Generalized primal topological spaces

Hanan Al-Saadi; Huda Al-Malki; Hanan Al-Saadi; Huda Al-Malki

doi:10.3934/math.20231232

AIMS Mathematics

2023, Volume 8, Issue 10: 24162-24175. doi: 10.3934/math.20231232

Previous Article Next Article

Research article

Generalized primal topological spaces

Hanan Al-Saadi ^{1
,
,},
Huda Al-Malki ²

1.
Department of Mathematics, Faculty of Applied Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
2.
Department of Basic Sciences, Adham University College, Umm Al-Qura University, Saudi Arabia
Correction on: AIMS Mathematics 9: 19068-19069.

Received: 17 June 2023 Revised: 16 July 2023 Accepted: 30 July 2023 Published: 10 August 2023
MSC : 54A05, 54A10, 54A20

In the present article, a new category of mathematical structure is described based on the topological structure "primal" and the notion of "generalized". Such a structure is discussed in detail in terms of topological properties and some basic theories. Also, we introduced some operators using the concepts "primal" and "generalized primal neighbourhood", which have a lot of nice properties.
- generalized primal topology,
- generalized primal neighbourhood,
- $ (\mathfrak{g}, \mathcal{P}) $-open sets,
- $ cl^\diamond $-operator,
- $ \Phi $-operator
Citation: Hanan Al-Saadi, Huda Al-Malki. Generalized primal topological spaces[J]. AIMS Mathematics, 2023, 8(10): 24162-24175. doi: 10.3934/math.20231232

Related Papers:

Abstract

In the present article, a new category of mathematical structure is described based on the topological structure "primal" and the notion of "generalized". Such a structure is discussed in detail in terms of topological properties and some basic theories. Also, we introduced some operators using the concepts "primal" and "generalized primal neighbourhood", which have a lot of nice properties.

