Topological automorphism-invariant subgroup

This article defines a property that can be evaluated for a subgroup of a semitopological group
ANALOGY: This is an analogue in topological group of a property encountered in group. Specifically, it is a topological subgroup property analogous to the subgroup property: characteristic subgroup
View other analogues of characteristic subgroup | View other analogues in topological groups of subgroup properties (OR, View as a tabulated list)

Definition

Symbol-free definition

A subgroup of a topological group is said to be topological automorphism-invariant or topologically characteristic if it is invariant under all the topological automorphisms of the topological group.

Definition with symbols

A subgroup $H$ of a topological group $G$ is said to be topological automorphism-invariant or topologically characteristic in $G$ if given a topological automorphism $\sigma:G \to G$ ( $\sigma$ is thus both a homeomorphism and an automorphism), $\sigma(H)=H$.

Importance

Any topological subgroup-defining function on topological groups, must always output a topologically characteristic subgroup.