A characterization of X for which spaces Cp(X) are distinguished and its applications

Jerzy Kakol, Arkady Leiderman

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Abstract

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--Mr\'owka 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.
Original languageEnglish GB
Pages (from-to)86-99
JournalProceedings of the American Mathematical Society
Volume8
StatePublished - 2021

Keywords

  • math.GN
  • math.FA
  • 54C35, 54G12, 54H05, 46A03

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