3737	3737	NUM
ejpam-5335	252	19	theorem	theorem	NOUN
ejpam-5335	252	20	5	5	NUM
ejpam-5335	252	21	.	.	X
ejpam-5335	252	22	for	for	ADP
ejpam-5335	252	23	a	a	DET
ejpam-5335	252	24	function	function	NOUN
ejpam-5335	252	25	f	f	NOUN
ejpam-5335	252	26	:	:	PUNCT
ejpam-5335	252	27	(	(	PUNCT
ejpam-5335	252	28	x	x	NOUN
ejpam-5335	252	29	,	,	PUNCT
ejpam-5335	252	30	τ1	τ1	NOUN
ejpam-5335	252	31	,	,	PUNCT
ejpam-5335	252	32	τ2	τ2	NOUN
ejpam-5335	252	33	)	)	PUNCT
ejpam-5335	252	34	→	→	SYM
ejpam-5335	252	35	(	(	PUNCT
ejpam-5335	252	36	y	y	PROPN
ejpam-5335	252	37	,	,	PUNCT
ejpam-5335	252	38	σ1	σ1	PROPN
ejpam-5335	252	39	,	,	PUNCT
ejpam-5335	252	40	σ2	σ2	NOUN
ejpam-5335	252	41	)	)	PUNCT
ejpam-5335	252	42	,	,	PUNCT
ejpam-5335	252	43	the	the	DET
ejpam-5335	252	44	following	follow	VERB
ejpam-5335	252	45	properties	property	NOUN
ejpam-5335	252	46	are	be	AUX
ejpam-5335	252	47	equivalent	equivalent	ADJ
ejpam-5335	252	48	:	:	PUNCT
ejpam-5335	252	49	(	(	PUNCT
ejpam-5335	252	50	1	1	X
ejpam-5335	252	51	)	)	PUNCT
ejpam-5335	252	52	f	f	NOUN
ejpam-5335	252	53	is	be	AUX
ejpam-5335	252	54	almost	almost	ADV
ejpam-5335	252	55	δ(τ1	δ(τ1	NOUN
ejpam-5335	252	56	,	,	PUNCT
ejpam-5335	252	57	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	252	58	;	;	PUNCT
ejpam-5335	252	59	(	(	PUNCT
ejpam-5335	252	60	2	2	X
ejpam-5335	252	61	)	)	PUNCT
ejpam-5335	252	62	δ(τ1	δ(τ1	NOUN
ejpam-5335	252	63	,	,	PUNCT
ejpam-5335	252	64	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	252	65	−1(v	−1(v	PROPN
ejpam-5335	252	66	)	)	PUNCT
ejpam-5335	252	67	)	)	PUNCT
ejpam-5335	253	1	⊆	⊆	NUM
ejpam-5335	253	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	253	3	-	-	PUNCT
ejpam-5335	253	4	cl(v	cl(v	NOUN
ejpam-5335	253	5	)	)	PUNCT
ejpam-5335	253	6	)	)	PUNCT
ejpam-5335	253	7	for	for	ADP
ejpam-5335	253	8	every	every	DET
ejpam-5335	253	9	(	(	PUNCT
ejpam-5335	253	10	σ1	σ1	PROPN
ejpam-5335	253	11	,	,	PUNCT
ejpam-5335	253	12	σ2)β	σ2)β	NOUN
ejpam-5335	253	13	-	-	PUNCT
ejpam-5335	253	14	open	open	NOUN
ejpam-5335	253	15	set	set	NOUN
ejpam-5335	253	16	v	v	NOUN
ejpam-5335	253	17	of	of	ADP
ejpam-5335	253	18	y	y	PROPN
ejpam-5335	253	19	;	;	PUNCT
ejpam-5335	253	20	(	(	PUNCT
ejpam-5335	253	21	3	3	X
ejpam-5335	253	22	)	)	PUNCT
ejpam-5335	253	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	253	24	,	,	PUNCT
ejpam-5335	253	25	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	253	26	−1(v	−1(v	PROPN
ejpam-5335	253	27	)	)	PUNCT
ejpam-5335	253	28	)	)	PUNCT
ejpam-5335	254	1	⊆	⊆	NUM
ejpam-5335	254	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	254	3	-	-	PUNCT
ejpam-5335	254	4	cl(v	cl(v	NOUN
ejpam-5335	254	5	)	)	PUNCT
ejpam-5335	254	6	)	)	PUNCT
ejpam-5335	254	7	for	for	ADP
ejpam-5335	254	8	every	every	DET
ejpam-5335	254	9	(	(	PUNCT
ejpam-5335	254	10	σ1	σ1	PROPN
ejpam-5335	254	11	,	,	PUNCT
ejpam-5335	254	12	σ2)s	σ2)s	NOUN
ejpam-5335	254	13	-	-	PUNCT
ejpam-5335	254	14	open	open	NOUN
ejpam-5335	254	15	set	set	NOUN
ejpam-5335	254	16	v	v	NOUN
ejpam-5335	254	17	of	of	ADP
ejpam-5335	254	18	y	y	PROPN
ejpam-5335	254	19	;	;	PUNCT
ejpam-5335	254	20	(	(	PUNCT
ejpam-5335	254	21	4	4	X
ejpam-5335	254	22	)	)	PUNCT
ejpam-5335	254	23	f−1(v	f−1(v	NOUN
ejpam-5335	254	24	)	)	PUNCT
ejpam-5335	255	1	⊆	⊆	NUM
ejpam-5335	255	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	255	3	,	,	PUNCT
ejpam-5335	255	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	255	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	255	6	-	-	PUNCT
ejpam-5335	255	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	255	8	-	-	PUNCT
ejpam-5335	255	9	cl(v	cl(v	NOUN
ejpam-5335	255	10	)	)	PUNCT
ejpam-5335	255	11	)	)	PUNCT
ejpam-5335	255	12	)	)	PUNCT
ejpam-5335	255	13	)	)	PUNCT
ejpam-5335	256	1	for	for	ADP
ejpam-5335	256	2	every	every	DET
ejpam-5335	256	3	(	(	PUNCT
ejpam-5335	256	4	σ1	σ1	PROPN
ejpam-5335	256	5	,	,	PUNCT
ejpam-5335	256	6	σ2)p	σ2)p	NOUN
ejpam-5335	256	7	-	-	PUNCT
ejpam-5335	256	8	open	open	NOUN
ejpam-5335	256	9	set	set	NOUN
ejpam-5335	256	10	v	v	NOUN
ejpam-5335	256	11	of	of	ADP
ejpam-5335	256	12	y	y	PROPN
ejpam-5335	256	13	.	.	PUNCT
ejpam-5335	257	1	proof	proof	NOUN
ejpam-5335	257	2	.	.	PUNCT
ejpam-5335	258	1	(	(	PUNCT
ejpam-5335	258	2	1	1	X
ejpam-5335	258	3	)	)	PUNCT
ejpam-5335	258	4	⇒	⇒	NOUN
ejpam-5335	258	5	(	(	PUNCT
ejpam-5335	258	6	2	2	NUM
ejpam-5335	258	7	):	):	PUNCT
ejpam-5335	258	8	let	let	VERB
ejpam-5335	258	9	v	v	PART
ejpam-5335	258	10	be	be	AUX
ejpam-5335	258	11	any	any	DET
ejpam-5335	258	12	(	(	PUNCT
ejpam-5335	258	13	σ1	σ1	PROPN
ejpam-5335	258	14	,	,	PUNCT
ejpam-5335	258	15	σ2)β	σ2)β	NOUN
ejpam-5335	258	16	-	-	PUNCT
ejpam-5335	258	17	open	open	ADJ
ejpam-5335	258	18	set	set	NOUN
ejpam-5335	258	19	of	of	ADP
ejpam-5335	258	20	y	y	PROPN
ejpam-5335	258	21	.	.	PUNCT
ejpam-5335	259	1	then	then	ADV
ejpam-5335	259	2	,	,	PUNCT
ejpam-5335	259	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	259	4	-	-	NUM
ejpam-5335	259	5	cl(v	cl(v	NOUN
ejpam-5335	259	6	)	)	PUNCT
ejpam-5335	259	7	is	be	AUX
ejpam-5335	259	8	(	(	PUNCT
ejpam-5335	259	9	σ1	σ1	NOUN
ejpam-5335	259	10	,	,	PUNCT
ejpam-5335	259	11	σ2)r	σ2)r	NOUN
ejpam-5335	259	12	-	-	PUNCT
ejpam-5335	259	13	closed	closed	ADJ
ejpam-5335	259	14	in	in	ADP
ejpam-5335	259	15	y	y	PROPN
ejpam-5335	259	16	.	.	PUNCT
ejpam-5335	260	1	by	by	ADP
ejpam-5335	260	2	theorem	theorem	NOUN
ejpam-5335	260	3	4	4	NUM
ejpam-5335	260	4	(	(	PUNCT
ejpam-5335	260	5	7	7	NUM
ejpam-5335	260	6	)	)	PUNCT
ejpam-5335	260	7	,	,	PUNCT
ejpam-5335	260	8	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	260	9	-	-	PUNCT
ejpam-5335	260	10	cl(v	cl(v	NOUN
ejpam-5335	260	11	)	)	PUNCT
ejpam-5335	260	12	)	)	PUNCT
ejpam-5335	260	13	is	be	AUX
ejpam-5335	260	14	δ(τ1	δ(τ1	NOUN
ejpam-5335	260	15	,	,	PUNCT
ejpam-5335	260	16	τ2)-closed	τ2)-close	VERB
ejpam-5335	260	17	in	in	ADP
ejpam-5335	260	18	x.	x.	NOUN
ejpam-5335	260	19	hence	hence	ADV
ejpam-5335	260	20	,	,	PUNCT
ejpam-5335	260	21	δ(τ1	δ(τ1	PROPN
ejpam-5335	260	22	,	,	PUNCT
ejpam-5335	260	23	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	260	24	−1(v	−1(v	PROPN
ejpam-5335	260	25	)	)	PUNCT
ejpam-5335	260	26	)	)	PUNCT
ejpam-5335	261	1	⊆	⊆	NUM
ejpam-5335	261	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	261	3	,	,	PUNCT
ejpam-5335	261	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	261	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	261	6	-	-	PUNCT
ejpam-5335	261	7	cl(v	cl(v	NOUN
ejpam-5335	261	8	)	)	PUNCT
ejpam-5335	261	9	)	)	PUNCT
ejpam-5335	261	10	)	)	PUNCT
ejpam-5335	262	1	=	=	PRON
ejpam-5335	262	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	262	3	-	-	PUNCT
ejpam-5335	262	4	cl(v	cl(v	NOUN
ejpam-5335	262	5	)	)	PUNCT
ejpam-5335	262	6	)	)	PUNCT
ejpam-5335	262	7	.	.	PUNCT
ejpam-5335	263	1	(	(	PUNCT
ejpam-5335	263	2	2	2	X
ejpam-5335	263	3	)	)	PUNCT
ejpam-5335	263	4	⇒	⇒	NOUN
ejpam-5335	263	5	(	(	PUNCT
ejpam-5335	263	6	3	3	NUM
ejpam-5335	263	7	):	):	PUNCT
ejpam-5335	263	8	the	the	DET
ejpam-5335	263	9	proof	proof	NOUN
ejpam-5335	263	10	is	be	AUX
ejpam-5335	263	11	obvious	obvious	ADJ
ejpam-5335	263	12	since	since	SCONJ
ejpam-5335	263	13	every	every	DET
ejpam-5335	263	14	(	(	PUNCT
ejpam-5335	263	15	σ1	σ1	PROPN
ejpam-5335	263	16	,	,	PUNCT
ejpam-5335	263	17	σ2)s	σ2)s	NOUN
ejpam-5335	263	18	-	-	PUNCT
ejpam-5335	263	19	open	open	ADJ
ejpam-5335	263	20	set	set	NOUN
ejpam-5335	263	21	is	be	AUX
ejpam-5335	263	22	(	(	PUNCT
ejpam-5335	263	23	σ1	σ1	PROPN
ejpam-5335	263	24	,	,	PUNCT
ejpam-5335	263	25	σ2)β	σ2)β	NOUN
ejpam-5335	263	26	-	-	PUNCT
ejpam-5335	263	27	open	open	ADJ
ejpam-5335	263	28	.	.	PUNCT
ejpam-5335	264	1	(	(	PUNCT
ejpam-5335	264	2	3	3	X
ejpam-5335	264	3	)	)	PUNCT
ejpam-5335	264	4	⇒	⇒	NOUN
ejpam-5335	264	5	(	(	PUNCT
ejpam-5335	264	6	1	1	NUM
ejpam-5335	264	7	):	):	PUNCT
ejpam-5335	264	8	let	let	VERB
ejpam-5335	264	9	f	f	PRON
ejpam-5335	264	10	be	be	AUX
ejpam-5335	264	11	any	any	DET
ejpam-5335	264	12	(	(	PUNCT
ejpam-5335	264	13	σ1	σ1	NOUN
ejpam-5335	264	14	,	,	PUNCT
ejpam-5335	264	15	σ2)r	σ2)r	NOUN
ejpam-5335	264	16	-	-	PUNCT
ejpam-5335	264	17	closed	close	VERB
ejpam-5335	264	18	set	set	NOUN
ejpam-5335	264	19	of	of	ADP
ejpam-5335	264	20	y	y	PROPN
ejpam-5335	264	21	.	.	PUNCT
ejpam-5335	265	1	then	then	ADV
ejpam-5335	265	2	,	,	PUNCT
ejpam-5335	265	3	f	f	PROPN
ejpam-5335	265	4	is	be	AUX
ejpam-5335	265	5	(	(	PUNCT
ejpam-5335	265	6	σ1	σ1	PROPN
ejpam-5335	265	7	,	,	PUNCT
ejpam-5335	265	8	σ2)s	σ2)s	NOUN
ejpam-5335	265	9	-	-	PUNCT
ejpam-5335	265	10	open	open	ADJ
ejpam-5335	265	11	in	in	ADP
ejpam-5335	265	12	y	y	PROPN
ejpam-5335	265	13	.	.	PUNCT
ejpam-5335	266	1	by	by	ADP
ejpam-5335	266	2	(	(	PUNCT
ejpam-5335	266	3	3	3	NUM
ejpam-5335	266	4	)	)	PUNCT
ejpam-5335	266	5	,	,	PUNCT
ejpam-5335	266	6	we	we	PRON
ejpam-5335	266	7	have	have	VERB
ejpam-5335	266	8	δ(τ1	δ(τ1	NOUN
ejpam-5335	266	9	,	,	PUNCT
ejpam-5335	266	10	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	266	11	−1(f	−1(f	NUM
ejpam-5335	266	12	)	)	PUNCT
ejpam-5335	266	13	)	)	PUNCT
ejpam-5335	267	1	⊆	⊆	NUM
ejpam-5335	267	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	267	3	-	-	PUNCT
ejpam-5335	267	4	cl(f	cl(f	NUM
ejpam-5335	267	5	)	)	PUNCT
ejpam-5335	267	6	)	)	PUNCT
ejpam-5335	268	1	=	=	SYM
ejpam-5335	268	2	f−1(f	f−1(f	PROPN
ejpam-5335	268	3	)	)	PUNCT
ejpam-5335	268	4	and	and	CCONJ
ejpam-5335	268	5	hence	hence	ADV
ejpam-5335	268	6	f−1(f	f−1(f	PROPN
ejpam-5335	268	7	)	)	PUNCT
ejpam-5335	268	8	is	be	AUX
ejpam-5335	268	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	268	10	,	,	PUNCT
ejpam-5335	268	11	τ2)-closed	τ2)-close	VERB
ejpam-5335	268	12	in	in	ADP
ejpam-5335	268	13	x.	x.	NOUN
ejpam-5335	268	14	by	by	ADP
ejpam-5335	268	15	theorem	theorem	NOUN
ejpam-5335	268	16	4	4	NUM
ejpam-5335	268	17	(	(	PUNCT
ejpam-5335	268	18	7	7	NUM
ejpam-5335	268	19	)	)	PUNCT
ejpam-5335	268	20	,	,	PUNCT
ejpam-5335	268	21	f	f	PROPN
ejpam-5335	268	22	is	be	AUX
ejpam-5335	268	23	almost	almost	ADV
ejpam-5335	268	24	δ(τ1	δ(τ1	NOUN
ejpam-5335	268	25	,	,	PUNCT
ejpam-5335	268	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	268	27	.	.	PUNCT
ejpam-5335	269	1	(	(	PUNCT
ejpam-5335	269	2	1	1	X
ejpam-5335	269	3	)	)	PUNCT
ejpam-5335	269	4	⇒	⇒	NOUN
ejpam-5335	269	5	(	(	PUNCT
ejpam-5335	269	6	4	4	NUM
ejpam-5335	269	7	):	):	PUNCT
ejpam-5335	269	8	let	let	VERB
ejpam-5335	269	9	v	v	PART
ejpam-5335	269	10	be	be	AUX
ejpam-5335	269	11	any	any	DET
ejpam-5335	269	12	(	(	PUNCT
ejpam-5335	269	13	σ1	σ1	PROPN
ejpam-5335	269	14	,	,	PUNCT
ejpam-5335	269	15	σ2)p	σ2)p	NOUN
ejpam-5335	269	16	-	-	PUNCT
ejpam-5335	269	17	open	open	ADJ
ejpam-5335	269	18	set	set	NOUN
ejpam-5335	269	19	of	of	ADP
ejpam-5335	269	20	y	y	PROPN
ejpam-5335	269	21	.	.	PUNCT
ejpam-5335	270	1	then	then	ADV
ejpam-5335	270	2	,	,	PUNCT
ejpam-5335	270	3	σ1σ2	σ1σ2	X
ejpam-5335	270	4	-	-	PUNCT
ejpam-5335	270	5	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	270	6	-	-	PUNCT
ejpam-5335	270	7	cl(v	cl(v	NOUN
ejpam-5335	270	8	)	)	PUNCT
ejpam-5335	270	9	)	)	PUNCT
ejpam-5335	270	10	is	be	AUX
ejpam-5335	270	11	(	(	PUNCT
ejpam-5335	270	12	σ1	σ1	NOUN
ejpam-5335	270	13	,	,	PUNCT
ejpam-5335	270	14	σ2)r	σ2)r	NOUN
ejpam-5335	270	15	-	-	PUNCT
ejpam-5335	270	16	open	open	ADJ
ejpam-5335	270	17	in	in	ADP
ejpam-5335	270	18	y	y	PROPN
ejpam-5335	270	19	.	.	PUNCT
ejpam-5335	271	1	by	by	ADP
ejpam-5335	271	2	theorem	theorem	ADJ
ejpam-5335	271	3	4	4	NUM
ejpam-5335	271	4	(	(	PUNCT
ejpam-5335	271	5	6	6	NUM
ejpam-5335	271	6	)	)	PUNCT
ejpam-5335	271	7	,	,	PUNCT
ejpam-5335	271	8	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	271	9	-	-	PUNCT
ejpam-5335	271	10	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	271	11	-	-	PUNCT
ejpam-5335	271	12	cl(v	cl(v	NOUN
ejpam-5335	271	13	)	)	PUNCT
ejpam-5335	271	14	)	)	PUNCT
ejpam-5335	271	15	)	)	PUNCT
ejpam-5335	271	16	is	be	AUX
ejpam-5335	271	17	δ(τ1	δ(τ1	NOUN
ejpam-5335	271	18	,	,	PUNCT
ejpam-5335	271	19	τ2)-open	τ2)-open	ADJ
ejpam-5335	271	20	in	in	ADP
ejpam-5335	271	21	x.	x.	PROPN
ejpam-5335	271	22	thus	thus	ADV
ejpam-5335	271	23	,	,	PUNCT
ejpam-5335	271	24	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	271	25	-	-	PUNCT
ejpam-5335	271	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	271	27	-	-	PUNCT
ejpam-5335	271	28	cl(v	cl(v	NOUN
ejpam-5335	271	29	)	)	PUNCT
ejpam-5335	271	30	)	)	PUNCT
ejpam-5335	271	31	)	)	PUNCT
ejpam-5335	272	1	=	=	SYM
ejpam-5335	272	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	272	3	,	,	PUNCT
ejpam-5335	272	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	272	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	272	6	-	-	PUNCT
ejpam-5335	272	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	272	8	-	-	PUNCT
ejpam-5335	272	9	cl(v	cl(v	NOUN
ejpam-5335	272	10	)	)	PUNCT
ejpam-5335	272	11	)	)	PUNCT
ejpam-5335	272	12	)	)	PUNCT
ejpam-5335	272	13	)	)	PUNCT
ejpam-5335	272	14	and	and	CCONJ
ejpam-5335	272	15	hence	hence	ADV
ejpam-5335	272	16	f−1(v	f−1(v	NOUN
ejpam-5335	272	17	)	)	PUNCT
ejpam-5335	272	18	⊆	⊆	NUM
ejpam-5335	272	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	272	20	-	-	PUNCT
ejpam-5335	272	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	272	22	-	-	PUNCT
ejpam-5335	272	23	cl(v	cl(v	NOUN
ejpam-5335	272	24	)	)	PUNCT
ejpam-5335	272	25	)	)	PUNCT
ejpam-5335	272	26	)	)	PUNCT
ejpam-5335	273	1	=	=	SYM
ejpam-5335	273	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	273	3	,	,	PUNCT
ejpam-5335	273	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	273	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	273	6	-	-	PUNCT
ejpam-5335	273	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	273	8	-	-	PUNCT
ejpam-5335	273	9	cl(v	cl(v	NOUN
ejpam-5335	273	10	)	)	PUNCT
ejpam-5335	273	11	)	)	PUNCT
ejpam-5335	273	12	)	)	PUNCT
ejpam-5335	273	13	)	)	PUNCT
ejpam-5335	273	14	.	.	PUNCT
ejpam-5335	274	1	(	(	PUNCT
ejpam-5335	274	2	4	4	X
ejpam-5335	274	3	)	)	PUNCT
ejpam-5335	274	4	⇒	⇒	NOUN
ejpam-5335	274	5	(	(	PUNCT
ejpam-5335	274	6	1	1	NUM
ejpam-5335	274	7	)	)	PUNCT
ejpam-5335	274	8	:	:	PUNCT
ejpam-5335	274	9	let	let	VERB
ejpam-5335	274	10	v	v	PART
ejpam-5335	274	11	be	be	AUX
ejpam-5335	274	12	any	any	DET
ejpam-5335	274	13	(	(	PUNCT
ejpam-5335	274	14	σ1	σ1	NOUN
