The countable type properties in free paratopological groups

TL;DR

This paper investigates the properties of free paratopological groups, establishing conditions under which these groups are of countable or subcountable type based on the properties of the underlying space.

Contribution

It characterizes when free paratopological groups are of countable or subcountable type in terms of the discreteness and pseudocharacter of the base space.

Findings

$FP(X)$ and $AP(X)$ are of countable type iff $X$ is discrete.

If $AP(X)$ is of subcountable type, then $X$ has countable pseudocharacter.

Locally compact $AP_2(X)$ or $FP_2(X)$ imply $X$ is a topological sum of a compact and a discrete space.

Abstract

A space $X$ is of countable type (resp. subcountable type) if every compact subspace $F$ of $X$ is contained in a compact subspace $K$ that is of countable character (resp. countable pseudocharacter) in $X$. In this paper, we mainly show that: (1) For a functionally Hausdorff space $X$, the free paratopological group $FP(X)$ and the free abelian paratopological group $AP(X)$ are of countable type if and only if $X$ is discrete; (2) For a functionally Hausdorff space $X$, if the free abelian paratopological group $AP(X)$ is of subcountable type then $X$ has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff $\mu$-space $X$, if $AP_{2}(X)$ or $FP_{2}(X)$ is locally compact, then $X$ is homeomorphic to the topological sum of a compact space and a discrete space.