Remark

Let $G'$ be another topological group operating continuously on $X$, and suppose that $s.(s'x) = s'(s.x)$ for all $s \in G, s' \in G'$ and x \in X. Then the equivalence relation S defined by G' is compatible with G, and since S is open (Lemma 2) it follows that G operates continuously on X/G'. Similarly, G' operates continuously on X/G. In these circumstances the two operations of the two groups G and G' on X are said to commute.