Contents

# Contents

## Definition

The action of a topological group $G$ on a topological space $X$ is called a proper action if the continuous function

$\array{ G \times X &\longrightarrow& X \times X \\ (g,x) &\mapsto& (g\cdot x, x) }$

## Examples

### Lie groups actions on smooth manifolds

###### Proposition

Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every smooth action of $G$ on $X$ is proper.

(e.g. Lee 12, Cor. 7.2)

For more see at equivariant differential topology.

For instance