Proposition 8

Let $f = \psi \circ g \circ \varphi$ be the canonical decomposition of a continuous mapping $f : X \to Y$, and let $R$ denote the equivalence relation

Then the following three conditions are equivalent: a) $g$ is a homeomorphism of $X/R$ onto $f(X)$. b) The image under $f$ of every open set which is saturated with respect to $R$ is an open set in the subspace $f(X)$. c) The image under $f$ of every closed set which is saturated with respect to $R$ is a closed set in the subspace $f(X)$.

For the condition b) [resp. c)] expresses that the image under $g$ of every open (resp. closed) set in $X/R$ is an open (resp. closed) set in $f(X)$.