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تسجيل مستخدم جديدBasic properties of $X$ for which spaces $C_p(X)$ are distinguished
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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.
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We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $Delta$-space in the sense of cite {Knight}. A
s an application of this characterization theorem we obtain the following results: 1) If $X$ is a v{C}ech-complete (in particular, compact) space such that $C_p(X)$ is distinguished, then $X$ is scattered. 2) For every separable compact space of the Isbell--Mrowka type $X$, the space $C_p(X)$ is distinguished. 3) If $X$ is the compact space of ordinals $[0,omega_1]$, then $C_p(X)$ is not distinguished. We observe that the existence of an uncountable separable metrizable space $X$ such that $C_p(X)$ is distinguished, is independent of ZFC. We explore also the question to which extent the class of $Delta$-spaces is invariant under basic topological operations.
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