Suitable sets for topological groups revisited

TL;DR

This paper investigates the existence of suitable sets in various classes of topological groups, providing conditions under which such sets exist or do not, and addressing several open problems in the field.

Contribution

It establishes new criteria for the existence of suitable sets in topological groups, including linearly orderable groups and groups with an 0-base, and solves related open problems.

Findings

Suitable sets do not exist in certain non-separable $k_{ ext{omega}}$-spaces without non-trivial convergent sequences.

Linearly orderable topological groups with an -base that are metrizable have suitable sets.

Topological groups with an -base and the $k$-space property possess suitable sets.

Abstract

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the existence of suitable sets in topological groups is studied. It is proved that, for a non-separable $k_{\omega}$-space $X$ without non-trivial convergent sequences, the $snf$-countability of $A(X)$ implies that $A(X)$ does not have a suitable set, which gives a partial answer to \cite[Problem 2.1]{TKA1997}. Moreover, the existence of suitable sets in some particular classes of linearly orderable topological groups is considered, where Theorem~\ref{t4} provides an affirmative answer to \cite[Problem 4.3]{ST2002}. Then, topological groups with an $\omega^{\omega}$-base are discussed, and every linearly orderable topological group with an $\omega^{\omega}$-base being metrizable is proved; thus it has a suitable set. Further, it follows that each topological group $G$ with an $\omega^{\omega}$-base has a suitable set whenever $G$ is a $k$-space, which gives a generalization of a well-known result in \cite{CM}. Finally, some cardinal invariant of topological groups with a suitable set are provided. Some results of this paper give some partial answers to some open problems posed in~\cite{DTA} and~\cite{TKA1997} respectively.