Exercise 19, § 1

An ordered set E is said to be without gaps if it contains two distinct comparable elements and if, for each pair of elements $x$, $y$ such that $x < y$, the open interval $]x, y[$ is not empty. Show that an ordinal sum $\sum_{\iota \in I} E_\iota$ (Exercise 3) is without gaps if and only if the following conditions are satisfied :

(I) Either I contains two distinct comparable elements, or else there exists $\iota \in I$ such that $E_\iota$ contains two distinct comparable elements.

(II) Each $E_\iota$ which contains at least two distinct comparable elements is without gaps.

(III) If $\alpha$, $\beta$ are two elements of I such that $\alpha < \beta$, and if the interval $]\alpha, \beta[$ in I is empty, then either $E_\alpha$ has no maximal element or else $E_\beta$ has no minimal element.

In particular, every ordinal sum $\sum_{\iota \in I} E_\iota$ of sets without gaps is itself without gaps provided that no $E_\iota$ has a maximal element (or provided that no $E_\iota$ has a minimal element). If I is without gaps, and if each $E_\iota$ is either without gaps or contains no two distinct comparable elements, then $\sum_{\iota \in I} E_\iota$ is without gaps.