Closure properties of function spaces

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 255–261 Closure properties of function spaces Ljubiˇ sa D.R. Koˇ cinac∗ Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. In this paper we investigate some closure properties of the space Ck(X) of continuous real-valued functions on a Tychonoff space Xendowed with the compact-open topology. 2000 AMS Classification: 54A25, 54C35, 54D20, 91A44. Keywords: Menger property, Rothberger property, Hurewicz property, Reznichenko property, k-cover, countable fan tightness, countable strong fan tightness, T-tightness, groupability, Ck(X), selection principles, game theory. 1. Introduction. All spaces in this paper are assumed to be Tychonoff. Our notation and terminology are standard, mainly the same as in [2]. Some notions will be defined when we need them. By Ck(X) we denote the space of continuous real-valued functions on a space Xin the compact-open topology. Basic open sets of Ck(X) are of the form W(K1, . . . , Kn;V1, . . . , Vn) := {f∈C(X) : f(Ki)⊂Vi, i = 1, . . . , n}, where K1, . . . , Knare compact subsets of Xand V1. . . , Vnare open in R. For a function f∈Ck(X), a compact set K⊂Xand a positive real number εwe let W(f;K;ε) := {g∈Ck(X) : |g(x)−f(x)|< ε, ∀x∈K}. The standard local base of a point f∈Ck(X) consists of the sets W(f;K;ε), where Kis a compact subset of Xand εis a positive real number. The symbol 0denotes the constantly zero function in Ck(X). The space Ck(X) is homogeneous so that we may consider the point 0when studying local properties of Ck(X). Many results in the literature show that for a Tychonoff space Xclosure properties of the function space Cp(X) of continuous real-valued functions on ∗Supported by the Serbian MSTD, grant 1233 256 Ljubiˇsa D.R. Koˇcinac Xendowed with the topology of pointwise convergence can be characterized by covering properties of X. We list here some properties which are expressed in this manner. (A) [Arhangel’skii–Pytkeev] Cp(X) has countable tightness if and only if all finite powers of Xhave the Lindel¨of property ([1]). (B) [Arhangel’skii] Cp(X) has countable fan tightness if and only if all finite powers of Xhave the Menger property ([1]). (C) [Sakai] Cp(X) has countable strong fan tightness if and only if all finite powers of Xhave the Rothberger property ([13]). (D) [Koˇcinac–Scheepers] Cp(X) has countable fan tightness and the Reznichenko property if and only if all finite powers of Xhave the Hurewicz property ([7]). In this paper we consider these properties in the context of spaces Ck(X). To get analogues for the compact-open topology one need modify the role of covers, preferably ω-covers from Cp-theory should be replaced by k-covers in Ck-theory. Recall that an open cover Uof Xis called a k-cover if for each compact set K⊂Xthere is a U∈ U such that K⊂U. The symbol Kdenotes the collection of all k-covers of a space. Let us mention one result of this sort which should be compared with the Arhangel’skii-Pytkeev theorem (A) above. In [9] it was remarked (without proof) that the following holds: Theorem 1.1. A space Ck(X)has countable tightness if and only if for each k-cover Uof Xthere is a countable set V ⊂ U which is a k-cover of X. The proof of this result can be found in Theorem 3.13 of [11]. 1.1. Selection principles and games. In this paper we shall need selection principles of the following two sorts: Let Sbe an infinite set and let Aand B both be sets whose members are families of subsets of S. Then S1(A,B) denotes the selection principle: For each sequence (An:n∈N) of elements of Athere is a sequence (bn:n∈N) such that for each n bn∈An, and {bn:n∈N}is an element of B. Sfin(A,B) denotes the selection principle: For each sequence (An:n∈N) of elements of Athere is a sequence (Bn:n∈N) of finite sets such that for each n Bn⊂An, and Sn∈NBnis an element of B. For a topological space X, let Odenote the collection of open covers of X. Then the property S1(O,O) is called the Rothberger property [12],[5], and the property Sfin(O,O) is known as the Menger property [10], [3],[5]. There is a natural game for two players, ONE and TWO, denoted Gfin(A,B), associated with Sfin(A,B). This game is played as follows: There is a round per positive integer. In the n-th round ONE chooses an An∈ A, and TWO Closure properties of function spaces 257 responds with a finite set Bn⊂An. A play A1, B1;· · · ;An, Bn;· · · is won by TWO if Sn∈NBnis an element of B; otherwise, ONE wins. Similarly, one defines the game G1(A,B), associated with S1(A,B). 2. Countable (strong) fan tightness of Ck(X). For Xa space and a point x∈Xthe symbol Ωxdenotes the set {A⊂ X\ {x}:x∈A}. A space Xhas countable fan tightness [1] if for each x∈Xand each sequence (An:n∈N) of elements of Ωxthere is a sequence (Bn:n∈N) of finite sets such that for each n Bn⊂Anand x∈Sn∈NBn, i.e. if Sfin(Ωx,Ωx) holds for each x∈X. A space Xhas countable strong fan tightness [13] if for each x∈Xthe selection principle S1(Ωx,Ωx) holds. In [8], it was shown (compare with the corresponding theorem (B) of Arhangel’- skii for Cp(X)): Theorem 2.1. The space Ck(X)has countable fan tightness if and only if X has property Sfin(K,K). We show Theorem 2.2. For a space Xthe following are equivalent: (a)Ck(X)has countable strong fan tightness; (b)Xhas property S1(K,K). Proof. (a)⇒(b): Let (Un:n∈N) be a sequence of k-covers of X. For a fixed n∈Nand a compact subset Kof Xlet Un,K := {U∈ Un:K⊂U}. For each U∈ Un,K let fK,U be a continuous function from Xinto [0,1] such that fK,U (K) = {0}and fK,U (X\U) = {1}. Let for each n, An={fK,U :Kcompact in X, U ∈ Un,K }. Then, as it is easily verified, 0is in the closure of Anfor each n∈N. Since Ck(X) has countable strong fan tightness there is a sequence (fKn,Un: n∈N) such that for each n,fKn,Un∈Anand 0∈ {fKn,Un:n∈N}. Consider the sets Un,n∈N. We claim that the sequence (Un:n∈N) witnesses that X has property S1(K,K). Let Cbe a compact subset of X. From 0∈ {fKn,Un:n∈N}it follows that there is i∈Nsuch that W=W(0;C; 1) contains the function fKi,Ui. Then C⊂Ui. Otherwise, for some x∈Cone has x /∈Uiso that fKi,Ui(x) = 1, which contradicts the fact fKi,Ui∈W. (b)⇒(a): Let (An:n∈N) be a sequence of subsets of Ck(X) the closures of which contain 0. Fix n. For every compact set K⊂Xthe neighborhood W=W(0;K; 1/n) of 0intersects Anso that there exists a function fK,n ∈An such that |fK,n(x)|<1/n for each x∈K. Since fK,n is a continuous function there are neighborhoods Ox,x∈K, such that for UK,n =Sx∈KOx⊃Kwe have fK,n(UK,n)⊂(−1/n, 1/n). Let Un={UK,n :Ka compact subset of X}. For each n∈N,Unis a k-cover of X. Apply that Xis an S1(K,K)-set: for each 258 Ljubiˇsa D.R. Koˇcinac m∈Nthere exists a sequence (UK,n :n≥m) such that for each n,UK,n ∈ Un and {UK,n :n≥m}is a k-cover for X. Look at the corresponding functions fK,n in An. Let us show 0∈ {fK,n :n∈N}. Let W=W(0;C;ε) be a neighborhood of 0in Ck(X) and let mbe a natural number such that 1/m < ε. Since Cis a compact subset of Xand Xis an S1(K,K)-set, there is j∈N,j≥msuch that one can find a UK,j with C⊂UK,j . We have fK,j(C)⊂fK,j (UK,j )⊂(−1/j, 1/j)⊂(−1/m, 1/m)⊂(−ε, ε), i.e. fK,j ∈W. 3. Countable T-tightness of Ck(X). The notion of T-tightness was introduced by I. Juh´asz at the IV International Conference “Topology and its Applications”, Dubrovnik, September 30 – October 5, 1985 (see [4]). A space Xhas countable T-tightness if for each uncountable regular cardinal ρand each increasing sequence (Aα:α < ρ) of closed subsets of Xthe set ∪{Aα:α < ρ}is closed. In [14], the T-tightness of Cp(X) was characterized by a covering property of X. The next theorem is an analogue of this result in the Ck(X) context. Theorem 3.1. For a space Xthe following are equivalent: (a)Ck(X)has countable T-tightness; (b)for each regular cardinal ρand each increasing sequence (Uα:α < ρ) of families of open subsets of Xsuch that Sα<ρ Uαis a k-cover of X there is a β < ρ so that Uβis a k-cover of X. Proof. (a)⇒(b): Let ρbe a regular uncountable cardinal and let (Uα:α < ρ) be an increasing sequence of families of open subsets of Xsuch that Sα<ρ Uα is a k-cover for X. For each α < ρ and each compact set K⊂Xlet Uα,K := {U∈ Uα:K⊂U}. For each U∈ Uα,K let fK,U be a continuous function from Xinto [0,1] such that fK,U (K) = {0}and fK,U (X\U) = {1}. For each α < ρ put Aα={fK,U :U∈ Uα,K }. Since the T-tightness of Ck(X) is countable, the set A=Sα<ρ Aαis closed in Ck(X). Let W(0;K;ε) be a standard basic neighborhood of 0. There is an α < ρ such that for some U∈ Uα,K⊂U. Then U∈ Uα,K and thus there is f∈Aα, hence f∈Aα∩W(0;K;ε). Therefore, each neighborhood of 0intersects some of the sets Aα,α < ρ, which means that 0belongs to the closure of the set Sα<ρ Aα. Since this set is actually the set Ait follows there exists a β < ρ with 0∈Aβ. We claim that the corresponding family Uβis a k-cover of X. Let Cbe a compact subset of X. Then the neighborhood W(0;C; 1) of 0 intersects Aβ; let fK,U ∈Aβ∩W(0;C; 1). By the definition of Aβ, then from fK,U (X\U) = 1 it follows C⊂U∈ Uβ. Closure properties of function spaces 259 (b)⇒(a): Let (Aα:α < ρ) be an increasing sequence of closed subsets of Ck(X), with ρa regular uncountable cardinal. We shall prove that the set A:= Sα<ρ Aαis closed. Let f∈A. For each n∈Nand each compact set K⊂Xthe neighborhood W(f;K; 1/n) of fintersects A. Put Un,α ={g←(−1/n, 1/n) : g∈Aα} and Un=[ α<ρ Un,α. Let us check that for each n∈N,Unis a k-cover of X. Let Kbe a compact subset of X. The neighborhood W:= W(f;K; 1/n) of fintersects A, i.e. there is g∈Asuch that |f(x)−g(x)|<1/n for all x∈K; this means K⊂g←(−1/n, 1/n)∈ Un. By (b) there is Un,βn⊂ Unwhich is a k-cover of X. Put β0= sup{βn:n∈ N}. Since ρis a regular cardinal, β0< ρ. It is easy to verify that for each nthe set Un,β0is a k-cover of X. Let us show that f∈Aβ0. Take a neighborhood W(f;C;ε) of fand let mbe a positive integer such that 1/m < ε. Since Um,β0 is a k-cover of Xone can find g∈Aβ0such that C⊂g←(−1/m, 1/m). Then g∈W(f;C; 1/m)∩Aβ0⊂W(f;C;ε)∩Aβ0, i.e. f∈Aβ0=Aβ0and thus f∈A. So, Ais closed.  4. The Reznichenko property of Ck(X). In this section we shall need the notion of groupability (see [7]). 1. A k-cover Uof a space Xis groupable if there is a partition (Un:n∈N) of Uinto pairwise disjoint finite sets such that: For each compact subset Kof X, for all but finitely many n, there is a U∈ Unsuch that K⊂U. 2. An element Aof Ωxis groupable if there is a partition (An:n∈N) of Ainto pairwise disjoint finite sets such that each neighborhood of x has nonempty intersection with all but finitely many of the An. We use the following notation: • Kgp – the collection of all groupable k-covers of a space; •Ωgp x– the family of all groupable elements of Ωx. In 1996 Reznichenko introduced (in a seminar at Moscow State University) the following property: Each countable element of Ωxis a member of Ωgp x. This property was further studied in [6] and [7] (see Introduction). In [6] it was called the Reznichenko property at x. When Xhas the Reznichenko property at each of its points, then Xis said to have the Reznichenko property. We study now the Reznichenko property in spaces Ck(X). Theorem 4.1. Let Xbe a Tychonoff space. If ONE has no winning strategy in the game G1(K,Kgp), then Ck(X)has property S1(Ω0,Ωgp 0)(i.e. Ck(X)has countable strong fan tightness and the Reznichenko property). 260 Ljubiˇsa D.R. Koˇcinac Proof. Evidently, from the fact that ONE has no winning strategy in the game G1(K,Kgp), it follows that Xsatisfies S1(K,Kgr) and consequently Xis in the class S1(K,K). By Theorem 2.2 Ck(X) has countable strong fan tightness, and thus countable tightness. Therefore, it remains to prove that each countable subset of Ω0is groupable (i.e. that Ck(X) has the Reznichenko property). Let Abe a countable subset of Ck(X) such that 0∈A. We define the following strategy σfor ONE in G1(K,Kgp). For a compact set K⊂Xthe neighborhood W=W(0;K; 1) of 0intersects A. Let fK∈A∩W. As fK is continuous, for every x∈Kthere is a neighborhood Oxof xsuch that fK(Ox)⊂(−1,1). From the open cover {Ox:x∈K}of Kchoose a finite subcover {Ox1, . . . , Oxm}and let UK=Ox1∪· · ·∪Oxm. Then fK(UK)⊂(−1,1) and the set U1={UK:Ka compact subset of X}is a k-cover of X. ONE’s first move, σ(∅), will be U1. Let TWO’s response be an element UK1∈ U1. ONE considers now the corresponding function fK1∈A(satisfying fK1(UK1)⊂ (−1,1)) and looks at the set A1=A\ {fK1}which obviously satisfies 0∈ A1. For every compact subset Kof XONE chooses a function fK∈A∩ W(0;K; 1/2) and a neighborhood UKof Ksuch that fK(UK)⊂(−1/2,1/2). The set U2={UK:Ka compact subset of X}\{UK1}is a k-cover of X. ONE plays σ(UK1) = U2. Suppose that UK2∈ U2is TWO’s response. ONE first considers the function fK2∈A1with fK2(UK2)⊂(−1/2,1/2) and then looks at the set A2=A1\ {fK2}the closure of which contains 0, and so on. The strategy σ, by definition, gives sequences (Un:n∈N), (Un:n∈N) and (fn:n∈N) having the following properties: (i) (Un:n∈N) is a sequence of k-covers of Xand for each nUn= σ(U1, . . . , Un−1); (ii) For each n,Un∈ Unand Un/∈ {U1, . . . , Un−1)}; (iii) For each n,fnis a member of A\ {f1, . . . , fn−1); (iv) For each n,fn(Un)⊂(−1/n, 1/n). Since σis not a winning strategy for ONE, the play U1, U1;. . . ;Un, Un;. . . is lost by ONE so that V:= {Un:n∈N}is a groupable k-cover of X. Therefore there is an increasing infinite sequence n1< n2<· · · < nk< . . . such that the sets Hk:= {Ui:nk≤i<nk+1},k= 1,2, . . . , are pairwise disjoint and for every compact set K⊂Xthere is k0such that for each k > k0there is H∈ Hk with K⊂H. Define also Mk:= {fi:nk≤i < nk+1}. Then the sets Mkare finite pairwise disjoint subsets of A. One can also suppose that A=Sk∈NMk; otherwise we distribute countably many elements of A\Sk∈NMkamong Mk’s so that after distribution new sets are still finite and pairwise disjoint. We claim that the sequence (Mk:k∈N) witnesses that Ais groupable (i.e. that Ck(X) has the Reznichenko property). Let W(0;K;ε) be a neighborhood of 0and let mbe the smallest natural number such that 1/m < ε. There is n0such that for each n>n0one can choose an element Hn∈ Hnwith K⊂Hn; choose also a corresponding function fn∈Mnsatisfying fn(Hn)⊂(−1/n, 1/n). So for each n > max{n0, m}we Closure properties of function spaces 261 have fn∈Mn∩W(0;K;ε), i.e for all but finitely many n W(0;K;ε)∩Mn6=∅. The theorem is shown.  In a similar way one can prove Theorem 4.2. For a space Xthe statement (a)below implies the statement (b): (a) ONE has no winning strategy in the game Gfin(K,Kgp); (b)Ck(X)has countable fan tightness and the Reznichenko property (i.e. Ck(X)has property Sfin(Ω0,Ωgp 0)). Problem 4.3. Is the converse in Theorem 4.1 and in Theorem 4.2 true? References [1] A.V. Arhangel’skiˇ i, Topological Function Spaces (Kluwer Academic Publishers, 1992). [2] R. Engelking, General Topology (PWN, Warszawa, 1977). [3] W. Hurewicz, ¨ Uber eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1925), 401–421. [4] I. Juh´asz, Variations on tightness, Studia Sci. Math. Hungar. 24 (1989), 179–186. [5] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of open covers (II), Topology Appl. 73 (1996), 241–266. [6] Lj.D. Koˇcinac and M. Scheepers, Function spaces and a property of Reznichenko, Topology Appl. 123 (2002), 135–143. [7] Lj.D.R. Koˇcinac and M. Scheepers, Combinatorics of open covers (VII): Groupability, Fund. Math. (to appear). [8] Shou Lin, Chuan Liu and Hui Teng, Fan tightness and strong Fr´echet property of Ck(X), Adv. Math. (China) 23:3 (1994), 234–237 (Chinese); MR. 95e:54007, Zbl. 808.54012. [9] R.A. McCoy, Function spaces which are k-spaces, Topology Proc. 5(1980), 139–146. [10] K. Menger, Einige ¨ Uberdeckungss¨atze der Punktmengenlehre, Sitzungsberischte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien) 133 (1924), 421–444. [11] A. Okuyama and T. Terada, Function spaces, In: Topics in General Topology, K. Morita and J. Nagata, eds. (Elsevier Science Publishers B.V., Amsterdam, 1989), 411–458. [12] F. Rothberger, Eine Versch¨arfung der Eigenschaft C, Fund. Math. 30 (1938), 50–55. [13] M. Sakai, Property C00 and function spaces, Proc. Amer. Math. Soc. 104 (1988), 917–919. [14] M. Sakai, Variations on tightness in function spaces, Topology Appl. 101 (2000), 273–280. Received November 2001 Revised November 2002 Ljubiˇ sa D.R. Koˇ cinac Faculty of Sciences, University of Niˇs, 18000 Niˇs, Serbia E-mail address:[email protected]