# Group Theory/Topological groups

**Definition (topological group)**:

A **topological group** is a group whose underlying set is endowed with a topology such that

- the group law is a continuous function and
- inversion is a continuous function .

Thus, a topological group is a group with structure in the category of topological spaces.

**Proposition (every topological group is a uniform space)**:

Let be a topological group, and let be a neighbourhood system of its identity. Then the sets

form an entourage system whose induced topology is identical to the topology of .

**Proof:**

**Theorem (Birkhoff‒Kakutani theorem)**:

**Proposition (the connected component of the identity of a topological group is one of its normal subgroups)**:

Let be a topological group, and let be the connected component of its identity. Then .

**Proof:**

**Proposition (each locally compact topological group is the disjoint union of translates of one of its σ-compact open subgroups)**:

Let be a locally compact topological group. Then there exists a σ-compact open subgroup , from which we may of course deduce that

- ,

where is a set that contains one element of each left coset of (the squared union symbol indicating that the union is disjoint). Moreover, each left coset of is σ-compact and open.

**Proof:** We shall denote the identity of by . Let be a compact neighbourhood of . We set . Since the image of a compact set via a continuous map is compact and the union of two compact sets is compact, is a compact neighbourhood of . Moreover, induction, the fact that the product of two compact sets is compact and the fact that the image of a compact set via a continuous map is compact (applied to the continuous group law map) yield that all the sets are compact. Yet, the group

- ,

ie. the group generated by the elements of , is the union of these sets, hence σ-compact.

It remains to show that is open. To this end, we may use that since is a neighbourhood of , there exists an open set such that . Since multiplication by a group element is an isomorphism, the set are open in whenever . Hence,

is open.

Finally, is open and σ-compact because multiplication by is an automorphism of in the category of topological spaces, whence it preserves openness and compactness.