Partial actions on quotient spaces and globalization

TL;DR

This paper investigates how certain topological properties of partial group actions on spaces can be extended to their globalizations, with applications to quotient spaces, invariant metrics, and inverse limits.

Contribution

It characterizes the extension of properties like Hausdorff, regular, and metrizable from partial actions to their globalizations and introduces partial actions of quotient groups on orbit spaces.

Findings

Properties like Hausdorff and metrizable can be extended to globalizations.

Partial actions of quotient groups on orbit spaces are studied.

Applications include invariant metrics and inverse limits.

Abstract

Given a partial action of a topological group $G$ on a space $X$, we determine properties $\mathcal P$ which can be extended from $X$ to its globalization. We treat the cases when $\mathcal P$ is any of the following: Hausdorff, regular, metrizable, second countable and having invariant metric. Further, for a normal subgroup $H$ we introduce and study a partial action of $G/H$ on the orbit space $X/\!\sim,$ applications to invariant metrics and inverse limits are presented.