  In this paper we present general techniques for characterising minimal and
maximal semigroup topologies on the endomorphism monoid
$\operatorname{End}(\mathbb{A})$ of a countable relational structure
$\mathbb{A}$. As applications, we show that the endomorphism monoids of several
well-known relational structures, including the random graph, the random
directed graph, and the random partial order, possess a unique Polish semigroup
topology. In every case this unique topology is the subspace topology induced
by the usual topology on the Baire space $\mathbb{N} ^ \mathbb{N}$. We also
show that many of these structures have the property that every homomorphism
from their endomorphism monoid to a second countable topological semigroup is
continuous; referred to as automatic continuity. Many of the results about
endomorphism monoids are extended to clones of polymorphisms on the same
structures.
