The topological structure of (homogeneous) spaces and groups with countable cs*-character

TL;DR

This paper introduces new topological invariants called cs*, cs-, and sb-characters, and explores their properties and implications for topological groups, especially those with countable cs*-character, revealing structural features like open $k_$-subgroups.

Contribution

It defines three new cardinal invariants, analyzes their stability under topological operations, and characterizes non-metrizable sequential topological groups with countable cs*-character.

Findings

Spaces with countable cs*-character include all aleph-spaces.

Non-metrizable sequential groups with this property have countable pseudo-character.

Such groups contain open $k_$-subgroups.

Abstract

In this paper we introduce and study three new cardinal topological invariants called the cs*, cs-, and sb-characters. The class of topological spaces with countable cs*-character is closed under many topological operations and contains all aleph-spaces and all spaces with point-countable cs*-network. Our principal result states that each non-metrizable sequential topological group with countable cs*-character has countable pseudo-character and contains an open $k_\omega$-subgroup.