On the Polishness of the inverse semigroup $\Gamma(X)$ on a compact metric space $X$

Abstract

Let $\Gamma(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $\tau_{hco}$, that makes $\Gamma(X)$ a topological inverse semigroup. We address the question of whether $\tau_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(\Gamma(X), \tau_{hco})$ is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup $I(\N)$. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open sets.