Intrinsically continuous automorphism

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 $$\tau$$.

Stronger properties

 * Intrinsically homeomorphic automorphism
 * Inner automorphism
 * Monomial automorphism

Metaproperties
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)