Basic properties of $X$ for which the space $C_p(X)$ is distinguished

Abstract:

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$ of $\Delta$-spaces is invariant under the basic topological operations.

We prove that if $X \in \Delta$ and $\varphi :X \to Y$ is a continuous surjection such that $\varphi (F)$ is an $F_{\sigma }$-set in $Y$ for every closed set $F \subset X$, then also $Y\in \Delta$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\in \Delta$, then also $X\in \Delta$. In particular, $\sigma$-product of any family of scattered Eberlein compact spaces is a $\Delta$-space and the product of a $\Delta$-space with a countable space is a $\Delta$-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].

Let $T:C_p(X) \longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $\widehat {T}$ from $\mathbb {R}^X$ onto $\mathbb {R}^Y$ and deduce that $Y$ is a $\Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is.

Making use of obtained results, we provide a very short proof for the claim that every compact $\Delta$-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact $\Delta$-space is sequential.

In the article we pose a dozen open questions.