Theory and Applications of Categories, Vol. 16, No. 21, 2006, pp. 558–584.

COMPACTIFICATIONS, C(X) AND RING EPIMORPHISMS

W.D. BURGESS AND R. RAPHAEL

Abstract. Given a topological space X, K(X) denotes the upper semi-lattice of its (Hausdorff) compactifications. Recent studies have asked when, for αX ∈ K(X), the restriction homomorphism ρ: C(αX) → C(X) is an epimorphism in the category of commutative rings. This article continues this study by examining the sub-semilattice, K_{epi}(X), of those compactifications where ρ is an epimorphism along with two of its subsets, and its complement K_{nepi}(X). The role of K_{z}(X) ⊆ K(X) of those αX where X is z-embedded in αX, is also examined. The cases where X is a P -space and, more particularly, where X is discrete, receive special attention.

1. Introduction

Throughout, “topological space” will be taken to mean a completely regular Hausdorff topological space. For a topological space X, C(X) denotes, as usual, the ring of continuous real valued functions on X. A compactification of a space X will be a compact Hausdorff space αX along with a continuous injection X → αX whose image is dense in αX. The space X will be identified with its image in αX. Following the terminology of [C], the complete upper semi-lattice of equivalence classes of (Hausdorff) compactifications of X will be denoted by K(X). When we write αX ∈ K(X) we mean that αX is a representative of a class in K(X). The maximal element of K(X) is the Tychonoff compactification, usually known as the Stone-Čech compactification, βX.

Several recent articles (e.g., [BBR], [BRW], [HM2], [S]) have discussed the question of when the restriction mapping

ρ: C(αX) → C(X) , for αX ∈ K(X) ,

COMPACTIFICATIONS, C(X) AND RING EPIMORPHISMS  559

is an epimorphism. (Recall that in a category C, a morphism f is an epimorphism if given gf = hf then g = h.) Of course, the answer depends on the category involved: the objects C(Y), Y a topological space, live in many important categories. Some of these are: CR, the category of commutative rings, the category R/N of reduced commutative rings (i.e., rings with no non-zero nilpotent elements), and various categories of partially ordered groups and rings, such as that of archimedean f-rings. Epimorphisms in this last category are called C-epics in [HM2]. If C(αX) → C(X) is a CR-epimorphism then it is also a R/N-epimorphism and a C-epic.

The authors each acknowledge the support of grants from the NSERC of Canada.

Received by the editors 2005-11-21 and, in revised form, 2006-08-25.

Transmitted by Walter Tholen. Published on 2006-09-05.

2000 Mathematics Subject Classification: 18A20, 54C45, 54B40.

Key words and phrases: epimorphism, ring of continuous functions, category of rings, compactifications.

\begin{matrix}c \\ ◯ \end{matrix} W.D. Burgess and R. Raphael, 2006. Permission to copy for private use granted.

It is easily seen that the question of when an inclusion X → Y of topological spaces, X dense in Y, induces a CR-epimorphism can be reduced to the case where Y is a compactification of X. To see this, suppose X is dense in Y. Then, βY is a compactification of X which we call αX. Now, C(αX) → C(Y) is a CR-epimorphism since Y is C^{∗}-embedded in αX = βY. The composition C(αX) → C(Y) → C(X) shows that C(Y) → C(X) is a CR-epimorphism if and only if C(αX) → C(X) is.

The aim of this article will be to look at when, for αX ∈ K(X), ρ: C(αX) → C(X) is a CR-epimorphism. The word “epimorphism” without qualifier will always mean CRepimorphism in what follows.

The collection of those αX ∈ K(X) for which X is z-embedded in αX is denoted K_{z}(X); those for which C(αX) → C(X) is an epimorphism is called K_{epi}(X). The main theme is to discuss K_{z}(X) and K_{epi}(X), along with two subsets of the latter. Of particular interest is the case where X is a P -space and, more specially, an uncountable discrete space. More details are found at the end of this introduction.

Before outlining the results, we establish some notation and terminology to be used throughout.

I. Compactifications. If αX ∈ K(X) for some space X, then the canonical continuous surjection βX → αX, fixing X, is denoted σ_{α}, or just σ; its restriction to βX − X is τ_{α} or τ: βX − X → αX − X. We denote by I_{α} = {a ∈ αX − X | |τ^{−1}(a)| > 1} and M_{α} = τ^{−1}(I_{α}). Note that X is C^{∗}-embedded in αX − I_{α}.

As usual, C^{∗}(X) is the subring of C(X) of bounded functions. If αX ∈ K(X) and f ∈ C^{∗}(X) extends to αX, the unique extension is denoted f^{α}. When αX ∈ K(X), C_{α} = {g ∈ C^{∗}(X) | g^{β} factorsas g^{β} = g̃σ, g̃ ∈ C(αX)} = {g ∈ C^{∗}(X) | g^{α} exists}. Then, C_{α} is a uniformly closed algebra which separates points from closed sets of X (see, e.g., [Wa]); it is the restriction of C(αX) to X. We will also encounter S_{α} = {h ∈ C_{α} | coz h = X}, a multiplicatively closed subset of the non-zero divisors of C_{α}.

The complete upper semi-lattice K(X) is a complete lattice exactly when X is locally compact ([C, Theorem 2.19]). In this case, the minimal element is the one-point compactification, ωX.

As already mentioned, K_{z}(X) = {αX ∈ K(X) | X z-embedded in αX} (i.e., every zeroset of X is the restriction of one in αX) and K_{epi}(X) = {αX ∈ K(X) | C(αX) → C(X) is an epimorphism }. Both are easily seen to be complete upper sub-semi-lattices of K(X) and both contain βX. The complement of K_{epi}(X) in K(X), when non-empty, is denoted K_{nepi}(X). When X is locally compact and ωX ∈ K_{epi}(X), then K(X) = K_{epi}(X). The situation when K(X) = K_{epi}(X) is studied in detail in [BRW]. In particular, when X = N, the countable discrete space, then ωN ∈ K_{epi}(N). In contrast, if X is uncountable and discrete, ωX ∉ K_{epi}(X), an observation which motivated the question: When X is uncountable discrete, to what extent can K_{epi}(X) be described? This question was the starting point of this article.

II. Rings of quotients. There is an extensive discussion about rings of quotients of rings of the form C(Y) in [FGL]. We follow the notation of that monograph and write Q(Y) for the complete ring of quotients Q(C(Y)), and Q_{cl}(Y) for the classical ring of quotients Q_{cl}(C(Y)). Of particular importance for us is the fact that if S is a multiplicatively closed set of non-zero divisors of a commutative ring R, the homomorphism R → RS^{−1} is a epimorphism. Of course, R → Q_{cl}(R) is an instance. Another example is the embedding C^{∗}(X) → C(X) which is always an epimorphism since, for any f ∈ C(X),

f = \frac{f}{1 + f^{2}} (\frac{1}{1 + f^{2}} )^{−1} ,

showing that C(X) = C^{∗}(X)S^{−1}, where S is the set of bounded continuous functions non-zero everywhere on X.

III. Epimorphisms and zig-zags. If A ⊆ B is an inclusion of commutative rings, then the inclusion is an epimorphism if and only if, for each b ∈ B there is, for some n ∈ N, an equation b = GHK, where (i) G, H and K are matrices over B of size 1 × n, n × n and n × 1, respectively, and (ii) GH, H and HK are matrices over A. Such a matrix equation is called an n×n zig-zag over A. (This is due to Mazet and quoted in [BBR] and [BRW].) It follows that if A is infinite then |B| = |A|; a fact which will be used below.

Any undefined terminology about C(X) conforms with that of the text by Gillman and Jerison ([GJ]) and about compactifications with that of the monograph by Chandler ([C]). Finally, if V ⊆ X, then χ_{V} is the characteristic function of V.

IV. A summary of results. Section 2 prepares the way for the paper by presenting tools concerning: (I) Reducing properties to C^{∗}-embedded subsets of X; (II) Constructing compactifications with desirable properties; (III) Relating αX ∈ K_{z}(X) to C_{α}S_{α}^{−1}; (IV) Relating αX ∈ K_{z}(X) to the Hewitt realcompactification υX of X, showing, in particular (2.8), that K_{z}(X) = K_{z}(υX); (V) Describing zero-sets of βX lying in βX − X (these zero-sets are an important feature in everything else). For αX ∈ K(X), a zero-set of αX lying in αX − X will be called an α-zero-set; these are the zero-sets z(g^{α}), for g ∈ S_{α}.

Section 3 specializes X to a P -space (a space in which zero-sets are open). The two key facts established here are that (i) K_{epi}(X) ⊆ K_{z}(X) (3.1), and (ii) αX ∈ K_{epi}(X) if and only if C(X) = Q_{cl}(αX), a regular ring, (3.4), a result to be strengthened in Section 4. As a consequence, when X is a P -space, K_{epi}(X) = K_{epi}(υX). It is also shown (3.5) that when X is a P -space and αX ∈ K(X), ρ: C(αX) → C(X) is an R/N-epimorphism if and only if it is a CR-epimorphism. It is not known if this statement is true for arbitrary spaces.

Section 4 looks at special subsets of K_{epi}(X) related to fractions. The easiest case is where αX ∈ K(X) has the property that for some h ∈ S_{α}, M_{α} ⊆ z(h^{β}). When this happens, αX ∈ K_{epi}(X) (4.1) and we call the set of all such compactifications K_{epi}^{1}(X). Another, potentially larger, family of elements of K_{epi}(X), called K_{epi}^{f}(X), is the set of compactifications αX so that C(X) = C_{α}S_{α}^{−1}. We always have K_{epi}^{f}(X) ⊆ K_{z}(X). When X is a P -space, K_{epi}(X) = K_{epi}^{f}(X) (4.5 (C)). The first of several results which give situations where αX ∈ K_{epi}^{f}(X) if and only if αX ∈ K_{epi}^{1}(X) is (4.9); in particular, if I_{α} is separable, then αX ∈ K_{epi}^{f}(X) if and only if it is in K_{epi}^{1}(X).

Section 5 looks at lattice properties of the various subsets of K(X) which are related to epimorphisms. It is shown (5.2) that, except in some trivial (from this point of view) cases the upper semi-lattices K_{z}(X), K_{epi}^{f}(X) and K_{epi}(X) do not have minimal elements. However, (5.3) presents a situation, with a locally compact X, in which a family of elements K_{epi}^{1}(X) has its meet also in K_{epi}^{1}(X). The section ends with a construction showing that, for X uncountable and discrete, K_{nepi}(X) is not closed under finite joins (5.6).

Section 6 deals mostly with the special case where X is uncountable and discrete. It starts with a description (6.2) of the β-zero-sets as closures of unions of sets of the form cl_{βX} V − V, V ⊆ X, countable. This tool allows us to show (6.3) that the join of a countable family from K_{epi}^{1}(X) is again in K_{epi}^{1}(X); examples ((5.4) and (6.5)) show that “countable” is neither necessary nor sufficient. The strongest statement about joins of elements of K_{epi}^{1}(X) is (6.6), a characterization of when a union of β-zero-sets is contained in a β-zero-set. It is followed by corollaries and examples. The section ends with a description of some elements of K_{nepi}(X) using cardinalities in various ways.

Section 7 asks some of the many questions that remain about K_{z}(X), K_{epi}(X), K_{epi}^{f}(X) and K_{nepi}(X).

2. On constructing compactifications, on fractions and on the realcompactification.

This section is divided into five subsections; it has various items which will be used as tools later in the article. Before giving some remarks on the construction of compactifications of general spaces and of discrete spaces, we show that the property of a compactification being in K_{z}(X) or in K_{epi}(X) is inherited by C^{∗}-embedded subsets.

