Proposition 7

Let $R$ be an open equivalence relation on a topological space $X$, and let $\varphi : X \to X/R$ be the canonical mapping. If $A$ is any subset of $X$ which is saturated with respect to $R$, then the closure (resp. the interior) of $\varphi(A)$ in $X/R$ is $\varphi(\overline{A})$ (resp. $\varphi(\overset{\circ}{A})$).

Each of the two assertions of the proposition can be deduced from the other by taking complements and using formula (2) of § 1, no. 6 and the fact that if $B$ is a saturated subset of $X$ then $\varphi(\complement B) = \complement_{\varphi}(B)$. By virtue of Proposition 6, $\overline{A}$ is saturated; hence $\varphi(\overline{A})$ is closed in $X/R$, and since $A \subset \overline{A}$ we have $\varphi(A) \subset \varphi(\overline{A})$, so that $\varphi(A) \subset \varphi(\overline{A})$. But since $\varphi$ is continuous, $\varphi(\overline{A}) \subset \varphi(A)$ (§ 2, no. 1, Theorem 1), and the result follows.