Topological monoid

Definition
A topological monoid is a set $$S$$ endowed with the structure of a monoid (with binary operation $$*$$) as well as the structure of a topological space, such that the monoid operation $$\! *:S \times S \to S$$ is a continuous map from the space $$S \times S$$ (endowed with the product topology) to $$S$$.

Stronger notions

 * Topological group

Weaker notions

 * Topological semigroup
 * Topological magma
 * Loop space