Condensations with extra properties

TL;DR

This paper explores the properties of condensations in topological spaces, demonstrating that certain locally compact spaces can be condensed onto separable but not compact spaces, and constructing groups with specific condensation properties.

Contribution

It provides new examples and answers to longstanding questions about condensation mappings in topology, especially regarding locally compact spaces and topological groups.

Findings

Existence of locally compact spaces condensed onto separable but not compact spaces.

Construction of locally compact topological groups condensed onto compact but not group spaces.

Answers to questions posed by Arhangel'skii and Buzyakova.

Abstract

We show that there are locally compact spaces that can be condensed onto separable spaces but not onto compact separable spaces. We also show that for every cardinal $\kappa$ there is a locally compact topological group of cardinality $2^\kappa$ that can be condensed onto a compact space but not onto a compact topological group. These answer some questions of Arhangel'skii and Buzyakova.