Corollary 4

Let $X$ be a topological space and $R$ a Hausdorff equivalence relation on $X$.

a) If there is a quasi-compact set $K$ in $X$ which meets every equivalence class mod $R$, then $X/R$ is compact and the canonical mapping of $K/R_K$ onto $X/R$ is a homeomorphism.

b) If $K$ also meets each equivalence class in only one point, then $K$ is a continuous section of $X$ with respect to the relation $R$ (§ 3, no. 5).

Let $f$ be the restriction to $K$ of the canonical mapping $X \to X/R$. Since $X/R$ is Hausdorff it follows from Corollary 1 that $X/R$ is compact and from Corollary 2 that $f$ is closed; hence the bijection $K/R_K \to X/R$ associated with $f$ is a homeomorphism (§ 5, no. 2, Proposition 3). This deals with a); b) follows immediately, since we now have $K/R_K = K$.