Theory and Applications of Categories, Vol. 11, No. 12, 2003, pp. 283–308.

RING EPIMORPHISMS AND C(X)

MICHAEL BARR, W.D. BURGESS AND R. RAPHAEL

ABSTRACT. This paper studies the homomorphism of rings of continuous functions ρ: C(X) → C(Y), Y a subspace of a Tychonoff space X, induced by restriction. We ask when ρ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When X is first countable and perfectly normal (e.g., a metric space), ρ is a CRepimorphism if and only if it is a R/N-epimorphism if and only if Y is locally closed in X. It is also shown that the restriction of ρ to C^{∗}(X) → C^{∗}(Y), when X is normal, is a CR-epimorphism if and only if it is a surjection. In general spaces the picture is more complicated, as is shown by various examples. Information about Spec ρ and Spec ρ restricted to the proconstructible set of prime zideals is given.

1. Introduction

During the 1960s considerable effort was expended in order to understand the meaning of “epimorphism” ([Mac Lane (1998), page 19]) in various concrete categories. These questions got particular attention in categories of rings. (See, for example, [Lazard (1968)], [Mazet (1968)], [Olivier (1968)] and [Storrer (1968)].) Much later the subject again received attention in categories of ordered rings (see [Schwartz & Madden (1999)] and its bibliography.) Taking the notation in [Schwartz & Madden (1999)] as a model, we will use the following symbols for the categories of interest to us (throughout, all rings referred to will be commutative and with 1); in each case, morphisms are assumed to preserve the relevant structures:

POR/N, the category of reduced partially ordered rings; R/N, the category of reduced rings; CR, the category of all (commutative) rings. The ring of continuous real-valued functions, C(X), on a topological space X is an

object in all three categories (such rings also live in several other important categories but our attention will be restricted to the above three). In each of these categories, it is easy to find examples of morphisms f: A → B which are epimorphisms without being surjective. However, if f is an epimorphism there must, nevertheless, be some close connections between the two rings.

The authors each acknowledge the support of grants from the NSERC of Canada. The authors would also like to thank the referee and Hans Storrer for several improvements to the paper.

Received by the editors 2003-01-10 and, in revised form, 2003-06-12. Transmitted by Walter Tholen. Published on 2003-06-22. 2000 Mathematics Subject Classification: 18A20, 54C45, 54B30. Key words and phrases: epimorphism, ring of continuous functions, category of rings. \begin{matrix}c \\ ◯ \end{matrix} Michael Barr, W.D. Burgess and R. Raphael, 2003. Permission to copy for private use granted.

It was observed in [Schwartz (2000)] that when f: A → B is a ring homomorphism between real closed rings then it is automatically a morphism in POR/N and R/N. (See [Schwartz & Madden (1999), §12] for basic properties of real closed rings and an entry into the literature.) Moreover, the criteria (quoted below) for being an epimorphism in the two categories are simultaneously satisfied in this situation. An epimorphism in CR is automatically an epimorphism in R/N, but not conversely. However, the question whether an R/N-epimorphism between real closed rings is also a CR-epimorphism remains open and will be discussed later in this article.

Our main topic will be the study of the morphism (in the categories) ρ induced by a subspace Y of a topological space X via restriction. Then ρ: C(X) → C(Y) is a ring homomorphism between two real closed rings ([Schwartz (1997), Theorem 1.2]). There are various ways in which a subspace can be related to the ambient space: closed, open, locally closed, zero-set, cozero-set, C-embedded, C^{∗}-embedded, z-embedded and (see below) Gembedded, among others. We will study when ρ is an epimorphism in the three categories. This, as we will show, has connections with the other sorts of embedding of Y in X. As one would expect, the results are more complete when the ambient space, X, is “nice” in topological terms.

The statement that ρ is an epimorphism in any of the categories connects not only Spec C(Y) with Spec C(X) but also the residue fields of the two rings as well. The question is also closely related to continuous extensions of elements of C(Y) to larger subsets of X.

The criteria for an epimorphism in POR/N and R/N are quoted in [Schwartz (2000), page 351]. When f: A → B is a morphism between real closed rings (as is the case for ρ) f is automatically a POR/N homomorphism because the positive cones are the sets of squares and the real spectrum ([Schwartz & Madden (1999), Proposition 12.4]) and the ordinary spectrum coincide. In the case of a homomorphism between real closed rings, it is easy to see that the criteria for an epimorphism in R/N and in POR/N coincide. This fact is summarized in Proposition 1.1. The symbol Q_{cl}(A) denotes the classical ring of quotients (or ring of fractions) of a ring A.

1.1. Proposition. Let f: A → B be a homomorphism between real closed rings. Then f is an epimorphism in POR/N and in R/N if and only if (i) Spec ρ: Spec B → Spec A is an injection, and (ii) for each 𝔭 ∈ Spec B, f induces an isomorphism Q_{cl}(A/f^{−1}(𝔭)) → Q_{cl}(B/𝔭).

There is an analogous characterization of epimorphisms in CR due to Ferrand, quoted in [Lazard (1968), Proposition IV 1.5]. However, a more down-to-earth criterion will be more convenient for us. The following is found in [Mazet (1968)] and in [Storrer (1968), page 73].

RING EPIMORPHISMS AND C(X)  285

1.2. Proposition. A homomorphism f: A → B in CR is an epimorphism if and only if for each b ∈ B there exist matrices C, D, E of sizes 1 × n, n × n and n × 1, respectively, where (i) C and E have entries in B, (ii) D has entries in f(A), (iii) the entries of CD and of DE are elements of f(A) and (iv) b = CDE. (Such an expression is called an n × n zig-zag for b over A.)

Section 2 deals with the category CR. Some easy cases where ρ: C(X) → C(Y) is a CR-epimorphism are dealt with first. For example, it suffices that Y be C^{∗}-embedded in X. The next steps are to show that the existence of a zig-zag for f ∈ C(Y) over C(X) says something about extending f to a larger domain; explicitly, f can be extended from a dense set to an open set (Proposition 2.6). The most complete result is for X first countable and perfectly normal (e.g., a metric space), in which case ρ is an epimorphism if and only if Y is locally closed (i.e., the intersection of an open and a closed, [Engelking (1968), page 96]) (Corollary 2.12).

Along the way, it is also shown that, when X is normal, the restriction ρ^{∗}: C^{∗}(X) → C^{∗}(Y) of ρ is an epimorphism if and only if it is surjective (Theorem 2.4); i.e., when Y is C^{∗}-embedded.

Section 3 looks at the categories R/N and POR/N. Since ρ is an epimorphism in one category if and only if it is an epimorphism in the other one, we will always talk about R/N-epimorphisms. Much of the section will deal with a necessary condition for ρ to be an R/N-epimorphism, namely that Y is G-embedded in X. This condition is strictly weaker than saying that ρ is an R/N-epimorphism; however, if the criteria (i) and (ii) of Proposition 1.1 are restricted to the z-spectra we get a characterization of when Y is G-embedded in X (Theorem 3.5). It is also related to a topic in N. Schwartz’s paper on C(X), [Schwartz (1997)], where the image of Spec ρ is studied. For X first countable and perfectly normal we have the attractive analogy to the result on (CR)-epimorphisms in such spaces: Y is G-embedded in X if and only if Y is a finite union of locally closed sets (Corollary 3.18). For these spaces, it is also shown that ρ is an R/N-epimorphism if and only if it is a CR-epimorphism.

Along the way, we look at some regular rings naturally related to C(X), namely T (X), Q(X), H(X) and G(X) and observe that they are all real closed, and, hence, epi-final in R/N and in POR/N (Proposition 3.1). This fact for G(X) is important in the rest of the section.

Section 4 has miscellaneous results and poses some vexing questions which remain unanswered.

Conventions of notation and terminology.

Throughout a topological space will mean a Tychonoff space. If Y is a subspace of a topological space X the restriction homomorphism ρ: C(X) → C(Y) will be the object of our study. “Regular ring” will always mean “von Neumann regular ring”.

Recall some of the basic terminology: a subset Z of X is a zero-set if there is f ∈ C(X) with Z = z(f) = {x ∈ X | f(x) = 0}; its complement, coz f, is a cozero-set. An ideal I of C(X) is a z-ideal if g ∈ I whenever f ∈ I and z(g) = z(f). A subset Y of X is locally closed if it is the intersection of an open and a closed subset. The general references for topology are [Engelking (1968)], [Gillman & Jerison (1960)] and [Willard (1970)], while that for rings of continuous functions is, of course, [Gillman & Jerison (1960)]. Some categorical material and properties of real closed rings are taken from [Schwartz & Madden (1999)] and [Schwartz (1997)].

2. Subspaces inducing epimorphisms in CR.

Recall ([Engelking (1968), page 60]) that a space is perfectly normal if every closed set is a zero-set. (A perfectly normal space is normal. If A and B are disjoint zero-sets of f and g, resp., then the function |f|/(|f| + |g|) is 0 on A and 1 on B.) One striking result in this section is that in a first countable perfectly normal space X (e.g., a metric space) ρ is an epimorphism in CR if and only if Y is locally closed. However, the picture is quite different when X does not have these strong conditions, as we shall see.

Let us first note that there are non-trivial cases where ρ is an epimorphism in CR, and, hence, in POR/N and R/N as well. In fact, to say that ρ is surjective is the same as saying that Y is C-embedded. See [Gillman & Jerison (1960), 1.18, 1.19, 1F] for examples.

We will see that ρ is an epimorphism if Y is only C^{∗}-embedded in X. This is part of the next observation which discusses the most elementary examples of subspaces where ρ is an epimorphism in CR but need not be surjective. It is only a start on the topic.

