- Email: [email protected]
The Samuel compactification for quasi-uniform biframes
Topology and its Applications 156 (2009) 2116–2122
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
The Samuel compactification for quasi-uniform biframes John Frith ∗ , Anneliese Schauerte Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa
a r t i c l e
i n f o
a b s t r a c t
Article history: Dedicated to Bob Lowen on the occasion of his 60th birthday MSC: 06D22 54E05 54E15
The paircover approach is used to explore the links between quasi-uniform and proximal biframes. The Samuel compactification for quasi-uniform biframes is constructed and its universal property discussed. © 2009 Elsevier B.V. All rights reserved.
Keywords: Quasi-uniformity Proximity Strong inclusion Frame Biframe Samuel compactification Proximal biframe Quasi-uniform biframe
1. Introduction In [7] it is mentioned that the uniform covers of a quasi-uniform space are not sufficient to characterize the given quasiuniformity. What is less well known is that the uniform paircovers of a quasi-uniform space do indeed characterize the quasi-uniformity. (See [17].) In his book [20], Isbell, in discussing the use of covers and entourages, says “. . . in this book each system is used where it is most convenient, with the result that Tukey’s system of uniform coverings is used nine-tenths of the time”. In contrast, the literature reveals that, in quasi-uniform spaces, the entourage approach is used almost exclusively. (See [12,22,23].) On the other hand, in the point-free setting (frames or locales, biframes), the theory of quasi-uniformities was broached using the paircover approach [13,15]; later [11,9] introduced an entourage-like approach and finally the so-called Weil uniformities of [19,26,27,29] provided the direct analogue of entourages. In this paper we exploit again the paircover approach to quasi-uniformities in the point-free setting of biframes. Our aims are two-fold. First we investigate the relationship between proximity and uniform structures on a biframe. Next, after a simple characterization of compact biframes in terms of paircovers, we construct a Samuel compactification for quasiuniform biframes and exhibit its universal property. We do so without using the existence of a completion.
*
Corresponding author. E-mail addresses: [email protected] (J. Frith), [email protected] (A. Schauerte).
0166-8641/$ – see front matter doi:10.1016/j.topol.2009.03.034
©
2009 Elsevier B.V. All rights reserved.
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2117
Samuel introduced his compactification of a uniform space using ultrafilters in [31]. Since then, the Samuel compactification, thought of as a reflection to the compact uniform spaces, has seen widespread use, and has been extended to different settings. See, for example, [18,6,24,5,28,37,1]. The Samuel compactification of a uniform frame is constructed in [4], using uniform covers. The authors then form the completion of a uniform frame using a quotient of this compactification. By contrast, in [16] a Samuel compactification of a quasi-uniform frame is formed by taking the completion of its totally bounded coreflection. In that paper, the universal properties of the compactification are not investigated. For a discussion of the differences between compactifications of frames and biframes, see [32,34]. 2. Preliminaries 2.1. Frames and biframes We have used [30] as a useful reference for frames; see also [21,36]. All the biframe notions in Definition 2.1 appear in the literature: see [3,32,33,2]. Definition 2.1. 1. A frame L is a complete lattice in which the distributive law x∧
2. 3.
4. 5. 6.
7. 8. 9. 10.
{ y: y ∈ Y } =
{x ∧ y: y ∈ Y }
holds for all x ∈ L , Y ⊆ L. A frame map is a set function between frames which preserves finite meets and arbitrary joins, and thus also the top (denoted 1) andthe bottom (denoted 0) of the frame. C = 1. For a frame L, C ⊆ L is a cover of L if (a) A biframe L is a triple L = ( L 0 , L 1 , L 2 ) in which L 0 is a frame, L 1 and L 2 are subframes of L 0 , and L 1 ∪ L 2 generates L 0 (by joins of finite meets). We call L 0 the total part, L 1 the first part and L 2 the second part of the biframe L. (b) A biframe map h : L → M is a frame map from L 0 to M 0 such that the image of L i under h is contained in M i for i = 1, 2. We call the restriction h| L 0 the total part of the map h and h| L 1 and h| L 2 its first and second parts, respectively. (c) BiFrm is the category of biframes and biframe maps. A biframe map h : L → M is dense if its total part is a dense frame map, i.e. a = 0 whenever h(a) = 0, for any a ∈ L 0 . A biframe map h is onto if its first and second parts are onto. If L isa biframe and x ∈ L i (i = 1, 2), we denote by x• the largest y ∈ L k (k = 1, 2, k = i ) for which x ∧ y = 0; that is, x• = { z ∈ L k | z ∧ x = 0}. We refer to this as the biframe pseudocomplement. We always have x ∧ x• = 0, but x ∨ x• need not equal 1. Note that, if x ∈ L 1 , then x• ∈ L 2 ; if x ∈ L 2 then x• ∈ L 1 . A biframe L is regular if each x ∈ L i (i = 1, 2) can be expressed as a join x = { y ∈ L i | y ≺i x} where y ≺i x means that there exists c ∈ L k (k = 1, 2, k = i ) such that y ∧ c = 0 and x ∨ c = 1. Equivalently, y ≺i x means that y • ∨ x = 1. A biframe is compact if its total part is a compact frame, that is, whenever an arbitrary join is the top element, a finite subjoin is already equal to the top. The category of compact regular biframes and biframe maps is denoted by CptRegBiFrm. A compactification of a biframe L is a dense, onto biframe map h : M → L from a compact, regular biframe M to L.
2.2. Paircovers and quasi-uniformities We now present the basic definitions for paircovers in biframes. Definition 2.2. Let L = ( L 0 , L 1 , L 2 ) be a biframe.
1. C ⊆ L 1 × L 2 is a paircover of L if {c ∧ c˜ : (c , c˜ ) ∈ C } = 1. 2. A paircover C of L is strong if, for any (c , c˜ ) ∈ C , whenever c ∧ c˜ = 0 then c ∨ c˜ = 0, that is, (c , c˜ ) = (0, 0). ˜ We then 3. For any paircovers, C and D of L we write C D if for any (c , c˜ ) ∈ C there is (d, d˜ ) ∈ D with c d and c˜ d. say C refines D. We also set C ∧ D = {(c ∧ d, c˜ ∧ d˜ ): (c , c˜ ) ∈ C , (d, d˜ ) ∈ D }. It is clear that C∧ D is a paircover of L. 4. For a ∈ L 0 and C , D paircovers of L, we set C 1 a = {c: (c , c˜ ) ∈ C and c˜ ∧ a = 0} and C 2 a = {˜c : (c , c˜ ) ∈ C and c ∧ a = 0}. We set C D = {(C 1 d, C 2 d˜ ): (d, d˜ ) ∈ D }. We say C star-refines D if C C D and write C ∗ D. The following lemma is easy to prove. Lemma 2.3. For C , D paircovers of L = ( L 0 , L 1 , L 2 ) and a ∈ L 0 we have: 1. a C i a (i = 1, 2). 2. If D ∗ C then D i D i a C i a (i = 1, 2).
