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Cauchy convergence topologies on the space of continuous functions
Topology and its Applications 161 (2014) 321–329
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Topology and its Applications www.elsevier.com/locate/topol
Cauchy convergence topologies on the space of continuous functions Zuquan Li Department of Mathematics, Hangzhou Normal University, Hangzhou, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 October 2013 Received in revised form 19 October 2013 Accepted 21 October 2013
In this paper, we investigate some cardinal invariants and variations on tightness in the space of continuous functions from a metric space X to real line R with Cauchy convergence topology. © 2013 Elsevier B.V. All rights reserved.
MSC: primary 54A25, 54C35 Keywords: Function space Cauchy convergence topology Cauchy cover c-Cover Cardinal function
1. Introduction Throughout the paper, X is a metric space, d is the metric of X. The topology of X is the induced topology by d. R is real line with the usual topology. N is the set of natural numbers. ω is countable cardinal. We denote C(X) the family of all the real-value continuous functions on X, and Ck (X) (Cu (X)) the space all the continuous functions of X with compact open topology (uniform convergent topology) [1,7,8]. We denote V (f, S, ε) = g ∈ C(X): f (x) − g(x) < ε, x ∈ S , where S ⊂ X, ε > 0. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.10.032
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A sequence {xn }n∈N of X is said to be a Cauchy sequence, if for any ε > 0, there exists N ∈ N such that d(xm , xn ) < ε with m, n > N . Let C(X) be the family of all the Cauchy sequences of X. To consider Cauchy convergence topology τc , for f ∈ C(X), we choose {V (f, C, ε): C ∈ C(X), ε > 0} as a neighborhood base of f . The space (C(X), τc ) is denoted by Cc (X). Cauchy convergence topology was first introduced by Michael H. Clapp and Ray C. Shiflett [4]. They defined the topological space so that they want to compare it with compact open topology τk and uniform convergent topology τu of C(X). The following theorem summarizes some of the results in [4]. Theorem 1.1. Let X be a space, then (1) τc τu , and τc = τu if and only if X is totally bounded. (2) τk τc , and τk = τc if and only if X is complete. In this paper, we construct the relations among w(Cc (X)), d(Cc (X)), c(Cc (X)), L(Cc (X)) and χ(Cc (X)), give the characterizations of tightness, countable fan tightness, countable strongly fan tightness, T-tightness and set-tightness of Cc (X). Let C = {xn }n∈N be a Cauchy sequence of X. We denote C=
C,
if {xn }n∈N is not convergent in X,
{x0 } ∪ C,
if {xn }n∈N → x0 .
C is said to be a Cauchy closed sequence of X. Each Cauchy closed sequence is a Cauchy sequence. Let C(X) be the family of all the Cauchy closed sequences of X. In this paper, each Cauchy sequence always means a Cauchy closed sequence. By V (f, C, 2ε ) ⊂ V (f, C, 2ε ) ⊂ V (f, C, ε) ⊂ V (f, C, ε), we can obtain the following lemma. Lemma 1.2. For every space X, f ∈ C(X), the neighborhood base of f generated by {V (f, C, ε): C ∈ C(X), ε > 0} is equivalent to the neighborhood base of f generated by {V (f, C, ε): C ∈ C(X), ε > 0}. In other words, we can choose {V (f, C, ε): C ∈ C(X), ε > 0} as a neighborhood base of f on Cauchy convergence topology τc . 2. Cauchy cover number of Cc (X) We primarily study five cardinal functions [5,9] in this section. The cardinality of a set S will be denoted by |S|. The character of a space X at a point x is defined by χ(X, x) = ω + min |Bx |: Bx is a base for X at x . Then the character of X is given by χ(X) = sup χ(X, x): x ∈ X . The weight of X is defined by w(X) = ω + min |B|: B is a base for X . The density of X has definition d(X) = ω + min |D|: D is a dense subset of X .
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The Lindelöf number of X is equal to the number L(X) = ω + sup{λ: every open cover of X has a subcover of cardinality λ}. Finally, the cellularity of X is defined by c(X) = ω + sup |U|: U is a pair disjoint family of nonempty open subsets of X . To study character of Cc (X) we must introduce a different cardinal function c(X). Definition 2.1. A family of Cauchy sequences {C λ : λ ∈ Λ} of X is said to be a Cauchy cover, if for any Cauchy sequence C, there is λ ∈ Λ such that C ⊂ C λ . Definition 2.2. The Cauchy cover number c(X) of X is defined by c(X) = ω + min |Λ|: {C λ }λ∈Λ is a Cauchy cover of X . Theorem 2.3. For every space X, χ(Cc (X)) = c(X). Proof. It is easy to see that Cc (X) is homogeneous, so we only consider the character of zero function f0 . Suppose that {V (f0 , C λ , ε): λ ∈ Λ, ε > 0} is a neighborhood base of f0 in Cc (X), where C λ is any Cauchy sequence of X, and |Λ| = χ(Cc (X)). If |Λ| < c(X), then {C λ : λ ∈ Λ} is not a Cauchy cover of X, hence there exists a Cauchy sequence C such that C ⊂ C λ for each λ ∈ Λ. Since V (f0 , C, 1) is a neighborhood of f0 , there exist λ ∈ Λ, ε > 0 such that V (f0 , C λ , ε) ⊂ V (f0 , C, 1). Taking c ∈ C − C λ , then there exists an open neighborhood U of c such that C λ ⊂ X − U . Let f ∈ C(X) with f (c) = 1, f (X − U ) ⊆ {0}, then f ∈ V (f0 , C λ , ε) − V (f0 , C, 1). This is a contradiction. So c(X) |Λ| = χ(Cc (X)). On the other hand, let {C λ : λ ∈ Λ} be a Cauchy cover of X with |Λ| = c(X). For each λ ∈ Λ, C λ is totally bounded, so Cc (C λ ) = Cu (C λ ). But Cu (C λ ) is a metric space, then Cc (C λ ) is a metrizable space. Let λ∈Λ C λ be the topological sum of {C λ : λ ∈ Λ}, p : λ∈Λ C λ → X the natural mapping, then the induced mapping ∗
p : Cc (X) → Cc
Cλ
λ∈Λ
is an embedding. So Cc ( λ∈Λ C λ ) is homeomorphic to λ∈Λ Cc (C λ ). We have χ Cc (X) χ Cc Cλ =χ
λ∈Λ
Cc (C λ )
λ∈Λ
|Λ| = c(X).
