THE OPEN-OPEN TOPOLOGY FOR FUNCTION SPACES

Let (X,T) and (Y,T’) be topological spaces and let F C yX. For each U _ T,V E T’, let (U,V) {f E F: f(U) C V}. Define the set SOD {(U,V): U Tand V T’}. Then SoD is a subbasis for a topology, Too on F, which is called the open-open topology. We compare Too with other topologies and discuss its properties. We also show that Too, on H(X), the collection of all self-homeomorphisms on X, is equivalent to the topology induced on H(X) by the Pervin quasi-uniformity on X.


INTRODUCTION.
The type of set-set topology which will be discussed here is one which can be defined as follows: Let (X, T) and (Y, T*) be topological spaces.Let U and V be collections of subsets of X and Y, respectively.Let F C yX, the collection of all functions from X into Y.Define, for U U and V V, (U,V)= {f E F: f(U) C Y}. Let S(U,V)= {(U,V): U U and Y e V}.If S(U, V) is a subbasis for a topology T(U, V) on F then T(U, V) is called a set-set topology.
One of the original set-set topologies is the compact-open topology, TeD, which was introduced in 1945 by R. Fox  [1].For this topology, as one may surmise from the name, U is the collection of all compact subsets of X and V T*, the collection of all open subsets of Y. Fox and Arens [2] developed and examined the properties of this now well-known topology.In particular, it was shown that if F C C(X, Y), the collection of all continuous functions on X into Y, then Tco on F is equivalent to the topology of uniform convergence on compacta; and, if in addition, X is compact, then TeD is equivalent to the topology of uniform convergence on F. Arens also defined the concept of admissible topology for function spaces and was instrumental in the study of groups of self-homeomorphisms and topological groups.
Other set-set topologies that have been of interest are: the point-open, topology, Tp, also known as the topology of pointwise convergence, in which U is the collection of all singletons in X and V T*; the closed-open topology, where U is the collection of all closed subsets of X and the set V T*; and the bounded-open topology (Lambrinos [3]), where U is the collection of all bounded subsets of X and again, V T*.
In section 2 of this paper, we shall introduce and discuss the open-open topology, To,,, for function spaces.It will be shown which of the desirable properties Too possesses.In section 3, the group of all self-homeomorphisms, H(X), endowed with Too, is discussed.
As will be proven in section 5, Too, on H(X), is actually equivalent to the Pervin topology of quasi-uniform convergence (Fletcher [4]).One of the advantages of the open-open topology is the set-set notation which provides us with simple notation and, hence, our proofs are more concise than those using the cumbersome notation of the quasi-uniformity.Pervin spaces will be discussed in section 4.   We assume a basic knowledge of quasi-uniform spaces.An introduction to quasi-uniform spaces may be found in Fletcher and Lindgren's [5] or in Murdeshwar and Naimpally's book [6].
Throughout this paper we shall assume (X, T) and (Y, T*) are topological spaces. 2.
THE OPEN-OPEN TOPOLOGY.
If we let U T and V T*, then So,, S(U, V) is the subbasis for the topology, T, on Let F C C(X, Y).If (Y, T*) is Ti for 0, 1, 2, then (F, Too) is Ti for =0,1,2.PROOF.We shall show the case 2; the other cases are done similarly.Let 2.
Let f,g (5 F such that f # g.Then there is some x (5 X such that f(x) g(x).IfY is T2 there exists disjoint open sets O and U in Y such that f(x) (5 U and g(x) (50.Both f and g are continuous, so there are open sets V and W in X with x (5 V f'l W, f(V) C U and g(W) C O.
A topology, T', on F C yx is called an admissible (Arens [2]) topology for F provided the evaluation map, E: (F, T') x (X,T) -(Y, T*), defined by E(f,x)= f(x), is continuous.
THEOREM 2. If F C C(X, Y) then Too is admissible for F.
PROOF.Let F C C(X,Y).Let O (5 T* and let (f,p) (5 E-l(O).Then f(p) (50.Since f is continuous, there exists some U (5 T such that p (5 U and f(U) C O. So, (f,p) (5 (U, O) x U. Therefore, Too is admissible for F.
Arens also has shown that if T' is admissible for F C C(X, Y), then T' is finer than Too.From this fact and Theorem 2, it follows that Tco C Too.

