Intrinsically continuous automorphism

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

Symbol-free definition

An automorphism of a group is termed intrinsically continuous if for any compatible topology on the group, the automorphism is a continuous map from the group to itself.

Definition with symbols

An automorphism \sigma of a group G is termed intrinsically continuous if the following holds:

Let \tau be any topology on G such that G forms a topological group under \tau, viz the multiplication maps and inversion map are continuous with respect to \tau. Then, \sigma is continuous on G with respect to \tau.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Template:Monoid-closed ap

A product of intrinsically continuous automorphisms is intrinsically continuous. This follows from two facts:

  • A product of automorphisms is an automorphism
  • A product of continuous maps is a continuous map (for any fixed topology)