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

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