Connected component of identity
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:
- 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.
- 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.
|Value of connected component of identity||What it says about the whole group|
|whole group||connected semitopological group|
|trivial subgroup||totally disconnected group|