id	sid	eid	entity	type
ma-246	1	1	2025	CARDINAL
ma-246	2	1	j. math	PERSON
ma-246	4	1	5 (	CARDINAL
ma-246	5	1	2doi	CARDINAL
ma-246	5	2	10.28924	CARDINAL
ma-246	5	3	junvon a. almocera∗	PERSON
ma-246	5	4	lezel m. tutanes department	PERSON
ma-246	5	5	bukidnon state university	ORG
ma-246	5	6	bukidnon	GPE
ma-246	5	7	1901103646@student.buksu.edu.ph	CARDINAL
ma-246	5	8	1901103646@student.buksu.edu.ph	CARDINAL
ma-246	7	1	∗η	GPE
ma-246	7	2	p(x	PERSON
ma-246	7	3	p(x	PERSON
ma-246	9	1	∈	ORG
ma-246	9	2	⊆	CARDINAL
ma-246	9	3	η-o(x	PERSON
ma-246	10	1	thisstudy	NORP
ma-246	10	2	a∗η	ORG
ma-246	10	3	γ∗	ORG
ma-246	11	1	⊆	CARDINAL
ma-246	13	1	1	CARDINAL
ma-246	13	2	first	ORDINAL
ma-246	14	1	4	CARDINAL
ma-246	14	2	9	CARDINAL
ma-246	16	1	3	CARDINAL
ma-246	18	1	al-omari	PERSON
ma-246	18	2	1	CARDINAL
ma-246	18	3	γ∗	ORG
ma-246	18	4	γ∗	ORG
ma-246	18	5	hatir et al	PERSON
ma-246	19	1	2	CARDINAL
ma-246	20	1	8	CARDINAL
ma-246	20	2	γ∗	ORG
ma-246	22	1	2	CARDINAL
ma-246	24	1	cl(a	NORP
ma-246	24	2	4	CARDINAL
ma-246	24	3	2024	DATE
ma-246	27	1	j. math	PERSON
ma-246	29	1	10.28924	CARDINAL
ma-246	29	2	int(a	GPE
ma-246	31	1	η	PERSON
ma-246	32	1	8	CARDINAL
ma-246	32	2	⊆	CARDINAL
ma-246	34	1	∩ int(cl(a	PERSON
ma-246	35	1	η-cl(a	ORG
ma-246	36	1	6	CARDINAL
ma-246	36	2	⊆	CARDINAL
ma-246	40	1	η-o(x	PERSON
ma-246	40	2	ro(x)).an ideal	ORG
ma-246	41	1	5	CARDINAL
ma-246	42	1	∈	ORG
ma-246	44	1	⊆	CARDINAL
ma-246	44	2	∈	ORG
ma-246	46	1	p(x	PERSON
ma-246	46	2	p(x	PERSON
ma-246	46	3	3,9	CARDINAL
ma-246	46	4	⊆	CARDINAL
ma-246	48	1	τ(x	ORG
ma-246	48	2	∈	ORG
ma-246	49	1	τ∗(i	ORG
ma-246	50	1	3	CARDINAL
ma-246	50	2	b(i	PERSON
ma-246	50	3	u \ j :	ORG
ma-246	50	4	b(i	PERSON
ma-246	52	1	5	CARDINAL
ma-246	52	2	3	CARDINAL
ma-246	56	1	∈	ORG
ma-246	58	1	1	CARDINAL
ma-246	59	1	3	CARDINAL
ma-246	59	2	1	CARDINAL
ma-246	61	1	∗η	GPE
ma-246	61	2	p(x	PERSON
ma-246	61	3	⊆	CARDINAL
ma-246	61	4	a∗η	PERSON
ma-246	61	5	η-o(x	PERSON
ma-246	63	1	∈	ORG
ma-246	64	1	η-o(x	PERSON
ma-246	64	2	∈ η-o(x	PERSON
ma-246	64	3	∈	ORG
ma-246	64	4	η-o(x	PERSON
ma-246	65	1	a∗η	PERSON
ma-246	65	2	η-o(x	PERSON
ma-246	65	3	a∗η	ORG
ma-246	67	1	1	CARDINAL
ma-246	68	1	x,∅	ORG
ma-246	69	1	η-o(x	PERSON
ma-246	71	1	1	CARDINAL
ma-246	71	2	a∗η	ORG
ma-246	72	1	1	CARDINAL
ma-246	74	1	⊆	CARDINAL
ma-246	74	2	a∗η ⊆	ORG
ma-246	74	3	a∗η ∩ b∗η	PRODUCT
ma-246	74	4	a∗η	ORG
ma-246	74	5	a∗η	ORG
ma-246	75	1	a∗η ∩	ORG
ma-246	75	2	6=	CARDINAL
ma-246	75	3	a∗η	ORG
ma-246	75	4	a∗η	ORG
ma-246	76	1	j. math	PERSON
