Exercise 24, § 1

(a) An ordered set E is said to be branched (on the right) if for each $x \in E$ there exist $y$, $z$ in E such that $x \leqslant y$, $x \leqslant z$ and the intervals $[y, \rightarrow[$ and $[z, \rightarrow[$ do not intersect. An antidirected set with no maximal elements (Exercise 23) is branched.

(b) Let E be the set of intervals in $\mathbf{R}$ of the form

ordered by the relation $\supset$. Show that E is antidirected and has no maximal elements.

(c) Give an example of a branched set in which there exists no antidirected cofinal subset. (Take the product of the set E defined in (b) with a well-ordered set which contains no countable cofinal subset, and use Exercise 22.)

(d) Give an example of an ordered set E which is not antidirected, but which has an antidirected cofinal subset (note that an ordinal sum $\sum_{\xi \in E} F_\xi$ contains a cofinal subset isomorphic to E). \*