Definition 5

Let X and Y be topological spaces and $f: X\rightarrow Y$ a mapping. Let R be the equivalence relation associated with $f$, and

the canonical decomposition of $f$ (E, II, p. 44). One says that the mapping $f$ is strict if $g$ is a homeomorphism, when $X/R$ is endowed with the quotient topology and $f(X)$ with the topology induced by that of Y.

A strict mapping is continuous.

Let us recall (TG, I, p. 22, prop. 8) that, in order that a mapping $f$ be strict, it is necessary and sufficient that $f$ be continuous and that for every open (resp. closed) saturated subset A of X, the set $f(A)$ be open (resp. closed) in $f(X)$.