2.1. Proposition. Let Y be a subspace of a topological space X and ρ: C(X) → C(Y) the restriction homomorphism.

(i) If Y is C^{∗}-embedded in X, then ρ is an epimorphism in CR .

(ii) If each f ∈ C^{∗}(Y) extends to a continuous function on a cozero-set of X, then ρ is an epimorphism in CR . This occurs, in particular, if Y is a cozero-set in X.

(iii) If for each f ∈ C^{∗}(Y) there are g, h ∈ C(X) with Y ⊆ coz g and fρ(g) = ρ(h), then ρ is an epimorphism.

Proof. (i) Given f ∈ C(Y) we wish to construct a zig-zag for f over C(X). Consider the two bounded functions f_{1} = f/(1 + f^{2}) and f_{2} = 1/(1 + f^{2}) and write f_{i} = ρ(g_{i}), for g_{i} ∈ C(X), i = 1, 2. Then, on Y, f_{2}^{−1}g_{1}g_{2}f_{2}^{−1} = f is the desired zig-zag (here n = 1).

(ii) Suppose first that f ∈ C^{∗}(Y) and that f extends to a cozero-set coz g, g ∈ C^{∗}(X) (we keep the symbol f). We get a zig-zag since fg extends to some h ∈ C^{∗}(X), and so, on Y, f = g^{−1}ghg^{−1}. Now, as above, we put f_{1} = f/(1 + f^{2}) and f_{2} = 1/(1 + f^{2}) and we have zig-zags with f_{i} = g_{i}^{−1}g_{i}h_{i}g_{i}^{−1} on Y, where h_{2} is non-zero on Y. These combine to give f = (g_{1}^{−1}g_{1}h_{1}g_{1}^{−1})(g_{2}^{−1}g_{2}h_{2}g_{2}^{−1})^{−1} = (g_{1}^{−1}h_{2}^{−1})(g_{1}g_{2}h_{1}h_{2})(g_{1}^{−1}h_{2}^{−1}) on Y.

(iii) Here ρ(g) is invertible and so f = ρ(g)^{−1}ρ(g)ρ(h)ρ(g)^{−1} is a (1 × 1) zig-zag.

The ideas in Proposition 2.1 will be pushed quite a bit further. Part (iii) will show up in Proposition 2.15, below. Notice that in each of the parts of the proposition, only a 1 × 1 zig-zag appears. This phenomenon will be discussed in Section 4.

2.2. Proposition. [Fine, Gillman, & Lambek (1965), page 24] Suppose that Y is a dense subset of X. Then for any f ∈ C(Y), there is a largest subset of X to which f can be continuously extended.

For any u ∈ C(Y), let us write dom(u) for the largest subspace of X to which u can be extended continuously.

2.3. Lemma. Suppose that X is a topological space and Y a dense subspace. Let f ∈ C(Y) and x ∈ X − Y. Assume that f = GAH is a zig-zag for f over C(Y ∪ {x}) (as in Proposition 1.2). Suppose, moreover, that for some neighbourhood U of x the entries of G and H are bounded in U ∩ Y. Then f can be extended continuously to x.

Proof. Let Â denote the constant matrix A(x) and f̂ = GÂH. Let U be a neighbourhood of x as in the statement. Suppose that M is a bound on both ∥G∥ and ∥H∥ in U ∩ Y. Then we have that in U,

|f − f̂| = ∥GAH − GÂH∥ ≤ ∥G∥ ∥A − Â∥ ∥H∥ ≤ M^{2}∥A − Â∥

and then lim_{y→x} A(y) = Â implies that

lim_{y→x} |f(y) − f̂(y)| = 0  (∗)

We also have that in U ∩ Y,

∥GA − GÂ∥ ≤ ∥G∥∥A − Â∥ ≤ M∥A − Â∥

from which we infer that lim_{y→x}(G(y)A(y) − G(y)Â) = 0 and hence that lim_{y→x} G(y)Â exists and equals lim_{y→x} G(y)A(y). Similarly, lim_{y→x} ÂH(y) = lim_{y→x} A(y)H(y) exists. Since Â is constant, there is a matrix B such that ÂBÂ = Â (the ring of matrices over R is regular) and then f̂ = GÂH = GÂBÂH is defined at x, and then so is f.

Lemma 2.3 gives us information about when ρ^{∗}: C^{∗}(X) → C^{∗}(Y), the restriction of ρ to C^{∗}(X), is an epimorphism. This behaviour is quite different from that of ρ itself.

2.4. Theorem. Suppose that Y is a proper dense subset of X. Then ρ^{∗}: C^{∗}(X) → C^{∗}(Y) is an epimorphism in CR if and only if it is surjective, that is, if and only if Y is C^{∗}- embedded in X. If, moreover, X is normal then the conclusion is valid even when Y is not dense.

Proof. If it is an epimorphism and f ∈ C^{∗}(Y), then there is a zig-zag f = GAH over C^{∗}(X), as in the lemma. Since G and H have entries in C^{∗}(Y) they are necessarily bounded and hence f can be extended to any point of X − Y. By bounding the extension with the bounds of f we may assume that the extension is in C^{∗}(X).

When X is normal and C^{∗}(X) → C^{∗}(Y) is an epimorphism then for f ∈ C^{∗}(Y) we can restrict a zig-zag for f over C^{∗}(X) to get one over C^{∗}(cl(Y)). This allows us to extend f to f̃ ∈ C^{∗}(cl(Y)). However, cl(Y) is C^{∗}-embedded in X, completing the proof.

When C^{∗}(Y) = C(Y), that is, when Y is pseudocompact, the same reasoning as in Theorem 2.4 gives us the following corollary.

2.5. Corollary. Suppose that Y is a pseudocompact proper dense subset of X. Then ρ^{∗}: C(X) → C(Y) is an epimorphism in CR if and only if it is surjective, that is, if and only if Y is C-embedded. If, moreover, X is normal then the conclusion is valid for pseudocompact Y even when not dense.

In light of the discussion in Section 3B, note that if Y is C^{∗}-embedded then it is zembedded. An almost compact space ([Gillman & Jerison (1960), 6J]) is C-embedded in any ambient space. If Y is pseudocompact but not almost compact then there is a compactification of Y in which Y is not z-embedded ([Blair & Hager (1974), Theorem 4.1]).

The next corollary is a partial converse to Proposition 2.1(ii).

2.6. Proposition. Suppose that X is a topological space and Y a dense subspace such that ρ: C(X) → C(Y) is an epimorphism in CR . Then for f ∈ C(Y), dom(f) contains an open set containing Y.

Proof. We know that there must exist G, A, and H as in Lemma 2.3 for which f = GAH. Suppose that y ∈ Y. Since G and H are defined and continuous on Y, there is an open neighbourhood U of y on which both G and H are bounded. For any x ∈ U, U is also a neighbourhood of x and it follows from 2.3 that f can be extended to x.

2.7. Proposition. [Gillman & Jerison (1960), 3.11(b)] In a space X, every first countable point is a zero-set.

2.8. Lemma. Suppose that t_{1} > t_{2} > t_{3} > ⋯ is a sequence of numbers in the open unit interval that converges to 0. Then there is a continuous function h : (0, ∞) → [0, 1] such that h(t) = 0 for t ≥ t_{1}, h(t_{n}) = 0 when n is odd and h(t) = 1 when n is even.

Proof. Just interpolate linearly between those values.

2.9. Proposition. The complement of a first countable, non-isolated point is not C^{∗}-embedded.

Proof. Let x ∈ X be a first countable non-isolated point. If Y = X − {x} were C^{∗}-embedded in X then βY = βX. Then by [Gillman & Jerison (1960), 9.6 and the remarks of 9.7], x is not a G_{δ}-set in βY but is first countable in βY. This is impossible.

2.10. Theorem. If a dense subset Y of a first countable space X induces an epimorphism in CR then Y is open in X.

Proof. If Y ⊆ X is not open, then since X is first countable, there must be a sequence of points x_{1}, x_{2}, … of X − Y which converges to some y ∈ Y. By using the device of Proposition 2.7, it can be assumed that the points are distinct and form a discrete subspace of X.

Choose, for each n, a function f_{n} : X − {x_{n}} → [0, 1] that cannot be extended to x_{n}. Then f_{n} ∈ C(Y) and so is f = ∑ 2^{−n}f_{n}, since the convergence is uniform. For each n there is a neighbourhood V of x_{n} excluding the other points of the sequence. Then f −2^{−n}f_{n} behaves like ∑_{m≠n} 2^{−m}f_{m} on V and so extends continuously to x_{n}. This implies that f cannot be extended continuously to x_{n}. Thus f ∈ C(Y) cannot be extended to any open set containing Y and the conclusion follows.

2.11. Corollary. Let Y be a subspace of a first countable space X. If Y induces an epimorphism in CR then Y is locally closed in X

Proof. Since C(cl(Y)) → C(Y) is an epimorphism (the second factor of an epimorphism), Theorem 2.10 applies.

2.12. Corollary. A subspace Y of a perfectly normal first countable space X induces an epimorphism in CR if and only if it is locally closed.

Proof. We already have one direction in Corollary 2.11. In the other direction, if Y = U ∩ V, where V is closed and U open, U is a cozero-set in X, and, hence, also in V. Then C(V) → C(Y) is an epimorphism. Moreover, V is C^{∗}-embedded in X; thus, C(X) → C(V) is also an epimorphism.

We will see in the next section, that there are dense open subsets of normal spaces which do not induce epimorphisms and a first countable space containing a zero-set which does not induce an epimorphism, even in R/N. There is a compact space with a dense subset which induces an epimorphism but is not even a finite union of locally closed subsets. See Examples 3.11, below. These show that there are limitations to generalizations of the last two corollaries. The following space shows that the assumption “first countable” cannot be removed from Corollary 2.12.

