Some classes of topological spaces extending the class of $\Delta$-spaces

Abstract:

A study of the class $\Delta$ consisting of topological $\Delta$-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 $\Delta _2 \subset \Delta _1$ properly containing $\Delta$.

We observe that for every first-countable $X$ the following equivalences hold: $X\in \Delta _1$ iff $X\in \Delta _2$ 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 $Y \in \Delta _1$ whenever $X \in \Delta _1$. 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–Čech compactification $\beta Y$ is scattered.