ma-246	78	1	10.28924	CARDINAL
ma-246	78	2	∈	ORG
ma-246	78	3	a∗η	ORG
ma-246	79	1	a∗η	ORG
ma-246	79	2	η-cl(a	PERSON
ma-246	79	3	p(x	PERSON
ma-246	79	4	a∗η	ORG
ma-246	79	5	η-cl(a	PERSON
ma-246	80	1	⊆	CARDINAL
ma-246	80	2	⊆	CARDINAL
ma-246	80	3	∈ η-o(x	PERSON
ma-246	81	1	⊆	CARDINAL
ma-246	81	2	∩	ORG
ma-246	81	3	⊆	CARDINAL
ma-246	82	1	/∈ a∗η	ORG
ma-246	83	1	⊆	CARDINAL
ma-246	84	1	⊆	CARDINAL
ma-246	84	2	⊆	CARDINAL
ma-246	84	3	1	CARDINAL
ma-246	84	4	a∗η	ORG
ma-246	84	5	⊆	CARDINAL
ma-246	84	6	b∗η	PRODUCT
ma-246	85	1	a∗η ∩ b∗η	PRODUCT
ma-246	86	1	⊆	CARDINAL
ma-246	87	1	⊆	CARDINAL
ma-246	87	2	⊆	CARDINAL
ma-246	87	3	1	CARDINAL
ma-246	87	4	a∗η ⊆	ORG
ma-246	88	1	a∗η	ORG
ma-246	89	1	a∗η 6=	LAW
ma-246	91	1	∈	ORG
ma-246	93	1	a∗η	ORG
ma-246	95	1	a∗η	ORG
ma-246	96	1	a∗η	ORG
ma-246	97	1	a∗η	ORG
ma-246	97	2	a∗η	PERSON
ma-246	99	1	∈	ORG
ma-246	99	2	u ∩ a∗η	LAW
ma-246	99	3	u ∩ a∗η 6= ∅.now	ORG
ma-246	100	1	∈ η-o(y	PERSON
ma-246	100	2	y ∈ a∗η	ORG
ma-246	103	1	∈ η-o(x	PERSON
ma-246	106	1	a∗η	ORG
ma-246	107	1	∈	ORG
ma-246	109	1	a∗η 6=	LAW
ma-246	109	2	∈ a∗η	ORG
ma-246	111	1	∈ η-o(x	PERSON
ma-246	112	1	⊆	CARDINAL
ma-246	112	2	∈	ORG
ma-246	115	1	a∗η 6=	ORG
ma-246	115	2	η-cl(a	GPE
ma-246	116	1	η-cl(a	PERSON
ma-246	116	2	a∗η	ORG
ma-246	117	1	⊆	CARDINAL
ma-246	118	1	∈	ORG
ma-246	119	1	6=	CARDINAL
ma-246	119	2	∈	ORG
ma-246	119	3	∈ η-o(x	PERSON
ma-246	120	1	⊆	CARDINAL
ma-246	123	1	⊆	CARDINAL
ma-246	123	2	kc	PERSON
ma-246	123	3	kc	PERSON
ma-246	123	4	kc	PERSON
ma-246	124	1	p(x	PERSON
ma-246	125	1	⊆	CARDINAL
ma-246	126	1	p(x	PERSON
ma-246	126	2	∈	ORG
ma-246	127	1	a∗η	ORG
ma-246	128	1	⊆	CARDINAL
ma-246	131	1	⊆	CARDINAL
ma-246	131	2	kc	PERSON
ma-246	131	3	kc ∈ η-o(x	PERSON
ma-246	133	1	kc ∈ η-o(x	PERSON
ma-246	135	1	j. math	PERSON
ma-246	137	1	10.28924	CARDINAL
ma-246	137	2	4	CARDINAL
ma-246	137	3	1	CARDINAL
ma-246	140	1	1	CARDINAL
ma-246	140	2	⊆	CARDINAL
ma-246	140	3	a∗η	ORG
ma-246	140	4	a∗η ⊆	ORG
ma-246	140	5	a∗η	ORG
ma-246	140	6	η-cl(a∗η	PERSON
ma-246	141	1	1	CARDINAL
ma-246	142	1	x,∅	ORG
ma-246	145	1	1	CARDINAL
ma-246	145	2	a∗η	ORG
ma-246	146	1	a∗η	ORG
ma-246	147	1	x,∅	ORG
ma-246	150	1	1	CARDINAL
ma-246	150	2	a∗η	ORG
ma-246	150	3	a * a∗η	ORG
ma-246	150	4	2	CARDINAL
ma-246	152	1	∈	ORG
ma-246	153	1	⊆	CARDINAL
ma-246	153	2	⊆	CARDINAL
ma-246	153	3	∩	PERSON
ma-246	153	4	⊆	CARDINAL
ma-246	153	5	u ∩ a∗η	ORG
ma-246	154	1	a∗η ⊆	ORG
ma-246	154	2	a ∩ a∗η	ORG
ma-246	154	3	∈	ORG
ma-246	155	1	a∗η	ORG
ma-246	155	2	η-cl(a∗η	PERSON
ma-246	155	3	⊆	CARDINAL
