Corollary

Let $(X_i)_{i \in I}$ be a family of topological spaces, and for each $i \in I$ let $R_i$ be an equivalence relation on $X_i$, and let $f_i$ be the canonical mapping $X_i \to X_i/R_i$. Let $R$ be the equivalence relation in $X = \prod_{i \in I} X_i$

"for each $i \in I$, $\operatorname{pr}_i(x) = \operatorname{pr}_i(y)$ (mod $R_i$)"

between $x$ and $y$, and let $f$ be the product mapping $(x_i) \to (f_i(x_i))$ of $X$ into $\prod_{i \in I} (X_i/R_i)$. If each of the relations $R_i$ is open, then the relation $R$ is open, and the bijection associated with $f$ is a homeomorphism of $X/R$ onto $\prod_{i \in I} (X_i/R_i)$.

$R$ is the relation $f(x) = f(y)$. Since $f$ is continuous and open by Proposition 8 above and § 4, no. 1, Corollary 1 of Proposition 1, the result follows from Proposition 3 of no. 2.

In particular, if $R$ (resp. $S$) is an open equivalence relation on a topological space $X$ (resp. $Y$), then the canonical bijection of

is a homeomorphism. If R and S are not assumed to be open, this bijection is continuous but is not necessarily a homeomorphism, even if one of the relations R, S is the relation of equality (Exercise 6).