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 is a Δ-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 Δ of Δ-spaces is invariant under the basic topological operations. We prove that if X Δ and ϕ : X → Y is a continuous surjection such that ϕ(F ) is an F_σ-set in Y for every closed set F ⊂ X, then also Y Δ. As a consequence, if X is a countable union of closed subspaces X_i such that each X_i Δ, then also X Δ. In particular, σ-product of any family of scattered Eberlein compact spaces is a Δ-space and the product of a Δ-space with a countable space is a Δ-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 ) be a continuous linear surjection. We observe that T admits an extension to a linear continuous operator T from R^X onto R^Y and deduce that Y is a Δ-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 Δ-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ-space is sequential. In the article we pose a dozen open questions.