On the topology of free paratopological groups

TL;DR

This paper extends Joiner's lemma to free paratopological groups on T_1 spaces, establishing key topological equivalences and properties that deepen understanding of their structure.

Contribution

It proves an analogue of Joiner's lemma for free paratopological groups and characterizes when these groups are T_1 based on subspace properties.

Findings

Equivalence of T_1 conditions for X and its free paratopological group

Characterization of subspace closure and discreteness in FP(X)

Conditions under which subspaces of FP(X) are closed or discrete

Abstract

The result often known as Joiner's lemma is fundamental in understanding the topology of the free topological group $F(X)$ on a Tychonoff space$X$. In this paper, an analogue of Joiner's lemma for the free paratopological group $\FP(X)$ on a $T_1$ space $X$ is proved. Using this, it is shown that the following conditions are equivalent for a space $X$: (1) $X$ is $T_1$; (2) $\FP(X)$ is $T_1$; (3) the subspace $X$ of $\FP(X)$ is closed; (4) the subspace $X^{-1}$ of $\FP(X)$ is discrete; (5) the subspace $X^{-1}$ is $T_1$; (6) the subspace $X^{-1}$ is closed; and (7) the subspace $\FP_n(X)$ is closed for all $n \in \N$, where $\FP_n(X)$ denotes the subspace of $\FP(X)$ consisting of all words of length at most $n$.