A semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.
A set equipped with both the algebraic structure of a semigroup and the structure of a topological Hausdorff space, such that the semi-group operation is continuous in the given topology.
Any semi-group is a topological semigroup in the discrete topology. There exist semigroups which admit only the discrete topology. Any Hausdorff space can be made into a topological semigroup, e.g. by giving it a left-singular or zero multiplication.
