%0 Journal Article %T A topological characterisation of endomorphism monoids of countable structures %A Manuel Bodirsky %A Friedrich Martin Schneider %J Mathematics %D 2015 %I arXiv %X A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable omega-categorical structures. Finally we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable omega-categorical structures. %U http://arxiv.org/abs/1508.07404v1