Theory and Applications of Categories, Vol. 22, No. 3, 2009, pp. 54–76.

SEARCHING FOR MORE ABSOLUTE CR-EPIC SPACES

MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL

Abstract. We continue our examination of absolute CR -epic spaces, or spaces with the property that any embedding induces an epimorphism, in the category of commutative rings, between their rings of continuous functions. We examine more closely the deleted plank construction, which generalizes the Dieudonné construction, and yields absolute CR -epic spaces which are not Lindelöf. For the Lindelöf case, an earlier paper has shown the usefulness of the countable neighbourhood property, CNP, and the Alster condition (where CNP means that the space is a P-space in any compactification and the Alster condition says that any cover of the space by G_{δ} sets has a countable subcover, provided each compact subset can be covered by a finite subset.) In this paper, we find further properties of Lindelöf CNP spaces and of Alster spaces, including constructions that preserve these properties and conditions equivalent to these properties. We explore the outgrowths of such spaces and find several examples that answer open questions in our previous work.

1. Introduction

Unless explicitly stated otherwise, all spaces in this paper are assumed to be Tychonoff (completely regular Hausdorff) and all functions are assumed continuous. We denote by C(X) the commutative unitary ring of real-valued continuous maps on X, with pointwise addition and multiplication and by C^{∗}(X) the subring of C(X) consisting of the bounded functions.

There is a functor β that is left adjoint to the inclusion of the category of Tychonoff spaces into all spaces. For any space X the inner adjunction X [figure] βX is a topological embedding if and only if X is Tychonoff. Since the unit interval [0, 1] is compact, it follows from the adjunction that Hom(βX, [0, 1]) [figure] Hom(X, [0, 1]) is an isomorphism from which one easily sees, from C^{∗}(X) = colim Hom(X, [−n, n]), that C^{∗}(βX) [figure] C^{∗}(X) is an isomorphism. The book [Gillman & Jerison (1960)] is devoted to studying the properties of the C and C^{∗} (contravariant) functors. We recall that a subset of X is called a zeroset (respectively, cozeroset) if it is the set of all points on which some real-valued function vanishes (respectively, is non-zero).

The first and third authors would like to thank NSERC of Canada for its support of this research. We would all like to thank McGill and Concordia Universities for partial support of the middle author’s visits to Montreal.

Received by the editors 2007-06-07 and, in revised form, 2008-08-05. Transmitted by Walter Tholen. Published on 2009-04-03. 2000 Mathematics Subject Classification: 18A20, 54C45, 54B30. Key words and phrases: absolute CR-epics, countable neighbourhood property, Alster’s condition

Dieudonné plank.

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr, John F. Kennison, and R. Raphael, 2007. Permission to copy for private use granted.

SEARCHING FOR MORE ABSOLUTE CR-EPIC SPACES  55

Before continuing, it will be useful to have some definitions. If a space X is embedded into a space Y, we say that X is C-embedded (respectively C^{∗}-embedded) in Y provided every function (respectively bounded function) on X extends to Y. These definitions are equivalent to saying that C(Y) [figure] C(X) (respectively C^{∗}(Y) [figure] C^{∗}(X)) is surjective. An embedding X [figure] Y is called CR -epic if the induced map C(Y) [figure] C(X) is an epimorphism in the category of commutative unitary rings. A space is called absolute CR -epic if every embedding into a larger space is CR -epic. If X [figure] K is a dense embedding and K is compact, we say that the inclusion X [figure] K (or, by abuse of notation, K alone) is a compactification of X. We will see in 2.4 below that a necessary and sufficient condition that a space be absolute CR -epic is that it be CR -epic in every compactification.

We have explored in several earlier papers the concept of CR -epic embeddings and absolute CR -epic spaces, [Barr, et al. (2003), Barr, et al. (2005), Barr, et al. (2007)]. In this paper we explore several classes of such spaces that generalize examples from the earlier paper. For example, in [Barr, et al. (2007)], we looked at the Tychonoff plank. In this paper, we consider the more general notion of deleted planks, see Section 3. These deleted planks give us further examples of absolute CR -epic spaces which are neither Lindelöf nor almost compact. Before outlining the other topics we consider, we will review the definition of CNP spaces, and note that Alster spaces proved very useful in our previous paper as reviewed in Section 2.

A space X is said to satisfy the Countable Neighbourhood Property (CNP) if a countable intersection of βX-neighbourhoods of X is a neighbourhood of X. We saw in [Barr, et al. (2007), Corollary 3.4] that every Lindelöf CNP space is absolute CR -epic. Since a locally compact space is open in any compactification [Gillman & Jerison (1960), 3.15D], it follows immediately that every locally compact Lindelöf space has the CNP and hence, if Lindelöf, is absolute CR -epic.

Section 3 deals with several generalizations of [Barr, et al. (2007), 7.15] in which it was shown that the Dieudonné plank was absolute CR -epic, even though it was not Lindelöf. Section 4 deals with properties of Alster spaces that were derived in [Barr, et al. (2007)] under additional, unnecessary, conditions. Section 5 shows that countable unions of Lindelöf absolute CR -epic spaces are absolute CR -epic. Section 6 describes counterexamples to certain plausible conjectures. Sections 7 and 9 give alternate characterizations of the CNP property, while Section 8 looks at spaces of the form βX − X.

1.1. Errata. We would like to mention two errors, one significant one not, in [Barr, et al. (2007)]. We claimed in the introduction to that paper that a perfect quotient of an absolute CR -epic space was absolute CR -epic. But we proved only that a perfect quotient of a Lindelöf CNP space is Lindelöf CNP. In Section 6 we give an example that shows that the claim is false.

The second is that we stated a weaker form of the converse part of Smirnov’s theorem ([Smirnov (1951)]), than is true or that we actually proved. The correct statement appears in 2.10 below.

2. General results

This section consists partly of material from [Barr, et al. (2007)]. The reason we have included it is that its contents, both definitions and theorems, are called upon frequently in the present paper. Since length is no longer a consideration in electronic publications, it seems like a good idea to make it as self-contained as feasible. See the original paper for proofs.

2.1. Standard definitions and notation. It is shown in [Gillman & Jerison (1960), Chapter 6] that βX is characterized as the unique compact space in which X is dense and C^{∗}-embedded. We denote by υX the Hewitt realcompactification of a Tychonoff space X; see Chapter 8 of [Gillman & Jerison (1960)] or 5.5(c) and 5.10 of [Porter & Woods (1988)]. A space X is called realcompact if X = υX. This space is characterized as the largest subspace of βX to which every real-valued function (including the unbounded ones) can be extended.

A continuous map θ : X [figure] Y is said to be perfect if it is closed and for all p ∈ Y, θ^{−1}(p) is compact. It can be shown that whenever K ⊆ Y is compact, so is θ^{−1}(K). The properties of perfect maps are explored in detail in [Porter & Woods (1988)]. However, be warned that they do not invariably assume that their functions are continuous, while we do.

In any topological space, a countable intersection of open sets is called a G_{δ} set. A cover by G_{δ} sets will be called a G_{δ} cover. A subset A of a space X will be called a P-set in X provided every G_{δ} set that contains A contains an open set containing A. If A = {x} is a singleton, then we will say that x is a P-point. If every element of X is a P-point, then X is called a P-space. We say that a cover of a space by not-necessarily-open sets is ample if each compact subset of the space is covered by finitely many of the sets. We say that a space is amply Lindelöf if every ample G_{δ} cover contains a countable subcover. We have subsequently discovered that an equivalent condition was first defined and exploited by K. Alster, [Alster (1988), Condition (∗)] in connection with the question of which spaces have a Lindelöf product with every Lindelöf space. Accordingly, we will rename this as Alster’s condition. (It must be admitted that we were never happy with “amply Lindelöf” in the first place.) A space that satisfies Alster’s condition will be called an Alster space. We will continue to call a cover of a space ample if every compact set is finitely covered. Since G_{δ} sets are closed under finite unions, we may and often will assume, without loss of generality, that our G_{δ} covers are closed under finite unions. For an ample G_{δ} cover, this means that every compact set is contained in some element of the cover.

We have studied Alster spaces in a related paper that concentrated on Alster’s original context, namely that the product of an Alster space and a Lindelöf space is Lindelöf. See [Barr, et al. (2007a)].

2.2. Notation. If θ : B [figure] A is a function, then we use the same θ for the direct image function P(B) [figure] P(A). This has a right adjoint θ^{−1} : P(A) [figure] P(B) and θ^{−1} itself has a right adjoint θ_{#} : P(B) [figure] P(A) that takes a set T ⊆ B to θ_{#}(T) = A − θ(B − T). It follows that if θ is a closed mapping between topological spaces, then θ_{#} takes open sets to open sets, a fact that will turn out to be important. Here are some properties of the adjunctions. Assume that S ⊆ A and T ⊆ B. Then

1. θ(T) ⊆ S if and only if T ⊆ θ^{−1}(S);

2. θ^{−1}(S) ⊆ T if and only if S ⊆ θ_{#}(T);

3. θ preserves unions, θ_{#} preserves intersections and θ^{−1} preserves both;

4. θ^{−1}(θ_{#}(T)) ⊆ T, with equality when θ is injective;

5. θ_{#}(θ^{−1}(S)) ⊇ S with equality when θ is surjective;

6. θ(θ^{−1}(S)) ⊆ S, with equality when θ is surjective;

7. θ^{−1}(θ(T)) ⊇ T with equality when θ is injective;

8. θ_{#}(T) ⊆ θ(T) if and only if θ is surjective;

9. θ^{−1}(θ(θ^{−1}(S))) = θ^{−1}(θ_{#}(θ^{−1}(S))) = θ^{−1}(S);

10. θ(θ^{−1}(θ(T))) = θ(T);

11. θ_{#}(θ^{−1}(θ_{#}(T))) = θ_{#}(T).

Incidentally, θ_{#} is called the universal image in topos theory and usually denoted ∀_{θ}. In contrast, the direct image is called the existential image and denoted ∃_{θ}.

We can use this to give an efficient proof of the following well-known folklore.

2.3. Proposition. A perfect preimage of a Lindelöf space is Lindelöf.

Proof. Let θ : Y [figure] X be a perfect map and suppose that X is Lindelöf. Let U be an open cover of Y. We can suppose without loss of generality that U is closed under finite unions. If p ∈ X, then θ^{−1}(p) is compact, hence there is some U ∈ U with θ^{−1}(p) ⊆ U, whence p ∈ θ_{#}(U). Thus {θ_{#}(U) | U ∈ U} is an open cover of X and has a countable refinement, say {θ_{#}(U_{n})}. Then {θ^{−1}(θ_{#}(U_{n}))} is an open cover of Y. Since θ^{−1}(θ_{#}(A)) ⊆ A for any A ⊆ Y, we see that {θ^{−1}(θ_{#}(U_{n}))} refines U.

If {X_{i} | i ∈ I} are disjoint open subspaces of X for which X = ⋃_{i∈I} X_{i} we will say that X is the sum of the X_{i} and write X = ∑_{i∈I} X_{i}. We will also write X = X_{1} +⋯+X_{n} for a finite sum. This is the sum in the category of topological spaces and continuous maps. The sum is called “free union” by topologists.

2.4. Theorem. A space is absolute CR -epic if and only if every dense embedding into a compact space is CR -epic.

Proof. Suppose σ : X [figure] Y is an embedding. We have a commutative diagram of embeddings

[figure: cl_{βY} (X); βY; ψ; X; cl_{βY} (X); τ; X; Y; σ; Y; βY; φ]

Apply the contravariant functor C to a commutative diagram

[figure: C(cl_{βY} (X)); C(βY); C(ψ); C(X); C(cl_{βY} (X)); C(τ); C(X); C(Y); C(σ); C(Y); C(βY); C(φ)]

The map C(φ) is always an epimorphism since Y [figure] βY is a C^{∗}-embedding ([Barr, et al. (2003), Proposition 2.1(i)]) and C(ψ) is surjective (hence an epimorphism) since cl_{βY} (X) is a closed subspace of the normal space βY. Then from elementary category theory, we see that C(σ) is epic if and only if C(τ) is.

2.5. Quotients. Since every Hausdorff compactification of a space X is a quotient space of βX modulo a closed equivalence relation, we will begin by looking at equivalence relations. Although the results are stated for βX, they are actually valid for any compactification of X.

2.6. Definition. Let X be a space. An equivalence relation E ⊆ βX × βX will be called admissible if it is a closed subspace of βX × βX and if (X × βX) ∩ E = Δ_{X} (the diagonal of X in X × X).

Throughout this paper, E will denote an admissible equivalence relation on the Stone-Čech compactification of a space, usually X, and θ : βX [figure] βX/E = K will denote the induced quotient map. The map θ, being continuous between compact sets, is closed. It is an immediate consequence that θ_{#} (see 2.2) takes open sets to open sets. Since θ is surjective, θ_{#}(U) ⊆ θ(U), so that when U is a βX-neighbourhood of X, both θ_{#}(U) and θ(U) are K-neighbourhoods of θ_{#}(X) and the admissibility of E implies that θ_{#}(X) = X.

2.7. Proposition. For any equivalence relation E on βX, the induced map θ : X [figure] K = βX/E is an embedding into a Tychonoff space if and only if E is admissible.

For the proof, see [Barr, et al. (2007), Proposition 2.5].

Central to our studies is the following theorem, which is essentially [Barr, et al. (2005), Corollary 2.14].

2.8. Theorem. A Lindelöf space X is absolute CR -epic if and only if, for every compactification K of X, every function in C(X) extends to a K-neighbourhood of X.

2.9. Lemma. Suppose E is an admissible equivalence relation on βX. Then for any f ∈ C(βX) and any n ∈ N, the set

U_{n} = {p ∈ βX | (p, q) ∈ E ⇒ |f(p) − f(q)| < 1/n}

is open in βX and contains X.

For the proof, see [Barr, et al. (2007), Lemma 2.7].

We will require both halves of the following result from general topology. It is well known; a readily accessible proof can be found in [Barr, et al. (2007), Theorem 2.8] except that to prove the full converse, you must replace every instance of βX by an arbitrary compactification.

2.10. Theorem. [Smirnov] If a Tychonoff space X is Lindelöf, then in any compactification K of X any open subset of K that contains X contains a cozeroset containing X. Conversely, if there is a compactification K of X with the property that every open subset of K that contains X contains a cozeroset containing X, then X is Lindelöf (and then the property holds for every compactification).

3. Some (punctured) planks are absolute CR -epic

In [Barr, et al. (2007), 7.15], we showed that the Tychonoff plank was absolute CR -epic. In this section, we extend this result to certain “generalized planks”.

3.1. Oscillation. We begin with a more detailed look at what is needed in order to extend a real-valued function from a subspace to a point of the containing space. Suppose X is densely embedded in Y, p ∈ Y − X, and f ∈ C(X). If U is a subset of Y, define

O(f, U) = sup{|f(x) − f(x′)| | x, x′ ∈ U ∩ X}

called the oscillation of f on U. Define

O(f, p) = inf{O(f, U) | U is a Y -neighbourhood of p}

called the oscillation of f at p.

3.2. Proposition. Let X be dense in Y, f ∈ C(X) and p ∈ Y − X. A necessary and sufficient condition that f be extendible continuously to p is that O(f, p) = 0.

Proof. The necessity of that condition is obvious. So suppose that O(f, p) = 0. For each positive integer n, there is a neighbourhood U_{n} of p such that x, x′ ∈ U ∩ X implies that |f(x) − f(x′)| < 1/n. Choose an arbitrary element x_{n} ∈ U_{n}. It is clear that the sequence f(x_{n}) is Cauchy and converges to a t ∈ R. Define f(p) = t. The proofs that this construction does not depend on the choice of the x_{n} and continuously extends f to p are routine and are left to the reader.

3.3. Proposition. Let X be dense in Y. Let A = {y ∈ Y | f extends to y}. Then the extension is continuous on A.

Proof. Let us continue to call the extension f. The construction of the preceding proposition implies that if U is a Y -open neighbourhood of y ∈ A, then for any x ∈ X ∩ U, |f(x)−f(y)| ≤ O(f, X ∩U). Thus if y′ ∈ U, we also see that |f(x)−f(y′)| ≤ O(f, X ∩U). Thus |f(y) − f(y′)| ≤ 2O(f, X ∩ U). It follows that O(f, A ∩ U) ≤ 2O(f, X ∩ U). Then O(f, X ∩ U) can be made arbitrarily small for appropriate choice of U and hence so can O(f, A ∩ U). This implies that the extended function is continuous on A.

A topological space that is a product of two spaces, often total orders, is sometimes called a plank. Sometimes the word is used to denote what we call a punctured plank, a product with one point removed. For example, let L denote the one-point Lindelöfization of the first uncountable ordinal, that is ω_{1} ∪ {ω_{1}} with the points of ω_{1} isolated and a neighbourhood of ω_{1} consisting of any set containing it whose complement is countable. The Dieudonné plank is the space (ω + 1) × L − {(ω, ω_{1})}. In [Barr, et al. (2007), 7.14, 7.15] we showed that the Dieudonné plank and certain other punctured planks are absolute CR -epic. Here we extend that result.

We will be dealing with complete lattices that are products of a finite number of complete chains (total orders). Each such lattice has a top and a bottom element that we will denote by ⊤ and ⊥, respectively. The topology on a chain is given by letting closed intervals be a subbasis of closed sets and the product topology is used for products.

3.4. Proposition. A complete chain is compact Hausdorff.

Proof. If X is a complete chain, then for any two elements x < y, suppose first that the open interval (x, y) is empty. In this case [⊥, x] and [y, ⊤] are disjoint clopen sets containing x and y respectively. If there is a p ∈ (x, y), then [⊥, p) and (p, ⊤] are disjoint open neighbourhoods of these points. It follows from [Kelley (1955), Theorem 5.6] that to establish compactness, it is sufficient to show that any family of closed intervals with the finite intersection property has non-empty intersection. Suppose we have such a family {[x_{α}, y_{α}]}. Let x = sup x_{α} and y = inf y_{α}. It will suffice to show that x ≤ y. If not, there will be some α for which x ≰ y_{α} and similarly, some γ with x_{γ} ≰ y_{α}. But then [x_{γ}, y_{γ}] ∩ [x_{α}, y_{α}] = ∅, a contradiction.

A space X is almost compact if βX − X has at most one point. This is equivalent to the fact that of any pair of disjoint zerosets, at least one is compact. A space is pseudocompact if every real-valued is bounded. An almost compact space is pseudocompact (see [Gillman & Jerison (1960), Exercise 6J]).

3.5. Proposition. Suppose X and Y are compact spaces and x_{0} ∈ X and y_{0} ∈ Y are such that X − {x_{0}} and Y − {y_{0}} are almost compact. Then X × Y − {(x_{0}, y_{0})} is almost compact.

Proof. According to [Gillman & Jerison (1960), 9.14] the product of a compact space and a pseudocompact space is pseudocompact. In particular, X × (Y − {y_{0}}) is pseudocompact. According to [Glicksberg, 1959, Theorem 1], this implies that

β(X × (Y − {y_{0}})) = X × Y

Since the β-compactification of any space between Z and βZ is βZ, we conclude in this case that

β(X × Y − {(x_{0}, y_{0}))} = X × Y

3.6. Proposition. Suppose that x and y are P-points in X and Y, respectively. Then (x, y) is a P-point of X × Y.

Proof. If {W_{n}} is a countable family of neighbourhoods of (x, y), there are for each n, neighbourhoods U_{n} and V_{n} of x and y, respectively, such that U_{n} × V_{n} ⊆ W_{n} and then ⋂ U_{n} × ⋂ V_{n} ⊆ ⋂ W_{n}, which is thereby a neighbourhood of (x, y).

We will say that a complete chain X has (respectively lacks) a proper countable cofinal subset when X − {⊤} does. Clearly X lacks a countable cofinal subset if and only if X − {⊤} is countably complete.

3.7. Theorem. Let X be a finite product of complete chains, each which lacks a proper countable cofinal subset. Then X − {⊤} is almost compact and ⊤ is a P-point of X.

Proof. First suppose X is a chain that lacks a proper countable cofinal set. We claim that β(X − {⊤}) = X. Since the β-compactification of a space is characterized as the unique compactification in which X is C^{∗}-embedded, this amounts to showing that every bounded function on X −{⊤} extends continuously to ⊤. Suppose f : X − {⊤} [figure] [0, 1]. We must show that O(f, ⊤) = 0. If not, there is some ϵ > 0 such that for every neighbourhood U of ⊤, there are elements x, x′ ∈ U −{⊤} for which |f(x)−f(x′)| > ϵ. A neighbourhood base at ⊤ consists of sets of the form (x, ⊤]. Choose one such neighbourhood, say (x_{0}, ⊤] and choose x_{1}, x′_{1} ∈ (x_{0}, ⊤) so that f(x_{1})−f(x′_{1}) > ϵ. Now choose elements x_{2}, x′_{2} ∈ (x_{1}∨x′_{1}, ⊤) so that f(x′_{2}) − f(x_{2}) > ϵ and so on by induction. We get two increasing sequences x_{1} < x_{2} < ⋯ < x_{n} < ⋯ and x′_{1} < x′_{2} < ⋯ < x′_{n} < ⋯ such that x_{n} < x′_{n+1}, x′_{n} < x_{n+1} and f(x_{n}) − f(x′_{n}) > ϵ. We see, in particular, that x_{n−1} < x′_{n} < x_{n+1} for all n, which implies that x = sup x_{n} = sup x′_{n}. The fact that X lacks a countable cofinal subset implies that x ≠ ⊤. Next we claim that the open intervals containing a point constitute a neighbourhood at that point. In fact, the complement of an open interval consists of at most two closed intervals and is therefore open. On the other hand, the complement of a finite union of closed intervals is the union of a finite set of open intervals, at most one of which will contain any given point. If there were a neighbourhood (x, x′) that contained none of the x′_{n}, then x′ would be a smaller upper bound for the sequence of x_{n} and similarly for the x′_{n}. Thus the fact that f(x_{n}) − f(x′_{n}) > ϵ contradicts the fact that f is well-defined and continuous at x.

If {U_{n}} is a countable family of neighbourhoods of ⊤, we can assume they have the form (x_{n}, ⊤], whose intersection is [sup x_{n}, ⊤]. Since sup x_{n} < ⊤, we see that (sup x_{n}, ⊤] is a open neighbourhood of ⊤.

An obvious induction based on the previous two propositions extends the conclusion from chains to finite products of them.

Now we turn to the case of spaces that do have countable cofinal sets.

3.8. Theorem. Suppose that X = ∏ X_{i} is a product of finitely many complete chains, each of which has a countable cofinal set. Then X − {⊤} is locally compact and σcompact.

Proof. The local compactness is obvious. It is obvious that a complete chain with a countable cofinal subset is σ-compact, so it suffices to show that if X and Y are spaces and x ∈ X and y ∈ Y are such that X − {x} and Y − {y} are σ-compact then the same is true of X × Y − {(x, y)}. Since the product and finite sum of σ-compact spaces is σ-compact, the conclusion follows from

X × Y − {(x, y)} = ((X − {x}) × (Y − {y})) ∪ ({x} × (Y − {y})) ∪ ((X − {x}) × {y})

A space is said to be weakly Lindelöf if from every open cover a countable subset can be found whose union is dense. Obviously a Lindelöf space is weakly Lindelöf and so is any space with a dense Lindelöf subspace.

3.9. Theorem. Suppose that Y is a space and y_{0} is a non-isolated P-point. Suppose Z is a space with a point z_{0} such that Z − {z_{0}} is weakly Lindelöf. Then Y × Z − {(y_{0}, z_{0})} is C-embedded in Y × Z.

Proof. The case that z_{0} is isolated is immediate from 3.7. Otherwise, let f ∈ C(Y ×Z − {(y_{0}, z_{0})}). For each z ∈ Z − {z_{0}}, f is continuous at (y_{0}, z). Thus for each n ∈ N, there is a neighbourhood V (z, n) of y_{0} in Y and a neighbourhood W (z, n) of z in Z − {z_{0}} such that O(f, V (z, n) × W (z, n)) < 1/n. Suppose z(1, n), z(2, n), … is a countable set of points of Z − {z_{0}} such that W (n) = ⋃_{m∈N} W (z(m, n), n) is dense in Z − {z_{0}}. Let V (n) = ⋂_{m∈N} V (z(m, n), n). It follows that O(f, V (n) × W (n)) ≤ 1/n. Since y_{0} is a P-point, V (n) is a neighbourhood of y_{0}. For any y ∈ Y − {y_{0}} both functions f(y, −) and f(y_{0}, −) are continuous on Z − {z_{0}} and hence, for any z ∈ Z − {z_{0}} and any m ∈ N, there is a neighbourhood T (m) of z such that the oscillation in T_{m} of both f(y, −) and f(y_{0}, −) is at most 1/m. There is some p ∈ W (n) ∩ T (m) and we have

\begin{aligned}|f(y, z) − f(y_{0}, z)| & ≤ |f(y, z) − f(y, p)| + |f(y, p) − f(y_{0}, p)| + |f(y_{0}, p) − f(y_{0}, z)| \\ & < 1/m + 1/n + 1/m = 2/m + 1/n \end{aligned}

Since the left hand side does not depend on m this implies that |f(y, z) − f(y_{0}, z)| ≤ 1/n. Finally, let V = ⋂_{n∈N} V (n). Then for y ∈ V, we have f(y, z) = f(y, z_{0}) and we can extend f by f(y_{0}, z_{0}) = f(y, z_{0}) for any y ∈ V.

This works, in particular, if Z − {z_{0}} is Lindelöf or if it contains a dense Lindelöf subspace.

From this, we get the following generalization of [Barr, et al. (2007), Theorem 7.14].

3.10. Theorem. Suppose the Lindelöf space Y is the union of a locally compact subspace and a non-isolated P-point y_{0}. Suppose Z is a compact space that has a proper dense Lindelöf subspace and z_{0} is a point not in that subspace. Then Y × Z − {(y_{0}, z_{0})} is absolute CR -epic.

Proof. Since D = Y × Z − {(y_{0}, z_{0})} is C-embedded in Y × Z, it follows that the realcompactification υ(D) = Y × Z. Since Y × Z is the union of a locally compact space and a compact space, the result follows from [Barr, et al. (2007), Theorem 7.11].

3.11. Theorem. Suppose that X = ∏_{i=1}^{n} X_{i} is a finite product of complete chains. Assume that ⊤ is not an isolated point of any of the chains. Then X − {⊤} is absolute CR -epic.

Proof. Divide the spaces into two classes, Y_{1}, Y_{2}, …, Y_{k} that lack proper countable cofinal sets and Z_{1}, Z_{2}, …, Z_{l} that have them. Let Y = ∏ Y_{i} and Z = ∏ Z_{j}. We know that ⊤ is a non-isolated P-point of Y. We know that Z − {⊤} is σ-compact, hence Lindelöf. It follows that Y × Z − {(⊤, ⊤)} is absolute CR -epic.

4. Alster’s condition

Recall from the introduction that a space is an Alster space if every ample cover by G_{δ} sets has a countable refinement.

Most of the following theorem was proved as [Barr, et al. (2007), Theorem 4.7] under the additional hypothesis that the spaces satisfied the CNP. That condition was not used in the proofs; it was simply that we never studied Alster’s condition separately.

4.1. Theorem.

1. The product of two Alster spaces is Alster space.

2. A closed subspace of an Alster space is Alster space.

3. If t : Y [figure] X is a continuous surjection and Y satisfies Alster’s condition, so does X.

4. A union of countably many Alster spaces is an Alster space.

5. A Lindelöf space is an Alster space if every point has a neighbourhood that satisfies Alster’s condition.

6. A cozero-subspace of an Alster space is an Alster space.

7. If θ : Y [figure] X is a perfect surjection (see 2.1) and X satisfies Alster’s condition so does Y.

Proof. We will mention briefly what, if any changes are needed from the proofs in [Barr, et al. (2007), Theorem 4.7].

1. This is just [Barr, et al. (2007), Theorem 4.5].

2. See [Barr, et al. (2007), proof of 4.7.2]

3. Let U be an ample G_{δ} cover of X. Since θ^{−1} preserves open sets and commutes with meets, it follows that θ^{−1}(U) = {θ^{−1}(U) | U ∈ U} is a G_{δ} cover of Y. It is also ample since if K is a compact subset of Y, θ(K) is a compact subset of X and is therefore finitely covered by U. But then θ^{−1}(θ(K)), which contains K, is finitely covered by θ^{−1}(U). It follows that Y is covered by some countable subset {θ^{−1}(U_{n})} and it is clear that {U_{n}} covers X.

4. A countable sum Alster spaces is obviously an Alster space and a union is a continuous image of a sum, so the preceding item finishes it.

5. Immediate from the preceding item.

6. Immediate from the second item and the preceding one since cozerosets are F_{σ}.

7. The proof of [Barr, et al. (2007), 4.7.5] does not use the CNP condition. However the first part of the proof contains an unneeded gap, so we give a complete proof here. Suppose that θ : Y [figure] X is perfect and X satisfies Alster’s condition. Let V be an ample cover of Y. We may assume that V is closed under finite unions. If K ⊆ X is compact—possibly a singleton—then θ^{−1}(K) is compact and hence there is a V ∈ V with vθ^{−1}(K) ⊆ V and hence K = θ_{#}(θ^{−1}(K)) ⊆ θ_{#}(V). This implies that U = {θ_{#}(V) | V ∈ V } is an ample cover of X and hence contains a finite subcover, say U_{0}. Clearly V_{0} = {θ^{−1}(U) | U ∈ U_{0}} is a countable cover of Y. Since θ^{−1}(t_{#}(V)) ⊆ V, we see that V_{0} refines V .

4.2. Theorem. Suppose that the Lindelöf space X has the property that, for each p ∈ X, whenever V is a G_{δ} set of βX that contains p, then V ∪ X is a βX-neighbourhood of p. Then X is a CNP space that satisfies Alster’s condition.

Proof. CNP is immediate. Let U be an ample G_{δ} cover of X of βX. We can suppose, without loss of generality, that U is closed under finite union. Let p be a point of X. We will show that there is a βX-neighbourhood of p that is covered by a countable subset of U. Let p ∈ U ∈ U. We claim there is a continuous f : βX [figure] [0, 1] such that p ∈ Z[f] ⊆ U. In fact, this is immediate when U is open and it follows for G_{δ} sets since a countable intersection of zerosets is a zeroset. Since V = Z[f] is a G_{δ} of βX containing p, the hypothesis implies that X ∪ V is a βX-neighbourhood of p. Thus there is a compact βXneighbourhood W of p with W ⊆ X ∪V. It follows that W ∩coz(f) ⊆ X. But W ∩coz(f) is σ-compact and hence there are U_{1}, U_{2}, … ∈ U such that W ∩ coz(f) ⊆ ⋃ U_{n} and then W ⊆ U ∪ ⋃ U_{n}. Having done this for each point p ∈ X, the Lindelöf property implies that there are countably many points for which the corresponding sets of W cover X and then so do the corresponding sequences of U_{n}.

For example, this theorem implies that when p ∈ βN − N is a P-point, then N ∪ {p} is absolute CR -epic. See also [Barr, et al. (2007), Theorem 5.4].

5. Countable unions of Lindelöf absolute CR -epic spaces

5.1. Proposition. A countable union of Lindelöf absolute CR -epic subspaces whose interiors cover it, is absolute CR -epic.

Proof. Let X = ⋃ X_{n} = ⋃ int_{X}(X_{n}). Since X is a countable union of Lindelöf spaces, it is Lindelöf. Let K be a compactification of X and f ∈ C^{∗}(X). Let U_{n} = int_{X}(X_{n}) and K_{n} = cl_{K}(X_{n}). Since K_{n} is a compactification of X_{n}, f|X_{n} extends to a K_{n}-open subset W_{n} ⊆ K_{n} that contains X_{n}. Since U_{n} is open in X there is a K-open set V_{n} such that X ∩ V_{n} = U_{n}. It is well known, as X is dense in K, that V_{n} ⊆ cl_{K}(U_{n}) ⊆ cl_{K}(X_{n}) = K_{n}. We claim that W_{n} ∩ V_{n} is open in K. In fact, W_{n} ∩ V_{n} is open in K_{n}, hence open in the subset V_{n}, while V_{n} is open in K. Since f|X_{n} extends to W_{n}, the conclusion follows from 3.3.

5.2. Theorem. The classes of Lindelöf absolute CR -epic spaces, Lindelöf CNP spaces, and Alster spaces are all closed under countable open unions.

Proof. The first case is included in the preceding proposition. The second is essentially [Barr, et al. (2007), Theorem 4.7.3] once we note that a countable union of Lindelöf spaces is Lindelöf. The case of Alster spaces follows from Theorem 4.1.4.

It is trivial to extend these conclusions to covers whose interiors also cover. We omit the details.

5.3. Theorem. A countable, locally finite union of closed Lindelöf CNP (respectively Alster) spaces is Lindelöf CNP (respectively Alster).

