Topology Atlas Document # taic-16.htm | Production Editor: R. Flagg
Topology Atlas Invited Contributions, vol. 2, issue 1 (1996), 27-30.

Weak forms of realcompactness

by

John Schommer

(University of Tennessee at Martin)

So much work has been done in the area of generalizing realcompactness that a short exhaustive introduction to this field is impossible. Realcompactness itself may be characterized in a number of different ways, and each of these invites a weakening of one kind or another. Perhaps a theorem listing just a few of the equivalences to realcompactness would be a nice starting point for a subsequent discussion of weakenings.

Theorem. Let $X$ be Tychonoff, let $\beta X$ represent the Stone-Cech compactification of $X$, and let $\upsilon X$ be its Hewitt-Nachbin realcompactification. Then the following are equivalent to realcompactness:

1. Every ultrafilter of zero-sets with the countable intersection property is fixed.
2. For each point $p$ in $\beta X - X$, there exists a continuous function $f$ on $\beta X$ such that $f\left(p\right)=0$ and $f$ is positive on $X$.
3. If $C$ρ (X) represents the collection of continuous functions with realcompact support, then $C$ρ(X) = C(X).
4. $X$ admits a compatible complete uniformity and contains no closed discrete subspace of measurable cardinality.
5. $X = \upsilon X$.
6. Each 2-valued Baire measure in $X$ is $\tau$-additive.

Our first group of weakenings of realcompactness derive from equivalence (1). If we replace the zero-sets of (1) with regular closed sets we get the weakening called almost realcompact. If we substitute closed sets for those zero-sets we arrive at a-realcompact. Frolik did the early work with almost realcompactness in the early 60's, and Dykes the work with a-realcompactness towards the end of that decade. In a 1980 paper Archangel'skii produced a generalization of a-realcompact spaces he called pure spaces. Pure spaces were further generalized by Sekai in 1986 and given the name neat spaces. The definitions of both pure and neat spaces are well beyond the scope of this little survey.

Another line of generalization derives from statement (2) of our theorem. In a 1970 paper Dykes defined c-realcompactness essentially by weakening the continuity criterion on $f$ to normal lower semicontinuity. Dykes demonstrates in that paper that almost realcompact spaces are c-realcompact. Swardson and Szeptycki announced a generalization of c-realcompactness in 1995, calling a space p-realcompact iff every zero-set of $\beta X$ that meets $\beta X - X$ meets $\beta X - \upsilon X$.

Rings of continuous functions were used successfully by Mandelker in 1971 and by Johnson and Mandelker in 1973 to explore still other ways to generalize realcompactness. If we let $C$K(X) denote the ring of continuous functions with compact support, $C$ψ(X) the ring of functions with pseudocompact support, and $I\left(X\right)$ the intersection of all free maximal ideals of $C\left(X\right)$, then $X$ is said to be $\mu$-compact if $C$K(X)=I(X), $\psi$-compact if $C$K(X)=Cψ(X), and $\eta$-compact if $C$ψ(X)=I(X). As promised, each of these turn out to generalize realcompactness, and Johnson and Mandelker showed that all these concepts are distinct.

Yet another trail of generalizations can be considered as descending from our fourth characterization of realcompactness. We call a space completely uniformizable (or Dieudonne complete or topologically complete), if it admits a compatible complete uniformity. Shirota demonstrated that realcompact spaces are completely uniformizable in 1952. In fact, if one assumes that there are no measurable cardinals, the notions are equivalent. A whole string of increasingly weaker "isocompact" properties turns out to generalize the concept of completely uniformizable. A space is said to be hyperisocompact [strongly isocompact] (isocompact) if every relatively pseudocompact [strongly relatively pseudocompact] (countably compact) closed subset is compact. The serious study of isocompact spaces appears to have begun with Bacon's work in the early 70's. Sekai has shown that neat spaces are isocompact.

In a posthumously published paper of 1992, Blair and van Douwen generalized property (5) of our theorem by defining a space to be nearly realcompact if $\beta X - \upsilon X$ is dense in $\beta X - X$. Nearly realcompact spaces prove to be precisely the $\eta$-compact spaces investigated by Mandelker. One of the alternate characterizations of nearly realcompactness -- a space in which every relatively pseudocompact cozero subset is $\sigma$-compact -- also makes nearly realcompactness a generalization of hyperisocompactness. (It is perhaps an interesting aside that if we assume MA+$¬$CH, we can claim that every perfectly normal space is nearly realcompact.)

Finally, we follow a measure-theoretic track away from the realcompactness description of (6). The result of parallel developments by Gardner (1975) and Rice and Reynolds (1972), this track is especially difficult to lay down in a brief survey article. Suffice it to say that a space is called weakly Borel-complete if each 2-valued Borel measure in $X$ is weakly $\tau$-additive, and is closed-complete (or $\alpha$-realcompact) if each 2-valued regular Borel measure in $X$ is $\tau$-additive. These measure-theoretic characterizations have rather nice equivalences involving ultrafilters. It can be shown that a space is weakly Borel-complete iff every Borel-ultrafilter with the countable intersection property converges, and closed-complete iff every closed ultrafilter with the countable intersection property is fixed. To put a fine point on it, closed-completeness is equivalent to the a-realcompactness we were discussing near the start of this article.

In a 1980 article of Blum and Swaminathan, a table was begun summarizing many of the relationships mentioned in this brief survey. Following their lead, all of the various weakenings of realcompactness have been collected into a website, where the properties themselves and the relationships they have to each other have been fully referenced. A dynamic site, new generalizations and new relationships will be added as they are discovered.

Received by the editors: February 15, 1996.