2.13. Example. There is a countable (hence, perfectly normal) space X such that every subspace induces an epimorphism in CR and not all subspaces are locally closed.

Proof. A space is called resolvable if it is the union of two disjoint dense subsets. Now suppose that X is a countable resolvable F -space ([Gillman & Jerison (1960), Theorem 14.25]). Since X is a countable F -space, every subset is C^{∗}-embedded ([ibid., 14 N5]).

Since it is resolvable, it has a dense subset which is not open. Since X is countable, it is perfectly normal ([Gillman & Jerison (1960), 3B(1)]). However, X has a dense non-open subset that induces an epimorphism.

We must now produce an example of such a space. We are grateful to R.G. Woods for pointing out the following. For the notion of “absolutes” and “perfectly irreducible surjections”, see [Porter & Woods (1988), Chapter 6]. We will need [ibid., 6.5(b)(4)]: if f: X → Y is irreducible and S is dense in Y, f^{−1}(S) is dense in X and f|_{f^{−1}(S)} is irreducible from f^{−1}(S) onto S.

Let E denote the absolute of the rationals Q and let A and B be complementary dense subsets of Q. Suppose that k: E → Q is the canonical perfect irreducible surjection and C = k^{−1}(A) and D = k^{−1}(B). Then C and D are dense in E. Since A and B are separable so are C and D. Now let R and S be countable dense subsets of C and D, respectively. Define X = R ∪ S; it is countable and resolvable. Since it is dense in E, it is extremally disconnected by [Gillman & Jerison (1960), 1H(4)] since E is. The space X has no isolated points because Q, and, hence, E does not have isolated points.

Proposition 2.9 is not the last word on complements of points.

2.14. Example. There are easy examples of non-isolated points in normal (even compact) spaces whose complements are C^{∗}-embedded. If x ∈ X is such an example, then ρ: C(X) → C(X − {x}) induces an epimorphism in CR but X − {x} is not a cozero-set.

Proof. Let Z be a non-compact space and a ∈ βZ − Z, X = βZ and Y = βZ − {a}. Since a is not isolated, Y is dense open in X. By [Gillman & Jerison (1960), 1H(6))], Y is C^{∗}-embedded in X and therefore Y induces an epimorphism. However, Y is not a cozero-set by [Gillman & Jerison (1960), 9.5]. Thus open sets can induce epimorphisms without being cozero-sets.

We finish this section with some special families of examples. They will appear again in Section 3. In particular, we will see that if Y ⊆ X and Y is homeomorphic to N, then ρ is an epimorphism.

2.15. Proposition. (i) Let Y be a subspace of a normal space X which has the following form: Y = ⋃_{n∈N} B_{n} where the B_{n} are closed sets of X and there is a family of pairwise disjoint open sets U_{n} from X with B_{n} ⊆ U_{n}, for each n ∈ N. Then Y induces an epimorphism in CR .

(ii) If, under the hypotheses of (i), each B_{n} is compact then then the conclusion of (i) holds for any X. This applies, in particular, if Y is homeomorphic to N.

(iii) Let Y be a subspace of a space X which has the following form: Y = ⋃_{n∈N} B_{n}, where there is a family of pairwise disjoint cozero-sets U_{n}, with B_{n} ⊆ U_{n}, for n ∈ N. Suppose, moreover, that each B_{n} is C^{∗}-embedded in U_{n}. Then Y ⊆ X induces an epimorphism in CR .

Proof. (i) Let f ∈ C^{∗}(Y). Set M = sup_{Y} |f(y)|. We use the sequence {U_{n}}. Now find h_{n} ∈ C(X) with h_{n} constantly 1 on B_{n} and zero on X − U_{n}. We may assume that values of h_{n} are in [0, 1]. Define g ∈ C^{∗}(Y) by g(y) = (1/n)f(y), for y ∈ B_{n}. It will be shown that g extends to X. Since X is normal and B_{n} closed, g_{n} = g|_{B_{n}} extends to l_{n} ∈ C(X) and we may assume that, for all x ∈ X, |l_{n}(x)| ≤ M/n.

The next step is to define a, b ∈ C(X), as follows. Put a(x) = h_{n}(x)l_{n}(x) if x ∈ U_{n}, and a(x) = 0 if x ∈ ⋂_{N}(X − U_{n}). Similarly, b(x) = (1/n)h_{n}(x) if x ∈ U_{n} and b(x) = 0 for x ∈ ⋂_{N}(X − U_{n}). We check that a is continuous (the continuity of b is similar). For an interval (u, v), a^{−1}((u, v)) = ⋃_{N}(h_{n}l_{n})^{−1}((u, v)), an open set, if 0 ∉ (u, v). If 0 ∈ (u, v), the inverse image includes all but finitely many of the U_{n}. Its complement is a finite union of sets of the form (h_{n}l_{n})^{−1}((−∞, u] ∪ [v, ∞)), which is closed. Finally, f = ρ(a)ρ(b)^{−1}. Then Proposition 2.1(iii) applies.

(ii) This is done by first embedding X in βX. There are open sets V_{n}, n ∈ N, in βX with U_{n} = X ∩ V_{n}; since X is dense in βX, the V_{n} are disjoint. The compact sets B_{n} are closed in βX. However, βX is normal and part (i) applies there. The functions constructed on βX, restricted to X, give the result.

(iii) Let U_{n} = coz φ_{n}, where we can suppose φ_{n}: X → [0, 1]. The proof here is similar to that of (i) with the following modifications. Here, g(y) = (1/n)f(y)φ_{n}(y), y ∈ B_{n}, and it extends to X since f|_{B} extends to U_{n}. From there, g|_{U_{n}} extends to l_{n} ∈ C(X) and a(x) = l_{n}(x), for x ∈ U_{n}. In addition, b is constructed via b(x) = (1/n)φ_{n}(x), for x ∈ U_{n}.

3. Epimorphisms in R/N and in POR/N, and generalizations.

A. Real closed regular rings.

It was mentioned earlier that the restriction homomorphism ρ: C(X) → C(Y) induces an epimorphism in one of these two categories if and only if it induces one in the other. In this section we will use the phrase “an epimorphism in R/N” or “R/N-epimorphism” to cover both categories.

As we saw in Proposition 1.1, the homomorphism ρ is an epimorphism in R/N if and only if Spec ρ: Spec C(Y) → Spec C(X) is injective and the residual fields are isomorphic via ρ. In many situations the prime z-ideals are easier to study. For that reason we will study a necessary condition for an epimorphism in R/N which only involves the prime z-ideals. The subspace of Spec C(X) consisting of prime z-ideals is called zSpec C(X); it is a proconstructible subset, i.e., it is closed in the constructible topology (see [Schwartz & Madden (1999), Section 4] for terminology and basic facts).

However, our first observations are about rings related to C(X). For any commutative ring there is a universal regular ring functor, often called T, which is the left adjoint to the inclusion of the category of (commutative von Neumann) regular rings into CR. For f: A → B in CR, where A and B are rings, if f is an epimorphism in CR then T (f) is surjective. However, we will see that for real closed rings T (f) surjective implies that f is an epimorphism in R/N.

An object A in a concrete category is called epi-final if each epimorphism f: A → B is a surjection. Note that a regular f-ring is real closed if and only if its residue fields (or, equivalently, its Pierce stalks) are all real closed (see [Lipshitz (1977), Definitions (vi)]; the conditions for a real closed regular ring go up and down from the Pierce stalks which are ordered fields).

3.1. Proposition. (1) A regular ring with factor fields of characteristic zero is epi-final in R/N . (2) A real closed regular ring is epi-final in POR/N .

Proof. The two proofs are similar. Part (2) is as follows. If f: (A, P) → (B, Q) is an epimorphism in POR/N and A is regular and real closed then the factor fields of A are real closed. If 𝔭 is the support of a prime cone in Sper(B, Q) then A/f^{−1}𝔭 is the real closure of Q_{cl}(B/𝔭). However, A/f^{−1}𝔭 order embeds in B/𝔭 and so we have equality. Therefore, 𝔭 is a maximal ideal. Moreover, every minimal prime of B is the support of a prime cone in Sper B ([Schwartz & Madden (1999), Proposition 4.4]) and it follows that the minimal primes are maximal; in other words, B is regular and real closed. In that case Sper B and Spec B are homeomorphic and Spec f(Spec B) is a closed subset of Spec A. Then B is isomorphic to A/I, where I is the intersection of all the maximal ideals not in the image of Spec f.

There are several regular rings in the literature related to a ring of the form C(X). Let us recall some of them. Firstly, there is T (C(X)), abbreviated to T (X), secondly F (X) = R^{X} and thirdly the complete ring of quotients, Q(X). The universal property of the functor T says that the inclusion of C(X) into F (X) factors through T (X) with image denoted G(X), and the inclusion of C(X) into Q(X) factors through T (X) with image H(X). The ring G(X) is the one of most interest in the rest of this section but the others all have been studied. See [Fine, Gillman, & Lambek (1965)] for Q(X), [Olivier (1968)] and [Wiegand (1971)] for the functor T, [Raphael & Woods (2000)] for H(X) and [Henriksen, Raphael, & Woods (2001)] for G(X).

3.2. Proposition. For a topological space X, the regular rings T (X), Q(X), G(X), and H(X) are all real closed.

Proof. The residue fields of T (X) are those of C(X) (i.e., the fields Q_{cl}(C(X)/𝔭), 𝔭 ∈ Spec C(X)). These fields are all real closed by [Schwartz (1997), Theorem 1.2]. As already remarked in [ibid., page 299] , since Q(X) is a direct limit of real closed rings, it is real closed. Finally, G(X) and H(X) are regular homomorphic images of real closed rings and are, hence, real closed (see the table [Schwartz & Madden (1999), page 255]).

B. G-embeddings and zSpec C(X).

For a space X, the elements of G(X) can be described explicitly ([Henriksen, Raphael, & Woods (2001), Theorem 1.1]) as follows. For a ∈ C(X), a^{∗}: X → R is defined by: a^{∗}(x) = 1/a(x) if a(x) ≠ 0, and is zero otherwise. Note that a^{∗} is continuous on the dense open set coz a ∪ (X − cl(coz a)). A typical element of G(X) has the form ∑_{i=1}^{m} a_{i}b_{i}^{∗} for some m ∈ N, where the a_{i}, b_{i} ∈ C(X). This shows that G(X) is generated, as a regular ring, by the elements of C(X). Notice that C(X) and G(X) have the same cardinality.

When Y is a subspace of X, there is an induced homomorphism G(ρ): G(X) → G(Y): if σ = ∑_{i=1}^{m} f_{i}h_{i}^{∗} ∈ G(X), then G(ρ)(σ) = ∑_{i=1}^{m} ρ(f_{i})(ρ(h_{i}))^{∗}. When ρ is an epimorphism in CR or in R/N then G(ρ) is a surjection. This follows since T is a left adjoint and thus preserves epimorpisms; moreover, T (X) → T (Y) is surjective (Proposition 3.1), while T (Y) → G(Y) is surjective by construction. Since the composite T (X) → T (Y) → G(Y) = T (X) → G(X) → G(Y), it follows that the second factor is surjective .

The surjectivity of G(ρ) is a necessary condition for an epimorphism in R/N and when this occurs we will say, following [Henriksen, Raphael, & Woods (2001), Proposition 2.1], that Y is G-embedded in X. (There are special cases where G-embedded suffices for a CR-epimorphism; see Proposition 4.1(ii), below.) The next observations will characterize when Y is G-embedded in X. The conditions are exactly those of Proposition 1.1, but restricted to the z-spectra of C(X) and C(Y).

3.3. Proposition. Let Y be a subspace of a topological space X. Then Y is G-embedded in X if and only if every f ∈ C(Y) is in the image of G(ρ). In other words, each f ∈ C(Y) can be written ∑_{i=1}^{m} ρ(a_{i})ρ(b_{i})^{∗} for some m ∈ N, with a_{i}, b_{i} ∈ C(X).

We next describe more explicitly the relation between T (X) and G(X). Recall that the maximal ideals of T (X) are in one-to-one correspondence with the primes of T (X). For P ∈ Spec C(X), let the corresponding maximal ideal be P̃.

3.4. Proposition. Let X be a topological space. (1) The kernel of the natural surjection ν_{X}: T (X) → G(X) is ker ν_{X} = ⋂_{P ∈zSpec C(X)} P̃. (2) Spec G(X) is zSpec C(X) as a subspace of Spec C(X) endowed with the constructible topology.

Proof. We will use the subscript ‘c’ when using the constructible topology, for example Spec_{c} C(X). As always with regular rings, we can view them (see [Pierce (1967)] or [Burgess & Stephenson (1976)]) as rings of sections of a sheaf of fields over a boolean space. The kernel of a homomorphism from T (X) will be the set of sections which vanish on an open subset U of the boolean space Spec_{c} C(X) and the image is the restriction of the sheaf to the complement of U. We will show that elements of ker ν_{X} have supports in Spec_{c} C(X) − zSpec_{c} C(X). We need the fact ([Schwartz (1997), Theorem 3.2] or [Montgomery (1973), Theorem 7.1]) that the fixed maximal ideals are constructibly dense in zSpec C(X).

In one direction, note what happens to elements of T (X) under ν_{X}. For x ∈ X denote by 𝔪_{x} the fixed maximal ideal for x. Then for α = ∑_{i=1}^{m} a_{i}b′_{i} and x ∈ X, ν_{X}(α)(x) = ∑_{i=1}^{m} a_{i}(𝔪_{x})(b′_{i}(𝔪_{x}) = ∑_{i=1}^{m} a_{i}(x)b′_{i}(x), where b′_{i}(x) = 1/b_{i}(x) or 0. Thus, if ν_{X}(α) = 0, α(𝔪_{x}) = 0 for all x ∈ X. Then, by the density of the of fixed maximal ideals, α is zero at all the prime z-ideals, since its support is both open and closed.

Next, suppose the support of α is in Spec_{c} C(X) − zSpec_{c} C(X). Take a basic set U = V (I) ∩ D(b) in the support of α. For any 𝔭 ∈ U, b ∉ 𝔭 and b ∉ 𝔪, for the unique maximal ideal containing 𝔭, because the maximal ideals are z-ideals. By [Gillman & Jerison (1960), 14.5], b(𝔭) is an infinitesimal for each 𝔭 ∈ U. If e is the idempotent in T (X) whose support is U, for all n ∈ Z, nbe ≤ 1. However, G(X) is archimedean and so ν_{X}(be) = 0. Moreover, e = (be)(be)′, showing that ν_{X}(e) = 0. It follows that any element of T (X) whose support does not meet zSpec_{c} C(X) is in the kernel of ν_{X}.

We can now characterize G-embedded subspaces in terms like those of Proposition 1.1.

3.5. Theorem. Let Y be a subspace of a topological space X. Then Y is G-embedded in X if and only if (i) zSpec ρ: zSpec C(Y) → zSpec C(X) is injective and (ii) for every 𝔭 ∈ zSpec C(Y), ρ induces an isomorphism Q_{cl}(C(X)/ρ^{−1}(𝔭)) → Q_{cl}(C(Y)/𝔭).

Proof. We need to know that, in fact, Spec ρ(zSpec C(Y)) ⊆ zSpec C(X). This is clear, for if 𝔭 ∈ zSpec C(Y), a ∈ ρ^{−1}𝔭 and b ∈ C(X) with z(b) = z(a), then ρ(b) ∈ 𝔭 as well. With this in hand, G(ρ) is surjective if and only if the two conditions hold, by Proposition 1.1 applied to G(ρ).

When the subspace Y is z-embedded in X then it is automatically the case that Spec ρ is injective on zSpec C(Y) by [Schwartz (1997), pages 288 and 289].

3.6. Definition. For a topological space X, the boolean algebra generated by the zerosets of X will be denoted ℨ(X).

3.7. Proposition. Let X be a normal space. Suppose that Y ⊆ X and Y ∈ ℨ(X). Then Y is G-embedded in X.

Proof. We know that the result is true if Y is a cozero-set (Proposition 2.1) and if it is a zero-set, since closed sets in a normal space are C^{∗}-embedded ([Gillman & Jerison (1960), 3D]). Recall that a closed subspace of a normal is space is normal. Moreover, every finite boolean combination of zeros sets in X is the support of an idempotent in G(X) (for a ∈ C(X), aa^{∗} ∈ G(X) is zero on Z(a) and constantly 1 on coz a).

Let Y ∈ ℨ(X). It is a boolean combination of zero-sets and so it can be expressed in conjunctive normal form, as a union of intersections of zero-sets and cozero-sets. By combining all the cozero-sets and all the zero-sets in each term, we get an expression Y = ⋃_{S}(coz g_{s} ∩ z(h_{s})).

The proof will be by induction on n = |S|. When n = 1, Y = coz g ∩ z(h) and ρ is an epimorphism in CR, a stronger property than G-embedding. Then, for f ∈ C^{∗}(Y) there is ĝ ∈ G(X) so that ĝ|_{Y} = f.

Suppose now that we have the result for n ≥ 1 and that Y = ⋃_{T} (coz g_{t} ∩ z(h_{t})), with |T | = n + 1. Write T = S ∪ {u}. Put A_{S} = ⋃_{S}(coz g_{s} ∩ z(h_{s}). For f ∈ C^{∗}(Y) there is, by the induction assumption, ĝ_{S} ∈ G(X) with ĝ_{S} = f|_{S}. Let e_{S} = e_{S}^{2} ∈ G(X) have support A_{S}. There is, also, ĝ_{u} ∈ G(X) with ĝ_{u}|_{coz g_{u}∩z(h_{u})} = f|_{coz g_{u}∩z(h_{u})}. Let e_{u} = e_{u}^{2} ∈ G(X) have support coz g_{u} ∩ z(h_{u}). Then, f = (ĝ_{S}e_{S} + ĝ_{u}e_{u}(1 − e_{S}))|_{Y}.

Proposition 3.7 shows that there are G-embedded subspaces which do not induce CR epimorphisms because there are subspaces, even of metric spaces X, which are in ℨ(X) but are not locally closed. As an example, we can take Y = ⋃_{n∈N}(1/n + 1, 1/n) ∪ {0} in R because Y is not locally compact.

Recall some notation from N. Schwartz, [Schwartz (1997), page 288]). For a subspace Y ⊆ X: (i) e(Y) ⊆ Spec C(X) is the set of fixed maximal ideals of C(X) attached to the elements of Y, (ii) Ỹ is the constructible closure of e(Y) in zSpec C(X). Schwartz then asks ([ibid., page 289]) whether the image of Spec ρ is the convex hull of Ỹ, cvx Ỹ, and gives a positive answer when Y is z-embedded in X. One key step in Schwartz’ proof is to show that when Y is z-embedded then Spec ρ is injective on zSpec C(Y). The conclusion does not, as we have seen, depend on Y being z-embedded. In fact the injectivity of Spec ρ on zSpec C(Y) will follow if G(ρ) sends the set of idempotents of G(X) onto that of G(Y). When Y is z-embedded, G(ρ) has this property. These remarks are summarized in the next two results.

3.8. Corollary. Let Y ⊆ X and suppose that G(ρ) sends the algebra of idempotents of G(X) onto that of G(Y). Then Spec ρ is injective on zSpec C(Y).

