Exercise 13, § 1

(a) Let E be a completely reticulated space. If $u \neq 0$ is an arbitrary element of E, now prove that the set of $x \in E$ such that there exists an integer $n$ for which $|x| \leqslant n|u|$, is a completely reticulated and coreticulated subspace $C_u$ (exerc. 3) of E. For every $x \in C_u$, denote by $\|x\|$ the greatest lower bound of the scalars $\lambda > 0$ such that $|x| \leqslant \lambda|u|$; now prove that $\|x\|$ is a norm on $C_u$ (use the fact that E is Archimedean (exerc. 5)), that, endowed with this norm, $C_u$ is complete, and that one has $\|x\| = \sup(\|x^+\|, \|x^-\|)$.

(b) Let $\mathscr{J}$ be the set of components of $u$ on the bands of the space $C_u$; now prove that $\mathscr{J}$ is the set of elements $c$ of $C_u$ such that $0 \leqslant c \leqslant u$ and $\inf(c, u - c) = 0$ (use the Riesz theorem). Now prove that $\mathscr{J}$ is closed in the normed space $C_u$ and that, for the order relation induced by that of $C_u$, $\mathscr{J}$ is a complete Boolean lattice (Ens., chap. III, 2nd ed., § 1, exerc. 11 and 17; it is enough to prove that, if $(c_i)$ is a family of elements of $\mathscr{J}$, sup $c_i$ belongs to $\mathscr{J}$).

(c) Let $x$ be an arbitrary element of $C_u$; for every $\lambda \in \mathbf{R}$, let $c(\lambda)$ be the component of $u$ on the band of $C_u$ generated by $(\lambda u - x)^+$. Now prove that if $\lambda \leqslant \mu$, one has $c(\lambda) \leqslant c(\mu)$; for $\hat{\lambda} < -\|x\|$, one has $c(\hat{\lambda}) = 0$, and for $\lambda > \|x\|$, one has $c(\lambda) = u$. Now prove that if $c \in \mathscr{J}$ is such that $c \leqslant c(\lambda)$, the component of $x$ on the band generated by $c$ is $\leqslant \lambda c$ (notice that the bands generated by $c(\lambda)$ and by $(\lambda u - x)^+$ are identical, and that the component of $\lambda u - x$ on this band is equal to that of $(\lambda u - x)^+$; deduce that the component of $x$ on this same band is $\leq \lambda c(\lambda)$). Prove similarly that if $c \in \mathscr{J}$ is such that $c \leq u - c(\lambda)$, the component of $x$ on the band generated by $c$ is $\geq \lambda c$.

$d$ ) We know (Top. gén., Chap. II, 3\textsuperscript{e} ed., § 4, exerc. 12) that there exists an isomorphism of ordered structure $c \mapsto \theta_c$ of the Boolean lattice $\mathscr{J}$ onto the Boolean lattice formed by the characteristic functions of the sets which are both open and closed in a totally disconnected compact space S. Show that the relation $\sum_i \lambda_i c_i = 0$ implies that the function $\sum_i \lambda_i \theta_{c_i}$ is zero in S (use the decomposition lemma); deduce from this remark and from $c$ that the mapping $c \mapsto \theta_c$ extends to an isomorphism $x \mapsto \theta_x$ of the normed space $C_u$ onto the normed space $\mathscr{C}(S; \mathbf{R})$ of finite continuous numerical functions in S, such that $(\theta_x)^+ = \theta_{x^+}$. (With the aid of $c$), show that for every $x \in C_u$ and every $\varepsilon > 0$, there exists a growing sequence $(\lambda_i)_{0 \leq i \leq n}$ of real numbers such that $\lambda_i - \lambda_{i-1} \leq \varepsilon$ and that

$e$ ) For S to be finite, it is necessary and sufficient that $C_u$ be of finite dimension; deduce from this (with the aid of exerc. 12) that every completely reticulated space of finite dimension is isomorphic to a product space $\mathbf{R}^n$ (cf. § 2, exerc. 7).

$f$ ) Show that S is an extremely disconnected space (Top. gén., Chap. I, 3\textsuperscript{e} ed., § 11, exerc. 21) (use the fact that $\mathscr{J}$ is a complete lattice). A compact extremely disconnected space is called a stonian space. Show that if X is a stonian space, for every numerical function $f$ lower semi-continuous in X, the upper semi-continuous regularization g of $f$ (Top. gén., Chap. IV, 3\textsuperscript{e} ed., § 6, No. 2) is continuous (for every $a < g(x)$, show that $x$ cannot be adherent to the (open) set of the $y$ such that $g(y) < a$, by observing that at a point $z$ adherent to this set, one has $f(z) \leq a$). Deduce that $\mathscr{C}(X; \mathbf{R})$ is then a completely reticulated space. Show that when the stonian space X is infinite, the supremum in $\mathscr{C}(X; \mathbf{R})$ of a bounded above set is not necessarily equal to its upper envelope (consider an open non-closed set in X); nevertheless these two functions are equal in the complement of a meagre set (consider the set of points where their difference is $\geq 1/n$).

$g$ ) Let X be a stonian space; show that if $f$ is a bounded numerical function, defined in the complement of a rare subset M of X, and continuous in $X - M$, then $f$ extends by continuity to all of X (use the fact that $\mathscr{C}(X; \mathbf{R})$ is completely reticulated). Deduce that the set $\mathscr{Q}(X; \overline{\mathbf{R}})$ of numerical functions $f$ finite or not, continuous in X, and such that $f^1(+\infty)$ and $f^1(-\infty)$ are rare in X is a completely reticulated vector space.

$h$ ) Show that the band $B_u$ generated by $u$ in E is isomorphic (as an ordered space) to an isolated subspace (exerc. 4) of $\mathscr{Q}(X; \overline{\mathbf{R}})$ (consider an element $f \geq 0$ of $\mathscr{Q}(X; \overline{\mathbf{R}})$ as the supremum of the elements $\inf(f, n)$).