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In this chapter we present a number of recent results related to the study of spaces \(C_{p}\left ( X\right ) \) of continuous real-valued functions on a Tychonoff spaces X that are distinguished. We prove, among the others, that the locally convex space \(C_{p}(X)\) is distinguished if and only if X is a \(\varDelta \) -space in the sense of Reed. This theorem apparently provides (with several applications) a significant ”bridge” between the still attractive problems from the set theory and related with \(\varDelta \) -sets, \(\lambda \) -sets, and \(\mathbb {Q}\) -sets X, and corresponding distinguished spaces \(C_p (X)\) .

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Distinguished Spaces \(C_{p}(X)\) and \(\Delta \) -Spaces X

  • 章节

  • 发表日期：

    2025年

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  • Jerzy Kąkol,
  • Wiesław Kubiś,
  • Manuel López-Pellicer,
  • Damian Sobota

摘要

In this chapter we present a number of recent results related to the study of spaces \(C_{p}\left ( X\right ) \) of continuous real-valued functions on a Tychonoff spaces X that are distinguished. We prove, among the others, that the locally convex space \(C_{p}(X)\) is distinguished if and only if X is a \(\varDelta \) -space in the sense of Reed. This theorem apparently provides (with several applications) a significant ”bridge” between the still attractive problems from the set theory and related with \(\varDelta \) -sets, \(\lambda \) -sets, and \(\mathbb {Q}\) -sets X, and corresponding distinguished spaces \(C_p (X)\) .