In our paper [18] we showed that a Tychonoff space $X$ is a $Delta$-space (in the sense of [20], [30]) 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 $Yin Delta$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_iin Delta$, then also $Xin Delta$. 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 in cite{KL}. 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 $R^X$ onto $R^Y$ 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 (PFA) every compact $Delta$-space is sequential. In the article we pose a dozen open questions.
تحميل البحث