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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 9 - 25

Fell type topologies of quasi-pseudo-metricspaces and the Kuratowski-Painlev�econvergenceJes�us Rodr��guez-L�opezAbstract. We study the double Fell topology when thishypertopology is constructed over a quasi-pseudo-metric space.In particular, its relationship with the Wijsman hypertopology isstudied. We also propose an extension of the Kuratowski-Painlev�econvergence in the bitopological setting.2000 AMS Classi�cation: 54B20, 54E35, 54E55.Keywords: quasi-pseudo-metric, double topological space, Fell topology, Wi-jsman topology, Kuratowski-Painlev�e convergence.1. Introduction and preliminariesRecently, the study of nonsymmetric structures has received a new drive asa consequence of its applications to Computer Science. This theory began withSmyth (see [20, 21]). He tried to �nd a convenient category for computation andhe proposed the quasi-uniform spaces as the suitable context. Continuing thework of Smyth, other authors have applied the nonsymmetric topology to thisarea (see [17, 18, 19]). Furthermore, some hypertopologies have been success-fully applied to several areas of Computer Science (see [21, 23]). All these factsmotivate our interest in the nonsymmetric study of several hypertopologies. Inthis paper, we continue the work developed by the author in [15].The Fell topology was introduced by Fell in [8]. In [15] it is introduceda de�nition for the Fell hypertopology in the nonsymmetric situation. Somesatisfactory results about the relationship of some hypertopologies with theFell topology are obtained in the quasi-uniform setting. We continue this workand obtain extensions of well-known results in the symmetric case about therelationship between the Fell and the Wijsman hypertopologies. We also studya de�nition for the Kuratowski-Painlev�e convergence in the bitopological settingand obtain extensions of interesting results as the Mrowka's Theorem.Our basic references for quasi-uniform and quasi-pseudo-metric spaces are[9] and [12]. Terms and unde�ned concepts may be found in such references.



10 J. Rodr��guez-L�opezA quasi-pseudo-metric on a set X is a nonnegative real valued function don X � X such that for all x;y;z 2 X : (i) d(x;x) = 0 and (ii) d(x;y) �d(x;z) + d(z;y):If, in addition, d satis�es the condition: (iii) d(x;y) = 0 ) x = y; then d issaid to be a quasi-metric on X:A quasi-(pseudo-)metric space is a pair (X;d) such that X is a nonempty setand d is a quasi-(pseudo-)metric on X:If d is a quasi-(pseudo-)metric on X; then the function d�1 de�ned on X �Xby d�1(x;y) = d(y;x) for all x;y 2 X, is also a quasi-(pseudo-)metric on X;called the conjugate quasi-pseudo-metric of d, and the function ds de�ned onX�X by ds(x;y) = maxfd(x;y);d�1(x;y)g for all x;y 2 X; is a (pseudo-)metricon X:Each quasi-pseudo-metric d on X generates a topology T (d) on X which hasas a base the family of balls of the form Bd(x;r) = fy 2 X : d(x;y) < rg; wherex 2 X and r > 0: Note that if d is a quasi-metric, then T (d) is a T1 topologyon X: We denote Bd(x;") = fy 2 X : d(x;y) � "g.The quasi-pseudo-metric ` on R de�ned by `(x;y) = maxfx � y;0g for allx;y 2 R, is called the lower quasi-pseudo-metric on R. Its conjugate quasi-pseudo-metric `�1 is denoted by u and is called the upper quasi-pseudo-metricon R. Note that `s = ` _ u is the usual metric on R. A function from atopological space (X;T ) to R is said to be lower semicontinuous (resp. uppersemicontinuous) if it is continuous when we consider the topology generated bythe lower (resp. upper) quasi-pseudo-metric on R.A quasi-uniformity on a set X is a �lter U on X � X which satis�es: (i)� � U for all U 2 U and (ii) given U 2 U there exists V 2 U such thatV 2 � U, where � = f(x;x) : x 2 Xg and V 2 = f(x;z) 2 X � X : exists y 2X such that (x;y) 2 V;(y;z) 2 V g. The elements of U are called entourages.The �lter U�1, formed by all sets of the form U�1 = f(x;y) 2 X � X :(y;x) 2 Ug where U 2 U, is a quasi-uniformity on X called the conjugatequasi-uniformity of U.If U is a quasi-uniformity on X, then the family fUs = U \ U�1 : U 2 Ugis a base for a quasi-uniformity Us (in fact, it is a uniformity), which is thecoarsest uniformity containing U. This uniformity is called the supremum ofthe quasi-uniformities U and U�1.Every quasi-uniformity U generates a topology T (U) on X. A neighborhoodbase for each point x 2 X is given by fU(x) : U 2 Ug where U(x) = fy 2 X :(x;y) 2 Ug.Each quasi-pseudo-metric d on X induces a quasi-uniformity Ud on X whichhas as a base the family of entourages of the form f(x;y) 2 X � X : d(x;y) <2�ng; n 2 N. Moreover, T (Ud) = T (d).In addition of a quasi-uniformity and a quasi-pseudo-metric, we can de�neon a space another structure which makes precise the concept of nearness. Thisstructure is a relation � in P0(X). We write A�B for (A;B) 2 � and A�Binstead of (A;B) 62 �.



