Proposition 7

Let $(\mathrm{E}_\alpha, f_{\beta\alpha})$ and $(\mathrm{E}'_\alpha, f'_{\beta\alpha})$ be two direct systems of sets relative to I, and for each $\alpha \in \mathrm{I}$ let $u_\alpha$ be a mapping of $\mathrm{E}_\alpha$ into $\mathrm{E}'_\alpha$ such that the $u_\alpha$ form a direct system of mappings. Let $u = \varinjlim u_\alpha$. If each $u_\alpha$ is injective (resp. surjective) then $u$ is injective (resp. surjective).

Let $\mathrm{E} = \varinjlim \mathrm{E}_\alpha$, $\mathrm{E}' = \varinjlim \mathrm{E}'_\alpha$, and let $f_\alpha : \mathrm{E}_\alpha \to \mathrm{E}$, $f'_\alpha : \mathrm{E}'_\alpha \to \mathrm{E}'$ be the canonical mappings. Suppose that each $u_\alpha$ is injective. To show that $u$ is injective it is enough, by Proposition 6, to verify that if $x \in \mathrm{E}_\alpha$ and $y \in \mathrm{E}_\alpha$ are such that $f'_\alpha(u_\alpha(x)) = f'_\alpha(u_\alpha(y))$, then there exists $\beta \geqslant \alpha$ such that $f_{\beta\alpha}(x) = f_{\beta\alpha}(y)$. Now the hypothesis implies (no. 6, Lemma 1) that there exists $\beta \geqslant \alpha$ such that

and hence $f_{\beta\alpha}(x) = f_{\beta\alpha}(y)$ since $u_\beta$ is injective.

Now suppose that the $u_\alpha$ are surjective. Then we have

With the notation of Proposition 7, let $\mathrm{M}_\alpha$ be a subset of $\mathrm{E}_\alpha$ for each $\alpha \in \mathrm{I}$; if we have $f_{\beta\alpha}(\mathrm{M}_\alpha) \in \mathrm{M}_\beta$ whenever $\alpha \leqslant \beta$, the family $(\mathrm{M}_\alpha)_{\alpha \in \mathrm{I}}$ is said to be a direct system of subsets of the $\mathrm{E}_\alpha$. Let $g_{\beta\alpha}$ (where $\alpha \leqslant \beta$) be the mapping of $\mathrm{M}_\alpha$ into $\mathrm{M}_\beta$ whose graph is the same as that of the restriction of $f_{\beta\alpha}$ to $\mathrm{M}_\alpha$. Then it is clear that $(\mathrm{M}_\alpha, g_{\beta\alpha})$ is a direct system of sets; and Proposition 7, applied to the canonical injections $j_\alpha : \mathrm{M}_\alpha \to \mathrm{E}_\alpha$, allows us to identify $\mathrm{M} = \varinjlim \mathrm{M}_\alpha$ with a subset of E by means of the injection $j = \varinjlim j_\alpha$.