On the topology of free paratopological groups. II

Abstract

Let $\FP(X)$ be the free paratopological group on a topological space $X$. For $n\in \N$, denote by $\FP_n(X)$ the subset of $\FP(X)$ consisting of all words of reduced length at most $n$, and by $i_n$ the natural mapping from $(X\oplus X^{-1}\oplus \{e\})^n$ to $\FP_n(X)$. In this paper a neighbourhood base at the identity $e$ in $\FP_2(X)$ is found. A number of characterisations are then given of the circumstances under which $i_2\colon (X\oplus X^{-1}_d\oplus \{e\})^2\to \FP_2(X)$ is a quotient map, where $X$ is a $T_1$ space and $X^{-1}_d$ denotes the set $X^{-1}$ equipped with the discrete topology. Further characterisations are given in the case where $X$ is a transitive $T_1$ space. Several specific spaces and classes of spaces are also examined. For example, $i_2$ is a quotient for every countable subspace of $\R$, $i_2$ is not a quotient for any uncountable compact subspace of $\R$, and it is undecidable in ZFC whether an uncountable subspace of $\R$ exists for which $i_2$ is a quotient.