ejpam-5335	274	15	,	,	PUNCT
ejpam-5335	274	16	σ2)r	σ2)r	NOUN
ejpam-5335	274	17	-	-	PUNCT
ejpam-5335	274	18	open	open	ADJ
ejpam-5335	274	19	set	set	NOUN
ejpam-5335	274	20	of	of	ADP
ejpam-5335	274	21	y	y	PROPN
ejpam-5335	274	22	.	.	PUNCT
ejpam-5335	275	1	then	then	ADV
ejpam-5335	275	2	,	,	PUNCT
ejpam-5335	275	3	v	v	NOUN
ejpam-5335	275	4	=	=	SYM
ejpam-5335	275	5	σ1σ2	σ1σ2	NOUN
ejpam-5335	275	6	-	-	PUNCT
ejpam-5335	275	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	275	8	-	-	PUNCT
ejpam-5335	275	9	cl(v	cl(v	NOUN
ejpam-5335	275	10	)	)	PUNCT
ejpam-5335	275	11	)	)	PUNCT
ejpam-5335	275	12	and	and	CCONJ
ejpam-5335	275	13	v	v	NOUN
ejpam-5335	275	14	is	be	AUX
ejpam-5335	275	15	(	(	PUNCT
ejpam-5335	275	16	σ1	σ1	PROPN
ejpam-5335	275	17	,	,	PUNCT
ejpam-5335	275	18	σ2)p	σ2)p	NOUN
ejpam-5335	275	19	-	-	PUNCT
ejpam-5335	275	20	open	open	ADJ
ejpam-5335	275	21	in	in	ADP
ejpam-5335	275	22	y	y	PROPN
ejpam-5335	275	23	.	.	PUNCT
ejpam-5335	276	1	by	by	ADP
ejpam-5335	276	2	(	(	PUNCT
ejpam-5335	276	3	4	4	NUM
ejpam-5335	276	4	)	)	PUNCT
ejpam-5335	276	5	,	,	PUNCT
ejpam-5335	276	6	f−1(v	f−1(v	PROPN
ejpam-5335	276	7	)	)	PUNCT
ejpam-5335	276	8	⊆	⊆	NUM
ejpam-5335	276	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	276	10	,	,	PUNCT
ejpam-5335	276	11	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	276	12	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	276	13	-	-	PUNCT
ejpam-5335	276	14	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	276	15	-	-	PUNCT
ejpam-5335	276	16	cl(v	cl(v	NOUN
ejpam-5335	276	17	)	)	PUNCT
ejpam-5335	276	18	)	)	PUNCT
ejpam-5335	276	19	)	)	PUNCT
ejpam-5335	276	20	)	)	PUNCT
ejpam-5335	277	1	=	=	SYM
ejpam-5335	277	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	277	3	,	,	PUNCT
ejpam-5335	277	4	τ2)-int(f	τ2)-int(f	NOUN
ejpam-5335	277	5	−1(v	−1(v	PROPN
ejpam-5335	277	6	)	)	PUNCT
ejpam-5335	277	7	)	)	PUNCT
ejpam-5335	277	8	.	.	PUNCT
ejpam-5335	278	1	therefore	therefore	ADV
ejpam-5335	278	2	,	,	PUNCT
ejpam-5335	278	3	f−1(v	f−1(v	PROPN
ejpam-5335	278	4	)	)	PUNCT
ejpam-5335	278	5	is	be	AUX
ejpam-5335	278	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	278	7	,	,	PUNCT
ejpam-5335	278	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	278	9	in	in	ADP
ejpam-5335	278	10	x.	x.	NOUN
ejpam-5335	278	11	by	by	ADP
ejpam-5335	278	12	theorem	theorem	NOUN
ejpam-5335	278	13	4	4	NUM
ejpam-5335	278	14	(	(	PUNCT
ejpam-5335	278	15	6	6	NUM
ejpam-5335	278	16	)	)	PUNCT
ejpam-5335	278	17	,	,	PUNCT
ejpam-5335	278	18	f	f	PROPN
ejpam-5335	278	19	is	be	AUX
ejpam-5335	278	20	almost	almost	ADV
ejpam-5335	278	21	δ(τ1	δ(τ1	NOUN
ejpam-5335	278	22	,	,	PUNCT
ejpam-5335	278	23	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	278	24	.	.	NOUN
ejpam-5335	278	25	5	5	NUM
ejpam-5335	278	26	.	.	X
ejpam-5335	279	1	on	on	ADP
ejpam-5335	279	2	weakly	weakly	ADJ
ejpam-5335	279	3	δ(τ1	δ(τ1	NOUN
ejpam-5335	279	4	,	,	PUNCT
ejpam-5335	279	5	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	279	6	functions	function	NOUN
ejpam-5335	279	7	in	in	ADP
ejpam-5335	279	8	this	this	DET
ejpam-5335	279	9	section	section	NOUN
ejpam-5335	279	10	,	,	PUNCT
ejpam-5335	279	11	we	we	PRON
ejpam-5335	279	12	introduce	introduce	VERB
ejpam-5335	279	13	and	and	CCONJ
ejpam-5335	279	14	investigate	investigate	VERB
ejpam-5335	279	15	the	the	DET
ejpam-5335	279	16	concept	concept	NOUN
ejpam-5335	279	17	of	of	ADP
ejpam-5335	279	18	weakly	weakly	ADJ
ejpam-5335	279	19	δ(τ1	δ(τ1	NOUN
ejpam-5335	279	20	,	,	PUNCT
ejpam-5335	279	21	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	279	22	functions	function	NOUN
ejpam-5335	279	23	.	.	PUNCT
ejpam-5335	280	1	furthermore	furthermore	ADV
ejpam-5335	280	2	,	,	PUNCT
ejpam-5335	280	3	we	we	PRON
ejpam-5335	280	4	discuss	discuss	VERB
ejpam-5335	280	5	the	the	DET
ejpam-5335	280	6	relationships	relationship	NOUN
ejpam-5335	280	7	between	between	ADP
ejpam-5335	280	8	almost	almost	ADV
ejpam-5335	280	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	280	10	,	,	PUNCT
ejpam-5335	280	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	280	12	functions	function	NOUN
ejpam-5335	280	13	and	and	CCONJ
ejpam-5335	280	14	weakly	weakly	ADJ
ejpam-5335	280	15	δ(τ1	δ(τ1	NOUN
ejpam-5335	280	16	,	,	PUNCT
ejpam-5335	280	17	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	280	18	functions	function	NOUN
ejpam-5335	280	19	.	.	PUNCT
ejpam-5335	281	1	definition	definition	NOUN
ejpam-5335	281	2	3	3	NUM
ejpam-5335	281	3	.	.	PUNCT
ejpam-5335	282	1	a	a	DET
ejpam-5335	282	2	function	function	NOUN
ejpam-5335	282	3	f	f	NOUN
ejpam-5335	282	4	:	:	PUNCT
ejpam-5335	282	5	(	(	PUNCT
ejpam-5335	282	6	x	x	NOUN
ejpam-5335	282	7	,	,	PUNCT
ejpam-5335	282	8	τ1	τ1	NOUN
ejpam-5335	282	9	,	,	PUNCT
ejpam-5335	282	10	τ2	τ2	NOUN
ejpam-5335	282	11	)	)	PUNCT
ejpam-5335	282	12	→	→	SYM
ejpam-5335	282	13	(	(	PUNCT
ejpam-5335	282	14	y	y	PROPN
ejpam-5335	282	15	,	,	PUNCT
ejpam-5335	282	16	σ1	σ1	PROPN
ejpam-5335	282	17	,	,	PUNCT
ejpam-5335	282	18	σ2	σ2	PROPN
ejpam-5335	282	19	)	)	PUNCT
ejpam-5335	282	20	is	be	AUX
ejpam-5335	282	21	said	say	VERB
ejpam-5335	282	22	to	to	PART
ejpam-5335	282	23	be	be	AUX
ejpam-5335	282	24	weakly	weakly	ADJ
ejpam-5335	282	25	δ(τ1	δ(τ1	NOUN
ejpam-5335	282	26	,	,	PUNCT
ejpam-5335	282	27	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	282	28	at	at	ADP
ejpam-5335	282	29	x	x	X
ejpam-5335	282	30	∈	∈	PROPN
ejpam-5335	282	31	x	x	PUNCT
ejpam-5335	282	32	if	if	SCONJ
ejpam-5335	282	33	for	for	ADP
ejpam-5335	282	34	each	each	DET
ejpam-5335	282	35	σ1σ2	σ1σ2	VERB
ejpam-5335	282	36	-	-	ADJ
ejpam-5335	282	37	open	open	ADJ
ejpam-5335	282	38	set	set	NOUN
ejpam-5335	282	39	v	v	NOUN
ejpam-5335	282	40	of	of	ADP
ejpam-5335	282	41	y	y	NOUN
ejpam-5335	282	42	containing	contain	VERB
ejpam-5335	282	43	f(x	f(x	PROPN
ejpam-5335	282	44	)	)	PUNCT
ejpam-5335	282	45	,	,	PUNCT
ejpam-5335	282	46	there	there	PRON
ejpam-5335	282	47	exists	exist	VERB
ejpam-5335	282	48	a	a	DET
ejpam-5335	282	49	δ(τ1	δ(τ1	NOUN
ejpam-5335	282	50	,	,	PUNCT
ejpam-5335	282	51	τ2)-open	τ2)-open	ADJ
ejpam-5335	282	52	set	set	ADJ
ejpam-5335	282	53	u	u	NOUN
ejpam-5335	282	54	of	of	ADP
ejpam-5335	282	55	x	x	PUNCT
ejpam-5335	282	56	containing	contain	VERB
ejpam-5335	282	57	x	x	PUNCT
ejpam-5335	282	58	such	such	ADJ
ejpam-5335	282	59	that	that	DET
ejpam-5335	282	60	f(u	f(u	PROPN
ejpam-5335	282	61	)	)	PUNCT
ejpam-5335	282	62	⊆	⊆	NUM
ejpam-5335	282	63	σ1σ2	σ1σ2	NOUN
ejpam-5335	282	64	-	-	NUM
ejpam-5335	282	65	cl(v	cl(v	NOUN
ejpam-5335	282	66	)	)	PUNCT
ejpam-5335	282	67	.	.	PUNCT
ejpam-5335	283	1	a	a	DET
ejpam-5335	283	2	function	function	NOUN
ejpam-5335	283	3	f	f	NOUN
ejpam-5335	283	4	:	:	PUNCT
ejpam-5335	283	5	(	(	PUNCT
ejpam-5335	283	6	x	x	NOUN
ejpam-5335	283	7	,	,	PUNCT
ejpam-5335	283	8	τ1	τ1	NOUN
ejpam-5335	283	9	,	,	PUNCT
ejpam-5335	283	10	τ2	τ2	NOUN
ejpam-5335	283	11	)	)	PUNCT
ejpam-5335	283	12	→	→	SYM
ejpam-5335	283	13	(	(	PUNCT
ejpam-5335	283	14	y	y	PROPN
ejpam-5335	283	15	,	,	PUNCT
ejpam-5335	283	16	σ1	σ1	PROPN
ejpam-5335	283	17	,	,	PUNCT
ejpam-5335	283	18	σ2	σ2	PROPN
ejpam-5335	283	19	)	)	PUNCT
ejpam-5335	283	20	is	be	AUX
ejpam-5335	283	21	called	call	VERB
ejpam-5335	283	22	weakly	weakly	ADJ
ejpam-5335	283	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	283	24	,	,	PUNCT
ejpam-5335	283	25	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	283	26	if	if	SCONJ
ejpam-5335	283	27	f	f	PROPN
ejpam-5335	283	28	is	be	AUX
ejpam-5335	283	29	weakly	weakly	ADJ
ejpam-5335	283	30	δ(τ1	δ(τ1	NOUN
ejpam-5335	283	31	,	,	PUNCT
ejpam-5335	283	32	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	283	33	at	at	ADP
ejpam-5335	283	34	each	each	DET
ejpam-5335	283	35	point	point	NOUN
ejpam-5335	283	36	of	of	ADP
ejpam-5335	283	37	x.	x.	NOUN
ejpam-5335	283	38	remark	remark	PROPN
ejpam-5335	283	39	2	2	NUM
ejpam-5335	283	40	.	.	PUNCT
ejpam-5335	284	1	for	for	ADP
ejpam-5335	284	2	a	a	DET
ejpam-5335	284	3	function	function	NOUN
ejpam-5335	284	4	f	f	NOUN
ejpam-5335	284	5	:	:	PUNCT
ejpam-5335	284	6	(	(	PUNCT
ejpam-5335	284	7	x	x	NOUN
ejpam-5335	284	8	,	,	PUNCT
ejpam-5335	284	9	τ1	τ1	NOUN
ejpam-5335	284	10	,	,	PUNCT
ejpam-5335	284	11	τ2	τ2	NOUN
ejpam-5335	284	12	)	)	PUNCT
ejpam-5335	284	13	→	→	SYM
ejpam-5335	284	14	(	(	PUNCT
ejpam-5335	284	15	y	y	PROPN
ejpam-5335	284	16	,	,	PUNCT
ejpam-5335	284	17	σ1	σ1	PROPN
ejpam-5335	284	18	,	,	PUNCT
ejpam-5335	284	19	σ2	σ2	NOUN
ejpam-5335	284	20	)	)	PUNCT
ejpam-5335	284	21	,	,	PUNCT
ejpam-5335	284	22	the	the	DET
ejpam-5335	284	23	following	follow	VERB
ejpam-5335	284	24	implication	implication	NOUN
ejpam-5335	284	25	holds	hold	VERB
ejpam-5335	284	26	:	:	PUNCT
ejpam-5335	284	27	almost	almost	ADV
ejpam-5335	284	28	δ(τ1	δ(τ1	NOUN
ejpam-5335	284	29	,	,	PUNCT
ejpam-5335	284	30	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	284	31	⇒	⇒	VERB
ejpam-5335	284	32	weak	weak	ADJ
ejpam-5335	284	33	δ(τ1	δ(τ1	NOUN
ejpam-5335	284	34	,	,	PUNCT
ejpam-5335	284	35	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	284	36	.	.	PUNCT
ejpam-5335	285	1	the	the	DET
ejpam-5335	285	2	converse	converse	NOUN
ejpam-5335	285	3	of	of	ADP
ejpam-5335	285	4	the	the	DET
ejpam-5335	285	5	implication	implication	NOUN
ejpam-5335	285	6	is	be	AUX
ejpam-5335	285	7	not	not	PART
ejpam-5335	285	8	true	true	ADJ
ejpam-5335	285	9	in	in	ADP
ejpam-5335	285	10	general	general	ADJ
ejpam-5335	285	11	.	.	PUNCT
ejpam-5335	286	1	we	we	PRON
ejpam-5335	286	2	give	give	VERB
ejpam-5335	286	3	an	an	DET
ejpam-5335	286	4	example	example	NOUN
ejpam-5335	286	5	for	for	ADP
ejpam-5335	286	6	the	the	DET
ejpam-5335	286	7	implication	implication	NOUN
ejpam-5335	286	8	as	as	SCONJ
ejpam-5335	286	9	follows	follow	VERB
ejpam-5335	286	10	.	.	PUNCT
ejpam-5335	287	1	c.	c.	PROPN
ejpam-5335	287	2	prachanpol	prachanpol	PROPN
ejpam-5335	287	3	,	,	PUNCT
ejpam-5335	287	4	c.	c.	PROPN
ejpam-5335	287	5	boonpok	boonpok	PROPN
ejpam-5335	287	6	,	,	PUNCT
ejpam-5335	287	7	c.	c.	PROPN
ejpam-5335	287	8	viriyapong	viriyapong	PROPN
ejpam-5335	287	9	/	/	SYM
ejpam-5335	287	10	eur	eur	PROPN
ejpam-5335	287	11	.	.	PUNCT
ejpam-5335	288	1	j.	j.	PROPN
ejpam-5335	288	2	pure	pure	PROPN
ejpam-5335	288	3	appl	appl	PROPN
ejpam-5335	288	4	.	.	PROPN
ejpam-5335	288	5	math	math	PROPN
ejpam-5335	288	6	,	,	PUNCT
ejpam-5335	288	7	17	17	NUM
ejpam-5335	288	8	(	(	PUNCT
ejpam-5335	288	9	4	4	NUM
ejpam-5335	288	10	)	)	PUNCT
ejpam-5335	288	11	(	(	PUNCT
ejpam-5335	288	12	2024	2024	NUM
ejpam-5335	288	13	)	)	PUNCT
ejpam-5335	288	14	,	,	PUNCT
ejpam-5335	288	15	3730	3730	NUM
ejpam-5335	288	16	-	-	SYM
ejpam-5335	288	17	3742	3742	NUM
ejpam-5335	288	18	3738	3738	NUM
ejpam-5335	288	19	example	example	NOUN
ejpam-5335	288	20	3	3	X
ejpam-5335	288	21	.	.	PUNCT
ejpam-5335	289	1	let	let	VERB
ejpam-5335	289	2	x	x	PUNCT
ejpam-5335	289	3	=	=	PRON
ejpam-5335	289	4	{	{	PUNCT
ejpam-5335	289	5	a	a	PRON
ejpam-5335	289	6	,	,	PUNCT
ejpam-5335	289	7	b	b	NOUN
ejpam-5335	289	8	,	,	PUNCT
ejpam-5335	289	9	c	c	NOUN
ejpam-5335	289	10	}	}	PUNCT
ejpam-5335	289	11	with	with	ADP
ejpam-5335	289	12	topologies	topology	NOUN
ejpam-5335	289	13	τ1	τ1	NOUN
ejpam-5335	289	14	=	=	SYM
ejpam-5335	289	15	{	{	PUNCT
ejpam-5335	289	16	∅	∅	NOUN
ejpam-5335	289	17	,	,	PUNCT
ejpam-5335	289	18	{	{	PUNCT
ejpam-5335	289	19	a	a	X
ejpam-5335	289	20	}	}	PUNCT
ejpam-5335	289	21	,	,	PUNCT
ejpam-5335	289	22	{	{	PUNCT
ejpam-5335	289	23	b	b	NOUN
ejpam-5335	289	24	}	}	PUNCT
ejpam-5335	289	25	,	,	PUNCT
ejpam-5335	289	26	{	{	PUNCT
ejpam-5335	289	27	a	a	DET
ejpam-5335	289	28	,	,	PUNCT
ejpam-5335	289	29	b	b	NOUN
ejpam-5335	289	30	}	}	PUNCT
ejpam-5335	289	31	,	,	PUNCT
ejpam-5335	289	32	{	{	PUNCT
ejpam-5335	289	33	a	a	X
ejpam-5335	289	34	,	,	PUNCT
ejpam-5335	289	35	c	c	NOUN
ejpam-5335	289	36	}	}	PUNCT
ejpam-5335	289	37	,	,	PUNCT
ejpam-5335	289	38	x	x	NOUN
ejpam-5335	289	39	}	}	PUNCT
ejpam-5335	289	40	and	and	CCONJ
ejpam-5335	289	41	τ2	τ2	NOUN
ejpam-5335	289	42	=	=	SYM
ejpam-5335	289	43	{	{	PUNCT
ejpam-5335	289	44	∅	∅	NOUN
ejpam-5335	289	45	,	,	PUNCT
ejpam-5335	289	46	{	{	PUNCT
ejpam-5335	289	47	a	a	X
ejpam-5335	289	48	}	}	PUNCT
ejpam-5335	289	49	,	,	PUNCT
ejpam-5335	289	50	{	{	PUNCT
ejpam-5335	289	51	b	b	NOUN
ejpam-5335	289	52	}	}	PUNCT
ejpam-5335	289	53	,	,	PUNCT
ejpam-5335	289	54	{	{	PUNCT
ejpam-5335	289	55	a	a	DET
ejpam-5335	289	56	,	,	PUNCT
ejpam-5335	289	57	b	b	NOUN
ejpam-5335	289	58	}	}	PUNCT
ejpam-5335	289	59	,	,	PUNCT
ejpam-5335	289	60	x	x	NOUN
ejpam-5335	289	61	}	}	PUNCT
ejpam-5335	289	62	.	.	PUNCT
ejpam-5335	290	1	let	let	VERB
ejpam-5335	290	2	y	y	PROPN
ejpam-5335	290	3	=	=	PUNCT
ejpam-5335	290	4	{	{	PUNCT
ejpam-5335	290	5	1	1	NUM
ejpam-5335	290	6	,	,	PUNCT
ejpam-5335	290	7	2	2	NUM
ejpam-5335	290	8	,	,	PUNCT
ejpam-5335	290	9	3	3	NUM
ejpam-5335	290	10	}	}	PUNCT
ejpam-5335	290	11	with	with	ADP
ejpam-5335	290	12	topologies	topology	NOUN
ejpam-5335	290	13	σ1	σ1	NOUN
ejpam-5335	290	14	=	=	SYM
ejpam-5335	290	15	{	{	PUNCT
ejpam-5335	290	16	∅	∅	NOUN
ejpam-5335	290	17	,	,	PUNCT
ejpam-5335	290	18	{	{	PUNCT
ejpam-5335	290	19	1	1	NUM
ejpam-5335	290	20	}	}	PUNCT
ejpam-5335	290	21	,	,	PUNCT
ejpam-5335	290	22	{	{	PUNCT
ejpam-5335	290	23	3	3	NUM
ejpam-5335	290	24	}	}	PUNCT
ejpam-5335	290	25	,	,	PUNCT
ejpam-5335	290	26	{	{	PUNCT
ejpam-5335	290	27	1	1	NUM
ejpam-5335	290	28	,	,	PUNCT
ejpam-5335	290	29	3	3	NUM
ejpam-5335	290	30	}	}	PUNCT
ejpam-5335	290	31	,	,	PUNCT
ejpam-5335	290	32	y	y	PROPN
ejpam-5335	290	33	}	}	PUNCT
ejpam-5335	290	34	and	and	CCONJ
ejpam-5335	290	35	σ2	σ2	PROPN
ejpam-5335	290	36	=	=	SYM
ejpam-5335	290	37	{	{	PUNCT
ejpam-5335	290	38	∅	∅	NOUN
ejpam-5335	290	39	,	,	PUNCT
ejpam-5335	290	40	{	{	PUNCT
ejpam-5335	290	41	1	1	NUM
ejpam-5335	290	42	}	}	PUNCT
ejpam-5335	290	43	,	,	PUNCT
ejpam-5335	290	44	{	{	PUNCT
ejpam-5335	290	45	3	3	NUM
ejpam-5335	290	46	}	}	PUNCT
ejpam-5335	290	47	,	,	PUNCT
ejpam-5335	290	48	{	{	PUNCT
ejpam-5335	290	49	1	1	NUM
ejpam-5335	290	50	,	,	PUNCT
ejpam-5335	290	51	2	2	NUM
ejpam-5335	290	52	}	}	PUNCT
ejpam-5335	290	53	,	,	PUNCT
ejpam-5335	290	54	{	{	PUNCT
ejpam-5335	290	55	1	1	NUM
ejpam-5335	290	56	,	,	PUNCT
ejpam-5335	290	57	3	3	NUM
ejpam-5335	290	58	}	}	PUNCT
ejpam-5335	290	59	,	,	PUNCT
