Topological Group

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.

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions. Adding the further requirement of Hausdorff (and cofibration) corresponds to refining to a model category.