Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space X X is a Δ \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 ) C_{p}(X) is distinguished. Continuing this research, we investigate whether the class Δ \Delta of Δ \Delta -spaces is invariant under the basic topological operations. We prove that if X ∈ Δ X \in \Delta and φ : X → Y \varphi :X \to Y is a continuous surjection such that φ ( F ) \varphi (F) is an F σ F_{\sigma } -set in Y Y for every closed set F ⊂ X F \subset X , then also Y ∈ Δ Y\in \Delta . As a consequence, if X X is a countable union of closed subspaces X i X_i such that each X i ∈ Δ X_i\in \Delta , then also X ∈ Δ X\in \Delta . In particular, σ \sigma -product of any family of scattered Eberlein compact spaces is a Δ \Delta -space and the product of a Δ \Delta -space with a countable space is a Δ \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 ) ⟶ C p ( Y ) T:C_p(X) \longrightarrow C_p(Y) be a continuous linear surjection. We observe that T T admits an extension to a linear continuous operator T ^ \widehat {T} from R X \mathbb {R}^X onto R Y \mathbb {R}^Y and deduce that Y Y is a Δ \Delta -space whenever X X is. Similarly, assuming that X X and Y Y are metrizable spaces, we show that Y Y is a Q Q -set whenever X X is. Making use of obtained results, we provide a very short proof for the claim that every compact Δ \Delta -space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ \Delta -space is sequential. In the article we pose a dozen open questions.