ejpam-5335	290	60	y	y	PROPN
ejpam-5335	290	61	}	}	PUNCT
ejpam-5335	290	62	.	.	PUNCT
ejpam-5335	291	1	a	a	DET
ejpam-5335	291	2	function	function	NOUN
ejpam-5335	291	3	f	f	NOUN
ejpam-5335	291	4	:	:	PUNCT
ejpam-5335	291	5	(	(	PUNCT
ejpam-5335	291	6	x	x	NOUN
ejpam-5335	291	7	,	,	PUNCT
ejpam-5335	291	8	τ1	τ1	NOUN
ejpam-5335	291	9	,	,	PUNCT
ejpam-5335	291	10	τ2	τ2	NOUN
ejpam-5335	291	11	)	)	PUNCT
ejpam-5335	291	12	→	→	SYM
ejpam-5335	291	13	(	(	PUNCT
ejpam-5335	291	14	y	y	PROPN
ejpam-5335	291	15	,	,	PUNCT
ejpam-5335	291	16	σ1	σ1	PROPN
ejpam-5335	291	17	,	,	PUNCT
ejpam-5335	291	18	σ2	σ2	PROPN
ejpam-5335	291	19	)	)	PUNCT
ejpam-5335	291	20	is	be	AUX
ejpam-5335	291	21	defined	define	VERB
ejpam-5335	291	22	as	as	SCONJ
ejpam-5335	291	23	follows	follow	VERB
ejpam-5335	291	24	:	:	PUNCT
ejpam-5335	291	25	f(a	f(a	NOUN
ejpam-5335	291	26	)	)	PUNCT
ejpam-5335	291	27	=	=	SYM
ejpam-5335	291	28	f(b	f(b	X
ejpam-5335	291	29	)	)	PUNCT
ejpam-5335	292	1	=	=	SYM
ejpam-5335	292	2	2	2	NUM
ejpam-5335	292	3	and	and	CCONJ
ejpam-5335	292	4	f(c	f(c	PROPN
ejpam-5335	292	5	)	)	PUNCT
ejpam-5335	292	6	=	=	SYM
ejpam-5335	292	7	1	1	X
ejpam-5335	292	8	.	.	PUNCT
ejpam-5335	292	9	then	then	ADV
ejpam-5335	292	10	f	f	PROPN
ejpam-5335	292	11	is	be	AUX
ejpam-5335	292	12	weakly	weakly	ADJ
ejpam-5335	292	13	δ(τ1	δ(τ1	NOUN
ejpam-5335	292	14	,	,	PUNCT
ejpam-5335	292	15	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	292	16	,	,	PUNCT
ejpam-5335	292	17	but	but	CCONJ
ejpam-5335	292	18	f	f	PROPN
ejpam-5335	292	19	is	be	AUX
ejpam-5335	292	20	not	not	PART
ejpam-5335	292	21	almost	almost	ADV
ejpam-5335	292	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	292	23	,	,	PUNCT
ejpam-5335	292	24	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5335	292	25	.	.	PUNCT
ejpam-5335	293	1	theorem	theorem	VERB
ejpam-5335	293	2	6	6	NUM
ejpam-5335	293	3	.	.	PUNCT
ejpam-5335	294	1	a	a	DET
ejpam-5335	294	2	function	function	NOUN
ejpam-5335	294	3	f	f	NOUN
ejpam-5335	294	4	:	:	PUNCT
ejpam-5335	294	5	(	(	PUNCT
ejpam-5335	294	6	x	x	NOUN
ejpam-5335	294	7	,	,	PUNCT
ejpam-5335	294	8	τ1	τ1	NOUN
ejpam-5335	294	9	,	,	PUNCT
ejpam-5335	294	10	τ2	τ2	NOUN
ejpam-5335	294	11	)	)	PUNCT
ejpam-5335	294	12	→	→	SYM
ejpam-5335	294	13	(	(	PUNCT
ejpam-5335	294	14	y	y	PROPN
ejpam-5335	294	15	,	,	PUNCT
ejpam-5335	294	16	σ1	σ1	PROPN
ejpam-5335	294	17	,	,	PUNCT
ejpam-5335	294	18	σ2	σ2	NOUN
ejpam-5335	294	19	)	)	PUNCT
ejpam-5335	294	20	is	be	AUX
ejpam-5335	294	21	weakly	weakly	ADJ
ejpam-5335	294	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	294	23	,	,	PUNCT
ejpam-5335	294	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	294	25	at	at	ADP
ejpam-5335	294	26	x	x	X
ejpam-5335	294	27	∈	∈	PROPN
ejpam-5335	294	28	x	x	SYM
ejpam-5335	294	29	if	if	SCONJ
ejpam-5335	294	30	and	and	CCONJ
ejpam-5335	294	31	only	only	ADV
ejpam-5335	294	32	if	if	SCONJ
ejpam-5335	294	33	x	x	PROPN
ejpam-5335	294	34	∈	∈	PROPN
ejpam-5335	294	35	δ(τ1	δ(τ1	PROPN
ejpam-5335	294	36	,	,	PUNCT
ejpam-5335	294	37	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	294	38	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	294	39	-	-	PUNCT
ejpam-5335	294	40	cl(v	cl(v	NOUN
ejpam-5335	294	41	)	)	PUNCT
ejpam-5335	294	42	)	)	PUNCT
ejpam-5335	294	43	)	)	PUNCT
ejpam-5335	295	1	for	for	ADP
ejpam-5335	295	2	every	every	DET
ejpam-5335	295	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	295	4	-	-	ADJ
ejpam-5335	295	5	open	open	ADJ
ejpam-5335	295	6	set	set	NOUN
ejpam-5335	295	7	v	v	NOUN
ejpam-5335	295	8	of	of	ADP
ejpam-5335	295	9	y	y	NOUN
ejpam-5335	295	10	containing	contain	VERB
ejpam-5335	295	11	f(x	f(x	PROPN
ejpam-5335	295	12	)	)	PUNCT
ejpam-5335	295	13	.	.	PUNCT
ejpam-5335	296	1	proof	proof	NOUN
ejpam-5335	296	2	.	.	PUNCT
ejpam-5335	297	1	let	let	VERB
ejpam-5335	297	2	x	x	PUNCT
ejpam-5335	297	3	∈	∈	PROPN
ejpam-5335	297	4	x	x	X
ejpam-5335	297	5	and	and	CCONJ
ejpam-5335	297	6	v	v	X
ejpam-5335	297	7	be	be	AUX
ejpam-5335	297	8	any	any	DET
ejpam-5335	297	9	σ1σ2	σ1σ2	NOUN
ejpam-5335	297	10	-	-	ADJ
ejpam-5335	297	11	open	open	ADJ
ejpam-5335	297	12	set	set	NOUN
ejpam-5335	297	13	of	of	ADP
ejpam-5335	297	14	y	y	PROPN
ejpam-5335	297	15	containing	contain	VERB
ejpam-5335	297	16	f(x	f(x	PROPN
ejpam-5335	297	17	)	)	PUNCT
ejpam-5335	297	18	.	.	PUNCT
ejpam-5335	298	1	since	since	SCONJ
ejpam-5335	298	2	f	f	PROPN
ejpam-5335	298	3	is	be	AUX
ejpam-5335	298	4	weakly	weakly	ADJ
ejpam-5335	298	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	298	6	,	,	PUNCT
ejpam-5335	298	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	298	8	at	at	ADP
ejpam-5335	298	9	x	x	X
ejpam-5335	298	10	,	,	PUNCT
ejpam-5335	298	11	so	so	SCONJ
ejpam-5335	298	12	there	there	PRON
ejpam-5335	298	13	exists	exist	VERB
ejpam-5335	298	14	a	a	DET
ejpam-5335	298	15	δ(τ1	δ(τ1	NOUN
ejpam-5335	298	16	,	,	PUNCT
ejpam-5335	298	17	τ2)-open	τ2)-open	ADJ
ejpam-5335	298	18	set	set	ADJ
ejpam-5335	298	19	u	u	NOUN
ejpam-5335	298	20	of	of	ADP
ejpam-5335	298	21	x	x	PUNCT
ejpam-5335	298	22	containing	contain	VERB
ejpam-5335	298	23	x	x	PUNCT
ejpam-5335	298	24	such	such	ADJ
ejpam-5335	298	25	that	that	DET
ejpam-5335	298	26	f(u	f(u	PROPN
ejpam-5335	298	27	)	)	PUNCT
ejpam-5335	298	28	⊆	⊆	NUM
ejpam-5335	298	29	σ1σ2	σ1σ2	NOUN
ejpam-5335	298	30	-	-	NUM
ejpam-5335	298	31	cl(v	cl(v	NOUN
ejpam-5335	298	32	)	)	PUNCT
ejpam-5335	298	33	.	.	PUNCT
ejpam-5335	299	1	therefore	therefore	ADV
ejpam-5335	299	2	,	,	PUNCT
ejpam-5335	299	3	x	x	PUNCT
ejpam-5335	299	4	∈	∈	PROPN
ejpam-5335	299	5	u	u	NOUN
ejpam-5335	299	6	⊆	⊆	NUM
ejpam-5335	299	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	299	8	-	-	PUNCT
ejpam-5335	299	9	cl(v	cl(v	NOUN
ejpam-5335	299	10	)	)	PUNCT
ejpam-5335	299	11	)	)	PUNCT
ejpam-5335	299	12	.	.	PUNCT
ejpam-5335	300	1	this	this	PRON
ejpam-5335	300	2	implies	imply	VERB
ejpam-5335	300	3	that	that	SCONJ
ejpam-5335	300	4	x	x	PROPN
ejpam-5335	300	5	∈	∈	PROPN
ejpam-5335	300	6	δ(τ1	δ(τ1	PROPN
ejpam-5335	300	7	,	,	PUNCT
ejpam-5335	300	8	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	300	9	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	300	10	-	-	PUNCT
ejpam-5335	300	11	cl(v	cl(v	NOUN
ejpam-5335	300	12	)	)	PUNCT
ejpam-5335	300	13	)	)	PUNCT
ejpam-5335	300	14	)	)	PUNCT
ejpam-5335	300	15	.	.	PUNCT
ejpam-5335	301	1	conversely	conversely	ADV
ejpam-5335	301	2	let	let	VERB
ejpam-5335	301	3	v	v	PART
ejpam-5335	301	4	be	be	AUX
ejpam-5335	301	5	any	any	DET
ejpam-5335	301	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	301	7	-	-	ADJ
ejpam-5335	301	8	open	open	ADJ
ejpam-5335	301	9	set	set	NOUN
ejpam-5335	301	10	of	of	ADP
ejpam-5335	301	11	y	y	PROPN
ejpam-5335	301	12	containing	contain	VERB
ejpam-5335	301	13	f(x	f(x	PROPN
ejpam-5335	301	14	)	)	PUNCT
ejpam-5335	301	15	.	.	PUNCT
ejpam-5335	302	1	by	by	ADP
ejpam-5335	302	2	the	the	DET
ejpam-5335	302	3	hypothesis	hypothesis	NOUN
ejpam-5335	302	4	,	,	PUNCT
ejpam-5335	302	5	we	we	PRON
ejpam-5335	302	6	obtain	obtain	VERB
ejpam-5335	302	7	that	that	SCONJ
ejpam-5335	302	8	x	x	PROPN
ejpam-5335	302	9	∈	∈	PROPN
ejpam-5335	302	10	δ(τ1	δ(τ1	PROPN
ejpam-5335	302	11	,	,	PUNCT
ejpam-5335	302	12	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	302	13	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	302	14	-	-	PUNCT
ejpam-5335	302	15	cl(v	cl(v	NOUN
ejpam-5335	302	16	)	)	PUNCT
ejpam-5335	302	17	)	)	PUNCT
ejpam-5335	302	18	)	)	PUNCT
ejpam-5335	302	19	.	.	PUNCT
ejpam-5335	303	1	then	then	ADV
ejpam-5335	303	2	,	,	PUNCT
ejpam-5335	303	3	there	there	PRON
ejpam-5335	303	4	exists	exist	VERB
ejpam-5335	303	5	a	a	DET
ejpam-5335	303	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	303	7	,	,	PUNCT
ejpam-5335	303	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	303	9	set	set	ADJ
ejpam-5335	303	10	u	u	NOUN
ejpam-5335	303	11	of	of	ADP
ejpam-5335	303	12	x	x	PUNCT
ejpam-5335	303	13	containing	contain	VERB
ejpam-5335	303	14	x	x	PUNCT
ejpam-5335	303	15	such	such	ADJ
ejpam-5335	303	16	that	that	SCONJ
ejpam-5335	303	17	u	u	PROPN
ejpam-5335	303	18	⊆	⊆	NUM
ejpam-5335	303	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	303	20	-	-	PUNCT
ejpam-5335	303	21	cl(v	cl(v	NOUN
ejpam-5335	303	22	)	)	PUNCT
ejpam-5335	303	23	)	)	PUNCT
ejpam-5335	303	24	.	.	PUNCT
ejpam-5335	304	1	thus	thus	ADV
ejpam-5335	304	2	,	,	PUNCT
ejpam-5335	304	3	f(u	f(u	PROPN
ejpam-5335	304	4	)	)	PUNCT
ejpam-5335	304	5	⊆	⊆	NUM
ejpam-5335	304	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	304	7	-	-	NUM
ejpam-5335	304	8	cl(v	cl(v	NOUN
ejpam-5335	304	9	)	)	PUNCT
ejpam-5335	304	10	.	.	PUNCT
ejpam-5335	305	1	this	this	PRON
ejpam-5335	305	2	shows	show	VERB
ejpam-5335	305	3	that	that	SCONJ
ejpam-5335	305	4	f	f	PROPN
ejpam-5335	305	5	is	be	AUX
ejpam-5335	305	6	weakly	weakly	ADJ
ejpam-5335	305	7	δ(τ1	δ(τ1	NOUN
ejpam-5335	305	8	,	,	PUNCT
ejpam-5335	305	9	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	305	10	at	at	ADP
ejpam-5335	305	11	x.	x.	NOUN
ejpam-5335	305	12	theorem	theorem	VERB
ejpam-5335	305	13	7	7	NUM
ejpam-5335	305	14	.	.	PUNCT
ejpam-5335	306	1	a	a	DET
ejpam-5335	306	2	function	function	NOUN
ejpam-5335	306	3	f	f	NOUN
ejpam-5335	306	4	:	:	PUNCT
ejpam-5335	306	5	(	(	PUNCT
ejpam-5335	306	6	x	x	NOUN
ejpam-5335	306	7	,	,	PUNCT
ejpam-5335	306	8	τ1	τ1	NOUN
ejpam-5335	306	9	,	,	PUNCT
ejpam-5335	306	10	τ2	τ2	NOUN
ejpam-5335	306	11	)	)	PUNCT
ejpam-5335	306	12	→	→	SYM
ejpam-5335	306	13	(	(	PUNCT
ejpam-5335	306	14	y	y	PROPN
ejpam-5335	306	15	,	,	PUNCT
ejpam-5335	306	16	σ1	σ1	PROPN
ejpam-5335	306	17	,	,	PUNCT
ejpam-5335	306	18	σ2	σ2	NOUN
ejpam-5335	306	19	)	)	PUNCT
ejpam-5335	306	20	is	be	AUX
ejpam-5335	306	21	weakly	weakly	ADJ
ejpam-5335	306	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	306	23	,	,	PUNCT
ejpam-5335	306	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	306	25	if	if	SCONJ
ejpam-5335	306	26	and	and	CCONJ
ejpam-5335	306	27	only	only	ADV
ejpam-5335	306	28	if	if	SCONJ
ejpam-5335	306	29	f−1(v	f−1(v	PROPN
ejpam-5335	306	30	)	)	PUNCT
ejpam-5335	307	1	⊆	⊆	NUM
ejpam-5335	307	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	307	3	,	,	PUNCT
ejpam-5335	307	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	307	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	307	6	-	-	PUNCT
ejpam-5335	307	7	cl(v	cl(v	NOUN
ejpam-5335	307	8	)	)	PUNCT
ejpam-5335	307	9	)	)	PUNCT
ejpam-5335	307	10	)	)	PUNCT
ejpam-5335	307	11	for	for	ADP
ejpam-5335	307	12	every	every	DET
ejpam-5335	307	13	σ1σ2	σ1σ2	NOUN
ejpam-5335	307	14	-	-	ADJ
ejpam-5335	307	15	open	open	ADJ
ejpam-5335	307	16	set	set	NOUN
ejpam-5335	307	17	v	v	NOUN
ejpam-5335	307	18	of	of	ADP
ejpam-5335	307	19	y	y	PROPN
ejpam-5335	307	20	.	.	PUNCT
ejpam-5335	308	1	proof	proof	NOUN
ejpam-5335	308	2	.	.	PUNCT
ejpam-5335	309	1	let	let	VERB
ejpam-5335	309	2	v	v	PART
ejpam-5335	309	3	be	be	AUX
ejpam-5335	309	4	any	any	DET
ejpam-5335	309	5	σ1σ2	σ1σ2	NOUN
ejpam-5335	309	6	-	-	ADJ
ejpam-5335	309	7	open	open	ADJ
ejpam-5335	309	8	set	set	NOUN
ejpam-5335	309	9	of	of	ADP
ejpam-5335	309	10	y	y	PROPN
ejpam-5335	309	11	and	and	CCONJ
ejpam-5335	309	12	x	x	PROPN
ejpam-5335	309	13	∈	∈	PROPN
ejpam-5335	309	14	f−1(v	f−1(v	NOUN
ejpam-5335	309	15	)	)	PUNCT
ejpam-5335	309	16	.	.	PUNCT
ejpam-5335	310	1	since	since	SCONJ
ejpam-5335	310	2	f	f	PROPN
ejpam-5335	310	3	is	be	AUX
ejpam-5335	310	4	weakly	weakly	ADJ
ejpam-5335	310	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	310	6	,	,	PUNCT
ejpam-5335	310	7	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	310	8	,	,	PUNCT
ejpam-5335	310	9	by	by	ADP
ejpam-5335	310	10	theorem	theorem	NOUN
ejpam-5335	310	11	6	6	NUM
ejpam-5335	310	12	,	,	PUNCT
ejpam-5335	310	13	x	x	PROPN
ejpam-5335	310	14	∈	∈	PROPN
ejpam-5335	310	15	δ(τ1	δ(τ1	PROPN
ejpam-5335	310	16	,	,	PUNCT
ejpam-5335	310	17	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	310	18	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	310	19	-	-	PUNCT
ejpam-5335	310	20	cl(v	cl(v	NOUN
ejpam-5335	310	21	)	)	PUNCT
ejpam-5335	310	22	)	)	PUNCT
ejpam-5335	310	23	)	)	PUNCT
ejpam-5335	310	24	.	.	PUNCT
ejpam-5335	311	1	therefore	therefore	ADV
ejpam-5335	311	2	,	,	PUNCT
ejpam-5335	311	3	f−1(v	f−1(v	PROPN
ejpam-5335	311	4	)	)	PUNCT
ejpam-5335	312	1	⊆	⊆	NUM
ejpam-5335	312	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	312	3	,	,	PUNCT
ejpam-5335	312	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	312	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	312	6	-	-	PUNCT
ejpam-5335	312	7	cl(v	cl(v	NOUN
ejpam-5335	312	8	)	)	PUNCT
ejpam-5335	312	9	)	)	PUNCT
ejpam-5335	312	10	)	)	PUNCT
ejpam-5335	312	11	.	.	PUNCT
ejpam-5335	313	1	conversely	conversely	ADV
ejpam-5335	313	2	,	,	PUNCT
ejpam-5335	313	3	let	let	VERB
ejpam-5335	313	4	x	x	X
ejpam-5335	313	5	∈	∈	PROPN
ejpam-5335	313	6	x	x	X
ejpam-5335	313	7	and	and	CCONJ
ejpam-5335	313	8	v	v	AUX
ejpam-5335	313	9	be	be	AUX
ejpam-5335	313	10	any	any	DET
ejpam-5335	313	11	σ1σ2	σ1σ2	NOUN
ejpam-5335	313	12	-	-	ADJ
ejpam-5335	313	13	open	open	ADJ
ejpam-5335	313	14	set	set	NOUN
ejpam-5335	313	15	of	of	ADP
ejpam-5335	313	16	y	y	PROPN
ejpam-5335	313	17	containing	contain	VERB
ejpam-5335	313	18	f(x	f(x	PROPN
ejpam-5335	313	19	)	)	PUNCT
ejpam-5335	313	20	.	.	PUNCT
ejpam-5335	314	1	by	by	ADP
ejpam-5335	314	2	the	the	DET
ejpam-5335	314	3	hypothesis	hypothesis	NOUN
ejpam-5335	314	4	,	,	PUNCT
ejpam-5335	314	5	x	x	PROPN
ejpam-5335	314	6	∈	∈	PROPN
ejpam-5335	314	7	f−1(v	f−1(v	NOUN
ejpam-5335	314	8	)	)	PUNCT
ejpam-5335	315	1	⊆	⊆	NUM
ejpam-5335	315	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	315	3	,	,	PUNCT
ejpam-5335	315	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	315	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	315	6	-	-	PUNCT
ejpam-5335	315	7	cl(v	cl(v	NOUN
ejpam-5335	315	8	)	)	PUNCT
ejpam-5335	315	9	)	)	PUNCT
ejpam-5335	315	10	)	)	PUNCT
ejpam-5335	315	11	.	.	PUNCT
ejpam-5335	316	1	by	by	ADP
ejpam-5335	316	2	theorem	theorem	NOUN
ejpam-5335	316	3	6	6	NUM
ejpam-5335	316	4	,	,	PUNCT
ejpam-5335	316	5	f	f	PROPN
ejpam-5335	316	6	is	be	AUX
ejpam-5335	316	7	weakly	weakly	ADJ
ejpam-5335	316	8	δ(τ1	δ(τ1	NOUN
ejpam-5335	316	9	,	,	PUNCT
ejpam-5335	316	10	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	316	11	at	at	ADP