Proof. By [Barr, et al. (2007), Theorem 4.7.4], a countable sum of Lindelöf CNP spaces is Lindelöf CNP and we show in [Barr, et al. (2007a)] that CNP spaces are closed under perfect image (and it is well known that being a Lindelöf space is as well.) Thus it is sufficient to show that the map from the sum to the union is perfect. The inverse image of each point is finite, hence compact. Let A = ∑ A_{n} be a closed subset of the sum with A_{n} closed in X_{n}. If p ∉ ⋃ A_{n} there is a neighbourhood U of p that meets only finitely many of the X_{n}, say X_{1}, X_{2}, …, X_{m}. For each n ≤ m, the set A_{n} is closed in X_{n}, which is closed in X and hence ⋃_{n=1}^{m} A_{n} is closed in X. Since p ∉ ⋃_{n=1}^{m} A_{n}, there is a neighbourhood V of p which misses that union. Since U does not meet any X_{n} for n > m, neither does V ∩ U so there is a neighbourhood of p that does not meet A. Since p was an arbitrary point not in A, we conclude that A is closed.

By contrast, we will see in the next section that even a finite union of closed Lindelöf absolute CR -epic, but not CNP spaces, need not be absolute CR -epic.

6. Three counter-examples

In this section we show by example that a Lindelöf absolute CR -epic space need not satisfy the CNP, that a perfect quotient of a Lindelöf absolute CR -epic space need not be absolute CR -epic, and that a Lindelöf space need not be absolute CR -epic even if it is the union of two closed absolute CR -epic subspaces.

6.1. The space X. Let {X_{n}} be any countable family of non-compact absolute CR - epic Lindelöf spaces. We can picture the space β (∑ X_{n}) as the union of three disjoint subspaces:

B = ∑ X_{n}  C = ∑(βX_{n} − X_{n}) A = β(∑ X_{n}) − ∑ βX_{n}

We let X be the space A ∪ B. Since X lies between ∑ X_{n} and its β-compactification, we see that βX = β (∑ X_{n}) and therefore βX − X = C.

Since X_{n} is a summand of ∑ X_{n}, βX_{n} is a summand of β (∑ X_{n}), so that ∑ βX_{n} is open in β (∑ X_{n}) which implies that A is compact. But then X is the union of a Lindelöf space and a compact space and is therefore Lindelöf.

6.2. X is absolute CR -epic, but does not satisfy the CNP. We define the level function ℓ : βX [figure] N ∪ {∞}, the one point compactification of N, as the continuous extension of the function that has the value n on X_{n}. It is clear that ℓ(p) = n if and only if p ∈ βX_{n} and ℓ(p) = ∞ otherwise.

Let E be an admissible equivalence relation on βX. This means that βX/E is Hausdorff and that X is embedded in it, see 2.6. Let θ : βX [figure] βX/E be the canonical projection. We will say that an element p is E-equivalent to an element q if (p, q) ∈ E, which is the same as θ(p) = θ(q).

6.3. Lemma. There is an N ∈ N such that whenever p is E-equivalent to q, then either p = q or ℓ(p), ℓ(q) < N.

Proof. Suppose we can find elements of arbitrarily high level that are E-equivalent to elements other than themselves. We will consider two cases. First suppose we can find a sequence of elements (p_{n}, q_{n}) ∈ E such that the levels of the p_{n} are unbounded while those of the q_{n} are bounded. Restricting to a subsequence, if necessary, we can assume that for some N ∈ N we have ℓ(q_{n}) ≤ N for all n, while N < ℓ(p_{1}) < ℓ(p_{2}) < · · ·. The sequence {(p_{n}, q_{n})} ⊆ E thus constructed is discrete since the βX_{ℓ(p_{n})} × βX_{ℓ(q_{n})} are a family of disjoint open sets of βX × βX each containing one element of the set. But the sequence has a limit point (p, q) and it is clear that ℓ(p) = ∞, which implies that p ∈ X, while ℓ(q) ≤ N. The result is that an element of X is E-equivalent to some other element of βX (which might or might not belong to X) which contradicts the fact that E is admissible.

In the other case, there are pairs (p_{n}, q_{n}) ∈ E in which p_{n} ≠ q_{n} and both ℓ(p_{n}) and ℓ(q_{n}) are unbounded. In that case, proceed as above, but assume that we have chosen subsequences so that ℓ(p_{n−1}) ∨ ℓ(q_{n−1}) < ℓ(p_{n}) ∧ ℓ(q_{n}). Again the sequence {(p_{n}, q_{n})} is discrete and hence has a limit point (p, q) but now both elements belong to X and we must show that p ≠ q. We do this as follows. Since p_{1} ≠ q_{1}, there is a function f_{1}, defined on all the elements ∑ βX_{n} of levels up to ℓ(p_{1})∨ℓ(q_{1}) such that f_{1}(p_{1}) = 0 and f_{1}(q_{1}) = 1. Since the levels of both p_{2} and q_{2} are above those of both p_{1} and q_{1}, this function can be extended to a function f_{2} defined on the summands of levels up to ℓ(p_{2}) ∨ ℓ(q_{2}) in such a way that f_{2}(p_{2}) = 0 and f_{2}(q_{2}) = 1. Continuing in this way, we define a function f on the sum of the finite levels such that f(p_{n}) = 0 and f(q_{n}) = 1. This function has a unique extension to the elements at infinite level and it is clear that f(p) = 0, while f(q) = 1. Again, this contradicts the admissibility of E.

6.4. Theorem. The space X is absolute CR -epic, but does not satisfy the CNP.

Proof. We will use [Barr, et al. (2007a), Theorem 2.4]. Let E be an admissible equivalence relation on βX so that K = βX/E is a compactification of X with canonical projection θ : βX [figure] K. For a fixed integer N, let L_{N} = {p ∈ βX | ℓ(p) ≤ N} and U_{N} = {p ∈ βX | ℓ(p) > N}. Then βX is the topological sum L_{N} +U_{N}. Let N be as in the preceding lemma. This forces θ to respect the decomposition so that K = θ(L_{N}) + θ(U_{n}). Let f : X [figure] [0, 1] be a given continuous function. We must show that f extends to an open subset of K by showing that it extends to an open subset of θ(L_{N}) and to an open subset of θ(U_{N}). Since ∑_{n=1}^{N} X_{n} is absolute CR -epic the restriction of f to that set extends to an open set in ∑_{n=1}^{N} βX_{n}, which is open in ∑_{n=1}^{∞} βX_{n}. Since there are no pairs of distinct E-equivalent elements in ∑_{n>N} βX_{n}, the restriction of f to elements of X of level above N extends to that open set. Thus f extends to an open set in K = βX/E that contains X. Since X is Lindelöf, it follows that X is absolute CR -epic.

To see that X is not CNP, choose a sequence of elements S = {p_{n}} such that p_{n} ∈ βX_{n} − X_{n} ⊆ βX − X. Let U_{n} = βX − {p_{n}}. Then each U_{n} is a neighbourhood of X in βX. But ⋂ U_{n} is not a neighbourhood of X since no point in the set D = cl_{βX}(S) − S has a neighbourhood in the intersection. Note that since ℓ is continuous, each point in D is at level ∞ and is therefore an element of X.

6.5. A perfect quotient of X need not be absolute CR -epic. Let S and D be as in the preceding paragraph. Each p_{n} has a neighbourhood, namely βX_{n}, that contains no other element of S so that D is closed and, as shown above, D ⊆ X. We then show that:

6.6. Proposition. If D is as above, the quotient space Y = X/D gotten by identifying the points of D to a single point is Tychonoff and the quotient map θ : X [figure] Y is perfect.

Proof. The set D is compact in βX, hence also closed in X. Let W ⊆ X be closed (possibly a single point) and p ∈ X be a point such that θ(p) ∉ θ(W). If W ∩ D = ∅, there is a function f : X [figure] [0, 1] that vanishes on D ∪ {p} and is 1 on W. If W ∩ D ≠ ∅, there is a function f that vanishes at p and is 1 on W ∪ D. In either case, f is constant on D and so descends to Y. Since every point not in θ(W) can be separated from it by a function, θ(W) is closed. Applied to a single point, we see that Y is Hausdorff.

6.7. Corollary. The space Y is not absolute CR -epic.

Proof. Let K be the space βX with the points of D identified to a point. There is an obvious dense embedding of Y in K. The sequence {p_{n}}, considered as sequence in K −Y, converges to the point θ(D). The conclusion follows from [Barr, et al. (2005), Theorem 2.22].

This gives a second argument that X is not CNP, since a perfect quotient of a CNP space is CNP, [Barr, et al. (2007), Theorem 3.6.5].

6.8. A Lindelöf space that is the union of two absolute CR -epic closed subspaces need not be absolute CR -epic. To get this example we use the space X of 6.2, with the additional assumption that all the X_{n} are locally compact. That implies that ∑ X_{n} is locally compact and hence open in β (∑ X_{n}). We will use ∔ to denote a disjoint union of subspaces that is not necessarily a topological sum. As above, we let X = A ∔ B, whence βX = A ∔ B ∔ C. Since A is compact and since each X_{n} is locally compact, it readily follows that B is open in βX. Since X_{n} is a topological summand of ∑ X_{n}, βX_{n} is a topological summand of β (∑ X_{n}) and it follows that B ∔ C = ∑ βX_{n} is open in β (∑ X_{n}). Now let X′ denote a space disjoint from and homeomorphic to X and let A′, B′, and C′ be the subspaces of βX′ corresponding to A, B, and C, respectively. We begin with:

6.9. Lemma. Suppose A can be embedded as a closed subset of the (Tychonoff) spaces X and Y. Then the pushout space Z = X +_{A} Y is Tychonoff provided either X = Y or one of X and Y is normal.

Proof. We begin with a closed subset B of Z and a point p ∉ B. We divide the proof into three cases. The first case, that p ∉ A, uses neither subsidiary hypothesis. We can suppose, without loss of generality that p ∈ X. Then p ∉ A ∪ (B ∩ X) so that there is a function f ∈ C(X) such that f(p) = 0 and f = 1 on A ∪ (B ∩ X). We can extend f to all of Z by letting f = 1 on all Y. Since f = 1 on A this is compatible with the previous values. Then f|X is continuous and f|Y is even constant. If θ : X + Y [figure] Z is the quotient mapping, θ^{−1}(X) = X ∪ A which is closed in X + Y and hence X is closed in Z. Similarly, Y is closed in Z and hence f is continuous on Z.

If p ∈ A and X = Y, then let φ_{0} and φ_{1} be the two maps of X [figure] Z, using the first summand and the second. Then one readily sees that p ∉ φ_{0}^{−1}(B) ∪ φ_{1}^{−1}(B) and hence there is an f ∈ C(X) such that g(p) = 0 and g = 1 on φ_{0}^{−1}(B) ∪ φ_{1}^{−1}(B). Let f ∈ C(Z) be the function such that f|X = g and f|Y = g. Since g is the same on both copies of X, this function is well defined and clearly is 1 on B.

Finally, suppose that p ∈ A and Y is normal. Let f ∈ C(X) such that f(p) = 0 and f = 1 on B ∩ X. Extend this to a continuous function on X ∪ B by letting f = 1 on all of B. Since this is 1 on B ∩ X the extension is continuous as above. Since Y is normal, the function f|(A ∪ (B ∩ Y)) can be extended continuously to all Y (see [Gillman & Jerison (1960), Exercise 3D]) and, as above, the total function is also continuous and is 1 on B.

6.8 continued. Define Y = X +_{A=A′} X′, that is the sum X + X′ with each point of A identified with the corresponding point of A′. The square

[figure: X′; Y; A ≅ A′; X′; A ≅ A′; X; X; Y]

is a pushout in the category of topological spaces. From the preceding lemma, we see that the facts that A is closed and X is Lindelöf, hence normal, imply that Y is Tychonoff. Now we can show that Y is not absolute CR -epic. Define f : Y [figure] [0, 1] by f(x) = 1/n if x ∈ X_{n} while f vanishes on all of X′. This is clearly continuous on the closed sets X and X′ and agrees on X ∩ X′ = A and so is continuous on Y. But it is clear that if f were to extend to a single y ∈ C = ∑(βX_{n} − X_{n}), then for βX_{n} − X_{n} we would have to have f(y) = 1/n on the one hand and f(y) = 0 on the other. Thus the maximal extension of f on K is to Y, which is not open in βY because any sequence of points of increasing level in C has all its limit points in A.

7. Characterizations of Lindelöf CNP spaces

In the following theorem, L denotes the convergent sequence 1, 1/2, 1/3, …, 0.

7.1. Theorem. A Lindelöf space satisfies the CNP if and only if its product with L is absolute CR -epic.

Proof. The product of a Lindelöf CNP space with a compact space is Lindelöf CNP ([Barr, et al. (2007), Theorem 4.6]) and therefore absolute CR -epic, so that direction is trivial. Conversely, assume that X × L is absolute CR -epic. Suppose {U_{n}} is a countable family of βX-open neighbourhoods of X. We may assume, without loss of generality, that U_{n} ⊇ U_{n+1}. We must show that U = ⋂ U_{n} is a βX-neighbourhood of X. Define an equivalence relation E on βX × L as follows. Let A_{n} = βX − U_{n} and E_{n} be the equivalence relation generated by ((p, 1/n), (p, 0)) ∈ E_{n} whenever p ∈ A_{n}. Then let E be the equivalence relation generated by ⋃ E_{n}.

7.2. Lemma. The set E is closed in (βX × L) × (βX × L).

Proof. The map βX ×L×L [figure] (βX ×L)×(βX ×L) that sends (p, s, t) to ((p, s), (p, t)) is clearly a closed embedding. Let D denote the subset consisting of the elements of the form (p, t, t). It is then readily verified that E is the direct image under this map of the set B that is the union of the following four sets (in which we denote the usual inf in Z by ∧):

1. {(p, 1/n, 1/m) | p ∈ A_{n∧m}};

2. {(p, 1/n, 0) | p ∈ A_{n}};

3. {(p, 0, 1/n) | p ∈ A_{n}};

4. D.

It suffices to show that B is a closed subset of βX × L × L. It is easily verified that t = (q, 1/n, 1/m) ∈ B if and only if n = m or q ∈ A_{n∧m}. It follows that if t = (q, 1/n, 1/m) ∉ B, then U_{n∧m} ×{1/n}×{1/m} is a neighbourhood of t that misses B. Similarly, s = (q, 1/n, 0) ∈ B if and only if q ∈ A_{n}, so that if s ∉ B then U_{n} × {1/n} × H_{n} is a neighbourhood of s which misses B where H_{n} = {0} ∪ {1/k | k > n}. A similar proof works for (q, 0, 1/n) and, since D is obviously closed, it readily follows that B is closed.

Proof of 7.1, concluded. Assume that X × L is absolute CR -epic. We see that (βX × L)/E is a compactification of X × L, see 2.7. Let f : X × L [figure] R be the second projection (recall that L ⊆ [0, 1]). Since X × L is absolute CR -epic, f extends to an open set W where W ⊆ (βX × L)/E. Consider the map βX [figure] βX × L which sends p to (p, 0). Let V be the inverse image of W under this map. Clearly V is an open subset of βX which contains X. It suffices to show that V is contained in ⋂ U_{n} . But if (p, 0) ∈ V then p must be in ⋂ U_{n} otherwise p ∈ A_{n} for some n and so ((p, 0), (p, 1/n)) ∈ E which shows that f cannot extend to (p, 0) as f(p, 0) = 0 but f(p, 1/n) = 1/n.

7.3. Remark. Since a closed subspace of a Lindelöf absolute CR -epic space is absolute CR -epic (use [Barr, et al. (2007), Theorem 6.1] in conjunction with the fact that a Lindelöf space is normal and hence any closed subspace is C^{∗}-embedded in it), one readily sees that if L is any space that contains a proper convergent sequence and X×L is Lindelöf absolute CR -epic, then X is Lindelöf CNP.

A space X is said to have the sequential bounded property or SBP at the point p if for any sequence {f_{n}} of functions in C(X) there is a neighbourhood of p on which each of the functions is bounded. A space has the SBP if it does so at every point. For example, every locally compact space has this property. So does every P-space. The easiest way to see this is to let p ∈ X and let U_{n} = {x ∈ X | |f_{n}(p) − f_{n}(x)| < 1}. Then ⋂ U_{n} is a G_{δ} containing p and in a P-space, every G_{δ} is open. Since both of these classes of spaces have the CNP, the following characterization comes as no surprise.

7.4. Theorem. A Lindelöf space is CNP if and only if it has the SBP at every point.

Proof. Suppose X is Lindelöf with the CNP. Let f_{1}, f_{2}, … be a sequence of functions in C(X). We can replace each f_{n} by 1 + |f_{n}| and assume that they are all positive and bounded away from 0. Then the functions g_{n} = 1/f_{n} are all bounded and hence extend to βX. Let U_{n} be the cozeroset of the extension of g_{n} to βX. The CNP implies that ⋂ U_{n} is a neighbourhood of X. Now let p ∈ X. There is a closed, hence compact, βX-neighbourhood V of p inside ⋂ U_{n}. Since every g_{n} is non-zero on V, it follows that every f_{n} is bounded there. In particular, every f_{n} is bounded in V ∩ X, which is an X-neighbourhood of p.

Conversely, suppose X satisfies the SBP. Let {U_{n}} be a sequence of K-neighbourhoods of X. By Smirnov’s theorem 2.10, the Lindelöf property allows us to choose, for each n, a function g_{n} : K [figure] [0, 1] such that X ⊆ coz(g_{n}) ⊆ U_{n}. Define f_{n} : K [figure] [1, ∞] by f_{n}(p) = 1/g_{n}(p) when g_{n}(p) ≠ 0 and f_{n}(p) = ∞ otherwise. Then X ⊆ fin(f_{n}) ⊆ U_{n}, where fin(f_{n}) = coz(g_{n}) is the set on which f_{n} is finite. For each p ∈ X, there is an Xopen set V_{p} on which each f_{n} is bounded, from which it is clear that each f_{n} is bounded on W_{p} = cl_{K}(V_{p}), which is a K-neighbourhood of p. Since f_{n} is bounded on W_{p}, it follows that W_{p} ⊆ fin(f_{n}) ⊆ U_{n} for each p and each n, so that W = ⋃_{p∈X} V_{p} is a K-neighbourhood of X contained in ⋂ U_{n}.

7.5. Definition. If f, g ∈ C(X) let us say that g surpasses f and write f ≺ g if there is a real number b > 0 such that f < bg.

7.6. Theorem. A Lindelöf space has CNP if and only if whenever f_{1}, f_{2}, ⋯ is a sequence of functions in C(X), there is a g ∈ C(X) that surpasses them all.

Proof. ⇐: Every f_{n} will be bounded on any neighbourhood on which g is bounded. ⇒: Let f_{1}, f_{2}, … be a sequence. We may assume, without loss of generality, that each f_{n} > 1. Using the SBP and the Lindelöf property, there is a countable cover U_{1}, U_{2}, … of X such that for all n, m ∈ N each f_{n} is bounded on each U_{m}. Since a Lindelöf space is paracompact, there is a partition of unity {t_{n}} subordinate to the cover. In fact, we may refine the cover and suppose that U_{n} = coz(t_{n}) (see [Kelley (1955), 5W and 5Y]). Let b_{n} be the sup of f_{n} on U_{1} ∪U_{2} ∪⋯∪U_{n−1}. Define h_{n} = f_{1} +f_{2} +⋯+f_{n} and g = ∑_{n∈N} h_{n}t_{n}. The local finiteness guarantees that this sum is actually finite in a neighbourhood of each point, so continuity is clear.

We next claim that x ∈ coz(t_{m}) implies that f_{n}(x) ≤ b_{n}h_{m}(x) for all n and m. In fact, if m < n, then f(x) ≤ b_{n} ≤ b_{n}h_{m} on U_{1} ∪ U_{2} ∪ ⋯ U_{n−1} ⊇ U_{m} = coz(t_{m}). If m ≥ n, then f_{n} is one of the summands of h_{m} so that f_{n} ≤ h_{m} ≤ b_{m}h_{m}. Note that the fact that each f_{n} ≥ 1 everywhere implies the same for every h_{n} and b_{n}.

We can now finish the proof. Given a point x ∈ X, let N(x) denote the finite set of indices n for which t_{n}(x) ≠ 0. Then for all m,

\begin{aligned}b_{n}g(x) & = ∑_{n∈N}b_{n}h_{n}(x)t_{n}(x) = ∑_{n∈N(x)} b_{n}h_{n}(x)t_{n}(x) \\ & ≥ ∑_{n∈N(x)} f_{m}(x)t_{n}(x) = ∑_{n∈N}f_{m}(x)t_{n}(x) = f_{m}(x) \end{aligned}

7.7. Example. Here is a nice application of Theorem 7.4. Say that a space satisfies the open refinement condition or ORC if the finite union closure of every ample G_{δ} cover has an open refinement. We explored this condition in some detail in [Barr, et al. (2007a)]. All P-spaces and all locally compact spaces satisfy the ORC and the class of spaces satisfying it is closed under finite products, closed subspaces and perfect images and preimages.

7.8. Theorem. For Lindelöf spaces, ORC implies CNP.

Proof. Let X be Lindelöf and satisfy the ORC and let f_{1}, f_{2}, … be a sequence of functions in C(X). For each compact set A ⊆ X and each n ∈ N, let b_{n}(A) = sup_{x∈A} |f_{n}(x)|. Let U_{n}(A) = {x ∈ X | |f_{n}(x)| < b_{n}(A) + 1} and U(A) = ⋂_{n∈N} U_{n}(A). Then U(A) is a G_{δ} containing A. The cover by the set of U(A), taken over all the compact subsets of X, is an ample G_{δ} cover. Each f_{n} is bounded on each U(A) and hence is bounded on the union of any finite number of them. Therefore each f_{n} is bounded on each set in an open refinement of {U(A)}.

8. Outgrowths

The outgrowth of a Tychonoff space X is the space βX − X. In this section, we explore some of the ways a space and its outgrowth influence each other. By a reduced outgrowth of a Tychonoff space, we mean any space of the form K − X where K is a compactification of X. Obviously a reduced outgrowth is a quotient of the outgrowth, but not every quotient is a reduced outgrowth.

8.1. Lemma. Let K be a compactification of X. Then K − X is countably compact if and only if βX − X is.

Proof. Let θ : βX [figure] K be the canonical quotient map. If βX − X is countably compact, then its image, K − X clearly is too. To go the other way, we have from [Kelley (1955), 5E(a)] that if βX − X is not countably compact, there is a sequence S = {p_{1}, p_{2}, p_{3}, …} that has no cluster point in βX − X. But then every cluster point in βX of the sequence lies in X. Either S has an infinite subsequence all of whose images in K are the same or S has an infinite subsequence all of whose images in K are distinct. We can suppose, without loss of generality that either θ(S) is a single point or that θ|S injective. The first case contradicts the fact that S is disjoint from X but has a cluster point in X . In the second case, θ(S) must have a cluster point in K − X. Let V be a non-principal (i.e, non-constant) ultrafilter on θ(S) which converges to a point of K − X. Then V = θ(U) where U is a non-principal ultrafilter on S, which must converge to a point in X, leading to a contradiction.

The proofs of the following are exercises.

8.2. Theorem. Of the following conditions on a space X and a compactification K,

1. X is a P-set in K;

2. The closure in K − X of every σ-compact subset of K − X is compact;

3. The closure in K − X of every countable subset of K − X is compact;

4. K − X is countably compact;

5. K − X is pseudocompact.

we have 1 ⇔ 2 ⇒ 3 ⇒ 4 ⇒ 5.

8.3. Theorem. If Y is a reduced outgrowth of a Lindelöf (respectively Alster) space, then any outgrowth of Y is Lindelöf (respectively Alster). Conversely, if some reduced outgrowth of Y is Lindelöf (respectively Alster), then Y is a reduced outgrowth of a Lindelöf (respectively Alster) space.

Proof. Suppose X is a Lindelöf space and K is a compactification of X with Y = K − X. Then L = cl_{K}(Y) is a compactification of Y and L − Y = L ∩ X is closed in X and hence Lindelöf. Since L − Y is a perfect image of βY − Y (see 2.1), it follows that βY − Y is Lindelöf (2.3) and hence the same is true for any compactification of Y. The Alster case goes the same way.

For the converse, suppose L is a compactification of Y with L − Y Lindelöf. Let N^{∗} be the one point compactification of N and K = N^{∗} × L. Embed Y as {∞} × Y and let X = K − Y. Clearly K is a compactification of X with reduced outgrowth Y. Finally, X = (N × L) ∪ {∞} × (L − Y) is the union of countably many Lindelöf spaces and is therefore Lindelöf. The argument with Alster’s condition is similar.

8.4. Theorem. A space with a locally compact reduced outgrowth is the union of a compact subset and a locally compact subset.

