Proposition 10

The canonical bijection $h : A/R_A \to f(A)$ is continuous. Furthermore, the following three statements are equivalent:

a) $h$ is a homeomorphism.

b) Every open subset of $A$ which is saturated with respect to $R_A$ is the intersection with $A$ of an open subset of $X$ which is saturated with respect to $R$.

c) Every closed subset of $A$ which is saturated with respect to $R_A$ is the intersection with $A$ of a closed subset of $X$ which is saturated with respect to $R$.

The first part of the proposition is immediate (no. 5). The second part follows from Proposition 8 of no. 5: if $B$ is an open (resp. closed) subset of $A$ which is saturated with respect to $R_A$, and $g(B) = f(B)$ is the intersection with $f(A)$ of an open (resp. closed) subset $C$ of $X/R$, then $B$ is the intersection with $A$ of the open (resp. closed) subset $f^{-1}(C)$

(*) This expression means that $f \circ j = \psi \circ h \circ \varphi$.

of X, which is saturated with respect to R; and conversely, if B is the intersection with A of an open (resp. closed) subset D which is saturated with respect to R, then $f(B)$ is the intersection of $f(A)$ and $f(D)$, which is open (resp. closed) in $X/R$.