ejpam-5335	316	12	x.	x.	NOUN
ejpam-5335	316	13	this	this	PRON
ejpam-5335	316	14	shows	show	VERB
ejpam-5335	316	15	that	that	SCONJ
ejpam-5335	316	16	f	f	PROPN
ejpam-5335	316	17	is	be	AUX
ejpam-5335	316	18	weakly	weakly	ADJ
ejpam-5335	316	19	δ(τ1	δ(τ1	NOUN
ejpam-5335	316	20	,	,	PUNCT
ejpam-5335	316	21	τ2)-continuous	τ2)-continuous	PROPN
ejpam-5335	316	22	.	.	PUNCT
ejpam-5335	317	1	theorem	theorem	VERB
ejpam-5335	317	2	8	8	NUM
ejpam-5335	317	3	.	.	PUNCT
ejpam-5335	318	1	for	for	ADP
ejpam-5335	318	2	a	a	DET
ejpam-5335	318	3	function	function	NOUN
ejpam-5335	318	4	f	f	NOUN
ejpam-5335	318	5	:	:	PUNCT
ejpam-5335	318	6	(	(	PUNCT
ejpam-5335	318	7	x	x	NOUN
ejpam-5335	318	8	,	,	PUNCT
ejpam-5335	318	9	τ1	τ1	NOUN
ejpam-5335	318	10	,	,	PUNCT
ejpam-5335	318	11	τ2	τ2	NOUN
ejpam-5335	318	12	)	)	PUNCT
ejpam-5335	318	13	→	→	SYM
ejpam-5335	318	14	(	(	PUNCT
ejpam-5335	318	15	y	y	PROPN
ejpam-5335	318	16	,	,	PUNCT
ejpam-5335	318	17	σ1	σ1	PROPN
ejpam-5335	318	18	,	,	PUNCT
ejpam-5335	318	19	σ2	σ2	NOUN
ejpam-5335	318	20	)	)	PUNCT
ejpam-5335	318	21	,	,	PUNCT
ejpam-5335	318	22	the	the	DET
ejpam-5335	318	23	following	follow	VERB
ejpam-5335	318	24	properties	property	NOUN
ejpam-5335	318	25	are	be	AUX
ejpam-5335	318	26	equivalent	equivalent	ADJ
ejpam-5335	318	27	:	:	PUNCT
ejpam-5335	318	28	(	(	PUNCT
ejpam-5335	318	29	1	1	X
ejpam-5335	318	30	)	)	PUNCT
ejpam-5335	318	31	f	f	PROPN
ejpam-5335	318	32	is	be	AUX
ejpam-5335	318	33	weakly	weakly	ADJ
ejpam-5335	318	34	δ(τ1	δ(τ1	NOUN
ejpam-5335	318	35	,	,	PUNCT
ejpam-5335	318	36	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	318	37	;	;	PUNCT
ejpam-5335	318	38	(	(	PUNCT
ejpam-5335	318	39	2	2	X
ejpam-5335	318	40	)	)	PUNCT
ejpam-5335	318	41	f−1(v	f−1(v	NOUN
ejpam-5335	318	42	)	)	PUNCT
ejpam-5335	319	1	⊆	⊆	NUM
ejpam-5335	319	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	319	3	,	,	PUNCT
ejpam-5335	319	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	319	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	319	6	-	-	PUNCT
ejpam-5335	319	7	cl(v	cl(v	NOUN
ejpam-5335	319	8	)	)	PUNCT
ejpam-5335	319	9	)	)	PUNCT
ejpam-5335	319	10	)	)	PUNCT
ejpam-5335	319	11	for	for	ADP
ejpam-5335	319	12	every	every	DET
ejpam-5335	319	13	σ1σ2	σ1σ2	NOUN
ejpam-5335	319	14	-	-	ADJ
ejpam-5335	319	15	open	open	ADJ
ejpam-5335	319	16	set	set	NOUN
ejpam-5335	319	17	v	v	NOUN
ejpam-5335	319	18	of	of	ADP
ejpam-5335	319	19	y	y	PROPN
ejpam-5335	319	20	;	;	PUNCT
ejpam-5335	319	21	(	(	PUNCT
ejpam-5335	319	22	3	3	X
ejpam-5335	319	23	)	)	PUNCT
ejpam-5335	319	24	δ(τ1	δ(τ1	NOUN
ejpam-5335	319	25	,	,	PUNCT
ejpam-5335	319	26	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	319	27	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	319	28	-	-	PUNCT
ejpam-5335	319	29	int(f	int(f	PROPN
ejpam-5335	319	30	)	)	PUNCT
ejpam-5335	319	31	)	)	PUNCT
ejpam-5335	319	32	)	)	PUNCT
ejpam-5335	320	1	⊆	⊆	NUM
ejpam-5335	320	2	f−1(f	f−1(f	PROPN
ejpam-5335	320	3	)	)	PUNCT
ejpam-5335	320	4	for	for	ADP
ejpam-5335	320	5	every	every	DET
ejpam-5335	320	6	σ1σ2	σ1σ2	NUM
ejpam-5335	320	7	-	-	PUNCT
ejpam-5335	320	8	closed	closed	ADJ
ejpam-5335	320	9	f	f	NOUN
ejpam-5335	320	10	of	of	ADP
ejpam-5335	320	11	y	y	PROPN
ejpam-5335	320	12	;	;	PUNCT
ejpam-5335	320	13	(	(	PUNCT
ejpam-5335	320	14	4	4	X
ejpam-5335	320	15	)	)	PUNCT
ejpam-5335	320	16	δ(τ1	δ(τ1	NOUN
ejpam-5335	320	17	,	,	PUNCT
ejpam-5335	320	18	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	320	19	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	320	20	-	-	PUNCT
ejpam-5335	320	21	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	320	22	-	-	PUNCT
ejpam-5335	320	23	cl(b	cl(b	NOUN
ejpam-5335	320	24	)	)	PUNCT
ejpam-5335	320	25	)	)	PUNCT
ejpam-5335	320	26	)	)	PUNCT
ejpam-5335	320	27	)	)	PUNCT
ejpam-5335	321	1	⊆	⊆	NUM
ejpam-5335	321	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	321	3	-	-	PUNCT
ejpam-5335	321	4	cl(b	cl(b	NOUN
ejpam-5335	321	5	)	)	PUNCT
ejpam-5335	321	6	)	)	PUNCT
ejpam-5335	321	7	for	for	ADP
ejpam-5335	321	8	every	every	DET
ejpam-5335	321	9	subset	subset	NOUN
ejpam-5335	321	10	b	b	PROPN
ejpam-5335	321	11	of	of	ADP
ejpam-5335	321	12	y	y	PROPN
ejpam-5335	321	13	;	;	PUNCT
ejpam-5335	321	14	(	(	PUNCT
ejpam-5335	321	15	5	5	X
ejpam-5335	321	16	)	)	PUNCT
ejpam-5335	321	17	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	321	18	-	-	PUNCT
ejpam-5335	321	19	int(b	int(b	NOUN
ejpam-5335	321	20	)	)	PUNCT
ejpam-5335	321	21	)	)	PUNCT
ejpam-5335	321	22	⊆	⊆	NUM
ejpam-5335	321	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	321	24	,	,	PUNCT
ejpam-5335	321	25	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	321	26	−1(σ1σ2	−1(σ1σ2	PROPN
ejpam-5335	321	27	-	-	PUNCT
ejpam-5335	321	28	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	321	29	-	-	PUNCT
ejpam-5335	321	30	int(b	int(b	NOUN
ejpam-5335	321	31	)	)	PUNCT
ejpam-5335	321	32	)	)	PUNCT
ejpam-5335	321	33	)	)	PUNCT
ejpam-5335	321	34	)	)	PUNCT
ejpam-5335	321	35	for	for	ADP
ejpam-5335	321	36	every	every	DET
ejpam-5335	321	37	subset	subset	NOUN
ejpam-5335	321	38	b	b	PROPN
ejpam-5335	321	39	of	of	ADP
ejpam-5335	321	40	y	y	PROPN
ejpam-5335	321	41	;	;	PUNCT
ejpam-5335	321	42	(	(	PUNCT
ejpam-5335	321	43	6	6	NUM
ejpam-5335	321	44	)	)	PUNCT
ejpam-5335	321	45	δ(τ1	δ(τ1	NOUN
ejpam-5335	321	46	,	,	PUNCT
ejpam-5335	321	47	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	321	48	−1(v	−1(v	PROPN
ejpam-5335	321	49	)	)	PUNCT
ejpam-5335	321	50	)	)	PUNCT
ejpam-5335	321	51	⊆	⊆	NUM
ejpam-5335	321	52	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	321	53	-	-	PUNCT
ejpam-5335	321	54	cl(v	cl(v	NOUN
ejpam-5335	321	55	)	)	PUNCT
ejpam-5335	321	56	)	)	PUNCT
ejpam-5335	321	57	for	for	ADP
ejpam-5335	321	58	every	every	DET
ejpam-5335	321	59	σ1σ2	σ1σ2	NOUN
ejpam-5335	321	60	-	-	ADJ
ejpam-5335	321	61	open	open	ADJ
ejpam-5335	321	62	set	set	NOUN
ejpam-5335	321	63	v	v	NOUN
ejpam-5335	321	64	of	of	ADP
ejpam-5335	321	65	y	y	PROPN
ejpam-5335	321	66	;	;	PUNCT
ejpam-5335	321	67	(	(	PUNCT
ejpam-5335	321	68	7	7	X
ejpam-5335	321	69	)	)	PUNCT
ejpam-5335	321	70	δ(τ1	δ(τ1	NOUN
ejpam-5335	321	71	,	,	PUNCT
ejpam-5335	321	72	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	321	73	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	321	74	-	-	PUNCT
ejpam-5335	321	75	int(f	int(f	PROPN
ejpam-5335	321	76	)	)	PUNCT
ejpam-5335	321	77	)	)	PUNCT
ejpam-5335	321	78	)	)	PUNCT
ejpam-5335	322	1	⊆	⊆	NUM
ejpam-5335	322	2	f−1(f	f−1(f	PROPN
ejpam-5335	322	3	)	)	PUNCT
ejpam-5335	322	4	for	for	ADP
ejpam-5335	322	5	every	every	DET
ejpam-5335	322	6	(	(	PUNCT
ejpam-5335	322	7	σ1	σ1	PROPN
ejpam-5335	322	8	,	,	PUNCT
ejpam-5335	322	9	σ2)r	σ2)r	NOUN
ejpam-5335	322	10	-	-	PUNCT
ejpam-5335	322	11	closed	close	VERB
ejpam-5335	322	12	set	set	ADJ
ejpam-5335	322	13	f	f	PROPN
ejpam-5335	322	14	of	of	ADP
ejpam-5335	322	15	y	y	PROPN
ejpam-5335	322	16	;	;	PUNCT
ejpam-5335	322	17	c.	c.	PROPN
ejpam-5335	322	18	prachanpol	prachanpol	PROPN
ejpam-5335	322	19	,	,	PUNCT
ejpam-5335	322	20	c.	c.	PROPN
ejpam-5335	322	21	boonpok	boonpok	PROPN
ejpam-5335	322	22	,	,	PUNCT
ejpam-5335	322	23	c.	c.	PROPN
ejpam-5335	322	24	viriyapong	viriyapong	PROPN
ejpam-5335	322	25	/	/	SYM
ejpam-5335	322	26	eur	eur	PROPN
ejpam-5335	322	27	.	.	PUNCT
ejpam-5335	323	1	j.	j.	PROPN
ejpam-5335	323	2	pure	pure	PROPN
ejpam-5335	323	3	appl	appl	PROPN
ejpam-5335	323	4	.	.	PROPN
ejpam-5335	323	5	math	math	PROPN
ejpam-5335	323	6	,	,	PUNCT
ejpam-5335	323	7	17	17	NUM
ejpam-5335	323	8	(	(	PUNCT
ejpam-5335	323	9	4	4	NUM
ejpam-5335	323	10	)	)	PUNCT
ejpam-5335	323	11	(	(	PUNCT
ejpam-5335	323	12	2024	2024	NUM
ejpam-5335	323	13	)	)	PUNCT
ejpam-5335	323	14	,	,	PUNCT
ejpam-5335	323	15	3730	3730	NUM
ejpam-5335	323	16	-	-	SYM
ejpam-5335	323	17	3742	3742	NUM
ejpam-5335	323	18	3739	3739	NUM
ejpam-5335	323	19	(	(	PUNCT
ejpam-5335	323	20	8)	8)	NUM
ejpam-5335	323	21	δ(τ1	δ(τ1	NOUN
ejpam-5335	323	22	,	,	PUNCT
ejpam-5335	323	23	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	323	24	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	323	25	-	-	PUNCT
ejpam-5335	323	26	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	323	27	-	-	PUNCT
ejpam-5335	323	28	cl(v	cl(v	NOUN
ejpam-5335	323	29	)	)	PUNCT
ejpam-5335	323	30	)	)	PUNCT
ejpam-5335	323	31	)	)	PUNCT
ejpam-5335	323	32	)	)	PUNCT
ejpam-5335	324	1	⊆	⊆	NUM
ejpam-5335	324	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	324	3	-	-	PUNCT
ejpam-5335	324	4	cl(v	cl(v	NOUN
ejpam-5335	324	5	)	)	PUNCT
ejpam-5335	324	6	)	)	PUNCT
ejpam-5335	324	7	for	for	ADP
ejpam-5335	324	8	every	every	DET
ejpam-5335	324	9	(	(	PUNCT
ejpam-5335	324	10	σ1	σ1	PROPN
ejpam-5335	324	11	,	,	PUNCT
ejpam-5335	324	12	σ2)β	σ2)β	NOUN
ejpam-5335	324	13	-	-	PUNCT
ejpam-5335	324	14	open	open	NOUN
ejpam-5335	324	15	set	set	NOUN
ejpam-5335	324	16	v	v	NOUN
ejpam-5335	324	17	of	of	ADP
ejpam-5335	324	18	y	y	PROPN
ejpam-5335	324	19	;	;	PUNCT
ejpam-5335	324	20	(	(	PUNCT
ejpam-5335	324	21	9	9	X
ejpam-5335	324	22	)	)	PUNCT
ejpam-5335	324	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	324	24	,	,	PUNCT
ejpam-5335	324	25	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	324	26	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	324	27	-	-	PUNCT
ejpam-5335	324	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	324	29	-	-	PUNCT
ejpam-5335	324	30	cl(v	cl(v	NOUN
ejpam-5335	324	31	)	)	PUNCT
ejpam-5335	324	32	)	)	PUNCT
ejpam-5335	324	33	)	)	PUNCT
ejpam-5335	324	34	)	)	PUNCT
ejpam-5335	325	1	⊆	⊆	NUM
ejpam-5335	325	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	325	3	-	-	PUNCT
ejpam-5335	325	4	cl(v	cl(v	NOUN
ejpam-5335	325	5	)	)	PUNCT
ejpam-5335	325	6	)	)	PUNCT
ejpam-5335	325	7	for	for	ADP
ejpam-5335	325	8	every	every	DET
ejpam-5335	325	9	(	(	PUNCT
ejpam-5335	325	10	σ1	σ1	PROPN
ejpam-5335	325	11	,	,	PUNCT
ejpam-5335	325	12	σ2)s	σ2)s	NOUN
ejpam-5335	325	13	-	-	PUNCT
ejpam-5335	325	14	open	open	NOUN
ejpam-5335	325	15	set	set	NOUN
ejpam-5335	325	16	v	v	NOUN
ejpam-5335	325	17	of	of	ADP
ejpam-5335	325	18	y	y	PROPN
ejpam-5335	325	19	;	;	PUNCT
ejpam-5335	325	20	(	(	PUNCT
ejpam-5335	325	21	10	10	NUM
ejpam-5335	325	22	)	)	PUNCT
ejpam-5335	325	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	325	24	,	,	PUNCT
ejpam-5335	325	25	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	325	26	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	325	27	-	-	PUNCT
ejpam-5335	325	28	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	325	29	-	-	PUNCT
ejpam-5335	325	30	cl(v	cl(v	NOUN
ejpam-5335	325	31	)	)	PUNCT
ejpam-5335	325	32	)	)	PUNCT
ejpam-5335	325	33	)	)	PUNCT
ejpam-5335	325	34	)	)	PUNCT
ejpam-5335	326	1	⊆	⊆	NUM
ejpam-5335	326	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	326	3	-	-	PUNCT
ejpam-5335	326	4	cl(v	cl(v	NOUN
ejpam-5335	326	5	)	)	PUNCT
ejpam-5335	326	6	)	)	PUNCT
ejpam-5335	326	7	for	for	ADP
ejpam-5335	326	8	every	every	DET
ejpam-5335	326	9	(	(	PUNCT
ejpam-5335	326	10	σ1	σ1	PROPN
ejpam-5335	326	11	,	,	PUNCT
ejpam-5335	326	12	σ2)p	σ2)p	NOUN
ejpam-5335	326	13	-	-	PUNCT
ejpam-5335	326	14	open	open	NOUN
ejpam-5335	326	15	set	set	NOUN
ejpam-5335	326	16	v	v	NOUN
ejpam-5335	326	17	of	of	ADP
ejpam-5335	326	18	y	y	PROPN
ejpam-5335	326	19	;	;	PUNCT
ejpam-5335	326	20	(	(	PUNCT
ejpam-5335	326	21	11	11	NUM
ejpam-5335	326	22	)	)	PUNCT
ejpam-5335	326	23	δ(τ1	δ(τ1	NOUN
ejpam-5335	326	24	,	,	PUNCT
ejpam-5335	326	25	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	326	26	−1(v	−1(v	PROPN
ejpam-5335	326	27	)	)	PUNCT
ejpam-5335	326	28	)	)	PUNCT
ejpam-5335	327	1	⊆	⊆	NUM
ejpam-5335	327	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	327	3	-	-	PUNCT
ejpam-5335	327	4	cl(v	cl(v	NOUN
ejpam-5335	327	5	)	)	PUNCT
ejpam-5335	327	6	)	)	PUNCT
ejpam-5335	327	7	for	for	ADP
ejpam-5335	327	8	every	every	DET
ejpam-5335	327	9	(	(	PUNCT
ejpam-5335	327	10	σ1	σ1	PROPN
ejpam-5335	327	11	,	,	PUNCT
ejpam-5335	327	12	σ2)p	σ2)p	NOUN
ejpam-5335	327	13	-	-	PUNCT
ejpam-5335	327	14	open	open	NOUN
ejpam-5335	327	15	set	set	NOUN
ejpam-5335	327	16	v	v	NOUN
ejpam-5335	327	17	of	of	ADP
ejpam-5335	327	18	y	y	PROPN
ejpam-5335	327	19	;	;	PUNCT
ejpam-5335	327	20	(	(	PUNCT
ejpam-5335	327	21	12	12	X
ejpam-5335	327	22	)	)	PUNCT
ejpam-5335	327	23	f−1(v	f−1(v	NOUN
ejpam-5335	327	24	)	)	PUNCT
ejpam-5335	328	1	⊆	⊆	NUM
ejpam-5335	328	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	328	3	,	,	PUNCT
ejpam-5335	328	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	328	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	328	6	-	-	PUNCT
ejpam-5335	328	7	cl(v	cl(v	NOUN
ejpam-5335	328	8	)	)	PUNCT
ejpam-5335	328	9	)	)	PUNCT
ejpam-5335	328	10	)	)	PUNCT
ejpam-5335	329	1	for	for	ADP
ejpam-5335	329	2	every	every	DET
ejpam-5335	329	3	(	(	PUNCT
ejpam-5335	329	4	σ1	σ1	PROPN
ejpam-5335	329	5	,	,	PUNCT
ejpam-5335	329	6	σ2)p	σ2)p	NOUN
ejpam-5335	329	7	-	-	PUNCT
ejpam-5335	329	8	open	open	NOUN
ejpam-5335	329	9	set	set	NOUN
ejpam-5335	329	10	v	v	NOUN
ejpam-5335	329	11	of	of	ADP
ejpam-5335	329	12	y	y	PROPN
ejpam-5335	329	13	.	.	PUNCT
ejpam-5335	330	1	proof	proof	NOUN
ejpam-5335	330	2	.	.	PUNCT
ejpam-5335	331	1	(	(	PUNCT
ejpam-5335	331	2	1	1	X
ejpam-5335	331	3	)	)	PUNCT
ejpam-5335	331	4	⇒	⇒	NOUN
ejpam-5335	331	5	(	(	PUNCT
ejpam-5335	331	6	2	2	NUM
ejpam-5335	331	7	):	):	PUNCT
ejpam-5335	331	8	let	let	VERB
ejpam-5335	331	9	v	v	PART
ejpam-5335	331	10	be	be	AUX
ejpam-5335	331	11	any	any	DET
ejpam-5335	331	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	331	13	-	-	ADJ
ejpam-5335	331	14	open	open	ADJ
ejpam-5335	331	15	set	set	NOUN
ejpam-5335	331	16	of	of	ADP
ejpam-5335	331	17	y	y	PROPN
ejpam-5335	331	18	.	.	PUNCT
ejpam-5335	332	1	it	it	PRON
ejpam-5335	332	2	follows	follow	VERB
ejpam-5335	332	3	from	from	ADP
ejpam-5335	332	4	theorem	theorem	ADJ
ejpam-5335	332	5	7	7	NUM
ejpam-5335	332	6	,	,	PUNCT
ejpam-5335	332	7	we	we	PRON
ejpam-5335	332	8	obtain	obtain	VERB
ejpam-5335	332	9	that	that	DET
ejpam-5335	332	10	f−1(v	f−1(v	NOUN
ejpam-5335	332	11	)	)	PUNCT
ejpam-5335	333	1	⊆	⊆	NUM
ejpam-5335	333	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	333	3	,	,	PUNCT
