GeoIli.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 6, No. 2, 2005 pp. 143-148 A note on locally ν-bounded spaces D. N. Georgiou and S. D. Iliadis Abstract. In this paper, on the family O(Y ) of all open subsets of a space Y (actually on a complete lattice) we define the so called strong ν-Scott topology, denoted by τs ν , where ν is an infinite cardinal. This topology defines on the set C(Y, Z) of all continuous functions on the space Y to a space Z a topology ts ν . The topology ts ν , is always larger than or equal to the strong Isbell topology (see [8]). We study the topology ts ν in the case where Y is a locally ν-bounded space. 2000 AMS Classification: 54C35 Keywords: Strong Scott topology, Strong Isbell topology, Function space, Admissible topology. 1. Basic Notions Let X be a space and G a map of X into C(Y, Z). By G̃ we denote the map of X × Y to Z such that G̃(x, y) = G(x)(y) for every (x, y) ∈ X × Y . A topology t on C(Y, Z) is called admissible if for every space X, the con- tinuity of a map G : X → Ct(Y, Z) implies that of the map G̃ : X × Y → Z. Equivalently, a topology t on C(Y, Z) is admissible if 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 L be a poset. The Scott topology τω (see, for example, [5]) is the family of all subsets IH of L such that: (α) IH =↑ IH, where ↑ IH = {y ∈ L : (∃x ∈ IH) x ≤ y}, and (β) for every directed subset D of L with sup D ∈ IH, D ∩ IH 6= ∅. Below, we consider the poset O(Y ) of all open subsets of the space Y on which the inclusion is considered as the order. The Isbell topology tω on C(Y, Z) (see, for example, [8], [11] and [9]) is the topology for which the family of all sets of the form (IH, U) = {f ∈ C(Y, Z) : f−1(U) ∈ IH}, 144 D. N. Georgiou and S. D. Iliadis where U ∈ O(Z) and IH ∈ τω, constitute a subbasis for this topology. The notion of a bounded subset was introduced in [3] and the notion of a locally bounded space in [7]. Some generalizations of locally bounded spaces are given in [10]. The notion of the strong Scott topology (defined on a complete lattice) was given in [8]. This topology determines on the set C(Y, Z) a topology called the strong Isbell topology (see [8]). It is proved that a space Y is locally bounded if and only if the strong Isbell topology on C(Y, 2), where 2 is the Sierpinski space, is admissible. In the case, where Y is locally bounded and Z is an arbitrary space, it is proved that the strong Isbell topology on C(Y, Z) is admissible. In this paper we denote by ν a fixed infinite cardinal. A subset D of a poset L is called ν-directed if every subset of D with cardinality less than ν has an upper bound in D (see [4]). Suppose that L is a complete lattice. We say that x is ν-way below y and write x <<ν y (see [4]) if for every ν-directed subset D of L the relation y ≤ sup D implies the existence of d ∈ D with x ≤ d. In particular, for two elements U and V of the complete lattice O(Y ) we have: U <<ν V if for every open cover {Wi : i ∈ I} of V there is a subcollection {Wi : i ∈ J ⊆ I} of this cover such that |J| < ν and U ⊆ ∪{Wi : i ∈ J}. It is clear that if U ⊆ V <<ν Y , then U <<ν Y . 2. Other notions Definition 2.1. A subset B of Y is called ν-bounded if every open cover of Y contains a cover of B of cardinality less than that of ν. (For the related notion of an (m, n)-bounded subset see [6].) A space is called locally ν-bounded if it has a basis for the open subsets consisting of ν-bounded sets. (For the related notion of a local P-space see [10].) Definition 2.2. Let (L, ≤) be a fixed complete lattice and 1 the maximal ele- ment of L. By τsν we denote the family of all subsets IH of L such that: (α) IH =↑ IH, where ↑ IH = {y ∈ L : (∃x ∈ IH) x ≤ y}, and (β) for every ν-directed subset D of L with sup D = 1 we have D ∩ IH 6= ∅. It is clear that, the family τsν is a T0 topology on L called the strong ν-Scott topology. In the case, where L = O(Y ), a subset IH of O(Y ) belongs to the strong ν-Scott topology if the following properties are true: Property (α). The conditions U ∈ IH, V ∈ O(Y ), and U ⊆ V imply V ∈ IH. Property (β). For every open cover {Ui : i ∈ I} of Y there exists a subset J of I of cardinality less than ν such that ∪{Ui : i ∈ J} ∈ IH. Remark 2.3. If µ is an infinite cardinal such that µ ≤ ν, then τsω ⊆ τ s µ ⊆ τ s ν , where ω is the first infinite cardinal. Definition 2.4. Let L be a complete lattice. An element x ∈ L is called ν-bounded if x <<ν 1. On locally ν-bounded spaces 145 The lattice L is called weakly ν-continuous if for all x ∈ L x = sup{u ∈ L : u ≤ x and u <<ν 1}. In the case, where L = O(Y ), a set U ∈ O(Y ) is ν-bounded if U <<ν Y . Notation. We denote by tsν the topology on the set C(Y, Z) for which the sets of the form: (IH, U) = {f ∈ C(Y, Z) : f−1(U) ∈ IH}, where U ∈ O(Z) and IH ∈ τsν , compose a subbasis. Obviously, if ω ≤ µ ≤ ν, then tsω ⊆ t s µ ⊆ t s ν. Remark 2.5. For ν = ω the notions of an ω-bounded subset, a locally ω- bounded space, and a weakly ω-continuous lattice coincide with the notions of a bounded subset, a locally bounded space, and a weakly continuous lattice, respectively. Also, the topologies τsω and t s ω coincide with the strong Scott topology and the strong Isbell topology, respectively. 3. The results Proposition 3.1. If Y is locally ν-bounded, then the topology tsν on C(Y, Z) is admissible. Proof. It is sufficient to prove that the evaluation map e : Cts ν (Y, Z) × Y → Z is continuous. Let (f, y) ∈ Cts ν (Y, Z) × Y , W ∈ O(Z), and e(f, y) = f(y) ∈ W. We need to prove that there exist IH ∈ τsν , U ∈ O(Z), and an open neighborhood V of y in Y such that f ∈ (IH, U) and e((IH, U) × V ) ⊆ W. Since Y is locally ν-bounded and y ∈ f−1(W) there exists an open ν- bounded set V such that: y ∈ V ⊆ f−1(W). We consider the set IH = {P ∈ O(Y ) : V ⊆ P} and prove that IH ∈ τsν , that is IH satisfies Properties (α) and (β). Property (α) is clear. Property (β). Let {Ui : i ∈ I} be an open cover of Y . Since V is ν- bounded there exists a subset J of I of cardinality less than of ν such that V ⊆ ⋃ {Ui : i ∈ J}. By the definition of IH we have ⋃ {Ui : i ∈ J} ∈ IH. Since V ⊆ f−1(W) we have f−1(W) ∈ IH and therefore f ∈ (IH, W). Thus, the subset (IH, W) × V is a neighborhood of (f, y) in Cτs ν (Y, Z) × Y . Now, we prove that e((IH, W) × V ) ⊆ W . Let (g, z) ∈ (IH, W) × V . Then g−1(W) ∈ IH, z ∈ V , and V ⊆ g−1(W). Therefore e((g, z)) = g(z) ∈ W. 146 D. N. Georgiou and S. D. Iliadis Thus, the map e is continuous which means that tsν is admissible. � Proposition 3.2. For the space Y the following statements are equivalent: (1) Y is locally ν-bounded. (2) For every space Z the evaluation map e : Cts ν (Y, Z) × Y → Z is con- tinuous. (3) The evaluation map e : Cts ν (Y, 2) × Y → 2 is continuous. (4) For every open neighborhood V of a point y of Y there is an open set IH ∈ τsν such that V ∈ IH and the set ∩{P : P ∈ IH} is a neighborhood of y in Y . (5) The lattice O(Y ) is weakly ν-continuous. Proof. (1) =⇒ (2) Follows by Proposition 3.1. (2) =⇒ (3) It is obvious. (3) =⇒ (4) Let V be an open neighborhood of y in Y . Consider the sets O(Y ) and C(Y, 2). We identify each element U of O(Y ) with the element fU of C(Y, 2) for which fU(U) ⊆ {0} and fU (Y \ U) ⊆ {1}. Then, each topology on one of the above sets can be considered as a topology on the other. In this case tsν = τ s ν and the map e : O(Y ) × Y → 2 is continuous. Since e(V, y) = e(fV , y) = fV (y) = 0, the continuity of e implies that for the open neighborhood {0} of e(V, y) in 2 there exist an open neighborhood IH ∈ τsν of V in O(Y ) and an open neighborhood V ′ of y in Y such that e(IH ×V ′) ⊆ {0}. Obviously, V ∈ IH. We need to prove that the relation V ′ ⊆ ∩{P : P ∈ IH} is true. Indeed, in the opposite case, there exist z ∈ V ′ and P ∈ IH such that z 6∈ P . Then, e(P, z) = e(fP , z) = fP (z) = 1 which contradicts the fact that e(IH × V ′) ⊆ {0}. Thus, the set ∩{P : P ∈ IH} is a neighborhood of y in Y . (4) =⇒ (5) Let V be an open subset of Y . It suffices to prove that for every y ∈ V there exists an open ν-bounded neighborhood U of y such that U ⊆ V . By assumption there exists a set IH ∈ τsν such that V ∈ IH and ∩{P : P ∈ IH} ≡ Q is a neighborhood of y in Y . We prove that the set Q is ν-bounded. Let {Ui : i ∈ I} be an open cover of Y . Since IH ∈ τ s ν , by the definition of τsν there exists a subset J of I of cardinality less than of ν such that ∪{Ui : i ∈ J} ∈ IH and therefore Q ⊆ ∪{Ui : i ∈ J}, which means that Q is ν-bounded. The required open neighborhood of y is an open subset U of Y such that y ∈ U ⊆ Q. (5) =⇒ (1) Let y ∈ Y and V be an open neighborhood of y. Since O(Y ) is weakly ν-continuous we have V = ∪{U ∈ O(Y ) : U ⊆ V and U <<ν Y } and therefore there exists an open ν-bounded subset U of Y such that y ∈ U ⊆ V. � On locally ν-bounded spaces 147 Proposition 3.3. If Y is ν-locally bounded, then the usual compositions oper- ations (see [2]) i) T : Ctco(X, Y ) × Cts ν (Y, Z) → Ctco(X, Z) and ii) T : Ctω (X, Y ) × Cts ν (Y, Z) → Ctω (X, Z), where tco and tω is the compact open and the Isbell topology, respectively, are continuous for arbitrary spaces X and Z. Proof. We prove only the statement ii). The proof of the case i) is similar. Let (f, g) ∈ Ctω (X, Y ) × Cts ν (Y, Z), IH a Scott open subset of X, and U ∈ O(Z) such that T (f, g) = g ◦ f ∈ (IH, U). It suffices to prove that there exist open neighborhoods IH1 and IH2 of f and g in Ctω (X, Y ) and Cts ν (Y, Z), respectively, such that T (IH1 × IH2) ⊆ (IH, U). We consider the open set g−1(U) of Y . By locally ν-boundedness of Y , for each point y ∈ g−1(U) ∈ O(Y ), there is an open set Vy of Y such that y ∈ Vy ⊆ g −1(U) and Vy <<ν Y . Therefore g−1(U) = ∪{Vy : y ∈ g −1(U)} and f−1(g−1(U)) = f−1(∪{Vy : y ∈ g −1(U)}) or (g ◦ f)−1(U) = ∪{f−1(Vy) : y ∈ g −1(U)}. Since g ◦ f ∈ (IH, U) we have (g ◦ f)−1(U) ∈ IH or ∪{f−1(Vy) : y ∈ g −1(U)} ∈ IH. Thus there exists a finite subset J of g−1(U) such that ∪{f−1(Vy) : y ∈ J} ∈ IH. Let V = ∪{Vy : y ∈ J}. Then f −1(V ) ∈ IH and V is a ν-bounded open set of Y . The set IH(V ) = {W ∈ O(Y ) : V ⊆ W} is strong ν-Scott open (see the proof of Proposition 3.1). Since V = ∪{Vy : y ∈ J ⊆ g −1(U)} and g −1(U) = ∪{Vy : y ∈ g −1(U)} we have that V ⊆ g−1(U) and therefore g−1(U) ∈ IH(V ). Setting IH1 = (IH, V ) and IH2 = (IH(V ), U) we have that the set IH1 × IH2 = (IH, V ) × (IH(V ), U) is an open neighborhood of (f, g) in Ctω (X, Y ) × Cts ν (Y, Z). Finally, we prove that T ((IH, V ) × (IH(V ), U)) ⊆ (IH, U). Let (p, q) ∈ (IH, V ) × (IH(V ), U). Then, p−1(V ) ∈ IH and q−1(U) ∈ IH(V ). Therefore V ⊆ q−1(U). Thus, p−1(V ) ⊆ p−1(q−1(U)) = (q ◦ p)−1(U). Since p−1(V ) ∈ IH, (q ◦ p)−1(U) ∈ IH, and therefore q ◦ p ∈ (IH, U). � 148 D. N. Georgiou and S. D. Iliadis References [1] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1(1951), 5-31. [2] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass. 1966. [3] S. Gagola and M. Gemignani, Absolutely bounded sets, Mathematica Japonicae, Vol. 13, No. 2 (1968), 129-132. [4] D. N. Georgiou and S. D. 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Poppe, On locally defined topological notions, Q and A in General Topology, Vol. 13 (1995), 39-53. [11] 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 October 2004 Accepted January 2005 D. N. Georgiou (georgiou@math.upatras.gr) Department of Mathematics, University of Patras, 265 00 Patras, Greece. S. D. Iliadis (iliadis@math.upatras.gr) Department of Mathematics, University of Patras, 265 00 Patras, Greece.