# Connected component of identity

From Groupprops

Template:Topological subgroup-defining function

## 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:

- 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.

## 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 |