Remark

Let $R$ be the equivalence relation $i(x) = i(x')$ on $X$. We have seen (no. 7, Proposition 12) that the graph $C$ of $R$ is the intersection of all the entourages of $X$. It is clear that every open set (and therefore also every closed set) in $X$ is saturated with respect to $R$; taking account of the definition of the inverse image of a topology, we conclude that the canonical bijection of the quotient space $X/R$ onto $i(X)$ induced by $i$ is a homeomorphism. The Hausdorff space associated with $X$ can therefore be identified, qua topological space, with $X/R$. The canonical mapping $i : X \to i(X)$ is open and closed, and even proper (Chapter I, $§ 10$, no. 2, Example).

Let $X'$ be another uniform space, $C'$ the intersection of all the entourages of $X'$, and $R'$ the equivalence relation whose graph is $C'$. Let $f : X \to X'$ be a continuous mapping. Since the inverse image under $f$ of any neighbourhood of $f(x)$ is a neighbourhood of $x$, it follows that the inverse image under $f$ of $C'(f(x))$ contains $C(x)$, and therefore $f$ is compatible with $R$ and $R'$, and induces a continuous mapping $X/R \to X'/R'$ (Chapter I, $§ 3$, no. 4, Corollary to Proposition 6). This generalizes the corollary to Proposition 16.