Proposition 8

Let I be a directed set, let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ be a direct system of sets relative to I, and let $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$ be its direct limit. Let J be a cofinal subset of I and let E′ be the direct limit of the direct system of sets obtained from $(\mathrm{E}_\alpha, f_{\beta\alpha})$ by restricting the index set to J. Then the canonical mapping $g$ of E′ into E is bijective.

J is necessarily a directed set (§ 1, no. 10). We shall use the criteria of Proposition 6 to show that $g$ is bijective. The condition for injectivity follows immediately from the definitions and from Lemma 1 of no. 5. To show that $g$ is surjective, we note that for each $\alpha \in \mathrm{J}$ we have

Now, for each $\beta \in I$, there exists $\gamma \in J$ such that $\beta \leqslant \gamma$, from which it follows that $g(E_\gamma) \supset g(f_{\gamma\beta}(E_\beta)) = f_\beta(E_\beta)$. E is therefore the union of the sets $g(E_\alpha)$ as $\alpha$ runs through J.