Order (2012) 29:105–118 DOI 10.1007/s11083-011-9200-x

Compact Hausdorff Approach Frames Bernhard Banaschewski · Robert Lowen · Christophe Van Olmen

Received: 14 September 2010 / Accepted: 2 February 2011 / Published online: 1 March 2011 © Springer Science+Business Media B.V. 2011

Abstract In this paper we provide a Heine–Borel type characterization for 0compactness in approach spaces (Lowen 1997). Since this requires making use of the so-called regular function frame the most natural setting to develop this in is approach frames (Banaschewski 1999; Banaschewski et al., Acta Math Hung 115(3):183–196, 2007, Topology Appl 153:3059–3070, 2006). We then go on to characterize Hausdorffness for approach frames which allows us to study some fundamental properties of compact Hausdorff approach frames. Keywords Compact · Hausdorff · Approach space · Approach frame · Prime · Nucleus · Order ideal

1 Introduction In approach spaces the most natural intrinsic notion of compactness, which has been characterized in various ways [6], is so-called 0-compactness. This follows not only from the fact that this concept satisfies natural fundamental properties such as e.g. a Tychonoff theorem [11] but also from the fact that it satisfies intrinsic good categorical properties in the setting of [5] such as a Kuratowski–Mrówka theorem. Until now however there was no characterization of 0-compactness like the Heine– Borel property. Such a characterization would of course require making use of the so-called regular function frame of an approach space [11]. In this paper we rectify this situation by going one step further and defining a notion of compactness in

B. Banaschewski Dept. of Mathematics & Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 R. Lowen (B) · C. Van Olmen Dept. of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020 Antwerp, Belgium e-mail: [email protected]

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the more general setting of approach frames [3, 4] which will turn out to give the required Heine–Borel type characterization of 0-compactness in the special case of approach spaces. Dualizing the Heine–Borel property it turns out that compactness can also be characterized by means of a special type of ideals, which we call small ideals. At the same time we characterize Hausdorfness of approach frames making use of the notion of (closed quotient, dense)-factorizations. Once we have obtained these required characterizations we go on to study some fundamental properties of compact Hausdorff approach frames.

2 Preliminaries 2.1 Approach Spaces Approach spaces form a topological construct which can be characterized in many ways [10, 11]. In this article we will use regular function frames, limit operators of filters and the distance. A distance on a set X is a function δ : X × 2 X → [0, ∞] with the following axioms: (D1) (D2) (D3) (D4)

∀x ∈ X : δ(x, {x}) = 0, ∀x ∈ X : δ(x, ∅) = ∞, ∀x ∈ X, ∀A, B ∈ 2 X : δ(x, A ∪ B) = min{δ(x, A), δ(x, B)}, ∀x ∈ X : ∀A ∈ 2 X , ∀ ∈ [0, ∞] : δ(x, A) ≤ δ(x, A() ) +  with A() = {x|δ(x, A) ≤ }.

A regular function frame is a collection R of [0, ∞]-valued functions on X fulfilling  (R1) ∀S ⊂ R : S ∈ R, (R2) ∀μ, ν ∈ R : μ ∧ ν ∈ R, (R3) ∀μ ∈ R, ∀α ∈ [0, ∞] : μ + α ∈ R, (R4) ∀μ ∈ R, ∀α ∈ [0, ∞] : (μ − α) ∨ 0 ∈ R. The morphisms between approach spaces are called contractions and have as defining property: f : (X, δ X ) → (Y, δY ) is a contraction if and only if ∀x ∈ X, A ⊂ X : δY ( f (x), f (A)) ≤ δ X (x, A). The regular function frame of an approach space X is the set of contractions from X to P := ([0, ∞], δP ) where δP is defined by δP (x, A) := (x − sup A) ∨ 0 for all x ∈ [0, ∞] and A ⊂ [0, ∞]. We denote this collection R X. F(X) stands for the set of all filters on X. Definition 2.1 [10] An approach space X is said to be 0-compact if sup

inf sup δ(x, F) = 0.

F ∈F(X) x∈X F∈F

We call this notion 0-compact to distinguish it from the stronger notion of compactness of approach spaces, which means that the topological coreflection of the space is compact. The notion of 0-compactness turns out to be the right notion also from the results of [6] where a Kuratowski-Mrowka theorem is shown to hold.

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Recall that a frame L is a complete lattice which satisfies the distributive law   a∧ S= {a ∧ t|t ∈ S} for all a ∈ L and S ⊆ L. The homomorphisms between frames are functions that commute with arbitrary joins (and thus preserve the bottom element 0) and finite meets (hence also preserving the top element 1). Compactness in frames is defined using an abstraction of the Heine–Borel property of compactness:   ∀S ⊂ L : S = 1 ⇒ ∃T ⊂ S, T finite : T = 1. For more information about this and about frames in general, see e.g. Johnstone [9], Pultr [13], or Vickers [16]. We note the familiar fact that the open subsets of a topological space form a frame which is compact if and only if the space is compact. Definition 2.2 [4] An approach frame L is a frame (with top  and bottom ⊥) equipped with two families of unary operations on L, addition and subtraction with α ∈ [0, ∞], denoted Aα and Sα respectively, which satisfy all identities valid for the frame operations and the addition and (truncated) subtraction by α in [0, ∞]. Note, however, that these identities are consequences of the following specific ones (1) Aα Aβ = Aα+β , (2) A∞ a = , (3) Sα Aα = Id if α  = ∞, (4)  Aα Sα = Aα ⊥ ∨ Id,  (5) α∈S Aα a =A S a, (6) Aα ( S) = Aα (S) if S  = ∅ (see [4] for the details). For convenience we will use the notation λ for Aλ ⊥ (and hence ⊥ = 0,  = ∞). The morphisms between approach frames are frame homomorphisms that commute with additions and subtractions, providing us with a category which we denote AFrm. It is clear that the regular function frame of an approach space is an approach frame and that contractions f : (X, R X) → (Y, RY) give approach frame homomorphisms from RY to R X. In fact, this determines a contravariant functor R : Ap → AFrm. This functor has an adjoint on the right, the so-called spectrum functor  defined in [4]. This sends an approach frame L to the approach space given by the set of homomorphisms from L to J, which is [0, ∞] with the obvious approach frame ˆ with aˆ :  L → [0, ∞] : ξ  → ξ(a). structure, and approach structure L, There is an alternative characterization of the spectrum using approach prime elements. These are the prime elements a such that λ  ≤ a for all λ > 0. The isomorphism between morphisms from L to J and the appraoch prime elements is given by sending a to ξa : b  → inf{λ|b ≤ Aλ a} and ξ to {a|ξ(a) = 0}.

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This pair of adjoint functors gives rise to the definition of sobriety and spatiality [4]. We say an approach space is sober if  X : X →  R X is an isomorphism and an approach frame is called spatial if η L : L → R L is an isomorphism where  X and η L are the adjunction maps for R and . Finally, we will also need the notion of a nucleus to define quotients. As in frames, onto homomorphisms can be described by nuclei. Definition 2.3 A nucleus on an approach frame L is a mapping ν : L → L such that (N1) (N2) (N3) (N4)

a ≤ ν(a), νν(a) = ν(a), ν(a ∧ b ) = ν(a) ∧ ν(b), ν(Aλ a) = Aλ (ν(a)).

Recall that the definition of a nucleus in frames is a mapping satisfying (N1) to (N3). For approach frames, we need to add stability for the extra operations and (N4) is sufficient for this. Note that, for any approach frame  homomorphism h : L → M the associated frame nucleus ν on L , given by ν(a) = {x ∈ L|h(x) = h(a)} is in fact an approach frame nucleus, and it will be seen below that any approach frame nucleus arises in this way. As is familiar for frames, Fix(ν) = ν[L] is a frame for any nucleus ν on a frame L such that the corestriction L → Fix(ν) of ν is a frame homomorphism. Moreover for any onto homomorphism h : L → M we get the (onto, one-one) factorization h

L

 Fix(ν)

/ M < y yy y yy yy

Similarly, we have that for any nucleus ν on an approach frame L, Fix(ν) is an approach frame such that its underlying frame is the usual one provided by ν, its addition by λ coincides with Aλ , and its subtraction by λ is a  → Sλ ν(a ∨ λ). In frame theory one defines the notion of a closed quotient. Expressed in terms of nuclei this means that for every element s ∈ L we have a nucleus a  → a ∨ s with associated quotient ↑ s. In the sense of the pointwise ordering of nuclei we see that this closed nucleus is the smallest nucleus ν for which ν(0) = s. In approach frames we cannot define an analogous nucleus for every element of L, since λ  ≤ ν(0) for all λ > 0 for every nucleus different from the trivial one, a  → ∞. We can however use the concept of the smallest nucleus for a given s = ν(0). This nucleus is not easily described, but we can define a pre-nucleus νs (in this context, a function satisfying (N1), (N4) and ν(x) ∧ y ≤ ν(x ∧ y)) which gives the desired nucleus by transfinite iteration or, equivalently, as the closure operator associated with Fix(νs ) = {a ∈ L|νs (a) = a}. Explicitly, νs is given by  νs (a) = (Aλ s ∧ (λ → a)) λ≥0

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and even satisfies (N3) as well. The associated nucleus will be denoted ν (s) , and we will denote Fix(νs ) by C(s).

3 Compact Approach Frames The top element plays a crucial role in defining compactness for frames. In an approach frame L, we will see that in analogous fashion the filter N L := {a ∈ L|∃λ > 0 : λ ≤ a} is of vital importance. It will turn out that we can generalise the concept of compactness in frames to approach frames in such a way that N L will then take over the role of the top element, analogous to the situation of regularity [3]. In the following, we put S L = L \ N L . Definition 3.1 In any approach frame L an ideal I ⊂ L is calleda small ideal if I ⊆ S L and L is called compact if for all small ideals I we have that I ∈ S L . Note that in frames we can alsocharacterize compactness using ideals, namely, a frame is compact if we have that I = 1 implies that 1 ∈ I for all ideals I . So we see that N L takes over the role of {1}, and the λ with λ > 0 correspond to the top element in this situation. Proposition 3.2 An approach space (X, δ) is 0-compact if and only if R X is compact. Proof We start by defining two constructions: take a small ideal I ⊂ R X and a filter F on X, we will define operations to transform an ideal into a filter and vice versa. ι(I ) :=< {μ ≤ α|μ ∈ I , α > 0} ∪ {{μ = 0}|∃x ∈ X : μ(x) = 0} >, ω(F ) := {ϕ|∃F ∈ F : ϕ ≤ δ F }.   It is clear that α F = ω(F ) and I ≤ αι(I ). Translating the definition of compactness for anapproach frame to this context gives that for all small ideals I we have that infx∈X i∈I i(x) = 0. Obviously we will then also have for all filters that infx∈X α F (x) = 0 and conversely, if the approach space is 0-compact we see that R X is a compact approach frame, using in both cases the remark of the relation between the adherence operator, ι and ω.   As a simple application, we have that J, being R X for the one-point space X, is compact. As noted before, we can express the compactness of approach frames in a fashion analogous to the frame concept of compactness; in fact we have more than one equivalent expression for compactness. Definition 3.3 I ⊂ L is a λ-ideal if I is a lattice ideal in L and μ  ∈ I for all μ > λ.

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Proposition 3.4 The following statements are equivalent: (1) (2) (3) (4) (5)

L is compact,   ∀H ⊂ L :  (∀H0 ⊂ H, H0 f inite : H0 ∈ S L ) ⇒  H ∈ SL , ∀H ⊂ L : H ∈ N L ⇒ ∃H0 ⊂ H, H0 f inite : H0 ∈ N L , The set N L is Scott-open.  ∀λ ∈ [0, ∞[: I λ-ideal ⇒ ∀μ > λ : μ  ≤ I .

Proof To show that 1 implies 2, take a set H which satisfies that the join of every finite subset is an element of S L . Then the ideal {a|∃H0 ⊂ H, H0 finite : a ≤  I :=  H0 } is a small ideal and hence by compactness I = H ∈ S L . It is clear that 2 and 3 are equivalent.  To show that 3 implies 4, takea directed set D and suppose D ∈ N L . By 3 we find afinite subset D such that D ∈ N L . Since D is directed, we find an element d ≥ D and thus D ∩ N L  = ∅. From 4 to 5, take a λ-ideal I , and consider Sλ I = {Sλ a|a ∈ I }, it is clear that this is  a directed set. Hence, Sλ I ∈ N L if and only if there exists an element i ∈ Sλ I such that i ∈ N L , but since I is a λ-ideal this cannot happen. Finally, to go from 5 to 1, we remark that small ideals are 0-ideals.   Note that an approach frame considered as a frame  will never be compact unless it is the trivial one element approach frame, since λ>0 λ = ∞ but the join of a finite subset T will never reach ∞. Given the similarity between compactness of frames and compactness of approach frames, we expect that some similar results can be shown. This is indeed the case, but nevertheless we see that results which are trivial for frames become non-trivial here, mainly due to the more complicated nature of closed sublocales in approach frames. Proposition 3.5 Every sub-approach frame of a compact approach frame is compact.  

Proof Trivial.

Corollary 3.6 Given a 0-compact approach space (X, R), every coarser approach structure on X is also 0-compact. Proposition 3.7 A closed sublocale C(s) of a compact approach frame L is compact. Proof Let ν be any closed nucleus and νs the prenucleus for s = ν(0) described earlier so that ν is the nucleus determined b νs . Then s ∈ S L and for any a ≥ s in S L and λ > 0, λ → a ∈ S L since μ ≤ λ → a if and only if λ ∧ μ ≤ a. Consequently, for any λ1 , . . . , λn > 0 s = A0 s ∧ (0 → a) ≤

n 

Aλi s ∧ (λi → a) ≤

i=1



 ≤

inf λi

i=1,...,n

n  (λi → a) i=1

→ a ∈ SL

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and by compactness it follows that νs (a) ∈ S L , showing that νs maps S L ∩ ↑ s into itself. Further, since a ≤ νs (a) and S L ∩ ↑ s is closed under joins of chains, ν(a) ∈ S L followsby transfinite iteration of ν s . Finally, for any small ideal I of C(s), I ⊂ C(s), hence I ∈ S L in L, and then ν( I ) ∈ S L ∩ C(s), saying that the join of I in C(s) is in SC(s) , and hence that C(s) is compact.   We also have a Tychonoff theorem for approach frames, proved making use of the axiom of choice, following a reasoning similar to that in Dowker [7]. For frames, the proof by Dowker was eventually superseded by Johnstone’s choice free and Vermeulen’s constructive proof. It is an open question whether the same is possible for approach frames. Lemma 3.8 In a compact approach frame L, each element in S L is below some approach prime element. of L, we see Proof Take a ∈ S L and set K = {x ∈ S L |a ≤ x}. Using the compactness  that every chain C in K has an upper bound in K, namely C. Hence, by Zorn’s Lemma, we find maximal elements in K. Any such element p is approach prime since p = b ∧ c implies that either b or c is an element of K and since p is maximal in K we have that b = p or c = p.   Remark 3.9 Actually, Zorn’s Lemma is not needed here, the weaker Prime Ideal Theorem, by a suitable modification of the argument in [2], will already suffice. Lemma 3.10 A maximal small ideal is prime. Proof Let I by any maximal small ideal and a, b  ∈ I . Then by maximality and since the λ are totally ordered there exist λ > 0 such that λ ∈ I ∨ ↓ a and λ ∈ I ∨ ↓ b ; hence λ ∈ (I ∨ ↓ a) ∩ (I ∨ ↓ b ) = I ∨ (↓ a∩ ↓ b ) = I ∨ ↓ (a ∧ b ) showing a ∧ b  ∈ I .

 

The following result is the counterpart of Tychonoff’s theorem for 0-compact approach spaces. Theorem 3.11 A coproduct of compact approach frames is compact. approach frame for each i ∈ I, and J Proof Let I be any index set, Li a compact  any small ideal in the coproduct L = i∈I Li . Since every small ideal is contained in a maximal small ideal (straightforward by Zorn’s lemma) it follows that we may suppose J to be a maximal small ideal. Further, let qi : Li → L be the coproduct maps. Now, recall that any a ∈ L is the join of elements qi1 (b i1 ) ∧ . . . ∧ qin (b in ) so that, for any a ∈ J , we also have qi j (b i j ) ∈ J for some j since J is maximal and therefore  prime by Lemma 3.10. It follows that a ≤ {qi (b )|i ∈ I, qi (b ) ∈ J }, showing J ≤  {ai (b i )|ai (b i ) ∈ J , i ∈ I}.

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 Next, let Ji = qi−1 [J ]. Since qi (λ) = λ this is a small ideal so that ui = Ii ∈ S Li by compactness. Further, by Lemma 3.8, there exists an approach prime pi ≥ ui , and we let ξi : Li → J be the associated homomorphism. Finally, if ξ : L → J is the homomorphism such that ξ ◦ qi = ξi then ξ



   J ≤ {ξ(qi (b i ))|qi (b i ) ∈ J , i ∈ I} = {ξi (b i )|b i ∈ Ji , i ∈ I})  ≤ {ξi ( pi )|i ∈ I} = 0.

This shows that



J ∈ S L , and hence L is compact.

 

Finally, we will obtain a compactification for any approach frame, derived from the well-known ideal compactification in frames. Proposition 3.12 For any approach frame L, the frame J L of all ideals of L can be equipped with an approach frame structure def ining a compact approach frame such that the map J L → L taking joins in L becomes an approach frame homomorphism. Further, the correspondence L  → J L is functorial and the maps J L → L are natural in L. Proof For any λ ∈ [0, ∞] and I ∈ J L, put {↓ Aλ a|a ∈ I }, Sλ I = {↓ Sλ a|a ∈ I } = {Sλ a|a ∈ I }. Aλ I =

It is an easy verification that this defines an approach frame such that L  →  J L is functorial and the maps I → I have the stated properties. Regarding compactness, note first that the zero of J L is the zero ideal {0} so that, in J L, λ = Aλ {0} =↓ {Aλ λ} =↓ λ. Hence NJ L consists of all I ∈ J L such that λ ∈ I for some λ > 0, and since updirected joins in J L are simply unions, NJ L is obviously Scott open.  

4 Compact Hausdorff Approach Frames The notion considered here is the exact counterpart of an important concept in the setting of frames which, in turn, was motivated by the role of compact Hausdorff spaces in classical topology. We begin by establishing some basic properties of Hausdorff approach frames in general which will then lead to the results in the compact setting that are our ultimate concern. As a starting point, we need a result concerning the interaction of homomorphisms with the closed quotient ν (s) : L → C(s) given by a prenucleus νs on L described at the end of Section 2. Lemma 4.1 For any approach frame homomorphism h : L → M, if ν (s) : L → C(s) and ν (u) : M → C(u) are closed quotients such that h(s) ≤ u then there exists an

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approach frame homomorphism h˜ : C(s) → C(u) such that the following diagram commutes h

L

/ M

 C(s)

 / C(u)

ν (s)

ν (u)

h˜

Proof First note that hνs ≤ νu h, that is hνs (a) ≤ νu h(a) for all a ∈ L: by the explicit formula for νs given earlier, hνs (a) = h(∨{Aλ s ∧ (λ → a) | λ ≥ 0}) = ∨{Aλ h(s) ∧ h(λ → a) | λ ≥ 0} ≤ ∨{Aλ u ∧ h(λ → a) | λ ≥ 0} ≤ νu h(a), the third step because h(s) ≤ u and the last since h(λ → a) ≤ λ → h(a): indeed, (λ → a) ∧ λ ≤ a by definition, hence h(λ → a) ∧ λ ≤ h(a) and therefore h(λ → a) ≤ λ → h(a). It follows that hνs ≤ ν (u) h because ν (u) is the nucleus determined by the prenucleus νu . Now let M be the set of all order-preserving ϕ : L → L such that hϕ ≤ ν (u) h, and note that (1) νs ∈ M, (2) ϕ ∈ M implies νs ϕ ∈ M, (3) ∨E ∈ M for any E ⊂ M. (1) is as just proved, (2) since hνs ϕ ≤ ν (u) hϕ ≤ ν (u) ν (u) h = ν (u) h, and (3) because h is a frame homomorphism. Next, take N ⊂ M to be the smallest subset of M for which these three conditions hold. Then ν := ∨N belongs to N so that νs ν ≤ ν, showing νs ν = ν. This says that ν(a) ∈ Fix(νs ) for each a ∈ L, hence ν (s) ν(a) = ν(a) by the definition of ν (s) , and finally ν (s) ≤ ν. On the other hand, L := {ϕ ∈ M | ϕ ≤ ν (s) } also satisfies the above three conditions, and consequently N ⊂ L showing ν ≤ ν (s) as well, which proves that ν = ν (s) . As a consequence, ν (s) ∈ M so that ν (s) h ≤ ν (u) h and hence ν (u) hν (s) = ν (u) h which says that the action of ν (u) h on C(s) provides the desired homomorphism h˜ : C(s) → C(u).   Corollary 4.2 Any approach frame homomorphism h : L → M has the factorization ν (s)

/ C(s) BB BB BB h BB h !  M  where s := h∗ (0) = {a ∈ L | h(a) = 0}. L

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Proof Clearly, s = ν(0) for the nucleus ν determined by h so that we do have the closed quotient ν (s) : L → C(s) as noted at the end of Section 2. On the other hand, since h(s) = 0 the lemma applies to the case u = 0 for which C(u) = M and ν (s) is the identity map.   Note that the above h is obviously dense by the definition of s. Therefore we shall refer to the decomposition h = hν (s) as the (closed quotient, dense)-factorization of h. With this preparation, we can now turn to the definition needed here. For any approach frame L, let i, j : L → L ⊕ L be the coproduct maps, γ : L ⊕ L → L the homomorphism such that γ i = id L = γ j, called the codiagonal map, and ∇ :=

 {i(a) ∧ j(b ) | a ∧ b = 0} = γ∗ (0).

Definition 4.3 An approach frame is called Hausdorff if γ in the (closed quotient, dense)-factorization γ = γ ν (∇) is an isomorphism. Note that this is the exact approach frame analogue of the notion for frames referred to as strongly Hausdorff [8]. Incidentally, it is obvious that both the initial and the final approach frame are Hausdorff, and hence any approach frame has Hausdorff approach subframes. For other separation concepts in approach spaces see [12]. The following characterization conveniently eliminates the reference to coproducts. Lemma 4.4 An approach frame L is Hausdorff if and only if, for any homomorphisms f, g : L → M we have that ν (u) f = ν (u) g where ν (u) is the nucleus on M given by the (closed quotient, dense)-factorization of the coequalizer k of f and g. Proof For any f, g : L → M Lemma 4.1 provides the commuting square L⊕L

h=γ M ( f ⊕g)

ν (∇)

 C(∇)

/ M ν (u)

h˜

 / C(u)

where γ M : M ⊕ M → M is the codiagonal map and f ⊕ g : L ⊕ L → M ⊕ M the homomorphism determined by f and g - the point here being that kγ M f ⊕ g(∇) = k(∇{ f (a) ∧ g(b ) | a ∧ b = 0}) =

 {kf (0) ∧ kg(b ) | a ∧ b = 0} = 0

so that h(∇) ≤ k∗ (0) = u, making Lemma 4.1 applicable. Now, if L is Hausdorff then ν (∇) i = ν (∇) j for the coproduct maps i, j : L → L ⊕ L since the (closed quotient, dense)-factorization γ L = γ L ν (∇) of the codiagonal

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map γ L : L ⊕ L → L permits the cancellation of γ L in the relation γ L i = γ L j. Consequently, ˜ (∇) = hν ˜ (∇) j = ν (u) hj = ν (u) g, ν (u) f = ν (u) hi = hν as claimed. Conversely, noting that the codiagonal map γ : L ⊕ L → L is the coequalizer of the coproduct maps i and j (a general categorical fact), one can apply the given condition to i and j, obtaining ν (∇) i = ν (∇) j. This, in turn, implies that ν (∇) (i(a) ∧ j(b )) = ν (∇) i(a ∧ b ) for any a, b ∈ L which shows C(∇) = Im(ν (∇) i), and since γ ∨ i = γ i = id L it follows that γ is one-one and hence an isomorphism.   Corollary 4.5 For Hausdorff approach frames any dense homomorphism is monic, and coequalizers are closed quotients. Proof For the first claim, consider the diagram f

//

L g

h

/ N

F

ν (u)