2
Theorem 2.4. For every space X, w(Cc (X)) = c(X) · d(Cc (X)). Proof. Obviously we have w Cc (X) χ Cc (X) · d Cc (X) = c(X) · d Cc (X) .
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On the other hand, let V0 = {V (f0 , C, ε): C ∈ C(X), ε > 0} be the neighborhood base of zero function f0 ≡ 0 in Cc (X) and |V0 | = c(X). Let B0 = {(C, ε): V (f0 , C, ε) ∈ V0 }, D be a dense set of Cc (X) with |D| = d(Cc (X)). Set B = V (f, C, ε): f ∈ D, (C, ε) ∈ B0 , we want to prove that B is a base of Cc (X). Suppose f ∈ Cc (X), C ∈ C(X), ε > 0. Because D is a dense set of Cc (X), there exist g ∈ D ∩ V (f, C, ε), C ∈ C(X), ε > 0 such that V (g, C , ε ) ⊂ V (f, C, ε). Thus there exists (C 1 , ε1 ) ∈ B0 such that V (f0 , C 1 , ε1 ) ⊂ V (f0 , C , ε ). If h ∈ V (g, C 1 , ε1 ), then h − g ∈ V (f0 , C 1 , ε1 ) ⊂ V (f0 , C , ε ). So h ∈ V (g, C , ε ) and V (g, C 1 , ε1 ) ⊂ V (g, C , ε ) ⊂ V (f, C, ε). Hence B is a base of Cc (X). Thus |B| |B0 | · |D| = |V0 | · |D|. Therefore w(Cc (X)) c(X) · d(Cc (X)). 2 Theorem 2.5. For every space X, d(Cc (X)) c(X) · c(Cc (X)). Proof. Let F be the family of Cauchy sequences of X with |F| = c(X). For each C ∈ F, n ∈ N, by Zorn (n) Lemma, we can define a maximal family UC satisfying: (n)
(1) For each U ∈ UC , U looks like V (fU , C, n1 ). (n)
(2) UC
is a pair disjoint family. (n)
It is clear that, |UC | c(Cc (X)). Let (n) D = fU : U ∈ UC , C ∈ F, n ∈ N , then |D| c(X) · c(Cc (X)). It suffices to show that D is dense in Cc (X). Let f ∈ Cc (X), ε > 0, C ∈ C(X), (n) there exist n ∈ N, C ∈ F with n2 < ε, C ⊂ C . If V (f, C , n1 ) is disjoint with the elements of UC , contradicting the maximality of UC , then there exists V (fU , C , n1 ) of UC such that V (f, C , n1 ) ∩ V (fU , C , n1 ) = ∅. Suppose g ∈ V (f, C , n1 ) ∩ V (fU , C , n1 ), then (n)
(n)
f (x) − fU (x) f (x) − g(x) + g(x) − fU (x) < 1 + 1 < ε, n n
x ∈ C ⊂ C .
Hence fU ∈ D ∩ V (f, C, ε). 2 Theorem 2.6. For every space X, d(Cc (X)) c(X) · L(Cc (X)). Proof. Let F be a family of Cauchy sequence covers of X with |F| = c(X). For each C ∈ F, n ∈ N, (n) (n) {V (f, C, n1 ): f ∈ Cc (X)} is an open cover of Cc (X), there exists a subcover VC such that |VC | L(Cc (X)). Let L=
1 f ∈ Cc (X): V f, C, n
∈
(n) VC
for some C ∈ F .
Then |L| c(X) · L(Cc (X)). It suffices to show that L is dense in Cc (X). We take f ∈ Cc (X), ε > 0, (n) (n) C ∈ C(X), n ∈ N with n1 < ε. Since VC is an open subcover of Cc (X), there exists V (g, C, n1 ) ∈ VC such that f ∈ V (g, C, n1 ) ⊂ V (g, C, ε). So g ∈ V (f, C, ε) and g ∈ L. 2
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3. Variations on tightness of Cc (X) There are some variations on tightness of Cp (X) and Ck (X) [1,7,10]. In this section we give the characterizations of the tightness, the countable fan tightness, the countable strongly fan tightness, the T-tightness and the set-tightness of Cc (X). The tightness of a space X is defined by t(X) = sup t(X, x): x ∈ X , where the tightness of X at x is defined by t(X, x) = ω + min λ: for each subset A of X, if x ∈ A, there exists B ⊂ A such that x ∈ B with |B| λ . Definition 3.1. A family U of subsets of a space X is called c-cover of X, if for each C of C(X), there exists U ∈ U such that C ⊂ U . If each element of U is open, U is called open c-cover of X. Definition 3.2. The cardinal number cL(X) of X is defined by cL(X) = ω + min λ: for each open c-cover U of X, there exists a c-subcover U with U λ . Theorem 3.3. For each space X, t(Cc (X)) = cL(X). Proof. Suppose that λ = t(Cc (X)), U is an open c-cover of X. For each C ∈ C(X), there exists UC ∈ U such that C ⊂ UC . Let F = fC : fC (C) = {0} and fC (X − UC ) ⊂ {1} . Then there exists F ⊂ F such that |F | λ and f0 ∈ F . Let W = {UC : fC ∈ F }, then W is a c-subcover of U. In fact, suppose C ∈ C(X), then V (f0 , C, 1) is a neighborhood of f0 , there exists C ∈ C(X) such that fC ∈ F ∩ V (f0 , C, 1). For x ∈ C, we have fC (x) < 1,
if x ∈ C ;
fC (x) = 1,
if x ∈ X − UC .
