Exercise 14, § 1

Let E be a Riesz space, equipped with a Hausdorff topology compatible with the ordered vector space structure of E.

a) Let K be a compact subset of E, H a subset of K that is directed for the relation $\leq$; show that the section filter $\mathfrak{F}$ of H is convergent. (First prove that the set L of upper bounds of H belonging to K is nonempty and compact, then that the set of cluster points of $\mathfrak{F}$ is contained in L, and finally that there cannot exist two distinct cluster points.)

b) Deduce from a) that if, in E, every interval $[a, b]$ is compact, then E is fully lattice-ordered.