Classifying the Polish semigroup topologies on the symmetric inverse monoid

TL;DR

This paper classifies all Polish semigroup topologies on the symmetric inverse monoid on natural numbers, revealing a rich lattice structure and characterizing topologies homeomorphic to the Baire space.

Contribution

It provides a complete classification of Polish semigroup topologies on the symmetric inverse monoid, answering an open question and describing their lattice structure.

Findings

Countably infinite Polish semigroup topologies exist.

The topologies form a join-semilattice with specific order properties.

Any second countable T_1 semigroup topology yields a space homeomorphic to the Baire space.

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 T_1 semigroup topology is homeomorphic to the Baire space.