Exercise 14, § 1

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

(a) Let K be a compact subset of E, H a subset of K filtering for the relation $\leq$; now prove that the filter $\mathscr{F}$ of the sections of H is convergent. (Proving first that the set L of the upper bounds of H belonging to K is nonempty and compact, then that the set of adherence values of $\mathscr{F}$ is contained in L, and finally that there cannot exist two distinct adherence values).

(b) Deduce from (a) that if, in E, every interval $[a, b]$ is compact, E is completely reticulated.