GeorgiouAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 10, No. 1, 2009 pp. 159-171 Topologies on function spaces and hyperspaces D. N. Georgiou ∗ Abstract. Let Y and Z be two fixed topological spaces, O(Z) the family of all open subsets of Z, C(Y, Z) the set of all continuous maps from Y to Z, and OZ (Y ) the set {f −1(U ) : f ∈ C(Y, Z) and U ∈ O(Z)}. In this paper, we give and study new topologies on the sets C(Y, Z) and OZ (Y ) calling (A, A0)-splitting and (A, A0)-admissible, where A and A0 families of spaces. 2000 AMS Classification: 54C35 Keywords: function space, hyperspace, splitting topology, admissible topol- ogy. 1. Preliminaries Let Y and Z be two fixed topological spaces. By C(Y, Z) we denote the set of all continuous maps from Y to Z. If t is a topology on the set C(Y, Z), then the corresponding topological space is denoted by Ct(Y, Z). Let X be a space. To each map g : X × Y → Z which is continuous in y ∈ Y for each fixed x ∈ X, we associate the map g∗ : X → C(Y, Z) defined as follows: for every x ∈ X, g∗(x) is the map from Y to Z such that g∗(x)(y) = g(x, y), y ∈ Y . Obviously, for a given map h : X → C(Y, Z), the map h⋄ : X × Y → Z defined by h⋄(x, y) = h(x)(y), (x, y) ∈ X × Y , satisfies (h⋄) ∗ = h and is continuous in y for each fixed x ∈ X. Thus, the above association (defined in [7]) between the mappings from X × Y to Z that are continuous in y for each fixed x ∈ X, and the mappings from X to C(Y, Z) is one-to-one. In 1946 R. Arens [1] introduced the notion of an admissible topology: a topology t on C(Y, Z) is called admissible if the map e : Ct(Y, Z) × Y → Z, called evaluation map, defined by e(f, y) = f (y), is continuous. In 1951 R. Arens and J. Dugundji [2] introduced the notion of a splitting topology: a topology t on C(Y, Z) is called splitting if for every space X, the continuity of a map g : X × Y → Z implies the continuity of the map ∗Work Supported by the Caratheodory programme of the University of Patras. 160 D. N. Georgiou g∗ : X → Ct(Y, Z). On the set C(Y, Z) there exists the greatest splitting topology, denoted here by tgs (see [2]). They also proved that a topology t on C(Y, Z) is admissible if and only if for every space X, the continuity of a map h : X → Ct(Y, Z) implies that of the map h ⋄ : X × Y → Z If in the above definitions it is assumed that the space X belongs to a fixed class A of topological spaces, then the topology t is called A-splitting or A-admissible, respectively (see [8]). In the case where A = {X} we write X-splitting (respectively, X-admissible) instead of {X}-splitting (respectively, {X}-admissible). Let X be a space. In what follows by O(X) we denote the family of all open subsets of X. Also, for two fixed topological spaces Y and Z we denote by OZ (Y ) the set {f −1(U ) : f ∈ C(Y, Z) and U ∈ O(Z)}. The Scott topology Ω(Y ) on O(Y ) (see, for example, [11]) is defined as follows: a subset IH of O(Y ) belongs to Ω(Y ) 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 strong Scott topology Ωs(Y ) on O(Y ) (see [12]) is defined as follows: a subset IH of O(Y ) belongs to Ωs(Y ) if: (α) the conditions U ∈ IH, V ∈ O(Y ), and U ⊆ V imply V ∈ IH, and (β) for every open cover of Y there are finitely many elements of this cover whose union also belongs to IH. The Isbell topology tIs (respectively, strong Isbell topology tsIs) on C(Y, Z) (see, for example, [13] and [12]) is the topology, which has as a subbasis the family of all sets of the form: (IH, U ) = {f ∈ C(Y, Z) : f −1(U ) ∈ IH}, where IH ∈ Ω(Y ) (respectively, IH ∈ Ωs(Y )) and U ∈ O(Z). The compact open topology (see [7]) on C(Y, Z), denoted here by tco, 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 is an open subset of Z, form a subbase. It is known that tco ⊆ tIs (see, for example, [13]). A subset K of a space X is said to be bounded if every open cover of X has a finite subcover for K (see [12]). A space X is called corecompact (see [11]) if for every x ∈ X and for every open neighborhood U of x, there exists an open neighborhood V of x such that the subset V is bounded in the space U (see [11]). Topologies on function spaces 161 Below, we give some well known results: (1) The Isbell topology and, hence, the compact open topology, and the point open topology (denoted here by tpo) on C(Y, Z) are always split- ting (see, for example, [2], [3], and [13]). (2) The compact open topology on C(Y, Z) is admissible if Y is a regular locally compact space. In this case the compact open topology is also the greatest splitting topology (see [2]). (3) The Isbell topology on C(Y, Z) is admissible if Y is a corecompact space. In this case the Isbell topology is also the greatest splitting topology (see, for example, [12] and [14]). (4) A topology larger than a admissible topology is also admissible (see [2]). (5) A topology smaller than a splitting topology is also splitting (see [2]). (6) The strong Isbell topology on C(Y, Z) is admissible if Y is a locally bounded space (see [12]). For a summary of all the above results and some open problems on function spaces see [10]. Also, [4] and [5] are other papers related to this area. In what follows if ϕ : X → Y is a map and X0 ⊆ X, then by ϕ|X0 : X0 → Y we denote the restriction of the map ϕ on the set X0. Also, if h : X × Y → Z is a map and X0 ⊆ X, then by h|X0×Y we denote the restriction of the map h on the set X0 × Y . In Sections 2 and 3 we give and study new topologies on the sets C(Y, Z) and OZ (Y ) calling (A, A0)-splitting and (A, A0)-admissible, where A and A0 families of spaces. 2. (A, A0)-splitting and (A, A0)-admissible topologies on the set C(Y, Z) Note 1. Let A be a family of topological spaces. For every X ∈ A we denote by X0 a subspace of X and by A0 the family of all such subspaces X0. In all paper by (A, A0) we denote the family of all pairs (X, X0) such that X ∈ A, X0 ∈ A0, and X0 is a subspace of X. Definition 2.1. A topology t on C(Y, Z) is called (A, A0)-splitting if for every pair (X, X0) ∈ (A, A0), the continuity of a map g : X × Y → Z implies the continuity of the map g∗|X0 : X0 → Ct(Y, Z), where g ∗ : X → Ct(Y, Z) the map which is defined in preliminaries. A topology t on C(Y, Z) is called (A, A0)-admissible if for every pair (X, X0) ∈ (A, A0), the continuity of a map h : X → Ct(Y, Z) implies that of the map h⋄|X0×Y : X0 × Y → Z, where h ⋄ : X × Y → Z the map which is defined in preliminaries. In the case where A = {X} and A0 = {X0}, where X0 is a subspace of X, we write (X, X0)-splitting (respectively, (X, X0)-admissible) instead of ({X}, {X0})-splitting (respectively, ({X}, {X0})-admissible). 