Some Types of Compactness in Topological Spaces

Fatima  Abdulameer Hamdan

doi:10.32792/jeps.v16i2.868

Authors

Fatima Abdulameer Hamdan University of Thi-Qar/College of Education for Pure Sciences

DOI:

https://doi.org/10.32792/jeps.v16i2.868

Keywords:

topological spaces, α-open sets, compactness, α-compactness, semi-compactness, strong compactness, regular compactness, generalized open sets, hierarchy of compactness

Abstract

This research explores the hierarchical structure among different generalizations of open sets and compactness within topological spaces. Specifically, it considers α-open, semi-open, pre-open, and regular open sets, along with their associated compactness concepts, including α- compactness, semi-compactness, strong compactness, and regular compactness. The study establishes several inclusion relations among these notions, supported by formal proofs and illustrative examples demonstrating that these implications are generally not reversible. Our results contribute to a deeper structural understanding of generalized topological properties and their interconnections, providing a clear reference for further research in general topology and related fields.

References

Levine, N., Semi-open Sets and Semi-Continuity in Topological Spaces, American Mathematical Monthly, vol. 70, no. 1, pp. 36–41, 1963.

DOI: https://doi.org/10.1080/00029890.1963.11990070

Munkres, J. R., Topology, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 2000.

DOI: https://doi.org/10.1002/9781118625392

Nasir, A. L., Some Kinds of Strongly Compact and Pairwise Compact Space, M.Sc. Thesis, University of Baghdad, 2005.

Nasser, M. A., New Types of Compact Spaces, Department of Mathematics, College of Education, Sana’a University, Yemen, 2007.

Oudetallah, J., Alharbi, R., and Jorge, G., On r-Compactness in Topological and Bitopological Spaces, MDPI, 2024.

DOI: https://doi.org/10.3390/axioms13030172

Al-Saadi, H. and Al-Hodieb, M., Sets Related to Openness and Continuity Decompo- sitions in Primal Topological Spaces, European Journal of Pure and Applied Mathe- matics, vol. 17, no. 2, pp. 1352–1368, 2024.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5127

Arkhangel’skii, A. V. and Fedorchuk, V. V., The Basic Concepts and Constructions of General Topology, Encyclopaedia of Mathematical Sciences, vol. 17, Springer, 1990.

DOI: https://doi.org/10.1007/978-3-642-61280-8

Dugundji, J., Topology, Allyn and Bacon, Boston, 1966.

DOI: https://doi.org/10.1007/978-1-4615-9514-8

Al-Omari, A. and Noiri, T., On -Compact Spaces, Acta Mathematica Hungarica, vol. 132, no. 3, pp. 275–283, 2011.

DOI: https://doi.org/10.1007/s10474-011-0123-1

Ganster, M., Preopen Sets and Resolvable Spaces, Kyungpook Mathematical Journal, vol. 27, no. 2, pp. 135–143, 1987.

DOI: https://doi.org/10.5666/KMJ.1987.27.2.135

Velicko, N. V., H-Closed Topological Spaces, American Mathematical Society Trans- lations, vol. 78, no. 2, pp. 103–118, 1968.

DOI: https://doi.org/10.1090/trans2/078/05

Stone, M. H., Applications of the Theory of Boolean Rings to General Topology, Transactions of the American Mathematical Society, vol. 41, no. 3, pp. 375–481, 1937.

DOI: https://doi.org/10.1090/S0002-9947-1937-1501905-7

Tychonoff, A., U¨ ber die topologische Erweiterung von R¨aumen, Mathematische An- nalen, vol. 102, no. 1, pp

. 544–561, 1930.

DOI: https://doi.org/10.1007/BF01782364

Singal, M. K. and Singal, A. R., Almost-Continuous Mappings, Yokohama Mathe- matical Journal, vol. 16, pp. 63–73, 1968.

DOI: https://doi.org/10.1007/BF02985823

Thompson, T., S-Closed Spaces, Proceedings of the American Mathematical Society, vol. 60, pp. 335–338, 1976.

DOI: https://doi.org/10.1090/S0002-9939-1976-0412326-7

Al-Saadi, H. and Al-Hodieb, M., New Forms of Compactness in Primal Topological Spaces, European Journal of Pure and Applied Mathematics, vol. 17, no. 3, pp. 1450–1465, 2024.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5180

Oudetallah, J., et al., On r-Compactness and s-Compactness in Bitopological Spaces,

Topology and its Applications, vol. 340, 108452, 2024.

DOI: https://doi.org/10.1016/j.topol.2023.108452

Al-Ghour, S., Certain Types of Mildly Compact Spaces, Filomat, vol. 37, no. 15, pp. 4985–4995, 2023.

DOI: https://doi.org/10.2298/FIL2315985A

Willard, S., General Topology, Dover Publications, 2004.

DOI: https://doi.org/10.1007/978-1-4612-0899-0

Engelking, R., General Topology (Revised ed.), Heldermann Verlag, 1989.

DOI: https://doi.org/10.1007/978-3-642-61715-7

Kelley, J. L., General Topology, Springer-Verlag, 1975.

DOI: https://doi.org/10.1007/978-1-4612-6315-2