Proposition 3

Let $f: X\rightarrow Y$ be a continuous and separated mapping. Every continuous section of $f$ induces a homeomorphism of Y onto a closed subset of X.

Let $s: Y\rightarrow X$ be a continuous section of $f$. The mapping $s$ induces a homeomorphism of Y onto the subspace $s(Y)$ of X (TG, I, p. 22, prop. 9). The identity mapping of X and the mapping $s\circ f$ are continuous and one has $f\circ$ Id$_X=f\circ (s\circ f)$. Since the mapping $f$ is separated, the set $s(Y)$, which is the set of points $x$ of X such that $x=s\circ f(x)$, is closed in X (prop. 1, (ii)).