Topological automorphism-invariant subgroup

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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.

Relation with other properties

Stronger properties

Weaker properties