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

Abstract

A study of the class Δ \Delta consisting of topological Δ \Delta -spaces was originated by Jerzy Ka̧kol 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 Δ 2 ⊂ Δ 1 \Delta _2 \subset \Delta _1 properly containing Δ \Delta . We observe that for every first-countable X X the following equivalences hold: X ∈ Δ 1 X\in \Delta _1 iff X ∈ Δ 2 X\in \Delta _2 iff each countable subset of X X is G δ G_{\delta } . Thus, new proposed concepts provide a natural extension of the family of all λ \lambda -sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X X belongs to the class Δ 1 \Delta _1 iff countable subsets of X X are scattered. (2) Every regular scattered space belongs to the class Δ 2 \Delta _2 . We investigate whether the classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the basic topological operations. Similarly to Δ \Delta , both classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to Δ \Delta , they are not preserved by closed continuous images. Let Y Y be l l -dominated by X X , i.e. C p ( X ) C_p(X) admits a continuous linear map onto C p ( Y ) C_p(Y) . We show that Y ∈ Δ 1 Y \in \Delta _1 whenever X ∈ Δ 1 X \in \Delta _1 . Moreover, we establish that if Y Y is l l -dominated by a compact scattered space X X , then Y Y is a pseudocompact space such that its Stone–Čech compactification β Y \beta Y is scattered.