ma-246	155	4	η-cl(a	GPE
ma-246	155	5	a∗η	ORG
ma-246	155	6	a∗η	ORG
ma-246	155	7	η-cl(a∗η	PERSON
ma-246	155	8	η-cl(a	PERSON
ma-246	157	1	⊆	CARDINAL
ma-246	158	1	⊆	CARDINAL
ma-246	158	2	a∗η	ORG
ma-246	161	1	∈	ORG
ma-246	162	1	∈	ORG
ma-246	169	1	j. math	PERSON
ma-246	171	1	10.28924	CARDINAL
ma-246	171	2	5or	CARDINAL
ma-246	173	1	∈	ORG
ma-246	175	1	a∗η ∪∅	PERSON
ma-246	176	1	∈ a∗η	ORG
ma-246	178	1	⊆	PRODUCT
ma-246	178	2	bytheorem 1	ORG
ma-246	178	3	a∗η ⊆	ORG
ma-246	179	1	a∗η	ORG
ma-246	181	1	∈	ORG
ma-246	183	1	∈ a∗η	ORG
ma-246	183	2	⊆	CARDINAL
ma-246	184	1	∈	ORG
ma-246	185	1	∈ η-o(x	PERSON
ma-246	185	2	∈	PERSON
ma-246	185	3	∩	PERSON
ma-246	185	4	⊆	CARDINAL
ma-246	185	5	∈	ORG
ma-246	188	1	∩	PERSON
ma-246	191	1	∈	ORG
ma-246	193	1	⊆	CARDINAL
ma-246	194	1	∈ η-o(x	PERSON
ma-246	195	1	⊆	CARDINAL
ma-246	196	1	\a	ORG
ma-246	196	2	⊆	CARDINAL
ma-246	196	3	⊆	CARDINAL
ma-246	196	4	⊆	CARDINAL
ma-246	196	5	b∗η	PRODUCT
ma-246	196	6	a∗η	ORG
ma-246	196	7	b∗η	PRODUCT
ma-246	197	1	1	CARDINAL
ma-246	197	2	a∗η	ORG
ma-246	198	1	⊆	CARDINAL
ma-246	199	1	⊆	CARDINAL
ma-246	200	1	u ∩a ⊆	ORG
ma-246	200	2	a∗η	ORG
ma-246	200	3	⊆	CARDINAL
ma-246	200	4	u ∩ a∗η	ORG
ma-246	203	1	∩	ORG
ma-246	203	2	⊆	CARDINAL
ma-246	203	3	∈	ORG
ma-246	207	1	1	CARDINAL
ma-246	209	1	a∗η	ORG
ma-246	209	2	η-cl(a	PERSON
ma-246	210	1	a∗η ⊆ a.	ORG
ma-246	210	2	⊆	CARDINAL
ma-246	211	1	1	CARDINAL
ma-246	211	2	⊆	CARDINAL
ma-246	212	1	a∗η	ORG
ma-246	213	1	a∗η	ORG
ma-246	216	1	a∪b ∈	PERSON
ma-246	217	1	∈	ORG
ma-246	218	1	theorem1	PERSON
ma-246	218	2	a∗η	ORG
ma-246	218	3	a∗η	ORG
ma-246	218	4	a∗η 6=	ORG
ma-246	218	5	η-cl(a∗η	GPE
ma-246	219	1	η-cl(a∗η	PERSON
ma-246	220	1	∈	ORG
ma-246	222	1	∈	ORG
ma-246	225	1	a∗η ⊆	ORG
ma-246	226	1	a∗η	ORG
ma-246	226	2	kc	PERSON
ma-246	226	3	kc ∩ a∗η	ORG
ma-246	226	4	kc	PERSON
ma-246	226	5	kc ∩ a∗η	PERSON
ma-246	226	6	kc ∈ η-o(x	PERSON
ma-246	226	7	kc ∈	PERSON
ma-246	226	8	a∗η	ORG
ma-246	226	9	η-cl(a∗η	PERSON
ma-246	226	10	1	CARDINAL
ma-246	226	11	a∗η	ORG
ma-246	227	1	a∗η	ORG
ma-246	227	2	η-cl(a∗η	PERSON
ma-246	227	3	a∗η	ORG
ma-246	227	4	η-cl(a∗η	PERSON
ma-246	227	5	⊆	CARDINAL
ma-246	227	6	η-cl(a	GPE
ma-246	229	1	∈	ORG
ma-246	230	1	∈	ORG
ma-246	230	2	k	PERSON
ma-246	230	3	⊆	CARDINAL
ma-246	230	4	⊆	CARDINAL
ma-246	230	5	theorem2	PERSON
ma-246	230	6	a∗η	ORG
ma-246	231	1	a∗η	ORG
ma-246	231	2	a∗η	ORG
ma-246	232	1	j. math	PERSON
ma-246	234	1	10.28924	CARDINAL
ma-246	234	2	{k : k	PERSON
ma-246	234	3	⊆	CARDINAL
ma-246	235	1	∈ a∗η	ORG
ma-246	236	1	η-cl(a	PERSON
ma-246	236	2	a∗η	ORG
ma-246	237	1	asa	ORG
ma-246	237	2	2	CARDINAL
ma-246	237	3	a∗η	ORG
ma-246	237	4	η-cl(a∗η	PERSON
ma-246	237	5	η-cl(a	PERSON
ma-246	238	1	3	CARDINAL
ma-246	239	1	η-o(x	FAC
ma-246	239	2	two	CARDINAL
ma-246	240	1	a∗η	ORG
ma-246	240	2	∈ η-o(x	PERSON
ma-246	240	3	⊆	CARDINAL
ma-246	240	4	∩	PERSON
ma-246	243	1	η-o(x	PERSON
ma-246	243	2	two	CARDINAL
ma-246	250	1	⊆	CARDINAL
ma-246	250	2	⊆	CARDINAL
ma-246	250	3	∩	PERSON
ma-246	250	4	∩	PERSON
ma-246	251	1	∩	PERSON
ma-246	252	1	⊆	CARDINAL
ma-246	253	1	∩	PERSON
ma-246	255	1	η-o(x	PERSON
ma-246	255	2	∩	PERSON
ma-246	258	1	⊆	CARDINAL
ma-246	258	2	a∗η	WORK_OF_ART
ma-246	259	1	1	CARDINAL
ma-246	259	2	a∗η	ORG
ma-246	260	1	η-o(x	PERSON
ma-246	260	2	two	CARDINAL
ma-246	261	1	∈ η-o(x	PERSON
ma-246	261	2	∈	ORG
ma-246	262	1	∈	ORG
ma-246	264	1	∈ η-o(x	PERSON
ma-246	264	2	∈ η-o(x	PERSON
ma-246	266	1	∩	PERSON
ma-246	268	1	u∩a)∩v	WORK_OF_ART
ma-246	271	1	u ∩a∗η ⊆	ORG
ma-246	272	1	u ∩a∗η ⊆	ORG
ma-246	272	2	∩	PERSON
ma-246	273	1	⊆	CARDINAL
ma-246	273	2	u ∩ a∗η	ORG
ma-246	273	3	u ∩ a∗η	ORG
ma-246	273	4	u ∩ a∗η ⊆	ORG
ma-246	273	5	∩	PERSON
ma-246	274	1	u ∩	ORG
ma-246	274	2	⊆	CARDINAL
ma-246	274	3	u ∩	ORG
ma-246	274	4	a∗η	ORG
ma-246	275	1	⊆	CARDINAL
ma-246	275	2	u ∩ a∗η	ORG
ma-246	277	1	j. math	PERSON
ma-246	279	1	10.28924	CARDINAL
ma-246	279	2	7result	ORG
ma-246	279	3	∩	PERSON
ma-246	280	1	∩	PERSON
ma-246	280	2	⊆	CARDINAL
ma-246	280	3	∩	PERSON
ma-246	281	1	u ∩ a∗η	LAW
ma-246	281	2	⊆	CARDINAL
ma-246	281	3	∩	PERSON
ma-246	282	1	⊆	CARDINAL
ma-246	284	1	a∗η	ORG
ma-246	285	1	η-o(x	PERSON
ma-246	285	2	two	CARDINAL
ma-246	285	3	a∗η	ORG
ma-246	285	4	a∗η	ORG
ma-246	286	1	a∗η	ORG
ma-246	286	2	a∗η	ORG
ma-246	286	3	a∗η	ORG
ma-246	288	1	∩	PERSON
ma-246	291	1	a∗η	ORG
ma-246	294	1	⊆	CARDINAL
ma-246	294	2	⊆	CARDINAL
ma-246	294	3	b∗η	PRODUCT
ma-246	294	4	a∗η	ORG
ma-246	294	5	a∗η	ORG
ma-246	298	1	a∗η	ORG
ma-246	299	1	4	CARDINAL
ma-246	300	1	⊆	CARDINAL
ma-246	301	1	a∗η ⊆	CARDINAL
ma-246	301	2	⊆	CARDINAL
ma-246	303	1	u ∈ τ(x	ORG
ma-246	304	1	∈ η-o(x	PERSON
ma-246	305	1	∈ η-o(x	PERSON
ma-246	308	1	∈	ORG
ma-246	308	2	a∗η ⊆	ORG
ma-246	308	3	∈	ORG
ma-246	309	1	∈ η-o(x	PERSON
ma-246	310	1	∈ η-o(x	PERSON
ma-246	311	1	∈	ORG
ma-246	311	2	∈	ORG
ma-246	312	1	a∗η ⊆	ORG
ma-246	314	1	j. math	PERSON
ma-246	316	1	10.28924	CARDINAL
ma-246	316	2	∈	ORG
ma-246	318	1	∈	ORG
ma-246	320	1	∈	ORG
ma-246	320	2	⊆	CARDINAL
ma-246	321	1	2	CARDINAL
ma-246	322	1	⊆	CARDINAL
ma-246	323	1	⊆	CARDINAL
ma-246	323	2	η-o(x	PERSON
ma-246	323	3	a∗η	ORG
ma-246	323	4	η-o(x	PERSON
ma-246	323	5	a∗η	ORG
ma-246	323	6	5	CARDINAL
ma-246	324	1	⊆	CARDINAL
ma-246	325	1	⊆	CARDINAL
ma-246	325	2	a∗η	ORG
ma-246	325	3	i2	ORG
ma-246	325	4	η-o(x	PERSON
ma-246	326	1	⊆	CARDINAL
ma-246	326	2	a∗η	ORG
ma-246	326	3	i1	ORG
ma-246	326	4	η-o(x	PERSON
ma-246	326	5	(ii) a∗η	ORG
ma-246	326	6	i1 ∩ i2	ORG
ma-246	326	7	η-o(x	PERSON
ma-246	327	1	i1	ORG
ma-246	327	2	η-o(x	PERSON
ma-246	327	3	a∗η	PERSON
ma-246	327	4	i2	ORG
ma-246	327	5	η-o(x	PERSON
ma-246	329	1	⊆	CARDINAL
ma-246	329	2	η-o(x	FAC
ma-246	330	1	∈	ORG
ma-246	331	1	i1 ⊆ i2	ORG
ma-246	331	2	∈	ORG
ma-246	332	1	η-o(x	PERSON
ma-246	333	1	⊆	CARDINAL
ma-246	333	2	a∗η	ORG
ma-246	333	3	i1	ORG
ma-246	333	4	η-o(x	PERSON
ma-246	335	1	i1 ∩	ORG
ma-246	335	2	⊆	CARDINAL
ma-246	336	1	⊆	CARDINAL
ma-246	337	1	a∗η	ORG
ma-246	337	2	i1	ORG
ma-246	337	3	η-o(x	PERSON
ma-246	338	1	⊆	CARDINAL
ma-246	338	2	i1 ∩ i2	ORG
ma-246	338	3	η-o(x	PERSON
ma-246	338	4	a∗η	ORG
ma-246	338	5	i2	ORG
ma-246	338	6	η-o(x	PERSON
ma-246	339	1	⊆	CARDINAL
ma-246	339	2	i1 ∩ i2	ORG
ma-246	339	3	η-o(x	PERSON
ma-246	339	4	a∗η	ORG
ma-246	339	5	i1	ORG
ma-246	339	6	η-o(x	PERSON
ma-246	339	7	a∗η	PERSON
ma-246	339	8	i2	ORG
ma-246	339	9	η-o(x	PERSON
ma-246	340	1	⊆	CARDINAL
ma-246	340	2	i1 ∩ i2	ORG
ma-246	340	3	η-o(x	PERSON
ma-246	341	1	η-o(x	PERSON
ma-246	341	2	∈	ORG
ma-246	345	1	∈	ORG
ma-246	345	2	η-o(x	PERSON
ma-246	345	3	η-o(x	FAC
ma-246	345	4	∈	ORG
ma-246	345	5	η-o(x	PERSON
ma-246	345	6	a∗η	PERSON
ma-246	345	7	i2	ORG
ma-246	345	8	η-o(x	PERSON
ma-246	345	9	a∗η	ORG
ma-246	345	10	i1 ∩ i2	ORG
ma-246	345	11	η-o(x	PERSON
ma-246	346	1	⊆	CARDINAL
ma-246	346	2	a∗η	ORG
ma-246	346	3	i1	ORG
ma-246	346	4	η-o(x	PERSON
ma-246	346	5	a∗η	PERSON
ma-246	346	6	i2	ORG
ma-246	346	7	η-o(x	PERSON
ma-246	347	1	a∗η	ORG
ma-246	347	2	i1 ∩ i2	ORG
ma-246	347	3	η-o(x	PERSON
ma-246	348	1	i1	ORG
ma-246	348	2	η-o(x	PERSON
ma-246	348	3	a∗η	PERSON
ma-246	348	4	i2	ORG
ma-246	348	5	η-o(x	PERSON
ma-246	350	1	j. math	PERSON
ma-246	352	1	10.28924	CARDINAL
ma-246	353	1	2	CARDINAL
ma-246	356	1	a∗η	PERSON
ma-246	356	2	⊆	CARDINAL
ma-246	357	1	3	CARDINAL
ma-246	360	1	1	CARDINAL
ma-246	360	2	2	CARDINAL
ma-246	360	3	a∗η	ORG
ma-246	361	1	6	CARDINAL
ma-246	362	1	⊆	CARDINAL
ma-246	363	1	⊆	CARDINAL
ma-246	363	2	⊆	CARDINAL
ma-246	363	3	⊆	CARDINAL
ma-246	363	4	η-cl(a	PERSON
ma-246	363	5	∈	ORG
ma-246	366	1	a∗η	ORG
ma-246	367	1	η-cl(a	PERSON
ma-246	367	2	∗	CARDINAL