Thus C ⊂ UC . So W is a c-subcover of U. Hence cL(X) t(Cc (X)). On the other hand, let λ = cL(X), we only prove t(Cc (X), f0 ) λ since Cc (X) is homogeneous. Let G ⊂ Cc (X) with f0 ∈ G. For n ∈ N, C ∈ C(X), we take g(n,C) ∈ G ∩ V (f0 , C, n1 ). Let U(n,C) = {x ∈ X: |g(n,C) (x)| < n1 }, then C ⊂ U(n,C) . So Un = {U(n,C) : C ∈ C(X)} is an open c-cover of X. Thus Un has a c-subcover Vn with |Vn | λ. Taking G = {g(n,C) : n ∈ N and U(n,C) ∈ Vn }, then G ⊂ G and |G | λ. For n ∈ N, C ∈ C(X), there exists C ∈ C(X) such that C ⊂ U(n,C ) ∈ Vn , hence g(n,C ) ∈ V (f0 , C, n1 ) ∩ G . So f0 ∈ G . Thus t(Cc (X), f0 ) λ. 2 The fan tightness of a space X is defined by f t(X) = sup f t(X, x): x ∈ X , where the fan tightness of X at x is defined by
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f t(X, x) = ω + min λ: for each sequence {An }n∈N of subsets of X and x ∈
n∈N
there exists Bn ⊂ An such that x ∈
An ,
Bn with |Bn | λ .
n∈N
Theorem 3.4. For each space X, the followings are equivalent: (1) f t(Cc (X)) = ω. (2) For each sequence of open c-covers {Un }n∈N of X, there exists a finite subfamily Un of Un such that n∈N Un is an open c-cover of X. Proof. (1) ⇒ (2) Let {Un }n∈N be a sequence of open c-covers of X. For each n ∈ N, let An = f ∈ Cc (X): there exists U ∈ Un such that f (X − U ) ⊂ {0} . Then An is dense in Cc (X). In fact, for any basic open set V (f, C, ε) of Cc (X), we take g ∈ V (f, C, ε). Since Un is an open c-cover of X, there exists U ∈ Un such that C ⊂ U . By Tietze’s theorem, there exists h ∈ Cc (X) with h(x) = g(x),
x ∈ C;
h(X − U ) ⊂ {0}.
∞ Then h ∈ An ∩ V (f, C, ε). We take h1 ∈ Cc (X) with h1 (x) ≡ 1, then h1 ∈ n=1 An . So there exists a finite ∞ set Bn of An such that h1 ∈ n=1 Bn . Let Bn = {f(n,j) : j i(n)}, there exists U(n,j) ∈ Un such that f(n,j) (X − U(n,j) ) ⊂ {0}. Let Un = {U(n,j) : j i(n)}, we want to show that n∈N Un is an open c-cover of X. For C ∈ C(X), there exist n ∈ N, j i(n) such that f(n,j) ∈ V (h1 , C, 1), then C ⊂ U(n,j) . So n∈N Un is an open c-cover of X. (2) ⇒ (1) Because Cc (X) is homogeneous, we need to show that f t(Cc (X), f0 ) = ω. Let f0 ∈ n∈N An , where An ⊂ Cc (X). For each n ∈ N, let Un = {f −1 ((− n1 , n1 )): f ∈ An }, then Un is an open c-cover of X. In fact, for each C ∈ C(X), there exists f ∈ V (f0 , C, n1 ) ∩ An such that C ⊂ f −1 ((− n1 , n1 )). Let M = {n ∈ N: X ∈ Un }. If M is infinite, for every base neighborhood V (f0 , C, ε) of f0 , ε > 0, there exists 1 1 1 −1 m ∈ M such that m < ε. By the construction of Um , there exists gm ∈ Am such that X = gm ((− m , m )). 1 1 Hence gm (X) ⊂ (− m , m ), gm ∈ V (f0 , C, ε). So {gm }m∈M is τc -convergent to f0 . If M is finite, there exists 1 1 n0 ∈ N such that g −1 ((− m , m )) = X whenever m n0 , g ∈ Am . But {Um }mn0 is a sequence of open c-covers of X, there exists a finite Um of Um such that mn0 Um is an open c-cover of X. We denote −1 1 1 Um = {U(m,j): ji(m) }, there exists f(m,j) ∈ Am such that U(m,j) = f(m,j) ((− m , m )). We need to show f0 ∈ {f(m,j) : m n0 , j i(m)}. For any base neighborhood V (f0 , C, ε) of f0 , let H = (m, j): m n0 , j i(m), C ⊂ U(m,j) . Then H = ∅. If H is finite, for each (m, j) ∈ H, by U(m,j) = X, taking x(m,j) ∈ X − U(m,j) , then {x(m,j) : (m, j) ∈ H}∪C ∈ C(X). But there exists no elements of {x(m,j) : (m, j) ∈ H}∪C in mn0 Um . With −1 1 1 this contradiction, H is infinite, so there exist m n0 , j i(m) such that C ⊂ U(m,j) = f(m,j) ((− m , m )), 1 m
< ε. Thus f(m,j) (C) ⊂ (−ε, ε), and f(m,j) ∈ V (f0 , C, ε). Therefore f0 ∈ {f(m,j) : m n0 , j i(m)}.
2
A space X is said to be countable strongly fan tightness, if for each x ∈ X, a sequence {An }n∈N of X with x ∈ n∈N An , then there exists xn ∈ An such that x ∈ {xn : n ∈ N}.
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Theorem 3.5. For each space X, the followings are equivalent: (1) Cc (X) is countable strongly fan tightness. (2) For each sequence of open c-covers {Un }n∈N of X, there exists Un ∈ Un such that {Un }n∈N is a c-cover of X. Proof. (1) ⇒ (2) Let {Un }n∈N be a sequence of open c-covers of X. For each n ∈ N, let An = f ∈ Cc (X): there exists U ∈ Un such that f (X − U ) ⊂ {2} . ∞ Analogously to Theorem 3.4, An = Cc (X). Let h(x) ≡ 1, then h ∈ n=1 An . So there exists fn ∈ An such that h ∈ {fn : n ∈ N}. By the definition of An , there exists Un ∈ Un such that fn (X − Un ) ⊂ {2}. For each C ∈ C(X), by h ∈ V (h, C, 1), there exists m ∈ N such that fm ∈ V (h, C, 1). Then C ⊂ Um . Thus {Un }n∈N is a c-cover of X. (2) ⇒ (1) We need to prove that Cc (X) is countable strongly fan tightness at f0 . Let An ⊂ Cc (X), f0 ∈ n∈N An . For each n ∈ N, let Un = {f −1 ((− n1 , n1 )): f ∈ An }, then Un is an open c-cover of X. Let M = {n ∈ N: X ∈ An }. If M is infinite, then for any base neighborhood V (f0 , C, ε) of f0 with 1 ε > 0, there exists m ∈ M such that m < ε. By the construction of Um , there exists gm ∈ Am such that 1 1 1 1 −1 , m ), gm ∈ V (f0 , C, ε). So {gm }m∈M is convergent to f0 . If X = g ((− m , m )). Therefore gm (X) ⊂ (− m M is finite, there exists n0 ∈ N such that {Um }mn0 is a sequence of open c-covers of X when m n0 , 1 1 , m )) = X for g ∈ Am . Thus there exists Um ∈ Um such that {Um }mn0 is an open c-cover of X. g −1 ((− m 1 1 −1 ((− m , m )). It now suffices to show f0 ∈ {fm : m n0 }. Therefore there exists fm ∈ Am such that Um = fm For any base neighborhood V (f0 , C, ε) of f0 , let UC = {Um : C ⊂ Um , m n0 }, then UC = ∅. If UC is finite, let UC = {Umj : j k}. For each j k, by Umj = X, we take xmj ∈ X − Umj , then {xmj : j k} ∪ C ∈ C(X), so Um ∩ ({xmj : j k} ∪ C) = ∅ when m n0 . With this contradiction, UC is infinite. Therefore 1 1 1 −1 ((− m , m )), m < ε. Thus |fm (x) − f0 (x)| < ε, x ∈ C, so there exists m n0 such that C ⊂ Um = fm f0 ∈ {fm : m n0 }. 2 The T-tightness T (X) of a space X [2,3,6] is defined by T (X) = ω + min τ : {Fα : α < κ} is an increasing family of closed subsets of X and cf (κ) > τ, then {Fα : α < κ} is closed in X . Definition 3.6. A space X satisfies property Tc (τ ) if whenever κ is a regular cardinal greater than τ and U = α<κ Uα an open c-cover of X such that Uα ⊂ Uβ (α < β), there exists α < κ such that Uα is an open c-cover of X. Theorem 3.7. For a space X, the followings are equivalent: (1) T (Cc (X)) τ ; (2) X satisfies the property Tc (τ ). Proof. (1) ⇒ (2) Let T (Cc (X)) τ , κ is a regular cardinal and τ < κ, U = X such that Uα ⊂ Uβ (α < β). For α < κ, let
α<κ
Uα is an open c-cover of
Fα = f ∈ Cc (X): there exists U ∈ Uα , such that f (X − U ) ≡ 1 .
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Then α<κ Fα = α<κ Fα . Indeed, α<κ Fα ⊂ α<κ Fα since Fα ⊂ α<κ Fα . On the other hand, Uα ⊂ Uβ (α < β), so {Fα : α < κ} is an increasing family of closed subsets. Because Cc (X) is a T-tightness space and {Fα : α < κ} is closed, so α<κ Fα ⊂ α<κ Fα = α<κ Fα . We want to prove that f0 ≡ 0 ∈ α<κ Fα . For each basic neighborhood V (f0 , C, ε) of f0 , there exist α < κ and U ∈ Uα such that C ⊂ U since U is an open c-cover of X. Take f ∈ Cc (X) with f (C) = {0}, f (X − U ) ⊂ 1, then f ∈ V (f0 , C, ε) ∩ Fα . So f0 ∈ α<κ Fα = α<κ Fα . We choose α < κ such that f0 ∈ Fα . Let C ∈ C(X), then V (f0 , C , 1) is a neighborhood f0 , so there exist α < κ, fα ∈ Fα , U ∈ Uα such that fα ∈ V (f0 , C , 1) ∩ Fα and fα | (X − U ) ≡ 1, fα (C ) ⊂ (−1, 1). Thus C ⊂ U , so Uα is an open c-cover of X. (2) ⇒ (1) Assume the condition (2). κ is a regular cardinal and τ < cf (κ). {Fα : α < κ} is an increasing family of closed subsets of Cc (X). Since Cc (X) is homogeneous, we suppose f0 ∈ α<κ Fα . For n ∈ N, α < κ, let Uα(n) =
f −1
1 1 − , : f ∈ Fα , n n
Un =
Uα(n) .
α<κ
Then Un is an open c-cover of X. Indeed, let C ∈ C(X), then V (f0 , C, n1 ) is a neighborhood of f0 . By (n) f0 ∈ α<κ Fα , there exists α < κ such that fα ∈ V (f0 , C, n1 ) ∩ Fα . So C ⊂ fα−1 ((− n1 , n1 )) ∈ Uα . For (n) n ∈ N, since X satisfies the property Tc (τ ), there exists αn < κ such that Uαn is an open c-cover of X. Let (n) β = sup{αn : n ∈ N}, then β < κ since κ is a regular cardinal. For n ∈ N, Uβ is an open c-cover. We claim f0 ∈ Fβ . Let V (f0 , C , ε) is a neighborhood of f0 , take n ∈ N such that n1 < ε. Since Uβ is an open c-cover of X, there exists f ∈ Fβ such that C ⊂ f −1 ((− n1 , n1 )). So f ∈ Fβ ∩ V (f0 , C , ε), hence f0 ∈ Fβ = Fβ . Thus f0 ∈ α<κ Fα . So {Fα : α < κ} is closed in Cc (X). 2 (n)
The set-tightness ts (X) of a space X [3,6,10] is defined by ts (X) = ω + min τ : if A ⊂ X, x ∈ A − A, there is {Aα : Aα ⊂ A, α < τ } such that for any α < τ, then x ∈ / Aα and x ∈
{Aα : α < τ } .
Definition 3.8. Let U be a collection of cozero subsets of a space X, and Z(U) = {Z(U ): U ∈ U} be a collection of zero-sets of a space of X indexed by U. (U, Z(U)) is called a tc -pair if (1) for every U ∈ U, Z(U ) ⊂ U = X; (2) Z(U) is a c-cover of X. Definition 3.9. A space X satisfies property tc (τ ) if for every tc -pair (U, Z(U)), there exists Uα ⊂ U(α < τ ) such that Z(Uα ) is not a c-cover of X, but α<τ Uα is a c-cover of X. Theorem 3.10. For a space X, the followings are equivalent: (1) ts (Cc (X)) τ ; (2) X satisfies the property tc (τ ). Proof. (1) ⇒ (2) Let ts (Cc (X)) τ , (U, Z(U)) is a tc -pair of X. For each U ∈ U, since Z(U ) is a zero-set, there exists f1 ∈ Cc (X), such that Z(U ) = f1−1 (0). Since U is a cozero set, there exists f2 ∈ Cc (X) such f1 (x) , then fU ∈ C(X) and fU (Z(U )) = 0, fU (X − U ) = 1. Set that X − U = f2−1 (0). Let fU (x) = f1 (x)+f 2 (x) F = {fU : U ∈ U}, then f0 ∈ F − F . By ts (Cc (X)) τ , there exists {Uα : Uα ⊂ U, α < τ } such that / {fU : U ∈ Uα }, and f0 ∈ {fU : U ∈ α<τ Uα }. If Z(Uα ) is a c-cover of X, for any α < τ , we have f0 ∈
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then f0 ∈ {fU : U ∈ Uα }. This is a contradiction. So each Z(Uα ) is not a c-cover of X. Let C ∈ C(X), then V (f , C, 1) is a neighborhood of f0 , there exists U ∈ α<τ Uα such that fU ∈ V (f0 , C, 1). So C ⊂ U , thus 0 α<τ Uα is a c-cover of X. (2) ⇒ (1) Let X be of the property tc (τ ), F ⊂ Cc (X), g ∈ F − F . Since Ck (X) is homogeneous, we can take g = f0 ≡ 0. For each f ∈ F , n ∈ N, let 1 1 − , ; =f n n (n) Un = Uf : f ∈ F ;
(n) Uf
−1
1 1 − , =f n+1 n+1 (n) Zn = Zf : f ∈ F . (n) Zf
−1
;
Let M = {n ∈ N: X ∈ Un }. If M is infinite, then {fn }n∈M ⊂ F uniformly converges f0 . So {fn }n∈M converges f0 in Cc (X). Let Fn = {fn }, then f0 ∈ / Fn , and f0 ∈ n∈M Fn . Thus ts (Cc (X)) ω τ . Without a loss of generality, suppose M = ∅. Then for any n ∈ N, we have X ∈ / Un . For each C ⊂ C(X), a basic 1 1 1 1 , n+1 ) ⊂ open neighborhood V (f0 , C, n+1 ) of f0 , there exists f ∈ V (f0 , C, n+1 ) ∩ F . So f (C) ⊂ (− n+1 1 1 1 1 [− n+1 , n+1 ], thus C ⊂ f −1 ([− n+1 , n+1 ]) = Zf . Obviously, Zf (n)
a tc -pair of X. By (2), there exists (n)
(n) {Fα
(n)
(n)
(n)
⊂ Uf , and Uf
= X. Hence (Un , Zn ) is
⊂ F : α < τ } such that
(n)
(a) {Zf : f ∈ Fα } is not a c-cover of X. (n) (n) (b) {Uf : f ∈ α<τ Fα } is a c-cover of X. (n)
(n)
The condition (a) means f0 ∈ / Fα . Indeed, there exists C ∈ C(X), such that for any n ∈ N, f ∈ Fα , (n) (n) (n) 1 ) ∩ Fα = ∅. The condition (b) means f0 ∈ {Fα : α < τ, n ∈ N}. we have C ⊂ Zf . Then V (f0 , C, n+1 (n) (n) Indeed, for any C ∈ C(X), ε > 0, there exists n ∈ N such that n1 < ε. Since {Uf : f ∈ α<τ Fα } is a (n) (n) (n) c-cover of X, there exists f ∈ α<τ Fα such that C ⊂ Uf . So f ∈ V (f0 , C, n1 ) ∩ ( α<τ Fα ). Hence ts (Cc (X)) τ . 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A.V. Arhangel’skii, Topological Function Spaces, Kluwer, Dordrecht, 1992. A.V. Arhangel’skii, R. Isler, G. Tironi, On pseudo-radial spaces, Comment. Math. Univ. Carol. 27 (1986) 137–154. A. Bella, On set tightness and T -tightness, Comment. Math. Univ. Carol. 27 (1986) 805–814. Michael H. Clapp, Ray C. Shiflett, Necessary and sufficient condition for the equivalence of the topologies of uniform and compact convergence, Can. Math. Bull. 22 (4) (1979) 467–470. R. Engelking, General Topology, Helderman, Berlin, 1989. I. Juhász, Variations on tightness, Studia Sci. Math. Hung. 24 (1989) 179–186. R.A. McCoy, I. Ntantu, Topological Properties of Spaces of Continuous Functions, Lect. Notes Math., vol. 1315, SpringerVerlag, New York, 1988. R.A. McCoy, I. Ntantu, Countability properties of function spaces with set-open topologies, Topol. Proc. 10 (1985) 329–345. G. Di Miao, L’. Holá, D. Holý, R.A. McCoy, Topologies on the spaces of continuous functions, Topol. Appl. 18 (1998) 105–122. M. Sakai, Variations on tightness in function spaces, Topol. Appl. 101 (2000) 273–280.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Cauchy convergence topologies on the space of continuous functions Zuquan Li Department of Mathematics, Hangzhou Normal University, Hangzhou, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 October 2013 Received in revised form 19 October 2013 Accepted 21 October 2013
In this paper, we investigate some cardinal invariants and variations on tightness in the space of continuous functions from a metric space X to real line R with Cauchy convergence topology. © 2013 Elsevier B.V. All rights reserved.
