Exercise 10, § 1

a) Let E be a fully lattice-ordered space; there exists a topology $\mathscr{T}_0(E)$ on E that is the finest of the topologies $\mathscr{T}$ on E for which every bounded filter $\mathscr{F}$ on E that converges in the sense of the order structure (Exer. 9) converges to the same limit for $\mathscr{T}$.

Let E and F be two fully lattice-ordered spaces, $f$ a mapping of E into F such that, for every bounded filter $\mathfrak{F}$ on E that is convergent for the order structure, $f(\mathfrak{F})$ is a base of a bounded filter on F that converges to $f(\lim \mathfrak{F})$ for the order structure. Show that, under these conditions, $f$ is continuous for the topologies $\mathscr{T}_0(E)$ and $\mathscr{T}_0(F)$.

b) Deduce from a) that the topology $\mathscr{T}_0(E)$ is compatible with the vector space structure of E and that the mapping $x \mapsto |x|$ is continuous for this topology. Prove that $\mathscr{T}_0(E)$ is compatible with the ordered vector space structure of E and, from this, deduce that this topology is Hausdorff.

c) Given any infinite set A, let $E = \mathscr{B}(A)$ be the fully lattice-ordered space of bounded real-valued functions on A. Let $\Phi$ be the set of numerical functions $\varphi$ (finite or not) defined on A such that $\varphi(t) > 0$ for all $t \in A$ and such that, for every integer $n > 0$, the set of $t \in A$ such that $\varphi(t) \leq n$ is finite. For every function $\varphi \in \Phi$, let $V_\varphi$ be the set of $x \in E$ such that $|x| \leq \varphi$; show that the sets $V_\varphi$ form a fundamental system of neighborhoods of 0 for a topology $\mathscr{T}_1(E)$ compatible with the ordered vector space structure of E, and for which E is Hausdorff and complete. Show that the topology $\mathscr{T}_0(E)$ is finer than $\mathscr{T}_1(E)$, but strictly coarser than the topology of uniform convergence in A (defined by the norm $\|x\| = \sup_{t \in A} |x(t)|$) (consider the elements $1 - \varphi_X$ of $\mathscr{B}(A)$, where X runs over the set of finite subsets of A, $\varphi_X$ being the characteristic function of X). Deduce from this that in order for a subset of E to be bounded (for the order structure), it is necessary and sufficient that it be bounded for the topology $\mathscr{T}_0(E)$ (TVS, III, §1, No. 2). Show that every filter on E that is bounded and convergent for the topology $\mathscr{T}_0(E)$ is convergent to the same limit for the order structure. Finally, show that there exist filters on E that are convergent for $\mathscr{T}_0(E)$ but are not bounded.