All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $. Our main result shows that, given a cardinal $\kappa $, the following conditions are equivalent:

• • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $.

• • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$.

$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$

$2^\kappa $

$\kappa $

$\kappa $

$\mathsf {Exp}(X)$

The precise condition on a completely regular space $X$ for every character on $C\left( X \right)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.

The paper examines the classes K1 and Γ1 of Hausdorff uniform spaces which are Gδ-closed in their Samuel compactifications, or completions. It is shown that the classes are epi-reflective, the reflections K1 and Γ are described, K1 and Γ1 are represented as epi-reflective hulls, membership in the classes is described by fixation of certain zero-set ultrafilters, and it is shown that k1 = Γ1 exactly on spaces without discrete sets of measurable power. The results include familiar facts about realcompact and topologically complete topological spaces and are closely connected with the theory of metric-fine uniform spaces.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 50, 54 E 15, 18 A 40; secondary 54 B 05, 54 B 10, 54 C 10, 54 C 30.

For a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.