2118
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
We can use paircovers to endow biframes with quasi-uniformities as follows: Definition 2.4. Let L = ( L 0 , L 1 , L 2 ) be a biframe. 1. A non-empty family, U , of paircovers of L is a quasi-uniformity on L if (a) The family of strong members of U is a filter-base for U with respect to ∧ and (Filter condition). (b) For any C ∈ U there is D ∈ U such that D ∗ C (Star-refinement condition). (c) For each x ∈ L i , x = { y ∈ L i : for some C ∈ U , C i y x}, (i = 1, 2) (Compatibility condition). 2. The pair ( L , U ) is called a quasi-uniform biframe. Members of U will be referred to as uniform paircovers. 3. B ⊆ U is a base for U if, for each C ∈ U , there is B ∈ B with B C . 4. Let ( L , U ) and ( M , V ) be quasi-uniform biframes. A biframe map f : L → M is uniform if for every C ∈ U , f [C ] ∈ V , where f [C ] = {( f (c ), f (˜c )): (c , c˜ ) ∈ C }. 5. Quasi-uniform biframes and uniform maps are the objects and arrows of the category QunBiFrm. The reader unfamiliar with the paircover approach to quasi-uniform spaces can consult [17]. The need for strong paircovers (called strong conjugate covers) is discussed there. We will see their importance in other situations in this paper. 2.3. Strong inclusions and proximal biframes Recall that a proximity space can be described in terms of a “nearness relation” (proximity) or, equivalently, in terms of an “inclusion relation” (strong inclusion). (See [25].) In what follows, we choose the inclusion relation approach in the biframe setting. Definition 2.5. ([13,32]) A strong inclusion on a biframe L is a pair = (1 , 2 ) of relations on L 1 and L 2 respectively such that (for i = 1, 2), (SI1) (SI2) (SI3) (SI4) (SI5) (SI6)
w x i y z implies that w i z. i is a sublattice of L i × L i . x i y implies that x ≺i y. x i z implies that there exists y ∈ L i with x i y i z. • • If x i y then y k x for k = 1, 2 and k = i. x = { y: y i x} for all x ∈ L i .
We write (a, b) (c , d) to mean a 1 c and b 2 d. Definition 2.6. A pair ( L , ) where is a strong inclusion on the biframe L will be called a proximal biframe. Let ( L , L ) and ( M , M ) be proximal biframes; a biframe map f : L → M is said to preserve the strong inclusion if a L i b ⇒ f (a) M i f (b) for i = 1, 2. We will denote the category whose objects are proximal biframes and whose maps are the strong inclusion preserving maps by ProxBiFrm. Note that the condition (SI3) guarantees that, for a proximal biframe ( L , ), L is a regular biframe. Proximal biframes are called “quasi-proximal frames” in [13]. We prefer the term “proximal” here, because our definition includes symmetry (SI5) and admissibility (SI6), in contrast to the notion of “quasi-proximity on a frame” used in [10]. 3. Proximal and quasi-uniform biframes In this section we explain the connection between proximal biframes and quasi-uniform biframes; we also provide functors between the two categories. The corresponding connections for uniform and proximal spaces and frames are well known; see [14,25,35,8]. The methods of proof in this paper are somewhat different; the interested reader will note that rather elegant proofs can be obtained using our methods in the frame case as well. In particular, the use of paircovers rather than entourages seems to contribute to the simplicity of the proof. We should mention that there are various equivalent notions of total boundedness for quasi-uniform biframes. The link between these and proximal biframes will be explored by the authors in a forthcoming paper. We now see how to use a quasi-uniformity on a biframe to induce (functorially) a strong inclusion on the same biframe. Define a functor P : QunBiFrm → ProxBiFrm as follows: For ( L , U L ) ∈ QunBiFrm, let P ( L , U L ) = ( L , L ) (= ( L , ) for brevity) where a i b iff C i a b for some C ∈ U L (i = 1, 2). We refer to L as the strong inclusion induced by U L . For f : ( L , U L ) → ( M , U M ) a uniform map, let P f : ( L , L ) → ( M , M ) have the same underlying map as f . We check that P ( L , U L ) ∈ ProxBiFrm: (SI1), (SI2) are straightforward.
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2119
For (SI3) suppose that C 1 x y for some x, y ∈ L 1 , C ∈ U L . Then
{ z ∈ L 2 : z ∧ x = 0} ∨ k: (k, k˜ ) ∈ C , k˜ ∧ x = 0 ˜ (k, k˜ ) ∈ C , k˜ ∧ x = 0 ∨ ˜ (k, k˜ ) ∈ C , k˜ ∧ x = 0 k ∧ k: k ∧ k: ˜ (k, k˜ ) ∈ C = 1. = k ∧ k:
x• ∨ y x• ∨ C 1 x =
For (SI4) suppose that x i y, that is, C i x y for some C ∈ U L . If D ∗ C for some D ∈ U L , then x D i x D i ( D i x) C i x y so x i D i x i y. For (SI5) suppose that C 1 x y for some C ∈ U L . If (k, k˜ ) ∈ C , then k ∧ y • = 0 ⇒ k y ⇒ k˜ ∧ x = 0 ⇒ k˜ x• , so C 2 y • x• . (SI6) is just the compatibility criterion for quasi-uniform biframes. We check that P f is a morphism in ProxBiFrm, that is, that it preserves the strong inclusion relation: if C i x y for some C ∈ U L then f [C ]i f (x) f ( y ) and f [C ] ∈ U M . The following theorem allows us to construct a functor in the other direction from ProxBiFrm to QunBiFrm. Theorem 3.1. Let L = ( L 0 , L 1 , L 2 ) be a biframe and = (1 , 2 ) a strong inclusion on L. There exists a quasi-uniformity on L which induces the strong inclusion . Proof. Let B be the collection of all finite paircovers
C = (c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )
such that there exists a finite paircover
˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n) W C = (w1, w
˜ j ) (c j , c˜ j ) for j = 1, 2, . . . , n. We claim that B forms a base for a quasi-uniformity U . of L such that ( w j , w Note that if C ∈ B then {(c , c˜ ) ∈ C : c ∧ c˜ = 0} ∈ B so there will be enough strong paircovers. B is closed under finite meets: (u , u˜ ) (c , c˜ ) and ( v , v˜ ) (d, d˜ ) implies (u ∧ v , u˜ ∧ v˜ ) (c ∧ d, c˜ ∧ d˜ ). Compatibility: consider x 1 a and take x 1 s 1 t 1 a with x, s, t , a ∈ L 1 . Then C = {(a, 1), (1, x• )} ∈ B since D = {(t , 1), (1, s• )} is a paircover ofL with (t , 1) (a, 1) and (1, s• ) (1, x• ). Note that C 1 x a. Since, for each a ∈ L 1 , a = {x ∈ L 1 : x 1 a}, we also now have that a = {x ∈ L 1 : for some C ∈ U , C 1 x a}. The argument for a ∈ L 2 is similar. ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} Star-refinement condition: start with C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )} ∈ B and W C = {( w 1 , w ˜ j ) (c j , c˜ j ) for j = 1, . . . , n. For each j, find d j ∈ L 1 and d˜ j ∈ L 2 with w j 1 d j 1 c j and w ˜ j 2 d˜ j 2 c˜ j . satisfying ( w j , w ˜ j = {(1, c˜ j ), (d˜ • , 1)}. Then D and each Let D = {(d1 , d˜ 1 ), (d2 , d˜ 2 ), . . . , (dn , d˜ n )} and for each j, let D j = {(c j , 1), (1, d•j )} and D j ˜ j is a member of B . Finally, let F = D ∧ D 1 ∧ D˜ 1 ∧ · · · ∧ D n ∧ D˜ n . We show that F ∗ C . D j, D A typical pair in F has the form ˜ 1 ∧ · · · ∧ z˜ n ∧ w ˜ n) (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n , d˜ j ∧ z˜ 1 ∧ w where
(z j , z˜ j ) ∈ D j ,
˜ j ) ∈ D˜ j , (w j , w
j = 1 . . . n.
For any j, F 1 (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n ) D j1 (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n ) D j1 (d j ∧ z j ) c j
˜ 1 ∧ · · · ∧ z˜ n ∧ w ˜ n ) c˜ j . and similarly F 2 (d˜ j ∧ z˜ 1 ∧ w For brevity denote the strong inclusion induced by U by . We show that =. We have already seen that if a 1 b then C = {(b, 1), (1, a• )} ∈ B and then C 1 a b, so a 1 b. Similarly a 2 b ⇒ a 2 b. Suppose now that a 1 b. Then there is ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} such that C 1 a b. C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )} ∈ B with its associated finite W C = {( w 1 , w Then a ( W C )1 a =
˜ j ) ∈ W C and w ˜ j ∧ a = 0 1 w j : (w j , w
Similarly a 2 b ⇒ a 2 b, so = as claimed.
2
˜ j ) ∈ W C and w ˜ j ∧ a = 0 C 1 a b. c j : (w j , w
2120
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
We finally define a functor U : ProxBiFrm → QunBiFrm as follows: For ( L , ) in ProxBiFrm, let U ( L , ) = ( L , U ) where U is the quasi-uniformity constructed in Theorem 3.1. For f : ( L , L ) → ( M , M ) a morphism in ProxBiFrm, let U f : ( L , UL ) → ( M , UM ) have the same underlying map as f . We check that U f is a morphism in QunBiFrm: Suppose D ∈ UL . There exist paircovers C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} of L such that ( w j , w ˜ j ) L (c j , c˜ j ) for j = 1, . . . , n and C D. Then (cn , c˜n )} and W C = {( w 1 , w ˜ j )) M ( f (c j ), f (˜c j )), which gives f [ D ] ∈ UM as needed. ( f ( w j ), f ( w Notice that P is a left inverse of U , that is, P U = idProxBiFrm . 4. Compactness We now characterize compactness for biframes in terms of paircovers. We then establish results needed to view CptRegBiFrm as a full subcategory of QunBiFrm. Lemma 4.1. A biframe L is compact iff every paircover of L has a finite sub-paircover.
Proof. Suppose that C is a paircover of L. Now use the fact that {c ∧ c˜ : (c , c˜ ) ∈ C } = 1 and the compactness of the total part L 0 to find a finite sub-paircover of C . Conversely, let D = {dβ : β ∈ B } be a cover of L 0 . Use the fact that each dβ can be expressed as a join of meets of elements from L 1 and L 2 to construct a paircover of L. The fact that this has a finite sub-paircover leads to the compactness of L 0 . 2 Lemma 4.2. Every compact regular biframe has a unique strong inclusion, namely the rather below relation (≺1 , ≺2 ). Proof. It is straightforward to check that (≺1 , ≺2 ) is a strong inclusion on a compact regular biframe. (See [32].) To show uniqueness, suppose that x ≺i y for some x, y ∈ L i (i = 1, 2). Then x ≺i y = that (1 , 2 ) is a strong inclusion on L and •∨ • ∨ ( z ∨ · · · ∨ z ) = 1. Then x ≺ z ∨ · · · ∨ z y, so x y. x z, x z = 1. Compactness gives a finite subjoin so 1 n n i i 1 i z i y z i y This is sufficient to show that (1 , 2 ) = (≺1 , ≺2 ). 2 Theorem 4.3. Every compact regular biframe has a unique compatible quasi-uniformity. Proof. We have already seen that every compact regular biframe has a strong inclusion (by Lemma 4.2) and hence a compatible quasi-uniformity (by Theorem 3.1). It remains to prove uniqueness. Let U be a quasi-uniformity on the compact regular biframe L. We show that any finite paircover is uniform. Then every paircover is uniform, since by compactness any paircover is refined by a finite paircover. This gives us uniqueness of U . So let C be a finite paircover of L. Let (1 , 2 ) be the unique strong inclusion on L which of course is induced by the quasi-uniformity U . Now set Z = {( z, z˜ ): z 1 c and z˜ 2 c˜ for some (c , c˜ ) ∈ C }. Z is easily seen to be a paircover of L; by compactness it has a finite sub-paircover, F = {( z1 , z˜ 1 ), ( z2 , z˜ 2 ), . . . , ( zn , z˜ n )}. Now for each j = 1, . . . , n there is (c j , c˜ j ) ∈ C j
j
such that z j 1 c j and z˜ j 2 c˜ j . Since z j 1 c j there is A j ∈ U with A 1 z j c j . Similarly there is B j ∈ U with B 2 z˜ j c˜ j . Let D ∈ U be a strong common refinement of the paircovers A 1 , A 2 , . . . , A n , B 1 , B 2 , . . . , B n . We claim that D C , and then, since D ∈ U , we have C ∈ U as needed. To see this, let (d, d˜ ) ∈ D with d ∧ d˜ = 0. There is j = 1, 2, . . . , n such that (d ∧ d˜ ) ∧ ( z j ∧ z˜ j ) = 0 since F is a paircover. Then d ∧ z˜ j = 0 and d˜ ∧ z j = 0, which implies that d D 1 z j c j and d˜ D 2 z˜ j c˜ j . So (d, d˜ ) (c j , c˜ j ) as we needed. On
the other hand, if d ∧ d˜ = 0, then (d, d˜ ) = (0, 0) since D is strong, and trivially (d, d˜ ) (c , c˜ ) for any (c , c˜ ) ∈ C .
2
Note that if f : L → M is a biframe map between compact regular biframes and U L , U M are the unique quasi-uniformities on L and M respectively, then f : ( L , U L ) → ( M , U M ) is uniform. We can therefore regard CptRegBiFrm as a full subcategory of QunBiFrm. 5. The Samuel compactification In [32], it was shown that a biframe has a strong inclusion if and only if it has a compactification. We recall here the method of constructing a compactification from a strong inclusion and will use it to construct a functor from proximal biframes to the compact regular biframes. The ideal biframe, J L, of a biframe L is J L = (J0 , J1 , J2 ) where Ji (i = 1, 2) consists of those ideals J of L 0 generated by J ∩ L i , and J0 is the subframe of the frame of all ideals of L 0 which is generated by J1 ∪ J2 . If h : L → M is a biframe map, J h : J L → J M is given by J h( J ) being the ideal generated by the image h( J ). Since h preserves first and second parts, so does J h. Now let L be a biframe with a strong inclusion . Call an ideal J in Ji (i = 1, 2) strongly regular if x ∈ J ∩ L i implies that there is y ∈ J ∩ L i with x i y. Let Ri consist of these strongly regular ideals, and let R0 be the subframe of the frame of all
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2121
ideals of L 0 generated by R1 ∪ R2 . Then R L = (R0 , R1 , R2 ) is a compact regular biframe and the join map L : R L → L provides the required compactification of L. For the proofs of these facts, see [32]. We can now define a functor R : ProxBiFrm → CptRegBiFrm as follows: For ( L , ) in ProxBiFrm, let R ( L , ) = R L. For h : ( L , ) → ( M , ) in ProxBiFrm, let Rh : R L → R M be the restriction of J h to R L. We check that, if J ∈ (R L )i (i = 1, 2) then J h( J ) ∈ (R M )i : if a ∈ J h( J ), then a h(x) for some x ∈ J ∩ L i , so there exists y ∈ J ∩ L i with x i y. So a h(x) i h( y ), as required. Definition 5.1. Let S : QunBiFrm → CptRegBiFrm be the composite S = R P . Lemma 5.2. For any quasi-uniform biframe ( L , U ), the join map
: S (L, U ) → (L, U )
is uniform. paircover of S ( L , U ). We have then that Proof. Let C = {( J 1 , ˜J 1 ), ( J 2 , ˜J 2 ), . . . , ( J n , ˜J n )} be a finite (and thus basic) uniform ( J 1 ∧ ˜J 1 ) ∨ · · · ∨ ( J n ∧ ˜J n ) = L 0 . We show that C = {( J 1 , ˜J 1 ), . . . , ( J n , ˜J n )} ∈ U . For j = 1, . . . , n take a j ∈ J j , a˜ j ∈ ˜J j such that (a1 ∧ a˜ 1 ) ∨ · · · ∨ (an ∧ a˜ n ) = 1. Using the definition of strongly regular
ideals, obtain b j ∈ J j , b˜ j ∈ ˜J j such that a j 1 b j and a˜ j 2 b˜ j . Take a strong paircover D ∈ U such that D 1 a j b j and D 2 a˜ j b˜ j for all j. Then D 1 a j
J j and D 2 a˜ j
˜J j for all j.