ma-246	369	1	⊆	CARDINAL
ma-246	369	2	⊆	CARDINAL
ma-246	369	3	2	CARDINAL
ma-246	370	1	a∗η	PERSON
ma-246	370	2	⊆	CARDINAL
ma-246	370	3	1	CARDINAL
ma-246	370	4	a∗η ⊆	ORG
ma-246	370	5	b∗η	PRODUCT
ma-246	371	1	cl∗η(a	ORG
ma-246	371	2	⊆	CARDINAL
ma-246	372	1	⊆	CARDINAL
ma-246	373	1	⊆	CARDINAL
ma-246	373	2	⊆	CARDINAL
ma-246	373	3	cl∗η(a ∩	ORG
ma-246	373	4	⊆	CARDINAL
ma-246	373	5	⊆	CARDINAL
ma-246	374	1	cl∗η(a ∩	ORG
ma-246	374	2	⊆	CARDINAL
ma-246	376	1	x /∈ a.	PERSON
ma-246	378	1	⊆	CARDINAL
ma-246	381	1	⊆	CARDINAL
ma-246	381	2	kc	PERSON
ma-246	381	3	kc ∈ η-o(x	PERSON
ma-246	383	1	kc ∈ η-o(x	PERSON
ma-246	383	2	a∗η ⊆ a.	ORG
ma-246	384	1	a∗η	PERSON
ma-246	385	1	η-cl(a	PERSON
ma-246	386	1	∈	ORG
ma-246	388	1	2	CARDINAL
ma-246	389	1	a∗η	PERSON
ma-246	389	2	cl∗η(a	ORG
ma-246	389	3	⊆	CARDINAL
ma-246	390	1	2	CARDINAL
ma-246	390	2	1	CARDINAL
ma-246	390	3	a∗η	ORG
ma-246	390	4	itfollows	PERSON
ma-246	391	1	a∗η	ORG
ma-246	393	1	j. math	PERSON
ma-246	395	1	10.28924	CARDINAL
ma-246	395	2	⊆	CARDINAL
ma-246	395	3	cl∗η(a	ORG
ma-246	395	4	6=	CARDINAL
ma-246	395	5	η-cl(a	GPE
ma-246	396	1	η-cl(a	PERSON
ma-246	396	2	⊂	GPE
ma-246	397	1	∈ cl∗η(a	ORG
ma-246	399	1	∈ cl∗η(a	ORG
ma-246	399	2	definition2	PERSON
ma-246	399	3	a∗η	ORG
ma-246	399	4	∈	ORG
ma-246	400	1	∈ a∗η	ORG
ma-246	401	1	∈	ORG
ma-246	404	1	⊆	CARDINAL
ma-246	404	2	⊆	CARDINAL
ma-246	404	3	kc	PERSON
ma-246	404	4	kc ∈ η-o(x	PERSON
ma-246	404	5	/∈ a∗η	ORG
ma-246	405	1	kc	PERSON
ma-246	407	1	⊆	CARDINAL
ma-246	408	1	2	CARDINAL
ma-246	408	2	1	CARDINAL
ma-246	408	3	∗ η	ORG
ma-246	408	4	a∗η∪	ORG
ma-246	408	5	a∗η	ORG
ma-246	408	6	∗ η .note	FAC
ma-246	408	7	1	CARDINAL
ma-246	408	8	a∗η	ORG
ma-246	408	9	a∗η	ORG
ma-246	409	1	a∗η ⊆	ORG
ma-246	409	2	∗ η	PERSON
ma-246	410	1	∈	ORG
ma-246	410	2	∗ η	ORG
ma-246	411	1	∈	ORG
ma-246	413	1	2	CARDINAL
ma-246	418	1	∈	ORG
ma-246	418	2	a∗η	ORG
ma-246	418	3	a∗η	ORG
ma-246	418	4	∈ a∗η	ORG
ma-246	419	1	∗ η ⊆ a∗η	ORG
ma-246	421	1	∗	CARDINAL
ma-246	423	1	⊆	CARDINAL
ma-246	424	1	⊆	CARDINAL
ma-246	424	2	a∗η ⊆ cl∗η(a	ORG
ma-246	424	3	⊆	CARDINAL
ma-246	424	4	∗ η ⊆ cl∗η(a	ORG
ma-246	425	1	cl∗η	PERSON
ma-246	427	1	⊆	CARDINAL
ma-246	428	1	⊆	CARDINAL
ma-246	428	2	a∗η	PERSON
ma-246	429	1	2	CARDINAL
ma-246	429	2	⊆	CARDINAL
ma-246	430	1	a∗η ⊆	ORG
ma-246	430	2	a∗η	ORG
ma-246	430	3	2	CARDINAL
ma-246	430	4	a∗η ⊆ cl∗η(a	ORG
ma-246	431	1	2	CARDINAL
ma-246	431	2	1	CARDINAL
ma-246	431	3	⊆	CARDINAL
ma-246	432	1	2	CARDINAL
ma-246	433	1	⊆	CARDINAL
ma-246	434	1	2	CARDINAL
