On the topology of free paratopological groups

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 $n$.