2021/04/21 by Kakol, Jerzy, Leiderman, Arkady
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2104.10506
In our paper [18] we showed that a Tychonoff space X is a Δ-space (in the sense of [20], [30]) if and only if the locally convex space Cp(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 Fσ-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 Xi such that each Xi∈ 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 in \citeKL. Let T:Cp(X) \longrightarrow Cp(Y) be a continuous linear surjection. We observe that T admits an extension to a linear continuous operator \widehatT from RX onto RY 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 (PFA) every compact Δ-space is sequential. In the article we pose a dozen open questions.