An amazingly useful mixture of two mathematical

concepts. Let

*G* be a

group. If

*G* is also a

topological space, we call it a

*topological group* if the two group operations

*f(x,y)*=

*xy* and

*g(x)*=

*x*^{-1} are

continuous functions. That is, the group structure and the topology of

*G* "agree".

Topological groups have much better properties than plain topological spaces. Many of the group properties extend easily (adding continuity where needed). Part of the reason is that *G* is a homogenous space: the map *f*_{a}(x)=*ax* is a homeomorphism, so the structure of the neighborhoods of the identity element of *G* completely specifies the topology.