References

[1]	S. Acharjee, M. Özkoç, F. Y. Issaka, Primal topological spaces, arXiv, 2022. https://doi.org/10.48550/arXiv.2209.12676
[2]	J. C. R. Alcantud, Convex soft geometries, J. Comput. Cognit. Eng., 1 (2022), 2–12. http://doi.org/10.47852/bonviewJCCE597820 doi: 10.47852/bonviewJCCE597820
[3]	A. Al-Omari, S. Acharjee, M. Özkoç, A new operator of primal topological spaces, arXiv, 2022. https://doi.org/10.48550/arXiv.2210.17278
[4]	A. Al-Omari, T. Noiri, On $ \widetilde\Psi_{\mathcal{G}}$-sets in grill topological spaces, Filomat, 25 (2011), 187–196. http://doi.org/10.2298/FIL1102187A doi: 10.2298/FIL1102187A
[5]	A. Al-Omari, T. Noiri, On $\Psi_{\ast}$-operator in ideal $m$-spaces, Bol. Soc. Parana. Math., 30 (2012), 53–66. http://doi.org/10.5269/bspm.v30i1.12787 doi: 10.5269/bspm.v30i1.12787
[6]	A. Al-Omari, T. Noiri, On $\Psi_G$-perator in grill topological spaces, Ann. Univ. Oradea Fasc. Mat., 2012.
[7]	T. M. Al-shami, M. Abo-Elhamayel, Novel class of ordered separation axioms using limit points, Appl. Math. Inf. Sci., 14 (2020), 1103–1111. http://doi.org/10.18576/amis/140617 doi: 10.18576/amis/140617
[8]	T. M. Al-shami, Z. A. Ameen, R. Abu-Gdairi, A. Mhemdi, On primal soft topology, Mathematics, 11 (2023), 2329. https://doi.org/10.3390/math11102329 doi: 10.3390/math11102329
[9]	K. C. Chattopadhyay, O. Njåstad, W. J. Thron, Merotopic spaces and extensions of closure spaces, Can. J. Math., 35 (1983), 613–629. https://doi.org/10.4153/CJM-1983-035-6 doi: 10.4153/CJM-1983-035-6
[10]	K. Chattopadhyay, W. J. Thron, Extensions of closure spaces, Can. J. Math., 29 (1977), 1277–1286. https://doi.org/10.4153/CJM-1977-127-6 doi: 10.4153/CJM-1977-127-6
[11]	G. Choquet, Sur les notions de filtre et de grille, Comptes Rendus Acad. Sci. Paris, 224 (1947), 171–173.
[12]	Á. Császár, Generalized open sets, Acta Math. Hung., 75 (1997), 65–87. https://doi.org/10.1023/A:1006582718102 doi: 10.1023/A:1006582718102
[13]	Á. Császár, Generalized topology, generalized continuity, Acta Math. Hung., 96 (2002), 351–357. https://doi.org/10.1023/A:1019713018007 doi: 10.1023/A:1019713018007
[14]	Á. Császár, Separation axioms for generalized topologies, Acta Math. Hung., 104 (2004), 63–69. https://doi.org/10.1023/B:AMHU.0000034362.97008.c6 doi: 10.1023/B:AMHU.0000034362.97008.c6
[15]	Á. Császár, Extremally disconnected generalized topologies, Ann. Univ. Sci. Budapest, 47 (2004), 151–161.
[16]	Á. Császár, $\delta$- and $\theta$-modifications of generalized topologies, Acta. Math. Hung., 120 (2008), 275–279. http://doi.org/10.1007/s10474-007-7136-9 doi: 10.1007/s10474-007-7136-9
[17]	T. R. Hamlett, Ideals in topological spaces and the set operator $\Psi$, Boll. Un. Mat. Ital., 7 (1990), 863–874.
[18]	E. Hatir, S. Jafari, On some new classes of sets and a new decomposition of continuity via grills, J. Adv. Math. Stud., 3 (2010), 33–40.
[19]	D. Janković, T. R. Hamlett, New topologies from old via ideals, Am. Math. Mon., 97 (1990), 295–310. https://doi.org/10.1080/00029890.1990.11995593 doi: 10.1080/00029890.1990.11995593
[20]	K. Kuratowski, Topology, Vol. I, Elsevier, 1966. https://doi.org/10.1016/C2013-0-11022-7
[21]	L. Mejías, J. Vielma, Á. Guale, E. Pineda, Primal topologies on finite-dimensional vector spaces induced by matrices, Int. J. Math. Math. Sci., 2023 (2023), 9393234. https://doi.org/10.1155/2023/9393234 doi: 10.1155/2023/9393234
[22]	S. Modak, Topology on grill-filter space and continuity, Bol. Soc. Parana. Mat., 31 (2013), 219–230. http://doi.org/10.5269/bspm.v31i2.16603 doi: 10.5269/bspm.v31i2.16603
[23]	S. Modak, Grill-filter space, J. Indian Math. Soc., 80 (2013), 313–320.
[24]	S. Modak, Decompositions of generalized continuity in grill topological spaces, Thai. J. Math., 13 (2015), 509–515.
[25]	L. Nachbin, Topology and order, D. Van Nostrand Inc., 1965.
[26]	B. Roy, M. N. Mukherjee, On a typical topology induced by a grill, Soochow J. Math., 33 (2007), 771–786.
[27]	B. Roy, M. N. Mukherjee, On a type of compactness via grills, Mat. Ves., 59 (2007), 113–120.
[28]	B. Roy, M. N. Mukherjee, S. K. Ghosh, On a new operator based on grill and its associated topology, Arab J. Math. Sci., 14 (2008), 21–32.
[29]	B. Roy, M. N. Mukherjee, Concerning topologies induced by principal grills, An. Stiint. Univ. Al. I. Cuza Iasi., 55 (2009), 285–294.
[30]	W. J. Thron, Proximity structure and grills, Math. Ann., 206 (1973), 35–62. https://doi.org/10.1007/BF01431527 doi: 10.1007/BF01431527
[31]	B. K. Tyagi, H. V. S. Chauhan, On generalized closed sets in generalized topological spaces, Cubo, 18 (2016), 27–45. https://doi.org/10.4067/S0719-06462016000100003 doi: 10.4067/S0719-06462016000100003
[32]	S. VibinSalimRaj, V. Senthilkumaran, Y. Palaniappan, On contra $G_{(gs)^\ast}$ continuous functions in grill topological spaces, Global. J. Pure Appl. Math., 17 (2021), 333–342.
[33]	G. E. Xun, G. E. Ying, $\mu$-separations in generalized topological spaces, Appl. Math., 25 (2010), 243–252. http://doi.org/10.1007/s11766-010-2274-1 doi: 10.1007/s11766-010-2274-1

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)