Epi-topology and Epi-convergence for Archimedean Lattice-ordered ...

Appl Categor Struct (2007) 15:81–107 DOI 10.1007/s10485-006-9021-z

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit Richard N. Ball · Anthony W. Hager

Received: 30 March 2006 / Accepted: 16 May 2006 / Published online: 21 July 2006 © Springer Science + Business Media B.V. 2006

Abstract W is the category of archimedean l-groups with distinguished weak order unit, with l-group homomorphisms which preserve unit. This category includes all rings of continuous functions C(X ) and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in W. There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each W– objectB, we define a topology τ B and a convergence B

−→. These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject A of B is dense iff A is epically embedded. The topology is T1 , but only sometimes Hausdorff or an l-group topology. The convergence is a Hausdorff l-group convergence, but only sometimes topological. The associations of B

B to τ B, and to −→, are functorial. Key words lattice-ordered group · epimorphism · topological group · convergence group · space with filter Mathematics Subject Classifications (2000) 06F20 · 18A20 · 22A30 · 46H15 · 54A20 · 54C35

Dedicated to Bernhard Banaschewski for his 80th birthday. R. N. Ball Department of Mathematics, University of Denver, Denver, CO 80208, USA e-mail: [email protected] A. W. Hager (B) Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT 06459, USA e-mail: [email protected]

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Introduction We continue, and elaborate on, the Abstract, and conclude this introduction with a Table of Contents accompanied by explanatory comments. B

The τ B, and the −→, are initially constructed by forming a supremum in the poset of closure operators, or convergences, on B, of a family of compact-open topological closures, and the associated convergences, initially defined with reference to Yosida’s Representation of W-objects as l-groups of continuous extended-realvalued functions. (That this supremum of closures is actually topological is, in fact, a Theorem, depending on “countable tightness”.) But early on these compact-open closures, and thus τ B, are reconstructed from algebra and order. This is a technical facilitation of the development, and also to some extent frees the theory from the Representation – in some circles, considered a virtue. Finally, the functor → : W → Haus l Conv is shown to have a strong and curious maximality property.

Table of Contents 1. Yosida Representation Each W– object B is, canonically, an l-group of extended-real-valued functions on a compact space YB. This representation, and the geometric insight thus proved, are exploited heavily throughout the paper, though, technically, Section 3 shows how to put the entire theory of Sections 3–8 on a representation-free orderalgebraic basis. 2. Dominion and R-dominion The dominion of A in B is d(A, B) = {b ∈ B| ∀ morphisms f, g out of B, f |A = g|A ⇒ f (b) = g(b)}. This is described in W, in generalizationSof our characterization of W– epics in [4]: there is the decomposition d(A, B)= {d R(A, B)|R∈ B ω }, where each d R is visibly closely related to a compact-open closure operator cl R defined with respect to a family K(R) S of compact subsets of YB, and d(A, B) is likewise closely related to cl B A ≡ {cl R A| R ∈ B ω } – a divisible subobject A has d(A, B) = cl B A. 3. Closures and topologies ˇ We summarize some basics about Cech closure operators for reference in subsequent sections. 4. R-closure and R-topology Here B ∈ |W|, and R ∈ B ω are fixed, and the details of the closure cl R are presented, including a description purely using the l-group operations and order. cl R is shown to be the closure operator for a topology τ R on B which is a countably tight, Hausdorff, topological l-group topology. The association (B, R) 7 → (B, τ R) is functorial. This section contains the most important technicalities of the paper. 5. Epi-topology W The epi-closure cl B from Section 2 re-appears as cl B ≡ {cl R | R ∈ B ω }. We then record various facts about (abstract) suprema of closure operators, including what might go wrong , and including a new theorem that certain such sups are topological. It then follows easily that cl B is the closure for a countably tight and T1 topology τ B, and that the association B 7 → (B, τ B) is functorial.

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6. The Hausdorff property A necessary and sufficient condition that τ B be Hausdorff is formulated, and this turns out to involve the theory of Spaces with Filter. We exhibit some B for which τ B is/is not Hausdorff. Thus τ B is not always a topological l-group topology. Most of what W can go wrong for abstract sups of closures, does go wrong for certain cl B = cl R. 7. Convergence and closure A convergence on a set is a stipulation of which filters converge to which points. Due to ignorance of a good reference, we sketch in some detail the situation φ

regarding the natural functors Clos  Conv, with particular attention to how ψ

sups are affected, and how φ may not preserve them - exemplified below. 8. Epi-convergence W B For B ∈ |W|, the epi-convergence is −→ ≡ {−→ | R ∈ B ω }, sup in the set R

Conv B of convergences on B, where −→ is the filter convergence for the topology τ R B

R

B

(−→ = φ(cl R), per the notation in 7. above). The closure for −→, R B

B

i.e. ψ(−→), is cl – in consequence of which, −→ “capture epics” – but for the B B convergence for cl B, i.e., φ(cl B)= −→ φ(cl B) can occur, in which case −→ is not topological. As a direct consequence of the abstract features of sups of B

convergences from Section 7, each (B, −→) is a Hausdorff convergence l-group, and → : W → Haus Conv is a functor with a curious maximality property. We B

describe the B for which −→ is topological.

1. Yosida Representation We describe the representation of W-objects as l-groups of extended-real-valued functions; see the discussion in Section 0. We take the notion of, and basics concerning, (archimedean) l-groups as familiar; see [1, 12, 23]. C(X ) = { f ∈ RX | f is continuous}, with pointwise + and ≤, is an example [16]. (Here, R is the reals, and X is a space.) For B any l-group, a weak unit – or just unit – in B is an e ∈ B+ (i.e., e ≥ 0) for which e ∧ |b| = 0 implies b = 0. It’s easily seen that f ∈ C(X )+ is a unit iff cozf is dense in X (where cozf = {x| f (x) 6 = 0}); the “canonical” unit is the constant function 1. W is the category: an object (B, e B) is an archimedean l-group B with a unit e B in ϕ B – and we usually just write B ∈ |W|, e B being understood; a morphism B −→ D is an l-homomorphism with ϕ(e B) = e D. R is the reals, and R ∪ {±∞} is given the obvious topology and order. For X a space, D(X ) = { f ∈ C( X, R ∪ {±∞})| f −1 R is dense in X}. D(X ) is a lattice under f ≤ g defined pointwise. For f, g, h ∈ D(X ), we say “ f + g = h in D(X )” if f (x) + g(x) = h(x) when all three are real. In general, this addition is only partly defined – given f, g, there may be no h. A subset B ⊆ D(X ) which is closed under +, and also binary pointwise ∨ and ∧, and with the constant function 1 ∈ B, is called a W-object in D(X ); one checks that (B, 1) ∈ |W|.

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Given abstract B ∈ |W|, the set of l-group ideals of B which are maximal for not containing e B, given the hull-kernel topology, is denoted YB. The following is from [29] and [19]. THEOREM 1.1. The Yosida Functor. (a) Let B ∈ |W|. The space YB is compact Hausdorff, and there is a W-isomorphism of B onto a W-object b B in D(YB) (so eˆ B = 1), with b B 0-1 separating the points of YB. If B ≈ e B ⊆ D(X ) is another such representation, then there is a homeomorphism h : X → YB for which b˜ = bˆ ◦ h ∀ b ∈ B. (b) Let ϕ : B → D be a W-morphism. Then, there is a unique continuous Yϕ : YB ← YD for which ϕ(b) ˆ = bˆ ◦ (Yϕ) ∀ b ∈ B. The map ϕ is surjective iff Yϕ is injective; if ϕ is injective, then Yϕ is surjective. (c) A functor Y : W → Comp Haus Top is defined by (a) and (b), called the Yosida Functor. REMARKS 1.2. (a) The representation of 1.1 (a) for C(X ) is C(X ) 3 f 7 → fˆ = β f ∈ D(β X), ˇ where β X is the Cech–Stone compactification and β f | X = f . (See [16] and [15].) Here, 1.1 (b) shows that any W-morphism (C(X ), 1) → (C(Y ), 1) is a ring homomorphism preserving the ring-identities 1, and we recover some of [16], ch.10. (See [19] for generalization.) ˆ ≤ n, and B∗ (b) For B ∈ |W|, B∗ = {b | ∃ n ∈ N with |b| ≤ ne B}. So b ∈ B∗ means |b| is the W-subobject of bounded functions. ˆ i ) = i. Now, 1.1 (a) and an easy ˆ i ) = i for i = 0, 1, then ((bˆ ∧ eˆ B) ∨ 0)(x (c) If b(x compactness argument shows: if Fi (i = 0, 1) are disjoint closed sets in YB, then ˆ i ) ⊆ {i}. there is b ∈ B∗ with b(F (d) Henceforth, whenever convenient, we view any B ∈ |W| as in its Yosida Representation, i.e., B ⊆ D(YB), i.e., we identify B and b B. ϕ

Yϕ

(e) In 1.1 (b), for surjective B  D, we have YB ←− YD injective, thus a homeomorphic embedding, thus essentially an inclusion of YD as a compact, or closed, subspace of YB, and, so regarded, the action of ϕ is b 7 → b| YD (the restriction), b and ϕ(b) with b ◦ Yϕ.) and D = B| YD. (We are identifying B with b B, D with D, But surjections of B are not created from arbitrary closed subspaces of YB. We address this point in 6.3, where some knowledge of the situation is needed.

2. Dominion and R-dominion Fix B ∈ |W|. We write A ⊆ B to mean just “A is a subset of B,” and hAi denotes the generated W-subobject; note that e B ∈ hAi (even for A = ∅). A ≤ B means A ⊆ B and A = hAi. For A ⊆ B, Isbell’s dominion of A in B [22] is d(A, B) ≡ {b ∈ B| ∀ W−morphisms f and g out of B f |A = g|A ⇒ f (b) = g(b)}. Thus, d(A, B) = B means that any two W-morphisms out of B will agree if they just agree on A, and thus, for A ≤ B, means that the inclusion morphism A ,→ B is an epimorphism in W. Note that

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d(A, B) = d(hAi, B), and this is always a subobject (e.g., d(∅, B) = {z e B| z ∈ Z}, since any morphism f : B → C has f (e B) = eC , thus f (z · e B) = z · eC ). In [4], the authors have characterized epimorphic embeddings A ≤ B in W, from the view of the Yosida Representation B ⊆ D(YB), in terms of how A and B relatively separate the points of YB. The basis of this paper is 2.2 below, which is the generalization of that result to describe all d(A, B). NOTATION 2.1. For B ∈ |W|, B+ ≡ {b | 0 ≤ b ∈ B}, and B ω ≡ {R ⊆ B+ | R is countable and non-empty }. For b ∈ B, b−1 R = {x ∈ YB | b(x) ∈ R}. Let R ∈ B ω : let T −1 SR = {r R | r ∈ R} – a dense subset of YB, by the Baire Category Theorem – and let BR = {b ∈ B | b−1 R ⊇ SR}.SNote that B ω is up-directed by inclusion, so is {BR | R ∈ B ω }, and note that B = {BR | R ∈ B ω }. A reading of the proof of 8.3.2 of [4] reveals the following more general result. THEOREM 2.2. With the view B ⊆ D(YB), let A ⊆ B and b ∈ B. b ∈ d(A, B) iff ∃ R ∈ B ω such that (∗) R [x, y ∈ SR, a(x) = a(y) ∀a ∈ A] ⇒ b(x) = b(y). DEFINITION 2.3. Let B ∈ |W|, and fix R ∈ B ω . For A ⊆ B, the R-dominion of A in S B is d R(A, B) ≡ {b ∈ B | (∗) R holds for b }. (Thus, by 2.2, d(A, B) = {d R(A, B) | R ∈ B ω }.) The condition (∗) R is reminiscent of the two-point condition appearing in some treatments of the Stone –Weierstrass Theorem, e.g., p. 310 in [13]. From that point of view we proceed to examine the condition. NOTATIONS AND DEFINITIONS 2.4. Let B ∈ |W|. (a) Fix R ∈ B ω : K(R) ≡ {K| K is a compact subset of SR}. For b ∈ B, K ∈ K(R), and  ∈ (0, 1): UR(K, , b) ≡ {b0 ∈ B | x ∈ K ⇒ |(b0 − b)(x)| ≤ }. For b ∈ B : N R(b) ≡ {UR(K, , b) | K ∈ K(R),  ∈ (0, 1)}. (b) Let A ⊆ B : For R ∈ B ω , cl R A ≡ {b ∈ B | U ∈ N R(b) ⇒ U ∩ A 6 = ∅}. Note that R1 ⊆ R2 implies cl R1 A ⊆ cl R2 A; we write cl R1 ≤ cl RS . So the family {cl R | R ∈ 2 B ω } is up-directed by inclusion in B ω . We set cl B A ≡ {cl R A| R ∈ B ω }. Note in 2.4 (a) the expression “|(b0 − b)(x)| ≤  .” Here b0 − b is of course the difference in the group B, and in D(YB) is that function c for which c(x) = b0 (x) − b(x) for all x ∈ b0 −1 R ∩ b−1 R. The point is that we can’t write “|b0 (x) − b(x)| ≤  ” since b0 (x) − b(x) could well be +∞ – (+∞), and undefined, while |c(x)| ≤ . Clearly, our intention is that for each b, N R(b) be a local neighborhood base at b (not open neighborhoods, due to the “≤”) for the associated closure operator cl R, and cl B is to be a closure operator. We fulfil that intention, and examine properties of these closures, in subsequent sections, but for now, lay those details aside and directly prove the “capturing of epis.” THEOREM 2.5. For B ∈ |W|, and fixed R ∈ B ω : (a) cl R A ⊆ d R(A, B), for any A ⊆ B;

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(b) cl R A ⊇ d R(A, B) ∩ BR, for A a W-subobject of B which contains rational multiples of e B. COROLLARY 2.6. For B ∈ |W|, and divisible A ≤ B: cl B A = d(A, B); Ais epimorphically embedded in B iff cl B A = B – A is “dense” in B. S Proof of 2.6. From 2.5, and the facts noted above that B = BR and d(A, B) = R S d R(A, B).  R

Proof of 2.5. We “expand” slightly the inclusions in (a) and (b): Let A ⊆ B, let b ∈ B, and consider these conditions. (1) b ∈ cl R A; (2) ∀ x, y ∈ SR, ∀ ∈ (0, 1), UR({x, y}, , b) ∩ A 6 = ∅ : (3) b ∈ d R(A, B), i.e., (∗) R holds. Then (a) (1) ⇒ (2) ⇒ (3); (b) If b ∈ BR and A ⊇ Q · e B , then (1) ⇐ (2) if also Ais a sub-lattice of B, and (2) ⇐ (3) if also A is a W-subobject of B. (a) (1) ⇒ (2) is just because {x, y} ∈ K(R). For (2) ⇒ (3), suppose (2), and let x1 , x2 ∈ SR with a(x1 ) = a(x2 ) ∀ a ∈ A. Fix  ∈ (0, 1) ; ∃ a() = a ∈ A with |(a − b)(xi )| ≤ . If a(xi ) = ±∞, then b(xi ) = ±∞, so b(x1 ) = b(x2 ). If a(xi ) ∈ R, then each b(xi ) ∈ R also, and the triangle inequality in R yields |b(x1 ) − b(x2 )| ≤ 2. This holds ∀  ∈ (0, 1), so b(x1 ) = b(x2 ). (b) Suppose in each case that A and b satisfy the extra hypotheses. (2) ⇐ (3). Suppose (3), let x1 , x2 ∈ SR and let  ∈ (0, 1). If b(x1 ) = b(x2 ), choose q ∈ Q with |b(xi ) − q| ≤  and let q e B = a ∈ A. Suppose b(x1 ) < b(x2 ). By (∗) R , there is a ∈ Awith a(x1 ) 6 = a(x2 ). By intelligently replacing this a by another element of A (using that A is a W-subobject containing Q · e B), we can suppose that a(x1 ) < b(x1 ) < b(x2 ) < a(x2 ), and then one more such replacement achieves b(x1 ) −  < a(x1 ) < b(x1 ) < a(x2 ) < b(x2 ) + . So (2) is proved. (1) ⇐ (2). This is just an application of the following “Stone–Weierstrass two-point lemma” quoted from [13], chapt. X, Section 4.2, Prop. 2 (p. 310), with inconsequential variation. LEMMA 2.7. Let X be compact, H a sublattice of C(X ), and f ∈ C(X ). Then, [ ∀  ∈ (0, 1) ∃ h ∈ H with |h(x) − f (x)| ≤  ∀ x ∈ X] iff [ ∀ x1 , x2 ∈ X ∀  ∈ (0, 1) ∃ h ∈ H with |h(xi ) − f (xi )| ≤  (i = 1, 2)]. To show (1) ⇐ (2): let b satisfy (2). Note that if A ⊇ Q · e B and A is a sublattice, then (2) holds replacing A by A∗ = the bounded functions in A, which is also a sublattice and A∗ ⊆ BR ∀ R. So b satisfies (2) using A∗ . Now take K ∈ K(R ), and apply 2.7 to X = K, H = A∗ | K (the restrictions), and f = b| K. Then 2.7 says that ∀ , UR(k, , b) ∩ A∗ 6 = ∅. Thus (1) holds. The proof of 2.5 is complete. 

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REMARKS 2.8. (a) Let B ∈ |W|, and suppose that e B is a strong unit in B, i.e., ∀ b ∈ B ∃ n ∈ N with |b| ≤ n e B; equivalently, in the Yosida Representation B ≤ C(YB). For R = {e B}, SR = YB, cl R is closure for the topology of uniform convergence on YB, and 2.5 is the Stone–Weierstrass Theorem – generalized slightly from just C(X )’ s , X compact – since d R(A, B) means every b satisfies (∗) R , which just means A separates points of YB. (b) 2.5 (b) is the implication b ∈ d R(A, B) ⇒ b ∈ cl R(A, B) if (i) b ∈ BR , and (ii) A ⊇ Q · e B , and (iii) A is a W-subobject of B. No one of these hypotheses can be dropped, T even in the presence of the other two: Not (i), since b ∈ cl R(A) implies SR ∩ a −1 R ⊆ b−1 R , while b ∈ d R(A, B) does not imply that condition – A

a specific example is easily manufactured; Not (ii), since B = Q or R , and R = {1} has cl R the usual closure in the topological space B, so Z = cl R Z , but d R(Z, B) = B; Not (iii), since d R(A, B) is always a subobject, but cl R(A, B) need not be – a specific example is easily manufactured. (c) The implication of 2.5 (b) is, in the proof, expanded to the two implications (1) ⇐ (2) ⇐ (3), each assuming A ⊇ Q · e B , the second assuming A is a subobject, the first just assuming A is a sublattice. The latter does not suffice for (3) ⇒ (2), indeed not for (3) ⇒ (1): Let B1 = B2 be any divisible W-object, and let B = B1 × B2 , which has YB = YB1 + YB2 . Let α : YB1 → YB2 be the map α(x) = x, and let A = {a ∈ B| either (a| Y B1 ) ◦ α = a| YB2 , or a| YB1 ≤ 0 ≤ a| YB2 }. Then : A ⊇ Q · e B and Ais a sublattice; A ⊇ B+ , hence hAi = B, hence d R(A, B) = B ∀R (since d R(A, B) is always a subobject); cl R(A, B) = A ∀R. This last requires some argument, which we omit.

3. Closures and Topologies We are going to show, for B ∈ |W| and R ∈ B ω , that cl R is a topological closure operator in B, with system of local bases {N R(b) | b ∈ B} (as defined in 2.3 and 2.4), and another such system {A R(b) | b ∈ B}, each with various salient features. We approach the situation via abstract closure operators, as described in Sections 14–16 of [14] – which seems to be the best reference. This section is a sketch of some of this material – it seems [14] is not so widely accessible, and that many mathematicians are not so familiar with abstract closures; indeed, initially, the authors were considerably confused by the situation: DEFINITIONS 3.1. (a) Let X be a set. A closure (operator) on X is a function u : P (X ) → P (X ) (P the power set) satisfying: uφ = φ; A ⊆ uA always; u(A1 ∪ A2 ) = uA1 ∪ uA2 always. (b) Let u be a closure on X. Let G, U ⊆ X, and x ∈ X. G is open if G = X − u(X − G ). U is a neighborhood (nbd) of x if x ∈ X − u(X − U). Nu (x) ≡ {U| U is a nbd of x}. (c) Let u be a closure on X, let x ∈ X, and let N ⊆ P (X ). N is a local base at x if N ⊆ Nu (x), and for each U ∈ Nu (x) there is V ∈ N with V ⊆ U. (Note: The sets V ∈ N need not be open.)

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THEOREM 3.2. (a) Let u be a closure on X, let x ∈ X, and let N (x) be a local base at x. Then T [nbd] N (x) 6 = ∅; x ∈ N (x); i f U1 , U2 ∈ N (x), then U1 ∩ U2 contains some U ∈ N (x) x ∈ uA iff U ∩ A 6 = ∅ for each U ∈ N (x). (b) Let X be a set, and suppose for each x ∈ X, N (x) ⊆ P (X ) satisfies the condition [nbd] in (a). Then, u : P (X ) → P (X ) defined as uA = {x| U ∩ A 6 = ∅ for each U ∈ N (x)} is the unique closure on X for which each N (x) is a local base at x. DEFINITION 3.3. The closure u : P (X ) → P (X ) is called topological if u(uA) = uA always (i.e., u is idempotent). THEOREM 3.4. (a) The following conditions on the closure u are equivalent: u is topological; [T-nbd] ∀x ∈ X (U ∈ Nu (x) ⇒ ∃ V ∈ Nu (x) such that (y ∈ V ⇒ U ∈ Nu (y))). (b) Let {N (x)| x ∈ X}, and then u, be as in 3.2 (b). If {N (x)| x ∈ X} satisfies the condition [T-nbd], then u is topological.

4. R-closure and R-topology In application of Section 3, we first show THEOREM 4.1. For B ∈ |W|, and fixed R ∈ B ω : cl R is a topological closure, with {N R(b)| b ∈ B} a system of local bases. The associated topology, τ R , is Hausdorff. Some preliminaries are needed for the verification: PROPOSITION 4.2. Let B ∈ |W|, R ∈ B ω , K and L ∈ K(R),  and δ ∈ (0, 1), and let b ∈ B. (a) K ⊇ L and  ≤ δ ⇒ UR(K, , b) ⊆ UR(L, δ, b). (b) UR(K, , b) ∩ UR(L, δ, b) ⊇ UR(K ∪ L,  ∧ δ, b). 4.2 is clear, the next proposition less so. PROPOSITION 4.3. (a) Any law of equality or inequality which holds in R holds in abelian l-groups, thus in W. (b) The triangle inequality, |a − b| ≤ |a − c| + |c − b|, holds in abelian l-groups, thus in W. Proof. The variety of abelian l-groups is HSP(R), by [27]. ((a) and (b) also follow from the Yosida Representation.) 

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PROPOSITION 4.4. Let B ∈ |W|, a1 , a2 ∈ B, x ∈ YB, and r, s ∈ R+ . Then, a1 (x) ≤ r and a2 (x) ≤ s ⇒ (a1 + a2 )(x) ≤ r + s. Proof. Under the hypotheses, for each real t > 0, choose by continuity of the ai , open Gt 3 X for which the restrictions satisfy a1 | Gt ≤ r + t and a2 | Gt ≤ s + t. For y ∈ Ht ≡ a1 −1 R ∩ a2 −1 R ∩ Gt , we have a1 (y) ≤ r + t, a2 (y) ≤ s + t, and thus (a1 + a2 )(y) = a1 (y) + a2 (y) ≤ (r + s) + 2t. Since Ht is dense in Gt , and a1 + a2 is continuous, we have (a1 + a2 )(x) ≤ (r + s) + 2t (regardless of whether the left side is real). Since t > 0 was arbitrary, (a1 + a2 )(x) ≤ (r + s).  Proof of 4.1. We first apply 3.2. The conditions [nbd] hold for N R(b): The first two are clear, and the third follows from 4.2. So cl R is a closure, with local bases N R(b). We now apply 3.4. The condition [T-nbd] for {N R(b)| b ∈ B} clearly is implied by c ∈ UR(K, /2, b) ⇒ UR(K, /2, c) ⊆ UR(K, , b).

(*)

To prove (*): First, |c0 − b| ≤ |c0 − c| + |c − b| by 4.3 (b). Second, in 4.4, take a1 = c − c, a2 = c − b, x ∈ K, and r = s = /2. This yields (*). Hausdorff: If b1 6 = b2 in B, then there is x0 ∈ b1 −1 R ∩ b2 −1 R ∩ SR with b1 (x0 ) 6 = b2 (x0 ). For K = {x0 } and  = n1 |b1 (x0 ) − b2 (x0 )| for suitably large n, we have UR(K, , b1 ) ∩ UR(K, , b2 ) = ∅.  0

Consider C(X ) in its Yosida Representation C(X ) = B ⊆ D(β X ), noted in 1.2 (a). Here, {b−1 R| b ∈ B} is exactly the collection of cozero-sets of β X which contain ˇ X (a topological exercise), so X itself is an SR iff X is Lindelöf and Cech-complete (see [15]), and then τ R is exactly the compact-open topology on C(X ). (Also, see below, after 4.11, and 8.8.) Further properties of cl R (and τ R), in particular, the interaction with the algebra, are best treated using a closure-equivalent system of local bases, formulated by inequalities. This may be only a technicality, but it seems very important. We note: We have not seen an algebraic treatment of compact-open topology for C(X ); it is remarked on p. 293 of [28] “. . . it cannot be easily derived from the algebraic structure . . .”. NOTATION AND DEFINITIONS 4.5. Let B ∈ |W|, R ∈ B ω and let P0 (R) ≡ {F| φ 6 = F ⊆ R , |F| < ω} (the non-void finite subsets of R). Let b ∈ B. (a) For F any non-void finite subset of B+ , and q ∈ N F (the functions from F to N), and  ∈ (0, 1), we set ^   0 0 VF (q, , b) = b ∈ B (q( f ) − f ) ∧ |b − b| ≤  . F

(b) For p ∈ N R and  ∈ (0, 1), we set VR( p, , b) =

[

 VF ( p |F, , b) F ∈ P0 (R) .

(c) A R(b) ≡ {VR( p, , b) | p ∈ N R ,  ∈ (0, 1)}.

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COMMENTS 4.6. (a) In the definition of VF ( ), q( f ) means the constant V function q( f )e B ∈ B. For x ∈ YB, if f (x) > q( f ) −  for some f ∈ F, then ( (q( f ) − f ))(x) < , so F V for any b0 ∈ B, (q( f ) − f )(x) ∧ |(b0 − b)(x)| ≤ . That is, the condition b0 ∈ F

VF (q, , b) is just a statement that b0 is pointwise near to b on a certain set where all f ∈ F areT appropriately bounded: (∗) b0 ∈ VF (q, , b) iff |(b0 − b)(x)| ≤  for each x ∈ {y| f (y) ≤ q( f ) − }. This last is a compact subset of YB and FT T given p ∈ N R, { {y| f (y) ≤ p( f ) − }| F ∈ P0 (R)} ∈ K(R). It is essentially F

this which is the connection with the earlier “compact-open” definition of τ R. (b) A cautionary note: In the definition of VR( p, , b) as a union, the various VF ( p | F, , b) are not chosen independently: p appears in each, as p|F. This is important. (c) A point, both technical and philosophical, is that the definitions in 4.5 involve only the algebra and order in B, via just the inequality in the definition of VF ( ). That inequality is written left-hand-side ≡ L ≤  because that is easiest to understand, though, strictly speaking, this need not make complete sense: Here  stands for  · e B which may not lie in B. However, one may always suppose  = n1 (n ∈ N), and then convert L ≤ n1 to nL ≤ 1 (= e B). In the sequel, “L ≤ ” and similar inequalities, are to be understood in that sense. THEOREM 4.7. Let B ∈ |W| and R ∈ B ω . Then {A R(b)| b ∈ B} is another system of local bases for cl R (or τ R). Some preliminaries are needed for the verification. Note that N R is a lattice in the pointwise partial ordering of functions: p ≤ q means p(r ) ≤ q(r ) ∀r ∈ R. PROPOSITION 4.8. Let B ∈ |W|, R ∈ B ω , p and q ∈ N R,  and δ ∈ (0, 1) and let b ∈ B. (a) (b)

p > q and  ≤ δ ⇒ VR( p, , b) ⊆ VR(q, δ, b). VR( p, , b) ∩ VR(q, δ, b) ⊇ VR( p ∧ q,  ∧ δ, b).

Proof. (a) follows from the corresponding inclusion for the VF ’s which becomes clear after a little thought about the inequality in 4.5 (a). (b) follows.  PROPOSITION 4.9. Let B ∈ |W| and R ∈ B ω (a) Let p ∈ N R. Define o o \n \n K( p) ≡ x r (x) ≤ p(r ) , L( p) ≡ x r (x) ≤ p(r ) − 1 R

R

Then K( p), L( p) ∈ K(R), and for each b ∈ B and  ∈ (0, 1),   (1)   (2) UR K( p), /2, b ⊆ VR( p, , b) ⊆ UR L( p), , b . (b) Let K ∈ K(R ). Define n n o o p ∈ N R : p(r ) ≡ min n ∈ N n ≥ sup r (x) x ∈ K + 1 .

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91

Then, K ⊆ L( p), and for each b ∈ B and  ∈ (0, 1),   (1)   (2)   VR p, , b ⊆ UR L( p), , b ⊆ UR K, , b . Proof. (a)

K( p)(resp., L( p)) is an intersection of closed subsets of YB, thus compact. At each point of K( p)(resp., L( p)) every r ∈ R is real; so K( p), L( p) ∈ K(R). For the inclusion (2), we need VF ( p, , b) ⊆ UR(L( p), , b) for each F ∈ P0 (R). This follows immediately from (*) in 4.6 (a), and the inclusion L( p) ⊆ T {y | f (y) ≤ p( f ) − }. F

For the inclusion (1), suppose that (†) x ∈ K( p) ⇒ |(b0 − b)(x)| ≤ /2. We want F ∈ P0 (R) for which b0 ∈ VF ( p, , b). It will suffice to find F for which y ∈ K( p | F) ⇒ |(b0 − b)(y)| ≤ , and for this it suffices that K( p | F) ⊆ {y | |(b0 − b)(y)| < } ≡ W. Note that W is open; its complement W0 is closed and compact. 0 Suppose there is no such F. Then, ∀F, K( p | F) K( p | F1 ∪ T ∩ W 6 = ∅. Since F2 ) = K( p | F1 ) ∩ K( p | F2 ) ∀F1 , F2 , there is x ∈ {K( p | F) ∩ W0 | F ∈ P0 (R)} by compactness. But this x ∈ K( p), and |(b0 − b)(x)| ≥ , contradicting (†). (b) The inclusion (2) is in (a). That K ⊆ L( p) results immediately from the definitions of p, and L( p). Inclusion (2) follows.  Proof of 4.7. It is enough to establish two claims: (a) ∀ b ∈ B, A R(b) satisfies the conditions [nbd] of 3.2 (a). Thus {A R(b) | b ∈ B} is a system of local bases for some closure, as in 3.2 (b). (b) Let b ∈ B. ∀ U ∈ N R(b) ∃ V ∈ A R(b) with V ⊆ U, and ∀ V ∈ A R(b) ∃ U ∈ N R(b) with U ⊆ V. Thus the closure in (a) is cl R. Here, 4.8 yields (a), and then 4.9 yields (b). So 4.7 is proved.  COROLLARY 4.10. At each b ∈ B, the local weight for τ R is ≤ 2ω . Proof. {VR( p, , b) | p ∈ N R,  ∈ (0, 1) ∩ Q} is a local base at b, by 4.8 (a), and the cardinality is ≤ |N R| · ω ≤ 2ω .  4.10 is just noted in passing; we shall have no further use. The next cardinality result is crucial, however; it stems directly from the definition 4.5 (b). THEOREM 4.11. τ R has countable tightness, i.e., b ∈ cl R A ⇒ b ∈ cl R A0 for some countable A0 ⊆ A. Proof. Let b ∈ cl R A, and let 1 0 ≡ {(q, n) | n ∈ N, q ∈ N F for some F ∈ P0 (R), VF (q, , b) ∩ A 6 = ∅}. n S Then 0 is countable, since 0 ⊆ {N F | F ∈ P0 (R)}× N and this latter is countable. For each (q, n)∈0, choose a(q, n)∈VF (q, n1 , b) ∩ A, and let A0 ≡{a(q, n)|(q, n)∈0}. Then, given p ∈ N R and  ∈ (0, 1), VR( p, , b) ∩ A0 6 = ∅ : choose n1 ≤ . Then,   S VR( p, , b) ⊇ VR( p, n1 , b) = VF p |F, n1 , b , and since b ∈ cl R A, there is F with P0 (R)       1 VF p|F, n , b ∩ A6 = ∅. This means p |F, n ∈0, and a p|F, n ∈VR( p, , b)∩ A0 . 

