Exercise 20, § 1

An ordered set E is said to be scattered if no ordered subset of E is without gaps (Exercise 19). Every subset of a scattered set is scattered. \* Every well-ordered set of more than one element is scattered. \*

(a) Suppose that E is scattered. Then, if $x$, $y$ are any two elements of E such that $x < y$, there exist two elements $x'$, $y'$ of E such that $x \leqslant x' < y' \leqslant y$ and such that the interval $]x', y'[$ is empty. \* Give an example of a totally ordered set which satisfies this condition and is not scattered (consider Cantor's triadic set). \*

(b) An ordinal sum $E = \sum_{\iota \in I} E_\iota$ (where neither I nor any $E_\iota$ is empty) is scattered if and only if I and each $E_\iota$ is scattered. (Note that E contains a subset isomorphic to I and that every subset F of E is the ordinal sum of those sets $F \cap E_\iota$ which are non-empty; finally use Exercise 19.)