Categories of Open Sets in Generalized Primal Topological Spaces
Abstract
:1. Introduction
1.1. Literature Review
1.2. Space
- (i)
- (ii)
- is -closed,
- (iii)
- (iv)
- whenever
- (v)
- (vi)
- (i)
- (ii)
- (iii)
- (iv)
- whenever
- (v)
2. Methodology
3. Main Results
3.1. Some Classes of -Open Sets
- (i)
- When E is named a -semi-open set.
- (ii)
- When E is named a -pre-open set.
- (iii)
- When E is named a -regular open set.
- (iv)
- When E is named a -β-open set.
- (v)
- When E is named a -α-open set.
- (i)
- Each -semi-open set is -semi-open.
- (ii)
- Each -α-open set is -α-open.
- (iii)
- Each -β-open set is -β-open.
- (iv)
- Each -pre-open set is -pre-open.
- (i)
- E forms a -α-open set iff E is -semi-open as well as -pre-open.
- (ii)
- Considering E as -semi-open, E is -β-open.
- (iii)
- Considering E as -pre-open, E is -β-open.
- (i)
- (ii)
- (iii)
- (i)
- E is -regular open;
- (ii)
- E is -semi-closed as well as -open;
- (iii)
- E is -pre-open as well as -semi-closed;
- (iv)
- E is -α-open as well as -β-closed;
- (v)
- E is -α-open as well as -semi-closed;
- (vi)
- E is -open as well as -β-closed.
- (i)
- E is -semi-open, where However, E is not -semi-open, where
- (ii)
- E is -β-open, where However, E is not -semi-open.
- (iii)
- E is -β-open. However, E is not -pre-open, where .
- (i)
- The countable union of -semi-open sets is -semi-open.
- (ii)
- The countable union of -pre-open sets is -pre-open.
- (iii)
- The countable union of -α-open sets is -α-open.
- (iv)
- The countable union of -β-open sets is -β-open.
3.2. Regular -Semi-Open and -Dense
- (i)
- E is regular -semi-open;
- (ii)
- E is -semi-open as well as -semi-closed;
- (iii)
- E is -β-open as well as -semi-closed;
- (iv)
- E is -semi-open as well as -β-closed.
- (i)
- E is -semi-open.
- (ii)
- E is -semi-open ⟺ ∃ a -open set F satisfying
- (iii)
- For Hence, G is -semi-open whenever E is -semi-open.
- (iv)
- Whenever E is -semi-open and F is -open, is -semi-open.
- (i)
- Whenever E is a -pre-closed set,
- (ii)
- Whenever E is a -α-closed set,
- (iii)
- Whenever E is a -β-closed set,
- (i)
- Whenever and
- (ii)
- Whenever and
- (iii)
- Whenever
4. Decomposition of -Continuity
- (i)
- is -α-continuous;
- (ii)
- and satisfy and there exists satisfying and
- (iii)
- is -closed and -closed.
- (i)
- is -α-continuous iff is -semi continuous as well as -pre continuous.
- (ii)
- Each -semi-continuous as well as each -pre-continuous set is -β-continuous.
- (i)
- Each -open function is -semi-open.
- (ii)
- Each -semi-open (respectively, -semi-closed) function is -semi-open (respectively, -semi-closed).
- (i)
- is -semi-continuous;
- (ii)
- is -semi-open;
- (iii)
- is -semi-closed.
- (i)
- is -regular continuous;
- (ii)
- is -pre-continuous as well as -semi-closed;
- (iii)
- is -α-continuous as well as -semi-closed.
- (i)
- is regular -semi-continuous;
- (ii)
- is -semi-continuous as well as -semi-closed;
- (iii)
- is -β-continuous as well as -semi-closed.
5. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Levine, N. Semi-open and semi-continuity in topological spaces. Am. Math. Mon. 1963, 70, 36–41. [Google Scholar] [CrossRef]
- Al-Ghour, S.; Mansur, K. Between open sets and semi-open sets. Univ. Sci. 2018, 23, 9–20. [Google Scholar] [CrossRef]
- Darwesh, H. A new type of semi-open sets and semi-continuity in topological spaces. Zan. J. Pur. Appl. Sci. 2011, 23, 82–94. [Google Scholar]
- Srinivasa, V. On Semi-open sets and semi-separability. Glob. J. Inc. 2013, 13, 17–20. [Google Scholar]
- Mashhour, A.; Abd El-Monsef, M.; El-Deeb, S. On precontinuous and weak precontinuous mappings. Proc. Math. Phys. Soc. Egypt 1982, 53, 47–53. [Google Scholar]
- Abd El-Monswf, M.; El-Deeb, S.; Mahmoud, R. β-open sets and β-continuous mappings. Bull. Fac. Sci. Assiut Univ. 1983, 12, 77–90. [Google Scholar]
- Najastad, O. On some classes of nearly open sets. Pac. J. Math. 1965, 15, 961–970. [Google Scholar] [CrossRef]
- Andrijevic, D. On b-open sets. Mat. Vesn. 1996, 48, 59–64. [Google Scholar]
- Császár, A. Generalized open sets. Acta Math. Hung. 1997, 75, 65–87. [Google Scholar] [CrossRef]
- Aponte, E.; Subramanian, V.; Macias, J.; Krishnan, M. On semi-continuous and clisquish functions in generalized topological spaces. Axioms 2023, 12, 130. [Google Scholar] [CrossRef]
- Korczak-Kubiak, E.; Loranty, A.; Pawlak, R.J. Baire generalized topological spaces, generalized metric spaces and infinite games. Acta Math. Hung. 2013, 140, 203–231. [Google Scholar] [CrossRef]
- Császár, A. Generalized topology, generalized continuity. Acta Math. Hung. 2002, 96, 351–357. [Google Scholar] [CrossRef]
- Császár, A. Remarks on quasi topologyies. Acta. Math. Hung. 2008, 119, 197–200. [Google Scholar] [CrossRef]
- Császár, A. Separation axioms for generalized topologies. Acta Math. Hung. 2004, 104, 63–69. [Google Scholar] [CrossRef]
- Ge, X.; Ge, Y. μ-Separations in generalized topological spaces. Appl. Math. J. Chin. Univ. 2010, 25, 243–252. [Google Scholar] [CrossRef]
- Császár, A. Generalized open sets in generalized topologies. Acta Math. Hung. 2005, 106, 53–66. [Google Scholar] [CrossRef]
- Császár, A. Extremally disconnected generalized topologies. Ann. Univ. Sci. Bp. 2004, 47, 91–96. [Google Scholar]
- Császár, A. δ- and θ-modifications of generalized topologies. Acta. Math. Hung. 2008, 120, 275–279. [Google Scholar] [CrossRef]
- Kuratowski, K. Topology, 1st ed.; Elsevier: Amsterdam, The Netherlands, 1966. [Google Scholar]
- Janković, D.; Hamlett, T. New topologies from old via ideals. Am. Math. Mon. 1990, 97, 295–310. [Google Scholar] [CrossRef]
- Choquet, G. Sur les notions de filtre et de grille. Comptes Rendus Acad. Sci. Paris 1947, 224, 171–173. [Google Scholar]
- Roy, B.; Mukherjee, M. On a typical topology induced by a grill. Soochow J. Math. 2007, 33, 771–786. [Google Scholar]
- Al-Omari, A.; Noiri, T. On ΨG-sets in grill topological spaces. Filomat 2011, 25, 187–196. [Google Scholar] [CrossRef]
- Al-Omari, A.; Noiri, T. On Ψ*-operator in ideal m-spaces. Bol. Soc. Paran. Math. 2012, 30, 53–66. [Google Scholar] [CrossRef]
- Al-Omari, A.; Noiri, T. On ΨG-operator in grill topological spaces. An. Univ. Oradea Fasc. Mat. 2012, 19, 187–196. [Google Scholar]
- Acharjee, S.; Özkoç, M.; Issaka, F. Primal topological spaces. arXiv 2022. [Google Scholar] [CrossRef]
- AL-Omari, A.; Acharjee, S.; Özkoç, M. A new operator of primal topological spaces. arXiv 2022. [Google Scholar] [CrossRef]
- Mejías, L.; Vielma, J.; Guale, A.; Pineda, E. Primal topologies on finite-dimensional vector spaces induced by matrices. Int. J. Math. Sci. 2023. [Google Scholar] [CrossRef]
- Al-Saadi, H.; Al-Malki, H. Generalized primal topological spaces. AIMS Math. 2023, 8, 24162–24175. [Google Scholar] [CrossRef]
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Al-Saadi, H.; Al-Malki, H. Categories of Open Sets in Generalized Primal Topological Spaces. Mathematics 2024, 12, 207. https://doi.org/10.3390/math12020207
Al-Saadi H, Al-Malki H. Categories of Open Sets in Generalized Primal Topological Spaces. Mathematics. 2024; 12(2):207. https://doi.org/10.3390/math12020207
Chicago/Turabian StyleAl-Saadi, Hanan, and Huda Al-Malki. 2024. "Categories of Open Sets in Generalized Primal Topological Spaces" Mathematics 12, no. 2: 207. https://doi.org/10.3390/math12020207
APA StyleAl-Saadi, H., & Al-Malki, H. (2024). Categories of Open Sets in Generalized Primal Topological Spaces. Mathematics, 12(2), 207. https://doi.org/10.3390/math12020207