I. Hereditary properties.

2.1. Proposition. For any space X and V ⊆ X, suppose that V is C^{∗}-embedded in X. (i) If αX ∈ K_{z}(X) then V is z-embedded in cl_{αX} V. (ii) If αX ∈ K_{epi}(X) and we write cl_{αX} V = γV ∈ K(V) then, γV ∈ K_{epi}(V).

Proof. (i) is clear since a zero-set in V is the intersection of a zero-set of X with V since V is C^{∗}-embedded in X.

To show (ii), we look at the diagram (with restriction maps):

C(αX) → C(X)
↓  ↓
C(γV) → C(V)

The upper arrow is an epimorphism and we need that the arrow on the right be an epimorphism to show that C(γV) → C(V) is one. However, C^{∗}(X) → C^{∗}(V) → C(V) is a surjection followed by an epimorphism, while this same homomorphism can also be written C^{∗}(X) → C(X) → C(V). Hence, C(X) → C(V) is an epimorphism, as required.

II. Constructions of compactifications. One of the ways in which compactifications are constructed is via subalgebras of C^{∗}(X), as described in [C, Chapter 2]. We call a subalgebra C of C^{∗}(X) a comp-algebra (“suitable for building a compactification”) if for each closed subset V ⊆ X and x ∈ X − V there is f ∈ C with f(x) ∉ cl_{R} f(V). (The algebra C “separates points from closed sets”.) Such algebras are easy to find when X is discrete but we first look at general spaces.

Finite collections of compact sets in βX − X (and, even, in αX − X, for αX ∈ K(X)) give rise to compactifications, as we will see. It follows from the next proposition that when T is a compact set in βX − X then there is a compactification so that σ(T) is one point and σ is one-to-one on βX −T; this compactification will be called α_{T} X. When the compact sets are β-zero-sets the compactifications constructed below will be in K_{epi}(X) (see Proposition 4.1). In the next proposition, “∧” means “meet” in the poset K(X).

2.2. Proposition. Let X be any space and αX ∈ K(X). Suppose T_{i} ⊆ αX, i = 1, …, k, are pairwise disjoint compact subsets of αX − X. (i) There is a compactification, γX ≤ αX, which identifies only the points of each T_{i}. (ii) Set K_{i} = τ^{−1}(T_{i}), i = 1, … k; then, γX = αX ∧ ⋀_{i=1}^{k} α_{K_{i}}X. (iii) Moreover, when each T_{i} is a zero-set and αX ∈ K_{z}(X), then γX ∈ K_{z}(X).

Proof. (i) is essentially [C, Lemma 5.18]. It follows by repeated application of the fact that αX is a normal space.

(ii) is [C, Theorem 2.18] since C_{γ} = C_{α} ∩ ⋂_{i=1}^{k} C_{K_{i}}.

(iii). We let T_{i} = z(f_{i}), f_{i} ∈ C(αX), i = 1, …, k. If V ⊆ X is a zero-set then there exists g ∈ C(αX) with z(g) ∩ X = V. Moreover, (gf_{1} ⋯ f_{k})|_{X} ∈ C_{γ} and z((gf_{1} ⋯ f_{k})|_{X})^{γ}) ∩ X = V, since f_{1} ⋯ f_{k} is non-zero on X,.

Given a compactification αX ∈ K(X) there is a way of constructing some compactifications above αX in K(X). Special cases of it will be used below.

2.3. Construction. For any space X, given αX ∈ K(X) and a subset B ⊆ I_{α}, there is a compactification α_{(B)}X with αX ≤ α_{(B)}X and a factorization τ_{α} = μτ_{α_{(B)}} such that (i) μ is one-to-one on μ^{−1}(B) and, for each b ∈ B, τ_{α_{(B)}}(τ_{α}^{−1}(b)) = {μ^{−1}(b)}; and (ii) μ^{−1}(B) is dense in I_{α_{(B)}}.

Proof. Let C = {f ∈ C^{∗}(X) | f^{β} is constant on τ_{α}^{−1}(b) for each b ∈ B}. It follows that C is a comp-algebra because C_{α} ⊆ C. Let α_{(B)}X be the compactification corresponding to C (in fact C = C_{α_{(B)}}, since it is uniformly closed). We have C_{α} ⊆ C and, hence, αX ≤ α_{(B)}X in K(X). Hence, there is a factorization τ_{α} = μτ_{α_{(B)}}.

By choice of C, for each b ∈ B, τ_{α_{(B)}}(τ^{−1}(b)) is a single point. Moreover, if b ≠ b′ in B, there is f ∈ C_{α} with f^{α}(b) ≠ f^{α}(b′). Hence, τ_{α_{(B)}}(τ_{α}^{−1}(b)) ≠ τ_{α_{(B)}}(τ_{α}^{−1}(b′)). In other words, μ is one-to-one on μ^{−1}(B).

Let c ∈ I_{α_{(B)}} − μ^{−1}(B). Suppose there is an open neighbourhood U of c in α_{(B)}X which does not meet μ^{−1}(B). Consider τ_{α_{(B)}}^{−1}(U), an open set of βX containing τ_{α_{(B)}}^{−1}(c) and not meeting L = ⋃_{b∈B} τ_{α_{(B)}}^{−1}(μ^{−1}(b)). Pick p ≠ q in τ_{α_{(B)}}^{−1}(c). There is g ∈ C(βX) such that g(p) = 1, g(q) = 2 and g(cl_{βX} L) = {0}. Then, g|_{X} ∈ C but g is not constant on τ_{α_{(B)}}^{−1}(c), which is impossible.

In what follows we have in mind the case where X is infinite discrete but some more generality is available. We will look at spaces X which have a base of compact open sets; for example, an infinite sum (disjoint union) of copies of ωN.

2.4. Proposition. Let X be an infinite space which has a basis of compact open sets. If C is any subalgebra of C^{∗}(X) which contains the characteristic functions of the compact open subsets of X, then C is a comp-algebra. Conversely, if αX ∈ K(X), C_{α} contains the characteristic functions of the compact open subsets of X.

Proof. Given a closed subset V ⊆ X and x ∈ X − V, then there is a compact open subset U ⊆ X − V containing x; its characteristic function, χ_{U}, separates V from x.

In the converse, for any clopen U in X, U is a clopen subset of αX since X is locally compact. Hence, χ_{U} ∈ C(αX).

At one extreme in Proposition 2.4, the subalgebra C of C^{∗}(X) generated by the characteristic functions of the compact open sets of X yields the compactification ωX, while C^{∗}(X) itself gives rise to βX. The proposition has a useful corollary. A family {S_{ν} | ν ∈ E} of non-empty subsets of a space Y is called separated ([KV, page 688]) if there is a family of pairwise disjoint open sets {U_{ν} | ν ∈ E} with S_{ν} ⊆ U_{ν} for ν ∈ E.

2.5. Corollary. Let X be an infinite space as in Proposition 2.4. Let {S_{ν} | ν ∈ E} be a separated family of closed sets in βX with each S_{ν} ⊆ βX − X and |S_{ν}| > 1. Then there is αX ∈ K(X) such that I_{α} = {a_{ν} | ν ∈ E} and τ^{−1}(a_{ν}) = S_{ν}, for ν ∈ E.

Proof. Put C = {f ∈ C^{∗}(X) | f^{β} is constant on each S_{ν}}. This algebra satisfies the criterion of Proposition 2.4. However, it must be shown that the resulting compactification, αX, behaves as predicted. Let {U_{ν} | ν ∈ E} be a family a open sets as in the definition of a separated family. Given μ ∈ E, the disjoint closed sets S_{μ} and βX − U_{μ} can be separated in the normal space βX, say by f_{μ} which is 1 on S_{μ} and zero on βX − U_{μ}; f_{μ}|_{X} ∈ C. Hence, the images of the S_{ν} are all distinct in αX. Next, if p ∉ ⋃_{E} S_{ν}, for each μ ∈ E we can separate p from S_{μ}. Indeed, if p ∉ U_{μ}, then f_{μ} ∈ C has f_{μ}(p) = 0 and f_{μ}(S_{μ}) = {1}. If p ∈ U_{μ}, then U_{μ} can be replaced by U_{μ} −{p} to get a function. Similarly, any two elements of βX not in ⋃_{E} S_{ν} can be separated by an element of C, showing that M_{α} = ⋃_{E} S_{ν}.

If X is as in Proposition 2.4 then any discrete family in βX of closed sets in βX − X ([E, page 193]) will work in Corollary 2.5 since βX is collectionwise normal ([E, page 214, definition and Theorem 6]).

III. Fractions and K_{z}(X). We will see that the ring Q_{cl}(αX) and its subring C_{α}S_{α}^{−1} play a role in our study of K_{epi}(X). However, fractions also show up when dealing with K_{z}(X), a topic we take up now. It will be seen in Section 3 that when X is a P -space, αX ∈ K_{epi}(X) implies αX ∈ K_{z}(X).

For any space X, there is a characterization of when αX ∈ K(X) is in K_{z}(X): [HM2, Theorem 8.2 (b) ⇔ (c)]. We suppose, as usual, that the compactification is given by τ: βX − X → αX − X. Recall that S_{α} = {g ∈ C_{α} | coz g = X}.

2.6. Lemma. [HM2, Theorem 8.2] A compactification αX is in K_{z}(X) if and only if for each h ∈ C(βX) there is a countable subset D_{h} ⊆ S_{α} so that if for p ≠ q in βX, h(p) ≠ h(q) while τ(p) = τ(q), then, for some g ∈ D_{h}, {p, q} ⊆ z(g^{β}).

Proof. Note that the countable family in [HM2, Theorem 8.2] has been expanded to be closed under finite products.

This result will now be translated into a statement about fractions.

2.7. Lemma. Let X be any space and suppose that αX ∈ K_{z}(X). For any f ∈ C(βX) and any finite subset {a_{1}, …, a_{m}} of I_{α}, there is g ∈ S_{α} such that fg^{β} is constant on each τ^{−1}(a_{i}), i = 1 …, m. In particular, there is g ∈ S_{α} such that ⋃_{i=1}^{m} τ^{−1}(a_{i}) ⊆ z(g). Hence, if B = {a_{1}, …, a_{m}}, as in (2.3), there is h ∈ C_{α(B)} with f|_{X} = hg^{−1}.

Proof. If f is already constant on each τ^{−1}(a_{i}), we can take g = 1. Otherwise, let F = {i | 1 ≤ i ≤ m and f is not constant on τ^{−1}(a_{i})}. For i ∈ F, let g_{i} ∈ S_{α} be such that g_{i}^{β} is zero on τ^{−1}(a_{i}) (using the criterion quoted above). Then g = ∏_{i∈F} g_{i} does what is required.

For the second part, it suffices to choose pairs of distinct elements p_{i}, q_{i} from τ^{−1}(a_{i}), i = 1, …, m, and f ∈ C(βX) so that f(p_{i}) ≠ f(q_{i}), i = 1, …, m.

IV. The Hewitt realcompactification and K_{z}(X). The following are tools to help us avoid requiring that a space be realcompact. Lemma 2.6 will play a role. A space X is C-embedded in its Hewitt realcompactification υX. If f ∈ S_{β} then f^{−1} ∈ C(X) and, hence, f^{−1} extends to υX. This shows that for any f ∈ S_{β}, z(f^{β}) ∩ υX = ∅ ([GJ, Theorem 8.4]).

2.8. Proposition. Let X be any space and αX ∈ K_{z}(X). (i) For any a ∈ I_{α}, τ^{−1}(a) ⊆ βX − υX. (ii) There is Υ, X ⊆ Υ ⊆ αX so that X → Υ is a copy of the Hewitt realcompactification of X in αX. (iii) The restriction of σ_{α} to υX is a homeomorphism onto Υ. Hence, Υ can be identified with υX. With this identification, αX ∈ K_{z}(υX). Moreover, K_{z}(X) = K_{z}(υX).

Proof. (i) Suppose, for some a ∈ I_{α}, that τ^{−1}(a) ∩ υX ≠ ∅. Pick p ∈ τ^{−1}(a) ∩ υX and q ∈ τ^{−1}(a), q ≠ p. By Lemma 2.6, there is a g ∈ S_{α} with {p, q} ⊆ z(g^{β}). However, this is impossible.

(ii) By [BH, Corollary 2.6 (a)], the intersection of all the cozero-sets of αX which contain X, call it Υ for now, is a copy of the realcompactification.

(iii) We give names to the various inclusions: X →^{j_{1}} υX →^{i_{1}} βX and X →^{j_{2}} Υ →^{i_{2}} αX. By [GJ, Theorem 8.7], there is a homeomorphism ζ: υX → Υ so that ζj_{1} = j_{2}. Since βX = β(υX), there is ξ: βX → αX with ξi_{1} = i_{2}ζ. Thus, ξi_{1}j_{1} = σ_{α}i_{1}j_{1}. The density of X in βX shows that ξ = σ_{α} and, thus, that ζ is the restriction of σ_{α} to υX.

The next part of (iii) is clear from [GJ, 8D 1.]. The last statement follows in the same way.

2.9. Remark. Let X be any space and αX ∈ K(X). If there is a subspace Y, X ⊆ Y ⊆ αX, so that X is C-embedded in Y, then, αX ∈ K_{epi}(X) if and only if αX ∈ K_{epi}(Y). If αX ∈ K_{z}(X) there is a copy of υX ⊆ αX to play the role of Y.

Proof. Consider the homomorphisms C(αX) → C(Y) → C(X). The second homomorphism is an isomorphism because X is C-embedded in Y. The second statement is from Proposition 2.8 (ii).

V. Zero-sets in βX −X. Zero-sets of βX lying in βX −X (i.e., β-zero-sets) will appear many times in what follows and thus it would be helpful to know more about them. The following is an adaptation of [Ca, Corollary 4.5].

2.10. Proposition. Let X be any space. Then, there is a one-to-one correspondence between the β-zero-sets of βX and the z-filters F of X such that (i) F has a countable base, (ii) F is the intersection of the z-ultrafilters containing it, and (iii) all the z-ultrafilters containing F are free.

In other words, the key players are countable families Z = {Z_{n}}_{N} of zero-sets of X which have the finite intersection property and ⋂_{N} Z_{n} = ∅. In one direction, with such Z where Z_{n} = z(f_{n}), 0 ≤ f_{n} ≤ 1, we can associate f = ∑_{n}(1/2^{n})f_{n} and f^{β} whose zero-set is non-empty and is in βX − X. In the other direction, if f ∈ C^{∗}(X) is such that ∅ ≠ z(f^{β}) ⊆ βX − X, we can find Z = {Z_{n}}_{N}, where Z_{n} = {x ∈ X | |f_{n}(x)| ≤ 1/n}.

The easiest case is where X is a P -space, then, the β-zero-sets can be attached to partitions of X. The following proposition will be used without mention, especially in Section 6.

2.11. Proposition. Let X be a P -space. Then, there is one-to-one correspondence between non-empty β-zero-sets of βX and partitions of X into countably many clopen sets.

Proof. First assume that {T_{n}}_{N} is a partition of X into clopen sets and let m_{1} < m_{2} < ⋯ be a sequence from N. Then f ∈ C^{∗}(X) may be defined by f(x) = 1/m_{n}, if x ∈ T_{n}. Then, z(f^{β}) ⊆ βX − X. In the other direction suppose, for g ∈ C(βX) that ∅ ≠ z(g) ⊆ βX − X. We may assume 0 < g ≤ 1. Set U_{n} = {x ∈ X | g(x) ≤ 1/n}. Put T_{n} = U_{n} − U_{n+1}. Each T_{n} is a clopen set of X and infinitely many of them are non-empty. Suppose that T_{m_{1}}, T_{m_{2}}, … are the non-empty ones, m_{1} < m_{2} < · · ·. We, thus, have a partition of X into countably many clopen sets.

We know from [GJ, Theorem 9.5] that a non-empty β-zero-set contains a copy of βN. However, these zero-sets can have the same cardinality as βX. Indeed, by [C, Lemma 5.2], if X is infinite and discrete, βX − X contains a copy of βX. It is clear from the proof of that lemma that the set \overline{T}, homeomorphic to βX, is in a β-zero-set. (If |X| is a measurable cardinal, the elements a_{i} chosen in the construction, need to be taken from βX − υX by [GJ, Theorem 12.2].)

When X is discrete, the β-zero-sets are studied in Proposition 6.2, below.

3. K_{epi}(X) vs K_{z}(X) in P -spaces.

Lemma 2.7 established, for general X, a connection between K_{epi}(X) and K_{z}(X). However, for general spaces, we have neither implication: “αX ∈ K_{z}(X) ⇒ αX ∈ K_{epi}(X)” nor “αX ∈ K_{epi}(X) ⇒ αX ∈ K_{z}(X)”. For the first we can use [BRW, Corollary 2.3] and the space Q: for the second, [BRW, Theorem 3.3] supplies examples. This section contains information when X is required to be a P -space. (See [GJ, 4J] for many equivalent conditions: we use that C(X) is a regular ring and that zero-sets in X are open.) Fractions will again play an important role and we will see that for a P -space X, αX ∈ K_{epi}(X) implies C(X) ≅ Q_{cl}(X). The first part of the next proposition could also be deduced from [Wa, Theorem 2.8].

3.1. Proposition. Let X be a P -space and αX ∈ K(X). Then: (i) αX ∈ K_{z}(X) if and only if the idempotents in C(X) are in Q_{cl}(C_{α}). In particular, if X is infinite and discrete then αX ∈ K_{z}(X) if and only if the characteristic functions of subsets of X are in Q_{cl}(C_{α}). (ii) If αX ∈ K_{epi}(X), then X is z-embedded in αX.

Proof. (i) Since X is a P -space, the zero-sets of X are open and the complement of a zero-set is a zero-set.

Suppose now that the idempotents of C(X) are in Q_{cl}(C_{α}). We first note that if g ∈ C_{α} is a non-zero divisor, then coz g = X. To see this, suppose coz g = V. Write χ_{X−V} = hk^{−1}, h, k ∈ C_{α} and k a non-zero divisor. Then χ_{X−V} · k = h shows that V ⊆ z(h), implying that gh = 0. Hence, h = 0 and V = X. Now let V be any zero-set of X. We write χ_{V} = hk^{−1} ∈ Q_{cl}(C_{α}). The equation χ_{V} · k = h says that z(h) = z(χ_{V}) = V and h ∈ C_{α}.

In the other direction, if V is a zero-set in X then there is f ∈ C_{α} with z(f) = V and g ∈ C_{α} with z(g) = X − V. Then f/(f + g) = χ_{V}.

(ii) This part is [BRW, Lemma 5.1].

In [HM2], the authors define the notion of “C-epic” which means epimorphism in the category of archimedean f-rings. As an illustration of the difference between this and our version of epimorphism, [HM2, Theorem 6.3] shows how to construct, in particular, a compactification αX of an uncountable discrete space X, where X is not z-embedded in αX but C(αX) → C(X) is C-epic. It suffices to take, in the construction, Y uncountable discrete. In fact, the criteria of [HM2, Theorems 7.1 and 8.2] show that “z-embedded” implies “C-epic” (explicitly [HM2, Corollary 2.5]), whereas this fails for CR-epimorphisms. As an example we take the space Q because Q is z-embedded in all spaces ([BH, Theorem 4.4]) but K_{z}(X) = K(X) ≠ K_{epi}(X) by [BRW, Corollary 2.23].

3.2. Lemma. Let X be a P -space and αX ∈ K_{z}(X). Then Q_{cl}(αX) is regular.

Proof. Consider f ∈ C(αX): then, V = z(f)∩X is a zero-set of X, and is, hence, clopen. It follows that X − V is also a zero-set and there is g ∈ C(αX) with z(g) ∩ X = X − V. Consider f + g. Since fg is zero on X, fg = 0. If, for some h ∈ C(αX), (f + g)h = 0, then h is zero on X, showing h = 0. Thus coz(f + g) is dense in αX and contains X. Put l = (f + g)^{−1} ∈ Q_{cl}(C(αX). It follows that f^{2}l coincides with f on X and extends continuously to coz(f + g). This shows that f = f^{2}l in Q_{cl}(αX) ([FGL, page 14]).

3.3. Proposition. Let X be a realcompact space. Suppose αX ∈ K_{z}(X). Then X is an intersection of cozero-sets of αX.

Proof. By [BH, Corollary 2.4], each f ∈ C(X) can be extended to a countable intersection of cozero-sets of αX, each containing X. Let Y be the intersection of all such cozero-sets of αX. Then, C(X) is C-embedded in C(Y). This says ([GJ, 8.14 and 8.9] along with [GJ, 8.10 (a)]) that Y is the realcompactification of X. However, X is realcompact and so Y = X.

Lemma 3.2 and Proposition 3.3 are used in the proof of the following theorem which says that for a P -space X, if αX ∈ K_{epi}(X) then C(X) = Q_{cl}(αX). (Theorem 4.5 (C) gives a sharper version of this.)

3.4. Theorem. Let X be a P -space. Suppose αX ∈ K_{epi}(X). Then, C(X) ≅ Q_{cl}(αX) via the natural inclusion C(αX) → C(X).

Proof. We first assume that X is realcompact. By Proposition 3.3, X is an intersection of cozero-sets of αX. From [RW, Definition 4.8], gY is the intersection of all dense cozerosets of a space Y. By [RW, Lemma 4.9(3)], X ⊆ g(αX), and, hence, X = g(αX). Then [RW, Lemma 4.7] says that the regular ring C(X) is a ring of quotients of C(αX). It follows that we have embeddings C(αX) → Q_{cl}(αX) → H(αX) → C(X), where (as in [RW, Section 2] and its references) H(αX) is the smallest regular subring between C(αX) and Q(αX), the complete ring of quotients of C(αX). However, Lemma 3.2 says that Q_{cl}(αX) is regular, showing that Q_{cl}(αX) = H(αX).

Since C(αX) → C(X) is an epimorphism, so is H(αX) → C(X). However, H(αX) is regular and so it has no proper epimorphic extensions ([St, Korollar 5.4]). This shows that Q_{cl}(αX) = C(X).

We now drop the assumption of realcompactness. By Proposition 3.1, αX ∈ K_{epi}(X) implies that αX ∈ K_{z}(X). Then Proposition 2.8 (ii) says that there is a copy of the realcompactification X → υX → αX. However, υX is also a P -space (because C(υX) ≅ C(X) is regular) and the first part of the proof says that C(υX) ≅ Q_{cl}(αX), via restriction, because αX ∈ K_{epi}(υX) by Remark 2.9. Restricting further to X gives C(X) ≅ Q_{cl}(αX).

When X is a P -space and αX ∈ K_{z}(X), then, [HW, Theorem 1.4 (d)] shows that, αX is cozero complemented. Hence, the minimum spectrum of Q_{cl}(C_{α}) = Q_{cl}(αX) is compact and all the equivalent conditions of [HW, Theorem 1.3] apply to αX.

When X is infinite discrete and αX ∈ K_{z}(X), then Q_{cl}(αX) and C(X) have the same idempotents. However, Q(αX) = C(X) (since X is the unique smallest dense open set of αX) so that Q_{cl}(αX) and Q(αX) have the same idempotents. Then the language of [HM1] applies and we can say that αX is fraction dense ([HM1, Theorem 1.1]). Moreover, when αX ∈ K_{epi}(X), Theorem 3.4, in this special case, shows that Q_{cl}(αX) = Q(αX), which implies that αX is strongly fraction dense ([HM1, page 979]).

The next topic is somewhat apart from the main theme of the article but it underlines the fact that, when talking about epimorphisms, P -spaces are easier to deal with than general spaces. Let X be a space and αX ∈ K(X). It was asked in [BBR, Section 3D] whether, in our context, R/N-epimorphisms were CR-epimorphisms. We cannot answer the question in general but can say that it is “yes” when X is a P -space. (See [BBR, Theorem 3.21] for another partial answer.)

3.5. Theorem. Let X be a P -space and αX ∈ K(X). Then, ρ: C(αX) → C(X) is an R/N -epimorphism if and only if it is CR -epimorphism.

Proof. We always have that ρ a CR-epic implies it is an R/N-epic.

By a result of Lazard, quoted in [S, page 351] and [BBR, Proposition 1.1], ρ is an R/N-epimorphism if and only if ρ^{−1} : Spec C(X) → Spec C(αX) is injective and, for each P ∈ Spec C(X) and Q = ρ^{−1}(P), Q_{cl}(C(αX)/Q) = Q_{cl}(C(X)/P), via ρ. We now assume that ρ is an R/N-epimorphism.

Now let T (αX) stand for T (C(αX)), the universal regular ring of C(αX), and μ: C(αX) → T (αX) the canonical injection (which is CR-epic). See [BBR, Section 3] for details. Since C(X) is a regular ring, the universal property of T (αX) defines a canonical ζ: T (αX) → C(X) extending ρ. Hence, ρ = ζμ and it will suffice to show that ζ is a surjection.

Recall that Spec T (αX) = Spec C(αX) as sets but Spec T (αX) has the constructible topology. In a regular ring S, for s ∈ S, s′ denotes the unique element where s^{2}s′ = s and (s′)^{2}s = s′. Moreover, Spec S is a totally disconnected compact Hausdorff space.

Fix f ∈ C(X). We will show that f is in the image of ζ. For P ∈ Spec C(X), let Q = ρ^{−1}(P). By Lazard’s criterion, there are u_{Q}, v_{Q} ∈ C(αX) with v_{Q} ∉ Q, such that f + P = (u_{Q} + P)(v_{Q} + P)^{−1} (we identify elements of C(αX) with their images under ρ). Then, ζ(μ(u_{Q})μ(v_{Q})′) and f coincide module P. Because we are dealing with regular rings there is a clopen neighbourhood U_{P} of P in Spec C(X) so that, for all R ∈ U_{P}, v_{Q} ∉ R and (u_{Q} + R)(v_{Q} + R)^{−1} = f + R.

For Q ∈ Spec C(αX) − ρ^{−1}(Spec C(X)) = U, an open subset of Spec T (αX), there is a clopen neighbourhood U_{Q} of Q in U. Then, Spec T (αX) is covered by ⋃_{P ∈Spec C(X)} ρ^{−1}(U_{P})∪⋃_{Q∈U} U_{Q}. There is, by compactness, a finite subcover and since the elements of the finite subcover are clopen, there is a refinement {U_{1}, …, U_{k}} where the clopen sets are now disjoint. Let the corresponding idempotents of T (αX) be e_{1}, …, e_{k}.

Divide this cover into two parts (the second part may be empty), {U_{1}, …, U_{m}} and {U_{m+1}, …, U_{k}}, where U_{i} ∩ ρ^{−1}(Spec C(X)) ≠ ∅, i = 1, …, m, and U_{j} ⊆ U, j = m + 1, …, k. Each U_{i}, i = 1, …, m, lies in one of the U_{P}, say U_{P_{i}} with ρ^{−1}(P_{i}) = Q_{i}. We now consider l = ∑_{i=1}^{m} e_{i}μ(u_{Q_{i}})μ(v_{Q_{i}})′ + ∑_{j=m+1}^{k} e_{j}. By construction, ζ(l) = f, since the two elements coincide modulo each P ∈ Spec C(X).

4. On K_{epi}(X)

This section contains tools to be used in later parts of the article; however, its main theme is to look at when C(X) = C_{α}S_{α}^{−1} (which implies that αX ∈ K_{epi}(X)). The collection of such compactifications is denoted K_{epi}^{f}(X) (“f” for “fractions”). These are the easiest sorts of elements of K_{epi}(X) to study. We always have K_{epi}^{f}(X) ⊆ K_{z}(X) and we will show that for P -spaces, K_{epi}(X) = K_{epi}^{f}(X).

We begin with a special kind of compactification defined as follows: K_{epi}^{1}(X) = {αX ∈ K(X) | there is h ∈ S_{β} with z(h^{β}) ⊇ M_{α}}. The next proposition will show that K_{epi}^{1}(X) ⊆ K_{epi}^{f}(X). The existence of compactifications of this type is guaranteed by Proposition 2.2 but other methods for finding them will be discussed later in this section and in Sections 5 and 6.

4.1. Proposition. Let X be any space and αX ∈ K(X). Suppose that αX ∈ K_{epi}^{1}(X). Then C(X) is the localization C_{α}S_{α}^{−1}; i.e., αX ∈ K_{epi}^{f}(X).

Proof. (The method is close to that of [BBR, Proposition 2.1(ii)].) Since h^{β} is zero on each τ^{−1}(a), a ∈ I_{α}, h ∈ S_{α}. Given f ∈ C^{∗}(X), f^{β}h^{β} is zero on all of M_{α} and, hence, l = fh ∈ C_{α}. Then f = lh^{−1} ∈ Q_{cl}(C_{α}). For general f ∈ C(X), we write

f = \frac{f}{1 + f^{2}} (\frac{1}{1 + f^{2}} )^{−1} = l_{1}h^{−1}(l_{2}h^{−1})^{−1} = l_{1}l_{2}^{−1} ,

for some l_{1}, l_{2} ∈ C_{α} and coz l_{2} = X. It follows that C(X) is an epimorphic extension of C_{α}.

The denominator set used in the proof is, in fact, smaller than that described since the only elements which are used are of the form l ∈ S_{α} where z(l^{β}) ⊇ M_{α}.

[BRW, Theorem 3.3] shows that, in general, K_{epi}^{1}(X) ≠ K_{epi}(X) and that K_{epi}(X) ⊈ K_{z}(X). It is clear that if αX ∈ K_{epi}^{1}(X), then so is every compactification above αX ∈ K(X). Recall ([Wk, Theorem, page 31]) that if X is realcompact then every p ∈ βX −X is contained in a zero-set, z(g^{β}), of some g ∈ S_{β}. The following is an immediate consequence. (See also Proposition 4.9 (iii).)

4.2. Corollary. Let X be any space. Suppose αX ∈ K(X) with M_{α} finite and M_{α} ⊆ βX − υX. Then αX ∈ K_{epi}^{1}(X). In particular, βX ∈ K_{epi}^{1}(X).

Proof. Let M_{α} = {p_{1}, …, p_{k}}. Each p_{i} ∈ z(h_{i}^{β}) for some h_{i} ∈ C^{∗}(υX) which are nowhere zero. Hence, h = h_{1}|_{X} ⋯ h_{k}|_{X} ∈ S_{β} can be used in Proposition 4.1.

There is another observation about υX and K_{epi}(X). A related result, [HM2, Corollary 2.7 (a)], is about C-epics; it requires that υX be Lindelöf.

4.3. Proposition. Let X be a space and αX ∈ K_{epi}(X). Then, τ|_{υX} is one-to-one, i.e., for each a ∈ I_{α}, |τ^{−1}(a) ∩ υX| ≤ 1.

Proof. Suppose that, for some a ∈ I_{α}, there are p ≠ q in τ^{−1}(a)∩υX. Let f ∈ C^{∗}(X) be such that f^{β}(p) ≠ f^{β}(q). By assumption, there is an n × n zig-zag for f over C_{α}, for some n, say f = GHK. However, all the functions in the zig-zag extend to υX. We use the same symbols for the extensions to υX. Then, 0 ≠ f(p) − f(q) = GH(p)K(p) − GH(q)K(q) = GH(p)(K(p) − K(q)) = G(p)(H(p)K(p) − H(q)K(q)) = G(p)(HK(p) − HK(q)) = 0, a contradiction.

4.4. Examples. There are examples of a space X and αX ∈ K_{epi}(X) so that for some a ∈ I_{α}, |τ^{−1}(a) ∩ υX| = 1.

Proof. One can use any of the examples found in [BRW, Theorem 3.3], where X is a sum of a Lindelöf absolute CR-epic space L (i.e., a space such that K_{epi}(L) = K(L)) and an almost compact space A. Indeed, K(X) = K_{epi}(X) by the quoted theorem. The single point s of βA − A is in υX − X and, when it is in M_{α} for some αX ∈ K(X), will be identified with some closed set of βL − L ⊆ βX − υX. As a specific example, we take any closed ∅ ≠ Y ⊆ βL − L, T = Y ∪ {s} and α_{T} X.

The following theorem contains observations about fractions. Some of the arguments can also be found in [BBR] and [BRW].

4.5. Theorem. Let X be any space and αX ∈ K(X). Then, (A) The following are equivalent.

(i) C^{∗}(X) ⊆ C_{α}S_{α}^{−1}.

(ii) Each f ∈ C^{∗}(X) extends to a cozero-set of αX containing X.

(iii) C(X) = C_{α}S_{α}^{−1}; i.e., αX ∈ K_{epi}^{f}(X). (B) If the conditions of (A) are satisfied then, αX ∈ K_{epi}(X) ∩ K_{z}(X). (C) Let X be a P -space. Then the following are equivalent for αX ∈ K(X):

(a) αX ∈ K_{epi}(X).

(b) αX ∈ K_{epi}^{f}(X).

Proof. (A) Assume (i). Then, each f ∈ C^{∗}(X) can be written f = gh^{−1}, where g ∈ C_{α} and h ∈ S_{α}. Since both g and h extend to αX, gh^{−1} extends to coz h^{α}.

Assume (ii). Suppose that f ∈ C^{∗}(X) extends to coz h, h ∈ C(αX), X ⊆ coz h with extension f̃. Then, f̃h|_{coz h} extends to, say, l ∈ C(αX). Then f = l|_{X}(h|_{X})^{−1} ∈ C_{α}S_{α}^{−1}. For any f ∈ C(X), we write f/(1 + f^{2}) = g_{1}h_{1}^{−1} and 1/(1 + f^{2}) = g_{2}h_{2}^{−1}, expressions from C_{α}S_{α}^{−1}. Then, f = g_{1}h_{2}(g_{2}h_{1})^{−1} ∈ C_{α}S_{α}^{−1}, since g_{2} ∈ S_{α}.

(iii) ⇒ (i) is obvious.

(B) Clearly, αX ∈ K_{epi}(X). Moreover, if V = z(f) is a zero-set of X, f ∈ C^{∗}(X) then, we write f = gh^{−1} ∈ C_{α}S_{α}^{−1} and note that z(g^{α}) ∩ X = V.

(C) Theorem 3.4 shows that (a) implies that C(X) = Q_{cl}(αX). To get (b) we need to verify that if g ∈ C(αX) is a non-zero divisor then g|_{X} ∈ S_{α}. Thus, coz g is dense in αX and we put V = coz g ∩ X, a clopen set of X. If V ≠ X there is an open U in αX with U ∩ X = X − V. We can find 0 ≠ h ∈ C(αX) with coz h ∩ U ≠ ∅. Then, coz h ∩ U ∩ X ≠ ∅. We get (g|_{X})(h|_{X}) = 0 and, hence, gh = 0, which is impossible. Hence, V = X and g|_{X} ∈ S_{α}.

