Exercise 5, § 7

Let I be a directed set and let $(E_\alpha)_{\alpha \in I}$ be a family of lattices such that each $E_\alpha$, endowed with the opposite ordering, is Noetherian (§ 6, no. 5). For each pair $(\alpha, \beta)$ of indices in I such that $\alpha \leqslant \beta$ let

be an increasing mapping, and suppose that $(E_\alpha, f_{\alpha\beta})$ is an inverse system of sets relative to I. For each $\alpha \in I$ let $G_\alpha$ be a non-empty subset of $E_\alpha$ such that (i) no two distinct elements of $G_\alpha$ are comparable, (ii) $f_{\alpha\beta}(G_\beta) = G_\alpha$ whenever $\alpha \leqslant \beta$, (iii) for each $\alpha \leqslant \beta$ and each $x_\alpha \in G_\alpha$, $\overset{-1}{f_{\alpha\beta}}(x_\alpha)$ has a greatest element $M_{\alpha\beta}(x_\alpha)$ in $E_\beta$, (iv) whenever $\alpha \leqslant \beta$, if $h_\beta \in E_\beta$ is such that there exists $y_\beta \in G_\beta$ such that $y_\beta \leqslant h_\beta$, then for each $x_\alpha \in G_\alpha$ such that $x_\alpha \leqslant f_{\alpha\beta}(h_\beta)$ there exists $x_\beta \in G_\beta$ such that $x_\beta \leqslant h_\beta$ and $x_\alpha = f_{\alpha\beta}(x_\beta)$. Under these conditions the inverse limit of the inverse system of subsets $(G_\alpha)$ is not empty. The proof runs as follows:

(a) Let J be a finite subset of I. A family $(x_\alpha)_{\alpha \in J}$, where $x_\alpha \in G_\alpha$ for all $\alpha \in J$, is said to be coherent if it satisfies the following two conditions :

(i) if $\alpha \in J$, $\beta \in J$, $\alpha \leqslant \beta$, then $x_\alpha = f_{\alpha\beta}(x_\beta)$; (ii) for each upper bound $\gamma$ of J in I there exists $x_\gamma \in G_\gamma$ such that $x_\alpha = f_{\alpha\gamma}(x_\gamma)$ for all $\alpha \in J$. Show that, for each upper bound $\gamma$ of J in I, the set $\bigcap\limits_{\alpha \in J} \overset{-1}{f_{\alpha\gamma}}(x_\alpha)$ has a greatest element equal to $\inf\limits_{\alpha \in J} (M_{\alpha\gamma}(x_\alpha))$; furthermore, the intersection of $G_\gamma$ and $\bigcap\limits_{\alpha \in J} \overset{-1}{f_{\alpha\gamma}}(x_\alpha)$ is the set (non-empty by hypothesis) of all $y_\gamma \in G_\gamma$ such that

(use condition (i)).

(b) Let J be any subset of I. A family $x_J = (x_\alpha)_{\alpha \in J}$, where $x_\alpha \in G_\alpha$ for all $\alpha \in J$, is said to be coherent if every finite subfamily of $x_J$ is coherent. If $J \neq I$ and if $\beta \in I - J$, show that there exists $x_\beta \in G_\beta$ such that the family $x_{J \cup \{\beta\}} = (x_\alpha)_{\alpha \in J \cup \{\beta\}}$ is coherent. (Using (a) and condition (iv), show that, for every finite subset F of J, if $\gamma$ is an upper bound of $F \cup \{\beta\}$, then $f_{\beta\gamma}\left(G_\gamma \cap \bigcap\limits_{\alpha \in F} \overset{-1}{f_{\alpha\gamma}}(x_\alpha)\right)$ is the (non-empty) set of all $y_\beta \in G_\beta$ which are $\leqslant f_{\beta\gamma}\left(\inf\limits_{\alpha \in F} (M_{\alpha\gamma}(x_\alpha))\right)$. Using the fact that $E_\beta$ endowed with the opposite ordering is Noetherian, show next that there exist a finite subset $F_0$ of J and an upper bound $\gamma_0$ of $F_0 \cup \{\beta\}$ such that for each finite subset F of J and each upper bound $\gamma$ of $F \cup \{\beta\}$ we have

Prove then that every element $x_\beta \in F_\beta$ which is $\leqslant f_{\beta\gamma_0}\left(\inf\limits_{\alpha \in F_0} (M_{\alpha\gamma_0}(x_\alpha))\right)$ satisfies the required conditions.

(c) Finally, complete the proof by showing that there exists a coherent family whose index set is the whole of I. (Order the set of coherent families $x_J$ by the relation "$x_J$ is a subfamily of $x_K$", and apply (b) and Zorn's lemma.)