Continuous multiplication

Remarks

Example: $\R$ with its usual topology and ordinary multiplication. The map $(x, y) \mapsto xy$ is a polynomial in the coordinates, so it is continuous on $\R^2$. Non-example: $\Z$ with the cofinite topology and addition. Every nonempty open set $U \times V$ of $\Z \times \Z$ meets the set $\set{(x, -x) : x \in \Z}$, because $U$ and $-V$ are cofinite and so they intersect. So the preimage of the open set $\Z \setminus \set{0}$ under addition contains no nonempty open set, and addition is not continuous. Continuity of $\mu$ implies that every left translation $y \mapsto xy$ and every right translation $x \mapsto xy$ is continuous. The converse fails: in the non-example above every translation is a bijection of $\Z$, hence a homeomorphism for the cofinite topology. In additive notation the same condition is called continuous addition.