0580514 - MÚ 2025 RIV US eng J - Journal Article
Kąkol, Jerzy - Kurka, Ondřej - Leiderman, A. G.
Some classes of topological spaces extending the class of Δ-spaces.
Proceedings of the American Mathematical Society. Roč. 152, č. 2 (2024), s. 883-898. ISSN 0002-9939. E-ISSN 1088-6826
R&D Projects: GA ČR(CZ) GF20-22230L; GA ČR(CZ) GF22-07833K
Institutional support: RVO:67985840
Keywords : scattered space * pseudocompact space * closed mapping
OECD category: Pure mathematics
Impact factor: 1, year: 2022
Method of publishing: Limited access
https://doi.org/10.1090/proc/16661

A study of the class Delta consisting of topological Delta-spaces was originated by Jerzy Kakol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86-99, Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267-280]. The main purpose of this paper is to introduce and investigate new classes Delta(2) subset of Delta(1) properly containing Delta.
We observe that for every first-countable X the following equivalences hold: X is an element of Delta(1) iff X is an element of Delta(2) iff each countable subset of X is G(delta). Thus, new proposed concepts provide a natural extension of the family of all lambda-sets beyond the separable metrizable spaces.
We prove that (1) A pseudocompact space X belongs to the class Delta(1) iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class Delta(2).
We investigate whether the classes Delta(1) and Delta(2) are invariant under the basic topological operations. Similarly to Delta, both classes Delta(1) and Delta(2) are invariant under the operation of taking countable unions of closed subspaces. In contrast to Delta, they are not preserved by closed continuous images.
Let Y be l-dominated by X, i.e. C-p(X) admits a continuous linear map onto C-p(Y). We show that Y is an element of Delta(1) whenever X is an element of Delta(1). Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone-Cech compactification beta Y is scattered.
Permanent Link: https://hdl.handle.net/11104/0349287