Notice also that when X is locally compact and Lindelöf, all compactifications fall under Proposition 4.1 since ωX ∈ K_{epi}^{1}(X) by [BRW, Theorem 2.15] and the proof of [BRW, Lemma 2.28]. The most obvious examples are X = N and X = R. Lemma 2.7 says that when X is locally compact and ωX ∈ K_{z}(X) then, in fact, K(X) = K_{epi}^{1}(X). To see this, by Lemma 2.7 there is a g ∈ S_{ω} with coz g^{ω} = X, which is what Proposition 4.1 requires. Now, [BRW, Theorem 2.29] implies that X is Lindelöf or almost compact. We summarize.

4.6. Corollary. [BRW] Let X be a locally compact space. Then the following are equivalent.

(i) ωX ∈ K_{z}(X). (ii) X is Lindelöf or almost compact. (iii) K(X) = K_{epi}^{1}(X).

The existence of compactifications in K_{epi}(X) where 1×1 zig-zags do not suffice ([BRW, Theorem 3.3] or Examples 4.4, above) shows that, in general, K_{epi}^{f}(X) can be strictly included in K_{epi}(X). However, we have seen that when X is a P -space, the two coincide. Moreover, we always have βX ∈ K_{epi}^{1}(X) ⊆ K_{epi}^{f}(X).

4.7. Remark. cf. [HM2, Corollary 2.9 (a)] A space X is pseudocompact space if and only if K_{epi}(X) = {βX}. In particular, if |βX − X| < 2^{𝔠} then K_{epi}(X) = {βX}.

Proof. If X is not pseudocompact then ([GJ, 6I 1.]) there is a non-empty β-zero-set Z, which is infinite by [GJ, Theorem 9.5]. Then, Proposition 4.1 says that α_{Z}X ∈ K_{epi}^{1}(X). On the other hand, if X is pseudocompact then the only C-epic compactification is βX by [HM2, Corollary 2.9 (a)], showing, a fortiori, that K_{epi}(X) = {βX}.

The second statement follows from [GJ, 9D 3.], which says that, under the hypothesis, X is pseudocompact.

We next find a situation in which αX ∈ K_{epi}^{f}(X) if and only if αX ∈ K_{epi}^{1}(X), and one where αX ∈ K_{z}(X) if and only if αX ∈ K_{epi}^{1}(X).

4.8. Lemma. [C, Theorem 5.32] Let X be any space. Suppose Y = {y_{n} | n ∈ N} is a countable subset. Then, there is f ∈ C^{∗}(X), 0 ≤ f, such that f(y_{i}) ≠ f(y_{j}), for all i ≠ j.

4.9. Proposition. Let X be any space. (i) Let αX ∈ K(X) be such that there is D ⊆ I_{α} which is dense in I_{α} and f ∈ C^{∗}(X) such that f^{β} is not constant on any τ^{−1}(a), a ∈ D. Then, αX ∈ K_{epi}^{f}(X) implies αX ∈ K_{epi}^{1}(X). (ii) If I_{α} is separable then αX ∈ K_{epi}^{f}(X) implies αX ∈ K_{epi}^{1}(X). (iii) Let αX ∈ K(X) be such that I_{α} is finite. If αX ∈ K_{z}(X) then αX ∈ K_{epi}^{1}(X).

Proof. (i) Since αX ∈ K_{epi}^{f}(X), there are g, h ∈ C_{α}, h ∈ S_{α} so that f = gh^{−1}. If follows from f^{β}h^{β} = g^{β} that h^{β} is zero on τ^{−1}(a), for each a ∈ D because g^{β} and h^{β} are both constant on each τ^{−1}(a), a ∈ D. Then h^{α}(D) = {0}. Since D is dense in I_{α}, h^{α}(I_{α}) = {0}, showing that M_{α} ⊆ z(h^{β}) and that αX ∈ K_{epi}^{1}(X) using the function h in Proposition 4.1.

(ii) Lemma 4.8 shows that there is f ∈ C^{∗}(X) as required in part (i). More precisely, we write D = {a_{1}, a_{2}, …}, dense in I_{α}, and pick p_{n} ≠ q_{n} from τ^{−1}(a_{n}), n ∈ N. Then l ∈ C(βX) is chosen so that l(p_{n}) ≠ l(q_{n}), for all n ∈ N. Hence, f = l|_{X} will work.

(iii) We can find f ∈ C(βX) such that f is not constant on any τ^{−1}(a), a ∈ I_{α}, since I_{α} is finite. By Lemma 2.7, there is h ∈ S_{α} so that h^{β} is zero on ⋃_{a∈I_{α}} τ^{−1}(a), showing that z(h^{β}) ⊇ M_{α}.

Construction 2.3 shows how to get spaces of the type found in Proposition 4.9 (ii). It suffices to start with αX ∈ K_{epi}^{f}(X) and apply the construction with B countable.

It can be seen that Proposition 4.5 (A) contains a stronger statement than [BBR, Theorem 2.6] since, if αX ∈ K_{epi}^{f}(X), the proof of (A) (ii) ⇒ (iii) shows that, here, every f ∈ C(X) extends to a cozero-set of αX containing X. We can now list three facts about extending functions.

1. If αX ∈ K_{epi}(X), each f ∈ C(X) extends to an open set of αX containing X ([BBR, Proposition 2.6]).

2. If αX ∈ K_{z}(X), for each f ∈ C(X) and finite B ⊆ I_{α}, f extends to a cozero-set of α_{(B)}X containing X (Lemma 2.7 and Proposition 4.9 (iii)).

3. If αX ∈ K_{epi}^{f}(X), each f ∈ C(X) extends to a cozero-set of αX containing X (Theorem 4.5 (A)).

5. Lattice properties

In this section it will be shown that while the subsets K_{z}(X) and K_{epi}^{f}(X) of K(X) have some lattice closure properties (they are both complete upper semilattices), they are not closed under all meets unless υX is locally compact and ω(υX) ∈ K_{z}(X), in which case K_{z}(X) = K_{epi}^{f}(X) = K(υX). In fact, if X is locally compact and realcompact, then Corollary 4.2 says that ωX is always a meet of elements of K_{epi}^{1}(X), but it need be neither in K_{epi}(X) nor in K_{z}(X). We will see that, in general, K_{z}(X) and K_{epi}^{f}(X) have no minimal elements. Moreover, K_{nepi}(X) need not be closed under finite joins. Recall the definition of α_{T} X from before Proposition 2.2.

5.1. Lemma. Let X be any space, αX ∈ K(X) and, for some g ∈ S_{α}, T = z(g^{β}). Let γX = αX ∧ α_{T} X.

(i) If αX ∈ K_{z}(X) then γX ∈ K_{z}(X).

(ii) If αX ∈ K_{epi}^{f}(X), then γX ∈ K_{epi}^{f}(X).

(iii) If αX ∈ K_{epi}(X), then γX ∈ K_{epi}(X).

Proof. In each case, the existence of γX is guaranteed by Proposition 2.2. (i) This is a special case of Proposition 2.2 (iii). (ii) If f ∈ C(X) and we write f = uv^{−1}, u ∈ C_{α}, v ∈ S_{α}, then f = ug(vg)^{−1}, where

ug ∈ C_{γ} and vg ∈ S_{γ}.

(iii) The method here is like that of (ii) but we need to use zig-zags. Suppose f ∈ C(X) and we write f as a n × n zig-zag over C_{α}:

f = [g_{1} ⋯ g_{n}] [h_{ij}] [k_{1} ; ⋮ ; k_{n}] , [h_{ij}] an n × n matrix .

Then, we can get an n × n zig-zag for f over C_{γ} as follows:

f = [g_{1}g^{−1} ⋯ g_{n}g^{−1}] [h_{ij}g^{2}] [k_{1}g^{−1} ; ⋮ ; k_{n}g^{−1}]

5.2. Theorem. Let X be any space.

(i) If K(υX) ≠ K_{z}(X), then K_{z}(X) has no minimal elements.

(ii) If K(υX) ≠ K_{epi}^{f}(X), then K_{epi}^{f}(X) has no minimal elements.

(iii) If K(υX) ≠ K_{epi}(υX), then K_{epi}(υX) has no minimal elements.

Proof. (i) (By [BH, Theorem 4.1], we have that υX is neither Lindelöf nor almost compact.) Let αX ∈ K_{z}(X) = K_{z}(υX) (by Proposition 2.8). If αX − υX were finite then υX would be locally compact and, by Lemma 5.1 (i) and Proposition 3.3, ω(υX) ∈ K_{z}(υX), contradicting the hypothesis. Hence, αX − υX is infinite and, given a_{1} ≠ a_{2} in αX − υX, there is, by Proposition 3.3 again, some g ∈ S_{α} with {a_{1}, a_{2}} ⊆ z(g^{α}). Put T = z(g^{β}). Then Lemma 5.1 (i) says that αX ∧ α_{T} X, which is strictly smaller than αX, is in K_{z}(X) = K_{z}(υX).

In part (ii), since K_{z}(X) ⊆ K_{epi}^{f}(X), we see that K_{epi}^{f}(X) = K_{epi}^{f}(υX). With this observation, parts (ii) and (iii) have proofs like that for part (i) using the corresponding parts of Lemma 5.1.

If X is locally compact and αX ∈ K(X), the set of compactifications containing αX, K_{≥α}(X), of K(X) forms a complete sublattice. Clearly, if αX is in one of K_{z}(X), K_{epi}(X), K_{epi}^{f}(X) or K_{epi}^{1}(X), then all of K_{≥α}(X) is. Moreover, if α_{1}X, …, α_{k}X ∈ K_{epi}^{1}(X), then, α_{1}X ∧ ⋯ ∧ α_{k}X ∈ K_{epi}^{1}(X) because there is some αX ∈ K_{epi}^{1}(X) contained in all of them.

Another situation is worthy of note. It shows a case where a meet of elements of K_{epi}^{1}(X) is in K_{epi}^{1}(X), if it is in K_{z}(X).

5.3. Proposition. Let X be a locally compact space. For some index set M, let {T_{μ}}_{μ∈M} be a family of closed sets in βX − X with each |T_{μ}| > 1. We write α_{μ}X for α_{T_{μ}}X and C_{μ} for C_{α_{Tμ}}. Put αX = ⋀_{μ∈M} α_{μ}X and T = cl_{βX}(⋃_{μ∈M} T_{μ}). Suppose, for some ∅ ≠ N ⊆ M, that (a) if ν_{1}, ν_{2} ∈ N then T_{ν_{1}} ∩T_{ν_{2}} ≠ ∅, and (b) for μ ∈ M −N, T_{μ} ∩[cl_{βX}(⋃_{ν∈N} T_{ν})] ≠ ∅. Then,

(i) C_{α} = {f ∈ C^{∗}(X) | f is constant on T }, showing αX = α_{T} X.

(ii) αX ∈ K_{z}(X) if and only if αX ∈ K_{epi}^{1}(X).

Proof. We have C_{α} = ⋂_{μ∈M} C_{μ} = {f ∈ C^{∗}(X) | f^{β} is constant on each T_{μ}}. However, if f ∈ C_{α} and f^{β} has value r on T_{ν}, some ν ∈ N, then it has value r on cl_{βX}(⋃_{ν∈N} T_{ν}), and, hence, on all T_{μ}, μ ∈ M, by the hypotheses on the intersections. In other words, C_{α} = {f ∈ C^{∗}(X) | f^{β} is constant on T }. This gives (i).

Having established (i), part (ii) is an instance of Proposition 4.9 (iii).

The following example shows that Proposition 5.3 (ii) can fail. It is a case where the T_{μ} form an uncountable chain.

5.4. Example. [HM1, page 988] There is an uncountable discrete space X where K_{epi}^{1}(X) is not closed under taking the meet of a descending chain of elements. The same example shows that neither K_{z}(X) nor K_{epi}(X) is closed under the meet of a descending chain.

Proof. In the cited example, |X| = ω_{1} and the compactifications, there called L(α_{i}), are in K_{epi}^{1}(X), while their meet is not even in K_{z}(X).

The situation is different when the meet takes place in some K_{≥α}(X), where αX ∈ K_{z}(X).

5.5. Proposition. Let X be any space and let αX ∈ K_{z}(X). Then αX is a meet of elements of K_{epi}^{1}(X) in K_{≥α}(X).

Proof. We use Construction 2.3 using each of the finite subsets F of I_{α}; let the resulting compactifications be α_{(F)}X. By Lemma 2.7, each α_{(F)}X ∈ K_{epi}^{1}(X) (since each is above an element of K_{epi}^{1}(X)). Recall that C_{α_{(F)}} = {f ∈ C^{∗}(X) | f is constant on each τ^{−1}(a), a ∈ F }. Hence, C_{α} = ⋂_{F ⊆I_{α},F finite} C_{α_{(F)}}. Then, [C, Theorem 2.18] shows that αX = ⋀_{F ⊆I_{α},F finite} α_{(F)}X.

Recall that the complement of K_{epi}(X) in K(X) is called K_{nepi}(X). If φ: A → B is any epimorphism in CR which is one-to-one and A is infinite then A and B have the same cardinality. This is one way to see that for X discrete and uncountable, ωX ∈ K_{nepi}(X) (although it is also a consequence of Corollary 4.6). Indeed, if X is uncountable and discrete, f ∈ C(ωX) is constant on a co-countable subset of X. Hence, |C(ωX)| ≤ |X| · 𝔠^{ℵ_{0}} < |C(X)| = 𝔠^{|X|}.

5.6. Proposition. When X is uncountable and discrete then K_{nepi}(X) is not closed under finite joins in K(X).

Proof. We first use Proposition 2.1 to show that if V ⊆ X is uncountable and T = cl_{βX} V − X = cl_{βX} V − V, then α_{T} X ∈ K_{nepi}(X). Indeed, cl_{α_{T} X} V is the one-point compactification of V. (This observation will be generalized in Proposition 6.9.)

Now let V_{1} and V_{2} be uncountable subsets of X with |V_{1} ∩V_{2}| ≤ ℵ_{0}. Put T_{i} = cl_{βX} V_{i} − X, i = 1, 2. Let αX = α_{T_{1}}X ∨ α_{T_{2}}X. In the construction of the join ([C, Theorem 2.7]) it is clear that for p ≠ q in βX − X, τ_{α}(p) = τ_{α}(q) if and only if {p, q} ⊆ T_{1} ∩ T_{2}. In other words, αX = α_{T_{1}∩T_{2}}X.

When V_{1} ∩ V_{2} is finite, T_{1} ∩ T_{2} = ∅ giving αX = βX. When V_{1} ∩ V_{2} is infinite, T_{1} ∩ T_{2} is a zero-set. Indeed, if we write V_{1} ∩ V_{2} = {x_{1}, x_{2}, …} and let f ∈ C^{∗}(X) be defined by f(x_{n}) = 1/n and f(x) = 1 for x ∉ V_{1} ∩ V_{2}, then T_{1} ∩ T_{2} = z(f^{β}). In either case, αX ∈ K_{epi}(X) while α_{T_{1}}X and α_{T_{2}}X are in K_{nepi}(X).

6. The special case where X is discrete

When X is uncountable and discrete, we know that K_{epi}(X) = K_{epi}^{f}(X) ⊆ K_{z}(X) (because X is a P -space) but we do not know if K_{epi}(X) = K_{epi}^{1}(X) or if K_{z}(X) = K_{epi}(X). The theme of the first part of this section will be to examine when a meet of compactifications of the form α_{Z}X, Z a zero-set of some g ∈ S_{β}, is again in K_{epi}^{1}(X). The second part finds elements of K_{nepi}(X).

For general spaces, if X is a dense subset of Y and ∅ ≠ U is open in Y, then U is contained in cl_{Y} (U ∩ X). In particular, if T ⊆ βX − X and U is open in βX with T ⊆ U, T ⊆ cl_{βX}(U ∩ X). This fact will often be used below. We recall also that if V ⊆ X, then cl_{βX} V = {p ∈ βX | V ∈ A^{p}}, where A^{p} is the ultrafilter attached to p ([GJ, 6.5]).

We begin by showing that a join of countably many elements of K_{epi}^{1}(X) is again in K_{epi}^{1}(X). (Example 5.4 shows that “countable” is essential.) A preliminary result about zero-sets of elements of S_{β} is needed; it relates them to almost disjoint families of countable subsets of X. We start with a lemma.

6.1. Lemma. Let X be an infinite discrete space, h ∈ S_{β}. Suppose that P a countable subset of X such that inf_{x∈P} h(x) = 0 and, for each n ∈ N, |h|^{−1}([1/n, ∞)) ∩ P is finite. Then, (i) cl_{βX} P − X ⊆ z(h^{β}); and, (ii) if, in addition, |h| is bounded away from 0 on X − P, then cl_{βX} P − X = z(h^{β}).

Proof. (i) For q ∈ cl_{βX} P − X and any V ∈ A^{q}, P ∩ V is infinite since P ∈ A^{q} and the ultrafilter is free. By hypothesis, inf_{x∈V ∩P} h(x) = 0 and, thus, h^{β}(q) = sup_{V ∈A^{q}} inf_{x∈V} h(x) = 0 ([GJ, 6C]).

(ii) If q ∈ z(h^{β}), then sup_{V ∈A^{q}} inf_{x∈V} h(x) = 0, which implies that V ∩ P is infinite for each V ∈ A^{q}. It follows that P ∈ A^{q} and, hence, q ∈ cl_{βX} P.

6.2. Theorem. (A) Let X be an uncountable discrete space and h ∈ S_{β} with z(h^{β}) ≠ ∅. Then there is a family of countable subsets of X, P = {P_{γ}}_{γ∈Γ}, such that (i) for γ, λ ∈ Γ, γ ≠ λ, P_{γ} ∩ P_{λ} is finite (i.e., the family P is almost disjoint); (ii) for each γ ∈ Γ, inf_{P_{γ}} h(x) = 0; and (iii) (⋃_{Γ} cl_{βX} P_{γ}) − X is dense in z(h^{β}).

(B) Let P = {P_{γ}}_{γ∈Γ}, Γ well-ordered, be an almost disjoint family of countable subsets of X. Then, there is a function h ∈ S_{β} such that for all γ ∈ Γ, inf_{x∈P_{γ}} h(x) = 0.

Proof. (A) We may assume that 0 < h ≤ 1. Put V_{n} = {x ∈ X | 1/(n+1) < h(x) ≤ 1/n}. Notice that V = {⋃_{n≥m} V_{n} | m ∈ N} is a filter base and that z(h^{β}) is the set of ultrafilters containing V.

In order to find an appropriate set P, we look at two cases. (a) If all but finitely many of the V_{n} are finite, then h is bounded away from zero on the union of those subsets V_{n} which are infinite. Put P = ⋃_{n∈N,V_{n} finite} V_{n}. Then, P = {P } gives the required set since z(h^{β}) = cl_{βX} P − X, by Lemma 6.1 (ii).

(b) In the contrary case, fix a subsequence {n(j)}_{j∈N} of N so that for each j ∈ N, V_{n(j)} is infinite and if there is i, n(j) < i < n(j + 1), then V_{i} is finite. For any sequence {x_{n(j)}}_{j∈N}, x_{n(j)} ∈ V_{n(j)}, we have that cl_{βX}{x_{n(j)}}_{j∈N} ⊆ z(h^{β}), by Lemma 6.1 (i). Any such choice of a sequence will serve as the starting point of an induction.

Suppose now that we have a well-ordered family of countable subsets of X, P_{γ}, γ < μ, μ an ordinal, satisfying conditions (i) and (ii). Then put T = (⋃_{γ<μ} cl_{βX} P_{γ})−X; T ⊆ z(h^{β}) since h^{β} is zero on each cl_{βX} P_{γ} − X, γ < μ. Assume that T is not dense in z(h^{β}). Let q ∈ z(h^{β}) − cl_{βX} T. There is a clopen set U of βX with q ∈ U and U ∩ cl_{βX} T = ∅. Put U = cl_{βX} V, for V ⊆ X. Then W_{j} = V ∩ (⋃_{k≥j} V_{n(k)}) ∈ A^{q}, for all j ∈ N.

Pick a sequence Q = {x_{j}}_{j∈N}, where x_{j} ∈ V_{n(j)} ∩ V; some indices may be missing if the intersection is empty, but infinitely many of the intersections must be non-empty because W_{j} ∈ A^{q}, for all j ∈ N. Suppose, for some γ < μ, that Q ∩ P_{γ} is infinite. Then ∅ ≠ cl_{βX}(Q∩P_{γ})−X ⊆ cl_{βX} P_{γ} −X ⊆ T; however, cl_{βX}(Q∩P_{γ})−X ⊆ U because Q ⊆ V. This is impossible. Hence, Q ∩ P_{γ} is finite for all γ < μ. This means that the collection P may be enlarged to satisfy (i); condition (ii) is satisfied by the choice of elements of Q. We, thus, have found P_{μ} = Q.

When the process above terminates, we have condition (iii).

(B) We build the function h by induction. The starting point is to well-order P_{1} = {x_{1,1}, x_{1,2}, …} and assign h(x_{1,n}) = 1/n.

Suppose now that γ ∈ Γ, γ > 1 and that values of h (in {1, 1/2, 1/3, …}) have been assigned to the elements of P_{ζ} for ζ < γ so that the condition on infima is satisfied. Put Q = P_{γ} ∩ ⋃_{ζ<γ} P_{ζ}. There are two cases to look at.

First suppose that Q is not cofinite in P_{γ}. Then we well-order the elements of P_{γ} − Q as q_{1}, q_{2}, … and assign h(q_{n}) = 1/n, while leaving h unchanged on P_{γ} ∩ Q.

If Q is cofinite in P_{γ}, there are again two cases to consider. If inf_{x∈Q} h(x) = 0, the value 1 is assigned to each element of P_{γ} − Q. Then inf_{x∈P_{γ}} h(x) = 0. If inf_{x∈Q} h(x) ≠ 0 then Q is well-ordered as Q = {q_{1}, q_{2}, …} and h is redefined on Q (and we recall that this changes only finitely many values on each P_{ζ}, ζ < γ) by h(q_{n}) = min{1/n, existing value}.

In either situation, h has now been defined on each P_{ζ}, ζ ≤ γ, so that h is never 0 but inf_{x∈P_{ζ}} h(x) = 0. The construction continues until Γ has been exhausted. The remaining elements of X are assigned value 1.

We can now deal with the meet of a countable set of elements of K_{epi}^{1}(X).

6.3. Corollary. Let X be an infinite discrete space and suppose that h_{n} ∈ S_{β}, n ∈ N, are such that h_{n}^{β}, n ∈ N, have non-empty and distinct zero-sets. Then, there is h ∈ S_{β} with T = z(h^{β}) ⊇ z(h_{n}^{β}), for all n ∈ N. Hence, if T_{n} = z(h_{n}^{β}), n ∈ N, then, ⋀_{n∈N} α_{T_{n}}X ∈ K_{epi}^{1}(X).

