Previous |  Up |  Next

Article

Title: Exponential separability is preserved by some products (English)
Author: Tkachuk, Vladimir V.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 63
Issue: 3
Year: 2022
Pages: 385-395
Summary lang: English
.
Category: math
.
Summary: We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma$-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $|Y|\leq 2^{\omega_1}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $|Y|\leq 2^{\mathfrak{c}} $ is scattered, then $X$ is exponentially separable. A Lindelöf $\Sigma$-space $X$ must be exponentially separable provided that every $Y\subset X$ with $|Y|\leq {\mathfrak{c}}$ is scattered. Under the Luzin axiom ($2^{\omega_1}>{\mathfrak{c}} $) we characterize weak exponential separability of $C_p(X,[0,1])$ for any metrizable space $X$. Our results solve several published open questions. (English)
Keyword: Lindelöf space
Keyword: scattered space
Keyword: $\sigma$-product
Keyword: function space
Keyword: $P$-space
Keyword: exponentially separable space
Keyword: product
Keyword: functionally countable space
Keyword: weakly exponentially separable space
MSC: 54C35
MSC: 54D65
MSC: 54G10
MSC: 54G12
idZBL: Zbl 07655808
idMR: MR4542797
DOI: 10.14712/1213-7243.2022.021
.
Date available: 2023-02-13T11:08:45Z
Last updated: 2024-10-04
Stable URL: http://hdl.handle.net/10338.dmlcz/151541
.
Reference: [1] Engelking R.: General Topology.Mathematical Monographs, 60, PWN—Polish Scientific Publishers, Warszawa, 1977. Zbl 0684.54001, MR 0500780
Reference: [2] Galvin F.: Problem 6444.Amer. Math. Monthly 90 (1983), no. 9, 648; solution: Amer. Math. Monthly 92 (1985), no. 6, 434. MR 1540672
Reference: [3] Gruenhage G., Tkachuk V. V., Wilson R. G.: Domination by small sets versus density.Topology Appl. 282 (2020), 107306, 10 pages. MR 4116835, 10.1016/j.topol.2020.107306
Reference: [4] Juhász I., van Mill J.: Countably compact spaces all countable subsets of which are scattered.Comment. Math. Univ. Carolin. 22 (1981), no. 4, 851–855. MR 0647031
Reference: [5] Levy R., Matveev M.: Functional separability.Comment. Math. Univ. Carolin. 51 (2010), no. 4, 705–711. Zbl 1224.54063, MR 2858271
Reference: [6] Moore J. T.: A solution to the $L$ space problem.J. Amer. Math. Soc. 19 (2006), no. 3, 717–736. Zbl 1107.03056, MR 2220104, 10.1090/S0894-0347-05-00517-5
Reference: [7] Pelczyński A., Semadeni Z.: Spaces of continuous functions. III. Spaces $C(\Omega)$ for $\Omega$ without perfect subsets.Studia Math. 18 (1959), 211–222. MR 0107806, 10.4064/sm-18-2-211-222
Reference: [8] Rudin W.: Continuous functions on compact spaces without perfect subsets.Proc. Amer. Math. Soc. 8 (1957), 39–42. Zbl 0077.31103, MR 0085475, 10.1090/S0002-9939-1957-0085475-7
Reference: [9] Tkachuk V. V.: A $C_p$-Theory Problem Book. Topological and Function Spaces.Problem Books in Mathematics, Springer, New York, 2011. MR 3024898
Reference: [10] Tkachuk V. V.: A $C_p$-Theory Problem Book. Special Features of Function Spaces.Problem Books in Mathematics, Springer, Cham, 2014. MR 3243753
Reference: [11] Tkachuk V. V.: A $C_p$-Theory Problem Book. Compactness in Function Spaces.Problem Books in Mathematics, Springer, Cham, 2015. MR 3364185
Reference: [12] Tkachuk V. V.: A nice subclass of functionally countable spaces.Comment. Math. Univ. Carolin. 59 (2018), no. 3, 399–409. MR 3861562
Reference: [13] Tkachuk V. V.: Exponential domination in function spaces.Comment. Math. Univ. Carolin. 61 (2020), no. 3, 397–408. MR 4186115
Reference: [14] Tkachuk V. V.: Some applications of discrete selectivity and Banakh property in function spaces.Eur. J. Math. 6 (2020), no. 1, 88–97. MR 4071459, 10.1007/s40879-019-00342-7
Reference: [15] Tkachuk V. V.: Some applications of exponentially separable spaces.Quaest. Math. 43 (2020), no. 10, 1391–1403. MR 4175405, 10.2989/16073606.2019.1623934
Reference: [16] Tkachuk V. V.: The extent of a weakly exponentially separable space can be arbitrarily large.Houston J. Math. 46 (2020), no. 3, 809–819. MR 4229084
Reference: [17] Vaughan J. E.: Countably compact and sequentially compact spaces.Handbook of Set-Theoretic Topology, North Holland, Amsterdam, 1984, pages 569–602. Zbl 0562.54031, MR 0776631
.

Files

Files Size Format View
CommentatMathUnivCarolRetro_63-2022-3_10.pdf 215.5Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo