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 fa(x)=ax is a homeomorphism, so the structure of the neighborhoods of the identity element of G completely specifies the topology.