Topological group

The connected component of the identity is a normal subgroup

Let be a Topological group and be the (path) connected component containing the identity , where is the (path) connectedness relation. Then is a normal subgroup of , called the (path) connected subgroup of group

Proof

by construction. Let , so . We will use The continuous image of a connected space is connected. so continuous . Right-multiplication by is continuous so . Thus , and is a subgroup by One step subgroup test. For any , conjugation by is continuous. Hence for any . Therefore is a normal subgroup.

Properties

1. The cosets of are the connected components of , i.e.

Proof of 1

Since multiplication is continuous and hence preserves connected components

proving ^P1.

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