ma-246	434	2	1	CARDINAL
ma-246	434	3	cl∗η(a ∪	ORG
ma-246	434	4	a∗η	ORG
ma-246	435	1	cl∗η(a	ORG
ma-246	435	2	⊆	CARDINAL
ma-246	437	1	j. math	PERSON
ma-246	439	1	10.28924	CARDINAL
ma-246	439	2	⊆	CARDINAL
ma-246	440	1	∗	CARDINAL
ma-246	440	2	a∗η	ORG
ma-246	440	3	a∗η ⊆	ORG
ma-246	440	4	∗ η ⊆ cl∗η(a	ORG
ma-246	441	1	2	CARDINAL
ma-246	442	1	∗ η	ORG
ma-246	443	1	∗ η ⊆ cl∗η(a	ORG
ma-246	443	2	cl∗η	PERSON
ma-246	444	1	∗	CARDINAL
ma-246	445	1	cl∗η(a	ORG
ma-246	446	1	cl∗η	PERSON
ma-246	447	1	3	CARDINAL
ma-246	450	1	4	CARDINAL
ma-246	453	1	1	CARDINAL
ma-246	453	2	a∗η	ORG
ma-246	454	1	2	CARDINAL
ma-246	455	1	cl∗η(a ∪	ORG
ma-246	457	1	cl∗η(a ∪	ORG
ma-246	457	2	cl∗η(a	ORG
ma-246	459	1	cl∗η(a ∪	ORG
ma-246	461	1	8	CARDINAL
ma-246	462	1	⊆	CARDINAL
ma-246	463	1	⊆	CARDINAL
ma-246	465	1	⊆	CARDINAL
ma-246	466	1	2	CARDINAL
ma-246	466	2	cl∗η(a	ORG
ma-246	467	1	a∗η	ORG
ma-246	468	1	a∗η ⊆	ORG
ma-246	468	2	∪	ORG
ma-246	468	3	a∗. hence	PERSON
ma-246	468	4	⊆	CARDINAL
ma-246	469	1	9	CARDINAL
ma-246	474	1	6	CARDINAL
ma-246	475	1	η-cl(a	PERSON
ma-246	478	1	6	CARDINAL
ma-246	478	2	η-cl(a	PERSON
ma-246	479	1	η-cl(a	PERSON
ma-246	481	1	three	CARDINAL
ma-246	484	1	4	CARDINAL
ma-246	485	1	cl∗η	PERSON
ma-246	485	2	η	PERSON
ma-246	487	1	j. math	PERSON
ma-246	489	1	10.28924	CARDINAL
ma-246	489	2	12	CARDINAL
ma-246	489	3	10	CARDINAL
ma-246	490	1	η-o(x	FAC
ma-246	490	2	two	CARDINAL
ma-246	490	3	⊆	CARDINAL
ma-246	491	1	cl∗η(a ∪	ORG
ma-246	493	1	η-o(x	PERSON
ma-246	493	2	two	CARDINAL
ma-246	494	1	2	CARDINAL
ma-246	494	2	3	CARDINAL
ma-246	494	3	i),it	ORG
ma-246	494	4	cl∗η(a ∪	ORG
ma-246	494	5	a∗η	ORG
ma-246	496	1	cl∗η(a ∪	ORG
ma-246	497	1	10	CARDINAL
ma-246	497	2	η-o(x	PERSON
ma-246	497	3	two	CARDINAL
ma-246	498	1	5	CARDINAL
ma-246	499	1	η-o(x	FAC
ma-246	499	2	two	CARDINAL
ma-246	499	3	cl∗η	PERSON
ma-246	499	4	η	GPE
ma-246	499	5	5	CARDINAL
ma-246	499	6	cl∗η	GPE
ma-246	499	7	η	PERSON
ma-246	499	8	η-o(x	PERSON
ma-246	501	1	η-o(x	PERSON
ma-246	501	2	two	CARDINAL
ma-246	503	1	1	CARDINAL
ma-246	504	1	η-o(x	PERSON
ma-246	504	2	two	CARDINAL
ma-246	505	1	iff cl∗η(a	ORG
ma-246	507	1	η-o(x	PERSON
ma-246	507	2	two	CARDINAL
ma-246	508	1	a∗η ⊆ a.	ORG
ma-246	508	2	a∗η = a. now	PERSON
ma-246	508	3	definiton2	PERSON
ma-246	510	1	2	CARDINAL
ma-246	511	1	a∗η = a. so	PERSON
ma-246	511	2	a∗η	PERSON
ma-246	511	3	a∗η ⊆	ORG
ma-246	512	1	11	CARDINAL
ma-246	513	1	η-o(x	FAC
ma-246	513	2	two	CARDINAL
ma-246	514	1	j ⊆	LAW
ma-246	516	1	τ∗η	GPE