ejpam-5335	333	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	333	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	333	6	-	-	PUNCT
ejpam-5335	333	7	cl(v	cl(v	NOUN
ejpam-5335	333	8	)	)	PUNCT
ejpam-5335	333	9	)	)	PUNCT
ejpam-5335	333	10	)	)	PUNCT
ejpam-5335	333	11	.	.	PUNCT
ejpam-5335	334	1	(	(	PUNCT
ejpam-5335	334	2	2	2	X
ejpam-5335	334	3	)	)	PUNCT
ejpam-5335	334	4	⇒	⇒	NOUN
ejpam-5335	334	5	(	(	PUNCT
ejpam-5335	334	6	3	3	NUM
ejpam-5335	334	7	):	):	PUNCT
ejpam-5335	334	8	let	let	VERB
ejpam-5335	334	9	f	f	PRON
ejpam-5335	334	10	be	be	AUX
ejpam-5335	334	11	any	any	DET
ejpam-5335	334	12	σ1σ2	σ1σ2	NUM
ejpam-5335	334	13	-	-	PUNCT
ejpam-5335	334	14	closed	closed	ADJ
ejpam-5335	334	15	set	set	NOUN
ejpam-5335	334	16	of	of	ADP
ejpam-5335	334	17	y	y	PROPN
ejpam-5335	334	18	.	.	PUNCT
ejpam-5335	335	1	then	then	ADV
ejpam-5335	335	2	,	,	PUNCT
ejpam-5335	335	3	y	y	PROPN
ejpam-5335	335	4	−	−	PROPN
ejpam-5335	336	1	f	f	PROPN
ejpam-5335	336	2	is	be	AUX
ejpam-5335	336	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	336	4	-	-	ADJ
ejpam-5335	336	5	open	open	ADJ
ejpam-5335	336	6	in	in	ADP
ejpam-5335	336	7	y	y	PROPN
ejpam-5335	336	8	.	.	PUNCT
ejpam-5335	337	1	thus	thus	ADV
ejpam-5335	337	2	by	by	ADP
ejpam-5335	337	3	(	(	PUNCT
ejpam-5335	337	4	2	2	NUM
ejpam-5335	337	5	)	)	PUNCT
ejpam-5335	337	6	,	,	PUNCT
ejpam-5335	337	7	we	we	PRON
ejpam-5335	337	8	have	have	VERB
ejpam-5335	337	9	x	x	PART
ejpam-5335	337	10	−	−	PROPN
ejpam-5335	337	11	f−1(f	f−1(f	PROPN
ejpam-5335	337	12	)	)	PUNCT
ejpam-5335	338	1	=	=	PUNCT
ejpam-5335	338	2	f−1(y	f−1(y	PROPN
ejpam-5335	339	1	−	−	PROPN
ejpam-5335	339	2	f	f	PROPN
ejpam-5335	339	3	)	)	PUNCT
ejpam-5335	339	4	⊆	⊆	NUM
ejpam-5335	339	5	δ(τ1	δ(τ1	NOUN
ejpam-5335	339	6	,	,	PUNCT
ejpam-5335	339	7	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	339	8	−1(σ1σ2	−1(σ1σ2	PROPN
ejpam-5335	339	9	-	-	PUNCT
ejpam-5335	339	10	cl(y	cl(y	NOUN
ejpam-5335	339	11	−	−	PROPN
ejpam-5335	339	12	f	f	PROPN
ejpam-5335	339	13	)	)	PUNCT
ejpam-5335	339	14	)	)	PUNCT
ejpam-5335	339	15	)	)	PUNCT
ejpam-5335	340	1	=	=	SYM
ejpam-5335	340	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	340	3	,	,	PUNCT
ejpam-5335	340	4	τ2)-int(f	τ2)-int(f	VERB
ejpam-5335	340	5	−1(y	−1(y	X
ejpam-5335	341	1	−	−	VERB
ejpam-5335	341	2	σ1σ2	σ1σ2	NOUN
ejpam-5335	341	3	-	-	NUM
ejpam-5335	341	4	int(f	int(f	NUM
ejpam-5335	341	5	)	)	PUNCT
ejpam-5335	341	6	)	)	PUNCT
ejpam-5335	341	7	)	)	PUNCT
ejpam-5335	342	1	=	=	SYM
ejpam-5335	342	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	342	3	,	,	PUNCT
ejpam-5335	342	4	τ2)-int(x	τ2)-int(x	NOUN
ejpam-5335	342	5	−	−	PRON
ejpam-5335	342	6	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	342	7	-	-	PUNCT
ejpam-5335	342	8	int(f	int(f	PROPN
ejpam-5335	342	9	)	)	PUNCT
ejpam-5335	342	10	)	)	PUNCT
ejpam-5335	342	11	)	)	PUNCT
ejpam-5335	343	1	=	=	PUNCT
ejpam-5335	344	1	x	x	X
ejpam-5335	344	2	−	−	PROPN
ejpam-5335	344	3	δ(τ1	δ(τ1	PROPN
ejpam-5335	344	4	,	,	PUNCT
ejpam-5335	344	5	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	344	6	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	344	7	-	-	PUNCT
ejpam-5335	344	8	int(f	int(f	PROPN
ejpam-5335	344	9	)	)	PUNCT
ejpam-5335	344	10	)	)	PUNCT
ejpam-5335	344	11	)	)	PUNCT
ejpam-5335	344	12	and	and	CCONJ
ejpam-5335	344	13	hence	hence	ADV
ejpam-5335	344	14	δ(τ1	δ(τ1	PROPN
ejpam-5335	344	15	,	,	PUNCT
ejpam-5335	344	16	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	344	17	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	344	18	-	-	PUNCT
ejpam-5335	344	19	int(f	int(f	PROPN
ejpam-5335	344	20	)	)	PUNCT
ejpam-5335	344	21	)	)	PUNCT
ejpam-5335	344	22	)	)	PUNCT
ejpam-5335	345	1	⊆	⊆	NUM
ejpam-5335	345	2	f−1(f	f−1(f	NOUN
ejpam-5335	345	3	)	)	PUNCT
ejpam-5335	345	4	.	.	PUNCT
ejpam-5335	346	1	(	(	PUNCT
ejpam-5335	346	2	3	3	X
ejpam-5335	346	3	)	)	PUNCT
ejpam-5335	346	4	⇒	⇒	NOUN
ejpam-5335	346	5	(	(	PUNCT
ejpam-5335	346	6	4	4	NUM
ejpam-5335	346	7	):	):	PUNCT
ejpam-5335	346	8	let	let	VERB
ejpam-5335	346	9	b	b	X
ejpam-5335	346	10	be	be	AUX
ejpam-5335	346	11	any	any	DET
ejpam-5335	346	12	subset	subset	NOUN
ejpam-5335	346	13	of	of	ADP
ejpam-5335	346	14	y	y	PROPN
ejpam-5335	346	15	.	.	PUNCT
ejpam-5335	347	1	since	since	SCONJ
ejpam-5335	347	2	σ1σ2	σ1σ2	NOUN
ejpam-5335	347	3	-	-	NOUN
ejpam-5335	347	4	cl(b	cl(b	NOUN
ejpam-5335	347	5	)	)	PUNCT
ejpam-5335	347	6	is	be	AUX
ejpam-5335	347	7	σ1σ2	σ1σ2	NOUN
ejpam-5335	347	8	-	-	ADJ
ejpam-5335	347	9	closed	closed	ADJ
ejpam-5335	347	10	in	in	ADP
ejpam-5335	347	11	y	y	PROPN
ejpam-5335	347	12	and	and	CCONJ
ejpam-5335	347	13	by	by	ADP
ejpam-5335	347	14	(	(	PUNCT
ejpam-5335	347	15	3	3	NUM
ejpam-5335	347	16	)	)	PUNCT
ejpam-5335	347	17	,	,	PUNCT
ejpam-5335	347	18	δ(τ1	δ(τ1	PROPN
ejpam-5335	347	19	,	,	PUNCT
ejpam-5335	347	20	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	347	21	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	347	22	-	-	PUNCT
ejpam-5335	347	23	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	347	24	-	-	PUNCT
ejpam-5335	347	25	cl(b	cl(b	NOUN
ejpam-5335	347	26	)	)	PUNCT
ejpam-5335	347	27	)	)	PUNCT
ejpam-5335	347	28	)	)	PUNCT
ejpam-5335	347	29	)	)	PUNCT
ejpam-5335	348	1	⊆	⊆	NUM
ejpam-5335	348	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	348	3	-	-	PUNCT
ejpam-5335	348	4	cl(b	cl(b	NOUN
ejpam-5335	348	5	)	)	PUNCT
ejpam-5335	348	6	)	)	PUNCT
ejpam-5335	348	7	.	.	PUNCT
ejpam-5335	349	1	(	(	PUNCT
ejpam-5335	349	2	4	4	X
ejpam-5335	349	3	)	)	PUNCT
ejpam-5335	349	4	⇒	⇒	NOUN
ejpam-5335	349	5	(	(	PUNCT
ejpam-5335	349	6	5	5	NUM
ejpam-5335	349	7	):	):	PUNCT
ejpam-5335	349	8	let	let	VERB
ejpam-5335	349	9	b	b	X
ejpam-5335	349	10	be	be	AUX
ejpam-5335	349	11	any	any	DET
ejpam-5335	349	12	subset	subset	NOUN
ejpam-5335	349	13	of	of	ADP
ejpam-5335	349	14	y	y	PROPN
ejpam-5335	349	15	.	.	PUNCT
ejpam-5335	350	1	then	then	ADV
ejpam-5335	350	2	by	by	ADP
ejpam-5335	350	3	(	(	PUNCT
ejpam-5335	350	4	4	4	NUM
ejpam-5335	350	5	)	)	PUNCT
ejpam-5335	350	6	,	,	PUNCT
ejpam-5335	350	7	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	350	8	-	-	PUNCT
ejpam-5335	350	9	int(b	int(b	NOUN
ejpam-5335	350	10	)	)	PUNCT
ejpam-5335	350	11	)	)	PUNCT
ejpam-5335	351	1	=	=	PUNCT
ejpam-5335	352	1	x	x	X
ejpam-5335	352	2	−	−	PRON
ejpam-5335	352	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	352	4	-	-	PUNCT
ejpam-5335	352	5	cl(y	cl(y	NOUN
ejpam-5335	352	6	−b	−b	NOUN
ejpam-5335	352	7	)	)	PUNCT
ejpam-5335	352	8	)	)	PUNCT
ejpam-5335	353	1	⊆	⊆	NUM
ejpam-5335	353	2	x	x	SYM
ejpam-5335	353	3	−	−	NOUN
ejpam-5335	353	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	353	5	,	,	PUNCT
ejpam-5335	353	6	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	353	7	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	353	8	-	-	PUNCT
ejpam-5335	353	9	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	353	10	-	-	PUNCT
ejpam-5335	353	11	cl(y	cl(y	NOUN
ejpam-5335	353	12	−b	−b	NOUN
ejpam-5335	353	13	)	)	PUNCT
ejpam-5335	353	14	)	)	PUNCT
ejpam-5335	353	15	)	)	PUNCT
ejpam-5335	353	16	)	)	PUNCT
ejpam-5335	354	1	=	=	SYM
ejpam-5335	354	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	354	3	,	,	PUNCT
ejpam-5335	354	4	τ2)-int(x	τ2)-int(x	NOUN
ejpam-5335	354	5	−	−	PRON
ejpam-5335	354	6	f−1(σ1σ2	f−1(σ1σ2	VERB
ejpam-5335	354	7	-	-	PUNCT
ejpam-5335	354	8	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	354	9	-	-	PUNCT
ejpam-5335	354	10	cl(y	cl(y	NOUN
ejpam-5335	354	11	−b	−b	NOUN
ejpam-5335	354	12	)	)	PUNCT
ejpam-5335	354	13	)	)	PUNCT
ejpam-5335	354	14	)	)	PUNCT
ejpam-5335	354	15	)	)	PUNCT
ejpam-5335	355	1	=	=	SYM
ejpam-5335	355	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	355	3	,	,	PUNCT
ejpam-5335	355	4	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	355	5	−1(σ1σ2	−1(σ1σ2	PROPN
ejpam-5335	355	6	-	-	PUNCT
ejpam-5335	355	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	355	8	-	-	PUNCT
ejpam-5335	355	9	int(b	int(b	NOUN
ejpam-5335	355	10	)	)	PUNCT
ejpam-5335	355	11	)	)	PUNCT
ejpam-5335	355	12	)	)	PUNCT
ejpam-5335	355	13	)	)	PUNCT
ejpam-5335	355	14	.	.	PUNCT
ejpam-5335	356	1	thus	thus	ADV
ejpam-5335	356	2	,	,	PUNCT
ejpam-5335	356	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	356	4	-	-	PUNCT
ejpam-5335	356	5	int(b	int(b	NOUN
ejpam-5335	356	6	)	)	PUNCT
ejpam-5335	356	7	)	)	PUNCT
ejpam-5335	356	8	⊆	⊆	NUM
ejpam-5335	356	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	356	10	,	,	PUNCT
ejpam-5335	356	11	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	356	12	−1(σ1σ2	−1(σ1σ2	PROPN
ejpam-5335	356	13	-	-	PUNCT
ejpam-5335	356	14	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	356	15	-	-	PUNCT
ejpam-5335	356	16	int(b	int(b	NOUN
ejpam-5335	356	17	)	)	PUNCT
ejpam-5335	356	18	)	)	PUNCT
ejpam-5335	356	19	)	)	PUNCT
ejpam-5335	356	20	)	)	PUNCT
ejpam-5335	356	21	.	.	PUNCT
ejpam-5335	357	1	(	(	PUNCT
ejpam-5335	357	2	5	5	X
ejpam-5335	357	3	)	)	PUNCT
ejpam-5335	357	4	⇒	⇒	NOUN
ejpam-5335	357	5	(	(	PUNCT
ejpam-5335	357	6	6	6	NUM
ejpam-5335	357	7	):	):	PUNCT
ejpam-5335	357	8	let	let	VERB
ejpam-5335	357	9	v	v	PART
ejpam-5335	357	10	be	be	AUX
ejpam-5335	357	11	any	any	DET
ejpam-5335	357	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	357	13	-	-	ADJ
ejpam-5335	357	14	open	open	ADJ
ejpam-5335	357	15	set	set	NOUN
ejpam-5335	357	16	of	of	ADP
ejpam-5335	357	17	y	y	PROPN
ejpam-5335	357	18	.	.	PUNCT
ejpam-5335	357	19	suppose	suppose	VERB
ejpam-5335	357	20	that	that	SCONJ
ejpam-5335	357	21	x	x	PROPN
ejpam-5335	357	22	/∈	/∈	PUNCT
ejpam-5335	357	23	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	357	24	-	-	PUNCT
ejpam-5335	357	25	cl(v	cl(v	NOUN
ejpam-5335	357	26	)	)	PUNCT
ejpam-5335	357	27	)	)	PUNCT
ejpam-5335	357	28	.	.	PUNCT
ejpam-5335	358	1	then	then	ADV
ejpam-5335	358	2	,	,	PUNCT
ejpam-5335	358	3	f(x	f(x	PROPN
ejpam-5335	358	4	)	)	PUNCT
ejpam-5335	358	5	̸∈	̸∈	PROPN
ejpam-5335	358	6	σ1σ2	σ1σ2	NOUN
ejpam-5335	358	7	-	-	NUM
ejpam-5335	358	8	cl(v	cl(v	NOUN
ejpam-5335	358	9	)	)	PUNCT
ejpam-5335	358	10	.	.	PUNCT
ejpam-5335	359	1	there	there	PRON
ejpam-5335	359	2	exists	exist	VERB
ejpam-5335	359	3	a	a	DET
ejpam-5335	359	4	σ1σ2	σ1σ2	NUM
ejpam-5335	359	5	-	-	ADJ
ejpam-5335	359	6	open	open	ADJ
ejpam-5335	359	7	set	set	NOUN
ejpam-5335	359	8	u	u	NOUN
ejpam-5335	359	9	of	of	ADP
ejpam-5335	359	10	y	y	PROPN
ejpam-5335	359	11	containing	contain	VERB
ejpam-5335	359	12	f(x	f(x	PROPN
ejpam-5335	359	13	)	)	PUNCT
ejpam-5335	359	14	such	such	ADJ
ejpam-5335	359	15	that	that	SCONJ
ejpam-5335	359	16	u	u	PROPN
ejpam-5335	359	17	∩	∩	NOUN
ejpam-5335	359	18	v	v	NOUN
ejpam-5335	359	19	=	=	PUNCT
ejpam-5335	359	20	∅.	∅.	VERB
ejpam-5335	359	21	hence	hence	ADV
ejpam-5335	359	22	,	,	PUNCT
ejpam-5335	359	23	σ1σ2	σ1σ2	NOUN
ejpam-5335	359	24	-	-	PUNCT
ejpam-5335	359	25	cl(u	cl(u	NOUN
ejpam-5335	359	26	)	)	PUNCT
ejpam-5335	359	27	∩	∩	NOUN
ejpam-5335	359	28	v	v	NOUN
ejpam-5335	359	29	=	=	PUNCT
ejpam-5335	359	30	∅.	∅.	X
ejpam-5335	359	31	by	by	ADP
ejpam-5335	359	32	(	(	PUNCT
ejpam-5335	359	33	5	5	NUM
ejpam-5335	359	34	)	)	PUNCT
ejpam-5335	359	35	,	,	PUNCT
ejpam-5335	359	36	x	x	PUNCT
ejpam-5335	359	37	∈	∈	PROPN
ejpam-5335	359	38	f−1(u	f−1(u	PROPN
ejpam-5335	359	39	)	)	PUNCT
ejpam-5335	359	40	=	=	SYM
ejpam-5335	359	41	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	359	42	-	-	PUNCT
ejpam-5335	359	43	int(u	int(u	PROPN
ejpam-5335	359	44	)	)	PUNCT
ejpam-5335	359	45	)	)	PUNCT
ejpam-5335	360	1	⊆	⊆	NUM
ejpam-5335	360	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	360	3	,	,	PUNCT
ejpam-5335	360	4	τ2)-int(f	τ2)-int(f	ADP
ejpam-5335	360	5	−1(σ1σ2	−1(σ1σ2	ADV
ejpam-5335	360	6	-	-	PUNCT
ejpam-5335	360	7	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	360	8	-	-	PUNCT
ejpam-5335	360	9	int(u	int(u	PROPN
ejpam-5335	360	10	)	)	PUNCT
ejpam-5335	360	11	)	)	PUNCT
ejpam-5335	360	12	)	)	PUNCT
ejpam-5335	360	13	)	)	PUNCT
ejpam-5335	361	1	=	=	SYM
ejpam-5335	361	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	361	3	,	,	PUNCT
ejpam-5335	361	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	361	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	361	6	-	-	PUNCT
ejpam-5335	361	7	cl(u	cl(u	NOUN
ejpam-5335	361	8	)	)	PUNCT
ejpam-5335	361	9	)	)	PUNCT
ejpam-5335	361	10	)	)	PUNCT
ejpam-5335	361	11	.	.	PUNCT
ejpam-5335	362	1	then	then	ADV
ejpam-5335	362	2	,	,	PUNCT
ejpam-5335	362	3	there	there	PRON
ejpam-5335	362	4	exists	exist	VERB
ejpam-5335	362	5	a	a	DET
ejpam-5335	362	6	δ(τ1	δ(τ1	NOUN
ejpam-5335	362	7	,	,	PUNCT
ejpam-5335	362	8	τ2)-open	τ2)-open	ADJ
ejpam-5335	362	9	set	set	VERB
ejpam-5335	362	10	g	g	NOUN
ejpam-5335	362	11	of	of	ADP
ejpam-5335	362	12	x	x	INTJ
ejpam-5335	362	13	such	such	ADJ
ejpam-5335	362	14	that	that	SCONJ
ejpam-5335	362	15	x	x	SYM
ejpam-5335	362	16	∈	∈	PROPN
ejpam-5335	362	17	g	g	PROPN
ejpam-5335	362	18	⊆	⊆	NUM
ejpam-5335	362	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	362	20	-	-	PUNCT
ejpam-5335	362	21	cl(u	cl(u	NOUN
ejpam-5335	362	22	)	)	PUNCT
ejpam-5335	362	23	)	)	PUNCT
ejpam-5335	362	24	.	.	PUNCT
ejpam-5335	363	1	therefore	therefore	ADV
ejpam-5335	363	2	,	,	PUNCT
ejpam-5335	363	3	f−1(v	f−1(v	PROPN
ejpam-5335	363	4	)	)	PUNCT
ejpam-5335	363	5	∩g	∩g	PROPN
ejpam-5335	364	1	⊆	⊆	NUM
ejpam-5335	364	2	f−1(v	f−1(v	NOUN
ejpam-5335	364	3	)	)	PUNCT
ejpam-5335	364	4	∩	∩	NOUN
ejpam-5335	364	5	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	364	6	-	-	PUNCT
ejpam-5335	364	7	cl(u	cl(u	NOUN
ejpam-5335	364	8	)	)	PUNCT
ejpam-5335	364	9	)	)	PUNCT
ejpam-5335	365	1	=	=	SYM
ejpam-5335	365	2	f−1(v	f−1(v	NOUN
ejpam-5335	365	3	∩	∩	VERB
ejpam-5335	365	4	σ1σ2	σ1σ2	NOUN
ejpam-5335	365	5	-	-	NOUN
ejpam-5335	365	6	cl(u	cl(u	NUM
ejpam-5335	365	7	)	)	PUNCT
ejpam-5335	365	8	)	)	PUNCT
ejpam-5335	366	1	=	=	PUNCT
ejpam-5335	366	2	∅.	∅.	VERB
ejpam-5335	366	3	thus	thus	ADV
ejpam-5335	366	4	,	,	PUNCT
ejpam-5335	366	5	x	x	PROPN
ejpam-5335	366	6	̸∈	̸∈	PROPN
ejpam-5335	366	7	δ(τ1	δ(τ1	PROPN
ejpam-5335	366	8	,	,	PUNCT
ejpam-5335	366	9	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	366	10	−1(v	−1(v	PROPN
ejpam-5335	366	11	)	)	PUNCT
ejpam-5335	366	12	)	)	PUNCT
ejpam-5335	366	13	and	and	CCONJ
ejpam-5335	366	14	hence	hence	ADV
ejpam-5335	366	15	δ(τ1	δ(τ1	PROPN
ejpam-5335	366	16	,	,	PUNCT
