Exercise 31, § 6

Let A be an infinite set and let E be the set of all infinite subsets of A, ordered by inclusion. Show that E is completely ramified (§ 2, Exercise 8) but not antidirected (§ 1, Exercise 23) and that E has an antidirected cofinal subset F. (Consider first the set $\mathfrak{D}(A)$ of countable infinite subsets of A (which is cofinal in E) and let $Z = R_0(\mathfrak{D}(A))$ (§ 1, Exercise 23). Write Z in the form $(z_\lambda)_{\lambda \in L}$, where L is a well-ordered set, and take F to be a set of countable subsets $X_n^\lambda$, where $\lambda$ runs through a suitable subset of L, $n \in \mathbf{N}$, $X_m^\lambda \supset X_n^\lambda$ whenever $m \leqslant n$, $X_n^\lambda - X_{n+1}^\lambda$ is infinite for all $n \geqslant 0$, and

the $X_n^\lambda$ are to be defined by transfinite induction in such a way that the images of the sets $X_n^\lambda$ under the canonical mapping $r : \mathfrak{D}(A) \to Z$ (§ 1, Exercise 23) are mutually disjoint and form a cofinal subset of Z.)