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 ,...

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Veröffentlicht in:	European journal of mathematics 2023-12, Vol.9 (4), Article 113
Hauptverfasser:	Pérez, Jerson, Uzcátegui, Carlos
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Sprache:	eng
Schlagworte:	Algebraic Geometry Isomorphism Lattices Mathematics Mathematics and Statistics Metric space Research Article Semigroups Topology
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description	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.
doi_str_mv	10.1007/s40879-023-00671-8
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title	On the Polishness of the inverse semigroup Γ(X) on a compact metric space x
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