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On Classes of Subcompact Spaces

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Abstract

This paper continues the study of P. S. Alexandroff’s problem: When can a Hausdorff space \(X\) be one-to-one continuously mapped onto a compact Hausdorff space?

For a cardinal number \(\tau\), the classes of \(a_\tau\)-spaces and strict \(a_\tau\)-spaces are defined. A compact space \(X\) is called an \(a_\tau\)-space if, for any \(C\in[X]^{\le\tau}\), there exists a one-to-one continuous mapping of \(X\setminus C\) onto a compact space. A compact space \(X\) is called a strict \(a_\tau\)-space if, for any \(C\in[X]^{\le\tau}\), there exits a one-to-one continuous mapping of \(X\setminus C\) onto a compact space \(Y\), and this mapping can be continuously extended to the whole space \(X\).

In this paper, we study properties of the classes of \(a_\tau\)- and strict \(a_\tau\)-spaces by using Raukhvarger’s method of special continuous paritions.

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Yekaterinburg, 620990, Russia

    V. I. Belugin, A. V. Osipov & E. G. Pytkeev

  2. Ural Federal University, Yekaterinburg, 620002, Russia

    A. V. Osipov & E. G. Pytkeev

  3. Ural State University of Economics, Yekaterinburg, 620219, Russia

    A. V. Osipov

Authors
  1. V. I. Belugin
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  2. A. V. Osipov
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  3. E. G. Pytkeev
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Correspondence to V. I. Belugin.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 810-820 https://doi.org/10.4213/mzm12499.

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Belugin, V.I., Osipov, A.V. & Pytkeev, E.G. On Classes of Subcompact Spaces. Math Notes 109, 849–858 (2021). https://doi.org/10.1134/S0001434621050187

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  • DOI: https://doi.org/10.1134/S0001434621050187

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