Proposition 4

Let R be an open (resp. closed) equivalence relation on a topological space X, and f the canonical mapping $X \to X/R$. Let A be a subset of X and suppose that one of the following two conditions is satisfied:

a) A is open (resp. closed) in X.

b) A is saturated with respect to R.

Then the relation $R_A$ induced on A is open (resp. closed) and the canonical mapping of $A/R_A$ onto $f(A)$ is a homeomorphism.

Consider the commutative diagram (i) of § 3, no. 6, which gives the canonical decomposition of $f \circ j$. Under condition a), $j$ is open (resp. closed) and so is $f$ by hypothesis; hence $f \circ j$ is open (resp. closed) [no. 1, Proposition 1 a)], and the result follows from Proposition 3. Under condition b) we have

and $h \circ \varphi$ is the mapping of $A$ into $f(A)$ which agrees with $f$ on $A$; by virtue of Proposition 2 a) of no. 1, $h \circ \varphi$ is open (resp. closed), and the result again follows from Proposition 3, applied to $h \circ \varphi$.