Polish topologies on endomorphism monoids of relational structures

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