Basic properties of ???? for which the space ????_{????}(????) 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 alttext="normal upper Delta"> ? encoding="application/x-tex">\Delta -space (in the sense of R. W. Knight [Trans. 339 (1993), 45–60], G. M. Reed [Fund. 110 (1980), 145–152]) if and only locally convex C Subscript p Baseline left-parenthesis X right-parenthesis"> C p ( stretchy="false">) encoding="application/x-tex">C_{p}(X) distinguished. Continuing this research, investigate whether class -spaces invariant under basic topological operations. We prove element-of normal ? encoding="application/x-tex">X \in \Delta alttext="phi colon right-arrow Y"> ?<!-- ? <mml:mo>: stretchy="false">?<!-- ? <mml:mi>Y encoding="application/x-tex">\varphi :X \to Y continuous surjection such F F (F) an sigma"> ?<!-- ? encoding="application/x-tex">F_{\sigma } -set in encoding="application/x-tex">Y for every closed set subset-of ?<!-- ? encoding="application/x-tex">F \subset X , then also Y encoding="application/x-tex">Y\in . As consequence, countable union subspaces i"> i encoding="application/x-tex">X_i each i encoding="application/x-tex">X_i\in encoding="application/x-tex">X\in particular, alttext="sigma"> encoding="application/x-tex">\sigma -product any family scattered Eberlein compact spaces product with -space. Our results give answers to several open problems posed by us 86–99]. Let T right-parenthesis long T stretchy="false">?<!-- ? encoding="application/x-tex">T:C_p(X) \longrightarrow C_p(Y) be linear surjection. observe T"> encoding="application/x-tex">T admits extension operator alttext="ModifyingAbove With caret"> ^<!-- ^ </mml:mover> encoding="application/x-tex">\widehat {T} from alttext="double-struck R Superscript mathvariant="double-struck">R encoding="application/x-tex">\mathbb {R}^X onto {R}^Y deduce whenever is. Similarly, assuming are metrizable spaces, show Q"> Q encoding="application/x-tex">Q Making use obtained results, provide very short proof claim has tightness. Proper Forcing Axiom sequential. article pose dozen questions.