Topological Groups

Topological Group

A topological group is a group that is also a topological space, such that the group operations of multiplication and inversion are both continuous maps. That is is a continuous mapping of the product space into .

Remark

In the language of category, topological groups can be defined concisely as group objects in the category of topological spaces .

Proposition

Let be a connected topological group and its identity element. If is any open neighborhood of , then is generated by .

Proof Let be an open neighborhood of . For each , let be the set of all products of at most elements of , and . Since each is open, is open. We now see that is also closed. For any , is an open neighbourhood of , so it must intersect . Let , then for some . Since , is a product of finitely many elements in , hence so is . Therefore, , and we have . Since is connected, we conclude that .