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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 101 - 112

Weak completeness of the Bourbakiquasi-uniformityM. A. S�anchez-Granero�Abstract. The concept of semicompleteness (weaker thanhalf-completeness) is de�ned for the Bourbaki quasi-uniformityof the hyperspace of a quasi-uniform space. It is proved thatthe Bourbaki quasi-uniformity is semicomplete in the space ofnonempty sets of a quasi-uniform space (X;U) if and only if eachstable �lter on (X;U�) has a cluster point in (X;U). As a conse-quence the space of nonempty sets of a quasi-pseudometric spaceis semicomplete if and only if the space itself is half-complete. Itis also given a characterization of semicompleteness of the space ofnonempty U�-compact sets of a quasi-uniform space (X;U) whichextends the well known Zenor-Morita theorem.2000 AMS Classi�cation: 54E15, 54E35, 54B20.Keywords: Bourbaki quasi-uniformity, Hausdor� quasi-uniformity, half com-pleteness. 1. IntroductionOur basic reference for quasi-uniform spaces is [8].A (base B of a) quasi-uniformity U on a set X is a (base B of a) �lter Uof binary relations (called entourages) on X such that (a) each element of Ucontains the diagonal �X of X � X and (b) for any U 2 U there is V 2 Usatisfying V � V � U.Let us recall that if U is a quasi-uniformity on a set X, then U�1 = fU�1 :U 2 Ug is also a quasi-uniformity on X called the conjugate of U. The unifor-mity U _ U�1 will be denoted by U�. If U 2 U, the entourage U \ U�1 of U�will be denoted by U�:Each quasi-uniformity U on X induces a topology T (U) on X; de�ned asfollows:T (U) = fA � X : for each x 2 A there is U 2 U such that U(x) � Ag:�The author acknowledges the support of the Spanish Ministry of Science and Technology,under grant BFM2000-1111.



102 M. A. S�anchez-GraneroIf (X;T ) is a topological space and U is a quasi-uniformity on X such thatT = T (U) we say that U is compatible with T .A quasi-uniform space (X;U) is precompact if for each U 2 U there existsa �nite subset F of X such that X = U(F). (X;U) is U�1-precompact if(X;U�1) is precompact and (X;U) is U�-precompact (totally bounded) if theuniform space (X;U�) is precompact.A sequence (xn)n2N in a quasi-pseudometric space (X;d) is called right K-Cauchy [12] if for each " > 0 there is k 2 N such that d(xn;xm) < " for eachn � m � k. (X;d) is said to be right K-sequentially complete if each right K-Cauchy sequence converges. A �lter F on a quasi-uniform space (X;U) is calledright K-Cauchy [13] if for each U 2 U there is an F 2 F such that U�1(x) 2 Ffor each x 2 F . (X;U) is said to be right K-complete if each right K-Cauchy�lter converges.Obviously a quasi-pseudometric space (X;d) is right K-sequentially completeif the quasi-uniformity Ud is right K-complete. It is known that the converseholds for regular spaces [2].A �lter F on a quasi-uniform space (X;U) is called left K-Cauchy [13] if foreach U 2 U there is an F 2 F such that U(x) 2 F for each x 2 F . (X;U) issaid to be left K-complete if each left K-Cauchy �lter converges.A quasi-uniform space (X;U) is half complete [7], if each Cauchy �lter on(X;U�) converges in (X;U).Let (X;U) and (Y;V) be two quasi-uniform spaces. A mapping f : (X;U) !(Y;V) is said to be quasi-uniformly continuous if for each V 2 V there is U 2 Usuch that (f(x);f(y)) 2 V whenever (x;y) 2 U.Let (X;U) be a quasi-uniform space and let P0(X) be the collection of allnonempty subsets of X. The Bourbaki (Hausdor�) quasi-uniformity on P0(X)is de�ned by UH = fUH : U 2 Ug, where UH is de�ned by UH = f(A;B) 2P0(X) : B � U(A) and A � U�1(B)g for each U 2 U (see [3] and [11]).Let (X;U) be a quasi-uniform space. Let denote by K0(X) (resp. K�10 (X),K�0(X)) the family of nonempty compact (resp. U�1-compact, U�-compact)subsets of X, by F0(X) the family of nonempty �nite subsets of X, by C0(X)(resp. C�10 (X), C�0(X)) the family of nonempty closed (resp. U�1-closed, U�-closed) subsets of X and by PC0(X) (resp. PC�10 (X), PC�0(X)) the family ofnonempty precompact (resp. U�1-precompact, U�-precompact) subsets of X.We will use the same symbol UH to denote the restriction of UH to any of theprevious subspaces.In this paper the concept of semicompleteness of the Bourbaki quasi uni-formity is introduced and used to extend the main theorems concerning com-pleteness in uniform (metric) spaces to the setting of quasi-uniform (quasi-pseudometric) spaces.The well-known Zenor-Morita theorem states that a uniform space (X;U)is complete if and only if (K0(X);UH) is complete. In [5] it is proved thata compactly symmetric quasi-uniform space (X;U) is complete if and only if(K0(X);UH) is complete, providing a generalization of the Zenor-Morita theo-rem for compactly symmetric quasi-uniform spaces. Here completeness is meant



Completeness of Bourbaki quasi-uniformity 103in the sense used by Fletcher and Lindgren in their monograph [8]. In section3 it is given a generalization of the Zenor-Morita theorem for quasi-uniformspaces in terms of semicompleteness.Burdick [4, Corollary 2], based on former work of Isbell [9], answered a ques-tion of Cs�asz�ar [6] in the a�rmative by proving the following characterization:The Hausdor� uniformity on P0(X) of a uniform space (X;U) is complete ifand only if each stable �lter on (X;U) has a cluster point. In [11] it is proved asatisfactory generalization of this result to the setting of quasi-uniform spaces,since it was proved that (P0(X);UH) is right K-complete if and only if eachstable �lter on the quasi-uniform space (X;U) has a cluster point in (X;U). Insection 3 it is given another generalization of Isbell-Burdick theorem for quasi-uniform spaces. In particular it is proved that (P0(X);UH) is semicomplete ifand only if each stable �lter on (X;U�) has a cluster point in (X;U). Moreover,a characterization of half completeness of (P0(X);UH) is obtained in terms ofdoubly stable �lters on (X;U).It is known (see e.g. [4, Corollary 6]) that the Hausdor� metric of a (bounded)metric space (X;d) is complete if and only if (X;d) is complete. In [11] itis proved a satisfactory generalization of this result to the setting of quasi-pseudometric spaces, since it was proved that (P0(X);dH) is right K-sequentiallycomplete if and only if (X;d) is right K-sequentially complete. In section 3 asimpler proof of this result is given. It is also proved that (P0(X);dH) is semi-complete if and only if (X;d) is half complete.2. Preliminary resultsLet us denote NPC�10 (X) = fA 2 P0(X) : for each U 2 U there exists a�nite subset F of X such that A � U�1(F)g.NPC�10 (X) can be used to describe the closure of F0(X) in (P0(X);UH).Proposition 2.1. Let (X;U) be a quasi-uniform space. Then ClT (UH)(F0(X))= NPC�10 (X).Proof. Let A 2 ClT (UH)(F0(X)), and let U 2 U. Then there exists F 2 F0(X)such that F 2 UH(A), and hence A � U�1(F). Therefore A 2 NPC�10 (X).Conversely, let A 2 NPC�10 (X) and let U 2 U. Then there exists F 2 F0(X)such that A � U�1(F). Let F 0 = F \U(A). It is easy to check that F 0 2 UH(A)and hence A 2 ClT (UH)(F0(X)). �Corollary 2.2. Let (X;U) be a quasi-uniform space such that (X;U�1) is pre-compact. Then K0(X) is dense in (P0(X);UH).Proof. It is clear that ClT (UH)(F0(X)) � ClT (UH)(K0(X)). Since (X;U�1) isprecompact then X 2 NPC�10 (X), and hence A 2 NPC�10 (X) for each A 2P0(X). By the previous result we conclude that ClT (UH)(K0(X)) = P0(X). �Proposition 2.3. Let (X;U) be a quasi-uniform space. Then it holds thatClT ((U�)H)(F0(X)) = PC�0(X) and hence ClT ((U�)H)(K�0(X)) = PC�0(X).



104 M. A. S�anchez-GraneroProof. Let A 2 ClT ((U�)H)(F0(X)), and let U 2 U. Then there exists F 2F0(X) such that F 2 (U�)H(A), and hence A � U�(F). Therefore A 2NPC�0(X) = PC�0(X).Conversely, let A 2 PC�0(X) and let U 2 U. Then there exists F 2 F0(X)such that F � A and A � U�(F). Then F 2 (U�)H(A) and hence A 2ClT ((U�)H)(F0(X)). �Corollary 2.4. Let (X;U) be a totally bounded quasi-uniform space. ThenK�0(X) is dense in (P0(X);(U�)H) and hence in (P0(X);(UH)�).Let us denote C�(F0(X)) = fA 2 P0(X) : there is a (U�)H-Cauchy net inF0(X) which T (UH)-converges to Ag, C(F0(X)) = fA 2 P0(X) : there is a leftK-Cauchy net in (F0(X);UH) which T (UH)-converges to Ag and C�1(F0(X)) =fA 2 P0(X) : there is a right K-Cauchy net in (F0(X);UH) which T (UH)-converges to Ag.The proof of the following result is a slight modi�cation of [10, Lemma 1].Proposition 2.5. Let (X;U) be a quasi-uniform space.(1) PC0(X) � C(F0(X)).(2) PC�10 (X) � C�1(F0(X)).(3) PC�0(X) = C�(F0(X)).Proof. Let us prove that PC0(X) � C(F0(X)). The proofs of PC�10 (X) �C�1(F0(X)) and PC�0(X) � C�(F0(X)) are analogous to this one.Let A 2 PC0(X). Let [A]<! be the set of nonempty �nite subsets of Adirected by set-theoretic inclusion. Then [A]<! can be considered a left K-Cauchy net in (F0(X);UH). Indeed, since A 2 PC0(X) for each U 2 U thereexists AU 2 [A]<! such that A � U(AU). Then for each B;C 2 [A]<! withAU � B � C we have that C � A � U(AU) � U(B) and B � C � U�1(C), andhence C 2 UH(B). On the other hand, it is clear that [A]<! T (UH)-convergesto A and hence A 2 C(F0(X)).Now, let us prove that C�(F0(X)) � PC�0(X). Let A 2 C�(F0(X)), thenthere exists a (U�)H-Cauchy net (Fi)i2I in F0(X) which T (UH)-converges toA. Given U 2 U, let V 2 U with V 2 � U, then there exists i 2 I such that Fi 2(V �)H(A). Then A � V �(Fi) and Fi � V �(A). Since Fi is �nite there existsB � A �nite and such that Fi � V �(B) and hence A � V � � V �(B) � U�(B).Therefore A is totally bounded, and hence A 2 PC�0(X). �3. Semicompleteness of the Bourbaki quasi-uniformityThe following concept is the main key of this paper.De�nition 3.1. Let (X;U) be a quasi-uniform space. (P0(X);UH) is said tobe semi-complete if each (U�)H-Cauchy net is T (UH)-convergent.Note that if (X;U) is a uniform space then (P0(X);UH) is semi-complete ifand only if it is complete.Since (UH)� � (U�)H ([11]), it follows that if (P0(X);U) is half completethen it is semi-complete.



Completeness of Bourbaki quasi-uniformity 105The following result shows that semicompleteness of (K0(X);UH) is a strongcondition.Proposition 3.2. Let (X;U) be a Hausdor� quasi-uniform space.(1) If (K0(X);UH) is semicomplete then C0(X) \ C�(F0(X)) � K0(X).(2) If (K�0(X);UH) is semicomplete then C0(X) \ C�(F0(X)) � K�0(X).(3) If (K0(X);UH) is left K-complete then C0(X) \ C(F0(X)) � K0(X).(4) If (K�0(X);UH) is left K-complete then C0(X) \ C(F0(X)) � K�0(X).(5) If (K0(X);UH) is right K-complete then C0(X)\C�1(F0(X)) � K0(X).(6) If (K�0(X);UH) is right K-complete then C0(X)\C�1(F0(X)) � K�0(X).Proof. Let us prove the �rst item. Suppose that (K0(X);UH) is semicompleteand let A 2 C0(X)\C�(F0(X)). Then there exists a (U�)H-Cauchy net (Fd)d2Din F0(X) which T (UH)-converges to A. Since (K0(X);UH) is semicomplete,there exists K 2 K0(X) such that the net T (UH)-converges also to K. Then itis easy to check that (Fd)d2D T (UH)-converges to A [ K.If A � K, since K is compact and A is closed then A is compact and henceA 2 K0(X). Suppose that A 6� K. Then there exists x 2 A n K. Since X isHausdor� and K is compact, there exists U 2 U such that U(x) \ U(K) = ?.Then UH(K [ A) \ UH(K) = ?. Indeed, if there is B 2 UH(K [ A) \ UH(K),then A � A [ K � U�1(B) � U�1 � U(K), and hence x 2 U�1 � U(K) whichcontradicts that U(x) \ U(K) = ?. Therefore UH(K [ A) \ UH(K) = ?, but(Fd)d2D converges to K [ A and converges to K, so there exists d0 2 D suchthat Fd0 2 UH(K [ A) \ UH(K). The contradiction shows that A � K, andA 2 K0(X) (note that this implies that A = K) .The rest of the items have an analogous proof. We only note that K�0(X) �K0(X) and C0(X) � C�0(X). �The next result provides a generalization of the Zenor-Morita theorem to thesetting of Hausdor� quasi-uniform spaces.Theorem 3.3. Let (X;U) be a Hausdor� quasi-uniform space. It follows that(K�0(X);UH) is semicomplete if and only if (X;U) is half-complete and C0(X)\C�(F0(X)) � K�0(X).Proof. Suppose that (K�0(X);UH) is semicomplete. It is easy to prove that(X;U) is half complete and C0(X) \ C�(F0(X)) � K�0(X) by Proposition 3.2.Conversely, suppose that (X;U) is half-complete. Let fC� : � 2 Dg be a(U�)H-Cauchy net in K�0(X). Let us show that fC� : � 2 Dg is convergent in(K�0(X);UH).For each � 2 D, let F� = S��� C�. Let F = filfF� : � 2 Dg.(1) Let F0 be an ultra�lter containing F. Let us prove that F0 is (U�)H-Cauchy. Let U 2 U and V 2 U with V 2 � U. Since fC� : � 2 Dg is(U�)H-Cauchy, there exists �0 2 D such that C� 2 (V �)H(C�0) for each� � �0 and hence C� � V �(C�0) for each � � �0. Since C�0 2 K�0(X)there exists a �nite subset B of C�0 such that C�0 � V �(B) and henceF�0 � V �(C�0) � V � �V �(B) � U�(B). Since F�0 2 F0, B is �nite and



106 M. A. S�anchez-GraneroF0 is an ultra�lter there exists b 2 B such that U�(b) 2 F0. ThereforeF0 is U�-Cauchy.Set C = T�2D F�. First, we note that C 6= ? by (1).(2) Let us prove that fC� : � 2 Dg T (UH)-converges to C. Let U 2 U (wecan suppose that U(x) is open for each x 2 X), then there exist V 2 Uwith V 2 � U and �0 2 D such that C � F� � V �1�V �(C�) � U�1(C�)for each � � �0. Suppose that F� 6� U(C) for each � � �0. LetG = filf(X n U(C)) \ F : F 2 Fg. It is clear that F � G. Let G0 be anultra�lter containing G. Analogous to (1), it can be proved that G0 is U�-Cauchy and hence it T (U)-converges to y0 2 X. Note that y0 2 C sinceF � G0 and y0 2 X n U(C) since X n U(C) is closed. The contradictionshows that there exists �1 � �0 such that C� � F�1 � U(C) for each� � �1. Therefore C� 2 UH(C) for each � � �1 and so fC� : � 2 DgT (UH)-converges to C.(3) Let us prove that C 2 K�0(X). For each U and � 2 D there existsF�;U � C� �nite such that C� 2 (U�)H(F�;U). Then the net fF�;U :(�;U) 2 D � Ug (where (�;U) � (�0;U 0) if and only if � � �0 andU 0 � U) is clearly (U�)H-Cauchy and T (UH)-convergent to C. ThenC 2 C0(X) \ C�(F0(X)) � K�0(X).Combining the previous arguments, (K�0(X);UH) is semicomplete. �Remark 3.4. In the proof of the previous theorem the hypothesis that X isHausdor� is only used in the only if part. Also note that K�0(X) � C0(X) \C�(F0(X)) if X is Hausdor�.Remark 3.5. Note that if (X;U) is a uniform space and (X;U) is complete thenC0(X)\C(F0(X)) = C0(X)\PC0(X) which is clearly a subset of K0(X), sincea closed totally bounded subspace in a complete uniform space is compact (alsonote that C(F0(X)) = C�1(F0(X)) = C�(F0(X)) and K0(X) = K�10 (X) =K�0(X)). Therefore Theorem 3.3 is a generalization of Zenor-Morita theorem.In order to generalize the Burdick-Isbell theorem, we will need the followingresult. Its proof is based on [11, Lemma 6].Lemma 3.6. Let (X;U) be a quasi-uniform space such that each stable �lteron (X;U�) has a cluster point in (X;U). Let F be a stable �lter on (X;U�) andlet C = TF2F F. Then U(C) 2 F for each U 2 U.Proof. Suppose that there exists U0 2 U such that E n U20 (C) 6= ? for eachE 2 F. Let HU;E = fa 2 X : There is V 2 U such that V 2 � U; V �(V �(a)) \U0(C) = ? and a 2 TF2F V �(F)\Eg. Given U 2 U and E 2 F, let V 2 U withV 2 � U0 \ U. Then it is easy to check that ? 6= (TF2F V �(F) \ E) n U20 (C) �HU;E, and hence HU;E 6= ? for each U 2 U and E 2 F.Note also that for any U1;U2 2 U such that U1 � U2 and any E1;E2 2 Fsuch that E1 � E2 we have that HU1;E1 � HU2;E2.Thus fHU;E : U 2 U;E 2 Fg is a base for a �lter H on X. In order toshow that H is stable on (X;U�), let U;V 2 U and E 2 F and let us provethat HU;X � U�(HV;E). Let a 2 HU;X, then there exists W 2 U such that



Completeness of Bourbaki quasi-uniformity 107W2 � U, W�(W�(a)) \ U0(C) = ? and a 2 TF2F W�(F). Choose Z 2 U suchthat Z2 � V \W . Since a 2 TF2F W�(F) and E \TF2F Z�(F) 2 F it followsthat there exists y 2 [E \ TF2F Z�(F)] \ W�(a). On the other hand, sinceZ�(Z�(y)) � W�(y) � W�(W�(a)) it follows that Z�(Z�(y)) \ U0(C) = ?. Weconclude that y 2 HV;E and since a 2 W�(y) � U�(y) we have that HU;X �U�(HV;E) and hence H is stable in (X;U�). Hence it has a cluster point x in(X;U) and since HX�X;F � F whenever F 2 F, it follows that x 2 C, but thisis a contradiction, since HV;E \ U0(C) = ? and hence HV;E \ C = ? for eachV 2 U and E 2 F. The contradiction shows that U20 (C) 2 F for each U0 2 Uand hence U(C) 2 F for each U 2 U. �Now, we generalize the Burdick-Isbell theorem to the setting of quasi-uniformspaces.Theorem 3.7. Let (X;U) be a quasi-uniform space. Then (P0(X);UH) issemicomplete if and only if each stable �lter on (X;U�) has a cluster point in(X;U).Proof. Suppose that (P0(X);UH) is semicomplete, and let F be a stable �lter on(X;U�). Consider the net (F)F2(F;�) on P0(X). Let U 2 U. Since F is stableon (X;U�) there exists FU 2 F such that FU � U�(F) for each F 2 F. Thus,for each F 2 F with F � FU, we have that FU � U�(F) and F � FU � U�(FU),so F 2 (U�)H(FU) for each F � FU and hence (F)F2(F;�) is a (U�)H-Cauchynet on P0(X). Since (P0(X);UH) is semicomplete, the net T (UH)-converges tosome C 2 P0(X). It is easy to see that x is a cluster point of F for each x 2 C.Conversely, suppose that each stable �lter on (X;U�) has a cluster point in(X;U). Let (Cd)d2D be a (U�)H-Cauchy net on P0(X). For each d 2 D, letFd = Se�d Ce and set F = filfFd : d 2 Dg on X. Let U 2 U and V 2 U withV 2 � U, then there exists dV 2 D such that Cd 2 (V �)H(CdV ) for each d � dV .Then Cd � V �(CdV ) and CdV � V �(Cd) for each d � dV . It follows thatFd � V �(CdV ) for each d � dV . In order to prove that FdV � Td2D U�(Fd), letx 2 FdV and d 2 D. Let h � d;dV , then x 2 FdV � V �(CdV ) � V �(V �(Ch)) �U�(Fd) and hence FdV � Td2D U�(Fd). Therefore F is a stable �lter on (X;U�).Let C = TF2F F . Since F is stable on (X;U�) it follows from the hypothesisthat C 6= ?. Let us prove that (Cd)d2D converges to C in (P0(X);UH). LetU 2 U and let V 2 U such that V 3 � U. It is clear that C � Fd � V �1(Fd) �V �1(V �(CdV )) � V �1(V �(V �(Cd))) � U�1(Cd) for each d � dV .On the other hand, by Lemma 3.6 we have that U(C) 2 F, and hence thereexists d0 � dV such that Fd0 � U(C), and hence Cd � U(C) for each d � d0.Therefore (Cd)d2D converges to C in (P0(X);UH). �The next result is a generalization of the following result: a metric space(X;d) is complete if and only if (P0(X);dH) is complete.Corollary 3.8. Let (X;d) be a quasi-pseudometric space. Then (X;d) is halfcomplete if and only if (P0(X);(U(d))H) is semicomplete (where U(d) denotesthe quasi-uniformity induced by d).



108 M. A. S�anchez-GraneroProof. Let Un = f(x;y) 2 X � X : d(x;y) < 12ng.By Theorem 3.7 we only have to prove that any stable �lter on (X;d�) has acluster point in (X;d) if X is half complete. Let F be a stable �lter on (X;d�),then for each n 2 N there exists Fn 2 F such that Fn � TF2F U�n(F). Letx1 2 F1, suppose that we have de�ned xn, and de�ne xn+1 as follows: SinceFn � U�n(Fn+1), let xn+1 2 Fn+1 \U�n(xn). Then it is easy to check that (xn) isa Cauchy sequence in (X;d�) and since X is half complete it converges to somepoint x 2 X. It is clear that x is a cluster point of F in (X;d). �Remark 3.9. Let (X;U) be a quasi-uniform space. From the Burdick-Isbelltheorem it follows that the uniform space (P0(X);(U�)H) is complete if andonly if each stable �lter on (X;U�) has a cluster point in (X;U�). Of course,(P0(X);UH) is semicomplete if (P0(X);(U�)H) is complete.On the other hand, for any half complete non-bicomplete quasi-metric space(X;d) it follows that (P0(X);(U(d)�)H) is not complete, but (P0(X);UH) issemicomplete by Corollary 3.8.The proof of the following proposition is analogous to the proof of [11, Propo-sition 7].Proposition 3.10. Let (X;U) and (Y;V) be quasi-uniform spaces and f :(X;U�) ! (Y;V�) be a uniformly continuous surjection that is T (U)-T (V)-perfect. If (P0(Y );VH) is semicomplete then (P0(X);UH) is semicomplete.Proof. Let F be a stable �lter on (X;U�) and let V 2 V. Since f is U�-V�-uniformly continuous there is U 2 U such that (f � f)(U�) � V �. Since Fis stable on (X;U�), there is F0 2 F such that F0 � U�(F) for each F 2 F,and hence f(F0) � V �(f(F)) for each F 2 F. Therefore the �lter f(F) :=ff(F) : F 2 Fg is stable on (Y;V�). Since (P0(Y );VH) is semicomplete, itfollows from Theorem 3.7 that f(F) has a cluster point y0 in (Y;V). Since fis T (U)-T (V)-perfect the �lter F has a cluster point x0 2 f�1(y0) in (X;U).Therefore (P0(X);UH) is semicomplete by Theorem 3.7. �Corollary 3.11. Let (X;U) and (Y;V) be quasi-uniform spaces and let f :(X;U) ! (Y;V) be a quasi-uniformly continuous surjection that is perfect. If(P0(Y );VH) is semicomplete then (P0(X);UH) is semicomplete.Corollary 3.12. Let (X;V) be a quasi-uniform space such that (P0(X);VH) issemicomplete. Then for any compatible quasi-uniformity U �ner than V on X,it follows that (P0(X);UH) is semicomplete.A condition for bicompleteness of (P0(X);UH) in terms of doubly stable�lters was given in [11].Recall that a �lter F on a quasi-uniform space (X;U) is said to be doublystable ([11]) provided that TF2F(U(F)\U�1(F)) belongs to F for each U 2 U.The next result characterizes half completeness of (P0(X);UH) in terms ofdoubly stable �lters.Proposition 3.13. Let (X;U) be a quasi-uniform space. Then (P0(X);UH)is half complete if and only if each doubly stable �lter on (X;U) veri�es that



Completeness of Bourbaki quasi-uniformity 109U(C) 2 F for each U 2 U (where C denotes the set of T (U)-cluster points ofF).Proof. Suppose that (P0(X);UH) is half complete, and let F be a doubly stable�lter on (X;U). Consider the net (F)F2(F;�) on P0(X). Let U 2 U. Since Fis doubly stable on (X;U) there exists FU 2 F such that FU � U(F) \U�1(F)for each F 2 F. Thus, for each F 2 F with F � FU, we have that FU �U(F) \ U�1(F) and F � FU � U(FU) \ U�1(FU), so F 2 (UH)�(FU) foreach F � FU and hence (F)F2(F;�) is a (UH)�-Cauchy net on P0(X). Since(P0(X);UH) is half complete, the net T (UH)-converges to some D 2 P0(X).It is easy to see that x is a cluster point of F in (X;U) whenever x 2 D, andhence, if C is the set of T (U)-cluster points of F we have that D � C. On theother hand, given U 2 U there exists F0 2 F such that F0 2 UH(D), and henceF0 � U(D) � U(C). Therefore U(C) 2 F for each U 2 U.Conversely, suppose that each doubly stable �lter on (X;U) veri�es U(C) 2 F(where C denotes the set of T (U)-cluster points of F). Let (Cd)d2D be a(UH)�-Cauchy net on P0(X). For each d 2 D, let Fd = Se�d Ce and setF = filfFd : d 2 Dg on X. Let U 2 U and V 2 U with V 2 � U, thenthere exists dV 2 D such that Cd 2 (VH)�(CdV ) for each d � dV . ThenCd � V (CdV ) \ V �1(CdV ) and CdV � V (Cd) \ V �1(Cd) for each d � dV . Itfollows that Fd � V (CdV ) \ V �1(CdV ) for each d � dV . In order to provethat FdV � Td2D U(Fd) \ U�1(Fd), let d 2 D, and h � d;dV , then FdV �V (CdV ) \ V �1(CdV ) � V � V (Ch) \ V �1 � V �1(Ch) � U(Fd) \ U�1(Fd) andhence FdV � Td2D U(Fd) \ U�1(Fd). Therefore F is a doubly stable �lter on(X;U).Let C = TF2F F (that is, C is the set of T (U)-cluster points of F). SinceF is doubly stable on (X;U) it follows from the hypothesis that C 6= ? andthat U(C) 2 F for each U 2 U. Let us prove that (Cd)d2D converges to C in(P0(X);UH). Let U 2 U and let V 2 U such that V 3 � U. It is clear that C �Fd � V �1(Fd) � V �1(V (CdV ) \V �1(CdV )) � V �1(V �1(V �1(Cd))) � U�1(Cd)for each d � dV .On the other hand, since U(C) 2 F, there exists d0 2 D such that Fd0 �U(C) and hence Cd � U(C) for each d � d0. Therefore (Cd)d2D converges toC in (P0(X);UH). �Example 3.14. Let Q be the rationals with the Sorgenfrey quasi-metric dS.Then (Q;dS ) is bicomplete and hence (P0(Q);UH ) is semicomplete by Corollary3.8 (in fact, (P0(Q); (U�)H) is complete), but (P0(X);UH) is not half complete.Indeed, by [11, Example 7] there is a doubly stable �lter F on (Q;dS ) withoutcluster point in (Q;dS ). Then (P0(X);UH) is not half complete by Proposition3.13.The next result is a simpler proof of [11, Proposition 5]. First, we prove alemma.Lemma 3.15. Let (X;d) be a quasi-pseudometric space. Then d is right K-sequentially complete if and only if whenever (U�1n (xn)) is a decreasing sequence



110 M. A. S�anchez-Granero(that is, xn+1 2 U�1n (xn) for each n 2 N) it follows that Tn2N U�1n�2(xn) 6= ?(where Un = f(x;y) 2 X � X : d(x;y) < 12ng).Proof. Suppose that whenever (U�1n (xn)) is a decreasing sequence it follows thatTn2N U�1n�2(xn) 6= ?, and let (xn) be a right K-Cauchy sequence. Then thereexists (xm(n)) a subsequence of (xn) such that xm(n+1) 2 U�1n (xm(n)) for eachn 2 N. By hypothesis there exists x 2 Tn2N U�1n�2(xm(n)), and hence (xm(n))converges to x. Since each right K-Cauchy sequence converges to its clusterpoints it follows that (xn) is convergent. Therefore d is right K-sequentiallycomplete.Suppose that d is right K-sequentially complete and let (xn) be a sequencewith xn+1 2 U�1n (xn). Then d(xn+h+l;xn+h) � Pl�1i=0 d(xn+h+i+1;xx+h+i) �12n�1 , so xk 2 U�1n�1(xm) for each k � m � n. Therefore (xn) is a right K-Cauchysequence and hence it converges to some x 2 X. Now, d(x;xn) � d(x;xk) +d(xk;xn) < 12n + 12n�1 < 12n�2 for a suitable k 2 N and hence x 2 U�1n�2(xn) foreach n 2 N. �Theorem 3.16. [11, Proposition 5] Let (X;d) be a quasi-pseudometric space.The following statements are equivalent:(1) (X;d) is right K-sequentially complete.(2) (P0(X);(U(d))H) is right K-sequentially complete.(3) (ClT (UH)(F0(X));(U(d))H) is right K-sequentially complete.Proof. Let Un = f(x;y) 2 X � X : d(x;y) < 12ng.2) implies 3) is clear.3) implies 1). Let (xn)n2N be a right K-Cauchy sequence in (X;d). Then itis clear that (fxng)n2N is a right K-Cauchy sequence in ClT (UH)(F0(X)), andsince it is right K-sequentially complete there exists C 2 ClT (UH)(F0(X)) suchthat (fxng)n2N T (UH)-converges to C. Let x0 2 C, then it is easy to check thatx0 is a cluster point of (xn)n2N, and since a right K-Cauchy sequence convergesto its cluster points, it follows that (xn)n2N is convergent and then (X;d) isright K-sequentially complete.1) implies 2). Suppose that X is right K-sequentially complete, and let (Fn)be a sequence in P0(X) such that Fn+1 2 (UH)�1n (Fn). Let F = Tn2N U�1n�2(Fn),then it is clear that F � U�1n�2(Fn) for each n 2 N. On the other hand,given xn 2 Fn, since Fn+1 2 (UH)�1n (Fn) and Fn 2 (UH)�1n�1(Fn�1), it followsthat Fn+1 � U�1n (Fn) and Fn�1 � Un(Fn), and then it is clear that we canconstruct a sequence (yk), with yn = xn, yk+1 2 U�1k (yk) and yk 2 Fk for eachk 2 N, and hence (U�1k (yk)) is a decreasing sequence, and since X is rightK-sequentially complete, by Lemma 3.15 there exists x 2 Tk2N U�1k�2(yk). Itis clear that x 2 F and xn = yn 2 Un�2(x). Therefore Fn � Un�2(F), andhence F 2 (UH)�1n�2(Fn) for each n 2 N, that is, F 2 Tn2N(UH)�1n�2(Fn), andby Lemma 3.15 (P0(X);UH) is right K-sequentially complete. �Remark 3.17. Let (X;U) be a quasi-uniform space. We have noted that if(P0(X);UH) is half complete or (P0(X);(U�)H) is complete then (P0(X);UH) is



Completeness of Bourbaki quasi-uniformity 111semicomplete. In Example 3.14 we proved that completeness of (P0(X), (U�)H)does not imply half completeness of (P0(X);UH).On the other hand, by Theorem 3.16, if (X;U) is any right K-sequentiallycomplete non-bicomplete quasi-metrizable quasi-uniform space it follows that(P0(X);UH) is half complete, but (P0(X);(U�)H) is not complete. Note ([14],[1]) that sequentially half-completeness and half-completeness are equivalent forquasi-pseudometrizable quasi-uniform spaces.Finally, note that bicompleteness of (P0(X);UH) implies that (P0(X);UH) ishalf complete, and in the light of Example 3.14 completeness of (P0(X), (U�)H)does not imply bicompleteness of (P0(X);UH).Acknowledgements. The author is grateful to the referee for his valuablesuggestions. References[1] E. Alemany and S. Romaguera, On half-completion and bicompletion of quasi-metric spaces,Comment. Math. Univ. Carolinae, 37, 4 (1996), 749-756.[2] E. Alemany and S.Romaguera, On right K-sequentially complete quasi-metric spaces, Acta Math.Hung. 75 (1997), 267-278.[3] G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977), 335-343.[4] B.S. Burdick, A note on completeness of hyperspaces, General Topology and Applications:FifthNortheast conference, ed. by S.J. Andima et al., Marcel Dekker, 1991, 19-24.[5] J. Cao, H.P.A. K�unzi, I.L. Reilly, S. Romaguera, Quasi-uniform hyperspaces of compact subsets,Topology and its Appl. 87 (1998), 117-126.[6] A. Cs�asz�ar, Strongly complete, supercomplete and ultracomplete spaces, Mathematical Structures-Computational Mathematics- Mathematical Modelling, Papers dedicated to Prof. L. Iliev's 60thAnniversary, So�a, 1975, p. 195-202.[7] J. De�ak, On the coincidence of some notions of quasi-uniform completeness de�ned by �lterspairs, Studia Sci. Math. Hungar. 26 (1991), 411-413.[8] P. Fletcher and W.F. Lindgren, Quasi-Uniform Spaces, Lecture Notes in Pure and Appl. Math.77, Marcel Dekker, New York, 1982.[9] J.R. Isbell, Uniform spaces, Mathematical Surveys, No. 12, American Mathematical Society,Providence, Rhode Island, 1964.[10] H.P.A. K�unzi, S. Romaguera, Left K-completeness of the Hausdor� quasi-uniformity, Rostock.Math. Kolloq. 51 (1997), 167-176.[11] H.P.A. K�unzi, C. Ryser, The Bourbaki quasi-uniformity, Top. Proc. 20 (1995), 161-182.[12] I.L. Reilly, P.V. Subrahmanyam and M.K. Vamanamurthy, Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 93 (1982), 127-140.[13] S. Romaguera, On hereditary precompactness and completeness in quasi-uniform spaces, ActaMath. Hungar. 73 (1996), 159-178.[14] S. Salbany, S. Romaguera, On countably compact quasi-pseudometrizable spaces, J. Austral.Math. Soc. (Series A) 49 (1990), 231-240. Received October 2000Revised version December 2000



112 M. A. S�anchez-GraneroM.A. S�anchez GraneroArea of Geometry and TopologyFaculty of ScienceUniversidad de Almer��a04120 Almer��aSpainE-mail address: misanche@ual.es