We show that D C which proves that C ∈ U . To this end, take (d, d˜ ) ∈ D. If d ∧ d˜ = 0 then (d, d˜ ) = (0, 0) and so (d, d˜ ) (c , c˜ ) for all (c , c˜ ) ∈ C . If d ∧ d˜ = 0 then d ∧ d˜ ∧ a j ∧ a˜ j = 0 for some j = 1, . . . , n. Then (d, d˜ ) ( D 1 a j , D 2 a˜ j ) ˜ ( J j , J j ). 2
Note that the join map, L , viewed as an underlying biframe map gives a compactification of L. We call the map L : S ( L , U ) → ( L , U ) the Samuel compactification of ( L , U ). This is justified by the next theorem. Theorem 5.3. CptRegBiFrm is a coreflective subcategory of QunBiFrm. The coreflection functor is S and the coreflection maps are given by
: S ( L , U ) → ( L , U ).
L
Proof. Let ( L , U ) and ( M , V ) be quasi-uniform biframes, with M a compact regular biframe, and h : ( M , V ) → ( L , U ) a uniform map. Consider the diagram:
∨L
S (L, U ) Sh
(L, U )
k
∨M
S (M , V )
h
(M , V )
1 ∨− M
Note that L ◦ Sh = h ◦ M since h preserves joins. The join map M restricted to thetotal parts is a frame compactification, since any biframe compactification gives a compactification on the total part. Hence M a frame isomorphism. Also, M is onto on first and second parts, so M is a biframe isomorphism. Let k = Sh ◦
◦k =
L
−1
L
M
. Then
◦ Sh ◦
−1 M
=h◦
−1 ◦ = h. M
M
The join map L is dense, because L J = 0 implies that J = {0}. Now a regular biframe has a regular total part (see [3]) and a dense map is left-cancellable in the category of regular frames (see [30]). This fact applies here, since ( L , U ) is a quasi-uniform biframe, so ( L , U ) is a proximal biframe and so L is a regular biframe. This all goes to show that k is unique. 2 References [1] D. Baboolal, R.G. Ori, Samuel compactification and uniform coreflection of nearness frames, in: Proceedings of the Symposium on Categorical Topology, 1994, University of Cape Town, 1999.
2122
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
[2] B. Banaschewski, Universal zero-dimensional compactifications, in: Categorical Topology and Its Relations to Modern Analysis, Algebra and Combinatorics, Prague, 1988, World Sci. Publishing, Teaneck, NJ, 1989, pp. 257–269. [3] B. Banaschewski, G.C.L. Brümmer, K.A. Hardie, Biframes and bispaces, Quaest. Math. 6 (1983) 13–25. [4] B. Banaschewski, A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Philos. Soc. 108 (1990) 63–78. [5] B. Banaschewski, A. Pultr, Paracompactness revisited, Appl. Categ. Structures 1 (1993) 181–190. [6] G.C.L. Brümmer, H.-P.A. Künzi, Bicompletion and Samuel bicompactification, Appl. Categ. Structures 10 (3) (2002) 317–330. [7] M.J. Ferreira, J. Picado, A method of constructing compatible frame quasi-uniformities, Kyungpook Math. J. 44 (2004) 415–442. [8] P. Fletcher, W. Hunsaker, Totally bounded uniformities for frames, Topology Proc. 17 (1992) 59–69. [9] P. Fletcher, W. Hunsaker, W. Lindgren, Characterizations of frame quasi-uniformities, Quaest. Math. 16 (1993) 371–383. [10] P. Fletcher, W. Hunsaker, W. Lindgren, Totally bounded frame quasi-uniformities, Comment. Math. Univ. Carolin. 34 (3) (1993) 529–537. [11] P. Fletcher, W. Hunsaker, W. Lindgren, Frame quasi-uniformities, Monatsh. Math. 117 (1994) 223–236. [12] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982. [13] J.L. Frith, Structured frames, PhD thesis, University of Cape Town, 1987. [14] J.L. Frith, The category of uniform frames, Cah. Topol. Geom. Differ. Categ. 31 (4) (1990) 305–313. [15] J.L. Frith, The category of quasi-uniform frames, Research Reports, Department of Mathematics, University of Cape Town, 1991. [16] J. Frith, W. Hunsaker, J. Walters-Wayland, The Samuel compactification of a quasi-uniform frame, Topology Proc. 23 (1998) 115–126. [17] T.E. Gantner, R.C. Steinlage, Characterizations of quasi-uniformities, J. London Math. Soc. 5 (2) (1972) 48–52. [18] S. Hernández, Uniformly continuous mappings defined by isometries of spaces of bounded uniformly continuous functions, Houston J. Math. 29 (1) (2003) 149–155. [19] W. Hunsaker, J. Picado, Frames with transitive structures, Appl. Categ. Structures 10 (2002) 63–79. [20] J.R. Isbell, Uniform Spaces, Math. Surveys, vol. 12, Amer. Math. Soc., 1964. [21] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [22] H.-P.A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: C.E. Aull, R. Lowen (Eds.), Handbook of the History of General Topology, vol. 3, 2001, pp. 853–968. [23] H.-P.A. Künzi, Uniform structures in the beginning of the third millenium, Topology Appl. 154 (2007) 2745–2756. [24] I. Naidoo, Samuel compactification and uniform coreflection of nearness σ -frames, Czechoslovak Math. J. 56 (131) (2006) 1229–1241. [25] S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge Tracts in Math. and Math. Phys., vol. 59, Cambridge University Press, Cambridge, 1970. [26] J. Picado, Weil entourages in pointfree topology, PhD thesis, University of Coimbra, 1995. [27] J. Picado, Frame quasi-uniformities by entourages, in: Proceedings of the Symposium on Categorical Topology, 1994, University of Cape Town, 1999. [28] J. Picado, Two theorems of Efremoviˇc in pointfree context, J. Pure Appl. Algebra 141 (1999) 289–298. [29] J. Picado, Structured frames by Weil entourages, Appl. Categ. Structures 8 (2000) 351–366. [30] A. Pultr, Frames, in: M. Hazewinkel (Ed.), Handbook of Algebra, vol. 3, North-Holland, Amsterdam, 2003, pp. 791–857. [31] P. Samuel, Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1) (1948) 100–132. [32] A. Schauerte, Biframe compactifications, Comment. Math. Univ. Carolin. 34 (3) (1993) 567–574. [33] A. Schauerte, Normality for biframes, Appl. Categ. Structures 3 (1995) 1–9. [34] A. Schauerte, Notions of local compactness and smallest compactifications of biframes, Appl. Categ. Structures 14 (2006) 259–272. [35] W.J. Thron, Topological Structures, Holt, Rinehart, Winston, New York, 1966. [36] S. Vickers, Topology via Logic, Cambridge Tracts Theoret. Comput. Sci., vol. 5, Cambridge University Press, Cambridge, 1989. [37] J. Walters, Compactifications and uniformities on sigma frames, Comment. Math. Univ. Carolin. 32 (1) (1991) 189–198.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
The Samuel compactification for quasi-uniform biframes John Frith ∗ , Anneliese Schauerte Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa
a r t i c l e
i n f o
a b s t r a c t
Article history: Dedicated to Bob Lowen on the occasion of his 60th birthday MSC: 06D22 54E05 54E15
The paircover approach is used to explore the links between quasi-uniform and proximal biframes. The Samuel compactification for quasi-uniform biframes is constructed and its universal property discussed. © 2009 Elsevier B.V. All rights reserved.
