Classifying the Polish semigroup topologies on the symmetric inverse monoid

2024/05/30 by Bardyla, Serhii, Elliott, Luna, Mitchell, James +1
#20M18 #20M20 #22A15 #54E35 #54H15 #FOS: Mathematics #General Topology (math.GN) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2405.20134

Abstract

We classify all Polish semigroup topologies on the symmetric inverse monoid on the natural numbers. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid endowed with any second countable T₁ semigroup topology is homeomorphic to the Baire space.

Related

• Topologies on the symmetric inverse semigroup
• Topological embeddings into transformation monoids
• Polish topologies on endomorphism monoids of relational structures
• Topological Transformation Monoids
• On the Polishness of the inverse semigroup Γ(X) on a compact metric space X