Continuous Group Action is by Homeomorphisms

Theorem

Let $G$ be a topological group acting continuously on a topological space $X$.

Then $G$ acts by homeomorphisms.

Proof

Let $\phi: G \times X \to X$ denote the group action.

Let $g \in G$.

We are given that $\phi$ is a bijection.

The map $\phi_g: X \to X : x \mapsto \map \phi {g, x}$ is continuous because $\phi$ is.

This article, or a section of it, needs explaining. In particular: Why? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

By Inverse of Bijection is Bijection, the inverse of $\phi$ is given by $\phi_{g^{-1} }$ is also a bijection.

There is believed to be a mistake here, possibly a typo. In particular: grammar in the above, inconsistency in linguistic presentation in the following You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mistake}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

So the inverse of $\phi$ is continuous.

So inverse of $\phi_g$ is continuous.

Thus $\phi_g$ is an homeomorphism of $X$.

$\blacksquare$

Also see

• Discrete Group Acts Continuously iff Acts by Homeomorphisms