Exercise 8, § 9

Let $X$ be a topological space and let $(G_\alpha)_{\alpha \in A}$ be a family of graphs of equivalence relations on $X$, whose index set $A$ is directed; let $R_\alpha$ be the relation $(x, y) \in G_\alpha$ and let $\varphi_\alpha$ be the canonical mapping $X \to X/R_\alpha$. Suppose that the relation $R_\beta$ implies $R_\alpha$ if $\alpha \leq \beta$ and let $\varphi_{\alpha \beta}$ be the canonical mapping $X/R_\beta \to X/R_\alpha$; then $(X/R_\alpha, \varphi_{\alpha \beta})$ is an inverse system of topological spaces. Let $Y = \varprojlim X/R_\alpha$.

(*) This example (unpublished) is due to D. Zelinsky.

a) Let $g : X \to Y$ be the continuous mapping $\varprojlim \varphi_\alpha$. Show that $g(x)$ is dense in $Y$ and that, for each $x \in X$, $\overline{g^{-1}}(g(x))$ is the class of $x$ with respect to the equivalence relation whose graph is $\bigcap_\alpha G_\alpha$.

b) Suppose that all the $R_\alpha$ are closed equivalence relations, and that for each $x \in X$ and each neighbourhood $V$ of $x$ there is an index $\alpha$ such that the class of $x$ with respect to $R_\alpha$ is contained in $V$. Show that under these conditions $g$ is an open mapping of $X$ onto $g(X)$.

c) Show that if the classes mod $R_\alpha$ are closed and quasi-compact for each $\alpha \in A$, then $g$ is surjective.