Keywords: Quasi-uniformity Proximity Strong inclusion Frame Biframe Samuel compactification Proximal biframe Quasi-uniform biframe
1. Introduction In [7] it is mentioned that the uniform covers of a quasi-uniform space are not sufficient to characterize the given quasiuniformity. What is less well known is that the uniform paircovers of a quasi-uniform space do indeed characterize the quasi-uniformity. (See [17].) In his book [20], Isbell, in discussing the use of covers and entourages, says “. . . in this book each system is used where it is most convenient, with the result that Tukey’s system of uniform coverings is used nine-tenths of the time”. In contrast, the literature reveals that, in quasi-uniform spaces, the entourage approach is used almost exclusively. (See [12,22,23].) On the other hand, in the point-free setting (frames or locales, biframes), the theory of quasi-uniformities was broached using the paircover approach [13,15]; later [11,9] introduced an entourage-like approach and finally the so-called Weil uniformities of [19,26,27,29] provided the direct analogue of entourages. In this paper we exploit again the paircover approach to quasi-uniformities in the point-free setting of biframes. Our aims are two-fold. First we investigate the relationship between proximity and uniform structures on a biframe. Next, after a simple characterization of compact biframes in terms of paircovers, we construct a Samuel compactification for quasiuniform biframes and exhibit its universal property. We do so without using the existence of a completion.
*
Corresponding author. E-mail addresses: [email protected] (J. Frith), [email protected] (A. Schauerte).
0166-8641/$ – see front matter doi:10.1016/j.topol.2009.03.034
©
2009 Elsevier B.V. All rights reserved.
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2117
Samuel introduced his compactification of a uniform space using ultrafilters in [31]. Since then, the Samuel compactification, thought of as a reflection to the compact uniform spaces, has seen widespread use, and has been extended to different settings. See, for example, [18,6,24,5,28,37,1]. The Samuel compactification of a uniform frame is constructed in [4], using uniform covers. The authors then form the completion of a uniform frame using a quotient of this compactification. By contrast, in [16] a Samuel compactification of a quasi-uniform frame is formed by taking the completion of its totally bounded coreflection. In that paper, the universal properties of the compactification are not investigated. For a discussion of the differences between compactifications of frames and biframes, see [32,34]. 2. Preliminaries 2.1. Frames and biframes We have used [30] as a useful reference for frames; see also [21,36]. All the biframe notions in Definition 2.1 appear in the literature: see [3,32,33,2]. Definition 2.1. 1. A frame L is a complete lattice in which the distributive law x∧
2. 3.
4. 5. 6.
7. 8. 9. 10.
{ y: y ∈ Y } =
{x ∧ y: y ∈ Y }
holds for all x ∈ L , Y ⊆ L. A frame map is a set function between frames which preserves finite meets and arbitrary joins, and thus also the top (denoted 1) andthe bottom (denoted 0) of the frame. C = 1. For a frame L, C ⊆ L is a cover of L if (a) A biframe L is a triple L = ( L 0 , L 1 , L 2 ) in which L 0 is a frame, L 1 and L 2 are subframes of L 0 , and L 1 ∪ L 2 generates L 0 (by joins of finite meets). We call L 0 the total part, L 1 the first part and L 2 the second part of the biframe L. (b) A biframe map h : L → M is a frame map from L 0 to M 0 such that the image of L i under h is contained in M i for i = 1, 2. We call the restriction h| L 0 the total part of the map h and h| L 1 and h| L 2 its first and second parts, respectively. (c) BiFrm is the category of biframes and biframe maps. A biframe map h : L → M is dense if its total part is a dense frame map, i.e. a = 0 whenever h(a) = 0, for any a ∈ L 0 . A biframe map h is onto if its first and second parts are onto. If L isa biframe and x ∈ L i (i = 1, 2), we denote by x• the largest y ∈ L k (k = 1, 2, k = i ) for which x ∧ y = 0; that is, x• = { z ∈ L k | z ∧ x = 0}. We refer to this as the biframe pseudocomplement. We always have x ∧ x• = 0, but x ∨ x• need not equal 1. Note that, if x ∈ L 1 , then x• ∈ L 2 ; if x ∈ L 2 then x• ∈ L 1 . A biframe L is regular if each x ∈ L i (i = 1, 2) can be expressed as a join x = { y ∈ L i | y ≺i x} where y ≺i x means that there exists c ∈ L k (k = 1, 2, k = i ) such that y ∧ c = 0 and x ∨ c = 1. Equivalently, y ≺i x means that y • ∨ x = 1. A biframe is compact if its total part is a compact frame, that is, whenever an arbitrary join is the top element, a finite subjoin is already equal to the top. The category of compact regular biframes and biframe maps is denoted by CptRegBiFrm. A compactification of a biframe L is a dense, onto biframe map h : M → L from a compact, regular biframe M to L.
2.2. Paircovers and quasi-uniformities We now present the basic definitions for paircovers in biframes. Definition 2.2. Let L = ( L 0 , L 1 , L 2 ) be a biframe.
1. C ⊆ L 1 × L 2 is a paircover of L if {c ∧ c˜ : (c , c˜ ) ∈ C } = 1. 2. A paircover C of L is strong if, for any (c , c˜ ) ∈ C , whenever c ∧ c˜ = 0 then c ∨ c˜ = 0, that is, (c , c˜ ) = (0, 0). ˜ We then 3. For any paircovers, C and D of L we write C D if for any (c , c˜ ) ∈ C there is (d, d˜ ) ∈ D with c d and c˜ d. say C refines D. We also set C ∧ D = {(c ∧ d, c˜ ∧ d˜ ): (c , c˜ ) ∈ C , (d, d˜ ) ∈ D }. It is clear that C∧ D is a paircover of L. 4. For a ∈ L 0 and C , D paircovers of L, we set C 1 a = {c: (c , c˜ ) ∈ C and c˜ ∧ a = 0} and C 2 a = {˜c : (c , c˜ ) ∈ C and c ∧ a = 0}. We set C D = {(C 1 d, C 2 d˜ ): (d, d˜ ) ∈ D }. We say C star-refines D if C C D and write C ∗ D. The following lemma is easy to prove. Lemma 2.3. For C , D paircovers of L = ( L 0 , L 1 , L 2 ) and a ∈ L 0 we have: 1. a C i a (i = 1, 2). 2. If D ∗ C then D i D i a C i a (i = 1, 2).
