Connected component of identity

Definition

In a semitopological group (and hence also in a topological group), the connected component of identity is the connected component of the identity element of the group with respect to the underlying topology. The following are true:

1. The connected component of identity is a topological automorphism-invariant subgroup of the whole group, and in particular, it is a normal subgroup of the whole group. In fact, it is a closed normal subgroup of the whole group.
2. The other connected components are all cosets of this subgroup. Since the connected component of identity is a normal subgroup, the left cosets are the same as the right cosets.

Particular cases

Value of connected component of identity	What it says about the whole group
whole group	connected semitopological group
trivial subgroup	totally disconnected group