id	sid	tid	token	lemma	pos
cana-551	1	1	communications	communication	NOUN
cana-551	1	2	on	on	ADP
cana-551	1	3	applied	apply	VERB
cana-551	1	4	nonlinear	nonlinear	ADJ
cana-551	1	5	analysis	analysis	NOUN
cana-551	1	6	issn	issn	NOUN
cana-551	1	7	:	:	PUNCT
cana-551	1	8	1074	1074	NUM
cana-551	1	9	-	-	PUNCT
cana-551	1	10	133x	133x	NUM
cana-551	1	11	vol	vol	NOUN
cana-551	1	12	31	31	NUM
cana-551	1	13	no	no	NOUN
cana-551	1	14	.	.	NOUN
cana-551	1	15	2	2	NUM
cana-551	1	16	(	(	PUNCT
cana-551	1	17	2024	2024	NUM
cana-551	1	18	)	)	PUNCT
cana-551	1	19	284	284	NUM
cana-551	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	1	21	on	on	ADP
cana-551	1	22	soft	soft	ADJ
cana-551	1	23	strongly	strongly	ADV
cana-551	1	24	ƅ∗	ƅ∗	NOUN
cana-551	1	25	−	−	PROPN
cana-551	1	26	𝐂𝐥𝐨𝐬𝐞𝐝	𝐂𝐥𝐨𝐬𝐞𝐝	PROPN
cana-551	1	27	via	via	ADP
cana-551	1	28	soft	soft	ADJ
cana-551	1	29	ideal	ideal	NOUN
cana-551	1	30	abdelaziz	abdelaziz	PROPN
cana-551	1	31	e.	e.	PROPN
cana-551	1	32	radwan1	radwan1	PROPN
cana-551	1	33	,	,	PUNCT
cana-551	1	34	essam	essam	PROPN
cana-551	1	35	el	el	PROPN
cana-551	1	36	-	-	PUNCT
cana-551	1	37	seidy2	seidy2	PROPN
cana-551	1	38	,	,	PUNCT
cana-551	1	39	saif	saif	PROPN
cana-551	1	40	z.	z.	PROPN
cana-551	1	41	hameed	hameed	PROPN
cana-551	1	42	*	*	PROPN
cana-551	1	43	,	,	PUNCT
cana-551	1	44	3	3	NUM
cana-551	1	45	1	1	NUM
cana-551	1	46	,	,	PUNCT
cana-551	1	47	2department	2department	NUM
cana-551	1	48	of	of	ADP
cana-551	1	49	mathematics	mathematic	NOUN
cana-551	1	50	,	,	PUNCT
cana-551	1	51	faculty	faculty	NOUN
cana-551	1	52	of	of	ADP
cana-551	1	53	science	science	NOUN
cana-551	1	54	,	,	PUNCT
cana-551	1	55	ain	ain	PROPN
cana-551	1	56	shams	shams	PROPN
cana-551	1	57	university	university	PROPN
cana-551	1	58	,	,	PUNCT
cana-551	1	59	cairo	cairo	PROPN
cana-551	1	60	,	,	PUNCT
cana-551	1	61	egypt	egypt	PROPN
cana-551	1	62	3department	3department	NUM
cana-551	1	63	of	of	ADP
cana-551	1	64	mathematics	mathematic	NOUN
cana-551	1	65	,	,	PUNCT
cana-551	1	66	college	college	NOUN
cana-551	1	67	of	of	ADP
cana-551	1	68	education	education	NOUN
cana-551	1	69	,	,	PUNCT
cana-551	1	70	mustansiriyah	mustansiriyah	NOUN
cana-551	1	71	university	university	NOUN
cana-551	1	72	,	,	PUNCT
cana-551	1	73	baghdad	baghdad	PROPN
cana-551	1	74	,	,	PUNCT
cana-551	1	75	iraq	iraq	PROPN
cana-551	1	76	email	email	NOUN
cana-551	1	77	address	address	NOUN
cana-551	1	78	:	:	PUNCT
cana-551	1	79	1zezoradwan@yahoo.com	1zezoradwan@yahoo.com	NUM
cana-551	1	80	,	,	PUNCT
cana-551	1	81	2esam_elsedy@hotmail.com	2esam_elsedy@hotmail.com	PROPN
cana-551	1	82	,	,	PUNCT
cana-551	1	83	3saif.zuhar.edbs@uomustansiriyah.edu.iq	3saif.zuhar.edbs@uomustansiriyah.edu.iq	NUM
cana-551	1	84	3https://orcid.org/0000-0001-6197-3525	3https://orcid.org/0000-0001-6197-3525	PROPN
cana-551	1	85	article	article	NOUN
cana-551	1	86	history	history	NOUN
cana-551	1	87	:	:	PUNCT
cana-551	1	88	received	receive	VERB
cana-551	1	89	:	:	PUNCT
cana-551	1	90	01	01	NUM
cana-551	1	91	-	-	PUNCT
cana-551	1	92	02	02	NUM
cana-551	1	93	-	-	PUNCT
cana-551	1	94	2024	2024	NUM
cana-551	1	95	revised	revise	VERB
cana-551	1	96	:	:	PUNCT
cana-551	1	97	15	15	NUM
cana-551	1	98	-	-	PUNCT
cana-551	1	99	04	04	NUM
cana-551	1	100	-	-	PUNCT
cana-551	1	101	2024	2024	NUM
cana-551	1	102	accepted	accept	VERB
cana-551	1	103	:	:	PUNCT
cana-551	1	104	04	04	NUM
cana-551	1	105	-	-	PUNCT
cana-551	1	106	05	05	NUM
cana-551	1	107	-	-	PUNCT
cana-551	1	108	2024	2024	NUM
cana-551	1	109	abstract	abstract	NOUN
cana-551	1	110	:	:	PUNCT
cana-551	1	111	in	in	ADP
cana-551	1	112	this	this	DET
cana-551	1	113	paper	paper	NOUN
cana-551	1	114	,	,	PUNCT
cana-551	1	115	we	we	PRON
cana-551	1	116	introduce	introduce	VERB
cana-551	1	117	the	the	DET
cana-551	1	118	soft	soft	ADJ
cana-551	1	119	strongly	strongly	ADV
cana-551	1	120	ƅ∗	ƅ∗	PROPN
cana-551	1	121	−closed	−close	VERB
cana-551	1	122	via	via	ADP
cana-551	1	123	soft	soft	ADJ
cana-551	1	124	ideal	ideal	NOUN
cana-551	1	125	and	and	CCONJ
cana-551	1	126	study	study	VERB
cana-551	1	127	the	the	DET
cana-551	1	128	behavior	behavior	NOUN
cana-551	1	129	of	of	ADP
cana-551	1	130	intersection	intersection	NOUN
cana-551	1	131	and	and	CCONJ
cana-551	1	132	union	union	NOUN
cana-551	1	133	of	of	ADP
cana-551	1	134	this	this	DET
cana-551	1	135	level	level	NOUN
cana-551	1	136	.	.	PUNCT
cana-551	2	1	also	also	ADV
cana-551	2	2	,	,	PUNCT
cana-551	2	3	we	we	PRON
cana-551	2	4	define	define	VERB
cana-551	2	5	the	the	DET
cana-551	2	6	soft	soft	ADJ
cana-551	2	7	strongly	strongly	ADV
cana-551	2	8	ƅ∗ῐ	ƅ∗ῐ	PUNCT
cana-551	2	9	−continuous	−continuous	ADJ
cana-551	2	10	,	,	PUNCT
cana-551	2	11	irresolute	irresolute	ADJ
cana-551	2	12	,	,	PUNCT
cana-551	2	13	soft	soft	ADJ
cana-551	2	14	strongly	strongly	ADV
cana-551	2	15	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	2	16	−open	−open	NOUN
cana-551	2	17	map	map	VERB
cana-551	2	18	and	and	CCONJ
cana-551	2	19	soft	soft	ADJ
cana-551	2	20	strongly	strongly	ADV
cana-551	2	21	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	2	22	−closed	−close	VERB
cana-551	2	23	map	map	NOUN
cana-551	2	24	with	with	ADP
cana-551	2	25	some	some	DET
cana-551	2	26	properties	property	NOUN
cana-551	2	27	.	.	PUNCT
cana-551	3	1	moreover	moreover	ADV
cana-551	3	2	,	,	PUNCT
cana-551	3	3	the	the	DET
cana-551	3	4	relationship	relationship	NOUN
cana-551	3	5	between	between	ADP
cana-551	3	6	another	another	DET
cana-551	3	7	closed	closed	ADJ
cana-551	3	8	sets	set	NOUN
cana-551	3	9	and	and	CCONJ
cana-551	3	10	soft	soft	ADJ
cana-551	3	11	strongly	strongly	ADV
cana-551	3	12	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	3	13	−closed	−close	VERB
cana-551	3	14	with	with	ADP
cana-551	3	15	counterexamples	counterexample	NOUN
cana-551	3	16	are	be	AUX
cana-551	3	17	discuss	discuss	ADJ
cana-551	3	18	.	.	PUNCT
cana-551	4	1	keywords	keyword	NOUN
cana-551	4	2	:	:	PUNCT
cana-551	4	3	soft	soft	ADJ
cana-551	4	4	ideal	ideal	NOUN
cana-551	4	5	,	,	PUNCT
cana-551	4	6	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	4	7	−closed	−close	VERB
cana-551	4	8	set	set	NOUN
cana-551	4	9	,	,	PUNCT
cana-551	4	10	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	4	11	−closed	−close	VERB
cana-551	4	12	,	,	PUNCT
cana-551	4	13	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	4	14	−continuous	−continuous	ADJ
cana-551	4	15	,	,	PUNCT
cana-551	4	16	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	4	17	−irresolute	−irresolute	NOUN
cana-551	4	18	.	.	PUNCT
cana-551	5	1	mathematics	mathematics	PROPN
cana-551	5	2	subject	subject	ADJ
cana-551	5	3	classification	classification	NOUN
cana-551	5	4	:	:	PUNCT
cana-551	5	5	54a05	54a05	NUM
cana-551	5	6	,	,	PUNCT
cana-551	5	7	54a10	54a10	NUM
cana-551	5	8	,	,	PUNCT
cana-551	5	9	54a20	54a20	NUM
cana-551	5	10	,	,	PUNCT
cana-551	5	11	54a40	54a40	NUM
cana-551	5	12	.	.	NOUN
cana-551	6	1	1	1	NUM
cana-551	6	2	.	.	X
cana-551	6	3	introduction	introduction	NOUN
cana-551	6	4	and	and	CCONJ
cana-551	6	5	preliminaries	preliminary	NOUN
cana-551	6	6	molodtsov	molodtsov	NOUN
cana-551	7	1	[	[	X
cana-551	7	2	1	1	NUM
cana-551	7	3	]	]	PUNCT
cana-551	7	4	,	,	PUNCT
cana-551	7	5	instigated	instigate	VERB
cana-551	7	6	the	the	DET
cana-551	7	7	concept	concept	NOUN
cana-551	7	8	of	of	ADP
cana-551	7	9	soft	soft	ADJ
cana-551	7	10	set	set	NOUN
cana-551	7	11	as	as	ADP
cana-551	7	12	a	a	DET
cana-551	7	13	new	new	ADJ
cana-551	7	14	mathematical	mathematical	ADJ
cana-551	7	15	tool	tool	NOUN
cana-551	7	16	to	to	PART
cana-551	7	17	deal	deal	VERB
cana-551	7	18	with	with	ADP
cana-551	7	19	uncertainties	uncertainty	NOUN
cana-551	7	20	problems	problem	NOUN
cana-551	7	21	in	in	ADP
cana-551	7	22	different	different	ADJ
cana-551	7	23	fields	field	NOUN
cana-551	7	24	of	of	ADP
cana-551	7	25	science	science	NOUN
cana-551	7	26	.	.	PUNCT
cana-551	8	1	i.	i.	PROPN
cana-551	8	2	arockiarani	arockiarani	PROPN
cana-551	8	3	and	and	CCONJ
cana-551	8	4	a.	a.	NOUN
cana-551	8	5	arokialancy	arokialancy	NOUN
cana-551	9	1	[	[	X
cana-551	9	2	2	2	NUM
cana-551	9	3	]	]	PUNCT
cana-551	9	4	studied	study	VERB
cana-551	9	5	the	the	DET
cana-551	9	6	soft	soft	ADJ
cana-551	9	7	𝛽	𝛽	NOUN
cana-551	9	8	−open	−open	NOUN
cana-551	9	9	sets	set	VERB
cana-551	9	10	and	and	CCONJ
cana-551	9	11	continuous	continuous	ADJ
cana-551	9	12	.	.	PUNCT
cana-551	10	1	akdag	akdag	PROPN
cana-551	10	2	and	and	CCONJ
cana-551	10	3	ozkan	ozkan	X
cana-551	11	1	[	[	X
cana-551	11	2	3	3	NUM
cana-551	11	3	,	,	PUNCT
cana-551	11	4	4	4	NUM
cana-551	11	5	]	]	PUNCT
cana-551	11	6	introduced	introduce	VERB
cana-551	11	7	the	the	DET
cana-551	11	8	soft	soft	ADJ
cana-551	11	9	𝛼-open	𝛼-open	NOUN
cana-551	11	10	and	and	CCONJ
cana-551	11	11	define	define	VERB
cana-551	11	12	soft	soft	ADJ
cana-551	11	13	b	b	NOUN
cana-551	11	14	-	-	PUNCT
cana-551	11	15	open	open	ADJ
cana-551	11	16	and	and	CCONJ
cana-551	11	17	continuous	continuous	ADJ
cana-551	11	18	.	.	PUNCT
cana-551	12	1	hameed	hameed	PROPN
cana-551	12	2	,	,	PUNCT
cana-551	12	3	s.	s.	PROPN
cana-551	12	4	z.	z.	PROPN
cana-551	12	5	,	,	PUNCT
cana-551	12	6	hussein	hussein	PROPN
cana-551	12	7	,	,	PUNCT
cana-551	12	8	a.	a.	PROPN
cana-551	12	9	k	k	PROPN
cana-551	13	1	[	[	X
cana-551	13	2	5	5	NUM
cana-551	13	3	]	]	PUNCT
cana-551	13	4	defined	define	VERB
cana-551	13	5	the	the	DET
cana-551	13	6	soft	soft	ADJ
cana-551	13	7	ƅ𝑐	ƅ𝑐	PRON
cana-551	13	8	−open	−open	NOUN
cana-551	13	9	set	set	VERB
cana-551	13	10	.	.	PUNCT
cana-551	14	1	the	the	DET
cana-551	14	2	soft	soft	ADJ
cana-551	14	3	ƅ∗	ƅ∗	NOUN
cana-551	14	4	−	−	NOUN
cana-551	14	5	closed	closed	ADJ
cana-551	14	6	,	,	PUNCT
cana-551	14	7	𝑠ƅ∗	𝑠ƅ∗	ADJ
cana-551	14	8	−continuous	−continuous	ADJ
cana-551	14	9	,	,	PUNCT
cana-551	14	10	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	14	11	−closed	−close	VERB
cana-551	14	12	sets	set	NOUN
cana-551	14	13	and	and	CCONJ
cana-551	14	14	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	14	15	−continuous	−continuous	ADJ
cana-551	14	16	functions	function	NOUN
cana-551	14	17	are	be	AUX
cana-551	14	18	studied	study	VERB
cana-551	14	19	by	by	ADP
cana-551	14	20	saif	saif	PROPN
cana-551	14	21	at	at	ADP
cana-551	14	22	el	el	PROPN
cana-551	14	23	.	.	PUNCT
cana-551	15	1	in	in	ADP
cana-551	15	2	[	[	X
cana-551	15	3	6	6	NUM
cana-551	15	4	]	]	PUNCT
cana-551	15	5	,	,	PUNCT
cana-551	15	6	[	[	X
cana-551	15	7	7	7	X
cana-551	15	8	]	]	PUNCT
cana-551	15	9	and	and	CCONJ
cana-551	15	10	[	[	X
cana-551	15	11	8	8	NUM
cana-551	15	12	]	]	PUNCT
cana-551	15	13	.	.	PUNCT
cana-551	16	1	kandil	kandil	PROPN
cana-551	16	2	et	et	PROPN
cana-551	16	3	al	al	PROPN
cana-551	16	4	.	.	PUNCT
cana-551	17	1	[	[	X
cana-551	17	2	9	9	NUM
cana-551	17	3	]	]	PUNCT
cana-551	17	4	define	define	VERB
cana-551	17	5	soft	soft	ADJ
cana-551	17	6	ideal	ideal	NOUN
cana-551	17	7	and	and	CCONJ
cana-551	17	8	introduced	introduce	VERB
cana-551	17	9	the	the	DET
cana-551	17	10	soft	soft	ADJ
cana-551	17	11	local	local	ADJ
cana-551	17	12	function	function	NOUN
cana-551	17	13	.	.	PUNCT
cana-551	18	1	these	these	DET
cana-551	18	2	concepts	concept	NOUN
cana-551	18	3	are	be	AUX
cana-551	18	4	discussed	discuss	VERB
cana-551	18	5	with	with	ADP
cana-551	18	6	a	a	DET
cana-551	18	7	view	view	NOUN
cana-551	18	8	to	to	PART
cana-551	18	9	find	find	VERB
cana-551	18	10	new	new	ADJ
cana-551	18	11	soft	soft	ADJ
cana-551	18	12	topologies	topology	NOUN
cana-551	18	13	from	from	ADP
cana-551	18	14	the	the	DET
cana-551	18	15	original	original	ADJ
cana-551	18	16	one	one	NOUN
cana-551	18	17	,	,	PUNCT
cana-551	18	18	called	call	VERB
cana-551	18	19	𝒮tss	𝒮tss	PROPN
cana-551	18	20	with	with	ADP
cana-551	18	21	soft	soft	ADJ
cana-551	18	22	ideal	ideal	NOUN
cana-551	18	23	(	(	PUNCT
cana-551	18	24	𝒵	𝒵	PROPN
cana-551	18	25	,	,	PUNCT
cana-551	18	26	𝔚	𝔚	PROPN
cana-551	18	27	,	,	PUNCT
cana-551	18	28	𝛥	𝛥	PROPN
cana-551	18	29	,	,	PUNCT
cana-551	18	30	ῐ	ῐ	PROPN
cana-551	18	31	)	)	PUNCT
cana-551	18	32	.	.	PUNCT
cana-551	19	1	mustafa	mustafa	PROPN
cana-551	19	2	and	and	CCONJ
cana-551	19	3	sleim	sleim	PROPN
cana-551	20	1	[	[	X
cana-551	20	2	10	10	NUM
cana-551	20	3	]	]	PUNCT
cana-551	20	4	studied	study	VERB
cana-551	20	5	the	the	DET
cana-551	20	6	notion	notion	NOUN
cana-551	20	7	of	of	ADP
cana-551	20	8	a	a	DET
cana-551	20	9	soft	soft	ADJ
cana-551	20	10	ideal	ideal	NOUN
cana-551	20	11	and	and	CCONJ
cana-551	20	12	they	they	PRON
cana-551	20	13	introduced	introduce	VERB
cana-551	20	14	the	the	DET
cana-551	20	15	soft	soft	ADJ
cana-551	20	16	generalized	generalized	ADJ
cana-551	20	17	closed	closed	ADJ
cana-551	20	18	sets	set	NOUN
cana-551	20	19	with	with	ADP
cana-551	20	20	respect	respect	NOUN
cana-551	20	21	to	to	ADP
cana-551	20	22	a	a	DET
cana-551	20	23	soft	soft	ADJ
cana-551	20	24	ideal	ideal	NOUN
cana-551	20	25	and	and	CCONJ
cana-551	20	26	studied	study	VERB
cana-551	20	27	their	their	PRON
cana-551	20	28	properties	property	NOUN
cana-551	20	29	in	in	ADP
cana-551	20	30	detail	detail	NOUN
cana-551	20	31	,	,	PUNCT
cana-551	20	32	which	which	PRON
cana-551	20	33	is	be	AUX
cana-551	20	34	extension	extension	NOUN
cana-551	20	35	of	of	ADP
cana-551	20	36	the	the	DET
cana-551	20	37	concept	concept	NOUN
cana-551	20	38	of	of	ADP
cana-551	20	39	soft	soft	ADJ
cana-551	20	40	generalized	generalize	VERB
cana-551	20	41	closed	closed	ADJ
cana-551	20	42	sets	set	NOUN
cana-551	20	43	.	.	PUNCT
cana-551	21	1	later	later	ADV
cana-551	21	2	,	,	PUNCT
cana-551	21	3	k.	k.	PROPN
cana-551	21	4	kannan	kannan	PROPN
cana-551	22	1	[	[	X
cana-551	22	2	11	11	NUM
cana-551	22	3	]	]	PUNCT
cana-551	22	4	introduced	introduce	VERB
cana-551	22	5	the	the	DET
cana-551	22	6	soft	soft	ADJ
cana-551	22	7	g	g	NOUN
cana-551	22	8	-	-	PUNCT
cana-551	22	9	closed	close	VERB
cana-551	22	10	soft	soft	ADJ
cana-551	22	11	sets	set	NOUN
cana-551	22	12	in	in	ADP
cana-551	22	13	a	a	DET
cana-551	22	14	𝒮ts	𝒮ts	PROPN
cana-551	22	15	.	.	PROPN
cana-551	22	16	in	in	ADP
cana-551	22	17	this	this	DET
cana-551	22	18	work	work	NOUN
cana-551	22	19	,	,	PUNCT
cana-551	22	20	we	we	PRON
cana-551	22	21	study	study	VERB
cana-551	22	22	the	the	DET
cana-551	22	23	concept	concept	NOUN
cana-551	22	24	of	of	ADP
cana-551	22	25	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	PROPN
cana-551	22	26	−closed	−close	VERB
cana-551	22	27	set	set	VERB
cana-551	22	28	via	via	ADP
cana-551	22	29	soft	soft	ADJ
cana-551	22	30	ideal	ideal	NOUN
cana-551	22	31	,	,	PUNCT
cana-551	22	32	also	also	ADV
cana-551	22	33	,	,	PUNCT
cana-551	22	34	we	we	PRON
cana-551	22	35	study	study	VERB
cana-551	22	36	the	the	DET
cana-551	22	37	relationship	relationship	NOUN
cana-551	22	38	between	between	ADP
cana-551	22	39	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	22	40	−closed	−close	VERB
cana-551	22	41	sets	set	NOUN
cana-551	22	42	and	and	CCONJ
cana-551	22	43	other	other	ADJ
cana-551	22	44	existing	exist	VERB
cana-551	22	45	soft	soft	ADJ
cana-551	22	46	sets	set	NOUN
cana-551	22	47	have	have	AUX
cana-551	22	48	been	be	AUX
cana-551	22	49	investigated	investigate	VERB
cana-551	22	50	.	.	PUNCT
cana-551	23	1	moreover	moreover	ADV
cana-551	23	2	,	,	PUNCT
cana-551	23	3	the	the	DET
cana-551	23	4	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	23	5	−continuous	−continuous	PROPN
cana-551	23	6	,	,	PUNCT
cana-551	23	7	irresolute	irresolute	ADJ
cana-551	23	8	,	,	PUNCT
cana-551	23	9	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	ADJ
cana-551	23	10	−open	−open	NOUN
cana-551	23	11	map	map	NOUN
cana-551	23	12	and	and	CCONJ
cana-551	23	13	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	ADJ
cana-551	23	14	−closed	−close	VERB
cana-551	23	15	map	map	NOUN
cana-551	23	16	with	with	ADP
cana-551	23	17	counterexamples	counterexample	NOUN
cana-551	23	18	are	be	AUX
cana-551	23	19	discuss	discus	VERB
cana-551	23	20	.	.	PUNCT
cana-551	24	1	communications	communication	NOUN
cana-551	24	2	on	on	ADP
cana-551	24	3	applied	apply	VERB
cana-551	24	4	nonlinear	nonlinear	ADJ
cana-551	24	5	analysis	analysis	NOUN
cana-551	24	6	issn	issn	NOUN
cana-551	24	7	:	:	PUNCT
cana-551	24	8	1074	1074	NUM
cana-551	24	9	-	-	PUNCT
cana-551	24	10	133x	133x	NUM
cana-551	24	11	vol	vol	NOUN
cana-551	24	12	31	31	NUM
cana-551	24	13	no	no	NOUN
cana-551	24	14	.	.	NOUN
cana-551	24	15	2	2	NUM
cana-551	24	16	(	(	PUNCT
cana-551	24	17	2024	2024	NUM
cana-551	24	18	)	)	PUNCT
cana-551	25	1	285	285	NUM
cana-551	25	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	25	3	definition	definition	NOUN
cana-551	25	4	1.1	1.1	NUM
cana-551	25	5	:	:	PUNCT
cana-551	26	1	[	[	X
cana-551	26	2	1	1	X
cana-551	26	3	]	]	PUNCT
cana-551	26	4	let	let	VERB
cana-551	26	5	𝒵	𝒵	PRON
cana-551	26	6	be	be	AUX
cana-551	26	7	an	an	DET
cana-551	26	8	initial	initial	ADJ
cana-551	26	9	universe	universe	NOUN
cana-551	26	10	set	set	VERB
cana-551	26	11	and	and	CCONJ
cana-551	26	12	𝐸	𝐸	PROPN
cana-551	26	13	be	be	VERB
cana-551	26	14	a	a	DET
cana-551	26	15	set	set	NOUN
cana-551	26	16	of	of	ADP
cana-551	26	17	parameters	parameter	NOUN
cana-551	26	18	.	.	PUNCT
cana-551	27	1	let	let	AUX
cana-551	27	2	𝑃(𝒵	𝑃(𝒵	NOUN
cana-551	27	3	)	)	PUNCT
cana-551	27	4	denote	denote	VERB
cana-551	27	5	the	the	DET
cana-551	27	6	power	power	NOUN
cana-551	27	7	set	set	NOUN
cana-551	27	8	of	of	ADP
cana-551	27	9	𝒵	𝒵	PROPN
cana-551	27	10	,	,	PUNCT
cana-551	27	11	and	and	CCONJ
cana-551	27	12	∆⊂	∆⊂	VERB
cana-551	27	13	𝐸.	𝐸.	VERB
cana-551	27	14	a	a	DET
cana-551	27	15	pair	pair	NOUN
cana-551	27	16	(	(	PUNCT
cana-551	27	17	𝛾	𝛾	NOUN
cana-551	27	18	,	,	PUNCT
cana-551	27	19	𝛥	𝛥	NOUN
cana-551	27	20	)	)	PUNCT
cana-551	27	21	is	be	AUX
cana-551	27	22	called	call	VERB
cana-551	27	23	a	a	DET
cana-551	27	24	soft	soft	ADJ
cana-551	27	25	set	set	NOUN
cana-551	27	26	over	over	ADP
cana-551	27	27	𝒵.	𝒵.	PROPN
cana-551	27	28	where	where	SCONJ
cana-551	27	29	𝛾	𝛾	NOUN
cana-551	27	30	is	be	AUX
cana-551	27	31	a	a	DET
cana-551	27	32	mapping	mapping	NOUN
cana-551	27	33	given	give	VERB
cana-551	27	34	by	by	ADP
cana-551	27	35	𝛾	𝛾	ADP
cana-551	27	36	:	:	PUNCT
cana-551	27	37	𝛥	𝛥	PROPN
cana-551	27	38	→	→	SYM
cana-551	27	39	𝑃(𝒵	𝑃(𝒵	NOUN
cana-551	27	40	)	)	PUNCT
cana-551	27	41	.	.	PUNCT
cana-551	28	1	the	the	DET
cana-551	28	2	family	family	NOUN
cana-551	28	3	of	of	ADP
cana-551	28	4	all	all	DET
cana-551	28	5	soft	soft	ADJ
cana-551	28	6	sets	set	NOUN
cana-551	28	7	over	over	ADP
cana-551	28	8	𝒵	𝒵	PROPN
cana-551	28	9	denote	denote	VERB
cana-551	28	10	by	by	ADP
cana-551	28	11	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	28	12	,	,	PUNCT
cana-551	28	13	𝛥	𝛥	PROPN
cana-551	28	14	)	)	PUNCT
cana-551	28	15	definition	definition	NOUN
cana-551	28	16	1.2	1.2	NUM
cana-551	28	17	:	:	PUNCT
cana-551	29	1	[	[	X
cana-551	29	2	12	12	NUM
cana-551	29	3	]	]	PUNCT
cana-551	29	4	the	the	DET
cana-551	29	5	soft	soft	ADJ
cana-551	29	6	set	set	NOUN
cana-551	29	7	(	(	PUNCT
cana-551	29	8	𝛿	𝛿	ADJ
cana-551	29	9	,	,	PUNCT
cana-551	29	10	𝛥	𝛥	NOUN
cana-551	29	11	)	)	PUNCT
cana-551	29	12	∈	∈	PROPN
cana-551	29	13	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	29	14	,	,	PUNCT
cana-551	29	15	𝛥	𝛥	PROPN
cana-551	29	16	)	)	PUNCT
cana-551	29	17	,	,	PUNCT
cana-551	29	18	where	where	SCONJ
cana-551	29	19	𝛿(𝑐	𝛿(𝑐	NOUN
cana-551	29	20	)	)	PUNCT
cana-551	29	21	=	=	NOUN
cana-551	29	22	∅	∅	NOUN
cana-551	29	23	,	,	PUNCT
cana-551	29	24	for	for	ADP
cana-551	29	25	every	every	DET
cana-551	29	26	c	c	NOUN
cana-551	29	27	∈	∈	PROPN
cana-551	29	28	𝛥	𝛥	PROPN
cana-551	29	29	is	be	AUX
cana-551	29	30	called	call	VERB
cana-551	29	31	a	a	DET
cana-551	29	32	-	-	PUNCT
cana-551	29	33	null	null	ADJ
cana-551	29	34	soft	soft	ADJ
cana-551	29	35	set	set	NOUN
cana-551	29	36	of	of	ADP
cana-551	29	37	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	29	38	,	,	PUNCT
cana-551	29	39	𝛥	𝛥	PROPN
cana-551	29	40	)	)	PUNCT
cana-551	29	41	and	and	CCONJ
cana-551	29	42	denoted	denote	VERB
cana-551	29	43	by	by	ADP
cana-551	29	44	∅̃.	∅̃.	NOUN
cana-551	29	45	the	the	DET
cana-551	29	46	soft	soft	ADJ
cana-551	29	47	set	set	NOUN
cana-551	29	48	(	(	PUNCT
cana-551	29	49	𝛿	𝛿	ADJ
cana-551	29	50	,	,	PUNCT
cana-551	29	51	𝛥	𝛥	NOUN
cana-551	29	52	)	)	PUNCT
cana-551	29	53	∈	∈	PROPN
cana-551	29	54	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	29	55	,	,	PUNCT
cana-551	29	56	𝛥	𝛥	PROPN
cana-551	29	57	)	)	PUNCT
cana-551	29	58	,	,	PUNCT
cana-551	29	59	where	where	SCONJ
cana-551	29	60	𝛿(𝑐	𝛿(𝑐	PROPN
cana-551	29	61	)	)	PUNCT
cana-551	30	1	=	=	SYM
cana-551	30	2	𝒵	𝒵	PROPN
cana-551	30	3	,	,	PUNCT
cana-551	30	4	for	for	ADP
cana-551	30	5	every	every	DET
cana-551	30	6	𝑐	𝑐	PROPN
cana-551	30	7	∈	∈	PROPN
cana-551	30	8	𝛥	𝛥	PROPN
cana-551	30	9	is	be	AUX
cana-551	30	10	called	call	VERB
cana-551	30	11	the	the	DET
cana-551	30	12	a	a	PRON
cana-551	30	13	-	-	PUNCT
cana-551	30	14	absolute	absolute	ADJ
cana-551	30	15	soft	soft	ADJ
cana-551	30	16	set	set	NOUN
cana-551	30	17	of	of	ADP
cana-551	30	18	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	30	19	,	,	PUNCT
cana-551	30	20	𝛥	𝛥	PROPN
cana-551	30	21	)	)	PUNCT
cana-551	30	22	and	and	CCONJ
cana-551	30	23	denoted	denote	VERB
cana-551	30	24	by	by	ADP
cana-551	30	25	𝒵.	𝒵.	PROPN
cana-551	30	26	definition	definition	NOUN
cana-551	30	27	1.3	1.3	NUM
cana-551	30	28	:	:	PUNCT
cana-551	31	1	[	[	X
cana-551	31	2	12	12	NUM
cana-551	31	3	]	]	PUNCT
cana-551	31	4	for	for	ADP
cana-551	31	5	two	two	NUM
cana-551	31	6	sets	set	NOUN
cana-551	31	7	(	(	PUNCT
cana-551	31	8	𝛾	𝛾	NOUN
cana-551	31	9	,	,	PUNCT
cana-551	31	10	𝛥	𝛥	NOUN
cana-551	31	11	)	)	PUNCT
cana-551	31	12	,	,	PUNCT
cana-551	31	13	(	(	PUNCT
cana-551	31	14	𝛿	𝛿	ADJ
cana-551	31	15	,	,	PUNCT
cana-551	31	16	𝐵	𝐵	NOUN
cana-551	31	17	)	)	PUNCT
cana-551	31	18	∈	∈	PROPN
cana-551	31	19	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	31	20	,	,	PUNCT
cana-551	31	21	𝛥	𝛥	PROPN
cana-551	31	22	)	)	PUNCT
cana-551	31	23	,	,	PUNCT
cana-551	31	24	we	we	PRON
cana-551	31	25	say	say	VERB
cana-551	31	26	that	that	SCONJ
cana-551	31	27	(	(	PUNCT
cana-551	31	28	𝛾	𝛾	NOUN
cana-551	31	29	,	,	PUNCT
cana-551	31	30	𝛥	𝛥	NOUN
cana-551	31	31	)	)	PUNCT
cana-551	31	32	is	be	AUX
cana-551	31	33	a	a	DET
cana-551	31	34	soft	soft	ADJ
cana-551	31	35	subset	subset	NOUN
cana-551	31	36	of	of	ADP
cana-551	31	37	(	(	PUNCT
cana-551	31	38	𝛿	𝛿	ADJ
cana-551	31	39	,	,	PUNCT
cana-551	31	40	𝐵	𝐵	NOUN
cana-551	31	41	)	)	PUNCT
cana-551	31	42	denoted	denote	VERB
cana-551	31	43	by	by	ADP
cana-551	31	44	(	(	PUNCT
cana-551	31	45	𝛾	𝛾	PROPN
cana-551	31	46	,	,	PUNCT
cana-551	31	47	𝛥	𝛥	NOUN
cana-551	31	48	)	)	PUNCT
cana-551	31	49	⊆	⊆	NUM
cana-551	31	50	(	(	PUNCT
cana-551	31	51	𝛿	𝛿	ADJ
cana-551	31	52	,	,	PUNCT
cana-551	31	53	𝐵	𝐵	NOUN
cana-551	31	54	)	)	PUNCT
cana-551	31	55	,	,	PUNCT
cana-551	31	56	if	if	SCONJ
cana-551	31	57	(	(	PUNCT
cana-551	31	58	1	1	X
cana-551	31	59	)	)	PUNCT
cana-551	31	60	𝛥	𝛥	PROPN
cana-551	32	1	⊆	⊆	NUM
cana-551	32	2	𝐵.	𝐵.	PROPN
cana-551	32	3	(	(	PUNCT
cana-551	32	4	2	2	NUM
cana-551	32	5	)	)	PUNCT
cana-551	32	6	𝛾(𝛻	𝛾(𝛻	NUM
cana-551	32	7	)	)	PUNCT
cana-551	32	8	⊆	⊆	NUM
cana-551	32	9	𝛿(𝛻	𝛿(𝛻	NUM
cana-551	32	10	)	)	PUNCT
cana-551	32	11	,	,	PUNCT
cana-551	32	12	∀	∀	X
cana-551	32	13	𝛻	𝛻	PROPN
cana-551	32	14	∈	∈	PROPN
cana-551	32	15	𝛥.	𝛥.	PROPN
cana-551	32	16	in	in	ADP
cana-551	32	17	this	this	DET
cana-551	32	18	case	case	NOUN
cana-551	32	19	,	,	PUNCT
cana-551	32	20	(	(	PUNCT
cana-551	32	21	𝛾	𝛾	NOUN
cana-551	32	22	,	,	PUNCT
cana-551	32	23	𝛥	𝛥	NOUN
cana-551	32	24	)	)	PUNCT
cana-551	32	25	is	be	AUX
cana-551	32	26	said	say	VERB
cana-551	32	27	to	to	PART
cana-551	32	28	be	be	AUX
cana-551	32	29	a	a	DET
cana-551	32	30	soft	soft	ADJ
cana-551	32	31	superset	superset	NOUN
cana-551	32	32	of	of	ADP
cana-551	32	33	(	(	PUNCT
cana-551	32	34	𝛿	𝛿	ADJ
cana-551	32	35	,	,	PUNCT
cana-551	32	36	𝐵	𝐵	NOUN
cana-551	32	37	)	)	PUNCT
cana-551	32	38	,	,	PUNCT
cana-551	32	39	if	if	SCONJ
cana-551	32	40	(	(	PUNCT
cana-551	32	41	𝛿	𝛿	ADJ
cana-551	32	42	,	,	PUNCT
cana-551	32	43	𝐵	𝐵	NOUN
cana-551	32	44	)	)	PUNCT
cana-551	32	45	is	be	AUX
cana-551	32	46	a	a	DET
cana-551	32	47	soft	soft	ADJ
cana-551	32	48	subset	subset	NOUN
cana-551	32	49	of	of	ADP
cana-551	32	50	(	(	PUNCT
cana-551	32	51	𝛾	𝛾	PROPN
cana-551	32	52	,	,	PUNCT
cana-551	32	53	𝛥	𝛥	NOUN
cana-551	32	54	)	)	PUNCT
cana-551	32	55	,	,	PUNCT
cana-551	32	56	(	(	PUNCT
cana-551	32	57	𝛾	𝛾	NOUN
cana-551	32	58	,	,	PUNCT
cana-551	32	59	𝛥	𝛥	NOUN
cana-551	32	60	)	)	PUNCT
cana-551	32	61	⊇	⊇	NOUN
cana-551	32	62	(	(	PUNCT
cana-551	32	63	𝛿	𝛿	ADJ
cana-551	32	64	,	,	PUNCT
cana-551	32	65	𝐵	𝐵	NOUN
cana-551	32	66	)	)	PUNCT
cana-551	32	67	.	.	PUNCT
cana-551	33	1	definition	definition	NOUN
cana-551	33	2	1.4	1.4	NUM
cana-551	33	3	:	:	PUNCT
cana-551	34	1	[	[	X
cana-551	34	2	13	13	NUM
cana-551	34	3	]	]	PUNCT
cana-551	34	4	let	let	AUX
cana-551	34	5	(	(	PUNCT
cana-551	34	6	𝛾	𝛾	NOUN
cana-551	34	7	,	,	PUNCT
cana-551	34	8	𝛥	𝛥	NOUN
cana-551	34	9	)	)	PUNCT
cana-551	34	10	be	be	VERB
cana-551	34	11	a	a	DET
cana-551	34	12	soft	soft	ADJ
cana-551	34	13	set	set	NOUN
cana-551	34	14	over	over	ADP
cana-551	34	15	𝒵	𝒵	PROPN
cana-551	34	16	and	and	CCONJ
cana-551	34	17	𝑧	𝑧	DET
cana-551	34	18	∈	∈	PROPN
cana-551	34	19	𝒵.	𝒵.	NOUN
cana-551	34	20	we	we	PRON
cana-551	34	21	say	say	VERB
cana-551	34	22	that	that	SCONJ
cana-551	34	23	𝑧	𝑧	PROPN
cana-551	34	24	∈	∈	PROPN
cana-551	34	25	(	(	PUNCT
cana-551	34	26	𝛾	𝛾	NOUN
cana-551	34	27	,	,	PUNCT
cana-551	34	28	𝛥	𝛥	NOUN
cana-551	34	29	)	)	PUNCT
cana-551	34	30	read	read	NOUN
cana-551	34	31	as	as	ADP
cana-551	34	32	𝑧	𝑧	PROPN
cana-551	34	33	belongs	belong	VERB
cana-551	34	34	to	to	ADP
cana-551	34	35	the	the	DET
cana-551	34	36	soft	soft	ADJ
cana-551	34	37	set	set	NOUN
cana-551	34	38	(	(	PUNCT
cana-551	34	39	𝛾	𝛾	NOUN
cana-551	34	40	,	,	PUNCT
cana-551	34	41	𝛥	𝛥	NOUN
cana-551	34	42	)	)	PUNCT
cana-551	34	43	whenever	whenever	SCONJ
cana-551	34	44	𝑧	𝑧	PRON
cana-551	34	45	∈	∈	PROPN
cana-551	34	46	𝛾	𝛾	X
cana-551	34	47	(	(	PUNCT
cana-551	34	48	∇	∇	X
cana-551	34	49	)	)	PUNCT
cana-551	34	50	for	for	ADP
cana-551	34	51	all	all	DET
cana-551	34	52	𝛻	𝛻	PROPN
cana-551	34	53	∈	∈	PROPN
cana-551	34	54	𝛥.	𝛥.	PROPN
cana-551	34	55	the	the	DET
cana-551	34	56	soft	soft	ADJ
cana-551	34	57	set	set	NOUN
cana-551	34	58	(	(	PUNCT
cana-551	34	59	𝛾	𝛾	NOUN
cana-551	34	60	,	,	PUNCT
cana-551	34	61	𝛥	𝛥	NOUN
cana-551	34	62	)	)	PUNCT
cana-551	34	63	over	over	ADP
cana-551	34	64	𝒵	𝒵	PROPN
cana-551	34	65	such	such	ADJ
cana-551	34	66	that	that	SCONJ
cana-551	34	67	𝛾	𝛾	AUX
cana-551	34	68	(	(	PUNCT
cana-551	34	69	𝛻	𝛻	NOUN
cana-551	34	70	)	)	PUNCT
cana-551	34	71	=	=	PRON
cana-551	34	72	{	{	PUNCT
cana-551	34	73	𝑧	𝑧	NOUN
cana-551	34	74	}	}	PUNCT
cana-551	34	75	∀	∀	X
cana-551	34	76	∇	∇	X
cana-551	34	77	∈	∈	NOUN
cana-551	34	78	𝛥	𝛥	PROPN
cana-551	34	79	is	be	AUX
cana-551	34	80	called	call	VERB
cana-551	34	81	singleton	singleton	PROPN
cana-551	34	82	soft	soft	ADJ
cana-551	34	83	point	point	NOUN
cana-551	34	84	and	and	CCONJ
cana-551	34	85	denoted	denote	VERB
cana-551	34	86	by	by	ADP
cana-551	34	87	𝑧𝛥	𝑧𝛥	NOUN
cana-551	34	88	or	or	CCONJ
cana-551	34	89	(	(	PUNCT
cana-551	34	90	𝑧	𝑧	PROPN
cana-551	34	91	,	,	PUNCT
cana-551	34	92	𝛥	𝛥	NOUN
cana-551	34	93	)	)	PUNCT
cana-551	34	94	.	.	PUNCT
cana-551	35	1	definition	definition	NOUN
cana-551	35	2	1.5	1.5	NUM
cana-551	35	3	:	:	PUNCT
cana-551	36	1	[	[	X
cana-551	36	2	13	13	NUM
cana-551	36	3	]	]	PUNCT
cana-551	36	4	let	let	VERB
cana-551	36	5	𝔚	𝔚	PRON
cana-551	36	6	be	be	AUX
cana-551	36	7	a	a	DET
cana-551	36	8	collection	collection	NOUN
cana-551	36	9	of	of	ADP
cana-551	36	10	soft	soft	ADJ
cana-551	36	11	sets	set	NOUN
cana-551	36	12	over	over	ADP
cana-551	36	13	𝒵	𝒵	PROPN
cana-551	36	14	,	,	PUNCT
cana-551	36	15	then	then	ADV
cana-551	36	16	𝔚	𝔚	PROPN
cana-551	36	17	is	be	AUX
cana-551	36	18	said	say	VERB
cana-551	36	19	to	to	PART
cana-551	36	20	be	be	AUX
cana-551	36	21	𝒮ts	𝒮ts	PROPN
cana-551	36	22	on	on	ADP
cana-551	36	23	𝒵	𝒵	PRON
cana-551	36	24	if	if	SCONJ
cana-551	36	25	(	(	PUNCT
cana-551	36	26	1	1	X
cana-551	36	27	)	)	PUNCT
cana-551	36	28	∅̃	∅̃	NOUN
cana-551	36	29	and	and	CCONJ
cana-551	36	30	𝒵	𝒵	PROPN
cana-551	36	31	belong	belong	VERB
cana-551	36	32	to	to	ADP
cana-551	36	33	𝔚.	𝔚.	PROPN
cana-551	36	34	(	(	PUNCT
cana-551	36	35	2	2	NUM
cana-551	36	36	)	)	PUNCT
cana-551	36	37	the	the	DET
cana-551	36	38	union	union	NOUN
cana-551	36	39	of	of	ADP
cana-551	36	40	any	any	DET
cana-551	36	41	subcollection	subcollection	NOUN
cana-551	36	42	of	of	ADP
cana-551	36	43	soft	soft	ADJ
cana-551	36	44	sets	set	NOUN
cana-551	36	45	of	of	ADP
cana-551	36	46	𝔚	𝔚	PROPN
cana-551	36	47	belongs	belong	VERB
cana-551	36	48	to	to	ADP
cana-551	36	49	𝔚.	𝔚.	PROPN
cana-551	36	50	(	(	PUNCT
cana-551	36	51	3	3	NUM
cana-551	36	52	)	)	PUNCT
cana-551	36	53	the	the	DET
cana-551	36	54	intersection	intersection	NOUN
cana-551	36	55	of	of	ADP
cana-551	36	56	any	any	DET
cana-551	36	57	two	two	NUM
cana-551	36	58	soft	soft	ADJ
cana-551	36	59	sets	set	NOUN
