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

Abstract

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