Exercise 7, § 7

Let I be a directed set and let $(E_\alpha, f_{\beta\alpha})$ be a direct system of sets relative to I. For each $\alpha \in I$, let $f_\alpha : E_\alpha \to E = \varinjlim E_\alpha$ be the canonical mapping. In each $E_\alpha$, let $R_\alpha$ be the equivalence relation $f_\alpha(x) = f_\alpha(y)$.

Show that, whenever $\alpha \leqslant \beta$, the mapping $f_{\beta\alpha}$ is compatible with the equivalence relations $\mathrm{R}_\alpha$ and $\mathrm{R}_\beta$. Let $\mathrm{E}'_\alpha = \mathrm{E}_\alpha/\mathrm{R}_\alpha$, and let $f'_{\beta\alpha}$ be the mapping of $\mathrm{E}'_\alpha$ into $\mathrm{E}'_\beta$ induced by $f_{\beta\alpha}$ on passing to the quotients. Show that $f'_{\beta\alpha}$ is injective and that $(\mathrm{E}'_\alpha, f'_{\beta\alpha})$ is a direct system of sets, and define a canonical bijection of E onto $\varinjlim \mathrm{E}'_\alpha$.