Proposition 9

Let $f : X \to Y$ be a continuous surjection, and let $R$ denote the equivalence relation $f(x) = f(y)$. If there is a continuous section $s : Y \to X$ associated with $f$ (Set Theory, Chapter II, § 3, no. 8, Definition 11), then the mapping $g : X/R \to Y$ associated with $f$ is a homeomorphism, and $s$ is a homeomorphism of $Y$ onto the subspace $s(Y)$ of $X$.

For if $\varphi : X \to X/R$ is the canonical mapping, then $g$ and $\varphi \circ s$ are bijective, continuous and inverse to each other; likewise $s$ and the restriction of $f$ to $s(Y)$ are bijective, continuous and inverse to each other.

If $R$ is an equivalence relation on a topological space $X$ and

is the canonical mapping, a continuous section $s : X/R \to X$ associated with $\varphi$ is also called a continuous section of $X$ with respect to $R$ (cf. Set Theory, Chapter II, § 6, no. 2); the subspace $s(X/R)$ of $X$ is then homeomorphic to $X/R$. If we are given $s(X/R)$, then $s$ is uniquely determined; $s(X/R)$ is often called, by abuse of language, a (continuous) section of $X$ with respect to $R$.

A continuous section of a topological space with respect to an equivalence relation need not exist (Exercise 12).