ejpam-5335	366	17	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	366	18	−1(v	−1(v	PROPN
ejpam-5335	366	19	)	)	PUNCT
ejpam-5335	366	20	)	)	PUNCT
ejpam-5335	366	21	⊆	⊆	NUM
ejpam-5335	366	22	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	366	23	-	-	PUNCT
ejpam-5335	366	24	cl(v	cl(v	NOUN
ejpam-5335	366	25	)	)	PUNCT
ejpam-5335	366	26	)	)	PUNCT
ejpam-5335	366	27	.	.	PUNCT
ejpam-5335	367	1	(	(	PUNCT
ejpam-5335	367	2	6	6	X
ejpam-5335	367	3	)	)	PUNCT
ejpam-5335	367	4	⇒	⇒	NOUN
ejpam-5335	367	5	(	(	PUNCT
ejpam-5335	367	6	7	7	NUM
ejpam-5335	367	7	):	):	PUNCT
ejpam-5335	367	8	let	let	VERB
ejpam-5335	367	9	f	f	PRON
ejpam-5335	367	10	be	be	AUX
ejpam-5335	367	11	any	any	DET
ejpam-5335	367	12	(	(	PUNCT
ejpam-5335	367	13	σ1	σ1	NOUN
ejpam-5335	367	14	,	,	PUNCT
ejpam-5335	367	15	σ2)r	σ2)r	NOUN
ejpam-5335	367	16	-	-	PUNCT
ejpam-5335	367	17	closed	close	VERB
ejpam-5335	367	18	set	set	NOUN
ejpam-5335	367	19	of	of	ADP
ejpam-5335	367	20	y	y	PROPN
ejpam-5335	367	21	.	.	PUNCT
ejpam-5335	368	1	then	then	ADV
ejpam-5335	368	2	,	,	PUNCT
ejpam-5335	368	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	368	4	-	-	PUNCT
ejpam-5335	368	5	int(f	int(f	X
ejpam-5335	368	6	)	)	PUNCT
ejpam-5335	368	7	is	be	AUX
ejpam-5335	368	8	σ1σ2	σ1σ2	NOUN
ejpam-5335	368	9	-	-	ADJ
ejpam-5335	368	10	open	open	ADJ
ejpam-5335	368	11	in	in	ADP
ejpam-5335	368	12	y	y	PROPN
ejpam-5335	368	13	.	.	PUNCT
ejpam-5335	369	1	by	by	ADP
ejpam-5335	369	2	(	(	PUNCT
ejpam-5335	369	3	6	6	NUM
ejpam-5335	369	4	)	)	PUNCT
ejpam-5335	369	5	,	,	PUNCT
ejpam-5335	369	6	δ(τ1	δ(τ1	PROPN
ejpam-5335	369	7	,	,	PUNCT
ejpam-5335	369	8	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	369	9	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	369	10	-	-	PUNCT
ejpam-5335	369	11	int(f	int(f	PROPN
ejpam-5335	369	12	)	)	PUNCT
ejpam-5335	369	13	)	)	PUNCT
ejpam-5335	369	14	)	)	PUNCT
ejpam-5335	370	1	⊆	⊆	NUM
ejpam-5335	370	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	370	3	-	-	PUNCT
ejpam-5335	370	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	370	5	-	-	PUNCT
ejpam-5335	370	6	int(f	int(f	PROPN
ejpam-5335	370	7	)	)	PUNCT
ejpam-5335	370	8	)	)	PUNCT
ejpam-5335	370	9	)	)	PUNCT
ejpam-5335	371	1	=	=	SYM
ejpam-5335	371	2	f−1(f	f−1(f	PROPN
ejpam-5335	371	3	)	)	PUNCT
ejpam-5335	371	4	.	.	PUNCT
ejpam-5335	372	1	c.	c.	PROPN
ejpam-5335	372	2	prachanpol	prachanpol	PROPN
ejpam-5335	372	3	,	,	PUNCT
ejpam-5335	372	4	c.	c.	PROPN
ejpam-5335	372	5	boonpok	boonpok	PROPN
ejpam-5335	372	6	,	,	PUNCT
ejpam-5335	372	7	c.	c.	PROPN
ejpam-5335	372	8	viriyapong	viriyapong	PROPN
ejpam-5335	372	9	/	/	SYM
ejpam-5335	372	10	eur	eur	PROPN
ejpam-5335	372	11	.	.	PUNCT
ejpam-5335	373	1	j.	j.	PROPN
ejpam-5335	373	2	pure	pure	PROPN
ejpam-5335	373	3	appl	appl	PROPN
ejpam-5335	373	4	.	.	PROPN
ejpam-5335	373	5	math	math	PROPN
ejpam-5335	373	6	,	,	PUNCT
ejpam-5335	373	7	17	17	NUM
ejpam-5335	373	8	(	(	PUNCT
ejpam-5335	373	9	4	4	NUM
ejpam-5335	373	10	)	)	PUNCT
ejpam-5335	373	11	(	(	PUNCT
ejpam-5335	373	12	2024	2024	NUM
ejpam-5335	373	13	)	)	PUNCT
ejpam-5335	373	14	,	,	PUNCT
ejpam-5335	373	15	3730	3730	NUM
ejpam-5335	373	16	-	-	SYM
ejpam-5335	373	17	3742	3742	NUM
ejpam-5335	373	18	3740	3740	NUM
ejpam-5335	373	19	(	(	PUNCT
ejpam-5335	373	20	7	7	NUM
ejpam-5335	373	21	)	)	PUNCT
ejpam-5335	373	22	⇒	⇒	NOUN
ejpam-5335	373	23	(	(	PUNCT
ejpam-5335	373	24	8)	8)	NUM
ejpam-5335	373	25	:	:	PUNCT
ejpam-5335	373	26	let	let	VERB
ejpam-5335	373	27	v	v	PART
ejpam-5335	373	28	be	be	AUX
ejpam-5335	373	29	any	any	DET
ejpam-5335	373	30	(	(	PUNCT
ejpam-5335	373	31	σ1	σ1	PROPN
ejpam-5335	373	32	,	,	PUNCT
ejpam-5335	373	33	σ2)β	σ2)β	NOUN
ejpam-5335	373	34	-	-	PUNCT
ejpam-5335	373	35	open	open	ADJ
ejpam-5335	373	36	set	set	NOUN
ejpam-5335	373	37	of	of	ADP
ejpam-5335	373	38	y	y	PROPN
ejpam-5335	373	39	.	.	PUNCT
ejpam-5335	374	1	then	then	ADV
ejpam-5335	374	2	,	,	PUNCT
ejpam-5335	374	3	we	we	PRON
ejpam-5335	374	4	have	have	VERB
ejpam-5335	374	5	v	v	ADP
ejpam-5335	374	6	⊆	⊆	NUM
ejpam-5335	374	7	σ1σ2	σ1σ2	NOUN
ejpam-5335	374	8	-	-	PUNCT
ejpam-5335	374	9	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	374	10	-	-	PUNCT
ejpam-5335	374	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	374	12	-	-	PUNCT
ejpam-5335	374	13	cl(v	cl(v	NOUN
ejpam-5335	374	14	)	)	PUNCT
ejpam-5335	374	15	)	)	PUNCT
ejpam-5335	374	16	)	)	PUNCT
ejpam-5335	375	1	and	and	CCONJ
ejpam-5335	375	2	so	so	ADV
ejpam-5335	375	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	375	4	-	-	NUM
ejpam-5335	375	5	cl(v	cl(v	NOUN
ejpam-5335	375	6	)	)	PUNCT
ejpam-5335	375	7	is	be	AUX
ejpam-5335	375	8	(	(	PUNCT
ejpam-5335	375	9	σ1	σ1	NOUN
ejpam-5335	375	10	,	,	PUNCT
ejpam-5335	375	11	σ2)r	σ2)r	NOUN
ejpam-5335	375	12	-	-	PUNCT
ejpam-5335	375	13	closed	closed	ADJ
ejpam-5335	375	14	.	.	PUNCT
ejpam-5335	376	1	by	by	ADP
ejpam-5335	376	2	(	(	PUNCT
ejpam-5335	376	3	7	7	NUM
ejpam-5335	376	4	)	)	PUNCT
ejpam-5335	376	5	,	,	PUNCT
ejpam-5335	376	6	δ(τ1	δ(τ1	PROPN
ejpam-5335	376	7	,	,	PUNCT
ejpam-5335	376	8	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	376	9	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	376	10	-	-	PUNCT
ejpam-5335	376	11	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	376	12	-	-	PUNCT
ejpam-5335	376	13	cl(v	cl(v	NOUN
ejpam-5335	376	14	)	)	PUNCT
ejpam-5335	376	15	)	)	PUNCT
ejpam-5335	376	16	)	)	PUNCT
ejpam-5335	376	17	)	)	PUNCT
ejpam-5335	376	18	⊆	⊆	NUM
ejpam-5335	376	19	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	376	20	-	-	PUNCT
ejpam-5335	376	21	cl(v	cl(v	NOUN
ejpam-5335	376	22	)	)	PUNCT
ejpam-5335	376	23	)	)	PUNCT
ejpam-5335	376	24	.	.	PUNCT
ejpam-5335	377	1	(	(	PUNCT
ejpam-5335	377	2	8)	8)	NUM
ejpam-5335	377	3	⇒	⇒	NOUN
ejpam-5335	377	4	(	(	PUNCT
ejpam-5335	377	5	9	9	NUM
ejpam-5335	377	6	):	):	PUNCT
ejpam-5335	377	7	the	the	DET
ejpam-5335	377	8	proof	proof	NOUN
ejpam-5335	377	9	is	be	AUX
ejpam-5335	377	10	obvious	obvious	ADJ
ejpam-5335	377	11	since	since	SCONJ
ejpam-5335	377	12	every	every	DET
ejpam-5335	377	13	(	(	PUNCT
ejpam-5335	377	14	σ1	σ1	PROPN
ejpam-5335	377	15	,	,	PUNCT
ejpam-5335	377	16	σ2)s	σ2)s	NOUN
ejpam-5335	377	17	-	-	PUNCT
ejpam-5335	377	18	open	open	ADJ
ejpam-5335	377	19	set	set	NOUN
ejpam-5335	377	20	is	be	AUX
ejpam-5335	377	21	(	(	PUNCT
ejpam-5335	377	22	σ1	σ1	PROPN
ejpam-5335	377	23	,	,	PUNCT
ejpam-5335	377	24	σ2)β	σ2)β	NOUN
ejpam-5335	377	25	-	-	PUNCT
ejpam-5335	377	26	open	open	ADJ
ejpam-5335	377	27	.	.	PUNCT
ejpam-5335	378	1	(	(	PUNCT
ejpam-5335	378	2	9)⇒	9)⇒	NUM
ejpam-5335	378	3	(	(	PUNCT
ejpam-5335	378	4	10	10	NUM
ejpam-5335	378	5	):	):	PUNCT
ejpam-5335	378	6	let	let	VERB
ejpam-5335	378	7	v	v	PART
ejpam-5335	378	8	be	be	AUX
ejpam-5335	378	9	any	any	DET
ejpam-5335	378	10	(	(	PUNCT
ejpam-5335	378	11	σ1	σ1	PROPN
ejpam-5335	378	12	,	,	PUNCT
ejpam-5335	378	13	σ2)p	σ2)p	NOUN
ejpam-5335	378	14	-	-	PUNCT
ejpam-5335	378	15	open	open	ADJ
ejpam-5335	378	16	set	set	NOUN
ejpam-5335	378	17	of	of	ADP
ejpam-5335	378	18	y	y	PROPN
ejpam-5335	378	19	.	.	PUNCT
ejpam-5335	379	1	then	then	ADV
ejpam-5335	379	2	,	,	PUNCT
ejpam-5335	379	3	v	v	ADP
ejpam-5335	379	4	⊆	⊆	NUM
ejpam-5335	379	5	σ1σ2	σ1σ2	NOUN
ejpam-5335	379	6	-	-	PUNCT
ejpam-5335	379	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	379	8	-	-	PUNCT
ejpam-5335	379	9	cl(v	cl(v	NOUN
ejpam-5335	379	10	)	)	PUNCT
ejpam-5335	379	11	)	)	PUNCT
ejpam-5335	379	12	and	and	CCONJ
ejpam-5335	379	13	σ1σ2	σ1σ2	NOUN
ejpam-5335	379	14	-	-	NUM
ejpam-5335	379	15	cl(v	cl(v	NOUN
ejpam-5335	379	16	)	)	PUNCT
ejpam-5335	379	17	⊆	⊆	NUM
ejpam-5335	379	18	σ1σ2	σ1σ2	X
ejpam-5335	379	19	-	-	PUNCT
ejpam-5335	379	20	cl(σ1σ2	cl(σ1σ2	VERB
ejpam-5335	379	21	-	-	PUNCT
ejpam-5335	379	22	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	379	23	-	-	PUNCT
ejpam-5335	379	24	cl(v	cl(v	NOUN
ejpam-5335	379	25	)	)	PUNCT
ejpam-5335	379	26	)	)	PUNCT
ejpam-5335	379	27	)	)	PUNCT
ejpam-5335	379	28	.	.	PUNCT
ejpam-5335	380	1	therefore	therefore	ADV
ejpam-5335	380	2	,	,	PUNCT
ejpam-5335	380	3	σ1σ2	σ1σ2	NOUN
ejpam-5335	380	4	-	-	NUM
ejpam-5335	380	5	cl(v	cl(v	NOUN
ejpam-5335	380	6	)	)	PUNCT
ejpam-5335	380	7	is	be	AUX
ejpam-5335	380	8	(	(	PUNCT
ejpam-5335	380	9	σ1	σ1	PROPN
ejpam-5335	380	10	,	,	PUNCT
ejpam-5335	380	11	σ2)s	σ2)s	NOUN
ejpam-5335	380	12	-	-	PUNCT
ejpam-5335	380	13	open	open	ADJ
ejpam-5335	380	14	in	in	ADP
ejpam-5335	380	15	y	y	PROPN
ejpam-5335	380	16	.	.	PUNCT
ejpam-5335	381	1	thus	thus	ADV
ejpam-5335	381	2	by	by	ADP
ejpam-5335	381	3	(	(	PUNCT
ejpam-5335	381	4	9	9	NUM
ejpam-5335	381	5	)	)	PUNCT
ejpam-5335	381	6	,	,	PUNCT
ejpam-5335	381	7	δ(τ1	δ(τ1	PROPN
ejpam-5335	381	8	,	,	PUNCT
ejpam-5335	381	9	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	381	10	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	381	11	-	-	PUNCT
ejpam-5335	381	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	381	13	-	-	PUNCT
ejpam-5335	381	14	cl(v	cl(v	NOUN
ejpam-5335	381	15	)	)	PUNCT
ejpam-5335	381	16	)	)	PUNCT
ejpam-5335	381	17	)	)	PUNCT
ejpam-5335	381	18	)	)	PUNCT
ejpam-5335	382	1	=	=	SYM
ejpam-5335	382	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	382	3	,	,	PUNCT
ejpam-5335	382	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	382	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	382	6	-	-	PUNCT
ejpam-5335	382	7	int(σ1σ2	int(σ1σ2	ADV
ejpam-5335	382	8	-	-	PUNCT
ejpam-5335	382	9	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	382	10	-	-	PUNCT
ejpam-5335	382	11	cl(v	cl(v	NOUN
ejpam-5335	382	12	)	)	PUNCT
ejpam-5335	382	13	)	)	PUNCT
ejpam-5335	382	14	)	)	PUNCT
ejpam-5335	382	15	)	)	PUNCT
ejpam-5335	382	16	)	)	PUNCT
ejpam-5335	383	1	⊆	⊆	NUM
ejpam-5335	383	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	383	3	-	-	PUNCT
ejpam-5335	383	4	cl(σ1σ2	cl(σ1σ2	NOUN
ejpam-5335	383	5	-	-	PUNCT
ejpam-5335	383	6	cl(v	cl(v	NOUN
ejpam-5335	383	7	)	)	PUNCT
ejpam-5335	383	8	)	)	PUNCT
ejpam-5335	383	9	)	)	PUNCT
ejpam-5335	384	1	=	=	PRON
ejpam-5335	384	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	384	3	-	-	PUNCT
ejpam-5335	384	4	cl(v	cl(v	NOUN
ejpam-5335	384	5	)	)	PUNCT
ejpam-5335	384	6	)	)	PUNCT
ejpam-5335	384	7	.	.	PUNCT
ejpam-5335	385	1	(	(	PUNCT
ejpam-5335	385	2	10)⇒	10)⇒	NUM
ejpam-5335	385	3	(	(	PUNCT
ejpam-5335	385	4	11	11	NUM
ejpam-5335	385	5	):	):	PUNCT
ejpam-5335	385	6	let	let	VERB
ejpam-5335	385	7	v	v	PART
ejpam-5335	385	8	be	be	AUX
ejpam-5335	385	9	any	any	DET
ejpam-5335	385	10	(	(	PUNCT
ejpam-5335	385	11	σ1	σ1	PROPN
ejpam-5335	385	12	,	,	PUNCT
ejpam-5335	385	13	σ2)p	σ2)p	NOUN
ejpam-5335	385	14	-	-	PUNCT
ejpam-5335	385	15	open	open	ADJ
ejpam-5335	385	16	set	set	NOUN
ejpam-5335	385	17	of	of	ADP
ejpam-5335	385	18	y	y	PROPN
ejpam-5335	385	19	.	.	PUNCT
ejpam-5335	386	1	then	then	ADV
ejpam-5335	386	2	by	by	ADP
ejpam-5335	386	3	(	(	PUNCT
ejpam-5335	386	4	10	10	NUM
ejpam-5335	386	5	)	)	PUNCT
ejpam-5335	386	6	,	,	PUNCT
ejpam-5335	386	7	δ(τ1	δ(τ1	PROPN
ejpam-5335	386	8	,	,	PUNCT
ejpam-5335	386	9	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	386	10	−1(v	−1(v	PROPN
ejpam-5335	386	11	)	)	PUNCT
ejpam-5335	386	12	)	)	PUNCT
ejpam-5335	387	1	⊆	⊆	NUM
ejpam-5335	387	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	387	3	,	,	PUNCT
ejpam-5335	387	4	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	387	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	387	6	-	-	PUNCT
ejpam-5335	387	7	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	387	8	-	-	PUNCT
ejpam-5335	387	9	cl(v	cl(v	NOUN
ejpam-5335	387	10	)	)	PUNCT
ejpam-5335	387	11	)	)	PUNCT
ejpam-5335	387	12	)	)	PUNCT
ejpam-5335	387	13	)	)	PUNCT
ejpam-5335	388	1	⊆	⊆	NUM
ejpam-5335	388	2	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	388	3	-	-	PUNCT
ejpam-5335	388	4	cl(v	cl(v	NOUN
ejpam-5335	388	5	)	)	PUNCT
ejpam-5335	388	6	)	)	PUNCT
ejpam-5335	388	7	.	.	PUNCT
ejpam-5335	389	1	(	(	PUNCT
ejpam-5335	389	2	11	11	NUM
ejpam-5335	389	3	)	)	PUNCT
ejpam-5335	389	4	⇒	⇒	NOUN
ejpam-5335	389	5	(	(	PUNCT
ejpam-5335	389	6	12	12	NUM
ejpam-5335	389	7	):	):	PUNCT
ejpam-5335	389	8	let	let	VERB
ejpam-5335	389	9	v	v	PART
ejpam-5335	389	10	be	be	AUX
ejpam-5335	389	11	any	any	DET
ejpam-5335	389	12	(	(	PUNCT
ejpam-5335	389	13	σ1	σ1	PROPN
ejpam-5335	389	14	,	,	PUNCT
ejpam-5335	389	15	σ2)p	σ2)p	NOUN
ejpam-5335	389	16	-	-	PUNCT
ejpam-5335	389	17	open	open	ADJ
ejpam-5335	389	18	set	set	NOUN
ejpam-5335	389	19	of	of	ADP
ejpam-5335	389	20	y	y	PROPN
ejpam-5335	389	21	.	.	PUNCT
ejpam-5335	390	1	thus	thus	ADV
ejpam-5335	390	2	by	by	ADP
ejpam-5335	390	3	(	(	PUNCT
ejpam-5335	390	4	11	11	NUM
ejpam-5335	390	5	)	)	PUNCT
ejpam-5335	390	6	,	,	PUNCT
ejpam-5335	390	7	f−1(v	f−1(v	PROPN
ejpam-5335	390	8	)	)	PUNCT
ejpam-5335	390	9	⊆	⊆	NUM
ejpam-5335	390	10	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	390	11	-	-	PUNCT
ejpam-5335	390	12	int(σ1σ2	int(σ1σ2	NOUN
ejpam-5335	390	13	-	-	PUNCT
ejpam-5335	390	14	cl(v	cl(v	NOUN
ejpam-5335	390	15	)	)	PUNCT
ejpam-5335	390	16	)	)	PUNCT
ejpam-5335	390	17	)	)	PUNCT
ejpam-5335	391	1	=	=	PUNCT
ejpam-5335	392	1	x	x	X
ejpam-5335	392	2	−	−	PRON
ejpam-5335	392	3	f−1(σ1σ2	f−1(σ1σ2	NOUN
ejpam-5335	392	4	-	-	PUNCT
ejpam-5335	392	5	cl(y	cl(y	NOUN
ejpam-5335	392	6	−	−	NOUN
ejpam-5335	392	7	σ1σ2	σ1σ2	NOUN
ejpam-5335	392	8	-	-	NUM
ejpam-5335	392	9	cl(v	cl(v	NOUN
ejpam-5335	392	10	)	)	PUNCT
ejpam-5335	392	11	)	)	PUNCT
ejpam-5335	392	12	)	)	PUNCT
ejpam-5335	393	1	⊆	⊆	NUM
ejpam-5335	393	2	x	x	SYM
ejpam-5335	393	3	−	−	PROPN
ejpam-5335	393	4	δ(τ1	δ(τ1	PROPN
ejpam-5335	393	5	,	,	PUNCT
ejpam-5335	393	6	τ2)-cl(f	τ2)-cl(f	PROPN
ejpam-5335	393	7	−1(y	−1(y	VERB
ejpam-5335	393	8	−	−	PUNCT
ejpam-5335	393	9	σ1σ2	σ1σ2	NOUN
ejpam-5335	393	10	-	-	NUM
ejpam-5335	393	11	cl(v	cl(v	NOUN
ejpam-5335	393	12	)	)	PUNCT
ejpam-5335	393	13	)	)	PUNCT
ejpam-5335	393	14	)	)	PUNCT
ejpam-5335	394	1	=	=	SYM
ejpam-5335	394	2	δ(τ1	δ(τ1	PROPN
ejpam-5335	394	3	,	,	PUNCT
ejpam-5335	394	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	394	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	394	6	-	-	PUNCT
ejpam-5335	394	7	cl(v	cl(v	NOUN
ejpam-5335	394	8	)	)	PUNCT
ejpam-5335	394	9	)	)	PUNCT
ejpam-5335	394	10	)	)	PUNCT
ejpam-5335	394	11	.	.	PUNCT
ejpam-5335	395	1	(	(	PUNCT
ejpam-5335	395	2	12	12	NUM
ejpam-5335	395	3	)	)	PUNCT
ejpam-5335	395	4	⇒	⇒	NOUN
ejpam-5335	395	5	(	(	PUNCT
ejpam-5335	395	6	1	1	NUM
ejpam-5335	395	7	):	):	PUNCT
ejpam-5335	395	8	let	let	VERB
ejpam-5335	395	9	v	v	PART
ejpam-5335	395	10	be	be	AUX
ejpam-5335	395	11	any	any	DET
ejpam-5335	395	12	σ1σ2	σ1σ2	NOUN
ejpam-5335	395	13	-	-	ADJ
ejpam-5335	395	14	open	open	ADJ
ejpam-5335	395	15	set	set	NOUN
ejpam-5335	395	16	of	of	ADP
ejpam-5335	395	17	y	y	PROPN
ejpam-5335	395	18	.	.	PUNCT
ejpam-5335	396	1	then	then	ADV
ejpam-5335	396	2	,	,	PUNCT
ejpam-5335	396	3	v	v	NOUN
ejpam-5335	396	4	is	be	AUX
ejpam-5335	396	5	(	(	PUNCT
ejpam-5335	396	6	σ1	σ1	PROPN
ejpam-5335	396	7	,	,	PUNCT
ejpam-5335	396	8	σ2)p	σ2)p	NOUN
ejpam-5335	396	9	-	-	PUNCT
ejpam-5335	396	10	open	open	ADJ
ejpam-5335	396	11	in	in	ADP
ejpam-5335	396	12	y	y	PROPN
ejpam-5335	396	13	and	and	CCONJ
ejpam-5335	396	14	by	by	ADP
ejpam-5335	396	15	(	(	PUNCT
ejpam-5335	396	16	12	12	NUM
ejpam-5335	396	17	)	)	PUNCT
ejpam-5335	396	18	,	,	PUNCT
ejpam-5335	396	19	f−1(v	f−1(v	PROPN
ejpam-5335	396	20	)	)	PUNCT
ejpam-5335	397	1	⊆	⊆	NUM
ejpam-5335	397	2	δ(τ1	δ(τ1	NOUN
ejpam-5335	397	3	,	,	PUNCT
ejpam-5335	397	4	τ2)-int(f	τ2)-int(f	ADV
ejpam-5335	397	5	−1(σ1σ2	−1(σ1σ2	NOUN
ejpam-5335	397	6	-	-	PUNCT
ejpam-5335	397	7	cl(v	cl(v	NOUN
ejpam-5335	397	8	)	)	PUNCT
ejpam-5335	397	9	)	)	PUNCT
ejpam-5335	397	10	)	)	PUNCT
ejpam-5335	397	11	.	.	PUNCT
ejpam-5335	398	1	it	it	PRON
ejpam-5335	398	2	follows	follow	VERB
ejpam-5335	398	3	from	from	ADP
ejpam-5335	398	4	theorem	theorem	ADJ
ejpam-5335	398	5	7	7	NUM
ejpam-5335	398	6	that	that	SCONJ
ejpam-5335	398	7	f	f	PROPN
ejpam-5335	398	8	is	be	AUX
ejpam-5335	398	9	weakly	weakly	ADJ
ejpam-5335	398	10	δ(τ1	δ(τ1	NOUN
ejpam-5335	398	11	,	,	PUNCT
ejpam-5335	398	12	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	398	13	.	.	NOUN
ejpam-5335	399	1	6	6	NUM
ejpam-5335	399	2	.	.	X
ejpam-5335	399	3	conclusion	conclusion	NOUN
ejpam-5335	399	4	this	this	DET
ejpam-5335	399	5	paper	paper	NOUN
ejpam-5335	399	6	deals	deal	NOUN
ejpam-5335	399	7	with	with	ADP
ejpam-5335	399	8	the	the	DET
ejpam-5335	399	9	concepts	concept	NOUN
ejpam-5335	399	10	of	of	ADP
ejpam-5335	399	11	δ(τ1	δ(τ1	NOUN
ejpam-5335	399	12	,	,	PUNCT
ejpam-5335	399	13	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	399	14	functions	function	NOUN
ejpam-5335	399	15	,	,	PUNCT
ejpam-5335	399	16	almost	almost	ADV
ejpam-5335	399	17	δ(τ1	δ(τ1	NOUN
ejpam-5335	399	18	,	,	PUNCT
ejpam-5335	399	19	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	399	20	functions	function	NOUN
ejpam-5335	399	21	,	,	PUNCT
ejpam-5335	399	22	and	and	CCONJ
ejpam-5335	399	23	weakly	weakly	ADJ
ejpam-5335	399	24	δ(τ1	δ(τ1	NOUN
ejpam-5335	399	25	,	,	PUNCT
ejpam-5335	399	26	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	399	27	functions	function	NOUN
ejpam-5335	399	28	.	.	PUNCT
ejpam-5335	400	1	moreover	moreover	ADV
ejpam-5335	400	2	,	,	PUNCT
ejpam-5335	400	3	some	some	DET
ejpam-5335	400	4	characterizations	characterization	NOUN
ejpam-5335	400	5	and	and	CCONJ
ejpam-5335	400	6	several	several	ADJ
ejpam-5335	400	7	properties	property	NOUN
ejpam-5335	400	8	concerning	concern	VERB
ejpam-5335	400	9	δ(τ1	δ(τ1	NOUN
ejpam-5335	400	10	,	,	PUNCT
ejpam-5335	400	11	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	400	12	functions	function	NOUN
ejpam-5335	400	13	,	,	PUNCT
ejpam-5335	400	14	almost	almost	ADV
ejpam-5335	400	15	δ(τ1	δ(τ1	NOUN
ejpam-5335	400	16	,	,	PUNCT
ejpam-5335	400	17	τ2)continuous	τ2)continuous	ADJ
ejpam-5335	400	18	functions	function	NOUN
ejpam-5335	400	19	,	,	PUNCT
ejpam-5335	400	20	and	and	CCONJ
ejpam-5335	400	21	weakly	weakly	ADJ
ejpam-5335	400	22	δ(τ1	δ(τ1	NOUN
ejpam-5335	400	23	,	,	PUNCT
ejpam-5335	400	24	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	400	25	functions	function	NOUN
ejpam-5335	400	26	are	be	AUX
ejpam-5335	400	27	obtained	obtain	VERB
ejpam-5335	400	28	.	.	PUNCT
ejpam-5335	401	1	for	for	ADP
ejpam-5335	401	2	a	a	DET
ejpam-5335	401	3	function	function	NOUN
ejpam-5335	401	4	f	f	NOUN
ejpam-5335	401	5	:	:	PUNCT
ejpam-5335	401	6	(	(	PUNCT
ejpam-5335	401	7	x	x	NOUN
ejpam-5335	401	8	,	,	PUNCT
ejpam-5335	401	9	τ1	τ1	NOUN
ejpam-5335	401	10	,	,	PUNCT
ejpam-5335	401	11	τ2	τ2	NOUN
ejpam-5335	401	12	)	)	PUNCT
ejpam-5335	401	13	→	→	SYM
ejpam-5335	401	14	(	(	PUNCT
ejpam-5335	401	15	y	y	PROPN
ejpam-5335	401	16	,	,	PUNCT
ejpam-5335	401	17	σ1	σ1	PROPN
ejpam-5335	401	18	,	,	PUNCT
ejpam-5335	401	19	σ2	σ2	NOUN
ejpam-5335	401	20	)	)	PUNCT
ejpam-5335	401	21	,	,	PUNCT
ejpam-5335	401	22	the	the	DET
ejpam-5335	401	23	following	follow	VERB
ejpam-5335	401	24	implications	implication	NOUN
ejpam-5335	401	25	hold	hold	VERB
ejpam-5335	401	26	:	:	PUNCT
ejpam-5335	401	27	δ(τ1	δ(τ1	NOUN
ejpam-5335	401	28	,	,	PUNCT
ejpam-5335	401	29	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	401	30	⇒	⇒	NOUN
ejpam-5335	401	31	almost	almost	ADV
ejpam-5335	401	32	δ(τ1	δ(τ1	VERB
ejpam-5335	401	33	,	,	PUNCT
ejpam-5335	401	34	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	401	35	⇒	⇒	VERB
ejpam-5335	401	36	weak	weak	ADJ
ejpam-5335	401	37	δ(τ1	δ(τ1	NOUN
ejpam-5335	401	38	,	,	PUNCT
ejpam-5335	401	39	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	401	40	.	.	PUNCT
ejpam-5335	402	1	acknowledgements	acknowledgement	VERB
ejpam-5335	402	2	this	this	DET
ejpam-5335	402	3	research	research	NOUN
ejpam-5335	402	4	project	project	NOUN
ejpam-5335	402	5	was	be	AUX
ejpam-5335	402	6	financially	financially	ADV
ejpam-5335	402	7	supported	support	VERB
ejpam-5335	402	8	by	by	ADP
ejpam-5335	402	9	mahasarakham	mahasarakham	PROPN
ejpam-5335	402	10	university	university	PROPN
ejpam-5335	402	11	.	.	PUNCT
ejpam-5335	403	1	references	reference	NOUN
ejpam-5335	403	2	3741	3741	NUM
ejpam-5335	403	3	references	reference	NOUN
ejpam-5335	403	4	[	[	X
ejpam-5335	403	5	1	1	X
ejpam-5335	403	6	]	]	PUNCT
ejpam-5335	403	7	s.	s.	PROPN
ejpam-5335	403	8	p.	p.	PROPN
ejpam-5335	403	9	arya	arya	PROPN
ejpam-5335	403	10	and	and	CCONJ
ejpam-5335	403	11	r.	r.	PROPN
ejpam-5335	403	12	gupta	gupta	PROPN
ejpam-5335	403	13	.	.	PUNCT
ejpam-5335	404	1	on	on	ADP
ejpam-5335	404	2	strongly	strongly	ADV
ejpam-5335	404	3	continuous	continuous	ADJ
ejpam-5335	404	4	mappings	mapping	NOUN
ejpam-5335	404	5	.	.	PUNCT
ejpam-5335	405	1	kyungpook	kyungpook	PROPN
ejpam-5335	405	2	mathematical	mathematical	PROPN
ejpam-5335	405	3	journal	journal	PROPN
ejpam-5335	405	4	,	,	PUNCT
ejpam-5335	405	5	14:131–143	14:131–143	PROPN
ejpam-5335	405	6	,	,	PUNCT
ejpam-5335	405	7	1974	1974	NUM
ejpam-5335	405	8	.	.	PUNCT
ejpam-5335	406	1	[	[	X
ejpam-5335	406	2	2	2	NUM
ejpam-5335	406	3	]	]	PUNCT
ejpam-5335	406	4	c.	c.	PROPN
ejpam-5335	406	5	w.	w.	PROPN
ejpam-5335	406	6	baker	baker	PROPN
ejpam-5335	406	7	.	.	PUNCT
ejpam-5335	407	1	on	on	ADP
ejpam-5335	407	2	super	super	ADJ
ejpam-5335	407	3	continuous	continuous	ADJ
ejpam-5335	407	4	functions	function	NOUN
ejpam-5335	407	5	.	.	PUNCT
ejpam-5335	408	1	bulletin	bulletin	NOUN
ejpam-5335	408	2	of	of	ADP
ejpam-5335	408	3	the	the	DET
ejpam-5335	408	4	korean	korean	PROPN
ejpam-5335	408	5	mathematical	mathematical	ADJ
ejpam-5335	408	6	society	society	NOUN
ejpam-5335	408	7	,	,	PUNCT
ejpam-5335	408	8	22(1):17–22	22(1):17–22	NUM
ejpam-5335	408	9	,	,	PUNCT
ejpam-5335	408	10	1985	1985	NUM
ejpam-5335	408	11	.	.	PUNCT
ejpam-5335	409	1	[	[	X
ejpam-5335	409	2	3	3	X
ejpam-5335	409	3	]	]	PUNCT
ejpam-5335	409	4	c.	c.	PROPN
ejpam-5335	409	5	boonpok	boonpok	PROPN
ejpam-5335	409	6	.	.	PUNCT
ejpam-5335	410	1	on	on	ADP
ejpam-5335	410	2	characterizations	characterization	NOUN
ejpam-5335	410	3	of	of	ADP
ejpam-5335	410	4	⋆-hyperconnected	⋆-hyperconnecte	VERB
ejpam-5335	410	5	ideal	ideal	ADJ
ejpam-5335	410	6	topological	topological	ADJ
ejpam-5335	410	7	spaces	space	NOUN
ejpam-5335	410	8	.	.	PUNCT
ejpam-5335	411	1	journal	journal	NOUN
ejpam-5335	411	2	of	of	ADP
ejpam-5335	411	3	mathematics	mathematic	NOUN
ejpam-5335	411	4	,	,	PUNCT
ejpam-5335	411	5	2020:9387601	2020:9387601	NUM
ejpam-5335	411	6	,	,	PUNCT
ejpam-5335	411	7	2020	2020	NUM
ejpam-5335	411	8	.	.	PUNCT
ejpam-5335	412	1	[	[	X
ejpam-5335	412	2	4	4	NUM
ejpam-5335	412	3	]	]	PUNCT
ejpam-5335	412	4	c.	c.	PROPN
ejpam-5335	412	5	boonpok	boonpok	PROPN
ejpam-5335	412	6	.	.	PUNCT
ejpam-5335	413	1	(	(	PUNCT
ejpam-5335	413	2	τ1	τ1	NOUN
ejpam-5335	413	3	,	,	PUNCT
ejpam-5335	413	4	τ2)δ	τ2)δ	ADJ
ejpam-5335	413	5	-	-	PUNCT
ejpam-5335	413	6	semicontinuous	semicontinuous	ADJ
ejpam-5335	413	7	multifunctions	multifunction	NOUN
ejpam-5335	413	8	.	.	PUNCT
ejpam-5335	414	1	heliyon	heliyon	NOUN
ejpam-5335	414	2	,	,	PUNCT
ejpam-5335	414	3	6	6	NUM
ejpam-5335	414	4	:	:	SYM
ejpam-5335	414	5	e05367	e05367	PROPN
ejpam-5335	414	6	,	,	PUNCT
ejpam-5335	414	7	2020	2020	NUM
ejpam-5335	414	8	.	.	PUNCT
ejpam-5335	415	1	[	[	X
ejpam-5335	415	2	5	5	X
ejpam-5335	415	3	]	]	PUNCT
ejpam-5335	415	4	c.	c.	PROPN
ejpam-5335	415	5	boonpok	boonpok	PROPN
ejpam-5335	415	6	.	.	PUNCT
ejpam-5335	416	1	on	on	ADP
ejpam-5335	416	2	some	some	DET
ejpam-5335	416	3	spaces	space	NOUN
ejpam-5335	416	4	via	via	ADP
ejpam-5335	416	5	topological	topological	ADJ
ejpam-5335	416	6	ideals	ideal	NOUN
ejpam-5335	416	7	.	.	PUNCT
ejpam-5335	417	1	open	open	ADJ
ejpam-5335	417	2	mathematics	mathematic	NOUN
ejpam-5335	417	3	,	,	PUNCT
ejpam-5335	417	4	21:20230118	21:20230118	NUM
ejpam-5335	417	5	,	,	PUNCT
ejpam-5335	417	6	2023	2023	NUM
ejpam-5335	417	7	.	.	PUNCT
ejpam-5335	418	1	[	[	X
ejpam-5335	418	2	6	6	NUM
ejpam-5335	418	3	]	]	PUNCT
ejpam-5335	418	4	c.	c.	PROPN
ejpam-5335	418	5	boonpok	boonpok	PROPN
ejpam-5335	418	6	.	.	PUNCT
ejpam-5335	419	1	θ(⋆)-precontinuity	θ(⋆)-precontinuity	NOUN
ejpam-5335	419	2	.	.	PUNCT
ejpam-5335	420	1	mathematica	mathematica	PROPN
ejpam-5335	420	2	,	,	PUNCT
ejpam-5335	420	3	65(1):31–42	65(1):31–42	NUM
ejpam-5335	420	4	,	,	PUNCT
ejpam-5335	420	5	2023	2023	NUM
ejpam-5335	420	6	.	.	PUNCT
ejpam-5335	421	1	[	[	X
ejpam-5335	421	2	7	7	X
ejpam-5335	421	3	]	]	X
ejpam-5335	421	4	c.	c.	PROPN
ejpam-5335	421	5	boonpok	boonpok	PROPN
ejpam-5335	421	6	and	and	CCONJ
ejpam-5335	421	7	c.	c.	PROPN
ejpam-5335	421	8	klanarong	klanarong	PROPN
ejpam-5335	421	9	.	.	PUNCT
ejpam-5335	422	1	on	on	ADP
ejpam-5335	422	2	weakly	weakly	ADJ
ejpam-5335	422	3	(	(	PUNCT
ejpam-5335	422	4	τ1	τ1	NOUN
ejpam-5335	422	5	,	,	PUNCT
ejpam-5335	422	6	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	422	7	functions	function	NOUN
ejpam-5335	422	8	.	.	PUNCT
ejpam-5335	423	1	european	european	ADJ
ejpam-5335	423	2	journal	journal	PROPN
ejpam-5335	423	3	of	of	ADP
ejpam-5335	423	4	pure	pure	ADJ
ejpam-5335	423	5	and	and	CCONJ
ejpam-5335	423	6	applied	applied	ADJ
ejpam-5335	423	7	mathematics	mathematic	NOUN
ejpam-5335	423	8	,	,	PUNCT
ejpam-5335	423	9	17(1):416–425	17(1):416–425	NUM
ejpam-5335	423	10	,	,	PUNCT
ejpam-5335	423	11	2024	2024	NUM
ejpam-5335	423	12	.	.	PUNCT
ejpam-5335	424	1	[	[	X
ejpam-5335	424	2	8	8	NUM
ejpam-5335	424	3	]	]	X
ejpam-5335	424	4	c.	c.	NOUN
ejpam-5335	424	5	boonpok	boonpok	PROPN
ejpam-5335	424	6	and	and	CCONJ
ejpam-5335	424	7	p.	p.	NOUN
ejpam-5335	424	8	pue	pue	NOUN
ejpam-5335	424	9	-	-	PUNCT
ejpam-5335	424	10	on	on	ADP
ejpam-5335	424	11	.	.	PUNCT
ejpam-5335	425	1	characterizations	characterization	NOUN
ejpam-5335	425	2	of	of	ADP
ejpam-5335	425	3	almost	almost	ADV
ejpam-5335	425	4	(	(	PUNCT
ejpam-5335	425	5	τ1	τ1	NOUN
ejpam-5335	425	6	,	,	PUNCT
ejpam-5335	425	7	τ2)-continuous	τ2)-continuous	ADJ
ejpam-5335	425	8	functions	function	NOUN
ejpam-5335	425	9	.	.	PUNCT
ejpam-5335	426	1	international	international	ADJ
ejpam-5335	426	2	journal	journal	NOUN
ejpam-5335	426	3	of	of	ADP
ejpam-5335	426	4	analysis	analysis	NOUN
ejpam-5335	426	5	and	and	CCONJ
ejpam-5335	426	6	applications	application	NOUN
ejpam-5335	426	7	,	,	PUNCT
ejpam-5335	426	8	22:33	22:33	NUM
ejpam-5335	426	9	,	,	PUNCT
ejpam-5335	426	10	2024	2024	NUM
ejpam-5335	426	11	.	.	PUNCT
ejpam-5335	427	1	[	[	X
ejpam-5335	427	2	9	9	NUM
ejpam-5335	427	3	]	]	PUNCT
ejpam-5335	427	4	c.	c.	NOUN
ejpam-5335	427	5	boonpok	boonpok	PROPN
ejpam-5335	427	6	and	and	CCONJ
ejpam-5335	427	7	n.	n.	PROPN
ejpam-5335	427	8	srisarakham	srisarakham	PROPN
ejpam-5335	427	9	.	.	PUNCT
ejpam-5335	428	1	(	(	PUNCT
ejpam-5335	428	2	τ1	τ1	NOUN
ejpam-5335	428	3	,	,	PUNCT
ejpam-5335	428	4	τ2)-continuity	τ2)-continuity	NOUN
ejpam-5335	428	5	for	for	ADP
ejpam-5335	428	6	functions	function	NOUN
ejpam-5335	428	7	.	.	PUNCT
ejpam-5335	429	1	asia	asia	PROPN
ejpam-5335	429	2	pacific	pacific	PROPN
ejpam-5335	429	3	journal	journal	PROPN
ejpam-5335	429	4	of	of	ADP
ejpam-5335	429	5	mathematics	mathematic	NOUN
ejpam-5335	429	6	,	,	PUNCT
ejpam-5335	429	7	11:21	11:21	NUM
ejpam-5335	429	8	,	,	PUNCT
ejpam-5335	429	9	2024	2024	NUM
ejpam-5335	429	10	.	.	PUNCT
ejpam-5335	430	1	[	[	X
ejpam-5335	430	2	10	10	NUM
ejpam-5335	430	3	]	]	X
ejpam-5335	430	4	c.	c.	PROPN
ejpam-5335	430	5	boonpok	boonpok	PROPN
ejpam-5335	430	6	,	,	PUNCT
ejpam-5335	430	7	c.	c.	PROPN
ejpam-5335	430	8	viriyapong	viriyapong	PROPN
ejpam-5335	430	9	,	,	PUNCT
ejpam-5335	430	10	and	and	CCONJ
ejpam-5335	430	11	m.	m.	NOUN
ejpam-5335	430	12	thongmoon	thongmoon	NOUN
ejpam-5335	430	13	.	.	PUNCT
ejpam-5335	431	1	on	on	ADP
ejpam-5335	431	2	upper	upper	ADJ
ejpam-5335	431	3	and	and	CCONJ
ejpam-5335	431	4	lower	low	ADJ
ejpam-5335	431	5	(	(	PUNCT
ejpam-5335	431	6	τ1	τ1	NOUN
ejpam-5335	431	7	,	,	PUNCT
ejpam-5335	431	8	τ2)precontinuous	τ2)precontinuous	ADJ
ejpam-5335	431	9	multifunctions	multifunction	NOUN
ejpam-5335	431	10	.	.	PUNCT
ejpam-5335	432	1	journal	journal	PROPN
ejpam-5335	432	2	of	of	ADP
ejpam-5335	432	3	mathematics	mathematics	PROPN
ejpam-5335	432	4	and	and	CCONJ
ejpam-5335	432	5	computer	computer	NOUN
ejpam-5335	432	6	science	science	NOUN
ejpam-5335	432	7	,	,	PUNCT
ejpam-5335	432	8	18:282–293	18:282–293	NUM
ejpam-5335	432	9	,	,	PUNCT
ejpam-5335	432	10	2018	2018	NUM
ejpam-5335	432	11	.	.	PUNCT
ejpam-5335	433	1	[	[	X
ejpam-5335	433	2	11	11	NUM
ejpam-5335	433	3	]	]	X
ejpam-5335	433	4	d.	d.	PROPN
ejpam-5335	433	5	carnahan	carnahan	PROPN
ejpam-5335	433	6	.	.	PUNCT
ejpam-5335	434	1	some	some	DET
ejpam-5335	434	2	properties	property	NOUN
