# Path-component of identity

From Groupprops

Template:Topological subgroup-defining function

## Definition

The **path-component of identity** in a topological group is defined as the path-component of the identity element of the group with respect to the topology. The following are true:

- The path-component of identity is a topological automorphism-invariant subgroup of the whole group. In particular, it is a normal subgroup of the whole group.
- The path-components are precisely the cosets of this subgroup. Since the subgroup is a normal subgroup, the left cosets coincide with the right cosets.