Exercise 4, § 7

Let I be a non-empty directed set with no greatest element, and let F be the set of all sequences $x = (\alpha_1, \alpha_2, \ldots, \alpha_{2n-1}, \alpha_{2n})$ of an even number $\geqslant 2$ of elements of I with the following properties : (i) $\alpha_{2i-1} < \alpha_{2i}$ for $1 \leqslant i \leqslant n$; (ii) $\alpha_{2i-1} \not\leqslant \alpha_{2j-1}$ for $1 \leqslant j < i \leqslant n$. The set F is not empty. Put $r(x) = \alpha_{2n-1}$, $s(x) = \alpha_{2n}$. The integer $n$ is called the length of $x$.

(a) For each $\alpha \in I$, let $E_\alpha$ be the set of all $x \in F$ such that $r(x) = \alpha$. Then $E_\alpha$ is not empty. For $\alpha \leqslant \beta$ in I, we define a mapping $f_{\alpha\beta}$ of $E_\beta$ into the set of all finite sequences of elements of I, as follows : if

let $j$ be the least index such that $\alpha \leqslant \alpha_{2j-1}$; then

Show that $f_{\alpha\beta}(E_\beta) = E_\alpha$ and that $(E_\alpha, f_{\alpha\beta})$ is an inverse system of sets relative to I.

(b) Show that if $x_\alpha \in E_\alpha$ and $x_\beta \in E_\beta$ are such that there exists an index $\gamma$ for which $\gamma \geqslant \alpha$ and $\gamma \geqslant \beta$, and an element $x_\gamma \in E_\gamma$ for which $x_\alpha = f_{\alpha\gamma}(x_\gamma)$ and $x_\beta = f_{\beta\gamma}(x_\gamma)$, then, provided also $x_\alpha$ and $x_\beta$ have the same length, we have $s(x_\alpha) = s(x_\beta)$.

(c) Deduce from (b) that, if $E = \varprojlim E_\alpha$ is not empty and if $y = (x_\alpha) \in E$, then the set of elements $s(x_\alpha)$ is countable and cofinal in I.

(d) Let I be the set of all finite subsets of an uncountable set A, ordered by inclusion. Show that I has no countable cofinal subset, and hence deduce from (c) an example of an inverse system of sets $(E_\alpha, f_{\alpha\beta})$ in which the $E_\alpha$ are non-empty and the $f_{\alpha\beta}$ are surjective, but for which $E = \varprojlim E_\alpha = \emptyset$.

(e) Deduce from (d) an example of an inverse system of mappings $u_\alpha : E_\alpha \to E'_\alpha$ such that each $u_\alpha$ is surjective but $\varprojlim u_\alpha$ is not surjective (let each $E'_\alpha$ consist of a single element).