A CHARACTERIZATION OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED AND ITS APPLICATIONS

Jerzy Ka¸Kol, Arkady Leiderman

Research output: Contribution to journalArticlepeer-review

16 Scopus citations

Abstract

We prove that the locally convex space Cp(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that Cp(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mrówka type X, the space Cp(X) is distinguished. 3) If X is the compact space of ordinals [0, ω1], then Cp(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that Cp(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

Original languageEnglish
Pages (from-to)86-99
Number of pages14
JournalProceedings of the American Mathematical Society, Series B
Volume8
DOIs
StatePublished - 1 Jan 2021

Keywords

  • Distinguished locally convex space
  • Isbell–Mrówka space
  • scattered compact space
  • Δ-set

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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