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.

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