  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.