cana-551	36	60	in	in	ADP
cana-551	36	61	𝔚	𝔚	PROPN
cana-551	36	62	belongs	belong	VERB
cana-551	36	63	to	to	PART
cana-551	36	64	𝔚.	𝔚.	PROPN
cana-551	36	65	it	it	PRON
cana-551	36	66	is	be	AUX
cana-551	36	67	denoted	denote	VERB
cana-551	36	68	by	by	ADP
cana-551	36	69	𝒮ts	𝒮ts	PROPN
cana-551	36	70	(	(	PUNCT
cana-551	36	71	𝒵	𝒵	PROPN
cana-551	36	72	,	,	PUNCT
cana-551	36	73	𝔚	𝔚	PROPN
cana-551	36	74	,	,	PUNCT
cana-551	36	75	𝛥	𝛥	PROPN
cana-551	36	76	)	)	PUNCT
cana-551	36	77	and	and	CCONJ
cana-551	36	78	briefly	briefly	ADV
cana-551	36	79	𝒵.	𝒵.	PROPN
cana-551	36	80	definition	definition	NOUN
cana-551	36	81	1.6	1.6	NUM
cana-551	36	82	:	:	PUNCT
cana-551	37	1	[	[	X
cana-551	37	2	13	13	NUM
cana-551	37	3	]	]	PUNCT
cana-551	37	4	let	let	VERB
cana-551	37	5	(	(	PUNCT
cana-551	37	6	𝒵	𝒵	PROPN
cana-551	37	7	,	,	PUNCT
cana-551	37	8	𝔚	𝔚	PROPN
cana-551	37	9	,	,	PUNCT
cana-551	37	10	𝛥	𝛥	PROPN
cana-551	37	11	)	)	PUNCT
cana-551	37	12	be	be	VERB
cana-551	37	13	a	a	DET
cana-551	37	14	soft	soft	ADJ
cana-551	37	15	space	space	NOUN
cana-551	37	16	over	over	ADP
cana-551	37	17	𝒵	𝒵	PROPN
cana-551	37	18	,	,	PUNCT
cana-551	37	19	then	then	ADV
cana-551	37	20	the	the	DET
cana-551	37	21	members	member	NOUN
cana-551	37	22	of	of	ADP
cana-551	37	23	𝔚	𝔚	PROPN
cana-551	37	24	are	be	AUX
cana-551	37	25	said	say	VERB
cana-551	37	26	to	to	PART
cana-551	37	27	be	be	AUX
cana-551	37	28	soft	soft	ADJ
cana-551	37	29	open	open	ADJ
cana-551	37	30	sets	set	NOUN
cana-551	37	31	in	in	ADP
cana-551	37	32	𝔚.	𝔚.	NOUN
cana-551	37	33	definition	definition	NOUN
cana-551	37	34	1.7	1.7	NUM
cana-551	37	35	:	:	PUNCT
cana-551	38	1	[	[	X
cana-551	38	2	13	13	NUM
cana-551	38	3	]	]	PUNCT
cana-551	38	4	let	let	VERB
cana-551	38	5	(	(	PUNCT
cana-551	38	6	𝒵	𝒵	PROPN
cana-551	38	7	,	,	PUNCT
cana-551	38	8	𝔚	𝔚	PROPN
cana-551	38	9	,	,	PUNCT
cana-551	38	10	𝛥	𝛥	PROPN
cana-551	38	11	)	)	PUNCT
cana-551	38	12	be	be	VERB
cana-551	38	13	a	a	DET
cana-551	38	14	soft	soft	ADJ
cana-551	38	15	space	space	NOUN
cana-551	38	16	over	over	ADP
cana-551	38	17	𝒵.	𝒵.	PROPN
cana-551	38	18	a	a	DET
cana-551	38	19	soft	soft	ADJ
cana-551	38	20	set	set	NOUN
cana-551	38	21	(	(	PUNCT
cana-551	38	22	p	p	NOUN
cana-551	38	23	,	,	PUNCT
cana-551	38	24	𝛥	𝛥	NOUN
cana-551	38	25	)	)	PUNCT
cana-551	38	26	over	over	ADP
cana-551	38	27	𝒵	𝒵	PROPN
cana-551	38	28	is	be	AUX
cana-551	38	29	said	say	VERB
cana-551	38	30	to	to	PART
cana-551	38	31	be	be	AUX
cana-551	38	32	a	a	DET
cana-551	38	33	soft	soft	ADJ
cana-551	38	34	closed	closed	ADJ
cana-551	38	35	set	set	NOUN
cana-551	38	36	in	in	ADP
cana-551	38	37	𝒵	𝒵	PROPN
cana-551	38	38	,	,	PUNCT
cana-551	38	39	if	if	SCONJ
cana-551	38	40	its	its	PRON
cana-551	38	41	relative	relative	ADJ
cana-551	38	42	complement	complement	NOUN
cana-551	38	43	(	(	PUNCT
cana-551	38	44	𝛾	𝛾	PROPN
cana-551	38	45	,	,	PUNCT
cana-551	38	46	𝛥)′	𝛥)′	PROPN
cana-551	38	47	belongs	belong	VERB
cana-551	38	48	to	to	ADP
cana-551	38	49	𝔚.	𝔚.	PROPN
cana-551	38	50	definition	definition	NOUN
cana-551	38	51	1.8	1.8	NUM
cana-551	38	52	:	:	PUNCT
cana-551	39	1	[	[	X
cana-551	39	2	14	14	NUM
cana-551	39	3	]	]	X
cana-551	39	4	let	let	NOUN
cana-551	39	5	(	(	PUNCT
cana-551	39	6	𝒵	𝒵	PROPN
cana-551	39	7	,	,	PUNCT
cana-551	39	8	𝔚	𝔚	PROPN
cana-551	39	9	,	,	PUNCT
cana-551	39	10	𝛥	𝛥	PROPN
cana-551	39	11	)	)	PUNCT
cana-551	39	12	be	be	VERB
cana-551	39	13	a	a	DET
cana-551	39	14	𝒮ts	𝒮ts	PROPN
cana-551	39	15	and	and	CCONJ
cana-551	39	16	(	(	PUNCT
cana-551	39	17	𝛾	𝛾	PROPN
cana-551	39	18	,	,	PUNCT
cana-551	39	19	𝛥	𝛥	NOUN
cana-551	39	20	)	)	PUNCT
cana-551	39	21	∈	∈	PROPN
cana-551	39	22	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	39	23	,	,	PUNCT
cana-551	39	24	𝛥	𝛥	PROPN
cana-551	39	25	)	)	PUNCT
cana-551	39	26	.	.	PUNCT
cana-551	40	1	then	then	ADV
cana-551	40	2	(	(	PUNCT
cana-551	40	3	1	1	X
cana-551	40	4	)	)	PUNCT
cana-551	40	5	the	the	DET
cana-551	40	6	soft	soft	ADJ
cana-551	40	7	closure	closure	NOUN
cana-551	40	8	of	of	ADP
cana-551	40	9	(	(	PUNCT
cana-551	40	10	𝛾	𝛾	PROPN
cana-551	40	11	,	,	PUNCT
cana-551	40	12	𝛥	𝛥	NOUN
cana-551	40	13	)	)	PUNCT
cana-551	40	14	is	be	AUX
cana-551	40	15	the	the	DET
cana-551	40	16	soft	soft	ADJ
cana-551	40	17	set	set	NOUN
cana-551	40	18	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NOUN
cana-551	40	19	,	,	PUNCT
cana-551	40	20	𝛥	𝛥	NOUN
cana-551	40	21	)	)	PUNCT
cana-551	40	22	=	=	NOUN
cana-551	40	23	∩	∩	NOUN
cana-551	40	24	{	{	PUNCT
cana-551	40	25	(	(	PUNCT
cana-551	40	26	𝐿	𝐿	PROPN
cana-551	40	27	,	,	PUNCT
cana-551	40	28	𝛥	𝛥	PROPN
cana-551	40	29	)	)	PUNCT
cana-551	40	30	∶	∶	NOUN
cana-551	40	31	(	(	PUNCT
cana-551	40	32	𝐿	𝐿	PROPN
cana-551	40	33	,	,	PUNCT
cana-551	40	34	𝛥	𝛥	PROPN
cana-551	40	35	)	)	PUNCT
cana-551	40	36	∈	∈	PROPN
cana-551	41	1	𝔚𝑐	𝔚𝑐	PROPN
cana-551	41	2	,	,	PUNCT
cana-551	41	3	(	(	PUNCT
cana-551	41	4	𝛾	𝛾	NOUN
cana-551	41	5	,	,	PUNCT
cana-551	41	6	𝛥	𝛥	NOUN
cana-551	41	7	)	)	PUNCT
cana-551	41	8	⊆	⊆	NUM
cana-551	41	9	(	(	PUNCT
cana-551	41	10	𝐿	𝐿	PROPN
cana-551	41	11	,	,	PUNCT
cana-551	41	12	𝛥	𝛥	PROPN
cana-551	41	13	)	)	PUNCT
cana-551	41	14	}	}	PUNCT
cana-551	41	15	.	.	PUNCT
cana-551	42	1	(	(	PUNCT
cana-551	42	2	2	2	X
cana-551	42	3	)	)	PUNCT
cana-551	42	4	the	the	DET
cana-551	42	5	soft	soft	ADJ
cana-551	42	6	interior	interior	NOUN
cana-551	42	7	of	of	ADP
cana-551	42	8	(	(	PUNCT
cana-551	42	9	𝛾	𝛾	PROPN
cana-551	42	10	,	,	PUNCT
cana-551	42	11	𝛥	𝛥	NOUN
cana-551	42	12	)	)	PUNCT
cana-551	42	13	is	be	AUX
cana-551	42	14	the	the	DET
cana-551	42	15	soft	soft	ADJ
cana-551	42	16	set	set	NOUN
cana-551	42	17	𝑖𝑛𝑡(𝛾	𝑖𝑛𝑡(𝛾	NOUN
cana-551	42	18	,	,	PUNCT
cana-551	42	19	𝛥	𝛥	NOUN
cana-551	42	20	)	)	PUNCT
cana-551	42	21	=	=	SYM
cana-551	42	22	∪	∪	X
cana-551	42	23	{	{	PUNCT
cana-551	42	24	(	(	PUNCT
cana-551	42	25	𝐻	𝐻	PROPN
cana-551	42	26	,	,	PUNCT
cana-551	42	27	𝛥	𝛥	PROPN
cana-551	42	28	)	)	PUNCT
cana-551	42	29	∶	∶	NOUN
cana-551	42	30	(	(	PUNCT
cana-551	42	31	𝐻	𝐻	PROPN
cana-551	42	32	,	,	PUNCT
cana-551	42	33	𝛥	𝛥	PROPN
cana-551	42	34	)	)	PUNCT
cana-551	42	35	∈	∈	PROPN
cana-551	42	36	𝔚	𝔚	PROPN
cana-551	42	37	,	,	PUNCT
cana-551	42	38	(	(	PUNCT
cana-551	42	39	𝐻	𝐻	PROPN
cana-551	42	40	,	,	PUNCT
cana-551	42	41	𝛥	𝛥	PROPN
cana-551	42	42	)	)	PUNCT
cana-551	42	43	⊆	⊆	NUM
cana-551	42	44	(	(	PUNCT
cana-551	42	45	𝛾	𝛾	NOUN
cana-551	42	46	,	,	PUNCT
cana-551	42	47	𝛥	𝛥	NOUN
cana-551	42	48	)	)	PUNCT
cana-551	42	49	}	}	PUNCT
cana-551	42	50	.	.	PUNCT
cana-551	43	1	definition	definition	NOUN
cana-551	43	2	1.9	1.9	NUM
cana-551	43	3	:	:	PUNCT
cana-551	44	1	[	[	X
cana-551	44	2	4	4	NUM
cana-551	44	3	,	,	PUNCT
cana-551	44	4	5	5	NUM
cana-551	44	5	,	,	PUNCT
cana-551	44	6	7	7	NUM
cana-551	44	7	,	,	PUNCT
cana-551	44	8	19	19	NUM
cana-551	44	9	]	]	PUNCT
cana-551	44	10	a	a	DET
cana-551	44	11	soft	soft	ADJ
cana-551	44	12	set	set	NOUN
cana-551	44	13	(	(	PUNCT
cana-551	44	14	𝛿	𝛿	ADJ
cana-551	44	15	,	,	PUNCT
cana-551	44	16	𝛥	𝛥	NOUN
cana-551	44	17	)	)	PUNCT
cana-551	44	18	of	of	ADP
cana-551	44	19	a	a	DET
cana-551	44	20	𝒮ts	𝒮ts	PROPN
cana-551	44	21	(	(	PUNCT
cana-551	44	22	𝒵	𝒵	PROPN
cana-551	44	23	,	,	PUNCT
cana-551	44	24	𝔚	𝔚	PROPN
cana-551	44	25	,	,	PUNCT
cana-551	44	26	𝛥	𝛥	PROPN
cana-551	44	27	)	)	PUNCT
cana-551	44	28	is	be	AUX
cana-551	44	29	said	say	VERB
cana-551	44	30	to	to	PART
cana-551	44	31	be	be	AUX
cana-551	44	32	(	(	PUNCT
cana-551	44	33	1	1	NUM
cana-551	44	34	)	)	PUNCT
cana-551	44	35	soft	soft	ADJ
cana-551	44	36	αopen	αopen	NOUN
cana-551	44	37	if	if	SCONJ
cana-551	44	38	(	(	PUNCT
cana-551	44	39	𝛿	𝛿	ADJ
cana-551	44	40	,	,	PUNCT
cana-551	44	41	𝛥	𝛥	NOUN
cana-551	44	42	)	)	PUNCT
cana-551	44	43	⊂	⊂	PROPN
cana-551	44	44	𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡((𝛿	𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡((𝛿	PROPN
cana-551	44	45	,	,	PUNCT
cana-551	44	46	𝛥	𝛥	PROPN
cana-551	44	47	)	)	PUNCT
cana-551	44	48	)	)	PUNCT
cana-551	44	49	)	)	PUNCT
cana-551	44	50	)	)	PUNCT
cana-551	44	51	.	.	PUNCT
cana-551	45	1	(	(	PUNCT
cana-551	45	2	2	2	X
cana-551	45	3	)	)	PUNCT
cana-551	45	4	soft	soft	ADJ
cana-551	45	5	preopen	preopen	NOUN
cana-551	45	6	if	if	SCONJ
cana-551	45	7	(	(	PUNCT
cana-551	45	8	𝛿	𝛿	ADJ
cana-551	45	9	,	,	PUNCT
cana-551	45	10	𝛥	𝛥	NOUN
cana-551	45	11	)	)	PUNCT
cana-551	45	12	⊂	⊂	PROPN
cana-551	45	13	𝑖𝑛𝑡(𝑐𝑙((𝛿	𝑖𝑛𝑡(𝑐𝑙((𝛿	PROPN
cana-551	45	14	,	,	PUNCT
cana-551	45	15	𝛥	𝛥	NOUN
cana-551	45	16	)	)	PUNCT
cana-551	45	17	)	)	PUNCT
cana-551	45	18	)	)	PUNCT
cana-551	45	19	.	.	PUNCT
cana-551	46	1	communications	communication	NOUN
cana-551	46	2	on	on	ADP
cana-551	46	3	applied	apply	VERB
cana-551	46	4	nonlinear	nonlinear	ADJ
cana-551	46	5	analysis	analysis	NOUN
cana-551	46	6	issn	issn	NOUN
cana-551	46	7	:	:	PUNCT
cana-551	46	8	1074	1074	NUM
cana-551	46	9	-	-	PUNCT
cana-551	46	10	133x	133x	NUM
cana-551	46	11	vol	vol	NOUN
cana-551	46	12	31	31	NUM
cana-551	46	13	no	no	NOUN
cana-551	46	14	.	.	NOUN
cana-551	46	15	2	2	NUM
cana-551	46	16	(	(	PUNCT
cana-551	46	17	2024	2024	NUM
cana-551	46	18	)	)	PUNCT
cana-551	46	19	286	286	NUM
cana-551	46	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	46	21	(	(	PUNCT
cana-551	46	22	3	3	X
cana-551	46	23	)	)	PUNCT
cana-551	46	24	soft	soft	ADJ
cana-551	46	25	semi	semi	ADV
cana-551	46	26	open	open	ADJ
cana-551	46	27	if	if	SCONJ
cana-551	46	28	(	(	PUNCT
cana-551	46	29	𝛿	𝛿	ADJ
cana-551	46	30	,	,	PUNCT
cana-551	46	31	𝛥	𝛥	NOUN
cana-551	46	32	)	)	PUNCT
cana-551	46	33	⊂	⊂	PROPN
cana-551	46	34	𝑐𝑙(𝑖𝑛𝑡((𝛿	𝑐𝑙(𝑖𝑛𝑡((𝛿	PROPN
cana-551	46	35	,	,	PUNCT
cana-551	46	36	𝛥	𝛥	PROPN
cana-551	46	37	)	)	PUNCT
cana-551	46	38	)	)	PUNCT
cana-551	46	39	)	)	PUNCT
cana-551	46	40	.	.	PUNCT
cana-551	47	1	(	(	PUNCT
cana-551	47	2	4	4	X
cana-551	47	3	)	)	PUNCT
cana-551	47	4	soft	soft	ADJ
cana-551	47	5	β	β	NOUN
cana-551	47	6	-	-	NOUN
cana-551	47	7	open	open	ADJ
cana-551	47	8	if	if	SCONJ
cana-551	47	9	(	(	PUNCT
cana-551	47	10	𝛿	𝛿	ADJ
cana-551	47	11	,	,	PUNCT
cana-551	47	12	𝛥	𝛥	NOUN
cana-551	47	13	)	)	PUNCT
cana-551	47	14	⊂	⊂	PROPN
cana-551	47	15	𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙((𝛿	𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙((𝛿	PROPN
cana-551	47	16	,	,	PUNCT
cana-551	47	17	𝛥	𝛥	NOUN
cana-551	47	18	)	)	PUNCT
cana-551	47	19	)	)	PUNCT
cana-551	47	20	)	)	PUNCT
cana-551	47	21	)	)	PUNCT
cana-551	47	22	.	.	PUNCT
cana-551	48	1	(	(	PUNCT
cana-551	48	2	5	5	X
cana-551	48	3	)	)	PUNCT
cana-551	48	4	soft	soft	ADJ
cana-551	48	5	ƅ	ƅ	NOUN
cana-551	48	6	−open	−open	VERB
cana-551	48	7	if	if	SCONJ
cana-551	48	8	(	(	PUNCT
cana-551	48	9	𝛿	𝛿	ADJ
cana-551	48	10	,	,	PUNCT
cana-551	48	11	𝛥	𝛥	NOUN
cana-551	48	12	)	)	PUNCT
cana-551	48	13	⊂	⊂	PROPN
cana-551	48	14	𝑖𝑛𝑡(𝑐𝑙((𝛿	𝑖𝑛𝑡(𝑐𝑙((𝛿	PROPN
cana-551	48	15	,	,	PUNCT
cana-551	48	16	𝛥	𝛥	PROPN
cana-551	48	17	)	)	PUNCT
cana-551	48	18	)	)	PUNCT
cana-551	48	19	)	)	PUNCT
cana-551	49	1	∪	∪	ADP
cana-551	49	2	𝑐𝑙(𝑖𝑛𝑡((𝛿	𝑐𝑙(𝑖𝑛𝑡((𝛿	PROPN
cana-551	49	3	,	,	PUNCT
cana-551	49	4	𝛥	𝛥	PROPN
cana-551	49	5	)	)	PUNCT
cana-551	49	6	)	)	PUNCT
cana-551	49	7	)	)	PUNCT
cana-551	49	8	)	)	PUNCT
cana-551	49	9	.	.	PUNCT
cana-551	50	1	definition	definition	NOUN
cana-551	50	2	1.15	1.15	NUM
cana-551	50	3	:	:	PUNCT
cana-551	51	1	[	[	X
cana-551	51	2	8	8	X
cana-551	51	3	]	]	PUNCT
cana-551	51	4	a	a	DET
cana-551	51	5	soft	soft	ADJ
cana-551	51	6	set	set	NOUN
cana-551	51	7	(	(	PUNCT
cana-551	51	8	𝛾	𝛾	NOUN
cana-551	51	9	,	,	PUNCT
cana-551	51	10	𝛥	𝛥	NOUN
cana-551	51	11	)	)	PUNCT
cana-551	51	12	of	of	ADP
cana-551	51	13	a	a	DET
cana-551	51	14	𝒮ts(𝒵	𝒮ts(𝒵	PROPN
cana-551	51	15	,	,	PUNCT
cana-551	51	16	𝔚	𝔚	PROPN
cana-551	51	17	,	,	PUNCT
cana-551	51	18	𝛥	𝛥	PROPN
cana-551	51	19	)	)	PUNCT
cana-551	51	20	is	be	AUX
cana-551	51	21	called	call	VERB
cana-551	51	22	a	a	DET
cana-551	51	23	soft	soft	ADJ
cana-551	51	24	strongly	strongly	ADV
cana-551	51	25	ƅ∗	ƅ∗	NOUN
cana-551	51	26	−closed	−close	VERB
cana-551	51	27	(	(	PUNCT
cana-551	51	28	briefly	briefly	ADV
cana-551	51	29	s𝑆ƅ∗	s𝑆ƅ∗	VERB
cana-551	51	30	−closed	−close	VERB
cana-551	51	31	)	)	PUNCT
cana-551	52	1	if	if	SCONJ
cana-551	52	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	52	3	,	,	PUNCT
cana-551	52	4	𝛥	𝛥	NOUN
cana-551	52	5	)	)	PUNCT
cana-551	52	6	)	)	PUNCT
cana-551	53	1	⊆	⊆	NUM
cana-551	53	2	(	(	PUNCT
cana-551	53	3	𝛿	𝛿	ADJ
cana-551	53	4	,	,	PUNCT
cana-551	53	5	𝛥	𝛥	NOUN
cana-551	53	6	)	)	PUNCT
cana-551	53	7	,	,	PUNCT
cana-551	53	8	whenever	whenever	SCONJ
cana-551	53	9	(	(	PUNCT
cana-551	53	10	𝛾	𝛾	NOUN
cana-551	53	11	,	,	PUNCT
cana-551	53	12	𝛥	𝛥	NOUN
cana-551	53	13	)	)	PUNCT
cana-551	53	14	⊂	⊂	PROPN
cana-551	53	15	(	(	PUNCT
cana-551	53	16	𝛿	𝛿	PROPN
cana-551	53	17	,	,	PUNCT
cana-551	53	18	𝛥	𝛥	NOUN
cana-551	53	19	)	)	PUNCT
cana-551	53	20	and	and	CCONJ
cana-551	53	21	(	(	PUNCT
cana-551	53	22	𝛿	𝛿	ADJ
cana-551	53	23	,	,	PUNCT
cana-551	53	24	𝛥	𝛥	NOUN
cana-551	53	25	)	)	PUNCT
cana-551	53	26	is	be	AUX
cana-551	53	27	soft	soft	ADJ
cana-551	53	28	ƅ	ƅ	X
cana-551	53	29	−open	−open	NOUN
cana-551	53	30	.	.	PUNCT
cana-551	54	1	the	the	DET
cana-551	54	2	complement	complement	NOUN
cana-551	54	3	of	of	ADP
cana-551	54	4	a	a	DET
cana-551	54	5	ƅ∗	ƅ∗	NOUN
cana-551	54	6	ƅ∗	ƅ∗	NOUN
cana-551	54	7	−closed	−close	VERB
cana-551	54	8	set	set	VERB
cana-551	54	9	is	be	AUX
cana-551	54	10	called	call	VERB
cana-551	54	11	ƅ∗	ƅ∗	PROPN
cana-551	54	12	ƅ∗	ƅ∗	PROPN
cana-551	54	13	−open	−open	VERB
cana-551	54	14	set	set	VERB
cana-551	54	15	.	.	PUNCT
cana-551	55	1	the	the	DET
cana-551	55	2	family	family	NOUN
cana-551	55	3	of	of	ADP
cana-551	55	4	all	all	DET
cana-551	55	5	ƅ∗	ƅ∗	PROPN
cana-551	55	6	ƅ∗	ƅ∗	NOUN
cana-551	55	7	−open	−open	VERB
cana-551	55	8	sets	set	NOUN
cana-551	55	9	denoted	denote	VERB
cana-551	55	10	by	by	ADP
cana-551	55	11	𝑠𝑆ƅ∗𝑂𝑆(𝒵	𝑠𝑆ƅ∗𝑂𝑆(𝒵	PROPN
cana-551	55	12	)	)	PUNCT
cana-551	55	13	.	.	PUNCT
cana-551	56	1	theorem	theorem	VERB
cana-551	56	2	1.16	1.16	NUM
cana-551	56	3	:	:	PUNCT
cana-551	57	1	[	[	X
cana-551	57	2	8	8	X
cana-551	57	3	]	]	X
cana-551	57	4	the	the	DET
cana-551	57	5	following	follow	VERB
cana-551	57	6	statements	statement	NOUN
cana-551	57	7	are	be	AUX
cana-551	57	8	true	true	ADJ
cana-551	57	9	.	.	PUNCT
cana-551	58	1	(	(	PUNCT
cana-551	58	2	i	i	NOUN
cana-551	58	3	)	)	PUNCT
cana-551	58	4	every	every	DET
cana-551	58	5	soft	soft	ADJ
cana-551	58	6	open	open	NOUN
cana-551	58	7	is	be	AUX
cana-551	58	8	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	58	9	−open	−open	ADJ
cana-551	58	10	.	.	PUNCT
cana-551	59	1	(	(	PUNCT
cana-551	59	2	ii	ii	NOUN
cana-551	59	3	)	)	PUNCT
cana-551	59	4	every	every	DET
cana-551	59	5	𝑠𝛼	𝑠𝛼	PROPN
cana-551	59	6	−open	−open	PROPN
cana-551	59	7	is	be	AUX
cana-551	59	8	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	59	9	−open	−open	ADJ
cana-551	59	10	.	.	PUNCT
cana-551	60	1	(	(	PUNCT
cana-551	60	2	iii	iii	X
cana-551	60	3	)	)	PUNCT
cana-551	60	4	every	every	PRON
cana-551	60	5	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	60	6	−open	−open	NOUN
cana-551	60	7	set	set	NOUN
cana-551	60	8	is	be	AUX
cana-551	60	9	𝑠ƅ	𝑠ƅ	PROPN
cana-551	60	10	−open	−open	PROPN
cana-551	60	11	.	.	PUNCT
cana-551	61	1	(	(	PUNCT
cana-551	61	2	iv	iv	X
cana-551	61	3	)	)	PUNCT
cana-551	61	4	every	every	DET
cana-551	61	5	𝑠𝜔	𝑠𝜔	NOUN
cana-551	61	6	−open	−open	PROPN
cana-551	61	7	is	be	AUX
cana-551	61	8	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	61	9	−open	−open	ADJ
cana-551	61	10	.	.	PUNCT
cana-551	62	1	definition	definition	NOUN
cana-551	62	2	1.17	1.17	NUM
cana-551	62	3	:	:	PUNCT
cana-551	63	1	[	[	X
cana-551	63	2	9	9	NUM
cana-551	63	3	]	]	PUNCT
cana-551	63	4	let	let	VERB
cana-551	63	5	ῐ	ῐ	PRON
cana-551	63	6	be	be	AUX
cana-551	63	7	a	a	DET
cana-551	63	8	non	non	ADJ
cana-551	63	9	-	-	ADJ
cana-551	63	10	null	null	ADJ
cana-551	63	11	collection	collection	NOUN
cana-551	63	12	of	of	ADP
cana-551	63	13	soft	soft	ADJ
cana-551	63	14	sets	set	NOUN
cana-551	63	15	over	over	ADP
cana-551	63	16	a	a	DET
cana-551	63	17	universe	universe	ADJ
cana-551	63	18	𝒵	𝒵	NOUN
cana-551	63	19	with	with	ADP
cana-551	63	20	the	the	DET
cana-551	63	21	same	same	ADJ
cana-551	63	22	set	set	NOUN
cana-551	63	23	of	of	ADP
cana-551	63	24	parameters	parameter	NOUN
cana-551	63	25	∆.	∆.	X
cana-551	63	26	then	then	ADV
cana-551	63	27	,	,	PUNCT
cana-551	63	28	ῐ	ῐ	PROPN
cana-551	63	29	∈	∈	PROPN
cana-551	63	30	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	63	31	,	,	PUNCT
cana-551	63	32	𝛥	𝛥	PROPN
cana-551	63	33	)	)	PUNCT
cana-551	63	34	is	be	AUX
cana-551	63	35	called	call	VERB
cana-551	63	36	a	a	DET
cana-551	63	37	soft	soft	ADJ
cana-551	63	38	ideal	ideal	NOUN
cana-551	63	39	on	on	ADP
cana-551	63	40	𝒵	𝒵	PROPN
cana-551	63	41	with	with	ADP
cana-551	63	42	the	the	DET
cana-551	63	43	same	same	ADJ
cana-551	63	44	set	set	NOUN
cana-551	63	45	∆	∆	PUNCT
cana-551	63	46	if	if	SCONJ
cana-551	63	47	(	(	PUNCT
cana-551	63	48	1	1	NUM
cana-551	63	49	)	)	PUNCT
cana-551	63	50	(	(	PUNCT
cana-551	63	51	𝛾	𝛾	PROPN
cana-551	63	52	,	,	PUNCT
cana-551	63	53	𝛥	𝛥	NOUN
cana-551	63	54	)	)	PUNCT
cana-551	63	55	∈	∈	PROPN
cana-551	63	56	ῐ	ῐ	PROPN
cana-551	63	57	and	and	CCONJ
cana-551	63	58	(	(	PUNCT
cana-551	63	59	𝛿	𝛿	ADJ
cana-551	63	60	,	,	PUNCT
cana-551	63	61	𝛥	𝛥	NOUN
cana-551	63	62	)	)	PUNCT
cana-551	63	63	∈	∈	PROPN
cana-551	63	64	ῐ	ῐ	PROPN
cana-551	63	65	⇒	⇒	NOUN
cana-551	63	66	(	(	PUNCT
cana-551	63	67	𝛾	𝛾	PROPN
cana-551	63	68	,	,	PUNCT
cana-551	63	69	𝛥	𝛥	NOUN
cana-551	63	70	)	)	PUNCT
cana-551	63	71	∪	∪	NOUN
cana-551	63	72	(	(	PUNCT
cana-551	63	73	𝛿	𝛿	ADJ
cana-551	63	74	,	,	PUNCT
cana-551	63	75	𝛥	𝛥	NOUN
cana-551	63	76	)	)	PUNCT
cana-551	63	77	∈	∈	PROPN
cana-551	63	78	ῐ	ῐ	PROPN
cana-551	63	79	,	,	PUNCT
cana-551	63	80	(	(	PUNCT
cana-551	63	81	2	2	NUM
cana-551	63	82	)	)	PUNCT
cana-551	63	83	(	(	PUNCT
cana-551	63	84	𝛾	𝛾	PROPN
cana-551	63	85	,	,	PUNCT
cana-551	63	86	𝛥	𝛥	NOUN
cana-551	63	87	)	)	PUNCT
cana-551	63	88	∈	∈	PROPN
cana-551	63	89	ῐ	ῐ	PROPN
cana-551	63	90	and	and	CCONJ
cana-551	63	91	(	(	PUNCT
cana-551	63	92	𝛿	𝛿	ADJ
cana-551	63	93	,	,	PUNCT
cana-551	63	94	𝛥	𝛥	NOUN
cana-551	63	95	)	)	PUNCT
cana-551	63	96	⊆	⊆	NUM
cana-551	63	97	(	(	PUNCT
cana-551	63	98	𝛾	𝛾	NOUN
cana-551	63	99	,	,	PUNCT
cana-551	63	100	𝛥	𝛥	NOUN
cana-551	63	101	)	)	PUNCT
cana-551	63	102	⇒	⇒	NOUN
cana-551	63	103	(	(	PUNCT
cana-551	63	104	𝛿	𝛿	ADJ
cana-551	63	105	,	,	PUNCT
cana-551	63	106	𝛥	𝛥	NOUN
cana-551	63	107	)	)	PUNCT
cana-551	63	108	∈	∈	PROPN
cana-551	63	109	ῐ.	ῐ.	NOUN
cana-551	63	110	i.e.	i.e.	X
cana-551	63	111	,	,	PUNCT
cana-551	63	112	ῐ	ῐ	PROPN
cana-551	63	113	is	be	AUX
cana-551	63	114	closed	close	VERB
cana-551	63	115	under	under	ADP
cana-551	63	116	finite	finite	ADJ
cana-551	63	117	soft	soft	ADJ
cana-551	63	118	unions	union	NOUN
cana-551	63	119	and	and	CCONJ
cana-551	63	120	soft	soft	ADJ
cana-551	63	121	subsets	subset	NOUN
cana-551	63	122	.	.	PUNCT
cana-551	64	1	definition	definition	NOUN
cana-551	64	2	1.18	1.18	NUM
cana-551	64	3	:	:	PUNCT
cana-551	65	1	[	[	X
cana-551	65	2	10	10	NUM
cana-551	65	3	]	]	X
cana-551	65	4	a	a	DET
cana-551	65	5	soft	soft	ADJ
cana-551	65	6	set	set	NOUN
cana-551	65	7	(	(	PUNCT
cana-551	65	8	𝛾	𝛾	NOUN
cana-551	65	9	,	,	PUNCT
cana-551	65	10	𝛥	𝛥	NOUN
cana-551	65	11	)	)	PUNCT
cana-551	65	12	∈	∈	PROPN
cana-551	65	13	𝑆𝑆(𝒵	𝑆𝑆(𝒵	PROPN
cana-551	65	14	,	,	PUNCT
cana-551	65	15	𝛥	𝛥	PROPN
cana-551	65	16	)	)	PUNCT
cana-551	65	17	is	be	AUX
cana-551	65	18	called	call	VERB
cana-551	65	19	soft	soft	ADJ
cana-551	65	20	generalized	generalize	VERB
cana-551	65	21	closed	close	VERB
cana-551	65	22	set	set	VERB
cana-551	65	23	with	with	ADP
cana-551	65	24	respect	respect	NOUN
cana-551	65	25	to	to	ADP
cana-551	65	26	soft	soft	ADJ
cana-551	65	27	ideal	ideal	ADJ
cana-551	65	28	ῐ	ῐ	PROPN
cana-551	65	29	(	(	PUNCT
cana-551	65	30	soft	soft	ADJ
cana-551	65	31	ῐg	ῐg	ADP
cana-551	65	32	−closed	−close	VERB
cana-551	65	33	set	set	NOUN
cana-551	65	34	)	)	PUNCT
cana-551	65	35	in	in	ADP
cana-551	65	36	𝒮ts(𝒵	𝒮ts(𝒵	PROPN
cana-551	65	37	,	,	PUNCT
cana-551	65	38	𝔚	𝔚	PROPN
cana-551	65	39	,	,	PUNCT
cana-551	65	40	𝛥	𝛥	PROPN
cana-551	65	41	)	)	PUNCT
cana-551	65	42	if	if	SCONJ
cana-551	65	43	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	65	44	,	,	PUNCT
cana-551	65	45	𝛥)\(𝛿	𝛥)\(𝛿	X
cana-551	65	46	,	,	PUNCT
cana-551	65	47	𝛥	𝛥	PROPN
cana-551	65	48	)	)	PUNCT
cana-551	65	49	∈	∈	PROPN
cana-551	65	50	ῐ	ῐ	PROPN
cana-551	65	51	whenever	whenever	SCONJ
cana-551	65	52	(	(	PUNCT
cana-551	65	53	𝛾	𝛾	NOUN
cana-551	65	54	,	,	PUNCT
cana-551	65	55	𝛥	𝛥	NOUN
cana-551	65	56	)	)	PUNCT
cana-551	65	57	⊂	⊂	PROPN
cana-551	65	58	(	(	PUNCT
cana-551	65	59	𝛿	𝛿	PROPN
cana-551	65	60	,	,	PUNCT
cana-551	65	61	𝛥	𝛥	NOUN
cana-551	65	62	)	)	PUNCT
cana-551	65	63	and	and	CCONJ
cana-551	65	64	(	(	PUNCT
cana-551	65	65	𝛿	𝛿	ADJ
cana-551	65	66	,	,	PUNCT
cana-551	65	67	𝛥	𝛥	NOUN
cana-551	65	68	)	)	PUNCT
cana-551	65	69	∈	∈	PROPN
cana-551	65	70	𝔚.	𝔚.	NOUN
cana-551	65	71	2	2	X
cana-551	65	72	.	.	PUNCT
cana-551	65	73	ss	ss	PROPN
cana-551	65	74	ƅ∗	ƅ∗	PROPN
cana-551	65	75	−closed	−close	VERB
cana-551	65	76	via	via	ADP
cana-551	65	77	soft	soft	ADJ
cana-551	65	78	ideal	ideal	NOUN
cana-551	65	79	in	in	ADP
cana-551	65	80	this	this	DET
cana-551	65	81	section	section	NOUN
cana-551	65	82	,	,	PUNCT
cana-551	65	83	we	we	PRON
cana-551	65	84	define	define	VERB
cana-551	65	85	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	65	86	−closed	−close	VERB
cana-551	65	87	set	set	VERB
cana-551	65	88	via	via	ADP
cana-551	65	89	soft	soft	ADJ
cana-551	65	90	ideal	ideal	NOUN
cana-551	65	91	and	and	CCONJ
cana-551	65	92	study	study	VERB
cana-551	65	93	some	some	PRON
cana-551	65	94	of	of	ADP
cana-551	65	95	their	their	PRON
cana-551	65	96	properties	property	NOUN
cana-551	65	97	.	.	PUNCT
cana-551	66	1	definition	definition	NOUN
cana-551	66	2	2.1	2.1	NUM
cana-551	66	3	:	:	PUNCT
cana-551	66	4	a	a	DET
cana-551	66	5	soft	soft	ADJ
cana-551	66	6	set	set	NOUN
cana-551	66	7	(	(	PUNCT
cana-551	66	8	𝛾	𝛾	PROPN
cana-551	66	9	,	,	PUNCT
cana-551	66	10	δ	δ	PROPN
cana-551	66	11	)	)	PUNCT
cana-551	66	12	of	of	ADP
cana-551	66	13	a	a	DET
cana-551	66	14	𝒮ts	𝒮ts	PROPN
cana-551	66	15	(	(	PUNCT
cana-551	66	16	𝒵	𝒵	PROPN
cana-551	66	17	,	,	PUNCT
cana-551	66	18	𝔚	𝔚	PROPN
cana-551	66	19	,	,	PUNCT
cana-551	66	20	𝛥	𝛥	PROPN
cana-551	66	21	)	)	PUNCT
cana-551	66	22	is	be	AUX
cana-551	66	23	called	call	VERB
cana-551	66	24	a	a	DET
cana-551	66	25	soft	soft	ADJ
cana-551	66	26	strongly	strongly	ADV
cana-551	66	27	ƅ∗	ƅ∗	NOUN
cana-551	66	28	−closed	−close	VERB
cana-551	66	29	with	with	ADP
cana-551	66	30	respect	respect	NOUN
cana-551	66	31	to	to	ADP
cana-551	66	32	soft	soft	ADJ
cana-551	66	33	ideal	ideal	ADJ
cana-551	66	34	ῐ	ῐ	NOUN
cana-551	66	35	(	(	PUNCT
cana-551	66	36	briefly	briefly	ADV
cana-551	66	37	s𝑆ƅ∗ῐ	s𝑆ƅ∗ῐ	PROPN
cana-551	66	38	−closed	−close	VERB
cana-551	66	39	)	)	PUNCT
cana-551	67	1	if	if	SCONJ
cana-551	67	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	67	3	,	,	PUNCT
cana-551	67	4	𝛥	𝛥	NOUN
cana-551	67	5	)	)	PUNCT
cana-551	67	6	)	)	PUNCT
cana-551	67	7	\	\	PUNCT
cana-551	67	8	(	(	PUNCT
cana-551	67	9	𝛿	𝛿	ADJ
cana-551	67	10	,	,	PUNCT
cana-551	67	11	𝛥	𝛥	NOUN
cana-551	67	12	)	)	PUNCT
cana-551	67	13	∈	∈	PROPN
cana-551	67	14	ῐ	ῐ	PROPN
cana-551	67	15	,	,	PUNCT
cana-551	67	16	whenever	whenever	SCONJ
cana-551	67	17	(	(	PUNCT
cana-551	67	18	𝛾	𝛾	NOUN
cana-551	67	19	,	,	PUNCT
cana-551	67	20	𝛥	𝛥	NOUN
cana-551	67	21	)	)	PUNCT
cana-551	67	22	⊂	⊂	PROPN
cana-551	67	23	(	(	PUNCT
cana-551	67	24	𝛿	𝛿	PROPN
cana-551	67	25	,	,	PUNCT
cana-551	67	26	𝛥	𝛥	NOUN
cana-551	67	27	)	)	PUNCT
cana-551	67	28	and	and	CCONJ
cana-551	67	29	(	(	PUNCT
cana-551	67	30	𝛿	𝛿	ADJ
cana-551	67	31	,	,	PUNCT
cana-551	67	32	𝛥	𝛥	NOUN
cana-551	67	33	)	)	PUNCT
cana-551	67	34	is	be	AUX
cana-551	67	35	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	67	36	−open	−open	PROPN
cana-551	67	37	.	.	PUNCT
cana-551	68	1	example	example	NOUN
cana-551	68	2	2.2	2.2	NUM
cana-551	68	3	.	.	PUNCT
cana-551	69	1	let	let	VERB
cana-551	69	2	𝒵	𝒵	PRON
cana-551	69	3	=	=	PUNCT
cana-551	69	4	{	{	PUNCT
cana-551	69	5	휀	휀	NOUN
cana-551	69	6	,	,	PUNCT
cana-551	69	7	𝜇	𝜇	ADP
cana-551	69	8	}	}	PUNCT
cana-551	69	9	and	and	CCONJ
cana-551	69	10	∆=	∆=	ADJ
cana-551	69	11	{	{	PUNCT
cana-551	69	12	∇1	∇1	NOUN
cana-551	69	13	,	,	PUNCT
cana-551	69	14	∇2	∇2	PROPN
cana-551	69	15	}	}	PUNCT
cana-551	69	16	.	.	PUNCT
cana-551	70	1	let	let	AUX
cana-551	70	2	(	(	PUNCT
cana-551	70	3	𝛾1	𝛾1	PROPN
cana-551	70	4	,	,	PUNCT
cana-551	70	5	δ	δ	PROPN
cana-551	70	6	)	)	PUNCT
cana-551	70	7	,	,	PUNCT
cana-551	70	8	(	(	PUNCT
cana-551	70	9	𝛾2	𝛾2	VERB
cana-551	70	10	,	,	PUNCT
cana-551	70	11	δ	δ	PROPN
cana-551	70	12	)	)	PUNCT
cana-551	70	13	and	and	CCONJ
cana-551	70	14	(	(	PUNCT
cana-551	70	15	𝛾3	𝛾3	PROPN
cana-551	70	16	,	,	PUNCT
cana-551	70	17	δ	δ	PROPN
cana-551	70	18	)	)	PUNCT
cana-551	70	19	be	be	VERB
cana-551	70	20	three	three	NUM
cana-551	70	21	soft	soft	ADJ
cana-551	70	22	sets	set	NOUN
cana-551	70	23	,	,	PUNCT
cana-551	70	24	where	where	SCONJ
cana-551	70	25	(	(	PUNCT
cana-551	70	26	𝛾1	𝛾1	PROPN
cana-551	70	27	,	,	PUNCT
cana-551	70	28	δ	δ	PROPN
cana-551	70	29	)	)	PUNCT
cana-551	71	1	=	=	PRON
cana-551	71	2	{	{	PUNCT
cana-551	71	3	(	(	PUNCT
cana-551	71	4	∇1	∇1	NOUN
cana-551	71	5	,	,	PUNCT
cana-551	71	6	∅	∅	NOUN
cana-551	71	7	)	)	PUNCT
cana-551	71	8	,	,	PUNCT
cana-551	71	9	(	(	PUNCT
cana-551	71	10	∇2	∇2	PROPN
cana-551	71	11	,	,	PUNCT
cana-551	71	12	{	{	PUNCT
cana-551	71	13	휀	휀	NOUN
cana-551	71	14	}	}	PUNCT
cana-551	71	15	)	)	PUNCT
cana-551	71	16	}	}	PUNCT
cana-551	71	17	,	,	PUNCT
cana-551	71	18	(	(	PUNCT
cana-551	71	19	𝛾2	𝛾2	VERB
cana-551	71	20	,	,	PUNCT
cana-551	71	21	δ	δ	NOUN
cana-551	71	22	)	)	PUNCT
cana-551	71	23	=	=	PRON
cana-551	71	24	{	{	PUNCT
cana-551	71	25	(	(	PUNCT
cana-551	71	26	∇1	∇1	PROPN
cana-551	71	27	,	,	PUNCT
cana-551	71	28	{	{	PUNCT
cana-551	71	29	𝜇	𝜇	X
cana-551	71	30	}	}	PUNCT
cana-551	71	31	)	)	PUNCT
cana-551	71	32	,	,	PUNCT
cana-551	71	33	(	(	PUNCT
cana-551	71	34	∇2	∇2	X
cana-551	71	35	,	,	PUNCT
cana-551	71	36	∅	∅	NOUN
cana-551	71	37	)	)	PUNCT
cana-551	71	38	}	}	PUNCT
cana-551	71	39	and	and	CCONJ
cana-551	71	40	(	(	PUNCT
cana-551	71	41	𝛾3	𝛾3	PROPN
cana-551	71	42	,	,	PUNCT
cana-551	71	43	δ	δ	PROPN
cana-551	71	44	)	)	PUNCT
cana-551	71	45	=	=	PRON
cana-551	71	46	{	{	PUNCT
cana-551	71	47	(	(	PUNCT
cana-551	71	48	∇1	∇1	PROPN
cana-551	71	49	,	,	PUNCT
cana-551	71	50	{	{	PUNCT
cana-551	71	51	𝜇	𝜇	X
cana-551	71	52	}	}	PUNCT
cana-551	71	53	)	)	PUNCT
cana-551	71	54	,	,	PUNCT
cana-551	71	55	(	(	PUNCT
cana-551	71	56	∇2	∇2	PROPN
cana-551	71	57	,	,	PUNCT
cana-551	71	58	{	{	PUNCT
cana-551	71	59	휀	휀	NOUN
cana-551	71	60	}	}	PUNCT
cana-551	71	61	)	)	PUNCT
cana-551	71	62	}	}	PUNCT
cana-551	71	63	.	.	PUNCT
cana-551	72	1	then	then	ADV
cana-551	72	2	(	(	PUNCT
cana-551	72	3	𝛾1	𝛾1	PROPN
cana-551	72	4	,	,	PUNCT
cana-551	72	5	δ	δ	PROPN
cana-551	72	6	)	)	PUNCT
cana-551	72	7	,	,	PUNCT
cana-551	72	8	(	(	PUNCT
cana-551	72	9	𝛾2	𝛾2	VERB
cana-551	72	10	,	,	PUNCT
cana-551	72	11	δ	δ	PROPN
cana-551	72	12	)	)	PUNCT
cana-551	72	13	and	and	CCONJ
cana-551	72	14	(	(	PUNCT
cana-551	72	15	𝛾3	𝛾3	PROPN
cana-551	72	16	,	,	PUNCT
cana-551	72	17	δ	δ	PROPN
cana-551	72	18	)	)	PUNCT
cana-551	72	19	are	be	AUX
cana-551	72	20	soft	soft	ADJ
cana-551	72	21	sets	set	NOUN
cana-551	72	22	over	over	ADP
cana-551	72	23	𝒵	𝒵	PROPN
cana-551	72	24	and	and	CCONJ
cana-551	72	25	𝔚	𝔚	NOUN
cana-551	72	26	=	=	NOUN
cana-551	72	27	{	{	PUNCT
cana-551	72	28	�	�	PROPN
cana-551	72	29	̃	̃	PROPN
cana-551	72	30	�	�	PROPN
cana-551	72	31	,	,	PUNCT
cana-551	72	32	∅̃	∅̃	NOUN
cana-551	72	33	,	,	PUNCT
cana-551	72	34	(	(	PUNCT
cana-551	72	35	𝛾1	𝛾1	PROPN
cana-551	72	36	,	,	PUNCT
cana-551	72	37	δ	δ	PROPN
cana-551	72	38	)	)	PUNCT
cana-551	72	39	,	,	PUNCT
cana-551	72	40	(	(	PUNCT
cana-551	72	41	𝛾2	𝛾2	VERB
cana-551	72	42	,	,	PUNCT
cana-551	72	43	δ	δ	PROPN
cana-551	72	44	)	)	PUNCT
cana-551	72	45	,	,	PUNCT
cana-551	72	46	(	(	PUNCT
cana-551	72	47	𝛾3	𝛾3	PROPN
cana-551	72	48	,	,	PUNCT
cana-551	72	49	δ	δ	PROPN
cana-551	72	50	)	)	PUNCT
cana-551	72	51	}	}	PUNCT
cana-551	72	52	is	be	AUX
cana-551	72	53	the	the	DET
cana-551	72	54	soft	soft	ADJ
cana-551	72	55	topology	topology	NOUN
cana-551	72	56	over	over	ADP
cana-551	72	57	𝒵.	𝒵.	PROPN
cana-551	72	58	let	let	VERB
cana-551	72	59	ῐ	ῐ	PROPN
cana-551	72	60	=	=	PRON
cana-551	72	61	{	{	PUNCT
cana-551	72	62	∅̃	∅̃	NOUN
cana-551	72	63	,	,	PUNCT
cana-551	72	64	(	(	PUNCT
cana-551	72	65	𝛿1	𝛿1	PROPN
cana-551	72	66	,	,	PUNCT
cana-551	72	67	δ	δ	PROPN
cana-551	72	68	)	)	PUNCT
cana-551	72	69	,	,	PUNCT
cana-551	72	70	(	(	PUNCT
cana-551	72	71	𝛿2	𝛿2	PROPN
cana-551	72	72	,	,	PUNCT
cana-551	72	73	δ	δ	PROPN
cana-551	72	74	)	)	PUNCT
cana-551	72	75	,	,	PUNCT
cana-551	72	76	(	(	PUNCT
cana-551	72	77	𝛿3	𝛿3	PROPN
cana-551	72	78	,	,	PUNCT
cana-551	72	79	δ	δ	PROPN
cana-551	72	80	)	)	PUNCT
cana-551	72	81	}	}	PUNCT
cana-551	72	82	be	be	AUX
cana-551	72	83	a	a	DET
cana-551	72	84	soft	soft	ADJ
cana-551	72	85	ideal	ideal	NOUN
cana-551	72	86	on	on	ADP
cana-551	72	87	𝒵	𝒵	PROPN
cana-551	72	88	,	,	PUNCT
cana-551	72	89	where	where	SCONJ
cana-551	72	90	(	(	PUNCT
cana-551	72	91	𝛿1	𝛿1	NOUN
cana-551	72	92	,	,	PUNCT
cana-551	72	93	δ	δ	PROPN
cana-551	72	94	)	)	PUNCT
cana-551	72	95	=	=	PRON
cana-551	72	96	{	{	PUNCT
cana-551	72	97	(	(	PUNCT
cana-551	72	98	∇1	∇1	PROPN
cana-551	72	99	,	,	PUNCT
cana-551	72	100	{	{	PUNCT
cana-551	72	101	𝜇	𝜇	X
cana-551	72	102	}	}	PUNCT
cana-551	72	103	)	)	PUNCT
cana-551	72	104	,	,	PUNCT
cana-551	72	105	(	(	PUNCT
cana-551	72	106	∇2	∇2	X
cana-551	72	107	,	,	PUNCT
cana-551	72	108	∅	∅	NOUN
cana-551	72	109	)	)	PUNCT
cana-551	72	110	}	}	PUNCT
cana-551	72	111	,	,	PUNCT
cana-551	72	112	communications	communication	NOUN
cana-551	72	113	on	on	ADP
cana-551	72	114	applied	apply	VERB
cana-551	72	115	nonlinear	nonlinear	ADJ
cana-551	72	116	analysis	analysis	NOUN
cana-551	72	117	issn	issn	NOUN
cana-551	72	118	:	:	PUNCT
cana-551	72	119	1074	1074	NUM
cana-551	72	120	-	-	PUNCT
cana-551	72	121	133x	133x	NUM
cana-551	72	122	vol	vol	NOUN
cana-551	72	123	31	31	NUM
cana-551	72	124	no	no	NOUN
cana-551	72	125	.	.	NOUN
cana-551	72	126	2	2	NUM
cana-551	72	127	(	(	PUNCT
cana-551	72	128	2024	2024	NUM
cana-551	72	129	)	)	PUNCT
cana-551	72	130	287	287	NUM
cana-551	72	131	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	72	132	(	(	PUNCT
cana-551	72	133	𝛿2	𝛿2	PROPN
cana-551	72	134	,	,	PUNCT
cana-551	72	135	δ	δ	PROPN
cana-551	72	136	)	)	PUNCT
cana-551	72	137	=	=	PRON
cana-551	72	138	{	{	PUNCT
cana-551	72	139	(	(	PUNCT
cana-551	72	140	∇1	∇1	PROPN
cana-551	72	141	,	,	PUNCT
cana-551	72	142	{	{	PUNCT
cana-551	72	143	𝜇	𝜇	X
cana-551	72	144	}	}	PUNCT
cana-551	72	145	)	)	PUNCT
cana-551	72	146	,	,	PUNCT
cana-551	72	147	(	(	PUNCT
cana-551	72	148	∇2	∇2	PROPN
cana-551	72	149	,	,	PUNCT
cana-551	72	150	{	{	PUNCT
cana-551	72	151	휀	휀	NOUN
cana-551	72	152	}	}	PUNCT
cana-551	72	153	)	)	PUNCT
cana-551	72	154	}	}	PUNCT
cana-551	72	155	and	and	CCONJ
cana-551	72	156	(	(	PUNCT
cana-551	72	157	𝛿3	𝛿3	PROPN
cana-551	72	158	,	,	PUNCT
cana-551	72	159	δ	δ	PROPN
cana-551	72	160	)	)	PUNCT
cana-551	72	161	=	=	PRON
cana-551	72	162	{	{	PUNCT
cana-551	72	163	(	(	PUNCT
cana-551	72	164	∇1	∇1	NOUN
cana-551	72	165	,	,	PUNCT
cana-551	72	166	∅	∅	NOUN
cana-551	72	167	)	)	PUNCT
cana-551	72	168	,	,	PUNCT
cana-551	72	169	(	(	PUNCT
cana-551	72	170	∇2	∇2	PROPN
cana-551	72	171	,	,	PUNCT
cana-551	72	172	{	{	PUNCT
cana-551	72	173	휀	휀	NOUN
cana-551	72	174	}	}	PUNCT
cana-551	72	175	)	)	PUNCT
cana-551	72	176	}	}	PUNCT
cana-551	72	177	.	.	PUNCT
cana-551	73	1	the	the	DET
cana-551	73	2	soft	soft	ADJ
cana-551	73	3	sets	set	NOUN
cana-551	73	4	(	(	PUNCT
cana-551	73	5	𝜗1	𝜗1	X
cana-551	73	6	,	,	PUNCT
cana-551	73	7	δ	δ	PROPN
cana-551	73	8	)	)	PUNCT
cana-551	73	9	,	,	PUNCT
cana-551	73	10	(	(	PUNCT
cana-551	73	11	𝜗2	𝜗2	PROPN
cana-551	73	12	,	,	PUNCT
cana-551	73	13	δ	δ	PROPN
cana-551	73	14	)	)	PUNCT
cana-551	73	15	,	,	PUNCT
cana-551	73	16	(	(	PUNCT
cana-551	73	17	𝜗3	𝜗3	ADJ
cana-551	73	18	,	,	PUNCT
cana-551	73	19	δ	δ	PROPN
cana-551	73	20	)	)	PUNCT
cana-551	73	21	are	be	AUX
cana-551	73	22	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	73	23	−closed	−close	VERB
cana-551	73	24	,	,	PUNCT
cana-551	73	25	where	where	SCONJ
cana-551	73	26	(	(	PUNCT
cana-551	73	27	𝜗1	𝜗1	X
cana-551	73	28	,	,	PUNCT
cana-551	73	29	δ	δ	NOUN
cana-551	73	30	)	)	PUNCT
cana-551	73	31	=	=	PRON
cana-551	73	32	{	{	PUNCT
cana-551	73	33	(	(	PUNCT
cana-551	73	34	∇1	∇1	NOUN
cana-551	73	35	,	,	PUNCT
cana-551	73	36	∅	∅	NOUN
cana-551	73	37	)	)	PUNCT
cana-551	73	38	,	,	PUNCT
cana-551	73	39	(	(	PUNCT
cana-551	73	40	∇2	∇2	PROPN
cana-551	73	41	,	,	PUNCT
cana-551	73	42	𝒵	𝒵	PROPN
cana-551	73	43	)	)	PUNCT
cana-551	73	44	}	}	PUNCT
cana-551	73	45	,	,	PUNCT
cana-551	73	46	(	(	PUNCT
cana-551	73	47	𝜗2	𝜗2	PROPN
cana-551	73	48	,	,	PUNCT
cana-551	73	49	δ	δ	PROPN
cana-551	73	50	)	)	PUNCT
cana-551	74	1	=	=	PRON
cana-551	74	2	{	{	PUNCT
cana-551	74	3	(	(	PUNCT
cana-551	74	4	∇1	∇1	PROPN
cana-551	74	5	,	,	PUNCT
cana-551	74	6	𝒵	𝒵	PROPN
cana-551	74	7	)	)	PUNCT
cana-551	74	8	,	,	PUNCT
cana-551	74	9	(	(	PUNCT
cana-551	74	10	∇2	∇2	PROPN
cana-551	74	11	,	,	PUNCT
cana-551	74	12	{	{	PUNCT
cana-551	74	13	휀	휀	NOUN
cana-551	74	14	}	}	PUNCT
cana-551	74	15	)	)	PUNCT
cana-551	74	16	}	}	PUNCT
cana-551	74	17	and	and	CCONJ
cana-551	74	18	(	(	PUNCT
cana-551	74	19	𝜗3	𝜗3	ADJ
cana-551	74	20	,	,	PUNCT
cana-551	74	21	δ	δ	NOUN
cana-551	74	22	)	)	PUNCT
cana-551	74	23	=	=	PRON
cana-551	74	24	{	{	PUNCT
cana-551	74	25	(	(	PUNCT
cana-551	74	26	∇1	∇1	PROPN
cana-551	74	27	,	,	PUNCT
cana-551	74	28	{	{	PUNCT
cana-551	74	29	𝜇	𝜇	X
cana-551	74	30	}	}	PUNCT
cana-551	74	31	)	)	PUNCT
cana-551	74	32	,	,	PUNCT
cana-551	74	33	(	(	PUNCT
cana-551	74	34	∇2	∇2	PROPN
cana-551	74	35	,	,	PUNCT
cana-551	74	36	{	{	PUNCT
cana-551	74	37	𝜇	𝜇	X
cana-551	74	38	}	}	PUNCT
cana-551	74	39	)	)	PUNCT
cana-551	74	40	}	}	PUNCT
cana-551	74	41	.	.	PUNCT
cana-551	75	1	and	and	CCONJ
cana-551	75	2	we	we	PRON
cana-551	75	3	see	see	VERB
cana-551	75	4	the	the	DET
cana-551	75	5	soft	soft	ADJ
cana-551	75	6	set	set	NOUN
cana-551	75	7	(	(	PUNCT
cana-551	75	8	𝜉	𝜉	X
cana-551	75	9	,	,	PUNCT
cana-551	75	10	δ	δ	PROPN
cana-551	75	11	)	)	PUNCT
cana-551	75	12	is	be	AUX
cana-551	75	13	not	not	PART
cana-551	75	14	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	ADJ
cana-551	75	15	−closed	−close	VERB
cana-551	75	16	,	,	PUNCT
cana-551	75	17	where	where	SCONJ
cana-551	75	18	(	(	PUNCT
cana-551	75	19	𝜉	𝜉	X
cana-551	75	20	,	,	PUNCT
cana-551	75	21	δ	δ	PROPN
cana-551	75	22	)	)	PUNCT
cana-551	75	23	=	=	PRON
cana-551	75	24	{	{	PUNCT
cana-551	75	25	(	(	PUNCT
cana-551	75	26	∇1	∇1	NOUN
cana-551	75	27	,	,	PUNCT
cana-551	75	28	∅	∅	NOUN
cana-551	75	29	)	)	PUNCT
cana-551	75	30	,	,	PUNCT
cana-551	75	31	(	(	PUNCT
cana-551	75	32	∇2	∇2	PROPN
cana-551	75	33	,	,	PUNCT
cana-551	75	34	{	{	PUNCT
cana-551	75	35	휀	휀	NOUN
cana-551	75	36	}	}	PUNCT
cana-551	75	37	)	)	PUNCT
cana-551	75	38	}	}	PUNCT
cana-551	75	39	.	.	PUNCT
cana-551	76	1	theorem	theorem	VERB
cana-551	76	2	2.3	2.3	NUM
cana-551	76	3	:	:	PUNCT
cana-551	76	4	(	(	PUNCT
cana-551	76	5	1	1	X
cana-551	76	6	)	)	PUNCT
cana-551	76	7	every	every	DET
cana-551	76	8	soft	soft	ADJ
cana-551	76	9	g	g	NOUN
cana-551	76	10	−closed	−close	VERB
cana-551	76	11	set	set	NOUN
cana-551	76	12	is	be	AUX
cana-551	76	13	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	76	14	−closed	−close	VERB
cana-551	76	15	.	.	PUNCT
cana-551	77	1	(	(	PUNCT
cana-551	77	2	2	2	X
cana-551	77	3	)	)	PUNCT
cana-551	77	4	every	every	DET
cana-551	77	5	closed	close	VERB
cana-551	77	6	soft	soft	ADJ
cana-551	77	7	set	set	NOUN
cana-551	77	8	is	be	AUX
cana-551	77	9	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	77	10	−closed	−close	VERB
cana-551	77	11	.	.	PUNCT
cana-551	78	1	(	(	PUNCT
cana-551	78	2	3	3	X
cana-551	78	3	)	)	PUNCT
cana-551	78	4	every	every	PRON
cana-551	78	5	soft	soft	ADJ
cana-551	78	6	ῐg	ῐg	ADP
cana-551	78	7	−closed	−closed	ADJ
cana-551	78	8	set	set	NOUN
cana-551	78	9	is	be	AUX
cana-551	78	10	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	78	11	−closed	−close	VERB
cana-551	78	12	.	.	PUNCT
cana-551	79	1	proof	proof	NOUN
cana-551	79	2	.	.	PUNCT
cana-551	80	1	(	(	PUNCT
cana-551	80	2	1	1	X
cana-551	80	3	)	)	PUNCT
cana-551	80	4	let	let	VERB
cana-551	80	5	(	(	PUNCT
cana-551	80	6	𝛾	𝛾	PROPN
cana-551	80	7	,	,	PUNCT
cana-551	80	8	δ	δ	PROPN
cana-551	80	9	)	)	PUNCT
cana-551	80	10	⊆	⊆	NUM
cana-551	80	11	(	(	PUNCT
cana-551	80	12	𝛿	𝛿	ADJ
cana-551	80	13	,	,	PUNCT
cana-551	80	14	δ	δ	PROPN
cana-551	80	15	)	)	PUNCT
cana-551	80	16	and	and	CCONJ
cana-551	80	17	(	(	PUNCT
cana-551	80	18	𝛿	𝛿	ADJ
cana-551	80	19	,	,	PUNCT
cana-551	80	20	δ	δ	PROPN
cana-551	80	21	)	)	PUNCT
cana-551	80	22	is	be	AUX
cana-551	80	23	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	80	24	−open	−open	ADJ
cana-551	80	25	.	.	PUNCT
cana-551	81	1	since	since	SCONJ
cana-551	81	2	(	(	PUNCT
cana-551	81	3	𝛾	𝛾	PROPN
cana-551	81	4	,	,	PUNCT
cana-551	81	5	δ	δ	PROPN
cana-551	81	6	)	)	PUNCT
cana-551	81	7	is	be	AUX
cana-551	81	8	soft	soft	ADJ
cana-551	81	9	g	g	ADP
cana-551	81	10	−closed	−close	VERB
cana-551	81	11	⇒	⇒	PROPN
cana-551	81	12	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	81	13	,	,	PUNCT
cana-551	81	14	δ	δ	PROPN
cana-551	81	15	)	)	PUNCT
cana-551	81	16	⊆	⊆	NUM
cana-551	81	17	(	(	PUNCT
cana-551	81	18	𝛿	𝛿	ADJ
cana-551	81	19	,	,	PUNCT
cana-551	81	20	δ	δ	NOUN
cana-551	81	21	)	)	PUNCT
cana-551	81	22	and	and	CCONJ
cana-551	81	23	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	81	24	,	,	PUNCT
cana-551	81	25	δ	δ	PROPN
cana-551	81	26	)	)	PUNCT
cana-551	81	27	)	)	PUNCT
cana-551	82	1	⊆	⊆	NUM
cana-551	82	2	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	82	3	,	,	PUNCT
cana-551	82	4	δ	δ	PROPN
cana-551	82	5	)	)	PUNCT
cana-551	82	6	.	.	PUNCT
cana-551	83	1	so	so	ADV
cana-551	83	2	,	,	PUNCT
cana-551	83	3	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	83	4	,	,	PUNCT
cana-551	83	5	δ))\(𝛿	δ))\(𝛿	PROPN
cana-551	83	6	,	,	PUNCT
cana-551	83	7	δ	δ	PROPN
cana-551	83	8	)	)	PUNCT
cana-551	83	9	=	=	PUNCT
cana-551	83	10	∅	∅	NOUN
cana-551	83	11	∈	∈	PROPN
cana-551	83	12	ῐ.	ῐ.	NOUN
cana-551	83	13	therefore	therefore	ADV
cana-551	83	14	,	,	PUNCT
cana-551	83	15	(	(	PUNCT
cana-551	83	16	𝛾	𝛾	PROPN
cana-551	83	17	,	,	PUNCT
cana-551	83	18	δ	δ	NOUN
cana-551	83	19	)	)	PUNCT
cana-551	83	20	is	be	AUX
cana-551	83	21	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	83	22	−closed	−close	VERB
cana-551	83	23	.	.	PUNCT
cana-551	84	1	(	(	PUNCT
cana-551	84	2	2	2	X
cana-551	84	3	)	)	PUNCT
cana-551	84	4	let	let	VERB
cana-551	84	5	(	(	PUNCT
cana-551	84	6	𝛾	𝛾	PROPN
cana-551	84	7	,	,	PUNCT
cana-551	84	8	δ	δ	PROPN
cana-551	84	9	)	)	PUNCT
cana-551	84	10	⊆	⊆	NUM
cana-551	84	11	(	(	PUNCT
cana-551	84	12	𝛿	𝛿	ADJ
cana-551	84	13	,	,	PUNCT
cana-551	84	14	δ	δ	PROPN
cana-551	84	15	)	)	PUNCT
cana-551	84	16	and	and	CCONJ
cana-551	84	17	(	(	PUNCT
cana-551	84	18	𝛿	𝛿	ADJ
cana-551	84	19	,	,	PUNCT
cana-551	84	20	δ	δ	PROPN
cana-551	84	21	)	)	PUNCT
cana-551	84	22	is	be	AUX
cana-551	84	23	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	84	24	−open	−open	ADJ
cana-551	84	25	.	.	PUNCT
cana-551	85	1	since	since	SCONJ
cana-551	85	2	(	(	PUNCT
cana-551	85	3	𝛾	𝛾	PROPN
cana-551	85	4	,	,	PUNCT
cana-551	85	5	δ	δ	PROPN
cana-551	85	6	)	)	PUNCT
cana-551	85	7	is	be	AUX
cana-551	85	8	soft	soft	ADJ
cana-551	85	9	closed	closed	ADJ
cana-551	85	10	,	,	PUNCT
cana-551	85	11	then	then	ADV
cana-551	85	12	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	85	13	,	,	PUNCT
cana-551	85	14	δ	δ	PROPN
cana-551	85	15	)	)	PUNCT
cana-551	85	16	)	)	PUNCT
cana-551	86	1	⊆	⊆	NUM
cana-551	86	2	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	86	3	,	,	PUNCT
cana-551	86	4	δ	δ	PROPN
cana-551	86	5	)	)	PUNCT
cana-551	86	6	=	=	PUNCT
cana-551	86	7	(	(	PUNCT
cana-551	86	8	𝛾	𝛾	PROPN
cana-551	86	9	,	,	PUNCT
cana-551	86	10	δ	δ	PROPN
cana-551	86	11	)	)	PUNCT
cana-551	86	12	⊆	⊆	NUM
cana-551	86	13	(	(	PUNCT
cana-551	86	14	𝛿	𝛿	ADJ
cana-551	86	15	,	,	PUNCT
cana-551	86	16	δ	δ	PROPN
cana-551	86	17	)	)	PUNCT
cana-551	86	18	.	.	PUNCT
cana-551	87	1	hence	hence	ADV
cana-551	87	2	,	,	PUNCT
cana-551	87	3	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	87	4	,	,	PUNCT
cana-551	87	5	δ))\(𝛿	δ))\(𝛿	PROPN
cana-551	87	6	,	,	PUNCT
cana-551	87	7	δ	δ	PROPN
cana-551	87	8	)	)	PUNCT
cana-551	87	9	=	=	PUNCT
cana-551	87	10	∅	∅	NOUN
cana-551	87	11	∈	∈	PROPN
cana-551	87	12	ῐ.	ῐ.	NOUN
cana-551	87	13	therefore	therefore	ADV
cana-551	87	14	,	,	PUNCT
cana-551	87	15	(	(	PUNCT
cana-551	87	16	𝛾	𝛾	PROPN
cana-551	87	17	,	,	PUNCT
cana-551	87	18	δ	δ	NOUN
cana-551	87	19	)	)	PUNCT
cana-551	87	20	is	be	AUX
cana-551	87	21	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	87	22	−closed	−close	VERB
cana-551	87	23	.	.	PUNCT
cana-551	88	1	(	(	PUNCT
cana-551	88	2	3	3	X
cana-551	88	3	)	)	PUNCT
cana-551	88	4	let	let	VERB
cana-551	88	5	(	(	PUNCT
cana-551	88	6	𝛾	𝛾	PROPN
cana-551	88	7	,	,	PUNCT
cana-551	88	8	δ	δ	PROPN
cana-551	88	9	)	)	PUNCT
cana-551	88	10	⊆	⊆	NUM
cana-551	88	11	(	(	PUNCT
cana-551	88	12	𝜉	𝜉	PROPN
cana-551	88	13	,	,	PUNCT
cana-551	88	14	δ	δ	PROPN
cana-551	88	15	)	)	PUNCT
cana-551	88	16	and	and	CCONJ
cana-551	88	17	(	(	PUNCT
cana-551	88	18	𝜉	𝜉	PROPN
cana-551	88	19	,	,	PUNCT
cana-551	88	20	δ	δ	PROPN
cana-551	88	21	)	)	PUNCT
cana-551	88	22	is	be	AUX
cana-551	88	23	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	88	24	−open	−open	ADJ
cana-551	88	25	.	.	PUNCT
cana-551	89	1	then	then	ADV
cana-551	89	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	89	3	,	,	PUNCT
cana-551	89	4	δ))\(𝜉	δ))\(𝜉	PROPN
cana-551	89	5	,	,	PUNCT
cana-551	89	6	δ	δ	PROPN
cana-551	89	7	)	)	PUNCT
cana-551	89	8	⊆	⊆	NUM
cana-551	89	9	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	89	10	,	,	PUNCT
cana-551	89	11	δ)\(𝜉	δ)\(𝜉	PROPN
cana-551	89	12	,	,	PUNCT
cana-551	89	13	δ	δ	PROPN
cana-551	89	14	)	)	PUNCT
cana-551	89	15	∈	∈	PROPN
cana-551	89	16	ῐ	ῐ	PROPN
cana-551	89	17	,	,	PUNCT
cana-551	89	18	(	(	PUNCT
cana-551	89	19	𝜉	𝜉	X
cana-551	89	20	,	,	PUNCT
cana-551	89	21	δ	δ	PROPN
cana-551	89	22	)	)	PUNCT
cana-551	89	23	∈	∈	PROPN
cana-551	89	24	ῐ	ῐ	PROPN
cana-551	89	25	hence	hence	ADV
cana-551	89	26	,	,	PUNCT
cana-551	89	27	(	(	PUNCT
cana-551	89	28	𝛾	𝛾	PROPN
cana-551	89	29	,	,	PUNCT
cana-551	89	30	δ	δ	NOUN
cana-551	89	31	)	)	PUNCT
cana-551	89	32	is	be	AUX
cana-551	89	33	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	89	34	−closed	−close	VERB
cana-551	89	35	.	.	PUNCT
cana-551	90	1	the	the	DET
cana-551	90	2	converse	converse	NOUN
cana-551	90	3	of	of	ADP
cana-551	90	4	the	the	DET
cana-551	90	5	above	above	ADJ
cana-551	90	6	theorem	theorem	NOUN
cana-551	90	7	is	be	AUX
cana-551	90	8	not	not	PART
cana-551	90	9	true	true	ADJ
cana-551	90	10	in	in	ADP
cana-551	90	11	general	general	ADJ
cana-551	90	12	.	.	PUNCT
cana-551	91	1	the	the	DET
cana-551	91	2	following	follow	VERB
cana-551	91	3	examples	example	NOUN
cana-551	91	4	support	support	VERB
cana-551	91	5	our	our	PRON
cana-551	91	6	claim	claim	NOUN
cana-551	91	7	.	.	PUNCT
cana-551	92	1	example	example	NOUN
cana-551	92	2	2.4	2.4	NUM
cana-551	92	3	:	:	PUNCT
cana-551	92	4	let	let	VERB
cana-551	92	5	𝒵	𝒵	PROPN
cana-551	92	6	=	=	PUNCT
cana-551	92	7	{	{	PUNCT
cana-551	92	8	휀	휀	NOUN
cana-551	92	9	,	,	PUNCT
cana-551	92	10	𝜇	𝜇	ADP
cana-551	92	11	}	}	PUNCT
cana-551	92	12	.	.	PUNCT
cana-551	93	1	let	let	VERB
cana-551	93	2	∆=	∆=	VERB
cana-551	93	3	{	{	PUNCT
cana-551	93	4	∇1	∇1	NOUN
cana-551	93	5	,	,	PUNCT
cana-551	93	6	∇2	∇2	PROPN
cana-551	93	7	}	}	PUNCT
cana-551	93	8	.	.	PUNCT
cana-551	94	1	let	let	VERB
cana-551	94	2	(	(	PUNCT
cana-551	94	3	𝛾1	𝛾1	PROPN
cana-551	94	4	,	,	PUNCT
cana-551	94	5	δ	δ	PROPN
cana-551	94	6	)	)	PUNCT
cana-551	94	7	,	,	PUNCT
cana-551	94	8	(	(	PUNCT
cana-551	94	9	𝛾2	𝛾2	VERB
cana-551	94	10	,	,	PUNCT
cana-551	94	11	δ	δ	PROPN
cana-551	94	12	)	)	PUNCT
cana-551	94	13	,	,	PUNCT
cana-551	94	14	(	(	PUNCT
cana-551	94	15	𝛾3	𝛾3	PROPN
cana-551	94	16	,	,	PUNCT
cana-551	94	17	δ	δ	PROPN
cana-551	94	18	)	)	PUNCT
cana-551	94	19	and	and	CCONJ
cana-551	94	20	(	(	PUNCT
cana-551	94	21	𝛾4	𝛾4	PROPN
cana-551	94	22	,	,	PUNCT
cana-551	94	23	δ	δ	PROPN
cana-551	94	24	)	)	PUNCT
cana-551	94	25	be	be	VERB
cana-551	94	26	four	four	NUM
cana-551	94	27	soft	soft	ADJ
cana-551	94	28	sets	set	NOUN
cana-551	94	29	,	,	PUNCT
cana-551	94	30	where	where	SCONJ
cana-551	94	31	(	(	PUNCT
cana-551	94	32	𝛾1	𝛾1	PROPN
cana-551	94	33	,	,	PUNCT
cana-551	94	34	δ	δ	PROPN
cana-551	94	35	)	)	PUNCT
cana-551	94	36	=	=	PRON
cana-551	94	37	{	{	PUNCT
cana-551	94	38	(	(	PUNCT
cana-551	94	39	∇1	∇1	PROPN
cana-551	94	40	,	,	PUNCT
cana-551	94	41	{	{	PUNCT
cana-551	94	42	휀	휀	NOUN
cana-551	94	43	}	}	PUNCT
cana-551	94	44	)	)	PUNCT
cana-551	94	45	,	,	PUNCT
cana-551	94	46	(	(	PUNCT
cana-551	94	47	∇2	∇2	PROPN
cana-551	94	48	,	,	PUNCT
cana-551	94	49	𝒵	𝒵	PROPN
cana-551	94	50	)	)	PUNCT
cana-551	94	51	}	}	PUNCT
cana-551	94	52	,	,	PUNCT
cana-551	94	53	(	(	PUNCT
cana-551	94	54	𝛾2	𝛾2	VERB
cana-551	94	55	,	,	PUNCT
cana-551	94	56	δ	δ	NOUN
cana-551	94	57	)	)	PUNCT
cana-551	94	58	=	=	PRON
cana-551	94	59	{	{	PUNCT
cana-551	94	60	(	(	PUNCT
cana-551	94	61	∇1	∇1	PROPN
cana-551	94	62	,	,	PUNCT
cana-551	94	63	{	{	PUNCT
cana-551	94	64	휀	휀	NOUN
cana-551	94	65	}	}	PUNCT
cana-551	94	66	)	)	PUNCT
cana-551	94	67	,	,	PUNCT
cana-551	94	68	(	(	PUNCT
cana-551	94	69	∇2	∇2	X
cana-551	94	70	,	,	PUNCT
cana-551	94	71	∅	∅	NOUN
cana-551	94	72	)	)	PUNCT
cana-551	94	73	}	}	PUNCT
cana-551	94	74	,	,	PUNCT
cana-551	94	75	(	(	PUNCT
cana-551	94	76	𝛾3	𝛾3	PROPN
cana-551	94	77	,	,	PUNCT
cana-551	94	78	δ	δ	PROPN
cana-551	94	79	)	)	PUNCT
cana-551	94	80	=	=	PRON
cana-551	94	81	{	{	PUNCT
cana-551	94	82	(	(	PUNCT
cana-551	94	83	∇1	∇1	PROPN
cana-551	94	84	,	,	PUNCT
cana-551	94	85	{	{	PUNCT
cana-551	94	86	휀	휀	NOUN
cana-551	94	87	}	}	PUNCT
cana-551	94	88	)	)	PUNCT
cana-551	94	89	,	,	PUNCT
cana-551	94	90	(	(	PUNCT
cana-551	94	91	∇2	∇2	PROPN
cana-551	94	92	,	,	PUNCT
cana-551	94	93	{	{	PUNCT
cana-551	94	94	𝜇	𝜇	X
cana-551	94	95	}	}	PUNCT
cana-551	94	96	)	)	PUNCT
cana-551	94	97	}	}	PUNCT
cana-551	94	98	and	and	CCONJ
cana-551	94	99	(	(	PUNCT
cana-551	94	100	𝛾4	𝛾4	PROPN
cana-551	94	101	,	,	PUNCT
cana-551	94	102	δ	δ	PROPN
cana-551	94	103	)	)	PUNCT
cana-551	94	104	=	=	PRON
cana-551	94	105	{	{	PUNCT
cana-551	94	106	(	(	PUNCT
cana-551	94	107	∇1	∇1	PROPN
cana-551	94	108	,	,	PUNCT
cana-551	94	109	{	{	PUNCT
cana-551	94	110	휀	휀	NOUN
cana-551	94	111	}	}	PUNCT
cana-551	94	112	)	)	PUNCT
cana-551	94	113	,	,	PUNCT
cana-551	94	114	(	(	PUNCT
cana-551	94	115	∇2	∇2	PROPN
cana-551	94	116	,	,	PUNCT
cana-551	94	117	{	{	PUNCT
cana-551	94	118	휀	휀	NOUN
cana-551	94	119	}	}	PUNCT
cana-551	94	120	)	)	PUNCT
cana-551	94	121	}	}	PUNCT
cana-551	94	122	.	.	PUNCT
cana-551	95	1	then	then	ADV
cana-551	95	2	,	,	PUNCT
cana-551	95	3	𝔚	𝔚	PROPN
cana-551	95	4	=	=	PRON
cana-551	95	5	{	{	PUNCT
cana-551	95	6	�	�	PROPN
cana-551	95	7	̃	̃	PROPN
cana-551	95	8	�	�	PROPN
cana-551	95	9	,	,	PUNCT
cana-551	95	10	∅̃	∅̃	NOUN
cana-551	95	11	,	,	PUNCT
cana-551	95	12	(	(	PUNCT
cana-551	95	13	𝛾1	𝛾1	PROPN
cana-551	95	14	,	,	PUNCT
cana-551	95	15	δ	δ	PROPN
cana-551	95	16	)	)	PUNCT
cana-551	95	17	,	,	PUNCT
cana-551	95	18	(	(	PUNCT
cana-551	95	19	𝛾2	𝛾2	VERB
cana-551	95	20	,	,	PUNCT
cana-551	95	21	δ	δ	PROPN
cana-551	95	22	)	)	PUNCT
cana-551	95	23	,	,	PUNCT
cana-551	95	24	(	(	PUNCT
cana-551	95	25	𝛾3	𝛾3	PROPN
cana-551	95	26	,	,	PUNCT
cana-551	95	27	δ	δ	PROPN
cana-551	95	28	)	)	PUNCT
cana-551	95	29	,	,	PUNCT
cana-551	95	30	(	(	PUNCT
cana-551	95	31	𝛾4	𝛾4	PROPN
cana-551	95	32	,	,	PUNCT
cana-551	95	33	δ	δ	PROPN
cana-551	95	34	)	)	PUNCT
cana-551	95	35	}	}	PUNCT
cana-551	95	36	is	be	AUX
cana-551	95	37	the	the	DET
cana-551	95	38	soft	soft	ADJ
cana-551	95	39	topology	topology	NOUN
cana-551	95	40	over	over	ADP
cana-551	95	41	𝒵.	𝒵.	PROPN
cana-551	95	42	let	let	VERB
cana-551	95	43	ῐ	ῐ	PROPN
cana-551	95	44	=	=	PRON
cana-551	95	45	{	{	PUNCT
cana-551	95	46	∅̃	∅̃	NOUN
cana-551	95	47	,	,	PUNCT
cana-551	95	48	(	(	PUNCT
cana-551	95	49	𝛿1	𝛿1	PROPN
cana-551	95	50	,	,	PUNCT
cana-551	95	51	δ	δ	PROPN
cana-551	95	52	)	)	PUNCT
cana-551	95	53	,	,	PUNCT
cana-551	95	54	(	(	PUNCT
cana-551	95	55	𝛿2	𝛿2	PROPN
cana-551	95	56	,	,	PUNCT
cana-551	95	57	δ	δ	PROPN
cana-551	95	58	)	)	PUNCT
cana-551	95	59	,	,	PUNCT
cana-551	95	60	(	(	PUNCT
cana-551	95	61	𝛿3	𝛿3	PROPN
cana-551	95	62	,	,	PUNCT
cana-551	95	63	δ	δ	PROPN
cana-551	95	64	)	)	PUNCT
cana-551	95	65	}	}	PUNCT
cana-551	95	66	be	be	AUX
cana-551	95	67	a	a	DET
cana-551	95	68	soft	soft	ADJ
cana-551	95	69	ideal	ideal	NOUN
cana-551	95	70	on	on	ADP
cana-551	95	71	𝒵	𝒵	PROPN
cana-551	95	72	,	,	PUNCT
cana-551	95	73	where	where	SCONJ
cana-551	95	74	(	(	PUNCT
cana-551	95	75	𝛿1	𝛿1	NOUN
cana-551	95	76	,	,	PUNCT
cana-551	95	77	δ	δ	PROPN
cana-551	95	78	)	)	PUNCT
cana-551	95	79	=	=	PRON
cana-551	95	80	{	{	PUNCT
cana-551	95	81	(	(	PUNCT
cana-551	95	82	∇1	∇1	PROPN
cana-551	95	83	,	,	PUNCT
cana-551	95	84	{	{	PUNCT
cana-551	95	85	휀	휀	NOUN
cana-551	95	86	}	}	PUNCT
cana-551	95	87	)	)	PUNCT
cana-551	95	88	,	,	PUNCT
cana-551	95	89	(	(	PUNCT
cana-551	95	90	∇2	∇2	X
cana-551	95	91	,	,	PUNCT
cana-551	95	92	∅	∅	NOUN
cana-551	95	93	)	)	PUNCT
cana-551	95	94	}	}	PUNCT
cana-551	95	95	,	,	PUNCT
cana-551	95	96	(	(	PUNCT
cana-551	95	97	𝛿2	𝛿2	PROPN
cana-551	95	98	,	,	PUNCT
cana-551	95	99	δ	δ	PROPN
cana-551	95	100	)	)	PUNCT
cana-551	96	1	=	=	PRON
cana-551	96	2	{	{	PUNCT
cana-551	96	3	(	(	PUNCT
cana-551	96	4	∇1	∇1	PROPN
cana-551	96	5	,	,	PUNCT
cana-551	96	6	{	{	PUNCT
cana-551	96	7	휀	휀	NOUN
cana-551	96	8	}	}	PUNCT
cana-551	96	9	)	)	PUNCT
cana-551	96	10	,	,	PUNCT
cana-551	96	11	(	(	PUNCT
cana-551	96	12	∇2	∇2	PROPN
cana-551	96	13	,	,	PUNCT
cana-551	96	14	{	{	PUNCT
cana-551	96	15	휀	휀	NOUN
cana-551	96	16	}	}	PUNCT
cana-551	96	17	)	)	PUNCT
cana-551	96	18	}	}	PUNCT
cana-551	96	19	and	and	CCONJ
cana-551	96	20	(	(	PUNCT
cana-551	96	21	𝛿3	𝛿3	PROPN
cana-551	96	22	,	,	PUNCT
cana-551	96	23	δ	δ	PROPN
cana-551	96	24	)	)	PUNCT
cana-551	96	25	=	=	PRON
cana-551	96	26	{	{	PUNCT
cana-551	96	27	(	(	PUNCT
cana-551	96	28	∇1	∇1	NOUN
cana-551	96	29	,	,	PUNCT
cana-551	96	30	∅	∅	NOUN
cana-551	96	31	)	)	PUNCT
cana-551	96	32	,	,	PUNCT
cana-551	96	33	(	(	PUNCT
cana-551	96	34	∇2	∇2	PROPN
cana-551	96	35	,	,	PUNCT
cana-551	96	36	{	{	PUNCT
cana-551	96	37	휀	휀	NOUN
cana-551	96	38	}	}	PUNCT
cana-551	96	39	)	)	PUNCT
cana-551	96	40	}	}	PUNCT
cana-551	96	41	.	.	PUNCT
cana-551	97	1	the	the	DET
cana-551	97	2	soft	soft	ADJ
cana-551	97	3	sets	set	NOUN
cana-551	97	4	(	(	PUNCT
cana-551	97	5	𝜗	𝜗	NOUN
cana-551	97	6	,	,	PUNCT
cana-551	97	7	δ	δ	PROPN
cana-551	97	8	)	)	PUNCT
cana-551	97	9	is	be	AUX
cana-551	97	10	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	97	11	−closed	−close	VERB
cana-551	97	12	but	but	CCONJ
cana-551	97	13	not	not	PART
cana-551	97	14	soft	soft	ADJ
cana-551	97	15	g	g	NOUN
cana-551	97	16	−closed	−close	VERB
cana-551	97	17	,	,	PUNCT
cana-551	97	18	where	where	SCONJ
cana-551	97	19	(	(	PUNCT
cana-551	97	20	𝜗	𝜗	NOUN
cana-551	97	21	,	,	PUNCT
cana-551	97	22	δ	δ	PROPN
cana-551	97	23	)	)	PUNCT
cana-551	97	24	=	=	PRON
cana-551	97	25	{	{	PUNCT
cana-551	97	26	(	(	PUNCT
cana-551	97	27	∇1	∇1	NOUN
cana-551	97	28	,	,	PUNCT
cana-551	97	29	∅	∅	NOUN
cana-551	97	30	)	)	PUNCT
cana-551	97	31	,	,	PUNCT
cana-551	97	32	(	(	PUNCT
cana-551	97	33	∇2	∇2	PROPN
cana-551	97	34	,	,	PUNCT
cana-551	97	35	{	{	PUNCT
cana-551	97	36	𝜇	𝜇	X
cana-551	97	37	}	}	PUNCT
cana-551	97	38	)	)	PUNCT
cana-551	97	39	}	}	PUNCT
cana-551	97	40	.	.	PUNCT
cana-551	98	1	communications	communication	NOUN
cana-551	98	2	on	on	ADP
cana-551	98	3	applied	apply	VERB
cana-551	98	4	nonlinear	nonlinear	ADJ
cana-551	98	5	analysis	analysis	NOUN
cana-551	98	6	issn	issn	NOUN
cana-551	98	7	:	:	PUNCT
cana-551	98	8	1074	1074	NUM
cana-551	98	9	-	-	PUNCT
cana-551	98	10	133x	133x	NUM
cana-551	98	11	vol	vol	NOUN
cana-551	98	12	31	31	NUM
cana-551	98	13	no	no	NOUN
cana-551	98	14	.	.	NOUN
cana-551	98	15	2	2	NUM
cana-551	98	16	(	(	PUNCT
cana-551	98	17	2024	2024	NUM
cana-551	98	18	)	)	PUNCT
cana-551	99	1	288	288	NUM
cana-551	99	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	99	3	example	example	NOUN
cana-551	99	4	2.5	2.5	NUM
cana-551	99	5	:	:	PUNCT
cana-551	99	6	in	in	ADP
cana-551	99	7	example	example	NOUN
cana-551	99	8	2.2	2.2	NUM
cana-551	99	9	,	,	PUNCT
cana-551	99	10	the	the	DET
cana-551	99	11	soft	soft	ADJ
cana-551	99	12	set	set	NOUN
cana-551	99	13	(	(	PUNCT
cana-551	99	14	γ	γ	X
cana-551	99	15	,	,	PUNCT
cana-551	99	16	δ	δ	PROPN
cana-551	99	17	)	)	PUNCT
cana-551	99	18	=	=	PRON
cana-551	99	19	{	{	PUNCT
cana-551	99	20	(	(	PUNCT
cana-551	99	21	∇1	∇1	PROPN
cana-551	99	22	,	,	PUNCT
cana-551	99	23	𝒵	𝒵	PROPN
cana-551	99	24	)	)	PUNCT
cana-551	99	25	,	,	PUNCT
cana-551	99	26	(	(	PUNCT
cana-551	99	27	∇2	∇2	PROPN
cana-551	99	28	,	,	PUNCT
cana-551	99	29	{	{	PUNCT
cana-551	99	30	휀	휀	NOUN
cana-551	99	31	}	}	PUNCT
cana-551	99	32	)	)	PUNCT
cana-551	99	33	}	}	PUNCT
cana-551	99	34	is	be	AUX
cana-551	99	35	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	99	36	−closed	−close	VERB
cana-551	99	37	but	but	CCONJ
cana-551	99	38	not	not	PART
cana-551	99	39	closed	close	VERB
cana-551	99	40	soft	soft	ADJ
cana-551	99	41	se	se	X
cana-551	99	42	.	.	PUNCT
cana-551	99	43	example	example	NOUN
cana-551	99	44	2.6	2.6	NUM
cana-551	99	45	:	:	PUNCT
cana-551	99	46	in	in	ADP
cana-551	99	47	example	example	NOUN
cana-551	99	48	2.4	2.4	NUM
cana-551	99	49	,	,	PUNCT
cana-551	99	50	the	the	DET
cana-551	99	51	soft	soft	ADJ
cana-551	99	52	set	set	NOUN
cana-551	99	53	(	(	PUNCT
cana-551	99	54	ζ	ζ	NOUN
cana-551	99	55	,	,	PUNCT
cana-551	99	56	δ	δ	PROPN
cana-551	99	57	)	)	PUNCT
cana-551	99	58	is	be	AUX
cana-551	99	59	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	99	60	−closed	−close	VERB
cana-551	99	61	but	but	CCONJ
cana-551	99	62	not	not	PART
cana-551	99	63	soft	soft	ADJ
cana-551	99	64	ῐg	ῐg	ADP
cana-551	99	65	−closed	−close	VERB
cana-551	99	66	set	set	NOUN
cana-551	99	67	,	,	PUNCT
cana-551	99	68	where	where	SCONJ
cana-551	99	69	(	(	PUNCT
cana-551	99	70	ζ	ζ	X
cana-551	99	71	,	,	PUNCT
cana-551	99	72	δ	δ	PROPN
cana-551	99	73	)	)	PUNCT
cana-551	100	1	=	=	PRON
cana-551	100	2	{	{	PUNCT
cana-551	100	3	(	(	PUNCT
cana-551	100	4	∇1	∇1	NOUN
cana-551	100	5	,	,	PUNCT
cana-551	100	6	∅	∅	NOUN
cana-551	100	7	)	)	PUNCT
cana-551	100	8	,	,	PUNCT
cana-551	100	9	(	(	PUNCT
cana-551	100	10	∇2	∇2	PROPN
cana-551	100	11	,	,	PUNCT
cana-551	100	12	{	{	PUNCT
cana-551	100	13	𝜇	𝜇	X
cana-551	100	14	}	}	PUNCT
cana-551	100	15	)	)	PUNCT
cana-551	100	16	}	}	PUNCT
cana-551	100	17	.	.	PUNCT
cana-551	101	1	remark	remark	VERB
cana-551	101	2	2.7	2.7	NUM
cana-551	101	3	:	:	PUNCT
cana-551	101	4	if	if	SCONJ
cana-551	101	5	a	a	DET
cana-551	101	6	soft	soft	ADJ
cana-551	101	7	subset	subset	NOUN
cana-551	101	8	(	(	PUNCT
cana-551	101	9	𝛾	𝛾	PROPN
cana-551	101	10	,	,	PUNCT
cana-551	101	11	δ	δ	PROPN
cana-551	101	12	)	)	PUNCT
cana-551	101	13	of	of	ADP
cana-551	101	14	a	a	DET
cana-551	101	15	𝒮ts	𝒮ts	PROPN
cana-551	101	16	(	(	PUNCT
cana-551	101	17	𝒵	𝒵	PROPN
cana-551	101	18	,	,	PUNCT
cana-551	101	19	𝔚	𝔚	PROPN
cana-551	101	20	,	,	PUNCT
cana-551	101	21	𝛥	𝛥	PROPN
cana-551	101	22	)	)	PUNCT
cana-551	101	23	is	be	AUX
cana-551	101	24	soft	soft	ADJ
cana-551	101	25	open	open	ADJ
cana-551	101	26	,	,	PUNCT
cana-551	101	27	then	then	ADV
cana-551	101	28	it	it	PRON
cana-551	101	29	is	be	AUX
cana-551	101	30	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	101	31	−closed	−close	VERB
cana-551	101	32	if	if	SCONJ
cana-551	102	1	and	and	CCONJ
cana-551	102	2	only	only	ADV
cana-551	102	3	if	if	SCONJ
cana-551	102	4	it	it	PRON
cana-551	102	5	is	be	AUX
cana-551	102	6	soft	soft	ADJ
cana-551	102	7	ῐg	ῐg	AUX
cana-551	102	8	−closed	−close	VERB
cana-551	102	9	.	.	PUNCT
cana-551	103	1	theorem	theorem	VERB
cana-551	103	2	2.8	2.8	NUM
cana-551	103	3	:	:	PUNCT
cana-551	103	4	a	a	DET
cana-551	103	5	soft	soft	ADJ
cana-551	103	6	set	set	NOUN
cana-551	103	7	(	(	PUNCT
cana-551	103	8	𝜗	𝜗	NOUN
cana-551	103	9	,	,	PUNCT
cana-551	103	10	δ	δ	PROPN
cana-551	103	11	)	)	PUNCT
cana-551	103	12	is	be	AUX
cana-551	103	13	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	103	14	−closed	−close	VERB
cana-551	103	15	in	in	ADP
cana-551	103	16	a	a	DET
cana-551	103	17	𝒮ts	𝒮ts	PROPN
cana-551	103	18	(	(	PUNCT
cana-551	103	19	𝒵	𝒵	PROPN
cana-551	103	20	,	,	PUNCT
cana-551	103	21	𝔚	𝔚	PROPN
cana-551	103	22	,	,	PUNCT
cana-551	103	23	𝛥	𝛥	NOUN
cana-551	103	24	)	)	PUNCT
cana-551	103	25	if	if	SCONJ
cana-551	103	26	and	and	CCONJ
cana-551	103	27	only	only	ADV
cana-551	103	28	if	if	SCONJ
cana-551	103	29	(	(	PUNCT
cana-551	103	30	𝛾	𝛾	NOUN
cana-551	103	31	,	,	PUNCT
cana-551	103	32	δ	δ	PROPN
cana-551	103	33	)	)	PUNCT
cana-551	104	1	⊆	⊆	NUM
cana-551	104	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	104	3	,	,	PUNCT
cana-551	104	4	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	104	5	,	,	PUNCT
cana-551	104	6	δ	δ	PROPN
cana-551	104	7	)	)	PUNCT
cana-551	104	8	and	and	CCONJ
cana-551	104	9	(	(	PUNCT
cana-551	104	10	𝛾	𝛾	PROPN
cana-551	104	11	,	,	PUNCT
cana-551	104	12	δ	δ	PROPN
cana-551	104	13	)	)	PUNCT
cana-551	104	14	is	be	AUX
cana-551	104	15	soft	soft	ADJ
cana-551	104	16	closed	closed	ADJ
cana-551	104	17	implies	implie	NOUN
cana-551	104	18	(	(	PUNCT
cana-551	104	19	𝛾	𝛾	PROPN
cana-551	104	20	,	,	PUNCT
cana-551	104	21	δ	δ	NOUN
cana-551	104	22	)	)	PUNCT
cana-551	104	23	∈	∈	PROPN
cana-551	104	24	ῐ.	ῐ.	NOUN
cana-551	104	25	proof	proof	NOUN
cana-551	104	26	.	.	PUNCT
cana-551	105	1	(	(	PUNCT
cana-551	105	2	⇒	⇒	NOUN
cana-551	105	3	)	)	PUNCT
cana-551	105	4	let	let	VERB
cana-551	105	5	(	(	PUNCT
cana-551	105	6	𝛾	𝛾	PROPN
cana-551	105	7	,	,	PUNCT
cana-551	105	8	δ	δ	PROPN
cana-551	105	9	)	)	PUNCT
cana-551	105	10	⊆	⊆	NUM
cana-551	105	11	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	105	12	,	,	PUNCT
cana-551	105	13	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	105	14	,	,	PUNCT
cana-551	105	15	δ	δ	PROPN
cana-551	105	16	)	)	PUNCT
cana-551	105	17	and	and	CCONJ
cana-551	105	18	(	(	PUNCT
cana-551	105	19	𝛾	𝛾	PROPN
cana-551	105	20	,	,	PUNCT
cana-551	105	21	δ	δ	PROPN
cana-551	105	22	)	)	PUNCT
cana-551	105	23	is	be	AUX
cana-551	105	24	soft	soft	ADJ
cana-551	105	25	closed	closed	ADJ
cana-551	105	26	.	.	PUNCT
cana-551	106	1	then	then	ADV
cana-551	106	2	,	,	PUNCT
cana-551	106	3	(	(	PUNCT
cana-551	106	4	𝜗	𝜗	NOUN
cana-551	106	5	,	,	PUNCT
cana-551	106	6	δ	δ	PROPN
cana-551	106	7	)	)	PUNCT
cana-551	106	8	⊆	⊆	NUM
cana-551	106	9	(	(	PUNCT
cana-551	106	10	𝛾	𝛾	PROPN
cana-551	106	11	,	,	PUNCT
cana-551	106	12	δ)𝑐.	δ)𝑐.	NOUN
cana-551	106	13	by	by	ADP
cana-551	106	14	hypothesis	hypothesis	NOUN
cana-551	106	15	,	,	PUNCT
cana-551	106	16	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	106	17	,	,	PUNCT
cana-551	106	18	δ))\(𝛾	δ))\(𝛾	NOUN
cana-551	106	19	,	,	PUNCT
cana-551	106	20	δ)𝑐	δ)𝑐	NOUN
cana-551	106	21	∈	∈	NOUN
cana-551	106	22	ῐ.	ῐ.	NOUN
cana-551	106	23	but	but	CCONJ
cana-551	106	24	(	(	PUNCT
cana-551	106	25	𝛾	𝛾	PROPN
cana-551	106	26	,	,	PUNCT
cana-551	106	27	δ	δ	PROPN
cana-551	106	28	)	)	PUNCT
cana-551	106	29	⊆	⊆	NUM
cana-551	106	30	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	106	31	,	,	PUNCT
cana-551	106	32	δ	δ	NOUN
cana-551	106	33	)	)	PUNCT
cana-551	106	34	)	)	PUNCT
cana-551	106	35	∩	∩	NOUN
cana-551	106	36	(	(	PUNCT
cana-551	106	37	𝛾	𝛾	PROPN
cana-551	106	38	,	,	PUNCT
cana-551	106	39	δ	δ	NOUN
cana-551	106	40	)	)	PUNCT
cana-551	106	41	=	=	SYM
cana-551	106	42	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	106	43	,	,	PUNCT
cana-551	106	44	δ))\(𝛾	δ))\(𝛾	NOUN
cana-551	106	45	,	,	PUNCT
cana-551	106	46	δ)𝑐.	δ)𝑐.	PROPN
cana-551	106	47	thus	thus	ADV
cana-551	106	48	,	,	PUNCT
cana-551	106	49	(	(	PUNCT
cana-551	106	50	𝛾	𝛾	PROPN
cana-551	106	51	,	,	PUNCT
cana-551	106	52	δ	δ	PROPN
cana-551	106	53	)	)	PUNCT
cana-551	106	54	∈	∈	PROPN
cana-551	106	55	ῐ	ῐ	PROPN
cana-551	106	56	from	from	ADP
cana-551	106	57	definition	definition	NOUN
cana-551	106	58	1.17	1.17	NUM
cana-551	106	59	.	.	PUNCT
cana-551	107	1	(	(	PUNCT
cana-551	107	2	⇐	⇐	ADJ
cana-551	107	3	)	)	PUNCT
cana-551	107	4	assume	assume	VERB
cana-551	107	5	that	that	SCONJ
cana-551	107	6	(	(	PUNCT
cana-551	107	7	𝜗	𝜗	NOUN
cana-551	107	8	,	,	PUNCT
cana-551	107	9	δ	δ	PROPN
cana-551	107	10	)	)	PUNCT
cana-551	107	11	⊆	⊆	NUM
cana-551	107	12	(	(	PUNCT
cana-551	107	13	𝛿	𝛿	ADJ
cana-551	107	14	,	,	PUNCT
cana-551	107	15	δ	δ	PROPN
cana-551	107	16	)	)	PUNCT
cana-551	107	17	and	and	CCONJ
cana-551	107	18	(	(	PUNCT
cana-551	107	19	𝛿	𝛿	ADJ
cana-551	107	20	,	,	PUNCT
cana-551	107	21	𝛥	𝛥	NOUN
cana-551	107	22	)	)	PUNCT
cana-551	107	23	is	be	AUX
cana-551	107	24	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	107	25	−open	−open	ADJ
cana-551	107	26	.	.	PUNCT
cana-551	108	1	then	then	ADV
cana-551	108	2	,	,	PUNCT
cana-551	108	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	108	4	,	,	PUNCT
cana-551	108	5	δ))\(𝛿	δ))\(𝛿	NOUN
cana-551	108	6	,	,	PUNCT
cana-551	108	7	δ	δ	PROPN
cana-551	108	8	)	)	PUNCT
cana-551	108	9	=	=	SYM
cana-551	108	10	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	108	11	,	,	PUNCT
cana-551	108	12	δ	δ	PROPN
cana-551	108	13	)	)	PUNCT
cana-551	108	14	)	)	PUNCT
cana-551	108	15	∩	∩	NOUN
cana-551	108	16	(	(	PUNCT
cana-551	108	17	𝛿	𝛿	ADJ
cana-551	108	18	,	,	PUNCT
cana-551	108	19	δ)𝑐	δ)𝑐	NOUN
cana-551	108	20	is	be	AUX
cana-551	108	21	a	a	DET
cana-551	108	22	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	108	23	−closed	−close	VERB
cana-551	108	24	set	set	NOUN
cana-551	108	25	and	and	CCONJ
cana-551	108	26	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	108	27	,	,	PUNCT
cana-551	108	28	δ))\(𝛿	δ))\(𝛿	NOUN
cana-551	108	29	,	,	PUNCT
cana-551	108	30	δ	δ	PROPN
cana-551	108	31	)	)	PUNCT
cana-551	108	32	⊆	⊆	NUM
cana-551	108	33	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	108	34	,	,	PUNCT
cana-551	108	35	δ))\	δ))\	PROPN
cana-551	108	36	(	(	PUNCT
cana-551	108	37	𝛿	𝛿	PROPN
cana-551	108	38	,	,	PUNCT
cana-551	108	39	δ	δ	PROPN
cana-551	108	40	)	)	PUNCT
cana-551	108	41	.	.	PUNCT
cana-551	109	1	by	by	ADP
cana-551	109	2	assumption	assumption	NOUN
cana-551	109	3	,	,	PUNCT
cana-551	109	4	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	109	5	,	,	PUNCT
cana-551	109	6	δ))\(𝛿	δ))\(𝛿	NOUN
cana-551	109	7	,	,	PUNCT
cana-551	109	8	δ	δ	PROPN
cana-551	109	9	)	)	PUNCT
cana-551	109	10	∈	∈	PROPN
cana-551	109	11	ῐ.	ῐ.	NOUN
cana-551	109	12	so	so	ADV
cana-551	109	13	,	,	PUNCT
cana-551	109	14	(	(	PUNCT
cana-551	109	15	𝜗	𝜗	NOUN
cana-551	109	16	,	,	PUNCT
cana-551	109	17	δ	δ	PROPN
cana-551	109	18	)	)	PUNCT
cana-551	109	19	is	be	AUX
cana-551	109	20	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	109	21	−closed	−close	VERB
cana-551	109	22	.	.	PUNCT
cana-551	110	1	theorem	theorem	VERB
cana-551	110	2	2.9	2.9	NUM
cana-551	110	3	:	:	PUNCT
cana-551	110	4	if	if	SCONJ
cana-551	110	5	(	(	PUNCT
cana-551	110	6	𝛾	𝛾	NOUN
cana-551	110	7	,	,	PUNCT
cana-551	110	8	δ	δ	NOUN
cana-551	110	9	)	)	PUNCT
cana-551	110	10	is	be	AUX
cana-551	110	11	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	110	12	−closed	−close	VERB
cana-551	110	13	in	in	ADP
cana-551	110	14	a	a	DET
cana-551	110	15	𝒮ts	𝒮ts	PROPN
cana-551	110	16	(	(	PUNCT
cana-551	110	17	𝒵	𝒵	PROPN
cana-551	110	18	,	,	PUNCT
cana-551	110	19	𝔚	𝔚	PROPN
cana-551	110	20	,	,	PUNCT
cana-551	110	21	𝛥	𝛥	PROPN
cana-551	110	22	)	)	PUNCT
cana-551	110	23	and	and	CCONJ
cana-551	110	24	(	(	PUNCT
cana-551	110	25	𝛾	𝛾	PROPN
cana-551	110	26	,	,	PUNCT
cana-551	110	27	δ	δ	PROPN
cana-551	110	28	)	)	PUNCT
cana-551	110	29	⊆	⊆	NUM
cana-551	110	30	(	(	PUNCT
cana-551	110	31	𝛿	𝛿	ADJ
cana-551	110	32	,	,	PUNCT
cana-551	110	33	δ	δ	PROPN
cana-551	110	34	)	)	PUNCT
cana-551	111	1	⊆	⊆	NUM
cana-551	111	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	111	3	,	,	PUNCT
cana-551	111	4	δ	δ	PROPN
cana-551	111	5	)	)	PUNCT
cana-551	111	6	)	)	PUNCT
cana-551	111	7	,	,	PUNCT
cana-551	111	8	then	then	ADV
cana-551	111	9	(	(	PUNCT
cana-551	111	10	𝛿	𝛿	ADJ
cana-551	111	11	,	,	PUNCT
cana-551	111	12	δ	δ	PROPN
cana-551	111	13	)	)	PUNCT
cana-551	111	14	is	be	AUX
cana-551	111	15	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	111	16	−closed	−close	VERB
cana-551	111	17	.	.	PUNCT
cana-551	112	1	proof	proof	NOUN
cana-551	112	2	.	.	PUNCT
cana-551	113	1	let	let	VERB
cana-551	113	2	(	(	PUNCT
cana-551	113	3	𝛿	𝛿	ADJ
cana-551	113	4	,	,	PUNCT
cana-551	113	5	δ	δ	PROPN
cana-551	113	6	)	)	PUNCT
cana-551	113	7	⊆	⊆	NUM
cana-551	113	8	(	(	PUNCT
cana-551	113	9	𝜉	𝜉	PROPN
cana-551	113	10	,	,	PUNCT
cana-551	113	11	δ	δ	PROPN
cana-551	113	12	)	)	PUNCT
cana-551	113	13	and	and	CCONJ
cana-551	113	14	(	(	PUNCT
cana-551	113	15	𝜉	𝜉	PROPN
cana-551	113	16	,	,	PUNCT
cana-551	113	17	δ	δ	PROPN
cana-551	113	18	)	)	PUNCT
cana-551	113	19	is	be	AUX
cana-551	113	20	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	113	21	−open	−open	ADJ
cana-551	113	22	.	.	PUNCT
cana-551	114	1	then	then	ADV
cana-551	114	2	,	,	PUNCT
cana-551	114	3	(	(	PUNCT
cana-551	114	4	𝛾	𝛾	PROPN
cana-551	114	5	,	,	PUNCT
cana-551	114	6	δ	δ	PROPN
cana-551	114	7	)	)	PUNCT
cana-551	114	8	⊆	⊆	NUM
cana-551	114	9	(	(	PUNCT
cana-551	114	10	𝜉	𝜉	PROPN
cana-551	114	11	,	,	PUNCT
cana-551	114	12	δ	δ	PROPN
cana-551	114	13	)	)	PUNCT
cana-551	114	14	.	.	PUNCT
cana-551	115	1	since	since	SCONJ
cana-551	115	2	(	(	PUNCT
cana-551	115	3	𝛾	𝛾	PROPN
cana-551	115	4	,	,	PUNCT
cana-551	115	5	δ	δ	NOUN
cana-551	115	6	)	)	PUNCT
cana-551	115	7	is	be	AUX
cana-551	115	8	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	115	9	−closed	−close	VERB
cana-551	115	10	,	,	PUNCT
cana-551	115	11	then	then	ADV
cana-551	115	12	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	115	13	,	,	PUNCT
cana-551	115	14	δ))\(𝜉	δ))\(𝜉	PROPN
cana-551	115	15	,	,	PUNCT
cana-551	115	16	δ	δ	PROPN
cana-551	115	17	)	)	PUNCT
cana-551	115	18	∈	∈	PROPN
cana-551	115	19	ῐ.	ῐ.	NOUN
cana-551	115	20	now	now	ADV
cana-551	115	21	,	,	PUNCT
cana-551	115	22	(	(	PUNCT
cana-551	115	23	𝛿	𝛿	ADJ
cana-551	115	24	,	,	PUNCT
cana-551	115	25	δ	δ	PROPN
cana-551	115	26	)	)	PUNCT
cana-551	116	1	⊆	⊆	NUM
cana-551	116	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	116	3	,	,	PUNCT
cana-551	116	4	δ	δ	PROPN
cana-551	116	5	)	)	PUNCT
cana-551	116	6	)	)	PUNCT
cana-551	116	7	implies	imply	VERB
cana-551	116	8	that	that	SCONJ
cana-551	116	9	𝑐𝑙(𝛿	𝑐𝑙(𝛿	ADP
cana-551	116	10	,	,	PUNCT
cana-551	116	11	δ	δ	PROPN
cana-551	116	12	)	)	PUNCT
cana-551	117	1	⊆	⊆	NUM
cana-551	117	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	117	3	,	,	PUNCT
cana-551	117	4	δ	δ	PROPN
cana-551	117	5	)	)	PUNCT
cana-551	117	6	)	)	PUNCT
cana-551	117	7	.	.	PUNCT
cana-551	118	1	thus	thus	ADV
cana-551	118	2	,	,	PUNCT
cana-551	118	3	𝑐𝑙(𝑖𝑛𝑡(𝛿	𝑐𝑙(𝑖𝑛𝑡(𝛿	NOUN
cana-551	118	4	,	,	PUNCT
cana-551	118	5	δ))\(𝜉	δ))\(𝜉	PROPN
cana-551	118	6	,	,	PUNCT
cana-551	118	7	δ	δ	PROPN
cana-551	118	8	)	)	PUNCT
cana-551	118	9	⊆	⊆	NUM
cana-551	118	10	𝑐𝑙(𝛿	𝑐𝑙(𝛿	NUM
cana-551	118	11	,	,	PUNCT
cana-551	118	12	δ)\(𝜉	δ)\(𝜉	PROPN
cana-551	118	13	,	,	PUNCT
cana-551	118	14	δ	δ	PROPN
cana-551	118	15	)	)	PUNCT
cana-551	118	16	⊆	⊆	NUM
cana-551	118	17	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	118	18	,	,	PUNCT
cana-551	118	19	δ))\	δ))\	PROPN
cana-551	118	20	(	(	PUNCT
cana-551	118	21	𝜉	𝜉	X
cana-551	118	22	,	,	PUNCT
cana-551	118	23	δ	δ	PROPN
cana-551	118	24	)	)	PUNCT
cana-551	118	25	.	.	PUNCT
cana-551	119	1	so	so	ADV
cana-551	119	2	,	,	PUNCT
cana-551	119	3	𝑐𝑙(𝑖𝑛𝑡(𝛿	𝑐𝑙(𝑖𝑛𝑡(𝛿	NOUN
cana-551	119	4	,	,	PUNCT
cana-551	119	5	δ))\(𝜉	δ))\(𝜉	PROPN
cana-551	119	6	,	,	PUNCT
cana-551	119	7	δ	δ	PROPN
cana-551	119	8	)	)	PUNCT
cana-551	119	9	∈	∈	PROPN
cana-551	119	10	ῐ	ῐ	PROPN
cana-551	119	11	from	from	ADP
cana-551	119	12	definition	definition	NOUN
cana-551	119	13	1.17	1.17	NUM
cana-551	119	14	.	.	PUNCT
cana-551	120	1	thus	thus	ADV
cana-551	120	2	,	,	PUNCT
cana-551	120	3	(	(	PUNCT
cana-551	120	4	𝛿	𝛿	ADJ
cana-551	120	5	,	,	PUNCT
cana-551	120	6	δ	δ	PROPN
cana-551	120	7	)	)	PUNCT
cana-551	120	8	is	be	AUX
cana-551	120	9	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	120	10	−closed	−close	VERB
cana-551	120	11	.	.	PUNCT
cana-551	121	1	the	the	DET
cana-551	121	2	intersection	intersection	NOUN
cana-551	121	3	of	of	ADP
cana-551	121	4	two	two	NUM
cana-551	121	5	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	ADJ
cana-551	121	6	−closed	−close	VERB
cana-551	121	7	sets	set	NOUN
cana-551	121	8	need	need	AUX
cana-551	121	9	not	not	PART
cana-551	121	10	be	be	AUX
cana-551	121	11	a	a	DET
cana-551	121	12	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	NOUN
cana-551	121	13	−closed	−close	VERB
cana-551	121	14	as	as	SCONJ
cana-551	121	15	shown	show	VERB
cana-551	121	16	by	by	ADP
cana-551	121	17	the	the	DET
cana-551	121	18	following	follow	VERB
cana-551	121	19	example	example	NOUN
cana-551	121	20	.	.	PUNCT
cana-551	122	1	example	example	NOUN
cana-551	122	2	2.10	2.10	NUM
cana-551	122	3	:	:	PUNCT
cana-551	122	4	in	in	ADP
cana-551	122	5	example	example	NOUN
cana-551	122	6	2.2	2.2	NUM
cana-551	123	1	,	,	PUNCT
cana-551	123	2	the	the	DET
cana-551	123	3	soft	soft	ADJ
cana-551	123	4	sets	set	NOUN
cana-551	123	5	(	(	PUNCT
cana-551	123	6	𝜗	𝜗	NOUN
cana-551	123	7	,	,	PUNCT
cana-551	123	8	δ	δ	PROPN
cana-551	123	9	)	)	PUNCT
cana-551	123	10	,	,	PUNCT
cana-551	123	11	(	(	PUNCT
cana-551	123	12	𝛿	𝛿	ADJ
cana-551	123	13	,	,	PUNCT
cana-551	123	14	δ	δ	NOUN
cana-551	123	15	)	)	PUNCT
cana-551	123	16	are	be	AUX
cana-551	123	17	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	123	18	−closed	−close	VERB
cana-551	123	19	.	.	PUNCT
cana-551	124	1	but	but	CCONJ
cana-551	124	2	(	(	PUNCT
cana-551	124	3	𝜉	𝜉	PROPN
cana-551	124	4	,	,	PUNCT
cana-551	124	5	δ	δ	PROPN
cana-551	124	6	)	)	PUNCT
cana-551	124	7	=	=	SYM
cana-551	124	8	(	(	PUNCT
cana-551	124	9	𝜗	𝜗	PROPN
cana-551	124	10	,	,	PUNCT
cana-551	124	11	δ	δ	NOUN
cana-551	124	12	)	)	PUNCT
cana-551	124	13	∩	∩	NOUN
cana-551	124	14	(	(	PUNCT
cana-551	124	15	𝛿	𝛿	ADJ
cana-551	124	16	,	,	PUNCT
cana-551	124	17	δ	δ	PROPN
cana-551	124	18	)	)	PUNCT
cana-551	124	19	is	be	AUX
cana-551	124	20	not	not	PART
cana-551	124	21	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	124	22	−closed	−close	VERB
cana-551	124	23	,	,	PUNCT
cana-551	124	24	where	where	SCONJ
cana-551	124	25	(	(	PUNCT
cana-551	124	26	𝜉	𝜉	X
cana-551	124	27	,	,	PUNCT
cana-551	124	28	δ	δ	PROPN
cana-551	124	29	)	)	PUNCT
cana-551	124	30	=	=	PRON
cana-551	124	31	{	{	PUNCT
cana-551	124	32	(	(	PUNCT
cana-551	124	33	∇1	∇1	NOUN
cana-551	124	34	,	,	PUNCT
cana-551	124	35	∅	∅	NOUN
cana-551	124	36	)	)	PUNCT
cana-551	124	37	,	,	PUNCT
cana-551	124	38	(	(	PUNCT
cana-551	124	39	∇2	∇2	PROPN
cana-551	124	40	,	,	PUNCT
cana-551	124	41	{	{	PUNCT
cana-551	124	42	휀	휀	NOUN
cana-551	124	43	}	}	PUNCT
cana-551	124	44	)	)	PUNCT
cana-551	124	45	}	}	PUNCT
cana-551	124	46	.	.	PUNCT
cana-551	125	1	theorem	theorem	VERB
cana-551	125	2	2.11	2.11	NUM
cana-551	125	3	:	:	PUNCT
cana-551	125	4	if	if	SCONJ
cana-551	125	5	(	(	PUNCT
cana-551	125	6	𝜗	𝜗	NOUN
cana-551	125	7	,	,	PUNCT
cana-551	125	8	δ	δ	PROPN
cana-551	125	9	)	)	PUNCT
cana-551	125	10	is	be	AUX
cana-551	125	11	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	125	12	−closed	−close	VERB
cana-551	125	13	and	and	CCONJ
cana-551	125	14	(	(	PUNCT
cana-551	125	15	𝛾	𝛾	PROPN
cana-551	125	16	,	,	PUNCT
cana-551	125	17	δ	δ	PROPN
cana-551	125	18	)	)	PUNCT
cana-551	125	19	is	be	AUX
cana-551	125	20	soft	soft	ADJ
cana-551	125	21	closed	closed	ADJ
cana-551	125	22	in	in	ADP
cana-551	125	23	a	a	DET
cana-551	125	24	𝒮ts	𝒮ts	PROPN
cana-551	125	25	(	(	PUNCT
cana-551	125	26	𝒵	𝒵	PROPN
cana-551	125	27	,	,	PUNCT
cana-551	125	28	𝔚	𝔚	PROPN
cana-551	125	29	,	,	PUNCT
cana-551	125	30	𝛥	𝛥	PROPN
cana-551	125	31	)	)	PUNCT
cana-551	125	32	.	.	PUNCT
cana-551	126	1	then	then	ADV
cana-551	126	2	,	,	PUNCT
cana-551	126	3	(	(	PUNCT
cana-551	126	4	𝜗	𝜗	NOUN
cana-551	126	5	,	,	PUNCT
cana-551	126	6	δ	δ	NOUN
cana-551	126	7	)	)	PUNCT
cana-551	126	8	∩	∩	NOUN
cana-551	126	9	(	(	PUNCT
cana-551	126	10	𝛾	𝛾	PROPN
cana-551	126	11	,	,	PUNCT
cana-551	126	12	δ	δ	PROPN
cana-551	126	13	)	)	PUNCT
cana-551	126	14	is	be	AUX
cana-551	126	15	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	126	16	−closed	−close	VERB
cana-551	126	17	.	.	PUNCT
cana-551	127	1	proof	proof	NOUN
cana-551	127	2	.	.	PUNCT
cana-551	128	1	let	let	VERB
cana-551	128	2	(	(	PUNCT
cana-551	128	3	𝜗	𝜗	NOUN
cana-551	128	4	,	,	PUNCT
cana-551	128	5	δ	δ	NOUN
cana-551	128	6	)	)	PUNCT
cana-551	128	7	∩	∩	NOUN
cana-551	128	8	(	(	PUNCT
cana-551	128	9	𝛾	𝛾	PROPN
cana-551	128	10	,	,	PUNCT
cana-551	128	11	δ	δ	PROPN
cana-551	128	12	)	)	PUNCT
cana-551	128	13	⊆	⊆	NUM
cana-551	128	14	(	(	PUNCT
cana-551	128	15	𝛿	𝛿	ADJ
cana-551	128	16	,	,	PUNCT
cana-551	128	17	δ	δ	PROPN
cana-551	128	18	)	)	PUNCT
cana-551	128	19	and	and	CCONJ
cana-551	128	20	(	(	PUNCT
cana-551	128	21	𝛿	𝛿	ADJ
cana-551	128	22	,	,	PUNCT
cana-551	128	23	𝛥	𝛥	NOUN
cana-551	128	24	)	)	PUNCT
cana-551	128	25	is	be	AUX
cana-551	128	26	a	a	DET
cana-551	128	27	ssƅ∗	ssƅ∗	ADJ
cana-551	128	28	−open	−open	NOUN
cana-551	128	29	.	.	PUNCT
cana-551	129	1	then	then	ADV
cana-551	129	2	(	(	PUNCT
cana-551	129	3	𝜗	𝜗	PROPN
cana-551	129	4	,	,	PUNCT
cana-551	129	5	δ	δ	PROPN
cana-551	129	6	)	)	PUNCT
cana-551	129	7	⊆	⊆	NUM
cana-551	129	8	(	(	PUNCT
cana-551	129	9	𝛿	𝛿	ADJ
cana-551	129	10	,	,	PUNCT
cana-551	129	11	δ	δ	NOUN
cana-551	129	12	)	)	PUNCT
cana-551	129	13	∪	∪	NOUN
cana-551	129	14	(	(	PUNCT
cana-551	129	15	𝛾	𝛾	PROPN
cana-551	129	16	,	,	PUNCT
cana-551	129	17	δ)𝑐.	δ)𝑐.	PROPN
cana-551	129	18	since	since	SCONJ
cana-551	129	19	(	(	PUNCT
cana-551	129	20	𝜗	𝜗	PROPN
cana-551	129	21	,	,	PUNCT
cana-551	129	22	δ	δ	PROPN
cana-551	129	23	)	)	PUNCT
cana-551	129	24	is	be	AUX
cana-551	129	25	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	129	26	−closed	−close	VERB
cana-551	129	27	,	,	PUNCT
cana-551	129	28	so	so	ADV
cana-551	129	29	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	129	30	,	,	PUNCT
cana-551	129	31	δ))\((𝛿	δ))\((𝛿	PROPN
cana-551	129	32	,	,	PUNCT
cana-551	129	33	δ	δ	PROPN
cana-551	129	34	)	)	PUNCT
cana-551	129	35	∪	∪	NOUN
cana-551	129	36	(	(	PUNCT
cana-551	129	37	𝛾	𝛾	NOUN
cana-551	129	38	,	,	PUNCT
cana-551	129	39	δ)𝑐	δ)𝑐	NOUN
cana-551	129	40	)	)	PUNCT
cana-551	129	41	∈	∈	PROPN
cana-551	129	42	ῐ.	ῐ.	NOUN
cana-551	129	43	now	now	ADV
cana-551	129	44	,	,	PUNCT
cana-551	129	45	𝑐𝑙	𝑐𝑙	X
cana-551	129	46	(	(	PUNCT
cana-551	129	47	𝑖𝑛𝑡((𝜗	𝑖𝑛𝑡((𝜗	NOUN
cana-551	129	48	,	,	PUNCT
cana-551	129	49	δ	δ	PROPN
cana-551	129	50	)	)	PUNCT
cana-551	129	51	∩	∩	NOUN
cana-551	129	52	(	(	PUNCT
cana-551	129	53	𝛾	𝛾	PROPN
cana-551	129	54	,	,	PUNCT
cana-551	129	55	δ	δ	NOUN
cana-551	129	56	)	)	PUNCT
cana-551	129	57	)	)	PUNCT
cana-551	129	58	)	)	PUNCT
cana-551	130	1	⊆	⊆	NUM
cana-551	130	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	130	3	,	,	PUNCT
cana-551	130	4	δ	δ	NOUN
cana-551	130	5	)	)	PUNCT
cana-551	130	6	)	)	PUNCT
cana-551	130	7	∩	∩	PROPN
cana-551	130	8	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	130	9	,	,	PUNCT
cana-551	130	10	δ	δ	PROPN
cana-551	130	11	)	)	PUNCT
cana-551	130	12	)	)	PUNCT
cana-551	131	1	⊆	⊆	NUM
cana-551	131	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	131	3	,	,	PUNCT
cana-551	131	4	δ	δ	NOUN
cana-551	131	5	)	)	PUNCT
cana-551	131	6	)	)	PUNCT
cana-551	131	7	∩	∩	NOUN
cana-551	131	8	𝑐𝑙(𝛾	𝑐𝑙(𝛾	NUM
cana-551	131	9	,	,	PUNCT
cana-551	131	10	δ	δ	PROPN
cana-551	131	11	)	)	PUNCT
cana-551	131	12	=	=	SYM
cana-551	131	13	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	131	14	,	,	PUNCT
cana-551	131	15	δ	δ	PROPN
cana-551	131	16	)	)	PUNCT
cana-551	131	17	)	)	PUNCT
cana-551	131	18	∩	∩	NOUN
cana-551	131	19	(	(	PUNCT
cana-551	131	20	𝛾	𝛾	PROPN
cana-551	131	21	,	,	PUNCT
cana-551	131	22	δ	δ	X
cana-551	131	23	)	)	PUNCT
cana-551	131	24	=	=	PUNCT
cana-551	132	1	[	[	X
cana-551	132	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	132	3	,	,	PUNCT
cana-551	132	4	δ	δ	NOUN
cana-551	132	5	)	)	PUNCT
cana-551	132	6	)	)	PUNCT
cana-551	133	1	∩	∩	NOUN
cana-551	133	2	(	(	PUNCT
cana-551	133	3	𝛾	𝛾	PROPN
cana-551	133	4	,	,	PUNCT
cana-551	133	5	δ)]\(𝛾	δ)]\(𝛾	PROPN
cana-551	133	6	,	,	PUNCT
cana-551	133	7	δ)𝑐.	δ)𝑐.	PROPN
cana-551	133	8	thus	thus	ADV
cana-551	133	9	,	,	PUNCT
cana-551	133	10	𝑐𝑙	𝑐𝑙	X
cana-551	133	11	(	(	PUNCT
cana-551	133	12	𝑖𝑛𝑡((𝜗	𝑖𝑛𝑡((𝜗	NOUN
cana-551	133	13	,	,	PUNCT
cana-551	133	14	δ	δ	PROPN
cana-551	133	15	)	)	PUNCT
cana-551	133	16	∩	∩	NOUN
cana-551	133	17	(	(	PUNCT
cana-551	133	18	𝛾	𝛾	PROPN
cana-551	133	19	,	,	PUNCT
cana-551	133	20	δ	δ	NOUN
cana-551	133	21	)	)	PUNCT
cana-551	133	22	)	)	PUNCT
cana-551	133	23	)	)	PUNCT
cana-551	133	24	\	\	PUNCT
cana-551	134	1	(	(	PUNCT
cana-551	134	2	𝛿	𝛿	ADJ
cana-551	134	3	,	,	PUNCT
cana-551	134	4	δ	δ	PROPN
cana-551	134	5	)	)	PUNCT
cana-551	135	1	⊆	⊆	NUM
cana-551	135	2	[	[	X
cana-551	135	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	135	4	,	,	PUNCT
cana-551	135	5	δ	δ	NOUN
cana-551	135	6	)	)	PUNCT
cana-551	135	7	)	)	PUNCT
cana-551	135	8	∩	∩	NOUN
cana-551	135	9	(	(	PUNCT
cana-551	135	10	𝛾	𝛾	PROPN
cana-551	135	11	,	,	PUNCT
cana-551	135	12	δ)]\((𝛿	δ)]\((𝛿	PROPN
cana-551	135	13	,	,	PUNCT
cana-551	135	14	δ	δ	PROPN
cana-551	135	15	)	)	PUNCT
cana-551	135	16	∪	∪	NOUN
cana-551	135	17	(	(	PUNCT
cana-551	135	18	𝛾	𝛾	NOUN
cana-551	135	19	,	,	PUNCT
cana-551	135	20	δ)𝑐	δ)𝑐	NOUN
cana-551	135	21	)	)	PUNCT
cana-551	135	22	communications	communication	NOUN
cana-551	135	23	on	on	ADP
cana-551	135	24	applied	apply	VERB
cana-551	135	25	nonlinear	nonlinear	ADJ
cana-551	135	26	analysis	analysis	NOUN
cana-551	135	27	issn	issn	NOUN
cana-551	135	28	:	:	PUNCT
cana-551	135	29	1074	1074	NUM
cana-551	135	30	-	-	PUNCT
cana-551	135	31	133x	133x	NUM
cana-551	135	32	vol	vol	NOUN
cana-551	135	33	31	31	NUM
cana-551	135	34	no	no	NOUN
cana-551	135	35	.	.	NOUN
cana-551	135	36	2	2	NUM
cana-551	135	37	(	(	PUNCT
cana-551	135	38	2024	2024	NUM
cana-551	135	39	)	)	PUNCT
cana-551	135	40	289	289	NUM
cana-551	135	41	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	135	42	⊆	⊆	NUM
cana-551	135	43	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	135	44	,	,	PUNCT
cana-551	135	45	δ))\((𝛿	δ))\((𝛿	PROPN
cana-551	135	46	,	,	PUNCT
cana-551	135	47	δ	δ	PROPN
cana-551	135	48	)	)	PUNCT
cana-551	135	49	∪	∪	NOUN
cana-551	135	50	(	(	PUNCT
cana-551	135	51	𝛾	𝛾	NOUN
cana-551	135	52	,	,	PUNCT
cana-551	135	53	δ)𝑐	δ)𝑐	NOUN
cana-551	135	54	)	)	PUNCT
cana-551	135	55	∈	∈	PROPN
cana-551	135	56	ῐ	ῐ	PROPN
cana-551	136	1	so	so	ADV
cana-551	136	2	,	,	PUNCT
cana-551	136	3	(	(	PUNCT
cana-551	136	4	𝜗	𝜗	NOUN
cana-551	136	5	,	,	PUNCT
cana-551	136	6	δ	δ	NOUN
cana-551	136	7	)	)	PUNCT
cana-551	136	8	∩	∩	NOUN
cana-551	136	9	(	(	PUNCT
cana-551	136	10	𝛾	𝛾	PROPN
cana-551	136	11	,	,	PUNCT
cana-551	136	12	δ	δ	PROPN
cana-551	136	13	)	)	PUNCT
cana-551	136	14	is	be	AUX
cana-551	136	15	ss	ss	PROPN
cana-551	136	16	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	136	17	−closed	−close	VERB
cana-551	136	18	.	.	PUNCT
cana-551	137	1	theorem	theorem	VERB
cana-551	137	2	2.12	2.12	NUM
cana-551	137	3	:	:	PUNCT
cana-551	137	4	let	let	VERB
cana-551	137	5	(	(	PUNCT
cana-551	137	6	𝒵	𝒵	PROPN
cana-551	137	7	,	,	PUNCT
cana-551	137	8	𝔚	𝔚	PROPN
cana-551	137	9	,	,	PUNCT
cana-551	137	10	𝛥	𝛥	NOUN
cana-551	137	11	)	)	PUNCT
cana-551	137	12	is	be	AUX
cana-551	137	13	𝒮ts	𝒮ts	PROPN
cana-551	137	14	and	and	CCONJ
cana-551	137	15	(	(	PUNCT
cana-551	137	16	𝛾	𝛾	PROPN
cana-551	137	17	,	,	PUNCT
cana-551	137	18	δ	δ	PROPN
cana-551	137	19	)	)	PUNCT
cana-551	137	20	,	,	PUNCT
cana-551	137	21	(	(	PUNCT
cana-551	137	22	𝛿	𝛿	ADJ
cana-551	137	23	,	,	PUNCT
cana-551	137	24	δ	δ	NOUN
cana-551	137	25	)	)	PUNCT
cana-551	137	26	are	be	AUX
cana-551	137	27	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	137	28	−closed	−close	VERB
cana-551	137	29	.	.	PUNCT
cana-551	138	1	then	then	ADV
cana-551	138	2	(	(	PUNCT
cana-551	138	3	𝛾	𝛾	PROPN
cana-551	138	4	,	,	PUNCT
cana-551	138	5	δ	δ	NOUN
cana-551	138	6	)	)	PUNCT
cana-551	138	7	∪	∪	NOUN
cana-551	138	8	(	(	PUNCT
cana-551	138	9	𝛿	𝛿	ADJ
cana-551	138	10	,	,	PUNCT
cana-551	138	11	δ	δ	NOUN
cana-551	138	12	)	)	PUNCT
cana-551	138	13	are	be	AUX
cana-551	138	14	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	138	15	−closed	−closed	ADJ
cana-551	138	16	.	.	PUNCT
cana-551	139	1	proof	proof	NOUN
cana-551	139	2	.	.	PUNCT
cana-551	140	1	let	let	VERB
cana-551	140	2	(	(	PUNCT
cana-551	140	3	𝛾	𝛾	PROPN
cana-551	140	4	,	,	PUNCT
cana-551	140	5	δ	δ	PROPN
cana-551	140	6	)	)	PUNCT
cana-551	140	7	and	and	CCONJ
cana-551	140	8	(	(	PUNCT
cana-551	140	9	𝛿	𝛿	ADJ
cana-551	140	10	,	,	PUNCT
cana-551	140	11	δ	δ	NOUN
cana-551	140	12	)	)	PUNCT
cana-551	140	13	are	be	AUX
cana-551	140	14	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	140	15	−closed	−close	VERB
cana-551	140	16	.	.	PUNCT
cana-551	141	1	suppose	suppose	VERB
cana-551	141	2	that	that	SCONJ
cana-551	141	3	(	(	PUNCT
cana-551	141	4	𝛾	𝛾	PROPN
cana-551	141	5	,	,	PUNCT
cana-551	141	6	δ	δ	NOUN
cana-551	141	7	)	)	PUNCT
cana-551	141	8	∪	∪	NOUN
cana-551	141	9	(	(	PUNCT
cana-551	141	10	𝛿	𝛿	ADJ
cana-551	141	11	,	,	PUNCT
cana-551	141	12	δ	δ	PROPN
cana-551	141	13	)	)	PUNCT
cana-551	141	14	⊆	⊆	NUM
cana-551	141	15	(	(	PUNCT
cana-551	141	16	𝜗	𝜗	PROPN
cana-551	141	17	,	,	PUNCT
cana-551	141	18	δ	δ	PROPN
cana-551	141	19	)	)	PUNCT
cana-551	141	20	and	and	CCONJ
cana-551	141	21	(	(	PUNCT
cana-551	141	22	𝜗	𝜗	PROPN
cana-551	141	23	,	,	PUNCT
cana-551	141	24	δ	δ	PROPN
cana-551	141	25	)	)	PUNCT
cana-551	141	26	is	be	AUX
cana-551	141	27	ssƅ∗	ssƅ∗	ADJ
cana-551	141	28	−open	−open	ADJ
cana-551	141	29	.	.	PUNCT
cana-551	142	1	then	then	ADV
cana-551	142	2	(	(	PUNCT
cana-551	142	3	𝛾	𝛾	PROPN
cana-551	142	4	,	,	PUNCT
cana-551	142	5	δ	δ	PROPN
cana-551	142	6	)	)	PUNCT
cana-551	142	7	⊆	⊆	NUM
cana-551	142	8	(	(	PUNCT
cana-551	142	9	𝜗	𝜗	PROPN
cana-551	142	10	,	,	PUNCT
cana-551	142	11	δ	δ	PROPN
cana-551	142	12	)	)	PUNCT
cana-551	142	13	and	and	CCONJ
cana-551	142	14	(	(	PUNCT
cana-551	142	15	𝛿	𝛿	ADJ
cana-551	142	16	,	,	PUNCT
cana-551	142	17	δ	δ	PROPN
cana-551	142	18	)	)	PUNCT
cana-551	142	19	⊆	⊆	NUM
cana-551	142	20	(	(	PUNCT
cana-551	142	21	𝜗	𝜗	PROPN
cana-551	142	22	,	,	PUNCT
cana-551	142	23	δ	δ	PROPN
cana-551	142	24	)	)	PUNCT
cana-551	142	25	.	.	PUNCT
cana-551	143	1	since	since	SCONJ
cana-551	143	2	(	(	PUNCT
cana-551	143	3	𝛾	𝛾	PROPN
cana-551	143	4	,	,	PUNCT
cana-551	143	5	δ	δ	PROPN
cana-551	143	6	)	)	PUNCT
cana-551	143	7	and	and	CCONJ
cana-551	143	8	(	(	PUNCT
cana-551	143	9	𝛿	𝛿	ADJ
cana-551	143	10	,	,	PUNCT
cana-551	143	11	δ	δ	NOUN
cana-551	143	12	)	)	PUNCT
cana-551	143	13	are	be	AUX
cana-551	143	14	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	143	15	−closed	−close	VERB
cana-551	143	16	sets	set	NOUN
cana-551	143	17	,	,	PUNCT
cana-551	143	18	then	then	ADV
cana-551	143	19	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	143	20	,	,	PUNCT
cana-551	143	21	δ))\(𝜗	δ))\(𝜗	PROPN
cana-551	143	22	,	,	PUNCT
cana-551	143	23	δ	δ	PROPN
cana-551	143	24	)	)	PUNCT
cana-551	143	25	∈	∈	PROPN
cana-551	143	26	ῐ	ῐ	PROPN
cana-551	143	27	and	and	CCONJ
cana-551	143	28	𝑐𝑙(𝑖𝑛𝑡(𝛿	𝑐𝑙(𝑖𝑛𝑡(𝛿	NOUN
cana-551	143	29	,	,	PUNCT
cana-551	143	30	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	143	31	,	,	PUNCT
cana-551	143	32	δ	δ	PROPN
cana-551	143	33	)	)	PUNCT
cana-551	143	34	∈	∈	PROPN
cana-551	143	35	ῐ.	ῐ.	NOUN
cana-551	143	36	therefore	therefore	ADV
cana-551	143	37	,	,	PUNCT
cana-551	143	38	𝑐𝑙	𝑐𝑙	PRON
cana-551	143	39	(	(	PUNCT
cana-551	143	40	𝑖𝑛𝑡((𝛾	𝑖𝑛𝑡((𝛾	PROPN
cana-551	143	41	,	,	PUNCT
cana-551	143	42	δ	δ	PROPN
cana-551	143	43	)	)	PUNCT
cana-551	143	44	∪	∪	NOUN
cana-551	143	45	(	(	PUNCT
cana-551	143	46	𝛿	𝛿	ADJ
cana-551	143	47	,	,	PUNCT
cana-551	143	48	δ	δ	NOUN
cana-551	143	49	)	)	PUNCT
cana-551	143	50	)	)	PUNCT
cana-551	143	51	)	)	PUNCT
cana-551	143	52	\	\	PUNCT
cana-551	144	1	(	(	PUNCT
cana-551	144	2	𝜗	𝜗	NOUN
cana-551	144	3	,	,	PUNCT
cana-551	144	4	δ	δ	PROPN
cana-551	144	5	)	)	PUNCT
cana-551	144	6	=	=	PUNCT
cana-551	145	1	[	[	X
cana-551	145	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	145	3	,	,	PUNCT
cana-551	145	4	δ))\(𝜗	δ))\(𝜗	PROPN
cana-551	145	5	,	,	PUNCT
cana-551	145	6	δ	δ	PROPN
cana-551	145	7	)	)	PUNCT
cana-551	145	8	]	]	PUNCT
cana-551	145	9	∪	∪	ADP
cana-551	145	10	[	[	X
cana-551	145	11	𝑐𝑙(𝑖𝑛𝑡(𝛿	𝑐𝑙(𝑖𝑛𝑡(𝛿	NOUN
cana-551	145	12	,	,	PUNCT
cana-551	145	13	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	145	14	,	,	PUNCT
cana-551	145	15	δ	δ	PROPN
cana-551	145	16	)	)	PUNCT
cana-551	145	17	]	]	PUNCT
cana-551	146	1	∈	∈	PROPN
cana-551	146	2	ῐ.	ῐ.	PROPN
cana-551	146	3	hence	hence	ADV
cana-551	146	4	,	,	PUNCT
cana-551	146	5	we	we	PRON
cana-551	146	6	obtain	obtain	VERB
cana-551	146	7	that	that	DET
cana-551	146	8	(	(	PUNCT
cana-551	146	9	𝛾	𝛾	PROPN
cana-551	146	10	,	,	PUNCT
cana-551	146	11	δ	δ	NOUN
cana-551	146	12	)	)	PUNCT
cana-551	146	13	∪	∪	NOUN
cana-551	146	14	(	(	PUNCT
cana-551	146	15	𝛿	𝛿	ADJ
cana-551	146	16	,	,	PUNCT
cana-551	146	17	δ	δ	NOUN
cana-551	146	18	)	)	PUNCT
cana-551	146	19	are	be	AUX
cana-551	146	20	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	146	21	−closed	−close	VERB
cana-551	146	22	.	.	PUNCT
cana-551	147	1	theorem	theorem	VERB
cana-551	147	2	2.13	2.13	NUM
cana-551	147	3	:	:	PUNCT
cana-551	147	4	let	let	VERB
cana-551	147	5	(	(	PUNCT
cana-551	147	6	𝒟	𝒟	NOUN
cana-551	147	7	,	,	PUNCT
cana-551	147	8	℧	℧	PROPN
cana-551	147	9	,	,	PUNCT
cana-551	147	10	𝛥	𝛥	PROPN
cana-551	147	11	)	)	PUNCT
cana-551	147	12	be	be	VERB
cana-551	147	13	a	a	DET
cana-551	147	14	soft	soft	ADJ
cana-551	147	15	subspace	subspace	NOUN
cana-551	147	16	of	of	ADP
cana-551	147	17	a	a	DET
cana-551	147	18	𝒮ts	𝒮ts	PROPN
cana-551	147	19	(	(	PUNCT
cana-551	147	20	𝒵	𝒵	PROPN
cana-551	147	21	,	,	PUNCT
cana-551	147	22	𝔚	𝔚	PROPN
cana-551	147	23	,	,	PUNCT
cana-551	147	24	𝛥	𝛥	PROPN
cana-551	147	25	)	)	PUNCT
cana-551	147	26	,	,	PUNCT
cana-551	147	27	and	and	CCONJ
cana-551	147	28	(	(	PUNCT
cana-551	147	29	ℜ	ℜ	PROPN
cana-551	147	30	,	,	PUNCT
cana-551	147	31	δ	δ	PROPN
cana-551	147	32	)	)	PUNCT
cana-551	147	33	is	be	AUX
cana-551	147	34	a	a	DET
cana-551	147	35	soft	soft	ADJ
cana-551	147	36	subset	subset	NOUN
cana-551	147	37	of	of	ADP
cana-551	147	38	(	(	PUNCT
cana-551	147	39	𝒟	𝒟	PROPN
cana-551	147	40	,	,	PUNCT
cana-551	147	41	℧	℧	PROPN
cana-551	147	42	,	,	PUNCT
cana-551	147	43	𝛥	𝛥	PROPN
cana-551	147	44	)	)	PUNCT
cana-551	147	45	and	and	CCONJ
cana-551	147	46	(	(	PUNCT
cana-551	147	47	𝛾	𝛾	PROPN
cana-551	147	48	,	,	PUNCT
cana-551	147	49	δ	δ	PROPN
cana-551	147	50	)	)	PUNCT
cana-551	147	51	⊆	⊆	NUM
cana-551	147	52	(	(	PUNCT
cana-551	147	53	ℜ	ℜ	PROPN
cana-551	147	54	,	,	PUNCT
cana-551	147	55	δ	δ	PROPN
cana-551	147	56	)	)	PUNCT
cana-551	147	57	and	and	CCONJ
cana-551	147	58	(	(	PUNCT
cana-551	147	59	𝛾	𝛾	PROPN
cana-551	147	60	,	,	PUNCT
cana-551	147	61	δ	δ	PROPN
cana-551	147	62	)	)	PUNCT
cana-551	147	63	is	be	AUX
cana-551	147	64	a	a	DET
cana-551	147	65	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	147	66	−closed	−close	VERB
cana-551	147	67	in	in	ADP
cana-551	147	68	(	(	PUNCT
cana-551	147	69	𝒵	𝒵	PROPN
cana-551	147	70	,	,	PUNCT
cana-551	147	71	𝔚	𝔚	PROPN
cana-551	147	72	,	,	PUNCT
cana-551	147	73	𝛥	𝛥	PROPN
cana-551	147	74	)	)	PUNCT
cana-551	147	75	.	.	PUNCT
cana-551	148	1	then	then	ADV
cana-551	148	2	,	,	PUNCT
cana-551	148	3	(	(	PUNCT
cana-551	148	4	𝛾	𝛾	PROPN
cana-551	148	5	,	,	PUNCT
cana-551	148	6	δ	δ	PROPN
cana-551	148	7	)	)	PUNCT
cana-551	148	8	is	be	AUX
cana-551	148	9	a	a	DET
cana-551	148	10	ssƅ∗ῐ𝒟	ssƅ∗ῐ𝒟	NOUN
cana-551	148	11	−closed	−close	VERB
cana-551	148	12	in	in	ADP
cana-551	148	13	(	(	PUNCT
cana-551	148	14	𝒟	𝒟	PROPN
cana-551	148	15	,	,	PUNCT
cana-551	148	16	℧	℧	PROPN
cana-551	148	17	,	,	PUNCT
cana-551	148	18	𝛥	𝛥	PROPN
cana-551	148	19	)	)	PUNCT
cana-551	148	20	.	.	PUNCT
cana-551	149	1	proof	proof	NOUN
cana-551	149	2	.	.	PUNCT
cana-551	150	1	assume	assume	VERB
cana-551	150	2	that	that	SCONJ
cana-551	150	3	(	(	PUNCT
cana-551	150	4	𝛾	𝛾	PROPN
cana-551	150	5	,	,	PUNCT
cana-551	150	6	δ	δ	PROPN
cana-551	150	7	)	)	PUNCT
cana-551	150	8	⊆	⊆	NUM
cana-551	150	9	(	(	PUNCT
cana-551	150	10	ℒ	ℒ	PROPN
cana-551	150	11	,	,	PUNCT
cana-551	150	12	δ	δ	NOUN
cana-551	150	13	)	)	PUNCT
cana-551	150	14	∩	∩	NOUN
cana-551	150	15	(	(	PUNCT
cana-551	150	16	ℜ	ℜ	PROPN
cana-551	150	17	,	,	PUNCT
cana-551	150	18	δ	δ	PROPN
cana-551	150	19	)	)	PUNCT
cana-551	150	20	and	and	CCONJ
cana-551	150	21	(	(	PUNCT
cana-551	150	22	ℒ	ℒ	PROPN
cana-551	150	23	,	,	PUNCT
cana-551	150	24	δ	δ	PROPN
cana-551	150	25	)	)	PUNCT
cana-551	150	26	∈	∈	PROPN
cana-551	150	27	𝔚.	𝔚.	PROPN
cana-551	150	28	then	then	ADV
cana-551	150	29	(	(	PUNCT
cana-551	150	30	ℒ	ℒ	PROPN
cana-551	150	31	,	,	PUNCT
cana-551	150	32	δ	δ	PROPN
cana-551	150	33	)	)	PUNCT
cana-551	150	34	∩	∩	NOUN
cana-551	150	35	(	(	PUNCT
cana-551	150	36	ℜ	ℜ	PROPN
cana-551	150	37	,	,	PUNCT
cana-551	150	38	δ	δ	PROPN
cana-551	150	39	)	)	PUNCT
cana-551	150	40	∈	∈	PROPN
cana-551	150	41	℧	℧	PROPN
cana-551	150	42	and	and	CCONJ
cana-551	150	43	(	(	PUNCT
cana-551	150	44	𝛾	𝛾	PROPN
cana-551	150	45	,	,	PUNCT
cana-551	150	46	δ	δ	PROPN
cana-551	150	47	)	)	PUNCT
cana-551	150	48	⊆	⊆	NUM
cana-551	150	49	(	(	PUNCT
cana-551	150	50	ℒ	ℒ	PROPN
cana-551	150	51	,	,	PUNCT
cana-551	150	52	δ	δ	PROPN
cana-551	150	53	)	)	PUNCT
cana-551	150	54	.	.	PUNCT
cana-551	151	1	since	since	SCONJ
cana-551	151	2	(	(	PUNCT
cana-551	151	3	𝛾	𝛾	PROPN
cana-551	151	4	,	,	PUNCT
cana-551	151	5	δ	δ	PROPN
cana-551	151	6	)	)	PUNCT
cana-551	151	7	is	be	AUX
cana-551	151	8	a	a	DET
cana-551	151	9	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	151	10	−closed	−close	VERB
cana-551	151	11	in	in	ADP
cana-551	151	12	(	(	PUNCT
cana-551	151	13	𝒵	𝒵	PROPN
cana-551	151	14	,	,	PUNCT
cana-551	151	15	𝔚	𝔚	PROPN
cana-551	151	16	,	,	PUNCT
cana-551	151	17	𝛥	𝛥	PROPN
cana-551	151	18	)	)	PUNCT
cana-551	151	19	,	,	PUNCT
cana-551	151	20	then	then	ADV
cana-551	151	21	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	151	22	,	,	PUNCT
cana-551	151	23	δ))\	δ))\	PROPN
cana-551	151	24	(	(	PUNCT
cana-551	151	25	ℒ	ℒ	PROPN
cana-551	151	26	,	,	PUNCT
cana-551	151	27	δ	δ	PROPN
cana-551	151	28	)	)	PUNCT
cana-551	151	29	∈	∈	PROPN
cana-551	151	30	ῐ.	ῐ.	NOUN
cana-551	151	31	now	now	ADV
cana-551	151	32	,	,	PUNCT
cana-551	151	33	[	[	X
cana-551	151	34	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	ADJ
cana-551	151	35	,	,	PUNCT
cana-551	151	36	δ	δ	PROPN
cana-551	151	37	)	)	PUNCT
cana-551	151	38	)	)	PUNCT
cana-551	151	39	∩	∩	NOUN
cana-551	151	40	(	(	PUNCT
cana-551	151	41	ℜ	ℜ	PROPN
cana-551	151	42	,	,	PUNCT
cana-551	151	43	δ	δ	PROPN
cana-551	151	44	)	)	PUNCT
cana-551	151	45	]	]	PUNCT
cana-551	151	46	\	\	PUNCT
cana-551	152	1	[	[	X
cana-551	152	2	(	(	PUNCT
cana-551	152	3	ℒ	ℒ	PROPN
cana-551	152	4	,	,	PUNCT
cana-551	152	5	δ	δ	NOUN
cana-551	152	6	)	)	PUNCT
cana-551	152	7	∩	∩	NOUN
cana-551	152	8	(	(	PUNCT
cana-551	152	9	ℜ	ℜ	PROPN
cana-551	152	10	,	,	PUNCT
cana-551	152	11	δ	δ	PROPN
cana-551	152	12	)	)	PUNCT
cana-551	152	13	]	]	PUNCT
cana-551	153	1	=	=	PUNCT
cana-551	154	1	[	[	X
cana-551	154	2	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	X
cana-551	154	3	,	,	PUNCT
cana-551	154	4	δ))\(ℒ	δ))\(ℒ	PROPN
cana-551	154	5	,	,	PUNCT
cana-551	154	6	δ	δ	PROPN
cana-551	154	7	)	)	PUNCT
cana-551	154	8	]	]	PUNCT
cana-551	154	9	∩	∩	NOUN
cana-551	154	10	(	(	PUNCT
cana-551	154	11	ℜ	ℜ	PROPN
cana-551	154	12	,	,	PUNCT
cana-551	154	13	δ	δ	PROPN
cana-551	154	14	)	)	PUNCT
cana-551	154	15	∈	∈	PROPN
cana-551	154	16	ῐ𝒟	ῐ𝒟	PROPN
cana-551	155	1	thus	thus	ADV
cana-551	155	2	,	,	PUNCT
cana-551	155	3	(	(	PUNCT
cana-551	155	4	𝛾	𝛾	PROPN
cana-551	155	5	,	,	PUNCT
cana-551	155	6	δ	δ	PROPN
cana-551	155	7	)	)	PUNCT
cana-551	155	8	is	be	AUX
cana-551	155	9	a	a	DET
cana-551	155	10	ssƅ∗ῐ𝒟	ssƅ∗ῐ𝒟	NOUN
cana-551	155	11	−closed	−close	VERB
cana-551	155	12	in	in	ADP
cana-551	155	13	(	(	PUNCT
cana-551	155	14	𝒟	𝒟	PROPN
cana-551	155	15	,	,	PUNCT
cana-551	155	16	℧	℧	PROPN
cana-551	155	17	,	,	PUNCT
cana-551	155	18	𝛥	𝛥	PROPN
cana-551	155	19	)	)	PUNCT
cana-551	155	20	.	.	PUNCT
cana-551	156	1	3	3	X
cana-551	156	2	.	.	X
cana-551	156	3	soft	soft	ADJ
cana-551	156	4	strongly	strongly	ADV
cana-551	156	5	ƅ∗	ƅ∗	NOUN
cana-551	156	6	−open	−open	ADJ
cana-551	156	7	via	via	ADP
cana-551	156	8	soft	soft	ADJ
cana-551	156	9	ideal	ideal	NOUN
cana-551	156	10	in	in	ADP
cana-551	156	11	this	this	DET
cana-551	156	12	section	section	NOUN
cana-551	156	13	,	,	PUNCT
cana-551	156	14	we	we	PRON
cana-551	156	15	define	define	VERB
cana-551	156	16	ssƅ∗	ssƅ∗	ADJ
cana-551	156	17	−open	−open	PROPN
cana-551	156	18	set	set	VERB
cana-551	156	19	via	via	ADP
cana-551	156	20	soft	soft	ADJ
cana-551	156	21	ideal	ideal	NOUN
cana-551	156	22	in	in	ADP
cana-551	156	23	𝒮tss	𝒮tss	PROPN
cana-551	156	24	.	.	PUNCT
cana-551	157	1	definition	definition	NOUN
cana-551	157	2	3.1	3.1	NUM
cana-551	157	3	:	:	PUNCT
cana-551	157	4	a	a	DET
cana-551	157	5	soft	soft	ADJ
cana-551	157	6	set	set	NOUN
cana-551	157	7	(	(	PUNCT
cana-551	157	8	𝛾	𝛾	PROPN
cana-551	157	9	,	,	PUNCT
cana-551	157	10	δ	δ	NOUN
cana-551	157	11	)	)	PUNCT
cana-551	157	12	in	in	ADP
cana-551	157	13	𝒮ts	𝒮ts	PROPN
cana-551	157	14	(	(	PUNCT
cana-551	157	15	𝒵	𝒵	PROPN
cana-551	157	16	,	,	PUNCT
cana-551	157	17	𝔚	𝔚	PROPN
cana-551	157	18	,	,	PUNCT
cana-551	157	19	𝛥	𝛥	PROPN
cana-551	157	20	)	)	PUNCT
cana-551	157	21	,	,	PUNCT
cana-551	157	22	is	be	AUX
cana-551	157	23	called	call	VERB
cana-551	157	24	a	a	DET
cana-551	157	25	soft	soft	ADJ
cana-551	157	26	stongly	stongly	ADV
cana-551	157	27	ƅ∗ῐ	ƅ∗ῐ	PUNCT
cana-551	157	28	−open	−open	ADV
cana-551	158	1	set	set	VERB
cana-551	158	2	with	with	ADP
cana-551	158	3	respect	respect	NOUN
cana-551	158	4	to	to	ADP
cana-551	158	5	soft	soft	ADJ
cana-551	158	6	ideal	ideal	ADJ
cana-551	158	7	ῐ	ῐ	PROPN
cana-551	158	8	(	(	PUNCT
cana-551	158	9	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	158	10	−open	−open	NOUN
cana-551	158	11	)	)	PUNCT
cana-551	159	1	if	if	SCONJ
cana-551	159	2	and	and	CCONJ
cana-551	159	3	only	only	ADV
cana-551	159	4	if	if	SCONJ
cana-551	159	5	its	its	PRON
cana-551	159	6	relative	relative	ADJ
cana-551	159	7	complement	complement	NOUN
cana-551	159	8	(	(	PUNCT
cana-551	159	9	𝛾	𝛾	NOUN
cana-551	159	10	,	,	PUNCT
cana-551	159	11	δ)𝑐	δ)𝑐	NOUN
cana-551	159	12	is	be	AUX
cana-551	159	13	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	159	14	−closed	−close	VERB
cana-551	159	15	in	in	ADP
cana-551	159	16	(	(	PUNCT
cana-551	159	17	𝒵	𝒵	PROPN
cana-551	159	18	,	,	PUNCT
cana-551	159	19	𝔚	𝔚	PROPN
cana-551	159	20	,	,	PUNCT
cana-551	159	21	𝛥	𝛥	PROPN
cana-551	159	22	)	)	PUNCT
cana-551	159	23	.	.	PUNCT
cana-551	160	1	example	example	NOUN
cana-551	160	2	3.2	3.2	NUM
cana-551	160	3	:	:	PUNCT
cana-551	160	4	in	in	ADP
cana-551	160	5	example	example	NOUN
cana-551	160	6	2.2	2.2	NUM
cana-551	160	7	.	.	PUNCT
cana-551	161	1	the	the	DET
cana-551	161	2	soft	soft	ADJ
cana-551	161	3	sets	set	NOUN
cana-551	161	4	(	(	PUNCT
cana-551	161	5	𝛾1	𝛾1	NOUN
cana-551	161	6	,	,	PUNCT
cana-551	161	7	δ)𝑐	δ)𝑐	NOUN
cana-551	161	8	,	,	PUNCT
cana-551	161	9	(	(	PUNCT
cana-551	161	10	𝛾2	𝛾2	NOUN
cana-551	161	11	,	,	PUNCT
cana-551	161	12	δ)𝑐	δ)𝑐	NOUN
cana-551	161	13	and	and	CCONJ
cana-551	161	14	(	(	PUNCT
cana-551	161	15	𝛾3	𝛾3	NOUN
cana-551	161	16	,	,	PUNCT
cana-551	161	17	δ)𝑐	δ)𝑐	NOUN
cana-551	161	18	are	be	AUX
cana-551	161	19	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	161	20	−open	−open	INTJ
cana-551	161	21	where	where	SCONJ
cana-551	161	22	(	(	PUNCT
cana-551	161	23	𝛾1	𝛾1	NOUN
cana-551	161	24	,	,	PUNCT
cana-551	161	25	δ)𝑐	δ)𝑐	NOUN
cana-551	161	26	,	,	PUNCT
cana-551	161	27	(	(	PUNCT
cana-551	161	28	𝛾2	𝛾2	NOUN
cana-551	161	29	,	,	PUNCT
cana-551	161	30	δ)𝑐	δ)𝑐	NOUN
cana-551	161	31	and	and	CCONJ
cana-551	161	32	(	(	PUNCT
cana-551	161	33	𝛾3	𝛾3	NOUN
cana-551	161	34	,	,	PUNCT
cana-551	161	35	δ)𝑐	δ)𝑐	NOUN
cana-551	161	36	are	be	AUX
cana-551	161	37	given	give	VERB
cana-551	161	38	by	by	ADP
cana-551	161	39	(	(	PUNCT
cana-551	161	40	𝛾1	𝛾1	PROPN
cana-551	161	41	,	,	PUNCT
cana-551	161	42	δ)𝑐	δ)𝑐	NOUN
cana-551	161	43	=	=	SYM
cana-551	161	44	{	{	PUNCT
cana-551	161	45	(	(	PUNCT
cana-551	161	46	∇1	∇1	PROPN
cana-551	161	47	,	,	PUNCT
cana-551	161	48	𝒵	𝒵	PROPN
cana-551	161	49	)	)	PUNCT
cana-551	161	50	,	,	PUNCT
cana-551	161	51	(	(	PUNCT
cana-551	161	52	∇2	∇2	X
cana-551	161	53	,	,	PUNCT
cana-551	161	54	∅	∅	NOUN
cana-551	161	55	)	)	PUNCT
cana-551	161	56	}	}	PUNCT
cana-551	161	57	,	,	PUNCT
cana-551	161	58	(	(	PUNCT
cana-551	161	59	𝛾1	𝛾1	PROPN
cana-551	161	60	,	,	PUNCT
cana-551	161	61	δ)𝑐	δ)𝑐	NOUN
cana-551	161	62	=	=	SYM
cana-551	161	63	{	{	PUNCT
cana-551	161	64	(	(	PUNCT
cana-551	161	65	∇1	∇1	NOUN
cana-551	161	66	,	,	PUNCT
cana-551	161	67	∅	∅	NOUN
cana-551	161	68	)	)	PUNCT
cana-551	161	69	,	,	PUNCT
cana-551	161	70	(	(	PUNCT
cana-551	161	71	∇2	∇2	PROPN
cana-551	161	72	,	,	PUNCT
cana-551	161	73	{	{	PUNCT
cana-551	161	74	𝜇	𝜇	X
cana-551	161	75	}	}	PUNCT
cana-551	161	76	)	)	PUNCT
cana-551	161	77	}	}	PUNCT
cana-551	161	78	and	and	CCONJ
cana-551	161	79	(	(	PUNCT
cana-551	161	80	𝛾3	𝛾3	NOUN
cana-551	161	81	,	,	PUNCT
cana-551	161	82	δ)𝑐	δ)𝑐	NOUN
cana-551	161	83	=	=	SYM
cana-551	161	84	{	{	PUNCT
cana-551	161	85	(	(	PUNCT
cana-551	161	86	∇1	∇1	PROPN
cana-551	161	87	,	,	PUNCT
cana-551	161	88	{	{	PUNCT
cana-551	161	89	휀	휀	NOUN
cana-551	161	90	}	}	PUNCT
cana-551	161	91	)	)	PUNCT
cana-551	161	92	,	,	PUNCT
cana-551	161	93	(	(	PUNCT
cana-551	161	94	∇2	∇2	PROPN
cana-551	161	95	,	,	PUNCT
cana-551	161	96	{	{	PUNCT
cana-551	161	97	𝜇	𝜇	X
cana-551	161	98	}	}	PUNCT
cana-551	161	99	)	)	PUNCT
cana-551	161	100	}	}	PUNCT
cana-551	161	101	.	.	PUNCT
cana-551	162	1	theorem	theorem	VERB
cana-551	162	2	3.3	3.3	NUM
cana-551	162	3	:	:	PUNCT
cana-551	162	4	a	a	DET
cana-551	162	5	soft	soft	ADJ
cana-551	162	6	set	set	NOUN
cana-551	162	7	(	(	PUNCT
cana-551	162	8	𝜗	𝜗	NOUN
cana-551	162	9	,	,	PUNCT
cana-551	162	10	δ	δ	PROPN
cana-551	162	11	)	)	PUNCT
cana-551	162	12	is	be	AUX
cana-551	162	13	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	162	14	−open	−open	ADJ
cana-551	162	15	in	in	ADP
cana-551	162	16	a	a	DET
cana-551	162	17	𝒮ts	𝒮ts	PROPN
cana-551	162	18	(	(	PUNCT
cana-551	162	19	𝒵	𝒵	PROPN
cana-551	162	20	,	,	PUNCT
cana-551	162	21	𝔚	𝔚	PROPN
cana-551	162	22	,	,	PUNCT
cana-551	162	23	𝛥	𝛥	NOUN
cana-551	162	24	)	)	PUNCT
cana-551	162	25	if	if	SCONJ
cana-551	163	1	and	and	CCONJ
cana-551	163	2	only	only	ADV
cana-551	163	3	if	if	SCONJ
cana-551	163	4	(	(	PUNCT
cana-551	163	5	𝛾	𝛾	PROPN
cana-551	163	6	,	,	PUNCT
cana-551	163	7	δ	δ	NOUN
cana-551	163	8	)	)	PUNCT
cana-551	163	9	\	\	PUNCT
cana-551	163	10	(	(	PUNCT
cana-551	163	11	ℒ	ℒ	PROPN
cana-551	163	12	,	,	PUNCT
cana-551	163	13	δ	δ	PROPN
cana-551	163	14	)	)	PUNCT
cana-551	163	15	⊆	⊆	NUM
cana-551	163	16	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	163	17	,	,	PUNCT
cana-551	163	18	δ	δ	PROPN
cana-551	163	19	)	)	PUNCT
cana-551	163	20	)	)	PUNCT
cana-551	163	21	for	for	ADP
cana-551	163	22	some	some	DET
cana-551	163	23	(	(	PUNCT
cana-551	163	24	ℒ	ℒ	PROPN
cana-551	163	25	,	,	PUNCT
cana-551	163	26	δ	δ	PROPN
cana-551	163	27	)	)	PUNCT
cana-551	163	28	∈	∈	PROPN
cana-551	163	29	ῐ	ῐ	PROPN
cana-551	163	30	,	,	PUNCT
cana-551	163	31	whenever	whenever	SCONJ
cana-551	163	32	(	(	PUNCT
cana-551	163	33	𝛾	𝛾	PROPN
cana-551	163	34	,	,	PUNCT
cana-551	163	35	δ	δ	PROPN
cana-551	163	36	)	)	PUNCT
cana-551	163	37	⊆	⊆	NUM
cana-551	163	38	(	(	PUNCT
cana-551	163	39	𝜗	𝜗	PROPN
cana-551	163	40	,	,	PUNCT
cana-551	163	41	δ	δ	PROPN
cana-551	163	42	)	)	PUNCT
cana-551	163	43	and	and	CCONJ
cana-551	163	44	(	(	PUNCT
cana-551	163	45	𝛾	𝛾	PROPN
cana-551	163	46	,	,	PUNCT
cana-551	163	47	δ	δ	PROPN
cana-551	163	48	)	)	PUNCT
cana-551	163	49	is	be	AUX
cana-551	163	50	soft	soft	ADJ
cana-551	163	51	closed	closed	ADJ
cana-551	163	52	in	in	ADP
cana-551	163	53	(	(	PUNCT
cana-551	163	54	𝒵	𝒵	PROPN
cana-551	163	55	,	,	PUNCT
cana-551	163	56	𝔚	𝔚	PROPN
cana-551	163	57	,	,	PUNCT
cana-551	163	58	𝛥	𝛥	PROPN
cana-551	163	59	)	)	PUNCT
cana-551	163	60	.	.	PUNCT
cana-551	164	1	proof	proof	NOUN
cana-551	164	2	.	.	PUNCT
cana-551	165	1	(	(	PUNCT
cana-551	165	2	⇒	⇒	NOUN
cana-551	165	3	)	)	PUNCT
cana-551	165	4	let	let	VERB
cana-551	165	5	(	(	PUNCT
cana-551	165	6	𝛾	𝛾	PROPN
cana-551	165	7	,	,	PUNCT
cana-551	165	8	δ	δ	PROPN
cana-551	165	9	)	)	PUNCT
cana-551	165	10	⊆	⊆	NUM
cana-551	165	11	(	(	PUNCT
cana-551	165	12	𝜗	𝜗	PROPN
cana-551	165	13	,	,	PUNCT
cana-551	165	14	δ	δ	PROPN
cana-551	165	15	)	)	PUNCT
cana-551	165	16	and	and	CCONJ
cana-551	165	17	(	(	PUNCT
cana-551	165	18	𝛾	𝛾	PROPN
cana-551	165	19	,	,	PUNCT
cana-551	165	20	δ	δ	PROPN
cana-551	165	21	)	)	PUNCT
cana-551	165	22	is	be	AUX
cana-551	165	23	soft	soft	ADJ
cana-551	165	24	closed	closed	ADJ
cana-551	165	25	.	.	PUNCT
cana-551	166	1	then	then	ADV
cana-551	166	2	(	(	PUNCT
cana-551	166	3	𝜗	𝜗	NOUN
cana-551	166	4	,	,	PUNCT
cana-551	166	5	δ)𝑐	δ)𝑐	NOUN
cana-551	166	6	⊆	⊆	NUM
cana-551	166	7	(	(	PUNCT
cana-551	166	8	𝛾	𝛾	NOUN
cana-551	166	9	,	,	PUNCT
cana-551	166	10	δ)𝑐	δ)𝑐	NOUN
cana-551	166	11	,	,	PUNCT
cana-551	166	12	(	(	PUNCT
cana-551	166	13	𝜗	𝜗	NOUN
cana-551	166	14	,	,	PUNCT
cana-551	166	15	δ)𝑐	δ)𝑐	NOUN
cana-551	166	16	is	be	AUX
cana-551	166	17	a	a	DET
cana-551	166	18	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	166	19	−closed	−close	VERB
cana-551	166	20	and	and	CCONJ
cana-551	166	21	(	(	PUNCT
cana-551	166	22	𝛾	𝛾	PROPN
cana-551	166	23	,	,	PUNCT
cana-551	166	24	δ)𝑐	δ)𝑐	NOUN
cana-551	166	25	∈	∈	PROPN
cana-551	166	26	𝔚.	𝔚.	NOUN
cana-551	166	27	by	by	ADP
cana-551	166	28	assumption	assumption	NOUN
cana-551	166	29	,	,	PUNCT
cana-551	166	30	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	166	31	,	,	PUNCT
cana-551	166	32	δ)𝑐	δ)𝑐	NOUN
cana-551	166	33	)	)	PUNCT
cana-551	166	34	\	\	PUNCT
cana-551	167	1	(	(	PUNCT
cana-551	167	2	𝛾	𝛾	NOUN
cana-551	167	3	,	,	PUNCT
cana-551	167	4	δ)𝑐	δ)𝑐	NOUN
cana-551	167	5	∈	∈	NOUN
cana-551	167	6	ῐ.	ῐ.	NOUN
cana-551	167	7	then	then	ADV
cana-551	167	8	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	167	9	,	,	PUNCT
cana-551	167	10	δ)𝑐	δ)𝑐	NOUN
cana-551	167	11	)	)	PUNCT
cana-551	167	12	\	\	PUNCT
cana-551	168	1	(	(	PUNCT
cana-551	168	2	𝛾	𝛾	NOUN
cana-551	168	3	,	,	PUNCT
cana-551	168	4	δ)𝑐	δ)𝑐	NOUN
cana-551	168	5	=	=	SYM
cana-551	168	6	(	(	PUNCT
cana-551	168	7	𝜂	𝜂	PROPN
cana-551	168	8	,	,	PUNCT
cana-551	168	9	δ	δ	NOUN
cana-551	168	10	)	)	PUNCT
cana-551	168	11	for	for	ADP
cana-551	168	12	some	some	DET
cana-551	168	13	(	(	PUNCT
cana-551	168	14	𝜂	𝜂	PROPN
cana-551	168	15	,	,	PUNCT
cana-551	168	16	δ	δ	NOUN
cana-551	168	17	)	)	PUNCT
cana-551	168	18	∈	∈	PROPN
cana-551	168	19	ῐ.	ῐ.	NOUN
cana-551	168	20	thus	thus	ADV
cana-551	168	21	,	,	PUNCT
cana-551	168	22	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	168	23	,	,	PUNCT
cana-551	168	24	δ)𝑐	δ)𝑐	NOUN
cana-551	168	25	)	)	PUNCT
cana-551	168	26	\	\	PUNCT
cana-551	169	1	(	(	PUNCT
cana-551	169	2	𝛾	𝛾	NOUN
cana-551	169	3	,	,	PUNCT
cana-551	169	4	δ)𝑐	δ)𝑐	NOUN
cana-551	169	5	=	=	SYM
cana-551	169	6	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	169	7	,	,	PUNCT
cana-551	169	8	δ)𝑐	δ)𝑐	NOUN
cana-551	169	9	)	)	PUNCT
cana-551	169	10	∩	∩	NOUN
cana-551	169	11	(	(	PUNCT
cana-551	169	12	𝛾	𝛾	PROPN
cana-551	169	13	,	,	PUNCT
cana-551	169	14	δ	δ	NOUN
cana-551	169	15	)	)	PUNCT
cana-551	169	16	=	=	PUNCT
cana-551	170	1	(	(	PUNCT
cana-551	170	2	𝜂	𝜂	PROPN
cana-551	170	3	,	,	PUNCT
cana-551	170	4	δ	δ	NOUN
cana-551	170	5	)	)	PUNCT
cana-551	170	6	∈	∈	PROPN
cana-551	170	7	ῐ.	ῐ.	NOUN
cana-551	170	8	so	so	ADV
cana-551	170	9	,	,	PUNCT
cana-551	170	10	communications	communication	NOUN
cana-551	170	11	on	on	ADP
cana-551	170	12	applied	apply	VERB
cana-551	170	13	nonlinear	nonlinear	ADJ
cana-551	170	14	analysis	analysis	NOUN
cana-551	170	15	issn	issn	NOUN
cana-551	170	16	:	:	PUNCT
cana-551	170	17	1074	1074	NUM
cana-551	170	18	-	-	PUNCT
cana-551	170	19	133x	133x	NUM
cana-551	170	20	vol	vol	NOUN
cana-551	170	21	31	31	NUM
cana-551	170	22	no	no	NOUN
cana-551	170	23	.	.	NOUN
cana-551	170	24	2	2	NUM
cana-551	170	25	(	(	PUNCT
cana-551	170	26	2024	2024	NUM
cana-551	170	27	)	)	PUNCT
cana-551	170	28	290	290	NUM
cana-551	170	29	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	170	30	e	e	X
cana-551	171	1	[	[	X
cana-551	171	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	ADJ
cana-551	171	3	,	,	PUNCT
cana-551	171	4	δ)𝑐	δ)𝑐	NOUN
cana-551	171	5	)	)	PUNCT
cana-551	171	6	∩	∩	NOUN
cana-551	171	7	(	(	PUNCT
cana-551	171	8	𝛾	𝛾	PROPN
cana-551	171	9	,	,	PUNCT
cana-551	171	10	δ	δ	PROPN
cana-551	171	11	)	)	PUNCT
cana-551	171	12	]	]	PUNCT
cana-551	171	13	∪	∪	X
cana-551	171	14	(	(	PUNCT
cana-551	171	15	𝛾	𝛾	NOUN
cana-551	171	16	,	,	PUNCT
cana-551	171	17	δ)𝑐	δ)𝑐	NOUN
cana-551	171	18	=	=	SYM
cana-551	171	19	(	(	PUNCT
cana-551	171	20	𝜂	𝜂	PROPN
cana-551	171	21	,	,	PUNCT
cana-551	171	22	δ	δ	NOUN
cana-551	171	23	)	)	PUNCT
cana-551	171	24	∪	∪	NOUN
cana-551	171	25	(	(	PUNCT
cana-551	171	26	𝛾	𝛾	PROPN
cana-551	171	27	,	,	PUNCT
cana-551	171	28	δ)𝑐.	δ)𝑐.	PROPN
cana-551	171	29	this	this	PRON
cana-551	171	30	implies	imply	VERB
cana-551	171	31	that	that	SCONJ
cana-551	171	32	,	,	PUNCT
cana-551	171	33	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	171	34	,	,	PUNCT
cana-551	171	35	δ)𝑐	δ)𝑐	NOUN
cana-551	171	36	)	)	PUNCT
cana-551	172	1	⊆	⊆	NUM
cana-551	172	2	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	172	3	,	,	PUNCT
cana-551	172	4	δ)𝑐	δ)𝑐	NOUN
cana-551	172	5	)	)	PUNCT
cana-551	172	6	∪	∪	NOUN
cana-551	172	7	(	(	PUNCT
cana-551	172	8	𝛾	𝛾	NOUN
cana-551	172	9	,	,	PUNCT
cana-551	172	10	δ)𝑐	δ)𝑐	NOUN
cana-551	172	11	=	=	SYM
cana-551	172	12	(	(	PUNCT
cana-551	172	13	ℒ	ℒ	PROPN
cana-551	172	14	,	,	PUNCT
cana-551	172	15	δ	δ	PROPN
cana-551	172	16	)	)	PUNCT
cana-551	172	17	∪	∪	NOUN
cana-551	172	18	(	(	PUNCT
cana-551	172	19	𝛾	𝛾	PROPN
cana-551	172	20	,	,	PUNCT
cana-551	172	21	δ)𝑐.	δ)𝑐.	PROPN
cana-551	172	22	hence	hence	ADV
cana-551	172	23	,	,	PUNCT
cana-551	172	24	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	172	25	,	,	PUNCT
cana-551	172	26	δ)𝑐	δ)𝑐	NOUN
cana-551	172	27	)	)	PUNCT
cana-551	172	28	⊆	⊆	NUM
cana-551	172	29	(	(	PUNCT
cana-551	172	30	𝜂	𝜂	NUM
cana-551	172	31	,	,	PUNCT
cana-551	172	32	δ	δ	NOUN
cana-551	172	33	)	)	PUNCT
cana-551	172	34	∪	∪	NOUN
cana-551	172	35	(	(	PUNCT
cana-551	172	36	𝛾	𝛾	NOUN
cana-551	172	37	,	,	PUNCT
cana-551	172	38	δ)𝑐	δ)𝑐	NOUN
cana-551	172	39	for	for	ADP
cana-551	172	40	some	some	DET
cana-551	172	41	(	(	PUNCT
cana-551	172	42	𝜂	𝜂	PROPN
cana-551	172	43	,	,	PUNCT
cana-551	172	44	δ	δ	NOUN
cana-551	172	45	)	)	PUNCT
cana-551	172	46	∈	∈	PROPN
cana-551	172	47	ῐ.	ῐ.	NOUN
cana-551	172	48	furthermore	furthermore	ADV
cana-551	172	49	,	,	PUNCT
cana-551	172	50	(	(	PUNCT
cana-551	172	51	𝜂	𝜂	NOUN
cana-551	172	52	,	,	PUNCT
cana-551	172	53	δ	δ	NOUN
cana-551	172	54	)	)	PUNCT
cana-551	172	55	∪	∪	NOUN
cana-551	172	56	(	(	PUNCT
cana-551	172	57	𝛾	𝛾	NOUN
cana-551	172	58	,	,	PUNCT
cana-551	172	59	δ)𝑐	δ)𝑐	NOUN
cana-551	172	60	⊆	⊆	NUM
cana-551	172	61	[	[	X
cana-551	172	62	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	ADJ
cana-551	172	63	,	,	PUNCT
cana-551	172	64	δ)𝑐)]𝑐	δ)𝑐)]𝑐	NOUN
cana-551	172	65	=	=	SYM
cana-551	172	66	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	172	67	,	,	PUNCT
cana-551	172	68	δ	δ	PROPN
cana-551	172	69	)	)	PUNCT
cana-551	172	70	)	)	PUNCT
cana-551	172	71	.	.	PUNCT
cana-551	173	1	therefore	therefore	ADV
cana-551	173	2	,	,	PUNCT
cana-551	173	3	(	(	PUNCT
cana-551	173	4	𝛾	𝛾	NOUN
cana-551	173	5	,	,	PUNCT
cana-551	173	6	δ)\(𝜂	δ)\(𝜂	NOUN
cana-551	173	7	,	,	PUNCT
cana-551	173	8	δ	δ	PROPN
cana-551	173	9	)	)	PUNCT
cana-551	173	10	=	=	PUNCT
cana-551	173	11	(	(	PUNCT
cana-551	173	12	𝛾	𝛾	PROPN
cana-551	173	13	,	,	PUNCT
cana-551	173	14	δ	δ	NOUN
cana-551	173	15	)	)	PUNCT
cana-551	173	16	∩	∩	NOUN
cana-551	173	17	(	(	PUNCT
cana-551	173	18	𝜂	𝜂	NOUN
cana-551	173	19	,	,	PUNCT
cana-551	173	20	δ)𝑐	δ)𝑐	NOUN
cana-551	173	21	⊆	⊆	NUM
cana-551	173	22	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	173	23	,	,	PUNCT
cana-551	173	24	δ	δ	PROPN
cana-551	173	25	)	)	PUNCT
cana-551	173	26	)	)	PUNCT
cana-551	173	27	.	.	PUNCT
cana-551	174	1	(	(	PUNCT
cana-551	174	2	⇐	⇐	ADJ
cana-551	174	3	)	)	PUNCT
cana-551	174	4	let	let	VERB
cana-551	174	5	(	(	PUNCT
cana-551	174	6	𝛾	𝛾	NOUN
cana-551	174	7	,	,	PUNCT
cana-551	174	8	δ)𝑐	δ)𝑐	NOUN
cana-551	174	9	⊆	⊆	NUM
cana-551	174	10	(	(	PUNCT
cana-551	174	11	𝛿	𝛿	ADJ
cana-551	174	12	,	,	PUNCT
cana-551	174	13	δ	δ	NOUN
cana-551	174	14	)	)	PUNCT
cana-551	174	15	such	such	ADJ
cana-551	174	16	that	that	SCONJ
cana-551	174	17	(	(	PUNCT
cana-551	174	18	𝛿	𝛿	ADJ
cana-551	174	19	,	,	PUNCT
cana-551	174	20	δ	δ	PROPN
cana-551	174	21	)	)	PUNCT
cana-551	174	22	is	be	AUX
cana-551	174	23	ssƅ∗	ssƅ∗	ADJ
cana-551	174	24	−open	−open	ADJ
cana-551	174	25	.	.	PUNCT
cana-551	175	1	then	then	ADV
cana-551	175	2	,	,	PUNCT
cana-551	175	3	(	(	PUNCT
cana-551	175	4	𝛿	𝛿	ADJ
cana-551	175	5	,	,	PUNCT
cana-551	175	6	δ)𝑐	δ)𝑐	NOUN
cana-551	175	7	⊆	⊆	NUM
cana-551	175	8	(	(	PUNCT
cana-551	175	9	𝜗	𝜗	PROPN
cana-551	175	10	,	,	PUNCT
cana-551	175	11	δ	δ	PROPN
cana-551	175	12	)	)	PUNCT
cana-551	175	13	.	.	PUNCT
cana-551	176	1	by	by	ADP
cana-551	176	2	assumption	assumption	NOUN
cana-551	176	3	,	,	PUNCT
cana-551	176	4	(	(	PUNCT
cana-551	176	5	𝛿	𝛿	ADJ
cana-551	176	6	,	,	PUNCT
cana-551	176	7	δ)𝑐\(γ	δ)𝑐\(γ	PROPN
cana-551	176	8	,	,	PUNCT
cana-551	176	9	δ	δ	PROPN
cana-551	176	10	)	)	PUNCT
cana-551	176	11	⊆	⊆	NUM
cana-551	176	12	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	176	13	,	,	PUNCT
cana-551	176	14	δ	δ	NOUN
cana-551	176	15	)	)	PUNCT
cana-551	176	16	)	)	PUNCT
cana-551	177	1	=	=	PUNCT
cana-551	178	1	[	[	X
cana-551	178	2	𝑐𝑙(𝑐𝑙(𝜗	𝑐𝑙(𝑐𝑙(𝜗	NOUN
cana-551	178	3	,	,	PUNCT
cana-551	178	4	δ)𝑐)]𝑐	δ)𝑐)]𝑐	NOUN
cana-551	178	5	for	for	ADP
cana-551	178	6	some	some	PRON
cana-551	178	7	(	(	PUNCT
cana-551	178	8	γ	γ	PROPN
cana-551	178	9	,	,	PUNCT
cana-551	178	10	δ	δ	PROPN
cana-551	178	11	)	)	PUNCT
cana-551	178	12	∈	∈	PROPN
cana-551	178	13	ῐ.	ῐ.	NOUN
cana-551	178	14	thus	thus	ADV
cana-551	178	15	,	,	PUNCT
cana-551	178	16	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	178	17	,	,	PUNCT
cana-551	178	18	δ)𝑐	δ)𝑐	NOUN
cana-551	178	19	)	)	PUNCT
cana-551	178	20	=	=	SYM
cana-551	178	21	𝑐𝑙(𝑐𝑙(𝜗	𝑐𝑙(𝑐𝑙(𝜗	NOUN
cana-551	178	22	,	,	PUNCT
cana-551	178	23	δ)𝑐	δ)𝑐	NOUN
cana-551	178	24	)	)	PUNCT
cana-551	178	25	⊆	⊆	NUM
cana-551	179	1	[	[	X
cana-551	179	2	(	(	PUNCT
cana-551	179	3	𝛿	𝛿	ADJ
cana-551	179	4	,	,	PUNCT
cana-551	179	5	δ)𝑐\(γ	δ)𝑐\(γ	NOUN
cana-551	179	6	,	,	PUNCT
cana-551	179	7	δ)]𝑐	δ)]𝑐	NOUN
cana-551	179	8	=	=	SYM
cana-551	179	9	(	(	PUNCT
cana-551	179	10	𝛿	𝛿	PROPN
cana-551	179	11	,	,	PUNCT
cana-551	179	12	δ	δ	NOUN
cana-551	179	13	)	)	PUNCT
cana-551	179	14	∪	∪	NOUN
cana-551	179	15	(	(	PUNCT
cana-551	179	16	γ	γ	X
cana-551	179	17	,	,	PUNCT
cana-551	179	18	δ	δ	PROPN
cana-551	179	19	)	)	PUNCT
cana-551	179	20	.	.	PUNCT
cana-551	180	1	so	so	ADV
cana-551	180	2	,	,	PUNCT
cana-551	180	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	180	4	,	,	PUNCT
cana-551	180	5	δ)𝑐)\(𝛿	δ)𝑐)\(𝛿	PROPN
cana-551	180	6	,	,	PUNCT
cana-551	180	7	δ	δ	PROPN
cana-551	180	8	)	)	PUNCT
cana-551	181	1	⊆	⊆	NUM
cana-551	181	2	[	[	X
cana-551	181	3	(	(	PUNCT
cana-551	181	4	𝛿	𝛿	ADJ
cana-551	181	5	,	,	PUNCT
cana-551	181	6	δ	δ	NOUN
cana-551	181	7	)	)	PUNCT
cana-551	181	8	∪	∪	NOUN
cana-551	181	9	(	(	PUNCT
cana-551	181	10	γ	γ	X
cana-551	181	11	,	,	PUNCT
cana-551	181	12	δ	δ	PROPN
cana-551	181	13	)	)	PUNCT
cana-551	181	14	]	]	PUNCT
cana-551	181	15	∩	∩	NOUN
cana-551	181	16	(	(	PUNCT
cana-551	181	17	𝛿	𝛿	ADJ
cana-551	181	18	,	,	PUNCT
cana-551	181	19	δ)𝑐	δ)𝑐	NOUN
cana-551	181	20	=	=	SYM
cana-551	181	21	(	(	PUNCT
cana-551	181	22	γ	γ	X
cana-551	181	23	,	,	PUNCT
cana-551	181	24	δ	δ	NOUN
cana-551	181	25	)	)	PUNCT
cana-551	181	26	∩	∩	NOUN
cana-551	181	27	(	(	PUNCT
cana-551	181	28	𝛿	𝛿	ADJ
cana-551	181	29	,	,	PUNCT
cana-551	181	30	δ)𝑐	δ)𝑐	NOUN
cana-551	181	31	⊆	⊆	NUM
cana-551	181	32	(	(	PUNCT
cana-551	181	33	γ	γ	X
cana-551	181	34	,	,	PUNCT
cana-551	181	35	δ	δ	PROPN
cana-551	181	36	)	)	PUNCT
cana-551	181	37	∈	∈	PROPN
cana-551	181	38	ῐ	ῐ	PROPN
cana-551	181	39	this	this	PRON
cana-551	181	40	shows	show	VERB
cana-551	181	41	that	that	SCONJ
cana-551	181	42	,	,	PUNCT
cana-551	181	43	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	181	44	,	,	PUNCT
cana-551	181	45	δ)𝑐)\(𝛿	δ)𝑐)\(𝛿	PROPN
cana-551	181	46	,	,	PUNCT
cana-551	181	47	δ	δ	PROPN
cana-551	181	48	)	)	PUNCT
cana-551	181	49	∈	∈	PROPN
cana-551	181	50	ῐ.	ῐ.	NOUN
cana-551	181	51	therefore	therefore	ADV
cana-551	181	52	,	,	PUNCT
cana-551	181	53	(	(	PUNCT
cana-551	181	54	𝜗	𝜗	NOUN
cana-551	181	55	,	,	PUNCT
cana-551	181	56	δ)𝑐is	δ)𝑐is	PROPN
cana-551	181	57	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	181	58	−closed	−close	VERB
cana-551	181	59	and	and	CCONJ
cana-551	181	60	hence	hence	ADV
cana-551	181	61	(	(	PUNCT
cana-551	181	62	𝜗	𝜗	PROPN
cana-551	181	63	,	,	PUNCT
cana-551	181	64	δ	δ	PROPN
cana-551	181	65	)	)	PUNCT
cana-551	181	66	is	be	AUX
cana-551	181	67	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	181	68	−open	−open	NOUN
cana-551	181	69	.	.	PUNCT
cana-551	182	1	theorem	theorem	VERB
cana-551	182	2	3.4	3.4	NUM
cana-551	182	3	:	:	PUNCT
cana-551	182	4	(	(	PUNCT
cana-551	182	5	1	1	X
cana-551	182	6	)	)	PUNCT
cana-551	182	7	every	every	DET
cana-551	182	8	open	open	ADJ
cana-551	182	9	soft	soft	ADJ
cana-551	182	10	set	set	NOUN
cana-551	182	11	is	be	AUX
cana-551	182	12	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	182	13	−open	−open	NOUN
cana-551	182	14	.	.	PUNCT
cana-551	183	1	(	(	PUNCT
cana-551	183	2	2	2	X
cana-551	183	3	)	)	PUNCT
cana-551	183	4	every	every	DET
cana-551	183	5	soft	soft	ADJ
cana-551	183	6	ῐg	ῐg	AUX
cana-551	183	7	−open	−open	NOUN
cana-551	183	8	set	set	VERB
cana-551	183	9	is	be	AUX
cana-551	183	10	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	183	11	−open	−open	NOUN
cana-551	183	12	.	.	PUNCT
cana-551	184	1	proof	proof	NOUN
cana-551	184	2	.	.	PUNCT
cana-551	185	1	immediate	immediate	ADJ
cana-551	185	2	from	from	ADP
cana-551	185	3	theorem	theorem	ADJ
cana-551	185	4	2.3	2.3	NUM
cana-551	185	5	.	.	PUNCT
cana-551	186	1	the	the	DET
cana-551	186	2	converse	converse	NOUN
cana-551	186	3	of	of	ADP
cana-551	186	4	the	the	DET
cana-551	186	5	above	above	ADJ
cana-551	186	6	theorem	theorem	NOUN
cana-551	186	7	is	be	AUX
cana-551	186	8	not	not	PART
cana-551	186	9	true	true	ADJ
cana-551	186	10	in	in	ADP
cana-551	186	11	general	general	ADJ
cana-551	186	12	as	as	SCONJ
cana-551	186	13	shall	shall	AUX
cana-551	186	14	show	show	VERB
cana-551	186	15	in	in	ADP
cana-551	186	16	the	the	DET
cana-551	186	17	following	follow	VERB
cana-551	186	18	examples	example	NOUN
cana-551	186	19	.	.	PUNCT
cana-551	187	1	example	example	NOUN
cana-551	187	2	3.5	3.5	NUM
cana-551	187	3	:	:	PUNCT
cana-551	187	4	in	in	ADP
cana-551	187	5	example	example	NOUN
cana-551	187	6	2.2	2.2	NUM
cana-551	187	7	,	,	PUNCT
cana-551	187	8	the	the	DET
cana-551	187	9	soft	soft	ADJ
cana-551	187	10	set	set	NOUN
cana-551	187	11	(	(	PUNCT
cana-551	187	12	𝜓	𝜓	PROPN
cana-551	187	13	,	,	PUNCT
cana-551	187	14	δ	δ	PROPN
cana-551	187	15	)	)	PUNCT
cana-551	187	16	is	be	AUX
cana-551	187	17	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	188	1	−open	−open	ADJ
cana-551	188	2	but	but	CCONJ
cana-551	188	3	not	not	PART
cana-551	188	4	open	open	VERB
cana-551	188	5	soft	soft	ADJ
cana-551	188	6	set	set	NOUN
cana-551	188	7	,	,	PUNCT
