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.
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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 we do not do so here, many authors^{[1]} require that the topology on G be Hausdorff. 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.
Homomorphisms
A homomorphism between two topological groups G and H is just a continuous group homomorphism G → H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any nondiscrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.
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