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

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.