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Topology Proceedings

Vol 17, 1992

TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

P. FLETCHER AND W. HUNSAKER

1. INTRODUCTION.

In 1975, J. R. Isbell [8] defined a uniformity for a locale, and subsequently A. Pultr defined a uniformity for a frame L in terms of covers of L [11,13]. B. Banaschewski introduced the concept of a strong inclusion for a frame [1 and 2] and proved that a frame L admits a strong inclusion if and only if L is compactifiable. In his doctoral thesis [4], J. L. Frith established a one-to-one correspondence between strong inclusions for a frame L and totally bounded covering uniformities for L. In this note we use an alternative characterization of a uniformity for a frame L in terms of order-preserving functions from L to L, which we established in [5], to obtain results analogous to those of Frith [4, p.63-67]. Our methods of proof differ from those of Frith in that his arguments use covering uniformities and for the most part ours do not. One positive aspect of our approach is that it affords an explicit construction of a base for the unique totally bounded frame uniformity associated with a given strong inclusion. The reader is referred to [9] for further information on frames and to [10] for further information on proximities.

2. PRELIMINARIES.

A frame is a complete lattice (L,~) in which the infinite distributivity condition x 1\ V{xo : 0 E A} = V{x 1\ X o : 0 E A} holds for each x ELand each subset {xo : 0 E A} of L. Given

59

60 P. FLETCHER AND W. HUNSAKER

a frame (L,~) the greatest element of L is denoted by 1 and ,the least element of L is denoted ,by o.

Let Land M be frames and let 9 : L -+ M be a function. Then 9 is a join homomorphism provided that 9 preserves the joins of arbitrary sets. A frame homomorphism is a join homo­morphism that also preserves meets of finite sets. If f : L -+ L is order preserving, an element b of L is f -small provided that b ~ f( a) whenever al\b =i 0 [5],[6]. For a given order-preserving function f, the collection of all f -small members of L is denot­ed by At. For any bEL the pseudocomplement of b, denoted by b, is V{a E L: a 1\ b = O}. Intuitively, b is the largest a E L such that a 1\ b= o.

Definition 1. A strong inclusion for a frame (L,~) is a binary relation <: on L satisfying the following axioms for a, b, c, d in L.

(1) 0 <: 0, 1 ~ 1. (2) If a ~ b, then a:5 b. (3) If a :5 b ~ c :5 d, then a ~ d. (4) If a ~ c and a ~ b, then a ~ b1\ c. (5) If a <: c and b ~ c, then a V b ~ c. (6) If a ~ b, then there exists c E L such that a ~ c ~ b. (7) For each a E L, a = V {b E L : b ~ a}. (8) If a ~ b there are a', b' E L such that al\a' = 0, bVb' = 1

and b' ~ a'.

A frame together with a specified strong inclusion will be called a proximity frame.

We call axioms (8) the Smyth symmetry axiom. It is taken from [14, Proposition 3]. A related concept has been inves­tigated by the authors in [6]. It is an easy exercise to show that the axioms (1)-(8) above are equivalent to the axioms for a strong inclusion defined in [1].

61 TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

3. THE TOTALLY BOUNDED FRAME UNIFORMITY FOR A

. STRONG INCLUSION.

We turn now to the alternative characterization of a totally bounded uniformity for a frame. Once we have verified th"at our alternative notion squares with the established concept of a to­tally bounded covering uniformity, we characterize the totally bounded uniformity compatible with a given strong inclusion on a frame L, and hence characterize the strong inclusion it­self, in terms of a collection of well-behaved functions from L to L.

Let (L,~) be a frame and let F be a collection of order­preserving functions form L to L. We define ~,A and V point­wise on F.

Definition. Let U be a collection of order-preserving functions on a frame (L, ~). For a, bEL we write a ~u b provided there is an 1 E U such that f(a) $ b. The collection U is a uniformity base on ~ [5] provided that for fEU and a, bEL :

(1) a $ f(a). (2) There exists a 9 E U such that 9 0 9 $ f. (3) a A I(b) = 0 if and only if f(a) 1\ b = O. (4) For any 1,9 E U there exists a join homomorphism

h E U such that h ~ fAg. (5) The collection Af of all f -small members of L is a cover

of L. (6) a = V{x E L : x ~u a}.

The uniformity U on L generated by U is the collection of all order-preserving functions 9 such that f ~ 9 for some fEU. Let U and V be uniformities on L then U is coarser then V if U~V.

It follows from the theorem in [5] that the theory of unifor­mities for frames is equivalent to the theory of covering unifor­mities for frames given in [3], [11], [12], and [13]. The following definitions and notations are taken from [3] and [5]. A cover of a frame L is a subset B of L such that V B = 1. For each cover A of L and each x E L,Ax = V{a E A: al\x ¥= OJ.

62 P. FLETCHER AND W. HUNSAKER

If A is a collection of covers and x, y E L, we write x ~.A y provided there is anA E A such that Ax.~ y. For covers A, B of L we write A -< B provided that for each a E A there is a b E B such that a ~ b. If A -< B, we say that A refines B, and if {Aa : a E A} -< B we write A -<* B. For covers A, B of L, A A B denotes {a A b : a E A, b E B}. A collection A of covers of L is a covering uniformity base on L provided that

(1) If A, B E A, there exists C E A such that C -< A A B. (2) For any A E A there is B E A such that B -<* A. (3) For any y E L, y = V{x E L : x ~.A y}.

The covering uniformity on L generated by A is the col­lection of all covers of L that are refined by some cover in A. A covering uniformity is totally bounded provided that each A E A is refined by a finite cover B E A [ 3, p. 67].

Let A be a covering uniformity base on L. It follows from Propositions 3.1, 3.2, and 3.3 below that ~.A is a strong inclu­sion on L. If W is a uniformity (base) on L, the corresponding covering uniformity Jl(W) has for a base the collection of all AJ for· fEW. If a is a covering uniformity (base), the corre­sponding uniformity U(a) has for a base the collection of all functions fA : L ---+ L defined by fA(X) = Ax for all A E a.

Proposition 3.1 Let W be a uniformity on a frame (L,~)

and let a, bEL. Then a ~w b if and only if a ¢:.A b where A = Jl(W).

Proof: For any fEW, f( a) :5 b if and only if A fa :5 b.

Proposition 3.2. Let a be a covering uniformity on a frame (L, <) and let a, bEL. Then a ~Q b if and only if a ¢:w b where W =U(a).

Proof: For any A E a and a, bEL Aa:5 b if and only if, fA(a) :5 b.

It is evident from the previous two propositions that the relation ~u used in the definition of a uniformity base on a frame L is the "uniformly below" given by B. Banaschewski and A. Pultr [3]. The following proposition, presented in terms

63 TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

of covering uniformities is given b~y Frith [ 4, proposition 4.19]. Since. the axioms for a strong inclusion given here differ from those of Frith [4], we outline a proof of his result. This proof illustrates the way in which our approach to uniformities and strong inclusions differs from the covering approach.

Proposition 3.3. Let U be a uniformity on a frame (L, ~);

then ~u is a strong inclusion on L.

Proof: The reader can easily verify' that the first seven axioms for a strong inclusion hold. To show that the Smyth symmetry axiom holds, let a, bEL such that a ~u b and let fEU such that f( a) $ b. There exists 9 E U such that gog ~ I. Let a' = a and let b' = g(a). Since b' 1\ g(a) = 0 g(b') 1\ a = 0, and g(b') ~ a'. Hence b' ~u a'. Evidently a 1\ a' = 0 and since Agg(a) ~ b, 1 = VAg = (V{x E Ag : x I\g(a) f:. OJ) V (V{x E

Ag : x 1\ g(a) = O}) ~ b V g(a) = b'v b'.

Definition. A uniformity U on a frame (L,~) is totally bound­ed provided that for each.9 E U there is a finite cover B of L such that IB E U and IB ~ 9. (Corollary 3.1 states that the assumption that IB E U may be omitted from the preceding definition. )

The definition of total boundedness of a uniformity is justi­fied by the following proposition.

Proposition 3.4. Let (L,~) be a frame. If 0 is a totally bounded covering uniformity on L, thenU{o) is a totally bound­ed uniformity; if W is a totally bounded uniformity on L, then Jl(W) is a totally bounded covering uniformity on L.

Proof: Let 0 be a totally bounded covering uniformity on L and let 9 E U(o). There exists A E 0 such that fA ~ 9 and there is a finite B E 0 such that B -< A. Then f B E U (0) and fB ~9·

Let W be a totally bounded uniformity and let A E Jl(W). There ia 9 E W such that Ag -<* A. Let fEW such that 1 0 1 0 1 ~ 9. There is a finite cover B of L such that IB E W

64 P. FLETCHER AND W. HUNSAKER

and fB $ f. For each b E B, AIBb $ fB(b) = Bb and so AlB -< .{Bb : b.E B}. It follows that {Bb : b E B} is a finite cover of L belonging to p(W). Moreover each b E B is fB-small and hence is both f-small and g-small. To see that {Bb :. b E B} refines Ag let b E B and let zEAl such that zAf(b) =F O. Since Bb = fB(b) $ f(b) it suffices to show that f(b) ~ Agb. It may h.e seen that f(b) is a g-small set, for if a ELand a A f(b) =F 0, then f( a) A b =F 0 and since b is f-small, b ~ f 0 f( a) and so f(b) ~ f 0 f 0 f(a) ~ g(a). Consequently f(b) ~ Agb.