MSC: primary 54A25, 54C35 Keywords: Function space Cauchy convergence topology Cauchy cover c-Cover Cardinal function
1. Introduction Throughout the paper, X is a metric space, d is the metric of X. The topology of X is the induced topology by d. R is real line with the usual topology. N is the set of natural numbers. ω is countable cardinal. We denote C(X) the family of all the real-value continuous functions on X, and Ck (X) (Cu (X)) the space all the continuous functions of X with compact open topology (uniform convergent topology) [1,7,8]. We denote V (f, S, ε) = g ∈ C(X): f (x) − g(x) < ε, x ∈ S , where S ⊂ X, ε > 0. E-mail address: [email protected]. 0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.10.032
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A sequence {xn }n∈N of X is said to be a Cauchy sequence, if for any ε > 0, there exists N ∈ N such that d(xm , xn ) < ε with m, n > N . Let C(X) be the family of all the Cauchy sequences of X. To consider Cauchy convergence topology τc , for f ∈ C(X), we choose {V (f, C, ε): C ∈ C(X), ε > 0} as a neighborhood base of f . The space (C(X), τc ) is denoted by Cc (X). Cauchy convergence topology was first introduced by Michael H. Clapp and Ray C. Shiflett [4]. They defined the topological space so that they want to compare it with compact open topology τk and uniform convergent topology τu of C(X). The following theorem summarizes some of the results in [4]. Theorem 1.1. Let X be a space, then (1) τc τu , and τc = τu if and only if X is totally bounded. (2) τk τc , and τk = τc if and only if X is complete. In this paper, we construct the relations among w(Cc (X)), d(Cc (X)), c(Cc (X)), L(Cc (X)) and χ(Cc (X)), give the characterizations of tightness, countable fan tightness, countable strongly fan tightness, T-tightness and set-tightness of Cc (X). Let C = {xn }n∈N be a Cauchy sequence of X. We denote C=
C,
if {xn }n∈N is not convergent in X,
{x0 } ∪ C,
if {xn }n∈N → x0 .
C is said to be a Cauchy closed sequence of X. Each Cauchy closed sequence is a Cauchy sequence. Let C(X) be the family of all the Cauchy closed sequences of X. In this paper, each Cauchy sequence always means a Cauchy closed sequence. By V (f, C, 2ε ) ⊂ V (f, C, 2ε ) ⊂ V (f, C, ε) ⊂ V (f, C, ε), we can obtain the following lemma. Lemma 1.2. For every space X, f ∈ C(X), the neighborhood base of f generated by {V (f, C, ε): C ∈ C(X), ε > 0} is equivalent to the neighborhood base of f generated by {V (f, C, ε): C ∈ C(X), ε > 0}. In other words, we can choose {V (f, C, ε): C ∈ C(X), ε > 0} as a neighborhood base of f on Cauchy convergence topology τc . 2. Cauchy cover number of Cc (X) We primarily study five cardinal functions [5,9] in this section. The cardinality of a set S will be denoted by |S|. The character of a space X at a point x is defined by χ(X, x) = ω + min |Bx |: Bx is a base for X at x . Then the character of X is given by χ(X) = sup χ(X, x): x ∈ X . The weight of X is defined by w(X) = ω + min |B|: B is a base for X . The density of X has definition d(X) = ω + min |D|: D is a dense subset of X .
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The Lindelöf number of X is equal to the number L(X) = ω + sup{λ: every open cover of X has a subcover of cardinality λ}. Finally, the cellularity of X is defined by c(X) = ω + sup |U|: U is a pair disjoint family of nonempty open subsets of X . To study character of Cc (X) we must introduce a different cardinal function c(X). Definition 2.1. A family of Cauchy sequences {C λ : λ ∈ Λ} of X is said to be a Cauchy cover, if for any Cauchy sequence C, there is λ ∈ Λ such that C ⊂ C λ . Definition 2.2. The Cauchy cover number c(X) of X is defined by c(X) = ω + min |Λ|: {C λ }λ∈Λ is a Cauchy cover of X . Theorem 2.3. For every space X, χ(Cc (X)) = c(X). Proof. It is easy to see that Cc (X) is homogeneous, so we only consider the character of zero function f0 . Suppose that {V (f0 , C λ , ε): λ ∈ Λ, ε > 0} is a neighborhood base of f0 in Cc (X), where C λ is any Cauchy sequence of X, and |Λ| = χ(Cc (X)). If |Λ| < c(X), then {C λ : λ ∈ Λ} is not a Cauchy cover of X, hence there exists a Cauchy sequence C such that C ⊂ C λ for each λ ∈ Λ. Since V (f0 , C, 1) is a neighborhood of f0 , there exist λ ∈ Λ, ε > 0 such that V (f0 , C λ , ε) ⊂ V (f0 , C, 1). Taking c ∈ C − C λ , then there exists an open neighborhood U of c such that C λ ⊂ X − U . Let f ∈ C(X) with f (c) = 1, f (X − U ) ⊆ {0}, then f ∈ V (f0 , C λ , ε) − V (f0 , C, 1). This is a contradiction. So c(X) |Λ| = χ(Cc (X)). On the other hand, let {C λ : λ ∈ Λ} be a Cauchy cover of X with |Λ| = c(X). For each λ ∈ Λ, C λ is totally bounded, so Cc (C λ ) = Cu (C λ ). But Cu (C λ ) is a metric space, then Cc (C λ ) is a metrizable space. Let λ∈Λ C λ be the topological sum of {C λ : λ ∈ Λ}, p : λ∈Λ C λ → X the natural mapping, then the induced mapping ∗
p : Cc (X) → Cc
Cλ
λ∈Λ
is an embedding. So Cc ( λ∈Λ C λ ) is homeomorphic to λ∈Λ Cc (C λ ). We have χ Cc (X) χ Cc Cλ =χ
λ∈Λ
Cc (C λ )
λ∈Λ
|Λ| = c(X).
2
Theorem 2.4. For every space X, w(Cc (X)) = c(X) · d(Cc (X)). Proof. Obviously we have w Cc (X) χ Cc (X) · d Cc (X) = c(X) · d Cc (X) .
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On the other hand, let V0 = {V (f0 , C, ε): C ∈ C(X), ε > 0} be the neighborhood base of zero function f0 ≡ 0 in Cc (X) and |V0 | = c(X). Let B0 = {(C, ε): V (f0 , C, ε) ∈ V0 }, D be a dense set of Cc (X) with |D| = d(Cc (X)). Set B = V (f, C, ε): f ∈ D, (C, ε) ∈ B0 , we want to prove that B is a base of Cc (X). Suppose f ∈ Cc (X), C ∈ C(X), ε > 0. Because D is a dense set of Cc (X), there exist g ∈ D ∩ V (f, C, ε), C ∈ C(X), ε > 0 such that V (g, C , ε ) ⊂ V (f, C, ε). Thus there exists (C 1 , ε1 ) ∈ B0 such that V (f0 , C 1 , ε1 ) ⊂ V (f0 , C , ε ). If h ∈ V (g, C 1 , ε1 ), then h − g ∈ V (f0 , C 1 , ε1 ) ⊂ V (f0 , C , ε ). So h ∈ V (g, C , ε ) and V (g, C 1 , ε1 ) ⊂ V (g, C , ε ) ⊂ V (f, C, ε). Hence B is a base of Cc (X). Thus |B| |B0 | · |D| = |V0 | · |D|. Therefore w(Cc (X)) c(X) · d(Cc (X)). 2 Theorem 2.5. For every space X, d(Cc (X)) c(X) · c(Cc (X)). Proof. Let F be the family of Cauchy sequences of X with |F| = c(X). For each C ∈ F, n ∈ N, by Zorn (n) Lemma, we can define a maximal family UC satisfying: (n)
(1) For each U ∈ UC , U looks like V (fU , C, n1 ). (n)
(2) UC
is a pair disjoint family. (n)
It is clear that, |UC | c(Cc (X)). Let (n) D = fU : U ∈ UC , C ∈ F, n ∈ N , then |D| c(X) · c(Cc (X)). It suffices to show that D is dense in Cc (X). Let f ∈ Cc (X), ε > 0, C ∈ C(X), (n) there exist n ∈ N, C ∈ F with n2 < ε, C ⊂ C . If V (f, C , n1 ) is disjoint with the elements of UC , contradicting the maximality of UC , then there exists V (fU , C , n1 ) of UC such that V (f, C , n1 ) ∩ V (fU , C , n1 ) = ∅. Suppose g ∈ V (f, C , n1 ) ∩ V (fU , C , n1 ), then (n)
(n)
f (x) − fU (x) f (x) − g(x) + g(x) − fU (x) < 1 + 1 < ε, n n
x ∈ C ⊂ C .
Hence fU ∈ D ∩ V (f, C, ε). 2 Theorem 2.6. For every space X, d(Cc (X)) c(X) · L(Cc (X)). Proof. Let F be a family of Cauchy sequence covers of X with |F| = c(X). For each C ∈ F, n ∈ N, (n) (n) {V (f, C, n1 ): f ∈ Cc (X)} is an open cover of Cc (X), there exists a subcover VC such that |VC | L(Cc (X)). Let L=
1 f ∈ Cc (X): V f, C, n
∈
(n) VC
for some C ∈ F .
Then |L| c(X) · L(Cc (X)). It suffices to show that L is dense in Cc (X). We take f ∈ Cc (X), ε > 0, (n) (n) C ∈ C(X), n ∈ N with n1 < ε. Since VC is an open subcover of Cc (X), there exists V (g, C, n1 ) ∈ VC such that f ∈ V (g, C, n1 ) ⊂ V (g, C, ε). So g ∈ V (f, C, ε) and g ∈ L. 2
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3. Variations on tightness of Cc (X) There are some variations on tightness of Cp (X) and Ck (X) [1,7,10]. In this section we give the characterizations of the tightness, the countable fan tightness, the countable strongly fan tightness, the T-tightness and the set-tightness of Cc (X). The tightness of a space X is defined by t(X) = sup t(X, x): x ∈ X , where the tightness of X at x is defined by t(X, x) = ω + min λ: for each subset A of X, if x ∈ A, there exists B ⊂ A such that x ∈ B with |B| λ . Definition 3.1. A family U of subsets of a space X is called c-cover of X, if for each C of C(X), there exists U ∈ U such that C ⊂ U . If each element of U is open, U is called open c-cover of X. Definition 3.2. The cardinal number cL(X) of X is defined by cL(X) = ω + min λ: for each open c-cover U of X, there exists a c-subcover U with U λ . Theorem 3.3. For each space X, t(Cc (X)) = cL(X). Proof. Suppose that λ = t(Cc (X)), U is an open c-cover of X. For each C ∈ C(X), there exists UC ∈ U such that C ⊂ UC . Let F = fC : fC (C) = {0} and fC (X − UC ) ⊂ {1} . Then there exists F ⊂ F such that |F | λ and f0 ∈ F . Let W = {UC : fC ∈ F }, then W is a c-subcover of U. In fact, suppose C ∈ C(X), then V (f0 , C, 1) is a neighborhood of f0 , there exists C ∈ C(X) such that fC ∈ F ∩ V (f0 , C, 1). For x ∈ C, we have fC (x) < 1,
if x ∈ C ;
fC (x) = 1,
if x ∈ X − UC .
Thus C ⊂ UC . So W is a c-subcover of U. Hence cL(X) t(Cc (X)). On the other hand, let λ = cL(X), we only prove t(Cc (X), f0 ) λ since Cc (X) is homogeneous. Let G ⊂ Cc (X) with f0 ∈ G. For n ∈ N, C ∈ C(X), we take g(n,C) ∈ G ∩ V (f0 , C, n1 ). Let U(n,C) = {x ∈ X: |g(n,C) (x)| < n1 }, then C ⊂ U(n,C) . So Un = {U(n,C) : C ∈ C(X)} is an open c-cover of X. Thus Un has a c-subcover Vn with |Vn | λ. Taking G = {g(n,C) : n ∈ N and U(n,C) ∈ Vn }, then G ⊂ G and |G | λ. For n ∈ N, C ∈ C(X), there exists C ∈ C(X) such that C ⊂ U(n,C ) ∈ Vn , hence g(n,C ) ∈ V (f0 , C, n1 ) ∩ G . So f0 ∈ G . Thus t(Cc (X), f0 ) λ. 2 The fan tightness of a space X is defined by f t(X) = sup f t(X, x): x ∈ X , where the fan tightness of X at x is defined by
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f t(X, x) = ω + min λ: for each sequence {An }n∈N of subsets of X and x ∈
n∈N
there exists Bn ⊂ An such that x ∈
An ,
Bn with |Bn | λ .
n∈N
Theorem 3.4. For each space X, the followings are equivalent: (1) f t(Cc (X)) = ω. (2) For each sequence of open c-covers {Un }n∈N of X, there exists a finite subfamily Un of Un such that n∈N Un is an open c-cover of X. Proof. (1) ⇒ (2) Let {Un }n∈N be a sequence of open c-covers of X. For each n ∈ N, let An = f ∈ Cc (X): there exists U ∈ Un such that f (X − U ) ⊂ {0} . Then An is dense in Cc (X). In fact, for any basic open set V (f, C, ε) of Cc (X), we take g ∈ V (f, C, ε). Since Un is an open c-cover of X, there exists U ∈ Un such that C ⊂ U . By Tietze’s theorem, there exists h ∈ Cc (X) with h(x) = g(x),
x ∈ C;
h(X − U ) ⊂ {0}.