Proof. Since G(X) and G(Y) are regular rings, every idempotent in the image of G(ρ) is the image of an idempotent in G(X). Moreover, the spectrum of a regular ring is the same as that of its algebra of idempotents. Hence, a surjection at the level of idempotents yields an injection of spectra.

3.9. Corollary. [cf. Schwartz (1997), Lemma 4.1 and Proposition 4.2] Let Y be a z-embedded subspace of a space X. Then Spec ρ is injective on zSpec C(Y).

Proof. By Corollary 3.8 it suffices to show that every idempotent of G(Y) is in the image of G(ρ). However, the idempotents of G(Y) are boolean combinations of idempotents of the form aa^{∗}, a ∈ C(Y). (This is the idempotent which is the characteristic function of coz a.) There is some b ∈ C(X) so that z(b) ∩ Y = z(a). Then aa^{∗} = G(ρ)(bb^{∗}).

Note that it is easy to find examples where G(ρ) restricted to the idempotents is surjective but G(ρ) is not. The following remark shows, as an illustration, that the set of irrationals in R is such an example, by [Fine, Gillman, & Lambek (1965), Theorem 3.10(2)].

3.10. Remark. (cf., Theorem 2.10) Let Y be a dense subspace of a space X which is G-embedded and suppose there is a function on Y which cannot be extended continuously to a larger subset. Then Y contains a dense open set.

Proof. Let f ∈ C(Y) be as in the statement. If there is ĝ ∈ G(X) with G(ρ)(ĝ) = f then ĝ is continuous on a dense open set D. As a result f can be extended to a continuous function on Y ∪ D, showing that D ⊆ Y.

There is a more complete statement than that in Corollary 3.9. It is easy to see that G(ρ) is surjective on idempotents if and only if for every zero-set Z of Y there is T ∈ ℨ(X) with Z = T ∩ Y. We do not yet have an example of such a subset which is not also z-embedded.

It is time for some more examples: in the first three, Y is not G-embedded and hence ρ is not an epimorphism even in R/N.

3.11. Examples. 1. Let X be a one-point compactification of an uncountable discrete set D, X = D ∪ {∗}. Then {∗} induces an epimorphism (in CR), while its complement, D, is not G-embedded.

2. There is a closed subset L of the space W of ordinals less than the first uncountable ordinal whose complement is not G-embedded.

3. There is a first countable space with a zero-set which is not G-embedded.

4. There is a space with a C^{∗}-embedded subset which is not a finite boolean combination of closed sets.

Proof. 1. The space X is functionally countable (quoted in [Levy & Rice (1981), Proposition 3.1]). Therefore every function in G(X) has countable range. If D were G-embedded, then elements of C(D) would be images of elements of G(X), by Proposition 3.3. That would mean that elements of C(D) also had countable range, but this is false. Therefore D is not G-embedded.

2. Take L to be the subspace of limit ordinals. By [Gillman & Jerison (1960), 5M (2)], L is closed in the normal space W and, hence, induces an epimorphism in CR. Its complement D is discrete ([ibid.; 5.11]) and uncountable. Moreover, elements of C(W) are constant on a “tail” of W and, hence, so are the functions in G(W). This is not true of C(D).

3. The famous example Γ (e.g., [Gillman & Jerison (1960), 3K] and also [Blair & Hager (1974), Remarks 2.5(a)]) is a completely regular space which is not normal (the closed upper half plane with the “tangent circle” topology). There is a subspace D which is uncountable and discrete. Then |C(D)| = 2^{𝔠} whereas |C(Γ)| = 𝔠. But then, also, |G(Γ)| = 𝔠. This shows that D, while a zero-set, cannot be G-embedded in Γ.

4. We take any space X for which βX − X is dense in βX (e.g., X = Q, [Gillman & Jerison (1960), 6.10]). Suppose X = ⋃_{i=1}^{n}(U_{i} ∩ V_{i}), where the U_{i} are open and the V_{i} are closed in βX. We will show that the U_{i} are pairwise disjoint and use this fact to get a contradiction.

Downward induction on n is used. Suppose first that U = ⋂_{i=1}^{n} U_{i} ≠ ∅. Then there is p ∈ U − X. Then p ∉ ⋃_{i=1}^{n} V_{i}. For any x ∈ X, x ∈ U_{i} ∩ V_{i} ⊆ U for some i. Hence X ⊆ ⋃_{i=1}^{n} V_{i} ∪ U^{c} = V. But V is a proper closed subset of βX containing X, which is impossible.

Next, suppose that for some r, 1 < r ≤ n, the intersection of any r of the U_{i} is empty. Suppose U = ⋂_{i=1}^{r−1} U_{i} ≠ ∅ (re-indexed if necessary). Then there is some p ∈ U −X. Again, p ∉ ⋃_{i=1}^{r−1} V_{i}. On the other hand, if x ∈ X then x ∈ ⋃_{i=1}^{r−1}(U_{i} ∩V_{i}) ⊆ U or x ∈ U_{j} ∩V_{j}, some r ≤ j ≤ n. But, by the induction hypothesis, x ∉ U. Hence, X ⊆ ⋃_{i=1}^{r−1} V_{i} ∪ U^{c}, a proper closed subset of βX. This is impossible and we see that the U_{i} are pairwise disjoint.

This shows that X is a disjoint union of a finite number of closed (in X) subsets U_{i}∩V_{i}, since U_{i} ∩ V_{i} = (⋃_{i=1}^{n} U_{i}) ∩ V_{i} = X ∩ V_{i}. These are clopen in X and, hence, C^{∗}-embedded. They are, thus, clopen in βX ([Gillman & Jerison (1960), 9.6(c)]), making X open in βX. This is not the case.

C. Necessary conditions for G-embeddings.

We saw in Proposition 3.7 that in a normal space X, a subset which is G-embedded is in ℨ(X). The present task is to find a converse. In Proposition 3.17, below, we will show that if X is first countable then a G-embedded subspace must be a boolean combination of closed sets. The two propositions together give a characterization of “G-embedded” in first countable, perfectly normal spaces.

We begin with a rather specialized situation. However, it will twice later turn up in two more general settings. As it stands, it give an example, for each n ∈ N, of a G-embedded subset Y in a perfectly normal first countable space X for which there is f ∈ C^{∗}(Y) so that when f = G(ρ)(ĝ) for some ĝ ∈ G(X), then ĝ has at least n + 1 terms.

3.12. Example. Suppose in a first countable space Y we have disjoint subsets A_{0}, A_{1}, …, A_{2n} with the following properties. (A specific example will be presented below.)

(i) A_{0} = {x_{0}}.

(ii) A_{1} = {x_{m(1)}}_{m(1)∈N}, a sequence of distinct points converging to x_{0}.

(iii) A_{2} = {x_{m(1),m(2)}}, (m(1), m(2)) ∈ N^{2}, where {x_{m(1),m(2)}}_{m(2)∈N} is a sequence of distinct points converging to x_{m(1)}, for each m(1) ∈ N.

(iv) For all k, 1 ≤ k ≤ 2n, A_{k} = {x_{m(1),...,m(k)}}, (m(1), …, m(k)) ∈ N^{k}, where, for each k − 1-tuple m(1), …, m(k − 1), {x_{m(1),...,m(k−1),m(k)}}_{m(k)∈N} is a sequence of distinct points converging to x_{m(1),...,m(k−1)}.

(v) For each k, 1 ≤ k ≤ 2n, cl(A_{k}) = ⋃_{i=0}^{k} A_{i}.

Define Z = ⋃_{i=0}^{n} A_{2i}. Let f ∈ C^{∗}(Z) have the following properties: f is non-zero on Z and cannot be extended to any point of any A_{2i−1}, i = 1, …, n. (As an example, let f(x_{0}) = 2 and f(x_{m(1),...,m(2k)}) = 2 + (−1)^{m(2)}/m(1) + ⋯ + (−1)^{m(2k)}/m(2k − 1).) We make two claims about Z and f. Claim 1: Z = ⋃_{i=0}^{n}(F_{i} ∩ G_{i}), where the F_{i} are closed and the G_{i} open in Y. (Hence, if Y is perfectly normal, Z is G-embedded in Y.) Claim 2: If f = G(ρ)(ĝ) for some ĝ ∈ G(Y), then ĝ has at least n + 1 terms and at least n of these terms are zero at x_{0}.

Proof. Claim 1: This is done by induction: A_{0} is closed and, by property (v), ⋃_{i=0}^{k+1} A_{2k} = ⋃_{i=0}^{k} A_{2i} ∪ [cl(A_{2k+2}) − cl(A_{2k+1})].

Claim 2: This will be by induction on n. When n = 1, suppose that our function f = ab^{∗}|_{Y} for some a, b ∈ C(Y). Since f(x_{0}) ≠ 0, ab^{∗} is continuous on a neighbourhood of x_{0} (on coz b, in fact) and so f can be extended to the points of A_{1} ∩ coz b. But this is not possible by the nature of f. Hence, there are at least two terms, say ĝ = ∑_{i=1}^{s} a_{i}b_{i}^{∗} whose restriction to Z is f. If all were non-zero at x_{0}, ĝ would be continuous on a neighbourhood of x_{0} and f would extend to some elements of A_{1}. This is not possible.

Now suppose that n > 1 and that the claim is true for n − 1. Suppose f can be expressed by ĝ|_{Y} where ĝ = ∑_{i=1}^{n} a_{i}b_{i}^{∗}, a_{i}, b_{i} ∈ C(Y). Assume that a_{i}b_{i}^{∗}(x_{0}) ≠ 0 for i = 1, …, r, while the other terms are zero at x_{0}. By replacing the various sequences in the A_{j} by cofinite subsequences, we may assume, by intersecting Y with ⋂_{i=1}^{r} coz a_{i}b_{i}, that the first r terms are non-zero everywhere on Z.

