Research Article
Year 2025,
Volume: 54 Issue: 3, 912 - 920, 24.06.2025
Abstract
Let $\mathcal{P}$ be any topological property of a space $X$. We say that $X$ is $\mathcal{P}$ at $x\in X$ if there exist an open set $U$ and a subspace $Y$ of $X$ satisfying $\mathcal{P}$ such that $x\in U\subseteq Y$. We also say that $X$ is locally $\mathcal{P}$ if $X$ is $\mathcal{P}$ at every point of $X$. We study this local property and obtain the following results under certain topological assumptions on $\mathcal{P}$.
(1) Every locally $\mathcal{P}$ Hausdorff $P$-space can be densely embedded in a $\mathcal{P}$ Hausdorff $P$-space.
(2) If a Hausdorff $P$-space $X$ is $\mathcal{P}$ at $x\in X$, then $\chi(x,X)\leq\psi(x,X)^\omega$.
(3) For a locally $\mathcal{P}$ Hausdorff $P$-space $X$, $w(X)\leq nw(X)^\omega\leq |X|^\omega$.
Besides, few separation like properties are obtained and preservation under certain topological operations are also investigated. Finally we present certain observations on remainders of locally $\mathcal{P}$ spaces.
References
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- [15] S. Mrówka, On completely regular spaces, Fund. Math. 41, 105-106, 1954.
- [16] K. Nagami, $\Sigma$-spaces, Fund. Math. 65, 169-192, 1969.
- [17] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69, 31-62, 1996.
- [18] H. Wang and W. He, On remainders of locally s-spaces, Topology Appl. 278, 107231, 2020.
- [19] S. Willard, General Topology, Addison Wesley Publishing Co., 1970.
Year 2025,
Volume: 54 Issue: 3, 912 - 920, 24.06.2025
Abstract
References
- [1] N. Alam and D. Chandra, On certain localized version of uniform selection principles, Filomat 36 (20), 6855-6865, 2022.
- [2] N. Alam and D. Chandra, On localization of the Menger property, Quaest. Math. 46 (6), 1069-1092, 2023.
- [3] A.V. Arhangel’skii, A class of spaces which contains all metric and all locally compact spaces, Mat. Sb. (N.S.) 67 (109), 55-88, 1965, English translation: Amer. Math. Soc. Transl. 92, 1-39, 1970.
- [4] A.V. Arhangel’skii, Mappings and spaces, Russian Math. Surveys 21 (4), 115-162, 1966.
- [5] A.V. Arhangel’skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150, 79-90, 2005.
- [6] A.V. Arhangel’skii and M.M. Choban, Some generalizations of the concept of a pspace, Topology Appl. 158, 1381-1389, 2011.
- [7] A.V. Arhangel’skii, A generalization of Cech-complete spaces and Lindelöf $\Sigma$-spaces, Comment. Math. Univ. Carolin. 54 (2), 121-139, 2013.
- [8] D. Chandra and N. Alam, On localization of the star-Menger selection principle, Hacet. J. Math. Stat. 50 (4), 1155-1168, 2021.
- [9] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
- [10] J. Gerlits and Zs. Nagy, Some properties of $C(X)$, I, Topology Appl. 14 (2), 151-161, 1982.
- [11] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics of open covers (II), Topology Appl. 73, 241-266, 1996.
- [12] Lj.D.R. Kocinac, Star-Menger and related spaces, Publ. Math. Debrecen 55, 421-431, 1999.
- [13] Lj.D.R. Kocinac, Selection principles in uniform spaces, Note Mat. 22 (2), 127-139, 2003.
- [14] E.A. Michael, Bi-quotient maps and cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble) 18 (2), 287-302, 1968.
- [15] S. Mrówka, On completely regular spaces, Fund. Math. 41, 105-106, 1954.
- [16] K. Nagami, $\Sigma$-spaces, Fund. Math. 65, 169-192, 1969.
- [17] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69, 31-62, 1996.
- [18] H. Wang and W. He, On remainders of locally s-spaces, Topology Appl. 278, 107231, 2020.
- [19] S. Willard, General Topology, Addison Wesley Publishing Co., 1970.
There are 19 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Topology |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | January 27, 2025 |
| Publication Date | June 24, 2025 |
| Submission Date | January 8, 2024 |
| Acceptance Date | August 3, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 3 |