On $\Gamma$-Paracompact Spaces

Abstract

We introduce the class of $\Gamma$-paracompact spaces as a stronger form of paracompactness. A space $X$ is said to be $\Gamma$-paracompact ($\Gamma$-P, for short) space if every open cover of $X$ has a strongly locally finite (SLF) open refinement. We give some characterizations of $\Gamma$-P spaces. We also define some weaker forms of $\Gamma$-P spaces as $\Gamma_{\sigma}$-paracompact and feebly $\Gamma$-P spaces We later introduce $\Gamma$-expandable spaces and study the relationships between $\Gamma$-expandable and $\Gamma$-P spaces. We also investigate some of topological properties of $\Gamma$-P spaces.

Keywords

• paracompact
• $\Gamma$-paracompact
• regular open set
• strongly locally finite collection

References

1. [1] S.P. Arya and R. Gupta, On strongly continuous mappings, Kyungpook Math. J., 14 (1974), 131-143.
2. [2] I˙. Demir, O¨ .B. O¨ zbakır, On b-paracompact spaces, Filomat, 27(6) (2013), 971-976.
3. [3] S. Al Ghour, Some generalizations of paracompactness, Mo. J. Math. Sci., 18 (2006), 64–77.
4. [4] S. Al Ghour, Decomposition, Mapping, and Sum Theorems of w-Paracompact Topological Spaces, Axioms, 10 (2021), 339.
5. [5] K.Y. Al-Zoubi, S-paracompact spaces, Acta Math. Hungar, 110 (2006), 165-174.
6. [6] K. Al-Zoubi, S. Al Ghour, On P3-paracompact spaces, Int. J. Math. Sci., 2007 (2007).
7. [7] N. Erg¨un, On nearly paracompact spaces, ˙Istanbul University Science Faculty the Journal of Mathematics Physics and Astronomy, 45 (1980), 65-87.
8. [8] L.L. Krajewski, On expanding locally finite collections, Can. J. Math., 23 (1971), 58-68.

9. [9] A. S. Mashhour, M. E. Abd El-Monsef, and I. A. Hasanein, On pretopological spaces, Bull. Math. Soc. Sci. Math. Roum, 28 (1984), 39-45.
10. [10] M. N. Mukherjee and A. Debray, On nearly paracompact spaces via regular even covers, Mat. Vesn., 50 (1998), 23-29.
11. [11] M. K. Singal and S. P. Arya, On nearly paracompact spaces, Mat. Vesn., 6 (1969), 3-16.
12. [12] D. Singh, cl-supercontinuous functions, Appl. Gen. Topol., 8 (2007), 293-300.
13. [13] L.A. Steen and J.A. Seebach, Counterexamples in Topology, Dover Publications, New York, 1995.
14. [14] E. Turanlı, O¨ .B. O¨ zbakır, On b1 􀀀I􀀀Paracompact Spaces, Konuralp J. Math., 7(1) (2019), 73-78.
15. [15] S. Willard, General Topology, Addison-Wesley, Massachusetts, 1970.