Some classes of topological spaces extending the class of Δ-spaces

0580514 - MÚ 2025 RIV US eng J - Článek v odborném periodiku
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
Grant CEP: GA ČR(CZ) GF20-22230L; GA ČR(CZ) GF22-07833K
Institucionální podpora: RVO:67985840
Klíčová slova: scattered space * pseudocompact space * closed mapping
Obor OECD: Pure mathematics
Impakt faktor: 1, rok: 2022
Způsob publikování: Omezený přístup
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.
Trvalý link: https://hdl.handle.net/11104/0349287