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

Abstract

Let $Γ(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $τ_{hco}$, that makes $Γ(X)$ a topological inverse semigroup. We address the question of whether $τ_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(Γ(X), τ_{hco})$ 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.