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Register a new userA characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications
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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}. As 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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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.
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X subset mathbb{R}^n$).
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.
A theorem by Norman L. Noble from 1970 asserts that every product of completely regular, locally pseudo-compact k_R-spaces is a k_R-space. As a consequence, all direct products of locally compact Hausdorff spaces are k_R-spaces. We provide a streamlined proof for this fact.
We study products of general topological spaces with Mengers covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming CH{}, every productively Lindelof space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.
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