Exercise 17, § 4

We retain the hypotheses and notation of Exercise 15.

a) A maximal ideal $\mathfrak{J}(x)$ is real if and only if, for every infinite sequence $(g_n)$ of elements of $\mathfrak{J}(x)$, the intersection of the sets $\overline{g_n}(0)$ is not empty. [Use Exercise 16 b); to show that the condition is necessary show that, if $\bigcap_n \overline{g_n}(0) = \varnothing$, the function $f(y) = \sum_{n=0}^\infty \inf (g_n(y), 2^{-n})$ is continuous, is a unit in $A$ and tends to 0 as $y$ tends to $x$ while remaining in $X$; note that $f(y) \leq 2^{-m}$ for $y \in \bigcap_{k \leq m} \overline{g_k}(0)$, and use Exercise 15 a) and the fact that the $V(g)$, where $g \in \mathfrak{J}(x)$, form a fundamental system of neighbourhoods of $x$ in $\beta X$.]

b) $X$ is said to be real-compact if the only real maximal ideals $\mathfrak{J}(x)$ are those for which $x \in X$. Show that every completely regular Lindelöf space (Chapter I, § 9, Exercise 15) is real-compact [use a)]. A completely regular pseudo-compact space is not real-compact unless it is compact.

c) Let $\nu X$ be the set of all $x \in \beta X$ such that $\mathfrak{J}(x)$ is real. Show that every continuous real-valued function $f \in A$ can be extended by continuity to $\upsilon X$, so that $C(X; \mathbf{R})$ and $C(\upsilon X; \mathbf{R})$ can be canonically identified. Show that the subspace $\upsilon X$ of $\beta X$ is real-compact [note that if $f$ is a unit in $C(X; \mathbf{R})$, then its continuous extension to $\upsilon X$ is a unit in $C(\upsilon X; \mathbf{R})$ and that $\beta(\upsilon X) = \beta X$]. $\upsilon X$ is called the real-compactification of $X$.

d) Show that $\upsilon X$ is the completion of $X$ with respect to the coarsest uniformity on $X$ for which all the functions $f \in A$ are uniformly continuous. (Note that a Cauchy filter $\mathfrak{F}$ on $X$ with respect to this uniformity converges to a point $x \in \beta X$; if $x \notin \upsilon X$, there would be a function $f \in A$ which was unbounded in a set of $F$.)

e) Deduce from $d$) that if $u : X \to Y$ is any continuous mapping of a completely regular space $X$ into a real-compact space $Y$, then $u$ can be extended by continuity to a mapping $\upsilon X \to Y$. The space $\upsilon X$ is therefore the solution of the universal mapping problem in which $\Sigma$ is the structure of a real-compact space, and the $\alpha$-mappings and morphisms are continuous mappings (Set Theory, Chapter IV, § 3, no. 1).

f) Show that every closed subspace of a real-compact space is real-compact, that every product of real-compact spaces is real-compact, and that in a Hausdorff space any intersection of real-compact subspaces is real-compact. [For example, to show that a closed subspace $X$ of a real-compact space $Y$ is real-compact, consider the extension $\bar{j}$ to $\upsilon X$ of the canonical injection $j : X \to Y$, and note that $\bar{j}^{-1}(X) = X$; deduce that $X$ is closed in $\upsilon X$. The proofs are similar in the other two cases.]

g) Deduce from $b$), $d$) and $f$) that a completely regular space is real-compact if and only if it is homeomorphic to a closed subspace of a product $\mathbf{R}^l$.