Bicompletion and Samuel bicompactification
Bicompletion and Samuel bicompactification G.C.L. Br¨ ummer Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa. E-mail: [email protected]
H.-P.A K¨ unzi Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa. E-mail: [email protected] Abstract. It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T0 -space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top0 of topological T0 -spaces and continuous maps is not lower K-true (in the sense of Br¨ ummer). It is also shown that a functorial quasi-uniformity F on Top0 is upper K-true if and only if F X is bicomplete whenever X is sober. Keywords: totally bounded, quasi-proximity, quasi-uniformity, Samuel bicompactification, bicompletion, functorial quasi-uniformity, lower K-true, upper K-true, sober. AMS (2000) Subject Classifications: 54B30, 54E05, 54E15, 54D35 :
To Professor Heinrich Kleisli in honour of his seventieth birthday Introduction The analogue of the first result mentioned in the Abstract is well known in the setting of uniform spaces (see [6] p. 239). However we could not find the corresponding fact discussed in the literature dealing with quasi-uniform spaces. It is one of the aims of this note to present a proof of that result in the realm of quasi-uniform spaces. The theorem will then be used to prove that the locally finite covering quasi-uniformity ummer [3], [4]) on the category is not lower K-true (in the sense of Br¨ Top0 of topological T0 -spaces and continuous maps. Here K denotes the bicompletion functor in the category of quasi-uniform T0 -spaces. For functorial covering quasi-uniformities in general we shall present a simple sufficient condition, in terms of the natural coverings, which will ensure that the corresponding functor is lower K-true. Along the way we shall prove that a functorial quasi-uniformity F operating on Top0 is upper K-true if and only if F X is bicomplete whenever X is sober, or equivalently, if and only if those spaces X for which F X is bicomplete form an epireflective subcategory of Top0 .
c 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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For basic facts about quasi-uniformities we refer the reader to [9]. Instead of the usual five-character notation (X, V) for a quasi-uniform space, we shall generally use a single character such as X, and then denote the filter V of entourages by entX. When really necessary we shall denote the point set of the space X by |X|, and otherwise indulge in a statement such as x ∈ X instead of x ∈ |X|. Likewise a topological space Y will have the notation OY for its family of open sets, and again |Y | for its point set. QU0 will denote the category of quasi-uniform T0 -spaces with the usual mappings which one may call quasi-uniform (q.u.). The forgetful functor T = QU0 → Top0 assigns to every space X in QU0 a topological T0 -space T X in which the neighbourhood system at any point x is {H(x) : H ∈ entX}, where H(x) = {y ∈ X|(x, y) ∈ H}. By a T -section we shall mean a functor F : Top0 → QU0 such that T F is the identity functor on Top0 . Such an F imposes compatible quasiuniformities on the topological T0 -spaces, and accordingly a T -section is also called a functorial quasi-uniformity on Top0 . (For examples of such functors see Section 2 and [9], [2], [11], [4].) We shall denote by SX the symmetrization (= uniform coreflection) of a quasi-uniform space X. Thus SX is the coarsest uniform space finer than X. We view S as an endofunctor S : QU0 → QU0 , the symmetrizer. A quasi-uniform mapping f : X → Y is called sup-dense if the image set f [X] is dense in the topological space T SY . This term stems from regarding the topology of T SY as the supremum of the topologies of T Y and T (Y −1 ), where Y −1 has the entourages H −1 (H ∈ entY ). A non-trivial folklore result is: 0.1. The epimorphisms in QU0 are precisely the sup-dense quasiuniform mappings. For a proof see [7]. 1. The Samuel bicompactification in QU0 . We recall that a quasi-uniform T0 -space X is called bicomplete [9] if the symmetrization SX is a complete uniform space. By a bicompletion of a space X in QU0 we mean a sup-dense quasi-uniform embedding f : X → Y with a Y a T0 bicomplete quasi-uniform space. For any two bicompletions f1 : X → Y1 and f2 : X → Y2 there exists a (unique) quasi-uniform isomorphism h : Y1 → Y2 with h ◦ f1 = f2 (see [9], Theorem 3.34. This result is known as the “Uniqueness of bicompletion”.) A particular bicompletion for any X in QU0 , which
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we shall denote by kX : X → KX, is constructed as follows in [9], Theorem 3.33 : The points of KX are the minimal Cauchy filters of the uniform e : U ∈ entX}, where space SX, and entKX has the base {U e = {(F, G) ∈ |KX| × |KX| : there exist F ∈ F and U G ∈ G such that F × G ⊆ U };
while the sup-dense quasi-uniform embedding kX : X → KX is given, for any x ∈ X, by kX (x) being the neighbourhood filter of x in T SX. The bicompletion has the following reflection property : Given any g : X → A in QU0 with A bicomplete, there exists a unique quasiuniform mapping gb : KX → A with gb ◦ kX = g (see [9], Theorem 3.29). It follows that KX is given by an endofunctor K : QU0 → QU0 . Moreover, kX is given by a natural transformation k : 1 → K, which means that for any f : X → Y in QU0 the diagram (1) commutes. f
X (1)
kX
KX
Kf
/Y
kY
/ KY
It is well known (and obvious from the above) that the bicompletion commutes with the symmetrizer, i.e. SKX = KSX for each X ∈ QU0 . The following two simple results, implicit in [5], (Remark 1.17), are not so well known, and are so useful and fundamental that we give explicit proofs. PROPOSITION 1.1 ([5]). Let f : X → Y be a morphism in QU0 . Then f is a sup-dense quasi-uniform embedding if and only if Kf : KX → KY is a quasi-uniform isomorphism. Proof. Let f in Diagram (1) be a sup-dense quasi-uniform embedding. Then kY ◦f : X → KY is a bicompletion of X. By the cited uniqueness of bicompletions there exists a q.u. isomorphism h : KX → KY with h ◦ kX = kY ◦ f , and thus h ◦ kX = (Kf ) ◦ kX . Since kX is epi, h = Kf , so that Kf is a quasi-uniform iso. Conversely, given f in Diagram (1) with Kf iso, it is clear that kY ◦ f is a sup-dense q.u. embedding. Being a right-hand factor of a q.u. embedding, f is a q.u. embedding. Thinking of the underlying topologies of SX, SY and SKY , for which f and kY are topological embeddings with kY ◦ f dense, one sees at once that f is dense into SY , hence a sup-dense q.u. embedding.
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COROLLARY 1.2 ([5]). Let f : X → Y and g : Y → Z be morphisms in QU0 such that g◦f : X → Z is a sup-dense quasi-uniform embedding. Then, f is a sup-dense quasi-uniform embedding if and only if g is one. Proof. By the previous result, Kg ◦ Kf = K(g ◦ f ) is an iso. Thus Kf is iso iff Kg is iso. A quasi-uniform space Y is called totally bounded iff SY is a totally bounded uniform space. For any quasi-uniform space (X, U ) there exists a finest totally bounded quasi-uniformity coarser than U, denoted by Uω in [9] and in much of the literature. Moreover, U and Uω induce the same topology (in fact the same bitopology). We shall denote (X, Uω ) by BX and then write bX : X → BX for the quasi-uniform mapping which is the identity function on the point set of X. Clearly this is a reflection to the totally bounded quasi-uniform spaces. We only need this on QU0 , so we have an endofunctor B : QU0 → QU0 and a natural transformation b : 1 → B. We note that T B = T and SB = BS. The bicompletion of a totally bounded T0 quasi-uniform space is totally bounded. This follows at once from the corresponding uniform result by the commutation of a bicompletion and symmetrization. Clearly also, a space Y in QU0 is bicomplete and totally bounded if and only if Y is sup-compact — that is, SY has compact topology. The Samuel bicompactification of X in QU0 is the following composite quasi-uniform mapping :
X
bX
/ BX kBX / KBX
This is clearly the reflection to the sup-compact spaces in QU0 . For brevity, let us for the moment denote the composite mapping by λX : X → KBX. We note that λX is a quasi-uniform embedding iff X is totally bounded. However, λX is always a topological embedding for the underlying sup-topologies and for the underlying topologies. The functors K and B do not commute (there are obvious examples even in Unif0 ), but there is the following quasi-uniform embedding in one direction. PROPOSITION 1.3. For X in QU0 there is a natural sup-dense quasiuniform embedding eX : BKX → KBX given by the commutative diagram (2).
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(a)
kX
(2)
bX
KX
5
/ BX
BkX
/ BKX ?
bKX
(b)
KbX
(c)
?
?
?
kBX
?
? eX ? ?
?
? *
?
?
KBX
Proof. The square (a) in Diagram (2) commutes by naturality of b. Since KBX is totally bounded, the reflection property of b at KX provides a unique q.u. morphism eX : BKX → KBX such that the triangle (b) commutes. To see that the triangle (c) commutes, we consider: eX ◦ BkX ◦ bX = eX ◦ bKX ◦ kX = KbX ◦ kX = kBX ◦ bX in which the last equality follows by naturality of k. The surjection bX is right-cancellable, giving eX ◦ BkX = kBX , so that (c) commutes. Moreover in (c), BkX is the right factor of the q.u. embedding kBX , hence a q.u. embedding, whereas in (a), BkX is a left factor of the sup-dense mapping bKX ◦ kX , so that BkX is sup-dense. Application of Corollary 1.2 to (c) now shows that eX is a sup-dense q.u. embedding. Naturality of e : BK → KB is a routine exercise. We shall sharpen Proposition 1.3 to the result that, literally, BKX is a quasi-uniform subspace of KBX. This is only true by virtue of the particular constructions used for the two functors involved, that is, for K in terms of minimal Cauchy filters and for B so as to preserve underlying sets. However, the literal inclusion will be convenient for the verification of our main example in Section 3. For this we still need a lemma. LEMMA 1.4. For every quasi-uniform space X, every minimal Cauchy filter on SX is a minimal Cauchy filter on SBX.
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Proof. Since SBX = BSX, the Lemma reduces to the assertion that every minimal Cauchy filter on any uniform space Y is minimal Cauchy on the totally bounded reflection BY . Now it is clear that any Cauchy filter F on Y is minimal Cauchy on Y if any only if F is a round filter with respect to the induced proximity relation δY . We apply the same observation to BY , noting that F Cauchy on Y implies F Cauchy on BY , and that δBY = δY . The assertion follows. (This argument is hinted at in [6], p. 239.) THEOREM 1.5. For any quasi-uniform T0 -space X, the space BKX is a sup-dense quasi-uniform subspace of the Samuel bicompactification KBX. (This is true subject to the given constructions of the functors K and B.) Proof. We consider the commutative triangle (b) in Diagram 2. The underlying point sets |KX| and |BKX| coincide, with bKX the identity function between these. Thus eX and KbX are the same function between point sets. By construction of K and by Lemma 1.4, eX is the inclusion function for |BKX| ⊆ |KBX|. The theorem now follows from Proposition 1.3. REMARKS 1.6. (1) The reader will have noticed that Proposition 1.3 is independent of any particular construction of the functors K and B. (2) All functorial quasi-uniformities on a given T0 topological space Y have the same codomain for their Samuel bicompactifications. Indeed, let F : Top0 → QU0 be any T -section. It is well known that BF = P, where P : Top0 → QU0 is the functor which imposes the Cs´asz´ ar-Pervin quasi-uniformity, i.e. the coarsest functorial quasiuniformity, on the T0 topological spaces. Thus the codomain of the Samuel bicompactification of F Y is KPY . (Compare [4], where P is denoted by C1∗ .) (3) For the categorically minded reader, the following may be of interest. In the terminology of [5], Proposition 1.1 says that the sup-dense q.u. embeddings form a firm class of morphisms in QU0 . Knowing that K reflects to the T0 bicomplete spaces, one then has from [5] that these spaces are precisely the injective objects in QU0 with respect to the class of sup-dense q.u. embeddings. (One can see this directly : That each T0 bicomplete space is such an injective, is precisely the content of Theorem 3.29 of [9]. Conversely, if Y is such an injective, there exists g : KY → Y such that g ◦ kY = 1Y . Since kY is epi, this implies that kY is iso, so that Y is T0 bicomplete.) Finally, Corollary 1.2 says that the class of sup-dense q.u. embeddings is both an essential and a coessential class; this in fact is true of any firm class (see e.g. [5]).
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2. New results on upper K-true functorial quasi-uniformities We consider a given T -section F : Top0 → QU0 and any space X in Top0 . The T0 quasi-uniform space F X has the bicompletion kF X : F X → KF X which, we recall, is a quasi-uniform embedding, epimorphic in QU0 . Since T F X = X, application of the forgetful functor T gives a dense topological embedding T kF X : X → T KF X, which for brevity we denote by rX : X → RX. Thus we have introduced an endofunctor: R = T KF : Top0 → Top0 and a natural transformation:
r = T kF : 1 → R given by rX = T kF X . We agree that these will be standing notations when any given T section F is considered. The following notation is standard: Fix(r) = {X ∈ Top0 : rX : X → RX is a homeomorphism } = {X ∈ Top0 : F X is bicomplete } and we shall regard this class of objects as a full subcategory of Top0 . The following are samples of the many basic questions that arise: (1) Is Fix(r) an epireflective subcategory of Top0 ? (2) Is rX an epimorphism in Top0 (i.e. Skula-dense or b-dense) for each X in Top0 ? (3) Is rX : X → RX a reflection to Fix(r) for every X in Top0 ? (4) Do the functors KF, F R : Top0 → QU0 coincide, i.e., is KF = F T KF ? Each of these four questions has a negative answer for some T sections F , and an affirmative answer for others. Most of the known published results around these questions are surveyed in [4]. To prove our new results, we have to define the key concepts. First we recall, for quasi-uniform spaces A and B, that one says A is coarser (finer) than B if the underlying sets are the same and entA ⊆ entB (entB ⊆ entA).
DEFINITION 2.1 ([3], [4]). For a T -section F : Top0 → QU0 we say that F is lower (upper) K-true if KF X is coarser (finer) than F T KF X for each X in Top0 . We say that F is bicompletion-true (more briefly, K-true) if F is both upper and lower K-true, that is, if KF = F T KF .
The above definition makes sense because the functor F T preserves the underlying point set (in fact also the underlying topological space)
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of KF X. Thus the equation KF = F T KF should be read as an actual equality of functors, not merely an isomorphism. As shown in [4], affirmative answers to the above four questions obey the following implications: (4) =⇒ (3) =⇒ (2) =⇒ (1); (2) ⇐⇒ F is upper K-true. An example by Z. Kimmie [10] (pp. 73-77) shows that (3) 6=⇒ (4). Thus, Kimmie’s example is upper K-true but not lower K-true. The 6 ⇒ (3). Below, present authors will show in a further paper that (2) = in Theorem 2.4, we shall see that (1) ⇐⇒ (2), that is, F is upper K-true ⇐⇒ Fix(T kF ) is epireflective in Top0 , thereby answering a question raised in [4] (Remark 4.11). (More characterizations of upper K-trueness are given in [4].) An open cover A of a topological space X is called well-monotone if A is well-ordered by the partial order of set inclusion. The wellmonotone quasi-uniformity on X has a subbase consisting of all binary relations of the form EA , defined by (∀p ∈ X)(EA (p) = ∩{G ∈ A : p ∈ G}), with A running through the well-monotone open covers of X. This quasi-uniformity is compatible on each X and is functorial ([3], [11]) and thus defines a T -section which we denote by W : Top0 → QU0 .
THEOREM 2.2 ([3], [11]). Let F : Top0 → QU0 be any T -section. Then, F is upper K-true ⇐⇒ F is finer than W .
(The implication “⇐=” was proved in [3], using results from [12], and the surprising “=⇒” was proved in [11].) EXAMPLES 2.3. (1) The functor W is K-true and Fix(T kW ) = Sob, the category of sober T0 -spaces ([12], Proposition 4 and Corollary 2). (2) The fine transitive quasi-uniformity, operating on Top0 , is Ktrue ([3], Example 6.4(4) or [4], Example 3.8). Descriptions of its Fix(r) are given in [12] (Corollary 2) and [10] (Remark 2.4.12). (3) The fine quasi-uniformity, operating on Top0 , is K-true ([12], Corollary 5). We do not know a complete characterization of its Fix(r). For the proof of Theorem 2.4 below we recall that the Skula topology (or b-topology, [13]) on a topological space X is generated by the open and the closed sets of X. It is obvious that the symmetrization of the Cs´az´ ar-Pervin quasi-uniformity on X carries the Skula topology of X. It then follows (e.g. as in [12], Lemma 2 or [4], p. 68) that for any functorial quasi-uniformity F the symmetrization of F X (for X in Top0 ) also carries the Skula topology of X.
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THEOREM 2.4. The following conditions are equivalent for a functorial quasi-uniformity F defined on the category Top0 of topological T0 -spaces and continuous maps: (a) F is upper K-true. (b) {X : F X is bicomplete} is closed under topological products and Skula-closed subspaces (equivalently, Fix(T kF ) is epireflective in Top0 – (see [1], Corollary 16.9). (c) F X is bicomplete whenever X is sober (that is, Sob ⊆ Fix(T kF )). (d) For each (infinite) regular cardinal α the space Xα defined by equipping the carrier set α with the lower topology {[0, β[: β ∈ α} ∪ {α} carries the well-monotone quasi-uniformity under F . Proof. (a) implies (b): See [4] (Proposition 4.2); for the application of [1] (Corollary 16.9) one also has to know that in Top0 the epimorphisms are the Skula-dense maps and the extremal subobjects are the Skulaclosed embeddings. (b) implies (c): Obviously the unique compatible quasi-uniformity of the Sierpi´ nski space is bicomplete. Since each sober space is a Skulaclosed subspace of a product of Sierpi´ nski spaces ([14]), the assertion follows from the hypothesis. (c) implies (d): One easily sees that the well-monotone quasi-uniformity coincides with the fine quasi-uniformity on Xα (for every ordinal α). The negation of (d) means that there is an (infinite) regular cardinal α such that F Xα does not carry the well-monotone (= fine) quasiuniformity. Note that Xα is sober, since it is a T0 -space in which the only nonempty closed sets are point-closures. Assuming (c) we thus have F Xα bicomplete. On the other hand, it is shown in [11] (Appendix) that for an (infinite) regular cardinal α, F Xα being strictly coarser than W Xα implies that the filter F generated by {[β, → [: β ∈ α} is a Cauchy filter on the symmetrization of F Xα . Since F does not contain a singleton, it cannot converge with respect to the discrete Skula topology on Xα . Thus F Xα is not bicomplete – a contradiction. We deduce that F Xα = W Xα for any (infinite) regular cardinal α. (d) implies (a): The assertion follows from the argument given in the last paragraph of the Appendix of [11]. REMARK 2.5. The condition (b) in Theorem 2.4 above is equivalent to the following version: (b0 ) Fix(T kF ) is embedding-reflective in Top0 . Clearly (b0 ) implies (b) (see [1], Proposition 6.3). However, the implication (b) =⇒ (b0 ) is not as trivial as it may seem, because the
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reflection morphisms into Fix(T kF ) need not be the embeddings of the form rX = T kF X (see our discussion following Definition 2.1, where we 6 ⇒ (3)”). Let us denote the reflection of any T0 -space X into said “(2) = Fix(T kF ) by ρX and the sobrification of X by σX : X → ΣX. Since (b) =⇒ (c) in Theorem 2.4, ΣX ∈ Fix(T kF ) and so ρX is a factor of σX . Since σX is known to be an embedding, so is ρX . 3. Transitivity and lower K-true functorial quasi-uniformities We recall a result due to P. Fletcher ([8]; [9], p. 29) which we have already once used in defining the well-monotone quasi-uniformity. Let C be a family of covers of a topological space X. For any A ∈ C denote by EA the transitive reflexive relation on X given by EA (p) = ∩{A ∈ A : p ∈ A} for all p ∈ X. Then (see [2], Proposition 2.7) {EA : A ∈ C} is a subbase for a compatible (transitive) quasi-uniformity on X if and only if C is a set of interior-preserving open covers of X and {EA (p) : A ∈ C and p ∈ X} is a subbase for OX (the topology of X). The transitive quasi-uniformity with subbase {EA : A ∈ C} is called the (covering) quasi-uniformity induced by C. We call the procedure the Fletcher construction and the relations EA the Fletcher entourages. For any compatible transitive quasi-uniformity V on any topological space X there exsits a family C of covers of X which induces V via the Fletcher construction ([8], [9]). As noted above, these covers are automatically interior-preserving and open. There is a functorial version of the Fletcher construction which accounts for all transitive-valued sections of the forgetful functor QU → Top, as follows. DEFINITION 3.1. ([2]). A natural kind of covers (of the topological spaces) is a function Γ = (Γ(X) : X ∈ Top) such that: (1) Γ(X) is a set of covers of X, for each X in Top; (2) Whenever f : X → Y is continuous and B ∈ Γ(Y ), then f −1 [B] ∈ Γ(X). We shall call a natural kind Γ of covers admissible if, for each X in Top, the covering quasi-uniformity induced by Γ(X) is compatible on X. Under the Fletcher construction, an admissible natural kind Γ of covers of the topological spaces induces a functor F : Top → QU which is a section of the forgetful functor QU → Top, and such that each F X is a transitive quasi-uniform space.
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The naturality condition (2) ensures functoriality: every continuous map f : X → Y becomes a quasi-uniform morphism for the induced quasi-uniformities, because one has (f × f )−1 [EB ] = Ef −1 [B] for all B ∈ Γ(Y ). Conversely, any functorial transitive quasi-uniformity F on Top (that is, transitive-valued section of the forgetful functor QU → Top) is induced by some natural (necessarily admissible) kind Γ of covers; there exists a largest natural kind Γ that does this, given by Γ(X) = {A : A covers X and EA ∈ entF X} for each X in Top (see [2], p. 169). (It is immaterial whether one restricts these results to the T0 -spaces.) Many familiar kinds of interior-preserving open covers are natural and admissible, e.g. the finite, the locally finite, the point-finite and the well-monotone open covers; the open spectra (which induce the semicontinuous quasi-uniformity, [9]); and the kind of all interior-preserving open covers (inducing the fine transitive quasi-uniformity). Some kinds of covers lend their names to the induced T -sections, often in abbreviated form, e.g. locally finite (open covering) quasi-uniformity. One notes that the finite open covers induce the Cs´asz´ar-Pervin quasi-uniformity. The proof of the next result uses our Theorem 1.5. EXAMPLE 3.2. The locally finite covering quasi-uniformity functor on Top0 is not lower K-true. Proof. For brevity we denote the above functor by L: Top0 → QU0 . For any X in Top0 , the Fletcher entourages EA determined by all locally finite open covers A of X form a base for the quasi-uniformity entLX. (The family is indeed a base, because if the covers A and B are locally finite open, so is A ∧ B = {A ∩ B : A ∈ A and B ∈ B}, and EA ∩ EB = EA∧B — see e.g. [2], Lemma 2.5.) To show that the functor L is not lower K-true it will suffice to find a topological T0 -space X such that entKLX is not contained in entLT KLX. As point set for the space X we take ω × {0, 1} where ω denotes the first infinite ordinal. By [0, n] resp. [n, → [ for n ∈ ω we denote the obvious intervals of ω. We let the topology of X be generated by the base {{(n, 0)} : n ∈ ω} ∪ {[0, n] × {0, 1} : n ∈ ω}. Thus X is a topological T0 -space. We denote by F the filter on X generated by the filter base {[n, → [×{1} : n ∈ ω}. We note that the members of this base are closed sets of X and that every non-void closed set of X is an element of F. We claim that F is a minimal Cauchy filter on the uniform space SLX (recalling that S denotes the symmetrizer).
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−1 For the uniformity of SLX we have the basic entourages EA ∩ EA where A ranges through all locally finite open covers of X. To see that F −1 )(p) ∈ is Cauchy it will suffice to find a point p ∈ X such that (EA ∩EA −1 (p) F. Now because EA is transitive, it is clear that for any p ∈ X, EA is closed in X, and therefore an element of F. Observing that the point (0, 1) ∈ X lies in just finitely many members of A, we conclude from the definition of the topology that X ∈ A and there exists m ∈ ω such that X is the only member of A that contains the point (m, 1). Then −1 EA ((m, 1)) = X. Thus (EA ∩ EA )((m, 1)) ∈ F, and F is Cauchy on SLX. We have a filter base of closed sets for F. For any such set F , G = X \ F is open, and the Pervin entourage SG = E{G,X} is a member of −1 entLX. Then SG ∩ SG = (F × F ) ∪ ((X \ F ) × (X \ F )) and (SG ∩ −1 SG )(F ) = F . Clearly then F is generated by the filter base {U (F ) : F ∈ F and U ∈ entSLX}. Thus by [9] (Proposition 3.30) F is a minimal Cauchy filter on SLX. Hence F is a point of KLX. Note next that C = {X} ∪ {{(n, 0)} : n ∈ ω} is a locally finite open cover of X. Thus we have the Fletcher entourage EC ∈ entLX. Recalling the notation used in constructing basic entourages in KLX fC ∈ (first part of Section 1, with notation from [9]) we have a set E entKLX. The proof that L is not lower K-true will be finished by fC 6∈ entLT KLX. showing that E Recall that the q.u. embedding kLX : LX → KLX sends any point p of X to the neighbourhood filter of p in T SLX. In particular, when p is the isolated point (n, 0) of X, one has
kLX ((n, 0)) = {A ⊆ |X| : (n, 0) ∈ A} for all n ∈ ω. We claim that, for all n ∈ ω, fC (kLX ((n, 0))) = {kLX ((n, 0))}. E
To see this, note that fC = {(H, H0 ) ∈ |KLX|2 : (∃ H ∈ H)(∃ H 0 ∈ H0 )(H × H 0 ⊆ EC )}. E fC (kLX ((n, 0))). Since {(n, 0)} ∈ kLX ((n, 0)), there Consider any H0 ∈ E 0 0 exists H ∈ H such that {(n, 0)} × H 0 ⊆ EC , and then H 0 ⊆ EC ((n, 0)) =
\
{C : (n, 0) ∈ C ∈ C} ⊆ {(n, 0)}
so that in fact H 0 = {(n, 0)}. Then, since H0 is a filter on |X| and H 0 ∈ H0 , one has H0 = kLX ((n, 0)), and the above claim is proved. By Theorem 1.5 and Remark 1.6(2), BKLX is a q.u. subspace of KBLX and moreover KBLX = KPX, where P : Top0 → QU0 is the Cs´asz´ ar-Pervin T -section. Applying the forgetful functor T and
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using the previously noted fact that T B = T , we see that T KLX is a topological subspace of T KPX. We return to the filter F generated by the non-void closed sets of X. We showed that F ∈ |KLX|. Thus F ∈ |KPX|. For any open set G of X we have SG ∈ entPX and thus (with obvious notation) (SG )∼ P ∈ entKPX, where 2 (SG )∼ P = {(M, N ) ∈ |KPX| : (∃ M ∈ M)(∃ N ∈ N )(M × N ⊆ SG )}.
We claim:
(SG )∼ P (F) = |KPX|.
It suffices to prove (∀N ∈ KPX)(∃ M ∈ F)(∃ N ∈ N )(M × N ⊆ SG ). Case 1 : G 6= X ; then X \ G ∈ F ; taking M = X \ G and N = X we have M × N ⊆ SG . Case 2 : G = X ; we take M = N = X, and the claim is proved. Noting that the sets SG (G open in X) form a subbase for entPX and that the operator (−)∼ P preserves finite intersections, we see that the ∼ form a subbase for entKPX. Since (S )∼ (F) = |KPX|, sets (SG )P G P it follows that |KPX| is the only neighbourhood of F in T KPX. Since F is also a point of the subspace T KLX, |KLX| is the only neighbourhood of F in T KLX. fC ∈ entLT KLX. To end the proof of our Example we assume that E fC Then there exists a locally finite open cover E of T KLX with EE ⊆ E (since, as noted at the beginning, the Fletcher entourages form a base for the locally finite covering quasi-uniformity). We have also seen that fC (kLX ((n, 0))) = {kLX ((n, 0))}, and so it follows from EE ⊆ E fC that E EE (kLX ((n, 0))) = {kLX ((n, 0))} for each n ∈ ω. Since the map kLX is injective, it follows that {EE (H) : H ∈ |KLX|} is an infinite collection. This collection is also a locally finite (open) cover of T KLX, as proved in ([2], Example 2.16 (2)). But this contradicts the fact that the point F has just one neighbourhood in T KLX (namely the whole space). We have thus proved that L fails to be lower K-true. A class A of (T0 ) quasi-uniform spaces is said to span a T -section F if, for each X in Top0 , F X has the initial quasi-uniformity for the class of all continuous maps from X into the spaces T A (A ∈ A) (see e.g. [4] or [3]). THEOREM 3.3 ([3]). A T -section F : Top0 → QU0 is lower K-true if and only if F is spanned by some class of bicomplete T0 quasi-uniform spaces. By virtue of this result, the Cs´asz´ar-Pervin quasi-uniformity and the semicontinuous quasi-uniformity are both lower K-true, since each
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can be spanned by a single bicomplete T0 quasi-uniform space (the Sierpi´ nski space with its unique quasi-uniformity, respectively the real line with the lower quasi-uniformity). Both functors fail to be upper K-true, by Theorem 2.2. REMARK 3.4. It can be proved that the sections of the forgetful functor QU → Top stand in a bijective correspondence with the sections of our functor T : QU0 → Top0 . The correspondence is given by restriction, its inverse being unique extension. Moreover, if a functorial quasi-uniformity G : Top → QU is spanned by a class B of bicomplete, not necessarily T0 , quasi-uniform spaces, then G is also spanned by the class of quasi-uniform T0 -reflections of the spaces in B. Thus the restriction of G to Top0 will be lower K-true by Theorem 3.3. PROPOSITION 3.5. Let F : Top0 → QU0 be a transitive T -section induced (via the Fletcher construction) by an admissible natural kind Γ of interior-preserving open covers of the topological spaces. For every X in Top0 and every A ∈ Γ(X) assume that A ∈ Γ(XA ), where XA denotes the topological space having the same point set as XTand such that each point p has as smallest XA -neighbourhood the set {A : p ∈ A ∈ A}. Then F is lower K-true.
Proof. Consider X ∈ Top and A ∈ Γ(X). For the Fletcher entourage T 0 EA one has EA (p) = {A : p ∈ A ∈ A} for every p ∈ X. Thus XA has a compatible quasi-uniformity generated by the single transitive entourage EA . Let Xq,A denote the quasi-uniform space thus obtained, so that T (Xq,A ) = XA . The symmetrization SXq,A has the uniformity −1 generated by the single equivalence relation EA ∩EA and is therefore a complete uniform space. Indeed, for any Cauchy filter F on SXq,A there −1 −1 exists p ∈ |X| such that (EA ∩ EA )(p) ∩ EA )(p) ∈ F , and since (EA is the smallest neighbourhood of p in T (SXq,A ), it is clear that F converges to p in the latter space. Thus Xq,A is bicomplete, but not necessarily T0 . Let B = {Xq,A : X ∈ Top0 and A ∈ Γ(X)} and let G : Top → QU denote the functor spanned by B. We shall show that G coincides with F on Top0 . Consider any Y in Top0 . The spanning of G means that entGY has a subbase consisting of all sets (g × g)−1 EA using all continuous maps g : Y → XA , X ∈ Top0 and A ∈ Γ(X). Now by hypothesis, A ∈ Γ(X) implies A ∈ Γ(XA ); then g −1 [A] ∈ Γ(Y ) by naturality of Γ, and then Eg−1 [A] ∈ entF Y by the Fletcher construction of F Y . Since Eg−1 [A] = (g × g)−1 EA we thus have entGY ⊆ entF Y . On the other hand, since entF Y has the subbase {EB : B ∈ Γ(Y )}, Yq,B is coarser than F Y , and then YB = T (Yq,B ) is coarser than T (F Y ) = Y , so that we have a continuous identity function i : Y → YB for each B ∈ Γ(Y ). By the spanning construction for G the map i lifts
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to a quasi-uniform morphism GY → Yq,B . Then EB ∈ entYq,B implies EB ∈ entGY , so that entF Y ⊆ entGY . Thus indeed F and G coincide on Top0 . Since B is a class of bicomplete spaces it now follows from Remark 3.4 that F is lower K-true. EXAMPLES 3.6. The sufficient condition on Γ in Proposition 3.5, namely (∀X ∈ Top0 )(∀A ∈ Γ(X))(A ∈ Γ(XA )),
is satisfied by the following admissible natural kinds of interior-preserving open covers: The finite; the point-finite; the well-monotone; the open spectra; and the kind of all interior-preserving open covers. Consequently the corresponding induced transitive functorial quasi-uniformities on Top0 are lower K-true. In the case of the point-finite covering quasi-uniformity this fact is new (to our knowledge). — We do not know whether the above sufficient condition is in some sense necessary. EXAMPLES 3.7. On Top0 , the point-finite covering quasi-uniformity is lower but not upper K-true, and the locally finite covering quasiuniformity is neither lower nor upper K-true. Proof. By Theorem 2.2, since the locally finite T -section is coarser than the point-finite one, it only remains to show that the point-finite T -section is not finer than the well-monotone T -section W . Let ωd be the ordinal ω equipped with the topology consisting of all downsets. The open cover A = {[0, n] : n ∈ ω} is well-monotone. Thus EA ∈ entW (ωd ). Observing that EA is the graph of the usual order on ω one sees at once that EA does not belong to the point-finite quasi-uniformity on ωd .
References 1. 2.
3.
4.
5.
J. Ad´ amek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, Wiley, New York etc., 1990. G.C.L. Br¨ ummer, Functorial transitive quasi-uniformities, In: H.L. Bentley et al., ed., Categorical Topology (Proc. Conf. Toledo, Ohio, 1983), Heldermann Verlag, Berlin, 1984, pp. 163–184. G.C.L. Br¨ ummer, Completions of functorial topological structures, In: W. G¨ ahler et al., ed., Recent Developments of General Topology and its Applications (Proc. Conf. Berlin, 1992), Akademie Verlag, Berlin, 1992, pp. 60–71. G.C.L. Br¨ ummer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste 30 Suppl. (1999), 45–74. Free online at:http://mathsun1.univ.trieste.it/Rendiconti/ G.C.L. Br¨ ummer and E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolinae 33 (1992), 131–147.
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6. 7. 8. 9. 10. 11. 12. 13. 14.
S.C. Carlson, Completely uniformizable proximity spaces, Topology Proc. 10 (1985), 237–250. D. Dikranjan and H.-P. K¨ unzi, Separation and epimorphisms in quasiuniform spaces, Appl. Categ. Structures 8 (2000), 175–207. P. Fletcher, On completeness of quasi-uniform spaces, Arch. Math. (Basel) 22 (1971), 200-204. P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982. Z. Kimmie, Functorial transitive quasi-uniformities and their bicompletions, PhD thesis, Univ. Cape Town, 1995. H.-P. A. K¨ unzi, Quasi-uniform spaces — eleven years later, Topology Proc. 18 (1993), 143–171. H.-P. A. K¨ unzi and N. Ferrario, Bicompleteness of the fine quasi-uniformity, Math. Proc. Cambridge Philos. Soc. 109 (1991), 167–186. L. Skula, On a reflective subcategory of the category of all topological spaces, Trans. Amer. Math. Soc. 142 (1969), 37–41. L.D. Nel and R.G. Wilson, Epireflections in the category of T0 -spaces, Fund. Math. 75 (1972), 69–74.
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H.-P.A K¨ unzi Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa. E-mail: [email protected] Abstract. It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T0 -space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top0 of topological T0 -spaces and continuous maps is not lower K-true (in the sense of Br¨ ummer). It is also shown that a functorial quasi-uniformity F on Top0 is upper K-true if and only if F X is bicomplete whenever X is sober. Keywords: totally bounded, quasi-proximity, quasi-uniformity, Samuel bicompactification, bicompletion, functorial quasi-uniformity, lower K-true, upper K-true, sober. AMS (2000) Subject Classifications: 54B30, 54E05, 54E15, 54D35 :
To Professor Heinrich Kleisli in honour of his seventieth birthday Introduction The analogue of the first result mentioned in the Abstract is well known in the setting of uniform spaces (see [6] p. 239). However we could not find the corresponding fact discussed in the literature dealing with quasi-uniform spaces. It is one of the aims of this note to present a proof of that result in the realm of quasi-uniform spaces. The theorem will then be used to prove that the locally finite covering quasi-uniformity ummer [3], [4]) on the category is not lower K-true (in the sense of Br¨ Top0 of topological T0 -spaces and continuous maps. Here K denotes the bicompletion functor in the category of quasi-uniform T0 -spaces. For functorial covering quasi-uniformities in general we shall present a simple sufficient condition, in terms of the natural coverings, which will ensure that the corresponding functor is lower K-true. Along the way we shall prove that a functorial quasi-uniformity F operating on Top0 is upper K-true if and only if F X is bicomplete whenever X is sober, or equivalently, if and only if those spaces X for which F X is bicomplete form an epireflective subcategory of Top0 .
c 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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For basic facts about quasi-uniformities we refer the reader to [9]. Instead of the usual five-character notation (X, V) for a quasi-uniform space, we shall generally use a single character such as X, and then denote the filter V of entourages by entX. When really necessary we shall denote the point set of the space X by |X|, and otherwise indulge in a statement such as x ∈ X instead of x ∈ |X|. Likewise a topological space Y will have the notation OY for its family of open sets, and again |Y | for its point set. QU0 will denote the category of quasi-uniform T0 -spaces with the usual mappings which one may call quasi-uniform (q.u.). The forgetful functor T = QU0 → Top0 assigns to every space X in QU0 a topological T0 -space T X in which the neighbourhood system at any point x is {H(x) : H ∈ entX}, where H(x) = {y ∈ X|(x, y) ∈ H}. By a T -section we shall mean a functor F : Top0 → QU0 such that T F is the identity functor on Top0 . Such an F imposes compatible quasiuniformities on the topological T0 -spaces, and accordingly a T -section is also called a functorial quasi-uniformity on Top0 . (For examples of such functors see Section 2 and [9], [2], [11], [4].) We shall denote by SX the symmetrization (= uniform coreflection) of a quasi-uniform space X. Thus SX is the coarsest uniform space finer than X. We view S as an endofunctor S : QU0 → QU0 , the symmetrizer. A quasi-uniform mapping f : X → Y is called sup-dense if the image set f [X] is dense in the topological space T SY . This term stems from regarding the topology of T SY as the supremum of the topologies of T Y and T (Y −1 ), where Y −1 has the entourages H −1 (H ∈ entY ). A non-trivial folklore result is: 0.1. The epimorphisms in QU0 are precisely the sup-dense quasiuniform mappings. For a proof see [7]. 1. The Samuel bicompactification in QU0 . We recall that a quasi-uniform T0 -space X is called bicomplete [9] if the symmetrization SX is a complete uniform space. By a bicompletion of a space X in QU0 we mean a sup-dense quasi-uniform embedding f : X → Y with a Y a T0 bicomplete quasi-uniform space. For any two bicompletions f1 : X → Y1 and f2 : X → Y2 there exists a (unique) quasi-uniform isomorphism h : Y1 → Y2 with h ◦ f1 = f2 (see [9], Theorem 3.34. This result is known as the “Uniqueness of bicompletion”.) A particular bicompletion for any X in QU0 , which
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we shall denote by kX : X → KX, is constructed as follows in [9], Theorem 3.33 : The points of KX are the minimal Cauchy filters of the uniform e : U ∈ entX}, where space SX, and entKX has the base {U e = {(F, G) ∈ |KX| × |KX| : there exist F ∈ F and U G ∈ G such that F × G ⊆ U };
while the sup-dense quasi-uniform embedding kX : X → KX is given, for any x ∈ X, by kX (x) being the neighbourhood filter of x in T SX. The bicompletion has the following reflection property : Given any g : X → A in QU0 with A bicomplete, there exists a unique quasiuniform mapping gb : KX → A with gb ◦ kX = g (see [9], Theorem 3.29). It follows that KX is given by an endofunctor K : QU0 → QU0 . Moreover, kX is given by a natural transformation k : 1 → K, which means that for any f : X → Y in QU0 the diagram (1) commutes. f
X (1)
kX
KX
Kf
/Y
kY
/ KY
It is well known (and obvious from the above) that the bicompletion commutes with the symmetrizer, i.e. SKX = KSX for each X ∈ QU0 . The following two simple results, implicit in [5], (Remark 1.17), are not so well known, and are so useful and fundamental that we give explicit proofs. PROPOSITION 1.1 ([5]). Let f : X → Y be a morphism in QU0 . Then f is a sup-dense quasi-uniform embedding if and only if Kf : KX → KY is a quasi-uniform isomorphism. Proof. Let f in Diagram (1) be a sup-dense quasi-uniform embedding. Then kY ◦f : X → KY is a bicompletion of X. By the cited uniqueness of bicompletions there exists a q.u. isomorphism h : KX → KY with h ◦ kX = kY ◦ f , and thus h ◦ kX = (Kf ) ◦ kX . Since kX is epi, h = Kf , so that Kf is a quasi-uniform iso. Conversely, given f in Diagram (1) with Kf iso, it is clear that kY ◦ f is a sup-dense q.u. embedding. Being a right-hand factor of a q.u. embedding, f is a q.u. embedding. Thinking of the underlying topologies of SX, SY and SKY , for which f and kY are topological embeddings with kY ◦ f dense, one sees at once that f is dense into SY , hence a sup-dense q.u. embedding.
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COROLLARY 1.2 ([5]). Let f : X → Y and g : Y → Z be morphisms in QU0 such that g◦f : X → Z is a sup-dense quasi-uniform embedding. Then, f is a sup-dense quasi-uniform embedding if and only if g is one. Proof. By the previous result, Kg ◦ Kf = K(g ◦ f ) is an iso. Thus Kf is iso iff Kg is iso. A quasi-uniform space Y is called totally bounded iff SY is a totally bounded uniform space. For any quasi-uniform space (X, U ) there exists a finest totally bounded quasi-uniformity coarser than U, denoted by Uω in [9] and in much of the literature. Moreover, U and Uω induce the same topology (in fact the same bitopology). We shall denote (X, Uω ) by BX and then write bX : X → BX for the quasi-uniform mapping which is the identity function on the point set of X. Clearly this is a reflection to the totally bounded quasi-uniform spaces. We only need this on QU0 , so we have an endofunctor B : QU0 → QU0 and a natural transformation b : 1 → B. We note that T B = T and SB = BS. The bicompletion of a totally bounded T0 quasi-uniform space is totally bounded. This follows at once from the corresponding uniform result by the commutation of a bicompletion and symmetrization. Clearly also, a space Y in QU0 is bicomplete and totally bounded if and only if Y is sup-compact — that is, SY has compact topology. The Samuel bicompactification of X in QU0 is the following composite quasi-uniform mapping :
X
bX
/ BX kBX / KBX
This is clearly the reflection to the sup-compact spaces in QU0 . For brevity, let us for the moment denote the composite mapping by λX : X → KBX. We note that λX is a quasi-uniform embedding iff X is totally bounded. However, λX is always a topological embedding for the underlying sup-topologies and for the underlying topologies. The functors K and B do not commute (there are obvious examples even in Unif0 ), but there is the following quasi-uniform embedding in one direction. PROPOSITION 1.3. For X in QU0 there is a natural sup-dense quasiuniform embedding eX : BKX → KBX given by the commutative diagram (2).
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(a)
kX
(2)
bX
KX
5
/ BX
BkX
/ BKX ?
bKX
(b)
KbX
(c)
?
?
?
kBX
?
? eX ? ?
?
? *
?
?
KBX
Proof. The square (a) in Diagram (2) commutes by naturality of b. Since KBX is totally bounded, the reflection property of b at KX provides a unique q.u. morphism eX : BKX → KBX such that the triangle (b) commutes. To see that the triangle (c) commutes, we consider: eX ◦ BkX ◦ bX = eX ◦ bKX ◦ kX = KbX ◦ kX = kBX ◦ bX in which the last equality follows by naturality of k. The surjection bX is right-cancellable, giving eX ◦ BkX = kBX , so that (c) commutes. Moreover in (c), BkX is the right factor of the q.u. embedding kBX , hence a q.u. embedding, whereas in (a), BkX is a left factor of the sup-dense mapping bKX ◦ kX , so that BkX is sup-dense. Application of Corollary 1.2 to (c) now shows that eX is a sup-dense q.u. embedding. Naturality of e : BK → KB is a routine exercise. We shall sharpen Proposition 1.3 to the result that, literally, BKX is a quasi-uniform subspace of KBX. This is only true by virtue of the particular constructions used for the two functors involved, that is, for K in terms of minimal Cauchy filters and for B so as to preserve underlying sets. However, the literal inclusion will be convenient for the verification of our main example in Section 3. For this we still need a lemma. LEMMA 1.4. For every quasi-uniform space X, every minimal Cauchy filter on SX is a minimal Cauchy filter on SBX.
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Proof. Since SBX = BSX, the Lemma reduces to the assertion that every minimal Cauchy filter on any uniform space Y is minimal Cauchy on the totally bounded reflection BY . Now it is clear that any Cauchy filter F on Y is minimal Cauchy on Y if any only if F is a round filter with respect to the induced proximity relation δY . We apply the same observation to BY , noting that F Cauchy on Y implies F Cauchy on BY , and that δBY = δY . The assertion follows. (This argument is hinted at in [6], p. 239.) THEOREM 1.5. For any quasi-uniform T0 -space X, the space BKX is a sup-dense quasi-uniform subspace of the Samuel bicompactification KBX. (This is true subject to the given constructions of the functors K and B.) Proof. We consider the commutative triangle (b) in Diagram 2. The underlying point sets |KX| and |BKX| coincide, with bKX the identity function between these. Thus eX and KbX are the same function between point sets. By construction of K and by Lemma 1.4, eX is the inclusion function for |BKX| ⊆ |KBX|. The theorem now follows from Proposition 1.3. REMARKS 1.6. (1) The reader will have noticed that Proposition 1.3 is independent of any particular construction of the functors K and B. (2) All functorial quasi-uniformities on a given T0 topological space Y have the same codomain for their Samuel bicompactifications. Indeed, let F : Top0 → QU0 be any T -section. It is well known that BF = P, where P : Top0 → QU0 is the functor which imposes the Cs´asz´ ar-Pervin quasi-uniformity, i.e. the coarsest functorial quasiuniformity, on the T0 topological spaces. Thus the codomain of the Samuel bicompactification of F Y is KPY . (Compare [4], where P is denoted by C1∗ .) (3) For the categorically minded reader, the following may be of interest. In the terminology of [5], Proposition 1.1 says that the sup-dense q.u. embeddings form a firm class of morphisms in QU0 . Knowing that K reflects to the T0 bicomplete spaces, one then has from [5] that these spaces are precisely the injective objects in QU0 with respect to the class of sup-dense q.u. embeddings. (One can see this directly : That each T0 bicomplete space is such an injective, is precisely the content of Theorem 3.29 of [9]. Conversely, if Y is such an injective, there exists g : KY → Y such that g ◦ kY = 1Y . Since kY is epi, this implies that kY is iso, so that Y is T0 bicomplete.) Finally, Corollary 1.2 says that the class of sup-dense q.u. embeddings is both an essential and a coessential class; this in fact is true of any firm class (see e.g. [5]).
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2. New results on upper K-true functorial quasi-uniformities We consider a given T -section F : Top0 → QU0 and any space X in Top0 . The T0 quasi-uniform space F X has the bicompletion kF X : F X → KF X which, we recall, is a quasi-uniform embedding, epimorphic in QU0 . Since T F X = X, application of the forgetful functor T gives a dense topological embedding T kF X : X → T KF X, which for brevity we denote by rX : X → RX. Thus we have introduced an endofunctor: R = T KF : Top0 → Top0 and a natural transformation:
r = T kF : 1 → R given by rX = T kF X . We agree that these will be standing notations when any given T section F is considered. The following notation is standard: Fix(r) = {X ∈ Top0 : rX : X → RX is a homeomorphism } = {X ∈ Top0 : F X is bicomplete } and we shall regard this class of objects as a full subcategory of Top0 . The following are samples of the many basic questions that arise: (1) Is Fix(r) an epireflective subcategory of Top0 ? (2) Is rX an epimorphism in Top0 (i.e. Skula-dense or b-dense) for each X in Top0 ? (3) Is rX : X → RX a reflection to Fix(r) for every X in Top0 ? (4) Do the functors KF, F R : Top0 → QU0 coincide, i.e., is KF = F T KF ? Each of these four questions has a negative answer for some T sections F , and an affirmative answer for others. Most of the known published results around these questions are surveyed in [4]. To prove our new results, we have to define the key concepts. First we recall, for quasi-uniform spaces A and B, that one says A is coarser (finer) than B if the underlying sets are the same and entA ⊆ entB (entB ⊆ entA).
DEFINITION 2.1 ([3], [4]). For a T -section F : Top0 → QU0 we say that F is lower (upper) K-true if KF X is coarser (finer) than F T KF X for each X in Top0 . We say that F is bicompletion-true (more briefly, K-true) if F is both upper and lower K-true, that is, if KF = F T KF .
The above definition makes sense because the functor F T preserves the underlying point set (in fact also the underlying topological space)
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of KF X. Thus the equation KF = F T KF should be read as an actual equality of functors, not merely an isomorphism. As shown in [4], affirmative answers to the above four questions obey the following implications: (4) =⇒ (3) =⇒ (2) =⇒ (1); (2) ⇐⇒ F is upper K-true. An example by Z. Kimmie [10] (pp. 73-77) shows that (3) 6=⇒ (4). Thus, Kimmie’s example is upper K-true but not lower K-true. The 6 ⇒ (3). Below, present authors will show in a further paper that (2) = in Theorem 2.4, we shall see that (1) ⇐⇒ (2), that is, F is upper K-true ⇐⇒ Fix(T kF ) is epireflective in Top0 , thereby answering a question raised in [4] (Remark 4.11). (More characterizations of upper K-trueness are given in [4].) An open cover A of a topological space X is called well-monotone if A is well-ordered by the partial order of set inclusion. The wellmonotone quasi-uniformity on X has a subbase consisting of all binary relations of the form EA , defined by (∀p ∈ X)(EA (p) = ∩{G ∈ A : p ∈ G}), with A running through the well-monotone open covers of X. This quasi-uniformity is compatible on each X and is functorial ([3], [11]) and thus defines a T -section which we denote by W : Top0 → QU0 .
THEOREM 2.2 ([3], [11]). Let F : Top0 → QU0 be any T -section. Then, F is upper K-true ⇐⇒ F is finer than W .
(The implication “⇐=” was proved in [3], using results from [12], and the surprising “=⇒” was proved in [11].) EXAMPLES 2.3. (1) The functor W is K-true and Fix(T kW ) = Sob, the category of sober T0 -spaces ([12], Proposition 4 and Corollary 2). (2) The fine transitive quasi-uniformity, operating on Top0 , is Ktrue ([3], Example 6.4(4) or [4], Example 3.8). Descriptions of its Fix(r) are given in [12] (Corollary 2) and [10] (Remark 2.4.12). (3) The fine quasi-uniformity, operating on Top0 , is K-true ([12], Corollary 5). We do not know a complete characterization of its Fix(r). For the proof of Theorem 2.4 below we recall that the Skula topology (or b-topology, [13]) on a topological space X is generated by the open and the closed sets of X. It is obvious that the symmetrization of the Cs´az´ ar-Pervin quasi-uniformity on X carries the Skula topology of X. It then follows (e.g. as in [12], Lemma 2 or [4], p. 68) that for any functorial quasi-uniformity F the symmetrization of F X (for X in Top0 ) also carries the Skula topology of X.
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THEOREM 2.4. The following conditions are equivalent for a functorial quasi-uniformity F defined on the category Top0 of topological T0 -spaces and continuous maps: (a) F is upper K-true. (b) {X : F X is bicomplete} is closed under topological products and Skula-closed subspaces (equivalently, Fix(T kF ) is epireflective in Top0 – (see [1], Corollary 16.9). (c) F X is bicomplete whenever X is sober (that is, Sob ⊆ Fix(T kF )). (d) For each (infinite) regular cardinal α the space Xα defined by equipping the carrier set α with the lower topology {[0, β[: β ∈ α} ∪ {α} carries the well-monotone quasi-uniformity under F . Proof. (a) implies (b): See [4] (Proposition 4.2); for the application of [1] (Corollary 16.9) one also has to know that in Top0 the epimorphisms are the Skula-dense maps and the extremal subobjects are the Skulaclosed embeddings. (b) implies (c): Obviously the unique compatible quasi-uniformity of the Sierpi´ nski space is bicomplete. Since each sober space is a Skulaclosed subspace of a product of Sierpi´ nski spaces ([14]), the assertion follows from the hypothesis. (c) implies (d): One easily sees that the well-monotone quasi-uniformity coincides with the fine quasi-uniformity on Xα (for every ordinal α). The negation of (d) means that there is an (infinite) regular cardinal α such that F Xα does not carry the well-monotone (= fine) quasiuniformity. Note that Xα is sober, since it is a T0 -space in which the only nonempty closed sets are point-closures. Assuming (c) we thus have F Xα bicomplete. On the other hand, it is shown in [11] (Appendix) that for an (infinite) regular cardinal α, F Xα being strictly coarser than W Xα implies that the filter F generated by {[β, → [: β ∈ α} is a Cauchy filter on the symmetrization of F Xα . Since F does not contain a singleton, it cannot converge with respect to the discrete Skula topology on Xα . Thus F Xα is not bicomplete – a contradiction. We deduce that F Xα = W Xα for any (infinite) regular cardinal α. (d) implies (a): The assertion follows from the argument given in the last paragraph of the Appendix of [11]. REMARK 2.5. The condition (b) in Theorem 2.4 above is equivalent to the following version: (b0 ) Fix(T kF ) is embedding-reflective in Top0 . Clearly (b0 ) implies (b) (see [1], Proposition 6.3). However, the implication (b) =⇒ (b0 ) is not as trivial as it may seem, because the
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reflection morphisms into Fix(T kF ) need not be the embeddings of the form rX = T kF X (see our discussion following Definition 2.1, where we 6 ⇒ (3)”). Let us denote the reflection of any T0 -space X into said “(2) = Fix(T kF ) by ρX and the sobrification of X by σX : X → ΣX. Since (b) =⇒ (c) in Theorem 2.4, ΣX ∈ Fix(T kF ) and so ρX is a factor of σX . Since σX is known to be an embedding, so is ρX . 3. Transitivity and lower K-true functorial quasi-uniformities We recall a result due to P. Fletcher ([8]; [9], p. 29) which we have already once used in defining the well-monotone quasi-uniformity. Let C be a family of covers of a topological space X. For any A ∈ C denote by EA the transitive reflexive relation on X given by EA (p) = ∩{A ∈ A : p ∈ A} for all p ∈ X. Then (see [2], Proposition 2.7) {EA : A ∈ C} is a subbase for a compatible (transitive) quasi-uniformity on X if and only if C is a set of interior-preserving open covers of X and {EA (p) : A ∈ C and p ∈ X} is a subbase for OX (the topology of X). The transitive quasi-uniformity with subbase {EA : A ∈ C} is called the (covering) quasi-uniformity induced by C. We call the procedure the Fletcher construction and the relations EA the Fletcher entourages. For any compatible transitive quasi-uniformity V on any topological space X there exsits a family C of covers of X which induces V via the Fletcher construction ([8], [9]). As noted above, these covers are automatically interior-preserving and open. There is a functorial version of the Fletcher construction which accounts for all transitive-valued sections of the forgetful functor QU → Top, as follows. DEFINITION 3.1. ([2]). A natural kind of covers (of the topological spaces) is a function Γ = (Γ(X) : X ∈ Top) such that: (1) Γ(X) is a set of covers of X, for each X in Top; (2) Whenever f : X → Y is continuous and B ∈ Γ(Y ), then f −1 [B] ∈ Γ(X). We shall call a natural kind Γ of covers admissible if, for each X in Top, the covering quasi-uniformity induced by Γ(X) is compatible on X. Under the Fletcher construction, an admissible natural kind Γ of covers of the topological spaces induces a functor F : Top → QU which is a section of the forgetful functor QU → Top, and such that each F X is a transitive quasi-uniform space.
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The naturality condition (2) ensures functoriality: every continuous map f : X → Y becomes a quasi-uniform morphism for the induced quasi-uniformities, because one has (f × f )−1 [EB ] = Ef −1 [B] for all B ∈ Γ(Y ). Conversely, any functorial transitive quasi-uniformity F on Top (that is, transitive-valued section of the forgetful functor QU → Top) is induced by some natural (necessarily admissible) kind Γ of covers; there exists a largest natural kind Γ that does this, given by Γ(X) = {A : A covers X and EA ∈ entF X} for each X in Top (see [2], p. 169). (It is immaterial whether one restricts these results to the T0 -spaces.) Many familiar kinds of interior-preserving open covers are natural and admissible, e.g. the finite, the locally finite, the point-finite and the well-monotone open covers; the open spectra (which induce the semicontinuous quasi-uniformity, [9]); and the kind of all interior-preserving open covers (inducing the fine transitive quasi-uniformity). Some kinds of covers lend their names to the induced T -sections, often in abbreviated form, e.g. locally finite (open covering) quasi-uniformity. One notes that the finite open covers induce the Cs´asz´ar-Pervin quasi-uniformity. The proof of the next result uses our Theorem 1.5. EXAMPLE 3.2. The locally finite covering quasi-uniformity functor on Top0 is not lower K-true. Proof. For brevity we denote the above functor by L: Top0 → QU0 . For any X in Top0 , the Fletcher entourages EA determined by all locally finite open covers A of X form a base for the quasi-uniformity entLX. (The family is indeed a base, because if the covers A and B are locally finite open, so is A ∧ B = {A ∩ B : A ∈ A and B ∈ B}, and EA ∩ EB = EA∧B — see e.g. [2], Lemma 2.5.) To show that the functor L is not lower K-true it will suffice to find a topological T0 -space X such that entKLX is not contained in entLT KLX. As point set for the space X we take ω × {0, 1} where ω denotes the first infinite ordinal. By [0, n] resp. [n, → [ for n ∈ ω we denote the obvious intervals of ω. We let the topology of X be generated by the base {{(n, 0)} : n ∈ ω} ∪ {[0, n] × {0, 1} : n ∈ ω}. Thus X is a topological T0 -space. We denote by F the filter on X generated by the filter base {[n, → [×{1} : n ∈ ω}. We note that the members of this base are closed sets of X and that every non-void closed set of X is an element of F. We claim that F is a minimal Cauchy filter on the uniform space SLX (recalling that S denotes the symmetrizer).
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−1 For the uniformity of SLX we have the basic entourages EA ∩ EA where A ranges through all locally finite open covers of X. To see that F −1 )(p) ∈ is Cauchy it will suffice to find a point p ∈ X such that (EA ∩EA −1 (p) F. Now because EA is transitive, it is clear that for any p ∈ X, EA is closed in X, and therefore an element of F. Observing that the point (0, 1) ∈ X lies in just finitely many members of A, we conclude from the definition of the topology that X ∈ A and there exists m ∈ ω such that X is the only member of A that contains the point (m, 1). Then −1 EA ((m, 1)) = X. Thus (EA ∩ EA )((m, 1)) ∈ F, and F is Cauchy on SLX. We have a filter base of closed sets for F. For any such set F , G = X \ F is open, and the Pervin entourage SG = E{G,X} is a member of −1 entLX. Then SG ∩ SG = (F × F ) ∪ ((X \ F ) × (X \ F )) and (SG ∩ −1 SG )(F ) = F . Clearly then F is generated by the filter base {U (F ) : F ∈ F and U ∈ entSLX}. Thus by [9] (Proposition 3.30) F is a minimal Cauchy filter on SLX. Hence F is a point of KLX. Note next that C = {X} ∪ {{(n, 0)} : n ∈ ω} is a locally finite open cover of X. Thus we have the Fletcher entourage EC ∈ entLX. Recalling the notation used in constructing basic entourages in KLX fC ∈ (first part of Section 1, with notation from [9]) we have a set E entKLX. The proof that L is not lower K-true will be finished by fC 6∈ entLT KLX. showing that E Recall that the q.u. embedding kLX : LX → KLX sends any point p of X to the neighbourhood filter of p in T SLX. In particular, when p is the isolated point (n, 0) of X, one has
kLX ((n, 0)) = {A ⊆ |X| : (n, 0) ∈ A} for all n ∈ ω. We claim that, for all n ∈ ω, fC (kLX ((n, 0))) = {kLX ((n, 0))}. E
To see this, note that fC = {(H, H0 ) ∈ |KLX|2 : (∃ H ∈ H)(∃ H 0 ∈ H0 )(H × H 0 ⊆ EC )}. E fC (kLX ((n, 0))). Since {(n, 0)} ∈ kLX ((n, 0)), there Consider any H0 ∈ E 0 0 exists H ∈ H such that {(n, 0)} × H 0 ⊆ EC , and then H 0 ⊆ EC ((n, 0)) =
\
{C : (n, 0) ∈ C ∈ C} ⊆ {(n, 0)}
so that in fact H 0 = {(n, 0)}. Then, since H0 is a filter on |X| and H 0 ∈ H0 , one has H0 = kLX ((n, 0)), and the above claim is proved. By Theorem 1.5 and Remark 1.6(2), BKLX is a q.u. subspace of KBLX and moreover KBLX = KPX, where P : Top0 → QU0 is the Cs´asz´ ar-Pervin T -section. Applying the forgetful functor T and
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using the previously noted fact that T B = T , we see that T KLX is a topological subspace of T KPX. We return to the filter F generated by the non-void closed sets of X. We showed that F ∈ |KLX|. Thus F ∈ |KPX|. For any open set G of X we have SG ∈ entPX and thus (with obvious notation) (SG )∼ P ∈ entKPX, where 2 (SG )∼ P = {(M, N ) ∈ |KPX| : (∃ M ∈ M)(∃ N ∈ N )(M × N ⊆ SG )}.
We claim:
(SG )∼ P (F) = |KPX|.
It suffices to prove (∀N ∈ KPX)(∃ M ∈ F)(∃ N ∈ N )(M × N ⊆ SG ). Case 1 : G 6= X ; then X \ G ∈ F ; taking M = X \ G and N = X we have M × N ⊆ SG . Case 2 : G = X ; we take M = N = X, and the claim is proved. Noting that the sets SG (G open in X) form a subbase for entPX and that the operator (−)∼ P preserves finite intersections, we see that the ∼ form a subbase for entKPX. Since (S )∼ (F) = |KPX|, sets (SG )P G P it follows that |KPX| is the only neighbourhood of F in T KPX. Since F is also a point of the subspace T KLX, |KLX| is the only neighbourhood of F in T KLX. fC ∈ entLT KLX. To end the proof of our Example we assume that E fC Then there exists a locally finite open cover E of T KLX with EE ⊆ E (since, as noted at the beginning, the Fletcher entourages form a base for the locally finite covering quasi-uniformity). We have also seen that fC (kLX ((n, 0))) = {kLX ((n, 0))}, and so it follows from EE ⊆ E fC that E EE (kLX ((n, 0))) = {kLX ((n, 0))} for each n ∈ ω. Since the map kLX is injective, it follows that {EE (H) : H ∈ |KLX|} is an infinite collection. This collection is also a locally finite (open) cover of T KLX, as proved in ([2], Example 2.16 (2)). But this contradicts the fact that the point F has just one neighbourhood in T KLX (namely the whole space). We have thus proved that L fails to be lower K-true. A class A of (T0 ) quasi-uniform spaces is said to span a T -section F if, for each X in Top0 , F X has the initial quasi-uniformity for the class of all continuous maps from X into the spaces T A (A ∈ A) (see e.g. [4] or [3]). THEOREM 3.3 ([3]). A T -section F : Top0 → QU0 is lower K-true if and only if F is spanned by some class of bicomplete T0 quasi-uniform spaces. By virtue of this result, the Cs´asz´ar-Pervin quasi-uniformity and the semicontinuous quasi-uniformity are both lower K-true, since each
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can be spanned by a single bicomplete T0 quasi-uniform space (the Sierpi´ nski space with its unique quasi-uniformity, respectively the real line with the lower quasi-uniformity). Both functors fail to be upper K-true, by Theorem 2.2. REMARK 3.4. It can be proved that the sections of the forgetful functor QU → Top stand in a bijective correspondence with the sections of our functor T : QU0 → Top0 . The correspondence is given by restriction, its inverse being unique extension. Moreover, if a functorial quasi-uniformity G : Top → QU is spanned by a class B of bicomplete, not necessarily T0 , quasi-uniform spaces, then G is also spanned by the class of quasi-uniform T0 -reflections of the spaces in B. Thus the restriction of G to Top0 will be lower K-true by Theorem 3.3. PROPOSITION 3.5. Let F : Top0 → QU0 be a transitive T -section induced (via the Fletcher construction) by an admissible natural kind Γ of interior-preserving open covers of the topological spaces. For every X in Top0 and every A ∈ Γ(X) assume that A ∈ Γ(XA ), where XA denotes the topological space having the same point set as XTand such that each point p has as smallest XA -neighbourhood the set {A : p ∈ A ∈ A}. Then F is lower K-true.
Proof. Consider X ∈ Top and A ∈ Γ(X). For the Fletcher entourage T 0 EA one has EA (p) = {A : p ∈ A ∈ A} for every p ∈ X. Thus XA has a compatible quasi-uniformity generated by the single transitive entourage EA . Let Xq,A denote the quasi-uniform space thus obtained, so that T (Xq,A ) = XA . The symmetrization SXq,A has the uniformity −1 generated by the single equivalence relation EA ∩EA and is therefore a complete uniform space. Indeed, for any Cauchy filter F on SXq,A there −1 −1 exists p ∈ |X| such that (EA ∩ EA )(p) ∩ EA )(p) ∈ F , and since (EA is the smallest neighbourhood of p in T (SXq,A ), it is clear that F converges to p in the latter space. Thus Xq,A is bicomplete, but not necessarily T0 . Let B = {Xq,A : X ∈ Top0 and A ∈ Γ(X)} and let G : Top → QU denote the functor spanned by B. We shall show that G coincides with F on Top0 . Consider any Y in Top0 . The spanning of G means that entGY has a subbase consisting of all sets (g × g)−1 EA using all continuous maps g : Y → XA , X ∈ Top0 and A ∈ Γ(X). Now by hypothesis, A ∈ Γ(X) implies A ∈ Γ(XA ); then g −1 [A] ∈ Γ(Y ) by naturality of Γ, and then Eg−1 [A] ∈ entF Y by the Fletcher construction of F Y . Since Eg−1 [A] = (g × g)−1 EA we thus have entGY ⊆ entF Y . On the other hand, since entF Y has the subbase {EB : B ∈ Γ(Y )}, Yq,B is coarser than F Y , and then YB = T (Yq,B ) is coarser than T (F Y ) = Y , so that we have a continuous identity function i : Y → YB for each B ∈ Γ(Y ). By the spanning construction for G the map i lifts
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to a quasi-uniform morphism GY → Yq,B . Then EB ∈ entYq,B implies EB ∈ entGY , so that entF Y ⊆ entGY . Thus indeed F and G coincide on Top0 . Since B is a class of bicomplete spaces it now follows from Remark 3.4 that F is lower K-true. EXAMPLES 3.6. The sufficient condition on Γ in Proposition 3.5, namely (∀X ∈ Top0 )(∀A ∈ Γ(X))(A ∈ Γ(XA )),
is satisfied by the following admissible natural kinds of interior-preserving open covers: The finite; the point-finite; the well-monotone; the open spectra; and the kind of all interior-preserving open covers. Consequently the corresponding induced transitive functorial quasi-uniformities on Top0 are lower K-true. In the case of the point-finite covering quasi-uniformity this fact is new (to our knowledge). — We do not know whether the above sufficient condition is in some sense necessary. EXAMPLES 3.7. On Top0 , the point-finite covering quasi-uniformity is lower but not upper K-true, and the locally finite covering quasiuniformity is neither lower nor upper K-true. Proof. By Theorem 2.2, since the locally finite T -section is coarser than the point-finite one, it only remains to show that the point-finite T -section is not finer than the well-monotone T -section W . Let ωd be the ordinal ω equipped with the topology consisting of all downsets. The open cover A = {[0, n] : n ∈ ω} is well-monotone. Thus EA ∈ entW (ωd ). Observing that EA is the graph of the usual order on ω one sees at once that EA does not belong to the point-finite quasi-uniformity on ωd .
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J. Ad´ amek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, Wiley, New York etc., 1990. G.C.L. Br¨ ummer, Functorial transitive quasi-uniformities, In: H.L. Bentley et al., ed., Categorical Topology (Proc. Conf. Toledo, Ohio, 1983), Heldermann Verlag, Berlin, 1984, pp. 163–184. G.C.L. Br¨ ummer, Completions of functorial topological structures, In: W. G¨ ahler et al., ed., Recent Developments of General Topology and its Applications (Proc. Conf. Berlin, 1992), Akademie Verlag, Berlin, 1992, pp. 60–71. G.C.L. Br¨ ummer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste 30 Suppl. (1999), 45–74. Free online at:http://mathsun1.univ.trieste.it/Rendiconti/ G.C.L. Br¨ ummer and E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolinae 33 (1992), 131–147.
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