Topological automorphism-invariant subgroup

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.

Stronger properties

 * Characteristic subgroup

Weaker properties

 * Normal subgroup