On the topology of free paratopological groups. II

TL;DR

This paper investigates the topological structure of free paratopological groups, especially focusing on neighborhoods at the identity and conditions under which certain natural maps are quotient maps, with specific results for subspaces of the real line.

Contribution

It provides new characterizations of when the natural map to the free paratopological group is a quotient map, including for various subspaces of the real line.

Findings

Neighborhood base at the identity in $ ext{FP}_2(X)$ is characterized.

Conditions for $i_2$ to be a quotient map are established for $T_1$ spaces.

$i_2$ is a quotient for countable subspaces of $ ext{R}$, but not for uncountable compact subspaces.

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.