Automatic continuity and uniqueness of Polish semigroup topologies

If S is a topological monoid such that every homomorphism from S to a second countable topological monoid T is continuous, then we say that S has automatic continuity. In this paper, we show that many well-known, and extensively studied, monoids have automatic continuity with respect to some natural semigroup topology. Namely, the following have automatic continuity: the monoid BN of all binary relations on the natural numbers N; the monoid PN of partial transformations; the full transformation monoid NN; the monoid Inj(N) of injective transformations on N; the symmetric inverse monoid IN; and the monoid C(2N) of continuous functions on the Cantor set 2N.
Additionally, we show that: BN has no Polish semigroup topologies; PN, C(2N), and the monoid C([0,1]N) of continuous functions on the Hilbert cube [0,1]N each have a unique Polish semigroup topology; the monoid End(Q,≤) of all order-endomorphisms of the rational numbers, and the monoid of endomorphisms of the countable random graph (the Rado graph) have at least one Polish semigroup topology; the monoid of self-embeddings of any Fraïssé limit which admits proper embeddings has at least two Polish semigroup topologies; IN has at least 3 Polish semigroup topologies, but a unique Polish inverse semigroup topology; and Inj(N) and the monoid Surj(N) of all surjective transformations of N have infinitely many distinct Polish semigroup topologies.
In proving the main results mentioned above we prove myriad ancillary results.