A TOPOLOGICAL LATTICE ON THE SET OF MULTIFUNCTIONS

Let X be a Wilker space and M(X,Y) the set of continuous multifunctions from X to a topological space Y equipped with the compact-open topology. Assuming that M(X,Y) is equipped with the partial order c we prove that (M(X,Y),c) is a topological V-semilattice. We also prove that if X is a Wilker normal space and U(X,Y) is the set of point-closed upper semi-continuous multifunctlons equipped with the compact-open topology, then (U(X,Y), c) is a topological lattice.


i. INTRODUCTION AND DEFINITIONS.
A mapping F from a set X to a set Y which maps each point of X to a subset of Y is called multifunction.
For any subset A of X, F (A)   , F(x).For any subset B of Y, F+(B) {x X:F(x) cB} and F-(B) {x X:F(x)oB 0}.Let X and Y be topological spaces.
A multifunction F from X to Y is upper semi-continuous (lower semi-continuous) if and only if F+(P) (F-(P)) is open for each open subset P of Y (see Smithson [I]).
A multifunction F:X Y is continuous if and only if it is both upper and lower semi-continuous [I].
A multifunction F:X/Y is polnt-closed [I] if and only if F(x) is a closed subset of Y, for each x X.
If FI, F 2 are two multlfunctlons from X to Y, by FIV F2, we denote the multifunctlon from X to Y defined by (FlY F2)(x) Fl(X) U F2(x).Also, by FIAF2, we denote the multifunction from X to Y defined by (FIA F2)(x) FI(X)NF2(x in Kuratowski [2]. In the following, by M(X,Y), we denote the set of continuous multifunctions.Also, by U(X,Y), we denote the set of polnt-closed upper semi-contlnuous multlfunctlons.
Let K be a compact subset of X and P an open subset of Y. Let <K,P> {F e M(X,Y):F(x) P # 0 for all x E K} and [K,P] {F e M(X,Y):F(K)C p}.The topology T on M(Y,Z) generated by the sets of the form <K,P> and [K,P], where K is co compact in X and P is open in Y, is called the compact open topology on M(X,Y) [I].* The topology T on U(X,Y) generated by the sets of the form co [K,P] {F e U(X,Y):F(K)c P}, where K is compact in X and P open in Y, is called the compact-open topology on U(X,Y).
For simplicity, in what follows, we use the symbols M(X,Y) (U(X,Y)) to denote the , topological spaces (M(X,Y), T ((U(X,Y),T )).

CO CO
We give now the definition of Wilker spaces that we will use in the following: A topological space X satisfies the Wilker's condition (D) For every compact subset KcX and for every pair of open subsets AI, A 2 e X with k AID A 2 there are compact subsets K c A1 and K2C A 2 such that KcKIU K 2 is called a Wilker space (Wilker [3]).
It can be easily proved that the class of Wilker spaces contains properly the class of T 2 spaces and also the class of basic locally compact spaces (i.e.those spaces every point of which has a neighborhood basis consisting of compact sets).In [4]   basic locally compact spaces are called locally quasl-compact spaces and in Murdehswar [5] they are called spaces which satisfy condition L 2.
In this paper we prove that if X is a Wilker space, then the V-semilattices (M(X,Y), c), (U(X,Y), are topological, i.e., we prove the continuity of the join operation V It is also noticed that if X is a normal space, (U(X,Y), is a semilattice [4,p.4].Finally, if X is a Wilker normal space, we prove that the meet operation A is continuous, i.e., (U(X,Y),) is a topological semilattice [4, p.274].
The worth of the above results relies on the fact that the space U(X,Y) (M(X,Y)) can be considered as a topological lattice (topological V-semilattlce [4,p.4]).
Let now FIV F 2 e <K,P>.Then (FIV F 2) (x) P # 0 for each x e K.So we have KC FI(P UF2(P ).But since X is a Wilker space there are compact subsets K I, K 2 of X, such that K iFi(P), i I, 2, and K CKIU K 2. So F e <K I, P>, F 2 e <K 2, P>.We prove now that (GI, G 2) e <KI,P> <K 2, P> implies that GIV G 2 e <K, P>.
The proof of the followlng Proposition is the same as that of Proposition 2.1 (first part) and It is omitted.
LEMMA 2.3.[2, p.179].Suppose X is a normal space.Let FI:X Y, F2: X Y be two point-closed upper semi-contlnuous multifunctions and P an open set in Y.Then, (FIA F2)+(P) U{F(V) O F(W)}, where V,W are open in Y, VOW P.
Let U(X,Y) be the set of point closed upper semi-continuous multlfunctions equipped with the compact-open topology.Then (U(X,Y), c) is a topological lattice.PROOF.It suffices to prove that (U(X,Y), c is a topological similattice, i.e., that the meet operation A is continuous.According to the previous lemma, it is obvious that the function (FI,F2) F F2:U(X,Y) U(X,Y) U(X,Y) is well defined, i.e. that (U(X,Y), c is a semilattlce.
We prove now that A continuous.
Let an arbitrary (FI,F2) U(X,Y) U(X,Y) and let F 1AF2 E [K,P], where K is compact in X and P is open in Y. Then by the previous lemma + K c(FIA F2)+(P) U(F(V) O F2(W)} where V,W are open in Y, V N W P. But since K is compact there are finitely many sets V i, W i, i l,...,n such that n + U__ {F(Vi) 0 F2(Wi) }, K i where Vi,Wi, are open in Y, V lowi P, I I,..., n.Moreover since X is a Wilker space there exist compact subsets of X, Ki, i-l,...,n, such that n + + U Ki K i Fi(Vi)0 F2(Wi) and KC i + + Thus, KicFl(Vl) KiCF2(Wi) i -I,..., n.
Hence F [KI,Vl] F 2 [Ki, Wi] i I,..., n and finally n n (FI,F2) e i [Ki, Vi] x i=l [Ki Wi]  It remains to prove that for each To prove this consider an aribtrary 2) e i must be shown that K _(G G2)+(P).Let an arbitrary x e K. Then x e K.,I for some It +(Vi), +(Wi) we have that Gl(X) Since KiCG KiCG2 GI(X) NG2(x) (GI AG 2) (x) ViN W.1 P" Thus, x e (GIAG2)+(P), which completes the proof. ACKNOWLEDGMENT.
I would like to thank the referee for his useful suggestions.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation