Classifying the Polish semigroup topologies on the symmetric inverse monoid
Serhii Bardyla, Luna Elliott, James D. Mitchell, Y. Péresse · Proceedings of the Edinburgh Mathematical Society · 2026
Abstract We classify all Polish semigroup topologies on the symmetric inverse monoid $I_{\mathbb N}$ on the natural numbers $\mathbb N$ . 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 $I_{\mathbb N}$ endowed with any second countable $T_1$ semigroup topology is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$ .