Exercise 9, § 9

Let $X$ be a topological space, $(G_\alpha)_{\alpha \in A}$ a family of equivalence graphs in $X$, whose index set is increasing filtered preordered. Let $R_\alpha$ be the relation $(x, y) \in G_\alpha$ and let $\varphi_\alpha$ be the canonical mapping $X \to X/R_\alpha$. We suppose that for $\alpha \leq \beta$, the relation $R_\beta$ implies $R_\alpha$; let then $\varphi_{\alpha\beta}$ be the canonical mapping $X/R_\beta \to X/R_\alpha$; $(X/R_\alpha, \varphi_{\alpha\beta})$ is a projective system of topological spaces; let $Y = \lim (X/R_\alpha)$.

(a) Let $g : X \to Y$ be the continuous mapping $\lim \varphi_\alpha$. Now prove that $g(X)$ is everywhere dense in $Y$, and that for every $x \in X$, $g^{-1}(g(x))$ is the class of $x$ for the equivalence relation whose graph is the intersection of the $G_\alpha$.

(b) We suppose that the equivalence relations $R_\alpha$ are closed, and that for every $x \in X$ and every neighbourhood $V$ of $x$, there exists an index $\alpha$ such that the class of $x$ for $R_\alpha$ is contained in $V$. Now prove that under these conditions $g$ is an open mapping of $X$ onto $g(X)$.

(c) If we suppose that the classes for the relations $R_\alpha$ are closed quasi-compact sets, then $g$ is surjective.