# topological group (obsolete)

This entry is obsolete, having been superseded by a new entry (http://planetmath.org/TopologicalGroup2). It is being retained for a short while because of the attached thread.

A topological group is a triple $(G,\cdot,\mathcal{T})$ where $(G,\cdot)$ is a group and $\mathcal{T}$ is a topology on $G$ such that under $\mathcal{T}$, the group operation $(x,y)\mapsto x\cdot y$ is continuous with respect to the product topology on $G\times G$ and the inverse map $x\mapsto x^{-1}$ is continuous on $G$.

Many authors require that the topology be Hausdorff.

Title topological group (obsolete) TopologicalGroupobsolete 2013-03-22 12:12:54 2013-03-22 12:12:54 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Definition msc 22A05 Group TopologicalRing BirkhoffKakutaniTheorem CategoryOfPolishGroups AlgebraicTopology