cana-551	188	8	where	where	SCONJ
cana-551	188	9	(	(	PUNCT
cana-551	188	10	𝜓	𝜓	PROPN
cana-551	188	11	,	,	PUNCT
cana-551	188	12	δ	δ	PROPN
cana-551	188	13	)	)	PUNCT
cana-551	188	14	=	=	PRON
cana-551	188	15	{	{	PUNCT
cana-551	188	16	(	(	PUNCT
cana-551	188	17	∇1	∇1	NOUN
cana-551	188	18	,	,	PUNCT
cana-551	188	19	∅	∅	NOUN
cana-551	188	20	)	)	PUNCT
cana-551	188	21	,	,	PUNCT
cana-551	188	22	(	(	PUNCT
cana-551	188	23	∇2	∇2	PROPN
cana-551	188	24	,	,	PUNCT
cana-551	188	25	{	{	PUNCT
cana-551	188	26	𝜇	𝜇	X
cana-551	188	27	}	}	PUNCT
cana-551	188	28	)	)	PUNCT
cana-551	188	29	}	}	PUNCT
cana-551	188	30	.	.	PUNCT
cana-551	189	1	example	example	NOUN
cana-551	189	2	3.6	3.6	NUM
cana-551	190	1	:	:	PUNCT
cana-551	190	2	in	in	ADP
cana-551	190	3	example	example	NOUN
cana-551	190	4	2.5	2.5	NUM
cana-551	190	5	,	,	PUNCT
cana-551	190	6	the	the	DET
cana-551	190	7	soft	soft	ADJ
cana-551	190	8	set	set	NOUN
cana-551	190	9	(	(	PUNCT
cana-551	190	10	𝜂	𝜂	NOUN
cana-551	190	11	,	,	PUNCT
cana-551	190	12	δ	δ	PROPN
cana-551	190	13	)	)	PUNCT
cana-551	190	14	is	be	AUX
cana-551	190	15	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	191	1	−open	−open	ADJ
cana-551	191	2	but	but	CCONJ
cana-551	191	3	not	not	PART
cana-551	191	4	soft	soft	ADJ
cana-551	191	5	ῐg	ῐg	ADP
cana-551	191	6	−open	−open	PROPN
cana-551	191	7	,	,	PUNCT
cana-551	191	8	where	where	SCONJ
cana-551	191	9	(	(	PUNCT
cana-551	191	10	𝜂	𝜂	NOUN
cana-551	191	11	,	,	PUNCT
cana-551	191	12	δ	δ	NOUN
cana-551	191	13	)	)	PUNCT
cana-551	192	1	=	=	PRON
cana-551	192	2	{	{	PUNCT
cana-551	192	3	(	(	PUNCT
cana-551	192	4	∇1	∇1	PROPN
cana-551	192	5	,	,	PUNCT
cana-551	192	6	𝒵	𝒵	PROPN
cana-551	192	7	)	)	PUNCT
cana-551	192	8	,	,	PUNCT
cana-551	192	9	(	(	PUNCT
cana-551	192	10	∇2	∇2	PROPN
cana-551	192	11	,	,	PUNCT
cana-551	192	12	{	{	PUNCT
cana-551	192	13	휀	휀	NOUN
cana-551	192	14	}	}	PUNCT
cana-551	192	15	)	)	PUNCT
cana-551	192	16	}	}	PUNCT
cana-551	192	17	.	.	PUNCT
cana-551	193	1	the	the	DET
cana-551	193	2	soft	soft	ADJ
cana-551	193	3	intersection	intersection	NOUN
cana-551	193	4	(	(	PUNCT
cana-551	193	5	resp	resp	NOUN
cana-551	193	6	.	.	PUNCT
cana-551	193	7	union	union	NOUN
cana-551	193	8	)	)	PUNCT
cana-551	193	9	of	of	ADP
cana-551	193	10	two	two	NUM
cana-551	193	11	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	193	12	−open	−open	NOUN
cana-551	193	13	sets	set	VERB
cana-551	193	14	need	need	AUX
cana-551	193	15	not	not	PART
cana-551	193	16	be	be	AUX
cana-551	193	17	a	a	DET
cana-551	193	18	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	193	19	−open	−open	NOUN
cana-551	193	20	as	as	SCONJ
cana-551	193	21	shown	show	VERB
cana-551	193	22	by	by	ADP
cana-551	193	23	the	the	DET
cana-551	193	24	following	follow	VERB
cana-551	193	25	example	example	NOUN
cana-551	193	26	.	.	PUNCT
cana-551	194	1	example	example	NOUN
cana-551	194	2	3.7	3.7	NUM
cana-551	194	3	:	:	PUNCT
cana-551	194	4	in	in	ADP
cana-551	194	5	example	example	NOUN
cana-551	194	6	2.2	2.2	NUM
cana-551	194	7	,	,	PUNCT
cana-551	194	8	the	the	DET
cana-551	194	9	soft	soft	ADJ
cana-551	194	10	sets	set	NOUN
cana-551	194	11	(	(	PUNCT
cana-551	194	12	𝛾1	𝛾1	NOUN
cana-551	194	13	,	,	PUNCT
cana-551	194	14	δ)𝑐	δ)𝑐	NOUN
cana-551	194	15	,	,	PUNCT
cana-551	194	16	(	(	PUNCT
cana-551	194	17	𝛾2	𝛾2	NOUN
cana-551	194	18	,	,	PUNCT
cana-551	194	19	δ)𝑐	δ)𝑐	NOUN
cana-551	194	20	,	,	PUNCT
cana-551	194	21	(	(	PUNCT
cana-551	194	22	𝛾3	𝛾3	NOUN
cana-551	194	23	,	,	PUNCT
cana-551	194	24	δ)𝑐	δ)𝑐	NOUN
cana-551	194	25	are	be	AUX
cana-551	194	26	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	194	27	−open	−open	ADJ
cana-551	194	28	.	.	PUNCT
cana-551	195	1	but	but	CCONJ
cana-551	195	2	(	(	PUNCT
cana-551	195	3	ℒ	ℒ	PROPN
cana-551	195	4	,	,	PUNCT
cana-551	195	5	δ	δ	PROPN
cana-551	195	6	)	)	PUNCT
cana-551	196	1	=	=	SYM
cana-551	196	2	(	(	PUNCT
cana-551	196	3	𝛾1	𝛾1	PROPN
cana-551	196	4	,	,	PUNCT
cana-551	196	5	δ)𝑐	δ)𝑐	NOUN
cana-551	196	6	∪	∪	NOUN
cana-551	196	7	(	(	PUNCT
cana-551	196	8	𝛾2	𝛾2	NOUN
cana-551	196	9	,	,	PUNCT
cana-551	196	10	δ)𝑐	δ)𝑐	NOUN
cana-551	196	11	is	be	AUX
cana-551	196	12	not	not	PART
cana-551	196	13	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	196	14	−open	−open	NOUN
cana-551	196	15	,	,	PUNCT
cana-551	196	16	where	where	SCONJ
cana-551	196	17	(	(	PUNCT
cana-551	196	18	ℒ	ℒ	PROPN
cana-551	196	19	,	,	PUNCT
cana-551	196	20	δ	δ	PROPN
cana-551	196	21	)	)	PUNCT
cana-551	196	22	=	=	PRON
cana-551	196	23	{	{	PUNCT
cana-551	196	24	(	(	PUNCT
cana-551	196	25	∇1	∇1	PROPN
cana-551	196	26	,	,	PUNCT
cana-551	196	27	𝒵	𝒵	PROPN
cana-551	196	28	)	)	PUNCT
cana-551	196	29	,	,	PUNCT
cana-551	196	30	(	(	PUNCT
cana-551	196	31	∇2	∇2	PROPN
cana-551	196	32	,	,	PUNCT
cana-551	196	33	{	{	PUNCT
cana-551	196	34	𝜇	𝜇	X
cana-551	196	35	}	}	PUNCT
cana-551	196	36	)	)	PUNCT
cana-551	196	37	}	}	PUNCT
cana-551	196	38	.	.	PUNCT
cana-551	197	1	theorem	theorem	VERB
cana-551	197	2	3.8	3.8	NUM
cana-551	197	3	:	:	PUNCT
cana-551	197	4	if	if	SCONJ
cana-551	197	5	(	(	PUNCT
cana-551	197	6	𝛾	𝛾	NOUN
cana-551	197	7	,	,	PUNCT
cana-551	197	8	δ	δ	PROPN
cana-551	197	9	)	)	PUNCT
cana-551	197	10	is	be	AUX
cana-551	197	11	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	197	12	−open	−open	ADJ
cana-551	197	13	in	in	ADP
cana-551	197	14	a	a	DET
cana-551	197	15	𝒮ts	𝒮ts	PROPN
cana-551	197	16	(	(	PUNCT
cana-551	197	17	𝒵	𝒵	PROPN
cana-551	197	18	,	,	PUNCT
cana-551	197	19	𝔚	𝔚	PROPN
cana-551	197	20	,	,	PUNCT
cana-551	197	21	𝛥	𝛥	PROPN
cana-551	197	22	)	)	PUNCT
cana-551	197	23	and	and	CCONJ
cana-551	197	24	𝑐𝑙(𝑖𝑛𝑡(𝛾	𝑐𝑙(𝑖𝑛𝑡(𝛾	PROPN
cana-551	197	25	,	,	PUNCT
cana-551	197	26	δ	δ	PROPN
cana-551	197	27	)	)	PUNCT
cana-551	197	28	)	)	PUNCT
cana-551	198	1	⊆	⊆	NUM
cana-551	198	2	(	(	PUNCT
cana-551	198	3	𝛿	𝛿	ADJ
cana-551	198	4	,	,	PUNCT
cana-551	198	5	δ	δ	PROPN
cana-551	198	6	)	)	PUNCT
cana-551	198	7	⊆	⊆	NUM
cana-551	198	8	(	(	PUNCT
cana-551	198	9	𝛾	𝛾	PROPN
cana-551	198	10	,	,	PUNCT
cana-551	198	11	δ	δ	PROPN
cana-551	198	12	)	)	PUNCT
cana-551	198	13	,	,	PUNCT
cana-551	198	14	then	then	ADV
cana-551	198	15	(	(	PUNCT
cana-551	198	16	𝛿	𝛿	ADJ
cana-551	198	17	,	,	PUNCT
cana-551	198	18	δ	δ	PROPN
cana-551	198	19	)	)	PUNCT
cana-551	198	20	is	be	AUX
cana-551	198	21	a	a	DET
cana-551	198	22	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	198	23	−open	−open	NOUN
cana-551	198	24	.	.	PUNCT
cana-551	199	1	proof	proof	NOUN
cana-551	199	2	.	.	PUNCT
cana-551	200	1	let	let	VERB
cana-551	200	2	(	(	PUNCT
cana-551	200	3	𝜂	𝜂	NOUN
cana-551	200	4	,	,	PUNCT
cana-551	200	5	δ	δ	PROPN
cana-551	200	6	)	)	PUNCT
cana-551	200	7	⊆	⊆	NUM
cana-551	200	8	(	(	PUNCT
cana-551	200	9	𝛿	𝛿	ADJ
cana-551	200	10	,	,	PUNCT
cana-551	200	11	δ	δ	PROPN
cana-551	200	12	)	)	PUNCT
cana-551	200	13	and	and	CCONJ
cana-551	200	14	(	(	PUNCT
cana-551	200	15	𝜂	𝜂	PROPN
cana-551	200	16	,	,	PUNCT
cana-551	200	17	δ	δ	PROPN
cana-551	200	18	)	)	PUNCT
cana-551	200	19	is	be	AUX
cana-551	200	20	a	a	DET
cana-551	200	21	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	200	22	−closed	−close	VERB
cana-551	200	23	.	.	PUNCT
cana-551	201	1	then	then	ADV
cana-551	201	2	,	,	PUNCT
cana-551	201	3	(	(	PUNCT
cana-551	201	4	𝜂	𝜂	NOUN
cana-551	201	5	,	,	PUNCT
cana-551	201	6	δ	δ	PROPN
cana-551	201	7	)	)	PUNCT
cana-551	201	8	⊆	⊆	NUM
cana-551	201	9	(	(	PUNCT
cana-551	201	10	𝛾	𝛾	PROPN
cana-551	201	11	,	,	PUNCT
cana-551	201	12	δ	δ	PROPN
cana-551	201	13	)	)	PUNCT
cana-551	201	14	.	.	PUNCT
cana-551	202	1	since	since	SCONJ
cana-551	202	2	(	(	PUNCT
cana-551	202	3	𝛾	𝛾	PROPN
cana-551	202	4	,	,	PUNCT
cana-551	202	5	δ	δ	PROPN
cana-551	202	6	)	)	PUNCT
cana-551	202	7	is	be	AUX
cana-551	202	8	ss	ss	ADP
cana-551	202	9	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	202	10	−open	−open	PROPN
cana-551	202	11	,	,	PUNCT
cana-551	202	12	then	then	ADV
cana-551	202	13	(	(	PUNCT
cana-551	202	14	𝛿	𝛿	ADJ
cana-551	202	15	,	,	PUNCT
cana-551	202	16	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	NUM
cana-551	202	17	,	,	PUNCT
cana-551	202	18	δ	δ	NOUN
cana-551	202	19	)	)	PUNCT
cana-551	202	20	)	)	PUNCT
cana-551	203	1	⊆	⊆	NUM
cana-551	203	2	(	(	PUNCT
cana-551	203	3	𝛾	𝛾	NOUN
cana-551	203	4	,	,	PUNCT
cana-551	203	5	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	NUM
cana-551	203	6	,	,	PUNCT
cana-551	203	7	δ	δ	NOUN
cana-551	203	8	)	)	PUNCT
cana-551	203	9	)	)	PUNCT
cana-551	204	1	∈	∈	PROPN
cana-551	204	2	ῐ.	ῐ.	NOUN
cana-551	204	3	it	it	PRON
cana-551	204	4	follows	follow	VERB
cana-551	204	5	that	that	SCONJ
cana-551	204	6	,	,	PUNCT
cana-551	204	7	(	(	PUNCT
cana-551	204	8	𝛿	𝛿	ADJ
cana-551	204	9	,	,	PUNCT
cana-551	204	10	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	δ)\𝑐𝑙(𝑖𝑛𝑡(𝜂	NUM
cana-551	204	11	,	,	PUNCT
cana-551	204	12	δ	δ	NOUN
cana-551	204	13	)	)	PUNCT
cana-551	204	14	)	)	PUNCT
cana-551	205	1	∈	∈	PROPN
cana-551	205	2	ῐ.	ῐ.	NOUN
cana-551	205	3	thus	thus	ADV
cana-551	205	4	,	,	PUNCT
cana-551	205	5	(	(	PUNCT
cana-551	205	6	𝛿	𝛿	ADJ
cana-551	205	7	,	,	PUNCT
cana-551	205	8	δ	δ	PROPN
cana-551	205	9	)	)	PUNCT
cana-551	205	10	is	be	AUX
cana-551	205	11	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	205	12	−open	−open	NOUN
cana-551	205	13	.	.	PUNCT
cana-551	206	1	theorem	theorem	VERB
cana-551	206	2	3.9	3.9	NUM
cana-551	206	3	:	:	PUNCT
cana-551	206	4	a	a	DET
cana-551	206	5	soft	soft	ADJ
cana-551	206	6	set	set	NOUN
cana-551	206	7	(	(	PUNCT
cana-551	206	8	𝜗	𝜗	NOUN
cana-551	206	9	,	,	PUNCT
cana-551	206	10	δ	δ	PROPN
cana-551	206	11	)	)	PUNCT
cana-551	206	12	is	be	AUX
cana-551	206	13	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	206	14	−closed	−close	VERB
cana-551	206	15	in	in	ADP
cana-551	206	16	a	a	DET
cana-551	206	17	𝒮ts	𝒮ts	PROPN
cana-551	206	18	(	(	PUNCT
cana-551	206	19	𝒵	𝒵	PROPN
cana-551	206	20	,	,	PUNCT
cana-551	206	21	𝔚	𝔚	PROPN
cana-551	206	22	,	,	PUNCT
cana-551	206	23	𝛥	𝛥	NOUN
cana-551	206	24	)	)	PUNCT
cana-551	207	1	if	if	SCONJ
cana-551	207	2	and	and	CCONJ
cana-551	207	3	only	only	ADV
cana-551	207	4	if	if	SCONJ
cana-551	207	5	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	207	6	,	,	PUNCT
cana-551	207	7	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	207	8	,	,	PUNCT
cana-551	207	9	δ	δ	PROPN
cana-551	207	10	)	)	PUNCT
cana-551	207	11	is	be	AUX
cana-551	207	12	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	207	13	−open	−open	NOUN
cana-551	207	14	.	.	PUNCT
cana-551	208	1	communications	communication	NOUN
cana-551	208	2	on	on	ADP
cana-551	208	3	applied	apply	VERB
cana-551	208	4	nonlinear	nonlinear	ADJ
cana-551	208	5	analysis	analysis	NOUN
cana-551	208	6	issn	issn	NOUN
cana-551	208	7	:	:	PUNCT
cana-551	208	8	1074	1074	NUM
cana-551	208	9	-	-	PUNCT
cana-551	208	10	133x	133x	NUM
cana-551	208	11	vol	vol	NOUN
cana-551	208	12	31	31	NUM
cana-551	208	13	no	no	NOUN
cana-551	208	14	.	.	NOUN
cana-551	208	15	2	2	NUM
cana-551	208	16	(	(	PUNCT
cana-551	208	17	2024	2024	NUM
cana-551	208	18	)	)	PUNCT
cana-551	208	19	291	291	NUM
cana-551	208	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	208	21	proof	proof	NOUN
cana-551	208	22	.	.	PUNCT
cana-551	209	1	(	(	PUNCT
cana-551	209	2	⇒	⇒	NOUN
cana-551	209	3	)	)	PUNCT
cana-551	209	4	let	let	VERB
cana-551	209	5	(	(	PUNCT
cana-551	209	6	𝛾	𝛾	PROPN
cana-551	209	7	,	,	PUNCT
cana-551	209	8	δ	δ	PROPN
cana-551	209	9	)	)	PUNCT
cana-551	209	10	⊆	⊆	NUM
cana-551	209	11	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	209	12	,	,	PUNCT
cana-551	209	13	δ	δ	PROPN
cana-551	209	14	)	)	PUNCT
cana-551	209	15	)	)	PUNCT
cana-551	210	1	and	and	CCONJ
cana-551	210	2	(	(	PUNCT
cana-551	210	3	𝛾	𝛾	AUX
cana-551	210	4	,	,	PUNCT
cana-551	210	5	δ	δ	PROPN
cana-551	210	6	)	)	PUNCT
cana-551	210	7	be	be	VERB
cana-551	210	8	a	a	DET
cana-551	210	9	soft	soft	ADJ
cana-551	210	10	closed	closed	ADJ
cana-551	210	11	set	set	NOUN
cana-551	210	12	.	.	PUNCT
cana-551	211	1	then	then	ADV
cana-551	211	2	,	,	PUNCT
cana-551	211	3	(	(	PUNCT
cana-551	211	4	𝛾	𝛾	PROPN
cana-551	211	5	,	,	PUNCT
cana-551	211	6	δ	δ	PROPN
cana-551	211	7	)	)	PUNCT
cana-551	211	8	∈	∈	PROPN
cana-551	211	9	ῐ	ῐ	PROPN
cana-551	211	10	from	from	ADP
cana-551	211	11	theorem	theorem	ADJ
cana-551	211	12	2.9	2.9	NUM
cana-551	211	13	so	so	SCONJ
cana-551	211	14	there	there	PRON
cana-551	211	15	exists	exist	VERB
cana-551	211	16	(	(	PUNCT
cana-551	211	17	𝜎	𝜎	PROPN
cana-551	211	18	,	,	PUNCT
cana-551	211	19	δ	δ	PROPN
cana-551	211	20	)	)	PUNCT
cana-551	211	21	∈	∈	PROPN
cana-551	212	1	ῐ	ῐ	PROPN
cana-551	212	2	such	such	ADJ
cana-551	212	3	that	that	PRON
cana-551	212	4	(	(	PUNCT
cana-551	212	5	𝛾	𝛾	PROPN
cana-551	212	6	,	,	PUNCT
cana-551	212	7	δ)\(𝜎	δ)\(𝜎	PROPN
cana-551	212	8	,	,	PUNCT
cana-551	212	9	δ	δ	PROPN
cana-551	212	10	)	)	PUNCT
cana-551	212	11	=	=	PUNCT
cana-551	212	12	∅̃.	∅̃.	NOUN
cana-551	212	13	thus	thus	ADV
cana-551	212	14	,	,	PUNCT
cana-551	212	15	that	that	SCONJ
cana-551	212	16	(	(	PUNCT
cana-551	212	17	𝛾	𝛾	PROPN
cana-551	212	18	,	,	PUNCT
cana-551	212	19	δ)\(𝜎	δ)\(𝜎	PROPN
cana-551	212	20	,	,	PUNCT
cana-551	212	21	δ	δ	PROPN
cana-551	212	22	)	)	PUNCT
cana-551	212	23	=	=	PUNCT
cana-551	212	24	∅̃	∅̃	NOUN
cana-551	212	25	⊆	⊆	NUM
cana-551	212	26	𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝜗	𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝜗	PROPN
cana-551	212	27	,	,	PUNCT
cana-551	212	28	δ)\(𝜗	δ)\(𝜗	NOUN
cana-551	212	29	,	,	PUNCT
cana-551	212	30	δ	δ	PROPN
cana-551	212	31	)	)	PUNCT
cana-551	212	32	]	]	PUNCT
cana-551	212	33	)	)	PUNCT
cana-551	212	34	.	.	PUNCT
cana-551	213	1	hence	hence	ADV
cana-551	213	2	,	,	PUNCT
cana-551	213	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	213	4	,	,	PUNCT
cana-551	213	5	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	213	6	,	,	PUNCT
cana-551	213	7	δ	δ	PROPN
cana-551	213	8	)	)	PUNCT
cana-551	213	9	is	be	AUX
cana-551	213	10	a	a	DET
cana-551	213	11	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	213	12	−open	−open	NOUN
cana-551	213	13	from	from	ADP
cana-551	213	14	theorem	theorem	ADJ
cana-551	213	15	3.3	3.3	NUM
cana-551	213	16	.	.	PUNCT
cana-551	214	1	(	(	PUNCT
cana-551	214	2	⇐	⇐	ADJ
cana-551	214	3	)	)	PUNCT
cana-551	214	4	let	let	VERB
cana-551	214	5	(	(	PUNCT
cana-551	214	6	𝜗	𝜗	NOUN
cana-551	214	7	,	,	PUNCT
cana-551	214	8	δ	δ	PROPN
cana-551	214	9	)	)	PUNCT
cana-551	215	1	⊆	⊆	NUM
cana-551	215	2	(	(	PUNCT
cana-551	215	3	𝛿	𝛿	ADJ
cana-551	215	4	,	,	PUNCT
cana-551	215	5	δ	δ	NOUN
cana-551	215	6	)	)	PUNCT
cana-551	215	7	such	such	ADJ
cana-551	215	8	that	that	SCONJ
cana-551	215	9	(	(	PUNCT
cana-551	215	10	𝛿	𝛿	ADJ
cana-551	215	11	,	,	PUNCT
cana-551	215	12	δ	δ	PROPN
cana-551	215	13	)	)	PUNCT
cana-551	215	14	is	be	AUX
cana-551	215	15	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	215	16	−open	−open	NOUN
cana-551	215	17	.	.	PUNCT
cana-551	216	1	then	then	ADV
cana-551	216	2	,	,	PUNCT
cana-551	216	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	PROPN
cana-551	216	4	,	,	PUNCT
cana-551	216	5	δ	δ	NOUN
cana-551	216	6	)	)	PUNCT
cana-551	216	7	)	)	PUNCT
cana-551	216	8	∩	∩	NOUN
cana-551	216	9	(	(	PUNCT
cana-551	216	10	𝛿	𝛿	ADJ
cana-551	216	11	,	,	PUNCT
cana-551	216	12	δ)𝑐	δ)𝑐	NOUN
cana-551	216	13	⊆	⊆	NUM
cana-551	216	14	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	216	15	,	,	PUNCT
cana-551	216	16	δ	δ	NOUN
cana-551	216	17	)	)	PUNCT
cana-551	216	18	)	)	PUNCT
cana-551	216	19	∩	∩	NOUN
cana-551	216	20	(	(	PUNCT
cana-551	216	21	𝜗	𝜗	NOUN
cana-551	216	22	,	,	PUNCT
cana-551	216	23	δ)𝑐	δ)𝑐	NOUN
cana-551	216	24	=	=	SYM
cana-551	216	25	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	216	26	,	,	PUNCT
cana-551	216	27	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	216	28	,	,	PUNCT
cana-551	216	29	δ	δ	PROPN
cana-551	216	30	)	)	PUNCT
cana-551	216	31	.	.	PUNCT
cana-551	217	1	by	by	ADP
cana-551	217	2	hypothesis	hypothesis	NOUN
cana-551	217	3	,	,	PUNCT
cana-551	217	4	[	[	X
cana-551	217	5	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	ADJ
cana-551	217	6	,	,	PUNCT
cana-551	217	7	δ	δ	NOUN
cana-551	217	8	)	)	PUNCT
cana-551	217	9	)	)	PUNCT
cana-551	217	10	∩	∩	NOUN
cana-551	217	11	(	(	PUNCT
cana-551	217	12	𝛿	𝛿	ADJ
cana-551	217	13	,	,	PUNCT
cana-551	217	14	δ)𝑐]\	δ)𝑐]\	PROPN
cana-551	217	15	(	(	PUNCT
cana-551	217	16	𝜎	𝜎	PROPN
cana-551	217	17	,	,	PUNCT
cana-551	217	18	δ	δ	PROPN
cana-551	217	19	)	)	PUNCT
cana-551	217	20	⊆	⊆	NUM
cana-551	217	21	𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑖𝑛𝑡(𝑐𝑙[𝑐𝑙(𝑖𝑛𝑡(𝜗	NUM
cana-551	217	22	,	,	PUNCT
cana-551	217	23	δ))\(𝜗	δ))\(𝜗	NOUN
cana-551	217	24	,	,	PUNCT
cana-551	217	25	δ	δ	PROPN
cana-551	217	26	)	)	PUNCT
cana-551	217	27	]	]	PUNCT
cana-551	217	28	)	)	PUNCT
cana-551	217	29	=	=	SYM
cana-551	217	30	∅̃	∅̃	NOUN
cana-551	217	31	,	,	PUNCT
cana-551	217	32	for	for	ADP
cana-551	217	33	some	some	PRON
cana-551	217	34	(	(	PUNCT
cana-551	217	35	𝜎	𝜎	PROPN
cana-551	217	36	,	,	PUNCT
cana-551	217	37	δ	δ	PROPN
cana-551	217	38	)	)	PUNCT
cana-551	217	39	∈	∈	PROPN
cana-551	217	40	ῐ	ῐ	PROPN
cana-551	217	41	from	from	ADP
cana-551	217	42	theorem	theorem	ADJ
cana-551	217	43	3.3	3.3	NUM
cana-551	217	44	.	.	PUNCT
cana-551	218	1	so	so	ADV
cana-551	218	2	,	,	PUNCT
cana-551	218	3	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	218	4	,	,	PUNCT
cana-551	218	5	δ	δ	NOUN
cana-551	218	6	)	)	PUNCT
cana-551	218	7	)	)	PUNCT
cana-551	219	1	∩	∩	NOUN
cana-551	219	2	(	(	PUNCT
cana-551	219	3	𝛿	𝛿	ADJ
cana-551	219	4	,	,	PUNCT
cana-551	219	5	δ)𝑐	δ)𝑐	NOUN
cana-551	219	6	⊆	⊆	NUM
cana-551	219	7	(	(	PUNCT
cana-551	219	8	𝜎	𝜎	PROPN
cana-551	219	9	,	,	PUNCT
cana-551	219	10	δ	δ	PROPN
cana-551	219	11	)	)	PUNCT
cana-551	219	12	∈	∈	PROPN
cana-551	219	13	ῐ.	ῐ.	NOUN
cana-551	219	14	thus	thus	ADV
cana-551	219	15	,	,	PUNCT
cana-551	219	16	𝑐𝑙(𝑖𝑛𝑡(𝜗	𝑐𝑙(𝑖𝑛𝑡(𝜗	NOUN
cana-551	219	17	,	,	PUNCT
cana-551	219	18	δ))\(𝛿	δ))\(𝛿	NOUN
cana-551	219	19	,	,	PUNCT
cana-551	219	20	δ	δ	PROPN
cana-551	219	21	)	)	PUNCT
cana-551	219	22	∈	∈	PROPN
cana-551	219	23	ῐ.	ῐ.	NOUN
cana-551	220	1	so	so	ADV
cana-551	220	2	,	,	PUNCT
cana-551	220	3	(	(	PUNCT
cana-551	220	4	𝜗	𝜗	NOUN
cana-551	220	5	,	,	PUNCT
cana-551	220	6	δ	δ	PROPN
cana-551	220	7	)	)	PUNCT
cana-551	220	8	is	be	AUX
cana-551	220	9	a	a	DET
cana-551	220	10	ss	ss	NOUN
cana-551	220	11	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	220	12	−closed	−close	VERB
cana-551	220	13	.	.	PUNCT
cana-551	221	1	4	4	X
cana-551	221	2	.	.	X
cana-551	221	3	ss	ss	PROPN
cana-551	221	4	ƅ∗	ƅ∗	PROPN
cana-551	221	5	−continuous	−continuous	ADJ
cana-551	221	6	via	via	ADP
cana-551	221	7	soft	soft	ADJ
cana-551	221	8	ideal	ideal	NOUN
cana-551	221	9	in	in	ADP
cana-551	221	10	this	this	DET
cana-551	221	11	section	section	NOUN
cana-551	221	12	,	,	PUNCT
cana-551	221	13	we	we	PRON
cana-551	221	14	introduce	introduce	VERB
cana-551	221	15	a	a	DET
cana-551	221	16	ssƅ∗	ssƅ∗	ADJ
cana-551	221	17	−continuous	−continuous	ADJ
cana-551	221	18	function	function	NOUN
cana-551	221	19	with	with	ADP
cana-551	221	20	respect	respect	NOUN
cana-551	221	21	to	to	ADP
cana-551	221	22	a	a	DET
cana-551	221	23	soft	soft	ADJ
cana-551	221	24	ideal	ideal	NOUN
cana-551	221	25	in	in	ADP
cana-551	221	26	𝒮tss	𝒮tss	PROPN
cana-551	221	27	.	.	PUNCT
cana-551	222	1	definition	definition	NOUN
cana-551	222	2	4.1	4.1	NUM
cana-551	222	3	:	:	PUNCT
cana-551	222	4	let	let	VERB
cana-551	222	5	𝛺	𝛺	VERB
cana-551	222	6	:	:	PUNCT
cana-551	222	7	(	(	PUNCT
cana-551	222	8	𝒵	𝒵	PROPN
cana-551	222	9	,	,	PUNCT
cana-551	222	10	𝔚	𝔚	PROPN
cana-551	222	11	,	,	PUNCT
cana-551	222	12	𝛥	𝛥	PROPN
cana-551	222	13	)	)	PUNCT
cana-551	222	14	→	→	SYM
cana-551	222	15	(	(	PUNCT
cana-551	222	16	𝒟	𝒟	PROPN
cana-551	222	17	,	,	PUNCT
cana-551	222	18	℧	℧	PROPN
cana-551	222	19	,	,	PUNCT
cana-551	222	20	θ	θ	PROPN
cana-551	222	21	)	)	PUNCT
cana-551	222	22	be	be	VERB
cana-551	222	23	a	a	DET
cana-551	222	24	soft	soft	ADJ
cana-551	222	25	mapping	mapping	NOUN
cana-551	222	26	.	.	PUNCT
cana-551	223	1	if	if	SCONJ
cana-551	223	2	𝛺−1((𝛿	𝛺−1((𝛿	PROPN
cana-551	223	3	,	,	PUNCT
cana-551	223	4	δ	δ	PROPN
cana-551	223	5	)	)	PUNCT
cana-551	223	6	)	)	PUNCT
cana-551	223	7	is	be	AUX
cana-551	223	8	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	223	9	−open	−open	ADJ
cana-551	223	10	in	in	ADP
cana-551	223	11	(	(	PUNCT
cana-551	223	12	𝒵	𝒵	PROPN
cana-551	223	13	,	,	PUNCT
cana-551	223	14	𝔚	𝔚	PROPN
cana-551	223	15	,	,	PUNCT
cana-551	223	16	𝛥	𝛥	NOUN
cana-551	223	17	)	)	PUNCT
cana-551	223	18	for	for	ADP
cana-551	223	19	each	each	DET
cana-551	223	20	soft	soft	ADJ
cana-551	223	21	open	open	ADJ
cana-551	223	22	set	set	NOUN
cana-551	223	23	(	(	PUNCT
cana-551	223	24	𝛿	𝛿	ADJ
cana-551	223	25	,	,	PUNCT
cana-551	223	26	δ	δ	PROPN
cana-551	223	27	)	)	PUNCT
cana-551	223	28	of	of	ADP
cana-551	223	29	(	(	PUNCT
cana-551	223	30	𝒟	𝒟	PROPN
cana-551	223	31	,	,	PUNCT
cana-551	223	32	℧	℧	PROPN
cana-551	223	33	,	,	PUNCT
cana-551	223	34	θ	θ	PROPN
cana-551	223	35	)	)	PUNCT
cana-551	223	36	,	,	PUNCT
cana-551	223	37	then	then	ADV
cana-551	223	38	𝛺	𝛺	PROPN
cana-551	223	39	is	be	AUX
cana-551	223	40	called	call	VERB
cana-551	223	41	soft	soft	ADJ
cana-551	223	42	strongly	strongly	ADV
cana-551	223	43	ƅ∗ῐ	ƅ∗ῐ	ADJ
cana-551	223	44	−continuous	−continuous	ADJ
cana-551	223	45	function	function	NOUN
cana-551	223	46	.	.	PUNCT
cana-551	224	1	corollary	corollary	ADJ
cana-551	224	2	4.2	4.2	NUM
cana-551	224	3	:	:	PUNCT
cana-551	224	4	let	let	VERB
cana-551	224	5	𝛺	𝛺	VERB
cana-551	224	6	:	:	PUNCT
cana-551	224	7	(	(	PUNCT
cana-551	224	8	𝒵	𝒵	PROPN
cana-551	224	9	,	,	PUNCT
cana-551	224	10	𝔚	𝔚	PROPN
cana-551	224	11	,	,	PUNCT
cana-551	224	12	𝛥	𝛥	PROPN
cana-551	224	13	)	)	PUNCT
cana-551	224	14	→	→	SYM
cana-551	224	15	(	(	PUNCT
cana-551	224	16	𝒟	𝒟	PROPN
cana-551	224	17	,	,	PUNCT
cana-551	224	18	℧	℧	PROPN
cana-551	224	19	,	,	PUNCT
cana-551	224	20	θ	θ	PROPN
cana-551	224	21	)	)	PUNCT
cana-551	224	22	be	be	VERB
cana-551	224	23	a	a	DET
cana-551	224	24	soft	soft	ADJ
cana-551	224	25	function	function	NOUN
cana-551	224	26	.	.	PUNCT
cana-551	225	1	then	then	ADV
cana-551	225	2	:	:	PUNCT
cana-551	225	3	1every	1every	NUM
cana-551	225	4	soft	soft	ADJ
cana-551	225	5	continuous	continuous	ADJ
cana-551	225	6	function	function	NOUN
cana-551	225	7	is	be	AUX
cana-551	225	8	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	225	9	−continuous	−continuous	ADJ
cana-551	225	10	functions	function	NOUN
cana-551	225	11	.	.	PUNCT
cana-551	226	1	2every	2every	NUM
cana-551	226	2	soft	soft	ADJ
cana-551	226	3	ῐg	ῐg	ADP
cana-551	226	4	−continuous	−continuous	ADJ
cana-551	226	5	is	be	AUX
cana-551	226	6	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	226	7	−continuous	−continuous	ADJ
cana-551	226	8	function	function	NOUN
cana-551	226	9	.	.	PUNCT
cana-551	227	1	proof	proof	NOUN
cana-551	227	2	.	.	PUNCT
cana-551	228	1	immediate	immediate	ADJ
cana-551	228	2	from	from	ADP
cana-551	228	3	theorem	theorem	ADJ
cana-551	228	4	3.4	3.4	NUM
cana-551	228	5	.	.	PUNCT
cana-551	229	1	the	the	DET
cana-551	229	2	converse	converse	NOUN
cana-551	229	3	of	of	ADP
cana-551	229	4	the	the	DET
cana-551	229	5	above	above	ADJ
cana-551	229	6	theorem	theorem	NOUN
cana-551	229	7	is	be	AUX
cana-551	229	8	not	not	PART
cana-551	229	9	true	true	ADJ
cana-551	229	10	in	in	ADP
cana-551	229	11	general	general	ADJ
cana-551	229	12	as	as	SCONJ
cana-551	229	13	shall	shall	AUX
cana-551	229	14	show	show	VERB
cana-551	229	15	in	in	ADP
cana-551	229	16	the	the	DET
cana-551	229	17	following	follow	VERB
cana-551	229	18	example	example	NOUN
cana-551	229	19	.	.	PUNCT
cana-551	230	1	example	example	NOUN
cana-551	230	2	4.3	4.3	NUM
cana-551	230	3	:	:	PUNCT
cana-551	230	4	let	let	VERB
cana-551	230	5	𝒵	𝒵	PROPN
cana-551	230	6	=	=	PUNCT
cana-551	230	7	{	{	PUNCT
cana-551	230	8	휀	휀	NOUN
cana-551	230	9	,	,	PUNCT
cana-551	230	10	𝜇	𝜇	ADP
cana-551	230	11	}	}	PUNCT
cana-551	230	12	and	and	CCONJ
cana-551	230	13	∆=	∆=	ADJ
cana-551	230	14	{	{	PUNCT
cana-551	230	15	∇1	∇1	NOUN
cana-551	230	16	,	,	PUNCT
cana-551	230	17	∇2	∇2	PROPN
cana-551	230	18	}	}	PUNCT
cana-551	230	19	.	.	PUNCT
cana-551	231	1	let	let	VERB
cana-551	231	2	(	(	PUNCT
cana-551	231	3	𝛾1	𝛾1	PROPN
cana-551	231	4	,	,	PUNCT
cana-551	231	5	δ	δ	PROPN
cana-551	231	6	)	)	PUNCT
cana-551	231	7	,	,	PUNCT
cana-551	231	8	(	(	PUNCT
cana-551	231	9	𝛾2	𝛾2	VERB
cana-551	231	10	,	,	PUNCT
cana-551	231	11	δ	δ	PROPN
cana-551	231	12	)	)	PUNCT
cana-551	231	13	be	be	VERB
cana-551	231	14	two	two	NUM
cana-551	231	15	soft	soft	ADJ
cana-551	231	16	sets	set	NOUN
cana-551	231	17	where	where	SCONJ
cana-551	231	18	(	(	PUNCT
cana-551	231	19	𝛾1	𝛾1	PROPN
cana-551	231	20	,	,	PUNCT
cana-551	231	21	δ	δ	PROPN
cana-551	231	22	)	)	PUNCT
cana-551	232	1	=	=	PRON
cana-551	232	2	{	{	PUNCT
cana-551	232	3	(	(	PUNCT
cana-551	232	4	∇1	∇1	PROPN
cana-551	232	5	,	,	PUNCT
cana-551	232	6	{	{	PUNCT
cana-551	232	7	휀	휀	NOUN
cana-551	232	8	}	}	PUNCT
cana-551	232	9	)	)	PUNCT
cana-551	232	10	}	}	PUNCT
cana-551	232	11	,	,	PUNCT
cana-551	232	12	(	(	PUNCT
cana-551	232	13	𝛾2	𝛾2	VERB
cana-551	232	14	,	,	PUNCT
cana-551	232	15	δ	δ	NOUN
cana-551	232	16	)	)	PUNCT
cana-551	232	17	=	=	PRON
cana-551	232	18	{	{	PUNCT
cana-551	232	19	(	(	PUNCT
cana-551	232	20	∇1	∇1	PROPN
cana-551	232	21	,	,	PUNCT
cana-551	232	22	{	{	PUNCT
cana-551	232	23	휀	휀	NOUN
cana-551	232	24	}	}	PUNCT
cana-551	232	25	)	)	PUNCT
cana-551	232	26	,	,	PUNCT
cana-551	232	27	(	(	PUNCT
cana-551	232	28	∇2	∇2	PROPN
cana-551	232	29	,	,	PUNCT
cana-551	232	30	{	{	PUNCT
cana-551	232	31	𝜇	𝜇	X
cana-551	232	32	}	}	PUNCT
cana-551	232	33	)	)	PUNCT
cana-551	232	34	}	}	PUNCT
cana-551	232	35	.	.	PUNCT
cana-551	233	1	𝔚	𝔚	NOUN
cana-551	233	2	=	=	PRON
cana-551	233	3	{	{	PUNCT
cana-551	233	4	�	�	PROPN
cana-551	233	5	̃	̃	PROPN
cana-551	233	6	�	�	PROPN
cana-551	233	7	,	,	PUNCT
cana-551	233	8	∅̃	∅̃	NOUN
cana-551	233	9	,	,	PUNCT
cana-551	233	10	(	(	PUNCT
cana-551	233	11	𝛾1	𝛾1	PROPN
cana-551	233	12	,	,	PUNCT
cana-551	233	13	δ	δ	PROPN
cana-551	233	14	)	)	PUNCT
cana-551	233	15	,	,	PUNCT
cana-551	233	16	(	(	PUNCT
cana-551	233	17	𝛾2	𝛾2	VERB
cana-551	233	18	,	,	PUNCT
cana-551	233	19	δ	δ	PROPN
cana-551	233	20	)	)	PUNCT
cana-551	233	21	}	}	PUNCT
cana-551	233	22	is	be	AUX
cana-551	233	23	the	the	DET
cana-551	233	24	soft	soft	ADJ
cana-551	233	25	topology	topology	NOUN
cana-551	233	26	over	over	ADP
cana-551	233	27	𝒵.	𝒵.	PROPN
cana-551	233	28	let	let	VERB
cana-551	233	29	ῐ	ῐ	PROPN
cana-551	233	30	=	=	PRON
cana-551	233	31	{	{	PUNCT
cana-551	233	32	∅̃	∅̃	NOUN
cana-551	233	33	}	}	PUNCT
cana-551	233	34	be	be	AUX
cana-551	233	35	a	a	DET
cana-551	233	36	soft	soft	ADJ
cana-551	233	37	ideal	ideal	NOUN
cana-551	233	38	on	on	ADP
cana-551	233	39	𝒵.	𝒵.	PROPN
cana-551	233	40	let	let	VERB
cana-551	233	41	𝒟	𝒟	NOUN
cana-551	233	42	=	=	SYM
cana-551	233	43	{	{	PUNCT
cana-551	233	44	𝜎	𝜎	PROPN
cana-551	233	45	,	,	PUNCT
cana-551	233	46	𝜌	𝜌	ADP
cana-551	233	47	}	}	PUNCT
cana-551	233	48	and	and	CCONJ
cana-551	233	49	θ	θ	NOUN
cana-551	233	50	=	=	SYM
cana-551	233	51	{	{	PUNCT
cana-551	233	52	ϱ1	ϱ1	NOUN
cana-551	233	53	,	,	PUNCT
cana-551	233	54	𝜚2	𝜚2	PROPN
cana-551	233	55	}	}	PUNCT
cana-551	233	56	,	,	PUNCT
cana-551	233	57	℧	℧	PROPN
cana-551	233	58	=	=	PRON
cana-551	233	59	{	{	PUNCT
cana-551	233	60	�	�	PROPN
cana-551	233	61	̃	̃	PROPN
cana-551	233	62	�	�	PROPN
cana-551	233	63	,	,	PUNCT
cana-551	233	64	∅̃	∅̃	NOUN
cana-551	233	65	,	,	PUNCT
cana-551	233	66	(	(	PUNCT
cana-551	233	67	𝛿	𝛿	ADJ
cana-551	233	68	,	,	PUNCT
cana-551	233	69	θ	θ	NOUN
cana-551	233	70	)	)	PUNCT
cana-551	233	71	}	}	PUNCT
cana-551	233	72	is	be	AUX
cana-551	233	73	soft	soft	ADJ
cana-551	233	74	topology	topology	NOUN
cana-551	233	75	on	on	ADP
cana-551	233	76	𝒟.	𝒟.	PROPN
cana-551	234	1	where	where	SCONJ
cana-551	234	2	(	(	PUNCT
cana-551	234	3	𝛿	𝛿	ADJ
cana-551	234	4	,	,	PUNCT
cana-551	234	5	θ	θ	NOUN
cana-551	234	6	)	)	PUNCT
cana-551	234	7	=	=	SYM
cana-551	234	8	{	{	PUNCT
cana-551	234	9	(	(	PUNCT
cana-551	234	10	ϱ1	ϱ1	NOUN
cana-551	234	11	,	,	PUNCT
cana-551	234	12	{	{	PUNCT
cana-551	234	13	𝜎	𝜎	NOUN
cana-551	234	14	}	}	PUNCT
cana-551	234	15	)	)	PUNCT
cana-551	234	16	,	,	PUNCT
cana-551	234	17	(	(	PUNCT
cana-551	234	18	ϱ2	ϱ2	NOUN
cana-551	234	19	,	,	PUNCT
cana-551	234	20	{	{	PUNCT
cana-551	234	21	𝜎	𝜎	NOUN
cana-551	234	22	}	}	PUNCT
cana-551	234	23	)	)	PUNCT
cana-551	234	24	}	}	PUNCT
cana-551	234	25	.	.	PUNCT
cana-551	235	1	then	then	ADV
cana-551	235	2	let	let	VERB
cana-551	235	3	𝛺	𝛺	NOUN
cana-551	235	4	:	:	PUNCT
cana-551	235	5	(	(	PUNCT
cana-551	235	6	𝒵	𝒵	PROPN
cana-551	235	7	,	,	PUNCT
cana-551	235	8	𝔚	𝔚	PROPN
cana-551	235	9	,	,	PUNCT
cana-551	235	10	𝛥	𝛥	PROPN
cana-551	235	11	)	)	PUNCT
cana-551	235	12	→	→	SYM
cana-551	235	13	(	(	PUNCT
cana-551	235	14	𝒟	𝒟	PROPN
cana-551	235	15	,	,	PUNCT
cana-551	235	16	℧	℧	PROPN
cana-551	235	17	,	,	PUNCT
cana-551	235	18	θ	θ	PROPN
cana-551	235	19	)	)	PUNCT
cana-551	235	20	be	be	VERB
cana-551	235	21	a	a	DET
cana-551	235	22	soft	soft	ADJ
cana-551	235	23	function	function	NOUN
cana-551	235	24	and	and	CCONJ
cana-551	235	25	𝑢	𝑢	NOUN
