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<p>For any cardinal number <i>κ</i> and an index set Γ, the Σ<sub><i>κ</i></sub>-product of real lines Σ<sub><i>κ</i></sub>(ℝ<sup>Γ</sup>) consists of all elements of ℝ<sup>Γ</sup> having &lt; <i>κ</i> many nonzero coordinates. A compact space <i>K</i> is <i>κ</i>-Corson compact if it can be embedded into Σ<sub><i>κ</i></sub>(ℝ<sup>Γ</sup>) for some Γ.</p><p>The class of (<i>ω</i><sub>1</sub>)-Corson compact spaces has been intensively studied over the last decades. We discuss properties of <i>κ</i>-Corson compacta for various cardinal numbers <i>κ</i> as well as properties of related Boolean algebras and spaces of continuous functions.</p><p>We present here a detailed discussion of the class of <i>ω</i>-Corson compacta extending the results of Nakhmanson and Yakovlev [NY]. For <i>κ</i> &gt; <i>ω</i>, our results on <i>κ</i>-Corson compact spaces are related to the line of research originated by Kalenda [Ka2] and Bell and Marciszewski [BM], and continued by Bonnet, Kubiś and Todorčević in their recent paper [BKT].</p>

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Digging into the classes of κ-Corson compact spaces

  • 期刊论文

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  • 发表日期：

    2025年9月4日

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  • Witold Marciszewski,
  • Grzegorz Plebanek,
  • Krzysztof Zakrzewski

摘要

For any cardinal number κ and an index set Γ, the Σκ-product of real lines Σκ(ℝΓ) consists of all elements of ℝΓ having < κ many nonzero coordinates. A compact space K is κ-Corson compact if it can be embedded into Σκ(ℝΓ) for some Γ.

The class of (ω1)-Corson compact spaces has been intensively studied over the last decades. We discuss properties of κ-Corson compacta for various cardinal numbers κ as well as properties of related Boolean algebras and spaces of continuous functions.

We present here a detailed discussion of the class of ω-Corson compacta extending the results of Nakhmanson and Yakovlev [NY]. For κ > ω, our results on κ-Corson compact spaces are related to the line of research originated by Kalenda [Ka2] and Bell and Marciszewski [BM], and continued by Bonnet, Kubiś and Todorčević in their recent paper [BKT].