Classifying the Polish semigroup topologies on the symmetric inverse monoid

We classify all Polish semigroup topologies on the symmetric inverse monoid IN on the natural numbers N . 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 IN endowed with any second countable T₁ semigroup topology is homeomorphic to the Baire space NN .