Suppose \(G\) is a topological group and \(X\) is a topological space . A left action of \(G\) on \(X\) is continuous if and only if the map \(G\times X\to X\) is continuous .

Examples Link to heading

• action of a topological group on itself
• action of a subgroup on its group
• \(\GL(n)\) on \(\mathbb{R}^n\)
• \(\O(n)\) on \(\mathbb{R}^n\)
• \(\O(n)\) on \(\mathbb{S}^{n-1}\)