Topological semigroup

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.

A left-topological semigroup requires only that every right translation $x\mapsto xa$ be continuous.

Every nonempty compact Hausdorff left-topological semigroup contains an idempotent. Choose a minimal nonempty compact subsemigroup $K$. For $a\in K$, minimality gives $Ka=K$; the nonempty compact subsemigroup $\{x\in K:xa=a\}$ is therefore all of $K$, so $a^2=a$.