Free Quasitopological Groups

TL;DR

This work constructs and analyzes the free quasitopological group $F_q(X)$ for a space $X$ by presenting an explicit quotient construction from the free semitopological monoid $M_{st}(X\sqcup X^{-1})$ using the cross topology $X\otimes Y$ and word reduction. It establishes a concrete filtration $F_q(X)_n$ (words of length $\le n$) and shows that for $T_1$ spaces $X$, $F_q(X)$ is the direct limit of these subspaces with quotient maps ${\bf i_n}$, providing a practical topology description and a criterion for when $F_q(X)$ is a topological group (iff $X$ is discrete). The paper proves that $F_q(X)$ is topological only in trivial cases, characterizes when inclusions of subspaces induce embeddings, and demonstrates that $F_q(Y)\to F_q(X)$ is a closed embedding exactly when $Y$ is closed in $X$ (under suitable separation axioms). These results yield a transparent, explicit framework for understanding free quasitopological groups and their subspaces, with implications for when such objects can form genuine topological groups and how subspace topology behaves.

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.

Figures (2)

• Figure 1: An open neighborhood of the origin in $\mathbb{R}\otimes\mathbb{R}$, which is not open in $\mathbb{R}\times \mathbb{R}$.
• Figure 2: The set $(X\times X\backslash \Delta)\cup \{(x,x)\}$ is open in $X\otimes X$ for all $x\in X$ since each intersection with a projection fiber is open.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• Theorem 3.2