Topologies on the symmetric inverse semigroup

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 $I(X)$.