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

Let $\mathrm{\Gamma }(X)$ 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 $\mathrm{\Gamma }(X)$ 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 $(\mathrm{\Gamma }(X),{\tau }_{hco})$ is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup $I(\mathbb{N})$. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open sets.