Proposition 9

Let X, Y and Z be topological spaces. Let $f: X\rightarrow Y$ be a continuous surjective mapping and $g: Y\rightarrow Z$ a mapping.

a) If $f$ is strict and if $g\circ f$ is continuous, the mapping $g$ is continuous.

b) If $g$ is continuous and if $g\circ f$ is strict, the mapping $g$ is strict.

c) If $f$ and $g\circ f$ are strict, $g$ is strict.

Let us prove assertion a). Let R denote the relation associated with $f$ in X; by hypothesis, the mapping of $X/R$ onto Y deduced from $f$ by passing to the quotient is a homeomorphism. The first assertion then follows from prop. 6 of I, p. 21.

Let us prove b). Let B be a closed saturated subset of Y for the relation defined by $g$ and let $A =\overset{-1}{f}(B)$. Since $f$ is continuous, A is closed in X, and A is saturated for the equivalence relation defined by $g\circ f$. Since $g\circ f$ is strict and $f$ is surjective, $g(B) =g\circ f(A)$ is therefore closed in Z. The continuous mapping $g$ is therefore strict.

Assertion c) results immediately from assertions a) and b).