Proposition 5

Let X, Y, Z be topological spaces and let $f: X\rightarrow Y,g: Y\rightarrow Z$ be continuous mappings. Suppose the mapping $g$ separated and the mapping $g\circ f$ proper. Then the mapping $f$ is proper.

Consider the following cartesian square:

$X\times_ZY^{pr_2}$ Y

pr$_1g$

X $^{g\circ f}Z$.

Let $s: X\rightarrow X\times_ZY$ be the mapping $x\mapsto (x, f(x))$. It is a continuous section of pr$_1: X\times_ZY\rightarrow X$. By prop. 4, the mapping pr$_1$ is separated; it follows that the mapping $s$ is proper (I, p. 27, prop. 3 and TG, I, p. 72, prop. 2). On the other hand, the mapping pr$_2$ is proper (I, p. 17, prop. 8). Hence the mapping $f$, which is equal to pr$_2\circ s$, is proper (TG, I, p. 73, prop. 5).