Free Quasitopological Groups

TL;DR

This paper investigates the topological structure of free quasitopological groups, providing explicit constructions, limit descriptions, and characterizations of subspace embeddings, advancing understanding of their properties and relationships to the underlying space.

Contribution

It introduces a direct quotient space construction of free quasitopological groups and characterizes their topology via direct limits and quotient maps, offering new insights into their structure.

Findings

Free quasitopological groups can be constructed as quotient spaces of free semitopological monoids.

The topology of $F_q(X)$ is the direct limit of subspaces $F_q(X)_n$ for $T_1$ spaces.

A subspace $Y$ is closed in $X$ iff $F_q(Y)$ embeds as a closed subspace in $F_q(X)$.

Abstract

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group $F_q(X)$ on a space $X$. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the "cross topology." Using this explicit description of $F_q(X)$, we show that for any $T_1$ space $X$, $F_q(X)$ is the direct limit of closed subspaces $F_q(X)_n$ of words of length at most $n$. We also prove that the natural map ${\bf i_n}:\coprod_{i=0}^{n}(X\sqcup X^{-1})^{\otimes i}\to F_q(X)_n$ is quotient for all $n\geq 0$. Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace $Y\subseteq X$ is closed if and only if the inclusion $Y\to X$ induces a closed embedding $F_q(Y)\to F_q(X)$ of free quasitopological groups.