162 D. N. Georgiou Clearly, the following theorem is true. Theorem 2.2. The following statements are true: (1) Every splitting (respectively, admissible) topology on C(Y, Z) is (A, A0)- splitting (respectively, (A, A0)-admissible), where A and A0 are arbi- trary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. (2) Every A-splitting (respectively, A-admissible) topology on C(Y, Z) is (A, A0)-splitting (respectively, (A, A0)-admissible), where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. Example 2.3. (1) The point-open, the compact open, and the Isbell topologies are (A, A0)- splitting, where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. (2) If Y is a regular locally compact space, then the compact-open topology is (A, A0)-admissible, where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. (3) If Y is a corecompact space, then the Isbell topology is (A, A0)-admissible, where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. (4) If Y is a locally bounded space, then the strong Isbell topology is (A, A0)-admissible, where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. (5) Let X be a space, x0 ∈ X, X0 the subspace {x0} of X, and t an arbitrary topology on C(Y, Z) which it is not X-splitting. Then, the topology t is (X, X0)-splitting. It is clear that this topology t is not splitting. (6) Let X be a space, x0 ∈ X, X0 the subspace {x0} of X, and t an arbitrary topology on C(Y, Z) which it is not X-admissible. Then, the topology t is (X, X0)-admissible. It is clear that this topology t is not admissible. Theorem 2.4. The following statements are true: (1) A topology smaller than an (A, A0)-splitting topology is also (A, A0)- splitting. (2) A topology larger than an (A, A0)-admissible topology is also (A, A0)- admissible. Proof. We prove only the statement (1). The proof of (2) is similar. Let t1 be an (A, A0)-splitting topology on C(Y, Z) and t2 a topology on C(Y, Z) such that t2 ⊆ t1. We prove that the topology t2 is a (A, A0)-splitting topology. Indeed, let (X, X0) ∈ (A, A0) and let g : X × Y → Z be a continuous map. Since the topology t1 is (A, A0)-splitting, the map g ∗|X0 : X0 → Ct1 (Y, Z) is continuous. Also, since t2 ⊆ t1, the identical map id : Ct1 (Y, Z) → Ct2 (Y, Z) is Topologies on function spaces 163 continuous. So, the map g∗|X0 : X0 → Ct2 (Y, Z) is continuous as a composition of continuous maps. Thus, the topology t2 is (A, A0)-splitting. � Definition 2.5. Let (A1, A10) and (A 2, A20) two pairs of spaces, where A 1 (re- spectively, A2) and A10 (respectively, A 2 0) are arbitrary families of spaces such that every element X0 ∈ A 1 0 (respectively, every element X0 ∈ A 2 0) is a sub- space of an element X ∈ A1 (respectively, of an element X ∈ A2). We say that the pairs (A1, A10) and (A 2, A20) are equivalent if a topology t on C(Y, Z) is (A1, A10)-splitting if and only if t is (A 2, A20)-splitting, and t is (A 1, A10)- admissible if and only if t is (A2, A20)-admissible. In this case we write (A1, A10) ∼ (A 2 , A20). Theorem 2.6. For every pair (A, A0), where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A, there exists a pair (X(A), X(A0)), where X(A) is a space and X(A0) is a subspace of X(A) such that (A, A0) ∼ (X(A), X(A0)). Proof. Let T csp be the set of all topologies on C(Y, Z) which are not (A, A0)- splitting and let T cad the set of all topologies on C(Y, Z) which are not (A, A0)- admissible. For each t ∈ T csp there exists in (A, A0) a pair (X sp t , X sp t,0) such that t is not (X sp t , X sp t,0)-splitting. Similarly, for each t ∈ T c ad there exists in (A, A0) a pair (X adt , X ad t,0) such that t is not (X ad t , X ad t,0)-admissible. Let A′ = {X sp t : t ∈ T c sp} ∪ {X ad t : t ∈ T c ad} and A′0 = {X sp t,0 : t ∈ T c sp} ∪ {X ad t,0 : t ∈ T c ad}. Of course, we can suppose that the spaces from A′ and A′0 are pair-wise disjoint. Let X(A) and X(A0) be the free union of all the spaces from A ′ and A′0, respectively. We prove that the pair (X(A), X(A0)) is the required pair. Let t be an (A, A0)-splitting topology on C(Y, Z). We prove that this topol- ogy is (X(A), X(A0))-splitting. Indeed, let g : X(A) × Y → Z be a continuous map. It suffices to prove that the map g∗|X(A0) : X(A0) → Ct(Y, Z) is continuous. Let X ∈ A′ ⊆ A. Then, the restriction g|X×Y of the map g on X × Y ⊆ X(A) × Y is also a continuous map and, therefore, since the topology t is (A, A0)-splitting we have that the map (g|X×Y ) ∗|X0 : X0 → Ct(Y, Z) is continuous. Since X(A0) is the free union of all the spaces from A ′ 0 and (g|X×Y ) ∗|X0 = (g ∗|X(A0))|X0 , it follows that the map g ∗|X(A0) : X(A0) → Ct(Y, Z) is continuous. Thus, the topology t on C(Y, Z) is (X(A), X(A0))- splitting. Now, let t be an (X(A), X(A0))-splitting topology on C(Y, Z). We prove that t is (A, A0)-splitting. We suppose that t is not (A, A0)-splitting. Then, t ∈ T csp and, therefore, t is not (X sp t , X sp t,0)-splitting for some pair (X sp t , X sp t,0) ∈ (A, A0). Thus, there exists a continuous map g : X sp t × Y → Z such that the 164 D. N. Georgiou map g∗|Xsp t,0 : X sp t,0 → Ct(Y, Z) is not continuous. Since the space X(A) is the free union of all the spaces from the family A′, the map g can be extended to a continuous map g1 : X(A) × Y → Z. Since the map g ∗|Xsp t,0 is not continuous, X sp t,0 ∈ A ′ 0, and the space X(A0) is the free union of all spaces from A ′ 0 we have that the map g∗|X(A0) : X(A0) → Ct(Y, Z) is not continuous, which contradicts our assumption that t is a (X(A), X(A0))- splitting topology. Thus, a topology t on C(Y, Z) is (A, A0)-splitting if and only if it is (X(A), X(A0))-splitting. Similarly, a topology t on C(Y, Z) is (A, A0)-admissible if and only if is (X(A), X(A0))-admissible. Hence, (A, A0) ∼ (X(A), X(A0)). � Theorem 2.7. There exists the greatest (A, A0)-splitting topology, where A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. Proof. Let {ti : i ∈ I} be the family of all (A, A0)-splitting topologies on C(Y, Z). We consider the topology t = ∨{ti : i ∈ I}. Clearly, t is (A, A0)- splitting and ti ⊆ t, for every i ∈ I. Thus, t is the greatest (A, A0)-splitting topology. � Note 2. In what follows we denote by t(A, A0) the greatest (A, A0)-splitting topology on C(Y, Z), Theorem 2.8. The following statements are true: (1) If (A, A0) = ∪{(A i, Ai0) : i ∈ I}, then t(A, A0) = ∩{t(A i , Ai0) : i ∈ I}. (2) t(A, A0) = ∩{t(X, X0) : (X, X0) ∈ (A, A0)}. (3) If (A, A0) = ∩{(A i, Ai0) : i ∈ I}, then ∨{t(Ai, Ai0) : i ∈ I} ⊆ t(A, A0). Proof. (1) Since (A, A0) = ∪{(A i, Ai0) : i ∈ I} we have that every topology which is (A, A0)-splitting is also (A i, Ai0)-splitting, for every i ∈ I. Thus, the topology t(A, A0) is (A i, Ai0)-splitting and, therefore, t(A, A0) ⊆ t(A i, Ai0), for every i ∈ I. So, we have t(A, A0) ⊆ ∩{t(A i, Ai0) : i ∈ I}. Now, we prove the converse relation, that is ∩{t(Ai, Ai0) : i ∈ I} ⊆ t(A, A0). Topologies on function spaces 165 For the above relation it suffices to prove that the topology ∩{t(Ai, Ai0) : i ∈ I} is (A, A0)-splitting. Let (X, X0) ∈ (A, A0) and let g : X × Y → Z be a continuous map. We prove that the map g ∗|X0 : X0 → C∩{t(Ai,Ai 0 ):i∈I}(Y, Z) is continuous. Since (X, X0) ∈ (A, A0), there exists i ∈ I such that (X, X0) ∈ (Ai, Ai0). This means that the map g∗|X0 : X0 → Ct(Ai,Ai 0 )(Y, Z) is continuous. Also, since ∩{t(Ai, Ai0) : i ∈ I} ⊆ t(A i, Ai0), the identical map id : Ct(Ai,Ai 0 )(Y, Z) → C∩{t(Ai,Ai 0 ):i∈I}(Y, Z) is continuous. So, the map g∗|X0 : X0 → C∩{t(Ai,Ai 0 ):i∈I}(Y, Z) is continuous as a composition of continuous maps. Thus, the topology ∩{t(Ai, Ai0) : i ∈ I} is (A, A0)-splitting. (2) The proof of this is a corollary of the statement (1). (3) The proof of this follows by the fact that the topology ∨{t(Ai, Ai0) : i ∈ I} is (A, A0)-splitting. � Theorem 2.9. Let t be an (A, A0)-admissible topology on C(Y, Z). If (Ct(Y, Z), Ct(Y, Z)) ∈ (A, A0), then t is admissible and t(A, A0) ⊆ t. Proof. Let id ≡ h : Ct(Y, Z) → Ct(Y, Z) be the identical map. Clearly, this map is continuous. Since (Ct(Y, Z), Ct(Y, Z)) ∈ (A, A0) and t is (A, A0)-admissible, the map h ⋄|Ct(Y,Z) ≡ h ⋄ : Ct(Y, Z) × Y → Z is continuous. Hence, the topology t is admissible. Now, since the map h⋄ ≡ g : Ct(Y, Z) × Y → Z is continuous, (Ct(Y, Z), Ct(Y, Z)) ∈ (A, A0), and the topology t(A, A0) is (A, A0)-splitting, the map g∗|Ct(Y,Z) = id : Ct(Y, Z) → Ct(A,A0)(Y, Z) is also continuous. Thus, t(A, A0) ⊆ t. � Corollary 2.10. Let t be an (A, A0)-splitting and (A, A0)-admissible topology on C(Y, Z). If (Ct(Y, Z), Ct(Y, Z)) ∈ (A, A0), then t(A, A0) = t. Proof. By Theorem 2.9, t(A, A0) ⊆ t. Also, since the topology t is (A, A0)- splitting, t ⊆ t(A, A0). Thus, t(A, A0) = t. � 166 D. N. Georgiou Theorem 2.11. Let Y be a regular locally compact space, A the family of all Ti- spaces, i = 0, 1, 2, 3, 3 1 2 , A0 an arbitrary family of spaces containing subspaces of spaces of A, Ctco (Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A, A0) = tco = tIs. Proof. Since Y is a regular locally compact space, the compact open topology coincides with the Isbell topology on C(Y, Z) and it is admissible. Hence, tco is (A, A0)-admissible. Also, the topology tco is splitting and, therefore, tco is (A, A0)-splitting. Since Z ∈ A, we have that Ctco (Y, Z) ∈ A (see preliminaries) and, therefore, (Ctco (Y, Z), Ctco (Y, Z)) ∈ (A, A0). Thus, by Corollary 2.10 we have that t(A, A0) = tco. � Theorem 2.12. Let Y be a regular locally compact space, A the family of all topological spaces whose weight is not greater than a certain fixed infinite cardinal, A0 an arbitrary family of spaces containing subspaces of spaces of A, Ctco (Y, Z) ∈ A0, and Y, Z ∈ A. Then, we have t(A, A0) = tco = tIs. Proof. The proof of this theorem is similar to the proof of Theorem 2.11 and follows by Corollary 2.10 and Theorem 3.4.16 of [6]. � Theorem 2.13. Let Y be a regular second-countable locally compact space, A the family of all metrizable spaces, A0 an arbitrary family of spaces contain- ing subspaces of spaces of A, Ctco (Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A, A0) = tco = tIs. Proof. The proof of this theorem is similar to the proof of Theorem 2.11 and follows by Corollary 2.10 and Exercices 4.2.H and 3.4.E(c) of [6]. � Theorem 2.14. Let Y be a regular locally compact Lindelöf space, A the family of all completely metrizable spaces, A0 an arbitrary family of spaces contain- ing subspaces of spaces of A, Ctco (Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A, A0) = tco = tIs. Proof. The proof of this theorem is similar to the proof of Theorem 2.11 and follows by Corollary 2.10 and Exercice 4.3.F(a) of [6]. � Theorem 2.15. Let Y be a corecompact space, A the family of all Ti-spaces, where i = 0, 1, 2, A0 an arbitrary family of spaces containing subspaces of spaces of A, CtIs (Y, Z) ∈ A0, and Z ∈ A. Then, we have t(A, A0) = tIs. Proof. Since Y is corecompact, the Isbell topology tIs on C(Y, Z) is admissible. Hence the topology tIs is (A, A0)-admissible. Also, the topology tIs is splitting and, therefore, tIs is (A, A0)-splitting. Since Z ∈ A, we have that CtIs (Y, Z) ∈ A (see preliminaries) and, therefore, (CtIs (Y, Z), CtIs (Y, Z)) ∈ (A, A0). Thus, by Corollary 2.10 we have that t(A, A0) = tIs. � Theorem 2.16. Let Y be a corecompact space, A the family of all second- countable spaces, A0 an arbitrary family of spaces containing subspaces of spaces of A, CtIs (Y, Z) ∈ A0, and Y, Z ∈ A. Then, we have t(A, A0) = tIs. Topologies on function spaces 167 Proof. The proof of this theorem is similar to the proof of Theorem 2.15 and follows by Corollary 2.10 and the fact that CtIs (Y, Z) ∈ A (see [12]). � 3. On dual topologies Note 3. Let Y and Z be two fixed topological spaces. By OZ (Y ) we denote the set {f −1(U ) : f ∈ C(Y, Z) and U ∈ O(Z)}. Let IH ⊆ OZ (Y ), H ⊆ C(Y, Z), and U ∈ O(Z). We set (IH, U ) = {f ∈ C(Y, Z) : f −1(U ) ∈ IH} and (H, U ) = {f −1(U ) : f ∈ H}. Definition 3.1. (See [9]) Let τ be a topology on OZ (Y ). The topology on C(Y, Z), for which the set {(IH, U ) : IH ∈ τ, U ∈ O(Z)} is a subbasis, is called dual to τ and is denoted by t(τ ). Now, let t be a topology on C(Y, Z). The topology on OZ (Y ), for which the set {(H, U ) : H ∈ t, U ∈ O(Z)} is a subbasis, is called dual to t and is denoted by τ (t). We observe that if τ is a topology on OZ (Y ) and σ a subbasis for τ , then the set {(IH, U ) : IH ∈ σ, U ∈ O(Z)} is a subbasis for t(τ ) (see Lemma 2.5 in [9]). Also, if t is a topology on C(Y, Z) and s a subbasis for t, then the set {(H, U ) : H ∈ s, U ∈ O(Z)} is a subbasis for τ (t) (see Lemma 2.6 in [9]). Note 4. Let X be a space and g : X ×Y → Z a continuous map. If gx : Y → Z is the map for which gx(y) = g(x, y), for every y ∈ Y , then by g we denote the map of X × O(Z) into OZ (Y ), for which g(x, U ) = g −1 x (U ) for every x ∈ X and U ∈ O(Z). Now, let h : X → C(Y, Z) be a map. By h we denote the map of X × O(Z) into OZ (Y ), for which h(x, U ) = (h(x)) −1(U ) for every x ∈ X and U ∈ O(Z). Definition 3.2. Let τ be a topology on OZ (Y ). We say that a map M : X × O(Z) → OZ (Y ) is continuous with respect to the first variable if for every fixed element U of O(Z), the map MU : X → (OZ (Y ), τ ), for which MU (x) = M (x, U ) for every x ∈ X, is continuous. Definition 3.3. A topology τ on OZ (Y ) is called (A, A0)-splitting if for every (X, X0) ∈ (A, A0) the continuity of a map g : X × Y → Z implies the conti- nuity with respect to the first variable of the map g|X0×O(Z) : X0 × O(Z) → (OZ (Y ), τ ). A topology τ on OZ (Y ) is called (A, A0)-admissible if for every (X, X0) ∈ (A, A0) and for every map h : X → C(Y, Z) the continuity with respect to the first variable of the map h : X × O(Z) → (OZ (Y ), τ ) implies the continuity of 168 D. N. Georgiou the map h⋄|X0×Y : X0 × Y → Z defined by h ⋄|X0×Y (x, y) = h(x)(y), (x, y) ∈ X0 × Y . Theorem 3.4. A topology τ on OZ (Y ) is (A, A0)-splitting if and only if the topology t(τ ) on C(Y, Z) is (A, A0)-splitting. Proof. Suppose that the topology τ on OZ (Y ) is (A, A0)-splitting, that is for every pair (X, X0) ∈ (A, A0) the continuity of a map g : X × Y → Z implies the continuity with respect to the first variable of the map g|X0×O(Z) : X0 × O(Z) → (OZ (Y ), τ ). We prove that the topology t(τ ) on C(Y, Z) is (A, A0)-splitting. Let (X, X0) ∈ (A, A0) and g : X × Y → Z be a continuous map. We need to prove that g∗|X0 : X0 → Ct(τ )(Y, Z) is a continuous map. Let x ∈ X0 and (IH, U ) be an open neighborhood of (g ∗|X0 )(x) in Ct(τ )(Y, Z). We must find an open neighborhood V of x in X0 such that (g ∗|X0 )(V ) ⊆ (IH, U ). We have that ((g∗|X0 )(x)) −1(U ) ∈ IH. Since (g∗|X0 )(x) = gx, we have g−1x (U ) ∈ IH, that is, g(x, U ) ∈ IH. Since the map g|X0×O(Z) : X0 × O(Z) → (OZ (Y ), τ ). is continuous with respect to the first variable, the map (g|X0×O(Z))U : X0 → (OZ (Y ), τ ) is continuous. Also, (g|X0×O(Z))U (x) ∈ IH. Thus, there exists an open neighborhood V of x in X0 such that (g|X0×O(Z))U (V ) ⊆ IH. Let x′ ∈ V . Then, (g|X0×O(Z))U (x ′) ∈ IH, that is, g−1 x′ (U ) ∈ IH or (g∗|X0 )(x ′) ∈ (IH, U ). Thus, (g∗|X0 )(V ) ⊆ (IH, U ), which means that the map g∗|X0 is continuous. Conversely, suppose that t(τ ) is (A, A0)-splitting. We prove that τ is (A, A0)- splitting. Let (X, X0) be an element of (A, A0) and g : X ×Y → Z a continuous map. It is sufficient to prove that g|X0×O(Z) : X0 × O(Z) → (OZ (Y ), τ ) is con- tinuous with respect to the first variable. Let U be a fixed element of O(Z). Consider the map (g|X0×O(Z))U : X0 → (OZ (Y ), τ ). Let x ∈ X0, IH ∈ τ , and (g|X0×O(Z))U (x) = g −1 x (U ) ∈ IH. We need to find an open neighborhood V of x in X0 such that (g|X0×O(Z))U (V ) ⊆ IH. Consider the open set (IH, U ) of the space Ct(τ )(Y, Z). Since (g|X0×O(Z))U (x) = g −1 x (U ) ∈ IH, we have gx ∈ (IH, U ). Since t(τ ) is (A, A0)-splitting, the map g ∗|X0 : X0 → Ct(τ )(Y, Z) is continuous. Hence, there exists an open neighborhood V of x in X0 such that (g ∗|X0 )(V ) ⊆ (IH, U ). Let x′ ∈ V . Then, (g∗|X0 )(x ′) = gx′ ∈ (IH, U ), that is, g −1 x′ (U ) ∈ IH or (g|X0×O(Z))U (x ′) ∈ IH. Thus, (g|X0×O(Z))U (V ) ⊆ IH, which means that the map (g|X0×O(Z))U is continuous. � Theorem 3.5. A topology t on C(Y, Z) is (A, A0)-splitting if and only if the topology τ (t) on OZ (Y ) is (A, A0)-splitting. Proof. The proof of this theorem is similar to the proof of Theorem 3.4. � Topologies on function spaces 169 Example 3.6. (1) The topologies τ (tco) and τ (tIs) are (A, A0)-splitting for every pair (A, A0). This follows by the fact that the topologies tco and tIs are splitting and, therefore, (A, A0)-splitting. (2) Let Z be the Sierpinski space, Ω(Y ) the Scott topology, and ΩZ (Y ) the relative topology of Ω(Y ) on OZ (Y ). Then, the topology t(ΩZ (Y )) coincides with the Isbell topology on C(Y, Z). Hence, the topology t(ΩZ (Y )) is splitting and, therefore, (A, A0)-splitting. Thus, the topol- ogy τ (t(ΩZ (Y ))) on OZ (Y ) is (A, A0)-splitting. Theorem 3.7. A topology τ on OZ (Y ) is (A, A0)-admissible if and only if the topology t(τ ) on C(Y, Z) is (A, A0)-admissible. Proof. Suppose that the topology τ on OZ (Y ) is (A, A0)-admissible, that is for every space (X, X0) ∈ (A, A0) and for every map h : X → C(Y, Z) the continuity with respect to the first variable of the map h : X × O(Z) → (OZ (Y ), τ ) implies the continuity of the map h ⋄|X0×Y : X0 ×Y → Z. We prove that t(τ ) is (A, A0)-admissible. Let (X, X0) ∈ (A, A0) and h : X → Ct(τ )(Y, Z) be a continuous map. It is sufficient to prove that the map h⋄|X0×Y : X0 ×Y → Z is continuous. Clearly, it suffices to prove that the map h : X × O(Z) → (OZ (Y ), τ ) is continuous with respect to the first variable. Let x ∈ X, U ∈ O(Z) and IH ∈ τ such that hU (x) = h(x, U ) = (h(x)) −1(U ) ∈ IH. We prove that there exists an open neighborhood V of x in X such that hU (V ) ⊆ IH. Consider the open set (IH, U ) of the space Ct(τ )(Y, Z). Then, h(x) ∈ (IH, U ). Since the map h : X → Ct(τ )(Y, Z) is continuous, there exists an open neighborhood V of x in X such that h(V ) ⊆ (IH, U ). Let x′ ∈ V . Then h(x′) ∈ (IH, U ), that is (h(x′))−1(U ) ∈ IH or hU (x ′) = h(x′, U ) ∈ IH. Thus, hU (V ) ⊆ IH, which means that hU is continuous. Conversely, suppose that the topology t(τ ) is (A, A0)-admissible. We prove that the topology τ is (A, A0)-admissible. Let (X, X0) be a pair of (A, A0) and h : X → C(Y, Z) a map such that h : X × O(Z) → (OZ (Y ), τ ) is continuous with respect to the first variable. We need to prove that the map h⋄|X0×Y : X0 × Y → Z is continuous. Since t(τ ) is (A, A0)-admissible, it is sufficient to prove that the map h : X → Ct(τ )(Y, Z) is continuous. Let x ∈ X, U ∈ O(Z), and IH ∈ τ such that h(x) ∈ (IH, U ). Then, (h(x))−1(U ) ∈ IH. Since the map hU : X → (OZ (Y ), τ ) is continuous, there exists an open neighborhood V of x in X such that hU (V ) ⊆ IH. Let x′ ∈ V . Then, hU (x ′) = (h(x′))−1(U ) ∈ IH or h(x′) ∈ (IH, U ). Thus, h(V ) ⊆ (IH, U ), which means that the map h is continuous. � Theorem 3.8. A topology t on C(Y, Z) is (A, A0)-admissible if and only if the topology τ (t) on OZ (Y ) is (A, A0)-admissible. Proof. The proof of this theorem is similar to the proof of Theorem 3.7. � 170 D. N. Georgiou Example 3.9. (1) If Y is a regular locally compact space, then the topology τ (tco) is (A, A0)-admissible for every pair (A, A0). (2) If Y is a corecompact space, then the topology τ (tIs) is (A, A0)-admissible for every pair (A, A0). (3) If Y is a locally bounded space, then the topology τ (tsIs) is (A, A0)- admissible for every pair (A, A0). (4) Let Ω(Y ) be the Scott topology on O(Y ). By ΩZ (Y ) we denote the relative topology of Ω(Y ) on ΩZ (Y ). If Y is corecompact, then the topology ΩZ (Y ) is admissible (see Corollary 3.12 of [9]) and, therefore, it is (A, A0)-admissible. Thus, the topology t(ΩZ (Y )) on C(Y, Z) is (A, A0)-admissible. Theorem 3.10. Let A and A0 are arbitrary families of spaces such that every element X0 ∈ A0 is a subspace of an element X ∈ A. Then in the set OZ (Y ) there exists the greatest (A, A0)-splitting topology. Proof. Let {τi : i ∈ I} be the set of all (A, A0)-splitting topologies on OZ (Y ). We consider the topology τ = ∨{τi : i ∈ I}. It is not difficult to prove that this topology is (A, A0)-splitting. By this fact we have that this topology is the required greatest (A, A0)-splitting topology. � 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] G. Di Maio, L. Holá, D. Holý and R. McCoy, Topologies on the set space of continuous functions, Topology Appl. 86 (1998), no. 2, 105–122. [5] G. Di Maio, E. Meccariello and S. Naimpally, Hyper-continuous convergence in function spaces, Quest. Answers Gen. Topology 22 (2004), no. 2, 157–162. [6] R. Engelking, General Topology, Warszawa 1977. [7] R. H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945), 429-432. [8] 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 (Russian). Translated in: J. Math. Sci., New York 81, (1996), no. 2, 2506–2514. [9] D. N. Georgiou, S. D. Iliadis and B. K. Papadopoulos, On dual topologies, Topology Appl. 140 (2004), 57–68. [10] D. N. Georgiou, S.D. Iliadis and F. Mynard, Function space topologies, Open Problems in Topology 2 (Elsevier), 15–23, 2007. [11] 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. [12] 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. [13] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lec- ture Notes in Mathematics, 1315, Springer Verlang. Topologies on function spaces 171 [14] F. Schwarz and S. Weck, Scott topology, Isbell topology, and continuous convergence, Lecture Notes in Pure and Appl. Math. No.101, Marcel Dekker, New York 1984, 251-271. Received February 2009 Accepted March 2009 D. N. Georgiou (georgiou@math.upatras.gr) Department of Mathematics, University of Patras, 265 04 Patras, Greece