Topological monoid

Remarks

Example: $(\R, \cdot)$ with the usual topology. Multiplication is associative with identity $1$, and $(x, y) \mapsto xy$ is continuous on $\R^2$. Non-example: the monoid $(\Z, +)$ with the cofinite topology. It has identity $0$, but addition is not jointly continuous, because every nonempty open set $U \times V$ of $\Z \times \Z$ contains a pair $(x, -x)$, so the preimage of $\Z \setminus \set{0}$ is not open. No condition on the identity element is needed beyond its existence. A topological monoid in which every element is invertible and inversion is continuous is a topological group.