Intrinsically continuous automorphism
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)
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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 of a group is termed intrinsically continuous if the following holds:
Let be any topology on such that forms a topological group under , viz the multiplication maps and inversion map are continuous with respect to . Then, is continuous on with respect to .
Relation with other properties
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)