Let ~ be a strong inclusion on a frame (L, ~). A cover A = {ai : i E A} is a proximity cover provided there is a cover B = {bi : i E A} such that bi ~ ai for each i E A. It is easy to see that if {Ai : 1 ~ i ~ n} is a finite collection of proximity covers then A{Ai : 1 ~ i ~ n} is also a proximity cover.

If D is a cover of L,D* = {VA: A ~ D and for all x,y E A, x A Y =F OJ. Although B* -< A and B -<* A say different things if C* -< Band B* -< A then C -<* A.

The proof of the following lemma is based· upon [3, Lemma 4]. _

Lemma 3.1. Let ~ be a strong inclusion on a frame (L,~)

and let E be a finite proximity cover. Then there is a finite proximity cover D such that D* -< E.

Proof. We first establish the lemma in the special case that E is a two-element cover of L. Let E = {bI'~} be a proximity cover.. There are covers {WI, W2}, {Xl, X2}, {ZI' Z2} such that for i = 1,2, Wi ~ Xi ~ Zi ~ bi. Since Xl ~ ZI by Smyth symmetry there are x~, z~ E L such that Xl AX~ = 0, Z1 V z~ = 1 and z~ ~ x~. Similarly there are x~, z~ E L such that X2 A x~ = 0, Z2 V z~ = 1 and z~ ~ x~. Set D = {Xl A b2 , X2 A bI , Xl A x~, x~ A X2}. Since 1 = Xl V X2 = [Xl A (Z2 V z~)] V [X2 A (ZI V z~)] ~ [Xl A (b2 V x~)] V [X2 A (b i V x~)] = VD, D is a cover. Because Xi :5 Zi :5 bi for i = 1, 2, is easy to verify that D* -< {bI, b2}. To see that D is a proximity cover, let C = {WI A Z2, W2 A ZI, WI A z~, z~ A W2}. Then WI A Z2 ~

65 TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

xl Ab2 , W2AzI ~ x2Abl' wIAz~ ~ xIAx~ and Z~AW2 ~ X~AX2. Moreover,l = WI VW2 = [WI A (Z2 V' z~)] V [W2 A(ZI Vz~)] = vC.

Now let A = {aI, a2, . .. ,an} be an arbitrary finite proximity cover and let X = {Xl,... ,xn } be a cover such that Xi' ~ ai for i = 1,2, ... ,n. Choose i, 1 ~ i ~ n. There are bi, Ci in L such that Xi ~ bi ~ Ci ~ at. Set Ai = {bi, ail. Then Ai .is a two element proximity cover and so we may choose a proximity cover Bi such that B: -<: Ai. Note that Xi A bi = O. For i = 1, 2, ... ,n define Ai as above. Since X is a cover,

n n n

/\ hi = °and so /\ Ai -< A. Let D = /\ Bi . As we have already i=l i=I i=l noted, D, being the finite meet of proximity covers is itself a proximity cover and D is the proximity cover we require since

n

D* -< /\A i -< A. i=l

Lemma 3.2 Let (L,~) be a frame and let ~ be a strong in­clusion ().n L. Let B = {fe : C is a finite proximity cover of L}. For a, bEL, a ~ b is and onl,y if there exists fEB such that f(a) ~ b.

Proof: Suppose there is a finite proximity cover C = {ei} i:l such that fe (a) ~ b. Let A = {ai} i=l be a cover such that ai ~ Ci for i = 1, ... ,n. Let j E {I, 2, ... ,n} such that aj A a =I O. Then aj ~ Cj ~ Ca. Hence Aa ~ Ca ~ b and so a ~ b.

Now suppose a ~ b. There exists C E L such that a ~ C ~

b. By Smyth symmetry there are a', e' , e" ,b" E L such that a A a' = 0, eVe' = 1, e' «: a', e A e" = 0, b" V b = 1, b" «: e". Let C = {b, a}. Since 1 = eVe' ~ c V a' ~ C V a ~ b V a, C is a finite cover. To see that C is a proximity cover note that A = {e', e} is a cover such that e «: band e' «: a' ~ a. Thus fe E Band fe{a) = b.

If ~ is any strong inclusion on a frame (L,~) we say that a uniformity U is compatible with «: provided that for all a, bEL, a ~ b is and only if a «:u b, and we say that U belongs to the proximity class 1r(«:).

66 P. FLETCHER AND W. HUNSAKER

Theorem 3.1. Let (L,~) be a frame and let <t:: be a strong inclusion on L. Let B = {Ie: C is a finite proximity cover of L}. Then B is a base for a uniformity P on L compatible with <t::. Moreover P is the coarsest member of 1r(<t::) and is the only totally bounded member of 1r(~).

Proof: It is evident that B satisfies conditions (1), (3), and (5) of the definition of a uniformity base and it follows from Lemma 3.2 that B satisfies condition (6) as well. Because the finite meet of proximity covers is a proximity cover, if fe and ID belong to B, then leAD E B. Hence condition (4) obtains. To see that condition (2) obtains let fEB, and let E be a finite proximity cover such that I = fEe By Lemma 3.1, there is a finite proximity cover D such that D* ~ E. By definition fD E B. We show that fD 0 fD ~ f. Let x E L. Then IDOfD(X) = V{d ED: there exists d' E D such that dAd' =I 0 and d' A x =1= O} ~ D*x ~ Ex = f(x).

We now show that P is the coarsest uniformity in 1r( <t::). It follows from Lemma .3.2 that 'P E 1r( <t::). Let V be any uniformity belonging to 1r( <t::) and let h E P. Then there is a finite proximity cover C = {Ci : 1 ~ i ~ n} such that fe ~ h. Let A = {ai : 1 ~ i ~ n} be a cover of L such that ai « Ci . for i = 1,2, ... ,n. There are gi E V,l ~ i ~ n, such that

n

9i (ai) ~ Ci· Let 9 = 1\ 9i· Since 9 E V, it suffices to show that n=l

9 ~ h. Let x ELand suppose that j is a positive integer, j ~n such that Cj Ax = o. Then 9j(aj) Ax ~ Cj Ax = 0 and so aj A g(x) ~ aj A gj(x) = o. Hence g(x) ~ V{ai E A : ai A x # O} ~ V{Ci E C: Ci A x =I O} = le(x) ~ h(x).

It is obvious that 'P is totally bounded and so it only remains to prove that P is the only totally bounded member of 1r(«). Let W be a totally bounded uniformity belonging to 1r(«). It suffices to show that W ~ 'P. Let 9 E W. Then Ag E Jl(W) and by Proposition 3.4 there is a finite cover C E Jl(W) such that C ~ Ag • In the comments preceding [3, Proposition 4] the authors prove that there is a finite cover D E Jl(W) such that

67 TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

D ~* C. For each d E D choose Cd E C such that Dd ~ Cd

and ·set R = {Cd: d ED}. Since D ~ R, R E p(W). Clearly R is a finite refinement of C and fR ~ fAg ~ g. We show that R is a proximity cover. Since D E p(W), it follows from the remarks preceding Proposition 3.1 that fD E U(Jl(W)) and by [5, proposition 2.3], U(Jl(W)) = W. Moreover for each d E D,jD(d) = Dd ~ Cd and so d ~ Cd. Therefore R is a proximity cover of L.

Corollary 3.1. Let (L,~) be a frame and let W be a unifor­mity on L. The following statements are equivalent:

(i) W is totally bounded, (ii) For each 9 E W, Ag has a finite subcover, and (iii) For each 9 E W, there exists a finite cover B of L such

that IB :5 g.

Proof: (i) => (ii). Let 9 E W. There is a finite cover B of L such that fB E U and fB :5 g. Let b E B and suppose that a A b # 0.. Then b ~ fB(a) ~ g(a) and so b is g-small.

(ii) => (iii). Let 9 E Wand let B be a finite subcover of Ag •

Let x E L; then fB(X) = V{b E B: bA x =I O} ~ g(x). (iii) => (i). Let 9 E W. There exists an hEW such that

h 0 h 0 h ~ 9 and h is a join homomorphism from L to L. Let B be a finite cover of L such that fB ~ h. Put C = {h(b) : b E B}. Then C is a finite proximity cover of L. It follows from Theorem 3.1 that fe E W; hence it suffices to show that f e ~ g. Let a E L; then

fe(a) - V{cEC:cAa#O}

- V{h(b): b E Band h(b) A a # O} V{h(b) : b E Band h(a) A b =I O}

- h(V{b E B: bA h(a) # O)} - h(Bh(a)) = h(IB(h(a))) :5 h 0 h 0 h(a) :5 g(a).

Corollary 3.2. Let (L,:5) be a frame and let Q' be a covering uniformity on L. Then Q' is totally bounded if and only if for each A E Q' there is a finite cover B of L such that B ~ A.

68 P. FLETCHER AND W. HUNSAKER

Proof: In view of Proposition 3.4 and the fact that 0 = J.l(U (0)) [5, Proposition 2.3] it suffices to show that U (0) is totally bounded. Let 9 E U(o). Then there is an A E ex such that fA $ g. Let B be a finite cover of L such that B -< A. Then fB $ fA and it follows from Corollary 3.1 that U(o) is totally bounded.

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69 TOTALLY BOUNDED UNIFORMITIES FOR FRAMES

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