Compact condensations of Hausdorff spaces

TL;DR

This paper investigates conditions under which Hausdorff spaces can be condensed onto compact spaces, focusing on the properties of subsets and their cardinalities, to extend classical topology results.

Contribution

It extends the understanding of condensations of Hausdorff spaces onto compacta, especially considering subsets with bounded cardinality.

Findings

Conditions for condensations onto compacta are characterized.

The role of subset cardinalities in condensation properties is clarified.

New classes of Hausdorff spaces with condensation properties are identified.

Abstract

In this paper, we continue to study one of the classic problems in general topology raised by P.S. Alexandrov: when a Hausdorff space $X$ has a continuous bijection (a condensation) onto a compactum? We concentrate on the situation when not only $X$ but also $X\setminus Y$ can be condensed onto a compactum whenever the cardinality of $Y$ does not exceed certain $\tau$.