Suppose that for all x_{m(1),m(2)} ∈ A_{2} we have a_{j}b_{j}^{∗}(x_{m(1),m(2)}) = 0, for j = r + 1, …, n. Then f and ĥ = ∑_{i=1}^{r} a_{i}b_{i}^{∗} coincide on A_{0} ∪ A_{2}. However, ĥ is continuous on a neighbourhood of x_{0} which must contain points of A_{1}. Hence, f can be extended continuously to some points of A_{1}, which is impossible. Hence, we may assume that a_{r+1}b_{r+1}^{∗}, say, is non-zero at some x_{u,v} ∈ A_{2}.

Consider the subset T of Z consisting of all points whose first two indices are u, v. Then f|_{T} and T are an example of the claim for n − 1. Hence, f|_{T} requires at least n terms, of which at least n − 1 are zero at x_{u,v}. Since r is at least 1, there are at least r + 1 terms non-zero at x_{u,v}; this give (n − 1) + (r + 1) = n + r terms. This is impossible.

We now know that ĝ has more than n terms, say s. Finally, suppose, again that r terms of ĝ are non-zero at x_{0}. If s − r < n, we see that, as above, we get some x_{u,v} ∈ A_{2} where ĝ is non-zero in r + 1 terms and zero on fewer than s − r − 1 < n − 1. This is a contradiction. Hence, s − r ≥ n.

3.13. Example. For each n ∈ N, there is an example of the construction of 3.12 in the interval [0, 1] ⊆ R.

Proof. Let A_{0} = {0}, A_{1} = {1/n}_{n>1}, A_{2} = {1/n + 1/mn(n + 1)}_{n,m>1}, A_{3} = {(1/n + 1/mn(n + 1)) + 1/km(m + 1)n(n + 1)}_{m,n,k>1}, and so on.

We now look for instances of the construction in other spaces. For a space X and a subspace Y we define, inductively, a sequence of subspaces as follows: Y_{0} = Y and Y_{1} = cl(Y) − Y. For k ≥ 1, once Y_{k} has been defined, Y_{k+1} = cl(Y_{k}) − Y_{k}. We need the following elementary facts.

3.14. Lemma. Let Y be a subspace of a topological space X. Consider the sequence of subspaces {Y_{k}}_{k∈N}. Then

1) For each k ≥ 0, Y_{2k+1} is a closed subspace of Y_{1} and Y_{1} ⊇ Y_{3} ⊇ Y_{5} ⊇ · · ·.

2) For each k ≥ 0, Y_{2k} is a closed subspace of Y and Y = Y_{0} ⊇ Y_{2} ⊇ Y_{4} ⊇ · · ·.

3) If, for some k ≥ 0, Y_{k} is closed, then Y is a finite boolean combination of closed sets. In fact, if, for some k > 0, Y_{2k} is closed, then Y is a union of k locally closed sets, one of which is closed. If, for some k ≥ 0, Y_{2k+1} is closed, then Y is a union of k locally closed sets.

Proof. 1) This is clear for k = 0. Suppose for some k ≥ 0 that Y_{1} ⊇ ⋯ ⊇ Y_{2k+1} and each Y_{2l+1},. 0 ≤ l ≤ k, is a closed subset of Y_{1}. Then

Y_{2k+3} = cl(Y_{2k+2}) − Y_{2k+2} = cl(Y_{2k+2}) ∩ Y_{2k+2}^{c}
= cl(cl(Y_{2k+1}) − Y_{2k+1}) ∩ (cl(Y_{2k+1}) − Y_{2k+1})^{c}
= cl(cl(Y_{2k+1}) ∩ Y_{2k+1}^{c}) ∩ (cl(Y_{2k+1})^{c} ∪ Y_{2k+1}) .

The first term is in cl(Y_{2k+1}) and so Y_{2k+3} ⊆ Y_{2k+1}, and it is closed in Y_{2k+1} and, hence, also in Y_{1}.

2) This is similar.

3) We need only apply the identity A = (cl(A) − A)^{c} ∩ cl(A) several times. Then

Y = Y_{1}^{c} ∩ cl(Y) = [Y_{2}^{c} ∩ cl(Y_{1})]^{c} ∩ cl(Y)
= [Y_{2} ∪ cl(Y_{1})^{c}] ∩ cl(Y) = [Y_{2} ∩ cl(Y)] ∪ [cl(Y_{1})^{c} ∩ cl(Y)]
= [(Y_{3}^{c} ∩ cl(Y_{2})) ∩ cl(Y)] ∪ [cl(Y_{1})^{c} ∩ cl(Y_{0})]
= [Y_{3}^{c} ∩ cl(Y_{2})] ∪ [cl(Y_{1})^{c} ∩ cl(Y)] , etc.

We need another technical lemma.

3.15. Lemma. Let X be a first countable space and F a countable compact subset. Then F is a zero-set in X.

Proof. For each y ∈ F we take a countable neighbourhood base for y and let U be the union of all of these. A subset of X is called good if it contains F and is the union of finitely many elements of U. For any x ∈ X − F there is a good subset excluding x; this is because X − {x} is a neighbourhood of each of the points of F. Compactness is then used to get a good subset. The good subsets can be indexed by N, say {V_{n}}_{n∈N}, and for each V_{n} there is, by the compactness of F ([Gillman & Jerison (1960), 3.11(a)]), a non-negative function f_{n} bounded by 1 which is zero on F and 1 on X − V_{n}. Now take f = ∑_{n∈N} 2^{−n}f_{n} whose zero-set is F.

3.16. Proposition. Let Y be a subspace of a first countable space X which is Gembedded. If, for some k > 0, Y_{2k} is non-empty, then there is f ∈ C^{∗}(Y), so that if f = G(ρ)(ĝ) then ĝ requires at least k + 1 terms.

Proof. We will build a closed subset F of Y which has the properties of the subset Z in Example 3.12. Suppose f is a function as in 3.12. If f = ĝ|_{F}, for some ĝ ∈ G(X), ĝ will require at least k + 1 terms because of Claim 2 in 3.12. However, as we shall see, F is C-embedded in Y and so an extension f̃ ∈ C^{∗}(Y) of f will also require k + 1 terms. The rest of the proof will be a construction of such a closed subset F and a verification that it is C-embedded in Y. I. Since Y_{2k} is non-empty, there is x_{0} ∈ Y_{2k}. It is in cl(Y_{2k−1})−Y_{2k−1} so there is a sequence {x_{m(1)}}_{m(1)∈N} from Y_{2k−1} converging to x_{0}. By, for example, [Gillman & Jerison (1960), 0.13] we may assume that {x_{m(1)}}_{m(1)∈N}, as a subset of X, is discrete. Pick a basis for the neighbourhoods of x_{0} which consists of a strictly decreasing sequence of open sets W_{n}, n ∈ N. By deleting some of the x_{m(1)} and some of the W_{n}, we may assume that each x_{m(1)} ∈ W_{m(1)} − W_{m(1)+1}.

This means that there are pairwise disjoint open sets U_{m(1)} with x_{m(1)} ∈ U_{m(1)}. We will choose each of the open sets U_{m(1)} so that U_{m(1)} ⊆ W_{m(1)}.

As a next step we note that each x_{m(1)}, m(1) ∈ N, is in cl(Y_{2k−2}) − Y_{2k−2}; thus there is a sequence {x_{m(1),m(2)}}_{m(2)∈N} from Y_{2k−2} converging to x_{m(1)}. We make sure that the terms are in U_{m(1)} and follow the same rules as for the construction of the first sequence.

Suppose, more generally, that we have constructed sequences as far as {x_{a}}_{a∈N^{l}}, 1 ≤ l < 2k, so that x_{a} ∈ Y_{2k−l} and that {x_{m(1),...,m(l)}}_{m(l)∈N} converges to x_{m(1),...,m(l−1)}. Moreover, we make the same provisos about the nature of these sequences; i.e., (i) for b = m(1), …, m(l − 1), the sequence {x_{b,m(l)}}_{m(l)∈N}, is discrete (as a set), (ii) there is a basic set of neighbourhoods of x_{b}, W_{bn} which is strictly decreasing by inclusion and x_{b,m(l)} ∈ W_{bm(l)} − W_{b m(l)+1} and (iii) there are pairwise disjoint open sets U_{b,m(l)} ⊆ W_{bm(l)} with x_{b,m(l)} ∈ U_{b,m(l)}.

The process stops when l = 2k. Let F = {x_{0}} ∪ {x_{a} | a ∈ N^{2l}, 1 ≤ l ≤ k}. By Lemma 3.14(2), F ⊆ Y and we must show that F is closed in Y.

The following observation will be needed. (∗) For any a ∈ N^{r}, 1 ≤ r < 2k, a sequence {v_{i}}_{i∈N} with v_{i} ∈ U_{a,i} will converge to x_{a}. This is because each U_{a,i} ⊆ W_{ai}.

Now suppose we have a sequence of distinct terms S = {x_{a_{i}}} from F converging to x. We will show that either x ∉ Y or x ∈ F.

We will work by induction on l, 1 ≤ l < 2k, by looking at terms of S which lie in the various U_{a}, a ∈ N^{2k−l}.

Case l = 1: if there are infinitely many terms of S in some U_{a}, a ∈ N^{2k−1}, then we have x = x_{a} since S has infinitely many terms of the sequence {x_{a,j}}, which, in turn, converges to x_{a}.

Induction Step: Now suppose that for some l, 1 ≤ l < 2k − 1, that if for some a ∈ N^{r}, with 2k − l ≤ r < 2k, there are infinitely many terms of S in U_{a} then x = x_{b}, where b = a or b = a, c, some c.