2118
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
We can use paircovers to endow biframes with quasi-uniformities as follows: Definition 2.4. Let L = ( L 0 , L 1 , L 2 ) be a biframe. 1. A non-empty family, U , of paircovers of L is a quasi-uniformity on L if (a) The family of strong members of U is a filter-base for U with respect to ∧ and (Filter condition). (b) For any C ∈ U there is D ∈ U such that D ∗ C (Star-refinement condition). (c) For each x ∈ L i , x = { y ∈ L i : for some C ∈ U , C i y x}, (i = 1, 2) (Compatibility condition). 2. The pair ( L , U ) is called a quasi-uniform biframe. Members of U will be referred to as uniform paircovers. 3. B ⊆ U is a base for U if, for each C ∈ U , there is B ∈ B with B C . 4. Let ( L , U ) and ( M , V ) be quasi-uniform biframes. A biframe map f : L → M is uniform if for every C ∈ U , f [C ] ∈ V , where f [C ] = {( f (c ), f (˜c )): (c , c˜ ) ∈ C }. 5. Quasi-uniform biframes and uniform maps are the objects and arrows of the category QunBiFrm. The reader unfamiliar with the paircover approach to quasi-uniform spaces can consult [17]. The need for strong paircovers (called strong conjugate covers) is discussed there. We will see their importance in other situations in this paper. 2.3. Strong inclusions and proximal biframes Recall that a proximity space can be described in terms of a “nearness relation” (proximity) or, equivalently, in terms of an “inclusion relation” (strong inclusion). (See [25].) In what follows, we choose the inclusion relation approach in the biframe setting. Definition 2.5. ([13,32]) A strong inclusion on a biframe L is a pair = (1 , 2 ) of relations on L 1 and L 2 respectively such that (for i = 1, 2), (SI1) (SI2) (SI3) (SI4) (SI5) (SI6)
w x i y z implies that w i z. i is a sublattice of L i × L i . x i y implies that x ≺i y. x i z implies that there exists y ∈ L i with x i y i z. • • If x i y then y k x for k = 1, 2 and k = i. x = { y: y i x} for all x ∈ L i .
We write (a, b) (c , d) to mean a 1 c and b 2 d. Definition 2.6. A pair ( L , ) where is a strong inclusion on the biframe L will be called a proximal biframe. Let ( L , L ) and ( M , M ) be proximal biframes; a biframe map f : L → M is said to preserve the strong inclusion if a L i b ⇒ f (a) M i f (b) for i = 1, 2. We will denote the category whose objects are proximal biframes and whose maps are the strong inclusion preserving maps by ProxBiFrm. Note that the condition (SI3) guarantees that, for a proximal biframe ( L , ), L is a regular biframe. Proximal biframes are called “quasi-proximal frames” in [13]. We prefer the term “proximal” here, because our definition includes symmetry (SI5) and admissibility (SI6), in contrast to the notion of “quasi-proximity on a frame” used in [10]. 3. Proximal and quasi-uniform biframes In this section we explain the connection between proximal biframes and quasi-uniform biframes; we also provide functors between the two categories. The corresponding connections for uniform and proximal spaces and frames are well known; see [14,25,35,8]. The methods of proof in this paper are somewhat different; the interested reader will note that rather elegant proofs can be obtained using our methods in the frame case as well. In particular, the use of paircovers rather than entourages seems to contribute to the simplicity of the proof. We should mention that there are various equivalent notions of total boundedness for quasi-uniform biframes. The link between these and proximal biframes will be explored by the authors in a forthcoming paper. We now see how to use a quasi-uniformity on a biframe to induce (functorially) a strong inclusion on the same biframe. Define a functor P : QunBiFrm → ProxBiFrm as follows: For ( L , U L ) ∈ QunBiFrm, let P ( L , U L ) = ( L , L ) (= ( L , ) for brevity) where a i b iff C i a b for some C ∈ U L (i = 1, 2). We refer to L as the strong inclusion induced by U L . For f : ( L , U L ) → ( M , U M ) a uniform map, let P f : ( L , L ) → ( M , M ) have the same underlying map as f . We check that P ( L , U L ) ∈ ProxBiFrm: (SI1), (SI2) are straightforward.
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2119
For (SI3) suppose that C 1 x y for some x, y ∈ L 1 , C ∈ U L . Then
{ z ∈ L 2 : z ∧ x = 0} ∨ k: (k, k˜ ) ∈ C , k˜ ∧ x = 0 ˜ (k, k˜ ) ∈ C , k˜ ∧ x = 0 ∨ ˜ (k, k˜ ) ∈ C , k˜ ∧ x = 0 k ∧ k: k ∧ k: ˜ (k, k˜ ) ∈ C = 1. = k ∧ k:
x• ∨ y x• ∨ C 1 x =
For (SI4) suppose that x i y, that is, C i x y for some C ∈ U L . If D ∗ C for some D ∈ U L , then x D i x D i ( D i x) C i x y so x i D i x i y. For (SI5) suppose that C 1 x y for some C ∈ U L . If (k, k˜ ) ∈ C , then k ∧ y • = 0 ⇒ k y ⇒ k˜ ∧ x = 0 ⇒ k˜ x• , so C 2 y • x• . (SI6) is just the compatibility criterion for quasi-uniform biframes. We check that P f is a morphism in ProxBiFrm, that is, that it preserves the strong inclusion relation: if C i x y for some C ∈ U L then f [C ]i f (x) f ( y ) and f [C ] ∈ U M . The following theorem allows us to construct a functor in the other direction from ProxBiFrm to QunBiFrm. Theorem 3.1. Let L = ( L 0 , L 1 , L 2 ) be a biframe and = (1 , 2 ) a strong inclusion on L. There exists a quasi-uniformity on L which induces the strong inclusion . Proof. Let B be the collection of all finite paircovers
C = (c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )
such that there exists a finite paircover
˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n) W C = (w1, w
˜ j ) (c j , c˜ j ) for j = 1, 2, . . . , n. We claim that B forms a base for a quasi-uniformity U . of L such that ( w j , w Note that if C ∈ B then {(c , c˜ ) ∈ C : c ∧ c˜ = 0} ∈ B so there will be enough strong paircovers. B is closed under finite meets: (u , u˜ ) (c , c˜ ) and ( v , v˜ ) (d, d˜ ) implies (u ∧ v , u˜ ∧ v˜ ) (c ∧ d, c˜ ∧ d˜ ). Compatibility: consider x 1 a and take x 1 s 1 t 1 a with x, s, t , a ∈ L 1 . Then C = {(a, 1), (1, x• )} ∈ B since D = {(t , 1), (1, s• )} is a paircover ofL with (t , 1) (a, 1) and (1, s• ) (1, x• ). Note that C 1 x a. Since, for each a ∈ L 1 , a = {x ∈ L 1 : x 1 a}, we also now have that a = {x ∈ L 1 : for some C ∈ U , C 1 x a}. The argument for a ∈ L 2 is similar. ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} Star-refinement condition: start with C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )} ∈ B and W C = {( w 1 , w ˜ j ) (c j , c˜ j ) for j = 1, . . . , n. For each j, find d j ∈ L 1 and d˜ j ∈ L 2 with w j 1 d j 1 c j and w ˜ j 2 d˜ j 2 c˜ j . satisfying ( w j , w ˜ j = {(1, c˜ j ), (d˜ • , 1)}. Then D and each Let D = {(d1 , d˜ 1 ), (d2 , d˜ 2 ), . . . , (dn , d˜ n )} and for each j, let D j = {(c j , 1), (1, d•j )} and D j ˜ j is a member of B . Finally, let F = D ∧ D 1 ∧ D˜ 1 ∧ · · · ∧ D n ∧ D˜ n . We show that F ∗ C . D j, D A typical pair in F has the form ˜ 1 ∧ · · · ∧ z˜ n ∧ w ˜ n) (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n , d˜ j ∧ z˜ 1 ∧ w where
(z j , z˜ j ) ∈ D j ,
˜ j ) ∈ D˜ j , (w j , w
j = 1 . . . n.
For any j, F 1 (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n ) D j1 (d j ∧ z1 ∧ w 1 ∧ · · · ∧ zn ∧ w n ) D j1 (d j ∧ z j ) c j
˜ 1 ∧ · · · ∧ z˜ n ∧ w ˜ n ) c˜ j . and similarly F 2 (d˜ j ∧ z˜ 1 ∧ w For brevity denote the strong inclusion induced by U by . We show that =. We have already seen that if a 1 b then C = {(b, 1), (1, a• )} ∈ B and then C 1 a b, so a 1 b. Similarly a 2 b ⇒ a 2 b. Suppose now that a 1 b. Then there is ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} such that C 1 a b. C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , (cn , c˜ n )} ∈ B with its associated finite W C = {( w 1 , w Then a ( W C )1 a =
˜ j ) ∈ W C and w ˜ j ∧ a = 0 1 w j : (w j , w
Similarly a 2 b ⇒ a 2 b, so = as claimed.