3.
THE OPEN-OPEN TOPOLOGY ON H(X).
We now consider Too on H(X), the collection of all self-homeomorphisms on X.Note that H(X) with the binary operation o, composition of functions, and identity element e, is a group.Some of the set-set topologies previously mentioned are equivalent under certain hypotheses.
For example, the closed-open topology is equal to the compact-open topology whenever X is com- pact T, the point-open topology is equivalent to the compact-open topology if all compact subsets of X are finite sets.It is always advantageous to know when topologies are or are not equivalent.
In particular, it is well known that if X is T1 then Tp C To and as we have shown To C Too.When are To and Too distinct?One hypothesis under which these two topologies are not equivalent is: "Let X be T and Galois." A topological space is Galois provided that for each closed set, C C X and each point p'E X\C, there is an h H(X) such that h(x) xforallx C and h(p) #p.Among the spaces which are T2 Galois are the topological vector spaces and, as Fletcher [7] has shown, locally euclidean T spaces or homogeneous 0-dimensional spaces which have no isolated points.THEOREM 3.
If X is a T Galois space then Too # To on H(X).

PROOF.
Let X be a T Galois space.Let x X; then X \ {x} is open in X so that (X \ {x},X \ {x}) is open in Too.But note that (X \ {x},X \ {x}) ({x}, {x}) in (H(X),To,,).
Let e be the identity map on X then e e ({x}, {x}).Claim: ({x}, {x}) Too and hence To # Too.Let (C,, U,) be a basic open set in To which contains e.So, C C U for all i----1 1,2, 3, n.
Set U0 X and Co .L et P {U}'=0 and Q {C, }'=0.Define Px t3 {U PIz U and Qx u{C QIx q C}.Let L Px \ Qx.Note that z L and L is open in X.
If q L then h(q) q C C U s.If q L then x Cj from which it follows that L C Uj. Thus, h(L) L C U s.In either case, h (C,,U,) and again (C,,U,) i'-I --1 Therefore, ({x }, {x}) To and To Too.
Effros' Theorem (Effros [8]) is a widely known and useful tool in the study of homogeneous spaces and continua theory.Of its several forms, the most popular is: If X is a compact homoge- neous metric space then for each x X, the evaluation map, E (H(X),Tco) X, defined by E:(h) h(x), is an open map.It follows that, if the conclusion holds when E is considered on the space (H(X),T.), and if T C T, on H(X), then the conclusion also holds on (H(X),T).Ancel  [9] has asked the following question: If the hypothesis of the Effros' Theorem is changed to "X is a compact, homogeneous, Hausdorff space,'is the evaluation map on (H(X),T,:o) still open *?To this end, since Too C Too, we could consider whether a form of Effros' Theorem would be true for Too on H(X).Unfortunately, we discover the following.Let (X,T) be a TI topological space.Then, for each z E X, the evaluation map, E: (H(X),Too) X defined by E:(h) h(z), is open only if T is the discrete topology.PROOF.Let z E X. Then the set O ((X\ {z}),(X \ {z}))is open in (H(X),Too).But ((X \ {z}),(X \ {z})) ({z}, {z}).So.E:(O) {z}.Thus E is open for each z fi X only if X is discrete.Let (G, o) be a group such that (G,T)is a topological space, then (G,T)is a topological group provided the following two maps are continuous.(1) m G x G G defined by re(g1, g2) gl o g2 and (I) G G defined by (I)(9) 9 -1.If only the first map is continuous, then we call (G, T) a quasi-topological group (Murdeshwar and Naimpally [6]).
A topological space, (X,T), is called a Pervin space (Fletcher [4]) provided that for each finite collection, ,4, of open sets in X, there exists some h c= H(X) such that h # e and h(U) C U for all U E ,4.
Topologies are rarely interesting if they are the trivial or discrete topology.To this end, we have: THEOREM 5.
(H(X), Too)is not discrete if and only if (X, T)is a Pervin space.
PROOF.First, assume that (X,T) is a Pervin space.Let W be a basic open set in Too *This was recently answered in the negative by Bellamy and Porter [10].
which contains e; i.e.W ](O,, U,) where O, C U, for each 1,2, 3, n and O, and U, are open in X. {O, 1,2,3,...,n} is a finite collection of open sets in X, and X is a Pervin space, hence, there exists some h (5 H(X) such that h e and h(O,) C O, C U,. So, h (5 W and h e.Since (H(X), Too) is a quasi-topological group, (H(X), Too) is not a discrete space.Now assume that (H(X),Too) is not discrete.Let V be a finite collection of open sets in X.Let O f"] (U,U).Then, O is a basic open set in (H(X),Too) which is not a discrete space.
Fletcher [4] proved that the Pervin topology of quasi-uniform convergence on H(X) is not discrete if and only if (X,T) is Pervin.In order to prove this, Fletcher had to first introduce numerous definitions along with some mind boggling notation.The above proof, along with the few needed definitions involving Too, is an example of the simplification that the definition of Too offers over the quasi-uniform definition and notation.
Recall that if Q is a quasi-uniformity on X, then the topology, TQ, on X, which has as its neighborhood base at x, B {U[x]: U (5 Q}, is called the topoloiy induced by Q.The ordered triple (X, Q, TQ) is called a quasi-uniform space.A topological space, (X, T) is quasi-uniformizable provided there exists a quasi-uniformity, Q, such that TQ T. In 1962, Pervin [11] proved that every topological space is quasi-uniformizable by giving the following construction.
Let (X,T)be a topological space.For each O (5 T, define the set So (OxO)t.J((X\O)xX).
Let S {So O (5 T}.Then S is a subbasis for a quasi-uniformity, P, for X, called the Pervin quasi-uniformity and, as is easily shown, Tp T. If (X, Q) is a quasi-uniform space then Q induces a topology on H(X) called the topology of quasi-uniform convergence w.r.tQ, as follows: For each set U (5 Q, let us define W(U) {(f,g)(5 H(X) x H(X): (f(x),g(x)) (5 U forall x (5 X}.Then, B(Q) {W(U): U (5 Q} is a basis for Q*, the quasi-uniformity of quasi-uniform convergence w.r.t.Q (Naimpally [12]).Let TO. denote the topology on H(X) induced by Q*.TQ. is called the topology of quasi-uniform convergence w.r.t.Q*.
If P is the Pervin quasi-uniformity on X Te. is the Pervin topology of quasi-uniform convergence.
At this time one could, once again, prove that Tp. is not discrete if and only if (X, T) is a Pervin space, this time using the quasi-uniform structure [4].We leave this to the reader.
We now show that the open-open topology is equivalent to the Pervin topology of quasi- uniform convergence.THEOREM 6.
Let (X,T) be a topological space and let G be a subgroup of H(X).
Then, Too Tp, on G.
Since U P, there is some finite collection, {U, 1,2, n} C T such that 5 Sv.C U. Define i----I A '(f-l(Ui) Ui).A is an open set in Too. and f E A. Assume g E A and let x for some j {1, 2, n}, then x f-I(U,).So, g(x) 6_.Us, hence, (f(x),g(x)) If f(x) U for some j E {1,2,...,n} then (f(x),g(x)) .(X\ U.i,X C 5 Su, C U, and it follows that g W(U)[f] C_ V and A C Y. So, Too T,..

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 any F C yx, which is called the open-open topology.We first examine some of the properties of function spaces the open-open topology possesses.THEOREM1.

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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