Path-component of identity

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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:

  1. 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.
  2. 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.

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