2
˜ j ) ∈ W C and w ˜ j ∧ a = 0 C 1 a b. c j : (w j , w
2120
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
We finally define a functor U : ProxBiFrm → QunBiFrm as follows: For ( L , ) in ProxBiFrm, let U ( L , ) = ( L , U ) where U is the quasi-uniformity constructed in Theorem 3.1. For f : ( L , L ) → ( M , M ) a morphism in ProxBiFrm, let U f : ( L , UL ) → ( M , UM ) have the same underlying map as f . We check that U f is a morphism in QunBiFrm: Suppose D ∈ UL . There exist paircovers C = {(c 1 , c˜ 1 ), (c 2 , c˜ 2 ), . . . , ˜ 1 ), ( w 2 , w ˜ 2 ), . . . , ( w n , w ˜ n )} of L such that ( w j , w ˜ j ) L (c j , c˜ j ) for j = 1, . . . , n and C D. Then (cn , c˜n )} and W C = {( w 1 , w ˜ j )) M ( f (c j ), f (˜c j )), which gives f [ D ] ∈ UM as needed. ( f ( w j ), f ( w Notice that P is a left inverse of U , that is, P U = idProxBiFrm . 4. Compactness We now characterize compactness for biframes in terms of paircovers. We then establish results needed to view CptRegBiFrm as a full subcategory of QunBiFrm. Lemma 4.1. A biframe L is compact iff every paircover of L has a finite sub-paircover.
Proof. Suppose that C is a paircover of L. Now use the fact that {c ∧ c˜ : (c , c˜ ) ∈ C } = 1 and the compactness of the total part L 0 to find a finite sub-paircover of C . Conversely, let D = {dβ : β ∈ B } be a cover of L 0 . Use the fact that each dβ can be expressed as a join of meets of elements from L 1 and L 2 to construct a paircover of L. The fact that this has a finite sub-paircover leads to the compactness of L 0 . 2 Lemma 4.2. Every compact regular biframe has a unique strong inclusion, namely the rather below relation (≺1 , ≺2 ). Proof. It is straightforward to check that (≺1 , ≺2 ) is a strong inclusion on a compact regular biframe. (See [32].) To show uniqueness, suppose that x ≺i y for some x, y ∈ L i (i = 1, 2). Then x ≺i y = that (1 , 2 ) is a strong inclusion on L and •∨ • ∨ ( z ∨ · · · ∨ z ) = 1. Then x ≺ z ∨ · · · ∨ z y, so x y. x z, x z = 1. Compactness gives a finite subjoin so 1 n n i i 1 i z i y z i y This is sufficient to show that (1 , 2 ) = (≺1 , ≺2 ). 2 Theorem 4.3. Every compact regular biframe has a unique compatible quasi-uniformity. Proof. We have already seen that every compact regular biframe has a strong inclusion (by Lemma 4.2) and hence a compatible quasi-uniformity (by Theorem 3.1). It remains to prove uniqueness. Let U be a quasi-uniformity on the compact regular biframe L. We show that any finite paircover is uniform. Then every paircover is uniform, since by compactness any paircover is refined by a finite paircover. This gives us uniqueness of U . So let C be a finite paircover of L. Let (1 , 2 ) be the unique strong inclusion on L which of course is induced by the quasi-uniformity U . Now set Z = {( z, z˜ ): z 1 c and z˜ 2 c˜ for some (c , c˜ ) ∈ C }. Z is easily seen to be a paircover of L; by compactness it has a finite sub-paircover, F = {( z1 , z˜ 1 ), ( z2 , z˜ 2 ), . . . , ( zn , z˜ n )}. Now for each j = 1, . . . , n there is (c j , c˜ j ) ∈ C j
j
such that z j 1 c j and z˜ j 2 c˜ j . Since z j 1 c j there is A j ∈ U with A 1 z j c j . Similarly there is B j ∈ U with B 2 z˜ j c˜ j . Let D ∈ U be a strong common refinement of the paircovers A 1 , A 2 , . . . , A n , B 1 , B 2 , . . . , B n . We claim that D C , and then, since D ∈ U , we have C ∈ U as needed. To see this, let (d, d˜ ) ∈ D with d ∧ d˜ = 0. There is j = 1, 2, . . . , n such that (d ∧ d˜ ) ∧ ( z j ∧ z˜ j ) = 0 since F is a paircover. Then d ∧ z˜ j = 0 and d˜ ∧ z j = 0, which implies that d D 1 z j c j and d˜ D 2 z˜ j c˜ j . So (d, d˜ ) (c j , c˜ j ) as we needed. On
the other hand, if d ∧ d˜ = 0, then (d, d˜ ) = (0, 0) since D is strong, and trivially (d, d˜ ) (c , c˜ ) for any (c , c˜ ) ∈ C .
2
Note that if f : L → M is a biframe map between compact regular biframes and U L , U M are the unique quasi-uniformities on L and M respectively, then f : ( L , U L ) → ( M , U M ) is uniform. We can therefore regard CptRegBiFrm as a full subcategory of QunBiFrm. 5. The Samuel compactification In [32], it was shown that a biframe has a strong inclusion if and only if it has a compactification. We recall here the method of constructing a compactification from a strong inclusion and will use it to construct a functor from proximal biframes to the compact regular biframes. The ideal biframe, J L, of a biframe L is J L = (J0 , J1 , J2 ) where Ji (i = 1, 2) consists of those ideals J of L 0 generated by J ∩ L i , and J0 is the subframe of the frame of all ideals of L 0 which is generated by J1 ∪ J2 . If h : L → M is a biframe map, J h : J L → J M is given by J h( J ) being the ideal generated by the image h( J ). Since h preserves first and second parts, so does J h. Now let L be a biframe with a strong inclusion . Call an ideal J in Ji (i = 1, 2) strongly regular if x ∈ J ∩ L i implies that there is y ∈ J ∩ L i with x i y. Let Ri consist of these strongly regular ideals, and let R0 be the subframe of the frame of all
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
2121
ideals of L 0 generated by R1 ∪ R2 . Then R L = (R0 , R1 , R2 ) is a compact regular biframe and the join map L : R L → L provides the required compactification of L. For the proofs of these facts, see [32]. We can now define a functor R : ProxBiFrm → CptRegBiFrm as follows: For ( L , ) in ProxBiFrm, let R ( L , ) = R L. For h : ( L , ) → ( M , ) in ProxBiFrm, let Rh : R L → R M be the restriction of J h to R L. We check that, if J ∈ (R L )i (i = 1, 2) then J h( J ) ∈ (R M )i : if a ∈ J h( J ), then a h(x) for some x ∈ J ∩ L i , so there exists y ∈ J ∩ L i with x i y. So a h(x) i h( y ), as required. Definition 5.1. Let S : QunBiFrm → CptRegBiFrm be the composite S = R P . Lemma 5.2. For any quasi-uniform biframe ( L , U ), the join map
: S (L, U ) → (L, U )
is uniform. paircover of S ( L , U ). We have then that Proof. Let C = {( J 1 , ˜J 1 ), ( J 2 , ˜J 2 ), . . . , ( J n , ˜J n )} be a finite (and thus basic) uniform ( J 1 ∧ ˜J 1 ) ∨ · · · ∨ ( J n ∧ ˜J n ) = L 0 . We show that C = {( J 1 , ˜J 1 ), . . . , ( J n , ˜J n )} ∈ U . For j = 1, . . . , n take a j ∈ J j , a˜ j ∈ ˜J j such that (a1 ∧ a˜ 1 ) ∨ · · · ∨ (an ∧ a˜ n ) = 1. Using the definition of strongly regular
ideals, obtain b j ∈ J j , b˜ j ∈ ˜J j such that a j 1 b j and a˜ j 2 b˜ j . Take a strong paircover D ∈ U such that D 1 a j b j and D 2 a˜ j b˜ j for all j. Then D 1 a j
J j and D 2 a˜ j
˜J j for all j.