cana-551	235	26	:	:	PUNCT
cana-551	235	27	𝒵	𝒵	PROPN
cana-551	235	28	→	→	SYM
cana-551	235	29	𝒟	𝒟	PROPN
cana-551	235	30	and	and	CCONJ
cana-551	235	31	𝑝	𝑝	NOUN
cana-551	235	32	:	:	PUNCT
cana-551	235	33	∆→	∆→	PROPN
cana-551	236	1	θ	θ	PROPN
cana-551	236	2	denoted	denote	VERB
cana-551	236	3	by	by	ADP
cana-551	236	4	𝑢(휀	𝑢(휀	NOUN
cana-551	236	5	)	)	PUNCT
cana-551	236	6	=	=	SYM
cana-551	236	7	𝜎	𝜎	PROPN
cana-551	236	8	,	,	PUNCT
cana-551	236	9	𝑢(𝜇	𝑢(𝜇	PROPN
cana-551	236	10	)	)	PUNCT
cana-551	236	11	=	=	SYM
cana-551	236	12	𝜌	𝜌	X
cana-551	236	13	,	,	PUNCT
cana-551	236	14	𝑝(∇1	𝑝(∇1	ADJ
cana-551	236	15	)	)	PUNCT
cana-551	236	16	=	=	SYM
cana-551	236	17	ϱ1	ϱ1	NOUN
cana-551	236	18	,	,	PUNCT
cana-551	236	19	𝑝(∇2	𝑝(∇2	ADJ
cana-551	236	20	)	)	PUNCT
cana-551	236	21	=	=	SYM
cana-551	236	22	ϱ2	ϱ2	NOUN
cana-551	236	23	.	.	PUNCT
cana-551	237	1	let	let	AUX
cana-551	237	2	take	take	VERB
cana-551	237	3	(	(	PUNCT
cana-551	237	4	𝜗	𝜗	NOUN
cana-551	237	5	,	,	PUNCT
cana-551	237	6	δ	δ	PROPN
cana-551	237	7	)	)	PUNCT
cana-551	237	8	=	=	PRON
cana-551	237	9	{	{	PUNCT
cana-551	237	10	(	(	PUNCT
cana-551	237	11	∇1	∇1	PROPN
cana-551	237	12	,	,	PUNCT
cana-551	237	13	{	{	PUNCT
cana-551	237	14	휀	휀	NOUN
cana-551	237	15	}	}	PUNCT
cana-551	237	16	)	)	PUNCT
cana-551	237	17	,	,	PUNCT
cana-551	237	18	(	(	PUNCT
cana-551	237	19	∇2	∇2	PROPN
cana-551	237	20	,	,	PUNCT
cana-551	237	21	{	{	PUNCT
cana-551	237	22	휀	휀	NOUN
cana-551	237	23	}	}	PUNCT
cana-551	237	24	)	)	PUNCT
cana-551	237	25	}	}	PUNCT
cana-551	237	26	.	.	PUNCT
cana-551	238	1	then	then	ADV
cana-551	238	2	𝛺	𝛺	PROPN
cana-551	238	3	is	be	AUX
cana-551	238	4	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	238	5	−continuous	−continuous	ADJ
cana-551	238	6	but	but	CCONJ
cana-551	238	7	not	not	PART
cana-551	238	8	soft	soft	ADJ
cana-551	238	9	continuous	continuous	ADJ
cana-551	238	10	.	.	PUNCT
cana-551	239	1	since	since	SCONJ
cana-551	239	2	if	if	SCONJ
cana-551	239	3	𝛺−1((𝛿	𝛺−1((𝛿	PROPN
cana-551	239	4	,	,	PUNCT
cana-551	239	5	θ	θ	NOUN
cana-551	239	6	)	)	PUNCT
cana-551	239	7	)	)	PUNCT
cana-551	239	8	=	=	SYM
cana-551	239	9	(	(	PUNCT
cana-551	239	10	𝜗	𝜗	PROPN
cana-551	239	11	,	,	PUNCT
cana-551	239	12	δ	δ	PROPN
cana-551	239	13	)	)	PUNCT
cana-551	239	14	is	be	AUX
cana-551	239	15	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	239	16	−open	−open	ADV
cana-551	239	17	set	set	VERB
cana-551	239	18	but	but	CCONJ
cana-551	239	19	not	not	PART
cana-551	239	20	soft	soft	ADJ
cana-551	239	21	open	open	ADJ
cana-551	239	22	set	set	NOUN
cana-551	239	23	.	.	PUNCT
cana-551	239	24	example	example	NOUN
cana-551	240	1	4.4	4.4	NUM
cana-551	240	2	:	:	PUNCT
cana-551	240	3	let	let	VERB
cana-551	240	4	𝒵	𝒵	PROPN
cana-551	240	5	=	=	PUNCT
cana-551	240	6	{	{	PUNCT
cana-551	240	7	휀	휀	NOUN
cana-551	240	8	,	,	PUNCT
cana-551	240	9	𝜇	𝜇	ADP
cana-551	240	10	,	,	PUNCT
cana-551	240	11	𝜔	𝜔	VERB
cana-551	240	12	}	}	PUNCT
cana-551	240	13	and	and	CCONJ
cana-551	240	14	∆=	∆=	ADJ
cana-551	240	15	{	{	PUNCT
cana-551	240	16	∇1	∇1	NOUN
cana-551	240	17	,	,	PUNCT
cana-551	240	18	∇2	∇2	PROPN
cana-551	240	19	}	}	PUNCT
cana-551	240	20	.	.	PUNCT
cana-551	241	1	let	let	VERB
cana-551	241	2	(	(	PUNCT
cana-551	241	3	𝛾1	𝛾1	PROPN
cana-551	241	4	,	,	PUNCT
cana-551	241	5	δ	δ	PROPN
cana-551	241	6	)	)	PUNCT
cana-551	241	7	,	,	PUNCT
cana-551	241	8	(	(	PUNCT
cana-551	241	9	𝛾2	𝛾2	VERB
cana-551	241	10	,	,	PUNCT
cana-551	241	11	δ	δ	PROPN
cana-551	241	12	)	)	PUNCT
cana-551	241	13	and	and	CCONJ
cana-551	241	14	(	(	PUNCT
cana-551	241	15	𝛾3	𝛾3	PROPN
cana-551	241	16	,	,	PUNCT
cana-551	241	17	δ	δ	PROPN
cana-551	241	18	)	)	PUNCT
cana-551	241	19	be	be	VERB
cana-551	241	20	soft	soft	ADJ
cana-551	241	21	sets	set	NOUN
cana-551	241	22	where	where	SCONJ
cana-551	241	23	:	:	PUNCT
cana-551	241	24	(	(	PUNCT
cana-551	241	25	𝛾1	𝛾1	PROPN
cana-551	241	26	,	,	PUNCT
cana-551	241	27	δ	δ	PROPN
cana-551	241	28	)	)	PUNCT
cana-551	241	29	=	=	PRON
cana-551	241	30	{	{	PUNCT
cana-551	241	31	(	(	PUNCT
cana-551	241	32	∇1	∇1	PROPN
cana-551	241	33	,	,	PUNCT
cana-551	241	34	{	{	PUNCT
cana-551	241	35	휀	휀	NOUN
cana-551	241	36	}	}	PUNCT
cana-551	241	37	,	,	PUNCT
cana-551	241	38	(	(	PUNCT
cana-551	241	39	∇2	∇2	PROPN
cana-551	241	40	,	,	PUNCT
cana-551	241	41	{	{	PUNCT
cana-551	241	42	휀	휀	NOUN
cana-551	241	43	}	}	PUNCT
cana-551	241	44	)	)	PUNCT
cana-551	241	45	)	)	PUNCT
cana-551	241	46	}	}	PUNCT
cana-551	241	47	,	,	PUNCT
cana-551	241	48	(	(	PUNCT
cana-551	241	49	𝛾2	𝛾2	VERB
cana-551	241	50	,	,	PUNCT
cana-551	241	51	δ	δ	NOUN
cana-551	241	52	)	)	PUNCT
cana-551	242	1	=	=	PRON
cana-551	242	2	{	{	PUNCT
cana-551	242	3	(	(	PUNCT
cana-551	242	4	∇1	∇1	PROPN
cana-551	242	5	,	,	PUNCT
cana-551	242	6	{	{	PUNCT
cana-551	242	7	𝜇	𝜇	X
cana-551	242	8	}	}	PUNCT
cana-551	242	9	)	)	PUNCT
cana-551	242	10	,	,	PUNCT
cana-551	242	11	(	(	PUNCT
cana-551	242	12	∇2	∇2	X
cana-551	242	13	,	,	PUNCT
cana-551	242	14	∅	∅	NOUN
cana-551	242	15	)	)	PUNCT
cana-551	242	16	}	}	PUNCT
cana-551	242	17	and	and	CCONJ
cana-551	242	18	communications	communication	NOUN
cana-551	242	19	on	on	ADP
cana-551	242	20	applied	apply	VERB
cana-551	242	21	nonlinear	nonlinear	ADJ
cana-551	242	22	analysis	analysis	NOUN
cana-551	242	23	issn	issn	NOUN
cana-551	242	24	:	:	PUNCT
cana-551	242	25	1074	1074	NUM
cana-551	242	26	-	-	PUNCT
cana-551	242	27	133x	133x	NUM
cana-551	242	28	vol	vol	NOUN
cana-551	242	29	31	31	NUM
cana-551	242	30	no	no	NOUN
cana-551	242	31	.	.	NOUN
cana-551	242	32	2	2	NUM
cana-551	242	33	(	(	PUNCT
cana-551	242	34	2024	2024	NUM
cana-551	242	35	)	)	PUNCT
cana-551	242	36	292	292	NUM
cana-551	242	37	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	242	38	(	(	PUNCT
cana-551	242	39	𝛾3	𝛾3	PROPN
cana-551	242	40	,	,	PUNCT
cana-551	242	41	δ	δ	PROPN
cana-551	242	42	)	)	PUNCT
cana-551	242	43	=	=	PRON
cana-551	242	44	{	{	PUNCT
cana-551	242	45	(	(	PUNCT
cana-551	242	46	∇1	∇1	PROPN
cana-551	242	47	,	,	PUNCT
cana-551	242	48	{	{	PUNCT
cana-551	242	49	휀	휀	NOUN
cana-551	242	50	,	,	PUNCT
cana-551	242	51	𝜔	𝜔	NOUN
cana-551	242	52	}	}	PUNCT
cana-551	242	53	)	)	PUNCT
cana-551	242	54	,	,	PUNCT
cana-551	242	55	(	(	PUNCT
cana-551	242	56	∇2	∇2	PROPN
cana-551	242	57	,	,	PUNCT
cana-551	242	58	{	{	PUNCT
cana-551	242	59	휀	휀	NOUN
cana-551	242	60	}	}	PUNCT
cana-551	242	61	)	)	PUNCT
cana-551	242	62	}	}	PUNCT
cana-551	242	63	and	and	CCONJ
cana-551	242	64	𝔚	𝔚	NOUN
cana-551	242	65	=	=	PRON
cana-551	242	66	{	{	PUNCT
cana-551	242	67	�	�	PROPN
cana-551	242	68	̃	̃	PROPN
cana-551	242	69	�	�	PROPN
cana-551	242	70	,	,	PUNCT
cana-551	242	71	∅̃	∅̃	NOUN
cana-551	242	72	,	,	PUNCT
cana-551	242	73	(	(	PUNCT
cana-551	242	74	𝛾1	𝛾1	PROPN
cana-551	242	75	,	,	PUNCT
cana-551	242	76	δ	δ	PROPN
cana-551	242	77	)	)	PUNCT
cana-551	242	78	,	,	PUNCT
cana-551	242	79	(	(	PUNCT
cana-551	242	80	𝛾2	𝛾2	VERB
cana-551	242	81	,	,	PUNCT
cana-551	242	82	δ	δ	PROPN
cana-551	242	83	)	)	PUNCT
cana-551	242	84	,	,	PUNCT
cana-551	242	85	(	(	PUNCT
cana-551	242	86	𝛾3	𝛾3	PROPN
cana-551	242	87	,	,	PUNCT
cana-551	242	88	δ	δ	PROPN
cana-551	242	89	)	)	PUNCT
cana-551	242	90	}	}	PUNCT
cana-551	242	91	is	be	AUX
cana-551	242	92	the	the	DET
cana-551	242	93	soft	soft	ADJ
cana-551	242	94	topology	topology	NOUN
cana-551	242	95	over	over	ADP
cana-551	242	96	𝒵.	𝒵.	PROPN
cana-551	242	97	let	let	VERB
cana-551	242	98	ῐ	ῐ	PROPN
cana-551	242	99	=	=	PRON
cana-551	242	100	{	{	PUNCT
cana-551	242	101	∅̃	∅̃	NOUN
cana-551	242	102	}	}	PUNCT
cana-551	242	103	be	be	AUX
cana-551	242	104	a	a	DET
cana-551	242	105	soft	soft	ADJ
cana-551	242	106	ideal	ideal	NOUN
cana-551	242	107	on	on	ADP
cana-551	242	108	𝒵.	𝒵.	PROPN
cana-551	242	109	let	let	VERB
cana-551	242	110	𝒟	𝒟	NOUN
cana-551	242	111	=	=	SYM
cana-551	242	112	{	{	PUNCT
cana-551	242	113	𝜎	𝜎	PROPN
cana-551	242	114	,	,	PUNCT
cana-551	242	115	𝜌	𝜌	X
cana-551	242	116	,	,	PUNCT
cana-551	242	117	𝜋	𝜋	NOUN
cana-551	242	118	}	}	PUNCT
cana-551	242	119	and	and	CCONJ
cana-551	242	120	θ	θ	NOUN
cana-551	242	121	=	=	SYM
cana-551	242	122	{	{	PUNCT
cana-551	242	123	ϱ1	ϱ1	NOUN
cana-551	242	124	,	,	PUNCT
cana-551	242	125	𝜚2	𝜚2	PROPN
cana-551	242	126	}	}	PUNCT
cana-551	242	127	,	,	PUNCT
cana-551	242	128	℧	℧	PROPN
cana-551	242	129	=	=	PRON
cana-551	242	130	{	{	PUNCT
cana-551	242	131	�	�	PROPN
cana-551	242	132	̃	̃	PROPN
cana-551	242	133	�	�	PROPN
cana-551	242	134	,	,	PUNCT
cana-551	242	135	∅̃	∅̃	NOUN
cana-551	242	136	,	,	PUNCT
cana-551	242	137	(	(	PUNCT
cana-551	242	138	𝛿	𝛿	ADJ
cana-551	242	139	,	,	PUNCT
cana-551	242	140	θ	θ	NOUN
cana-551	242	141	)	)	PUNCT
cana-551	242	142	}	}	PUNCT
cana-551	242	143	is	be	AUX
cana-551	242	144	soft	soft	ADJ
cana-551	242	145	topology	topology	NOUN
cana-551	242	146	on	on	ADP
cana-551	242	147	𝒟.	𝒟.	PROPN
cana-551	242	148	where	where	SCONJ
cana-551	242	149	(	(	PUNCT
cana-551	242	150	𝛿	𝛿	ADJ
cana-551	242	151	,	,	PUNCT
cana-551	242	152	θ	θ	NOUN
cana-551	242	153	)	)	PUNCT
cana-551	242	154	=	=	SYM
cana-551	242	155	{	{	PUNCT
cana-551	242	156	(	(	PUNCT
cana-551	242	157	ϱ1	ϱ1	NOUN
cana-551	242	158	,	,	PUNCT
cana-551	242	159	{	{	PUNCT
cana-551	242	160	𝜎	𝜎	NOUN
cana-551	242	161	}	}	PUNCT
cana-551	242	162	)	)	PUNCT
cana-551	242	163	,	,	PUNCT
cana-551	242	164	(	(	PUNCT
cana-551	242	165	ϱ1	ϱ1	NOUN
cana-551	242	166	,	,	PUNCT
cana-551	242	167	∅	∅	NOUN
cana-551	242	168	)	)	PUNCT
cana-551	242	169	}	}	PUNCT
cana-551	242	170	.	.	PUNCT
cana-551	243	1	then	then	ADV
cana-551	243	2	let	let	VERB
cana-551	243	3	𝛺	𝛺	NOUN
cana-551	243	4	:	:	PUNCT
cana-551	243	5	(	(	PUNCT
cana-551	243	6	𝒵	𝒵	PROPN
cana-551	243	7	,	,	PUNCT
cana-551	243	8	𝔚	𝔚	PROPN
cana-551	243	9	,	,	PUNCT
cana-551	243	10	𝛥	𝛥	PROPN
cana-551	243	11	)	)	PUNCT
cana-551	243	12	→	→	SYM
cana-551	243	13	(	(	PUNCT
cana-551	243	14	𝒟	𝒟	PROPN
cana-551	243	15	,	,	PUNCT
cana-551	243	16	℧	℧	PROPN
cana-551	243	17	,	,	PUNCT
cana-551	243	18	θ	θ	PROPN
cana-551	243	19	)	)	PUNCT
cana-551	243	20	be	be	VERB
cana-551	243	21	a	a	DET
cana-551	243	22	soft	soft	ADJ
cana-551	243	23	function	function	NOUN
cana-551	243	24	and	and	CCONJ
cana-551	243	25	𝑢	𝑢	NOUN
cana-551	243	26	:	:	PUNCT
cana-551	243	27	𝒵	𝒵	PROPN
cana-551	243	28	→	→	SYM
cana-551	243	29	𝒟	𝒟	PROPN
cana-551	243	30	and	and	CCONJ
cana-551	243	31	𝑝	𝑝	NOUN
cana-551	243	32	:	:	PUNCT
cana-551	243	33	∆→	∆→	PROPN
cana-551	244	1	θ	θ	PROPN
cana-551	244	2	denoted	denote	VERB
cana-551	244	3	by	by	ADP
cana-551	244	4	𝑢(휀	𝑢(휀	NOUN
cana-551	244	5	)	)	PUNCT
cana-551	244	6	=	=	SYM
cana-551	244	7	𝜎	𝜎	PROPN
cana-551	244	8	,	,	PUNCT
cana-551	244	9	𝑢(𝜇	𝑢(𝜇	PROPN
cana-551	244	10	)	)	PUNCT
cana-551	244	11	=	=	SYM
cana-551	244	12	𝜌	𝜌	X
cana-551	244	13	,	,	PUNCT
cana-551	244	14	𝑢(𝜔	𝑢(𝜔	PROPN
cana-551	244	15	)	)	PUNCT
cana-551	244	16	=	=	SYM
cana-551	244	17	𝜋	𝜋	NOUN
cana-551	244	18	,	,	PUNCT
cana-551	244	19	𝑝(∇1	𝑝(∇1	ADJ
cana-551	244	20	)	)	PUNCT
cana-551	244	21	=	=	SYM
cana-551	244	22	ϱ1	ϱ1	NOUN
cana-551	244	23	,	,	PUNCT
cana-551	244	24	𝑝(∇2	𝑝(∇2	ADJ
cana-551	244	25	)	)	PUNCT
cana-551	244	26	=	=	SYM
cana-551	244	27	ϱ2	ϱ2	NOUN
cana-551	244	28	.	.	PUNCT
cana-551	245	1	let	let	AUX
cana-551	245	2	take	take	VERB
cana-551	245	3	(	(	PUNCT
cana-551	245	4	𝜗	𝜗	NOUN
cana-551	245	5	,	,	PUNCT
cana-551	245	6	δ	δ	PROPN
cana-551	245	7	)	)	PUNCT
cana-551	245	8	=	=	PRON
cana-551	245	9	{	{	PUNCT
cana-551	245	10	(	(	PUNCT
cana-551	245	11	∇1	∇1	PROPN
cana-551	245	12	,	,	PUNCT
cana-551	245	13	{	{	PUNCT
cana-551	245	14	휀	휀	NOUN
cana-551	245	15	}	}	PUNCT
cana-551	245	16	)	)	PUNCT
cana-551	245	17	,	,	PUNCT
cana-551	245	18	(	(	PUNCT
cana-551	245	19	∇2	∇2	X
cana-551	245	20	,	,	PUNCT
cana-551	245	21	∅	∅	NOUN
cana-551	245	22	)	)	PUNCT
cana-551	245	23	}	}	PUNCT
cana-551	245	24	.	.	PUNCT
cana-551	246	1	then	then	ADV
cana-551	246	2	ω	ω	PROPN
cana-551	246	3	is	be	AUX
cana-551	246	4	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	246	5	−continuous	−continuous	ADJ
cana-551	246	6	but	but	CCONJ
cana-551	246	7	not	not	PART
cana-551	246	8	soft	soft	ADJ
cana-551	246	9	ῐg	ῐg	ADP
cana-551	246	10	−continuous	−continuous	PROPN
cana-551	246	11	.	.	PUNCT
cana-551	247	1	since	since	SCONJ
cana-551	247	2	if	if	SCONJ
cana-551	247	3	𝛺−1((𝛿	𝛺−1((𝛿	PROPN
cana-551	247	4	,	,	PUNCT
cana-551	247	5	θ	θ	NOUN
cana-551	247	6	)	)	PUNCT
cana-551	247	7	)	)	PUNCT
cana-551	247	8	=	=	SYM
cana-551	247	9	(	(	PUNCT
cana-551	247	10	𝜗	𝜗	PROPN
cana-551	247	11	,	,	PUNCT
cana-551	247	12	δ	δ	PROPN
cana-551	247	13	)	)	PUNCT
cana-551	247	14	is	be	AUX
cana-551	247	15	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	247	16	−closed	−close	VERB
cana-551	247	17	set	set	NOUN
cana-551	247	18	but	but	CCONJ
cana-551	247	19	not	not	PART
cana-551	247	20	soft	soft	ADJ
cana-551	247	21	ῐg	ῐg	ADP
cana-551	247	22	−closed	−close	VERB
cana-551	247	23	set	set	NOUN
cana-551	247	24	.	.	PUNCT
cana-551	248	1	definition	definition	NOUN
cana-551	248	2	4.5	4.5	NUM
cana-551	248	3	:	:	PUNCT
cana-551	248	4	let	let	VERB
cana-551	248	5	𝛺	𝛺	VERB
cana-551	248	6	:	:	PUNCT
cana-551	248	7	(	(	PUNCT
cana-551	248	8	𝒵	𝒵	PROPN
cana-551	248	9	,	,	PUNCT
cana-551	248	10	𝔚	𝔚	PROPN
cana-551	248	11	,	,	PUNCT
cana-551	248	12	𝛥	𝛥	PROPN
cana-551	248	13	)	)	PUNCT
cana-551	248	14	→	→	SYM
cana-551	248	15	(	(	PUNCT
cana-551	248	16	𝒟	𝒟	PROPN
cana-551	248	17	,	,	PUNCT
cana-551	248	18	℧	℧	PROPN
cana-551	248	19	,	,	PUNCT
cana-551	248	20	θ	θ	PROPN
cana-551	248	21	)	)	PUNCT
cana-551	248	22	be	be	VERB
cana-551	248	23	a	a	DET
cana-551	248	24	soft	soft	ADJ
cana-551	248	25	mapping	mapping	NOUN
cana-551	248	26	.	.	PUNCT
cana-551	249	1	if	if	SCONJ
cana-551	249	2	𝛺−1((𝛿	𝛺−1((𝛿	PROPN
cana-551	249	3	,	,	PUNCT
cana-551	249	4	δ	δ	PROPN
cana-551	249	5	)	)	PUNCT
cana-551	249	6	)	)	PUNCT
cana-551	249	7	is	be	AUX
cana-551	249	8	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	249	9	−closed	−close	VERB
cana-551	249	10	in	in	ADP
cana-551	249	11	(	(	PUNCT
cana-551	249	12	𝒵	𝒵	PROPN
cana-551	249	13	,	,	PUNCT
cana-551	249	14	𝔚	𝔚	PROPN
cana-551	249	15	,	,	PUNCT
cana-551	249	16	𝛥	𝛥	NOUN
cana-551	249	17	)	)	PUNCT
cana-551	249	18	for	for	ADP
cana-551	249	19	each	each	DET
cana-551	249	20	ssƅ∗	ssƅ∗	NOUN
cana-551	249	21	−closed	−close	VERB
cana-551	249	22	set	set	NOUN
cana-551	249	23	(	(	PUNCT
cana-551	249	24	𝛿	𝛿	ADJ
cana-551	249	25	,	,	PUNCT
cana-551	249	26	δ	δ	PROPN
cana-551	249	27	)	)	PUNCT
cana-551	249	28	of	of	ADP
cana-551	249	29	(	(	PUNCT
cana-551	249	30	𝒟	𝒟	PROPN
cana-551	249	31	,	,	PUNCT
cana-551	249	32	℧	℧	PROPN
cana-551	249	33	,	,	PUNCT
cana-551	249	34	θ	θ	PROPN
cana-551	249	35	)	)	PUNCT
cana-551	249	36	,	,	PUNCT
cana-551	249	37	then	then	ADV
cana-551	249	38	𝛺	𝛺	PROPN
cana-551	249	39	is	be	AUX
cana-551	249	40	said	say	VERB
cana-551	249	41	to	to	PART
cana-551	249	42	be	be	AUX
cana-551	249	43	soft	soft	ADJ
cana-551	249	44	strongly	strongly	ADV
cana-551	249	45	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	249	46	−irresolute	−irresolute	NOUN
cana-551	249	47	function	function	NOUN
cana-551	249	48	.	.	PUNCT
cana-551	250	1	theorem	theorem	VERB
cana-551	250	2	4.6	4.6	NUM
cana-551	250	3	:	:	PUNCT
cana-551	250	4	a	a	DET
cana-551	250	5	map	map	NOUN
cana-551	250	6	𝛺	𝛺	NOUN
cana-551	250	7	:	:	PUNCT
cana-551	250	8	(	(	PUNCT
cana-551	250	9	𝒵	𝒵	PROPN
cana-551	250	10	,	,	PUNCT
cana-551	250	11	𝔚	𝔚	PROPN
cana-551	250	12	,	,	PUNCT
cana-551	250	13	𝛥	𝛥	PROPN
cana-551	250	14	)	)	PUNCT
cana-551	250	15	→	→	SYM
cana-551	250	16	(	(	PUNCT
cana-551	250	17	𝒟	𝒟	PROPN
cana-551	250	18	,	,	PUNCT
cana-551	250	19	℧	℧	PROPN
cana-551	250	20	,	,	PUNCT
cana-551	250	21	θ	θ	PROPN
cana-551	250	22	)	)	PUNCT
cana-551	250	23	is	be	AUX
cana-551	250	24	ssƅ∗	ssƅ∗	ADJ
cana-551	250	25	−	−	PROPN
cana-551	250	26	𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒	𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒	PROPN
cana-551	251	1	if	if	SCONJ
cana-551	251	2	and	and	CCONJ
cana-551	251	3	only	only	ADV
cana-551	251	4	if	if	SCONJ
cana-551	251	5	the	the	DET
cana-551	251	6	inverse	inverse	ADJ
cana-551	251	7	image	image	NOUN
cana-551	251	8	of	of	ADP
cana-551	251	9	every	every	DET
cana-551	251	10	soft	soft	ADJ
cana-551	251	11	strongly	strongly	ADV
cana-551	251	12	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	251	13	−	−	NOUN
cana-551	251	14	𝑜𝑝𝑒𝑛	𝑜𝑝𝑒𝑛	NOUN
cana-551	251	15	set	set	VERB
cana-551	251	16	in	in	ADP
cana-551	251	17	𝒟	𝒟	PROPN
cana-551	251	18	is	be	AUX
cana-551	251	19	soft	soft	ADJ
cana-551	251	20	strongly	strongly	ADV
cana-551	251	21	ƅ∗	ƅ∗	NOUN
cana-551	251	22	−	−	NOUN
cana-551	251	23	𝑜𝑝𝑒𝑛	𝑜𝑝𝑒𝑛	NOUN
cana-551	251	24	in	in	ADP
cana-551	251	25	𝒵.	𝒵.	NOUN
cana-551	251	26	proof	proof	NOUN
cana-551	251	27	.	.	PUNCT
cana-551	252	1	clearly	clearly	ADV
cana-551	252	2	.	.	PUNCT
cana-551	253	1	theorem	theorem	VERB
cana-551	253	2	4.8	4.8	NUM
cana-551	253	3	:	:	PUNCT
cana-551	253	4	every	every	DET
cana-551	253	5	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	253	6	−irresolute	−irresolute	VERB
cana-551	253	7	mapping	mapping	NOUN
cana-551	253	8	is	be	AUX
cana-551	253	9	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	253	10	−continuous	−continuous	ADJ
cana-551	253	11	functions	function	NOUN
cana-551	253	12	.	.	PUNCT
cana-551	254	1	proof	proof	NOUN
cana-551	254	2	.	.	PUNCT
cana-551	255	1	let	let	VERB
cana-551	255	2	𝛺	𝛺	VERB
cana-551	255	3	:	:	PUNCT
cana-551	255	4	(	(	PUNCT
cana-551	255	5	𝒵	𝒵	PROPN
cana-551	255	6	,	,	PUNCT
cana-551	255	7	𝔚	𝔚	PROPN
cana-551	255	8	,	,	PUNCT
cana-551	255	9	𝛥	𝛥	PROPN
cana-551	255	10	)	)	PUNCT
cana-551	255	11	→	→	SYM
cana-551	255	12	(	(	PUNCT
cana-551	255	13	𝒟	𝒟	PROPN
cana-551	255	14	,	,	PUNCT
cana-551	255	15	℧	℧	PROPN
cana-551	255	16	,	,	PUNCT
cana-551	255	17	θ	θ	PROPN
cana-551	255	18	)	)	PUNCT
cana-551	255	19	be	be	VERB
cana-551	255	20	a	a	DET
cana-551	255	21	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	255	22	−irresolute	−irresolute	NOUN
cana-551	256	1	mapping	mapping	NOUN
cana-551	256	2	.	.	PUNCT
cana-551	257	1	let	let	AUX
cana-551	257	2	(	(	PUNCT
cana-551	257	3	𝛿	𝛿	ADJ
cana-551	257	4	,	,	PUNCT
cana-551	257	5	δ	δ	PROPN
cana-551	257	6	)	)	PUNCT
cana-551	257	7	be	be	VERB
cana-551	257	8	a	a	DET
cana-551	257	9	soft	soft	ADJ
cana-551	257	10	closed	closed	ADJ
cana-551	257	11	set	set	NOUN
cana-551	257	12	in	in	ADP
cana-551	257	13	𝒟.	𝒟.	PROPN
cana-551	257	14	then	then	ADV
cana-551	257	15	(	(	PUNCT
cana-551	257	16	𝛿	𝛿	ADJ
cana-551	257	17	,	,	PUNCT
cana-551	257	18	δ	δ	PROPN
cana-551	257	19	)	)	PUNCT
cana-551	257	20	is	be	AUX
cana-551	257	21	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	257	22	−closed	−close	VERB
cana-551	257	23	set	set	VERB
cana-551	257	24	in	in	ADP
cana-551	257	25	𝒟.	𝒟.	PROPN
cana-551	257	26	since	since	SCONJ
cana-551	257	27	𝛺	𝛺	PROPN
cana-551	257	28	is	be	AUX
cana-551	257	29	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	257	30	−irresolute	−irresolute	NOUN
cana-551	257	31	mapping	mapping	NOUN
cana-551	257	32	,	,	PUNCT
cana-551	257	33	𝛺−1((𝛿	𝛺−1((𝛿	PROPN
cana-551	257	34	,	,	PUNCT
cana-551	257	35	δ	δ	PROPN
cana-551	257	36	)	)	PUNCT
cana-551	257	37	)	)	PUNCT
cana-551	258	1	is	be	AUX
cana-551	258	2	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	258	3	−closed	−close	VERB
cana-551	258	4	set	set	VERB
cana-551	258	5	in	in	ADP
cana-551	258	6	𝒵.	𝒵.	PROPN
cana-551	258	7	hence	hence	ADV
cana-551	258	8	,	,	PUNCT
cana-551	258	9	𝛺	𝛺	PROPN
cana-551	258	10	is	be	AUX
cana-551	258	11	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	258	12	−ccontinuous	−ccontinuous	ADJ
cana-551	258	13	function	function	NOUN
cana-551	258	14	.	.	PUNCT
cana-551	259	1	definition	definition	NOUN
cana-551	259	2	4.9	4.9	NUM
cana-551	259	3	:	:	PUNCT
cana-551	259	4	a	a	DET
cana-551	259	5	soft	soft	ADJ
cana-551	259	6	mapping	mapping	NOUN
cana-551	259	7	𝛺	𝛺	NOUN
cana-551	259	8	:	:	PUNCT
cana-551	259	9	(	(	PUNCT
cana-551	259	10	𝒵	𝒵	PROPN
cana-551	259	11	,	,	PUNCT
cana-551	259	12	𝔚	𝔚	PROPN
cana-551	259	13	,	,	PUNCT
cana-551	259	14	𝛥	𝛥	PROPN
cana-551	259	15	)	)	PUNCT
cana-551	259	16	→	→	SYM
cana-551	259	17	(	(	PUNCT
cana-551	259	18	𝒟	𝒟	PROPN
cana-551	259	19	,	,	PUNCT
cana-551	259	20	℧	℧	PROPN
cana-551	259	21	,	,	PUNCT
cana-551	259	22	θ	θ	PROPN
cana-551	259	23	)	)	PUNCT
cana-551	259	24	is	be	AUX
cana-551	259	25	said	say	VERB
cana-551	259	26	to	to	PART
cana-551	259	27	be	be	AUX
cana-551	259	28	soft	soft	ADJ
cana-551	259	29	strongly	strongly	ADV
cana-551	259	30	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	259	31	−open	−open	ADJ
cana-551	259	32	(	(	PUNCT
cana-551	259	33	soft	soft	ADJ
cana-551	259	34	strongly	strongly	ADV
cana-551	259	35	ƅ∗ῐ	ƅ∗ῐ	NUM
cana-551	259	36	−closed	−closed	ADJ
cana-551	259	37	)	)	PUNCT
cana-551	259	38	map	map	VERB
cana-551	259	39	if	if	SCONJ
cana-551	259	40	the	the	DET
cana-551	259	41	image	image	NOUN
cana-551	259	42	of	of	ADP
cana-551	259	43	every	every	DET
cana-551	259	44	soft	soft	ADJ
cana-551	259	45	𝑜𝑝𝑒𝑛	𝑜𝑝𝑒𝑛	NOUN
cana-551	259	46	(	(	PUNCT
cana-551	259	47	soft	soft	ADJ
cana-551	259	48	𝑐𝑙𝑜𝑠𝑒𝑑	𝑐𝑙𝑜𝑠𝑒𝑑	NOUN
cana-551	259	49	)	)	PUNCT
cana-551	259	50	set	set	VERB
cana-551	259	51	in	in	ADP
cana-551	259	52	𝒵	𝒵	PROPN
cana-551	259	53	is	be	AUX
cana-551	259	54	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	259	55	−open	−open	VERB
cana-551	259	56	(	(	PUNCT
cana-551	259	57	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	259	58	−closed	−close	VERB
cana-551	259	59	)	)	PUNCT
cana-551	259	60	set	set	VERB
cana-551	259	61	in	in	ADP
cana-551	259	62	𝒟.	𝒟.	PROPN
cana-551	259	63	remark	remark	NOUN
cana-551	259	64	4.10	4.10	NUM
cana-551	259	65	:	:	PUNCT
cana-551	259	66	(	(	PUNCT
cana-551	259	67	1	1	X
cana-551	259	68	)	)	PUNCT
cana-551	259	69	every	every	DET
cana-551	259	70	soft	soft	ADJ
cana-551	259	71	open	open	ADJ
cana-551	259	72	map	map	NOUN
cana-551	259	73	is	be	AUX
cana-551	259	74	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	259	75	−open	−open	NOUN
cana-551	259	76	.	.	PUNCT
cana-551	260	1	(	(	PUNCT
cana-551	260	2	2	2	X
cana-551	260	3	)	)	PUNCT
cana-551	260	4	every	every	DET
cana-551	260	5	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	260	6	−open	−open	NOUN
cana-551	260	7	map	map	NOUN
cana-551	260	8	is	be	AUX
cana-551	260	9	ssƅ∗	ssƅ∗	ADJ
cana-551	260	10	−open	−open	ADJ
cana-551	260	11	.	.	PUNCT
cana-551	261	1	in	in	ADP
cana-551	261	2	the	the	DET
cana-551	261	3	following	follow	VERB
cana-551	261	4	examples	example	NOUN
cana-551	261	5	as	as	SCONJ
cana-551	261	6	observed	observe	VERB
cana-551	261	7	the	the	DET
cana-551	261	8	converses	converse	NOUN
cana-551	261	9	are	be	AUX
cana-551	261	10	not	not	PART
cana-551	261	11	true	true	ADJ
cana-551	261	12	.	.	PUNCT
cana-551	262	1	example	example	NOUN
cana-551	262	2	4.11	4.11	NUM
cana-551	262	3	:	:	PUNCT
cana-551	262	4	let	let	VERB
cana-551	262	5	𝒵	𝒵	PROPN
cana-551	262	6	=	=	PUNCT
cana-551	262	7	{	{	PUNCT
cana-551	262	8	휀	휀	NOUN
cana-551	262	9	,	,	PUNCT
cana-551	262	10	𝜇	𝜇	ADP
cana-551	262	11	}	}	PUNCT
cana-551	262	12	and	and	CCONJ
cana-551	262	13	∆=	∆=	ADJ
cana-551	262	14	{	{	PUNCT
cana-551	262	15	∇1	∇1	NOUN
cana-551	262	16	,	,	PUNCT
cana-551	262	17	∇2	∇2	PROPN
cana-551	262	18	}	}	PUNCT
cana-551	262	19	.	.	PUNCT
cana-551	263	1	𝔚	𝔚	NOUN
cana-551	263	2	=	=	PRON
cana-551	263	3	{	{	PUNCT
cana-551	263	4	�	�	PROPN
cana-551	263	5	̃	̃	PROPN
cana-551	263	6	�	�	PROPN
cana-551	263	7	,	,	PUNCT
cana-551	263	8	∅̃	∅̃	NOUN
cana-551	263	9	,	,	PUNCT
cana-551	263	10	(	(	PUNCT
cana-551	263	11	𝛾	𝛾	PROPN
cana-551	263	12	,	,	PUNCT
cana-551	263	13	δ	δ	PROPN
cana-551	263	14	)	)	PUNCT
cana-551	263	15	}	}	PUNCT
cana-551	263	16	is	be	AUX
cana-551	263	17	the	the	DET
cana-551	263	18	soft	soft	ADJ
cana-551	263	19	topology	topology	NOUN
cana-551	263	20	over	over	ADP
cana-551	263	21	𝒵	𝒵	PROPN
cana-551	263	22	where	where	SCONJ
cana-551	263	23	(	(	PUNCT
cana-551	263	24	𝛾	𝛾	PROPN
cana-551	263	25	,	,	PUNCT
cana-551	263	26	δ	δ	NOUN
cana-551	263	27	)	)	PUNCT
cana-551	263	28	=	=	PRON
cana-551	263	29	{	{	PUNCT
cana-551	263	30	(	(	PUNCT
cana-551	263	31	∇1	∇1	PROPN
cana-551	263	32	,	,	PUNCT
cana-551	263	33	{	{	PUNCT
cana-551	263	34	휀	휀	NOUN
cana-551	263	35	}	}	PUNCT
cana-551	263	36	)	)	PUNCT
cana-551	263	37	}	}	PUNCT
cana-551	263	38	.	.	PUNCT
cana-551	264	1	let	let	VERB
cana-551	264	2	𝐽	𝐽	PRON
cana-551	264	3	=	=	PRON
cana-551	264	4	{	{	PUNCT
cana-551	264	5	∅̃	∅̃	NOUN
cana-551	264	6	}	}	PUNCT
cana-551	264	7	be	be	AUX
cana-551	264	8	a	a	DET
cana-551	264	9	soft	soft	ADJ
cana-551	264	10	ideal	ideal	NOUN
cana-551	264	11	on	on	ADP
cana-551	264	12	𝒵.	𝒵.	PROPN
cana-551	264	13	also	also	ADV
cana-551	264	14	,	,	PUNCT
cana-551	264	15	let	let	VERB
cana-551	264	16	𝒟	𝒟	NOUN
cana-551	264	17	=	=	PUNCT
cana-551	264	18	{	{	PUNCT
cana-551	264	19	𝜎	𝜎	PROPN
cana-551	264	20	,	,	PUNCT
cana-551	264	21	𝜌	𝜌	ADP
cana-551	264	22	}	}	PUNCT
cana-551	264	23	and	and	CCONJ
cana-551	264	24	θ	θ	NOUN
cana-551	264	25	=	=	SYM
cana-551	264	26	{	{	PUNCT
cana-551	264	27	ϱ1	ϱ1	NOUN
cana-551	264	28	,	,	PUNCT
cana-551	264	29	𝜚2	𝜚2	PROPN
cana-551	264	30	}	}	PUNCT
cana-551	264	31	,	,	PUNCT
cana-551	264	32	℧	℧	PROPN
cana-551	264	33	=	=	PRON
cana-551	264	34	{	{	PUNCT
cana-551	264	35	�	�	PROPN
cana-551	264	36	̃	̃	PROPN
cana-551	264	37	�	�	PROPN
cana-551	264	38	,	,	PUNCT
cana-551	264	39	∅̃	∅̃	NOUN
cana-551	264	40	,	,	PUNCT
cana-551	264	41	(	(	PUNCT
cana-551	264	42	𝛿1	𝛿1	PROPN
cana-551	264	43	,	,	PUNCT
cana-551	264	44	θ	θ	PROPN
cana-551	264	45	)	)	PUNCT
cana-551	264	46	,	,	PUNCT
cana-551	264	47	(	(	PUNCT
cana-551	264	48	𝛿2	𝛿2	NOUN
cana-551	264	49	,	,	PUNCT
cana-551	264	50	θ	θ	NOUN
cana-551	264	51	)	)	PUNCT
cana-551	264	52	}	}	PUNCT
cana-551	264	53	is	be	AUX
cana-551	264	54	soft	soft	ADJ
cana-551	264	55	topology	topology	NOUN
cana-551	264	56	on	on	ADP
cana-551	264	57	𝒟.	𝒟.	PROPN
cana-551	265	1	where	where	SCONJ
cana-551	265	2	(	(	PUNCT
cana-551	265	3	𝛿1	𝛿1	NOUN
cana-551	265	4	,	,	PUNCT
cana-551	265	5	θ	θ	NOUN
cana-551	265	6	)	)	PUNCT
cana-551	265	7	=	=	SYM
cana-551	265	8	{	{	PUNCT
cana-551	265	9	(	(	PUNCT
cana-551	265	10	ϱ1	ϱ1	NOUN
cana-551	265	11	,	,	PUNCT
cana-551	265	12	{	{	PUNCT
cana-551	265	13	𝜎	𝜎	NOUN
cana-551	265	14	}	}	PUNCT
cana-551	265	15	)	)	PUNCT
cana-551	265	16	,	,	PUNCT
cana-551	265	17	(	(	PUNCT
cana-551	265	18	ϱ2	ϱ2	NOUN
cana-551	265	19	,	,	PUNCT
cana-551	265	20	{	{	PUNCT
cana-551	265	21	𝜌	𝜌	X
cana-551	265	22	}	}	PUNCT
cana-551	265	23	)	)	PUNCT
cana-551	265	24	}	}	PUNCT
cana-551	265	25	and	and	CCONJ
cana-551	265	26	(	(	PUNCT
cana-551	265	27	𝛿2	𝛿2	NOUN
cana-551	265	28	,	,	PUNCT
cana-551	265	29	θ	θ	NOUN
cana-551	265	30	)	)	PUNCT
cana-551	265	31	=	=	SYM
cana-551	265	32	{	{	PUNCT
cana-551	265	33	(	(	PUNCT
cana-551	265	34	ϱ1	ϱ1	NOUN
cana-551	265	35	,	,	PUNCT
cana-551	265	36	{	{	PUNCT
cana-551	265	37	𝜌	𝜌	X
cana-551	265	38	}	}	PUNCT
cana-551	265	39	)	)	PUNCT
cana-551	265	40	,	,	PUNCT
cana-551	265	41	(	(	PUNCT
cana-551	265	42	ϱ2	ϱ2	NOUN
cana-551	265	43	,	,	PUNCT
cana-551	265	44	{	{	PUNCT
cana-551	265	45	𝜎	𝜎	NOUN
cana-551	265	46	}	}	PUNCT
cana-551	265	47	)	)	PUNCT
cana-551	265	48	}	}	PUNCT
cana-551	265	49	.	.	PUNCT
cana-551	266	1	then	then	ADV
cana-551	266	2	the	the	DET
cana-551	266	3	soft	soft	ADJ
cana-551	266	4	function	function	NOUN
cana-551	266	5	𝛺	𝛺	PROPN
cana-551	266	6	:	:	PUNCT
cana-551	266	7	(	(	PUNCT
cana-551	266	8	𝒵	𝒵	PROPN
cana-551	266	9	,	,	PUNCT
cana-551	266	10	𝔚	𝔚	PROPN
cana-551	266	11	,	,	PUNCT
cana-551	266	12	𝛥	𝛥	PROPN
cana-551	266	13	)	)	PUNCT
cana-551	266	14	→	→	SYM
cana-551	266	15	(	(	PUNCT
cana-551	266	16	𝒟	𝒟	PROPN
cana-551	266	17	,	,	PUNCT
cana-551	266	18	℧	℧	PROPN
cana-551	266	19	,	,	PUNCT
cana-551	266	20	θ	θ	PROPN
cana-551	266	21	)	)	PUNCT
cana-551	266	22	where	where	SCONJ
cana-551	266	23	𝑢	𝑢	X
cana-551	266	24	:	:	PUNCT
cana-551	266	25	𝒵	𝒵	PROPN
cana-551	266	26	→	→	SYM
cana-551	266	27	𝒟	𝒟	PROPN
cana-551	266	28	and	and	CCONJ
cana-551	266	29	𝑝	𝑝	NOUN
cana-551	266	30	:	:	PUNCT
cana-551	266	31	∆→	∆→	PROPN
cana-551	267	1	θ	θ	PROPN
cana-551	267	2	denoted	denote	VERB
cana-551	267	3	by	by	ADP
cana-551	267	4	𝑢(휀	𝑢(휀	NOUN
cana-551	267	5	)	)	PUNCT
cana-551	267	6	=	=	SYM
cana-551	267	7	𝜎	𝜎	PROPN
cana-551	267	8	,	,	PUNCT
cana-551	267	9	𝑢(𝜇	𝑢(𝜇	PROPN
cana-551	267	10	)	)	PUNCT
cana-551	267	11	=	=	SYM
cana-551	267	12	𝜌	𝜌	X
cana-551	267	13	,	,	PUNCT
cana-551	267	14	𝑢(𝜔	𝑢(𝜔	PROPN
cana-551	267	15	)	)	PUNCT
cana-551	267	16	=	=	SYM
cana-551	267	17	𝜋	𝜋	NOUN
cana-551	267	18	,	,	PUNCT
cana-551	267	19	𝑝(∇1	𝑝(∇1	ADJ
cana-551	267	20	)	)	PUNCT
cana-551	267	21	=	=	SYM
cana-551	267	22	ϱ1	ϱ1	NOUN
cana-551	267	23	,	,	PUNCT
cana-551	267	24	𝑝(∇2	𝑝(∇2	ADJ
cana-551	267	25	)	)	PUNCT
cana-551	267	26	=	=	VERB
cana-551	267	27	ϱ2	ϱ2	NOUN
cana-551	267	28	is	be	AUX
cana-551	267	29	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	267	30	−open	−open	ADJ
cana-551	267	31	but	but	CCONJ
cana-551	267	32	not	not	PART
cana-551	267	33	soft	soft	ADJ
cana-551	267	34	open	open	ADJ
cana-551	267	35	.	.	PUNCT
cana-551	268	1	since	since	SCONJ
cana-551	268	2	for	for	ADP
cana-551	268	3	each	each	DET
cana-551	268	4	soft	soft	ADJ
cana-551	268	5	open	open	ADJ
cana-551	268	6	(	(	PUNCT
cana-551	268	7	𝛾	𝛾	PROPN
cana-551	268	8	,	,	PUNCT
cana-551	268	9	δ	δ	NOUN
cana-551	268	10	)	)	PUNCT
cana-551	268	11	in	in	ADP
cana-551	268	12	𝒵	𝒵	PROPN
cana-551	268	13	,	,	PUNCT
cana-551	268	14	𝛺((𝛾	𝛺((𝛾	PROPN
cana-551	268	15	,	,	PUNCT
cana-551	268	16	δ	δ	PROPN
cana-551	268	17	)	)	PUNCT
cana-551	268	18	)	)	PUNCT
cana-551	269	1	=	=	PUNCT
cana-551	269	2	(	(	PUNCT
cana-551	269	3	𝜗	𝜗	NOUN
cana-551	269	4	,	,	PUNCT
cana-551	269	5	θ	θ	NOUN
cana-551	269	6	)	)	PUNCT
cana-551	269	7	=	=	SYM
cana-551	269	8	{	{	PUNCT
cana-551	269	9	(	(	PUNCT
cana-551	269	10	ϱ1	ϱ1	NOUN
cana-551	269	11	,	,	PUNCT
cana-551	269	12	{	{	PUNCT
cana-551	269	13	𝜎	𝜎	NOUN
cana-551	269	14	}	}	PUNCT
cana-551	269	15	)	)	PUNCT
cana-551	269	16	}	}	PUNCT
cana-551	269	17	is	be	AUX
cana-551	269	18	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	269	19	−open	−open	ADV
cana-551	269	20	set	set	VERB
cana-551	269	21	but	but	CCONJ
cana-551	269	22	not	not	PART
cana-551	269	23	soft	soft	ADJ
cana-551	269	24	open	open	ADJ
cana-551	269	25	set	set	NOUN
cana-551	269	26	.	.	PUNCT
cana-551	270	1	communications	communication	NOUN
cana-551	270	2	on	on	ADP
cana-551	270	3	applied	apply	VERB
cana-551	270	4	nonlinear	nonlinear	ADJ
cana-551	270	5	analysis	analysis	NOUN
cana-551	270	6	issn	issn	NOUN
cana-551	270	7	:	:	PUNCT
cana-551	270	8	1074	1074	NUM
cana-551	270	9	-	-	PUNCT
cana-551	270	10	133x	133x	NUM
cana-551	270	11	vol	vol	NOUN
cana-551	270	12	31	31	NUM
cana-551	270	13	no	no	NOUN
cana-551	270	14	.	.	NOUN
cana-551	270	15	2	2	NUM
cana-551	270	16	(	(	PUNCT
cana-551	270	17	2024	2024	NUM
cana-551	270	18	)	)	PUNCT
cana-551	271	1	293	293	NUM
cana-551	271	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-551	271	3	example	example	NOUN
cana-551	271	4	4.12	4.12	NUM
cana-551	271	5	:	:	PUNCT
cana-551	271	6	let	let	VERB
cana-551	271	7	𝒵	𝒵	PROPN
cana-551	271	8	=	=	PUNCT
cana-551	271	9	{	{	PUNCT
cana-551	271	10	휀	휀	NOUN
cana-551	271	11	,	,	PUNCT
cana-551	271	12	𝜇	𝜇	ADP
cana-551	271	13	}	}	PUNCT
cana-551	271	14	and	and	CCONJ
cana-551	271	15	∆=	∆=	ADJ
cana-551	271	16	{	{	PUNCT
cana-551	271	17	∇1	∇1	NOUN
cana-551	271	18	,	,	PUNCT
cana-551	271	19	∇2	∇2	PROPN
cana-551	271	20	}	}	PUNCT
cana-551	271	21	.	.	PUNCT
cana-551	272	1	𝔚	𝔚	NOUN
cana-551	272	2	=	=	PRON
cana-551	272	3	{	{	PUNCT
cana-551	272	4	�	�	PROPN
cana-551	272	5	̃	̃	PROPN
cana-551	272	6	�	�	PROPN
cana-551	272	7	,	,	PUNCT
cana-551	272	8	∅̃	∅̃	NOUN
cana-551	272	9	,	,	PUNCT
cana-551	272	10	(	(	PUNCT
cana-551	272	11	𝛾	𝛾	PROPN
cana-551	272	12	,	,	PUNCT
cana-551	272	13	δ	δ	PROPN
cana-551	272	14	)	)	PUNCT
cana-551	272	15	}	}	PUNCT
cana-551	272	16	is	be	AUX
cana-551	272	17	the	the	DET
cana-551	272	18	soft	soft	ADJ
cana-551	272	19	topology	topology	NOUN
cana-551	272	20	over	over	ADP
cana-551	272	21	𝒵	𝒵	PROPN
cana-551	272	22	where	where	SCONJ
cana-551	272	23	(	(	PUNCT
cana-551	272	24	𝛾	𝛾	PROPN
cana-551	272	25	,	,	PUNCT
cana-551	272	26	δ	δ	NOUN
cana-551	272	27	)	)	PUNCT
cana-551	272	28	=	=	PRON
cana-551	272	29	{	{	PUNCT
cana-551	272	30	(	(	PUNCT
cana-551	272	31	∇1	∇1	PROPN
cana-551	272	32	,	,	PUNCT
cana-551	272	33	𝒵	𝒵	PROPN
cana-551	272	34	)	)	PUNCT
cana-551	272	35	,	,	PUNCT
cana-551	272	36	(	(	PUNCT
cana-551	272	37	∇1	∇1	PROPN
cana-551	272	38	,	,	PUNCT
cana-551	272	39	{	{	PUNCT
cana-551	272	40	휀	휀	NOUN
cana-551	272	41	}	}	PUNCT
cana-551	272	42	)	)	PUNCT
cana-551	272	43	}	}	PUNCT
cana-551	272	44	.	.	PUNCT
cana-551	273	1	also	also	ADV
cana-551	273	2	,	,	PUNCT
cana-551	273	3	let	let	VERB
cana-551	273	4	𝒟	𝒟	NOUN
cana-551	273	5	=	=	PUNCT
cana-551	273	6	{	{	PUNCT
cana-551	273	7	𝜎	𝜎	PROPN
cana-551	273	8	,	,	PUNCT
cana-551	273	9	𝜌	𝜌	ADP
cana-551	273	10	}	}	PUNCT
cana-551	273	11	and	and	CCONJ
cana-551	273	12	θ	θ	NOUN
cana-551	273	13	=	=	SYM
cana-551	273	14	{	{	PUNCT
cana-551	273	15	ϱ1	ϱ1	NOUN
cana-551	273	16	,	,	PUNCT
cana-551	273	17	𝜚2	𝜚2	PROPN
cana-551	273	18	}	}	PUNCT
cana-551	273	19	,	,	PUNCT
cana-551	273	20	℧	℧	PROPN
cana-551	273	21	=	=	PRON
cana-551	273	22	{	{	PUNCT
cana-551	273	23	�	�	PROPN
cana-551	273	24	̃	̃	PROPN
cana-551	273	25	�	�	PROPN
cana-551	273	26	,	,	PUNCT
cana-551	273	27	∅̃	∅̃	NOUN
cana-551	273	28	,	,	PUNCT
cana-551	273	29	(	(	PUNCT
cana-551	273	30	𝛿1	𝛿1	PROPN
cana-551	273	31	,	,	PUNCT
cana-551	273	32	θ	θ	PROPN
cana-551	273	33	)	)	PUNCT
cana-551	273	34	,	,	PUNCT
cana-551	273	35	(	(	PUNCT
cana-551	273	36	𝛿2	𝛿2	NOUN
cana-551	273	37	,	,	PUNCT
cana-551	273	38	θ	θ	NOUN
cana-551	273	39	)	)	PUNCT
cana-551	273	40	}	}	PUNCT
cana-551	273	41	is	be	AUX
cana-551	273	42	soft	soft	ADJ
cana-551	273	43	topology	topology	NOUN
cana-551	273	44	on	on	ADP
cana-551	273	45	𝒟	𝒟	PROPN
cana-551	273	46	,	,	PUNCT
cana-551	273	47	where	where	SCONJ
cana-551	273	48	(	(	PUNCT
cana-551	273	49	𝛿1	𝛿1	NOUN
cana-551	273	50	,	,	PUNCT
cana-551	273	51	θ	θ	NOUN
cana-551	273	52	)	)	PUNCT
cana-551	274	1	=	=	SYM
cana-551	274	2	{	{	PUNCT
cana-551	274	3	(	(	PUNCT
cana-551	274	4	ϱ1	ϱ1	NOUN
cana-551	274	5	,	,	PUNCT
cana-551	274	6	{	{	PUNCT
cana-551	274	7	𝜎	𝜎	NOUN
cana-551	274	8	}	}	PUNCT
cana-551	274	9	)	)	PUNCT
cana-551	274	10	}	}	PUNCT
cana-551	274	11	and	and	CCONJ
cana-551	274	12	(	(	PUNCT
cana-551	274	13	𝛿2	𝛿2	NOUN
cana-551	274	14	,	,	PUNCT
cana-551	274	15	θ	θ	NOUN
cana-551	274	16	)	)	PUNCT
cana-551	274	17	=	=	SYM
cana-551	274	18	{	{	PUNCT
cana-551	274	19	(	(	PUNCT
cana-551	274	20	ϱ1	ϱ1	NOUN
cana-551	274	21	,	,	PUNCT
cana-551	274	22	{	{	PUNCT
cana-551	274	23	𝜎	𝜎	NOUN
cana-551	274	24	}	}	PUNCT
cana-551	274	25	)	)	PUNCT
cana-551	274	26	,	,	PUNCT
cana-551	274	27	(	(	PUNCT
cana-551	274	28	ϱ2	ϱ2	NOUN
cana-551	274	29	,	,	PUNCT
cana-551	274	30	{	{	PUNCT
cana-551	274	31	𝜌	𝜌	X
cana-551	274	32	}	}	PUNCT
cana-551	274	33	)	)	PUNCT
cana-551	274	34	}	}	PUNCT
cana-551	274	35	.	.	PUNCT
cana-551	275	1	let	let	VERB
cana-551	275	2	𝐽	𝐽	PRON
cana-551	275	3	=	=	PUNCT
cana-551	275	4	𝑃(𝒟	𝑃(𝒟	PROPN
cana-551	275	5	)	)	PUNCT
cana-551	275	6	be	be	AUX
cana-551	275	7	a	a	DET
cana-551	275	8	soft	soft	ADJ
cana-551	275	9	ideal	ideal	NOUN
cana-551	275	10	on	on	ADP
cana-551	275	11	𝒟.	𝒟.	PROPN
cana-551	275	12	then	then	ADV
cana-551	275	13	the	the	DET
cana-551	275	14	soft	soft	ADJ
cana-551	275	15	function	function	NOUN
cana-551	275	16	𝛺	𝛺	PROPN
cana-551	275	17	:	:	PUNCT
cana-551	275	18	(	(	PUNCT
cana-551	275	19	𝒵	𝒵	PROPN
cana-551	275	20	,	,	PUNCT
cana-551	275	21	𝔚	𝔚	PROPN
cana-551	275	22	,	,	PUNCT
cana-551	275	23	𝛥	𝛥	PROPN
cana-551	275	24	)	)	PUNCT
cana-551	275	25	→	→	SYM
cana-551	275	26	(	(	PUNCT
cana-551	275	27	𝒟	𝒟	PROPN
cana-551	275	28	,	,	PUNCT
cana-551	275	29	℧	℧	PROPN
cana-551	275	30	,	,	PUNCT
cana-551	275	31	θ	θ	PROPN
cana-551	275	32	)	)	PUNCT
cana-551	275	33	where	where	SCONJ
cana-551	275	34	𝑢	𝑢	X
cana-551	275	35	:	:	PUNCT
cana-551	275	36	𝒵	𝒵	PROPN
cana-551	275	37	→	→	SYM
cana-551	275	38	𝒟	𝒟	PROPN
cana-551	275	39	and	and	CCONJ
cana-551	275	40	𝑝	𝑝	NOUN
cana-551	275	41	:	:	PUNCT
cana-551	275	42	∆→	∆→	PROPN
cana-551	276	1	θ	θ	PROPN
cana-551	276	2	denoted	denote	VERB
cana-551	276	3	by	by	ADP
cana-551	276	4	𝑢(휀	𝑢(휀	NOUN
cana-551	276	5	)	)	PUNCT
cana-551	276	6	=	=	SYM
cana-551	276	7	𝜎	𝜎	PROPN
cana-551	276	8	,	,	PUNCT
cana-551	276	9	𝑢(𝜇	𝑢(𝜇	PROPN
cana-551	276	10	)	)	PUNCT
cana-551	276	11	=	=	SYM
cana-551	276	12	𝜌	𝜌	X
cana-551	276	13	,	,	PUNCT
cana-551	276	14	𝑢(𝜔	𝑢(𝜔	PROPN
cana-551	276	15	)	)	PUNCT
cana-551	276	16	=	=	SYM
cana-551	276	17	𝜋	𝜋	NOUN
cana-551	276	18	,	,	PUNCT
cana-551	276	19	𝑝(∇1	𝑝(∇1	ADJ
cana-551	276	20	)	)	PUNCT
cana-551	276	21	=	=	SYM
cana-551	276	22	ϱ1	ϱ1	NOUN
cana-551	276	23	,	,	PUNCT
cana-551	276	24	𝑝(∇2	𝑝(∇2	ADJ
cana-551	276	25	)	)	PUNCT
cana-551	276	26	=	=	VERB
cana-551	276	27	ϱ2	ϱ2	NOUN
cana-551	276	28	is	be	AUX
cana-551	276	29	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	276	30	−open	−open	ADJ
cana-551	276	31	but	but	CCONJ
cana-551	276	32	not	not	PART
cana-551	276	33	ssƅ∗	ssƅ∗	ADJ
cana-551	276	34	−open	−open	ADJ
cana-551	276	35	.	.	PUNCT
cana-551	277	1	since	since	SCONJ
cana-551	277	2	for	for	ADP
cana-551	277	3	each	each	DET
cana-551	277	4	soft	soft	ADJ
cana-551	277	5	open	open	ADJ
cana-551	277	6	(	(	PUNCT
cana-551	277	7	𝛾	𝛾	PROPN
cana-551	277	8	,	,	PUNCT
cana-551	277	9	δ	δ	NOUN
cana-551	277	10	)	)	PUNCT
cana-551	277	11	in	in	ADP
cana-551	277	12	𝒵	𝒵	PROPN
cana-551	277	13	,	,	PUNCT
cana-551	277	14	𝛺((𝛾	𝛺((𝛾	PROPN
cana-551	277	15	,	,	PUNCT
cana-551	277	16	δ	δ	PROPN
cana-551	277	17	)	)	PUNCT
cana-551	277	18	)	)	PUNCT
cana-551	278	1	=	=	PUNCT
cana-551	278	2	(	(	PUNCT
cana-551	278	3	𝜗	𝜗	NOUN
cana-551	278	4	,	,	PUNCT
cana-551	278	5	θ	θ	NOUN
cana-551	278	6	)	)	PUNCT
cana-551	278	7	=	=	SYM
cana-551	278	8	{	{	PUNCT
cana-551	278	9	(	(	PUNCT
cana-551	278	10	ϱ1	ϱ1	NOUN
cana-551	278	11	,	,	PUNCT
cana-551	278	12	𝒟	𝒟	PROPN
cana-551	278	13	)	)	PUNCT
cana-551	278	14	,	,	PUNCT
cana-551	278	15	(	(	PUNCT
cana-551	278	16	ϱ1	ϱ1	NOUN
cana-551	278	17	,	,	PUNCT
cana-551	278	18	{	{	PUNCT
cana-551	278	19	𝜎	𝜎	NOUN
cana-551	278	20	}	}	PUNCT
cana-551	278	21	)	)	PUNCT
cana-551	278	22	}	}	PUNCT
cana-551	278	23	is	be	AUX
cana-551	278	24	𝑠𝑆ƅ∗ῐ	𝑠𝑆ƅ∗ῐ	PROPN
cana-551	278	25	−open	−open	ADV
cana-551	278	26	set	set	VERB
cana-551	278	27	but	but	CCONJ
cana-551	278	28	not	not	PART
cana-551	278	29	𝑠𝑆ƅ∗	𝑠𝑆ƅ∗	ADJ
cana-551	278	30	−open	−open	VERB
cana-551	278	31	set	set	VERB
cana-551	278	32	.	.	PUNCT
cana-551	279	1	5	5	X
cana-551	279	2	.	.	X
cana-551	279	3	conclusions	conclusion	NOUN
cana-551	279	4	in	in	ADP
cana-551	279	5	this	this	DET
cana-551	279	6	work	work	NOUN
cana-551	279	7	,	,	PUNCT
cana-551	279	8	we	we	PRON
cana-551	279	9	study	study	VERB
cana-551	279	10	the	the	DET
cana-551	279	11	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	279	12	−closed	−close	VERB
cana-551	279	13	sets	set	NOUN
cana-551	279	14	and	and	CCONJ
cana-551	279	15	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	279	16	−open	−open	NOUN
cana-551	279	17	sets	set	VERB
cana-551	279	18	and	and	CCONJ
cana-551	279	19	some	some	PRON
cana-551	279	20	of	of	ADP
cana-551	279	21	their	their	PRON
cana-551	279	22	properties	property	NOUN
cana-551	279	23	and	and	CCONJ
cana-551	279	24	investigated	investigate	VERB
cana-551	279	25	.	.	PUNCT
cana-551	280	1	also	also	ADV
cana-551	280	2	,	,	PUNCT
cana-551	280	3	we	we	PRON
cana-551	280	4	define	define	VERB
cana-551	280	5	the	the	DET
cana-551	280	6	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	280	7	−continuous	−continuous	ADJ
cana-551	280	8	and	and	CCONJ
cana-551	280	9	ssƅ∗ῐ	ssƅ∗ῐ	ADJ
cana-551	280	10	−irresolute	−irresolute	NOUN
cana-551	280	11	.	.	PUNCT
cana-551	281	1	in	in	ADP
cana-551	281	2	future	future	NOUN
cana-551	281	3	,	,	PUNCT
cana-551	281	4	more	more	ADV
cana-551	281	5	general	general	ADJ
cana-551	281	6	types	type	NOUN
cana-551	281	7	of	of	ADP
cana-551	281	8	ssƅ∗ῐ	ssƅ∗ῐ	NOUN
cana-551	281	9	−closed	−close	VERB
cana-551	281	10	sets	set	NOUN
cana-551	281	11	may	may	AUX
cana-551	281	12	be	be	AUX
cana-551	281	13	defined	define	VERB
cana-551	281	14	and	and	CCONJ
cana-551	281	15	using	use	VERB
cana-551	281	16	of	of	ADP
cana-551	281	17	them	they	PRON
cana-551	281	18	characterizations	characterization	NOUN
cana-551	281	19	related	relate	VERB
cana-551	281	20	with	with	ADP
cana-551	281	21	soft	soft	ADJ
cana-551	281	22	separation	separation	NOUN
cana-551	281	23	axioms	axiom	NOUN
cana-551	281	24	and	and	CCONJ
cana-551	281	25	soft	soft	ADJ
cana-551	281	26	continuity	continuity	NOUN
cana-551	281	27	may	may	AUX
cana-551	281	28	be	be	AUX
cana-551	281	29	studied	study	VERB
cana-551	281	30	.	.	PUNCT
cana-551	282	1	references	reference	NOUN
cana-551	282	2	[	[	X
cana-551	282	3	1	1	NUM
cana-551	282	4	]	]	X
cana-551	282	5	d.	d.	PROPN
cana-551	282	6	molodtsov	molodtsov	PROPN
cana-551	282	7	,	,	PUNCT
cana-551	282	8	"	"	PUNCT
cana-551	282	9	soft	soft	ADJ
cana-551	282	10	set	set	NOUN
cana-551	282	11	theory	theory	NOUN
cana-551	282	12	-	-	PUNCT
cana-551	282	13	first	first	ADJ
cana-551	282	14	results	result	NOUN
cana-551	282	15	,	,	PUNCT
cana-551	282	16	"	"	PUNCT
cana-551	282	17	computers	computer	NOUN
cana-551	282	18	and	and	CCONJ
cana-551	282	19	mathematics	mathematic	NOUN
cana-551	282	20	with	with	ADP
cana-551	282	21	applications	application	NOUN
cana-551	282	22	,	,	PUNCT
cana-551	282	23	vol	vol	NOUN
cana-551	282	24	.	.	PROPN
cana-551	282	25	37	37	NUM
cana-551	282	26	,	,	PUNCT
cana-551	282	27	no	no	INTJ
cana-551	282	28	.	.	NOUN
cana-551	282	29	4	4	NUM
cana-551	282	30	-	-	SYM
cana-551	282	31	5	5	NUM
cana-551	282	32	,	,	PUNCT
cana-551	282	33	pp	pp	ADJ
cana-551	282	34	.	.	PUNCT
cana-551	283	1	19	19	NUM
cana-551	283	2	-	-	SYM
cana-551	283	3	31	31	NUM
cana-551	283	4	,	,	PUNCT
cana-551	283	5	1999	1999	NUM
cana-551	283	6	.	.	PUNCT
cana-551	284	1	[	[	X
cana-551	284	2	2	2	NUM
cana-551	284	3	]	]	PUNCT
cana-551	284	4	i.	i.	NOUN
cana-551	284	5	arockiarani	arockiarani	PROPN
cana-551	284	6	and	and	CCONJ
cana-551	284	7	a.	a.	NOUN
cana-551	284	8	arokialancy	arokialancy	NOUN
cana-551	284	9	,	,	PUNCT
cana-551	284	10	"	"	PUNCT
cana-551	284	11	generalized	generalize	VERB
cana-551	284	12	soft	soft	ADJ
cana-551	284	13	gβ	gβ	NOUN
cana-551	284	14	-	-	PUNCT
cana-551	284	15	closed	closed	ADJ
cana-551	284	16	sets	set	NOUN
cana-551	284	17	and	and	CCONJ
cana-551	284	18	soft	soft	ADJ
cana-551	284	19	gsβ	gsβ	ADV
cana-551	284	20	-	-	PUNCT
cana-551	284	21	closed	close	VERB
cana-551	284	22	sets	set	NOUN
cana-551	284	23	in	in	ADP
cana-551	284	24	𝒮tss	𝒮tss	PROPN
cana-551	284	25	,	,	PUNCT
cana-551	284	26	"	"	PUNCT
cana-551	284	27	international	international	ADJ
cana-551	284	28	journal	journal	NOUN
cana-551	284	29	of	of	ADP
cana-551	284	30	mathematical	mathematical	ADJ
cana-551	284	31	archive	archive	NOUN
cana-551	284	32	,	,	PUNCT
cana-551	284	33	vol	vol	NOUN
cana-551	284	34	.	.	PROPN
cana-551	284	35	4	4	NUM
cana-551	284	36	,	,	PUNCT
cana-551	284	37	no	no	INTJ
cana-551	284	38	.	.	NOUN
cana-551	284	39	2	2	NUM
cana-551	284	40	,	,	PUNCT
cana-551	284	41	pp	pp	ADJ
cana-551	284	42	.	.	PUNCT
cana-551	285	1	1	1	NUM
cana-551	285	2	-	-	SYM
cana-551	285	3	7	7	NUM
cana-551	285	4	,	,	PUNCT
cana-551	285	5	2013	2013	NUM
cana-551	285	6	.	.	PUNCT
cana-551	286	1	[	[	X
cana-551	286	2	3	3	X
cana-551	286	3	]	]	X
cana-551	286	4	m.	m.	NOUN
cana-551	286	5	akdag	akdag	PROPN
cana-551	286	6	and	and	CCONJ
cana-551	286	7	a.	a.	NOUN
cana-551	286	8	ozkan	ozkan	PROPN
cana-551	286	9	,	,	PUNCT
cana-551	286	10	"	"	PUNCT
cana-551	286	11	soft	soft	ADJ
cana-551	286	12	𝛼-open	𝛼-open	NOUN
cana-551	286	13	sets	set	NOUN
cana-551	286	14	and	and	CCONJ
cana-551	286	15	soft	soft	ADJ
cana-551	286	16	𝛼-continuous	𝛼-continuous	ADJ
cana-551	286	17	functions	function	NOUN
cana-551	286	18	,	,	PUNCT
cana-551	286	19	"	"	PUNCT
cana-551	286	20	abstr	abstr	NOUN
cana-551	286	21	.	.	PUNCT
cana-551	287	1	anal	anal	PROPN
cana-551	287	2	.	.	PUNCT
cana-551	288	1	appl	appl	PROPN
cana-551	288	2	.	.	PROPN
cana-551	288	3	,	,	PUNCT
cana-551	289	1	pp	pp	PROPN
cana-551	289	2	.	.	PUNCT
cana-551	290	1	1	1	NUM
cana-551	290	2	-	-	SYM
cana-551	290	3	7	7	NUM
cana-551	290	4	,	,	PUNCT
cana-551	290	5	2014	2014	NUM
cana-551	290	6	.	.	PUNCT
cana-551	291	1	[	[	X
cana-551	291	2	4	4	NUM
cana-551	291	3	]	]	X
cana-551	291	4	m.	m.	NOUN
cana-551	291	5	akdag	akdag	PROPN
cana-551	291	6	,	,	PUNCT
cana-551	291	7	a.	a.	NOUN
cana-551	291	8	ozkan	ozkan	PROPN
cana-551	291	9	,	,	PUNCT
cana-551	291	10	"	"	PUNCT
cana-551	291	11	soft	soft	ADJ
cana-551	291	12	b	b	NOUN
cana-551	291	13	-	-	PUNCT
cana-551	291	14	open	open	ADJ
cana-551	291	15	sets	set	NOUN
cana-551	291	16	and	and	CCONJ
cana-551	291	17	soft	soft	ADJ
cana-551	291	18	b	b	NOUN
cana-551	291	19	-	-	PUNCT
cana-551	291	20	continuous	continuous	ADJ
cana-551	291	21	functions	function	NOUN
cana-551	291	22	,	,	PUNCT
cana-551	291	23	"	"	PUNCT
cana-551	291	24	math	math	PROPN
cana-551	291	25	sci	sci	PROPN
cana-551	291	26	,	,	PUNCT
cana-551	291	27	vol	vol	NOUN
cana-551	291	28	.	.	PROPN
cana-551	291	29	8	8	NUM
cana-551	291	30	,	,	PUNCT
cana-551	291	31	no	no	INTJ
cana-551	291	32	.	.	NOUN
cana-551	291	33	127	127	NUM
cana-551	291	34	,	,	PUNCT
cana-551	291	35	pp	pp	ADJ
cana-551	291	36	.	.	PROPN
cana-551	291	37	19	19	NUM
cana-551	291	38	,	,	PUNCT
cana-551	291	39	2014	2014	NUM
cana-551	291	40	.	.	PUNCT
cana-551	292	1	[	[	X
cana-551	292	2	5	5	NUM
cana-551	292	3	]	]	PUNCT
cana-551	292	4	hameed	hameed	NOUN
cana-551	292	5	,	,	PUNCT
cana-551	292	6	s.	s.	PROPN
cana-551	292	7	z.	z.	PROPN
cana-551	292	8	and	and	CCONJ
cana-551	292	9	a.	a.	PROPN
cana-551	292	10	k.	k.	PROPN
cana-551	292	11	hussein	hussein	PROPN
cana-551	292	12	,	,	PUNCT
cana-551	292	13	"	"	PUNCT
cana-551	292	14	on	on	ADP
cana-551	292	15	soft	soft	ADJ
cana-551	292	16	ƅ𝑐	ƅ𝑐	NOUN
cana-551	292	17	−	−	PUNCT
cana-551	292	18	open	open	ADJ
cana-551	292	19	sets	set	NOUN
cana-551	292	20	in	in	ADP
cana-551	292	21	𝒮tss	𝒮tss	PROPN
cana-551	292	22	,	,	PUNCT
cana-551	292	23	"	"	PUNCT
cana-551	292	24	iraqi	iraqi	ADJ
cana-551	292	25	journal	journal	NOUN
cana-551	292	26	of	of	ADP
cana-551	292	27	science	science	NOUN
cana-551	292	28	,	,	PUNCT
cana-551	292	29	pp	pp	ADJ
cana-551	292	30	.	.	PUNCT
cana-551	293	1	238242	238242	NUM
cana-551	293	2	,	,	PUNCT
cana-551	293	3	2020	2020	NUM
cana-551	293	4	.	.	PUNCT
cana-551	294	1	[	[	X
cana-551	294	2	6	6	NUM
cana-551	294	3	]	]	SYM
cana-551	294	4	hameed	hameed	PROPN
cana-551	294	5	,	,	PUNCT
cana-551	294	6	s.	s.	PROPN
cana-551	294	7	z.	z.	PROPN
cana-551	294	8	,	,	PUNCT
cana-551	294	9	f.	f.	PROPN
cana-551	294	10	a.	a.	PROPN
cana-551	294	11	ibrahem	ibrahem	PROPN
cana-551	294	12	and	and	CCONJ
cana-551	294	13	and	and	CCONJ
cana-551	294	14	essam	essam	PROPN
cana-551	294	15	a.	a.	PROPN
cana-551	294	16	el	el	PROPN
cana-551	294	17	-	-	PUNCT
cana-551	294	18	seidy	seidy	ADJ
cana-551	294	19	,	,	PUNCT
cana-551	294	20	"	"	PUNCT
cana-551	294	21	on	on	ADP
cana-551	294	22	soft	soft	ADJ
cana-551	294	23	b*-closed	b*-close	VERB
cana-551	294	24	sets	set	NOUN
cana-551	294	25	in	in	ADP
cana-551	294	26	𝒮ts	𝒮ts	PROPN
cana-551	294	27	,	,	PUNCT
cana-551	294	28	"	"	PUNCT
cana-551	294	29	international	international	ADJ
cana-551	294	30	journal	journal	NOUN
cana-551	294	31	of	of	ADP
cana-551	294	32	nonlinear	nonlinear	ADJ
cana-551	294	33	analysis	analysis	NOUN
cana-551	294	34	and	and	CCONJ
cana-551	294	35	applications	application	NOUN
cana-551	294	36	,	,	PUNCT
cana-551	294	37	vol	vol	NOUN
cana-551	294	38	.	.	PROPN
cana-551	294	39	12	12	NUM
cana-551	294	40	,	,	PUNCT
cana-551	294	41	no	no	INTJ
cana-551	294	42	.	.	NOUN
cana-551	294	43	1	1	NUM
cana-551	294	44	,	,	PUNCT
cana-551	294	45	pp	pp	ADJ
cana-551	294	46	.	.	PUNCT
cana-551	295	1	1235	1235	NUM
cana-551	295	2	-	-	SYM
cana-551	295	3	1242	1242	NUM
cana-551	295	4	,	,	PUNCT
cana-551	295	5	2021	2021	NUM
cana-551	295	6	.	.	PUNCT
cana-551	296	1	[	[	X
cana-551	296	2	7	7	NUM
cana-551	296	3	]	]	X
cana-551	296	4	hameed	hameed	NOUN
cana-551	296	5	,	,	PUNCT
cana-551	296	6	s.	s.	PROPN
cana-551	296	7	z.	z.	PROPN
cana-551	296	8	,	,	PUNCT
cana-551	296	9	a.	a.	PROPN
cana-551	296	10	e.	e.	PROPN
cana-551	296	11	radwan	radwan	PROPN
cana-551	296	12	and	and	CCONJ
cana-551	296	13	essam	essam	PROPN
cana-551	296	14	a.	a.	PROPN
cana-551	296	15	el	el	PROPN
cana-551	296	16	-	-	PUNCT
cana-551	296	17	seidy	seidy	ADJ
cana-551	296	18	,	,	PUNCT
cana-551	296	19	"	"	PUNCT
cana-551	296	20	on	on	ADP
cana-551	296	21	soft	soft	ADJ
cana-551	296	22	b	b	NOUN
cana-551	296	23	*	*	ADJ
cana-551	296	24	continuous	continuous	ADJ
cana-551	296	25	functions	function	NOUN
cana-551	296	26	in	in	ADP
cana-551	296	27	𝒮tss	𝒮tss	PROPN
cana-551	296	28	,	,	PUNCT
cana-551	296	29	"	"	PUNCT
cana-551	296	30	measurement	measurement	NOUN
cana-551	296	31	:	:	PUNCT
cana-551	296	32	sensors	sensor	NOUN
cana-551	296	33	,	,	PUNCT
cana-551	296	34	vol	vol	NOUN
cana-551	296	35	.	.	PROPN
cana-551	296	36	27	27	NUM
cana-551	296	37	,	,	PUNCT
cana-551	296	38	pp	pp	ADJ
cana-551	296	39	.	.	PUNCT
cana-551	297	1	1	1	NUM
cana-551	297	2	-	-	SYM
cana-551	297	3	5	5	NUM
cana-551	297	4	,	,	PUNCT
cana-551	297	5	2023	2023	NUM
cana-551	297	6	.	.	PUNCT
cana-551	298	1	[	[	X
cana-551	298	2	8	8	NUM
cana-551	298	3	]	]	X
cana-551	298	4	hameed	hameed	NOUN
cana-551	298	5	,	,	PUNCT
cana-551	298	6	saif	saif	PROPN
cana-551	298	7	z.	z.	PROPN
cana-551	298	8	,	,	PUNCT
cana-551	298	9	a.	a.	PROPN
cana-551	298	10	e.	e.	PROPN
cana-551	298	11	radwan	radwan	PROPN
cana-551	298	12	and	and	CCONJ
cana-551	298	13	essam	essam	PROPN
cana-551	298	14	a.	a.	PROPN
cana-551	298	15	el	el	PROPN
cana-551	298	16	-	-	PUNCT
cana-551	298	17	seidy	seidy	ADJ
cana-551	298	18	,	,	PUNCT
cana-551	298	19	"	"	PUNCT
cana-551	298	20	on	on	ADP
cana-551	298	21	ss	ss	ADP
cana-551	298	22	ƅ*-closed	ƅ*-close	VERB
cana-551	298	23	sets	set	NOUN
cana-551	298	24	and	and	CCONJ
cana-551	298	25	ss	ss	ADP
cana-551	298	26	ƅ	ƅ	NOUN
cana-551	298	27	*	*	NOUN
cana-551	298	28	−	−	PROPN
cana-551	298	29	continuous	continuous	ADJ
cana-551	298	30	functions	function	NOUN
cana-551	298	31	in	in	ADP
cana-551	298	32	𝒮tss	𝒮tss	PROPN
cana-551	298	33	,	,	PUNCT
cana-551	298	34	"	"	PUNCT
cana-551	298	35	under	under	ADP
cana-551	298	36	the	the	DET
cana-551	298	37	publication	publication	NOUN
cana-551	298	38	,	,	PUNCT
cana-551	298	39	2023	2023	NUM
cana-551	298	40	.	.	PUNCT
cana-551	299	1	[	[	X
cana-551	299	2	9	9	NUM
cana-551	299	3	]	]	PUNCT
cana-551	299	4	a.	a.	NOUN
cana-551	299	5	kandil	kandil	PROPN
cana-551	299	6	,	,	PUNCT
cana-551	299	7	o.	o.	PROPN
cana-551	299	8	a.	a.	PROPN
cana-551	299	9	e.	e.	PROPN
cana-551	299	10	tantawy	tantawy	PROPN
cana-551	299	11	,	,	PUNCT
cana-551	299	12	s.	s.	PROPN
cana-551	299	13	a.	a.	PROPN
cana-551	299	14	el	el	PROPN
cana-551	299	15	-	-	PUNCT
cana-551	299	16	sheikh	sheikh	PROPN
cana-551	299	17	and	and	CCONJ
cana-551	299	18	a.	a.	NOUN
cana-551	299	19	m.	m.	NOUN
cana-551	299	20	abd	abd	PROPN
cana-551	299	21	el	el	PROPN
cana-551	299	22	-	-	PROPN
cana-551	299	23	latif	latif	PROPN
cana-551	299	24	,	,	PUNCT
cana-551	299	25	"	"	PUNCT
cana-551	299	26	soft	soft	ADJ
cana-551	299	27	ideal	ideal	ADJ
cana-551	299	28	theory	theory	NOUN
cana-551	299	29	,	,	PUNCT
cana-551	299	30	soft	soft	ADJ
cana-551	299	31	local	local	ADJ
cana-551	299	32	function	function	NOUN
cana-551	299	33	and	and	CCONJ
cana-551	299	34	generated	generate	VERB
cana-551	299	35	𝒮tss	𝒮tss	PROPN
cana-551	299	36	,	,	PUNCT
cana-551	299	37	"	"	PUNCT
cana-551	299	38	appl	appl	NOUN
cana-551	299	39	.	.	PROPN
cana-551	299	40	math	math	PROPN
cana-551	299	41	.	.	PUNCT
cana-551	300	1	inf	inf	PROPN
cana-551	300	2	.	.	PUNCT
cana-551	301	1	sci	sci	PROPN
cana-551	301	2	.	.	PROPN
cana-551	301	3	,	,	PUNCT
cana-551	301	4	vol	vol	NOUN
cana-551	301	5	.	.	PROPN
cana-551	301	6	8	8	NUM
cana-551	301	7	,	,	PUNCT
cana-551	301	8	no	no	INTJ
cana-551	301	9	.	.	NOUN
cana-551	301	10	4	4	NUM
cana-551	301	11	,	,	PUNCT
cana-551	301	12	pp	pp	ADJ
cana-551	301	13	.	.	PUNCT
cana-551	302	1	1595	1595	NUM
cana-551	302	2	-	-	SYM
cana-551	302	3	1603	1603	NUM
cana-551	302	4	,	,	PUNCT
cana-551	302	5	2014	2014	NUM
cana-551	302	6	.	.	PUNCT
cana-551	303	1	[	[	X
cana-551	303	2	10	10	NUM
cana-551	303	3	]	]	X
cana-551	303	4	h.	h.	PROPN
cana-551	303	5	i.	i.	PROPN
cana-551	303	6	mustafa	mustafa	PROPN
cana-551	303	7	and	and	CCONJ
cana-551	303	8	f.	f.	PROPN
cana-551	303	9	m.	m.	PROPN
cana-551	303	10	sleim	sleim	PROPN
cana-551	303	11	,	,	PUNCT
cana-551	303	12	"	"	PUNCT
cana-551	303	13	soft	soft	ADJ
cana-551	303	14	generalized	generalize	VERB
cana-551	303	15	closed	closed	ADJ
cana-551	303	16	sets	set	NOUN
cana-551	303	17	with	with	ADP
cana-551	303	18	respect	respect	NOUN
cana-551	303	19	to	to	ADP
cana-551	303	20	an	an	DET
cana-551	303	21	ideal	ideal	NOUN
cana-551	303	22	in	in	ADP
cana-551	303	23	𝒮tss	𝒮tss	PROPN
cana-551	303	24	,	,	PUNCT
cana-551	303	25	"	"	PUNCT
cana-551	303	26	applied	apply	VERB
cana-551	303	27	mathematics	mathematics	PROPN
cana-551	303	28	&	&	CCONJ
cana-551	303	29	information	information	NOUN
cana-551	303	30	sciences	sciences	PROPN
cana-551	303	31	,	,	PUNCT
cana-551	303	32	vol	vol	NOUN
cana-551	303	33	.	.	PROPN
cana-551	303	34	8	8	NUM
cana-551	303	35	,	,	PUNCT
cana-551	303	36	no	no	INTJ
cana-551	303	37	.	.	NOUN
cana-551	303	38	2	2	NUM
cana-551	303	39	,	,	PUNCT
cana-551	303	40	p.	p.	NOUN
cana-551	303	41	665–671	665–671	NUM
cana-551	303	42	,	,	PUNCT
cana-551	303	43	2014	2014	NUM
cana-551	303	44	.	.	PUNCT
cana-551	304	1	[	[	X
cana-551	304	2	11	11	NUM
cana-551	304	3	]	]	PUNCT
cana-551	304	4	k.	k.	PROPN
cana-551	304	5	kannan	kannan	PROPN
cana-551	304	6	,	,	PUNCT
cana-551	304	7	"	"	PUNCT
cana-551	304	8	soft	soft	ADJ
cana-551	304	9	generalized	generalize	VERB
cana-551	304	10	closed	closed	ADJ
cana-551	304	11	sets	set	NOUN
cana-551	304	12	in	in	ADP
cana-551	304	13	𝒮tss	𝒮tss	PROPN
cana-551	304	14	,	,	PUNCT
cana-551	304	15	"	"	PUNCT
cana-551	304	16	journal	journal	NOUN
cana-551	304	17	of	of	ADP
cana-551	304	18	theoretical	theoretical	ADJ
cana-551	304	19	and	and	CCONJ
cana-551	304	20	appl	appl	NOUN
cana-551	304	21	.	.	PUNCT
cana-551	305	1	inform	inform	NOUN
cana-551	305	2	.	.	PUNCT
cana-551	306	1	technology	technology	NOUN
cana-551	306	2	,	,	PUNCT
cana-551	306	3	vol	vol	NOUN
cana-551	306	4	.	.	PROPN
cana-551	307	1	37	37	NUM
cana-551	307	2	,	,	PUNCT
cana-551	307	3	no	no	INTJ
cana-551	307	4	.	.	NOUN
cana-551	307	5	1	1	NUM
cana-551	307	6	,	,	PUNCT
cana-551	307	7	p.	p.	NOUN
cana-551	307	8	17	17	NUM
cana-551	307	9	–	–	PUNCT
cana-551	307	10	21	21	NUM
cana-551	307	11	,	,	PUNCT
cana-551	307	12	2012	2012	NUM
cana-551	307	13	.	.	PUNCT
cana-551	308	1	[	[	X
cana-551	308	2	12	12	NUM
cana-551	308	3	]	]	X
cana-551	308	4	maji	maji	NOUN
cana-551	308	5	,	,	PUNCT
cana-551	308	6	p.	p.	PROPN
cana-551	308	7	k.	k.	PROPN
cana-551	308	8	,	,	PUNCT
cana-551	308	9	r.	r.	PROPN
cana-551	308	10	biswas	biswas	PROPN
cana-551	308	11	and	and	CCONJ
cana-551	308	12	a.	a.	PROPN
cana-551	308	13	r.	r.	PROPN
cana-551	308	14	roy	roy	PROPN
cana-551	308	15	,	,	PUNCT
cana-551	308	16	"	"	PUNCT
cana-551	308	17	soft	soft	ADJ
cana-551	308	18	set	set	NOUN
cana-551	308	19	theory	theory	NOUN
cana-551	308	20	,	,	PUNCT
cana-551	308	21	"	"	PUNCT
cana-551	308	22	comput	comput	NOUN
cana-551	308	23	.	.	PUNCT
cana-551	309	1	math	math	NOUN
cana-551	309	2	.	.	PUNCT
cana-551	310	1	appl	appl	PROPN
cana-551	310	2	.	.	PROPN
cana-551	310	3	,	,	PUNCT
cana-551	310	4	vol	vol	NOUN
cana-551	310	5	.	.	PROPN
cana-551	310	6	45	45	NUM
cana-551	310	7	,	,	PUNCT
cana-551	310	8	p.	p.	NOUN
cana-551	310	9	555–562	555–562	NUM
cana-551	310	10	,	,	PUNCT
cana-551	310	11	2003	2003	NUM
cana-551	310	12	.	.	PUNCT
cana-551	311	1	[	[	X
cana-551	311	2	13	13	NUM
cana-551	311	3	]	]	X
cana-551	311	4	shabir	shabir	NOUN
cana-551	311	5	,	,	PUNCT
cana-551	311	6	m.	m.	NOUN
cana-551	311	7	,	,	PUNCT
cana-551	311	8	naz	naz	PROPN
cana-551	311	9	and	and	CCONJ
cana-551	311	10	m.	m.	NOUN
cana-551	311	11	,	,	PUNCT
cana-551	311	12	"	"	PUNCT
cana-551	311	13	on	on	ADP
cana-551	311	14	𝒮tss	𝒮tss	PROPN
cana-551	311	15	,	,	PUNCT
cana-551	311	16	"	"	PUNCT
cana-551	311	17	comput	comput	NOUN
cana-551	311	18	.	.	PUNCT
cana-551	312	1	math	math	NOUN
cana-551	312	2	.	.	PUNCT
cana-551	313	1	appl	appl	PROPN
cana-551	313	2	.	.	PROPN
cana-551	313	3	,	,	PUNCT
cana-551	313	4	vol	vol	NOUN
cana-551	313	5	.	.	PROPN
cana-551	313	6	61	61	NUM
cana-551	313	7	,	,	PUNCT
cana-551	313	8	p.	p.	NOUN
cana-551	313	9	1786–1799	1786–1799	NUM
cana-551	313	10	,	,	PUNCT
cana-551	313	11	2011	2011	NUM
cana-551	313	12	.	.	PUNCT
cana-551	314	1	[	[	X
cana-551	314	2	14	14	NUM
cana-551	314	3	]	]	X
cana-551	314	4	i.	i.	PROPN
cana-551	314	5	zorlutuna	zorlutuna	PROPN
cana-551	314	6	,	,	PUNCT
cana-551	314	7	m.	m.	NOUN
cana-551	314	8	akdag	akdag	PROPN
cana-551	314	9	,	,	PUNCT
cana-551	314	10	w.	w.	PROPN
cana-551	314	11	k.	k.	PROPN
cana-551	314	12	min	min	PROPN
cana-551	314	13	and	and	CCONJ
cana-551	314	14	s.	s.	PROPN
cana-551	314	15	atmaca	atmaca	PROPN
cana-551	314	16	,	,	PUNCT
cana-551	314	17	"	"	PUNCT
cana-551	314	18	remarks	remark	NOUN
cana-551	314	19	on	on	ADP
cana-551	314	20	𝒮tss	𝒮tss	PROPN
cana-551	314	21	,	,	PUNCT
cana-551	314	22	"	"	PUNCT
cana-551	314	23	annals	annal	NOUN
cana-551	314	24	of	of	ADP
cana-551	314	25	fuzzy	fuzzy	ADJ
cana-551	314	26	mathematics	mathematic	NOUN
cana-551	314	27	and	and	CCONJ
cana-551	314	28	informatics	informatic	NOUN
cana-551	314	29	,	,	PUNCT
cana-551	314	30	vol	vol	NOUN
cana-551	314	31	.	.	PROPN
cana-551	315	1	3	3	NUM
cana-551	315	2	,	,	PUNCT
cana-551	315	3	no	no	INTJ
cana-551	315	4	.	.	NOUN
cana-551	315	5	2	2	NUM
cana-551	315	6	,	,	PUNCT
cana-551	315	7	pp	pp	ADJ
cana-551	315	8	.	.	PUNCT
cana-551	316	1	171	171	NUM
cana-551	316	2	-	-	SYM
cana-551	316	3	185	185	NUM
cana-551	316	4	,	,	PUNCT
cana-551	316	5	2012	2012	NUM
cana-551	316	6	.	.	PUNCT