Take a ∈ N^{2k−l−1} and suppose that infinitely many terms of S lie in U_{a}. There are two situations to explore. Firstly, if for some c ∈ N^{s}, s ≥ 1, infinitely many terms of S lie in U_{a,c} then, by the induction hypothesis, x = x_{b} where b = a, c or b = a, c, d. Secondly, the first case does not occur and then there are only finitely many terms of S in each U_{a,j}, j ∈ N. This means that there are terms of S in infinitely many of the sets U_{a,j} and (∗) tells us that S converges to x_{a}.

There is only one remaining situation to explore: no U_{j}, j ∈ N, has infinitely many terms of S. Here, the above induction does not apply but we then know that there are infinitely many U_{j}, j ∈ N, containing terms of S. Then (∗) tells us that S converges to x_{0}.

Hence, in all cases S converges to some x_{a} or to x_{0}. Thus either x ∈ F or x ∉ Y, as required. II. The next step will be to show that cl(F) is, in fact, compact. Let the space in Example 3.13 be S with closure T found in the interval [0, 1]. Then T is a compact space which is in one-to-one correspondence with cl(F) and the obvious function T → cl(F) preserves sequential convergence. It is then continuous ([Willard (1970), Corollary 10.5(c)]), showing that cl(F) is compact. III. Since cl(F) is countable and compact, it is a zero-set in X by Lemma 3.15. Then F = cl(F) ∩ Y is a zero-set in Y. Since it is Lindelöf, it is z-embedded ([Blair & Hager (1974), Theorem 4.1]) and then [ibid., Proposition 4.4] says that F is C-embedded in Y.

The next theorem is a converse to Proposition 3.7 but “normal” can be dropped while “first countable” is added.

3.17. Theorem. Let X be a first countable space. If a subspace Y is G-embedded then it is a finite boolean combination of closed sets.

Proof. Suppose, in a general first countable space, that Y is G-embedded. We will show that for some k, Y_{k} = ∅ (with sets Y_{k} as in Lemma 3.14) and, hence, that Y is a finite boolean combination of closed sets. One possibility is that all the Y_{k} are non-empty; then the subsets Y = Y_{0} ⊃ Y_{2} ⊃ Y_{4} ⊃ ⋯ form an infinite chain of proper inclusions. Choose a sequence of points T = {x_{i}} with x_{i} ∈ Y_{2i} − Y_{2i−2}. The set T contains a countable subset which is discrete ([Gillman & Jerison (1960), 0.13]), say S = {x_{i_{j}}}, which may be ordered in the same way as T. There is, then, a family of pairwise disjoint open sets U_{i_{j}} with x_{i_{j}} ∈ U_{i_{j}}.

We proceed as in Proposition 3.16 to construct sets F_{i_{j}} ⊆ U_{i_{j}} ∩ Y which are Cembedded in U_{i_{j}} ∩ Y and which cannot be expressed as the restriction of any ĝ ∈ G(X) with fewer than i_{j} + 1 terms. Then we can use Proposition 2.15 (iii) within Y, as follows. The role of the B_{n} is taken by the F_{i_{j}}, which are closed in Y, and that of the U_{n} by U_{i_{j}} ∩ Y. Put W = ⋃_{j∈N} F_{i_{j}} and define f ∈ C^{∗}(W) by having its restriction to F_{i_{j}} be f_{i_{j}}. Then Proposition 2.15 says that f = ab^{∗}|_{W} for some a, b ∈ C(Y). In turn a = ĝ|_{Y} and b = ĥ|_{Y} for some ĝ, ĥ ∈ G(X), because Y ⊆ X induces an epi. This means that f is the restriction of some element of G(X) to W. This is impossible.

The other possibility is that for some k, all the subsets Y_{2l}, l ≥ k, are equal. Since the sequence does not stop, Y_{2k} is infinite. We then pick our sequence from Y_{2k} and proceed as above; since the elements of Y_{2k} are in all the later ones, any such can be used for the construction with as many steps as we like.

3.18. Corollary. Let Y be a subspace of a first countable perfectly normal space X. Then Y is G-embedded if and only if Y ∈ ℨ(X).

Proof. We combine Theorem 3.17 and Proposition 3.7.

D. Comparing epimorphisms in CR and in R/N.

We are now in a position to compare CR and R/N epimorphisms in some cases. We will show that for a first countable perfectly normal space X, ρ is an R/N-epimorphism if and only if it is a CR-epimorphism, if and only if Y is locally closed in X.

Consider the space in Example 3.12 with n = 1, namely X = {x_{0}} ∪ {x_{i}}_{i∈N} ∪ {x_{ij}}_{ij∈N^{2}} and Y = {x_{0}}∪{x_{ij}}_{ij∈N^{2}}. We will show that ρ: C(X) → C(Y) is not an R/Nepimorphism. To do this we will exhibit a semiprime ring A and ring homomorphisms α, β: C(Y) → A with αρ = βρ but α ≠ β.

Let B ⊆ R^{N^{2}} where (b_{ij}) ∈ B if there is a neighbourhood U of x_{0} ∈ X and M > 0 such that for all x_{ij} ∈ U, |b_{ij}| < M. Then B is a subring of the product. Define I ⊆ B as follows: (b_{ij}) ∈ I if and only if there is a neighbourhood U of x_{0} in X so that for all i ∈ N with x_{i} ∈ U, the sequence {b_{ij}}_{j∈N} converges to 0.

3.19. Lemma. The subset I of B is a semiprime ideal.

Proof. The subset I is clearly closed under subtraction. Let (b_{ij}) ∈ B and (a_{ij}) ∈ I. Then there is a neighbourhood U of x_{0} in X and M > 0 so that for x_{ij} ∈ U, |b_{ij}| < M and for all x_{i} ∈ U each sequence {a_{ij}} converges to 0. Then for all x_{i} ∈ U, the sequence {a_{ij}b_{ij}} also converges to 0. Moreover, if (a_{ij})^{2} ∈ I, so is (a_{ij}).

Define A = B/I. The homomorphisms α, β: C(Y) → A are defined as follows. For f ∈ C(Y), α(f) = (u_{ij})+I where u_{ij} = f(x_{i 2j}); and β(f) = (v_{ij})+I where v_{ij} = f(x_{i 2j+1}).

3.20. Proposition. With the above notation: (i) α and β are homomorphisms; (ii) αρ = βρ; and (iii) α ≠ β. Hence, C(X) → C(Y) is not an R/N -epimorphism.

Proof. (i) We need only check that (u_{ij}) and (v_{ij}) are elements of B. However, by the continuity of f at x_{0} ∈ Y there is a neighbourhood U of x_{0} in X such that f is bounded on U ∩ Y. (ii) When f = ρ(g), the sequences {f(x_{i 2j})} and {f(x_{i 2j+1})} both converge to g(x_{i}). Then {u_{ij} − v_{ij}}_{j∈N} converges to 0, for each i. This shows (u_{ij}) − (v_{ij}) ∈ I and α(f) = β(f). (iii) We need only exhibit f ∈ C(Y) so that α(f) ≠ β(f). Take f such that f(x_{ij}) = 1/i, if j is even and 2/i if j is odd, and f(x_{0}) = 0. Then (u_{ij})−(v_{ij}) = (−1/i) ∉ I.

It would be interesting to know how the criteria of Proposition 1.1 fail in this case. Proposition 3.20 will enable us to look at more general situations.

3.21. Theorem. Let X be a first countable space and Y a subspace of X.

(i) (Cf. Corollary 2.11) If ρ: C(X) → C(Y) is an R/N -epimorphism, then Y is locally closed in X.

(ii) Suppose, in addition, that X is perfectly normal. Then ρ is an R/N -epimorphism if and only if it is a CR -epimorphism.

Proof. (i) Assume that Y is not locally closed. Then cl(Y) − Y is not closed and so there exists x_{0} ∈ Y and a sequence {x_{i}} of distinct points in cl(Y) − Y converging to x_{0}. Hence, there is a subset Z = {x_{0}}∪{x_{ij}}_{ij∈N^{2}} of Y, constructed as in Example 3.12 (with n = 1), whose closure is cl(Z) = Z ∪ {x_{i}}_{i∈N}. Now consider the diagram

\begin{aligned}C(X) & → & C(Y) \\ ↓ & & ↓ \\ C(cl(Z)) & → & C(Z) \end{aligned} .

As shown in part III of the proof of Proposition 3.16, Z is C-embedded in Y so that C(Y) → C(Z) is a surjection. We have seen (Proposition 3.20) that C(cl(Z)) → C(Z) is not an R/N-epimorphism. Hence, C(X) → C(Z) cannot be an R/N-epimorphism. This shows that C(X) → C(Y) is not an R/N-epimorphism.

(ii) If ρ is a CR-epimorphism, it is automatically an R/N-epimorphism. If it is an R/N-epimorphism, the first part of this theorem shows that Y is locally closed. Then ρ is a CR-epimorphism by Corollary 2.12.

4. Remarks and questions.

A. On 1 × 1 zig-zags and P-spaces.

In all the examples above where ρ is a CR-epimorphism, the existence of an epimorphism can be demonstrated using a 1 × 1 zig-zag. In particular, this is true for the subspaces found in Propositions 2.1 and 2.15, and where Corollary 2.12 applies. The question naturally arises whether or not there can be a case where ρ is a CR-epimorphism but there is an f ∈ C(Y) not expressible as a 1 × 1 zig-zag over C(X). The following result gives information about the case where Y is a P -space and suggests where one could look for examples where bigger zig-zags are needed and where they are not to be found. However, the question has not been answered.

