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On the class of NY compact spaces of finitely supported elements and related classes

Antonio Avilés, Mikołaj Krupski

TL;DR

The paper characterizes when a compact space embeds into a $σ$-product of metrizable compacta, showing this occurs precisely for $K$ that are strongly countable-dimensional, hereditarily metalindelöf, and every subspace has a nonempty relatively open second-countable subset. It proves that $NY$ compacta are exactly Corson compacta that are $M$-scattered, and provides parallel results for $ω$-Corson, with several interlinked equivalent conditions including Eberlein counterparts and hereditary covering properties. The authors also analyze Alexandroff duplicates, proving that pivotal embedding properties are preserved by duplicates and constructing notable examples (e.g., a uniform Eberlein compact not $NY$-Valdivia) that resolve open questions. Finally, they establish that $NY$ compactness is invariant under homeomorphisms of $C_p$-spaces, extending prior work and highlighting the robustness of these topological distinctions under function-space transformations. Together, these results deepen the understanding of the structure of NY and Corson-type compacta and their behavior under standard topological constructions and representations.

Abstract

We prove that a compact space $K$ embeds into a $σ$-product of compact metrizable spaces ($σ$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindelöf and every subspace of $K$ has a nonempty relative open second-countable subset. This provides novel characterizations of $ω$-Corson and $NY$ compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the $σ$-product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space $K$ the property of being $NY$ compact is determined by the topological structure of the space $C_p(K)$ of continuous real-valued functions of $K$ equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

On the class of NY compact spaces of finitely supported elements and related classes

TL;DR

The paper characterizes when a compact space embeds into a -product of metrizable compacta, showing this occurs precisely for that are strongly countable-dimensional, hereditarily metalindelöf, and every subspace has a nonempty relatively open second-countable subset. It proves that compacta are exactly Corson compacta that are -scattered, and provides parallel results for -Corson, with several interlinked equivalent conditions including Eberlein counterparts and hereditary covering properties. The authors also analyze Alexandroff duplicates, proving that pivotal embedding properties are preserved by duplicates and constructing notable examples (e.g., a uniform Eberlein compact not -Valdivia) that resolve open questions. Finally, they establish that compactness is invariant under homeomorphisms of -spaces, extending prior work and highlighting the robustness of these topological distinctions under function-space transformations. Together, these results deepen the understanding of the structure of NY and Corson-type compacta and their behavior under standard topological constructions and representations.

Abstract

We prove that a compact space embeds into a -product of compact metrizable spaces (-product of intervals) if and only if is (strongly countable-dimensional) hereditarily metalindelöf and every subspace of has a nonempty relative open second-countable subset. This provides novel characterizations of -Corson and compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the -product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space the property of being compact is determined by the topological structure of the space of continuous real-valued functions of equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.
Paper Structure (7 sections, 18 theorems, 65 equations, 2 figures)

This paper contains 7 sections, 18 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. $\sigma$-products and $\Sigma$-product
  4. Point-finite and point-countable families
  5. Characterizing the class of NY and $\omega$-Corson compacta
  6. Duplicates
  7. $NY$ compacta are invariant under homeomorphisms of $C_p$-spaces

Key Result

Proposition 2.1

MPZ Given a set $\Gamma$ and $\gamma\in \Gamma$, we let $Q_\gamma=[0,1]^\omega$ to be the Hilbert cube. A compact space $K$ is $NY$ compact if and only if $K$ embeds into $\sigma(\prod_{\gamma\in \Gamma}Q_\gamma)$, for some $\Gamma$.

Figures (2)

  • Figure 1: Relations between certain classes of compacta. The symbols $\mathcal{MSCD}$, $\mathcal{M}$ and $\mathcal{NY}$ stand for the class of strongly countable-dimensional metrizable, metrizable and $NY$ compacta, respectively.
  • Figure :

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • ...and 33 more