Proof. Theorem 6.2 is applied repeatedly where we assume, without loss of generality since we are dealing with zero-sets, that each h_{n} satisfies 0 < h_{n} ≤ 1. The case where all the zero-sets are in a union of finitely many of them is easy. Hence, we assume this is not the case and we delete any h_{n}, n > 1 for which z(h_{n}^{β}) ⊆ ⋃_{i<n} z(h_{i}^{β}). With renumbering, we then have, for n > 1, z(h_{n}^{β}) ⊈ ⋃_{i<n} z(h_{i}^{β}).

We begin by finding a well-ordered family of countable subsets as in Theorem 6.2 (A) for the zero-set T_{1}, say P_{1} = {P_{γ}}_{γ∈Γ_{1}}. Then T_{1} = cl_{βX}(⋃_{γ∈Γ_{1}} cl_{βX} P_{γ} − X). Since T_{1} ≠ T_{2}, we look at T_{1} ∪ T_{2} = z((h_{1}h_{2})^{β}) and continue the induction in the proof of the proposition to enlarge the set P_{1} to P_{2} = {P_{γ}}_{γ∈Γ_{2}}, where Γ_{2} ⊇ Γ_{1} and the conditions of Theorem 6.2 (A) are satisfied for T_{1} ∪ T_{2}. Notice that P_{1} satisfies condition (A) (ii) in Theorem 6.2 for h_{1} and, hence, it also satisfies that condition for h_{1}h_{2} because h_{1}h_{2} ≤ h_{1}.

This process continues, giving index sets Γ_{1} ⊆ Γ_{2} ⊆ ⋯ ⊆ Γ_{n} and P_{n} = {P_{γ}}_{γ∈Γ_{n}} satisfying the conditions for T_{1} ∪ ⋯ ∪ T_{n}, for each n ∈ N. Put Γ = ⋃_{n∈N} Γ_{n} and P = ⋃_{n∈N} P_{n}. We now apply Theorem 6.2 (B) to P to get h ∈ S_{β}.

The set R = (⋃_{γ∈Γ} cl_{βX} P_{γ}) − X contains subsets dense in each z(h_{n}^{β}), n ∈ N and, moreover, by the construction of h in Theorem 6.2 B, T = z(h^{β}) ⊇ cl_{βX} R. Hence, α_{T} X ≤ α_{T_{n}}X, for each n ∈ N. The conclusion follows.

The following is in the same vein as Proposition 4.9. When X is discrete we can go from K_{z}(X) to K_{epi}^{1}(X); however, a stronger condition on the countable subset of I_{α} is needed.

6.4. Corollary. Let X be an uncountable discrete space and αX ∈ K_{z}(X). Let f ∈ C(βX) and B = {b ∈ I_{α} | f is not constant on τ^{−1}(b)}. Suppose that ⋃_{b∈B} τ^{−1}(b) is dense in M_{α}. Then, αX ∈ K_{epi}^{1}(X).

In particular, if there is a countable subset D of I_{α} with τ^{−1}(D) dense in M_{α}, then αX ∈ K_{epi}^{1}(X).

Proof. By Lemma 2.6 there is a countable subset G = {g_{n}}_{n∈N} of S_{α} so that for each b ∈ B, there is m ∈ N with τ^{−1}(b) ⊆ z(g_{m}^{β}). According to Corollary 6.3 there is g ∈ S_{β} so that z(g_{n}^{β}) ⊆ z(g^{β}), for each n ∈ N. Then, cl_{βX}(⋃_{b∈B} τ^{−1}(b)) ⊆ z(g^{β}). In other words, M_{α} ⊆ z(g^{β}) = Z. Hence, α_{Z}X ≤ αX and so αX ∈ K_{epi}^{1}(X).

For the second part, from each τ^{−1}(d), d ∈ D, choose a pair of distinct points and then apply Lemma 4.8 to the resulting countable set to get f ∈ C(βX) not constant on any τ^{−1}(d), d ∈ D. Then, the first part of the statement applies.

We know from Example 5.4 that the condition “countable” cannot simply be dropped from the second part of Corollary 6.4; however, we will next show that the condition is not necessary either.

6.5. Example. Let X be an uncountable discrete space which is partitioned into disjoint countable subsets, P = {Q_{γ}}_{γ∈Γ}, where Q_{γ} = {x_{γ,1}, x_{γ,2}, …}, γ ∈ Γ, and we put T_{γ} = cl_{βX} Q_{γ}. Then, αX = ⋀_{γ∈Γ} α_{T_{γ}}X ∈ K_{epi}^{1}(X) while no countable subset of I_{α} satisfies the requirements of Corollary 6.4.

Proof. We note first that each T_{γ} is a zero-set, namely of f_{γ}^{β}, where f_{γ}(x_{λ,n}) = 1/n, if λ = γ, and 1, otherwise. Then, Corollary 2.5 shows that I_{α} has one element for each γ ∈ Γ, say τ(T_{γ}) = {a_{γ}}. Let f ∈ C^{∗}(X) be defined by f(x_{γ,n}) = 1/n, for all γ ∈ Γ and n ∈ N. Then, Z = z(f^{β}) contains each T_{γ}, showing that α_{Z}(X) ≤ αX in K(X). However, since α_{Z}X ∈ K_{epi}^{1}(X), so is αX and the last statement is clear.

The next theorem and its corollary look at arbitrary families of β-zero-sets. The aim is still to find meets of families of elements of K_{epi}^{1}(X) which are in K_{epi}^{1}(X).

6.6. Theorem. Let X be an uncountable discrete space and {h_{ν}}_{ν∈E} a family of elements of S_{β}. Then, the following statements are equivalent.

(i) There is an h ∈ S_{β} such that z(h^{β}) ⊇ ⋃_{ν∈E} z(h_{ν}^{β}).

(ii) There is a family {g_{ν}}_{ν∈E} of positive elements of S_{β}, so that (a) z(g_{ν}^{β}) = z(h_{ν}^{β}) for all ν ∈ E, and (b) the function defined by h̃(x) = inf_{ν∈E} g_{ν}(x) is in S_{β} and z(h̃^{β}) ⊇ ⋃_{ν∈E} z(h_{ν}^{β}).

(iii) ⋀_{ν∈E} α_{T_{ν}}X ∈ K_{epi}^{1}(X).

Proof. The equivalence of (i) and (iii) is clear, as is the implication (ii) ⇒ (i).

Assume (i). As usual, we assume that, for ν ∈ E, 0 < h_{ν} ≤ 1 and 0 < h ≤ 1. Define, for each n ∈ N and each ν ∈ E, V_{n} = {x ∈ X | h(x) ≤ 1/n} and V_{ν,n} = {x ∈ X | h_{ν}(x) ≤ 1/n}. Let H = {V_{n}}_{n∈N} and H_{ν} = {V_{ν,n}}_{n∈N}; these are filter bases. Then, z(h^{β}) = {p ∈ βX | A^{p} ⊇ H} and, for ν ∈ E, z(h_{ν}^{β}) = {p ∈ βX | A^{p} ⊇ H_{ν}}.

For each ν ∈ E, define g_{ν} = h + h_{ν}. Since z(h_{ν}^{β}) ⊆ z(h^{β}), z(g_{ν}^{β}) = z(h_{ν}^{β}), for each ν ∈ E, because all the functions are non-negative. Now h̃ ∈ C^{∗}(X) is defined as follows: for each x ∈ X, h̃(x) = h(x) + inf_{ν∈E} h_{ν}(x) = inf_{ν∈E} g_{ν}(x). Since h > 0, h̃ > 0 (i.e., h̃ ∈ S_{β}) and so it remains to be shown that, for each ν ∈ E, z(h_{ν}^{β}) ⊆ z(h̃^{β}).

For some ν ∈ E, let p ∈ z(h_{ν}^{β}) = z(g_{ν}^{β}). Then A^{p} ⊇ H ∪ H_{ν}. Hence, for each V ∈ A^{p} and every m, n ∈ N, V ∩ V_{m} ∩ V_{ν,n} ≠ ∅. Then,

inf_{x∈V} h̃(x) ≤ inf_{x∈V ∩V_{m}∩V_{ν,n}} h̃(x) ≤ 1/m + inf_{μ∈E} h_{μ}(x) ≤ 1/m + 1/n .

Since this is true for all m, n ∈ N, inf_{x∈V} h̃(x) = 0. Thus, h̃^{β}(p) = 0, as required.

It can be remarked that, in the proof of (i) ⇒ (ii) in the proposition, each g_{ν} > h_{ν}. The pointwise infimum of the original family of functions is not necessarily in S_{β}, while that of the new family is.

The next result presents two situations where Theorem 6.6 can be applied.

6.7. Corollary. Let X be uncountable and discrete and {h_{ν}}_{ν∈E} be a family of elements of S_{β}. Put L = {x ∈ X | inf_{ν∈E} |h_{ν}|(x) = 0}. If one of the following holds then there is an h ∈ S_{β} such that z(h^{β}) ⊇ ⋃_{ν∈E} z(h_{ν}^{β}) so that Theorem 6.6 applies.

(i) A = {ν ∈ E | z(h_{ν}^{β}) ∩ cl_{βX} L ≠ ∅} is countable.

(ii) L is countable.

Moreover, if there is J ⊆ L so that, for each ν ∈ E, inf_{x∈J} |h_{ν}|(x) > 0; then, there is an h ∈ S_{β} with Z(h^{β}) ⊇ ⋃_{ν∈E} z(h_{ν}^{β}) if and only if the conclusion holds for {h_{ν}|_{L−J}}_{ν∈E} in the space β(L − J).

Proof. We need the following remark: if f ∈ C^{∗}(X) and W ⊆ X we define f_{W} = f · χ_{W} and f_{W′} = f · χ_{X−W}, then z(f^{β}) = z(f_{W}^{β}) ∪ z(f_{W′}^{β}).

We normalize to get 0 < h_{ν} ≤ 1, for all ν ∈ E.

(i) Assume that L is uncountable since the contrary case is covered in part (ii). We use Corollary 6.3 applied to {h_{ν}|_{L}}_{ν∈A}. There is u ∈ C^{∗}(L) with coz u = L and 0 < u ≤ 1, so that z(u^{β}) ⊇ ⋃_{ν∈A} z((h_{ν}|_{L})^{β}) (in cl_{βX} L = βL). We extend u to X, without changing its name, by making it zero on X − L. We now define, for ν ∈ E, g_{ν} = h_{ν} + u and h = inf_{ν∈E} g_{ν}. The first thing to note is that on X − L, g_{ν} and h_{ν} coincide, and that, for x ∈ X − L, h(x) = inf_{ν∈E} h_{ν}(x) ≠ 0, while for x ∈ L, h(x) = u(x) ≠ 0.

For ν ∈ E − A, z(g_{ν}^{β}) = z(h_{ν}^{β}) because the zero-set is in cl_{βX}(X − L). For ν ∈ A,

z(g_{ν}^{β}) = (z(g_{ν}^{β}) ∩ cl_{βX} L) ∪ (z(g_{ν}^{β}) ∩ cl_{βX}(X − L))
= (z(h_{ν}^{β} + u^{β}) ∩ cl_{βX} L) ∪ (z(h_{ν}^{β}) ∩ cl_{βX}(X − L)) = z(h_{ν}^{β}) ,

since z(u^{β}) ⊇ z((h_{ν}|_{L})^{β}) on cl_{βX} L.

Since h ≤ g_{ν}, for each ν ∈ E, z(h^{β}) ⊇ ⋃_{ν∈E} z(h_{ν}^{β}).

(ii) This part is similar except that we order L = {x_{1}, x_{2}, …} and define u(x_{n}) = 1/n on L and 0 elsewhere. Then, z(u^{β}) = βX − L. Again g_{ν} is defined as h_{ν} + u, for each ν ∈ E and h = inf_{ν∈E} g_{ν}.

For the last part, it suffices to notice that the zero-sets do not change if one makes each h_{ν} constantly 1 on J.

Examples can be constructed to illustrate each of the parts of Corollary 6.7.

6.8. Examples. Let X be an uncountable discrete space. Then there are examples of families of functions satisfying Corollary 6.7 (i) and families satisfying Corollary 6.7 (ii). The last part of the corollary can be illustrated by using one of the examples in a subset of X.

Proof. (i) We partition X into uncountably many uncountable subsets {Y_{ν}}_{ν∈E} and single out special elements ν_{0}, ν_{1}, ν_{2}, … of E. Partition each Y_{ν} into countably many infinite subsets {Y_{ν,n}}_{n∈N}. For ν not in the special subset of E, we define h_{ν}: X → R by

h_{ν}(x) = \begin{cases}1/n & for x ∈ Y_{ν,n} \\ 1 & for x ∉ Y_{ν} \end{cases} .

For m > 0, we define h_{ν_{m}} by

h_{ν_{m}}(x) = \begin{cases}1/n & for x ∈ Y_{ν_{m},n} \\ 1/(m + k) & for x ∈ Y_{ν_{0},k} \\ 1 & for x ∉ Y_{ν_{m}} ∪ Y_{ν_{0}} \end{cases} .

Here, L = Y_{ν_{0}} is uncountable but A = {h_{ν_{m}}}_{m>0} is countable.

(ii) We divide X into uncountably many disjoint uncountable subsets Y_{ν}, ν ∈ E, and one countably infinite set P = {x_{1}, x_{2}, …}. Each Y_{ν} is further partitioned into countably many infinite subsets {Y_{ν,n}}_{n∈N}. We need a family of bijections θ_{ν}: N → N, ν ∈ E, which may be chosen in any way but with the proviso that for each pair m, n ∈ N, there is ν ∈ E so that θ_{ν}(n) = m.

For each ν ∈ E we define h_{ν}: X → R by

h_{ν}(x) = \begin{cases}1/n & if x ∈ Y_{ν,n} \\ 1/θ_{ν}(n) & if x = x_{n} \\ 1 & if x ∉ Y_{ν} ∪ P \end{cases} .

In this case L = P is countable.

Not much has been said so far about K_{nepi}(X), except that it is not closed under finite joins (Proposition 5.6); we give here some of its elements. The example from [HM1] quoted as Example 5.4 gives an instance of the following result which will give compactifications in K_{nepi}(X). Notice that the cardinality of αX −X is not involved since in these examples we can have |αX − X| = |βX − X|.

6.9. Proposition. Let X be an uncountable discrete space and αX ∈ K(X). Suppose that for some uncountable V ⊆ X there is a ∈ I_{α} such that for every neighbourhood U of a in αX, |V − U| < |V |. Then, αX ∉ K_{z}(X), and, hence, αX ∈ K_{nepi}(X).

Proof. By Proposition 2.1, it suffices to show that V is not z-embedded in V ∪ {a} (as a subspace of cl_{αX} V ⊆ αX). To establish this, consider W ⊆ V with |W | = |V − W | = |V | and f ∈ C(V ∪ {a}) so that W = z(f) ∩ V. There are two cases. (i) If f(a) = 0 then W = ⋂_{n∈N} f^{−1}((−1/n, 1/n)) ∩ V and V − W = ⋃_{n∈N}(V − f^{−1}((−1/n, 1/n)), a set of cardinality < |V |. This is not possible. (ii) If f(a) = r ≠ 0. For any n ∈ N with 1/n < |r|, put f^{−1}((r − 1/n, r + 1/n)) ∩ V = Y and repeat the argument from (i). Hence, V is not z-embedded in cl_{αX} V.

Proposition 6.9 can be used to give a K_{nepi} counterpart to Corollary 6.7. The notation of that corollary is used.

6.10. Corollary. Let X be uncountable and discrete and {h_{ν}}_{ν∈E} be a family of elements of S_{β}, E uncountable. Suppose that (i) L = {x ∈ X | inf_{ν∈E} h_{ν}(x) = 0} is uncountable and (ii) for every W ⊆ L with |W | = |L − W | there is some ν ∈ E with inf_{x∈W} h_{ν}(x) = 0. Then, for any closed set K, ⋃_{ν∈E} z(h_{ν}^{β}) ⊆ K ⊆ βX − X, α_{K}X ∉ K_{z}(X). In particular, if, for each ν ∈ E, we write T_{ν} = z(h_{ν}^{β}), then ⋀_{ν∈E} α_{T_{ν}}X ∉ K_{epi}(X).

Proof. We use Proposition 2.1 to assume that X = L. Let K be a closed set as in the statement. Suppose, if possible, that for some open neighbourhood U of K, |U ∩ X| = |X − U|. We have that K ⊆ cl_{βX}(U ∩ X). However, |X − U| = |X| and so, by hypothesis, there is ν ∈ E so that inf_{x∈X−U} h_{ν}(x) = 0, and, hence, z(h_{ν}^{β}) meets cl_{βX}(X − U). This is impossible. Hence, for each neighbourhood U of K, |X − U| < |X|, showing, by Proposition 6.9, that α_{K}X ∉ K_{z}(X) and, hence, not in K_{epi}(X).

The “size” of αX − X may, however, come into play. Recall that if αX ∈ K_{epi}(X) then |C_{α}| = |C(X)| and, when X is discrete, this is 𝔠^{|X|}.

[C, Lemma 7.6] says that given a compact space Y with a dense subset D where |D| ≤ |X| and a map ρ: X → D such that ρ(V) is dense in Y for each cofinite subset of X, there is an explicit construction of a compactification αX so that αX − X = Y. This construction gives examples of the compactifications which appear in the following.

6.11. Proposition. Let X be an infinite P -space and αX ∈ K(X).

(A) Suppose (i) there is D dense in αX−X with |D| < |X|, and (ii) |C(X)| > 𝔠^{|D|}·|X|. Then, αX ∉ K_{z}(X), and, thus, αX ∈ K_{nepi}(X).

(B) Let X be uncountable and discrete and suppose there is D dense in αX − X with 2^{|D|} < 2^{|X|}. Then, αX ∉ K_{z}(X).

Proof. (A) We first need an observation about elements of C(βX). Suppose f, g ∈ C(βX) and they agree on βX − X. For n ∈ N, put U_{n} = {x ∈ X | |(f − g)|(x) ≥ 1/n}. If U_{n} were in a free z-ultrafilter, p, then |f − g|(p) ≥ 1/n. Since this is impossible, U_{n} must be compact and, therefore, finite ([GJ, 4 K 2.]). Hence, coz(f −g) is countable. This same remark then applies to elements of C(γX), for any γX ∈ K(X), by lifting to βX.

We now get an upper bound on |C(αX)|: In the light of the above, an element f ∈ C(αX) is completely determined by its action on D and on a countable subset of X. Thus, |C(αX)| ≤ 𝔠^{|D|} · |X| · 𝔠^{ω}, since X is uncountable. On the other hand, 𝔠^{ω+|D|} · |X| = 𝔠^{|D|}|X| < |C(X)|, by hypothesis. Since, for any infinite space Y, |C(Y)| equals the cardinality of the set of zero-sets of Y ([DZ, Corollary 1,3])), we see that αX cannot be in K_{z}(X).

(B) We show that part (A) applies here. This is easy since X has 2^{|X|} zero-sets (since every subset of X is a zero-set) and |C(αX)| ≤ 𝔠^{|D|}|X| = 2^{ω|D|}|X| = 2^{|D|}|X| < 2^{|X|}. Hence, X has more zero-sets than there are elements of C(αX).

The cardinality of C(αX) is too coarse a measure to determine elements of K_{z}(X). When X is uncountable and discrete, it suffices to partition it into two parts, V_{1} and V_{2}, of equal cardinality and put T = cl_{βX} V_{1} − X. Then, |C(α_{T} X)| = |C(X)|, but, by Proposition 6.9, α_{T} X ∉ K_{z}(X).

Let us apply the construction of [C, Lemma 7.6] (using its terminology) to the case where X is uncountable and discrete, K = ωX, D = X ⊂ ωX = K and f: X → D described as follows. We partition X = ⋃_{d∈D} X_{d}, where |X_{d}| = |X|, for all d ∈ D and set f(x) = d when x ∈ X_{d}. Then, for finite F ⊆ X, f(X −F) = D is dense in K, as required. Hence, there is a compactification αX ∈ K(X) with αX − X homeomorphic to ωX. Proposition 6.11 does not apply here. But the next result will show that αX ∈ K_{nepi}(X). Recall that a scattered space is one in which every subspace has an isolated point; ωX is an example.

6.12. Proposition. Let X be a scattered space and let αX ∈ K(X) be such that αX − X is scattered. If X is not functionally countable then αX ∈ K_{nepi}(X). The result applies, in particular, if X is uncountable and discrete.

Proof. We prove this separately for X a P -space because in this case the proof just uses Theorem 4.5. Since X is scattered and αX−X is scattered, then αX is scattered. By [LR, Proposition 3.1], αX is functionally countable (every f ∈ C(αX) has countable range). If αX ∈ K_{epi}(X) = K_{epi}^{f}(X), we would have C(X) = C_{α}S_{α}^{−1} (Theorem 4.5 (C)). This would say that every element of C(X) had countable range, since it is true of elements of C_{α}. This is not the case.

The general case is done using the material from [BBR, Section 3] and the regular ring G(X). By [BBR, Proposition 3.3], if αX is functionally countable, so is G(X). If αX ∈ K_{epi}(X), then the same result says that X is functionally countable

7. Questions

Many questions remain unanswered and we only list a few of them. The case where X is uncountable and discrete remains a challenge.

7.1. Question. (i) Let X be an uncountable discrete space. We know that K_{z}(X) ⊆ K_{epi}(X) = K_{epi}^{f}(X). (a) Does K_{z}(X) = K_{epi}^{f}(X)? (b) Does K_{epi}^{f}(X) = K_{epi}^{1}(X). In more general form: Is there a P -space X where K_{z}(X) ≠ K_{epi}^{f}(X)? (ii) Is there a space where K_{epi}^{f}(X) ≠ K_{epi}^{1}(X)?

Various results above restrict where one might look for a positive answer to Question 7.1 (i), (a) or (b). In particular, the examples of elements of K_{nepi}(X) in Section 6 are found by showing that αX ∉ K_{z}(X). Another constraint is given in [HVW, Proposition 3.2] which says that if αX is a quasi-F space then αX ∈ K_{z}(X) if and only if αX = βX. Moreover, [BBR, Proposition 4.1 (iii)] and Theorem 4.5 (C) show that if X is a P -space and X is functionally countable then K_{z}(X) = K_{epi}(X) = K_{epi}^{f}(X).

7.2. Question. Is there a proof of Lemma 2.7 (ii) ([BRW, Lemma 5.1]) just using the sorts of methods in this paper? Is there a theorem in the spirit of [HM2, Theorem 8.2] for αX ∈ K_{epi}(X) (rather than αX ∈ K_{z}(X))?

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Department of Mathematics and Statistics University of Ottawa, Ottawa, ON, K1N 6N5

Department of Mathematics and Statistics Concordia University, Montreal, QC, H4B 1R6

Email: wburgess@uottawa.ca
      raphael@alcor.concordia.ca

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