Exercise 17, § 4

The hypotheses and notations are those of Ex. 15 (X, p. 58).

a) For a maximal ideal $\mathscr{F}(x)$ to be real, it is necessary and sufficient that for every infinite sequence $(g_n)$ of elements of $\mathscr{F}(x)$, the intersection of the sets $g_n^{-1}(0)$ is nonempty. (Use Ex. 16 b); to see that the condition is necessary, prove that if the intersection of the $g_n^{-1}(0)$ is empty, the function $f(y) = \sum_{n=0}^\infty \inf(g_n(y), 2^{-n})$ is continuous, invertible in $A$ and tends to 0 when $y$ tends to $x$ while remaining in $X$; note that $f(y) \leq 2^{-m}$ in the intersection of the $g_k^{-1}(0)$ for $k \leq m$, and use Ex. 15 a) (X, p. 58) and the fact that the $V(g)$, where $g \in \mathscr{F}(x)$, form a fundamental system of neighbourhoods of $x$ in $\beta X$.

b) One says that $X$ is replete if the only real maximal ideals $\mathscr{F}(x)$ are those for which $x \in X$. Prove that every completely regular Lindelöf space (IX, p. 75) is replete (use a)). A completely regular weierstrassian space is replete only if it is compact.

c) Let $\upsilon X$ be the set of the $x \in \beta X$ such that $\mathscr{F}(x)$ is real; prove that every numerical function $f \in A$ can be extended by continuity to $\upsilon X$, in such a way that $C(X; \mathbf{R})$ and $C(\upsilon X; \mathbf{R})$ can be canonically identified. Prove that the subspace $\upsilon X$ of $\beta X$ is replete (notice that if $f$ is invertible in $C(X; \mathbf{R})$, its extension by continuity to $\upsilon X$ is invertible in $C(\upsilon X; \mathbf{R})$, and that one has $\beta(\upsilon X) = \beta X$); one says that $\upsilon X$ is the repletion of $X$. For $\beta X = \upsilon X$, it is necessary and sufficient that $X$ be weierstrassian.

d) Prove that $\upsilon X$ is the completion space of $X$ when $X$ is endowed with the least fine uniform structure making all the functions $f \in A$ uniformly continuous. (Notice that a Cauchy filter $\mathscr{F}$ on $X$ for this uniform structure converges to a point $x \in \beta X$; if $x$ did not belong to $\upsilon X$, there would exist a function $f \in A$ which is unbounded in a set of $\mathscr{F}$).

e) Deduce from d) that for every continuous mapping $u : X \to Y$ from a completely regular space $X$ into a replete space $Y$, $u$ can be extended by continuity to a mapping from $\upsilon X$ into $Y$. The space $\upsilon X$ is therefore a solution of a universal mapping problem where $\Sigma$ is the structure of replete space, the $\alpha$-mappings and the morphisms being the continuous mappings (E, IV, p. 22).

f) Now prove that every closed subspace of a replete space is replete; every product of replete spaces is replete; in a separated space, every intersection of replete subspaces is a replete subspace. (For proving, for example, that a closed subspace $X$ of a replete space $Y$ is replete, consider the extension $\bar{j}$ to $\upsilon X$ of the canonical injection $j : X \to Y$, and remark that $\bar{j}^{-1}(X) = X$; deduce that $X$ is closed in $\upsilon X$. Analogous reasonings in the two other cases).

g) Deduce from b), d) and f) that for a completely regular space to be replete, it is necessary and sufficient that it be homeomorphic to a closed subspace of a product space $\mathbf{R}^l$.