Topologies on the symmetric inverse semigroup

TL;DR

This paper investigates minimal Hausdorff topologies on the symmetric inverse semigroup, exploring conditions for Polish semigroup topologies when the underlying set is countable.

Contribution

It characterizes minimal Hausdorff inverse semigroup topologies on $I(X)$ and identifies Polish semigroup topologies for countable sets.

Findings

Identification of minimal Hausdorff topologies on $I(X)$

Existence of Polish semigroup topologies for countable $X$

Conditions under which these topologies exist

Abstract

The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When $X$ is countable, we show some Polish semigroup topologies on $I(X)$.