Topological group

Remarks

Example: $(\R, +)$ with the usual topology. Addition $(x, y) \mapsto x + y$ is continuous on $\R^2$, and $x \mapsto -x$ is continuous. Non-example: $(\R, +)$ with the lower limit topology, generated by the intervals $[a, b)$. Addition is still continuous, but inversion fails clause 2: the preimage of the open set $[0, 1)$ under $x \mapsto -x$ is $(-1, 0]$, which is not open. The two clauses together are equivalent to the single condition that $(x, y) \mapsto x y^{-1}$ is continuous from $G \times G$ to $G$. Every group with the discrete topology, and every group with the indiscrete topology, is a topological group. Many authors also require the topology to be Hausdorff.