4.1. Proposition. Let Y be a subspace of a space X, where Y is a P -space.

(i) If every element of C(Y) satisfies a 1 × 1 zig-zag over C(X), then Y is z-embedded in X.

(ii) If Y is z-embedded and G-embedded in X then every element of C(Y) can be expressed as a 1 × 1 zig-zag over C(X); and, hence, ρ is a CR -epimorphism.

(iii) If Y z-embedded in X, and, moreover, Y is functionally countable (e.g., Y is Lindelöf), then each element of C(Y) satisfies a 1 × 1 zig-zag over C(X).

Proof. (i) If f ∈ C(Y) and there are a, c ∈ C(Y) with b ∈ C(X) so that f = aρ(b)c is a 1 × 1 zig-zag then, since C(Y) is a regular ring, f = aρ(b)ρ(b)^{∗}ρ(b)c = ρ(b)^{∗}aρ(b)ρ(b)c. Hence, fρ(b)^{∗}ρ(d), where d ∈ C(X) is an extension of aρ(b) ρ(b)c. It follows that z(f) = z(bd) ∩ Y.

(ii) It would suffice to have the conclusion for Y in βX, since a 1 × 1 zig-zag over C(βX) restricts to one over C(X). However, X is C^{∗}-embedded in βX; hence, it is both z-embedded and G-embedded. By replacing X by βX, we may assume that X is normal. Then, cl(Y) is C-embedded in X and so we may assume, in addition to the other hypotheses, that Y is dense in X.

Since G(ρ) is surjective, every f ∈ C(Y) can be put in the form f = ∑_{i=1}^{m} ρ(a_{i})ρ(b_{i})^{∗} for some a_{i}, b_{i} ∈ C(X). By Proposition 2.1(ii), it suffices to show that any f ∈ C^{∗}(Y) can be extended continuously to a cozero-set containing Y. We will show that each ρ(b_{i})^{∗} can be so extended, which will give the result. Consider ρ(b)^{∗} for some b ∈ C(X). Its zero-set and cozero-set are both clopen in Y and so are both zero-sets. Clearly coz(b) ∩ Y = coz(ρ(b)^{∗}) and there is some c ∈ C(X) so that coz(c) ∩ Y = z(ρ(b)^{∗}). By density of Y, coz(b) ∩ coz(c) = ∅. Then ρ(b)^{∗} can be extended to g on coz(b) ∪ coz(c) by g(x) = 1/b(x) for x ∈ coz(b) and g(x) = 0 for x ∈ coz(c).

(iii) As in (ii), it suffices to assume that Y is dense in X. If f ∈ C(Y) then its image is countable, say {r_{i}}_{i∈N}. Then, as in (ii), each f^{−1}(r_{i}) = B_{i} is a cozero-set in Y. Write it as coz(a_{i}) ∩ Y = B_{i}, for some a_{i} ∈ C(X). By the density, the coz(a_{i}) are disjoint and ⋃_{i∈N} coz(a_{i}) is a dense cozero-set to which f can be extended (by letting the new function be constantly r_{i} on coz(a_{i})).

For an example of a P -subspace Y of X which is functionally countable, but not Lindelöf, see [Levy & Rice (1981), Example 4]. It induces an epimorphism via 1 × 1 zig-zags.

It would be interesting to find an example with the following properties: A space X and subspace Y such that

(i) Y is an uncountable discrete space not z-embedded in X;

(ii) Y is the intersection of the dense cozero sets of X;

(iii) X − Y is strongly discrete (see [Henriksen, Raphael, & Woods (2001), Proposition 2.9]). In such a situation ρ would be an epimorphism but there would be an element of C(Y) which is not expressible as a 1 × 1 zig-zag over C(X), and Y would be G-embedded but not z-embedded.

B. On nilpotents in C(Y) ⊗_{C(X)} C(Y).

It is clear from the proof of the spectrum characterization of epimorphisms in CR ([Lazard (1968), Proposition IV 1.5]) that if ρ is an R/N-epimorphism then it would be a CR-epimorphism, if we knew that C(Y) ⊗_{C(X)} C(Y) had no non-zero nilpotent elements. On the one hand, the fact that all the residue fields here are of characteristic zero and the corresponding extensions are trivial or purely transcendental would suggest that there are no nilpotents in the tensor product. On the other hand, it is easy to find G-embedded subspaces Y of X, where there are elements f ∈ C(Y) not extendible to X whose squares do extend (for example, in the space of Example 3.12, with n = 1). This makes the construction of a nilpotent element seem plausible.

We know that for first countable, perfectly normal spaces X, the two kinds of epimorphisms coincide. The question is still open for more general kinds of spaces. If it turns out that the two kinds of epimorphisms differ for ρ, the relationship between “ρ an R/N-epimorphism” and “Y is G-embedded” still remains to be clarified.

C. On the embedding of Spec C(Y) in Spec C(X).

In [Schwartz (1997)] it is shown that when Y is z-embedded in X, the image of Spec ρ can be described as cvx Ỹ. There are two ingredients: (1) the injectivity of Spec ρ on zSpec C(Y) and (2) that Spec ρ reflects inclusions in Spec ρ (zSpec C(Y)). The first part has been discussed in detail in Section 3. Is there a version of [Schwartz (1997), Lemma 4.1] if, instead of assuming that Y is z-embedded in X, we only say that G(ρ) is surjective on the set of idempotents of G(Y)?

D. Remarks on H(X) and Q_{cl}(X).

The ring H(X), the epimorphic hull of C(X) in CR, was mentioned briefly in Section 3. It is the smallest regular ring between C(X) and the complete ring of quotients, Q(X), and satisfies the universal property of being the largest essential extension which is an epimorphism in CR ([Storrer (1968), Definition 8.3]). Schwartz has shown ([Schwartz (2000), Theorem 1.2]) that if a real closed ring A has an epimorphic hull in POR/N then it coincides with that in R/N and in CR. Moreover, [ibid., Theorem 4.4] shows that C(X) does have an epimorphic hull in POR/N; it is, hence, H(X). We always have Q_{cl}(X) ⊆ H(X). In [Raphael & Woods (2000), Open question 8.1] it was asked when Q_{cl}(X) and H(X) are, themselves, of the form C(Z), for a space Z. In Theorem 4.2, we will answer this question for some spaces. These results are closely related to our theme because in each case discussed the subspace gX of X induces an epimorphism of rings. Recall (e.g., [Raphael & Woods (2000)]) that gX denotes the intersection of the dense cozero sets of X.

A space is called a quasi-F space if each dense cozero-set is C^{∗}-embedded. Much is known about these spaces (for example, [Dashiell, Hager & Henriksen (1980), Theorem 5.1] and [Schwartz, 1989b, Theorem 6.2]). By [Gillman & Jerison (1960), 14N], basically disconnected spaces are F -spaces and therefore quasi-F spaces. In particular, extremally disconnected spaces are quasi-F spaces.

4.2. Theorem. (i) Let X be a quasi-F space. Then Q_{cl}(X) is a ring of continuous functions if and only if gX is dense and C^{∗}-embedded in X.

(ii) Let X be basically disconnected. Then the epimorphic hull, H(X), of C(X) is a ring of continuous functions if and only if its set of P-points is dense and C^{∗}-embedded in X.

(iii) Let X be an extremally disconnected space in which gX is of non-measurable cardinality. Then H(X) is a ring of continuous functions if and only if the set of isolated points of X is dense and C^{∗}-embedded.

Proof. i) Suppose Q_{cl}(X) is a ring of continuous functions. By Theorem 7.2 of [Raphael & Woods (2000)], gX is dense in X, Q_{cl}(X) = C(gX) and every function in C(gX) extends to a dense cozero set of X. Let f ∈ C^{∗}(gX). It extends to a bounded function on a dense cozero set of X and from it to a bounded function on X. Thus C^{∗}(X) → C^{∗}(gX) is surjective and gX is C^{∗}-embedded in X.

Conversely let gX be dense and C^{∗}-embedded in X. By [ibid., Theorem 7.2] again, it suffices to show that each f ∈ C(gX) extends to a dense cozero set of X. As usual, let f_{1} = f/(f^{2} +1) and f_{2} = 1/(f^{2} +1). Let them extend to F_{1} and F_{2} in C(X), respectively. Then f extends to F_{1}/F_{2} on coz(F_{2}).

(ii) This becomes an instance of (i) if one notes that when X is basically disconnected gX is a P -space ([Raphael & Woods (2000), Corollary 4.12]), and that Q_{cl}(X) and H(X) coincide because in a basically disconnected space the annihilator on an element is a principal ideal. This says ([Raphael (1972), Corollary 1.6]) that Q_{cl}(X) is regular and, hence, equals H(X)).

(iii) This is also an instance of (i), but one needs [Gillman & Jerison (1960), 12H]. When X is extremally disconnected then gX is, as well, if it is dense in X. It is a P -space because X is basically disconnected. Thus it is the set of isolated points of X, if gX is of non-measurable cardinality.

Since almost-P spaces are quasi-F spaces [Dashiell, Hager & Henriksen (1980), p. 681], there are instances where case (i) holds but case (ii) fails. An example is the one point compactification of an uncountable discrete space.

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

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

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

Email: barr@barrs.org
      wburgess@uottawa.ca
      raphael@alcor.concordia.ca

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mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/12/11-12.{dvi,ps}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi, Postscript and PDF. Details will be e-mailed to new subscribers. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh,

rrosebrugh@mta.ca.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX is the preferred flavour. TEX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal’s WWW server at http://www.tac.mta.ca/tac/. You may also write to tac@mta.ca to receive details by e-mail.

Editorial board.

John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org, Associate Managing Editor Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of Wales Bangor: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Palo Alto Research Center: paiva@parc.xerox.com Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca, Managing Editor Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca