# 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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## 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 $\tau$.

## 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)