A note on locally v-bounded spaces

@Applied General Topology c Universidad Polit´ecnica 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 νis an infinite cardinal. This topology defines on the set C(Y, Z) of all continuous functions on the space Yto a space Za 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 νin the case where Yis a locally ν-bounded space. 2000 AMS Classification: 54C35 Keywords: Strong Scott topology, Strong Isbell topology, Function space, Admissible topology. 1. Basic Notions Let Xbe a space and Ga map of Xinto C(Y, Z). By e Gwe denote the map of X×Yto Zsuch that e G(x, y) = G(x)(y) for every (x, y)∈X×Y. A topology ton C(Y, Z) is called admissible if for every space X, the continuity of a map G:X→Ct(Y, Z) implies that of the map e G:X×Y→Z. Equivalently, a topology ton C(Y, Z) is admissible if the evaluation map e:Ct(Y, Z)×Y→Zdefined by relation e(f, y) = f(y), (f, y)∈C(Y, Z)×Y, is continuous (see [1]). Let Lbe a poset. The Scott topology τω(see, for example, [5]) is the family of all subsets IH of Lsuch that: (α)IH =↑IH, where ↑IH ={y∈L: (∃x∈IH)x≤y}, and (β) for every directed subset Dof Lwith sup D∈IH,D∩IH 6=∅. Below, we consider the poset O(Y) of all open subsets of the space Yon which the inclusion is considered as the order. The Isbell topology tωon 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 Yis locally bounded if and only if the strong Isbell topology on C(Y, 2), where 2is the Sierpinski space, is admissible. In the case, where Yis 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 νa fixed infinite cardinal. A subset Dof a poset Lis called ν-directed if every subset of Dwith cardinality less than νhas an upper bound in D(see [4]). Suppose that Lis a complete lattice. We say that xis ν-way below yand write x <<νy(see [4]) if for every ν-directed subset Dof Lthe relation y≤sup Dimplies the existence of d∈Dwith x≤d. In particular, for two elements Uand Vof the complete lattice O(Y) we have: U <<νVif for every open cover {Wi:i∈I}of Vthere 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 <<νY, then U <<νY. 2. Other notions Definition 2.1. A subset Bof Yis called ν-bounded if every open cover of Y contains a cover of Bof 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 1the maximal element of L. By τs νwe denote the family of all subsets IH of Lsuch that: (α)IH =↑IH, where ↑IH ={y∈L: (∃x∈IH)x≤y}, and (β)for every ν-directed subset Dof Lwith sup D= 1 we have D∩IH 6=∅. It is clear that, the family τs νis a T0topology on Lcalled 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⊆Vimply V∈IH. Property (β). For every open cover {Ui:i∈I}of Ythere exists a subset J of Iof cardinality less than νsuch that ∪{Ui:i∈J} ∈ IH. Remark 2.3. If µis an infinite cardinal such that µ≤ν, then τs ω⊆τs µ⊆τs ν, where ωis the first infinite cardinal. Definition 2.4. Let Lbe a complete lattice. An element x∈Lis called ν-bounded if x <<ν1. On locally ν-bounded spaces 145 The lattice Lis called weakly ν-continuous if for all x∈L x= sup{u∈L:u≤xand u <<ν1}. In the case, where L=O(Y), a set U∈ O(Y)is ν-bounded if U <<νY. Notation. We denote by ts νthe 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 ∈τs ν, compose a subbasis. Obviously, if ω≤µ≤ν, then ts ω⊆ts µ⊆ts ν. Remark 2.5. For ν=ωthe notions of an ω-bounded subset, a locally ωbounded 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 τs ωand ts ωcoincide with the strong Scott topology and the strong Isbell topology, respectively. 3. The results Proposition 3.1. If Yis locally ν-bounded, then the topology ts νon 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 ∈τs ν,U∈ O(Z), and an open neighborhood Vof y in Ysuch that f∈(IH, U) and e((IH, U)×V)⊆W. Since Yis locally ν-bounded and y∈f−1(W) there exists an open νbounded set Vsuch that: y∈V⊆f−1(W). We consider the set IH ={P∈ O(Y) : V⊆P} and prove that IH ∈τs ν, 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 Vis νbounded there exists a subset Jof Iof cardinality less than of νsuch that V⊆S{Ui:i∈J}. By the definition of IH we have S{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 )×Vis 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 eis continuous which means that ts νis admissible.  Proposition 3.2. For the space Ythe following statements are equivalent: (1) Yis locally ν-bounded. (2) For every space Zthe evaluation map e:Cts ν(Y, Z)×Y→Zis continuous. (3) The evaluation map e:Cts ν(Y, 2)×Y→2is continuous. (4) For every open neighborhood Vof a point yof Ythere is an open set IH ∈τs νsuch that V∈IH and the set ∩{P:P∈IH}is a neighborhood of yin 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 Vbe an open neighborhood of yin Y. Consider the sets O(Y) and C(Y, 2). We identify each element Uof O(Y) with the element fUof 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 ts ν=τs νand the map e:O(Y)×Y→2is continuous. Since e(V, y) = e(fV, y) = fV(y) = 0, the continuity of eimplies that for the open neighborhood {0}of e(V, y) in 2there exist an open neighborhood IH ∈τs νof Vin O(Y) and an open neighborhood V′of yin Ysuch 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 z6∈ 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 yin Y. (4) =⇒(5) Let Vbe an open subset of Y. It suffices to prove that for every y∈Vthere exists an open ν-bounded neighborhood Uof ysuch that U⊆V. By assumption there exists a set IH ∈τs νsuch that V∈IH and ∩{P:P∈IH} ≡ Qis a neighborhood of yin Y. We prove that the set Q is ν-bounded. Let {Ui:i∈I}be an open cover of Y. Since IH ∈τs ν, by the definition of τs νthere exists a subset Jof Iof cardinality less than of νsuch that ∪{Ui:i∈J} ∈ IH and therefore Q⊆ ∪{Ui:i∈J}, which means that Q is ν-bounded. The required open neighborhood of yis an open subset Uof Y such that y∈U⊆Q. (5) =⇒(1) Let y∈Yand Vbe an open neighborhood of y. Since O(Y) is weakly ν-continuous we have V=∪{U∈ O(Y) : U⊆Vand U <<νY} and therefore there exists an open ν-bounded subset Uof Ysuch that y∈U⊆V.  On locally ν-bounded spaces 147 Proposition 3.3. If Yis ν-locally bounded, then the usual compositions operations (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 tωis the compact open and the Isbell topology, respectively, are continuous for arbitrary spaces Xand 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 IH1and IH2of fand gin 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 Vyof Ysuch that y∈Vy⊆g−1(U) and Vy<<νY. 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 Jof g−1(U) such that ∪{f−1(Vy) : y∈J} ∈ IH. Let V=∪{Vy:y∈J}. Then f−1(V)∈IH and Vis 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. 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Math. No. 101, Marcel Dekker, New York 1984, 191-211. [9] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notes in Mathematics, 1315, Springer Verlang (1988). [10] H. 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 ([email protected]r) Department of Mathematics, University of Patras, 265 00 Patras, Greece. S. D. Iliadis (iliadi[email protected]r) Department of Mathematics, University of Patras, 265 00 Patras, Greece.