C(u)

h



k 

M

K where L is Hausdorff, h is dense, hf = hg, kν (u) is the (closed quotient, dense)factorization of the coequalizer k : M → K of f and g, and h = hk. Now, u = k∗ (0) so that h(u) = hkk∗ (0) = 0, consequently u = 0 by denseness and therefore C(u) = M and ν (u) = id M . On the other hand, ν (u) f = ν (u) g by the lemma so that f = g, as desired. To prove the second claim, consider the diagram f

//

L

M

g ν (u)

B l

 C(u)  K

k

k

where L is Hausdorff and kν (u) is the (closed quotient, dense)-factorization of the coequalizer k of f and g. Now, ν (u) f = ν (u) g by the lemma, hence there exists l : K → C(u) such that ν (u) = lk = lkν (u) , and since ν (u) and k are onto this makes k an isomorphism.  

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Next we have two further applications of Lemma 4.4. Lemma 4.6 The following hold: (1) If L is Hausdorff and h : L → M is an epimorphism then M is Hausdorff. (2) Any coproduct of Hausdorf f approach frames is Hausdorff. Proof (1) Consider the diagram h

H

/ L

f

ν (u)

//

M

/ C(u)

k

g

/ K 8

k

where H is Hausdorff, h is an epimorphism, and kν (u) is the (closed quotient, dense)-factorization of the coequalizer k of f h and gh. Then ν (u) f h = ν (u) gh by Lemma 4.4 and hence ν (u) f = ν (u) g as h is epic. On the other hand, for the same reason, k is also the coequalizer of f and g, and hence this identity shows that L is Hausdorff. (2) For any family (Lα )α∈I of Hausdorff approach frames, let L be its coproduct with coproduct maps iα : Lα → L and let f, g : L → M be homomorphisms, and kα : M → Kα the coequalizer of f iα and giα for each α ∈ I. Further let k : M → K be the coequalizer of f and g, u := k∗ (0) and uα := (kα )∗ (0) as depicted in the following diagram. k iα

Lα

/ L

f g

kα

& k / C(u) / K
ν (u)

M

Then by hypothesis and Lemma 4.4 ν (uα ) f iα = ν (uα ) giα for each α ∈ I. On the other hand, since kf = kg implies kf iα = kgiα we have k = lα kα for some lα so that k(uα ) = lα kα (uα ) = 0 and therefore uα ≤ u. Consequently, ν (u) = mα ν (uα ) for some mα by Lemma 4.1 so that ν (u) f iα = mα ν (uα ) f iα = mα ν (uα ) giα = ν (u) giα hence ν (u) f = ν (u) g showing that L is Hausdorff by Lemma 4.4.

 

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Corollary 4.7 Any approach frame L has a largest Hausdorff approach subframe HdL providing the Hausdorff coreflection of L. Proof Given Lemma 4.6 HdL is just the image of the coproduct of all Hausdorff approach subframes M of L by the homomorphism resulting from the identical embeddings M → L.   We denote by KHAFrm the subcategory of AFrm with objects all compact Hausdorff approach frames. Theorem 4.8 KHAFrm is stable in AFrm under coproducts and coequalizers so that it is cocomplete and its identical embedding into AFrm preserves colimits. Proof For coproducts this is immediate by Theorem 3.11 and Lemma 4.6 and for coequalizers it follows from Proposition 3.7, Corollary 4.5 and Lemma 4.6.   Remark 4.9 It is natural to ask whether the counterpart of Corollary 4.7 holds in the present situation because that would provide an analogue to the corresponding result in Frm, a consequence of the (perhaps more familiar) coreflectiveness of the compact regular frames and the fact that compact, strongly Hausdorff frames are regular, and conversely. Of course, that result, in turn, provides the pointfree Stone– ˇ Cech compactification of which the classical case is then a consequence. Obviously, given Theorem 4.8, the missing ingredient here is the Solution Set Condition which would say that, for any approach frame L there is only a set, up to isomorphism, of compact Hausdorff approach frames M with a dense homomorphism M → L. On the other hand, one may consider the functor L → Hd(J L), J L the approach frame of ideals of L as in Proposition 3.12, together with the natural homomorphisms Hd(J L) → L given by taking joins in L. In this situation, the question is whether the latter is an isomorphism whenever L itself is compact Hausdorff. Neither of these points have been settled so far, and for the time being the question whether KHAFrm is coreflective in AFrm remains open.

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