We show that D C which proves that C ∈ U . To this end, take (d, d˜ ) ∈ D. If d ∧ d˜ = 0 then (d, d˜ ) = (0, 0) and so (d, d˜ ) (c , c˜ ) for all (c , c˜ ) ∈ C . If d ∧ d˜ = 0 then d ∧ d˜ ∧ a j ∧ a˜ j = 0 for some j = 1, . . . , n. Then (d, d˜ ) ( D 1 a j , D 2 a˜ j ) ˜ ( J j , J j ). 2
Note that the join map, L , viewed as an underlying biframe map gives a compactification of L. We call the map L : S ( L , U ) → ( L , U ) the Samuel compactification of ( L , U ). This is justified by the next theorem. Theorem 5.3. CptRegBiFrm is a coreflective subcategory of QunBiFrm. The coreflection functor is S and the coreflection maps are given by
: S ( L , U ) → ( L , U ).
L
Proof. Let ( L , U ) and ( M , V ) be quasi-uniform biframes, with M a compact regular biframe, and h : ( M , V ) → ( L , U ) a uniform map. Consider the diagram:
∨L
S (L, U ) Sh
(L, U )
k
∨M
S (M , V )
h
(M , V )
1 ∨− M
Note that L ◦ Sh = h ◦ M since h preserves joins. The join map M restricted to thetotal parts is a frame compactification, since any biframe compactification gives a compactification on the total part. Hence M a frame isomorphism. Also, M is onto on first and second parts, so M is a biframe isomorphism. Let k = Sh ◦
◦k =
L
−1
L
M
. Then
◦ Sh ◦
−1 M
=h◦
−1 ◦ = h. M
M
The join map L is dense, because L J = 0 implies that J = {0}. Now a regular biframe has a regular total part (see [3]) and a dense map is left-cancellable in the category of regular frames (see [30]). This fact applies here, since ( L , U ) is a quasi-uniform biframe, so ( L , U ) is a proximal biframe and so L is a regular biframe. This all goes to show that k is unique. 2 References [1] D. Baboolal, R.G. Ori, Samuel compactification and uniform coreflection of nearness frames, in: Proceedings of the Symposium on Categorical Topology, 1994, University of Cape Town, 1999.
2122
J. Frith, A. Schauerte / Topology and its Applications 156 (2009) 2116–2122
[2] B. Banaschewski, Universal zero-dimensional compactifications, in: Categorical Topology and Its Relations to Modern Analysis, Algebra and Combinatorics, Prague, 1988, World Sci. Publishing, Teaneck, NJ, 1989, pp. 257–269. [3] B. Banaschewski, G.C.L. Brümmer, K.A. Hardie, Biframes and bispaces, Quaest. Math. 6 (1983) 13–25. [4] B. Banaschewski, A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Philos. Soc. 108 (1990) 63–78. [5] B. Banaschewski, A. Pultr, Paracompactness revisited, Appl. Categ. Structures 1 (1993) 181–190. [6] G.C.L. Brümmer, H.-P.A. Künzi, Bicompletion and Samuel bicompactification, Appl. Categ. Structures 10 (3) (2002) 317–330. [7] M.J. Ferreira, J. Picado, A method of constructing compatible frame quasi-uniformities, Kyungpook Math. J. 44 (2004) 415–442. [8] P. Fletcher, W. Hunsaker, Totally bounded uniformities for frames, Topology Proc. 17 (1992) 59–69. [9] P. Fletcher, W. Hunsaker, W. Lindgren, Characterizations of frame quasi-uniformities, Quaest. Math. 16 (1993) 371–383. [10] P. Fletcher, W. Hunsaker, W. Lindgren, Totally bounded frame quasi-uniformities, Comment. Math. Univ. Carolin. 34 (3) (1993) 529–537. [11] P. Fletcher, W. Hunsaker, W. Lindgren, Frame quasi-uniformities, Monatsh. Math. 117 (1994) 223–236. [12] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982. [13] J.L. Frith, Structured frames, PhD thesis, University of Cape Town, 1987. [14] J.L. Frith, The category of uniform frames, Cah. Topol. Geom. Differ. Categ. 31 (4) (1990) 305–313. [15] J.L. Frith, The category of quasi-uniform frames, Research Reports, Department of Mathematics, University of Cape Town, 1991. [16] J. Frith, W. Hunsaker, J. Walters-Wayland, The Samuel compactification of a quasi-uniform frame, Topology Proc. 23 (1998) 115–126. [17] T.E. Gantner, R.C. Steinlage, Characterizations of quasi-uniformities, J. London Math. Soc. 5 (2) (1972) 48–52. [18] S. Hernández, Uniformly continuous mappings defined by isometries of spaces of bounded uniformly continuous functions, Houston J. Math. 29 (1) (2003) 149–155. [19] W. Hunsaker, J. Picado, Frames with transitive structures, Appl. Categ. Structures 10 (2002) 63–79. [20] J.R. Isbell, Uniform Spaces, Math. Surveys, vol. 12, Amer. Math. Soc., 1964. [21] P.T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982. [22] H.-P.A. Künzi, Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, in: C.E. Aull, R. Lowen (Eds.), Handbook of the History of General Topology, vol. 3, 2001, pp. 853–968. [23] H.-P.A. Künzi, Uniform structures in the beginning of the third millenium, Topology Appl. 154 (2007) 2745–2756. [24] I. Naidoo, Samuel compactification and uniform coreflection of nearness σ -frames, Czechoslovak Math. J. 56 (131) (2006) 1229–1241. [25] S.A. Naimpally, B.D. Warrack, Proximity Spaces, Cambridge Tracts in Math. and Math. Phys., vol. 59, Cambridge University Press, Cambridge, 1970. [26] J. Picado, Weil entourages in pointfree topology, PhD thesis, University of Coimbra, 1995. [27] J. Picado, Frame quasi-uniformities by entourages, in: Proceedings of the Symposium on Categorical Topology, 1994, University of Cape Town, 1999. [28] J. Picado, Two theorems of Efremoviˇc in pointfree context, J. Pure Appl. Algebra 141 (1999) 289–298. [29] J. Picado, Structured frames by Weil entourages, Appl. Categ. Structures 8 (2000) 351–366. [30] A. Pultr, Frames, in: M. Hazewinkel (Ed.), Handbook of Algebra, vol. 3, North-Holland, Amsterdam, 2003, pp. 791–857. [31] P. Samuel, Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1) (1948) 100–132. [32] A. Schauerte, Biframe compactifications, Comment. Math. Univ. Carolin. 34 (3) (1993) 567–574. [33] A. Schauerte, Normality for biframes, Appl. Categ. Structures 3 (1995) 1–9. [34] A. Schauerte, Notions of local compactness and smallest compactifications of biframes, Appl. Categ. Structures 14 (2006) 259–272. [35] W.J. Thron, Topological Structures, Holt, Rinehart, Winston, New York, 1966. [36] S. Vickers, Topology via Logic, Cambridge Tracts Theoret. Comput. Sci., vol. 5, Cambridge University Press, Cambridge, 1989. [37] J. Walters, Compactifications and uniformities on sigma frames, Comment. Math. Univ. Carolin. 32 (1) (1991) 189–198.