Topological group

In mathematics, a topological group is a group, G, together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics.

Formal definition

A topological group, G, is a topological space and group such that the group operations of product:

and taking inverses:

are continuous functions. Here, G × G is viewed as a topological space by using the product topology.

Although not part of this definition, many authors require that the topology on G be Hausdorff; this corresponds to the identity map being a closed inclusion (hence also a cofibration). The reasons, and some equivalent conditions, are discussed below. In the end, this is not a serious restriction—any topological group can be made Hausdorff in a canonical fashion.