Fell type topologies and the Kuratowski-Painlev�e convergence 11De�nition 1.1. Let X be a nonempty set. A relation � in P0(X) is a quasi-proximity for X if it satis�es the following conditions:i) X�? and ?�X:ii) C�(A [ B) if and only if C�A or C�B(A [ B)�C if and only if A�C or B�C.iii) fxg�fxg for each x 2 X.iv) If A�B, there exists C 2 P0(X) such that A�C and (XnC)�B.The pair (X;�) is called a quasi-proximity space.Obviously, if � is a quasi-proximity on X, then so is the opposite relation��1. This quasi-proximity is called the conjugate quasi-proximity of �. A quasi-proximity � is a proximity if � = ��1.Let A and B be subsets of a quasi-proximity space (X;�). If A�B , then Ais said to be near B and if A�B, then A is said to be far from B. A set B issaid to be a �-neighborhood of a set A if A�(XnB).Every quasi-proximity � on a space X induces in a natural way a topologyon X. If x 2 X, the neighborhoods of x are the �-neighborhoods of x.Furthermore, if (X;U) is a quasi-uniform space, then U induces a quasi-proximity �U such that A�UB if and only if (A � B) \ U 6= ? for all U 2 U.A bitopological space (see [10, 13]) is a triple (X;P;Q) where X is a set andP and Q are topologies on X. A bitopological space is said to be quasi-pseudo-metrizable (resp. quasi-uniformizable) if there exists a quasi-pseudo-metric d(resp. a quasi-uniformity U) on X such that T (d) = P and T (d�1) = Q (resp.T (U) = P and T (U�1) = Q). In this case we say that d (resp. U) is a quasi-pseudo-metric (resp. quasi-uniformity) compatible with the bitopological space(X;P;Q).Given a topological space (X;T ) we denote by P0(X) the family of nonemptysubsets of X and by CL0(X) we denote the family of nonempty closed subsetsof X: We also shall use P(X) = P0(X) [ ?. If (X;P;Q) is a bitopologicalspace we denote by CLP0 (X) (resp. CLQ0 (X), CLs0(X)) the family of nonemptyP-closed (resp. Q-closed, P _ Q-closed) subsets of X:2. Fell type topologies of quasi-pseudo-metric spacesIn [15] it can be found a discussion about the de�nition of the Fell hyper-topology in the nonsymmetric case. We propose the use of double topologicalspaces rather than bitopological spaces. We recall some de�nitions.De�nition 2.1 ([15]). A double topological space is simply a pair of topologicalspaces ((X;�);(Y;�)).In the following, we will also use double space.De�nition 2.2 ([15]). Let (X;P;Q) be a bitopological space. We de�ne thedouble upper Fell topological space as the double topological space ((CLQ0 (X),F+P ), (CLP0 (X), F+Q )) where F+P is the topology generated by all sets of theform G+ = fA 2 CLQ0 (X) : A � Gg where G is a P-open set and XnG is



12 J. Rodr��guez-L�opezP _ Q-compact; F+Q is de�ned in a similar way by writing P instead of Q andQ instead of P. The pair (F+P ;F+Q ) is called the double upper Fell topology.The double lower Fell topological space is de�ned as the double topologicalspace ((CLQ0 (X);F�P ),(CLP0 (X);F�Q )) where F�P is generated by all sets of theform G� = fA 2 CLQ0 (X) : A\G 6= ?g where G is P-open; F�Q is de�ned in asimilar way by writing P instead of Q and Q instead of P. The pair (F�P ;F�Q )is called the double lower Fell topology.The double Fell topological space is the double topological space ((CLQ0 (X),FP ), (CLP0 (X), FQ)) where FP = F+P _ F�P and FQ = F+Q _ F�Q . The pair(FP ;FQ) is called the double Fell topology.In this section we study the relationship between the double Fell topologyand the double Wijsman topology. We also motivate the fact of consideringP _ Q-compact sets in the de�nition of the double Fell topology (see Remark2.10). The results in the symmetric case can be found in [3] and [4].The following is an extension of the Wijsman hypertopology de�nition whend is a quasi-pseudo-metric (see [16]).De�nition 2.3. Let (X;d) be a quasi-pseudo-metric space. Let P = T (d)and Q = T (d�1). The double upper Wijsman topological space is the doubletopological space ((CLQ0 (X);T +(Wd));(CLP0 (X);T +(Wd�1))) where T +(Wd) isthe weakest topology on CLQ0 (X) such that for each x 2 X, the functionald(�;x) is lower semicontinuous on CLQ0 (X). The de�nition for T +(Wd�1) issymmetric. The pair (T +(Wd);T +(Wd�1)) is called the double upper Wijsmantopology.The double lower Wijsman topological space is the double topological space((CLQ0 (X);T �(Wd)), (CLP0 (X);T �(Wd�1))) where T �(Wd) is the weakest top-ology on CLQ0 (X) such that for each x 2 X, the functional d(x; �) is uppersemicontinuous on CLQ0 (X). The de�nition for T �(Wd�1) is symmetric. Thepair (T �(Wd);T �(Wd�1)) is called the double lower Wijsman topology.The double topological space ((CLQ0 (X);T (Wd));(CLP0 (X);T (Wd�1))) whereT (Wd) = T +(Wd) _ T �(Wd) and T(Wd�1) = T +(Wd�1) _ T �(Wd�1) is calledthe double Wijsman topological space. The pair (T (Wd);T (Wd�1)) is calledthe double Wijsman topology.Proposition 2.4. Let (X;P;Q) be a quasi-pseudo-metrizable bitopological space.Then F�P = T �(Wd), F�Q = T �(Wd�1) on P0(X) and F+P � T +(Wd) onCLQ0 (X) and F+Q � T +(Wd�1) on CLP0 (X) where d is a quasi-pseudo-metriccompatible with the bitopological space.Proof. It is easy to show that d(x; �)�1(�1;�) = Bd(x;�)� so we obtain thatF�P = T �(Wd) on P0(X): In a similar way, it can be proved F�Q = T �(Wd�1)on P0(X):Now, we show that F+P � T +(Wd) on CLQ0 (X). Let G be a P-open set suchthat XnG is P _ Q-compact, and A 2 G+. For all x 2 XnG, let us consider



Fell type topologies and the Kuratowski-Painlev�e convergence 13"x = d(A;x). Since A is a Q-closed set then "x > 0. Choose 0 < �x < "x:Thus fBd�1(x;�x) : x 2 XnGg is a Q-open cover of XnG so there existsfx1; : : : ;xng � XnG such thatXnG � n[i=1 Bd�1(xi;�xi):Consider the T +(Wd)-open set C = \ni=1d(�;xi)�1(�xi;+1). Clearly, A 2 C.Let us see that C � G+. Let B 2 C and suppose that B \ (XnG) 6= ?.Given b 2 B \ (XnG) there exists i 2 f1; : : : ;ng such that d�1(xi;b) < �xi. Acontradiction with d(B;xi) > �xi. Thus, B 2 C � G+, i.e. G+ 2 T +(Wd).Similarly, we prove F+Q � T +(Wd�1) on CLP0 (X). �Remark 2.5. We give an example showing that the above Proposition is nottrue when we de�ne F+P and T +(Wd) on CLP0 (X) and F+Q and T +(Wd�1) onCLQ0 (X).We consider the set N [ f1g with the following quasi-metric:8>>>>>><>>>>>>:
d(n;m) = 1 if n 6= md(n;n) = 0 for all n 2 Nd(n;1) = 1n for all n 2 Nd(1;n) = 1 for all n 2 Nd(1;1) = 0 :It is evident that T (d) = P is the discrete topology. Therefore N is a T (d)-clopen set, and its complement is obviously a T (ds)-compact set, so we considerthe F+P -open set N+. We shall prove that the set N 2 N+ has not a T +(Wd)-neighborhood contained in N+. If n 2 N and N 2 d(�;n)�1(�;+1) where� 2 R we deduce that � < 0 but f1g is a T (d)-closed set which belongs tod(�;n)�1(�;+1) = CLP0 (X) so this set is not contained in N+. On the otherhand, if N 2 d(�;1)�1(�;+1) we obtain the same contradiction.It is natural to wonder when the Wijsman and Fell hypertopologies agree.The following extension of a concept introduced by Beer in [1] and reformulatedin [2], gives us the answer.De�nition 2.6. Let (X;d) be a quasi-pseudo-metric space. We say that it hasnice closed balls if the proper closed d-balls and the proper closed d�1-balls areT (ds)-compact.Now, we can extend a result which can be found in [3].Theorem 2.7. Let (X;d) be a quasi-pseudo-metric space, P = T (d) and Q =T (d�1). Then FP = T (Wd) on CLQ0 (X) and FQ = T (Wd�1) on CLP0 (X) ifand only if (X;d) has nice closed balls.Proof. Let us suppose that there exists a proper closed d�1-ball Bd�1(x;�)which is not P _ Q-compact. Therefore, there is y0 2 X such that d(y0;x) > �



14 J. Rodr��guez-L�opezand there exists a sequence fxngn2N � Bd�1(x;�) which does not admit a P_Q-cluster point in X since Bd�1(x;�) is a P-closed set. Let An = fxngQ [ fy0gQfor all n 2 N. Let us prove that the sequence fAngn2N FP -converges to fy0gQ.Let V + \ V �1 \ : : : \ V �n be an FP -open set containing fy0gQ. Clearly An 2V �1 \ : : : \ V �n . Suppose now, to obtain a contradiction, that given k 2 N wecan �nd nk � k such that Ank 62 V +. Choose ynk 2 Ank such that ynk 62 V forall k 2 N. Since Ank = fxnkgQ [ fy0gQ and fy0gQ 2 V +, it is easy to showthat xnk 62 V . Hence, since XnV is a P _ Q-compact set, fxnkgk2N admits aP _Q-cluster point z. A contradiction, so fAngn2N is FP -convergent to fy0gQ.Let us show now that fd(An;x)gn2N does not converge to d(fy0gQ;x) in thelower topology of R, i.e. fAngn2N is not T (Wd)-convergent to fy0gQ. We havethat d(An;x) � d(xn;x) � �. Moreover, d(fy0gQ;x) > � since if z 2 fy0gQthen d(y0;z) = 0, so � < d(y0;x) � d(y0;z) + d(z;x) = d(z;x). Therefored(fy0gQ;x) � d(An;x) � d(fy0gQ;x) � � > 0:Consequently, FP 6= T (Wd) on CLQ0 (X). A contradiction.If there is a proper closed d-ball Bd(x;�) which is not T (ds)-compact, we canprove the statement in a similar way.Suppose now that (X;d) has nice closed balls. By Proposition 2.4, weonly have to show that T +(Wd) � F+P on CLQ0 (X) and T +(Wd�1) � F+Qon CLP0 (X). Let x 2 X and � � 0. Let us consider the T +(Wd)-open setd(�;x)�1(�;+1). We �rst suppose that Bd�1(x;�) 6= X. Fix � > � such thatBd�1(x;�) is not equal to X. Then Bd�1(x;�) is P_Q-compact. If A 2 CLQ0 (X)and d(A;x) = �, we can �nd a sequence fangn2N � Bd�1(x;�) \ A such thatfd(an;x)gn2N converges to d(A;x). Since Bd�1(x;�) is P _ Q-compact, thereis a P _ Q-convergent subsequence fankgk2N of fangn2N. If we denote by a itslimit we obtain that a 2 A and d(a;x) = �. Therefore, d(�;x)�1(�;+1) =(XnBd�1(x;�))+ which is a FP -open set.On the other hand, if Bd�1(x;�) = X then d(�;x)�1(�;+1) = ? 2 FP .In a similar way it can be proved T +(Wd�1) � F+Q on CLP0 (X): �Remark 2.8. We observe that by using the above proof, it can be shown:F+P = T +(Wd) on CLQ0 (X) and F+Q = T (W+d�1) on CLP0 (X) if and only if(X;d) has nice closed balls. Therefore, we deduce that the double Fell topologyagrees with the double Wijsman topology if and only if the double upper Felltopology agrees with the double upper Wijsman topology.In the following Remark, we give an example where the above theorem doesnot work if we change either the de�nition of the double Fell topology or thede�nition of nice closed balls. We will use the following de�nition.De�nition 2.9. Let (X;P;Q) be a bitopological space. We de�ne the doubleupper Vietoris topological space as the double topological space ((CLQ0 (X);V +P ),(CLP0 (X);V +Q )) where V +P is the topology generated by all sets of the form



Fell type topologies and the Kuratowski-Painlev�e convergence 15G+ = fA 2 CLQ0 (X) : A � Gg where G is a P-open set; V +Q is de�ned in asimilar way by writing P instead of Q and Q instead of P.The double lower Vietoris topological space is de�ned as the double topologicalspace ((CLQ0 (X);V �P );(CLP0 (X);V �Q )) where V �P is generated by all sets of theform G� = fA 2 CLQ0 (X) : A \ G 6= ?g where G is P-open; V �Q is de�ned ina similar way by writing P instead of Q and Q instead of P.The double Vietoris topological space is de�ned as the double topological space((CLQ0 (X);VP ), (CLP0 (X);VQ)) where VP = V +P _ V �P and VQ = V +Q _ V �Q .Remark 2.10. Now we motivate one fact about the de�nition of the doubleFell topology. We think that, maybe, the natural de�nition for FP is to bethe topology generated by the sets of the form G+ and V � where G and Vare P-open sets and XnG is Q-compact. In a similar way, we de�ne the FQhypertopology. We give an example where, with this de�nition, Theorem 2.7 isnot true.Let d be the quasi-metric on N given byd(n;m) = 8><>: 1m if n < m1 if n > m0 if n = m :We consider the quasi-metric space (N;d). Let P = T (d) and Q = T (d�1):We claim that FP = VP on P0(X) and FQ = VQ on P0(X) . Since Proposition2.4 is also true with this de�nition for the double Fell topology, we can deduce,using that T (Wd) � VP and T (Wd�1) � VQ on P0(X) (see [16]), that FP =T (Wd) on CLQ0 (X) and FQ = T (Wd�1) on CLP0 (X) but (X;d) has not niceclosed balls. We only have to prove that V +P � F+P and V +Q � F+Q .Let G 2 P and we consider the V +P -open set G+. Since G is a T (d)-openset, it easy to prove that NnG is a �nite set, so it is Q-compact. Therefore,G+ 2 F+P so V +P = F+P on P0(X) .On the other hand, let us suppose that G 2 Q and we consider the V +Q -openset G+. It is clear that every subset of X is P-compact. Hence, F+Q = V +Q onP0(X) . We observe that this statement is not true if we consider the topologyP _ Q, since it is the discrete topology.We consider the closed ball Bd(n;1=n) = fn;n + 1; : : :g. It is evident thatthis set is not P _ Q-compact.We notice that if we change the de�nition of a quasi-pseudo-metric spacehaving nice closed balls by saying that a quasi-pseudo-metric space has thisproperty if the proper closed d-balls are T (d�1)-compact and the proper closedd�1-balls are T (d)-compact the result is not true either. The preceding exampleshows that. The above ball is not Q-compact, since it is an in�nite set and Qis the discrete topology.Remark 2.11. We claim that if (X;d) is a quasi-metric space having niceclosed balls then T (d) = T (d�1). Let us show this.



16 J. Rodr��guez-L�opezLet fxngn2N be a T (d�1)-convergent sequence to x. Then, if m 2 N thereexists n0 2 N such that d(xn;x) < 1=2m for all n � n0. In addition, wecan �nd a proper closed d�1-ball with center x (otherwise, since (X;T (d�1))is a T1 space, we would have that X = fxg and the result is obvious). LetBd�1(x;�) be such a ball. Then, xn 2 Bd�1(x;�) if n is greater or equal thana certain natural number n1. Hence, fxngn2N admits a T (d)-cluster point y,and, furthermore for each m 2 Nd(y;x) � d(y;xn) + d(xn;x) < 1mfor a su�cient large n, so x = y and, therefore, T (d) � T (d�1).The other inclusion is similar.In general, the equality T (d) = T (d�1) is not true in a quasi-pseudo-metricspace (X;d) having nice closed balls. Let Z be the set of integers. The Khal-imsky line consists of Z with the topology generated by all sets of the formf2n � 1;2n;2n + 1g, n 2 Z. It is introduced in image processing in [11]. Thenthe quasi-pseudo-metric d de�ned on Z by d(2n;2n � 1) = d(2n;2n + 1) =d(n;n) = 0 for all n 2 N and d(x;y) = 1 otherwise, generates the topology ofthe Khalimsky line. It is clear that the proper closed d-balls and the properclosed d�1-balls are �nite so they are T (ds)-compact. Furthermore, it is obviousthat T (d) 6= T (d�1).When we consider a quasi-metric space we obtain the following result.Corollary 2.12. Let (X;d) be a quasi-metric space and T (d) = P, T (d�1) =Q. The following statements are equivalent.i) FP = T (Wd) and FQ = T (Wd�1) on CLs0(X).ii) FP = T (Wd) on CLP0 (X) and FQ = T (Wd�1) on CLQ0 (X).iii) FP = T (Wd) on CLQ0 (X) and FQ = T (Wd�1) on CLP0 (X).iv) (X;d) has nice closed balls.v) P = Q and (X;d) has nice closed balls.Proof. i) ) ii) and i) ) iii) are obvious. ii) implies iv) can be shown asabove, taking into account that (X;P) and (X;Q) are T1 spaces. By the aboveTheorem we obtain iii) ) iv). iv) ) v) is the above Remark. The implicationv) ) i) is [3, Theorem 5.1.10]. �Remark 2.13. Let us observe that the above Corollary is not true when weconsider a quasi-pseudo-metric space. Let us show that ii) ) iv) fails. Considerthe quasi-pseudo-metric space (R;`) where ` denotes the lower quasi-pseudo-metric. Clearly, we have that FP = T (W`) on CLP0 (X) and FQ = T (Wu) onCLQ0 (X) where P = T (`) and Q = T (u). However, (R; `) does not have niceclosed balls, since the closed `-balls and closed u-balls are not bounded.3. Other Fell type topologiesAs we have already observed, we have various possibilities in order to de�nethe Fell hypertopology in the nonsymmetric situation. This section is devoted



Fell type topologies and the Kuratowski-Painlev�e convergence 17to describe the advantages and disadvantages of our de�nition compared withother ones.We begin giving the de�nition that we think is more natural.De�nition 3.1. Let (X;P;Q) be a bitopological space. We de�ne the doubleupper �ne Fell space as the double space ((CLQ0 (X);FF+P );(CLP0 (X);FF+Q ))where FF+P is the topology generated by all sets of the form G+ where G is aP-open set and XnG is Q-compact; the topology FF+Q is de�ned in the corre-sponding natural way.The double �ne Fell space is the double space ((CLQ0 (X);FFP );(CLP0 (X);FFQ)) where FFP = FF+P _ F�P and FFQ = FF+Q _ F�Q .With this de�nition, not all the results proved in the previous section work(see Remark 2.10).Another possible de�nition is suggested by Burdick's investigations ([5, 6, 7]).He looked for a context in which he considered separately the upper and lowerVietoris topologies on a hyperspace and explored the interactions between them.De�nition 3.2. Let (X;P;Q) be a bitopological space. The double mixed Fellspace is the double space ((CLQ0 (X);MFP );(CLP0 (X);MFQ)) where MFP =F+P _ F�Q and MFQ = F+Q _ F�P .We call this hypertopology mixed, because we interchange the natural lowerhypertopologies between the two hyperspaces that we construct. We noticethat all results obtained in the previous section are true using this de�nitionwhenever we change the de�nition of the Wijsman lower hypertopology. Letus observe that our main results only use the upper hypertopologies since thedouble lower Wijsman topology always coincides with the double lower Felltopology. However, we think that is not a natural de�nition, although it pro-vides a nontrivial topology on the bitopological space (R;T (`);T (u)).Furthermore, we can give another de�nition.De�nition 3.3. Let (X;P;Q) be a bitopological space. The double mixed �neFell space is the double space ((CLQ0 (X);MFFP );(CLP0 (X);MFFQ)) whereMFFP = FF+P _ F�Q and MFFQ = FF+Q _ F�P .Unfortunately, the double mixed �ne Fell space has the same problems ofgeneralization as the double �ne Fell space. However, it is an appropriate Felltype topology to study epiconvergence of lower semicontinuous functions in thedouble setting, which will be discussed elsewhere.4. The Kuratowski-Painlev�e convergenceIn this section, we propose a de�nition for the Kuratowski-Painlev�e conver-gence in the nonsymmetric case and obtain some results about the relationshipsof this type of convergence and some hypertopologies.The Kuratowski-Painlev�e convergence was introduced to describe the limitof a net in terms of the members of the net itself. We propose the followingde�nitions.



18 J. Rodr��guez-L�opezDe�nition 4.1. Let (X;P;Q) be a bitopological space and fA�g�2�a net ofsubsets of X.i) A point x0 belongs to P-LiA� (resp. Q-LiA�) and we say that x0 is aP-limit point (resp. Q-limit point) of fA�g�2�if each Q-neighborhood(resp. P-neighborhood) of x0 intersects A� for all � in some residualsubset of �.ii) A point x0 belongs to P-LsA� (resp. Q-LsA�) and we say that x0is a P-cluster point (resp. Q-cluster point) of fA�g�2�if each Q-neighborhood (resp. P-neighborhood) of x0 intersects A� for all � insome co�nal subset of �.The proof of the following proposition is straightforward.Proposition 4.2. Let (X;P;Q) be a bitopological space. If fA�g�2�is a net ofsubsets of X then P-LiA� (resp. Q-LiA�) and P-LsA� (resp. Q-LsA�) areQ-closed sets (resp. P-closed sets).De�nition 4.3. Let (X;P;Q) be a bitopological space and let fA�g�2�be a netof subsets of X and A 2 P(X).We say that fA�g�2�is P-Kuratowski-Painlev�e upper convergent (resp. Q-Kuratowski-Painlev�e upper convergent) to A if P-LsA� � A (resp. Q-LsA� �A). We write A = K+P -limA� (resp. A = K+Q-limA�).We say that fA�g�2�is P-Kuratowski-Painlev�e lower convergent (resp. Q-Kuratowski-Painlev�e lower convergent) to A if A � P-LiA� (resp. A � Q-LiA�). We write A = K�P -limA� (resp. A = K�Q-limA�).We say that fA�g�2�is P-Kuratowski-Painlev�e convergent (resp. Q-Kur-atowski-Painlev�e convergent) to A if A = P-LiA� = P-LsA� (resp. A = Q-LiA� = Q-LsA�). We write A = KP -limA� (resp. A = KQ-limA�).With these de�nitions, we can extend a classical result which gives a rela-tionship between the Kuratowski-Painlev�e convergence and the convergence inthe Fell topology. We need the following de�nition.De�nition 4.4 ([15]). A bitopological space (X;P;Q) is said to be locallybicompact if every point has a neighborhood base in P and a neighborhood basein Q whose elements are P _ Q-compact sets.Theorem 4.5. Let (X;P;Q) be a bitopological space, A 2 CLQ0 (X) (resp.A 2 CLP0 (X)) and fA�g�2�a net in CLQ0 (X) (resp. CLP0 (X)).i) A = K�P -limA� (resp. A = K�Q-limA�) if and only if A = F�Q -limA�(resp. A = F�P -limA�).ii) If A = K+P -limA� (resp. A = K+Q-limA�) then A = F+P -limA� (resp.A = F+Q -limA�).iii) If (X;P;Q) is a locally bicompact quasi-uniformizable bitopological spaceand A = F+P � limA� (resp. A = F+Q � limA�) then A = K+P � limA�(resp. A = K+Q � limA�).



Fell type topologies and the Kuratowski-Painlev�e convergence 19Proof. i) This statement is straightforward.ii) Let us suppose that A = K+P -limA�. ThenP � Ls(K \ A�) � P � LsA� � Afor all P _Q-compact set K. Let K0 be a P _Q-compact and P-closed set suchthat K \ A� 6= ? for a co�nal subset of �. Let us prove that P-LsA� 6= ?.Choose k� 2 K \ A� for all � belonging to a co�nal subset �0 of �. We obtainthat fk�g�2�0 admits a P _ Q-cluster point k 2 K. It is evident that k 2 P-LsA� \ K � A so A \ K 6= ?. Therefore, if A \ K = ? then A� \ K = ?eventually. Hence, A = F+P -limA�.The other statement can be proved in a similar way.iii) Let us suppose that P-LsA� 6� A. Let x 2 (P-LsA�)nA. Since (X;P;Q)is a locally bicompact quasi-uniformizable bitopological space, we can �nd aP-closed and P _Q-compact Q-neighborhood V of x such that V \A = ? butA� \ V 6= ? frequently. Therefore, A 6= F+P -limA� which is a contradiction.Consequently, P-LsA� � A.The same reasoning proves the statement for Q. �We characterize the Kuratowski-Painlev�e convergence in terms of sequencesof points.Proposition 4.6. Let (X;P;Q) be a quasi-pseudo-metrizable bitopological spaceand d a quasi-pseudo-metric on X compatible with the bitopological space. A se-quence fAngn2N � P(X) is P-Kuratowski-Painlev�e lower convergent to a set Aif and only if each point a 2 A is the limit of some T (d�1)-convergent sequencefangn2N such that an 2 An for all n 2 N.Proof. Let us suppose that fAngn2N is F�Q -convergent to A 2 P(X). Picka 2 A (if A = ? the result is evident). Given k 2 N, we have that A 2Bd�1(a;1=k)� so there exists nk 2 N such that An 2 Bd�1(a;1=k)� for alln � nk. We can suppose that n1 < n2 < ::: < nk < :::. Therefore, we can �ndan 2 An \ Bd�1(a;1=k) for all nk+1 � n � nk + 1. If we consider the sequencefangn2N where if n 2 f1; : : : ;n1g we consider a �xed point an 2 An, we havethat this sequence is T (d�1)-convergent to a.Conversely, let fAngn2N be a sequence and A � X satisfying our assumption.If a 2 A \ G where G is a T (d�1)-open set, there exists a sequence fangn2NT (d�1)-convergent to a verifying that an 2 An for all n 2 N. On the other hand,we can �nd " > 0 and n0 2 N such that Bd�1(a;") � G and d�1(a;an) < " forall n � n0. Therefore, An 2 G� for all n � n0. �We can also obtain a characterization of the Kuratowski-Painlev�e upper con-vergence in terms of sequences.Proposition 4.7. Let (X;P;Q) be a quasi-pseudo-metrizable bitopological spaceand d a quasi-pseudo-metric on X compatible with the bitopological space. A se-quence fAngn2N � P(X) is P-Kuratowski-Painlev�e upper convergent to a set Aif and only if whenever there exist positive integers n1 < n2 < ::: and ak 2 Ankfor all k 2 N such that fakgk2N is T (d�1)-convergent to a then a 2 A.



20 J. Rodr��guez-L�opezProof. Let us suppose that fAngn2N � P(X) is P-Kuratowski-Painlev�e con-vergent to A. If fangn2N is a sequence as in the statement, it is evident thata 2 P-LsAn � A.Now, let a 2 P-LsAn and fBd�1(a;1=n) : n 2 Ng a countable T (d�1)-neighborhood base of a. Choose n1 2 N such that Bd�1(a;1) \ An1 6= ?.Since fn 2 N : Bd�1(a;1=2) \ Ang is in�nite, we can �nd n2 > n1 verifyingBd�1(a;1=2) \ An2 6= ?. Following this procedure, we can construct a strictlyincreasing sequence of positive integers fnkgk2N such that Bd�1(a;1=k)\Ank 6=? for all k 2 N. If ak 2 Bd�1(a;1=k) \ Ank for all k 2 N, it is evident that thissequence is T (d�1)-convergent to a, so by assumption a 2 A. �Now, we can extend an interesting result due to Mrowka (see [14]).Theorem 4.8 (Mrowka). Let (X;P;Q) be a bitopological space and let fA�g�2�be a net in P(X). Then fA�g�2�has a P-Kuratowski-Painlev�e convergent sub-net and a Q-Kuratowski-Painlev�e convergent subnet.Proof. Let B be a base for the topology Q. Let us consider the space f0;1gwith the discrete topology. For each � 2 �, we de�ne f� : B ! f0;1g as follows:f�(V ) = (1 if A� \ V 6= ?0 if A� \ V = ? :By the Tychono�'s theorem, ff�g�2� has a convergent subnet ff�0g�02�0. Foreach V 2 B, we obtain that f�0(V ) = 1 eventually if and only if f�0(V ) = 1frequently. Therefore, if A� \ V 6= ? frequently then A� \ V 6= ? eventually,so fA�0g�02�0 is P-Kuratowski-Painlev�e convergent.The reasoning for P is similar. �Theorem 4.9. Let (X;d) be a quasi-pseudo-metric space. Let P = T (d) andQ = T (d�1). Let us consider fA�g�2�a net in CLQ0 (X) and fB
g
2� a net inCLP0 (X).i) If A = T +(Wd)-limA� and B = T +(Wd�1)-limB
 then A = K+P -limA� and B = K+Q-limB
.ii) A = K+P -limA� and B = K+Q-limB
 implies A = T +(Wd)-limA� andB = T +(Wd�1)-limB
 if and only if (X;d) has nice closed balls.Proof. i) Let us suppose that A = T +(Wd)-limA� and B = T +(Wd�1)-limB
.Let a 2 P-LsA� and suppose that a 62 A. Thus d(A;a) > 0. Therefore, given0 < � < d(A;a) since a 2 P-LsA� we obtain that d(A�;a) < � frequently sod(A;a) � d(A�;a) > d(A;a) � � > 0frequently which contradicts that d(A;a) = T (`)-limd(A�;a). The same rea-soning shows that B � Q-LsB
.ii) This statement is similar to the proof of Theorem 2.7. �We recall the following de�nitions (see [15]).



Fell type topologies and the Kuratowski-Painlev�e convergence 21De�nition 4.10. Let (X;P;Q) be a quasi-uniformizable bitopological spaceand U a quasi-uniformity compatible with the bitopological space. The dou-ble upper U-proximal topological space is de�ned as the double topologicalspace ((CLQ0 (X);T +(�U));(CLP0 (X);T +(�U�1))) where T +(�U) is the topologygenerated by all sets of the form G++ = fA 2 CLQ0 (X) : there exists U 2U such that U(A) � Gg where G is a P-open set. The topology T (�U�1) isde�ned in a similar way by writing P instead of Q, Q instead of P and U�1 in-stead of U. The pair (T +(�U);T +(�U�1)) is called the double upper U-proximaltopology.The double lower U-proximal topological space is de�ned as the double topo-logical space ((CLQ0 (X);T �(�U));(CLP0 (X);T �(�U�1))) where this double topo-logical space coincides with the double lower Fell topological space. The pair(T �(�U);T �(�U�1)) is called the double lower U-proximal topology.The double U-proximal topological space is de�ned as the double topologicalspace ((CLQ0 (X);T (�U));(CLP0 (X);T (�U�1))) where T (�U) = T +(�U)_T �(�U)and T (�U�1) = T +(�U�1) _ T �(�U�1). The pair (T (�U);T (�U�1)) is called thedouble U-proximal topology.De�nition 4.11. Let (X;P;Q) be a quasi-uniformizable bitopological space andU a quasi-uniformity compatible with (X;P;Q). We say that (X;P;Q) has theproperty pairwise star ifi) given A 2 CLQ0 (X) and B 2 CLP0 (X) with A�UB there exist fx1; : : : ;xng � X and U1; : : : ;Un 2 U such that A \ ([ni=1U�1i (xi)P ) = ? andB � [ni=1U�1i (xi).ii) given A 2 CLP0 (X) and B 2 CLQ0 (X) with A�U�1B there exist fx1; : : : ;xng � X and U1; : : : ;Un 2 U such that A \ ([ni=1Ui(xi)Q) = ? andB � [ni=1Ui(xi).De�nition 4.12. Let (X;U) be a quasi-uniform space. We say that it hasnice closed balls if every proper set of the form U(x)T (U�1) or U�1(x)T (U) isT (Us)-compact, where U 2 U and x 2 X:Theorem 4.13. Let (X;P;Q) be a quasi-uniformizable bitopological space andU a quasi-uniformity compatible with the bitopological space. Let us considerfA�g�2�a net in CLQ0 (X) and fB
g
2� a net in CLP0 (X).i) If A = T +(�U)-limA� and B = T +(�U�1)-limB
 then A = K+P -limA�and B = K+Q-limB
.ii) A = K+P -limA� and B = K+Q-limB
 implies A = T +(�U)-limA� andB = T +(�U�1)-limB
 if and only if (X;P;Q) has the property pairwisestar and (X;U) has nice closed balls.Proof. i) Suppose that A = T +(�U)-limA� and B = T +(�U�1)-limB
. Ifthere exists a 2 P-LsA�nA, we can �nd U 2 U such that U�1(a) \ A =?. It is easy to prove that A 2 (intPV (A))++, where V 2 U and V 2 � U.Therefore, A� � (intPV (A))++ for all � in a residual subset of �. Furthermore,



22 J. Rodr��guez-L�opezV (A) \ V �1(a) = ?. On the other hand, since a 2 P-LsA� we obtain thatV �1(a) \ A� 6= ? for a co�nal subset of � which is not possible. Therefore,LsA� � A. B = T +(�U�1)-limB
 implies B = K+Q-limB
 can be proved in asimilar way.ii) Let us suppose that there exist U 2 U and x 2 X such that U�1(x)P is aproper set and is not P _ Q-compact. Then, we can �nd y0 2 XnU�1(x)P anda net fx�g�2� � U�1(x)P such that it does not admit a P _ Q-cluster point.We can easily deduce that the net fA�g�2� is P-Kuratowski-Painlev�e upperconvergent to fy0gQ, where A� = fx�gQ [ fy0gQ for all � 2 �. On the otherhand, if we consider the T +(�U)-open set (intPV (fy0gQ))++ where V;U0 2 U,V 2 � U0 and U0(y0) \ U�1(x)P = ?, we have that x� 62 intPV (fy0gQ) for all� 2 � since if there exists z 2 fy0gQ verifying (z;x�) 2 V , for some � 2 �,we obtain that (y0;x�) 2 U0 which is not possible. Consequently, fA�g�2� isnot T +(�U)-convergent to fy0gQ. A contradiction. In a similar way, it can beproved that the proper sets of the form U(x)Q are P_Q-compact. Consequently,(X;U) has nice closed balls.Therefore, (X;P;Q) is a locally bicompact space. Applying Theorem 4.5, wededuce that the double Fell topology agrees with the double proximal topologywhich implies (see [15]) that the bitopological space has the property pairwisestar.Conversely, if (X;P;Q) has the property pairwise star and (X;U) has niceclosed balls it can be proved (see [15]) that the double upper Fell topologyagrees with the double upper proximal topology, and the statement followsdirectly. �At last, we establish the relationship of the Kuratowski-Painlev�e convergenceand the convergence in the Vietoris hypertopology.Theorem 4.14. Let (X;P;Q) be a quasi-uniformizable bitopological space. Letus consider fA�g�2�a net in CLQ0 (X) and fB
g
2� a net in CLP0 (X). Theni) If A = V +P -limA� and B = V +Q -limB
 then A = K+P -limA� and B =K+Q-limB
.ii) A = K+P -limA� and B = K+Q-limB
 implies A = V +P -limA� and B =V +Q -limB
 if and only if (X;P _ Q) is a compact space.Proof. i) The proof is similar to the part i) of the above theorem.ii) Let us suppose that (X;P _ Q) is not a compact space. Therefore, thereexists a net fx�g�2� that does not admit a P _Q-cluster point. If we �x y0 2 Xit is easy to prove that ffx�gQ [ fy0gQg�2� is P-Kuratowski-Painlev�e upperconvergent to fy0gQ. We can also prove that the net ffx�gP [ fy0gPg�2� isQ-Kuratowski-Painlev�e upper convergent to fy0gP .On the other hand, since y0 is not a P _Q-cluster point of fx�g�2�, we can �nd



Fell type topologies and the Kuratowski-Painlev�e convergence 23U 2 U such that x� 62 Us(y0) for all � in whatever co�nal subset �0 of �. Wecan choose �0 in such way that we only have to distinguish two possibilities:i) x� 62 U(y0) for all � 2 �0. Therefore, it is evident that if we considerthe P-open set intPV (fy0gQ) where V 2 U and V 2 � U, we havethat x� 62 V (fy0gQ) for all � 2 �0 so ffx�gQ [ fy0gQg�2� is not V +P -convergent to fy0gQ. A contradiction.ii) x� 62 U�1(y0) for all � 2 �0. Reasoning as above we obtain thatffx�gP [ fy0gPg�2� is not V +Q -convergent to fy0gP . A contradiction.Conversely, since (X;P _ Q) is a compact space then F+P = V +P on CLQ0 (X)and F+Q = V +Q on CLP0 (X) (see [15] ), so the proof is evident. �We can also de�ne the Kuratowski-Painlev�e convergence in a bitopologicalsense in a di�erent way.De�nition 4.15. Let (X;P;Q) be a bitopological space and let fA�g�2�be anet of subsets of X and A 2 P(X).We say that fA�g�2�is P-mixed Kuratowski-Painlev�e convergent (resp. Q-mixed Kuratowski-Painlev�e convergent) to A if A = K�Q-limA� and A = K+P -limA� (resp. A = K�P -limA� and A = K+Q-limA�). We write A = MKP -limA� (resp. A = MKQ-limA�).With this de�nition, we can also wonder under which conditions we cantopologize this convergence. For the other de�nition, the condition was tomake the space locally bicompact. We observe that we use this condition onlyto reconcile the Kuratowski-Painlev�e upper convergence with the convergencein the upper Fell topology. So we have that this is an appropriate concept towork with the double Fell topology.Proposition 4.16. Let (X;P;Q) be a locally bicompact bitopological space.Then the mixed Kuratowski-Painlev�e convergence agrees with the convergencein the double Fell topology.Consequently, the concept of Kuratowski-Painlev�e convergence is suitable toobtain relationships with the double mixed Fell topology and the topologizationof the mixed Kuratowski-Painlev�e convergence is the double Fell topology.It is natural to wonder if we can obtain conditions for the bitopologicalspace (X;P;Q) in order to obtain the coincidence of the Kuratowski-Painlev�econvergence and the other Fell topologies de�ned. We give a positive answerto this question. It is natural to look for other de�nitions of local compactnessin bitopological spaces. The next de�nition is due to Stoltenberg.De�nition 4.17 ([22]). Let (X;P;Q) be a bitopological space. We say that P islocally compact with respect to Q if for all x 2 X there exists a P-neighborhoodG of x such that the Q-closure of G is Q-compact.We say that (X;P;Q) is pairwise locally compact if P is locally compact withrespect to Q and Q is locally compact with respect to P.



24 J. Rodr��guez-L�opezThis de�nition is not suitable here. If we want that our techniques work withthis de�nition, we have to de�ne another upper Fell topology for P consideringthat this topology is generated by the sets of the form G+ where G is P-openand XnG is P-compact. The upper Fell topology for Q would be de�ned in asimilar way. But this topology does not give good results.Taking into account this, we propose the following de�nition.De�nition 4.18. Let (X;P;Q) be a bitopological space. We say that (X;P;Q)is bilocally compact if (X;P) and (X;Q) are locally compact spaces.It is clear that if (X;P;Q) is locally bicompact then it is bilocally compact.With this de�nition we have the following obvious result.Proposition 4.19. Let (X;P;Q) be a bitopological space and suppose that A 2CLQ0 (X) (resp. A 2 CLP0 (X)) and that fA�g�2�is a net in CLQ0 (X) (resp.CLP0 (X)). Theni) If A = K+P -limA� (resp. A = K+Q-limA�) then A = FF+P -limA� (resp.A = FF+Q -limA�).ii) If (X;P;Q) is a bilocally compact pairwise Hausdor� bitopological spaceand A = FF+P -limA� (resp. A = FF+Q -limA�) then A = K+P -limA�(resp. A = K+Q-limA�).Proof. The proof is similar to Theorem 4.5. �Acknowledgements. The author is very grateful to Professor S. Romaguerafor his help, advice and encouragement.References[1] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull.Aust. Math. Soc. 31 (1985), 421{432.[2] , Metric spaces with nice closed balls and distance functions for closed sets, Bull.Aust. Math. Soc. 35 (1987), 81{96.[3] , Topologies on Closed and Closed Convex Sets, vol. 268, Kluwer Academic Pub-lishers, 1993.[4] G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema ofhyperspace topologies, Ann. Mat. pura ed Appl. 162 (1992), 367{381.[5] B. S. Burdick, Separation properties of the asymmetric hyperspace of a bitopological space,Proceedings of the Tennessee Topology Conference, P. R. Misra and M. Rajagopalan, eds.,World Scienti�c, Singapore, 1997.[6] , Compactness and sobriety in bitopological spaces, Topology Proc. 22 (1997),43{61.[7] , Characterizations of hyperspaces of bitopological spaces, Topology Proc. 23(1998), 27{43.[8] J. M. G. Fell, A Hausdor� topology for the closed subsets of a locally compact non-Hausdor� space, Proc. Amer. Math. Soc. 13 (1962), 472{476.[9] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.[10] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13 (1963), 71{89.[11] R. Kopperman, The Khalimsky line as a foundation for digital topology, Proceedings in ofNATO Advanced Research Workshop "Shape in Picture", Driebergen, The Netherlands126 (1994), 3{20.



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J. Rodr��guez-L�opezEscuela Polit�ecnica Superior de AlcoyDepartamento de Matem�atica AplicadaUniversidad Polit�ecnica de ValenciaPza. Ferr�andiz-Carbonell, 203801 Alcoy (Alicante)SpainE-mail address: jerodlo@mat.upv.es