Proof. Let X ⊆ K be a compactification such that Y = K − X is locally compact. Let L = cl_{K}(Y). Since Y is locally compact it is open in L and hence L − Y is a compact subset of X. If p ∉ L − Y, then p has a K-neighbourhood that does not meet Y and a Kclosed K-neighbourhood inside it. Such a neighbourhood is a compact K-neighbourhood of p inside X.

8.5. Theorem. A locally Alster reduced outgrowth of a Lindelöf CNP space is locally compact.

Proof. Suppose that X is a Lindelöf CNP space, K is a compactification of X and Y = K − X. Assume that each p ∈ Y has an Alster neighbourhood. Let F denote the family of all f ∈ C(K) that vanish nowhere on X. Then {Z[f] | f ∈ F } is readily seen to be an ample G_{δ} cover of Y, since for each compact set A ⊆ Y there is an f ∈ F that vanishes on A (Theorem 2.10). If p ∈ Y, there is a neighbourhood U of p that is covered by a countable family of Z[f]. This means that U ⊆ ⋃ Z[f_{n}]. But then ⋂ coz(f_{n}) is a G_{δ} that contains X and, by the CNP, there is an open V ⊆ K such that X ⊆ V ⊆ ⋂ coz(f_{n}) and then K − V ⊇ U is a compact neighbourhood of p.

9. SCZ spaces

We will say that a space satisfies the SCZ condition if every σ-compact subset is contained in a compact zeroset. This can be usefully broken into two separate conditions:

SCZ-1. The closure of any σ-compact subset is compact;

SCZ-2. Every compact subset is contained in a compact zeroset.

The following claim is immediate.

9.1. Proposition. A reduced outgrowth of a space satisfies SZC-1 if and only if the space satisfies the CNP.

9.2. Proposition. A reduced outgrowth of a Lindelöf space satisfies SCZ-2.

Proof. If X is Lindelöf and K is a compactification, let Y = K − X. If A is a compact set in Y, then K − A is an open set that contains X. Since X is Lindelöf, there is a cozeroset U ⊆ K such that X ⊆ U ⊆ K − A (Theorem 2.10). Then K − U is a zeroset, hence compact, in K and evidently A ⊆ K − U ⊆ Y.

9.3. Proposition. A reduced outgrowth of a space that satisfies SCZ-1 satisfies the CNP.

Proof. Suppose X satisfies SCZ-1 and K is a compactification of X. Let Y = K −X and L = cl_{K}(Y). Then L is a compactification of Y and hence L−Y is a reduced outgrowth of Y. But L − Y is a closed subspace of X and hence also satisfies SCZ-1. Let U = ⋂_{n∈N} U_{n} (each U_{n} open in L), be a G_{δ} set in L that contains Y. It follows that for each n, L − U_{n} is a compact subset of L − Y. According to SCZ-1, there is a compact set A ⊆ L − Y with L − U_{n} ⊆ A for all n ∈ N. But then L − A is an open set with Y ⊆ L − A ⊆ U.

9.4. Proposition. A space that satisfies SCZ-1 is pseudocompact.

Proof. Suppose Y satisfies SCZ-1. If f were an unbounded function in C(Y) we could choose points p_{1}, p_{2}, …, p_{n}, … such that |f(p_{n})| > n. The set {p_{1}, p_{2}, …} is discrete and not compact, but its closure is compact. If p is any point in its frontier, it is clear that f(p) cannot be defined.

9.5. Proposition. A reduced outgrowth of any space that satisfies SCZ is Lindelöf.

Proof. Let X satisfy SCZ. We will begin by showing that Y = βX − X is Lindelöf. By Smirnov’s Theorem (see 2.10), it will suffice to show that any open subset of cl_{βX}(Y) that contains Y contains a cozeroset containing Y. It will be sufficient to show the same with βX replacing cl_{βX}(Y). If U is an open subset of βX containing Y, βX − U is closed in βX and hence compact and βX − U ⊆ X. Then there is a function f : X [figure] [0, 1] such that Z[f] is compact and βX − U ⊆ Z[f]. The function f extends to a function f̂ : βX [figure] [0, 1]. We claim that Z[f̂] does not meet Y. For suppose that p ∈ Y with f̂(p) = 0. Since Z[f] is compact and therefore closed in βX, there is a function g : βX [figure] [0, 1] such that g(p) = 0 and g(Z[f]) = 1. Then f̂+ g vanishes nowhere on X since g = 1 wherever f = 0. The previous proposition implies that 1/(f̂+ g) is bounded on X and hence bounded on βX. Thus f̂+ g cannot vanish anywhere on Y, in particular at p. It follows that Z[f̂] is a compact zeroset in βX that does not meet Y and then Y ⊆ βX − Z[f̂] ⊆ U.

In the general case of a reduced outgrowth, any space of the form K − X, for a compactification K of X, is a quotient of βX − X.

In the next two propositions, X denotes a space that satisfies the SCZ, K is a compactification of X, Y = K − X, and L = N^{∗} ×K. Embed K into L as {∞}×K and similarly embed X and Y into the infinity slice. Let Z = (N × K) ∪ ({∞} × Y) = L − ({∞} × X).

9.6. Proposition. The space Z is Lindelöf.

Proof. The Since X satisfies the SZC, Y is Lindelöf by the preceding proposition. Obviously N × K is Lindelöf and hence so is Z.

9.7. Proposition. If Y satisfies the CNP, so does Z.

Proof. Suppose Z ⊆ U = ⋂ U_{n} with each U_{n} open in L. Then Y ⊆ ⋂(K ∩ U_{n}) and each K ∩ U_{n} is open in K. Since Y satisfies the CNP, there is an open V ⊆ K such that Y ⊆ V ⊆ U ∩ K. But then N^{∗} × V and N × K are open in L and

Z ⊆ (N^{∗} × V) ∪ (N × K) ⊆ U

9.8. Theorem. A reduced outgrowth of a Lindelöf CNP space satisfies SCZ; a space that satisfies the SCZ is the reduced outgrowth of a Lindelöf CNP space.

Proof. Propositions 9.1 and 9.2 give one direction. The previous proposition gives the converse.

References

[Alster (1988)] K. Alster (1988), On the class of all spaces of weight not greater than ω_{1} whose Cartesian product with every Lindelöf space is Lindelöf. Fund. Math. 129, 133–140.

[Barr, et al. (2003)] M. Barr, W. Burgess, and R. Raphael (2003), Ring Epimorphisms and C(X). Theory Appl. Categories 11, 283–308. ftp://ftp.math.mcgill.ca/pub/

  barr/pdffiles/bbr.pdf

[Barr, et al. (2007)] M. Barr, J.F. Kennison, and R. Raphael (2007), Searching for absolute CR -epic spaces. Canadian J. Math., 59, 465–487. ftp://ftp.math.mcgill.ca/

  pub/barr/pdffiles/bkr.pdf

[Barr, et al. (2007a)] M. Barr, J.F. Kennison and R. Raphael, On productively Lindelöf spaces. Scientia Math. Japonicae, 20 (2007). ftp://ftp.math.mcgill.ca/pub/barr/

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[Barr, et al. (2005)] M. Barr, R. Raphael, and R.G. Woods, On CR -epic embeddings and absolute CR -epic spaces. Canad. J. Math. 57 (2005), 1121–1138. ftp://ftp.math.

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[Fine, Gillman, & Lambek (1965)] N. Fine, L. Gillman and J. Lambek (1965), Rings of Quotients of Rings of Functions. McGill University Press, Montreal. See also ftp://

  ftp.math.mcgill.ca/pub/barr/pdffiles/fgl.pdf

[Gillman & Jerison (1960)] L. Gillman and M. Jerison (1960), Rings of Continuous Functions. D. Van Nostrand, Princeton.

[Glicksberg, 1959] I. Glicksberg (1959), Stone-Čech compactifications of products. Trans Amer. Math. Soc. 90, 369–382.

[Kelley (1955)] J.L. Kelley (1955), General Topology. Van Nostrand, New York.

[Porter & Woods (1988)] J.R. Porter and R.G. Woods (1988), Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag, New York.

[Smirnov (1951)] Ju. Smirnov (1951), On normally placed sets in normal spaces. Mat. Sb. (N.S.) 29 (1951), 173-176

Department of Mathematics and Statistics McGill University, Montreal, QC, H3A 2K6

Department of Mathematics and Computer Science Clark University, Worcester, MA 01610

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

Email: barr@barrs.org jkennison@clarku.edu raphael@alcor.concordia.ca

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