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A generalization of core compact spaces
Topology and its Applications 153 (2005) 868–873 www.elsevier.com/locate/topol
A generalization of core compact spaces D.N. Georgiou ∗ , S.D. Iliadis University of Patras, Department of Mathematics, 265 00 Patras, Greece Received 30 November 2003; received in revised form 15 June 2004
Abstract In this paper, for a fixed infinite cardinal ν, we give the notion of a ν-core compact space generalizing the notion of a core compact space. Also, on the family of all open subsets of a space Y (actually on a partially ordered set) we define a topology denoted by τν and calling ν-Scott topology. This topology defines on the set C(Y, Z) of all continuous functions of the space Y into a space Z a topology denoted by tν . Some relations between ν-core compact spaces and topologies τν and tν are given. 2005 Published by Elsevier B.V. MSC: 54C35 Keywords: Scott topology; Isbell topology; Function space; Splitting topology; Admissible topology
1. Introduction It is well known that the Scott topology (which is defined originally in the family of all open subsets of a space Y (see [2]) and then on a partially ordered set (see [7])) determines on the set C(Y, Z) of all continuous functions of Y into a space Z a topology calling (by some authors) the Isbell topology (see [8,10,11,14]). The Isbell topology is always splitting (see [3,8,10,11,14]). We note that in [4] it is proved that on the set C(Nω , N), where N is the (discrete) non-negative integers and Nω the Baire space, the Isbell topology does not coincide with the greatest splitting topology (which is defined as the union of all splitting * Corresponding author.
E-mail addresses: [email protected] (D.N. Georgiou), [email protected] (S.D. Iliadis). 0166-8641/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.topol.2005.01.016
D.N. Georgiou, S.D. Iliadis / Topology and its Applications 153 (2005) 868–873
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topologies). This gives a negative answer to Problem 1 of [6]. In the case, where Y is core compact and Z an arbitrary space the Isbell topology is admissible (see [8,10,11,14]) which means that it is the greatest splitting topology. In the case, where Z is the Sierpinski space 2, and Y an arbitrary space the Isbell topology is also the greatest splitting topology (see [13,14]). Moreover, the Isbell topology on C(Y, 2) is admissible if and only if the space Y is core compact (see [10,14]). In what follows two spaces Y and Z are fixed. If t is a topology on C(Y, Z), then the corresponding space is denoted by Ct (Y, Z). We recall some notions and notations. Let X be a space and F : X × Y → Z be a continuous map. By Fx , where x ∈ X, we we denote denote the continuous map of Y into Z such that Fx (y) = F (x, y), y ∈ Y . By F (x) = Fx , x ∈ X. Let G be a map of X into the map of X into the set C(Y, Z) such that F we denote the map of X × Y into Z such that G(x, y) = G(x)(y) for every C(Y, Z). By G (x, y) ∈ X × Y . A topology t on C(Y, Z) is called splitting if for every space X, the continuity of a map : X → Ct (Y, Z). A topology t on C(Y, Z) is F : X × Y → Z implies that of the map F called admissible if for every space X, the continuity of a map G : X → Ct (Y, Z) implies : X × Y → Z. Equivalently, a topology t on C(Y, Z) is admissible if that of the map G the evaluation map e : Ct (Y, Z) × Y → Z defined by relation e(f, y) = f (y), (f, y) ∈ C(Y, Z) × Y is continuous (see [1]). Let A be a family of spaces. If in the above definitions it is assumed that the space X belongs to A, then the topology t is called A-splitting (respectively, A-admissible) (see [5]). A subset B of a space X is called bounded if every open cover of X contains finite elements which cover the subset B (see [9]). A space X is called core compact (see, for example, [10,14]) if for every open neighborhood U of a point x there exists an open neighborhood V of x such that V ⊆ U and V is bounded in the space U . A space X is called Pν -space, where ν is an infinite cardinal, if the intersection of not more than ν many open sets of X is open (see [12]). In what follows we denote by ω the first infinite cardinal and by ν an arbitrary fixed infinite cardinal.
2. The ν-Scott topology on a partially ordered set and the topology tν on C(Y, Z) Below by (L, ) we denote a fixed partially ordered set. Definition. A subset D of L is called ν-directed if every subset of D with cardinality less than ν has an upper bound in D. By τν we denote the family of all subsets H of L such that: (α) H = ↑H, where ↑H = {y ∈ L: (∃x ∈ H) x y}, and (β) for every ν-directed subset D of L with sup D ∈ H, D ∩ H = ∅.
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Remarks. It is clear that: (1) The family τν is a T0 -topology on L calling ν-Scott topology. (2) The Scott topology and topology τν , ν = ω, coincide. For ν > ω, It is easy to construct an example such that τω = τν . (3) If µ is an infinite cardinal such that µ ν, then τω ⊆ τµ ⊆ τν . (4) The space (L, τν ) is a Pν -space. Definition. Let L be a complete lattice and x, y ∈ L. We write x ν y if for every νdirected subset D of L the relation y sup D implies the existence of d ∈ D with x d. Obviously, we have: (1) If z x ν y w, then z ν w. (2) If xj ν yj for every element j of a set J with |J | < ν, then sup{xj : j ∈ J } ν sup{yj : j ∈ J }. Remarks. Let L be the set O(Y ) of all open subsets of Y with the inclusion as order. Then, by the above we have: (1) A subset H of O(Y ) belongs to τν if (α) the conditions U ∈ H, V ∈ O(Y ), and U ⊆ V imply V ∈ H, and (β) for every collection {Ui : i ∈ I } of open sets of Y whose union belongs to H there exists J ⊆ I with |J | < ν such that {Ui : i ∈ J } ∈ H. (2) For two elements U and V of O(Y ), U ν V if for every open cover{Wi : i ∈ I } of V there is a subcollection {Wi : i ∈ J ⊆ I } such that |J | < ν and U ⊆ {Wi : i ∈ J }. Definition. A space Y is called ν-core compact if for every open neighborhood U of a point y of Y there exists an open neighborhood V of y such that V ν U . Obviously, an ω-core compact space is core compact and conversely. Definition. On the set C(Y, Z) we define a topology, denoted by tν , such that the sets of the form: (H, U ) = f ∈ C(Y, Z): f −1 (U ) ∈ H , where U ∈ O(Z) and H ∈ τν , compose a subbasis (calling the standard subbasis) for this topology. It is clear that the Isbell topology coincides with tω . Also, if µ is an infinite cardinal such that µ ν, then tω ⊆ tµ ⊆ tν . Proposition 1. The following statements are true: (1) The topology tν is A-splitting, where A is the family of all Pν -spaces. (2) If Y is ν-core compact, then tν is admissible.
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Proof. We prove only the statement (2). It is sufficient to prove that the evaluation map e : Ctν (Y, Z) × Y → Z is continuous. Let (f, y) ∈ Ctν (Y, Z) × Y and W ∈ O(Z) such that e(f, y) ∈ W. We must prove that there exist H ∈ τν , U ∈ O(Z) and an open neighborhood V of y in Y such that e((H, U ) × V ) ⊆ W. Since e(f, y) = f (y) ∈ W , by the ν-core compactness of Y it follows that there exists an open set V such that y ∈ V ν f −1 (W ). We prove that the set H = {P ∈ O(Y ): V ν P } belongs to τν . Property (α). Suppose that P ∈ H, U ∈ O(Y ) and P ⊆ U . Then V ν P ⊆ U and therefore we have V ν U which means that U ∈ H. Property (β). Let {Ui : i ∈ I } be a collection of open sets of Y such that Q ≡ {Ui : i ∈ I } ∈ H, that is V ν Q. Since the space Y is ν-core compact, for every z ∈ Q there exist an element i(z) ofI and an open set Vi(z) of Y such that z ∈ Ui(z) and z ∈ Vi(z) ν Ui(z) . : z ∈ Q} and therefore there is a Since V ν Q ⊆ {Vi(z) : z ∈ Q} we have V ν {Vi(z) subset K of Q of cardinality less than of ν such that V ⊆ {Vi(z) : z ∈ K}. Let J = {i(z): z ∈ K}. By the above we have: Ui(z) : i(z) ∈ J = Ui : i ∈ J , V⊆ Vi(z) : i(z) ∈ J ⊆ and therefore Vi(z) : i(z) ∈ J ν Ui : i ∈ J and V ν
{Ui : i ∈ J }.
Thus the set H belongs to τν . Since V ν f −1 (W ) we have f −1 (W ) ∈ H and therefore f ∈ (H, W ). This mans that (H, W ) × V is a neighborhood of (f, y) in Cτν (Y, Z) × Y . Finally, we prove that e((H, W ) × V ) ⊆ W. Let (g, z) ∈ (H, W ) × V . Then z ∈ V and g −1 (W ) ∈ H. Therefore z ∈ V and V ν g −1 (W ). Thus z ∈ V ⊆ g −1 (W ). Hence e((g, z)) = g(z) ∈ W completing the proof of the proposition. 2 Proposition 2. The following statements are equivalent: (1) (2) (3) (4)
Y is ν-core compact. For every space Z the evaluation map e : Ctν (Y, Z) × Y → Z is continuous. The evaluation map e : Ctν (Y, 2) × Y → 2 is continuous. For every open neighborhood V of a point y of Y there is an open set H ∈ τν such that V ∈ H and the set {P : P ∈ H} is a neighborhood of y in Y .
Proof. (1) ⇒ (2) Follows by Proposition 1. (2) ⇒ (3) Follows by (2) for the case Z = 2. (3) ⇒ (4) Let V be an arbitrary open neighborhood of a point y in Y . For every element U of O(Y ) we denote by fU the element of C(Y, 2) for which fU (U ) ⊆ {0} and fU (Y \ U ) ⊆ {1}. Identifying U with fU , each topology on one of the above sets can be considered as a topology on the other. In this case tν = τν . Then the map e : O(Y ) × Y → 2 is continuous. We have e(V , y) = e(fV , y) = fV (y) = 0. For the open neighborhood {0}
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of e(V , y) = 0 in 2 there exist an open neighborhood H ∈ τν of V in O(Y ) and an open neighborhood V of y in Y such that e(H × V ) ⊆ {0}. Obviously, V ∈ H. We prove that V ⊆ ∩{P : P ∈ H}. In the opposite case, there exist / P . Then, e(P , z) = e(fP , z) = fP (z) = 1. Since P ∈ H, z ∈ V and P ∈ H such that z ∈ z ∈ V and e(H × V ) ⊆ {0} the above is a contradiction. Thus, the set {P : P ∈ H} is a neighborhood of y in Y . (4) ⇒ (1) Let V be an open neighborhood of y in Y . We prove that there exists an open neighborhood U of y such that U ν V . By assumption there exists a set H ∈ τν such that V ∈ H and {P : P ∈ H} is a neighborhood of y in Y . We prove that {P : P ∈ H} ν V . Let {Ui : i ∈ I } be an open cover of V . Then {Ui : i ∈ I } ∈ H. Since H ∈ τν there exists a subset J of I with |J | < ν such that {Ui : i ∈ J } ∈ H. Then {P : P ∈ H} ⊆ {Ui : i ∈ J } and therefore {P : P ∈ H} ν V . Since the set {P : P ∈ H} is a neighborhood of y there exists an open set U of Y such that y ∈ U ⊆ {P : P ∈ H}. Clearly, U ν V which means that the space Y is ν-core compact. 2 Proposition 3. If Y is ν-core compact, then the usual compositions operations (1) T : Ctν (X, Y ) × Ctν (Y, Z) → Ctν (X, Z), (2) T : Ctω (X, Y ) × Ctν (Y, Z) → Ctω (X, Z), and (3) T : Ctco (X, Y ) × Ctν (Y, Z) → Ctco (X, Z), where tco is the compact open topology, are continuous for arbitrary spaces X and Z. The proof of this proposition is omitted. This proof is similar to Proposition 2.1 of [10].
Acknowledgements The authors thank the referee for making several suggestions which improved this paper.
References [1] R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31. [2] B.J. Day, G.M. Kelly, On topological quotient maps preserved by pull backs or products, Proc. Cambridge Philos. Soc. 67 (1970) 553–558. [3] M.H. Escardo, R. Heckmann, Topologies on spaces of continuous functions, Topology Proc. 26 (2) (2001– 2002) 545–564.
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[4] M.H. Escardo, J. Lawson, A. Simpson, Comparing Cartesian closed categories of (core) compactly generated spaces, Topology Appl., submitted for publication. [5] D.N. Georgiou, S.D. Iliadis, B.K. Papadopoulos, Topologies on function spaces, in: Studies in Topology, VII, Zap. Nauchn. Sem. S.-Peterburg Otdel. Mat. Inst. Steklov (POMI) 208 (1992) 82–97 (in Russian); J. Math. Sci. New York 81 (2) (1996) 2506–2514. [6] D.N. Georgiou, S.D. Iliadis, B.K. Papadopoulos, On dual topologies, Topology Appl. 140 (2004) 57–68. [7] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, A Compendium of Continuous Lattices, Springer, Berlin, 1980. [8] J.R. Isbell, Function spaces and adjoints, Symposia Math. 36 (1975) 317–339. [9] P. Lambrinos, Subsets (m, n)-bounded in a topological space, Math. Balkanica 4 (70) (1974) 391–397. [10] P. Lambrinos, B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, in: Continuous Lattices and Applications, in: Lecture Notes in Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 191–211. [11] R. McCoy, I. Ntantu, Topological Properties of Spaces of Continuous Functions, in: Lecture Notes in Math., vol. 1315, Springer, Berlin, 1988. [12] W.W. Comfort, S. Negrepontis, The Theory of Ultrafilters, Springer, Berlin, 1974. [13] B.K. Papadopoulos, Proper topologies on the set SY , Glas. Mat. 23 (43) (1988) 143–146. [14] F. Schwarz, S. Weck, Scott topology, Isbell topology and continuous convergence, in: Lecture Notes in Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1984, pp. 251–271.
Further reading [15] M. Mislove, Topology, domain theory and theoretical computer science, Topology Appl. 89 (1998) 3–59.
A generalization of core compact spaces D.N. Georgiou ∗ , S.D. Iliadis University of Patras, Department of Mathematics, 265 00 Patras, Greece Received 30 November 2003; received in revised form 15 June 2004
Abstract In this paper, for a fixed infinite cardinal ν, we give the notion of a ν-core compact space generalizing the notion of a core compact space. Also, on the family of all open subsets of a space Y (actually on a partially ordered set) we define a topology denoted by τν and calling ν-Scott topology. This topology defines on the set C(Y, Z) of all continuous functions of the space Y into a space Z a topology denoted by tν . Some relations between ν-core compact spaces and topologies τν and tν are given. 2005 Published by Elsevier B.V. MSC: 54C35 Keywords: Scott topology; Isbell topology; Function space; Splitting topology; Admissible topology
1. Introduction It is well known that the Scott topology (which is defined originally in the family of all open subsets of a space Y (see [2]) and then on a partially ordered set (see [7])) determines on the set C(Y, Z) of all continuous functions of Y into a space Z a topology calling (by some authors) the Isbell topology (see [8,10,11,14]). The Isbell topology is always splitting (see [3,8,10,11,14]). We note that in [4] it is proved that on the set C(Nω , N), where N is the (discrete) non-negative integers and Nω the Baire space, the Isbell topology does not coincide with the greatest splitting topology (which is defined as the union of all splitting * Corresponding author.
E-mail addresses: [email protected] (D.N. Georgiou), [email protected] (S.D. Iliadis). 0166-8641/$ – see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.topol.2005.01.016
D.N. Georgiou, S.D. Iliadis / Topology and its Applications 153 (2005) 868–873
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topologies). This gives a negative answer to Problem 1 of [6]. In the case, where Y is core compact and Z an arbitrary space the Isbell topology is admissible (see [8,10,11,14]) which means that it is the greatest splitting topology. In the case, where Z is the Sierpinski space 2, and Y an arbitrary space the Isbell topology is also the greatest splitting topology (see [13,14]). Moreover, the Isbell topology on C(Y, 2) is admissible if and only if the space Y is core compact (see [10,14]). In what follows two spaces Y and Z are fixed. If t is a topology on C(Y, Z), then the corresponding space is denoted by Ct (Y, Z). We recall some notions and notations. Let X be a space and F : X × Y → Z be a continuous map. By Fx , where x ∈ X, we we denote denote the continuous map of Y into Z such that Fx (y) = F (x, y), y ∈ Y . By F (x) = Fx , x ∈ X. Let G be a map of X into the map of X into the set C(Y, Z) such that F we denote the map of X × Y into Z such that G(x, y) = G(x)(y) for every C(Y, Z). By G (x, y) ∈ X × Y . A topology t on C(Y, Z) is called splitting if for every space X, the continuity of a map : X → Ct (Y, Z). A topology t on C(Y, Z) is F : X × Y → Z implies that of the map F called admissible if for every space X, the continuity of a map G : X → Ct (Y, Z) implies : X × Y → Z. Equivalently, a topology t on C(Y, Z) is admissible if that of the map G the evaluation map e : Ct (Y, Z) × Y → Z defined by relation e(f, y) = f (y), (f, y) ∈ C(Y, Z) × Y is continuous (see [1]). Let A be a family of spaces. If in the above definitions it is assumed that the space X belongs to A, then the topology t is called A-splitting (respectively, A-admissible) (see [5]). A subset B of a space X is called bounded if every open cover of X contains finite elements which cover the subset B (see [9]). A space X is called core compact (see, for example, [10,14]) if for every open neighborhood U of a point x there exists an open neighborhood V of x such that V ⊆ U and V is bounded in the space U . A space X is called Pν -space, where ν is an infinite cardinal, if the intersection of not more than ν many open sets of X is open (see [12]). In what follows we denote by ω the first infinite cardinal and by ν an arbitrary fixed infinite cardinal.
2. The ν-Scott topology on a partially ordered set and the topology tν on C(Y, Z) Below by (L, ) we denote a fixed partially ordered set. Definition. A subset D of L is called ν-directed if every subset of D with cardinality less than ν has an upper bound in D. By τν we denote the family of all subsets H of L such that: (α) H = ↑H, where ↑H = {y ∈ L: (∃x ∈ H) x y}, and (β) for every ν-directed subset D of L with sup D ∈ H, D ∩ H = ∅.
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Remarks. It is clear that: (1) The family τν is a T0 -topology on L calling ν-Scott topology. (2) The Scott topology and topology τν , ν = ω, coincide. For ν > ω, It is easy to construct an example such that τω = τν . (3) If µ is an infinite cardinal such that µ ν, then τω ⊆ τµ ⊆ τν . (4) The space (L, τν ) is a Pν -space. Definition. Let L be a complete lattice and x, y ∈ L. We write x ν y if for every νdirected subset D of L the relation y sup D implies the existence of d ∈ D with x d. Obviously, we have: (1) If z x ν y w, then z ν w. (2) If xj ν yj for every element j of a set J with |J | < ν, then sup{xj : j ∈ J } ν sup{yj : j ∈ J }. Remarks. Let L be the set O(Y ) of all open subsets of Y with the inclusion as order. Then, by the above we have: (1) A subset H of O(Y ) belongs to τν if (α) the conditions U ∈ H, V ∈ O(Y ), and U ⊆ V imply V ∈ H, and (β) for every collection {Ui : i ∈ I } of open sets of Y whose union belongs to H there exists J ⊆ I with |J | < ν such that {Ui : i ∈ J } ∈ H. (2) For two elements U and V of O(Y ), U ν V if for every open cover{Wi : i ∈ I } of V there is a subcollection {Wi : i ∈ J ⊆ I } such that |J | < ν and U ⊆ {Wi : i ∈ J }. Definition. A space Y is called ν-core compact if for every open neighborhood U of a point y of Y there exists an open neighborhood V of y such that V ν U . Obviously, an ω-core compact space is core compact and conversely. Definition. On the set C(Y, Z) we define a topology, denoted by tν , such that the sets of the form: (H, U ) = f ∈ C(Y, Z): f −1 (U ) ∈ H , where U ∈ O(Z) and H ∈ τν , compose a subbasis (calling the standard subbasis) for this topology. It is clear that the Isbell topology coincides with tω . Also, if µ is an infinite cardinal such that µ ν, then tω ⊆ tµ ⊆ tν . Proposition 1. The following statements are true: (1) The topology tν is A-splitting, where A is the family of all Pν -spaces. (2) If Y is ν-core compact, then tν is admissible.
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Proof. We prove only the statement (2). It is sufficient to prove that the evaluation map e : Ctν (Y, Z) × Y → Z is continuous. Let (f, y) ∈ Ctν (Y, Z) × Y and W ∈ O(Z) such that e(f, y) ∈ W. We must prove that there exist H ∈ τν , U ∈ O(Z) and an open neighborhood V of y in Y such that e((H, U ) × V ) ⊆ W. Since e(f, y) = f (y) ∈ W , by the ν-core compactness of Y it follows that there exists an open set V such that y ∈ V ν f −1 (W ). We prove that the set H = {P ∈ O(Y ): V ν P } belongs to τν . Property (α). Suppose that P ∈ H, U ∈ O(Y ) and P ⊆ U . Then V ν P ⊆ U and therefore we have V ν U which means that U ∈ H. Property (β). Let {Ui : i ∈ I } be a collection of open sets of Y such that Q ≡ {Ui : i ∈ I } ∈ H, that is V ν Q. Since the space Y is ν-core compact, for every z ∈ Q there exist an element i(z) ofI and an open set Vi(z) of Y such that z ∈ Ui(z) and z ∈ Vi(z) ν Ui(z) . : z ∈ Q} and therefore there is a Since V ν Q ⊆ {Vi(z) : z ∈ Q} we have V ν {Vi(z) subset K of Q of cardinality less than of ν such that V ⊆ {Vi(z) : z ∈ K}. Let J = {i(z): z ∈ K}. By the above we have: Ui(z) : i(z) ∈ J = Ui : i ∈ J , V⊆ Vi(z) : i(z) ∈ J ⊆ and therefore Vi(z) : i(z) ∈ J ν Ui : i ∈ J and V ν
{Ui : i ∈ J }.
Thus the set H belongs to τν . Since V ν f −1 (W ) we have f −1 (W ) ∈ H and therefore f ∈ (H, W ). This mans that (H, W ) × V is a neighborhood of (f, y) in Cτν (Y, Z) × Y . Finally, we prove that e((H, W ) × V ) ⊆ W. Let (g, z) ∈ (H, W ) × V . Then z ∈ V and g −1 (W ) ∈ H. Therefore z ∈ V and V ν g −1 (W ). Thus z ∈ V ⊆ g −1 (W ). Hence e((g, z)) = g(z) ∈ W completing the proof of the proposition. 2 Proposition 2. The following statements are equivalent: (1) (2) (3) (4)
Y is ν-core compact. For every space Z the evaluation map e : Ctν (Y, Z) × Y → Z is continuous. The evaluation map e : Ctν (Y, 2) × Y → 2 is continuous. For every open neighborhood V of a point y of Y there is an open set H ∈ τν such that V ∈ H and the set {P : P ∈ H} is a neighborhood of y in Y .
Proof. (1) ⇒ (2) Follows by Proposition 1. (2) ⇒ (3) Follows by (2) for the case Z = 2. (3) ⇒ (4) Let V be an arbitrary open neighborhood of a point y in Y . For every element U of O(Y ) we denote by fU the element of C(Y, 2) for which fU (U ) ⊆ {0} and fU (Y \ U ) ⊆ {1}. Identifying U with fU , each topology on one of the above sets can be considered as a topology on the other. In this case tν = τν . Then the map e : O(Y ) × Y → 2 is continuous. We have e(V , y) = e(fV , y) = fV (y) = 0. For the open neighborhood {0}
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of e(V , y) = 0 in 2 there exist an open neighborhood H ∈ τν of V in O(Y ) and an open neighborhood V of y in Y such that e(H × V ) ⊆ {0}. Obviously, V ∈ H. We prove that V ⊆ ∩{P : P ∈ H}. In the opposite case, there exist / P . Then, e(P , z) = e(fP , z) = fP (z) = 1. Since P ∈ H, z ∈ V and P ∈ H such that z ∈ z ∈ V and e(H × V ) ⊆ {0} the above is a contradiction. Thus, the set {P : P ∈ H} is a neighborhood of y in Y . (4) ⇒ (1) Let V be an open neighborhood of y in Y . We prove that there exists an open neighborhood U of y such that U ν V . By assumption there exists a set H ∈ τν such that V ∈ H and {P : P ∈ H} is a neighborhood of y in Y . We prove that {P : P ∈ H} ν V . Let {Ui : i ∈ I } be an open cover of V . Then {Ui : i ∈ I } ∈ H. Since H ∈ τν there exists a subset J of I with |J | < ν such that {Ui : i ∈ J } ∈ H. Then {P : P ∈ H} ⊆ {Ui : i ∈ J } and therefore {P : P ∈ H} ν V . Since the set {P : P ∈ H} is a neighborhood of y there exists an open set U of Y such that y ∈ U ⊆ {P : P ∈ H}. Clearly, U ν V which means that the space Y is ν-core compact. 2 Proposition 3. If Y is ν-core compact, then the usual compositions operations (1) T : Ctν (X, Y ) × Ctν (Y, Z) → Ctν (X, Z), (2) T : Ctω (X, Y ) × Ctν (Y, Z) → Ctω (X, Z), and (3) T : Ctco (X, Y ) × Ctν (Y, Z) → Ctco (X, Z), where tco is the compact open topology, are continuous for arbitrary spaces X and Z. The proof of this proposition is omitted. This proof is similar to Proposition 2.1 of [10].
Acknowledgements The authors thank the referee for making several suggestions which improved this paper.
References [1] R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5–31. [2] B.J. Day, G.M. Kelly, On topological quotient maps preserved by pull backs or products, Proc. Cambridge Philos. Soc. 67 (1970) 553–558. [3] M.H. Escardo, R. Heckmann, Topologies on spaces of continuous functions, Topology Proc. 26 (2) (2001– 2002) 545–564.
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Further reading [15] M. Mislove, Topology, domain theory and theoretical computer science, Topology Appl. 89 (1998) 3–59.