 Topological semigroup

A topological semigroup is a semigroup equipped with a topology for which multiplication $S\times S\to S$ is a continuous map, using the product topology on $S\times S$. Thus the two inputs vary together. The real numbers under addition are an example.

Continuity of each translation separately is weaker. A left-topological semigroup requires continuous maps $x\mapsto ax$ for fixed $a$; a right-topological semigroup instead requires continuous maps $x\mapsto xa$. Stating which factor is fixed avoids confusing the two conventions.

A **topological semigroup** is a mathematical structure that combines elements of both semigroup theory and topology. Specifically, it is a set equipped with a binary operation that is associative and is also endowed with a topology that makes the operation continuous.