Free paratopological groups

TL;DR

This paper investigates the properties of free paratopological groups on specific topological spaces, establishing conditions under which these groups are Alexandroff, $T_0$, or possess the inductive limit property.

Contribution

It characterizes the topological properties of free paratopological groups on $P_eta$-spaces, including their Alexandroff and $T_0$ nature, and describes neighborhood bases at the identity.

Findings

$ ext{FP}(X)$ is an Alexandroff space if $X$ is Alexandroff.

$ ext{FP}(X)$ is $T_0$ whenever $X$ is $T_0$.

Identifies classes of spaces where $ ext{FP}(X)$ has the inductive limit property.

Abstract

Let $\FP(X)$ be the free paratopological group on a topological space $X$ in the sense of Markov. In this paper, we study the group $\FP(X)$ on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group $\FP(X)$ is an Alexandroff space if $X$ is an Alexandroff space. Moreover, we introduce a neighborhood base at the identity of the group $\FP(X)$ when the space $X$ is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group $\FP(X)$ is $T_0$ if $X$ is $T_0$, we characterize the spaces $X$ for which the group $\FP(X)$ is a topological group and then we give a class of spaces $X$ for which the group $\FP(X)$ has the inductive limit property.