ejpam-5335	434	3	of	of	ADP
ejpam-5335	434	4	related	relate	VERB
ejpam-5335	434	5	to	to	ADP
ejpam-5335	434	6	compactness	compactness	NOUN
ejpam-5335	434	7	in	in	ADP
ejpam-5335	434	8	topological	topological	ADJ
ejpam-5335	434	9	spaces	space	NOUN
ejpam-5335	434	10	.	.	PUNCT
ejpam-5335	435	1	ph	ph	PROPN
ejpam-5335	435	2	.	.	PROPN
ejpam-5335	435	3	d.	d.	PROPN
ejpam-5335	435	4	thesis	thesis	PROPN
ejpam-5335	435	5	,	,	PUNCT
ejpam-5335	435	6	university	university	NOUN
ejpam-5335	435	7	of	of	ADP
ejpam-5335	435	8	arkansas	arkansas	PROPN
ejpam-5335	435	9	,	,	PUNCT
ejpam-5335	435	10	1973	1973	NUM
ejpam-5335	435	11	.	.	PUNCT
ejpam-5335	436	1	[	[	X
ejpam-5335	436	2	12	12	NUM
ejpam-5335	436	3	]	]	X
ejpam-5335	436	4	e.	e.	PROPN
ejpam-5335	436	5	ekici	ekici	PROPN
ejpam-5335	436	6	.	.	PUNCT
ejpam-5335	437	1	on	on	ADP
ejpam-5335	437	2	δ	δ	PROPN
ejpam-5335	437	3	-	-	PUNCT
ejpam-5335	437	4	semiopen	semiopen	ADJ
ejpam-5335	437	5	sets	set	NOUN
ejpam-5335	437	6	and	and	CCONJ
ejpam-5335	437	7	a	a	DET
ejpam-5335	437	8	generalization	generalization	NOUN
ejpam-5335	437	9	of	of	ADP
ejpam-5335	437	10	functions	function	NOUN
ejpam-5335	437	11	.	.	PUNCT
ejpam-5335	438	1	boletim	boletim	PROPN
ejpam-5335	438	2	da	da	PROPN
ejpam-5335	438	3	sociedade	sociedade	PROPN
ejpam-5335	438	4	paranaense	paranaense	PROPN
ejpam-5335	438	5	de	de	PROPN
ejpam-5335	438	6	matemática	matemática	PROPN
ejpam-5335	438	7	,	,	PUNCT
ejpam-5335	438	8	23(1	23(1	NUM
ejpam-5335	438	9	-	-	PUNCT
ejpam-5335	438	10	2):73–84	2):73–84	NOUN
ejpam-5335	438	11	,	,	PUNCT
ejpam-5335	438	12	2005	2005	NUM
ejpam-5335	438	13	.	.	PUNCT
ejpam-5335	439	1	[	[	X
ejpam-5335	439	2	13	13	NUM
ejpam-5335	439	3	]	]	X
ejpam-5335	439	4	e.	e.	PROPN
ejpam-5335	439	5	ekici	ekici	PROPN
ejpam-5335	439	6	and	and	CCONJ
ejpam-5335	439	7	g.	g.	PROPN
ejpam-5335	439	8	navalagi	navalagi	PROPN
ejpam-5335	439	9	.	.	PUNCT
ejpam-5335	440	1	δ	δ	NOUN
ejpam-5335	440	2	-	-	PUNCT
ejpam-5335	440	3	semicontinuous	semicontinuous	ADJ
ejpam-5335	440	4	functions	function	NOUN
ejpam-5335	440	5	.	.	PUNCT
ejpam-5335	441	1	mathematical	mathematical	ADJ
ejpam-5335	441	2	forum	forum	PROPN
ejpam-5335	441	3	,	,	PUNCT
ejpam-5335	441	4	17:29–42	17:29–42	NUM
ejpam-5335	441	5	,	,	PUNCT
ejpam-5335	441	6	2005	2005	NUM
ejpam-5335	441	7	.	.	PUNCT
ejpam-5335	442	1	[	[	X
ejpam-5335	442	2	14	14	NUM
ejpam-5335	442	3	]	]	PUNCT
ejpam-5335	443	1	p.	p.	PROPN
ejpam-5335	443	2	e.	e.	PROPN
ejpam-5335	444	1	long	long	PROPN
ejpam-5335	444	2	and	and	CCONJ
ejpam-5335	444	3	l.	l.	PROPN
ejpam-5335	444	4	l.	l.	PROPN
ejpam-5335	444	5	herrington	herrington	PROPN
ejpam-5335	444	6	.	.	PUNCT
ejpam-5335	445	1	strongly	strongly	ADV
ejpam-5335	445	2	θ	θ	ADJ
ejpam-5335	445	3	-	-	ADJ
ejpam-5335	445	4	continuous	continuous	ADJ
ejpam-5335	445	5	functions	function	NOUN
ejpam-5335	445	6	.	.	PUNCT
ejpam-5335	446	1	journal	journal	NOUN
ejpam-5335	446	2	of	of	ADP
ejpam-5335	446	3	the	the	DET
ejpam-5335	446	4	korean	korean	PROPN
ejpam-5335	446	5	mathematical	mathematical	ADJ
ejpam-5335	446	6	society	society	NOUN
ejpam-5335	446	7	,	,	PUNCT
ejpam-5335	446	8	18:21–28	18:21–28	NUM
ejpam-5335	446	9	,	,	PUNCT
ejpam-5335	446	10	1981	1981	NUM
ejpam-5335	446	11	.	.	PUNCT
ejpam-5335	447	1	[	[	X
ejpam-5335	447	2	15	15	NUM
ejpam-5335	447	3	]	]	X
ejpam-5335	447	4	a.	a.	NOUN
ejpam-5335	447	5	s.	s.	PROPN
ejpam-5335	447	6	mashhour	mashhour	PROPN
ejpam-5335	447	7	,	,	PUNCT
ejpam-5335	447	8	m.	m.	PROPN
ejpam-5335	447	9	e.	e.	PROPN
ejpam-5335	447	10	abd	abd	PROPN
ejpam-5335	447	11	el	el	PROPN
ejpam-5335	447	12	-	-	PROPN
ejpam-5335	447	13	monsef	monsef	ADJ
ejpam-5335	447	14	,	,	PUNCT
ejpam-5335	447	15	and	and	CCONJ
ejpam-5335	447	16	s.	s.	PROPN
ejpam-5335	447	17	n.	n.	PROPN
ejpam-5335	447	18	el	el	PROPN
ejpam-5335	447	19	-	-	PROPN
ejpam-5335	447	20	deeb	deeb	PROPN
ejpam-5335	447	21	.	.	PUNCT
ejpam-5335	448	1	on	on	ADP
ejpam-5335	448	2	precontinuous	precontinuous	ADJ
ejpam-5335	448	3	and	and	CCONJ
ejpam-5335	448	4	weak	weak	ADJ
ejpam-5335	448	5	precontinuous	precontinuous	ADJ
ejpam-5335	448	6	mappings	mapping	NOUN
ejpam-5335	448	7	.	.	PUNCT
ejpam-5335	449	1	proceedings	proceeding	NOUN
ejpam-5335	449	2	of	of	ADP
ejpam-5335	449	3	the	the	DET
ejpam-5335	449	4	mathematical	mathematical	ADJ
ejpam-5335	449	5	and	and	CCONJ
ejpam-5335	449	6	physical	physical	ADJ
ejpam-5335	449	7	society	society	NOUN
ejpam-5335	449	8	of	of	ADP
ejpam-5335	449	9	egypt	egypt	PROPN
ejpam-5335	449	10	,	,	PUNCT
ejpam-5335	449	11	53:47–53	53:47–53	NUM
ejpam-5335	449	12	,	,	PUNCT
ejpam-5335	449	13	1982	1982	NUM
ejpam-5335	449	14	.	.	PUNCT
ejpam-5335	450	1	[	[	X
ejpam-5335	450	2	16	16	NUM
ejpam-5335	450	3	]	]	X
ejpam-5335	450	4	b.	b.	PROPN
ejpam-5335	450	5	m.	m.	PROPN
ejpam-5335	450	6	munshi	munshi	PROPN
ejpam-5335	450	7	and	and	CCONJ
ejpam-5335	450	8	d.	d.	PROPN
ejpam-5335	450	9	s.	s.	PROPN
ejpam-5335	450	10	bassan	bassan	PROPN
ejpam-5335	450	11	.	.	PUNCT
ejpam-5335	451	1	super	super	ADJ
ejpam-5335	451	2	-	-	ADJ
ejpam-5335	451	3	continuous	continuous	ADJ
ejpam-5335	451	4	mappings	mapping	NOUN
ejpam-5335	451	5	.	.	PUNCT
ejpam-5335	452	1	indian	indian	ADJ
ejpam-5335	452	2	journal	journal	PROPN
ejpam-5335	452	3	of	of	ADP
ejpam-5335	452	4	pure	pure	ADJ
ejpam-5335	452	5	and	and	CCONJ
ejpam-5335	452	6	applied	applied	ADJ
ejpam-5335	452	7	mathematics	mathematic	NOUN
ejpam-5335	452	8	,	,	PUNCT
ejpam-5335	452	9	13:229–239	13:229–239	NUM
ejpam-5335	452	10	,	,	PUNCT
ejpam-5335	452	11	1982	1982	NUM
ejpam-5335	452	12	.	.	PUNCT
ejpam-5335	453	1	references	reference	NOUN
ejpam-5335	453	2	3742	3742	NUM
ejpam-5335	454	1	[	[	X
ejpam-5335	454	2	17	17	NUM
ejpam-5335	454	3	]	]	PUNCT
ejpam-5335	454	4	t.	t.	PROPN
ejpam-5335	454	5	noiri	noiri	PROPN
ejpam-5335	454	6	.	.	PUNCT
ejpam-5335	455	1	on	on	ADP
ejpam-5335	455	2	δ	δ	PROPN
ejpam-5335	455	3	-	-	ADJ
ejpam-5335	455	4	continuous	continuous	ADJ
ejpam-5335	455	5	functions	function	NOUN
ejpam-5335	455	6	.	.	PUNCT
ejpam-5335	456	1	journal	journal	NOUN
ejpam-5335	456	2	of	of	ADP
ejpam-5335	456	3	the	the	DET
ejpam-5335	456	4	korean	korean	PROPN
ejpam-5335	456	5	mathematical	mathematical	ADJ
ejpam-5335	456	6	society	society	NOUN
ejpam-5335	456	7	,	,	PUNCT
ejpam-5335	456	8	16:161–166	16:161–166	PROPN
ejpam-5335	456	9	,	,	PUNCT
ejpam-5335	456	10	1980	1980	NUM
ejpam-5335	456	11	.	.	PUNCT
ejpam-5335	457	1	[	[	X
ejpam-5335	457	2	18	18	NUM
ejpam-5335	457	3	]	]	PUNCT
ejpam-5335	457	4	t.	t.	PROPN
ejpam-5335	457	5	noiri	noiri	PROPN
ejpam-5335	457	6	.	.	PUNCT
ejpam-5335	458	1	supercontinuity	supercontinuity	NOUN
ejpam-5335	458	2	and	and	CCONJ
ejpam-5335	458	3	some	some	DET
ejpam-5335	458	4	strong	strong	ADJ
ejpam-5335	458	5	forms	form	NOUN
ejpam-5335	458	6	of	of	ADP
ejpam-5335	458	7	continuity	continuity	NOUN
ejpam-5335	458	8	.	.	PUNCT
ejpam-5335	459	1	indian	indian	ADJ
ejpam-5335	459	2	journal	journal	PROPN
ejpam-5335	459	3	of	of	ADP
ejpam-5335	459	4	pure	pure	ADJ
ejpam-5335	459	5	and	and	CCONJ
ejpam-5335	459	6	applied	applied	ADJ
ejpam-5335	459	7	mathematics	mathematic	NOUN
ejpam-5335	459	8	,	,	PUNCT
ejpam-5335	459	9	15:241–250	15:241–250	NUM
ejpam-5335	459	10	,	,	PUNCT
ejpam-5335	459	11	1984	1984	NUM
ejpam-5335	459	12	.	.	PUNCT
ejpam-5335	460	1	[	[	X
ejpam-5335	460	2	19	19	NUM
ejpam-5335	460	3	]	]	X
ejpam-5335	460	4	j.	j.	PROPN
ejpam-5335	460	5	h.	h.	PROPN
ejpam-5335	460	6	park	park	PROPN
ejpam-5335	460	7	,	,	PUNCT
ejpam-5335	460	8	b.	b.	PROPN
ejpam-5335	460	9	y.	y.	PROPN
ejpam-5335	460	10	lee	lee	PROPN
ejpam-5335	460	11	,	,	PUNCT
ejpam-5335	460	12	and	and	CCONJ
ejpam-5335	460	13	m.	m.	PROPN
ejpam-5335	460	14	j.	j.	PROPN
ejpam-5335	460	15	son	son	PROPN
ejpam-5335	460	16	.	.	PUNCT
ejpam-5335	461	1	on	on	ADP
ejpam-5335	461	2	δ	δ	PROPN
ejpam-5335	461	3	-	-	PUNCT
ejpam-5335	461	4	semiopen	semiopen	ADJ
ejpam-5335	461	5	sets	set	NOUN
ejpam-5335	461	6	in	in	ADP
ejpam-5335	461	7	topological	topological	ADJ
ejpam-5335	461	8	spaces	space	NOUN
ejpam-5335	461	9	.	.	PUNCT
ejpam-5335	462	1	journal	journal	PROPN
ejpam-5335	462	2	of	of	ADP
ejpam-5335	462	3	indian	indian	PROPN
ejpam-5335	462	4	academy	academy	PROPN
ejpam-5335	462	5	of	of	ADP
ejpam-5335	462	6	mathematics	mathematic	NOUN
ejpam-5335	462	7	,	,	PUNCT
ejpam-5335	462	8	19(1):59–67	19(1):59–67	NUM
ejpam-5335	462	9	,	,	PUNCT
ejpam-5335	462	10	1997	1997	NUM
ejpam-5335	462	11	.	.	PUNCT
ejpam-5335	463	1	[	[	X
ejpam-5335	463	2	20	20	NUM
ejpam-5335	463	3	]	]	PUNCT
ejpam-5335	463	4	s.	s.	PROPN
ejpam-5335	463	5	raychaudhuri	raychaudhuri	PROPN
ejpam-5335	463	6	and	and	CCONJ
ejpam-5335	463	7	n.	n.	PROPN
ejpam-5335	463	8	mukherjee	mukherjee	PROPN
ejpam-5335	463	9	.	.	PUNCT
ejpam-5335	464	1	on	on	ADP
ejpam-5335	464	2	δ	δ	PROPN
ejpam-5335	464	3	-	-	PUNCT
ejpam-5335	464	4	almost	almost	ADV
ejpam-5335	464	5	continuity	continuity	NOUN
ejpam-5335	464	6	and	and	CCONJ
ejpam-5335	464	7	δ	δ	NOUN
ejpam-5335	464	8	-	-	PUNCT
ejpam-5335	464	9	preopen	preopen	ADJ
ejpam-5335	464	10	sets	set	NOUN
ejpam-5335	464	11	.	.	PUNCT
ejpam-5335	465	1	bulletin	bulletin	NOUN
ejpam-5335	465	2	of	of	ADP
ejpam-5335	465	3	the	the	DET
ejpam-5335	465	4	institute	institute	NOUN
ejpam-5335	465	5	of	of	ADP
ejpam-5335	465	6	mathematics	mathematics	PROPN
ejpam-5335	465	7	,	,	PUNCT
ejpam-5335	465	8	academia	academia	PROPN
ejpam-5335	465	9	sinica	sinica	PROPN
ejpam-5335	465	10	,	,	PUNCT
ejpam-5335	465	11	21:357–366	21:357–366	PROPN
ejpam-5335	465	12	,	,	PUNCT
ejpam-5335	465	13	1993	1993	NUM
ejpam-5335	465	14	.	.	PUNCT
ejpam-5335	466	1	[	[	X
ejpam-5335	466	2	21	21	NUM
ejpam-5335	466	3	]	]	X
ejpam-5335	466	4	i.	i.	PROPN
ejpam-5335	466	5	l.	l.	PROPN
ejpam-5335	466	6	reilly	reilly	PROPN
ejpam-5335	466	7	and	and	CCONJ
ejpam-5335	466	8	m.	m.	PROPN
ejpam-5335	466	9	k.	k.	PROPN
ejpam-5335	466	10	vamanamurthy	vamanamurthy	PROPN
ejpam-5335	466	11	.	.	PUNCT
ejpam-5335	467	1	on	on	ADP
ejpam-5335	467	2	super	super	ADJ
ejpam-5335	467	3	-	-	ADJ
ejpam-5335	467	4	continuous	continuous	ADJ
ejpam-5335	467	5	mappings	mapping	NOUN
ejpam-5335	467	6	.	.	PUNCT
ejpam-5335	468	1	indian	indian	ADJ
ejpam-5335	468	2	journal	journal	PROPN
ejpam-5335	468	3	of	of	ADP
ejpam-5335	468	4	pure	pure	ADJ
ejpam-5335	468	5	and	and	CCONJ
ejpam-5335	468	6	applied	applied	ADJ
ejpam-5335	468	7	mathematics	mathematic	NOUN
ejpam-5335	468	8	,	,	PUNCT
ejpam-5335	468	9	14:767–772	14:767–772	NUM
ejpam-5335	468	10	,	,	PUNCT
ejpam-5335	468	11	1983	1983	NUM
ejpam-5335	468	12	.	.	PUNCT
ejpam-5335	469	1	[	[	X
ejpam-5335	469	2	22	22	NUM
ejpam-5335	469	3	]	]	X
ejpam-5335	469	4	n.	n.	NOUN
ejpam-5335	469	5	v.	v.	ADP
ejpam-5335	469	6	veličko	veličko	PROPN
ejpam-5335	469	7	.	.	PUNCT
ejpam-5335	470	1	h	h	NOUN
ejpam-5335	470	2	-	-	PUNCT
ejpam-5335	470	3	closed	close	VERB
ejpam-5335	470	4	topological	topological	ADJ
ejpam-5335	470	5	spaces	space	NOUN
ejpam-5335	470	6	.	.	PUNCT
ejpam-5335	471	1	american	american	PROPN
ejpam-5335	471	2	mathematical	mathematical	ADJ
ejpam-5335	471	3	society	society	NOUN
ejpam-5335	471	4	translations	translation	NOUN
ejpam-5335	471	5	,	,	PUNCT
ejpam-5335	471	6	78(2):102–118	78(2):102–118	NUM
ejpam-5335	471	7	,	,	PUNCT
ejpam-5335	471	8	1968	1968	NUM
ejpam-5335	471	9	.	.	PUNCT
ejpam-5335	472	1	[	[	X
ejpam-5335	472	2	23	23	NUM
ejpam-5335	472	3	]	]	X
ejpam-5335	472	4	c.	c.	PROPN
ejpam-5335	472	5	viriyapong	viriyapong	PROPN
ejpam-5335	472	6	and	and	CCONJ
ejpam-5335	472	7	c.	c.	PROPN
ejpam-5335	472	8	boonpok	boonpok	PROPN
ejpam-5335	472	9	.	.	PUNCT
ejpam-5335	473	1	(	(	PUNCT
ejpam-5335	473	2	τ1	τ1	NOUN
ejpam-5335	473	3	,	,	PUNCT
ejpam-5335	473	4	τ2)α	τ2)α	NOUN
ejpam-5335	473	5	-	-	PUNCT
ejpam-5335	473	6	continuity	continuity	NOUN
ejpam-5335	473	7	for	for	ADP
ejpam-5335	473	8	multifunctions	multifunction	NOUN
ejpam-5335	473	9	.	.	PUNCT
ejpam-5335	474	1	journal	journal	PROPN
ejpam-5335	474	2	of	of	ADP
ejpam-5335	474	3	mathematics	mathematic	NOUN
ejpam-5335	474	4	,	,	PUNCT
ejpam-5335	474	5	2020:6285763	2020:6285763	NUM
ejpam-5335	474	6	,	,	PUNCT
ejpam-5335	474	7	2020	2020	NUM
ejpam-5335	474	8	.	.	PUNCT
ejpam-5335	475	1	[	[	X
ejpam-5335	475	2	24	24	NUM
ejpam-5335	475	3	]	]	X
ejpam-5335	475	4	n.	n.	PROPN
ejpam-5335	475	5	viriyapong	viriyapong	PROPN
ejpam-5335	475	6	,	,	PUNCT
ejpam-5335	475	7	s.	s.	PROPN
ejpam-5335	475	8	sompong	sompong	PROPN
ejpam-5335	475	9	,	,	PUNCT
ejpam-5335	475	10	and	and	CCONJ
ejpam-5335	475	11	c.	c.	PROPN
ejpam-5335	475	12	boonpok	boonpok	PROPN
ejpam-5335	475	13	.	.	PUNCT
ejpam-5335	476	1	(	(	PUNCT
ejpam-5335	476	2	τ1	τ1	NOUN
ejpam-5335	476	3	,	,	PUNCT
ejpam-5335	476	4	τ2)-extremal	τ2)-extremal	ADJ
ejpam-5335	476	5	disconnectedness	disconnectedness	NOUN
ejpam-5335	476	6	in	in	ADP
ejpam-5335	476	7	bitopological	bitopological	ADJ
ejpam-5335	476	8	spaces	space	NOUN
ejpam-5335	476	9	.	.	PUNCT
ejpam-5335	477	1	international	international	ADJ
ejpam-5335	477	2	journal	journal	PROPN
ejpam-5335	477	3	of	of	ADP
ejpam-5335	477	4	mathematics	mathematic	NOUN
ejpam-5335	477	5	and	and	CCONJ
ejpam-5335	477	6	computer	computer	NOUN
ejpam-5335	477	7	science	science	NOUN
ejpam-5335	477	8	,	,	PUNCT
ejpam-5335	477	9	19(3):855–860	19(3):855–860	PROPN
ejpam-5335	477	10	,	,	PUNCT
ejpam-5335	477	11	2024	2024	NUM
ejpam-5335	477	12	.	.	PUNCT
ejpam-5335	478	1	[	[	X
ejpam-5335	478	2	25	25	NUM
ejpam-5335	478	3	]	]	X
ejpam-5335	478	4	s.	s.	PROPN
ejpam-5335	478	5	yüksel	yüksel	PROPN
ejpam-5335	478	6	,	,	PUNCT
ejpam-5335	478	7	a.	a.	NOUN
ejpam-5335	478	8	açikgöz	açikgöz	PROPN
ejpam-5335	478	9	,	,	PUNCT
ejpam-5335	478	10	and	and	CCONJ
ejpam-5335	478	11	t.	t.	PROPN
ejpam-5335	478	12	noiri	noiri	PROPN
ejpam-5335	478	13	.	.	PUNCT
ejpam-5335	479	1	on	on	ADP
ejpam-5335	479	2	δ	δ	PROPN
ejpam-5335	479	3	-	-	PUNCT
ejpam-5335	479	4	i	i	NOUN
ejpam-5335	479	5	-	-	PUNCT
ejpam-5335	479	6	continuous	continuous	ADJ
ejpam-5335	479	7	functions	function	NOUN
ejpam-5335	479	8	.	.	PUNCT
ejpam-5335	480	1	turkish	turkish	ADJ
ejpam-5335	480	2	joutnal	joutnal	NOUN
ejpam-5335	480	3	of	of	ADP
ejpam-5335	480	4	mathematics	mathematic	NOUN
ejpam-5335	480	5	,	,	PUNCT
ejpam-5335	480	6	29:39–51	29:39–51	NUM
ejpam-5335	480	7	,	,	PUNCT
ejpam-5335	480	8	2005	2005	NUM
ejpam-5335	480	9	.	.	PUNCT