Scholarlink

<p>Δ-spaces have been defined by a natural generalization of a classical notion of Δ-sets of reals to Tychonoff topological spaces; moreover, the class Δ of all Δ-spaces consists precisely of those <i>X</i> for which the locally convex space <i>C</i><sub><i>p</i></sub>(<i>X</i>) is distinguished [<CitationRef CitationID="CR25">25</CitationRef>].</p><p>The aim of this article is to better understand the boundaries of the class Δ, by presenting new examples and counterexamples.</p><p>(1) We examine when trees considered as topological spaces equipped with the interval topology belong to Δ. In particular, we prove that no Souslin tree is a Δ-space. Other main results are connected with the study of (2) Ψ-spaces built on maximal almost disjoint families of countable sets; and (3) Ladder system spaces.</p><p>It is consistent with CH that all ladder system spaces on ω<sub>1</sub> are in Δ. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on ω<sub>1</sub> which is not in Δ.</p><p>We resolve several open problems posed in [<CitationRef CitationID="CR12">12</CitationRef>], [<CitationRef CitationID="CR25">25</CitationRef>], [<CitationRef CitationID="CR31">31</CitationRef>], [<CitationRef CitationID="CR32">32</CitationRef>].</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On Δ-spaces

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年2月17日

查看全文
  • Arkady Leiderman,
  • Paul Szeptycki

摘要

Δ-spaces have been defined by a natural generalization of a classical notion of Δ-sets of reals to Tychonoff topological spaces; moreover, the class Δ of all Δ-spaces consists precisely of those X for which the locally convex space Cp(X) is distinguished [25].

The aim of this article is to better understand the boundaries of the class Δ, by presenting new examples and counterexamples.

(1) We examine when trees considered as topological spaces equipped with the interval topology belong to Δ. In particular, we prove that no Souslin tree is a Δ-space. Other main results are connected with the study of (2) Ψ-spaces built on maximal almost disjoint families of countable sets; and (3) Ladder system spaces.

It is consistent with CH that all ladder system spaces on ω1 are in Δ. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on ω1 which is not in Δ.

We resolve several open problems posed in [12], [25], [31], [32].