@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 2, 2003 pp. 467–474 n-Tuple relations and topologies on function spaces D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. In [7] some results concerning S-splitting, S-jointly con- tinuous, D-splitting and D-jointly continuous topologies are considered, where S and D are the Sierpinski space and the double-point space, re- spectively. Here we generalize these results replacing the spaces S and D by any finite space. 2000 AMS Classification: 54C35. Keywords: function space, A-splitting topology, A-jointly continuous topology. 1. Introduction. By Y and Z we denote two fixed topological spaces and by tZ the topology of Z. By C(Y,Z) we denote the set of all continuous maps of Y into Z. If τ is a topology on the set C(Y,Z), then the corresponding topological space is denoted by Cτ (Y,Z). Let X be a space and F : X ×Y → Z be a continuous map. By Fx, where x ∈ X, we denote the continuous map of Y into Z, for which Fx(y) = F(x,y), for every y ∈ Y . By F̂ we denote the map of X into the set C(Y,Z), for which F̂(x) = Fx for every x ∈ X. Let G be a map of the space X into the set C(Y,Z). By G̃ we denote the map of the space X × Y into the space Z, for which G̃(x,y) = G(x)(y) for every (x,y) ∈ X ×Y . A topology t on C(Y,Z) is called splitting if for every space X, the continuity of a map F : X×Y → Z implies that of the map F̂ : X → Ct(Y,Z). A topology t on C(Y,Z) is called jointly continuous if for every space X, the continuity of a map G : X → Ct(Y,Z) implies that of the map G̃ : X ×Y → Z (see [5], [1], [2] and [3]). If in the above definitions it is assumed that the space X belongs to a given family A of spaces, then the topology τ is called A−splitting (respectively, 468 D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos A−jointly continuous) (see [6]). In the present paper we shall considered only the case A = {F}, where F is a space, and instead of A-splitting and A-jointly continuous we write F-splitting and F-jointly continuous. Let X be a space with a topology τ. We denote (see, for example, [8]) by ≤ τ (respectively, by ∼ τ ) a preorder (respectively, an equivalence relation) on X defined as follows: if x,y ∈ X, then we write x ≤ τ y (respectively, x∼ τ y) if and only if x ∈ ClX({y}) (respectively, x ∈ ClX({y}) and y ∈ ClX({x})). (By ClX(Q) we denote the closure of a set Q in the space X). On the set C(Y,Z) we denote a preorder ≤ (respectively, an equivalence relation ∼ ) as follows: if g,f ∈ C(Y,Z), then we write g ≤ f (respectively, g ∼ f) if g(y) ≤ τZ f(y) (respectively, g(y) ∼ τZ f(y)) for every y ∈ Y (see, for example, [7]). By S we denote the Sierpinski space, that is, the set {0, 1} equipped with the topology τ(S) ≡ {∅,{0, 1},{1}}, and by D the set {0, 1} with the trivial topology. In [7] the notions of S-splitting and S-jointly continuous (respec- tively, D-splitting and D-jointly continuous) topologies are characterized by the above preorders (respectively, equivalence relations) on C(Y,Z). By the trivial topology on a set X we mean the topology {∅,X}. Let U be a quasi-uniformity on the space Z (see, for example, [4]). This quasi-uniformity defines on the set C(Y,Z) a quasi-uniformity Q(U) as follows (see [11]): the set of all subsets of C(Y,Z) of the form (Y,U) = {(f,g) ∈ C(Y,Z) ×C(Y,Z) : (f(y),g(y)) ∈ U, for every y ∈ Y}, where U ∈ U, is a basis for the quasi-uniformity Q(U). We denote by τQ(U) (see [11]) the topology on C(Y,Z), which is defined by the quasi-uniformity Q(U), that is: the subbasic neighborhoods of an arbitrary element f ∈ C(Y,Z) in τQ(U) are of the form: (Y,U)[f] = {g ∈ C(Y,Z) : (f,g) ∈ (Y,U)}, where U ∈ U. In this case we shall say also that τQ(U) is generated by the quasi- uniformity U. Let O(Y ) be the family of all open sets of the space Y . The Scott topology on O(Y ) (see, for example, [8]) is defined as follows: a subset IH of O(Y ) is open if: (α) the conditions U ∈ IH, V ∈O(Y ), and U ⊆ V imply V ∈ IH, and (β) for every collection of open sets of Y , whose union belongs to IH, there are finitely many elements of this collection whose union also belongs to IH. The Isbell topology τis on C(Y,Z) (see [9] and [10]) is the topology for which the family of all sets of the form (IH,U) = {f ∈ C(Y,Z) : f−1(U) ∈ IH}, where IH is Scott open in O(Y ) and U ∈O(Z), is a subbasis. The pointwise topology (see, for example, [3]) τp on C(Y,Z) is the topology for which the family of all sets of the form n-Tuple relations and topologies on function spaces 469 ({y},U) = {f ∈ C(Y,Z) : f(y) ∈ U}, where y ∈ Y and U ∈O(Z), is a subbasis. The compact open (see [5]) topology τc on C(Y,Z) is the topology for which the family of all sets of the form (K,U) = {f ∈ C(Y,Z) : f(K) ⊆ U}, where K is a compact subset of Y and U ∈O(Z), is a subbasis. Below, we recall some well known results: (1) The pointwise topology, the compact open topology and the Isbell topol- ogy on C(Y,Z) are always splitting (see, for example, [1], [2], [3], [5], [9] and [10]). (2) The compact open topology on C(Y,Z) is jointly continuous if Y is locally compact (see [5] and [2]). (3) The Isbell topology on C(Y,Z) is jointly continuous if Y is corecompact (see, for example, [9]). (4) The topology τQ(U) is jointly continuous (see [11]). 2. F-splitting and F-jointly continuous topologies. In the paper we denote by F a non-discrete space which is the set {0, 1, ...,n}, n > 0, equipped with an arbitrary fixed topology. By Uj, j = 0, 1, ...,n, we denote the intersection of all open neighborhoods of j in F. It is clear that if F is the discrete space, then every topology τ on C(Y,Z) is F-splitting and F-jointly continuous. Theorem 2.1. The trivial topology and, hence, every topology on the set C(Y,Z) is F-jointly continuous if and only if the topology of Z is trivial. Proof. Suppose that the topology of Z is trivial. Then for any topology τ on C(Y,Z) and any continuous map G : F → Cτ (Y,Z), the map G̃ : F ×Y → Z is trivially continuous, that is τ is F-jointly continuous. Conversely, suppose that the trivial topology τ on C(Y,Z) is F-jointly con- tinuous. We prove that the topology of Z is trivial. Indeed, in the opposite case, there exist two distinct elements z1, z2 of Z and an open subset U of Z such that z1 ∈ U and z2 6∈ U. We consider the maps f,g ∈ C(Y,Z) such that f(Y ) = {z1} and g(Y ) = {z2}. Denote by i, the element of F such that Ui 6= {i}. Let G : F → Cτ (Y,Z) be a map such that G(i) = f and G(j) = g, for every j ∈ F \ {i}. Since τ is trivial, the map G is continuous. Since τ is F-jointly continuous, the map G̃ : F × Y → Z is also continuous. By the definition of G̃, G̃(i,y) = G(i)(y) = f(y) = z1 ∈ U, y ∈ Y . Therefore for a fixed y ∈ Y there exists an open neighborhood Vy such that G̃(Ui ×Vy) ⊆ U. Let j ∈ Ui \{i}. Then, we have G̃(j,y) = G(j)(y) = g(y) = z2 6∈ U which is a contradiction. Thus the topology of Z is trivial. � Theorem 2.2. If the discrete topology, and hence, every topology on C(Y,Z) is F-splitting, then Z is a T0 space. 470 D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos Proof. Suppose that the discrete topology τ on C(Y,Z) is F-splitting and Z is not T0 space. We shall construct a continuous map F : F ×Y → Z such that F̂ is not continuous, which will be a contradiction. There exist two distinct elements z1, z2 of Z such that either z1,z2 ∈ V or z1,z2 6∈ V for every open subset V of Z. Let i be an element of F such that Ui 6= {i}. We consider the map F : F × Y → Z such that F(i,y) = z1 for every y ∈ Y , and F(j,y) = z2 for every j ∈ F \{i} and y ∈ Y . Let V be an open subset of Z. Then, either F−1(V ) = F×Y or F−1(V ) = ∅, which means that F is continuous. By the definition of F̂ : F → Cτ (Y,Z) we have F̂(i)(Y ) = {z1}, and F̂(j)(Y ) = {z2} for every j ∈ F \{i}. Let j ∈ Ui \{i}. Then F̂(j) 6∈ {F̂(i)}, that is, F̂(Ui) 6⊆ {F̂(i)}, which means that F̂ is not continuous. � Theorem 2.3. Let Z be a T1 space. Then, the discrete topology, and hence, every topology on C(Y,Z) is F-splitting. Proof. Let τ be the discrete topology on C(Y,Z) and F : F × Y → Z a continuous map. We prove that the map F̂ : F → Cτ (Y,Z) is continuous. Let i ∈ F and F̂(i) = f. Then f ∈ {f} ∈ τ. It is suffices to prove that F̂(Ui) ⊆ {f}, that is, F̂(j) = f for every j ∈ Ui. Let j ∈ Ui and y be an arbitrary point of Y . We need to prove that F̂(j)(y) = f(y). Let U be an arbitrary open neighborhood of f(y) = F̂(i)(y) = F(i,y) in Z. Since the map F is continuous there exists an open neighborhood Vy of y in Y such that F(Ui × Vy) ⊆ U. Therefore, F(j,y) = F̂(j)(y) ∈ U, which means that f(y) ∈ ClZ({F̂(j)(y)}). Since Z is a T1 space, f(y) = F̂(j)(y). Hence, F̂(j) = f. Thus, the map F̂ : F → Cτ (Y,Z) is continuous and therefore the topology τ on C(Y,Z) is F-splitting. � Theorem 2.4. The pointwise topology τp, the compact-open topology τc, and the Isbell topology τis on C(Y,Z) are F-splitting and F-jointly continuous. Proof. First, we prove that τp is F-jointly continuous. Let G : F → Cτp(Y,Z) be a continuous map. We need to prove that the map G̃ : F × Y → Z is continuous. Let (i,y) ∈ F × Y and U be an arbitrary open neighborhood of G̃(i,y) = G(i)(y) in Z. Then G(i) ∈ ({y},U). Since G is continuous, G(Ui) ⊆ ({y},U). Also, since the map G(j), j ∈ Ui, is continuous and G(j)(y) ∈ U there exists an open neighborhood V jy of y in Y such that G(j)(V j y ) ⊆ U. Let Vy = ∩{V jy : j ∈ Ui}. Then, G̃(Ui × Vy) ⊆ U. Thus, the map G̃ is continuous and the therefore the topology τp is F-jointly continuous. Since τp ⊆ τc and τp ⊆ τis (see [10]) the topologies τc and τis are also F-jointly continuous. Finally, since the topologies τp, τc and τis are splitting, they are also F- splitting. � n-Tuple relations and topologies on function spaces 471 Theorem 2.5. The topology τQ(U) on the set C(Y,Z) generated by a quasi- uniformity U on the space Z is F-splitting and F-jointly continuous. Proof. Let U be a quasi-uniformity on the space Z. Since τQ(U) is jointly continuous (see [11]), this topology is also F-jointly continuous. We prove that τQ(U) is F-splitting. Let F : F × Y → Z be a continuous map. We need to prove that F̂ : F → CτQ(U) (Y,Z) is continuous. Let i ∈ F and F̂(i) = fi. The set (Y,U)[fi] = {h ∈ C(Y,Z) : (fi,h) ∈ (Y,U)}, where U is an element of U, is an open neighborhood of fi in CτQ(U) (Y,Z). We prove that F̂(Ui) ⊆ (Y,U)[fi]. Let j ∈ Ui. It is suffices to prove that F̂(j) = fj ∈ (Y,U)[fi], that is fj ∈ (Y,U)[fi] or (fi(y),fj(y)) ∈ U for every y ∈ Y . Let y ∈ Y and U[fi(y)] = {z ∈ Z : (fi(y),z) ∈ U}. Since F is continuous there exists an open neighborhood Vy of y in Y such that F(Ui × Vy) ⊆ U[fi(y)]. So, for the element (j,y) of Ui × Vy we have F(j,y) = fj(y) ∈ U[fi(y)] or (fi(y),fj(y)) ∈ U. Thus, the map F̂ is continuous and therefore the topology τQ(U) is F-splitting. � Definition 2.6. For every space X with a topology t we define an (n + 1)- tuple relation denoted by Rt in X as follows: an (n + 1)-tuple (x0,x1, ...,xn) of elements of X belongs to Rt if for every i,j ∈ F, xi ∈ ClX({xj}) provided that i ∈ ClF({j}). We observe that if t1, t2 are two topologies on a set X such that t1 ⊆ t2, then Rt2 ⊆ Rt1 . Definition 2.7. On the set C(Y,Z) we define an (n+1)-tuple relation denoted by R as follows: an (n + 1)-tuple (f0,f1, ...,fn) of elements of C(Y,Z) belongs to R if (f0(y),f1(y), ...,fn(y)) ∈ RtZ for every y ∈ Y . Below we give necessary and sufficient conditions for an arbitrary topology τ on C(Y,Z) to be F-splitting or F-jointly continuous. Theorem 2.8. A topology τ on C(Y,Z) is F-splitting if and only if R ⊆ Rτ . Proof. Let τ be an F-splitting topology on C(Y,Z). Suppose that (f0,f1, ...,fn) ∈ R. We need to prove that (f0,f1, ...,fn) ∈ Rτ . Let F : F×Y → Z be a map for which F(i,y) = fi(y), for every i ∈ F and y ∈ Y . This map is continuous. Indeed, let U be an open neighborhood of fi(y) in Z. Since fi is continuous, the set f −1 i (U) is open neighborhood of y in Y . Therefore it is sufficient to prove that: F(Ui ×f−1i (U)) ⊆ U. Let (j,y′) ∈ Ui × f−1i (U). By the definition of F, F(j,y ′) = fj(y′). Since j ∈ Ui we have i ∈ ClF({j}). Also, by the definition of the (n + 1)-tuple relation R we have fi(y′) ∈ ClZ(fj(y′)). Since fi(y′) ∈ U we have fj(y′) ∈ U. Thus, F(Ui ×f−1i (U)) ⊆ U, that is F is continuous. Furthermore, since τ is F-splitting, the map F̂ : F→ Cτ (Y,Z) is continuous. 472 D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos Now, we prove that (f0,f1, ...,fn) ∈ Rτ . Let i,j ∈ F such that i ∈ ClF({j}). We need to prove that fi ∈ ClCτ (Y,Z)({fj}). Let W be an open neighborhood of fi in Cτ (Y,Z). Then, F̂−1(W) is an open neighborhood of i in F and therefore j ∈ F̂−1(W). This means that F̂(j) = fj ∈ W and therefore fi ∈ ClCτ (Y,Z)({fj}). Hence, (f0,f1, ...,fn) ∈ R τ . Conversely, let τ be a topology on C(Y,Z) such that the condition (f0,f1, ..,fn) ∈ R implies (f0,f1, ...,fn) ∈ Rτ . We prove that τ is F-splitting. Let F : F×Y → Z be a continuous map. Consider the map F̂ : F→ Cτ (Y,Z) and let F̂(i) = fi, i ∈ F. First, we prove that (f0,f1, ...,fn) ∈ R. Indeed, let y ∈ Y . Consider the (n + 1)-tuple (f0(y),f1(y), ...,fn(y)) and suppose that i ∈ ClF({j}). Let U be an open neighborhood of fi(y) in Z. Since F(i,y) = fi(y) and F is continuous, the set F−1(U) is an open neighborhood of (i,y) in F×Y . Therefore there exist open sets V and W of F and Y , respectively, such that (i,y) ∈ V × W ⊆ F−1(U). This means that j ∈ V and F(j,y) = fj(y) ∈ U and therefore fi(y) ∈ ClZ(fj(y)), that is (f0(y),f1(y), ...,fn(y)) ∈ RtZ . Hence (f0,f1, ...,fn) ∈ R. By the assumption, (f0,f1, ...,fn) ∈ Rτ . Now, we prove that F̂ is continuous. Let F̂(i) = fi and H be an open neighborhood of fi in Cτ (Y,Z). It suffices to prove that F̂(Ui) ⊆ H. Let j ∈ Ui. Then i ∈ ClF({j}). Since (f0,f1, ...,fn) ∈ R we have (f0(y),f1(y) , ...,fn(y)) ∈ RtZ for every y ∈ Y . Therefore fi(y) ∈ ClZ({fj(y)}) for every y ∈ Y , that is fi ∈ ClCτ (Y,Z)({fj}) which means that F̂(j) = fj ∈ H. Hence the map F̂ is continuous and the topology τ is F-splitting. � The next corollary follows by the fact that for F=S (respectively, for F=D) then the 2-tuple relations R and Rτ on C(Y,Z) coincide with the relations ≤ and ≤ τ (respectively, with the relations ∼ and ∼ τ ). Corollary 2.9. The following (see [7]) are true: (1) A topology τ on C(Y,Z) is S-splitting if and only if the condition f ≤ g implies f ≤ τ g. (2) A topology τ on C(Y,Z) is D-splitting if and only if the condition f ∼ g implies f ∼ τ g. Theorem 2.10. A topology τ on C(Y,Z) is F-jointly continuous if and only if Rτ ⊆ R. Proof. Let τ be an F-jointly continuous topology on C(Y,Z). Suppose that (f0,f1, ...,fn) ∈ Rτ . We need to prove that (f0,f1, ...,fn) ∈ R. Let G : F→ Cτ (Y,Z) be a map for which G(i) = fi for every i ∈ F. We prove that G is continuous. Let H be an open subset of Cτ (Y,Z) such that fi ∈ H. It is suffices to prove that G(Ui) ⊆ H. Let j ∈ Ui. Since, i ∈ ClF({j}). and (f0,f1, ...,fn) ∈ Rτ we have fi ∈ ClCτ (Y,Z)({fj}). Therefore G(j) = fj ∈ H, that is the map G is continuous. n-Tuple relations and topologies on function spaces 473 Moreover, since τ is F-jointly continuous, the map G̃ : F×Y → Z is also continuous. Now, we prove that (f0,f1, ..,fn) ∈ R. Let y ∈ Y . Consider the (n+1)-tuple (f0(y),f1(y), ...,fn(y)) and let i ∈ ClF({j}). It is suffices to prove prove that fi(y) ∈ ClZ({fj(y)}). Let U be an open neighborhood of fi(y) in Z. Since G̃(i,y) = fi(y) we have G̃−1(U) is an open subset of F×Y containing the point (i,y). There exist an open neighborhood V of i in F and an open neighborhood W of y in Y such that V × W ⊆ G̃−1(U). Since i ∈ ClF({j}) we have that j ∈ V and therefore (j,y) ∈ G̃−1(U), which means that G̃(j,y) = fj(y) ∈ U. Thus, fi(y) ∈ ClZ({fj(y)}). Hence, (f0,f1, ...,fn) ∈ R. Conversely, let τ be a topology on C(Y,Z) such that the condition (f0,f1, ...,fn) ∈ Rτ implies (f0,f1, ...,fn) ∈ R. We prove that τ is F-jointly continuous. Let G : F→ Cτ (Y,Z) be a continuous map such that G(i) = fi for every i ∈ F. Then the (n+1)-tuple (f0,f1, ...,fn) belongs to Rτ . Indeed, let i ∈ ClF({j}) and H be an open neighborhood of fi in Cτ (Y,Z). Since G is continuous, the set G−1(H) is an open subset of F containing the point i. Hence, j ∈ G−1(H) and, therefore, G(j) = fj ∈ H, which means that fi ∈ ClCτ (Y,Z)({fj}). Thus, (f0,f1, ...,fn) ∈ Rτ . Now, we consider the map G̃ : F×Y → Z and prove that this map is continuous. Let (i,y) ∈ F×Y . Suppose that U is an open subset of Z such that G̃(i,y) = G(i)(y) = fi(y) ∈ U. Since the map fi is continuous and fi(y) ∈ U, there exists an open neighborhood W of y in Y such that fi(W) ⊆ U. To prove that G̃ is continuous it is suffices to prove that G̃(Ui ×W) ⊆ U. Indeed, let (j,y′) ∈ Ui × W . Then j ∈ Ui, that is i ∈ ClF({j}). By the above fi ∈ ClCτ (Y,Z)({fj}). Since (f0,f1, ...,fn) ∈ R τ , by assumption we have (f0,f1, ...,fn) ∈ R. Thus fi(y) ∈ ClZ({fj(y)}) for every y ∈ Y and therefore fi(y′) ∈ ClZ({fj(y′)}). Hence G̃(j,y′) = G(j)(y′) = fj(y′) ∈ U. Thus, G̃ is continuous and therefore τ is an F-jointly continuous topology. � Corollary 2.11. The following (see [7]) are true: (1) A topology τ on C(Y,Z) is S-jointly continuous if and only if the con- dition g ≤ τ f implies g ≤ f. (2) A topology τ on C(Y,Z) is D-jointly continuous if and only if the con- dition f ∼ τ g implies f ∼ g. Remark 2.12. The first five Theorems of this paper can be obtained by the last two Theorems provided that: (1) For the trivial topology, and hence, for every topology τ on the set C(Y,Z) we have Rτ ⊆ R if and only if the topology of Z is trivial. (2) If for the discrete topology, and hence, for every topology τ on C(Y,Z) we have R ⊆ Rτ , then Z is T0 space. 474 D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos (3) Let Z be a T1 space. Then, for the discrete topology, and hence, for every topology τ on C(Y,Z) we have R ⊆ Rτ . (4) For the pointwise topology, for the compact open topology, and for the Isbell topology τ on C(Y,Z) we have Rτ = R. (5) For the topology τQ(U) on the set C(Y,Z) which generated by a quasi- uniformity U we have R = RτQ(U) . The above statements can be easily proved. References [1] R. Arens, A topology of spaces of transformations, Annals of Math., 47(1946), 480-495. [2] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1(1951), 5-31. [3] J. Dugundji, Topology, (Allyn and Bacon, Inc., Boston 1968). [4] P. Fletcher and W. Lindgren, Quasi-uniform spaces, (Lecture Notes in Pure and Applied Mathematics; Vol. 77 (1982)). [5] R. H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51(1945), 429-432. [6] D. N. Georgiou, S.D. Iliadis and B. K. Papadopoulos, Topologies on function spaces, Studies in Topology, VII, Zap. Nauchn. Sem. S.-Peterburg Otdel. Mat. Inst. Steklov (POMI), 208(1992), 82-97. J. Math. Sci., New York 81(1996), No. 2, 2506-2514. [7] D. N. Georgiou, S.D. Iliadis and B. K. Papadopoulos, Topologies and orders on function spaces, Publ. Math. Debrecen, 46/ 1-2 (1995), 1-10. [8] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A Compendium of Continuous Lattices, (Springer, Berlin-Heidelberg-New York 1980). [9] P. Lambrinos and B. K. Papadopoulos, The (strong) Isbell topology and (weakly) con- tinuous lattices, Continuous Lattices and Applications, Lecture Notes in pure and Appl. Math. No. 101, Marcel Dekker, New York 1984, 191-211. [10] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, (Lec- ture Notes in Mathematics, 1315, Springer Verlang). [11] M. G. Murdershwar and S. A. Naimpaly, Quasi uniform spaces, (Noordhoff, 1966). Received January 2002 Revised September 2002 D. N. Georgiou Department of Mathematics, University of Patras, 265 00 Patras, Greece E-mail address : georgiou@math.upatras.gr S. D. Iliadis Department of Mathematics, University of Patras, 265 00 Patras, Greece E-mail address : iliadis@math.upatras.gr B. K. Papadopoulos Department of Civil Engineering, Democritus University of Thrace, 67100 Xan- thi, Greece E-mail address : papadob@civil.duth.gr