ma-246	516	2	⊆	CARDINAL
ma-246	516	3	η-o(x	PERSON
ma-246	516	4	⊆	CARDINAL
ma-246	518	1	η-o(x	PERSON
ma-246	518	2	two	CARDINAL
ma-246	519	1	5	CARDINAL
ma-246	519	2	cl∗η	GPE
ma-246	519	3	η	GPE
ma-246	519	4	cl∗η	PERSON
ma-246	520	1	⊆	CARDINAL
ma-246	521	1	ac	ORG
ma-246	522	1	τ∗-closed	ORG
ma-246	523	1	j. math	PERSON
ma-246	525	1	10.28924	CARDINAL
ma-246	525	2	∗	CARDINAL
ma-246	525	3	⊆	CARDINAL
ma-246	527	1	ac	ORG
ma-246	527	2	∗	CARDINAL
ma-246	529	1	theorem8	ORG
ma-246	529	2	⊆	CARDINAL
ma-246	529	3	ac	ORG
ma-246	530	1	⊆	CARDINAL
ma-246	530	2	ac	ORG
ma-246	531	1	lemma1	ORG
ma-246	531	2	ac	ORG
ma-246	532	1	⊆	CARDINAL
ma-246	533	1	η-o(x	PERSON
ma-246	533	2	⊆	CARDINAL
ma-246	534	1	ac	ORG
ma-246	534	2	9	CARDINAL
ma-246	534	3	ac	ORG
ma-246	535	1	lemma 1	ORG
ma-246	535	2	ac	ORG
ma-246	536	1	η-o(x	PERSON
ma-246	536	2	⊆	CARDINAL
ma-246	537	1	5	CARDINAL
ma-246	539	1	almocera closureoperator	ORG
ma-246	539	2	η-o(x	PERSON
ma-246	539	3	two	CARDINAL
ma-246	540	1	η-o(x	ORG
ma-246	541	1	1	CARDINAL
ma-246	541	2	a. al-omari	PERSON
ma-246	541	3	t. noiri	PERSON
ma-246	541	4	γ∗	ORG
ma-246	541	5	sci.	ORG
ma-246	544	1	26	CARDINAL
ma-246	544	2	1	CARDINAL
ma-246	544	3	2016)5-16.[2	DATE
ma-246	544	4	e. hatir	PERSON
ma-246	544	5	a. al-omari	PERSON
ma-246	544	6	s. jafari	PERSON
ma-246	545	1	53(2014	CARDINAL
ma-246	545	2	53-64.[3	CARDINAL
ma-246	545	3	d. jankovic	PERSON
ma-246	545	4	hamlett	PERSON
ma-246	545	5	amer	PERSON
ma-246	546	1	monthly 97	DATE
ma-246	546	2	4	CARDINAL
ma-246	546	3	1990	DATE
ma-246	546	4	295	CARDINAL
ma-246	546	5	k. kuratowski	PERSON
ma-246	546	6	1933.[5	CARDINAL
ma-246	546	7	k. kuratowski	PERSON
ma-246	546	8	new york	GPE
ma-246	546	9	1966.[6	CARDINAL
ma-246	546	10	n. levine	PERSON
ma-246	546	11	amer	PERSON
ma-246	547	1	monthly	DATE
ma-246	547	2	70	DATE
ma-246	547	3	1963	DATE
ma-246	547	4	36-41.[7	CARDINAL
ma-246	547	5	k. rajak	PERSON
ma-246	548	1	j. com	PERSON
ma-246	549	1	38	CARDINAL
ma-246	549	2	5)(2012	CARDINAL
ma-246	549	3	12-17.[8	CARDINAL
ma-246	549	4	d. subbulakshmi	PERSON
ma-246	549	5	k. sumathi	PERSON
ma-246	549	6	k. indiran	PERSON
ma-246	549	7	η	GPE
ma-246	550	1	j. innov	PERSON
ma-246	553	1	eng	PERSON
ma-246	553	2	8	CARDINAL
ma-246	553	3	276	CARDINAL
ma-246	553	4	r. vaidyanathaswamy	PERSON
ma-246	554	1	indian	NORP
ma-246	555	1	sci.	ORG
ma-246	556	1	20 (1944	DATE
ma-246	556	2	51	CARDINAL
ma-246	557	1	1	CARDINAL
ma-246	557	2	2	CARDINAL
ma-246	557	3	3	CARDINAL
ma-246	557	4	4	CARDINAL
ma-246	558	1	5	CARDINAL