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ˇ Consider C(X ) for X Lindelöf and Cech-complete, as pointed at before 4.5: So X is an SR, τ R is the compact-open topology on C(X ), and by 4.11, this topology is countably tight. This may be a new theorem: cf. [25], and see our generalization in [8]. We turn to the interaction of τ R with the algebra and order in B. On a (general) l-group B, a topology τ is called an l-topology if the group and lattice operations are continuous, as – : (B, τ ) → (B, τ ) and +, ∨, ∧ : (B, τ ) × (B, τ ) → (B, τ ). Then (B, τ ) is called a topological l-group. Basic theory is described in [2] and other literature given there; but we shall require almost no reference to these works. THEOREM 4.12. For any B ∈ |W|, and R ∈ B ω , (B, τ R) is a topological l-group. The verification comes immediately from these three lemmas. LEMMA 4.13. Let b ∈ B, p ∈ N R,  ∈ (0, 1), and F2 ⊆ F1 ∈ P0 (R). Then VF2 ( p | F2 , , b) ⊆ VF1 ( p | F1 , , b). LEMMA 4.14. The following hold in any B ∈ |W|. (a) a ∧ (c1 + c2 ) ≤ a ∧ c1 + a ∧ c2 , for a, ci ∈ B+ . (b) |b1 ∨ b2 − b1 0 ∨ b2 0 | ≤ |b1 − b1 0 | ∨ |b2 − b2 0 |, for bi , bi 0 ∈ B, and likewise for ∧. 4.13 follows from the definition of the VF ( ) – 4.6(a)(*) may be helpful. 4.14 is easily checked using 4.3 (a). The next lemma proves 4.12. LEMMA 4.15. The following hold for bi ∈ B, p ∈ N R, and  ∈ (0, 1). (a) (b) (c)

VR( p, , b1 ) = −VR( p, , b1 ). VR( p, , b1 ) + VR( p, , b2 ) ⊆ VR( p, 2, b1 + b2 ). VR( p, , b1 ) ∨ VR( p, , b2 ) ⊆ VR( p, , b1 ∨ b2 ), and likewise for ∧.

Proof of 4.15. (a) is clear. For (b) and (c), let bi 0 ∈ VR( p, , bi ), so that bi 0 ∈ VFi ( p | Fi , , bi ) ≡ VVi for some F1 , F2 ∈ P0 (R). Let F = F1 ∪ F2 . By 4.13, Vi ⊆ VF ( p | F, , bi ). Let a = ( p( f ) − f ). F

By 4.14, we have a ∧ |(b1 + b2 ) − (b1 0 + b2 0 )| ≤ a ∧ |b1 − b1 0 | + a ∧ |b2 − b2 0 | ≤ 2 , a ∧ |(b1 ∨ b2 ) − (b1 0 ∨ b2 0 )| ≤ (a ∧ |b1 − b1 0 |) ∨ (a ∧ |b2 − b2 0 |) ≤ . (and likewise for ∧).



REMARK 4.16. We note in passing three more features of a topology τ R. A subset S of the l-group B is convex if s1 ≤ b ≤ s2 , si ∈ S implies b ∈ S. (a) All basic sets UR(K, , b) and VR( p, , b) are convex. (b) If S is a convex sublattice of B, then so is cl R S. (c) If A is a W-subobject of B, then so is cl R A. Here, (a) is clear, and (b) and (c) are consequences of 4.12 - see 1.8 of [2].

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93 ϕ

We turn to W-homomorphisms B −→ D. For R ∈ B ω , we have the topology τ R on B. Of course, ϕ(R) ∈ Dω , so we have the topology τϕ(R) on D. ϕ

THEOREM 4.17. Let B −→ D be a W-homomorphism, and let R ∈ B ω . Then ϕ (B, τ R) −→(D, τϕ(R) ) is continuous. Proof. We are to show: If b ∈ B, and G is a τϕ(R) -neighborhood of ϕ(b), then there is a τ R-neighborhood H of b with ϕ(H) ⊆ G. We take q ∈ Nϕ(R) , and G = S R Vϕ(R) (q, , ϕ(b)) = {VS E (q | E, , ϕ(b)) | E ∈ P0 (ϕ(R))}, define q ◦ ϕ ≡ p ∈ N , and set H = VR( p, , b) = {VF ( p | F, , b) | F ∈ P0 (R)}. Then, ϕ(H) ⊆ G, because for each F ∈ P0 (R), with E = ϕ(F), ϕ(VF ( p | F, , b)) ⊆ VE (q | E, , ϕ(b)), because: if V ( p( f ) − f ) ∧ |b0 − b| ≤ , then applying ϕ, and noting ϕ( p( f )) = p( f ) = q(ϕ( f )), F V  and ϕ() = , we obtain (q(e) − e) ∧ |ϕ(b0 ) − ϕ(b)| ≤ . E

REMARKS 4.18. (a) It is (also) easy to prove 4.17 using the compact-open description of the topologies, i.e., with neighborhoods UR(k, , b). For, the action of ϕ is ϕ(b) = σ b ◦ σ for certain continuous YB ←− YD, as discussed in Section 1. Then, given K ∈ K(Sϕ(R) ), σ (K) ∈ K(SR), and ϕ(UR(σ (K), , b)) ⊆ Uϕ(R) (K, , ϕ(b)). (b) However, using the neighborhoods UR(K, , b), a proof of 4.12 appears tedious, and we don’t see how to prove at all the important 4.11.

5. Epi-topology The following definition was already made in 2.4, and motivated in 2.5 and 2.6. It now becomes the focus of attention. DEFINITION 5.1. Let S B ∈ |W|. The epi-closure for B is the operator cl B : P (B) → B P (B) given by cl A ≡ {cl R A| R ∈ B ω } (A ⊆ B). For the same reasons that produced Section 3, we now synopsize from Section 31 of [14] some basic theory regarding this kind of definition. SUPREMA OF CLOSURES 5.2. (a) For X a set, let Clos X denote the set of closure operators on X. For u, v ∈ Clos X, u ≤ v means uA ⊆ v A ∀A ⊆ X, and this is a partial order. u ≤ v iff Nu (x) ⊇ Nv (x) ∀x ∈ X. S (b) Let CW⊆ Clos X. For A ⊆ X, let s A ≡ {c A| c ∈ C }. This defines s ∈ Clos X, and s = C (supremum) in Clos X. Thus Clos X is a complete lattice. Let x ∈ X,Sand suppose that for each c ∈ C , Nc (x) is a local base at x for c. Then, Ns (x) ≡ { Uc (x) | c ∈ C ; Uc (x) ∈ Nc (x)} is a local base at x for s. (c) For c ∈ Clos X, d ∈ Clos Y, and f : X → Y a function, “ f : (X, c) → (Y, d) is continuous” means f (c A) ⊆ d f (A) for A ⊆ X, equivalently, for each x ∈ X and G in a neighborhood base for f (x), there is H in a neighborhood base for x, with f (H) ⊆ G.

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(d) Let C ⊆ Clos X, D ⊆ Clos Y, and let f : X → Y be a function for which, W ∀c ∈ C ∃W d ∈ D such that f : (X, c) → (Y, d) is continuous. Then f : (X, C ) → (Y, D) is continuous. (e) Let u ∈ Clos X, v ∈ Clos Y. Then, u × v ∈ Clos (X × Y) is defined by the local bases N ((x, y)) = {U × V| U ∈ Nu (x), V ∈ Nv (y)}. A function f from a closure space to X × Y is continuous for u × v iff π X ◦ f is continuous for u, and πY ◦ f for v. (X × Y, u × v) is the categorical product for Clos. (f) Let C ⊆ Clos X, and let g : X × X → X be a functionWfor which, ∀ c ∈ C , g : (X W× X, c × c) → (X, c) is continuous. Then g : (X × X, {c × c| c ∈ C }) → (X, C ) is continuous. W W W But: WwhileW C × C ≥ W {c × c| c ∈ C } always, (∗) [6 = can occur, and g : (X × X, C × C ) → (X, C ) can fail to be continuous]. (g) Let C ⊆WClos X, and suppose that each c ∈ C is topological, with topology V τc , and let s = C . (s can fail to be topological.) If s is topological, then τs = {τc | c ∈ C } in the lattice of topologies on X (p.o.’d by inclusion), and if each τc is T1 , then so is τs ; but (∗) [τs need not be Hausdorff when each τc is]. A few comments on the situations in 5.2. About proofs: These are in [14], W but, (d) is obvious, (d) implies the first part of (f), and then that and (e) yield {c × c| c ∈ W W C } ≤ C × C . Regarding (∗) in (f): [14] shows that an example of 6 = occurs in any Clos X when X has at least 3 points (p. 565), and more interestingly, illustrates “g is not continuous” by constructing in Clos R a sequence u1 W ≤ u2 ≤ . . . for which each + : (R × R, un × un ) → (R, un ) is continuous, but for u = un , + : (R × R, u × u) → W (R, u) is not. Exactly this sort of thing occurs in W, for certain ’s cl B; see 5.6. This motivates us to analyze in Section 7, associated convergence structures, where these “defects” evaporate. For now, we turn to what is true. W THEOREM 5.3. Let B ∈ |W|. The epi-closure cl B is the closure operator {cl R | R ∈ B ω }. It is topological; its topology τ B is called the epi-topology, and τ B is T1 and countably tight. Inversion b 7 → −b, and any translation b 7 → b + c (c ∈ B) are homeomorphisms of (B, τ B). ϕ ϕ Any W-homomorphism B −→ D is continuous for the epi-topologies (B, τ B) −→ (D, τ D). The associated “τ ”: W → Top is a set-preserving functor. In 5.3, once we know cl B is topological, the second and third paragraphs are consequences of the corresponding properties of each (B, τ R) established in Section 4, and 5.2 (d) – the functions here are functions of just one variable. The existence and properties of τ B are immediate from the following theorem – it being evident that {cl R | R ∈ B ω } satisfies the hypotheses (noting 4.11!). THEOREM 5.4. Let X be a set, C ⊆ Clos X, and let s = on C .

W

C . Consider the conditions

(a) ∀ c ∈ C ∃ c∗ ∈ C which is countably tight and c∗ ≥ c. (b) ∀ {cn | n ∈ N} ⊆ C ∃ c∗ ∈ C which is topological and c∗ ≥ cn ∀n. Then: If (a) holds, then s is countably tight. If (a) and (b) hold, then s is topological.

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Proof. If x ∈ s A =

S

95

c A, then x ∈ some c A, and then there is countably tight

C

c∗ ≥ c, so x ∈ c∗ A.SThen there is countable A1 ⊆ A with x ∈ c∗ A1 ⊆ s A1 . If x ∈ s(s A) = c (s A), then x ∈ some cx (s A), and we choose countably tight C

∗ ∗ cx∗ ≥ cx . So x ∈ cS x (s A) ⊆ cx (s A), and then there is countable B ⊆ s A with x ∈ cx B. Now B ⊆ s A = c A, so ∀ b ∈ B ∃ cb with b ∈ cb A. Since B is countable, there

C

is topological c∗ ≥ all cb , cx∗ . So b ∈ c∗ A ∀ b, i.e., B ⊆ c∗ A, so x ∈ cx∗ B ⊆ c∗ B ⊆ c∗ (c∗ A) = c∗ A ⊆ s A.  REMARK 5.5. In continuation of 4.16, but now concerning τ B, we note: S (a) There seems no reason why basic neighborhoods, e.g., the U = {UR(kR,  R, b)| ω R ∈ B }, are convex. (b) If S is a convex sublattice of B, then so is cl B S. (c) If A is a W-subobject of B, then so is cl B A. (b) and (c) are proved easily using the “up-directedness” of {cl R | R ∈ B ω }, i.e., for R = R1 ∪ R2 , cl Ri ≤ cl R. REMARK 5.6. Again regarding the “defects” (∗) in 5.2 (f) and (g), more specifically, their occurrence in the context of our functor τ : W → Top : In the next section, we shall see B ∈ |W| for which τ B is ( T1 but) not Hausdorff. Thus (B, τ B) cannot be a topological group, and thus g = + : (B × B, τ B × τ B) → (B, τ B) cannot be continuous. This is quite indirect, not very clear, and an object of our continued study; but it’s the state of our art.

6. The Hausdorff Property We give a criterion that an epi-topology τ B be Hausdorff. It is not especially transparent, but at least suffices to (1) identify some B for which τ B is not Hausdorff, thus not a topological l-group topology, and (2) show that for any C(X ), τ C(X ) is Hausdorff. But the question remains open of exactly when τ B is a topological l-group topology, in particular, for B = C(X ). A minor contribution appears at 8.8. We summarize some basic data. 6.1. (a) Basic τ B-neighborhoods of b, in the “compact-open” description, are of S the form U(b) = {UR(b) | R ∈ B ω , UR(b) ∈ N R(b)}. Recall that UR(b) = UR(M, , b) for M ∈ K(R),  ∈ (0, 1). Here M and  depend on R, so in fact maps, or ND-maps, M, : M : B ω → S we have “neighborhood-defining” {K(R) | R ∈ B ω } with M(R) ∈ K(R) ∀R,  : B ω → (0, 1). So we write U(b) = U(M, , b) for ND-maps M,  as above, and N (b) = {U(M, , b) | M,  are ND-maps } is a local base. (b) Let b, c ∈ B. If b and c have disjoint neighborhoods, as U(P, δ, b) ∩ U(Q, γ , b) = ∅ for ND-maps (P, δ) and (Q, γ ), then setting M(R) = P(R) ∪ Q(R), and (R) = δ(R) ∧ γ (R), we find U(M, , b) ∩ U(M, , c) = ∅, which means (∗) ∀ R1 , R2 ∈ B ω , UR1 (M(R1 ), (R1 ), b) ∩ UR2 (M(R2 ), (R2 ), c) = ∅.

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And conversely, if M,  satisfy (∗), then U(M, , b) ∩ U(M, , c) = ∅, so b and c have disjoint neighborhoods. (c) For any c ∈ B, the translation TC (b) ≡ b + c is a homeomorphism of (B, τ B). Thus, b and c have disjoint neighborhoods iff b − c and 0 do; τ B is Hausdorff iff ∀ b 6 = 0, b and 0 have disjoint neighborhoods. (d) For reasons that will be apparent quickly, an ND-map M for which (CS) ∀ R1 , R2 ∈ B ω , M(R1 ) ∩ M(R2 ) 6 = ∅ will be called a “constant-separating” map, or CS-map, for B. The following might be construed as some version, in our context, of a separation axiom T1.5 . PROPOSITION 6.2. For B ∈ |W|, the following are equivalent. (a) Any two different constants in B have disjoint neighborhoods. (b) e B(= 1) and 0 have disjoint neighborhoods. (c) There is a CS-map for B. Proof. (a)⇒(b) of course. (b)⇒(c). Per 6.1 (b), there are ND-maps M,  with (∗) UR1 (M(R1 ), (R1 ), e B) ∩ UR2 (M(R2 ), (R2 ), 0) = ∅ ∀ R1 , R2 . Then M satisfies (CS) in 6.1 (d) if some M(R1 ) ∩ M(R2 ) = ∅, then, since these are disjoint closed sets of YB, there is f ∈ B with f = (1 on M(R1 ); 0 on M(R2 )) (by 1.1). But then f contradicts (∗). (c)⇒(a). Let M be a CS-map, and b 6 = c constants in B. Let  = (R) = 31 |b − c| ∀ R. Then, (M, ) satisfies 6.1 (b) (∗): Otherwise some f ∈ some UR1 (M(R1 ), , b) ∩ UR2 (M(R2 ), , c), and there is x ∈ M(R1 ) ∩ M(R2 ), where | f (x) − b| ≤  and | f (x) − c| ≤ , which leads to a contradiction.  We shortly exhibit some B ∈ |W| for which there is no CS-map; thus τ B is not Hausdorff, and is no group topology. We need to recall the interaction of W with the category of “Spaces with Filter,” called SpFi. Here is a sketch of the situation as described in [6] and [11]; we are recalling more than we need, in order that the reader might get the larger picture. 6.3. (a) An object of SpFi is an (X, X ), X a compact Hausdorff space and X a filter of f

dense open sets in X. A morphism (X, X ) −→(Y, Y ) is a continuous function f

X −→ Y for which f −1 (G) ∈ X ∀ G ∈ Y . (b) Let B ∈ |W|, and let B−1 R be the filter of dense open sets with base {b−1 R | b ∈ B}. Then S YB ≡ (YB, B−1 R) ∈ |SpFi|. ϕ

Yϕ

Let B −→ D be a W-morphism, induced by YB ←− YD as in 1.1 (b): ϕ(b) = Yϕ

b ◦ (Yϕ). Then, (Yϕ)−1 b−1 (R) = ϕ(b)−1 R, so that (YB, B−1 R) ←−(YD, D−1 R) is a SpFi-morphism, denoted SYϕ. This defines The SpFic Yosida Functor SY : W → SpFi(for which, by the way, ϕ is W-epic iff SYϕ is SpFi-monic; see [6, 7]). (c) For general (X, X ) ∈ |SpFi|, let Y be closed in X with Y ∩ F dense in Y ∀ F ∈ X . Then (Y, Y ∩ X ) ∈ SpFi, and Y is called a SpFi-set for (X, X ). Any closed T in X contains a largest SpFi-set. Any regular closed set is SpFi.

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T (d) Let B ∈ |W|. For T closed in YB, let T 0 = {T ∩ SR | R ∈ B ω }. Then T 0 is SpFi for SYB, is the largest SpFi-set contained in T, T is SpFi iff T = T 0 ; T 0 = ∅ iff ∃ R ∈ B ω with T ∩ SR = ∅. (This process of constructing the largest SpFi-set contained in T is not completely general — it depends on the fact that each b−1 R is a cozero-set; see [7].) ϕ (e) (This is in continuation of 1.2 (e).) If B −→ D is a W-surjection, thus induced by the inclusion Yϕ : YB ⊇ YD as b 7 → ϕ(b) = b | YD, then YD is SpFi in SYB, and kerϕ = {b | b(T) ⊆ {0}}. Conversely, if T is SpFi in SYB, then B|T ∈ |W| and the restriction ρT : B → B|T (ρT (b) = b|T) is a W-surjection with kerρT = {b | b(T) ⊆ {0}}. Thus, we have an order-reversing bijection between the co-frame of SpFi-sets of SYB and the frame of “W-kernels” of B, T 7 → I (T) = {b | b(T) ⊆ {0}}. (Thus, any statement about SpFi-sets is a statement about frames of W-kernels; we have not analyzed our present ideas from that perspective.) (f) The correspondence in (e) was described for C(X )’s in [26], for archimedean f -algebras with identity in [20] and in full generality in [6]. COROLLARY 6.4. Let B ∈ |W| (a) If M is a CS-map for B, then M(R)0 6 = ∅ ∀ R ∈ B ω . (b) If B has a CS-map, then ∀ R ∈ B ω , SR contains a non-void SpFi-set. Proof. (a) Given R, M(R) ∩ M(R1 ) 6 = ∅ ∀ R1 . So M(R) ∩ SR1 6 = ∅ ∀ R1 . So by 6.3, M(R)0 6 = ∅. (b) ∅ 6 = M(R)0 ⊆ SR ∀ R.  EXAMPLE 6.5. B ∈ |W| with no CS-map; τ B is not Hausdorff. The idea is to produce B with all SpFi-sets fat, but some SR thin, to violate 6.4 (b). (a) (See [5].) Let X be compact and basically disconnected. Then D(X ) ∈ |W|, and for S ⊆ X, S = b−1 R for some b ∈ D(X ) iff S is a dense cozero-set, and S = SR for some R ∈ B ω iff S is a countable intersection of dense cozero-sets, a dense “Cδ .” Let T be closed in X. Then, T is SpFi for D(X ) iff T is a P-set, i.e., any zeroset of X which contains T is a neighborhood of T. (See [10].) If ST (For, R contains a non-void SpFi-set, then SR has a non-void interior. T SR = r −1 R, and if SR ⊇ compact T, then for some p ∈ N R, T ⊆ {x | r (x) ≤ R

R

p(r )} ≡ Z ⊆ SR and Z is a countable intersection of zero-sets, thus itself a zeroset. But, if T is SpFi, thus a P-set, then T ⊆ intZ ⊆ SR, and so, if T 6 = ∅, SR has interior.) Thus: If compact basically disconnected X contains a dense Cδ with no interior, then D(X ) has no CS-map. (b) Examples. If Y is any compact space which contains a dense Cδ with no interior, then X = the basically disconnected cover, or the extremally disconnected cover (= absolute, projective cover) of Y also contains such a Cδ — so D(X ) has no CS-map. (There is the irreducible map ρ : X  Y, which inversely preserves

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Cδ ’s by continuity, and by irreducibility, inversely preserves density and directly

preserves non-voidness of interior. See [17], among many possible references.) Specifically, we can take Y = [0, 1], in which the set of irrational points is a Cδ with no interior. We now sharpen 6.2 to a characterization of Hausdorff. THEOREM 6.6. For B ∈ |W|, the following are equivalent. (a) τ B is Hausdorff. (b) For each b ∈ B with 0 < b ≤ 1, b and 0 have disjoint neighborhoods. (c) For each non-void open G in YB, there is a CS-map L (depending on G) with L(R) ⊆ G for each R ∈ B ω . Proof. (a)⇒(b) of course. (b)⇒(c). Consider open G 6 = ∅. Take open H and G1 with ∅ 6 = H ⊆ G1 ⊆ G1 ⊆ G (by regularity of YB), and then choose b ∈ B, 0 ≤ b ≤ 1, with b = (1 on H; 0 on YB − G1 ) (by 1.2 (c)). Assuming (b), b and 0 have disjoint neighborhoods, which means (by 6.1) that there are ND-maps M,  for which (∗) U(M, , b) ∩ U(M, , 0) = ∅. Set L(R) ≡ M(R) ∩ G1 . We need to show only ∀ R1 , R2 L(R1 ) ∩ L(R2 ) 6 = ∅. First note: ∀ R, L(R) 6 = ∅. (For, if L(R) = M(R) ∩ G1 = ∅, then, since b = 0 on YB−G1 , we would have UR1 (M(R1 ), (R1 ), b)=UR1 (M(R1 ), (R1 ), 0), violating (∗) .) Now, if L(R1 ) ∩ L(R2 ) = ∅, then M(R1 ) ∩ (M(R2 ) ∩ G1 ) = ∅, and there is f ∈ B with f = (b on M(R1 ); 0 on M(R2 )). (Take h ∈ B with 0 ≤ h ≤ 1 and h = (1 on M(R1 ); 0 on M(R2 ) ∩ G1 ). Then let f = h ∧ b. That f | M(R2 ) = 0 results from b | YB − G1 = 0.) Then, f ∈ UR1 (M(R1 ), (R1 ), b) ∩ UR2 (M(R2 ), (R2 ), 0), violating (∗). (c)⇒(a). Suppose b 6 = c in B. Then there is p ∈ YB where b( p) and c( p) are real and b( p) 6 = c( p). Let  = 15 |b( p) − c( p)| and choose open G containing p such that |b(x) − b( p)| ≤  and |c(x) − c( p)| ≤  for all x ∈ G. Now, referring to 6.3 (c) and (e) restriction of functions, ρ(b) = b | G, defines a W-homomorphism ρ : B → B | G ≡ {b | G | b ∈ B}. Assume (c): There is a CS-map L for B with L(R) ⊆ G ∀ R. This produces a ω CS-map M for B | G: the members of (B | G) are the sets ρ(R) for R ∈ B ω , and M(ρ(R)) ≡ L(R). Let β = b( p) and γ = c( p) be the constant functions in B | G. Since M is a CS-map in B | G, we have U(M, , β) ∩ U(M, , γ ) = ∅ in B | G. The triangle inequality yields U(M, , b | G) ∩ U(M, , c | G) = ∅ in B | G. Thus, ρ −1 U(M, , b | G) ∩ ρ −1 U(M, , c | G) = ∅ in B, which is a display of disjoint τ B-neighborhoods of b and c since ρ is continuous (Section 5).  REMARK 6.7. For later reference, we formalize the idea of the last paragraphs ϕ of the proof above, regarding a W-surjection B −→ D and the resulting association ϕ between CS-maps in B and D. By 6.3, a surjection B −→ D is in effect a restriction ρ B −→ B | T for some SpFi-set T, and, as above, a CS-map L for B for which each L(R) ⊆ T, produces a CS-map M(ρ(R)) ≡ L(R) T for B | T. Also conversely: If M is a CS-map for B | T, then M(ρ(R)) ⊆ Sρ(R) = {T ∩ ρ(r )−1 R | r ∈ R} = T ∩ SR, so L(R) ≡ M(ρ(R)) defines the CS-map L for B.

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We now show that any T τ C(X ) (among other τ B) is Hausdorff. For B ∈ |W|, let R B ≡ {b−1 R | b ∈ B} ⊆ YB. This is the space of real ideals of B: If p ∈ R B, then e p (b) ≡ b( p) defines the W-morphism e p : B → R, and any e : B → R in W is an e p . We always have the W-morphism of restriction ρ : B → B | R B ⊆ C(R B), and ρ is one-to-one iff R B is dense in YB iff there is some W-injection ϕ : B → R X. When this occurs, B is sometimes called a W-object of real-valued functions. ˇ Obviously, any C(X ) is such; here, YC(X ) = β X (Cech–Stone compactification) and RC(X ) = υ X (Hewitt realcompactification). (See [16].) COROLLARY 6.8. (a) If p ∈ R B, then M(R) = { p} ∀ R ∈ B ω defines a CS-map. (b) If R B is dense in YB, then τ B is Hausdorff. (c) Any τ C(X ) is Hausdorff. Proof. (a) is obvious. (b) Using (a), 6.6 (c) holds. (c) C(X ) satisfies (b).



REMARKS 6.9. (a) We have produced recently an l-group B with τ B Hausdorff and R B = ∅, for which, assuming the continuum hypothesis, τ B is not a group topology. We hope to return to this still murky subject in a later paper. (b) ∃ CS-map 6 ⇒ τ B Hausdorff: Let A ∈ |W| have no CS-map, e.g., any example as in 6.5. Let X = YA, B = A× C(X ), for which YB = X1 + X2 , Xi = X, and the Yosida Representation is b = (a, f ) = (a on X1 ; f on X2 ). Here, X1 and X2 are clopen, thus SpFi-sets in YB, with A = B | X1 , and C(X ) = B | X2 , and we have the restriction morphism ϕi : B → B | Xi , and thus the association described in 6.7 between CS-maps of B ranging inside Xi , and CS-maps of B | Xi . By 6.8, B | X2 has CS-maps, thus so does B. But B | X, has no CS-maps, thus B has none ranging inside X1 . Since X1 is open in YB, τ B is not Hausdorff (6.6). (c) The argument in (b) above shows this: If B has a SpFi-set T 6 = ∅ for which R B ∩ T is dense in T, then B has a CS-map; but τ B need not be Hausdorff, exemplified in (b). ϕ (d) A curious observation: Suppose τ B is Hausdorff and B −→ D is a W-surjection. If ϕ preserves arbitrary suprema (resp., countable suprema) then τ D is (resp., need not be) Hausdorff. Here, D = B | T for a SpFi-set T, and ϕ preserves arbitrary suprema iff T = G for some open G [5]. Then τ B Hausdorff implies τ D Hausdorff, using 6.6. On the other hand, the examples D(X ) in 6.5 (b) have the form D(X ) = B(X )/N, where B(X ) is the real-valued Baire functions on X and N is a certain σ -ideal — this is a version of the Loomis–Sikorsky–Stone theorem; see [5], for ϕ example. Then, the quotient B(X ) −→ B(X )/N preserves countable suprema because N is a σ -ideal, τ B(X ) is Hausdorff by 6.8, but τ D(X ) is not by 6.5.

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7. Convergence and Closure We begin with a short “Abstract” of this section and the next, whose intuitive content is intended to be clear; the various definitions will be provided shortly. We have seen that the epi-topology functor τ : W → Top (or the epi-closure functor cl : W → Clos) has its shortcomings. We shall provide a smoother theory with the “epi-convergence” functor → : W → Conv ≡ the category of convergence spaces: Every closure has its convergence and every convergence has its closure. So, for B ∈ |W|: For each R, cl R has its −→; and, the epi-closure cl B has its convergence, R

B

B

for the nonce =⇒ , whose closure is cl B. We define the epi-convergence: −→ ≡ W W B B {−→ | R ∈ B ω } ( in the poset of convergences on B). Then −→ ≤ =⇒ always, 6 = R

B

B

can occur, but the closure for −→ is again cl B – so −→ still “capture W-epis.” And, B each (B, −→) is a convex Hausdorff convergence l-group, and → : W → Haus Conv is a functor with an intriguing maximality property. The superiority of → over τ can be viewed as tracable to the facts that in Conv, the sup of products is the product of sups, while that fails in Clos. We now sketch out the relevant basic definitions and facts about Conv and its connections with Clos, with a few short proofs inserted. As with Section 3 we flag certain “imperfections” relevant to W with (∗). We do not know an adequate reference for this material. CONVERGENCE 7.1. (a) Definitions. Let X be a set. A convergence on X is a set → of ordered pairs (X , x), where x ∈ X and X is a filter of subsets of X - and (X , x) ∈ → will be indicated as X → x – which satisfies x˙ → x ∀x (where x˙ ≡ {A ⊆ X | x ∈ A}), X → x and Y → x ⇒ X ∩ Y → x (where X ∩ Y ≡ {A ⊆ X | A ∈ X , A ∈ Y }), Y ⊇ X → x ⇒ Y ⊇ X . Then, (X, →) is called a convergence space. Sometimes we shall write “X is a convergence space,” with implied convergence X

denoted −→. Sometimes, when the typography seems to demand it, we may write: λ ∈ ConvX, and then X λx. A function f : X → Y between convergence spaces is continuous if X

Y

[X −→ x ⇒ f (X ) −→ f (x)] ( where f (X )≡{A ⊆ Y | A ⊇ f (F) for some F ∈ X }, i.e., the filter generated by { f (F) | F ∈ X }). A category, Conv, is formed from objects convergence spaces, and morphisms continuous functions. X

X

X ∈ |Conv| is Hausdorff if [X −→ x and X −→ x 0 ⇒ x = x 0 ]. The full subcategory of Conv whose objects are Hausdorff is denoted Haus Conv. (b) Suprema. For X a set, ConvX denotes the set of all convergences on X. ConvX is 0

0

partially ordered by: → ≤ → means [X → x ⇒ X → x]. If {−→ | i ∈ I} is up-directed in ConvX, then the definition [X → x means i

∃ i ∈ I with X −→ x] yields → = sup −→ (supremum in ConvX). (The second i

I

i

axiom for → requires the up-direction.) If each −→ is Hausdorff, then so is sup −→. i

I

i

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If {−→} ⊆ ConvX and {−→} ⊆ ConvY are each up-directed, and f : X → Y i

j

I

J

is a function such that, ∀ i ∃ j for which f : (X, −→) → (Y, −→) is continuous, i

j

then f : (X, sup −→) → (Y, sup −→) is continuous. I

i

j

J

(c) Products. Given X, Y ∈ |Conv|, a convergence → on the set X × Y is defined by X

Y

[H → (x, y) means ∃ X −→ x and Y −→ y with X × Y ⊆ H] (where X × Y is the filter generated by {F × G | F ∈ X , G ∈ Y }). This → is the greatest element in Conv(X × Y ) which makes the projections π X, πY continuous. 0

0

(Proof. Suppose → ∈ Conv(X × Y) has projections continuous, and let H → X

Y

(X, Y). Take X = π X(H) −→ x and Y = πY (H) −→ y, so H ⊇ X × Y just be0

cause H ⊆ π X(H) × πY (H) ∀ H ⊆ X × Y. So → ≤ →. ) X×Y

(X × Y, →) may be abbreviated to X × Y, and → may be denoted −→ or X

Y

−→ × −→, as convenient. (X × Y, {π X, πY }) is the categorical product in Conv. (d) Sups of products. Let {−→} ⊆ ConvX be up-directed. Then i

I

(1) If f : X × X → X is a function for which f : (X × X, −→ × −→) → (X, −→) is continuous ∀ i ∈ I, then i

i

i

f : (X × X, sup (−→ × −→)) → (X, sup −→) is continuous, and I

i

i

I

i

(2) ( ! : contrasting with the situation in Clos) sup (−→ × −→) = (sup −→) × (sup −→). I

i

i

I

i

I

i

(Proof. Let σ = sup −→ × −→ and λ = sup −→. By (i) applied to the projecI

i

i

I

i

tions πa (a = 1, 2), there are “σ – λ continuous,” hence σ ≤ λ × λ. Now suppose H λ × λ(x1 , x2 ). So H ⊇ some X1 × X2 with Xa λ xa . So ∃ i(a) with Xa −→ x(a), i(a)

and then ∃ j with −→ ≤ −→ so Xa −→ x(a) (a = 1, 2). Since H ⊇ X1 × X2 , i(a)

j

j

H −→ × −→(x1 , x2 ), and therefore H σ (x1 , x2 ). Thus λ × λ ≤ σ . ) j

j

CLOSURE AND CONVERGENCE 7.2. φ

(a) Set-preserving functors Clos  Conv are defined by: ψ

For (X, u) ∈ Clos, φ((X, u)) = (X,→) is: X → x means Nu (x) ⊆ X (Nu (x) being the filter of all u-neighborhoods of x (Section 3)). This → may be denoted φ(u), and if we write X ∈ Clos, we may write φ(X ). For (X, →) ∈ Conv, ψ((X, →)) = (X, u) is: x ∈ uA means ∃ X → x with A ∈ X . This u may be denoted ψ(→), and if we write X ∈ Conv, we may write ψ(X ). φ and ψ are order-preserving in the following sense. If u ≤ u0 in ClosX, then 0

0

φ(u) ≤ φ(u0 ) in ConvX. If →≤ → in ConvX, then ψ(→) ≤ ψ(→) in ClosX. For each closure u, u = ψ(φ(u)); so φ is one-to-one and ψ is onto. For each convergence →, →≤ φ(ψ(→)), and (∗) 6 = can occur; so ψ is not one-to-one and

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φ is not onto. We can say ψ φ = Id and φ ψ ≥ Id, the “Id’s” being the identity functors in Clos and Conv, respectively. The →∈ φ(Clos) may be called “closurical.” Note that, if →= φ(u), then u is unique (φ is one-to-one), and →∈ φ(Clos) iff →= φ(ψ(→)). If →= φ(u) and the closure u is topological, then → is called topological. Thus, → is topological iff →= φ(ψ(→)) and ψ(→) is topological. (b) For X ∈ Clos, X is Hausdorff iff φ(X ) is. For X ∈ Conv, if ψ(X ) is Hausdorff, then X is; if X is Hausdorff, then ψ(X ) is T1 , not necessarily Hausdorff (∗). (Proof. Let (X, →) be Hausdorff and let u = ψ(→). To show that points are closed, let x 0 ∈ u{x}. By definition of u, there is {x} ∈ X → x 0 . Since {x} ∈ X , x˙ ⊇ X , so x˙ → x 0 . Since x˙ → x and → is Hausdorff, x 0 = x. ) (c) For X, Y ∈ Clos and f : X → Y a function between the sets: if f is Closcontinuous, then φ( f ) is Conv-continuous (since φ is a functor). Also, if f : φ(X ) → φ(Y ) is Conv-continuous, then f : X → Y is Clos-continuous. (The functor φ is full onto its range [21].) For X, Y ∈ Conv and g : X → Y a function: if g is Conv-continuous, then ψ(g) is Clos-continuous (since ψ is a functor). But, we can have continuous 0

f : ψ(X ) → ψ(Y ) and f is no ψ(g), e.g., whenever → → in a ClosX with 0

ψ(→) = ψ(→), use f (x) = x (∗). (ψ is not “full onto its range.”) (d) (i) For X1 , X2 ∈ Clos, φ(X1 × X2 ) = φ(X1 ) × φ(X2 ). (Proof. Denote by λ the convergence on the left, and by ρi that for φ(Xi ). Each projection πi ∈ Clos, and φ(πi ) ∈ Conv. That shows λ ≤ ρ1 × ρ2 . For the reverse, suppose H ρ1 × ρ2 (X1 , X2 ), so H ⊇ X1 × X2 for Xi ρi xi . But Xi ρi xi means X i ⊇ the Xi – neighborhood filter for xi , say Ni . Thus, H ⊇ N1 × N2 , which means H λ (X1 , X2 ). ) (ii) For X1 , X2 ∈ Conv, ψ(X1 × X2 ) ≤ ψ(X1 ) × ψ(X2 ), and 6 = can occur (∗). (Here, ≤ is shown as ≤ in (i). The 6 = can be illustrated in W, borrowB

ing from 7.3 and Section 8 the fact that ∀ B ∈ W, (B, −→) is a converB

B

gence l-group, which includes continuity of + : (B × B, −→ × −→) → B

B

B

B

(B, −→); thus ψ(+) : (B × B, ψ(−→ × −→)) → (B, ψ(−→)) is continuous. But from Section 6, ∃ B for which + : (B × B, τ B × τ B) → (B, τ B) B is not continuous; but in Clos this function is + : (B × B, ψ(−→) × B

B

ψ(−→)) → (B, ψ(−→)).) (e) In the following, suppose U ⊆ ClosX and L ⊆ ConvX are each up-directed. Then (i) sup φ(U ) ≤ φ(sup U ), and 6 = can occur (∗); (ii) sup ψ(L) = ψ(sup L); (iii) sup U = ψ(sup φ(U )). (Proofs. (i). If u ∈ U , then u ≤ sup U , so φ(u) ≤ φ(sup U ), so sup φ(U ) ≤ φ (sup U ). Now “=” just means sup φ(u) is closurical; see 8.8 below for the necessary and sufficient condition for our situation: B ∈ |W| and U = {cl R | R ∈ B ω }. (ii). ≤ as in (i). For ≥: φψ(sup L) ≥ sup L (since φψ ≥ Id), so ψφψ(sup L) ≥ ψ(sup L). But the left side is ψ(sup L) since ψφ = Id.

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(iii). In (ii), take L = φ(U ), so sup ψφ(U ) = sup U (since ψφ = Id). )

l-GROUP CONVERGENCE 7.3. (a) Definition. (See [3]). Let B be an l-group, and →∈ ConvB. Then, → is an l-group convergence if (l1 ) b 7 → −b is continuous (B, →) → (B, →), (l2 ) (b1 , b2 ) 7 → b1 ⊗ b2 is continuous (B × B, → × →) → (B, →), for each ⊗ = +, ∨, ∧. And, → is called convex if also e ≡ {b | ∃ f1 , f2 ∈ F with e → b where: For F ⊆ B, F (l3 ) X → b ⇒ X e e f1 ≤ b ≤ f2 }; then X ≡ { F | F ∈ X }.

(b) (c)

(d)

(e)

The set of l-group convergences on B is denoted lConvB. When →∈ lConvB, we call (B, →) a convergence l-group, or lc-group. Let B be an l-group, and suppose L ⊆ lConvB is up-directed. Then σ ≡ sup L ∈ lConvB, and if each λ ∈ L is convex, so is σ . Let B1 , B2 ∈ |W|, suppose Li ⊆ ConvB is up-directed, and let σi = sup Li (i = ϕ 1, 2). Suppose that (†) ∀ W-morphism B1 −→ B2 , ∀ λ1 ∈ L1 , ∃ λ2 ∈ L2 such that ϕ ϕ (B1 , λ1 ) −→(B2 , λ2 ) is continuous. Then, (B1 , σ1 ) −→(B2 , σ2 ) is continuous. Suppose ∀ B ∈ |W| is given up-directed L(B) ⊆ ConvB; let σ (B) = sup L(B), for each B. Suppose that {L(B) | B ∈ |W|} has the property: ∀ B1 , B2 , (†) in (c) obϕ ϕ tains. Then, any W-morphism B −→ D has (B, σ (B)) −→(D, σ (D)) continuous. Thus a set-preserving functor σ : W → Conv is defined. (See [3].) Let B be an l-group. Let u ∈ ClosB be the closure for a topological lgroup topology. Then φ(u) ∈ ConvB is a topological l-group convergence, which is Hausdorff if u is Hausdorff if also each b ∈ B has a local base for u whose members are convex, then ϕ(u) is convex.

Proofs. In (b), 7.1 (b) yields (l1 ), 7.1 (c) yields (l2 ), and just the definition of σ yields (l3 ). (c) follows from the definition of sup, and then (d) is immediate. (e). φ(u) is topological by definition, and Hausdorff by 7.2 (b). Now, the l-group operations ⊗ are continuous as ⊗ : (B × B, u × u) → (B, u). Thus the ϕ(⊗) are continuous as ϕ(⊗) : (B × B, ϕ(u × u)) → (B, ϕ(u)). By 7.2 (d), ϕ(u × u) = ϕ(u) × ϕ(u). So ϕ(u) is an l-group convergence. Now suppose b has the local base N (b) ] e⊇ N (b), with convex members, and let X → b (→= φ(u)). Then X ⊇ N (b), so X e → b. and X 

8. Epi-convergence We now apply the generalities of Section 7 to the following array of data in W, whose properties have been given in Sections 2, 4–6. The process is straightforward.

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THE DATA 8.1. Let B ∈ |W|. For each R ∈ B ω , τ R is a Hausdorff topological lgroup topology which is convex. The family {cl R | R ∈ B ω } ⊆ Clos B is up-directed, W B and the epi-closure was defined: cl ≡ {cl R | R ∈ B ω }; cl B is topological and T1 . For each R ∈ B ω , φ(cl R) ∈ Conv B is a Hausdorff topological l-group convergence which is convex. {φ(cl R) | R ∈ B ω } ⊆ Haus l Conv(B) is up-directed. So the following definition makes sense, and the two corollaries are immediate. W B DEFINITION 8.2. For B ∈ |W|, the epi-convergence on B is −→ ≡ {φ(cl R)|R∈B ω } B

COROLLARY 8.3. (local at B). For B ∈ |W| (a) −→ ∈ Haus l Conv B, and is convex, B

B

(b) ψ(−→) = cl B, (c) −→ ≤ φ(cl B), and there are B for which 6 = occurs. Proof. (a) follows from the properties of the φ(cl R) in 8.1, 7.1 (b), and 7.3 (b). B

(b) and ≤ in (c) come from 7.2 (b). The example B = D(X ) in 6.5 has −→ 6 = B

φ(cl B), since −→ is Hausdorff but cl B is not, thus φ(cl B) is not (by 7.2 (b)). ϕ

 B

COROLLARY 8.4 (global for W). Any W-homomorphism B −→ D has (B, −→) ϕ

D

−→(D, −→) continuous. Thus, a set-preserving functor is defined: −→ : W −→ Haus l Conv, the epi-convergence functor. We have ψ ◦ → = cl, the epi-closure functor. Proof. This follows from 8.3, 4.17, then 7.3 (c) and (d) (or, from 8.3 (b) (which shows ψ ◦ → = cl). Then 5.3 and 7.3 (d)).  We now show a curious maximality property of the functor → . Recall from Section 2 the definition of the dominion d(A, B), for A ⊆ B ∈ |W|, and specifically cl B A ⊆ d(A, B), and for divisible subobjects A, cl B A = d(A, B). We interpret dominion in B as a function d(· , B) : P (B) → P (B). The definition of ≤ in Clos B is really just the element-wise order on all f : P (B) → P (B). In particular, for u ∈ Clos B, B

u ≤ d(· , B) means uA ⊆ d(A, B) ∀ A ⊆ B; thus ψ(−→) = cl B ≤ d(· , B) ∀B ∈ |W|. THEOREM 8.5. Let λ : W → Haus Conv be any set-preserving functor. Then ψ(λ(B)) ≤ d(· , B) ∀B. Thus, ∀B, for divisible subobjects A of B, ψ(λ(B))(A) ⊆ B

cl B A = ψ(−→)(A). Proof. Let A ⊆ B. We are to show ψ(λ(B))(A) ⊆ d(A, B). So let b ∈ ψ(λ(B))(A); this means there is a filter X with A ∈ X and X λ(B)b (read λ(B) here as a →). To show b ∈ d(A, B), let ϕi : B → D have ϕ1 | A = ϕ2 | A. Now, X is generated as a filter by {F ∩ A| F ∈ X }, so ϕi (X ) is generated by {ϕi (A∩ F)| F ∈ X }. Since ϕ1 | A = ϕ2 | A, it follows that ϕ1 (X ) = ϕ2 (X ). By hypothesis, ϕi : (B, λ(B)) → (D, λ(D)) is continuous, so ϕi (X ) λ(D) ϕi (b). I.e., the filter ϕ1 (X ) = ϕ2 (X ) λ(D)-converges to both ϕ1 (b) and ϕ2 (b). Since λ(D) is Hausdorff, ϕ1 (b) = ϕ2 (b). Thus b ∈ d(A, B).  REMARKS 8.6. Our attempts to improve 8.5, or bring the idea into sharper focus, have not been succesful. Here are some specific questions in that regard. In the following, Clos∗ stands for some subclass of Clos which might “work,” e.g., Clos, Top Clos, T1 Clos, Haus Clos, l Clos, · · · ; likewise, Conv∗ . We abbreviate “setpreserving functor” to SPF.

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(a) In 8.5, can we conclude ψ(λ(B)) ≤ cl B ∀B? If ψ ranges in Conv∗ ? W B (b) Is it true that −→ =W {λ(B)| λ : W → Conv∗ SPF} ∀B? (c) Is it true that cl B = W{u ∈ Clos∗ B| u ≤ d(· , B)} ∀B? (d) Is it true that cl B = {u(B)| u : W → Clos∗ SPF} ∀B? Having failed to come to grips with these questions, let us be more ambitious. (e) Let C be more-or-less any, or the readers’s favorite, set-based category, so W that ∀B ∈ |C |, dominion d(· , B) is defined. What is s(B) ≡ {u ∈ Clos∗ B|u ≤ d (· , B)} (B ∈ |C |)? This is, of course, a candidate for epi-closure in C . Etc. SOME EXAMPLES 8.7. Here are two classes of examples of set-preserving functors λ : W → Haus Conv, to which 8.5 applies. It will be clear that the first class does not provide “no” answers to the questions in 8.6, and for the second class, it seems doubtful, but the analysis escapes us. (1) Suppose ∀B ∈ |W| is given R(B) ⊆ B ω with the properties: ∀B, ∀R1 , R2 ∈ ϕ R(B), R1 ∪ R2 ∈ R(B); ∀ W-morphism B −→ D, ∀R ∈ RW (B), we have ϕ(R) ∈ R(D). Then, let λ(R) = λ : W → Haus Convbe: λ(B) ≡ {φ(cl R) | R ∈ R(B)}. B

Clearly, λ(B) ≤ −→ ∀B. Two examples of this are: R(B) = {e B}, for which λ(B) is ordinary uniform convergence on YB. And, R(B) = all finite subsets of B, for which λ(B) is certainly related to the considerations of [9]. (2) Let r : W → W be an “Extension Functor”, or EF: To each B, there is a ϕ monic r B : B → r B, and for each W-morphism B −→ D, there is a W-morphism rϕ r B −→ r D for which r D ◦ ϕ = (r ϕ) ◦ r B. The first examples of EF’s are monoreflections, (which are abundant in W- see [24]), which are characterized by the two conditions: r is idempotent (meaning rγ B is an isomorphism ∀B); r B is epic ∀B. But we are requiring neither condition, and an example of an EF satisfying neither is below. ([18] discusses some aspects of a general theory of EF’s.) Define λ(r ) = λ : W → Haus Conv by: λ(B) is the restriction of the epi-converrB

ϕ

ϕ

gence −→ to B. Then any W-morphism B −→ D has (B, λ(B)) −→(D, λ(D)) continuous; λ is an SPF. For EF’s which are monoreflections s ≤ r means: ∀B ∃ monic µ : s B → r B for which µ ◦ s B = r B. It is not hard to see that s ≤ r implies λ(s)(B) ≤ λ(r )(B) in Conv B, B

B

∀B. This includes s = Id, for which λ(s)(B) = −→. So −→ ≤ λ(r )(B), so the best chance of producing “no” answers in 8.6 with such λ(r ) has r as big as possible. In fact, W has the maximum monoreflection, the W-epicompletion β, studied in [5]. But, B

we have not penetrated λ(β); we can’t even tell if −→ = λ(β)(B) ∀B. (An example of an EF r which is neither idempotent, nor has r B’s epic is this: Let r (B) ≡ { f ∈ β B | | f | ≤ β B(b) for some b ∈ B}. Then, for example, for X compact and C(X ) ∈ |W|, βC(X ) = B(X ), the Baire functions, and rC(X ) = B∗ (X ), the bounded Baire functions, C(X ) ≤ B∗ (X ) is not epic when X is infinite, and the tower C(X ) rC(X ) rrC(X ) · · · r α C(X ) r α+1 C(X ) · · · (α is an ordinal) goes on forever. For C(X ), X compact, this is not too interesting for present purposes, because λ(rC(X )) is ordinary uniform convergence. For other B, who knows?)

106

Appl Categor Struct (2007) 15:81–107 B

We now describe those special B ∈ |W| for which −→ is topological and τ B is an l-group topology (contrasting with 8.3.1). In general, → is topological iff → = φ(ψ(→)) (i.e., → is closurical) and ψ(→) is B

B

topological (Section 7). But any ψ(−→) = cl B is topological (Section 5) so (∗) −→ B

B

is topological iff −→ = φ(cl B), i.e., −→ is the convergence for the epi-topology τ B. Again in general, → = φ(u) implies ∀x ∃ minimum filter → x, namely Nu (x), the filter of all u-neighborhoods of x. In general, the converse fails, but it holds if → is an l-group convergence, and for such a convergence, ∀ b ∃ minimum X → b iff ∃ B

B

minimum X0 → 0 ([3], p. 358). This applies to any −→, so (∗∗) −→ is topological iff B

∃ minimum X0 −→ 0. It is now easy to prove the B

THEOREM 8.8. Let B ∈ |W|. −→ is topological iff there is R0 ∈ B ω such that SR0 ⊆ SR ∀R ∈ B ω ; then, τ B = τ R0 is a Hausdorff topological l-group topology. B

Proof. SR0 ⊆ SR ∀ R implies that N R0 (0) is the minimum X −→ 0. This follows from the definitions. B B B Conversely, suppose X 0 −→ 0 and Y −→ 0 implies Y ⊇ X 0 . Now, X 0 −→ 0 B means there is R0 with X 0 −→ 0, which means N R0 (0) ⊆ X0 . Since N R0 (0) −→ 0, R0

B

X 0 = N R0 (0). Now let R ∈ B ω . Since N R( 0) −→ 0, we have N R( 0) ⊇ N R0 (0). This last means: ∀ K ∈ K(R0 ),  ∈ (0, 1), ∃ K0 ∈ K(R), δ ∈ (0, 1) with UR(K0 , δ, 0) ⊆

UR0 (K, , 0); i.e., |b(x)| ≤ δ ∀ x ∈ K0 implies |b(x)| ≤  ∀ x ∈ K. To show SR0 ⊆ SR, take x0 ∈ SR0 . Let K = {x0 } and f = 12 ; there are K0 ∈ K(R) and δ for which (|b(x)| ≤ δ ∀ x ∈ K0 ) implies |b(x0 )| ≤ 21 . Then, x0 ∈ K0 ⊆ SR (for if x0 6 ∈ K0 there would be b with b = (0 on K0 ; 1 at x0 )).  For any B, R B =

T B

b−1 R =

T

ˇ SR. Any SR is Lindelöf and Cech-complete, in

R

consequence of [15], p. 201, and is, of course, dense in YB. The condition in 8.8 can be ˇ put: ∃ R0 with R B = SR0 ; then R B is Lindelöf, Cech-complete, and dense in YB. It is ˇ straightforward to construct B for which R B is Lindelöf, Cech-complete, and dense ˇ in YB, but R B ( SR ∀ R: Just start with X Lindelöf Cech-complete andTnot compact, take a filter base E consisting of cozero-sets of β X which contain X, E = X, and T ∀ E1 , E2 , · · · ∈ E , X 6 = En ; then let B = {b ∈ D(β X ) | b|E ∈ C(E) for some E ∈ E }. Here, {b−1 R | b ∈ B} = E , so R B = X. On the other hand: B

COROLLARY 8.9. Let B = C(X ). Then −→ is topological iff R B (which is υ X, the ˇ Hewitt realcompactification) is Lindelöf and Cech-complete. Proof. ⇒ is shown above. ⇐ follows from T ˇ (1) Y is Lindelöf and Cech-complete iff Y = En for E1 , E2 , · · · cozero-sets in βY, (2) For B = C(X ) ⊆ D(β X), {b−1 R | b ∈ C(X )} is all cozero-sets of β X which contain X. 

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Acknowledgments We are very pleased to thank Vasil Gochev for his careful reading of the paper, with resulting improvements. We are very pleased to thank Mauricio Gendelman for his expert preparation of the manuscript.

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