  We investigate semigroup topologies on the full transformation monoid T(X) of
an infinite set X. We show that the standard pointwise topology is the weakest
Hausdorff semigroup topology on T(X), show that the pointwise topology is the
unique Hausdorff semigroup topology on T(X) that induces the pointwise topology
on the group of all permutations of X, and construct |X| distinct Hausdorff
semigroup topologies on T(X). In the case where X is countable, we prove that
the pointwise topology is the only Polish semigroup topology on T(X). We also
show that every separable semigroup topology on T(X) is perfect, describe the
compact sets in an arbitrary Hausdorff semigroup topology on T(X), and show
that there are no locally compact perfect Hausdorff semigroup topologies on
T(X) when |X| has uncountable cofinality.