∞ Then h ∈ An ∩ V (f, C, ε). We take h1 ∈ Cc (X) with h1 (x) ≡ 1, then h1 ∈ n=1 An . So there exists a finite ∞ set Bn of An such that h1 ∈ n=1 Bn . Let Bn = {f(n,j) : j i(n)}, there exists U(n,j) ∈ Un such that f(n,j) (X − U(n,j) ) ⊂ {0}. Let Un = {U(n,j) : j i(n)}, we want to show that n∈N Un is an open c-cover of X. For C ∈ C(X), there exist n ∈ N, j i(n) such that f(n,j) ∈ V (h1 , C, 1), then C ⊂ U(n,j) . So n∈N Un is an open c-cover of X. (2) ⇒ (1) Because Cc (X) is homogeneous, we need to show that f t(Cc (X), f0 ) = ω. Let f0 ∈ n∈N An , where An ⊂ Cc (X). For each n ∈ N, let Un = {f −1 ((− n1 , n1 )): f ∈ An }, then Un is an open c-cover of X. In fact, for each C ∈ C(X), there exists f ∈ V (f0 , C, n1 ) ∩ An such that C ⊂ f −1 ((− n1 , n1 )). Let M = {n ∈ N: X ∈ Un }. If M is infinite, for every base neighborhood V (f0 , C, ε) of f0 , ε > 0, there exists 1 1 1 −1 m ∈ M such that m < ε. By the construction of Um , there exists gm ∈ Am such that X = gm ((− m , m )). 1 1 Hence gm (X) ⊂ (− m , m ), gm ∈ V (f0 , C, ε). So {gm }m∈M is τc -convergent to f0 . If M is finite, there exists 1 1 n0 ∈ N such that g −1 ((− m , m )) = X whenever m n0 , g ∈ Am . But {Um }mn0 is a sequence of open c-covers of X, there exists a finite Um of Um such that mn0 Um is an open c-cover of X. We denote −1 1 1 Um = {U(m,j): ji(m) }, there exists f(m,j) ∈ Am such that U(m,j) = f(m,j) ((− m , m )). We need to show f0 ∈ {f(m,j) : m n0 , j i(m)}. For any base neighborhood V (f0 , C, ε) of f0 , let H = (m, j): m n0 , j i(m), C ⊂ U(m,j) . Then H = ∅. If H is finite, for each (m, j) ∈ H, by U(m,j) = X, taking x(m,j) ∈ X − U(m,j) , then {x(m,j) : (m, j) ∈ H}∪C ∈ C(X). But there exists no elements of {x(m,j) : (m, j) ∈ H}∪C in mn0 Um . With −1 1 1 this contradiction, H is infinite, so there exist m n0 , j i(m) such that C ⊂ U(m,j) = f(m,j) ((− m , m )), 1 m
< ε. Thus f(m,j) (C) ⊂ (−ε, ε), and f(m,j) ∈ V (f0 , C, ε). Therefore f0 ∈ {f(m,j) : m n0 , j i(m)}.
2
A space X is said to be countable strongly fan tightness, if for each x ∈ X, a sequence {An }n∈N of X with x ∈ n∈N An , then there exists xn ∈ An such that x ∈ {xn : n ∈ N}.
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Theorem 3.5. For each space X, the followings are equivalent: (1) Cc (X) is countable strongly fan tightness. (2) For each sequence of open c-covers {Un }n∈N of X, there exists Un ∈ Un such that {Un }n∈N is a c-cover of X. Proof. (1) ⇒ (2) Let {Un }n∈N be a sequence of open c-covers of X. For each n ∈ N, let An = f ∈ Cc (X): there exists U ∈ Un such that f (X − U ) ⊂ {2} . ∞ Analogously to Theorem 3.4, An = Cc (X). Let h(x) ≡ 1, then h ∈ n=1 An . So there exists fn ∈ An such that h ∈ {fn : n ∈ N}. By the definition of An , there exists Un ∈ Un such that fn (X − Un ) ⊂ {2}. For each C ∈ C(X), by h ∈ V (h, C, 1), there exists m ∈ N such that fm ∈ V (h, C, 1). Then C ⊂ Um . Thus {Un }n∈N is a c-cover of X. (2) ⇒ (1) We need to prove that Cc (X) is countable strongly fan tightness at f0 . Let An ⊂ Cc (X), f0 ∈ n∈N An . For each n ∈ N, let Un = {f −1 ((− n1 , n1 )): f ∈ An }, then Un is an open c-cover of X. Let M = {n ∈ N: X ∈ An }. If M is infinite, then for any base neighborhood V (f0 , C, ε) of f0 with 1 ε > 0, there exists m ∈ M such that m < ε. By the construction of Um , there exists gm ∈ Am such that 1 1 1 1 −1 , m ), gm ∈ V (f0 , C, ε). So {gm }m∈M is convergent to f0 . If X = g ((− m , m )). Therefore gm (X) ⊂ (− m M is finite, there exists n0 ∈ N such that {Um }mn0 is a sequence of open c-covers of X when m n0 , 1 1 , m )) = X for g ∈ Am . Thus there exists Um ∈ Um such that {Um }mn0 is an open c-cover of X. g −1 ((− m 1 1 −1 ((− m , m )). It now suffices to show f0 ∈ {fm : m n0 }. Therefore there exists fm ∈ Am such that Um = fm For any base neighborhood V (f0 , C, ε) of f0 , let UC = {Um : C ⊂ Um , m n0 }, then UC = ∅. If UC is finite, let UC = {Umj : j k}. For each j k, by Umj = X, we take xmj ∈ X − Umj , then {xmj : j k} ∪ C ∈ C(X), so Um ∩ ({xmj : j k} ∪ C) = ∅ when m n0 . With this contradiction, UC is infinite. Therefore 1 1 1 −1 ((− m , m )), m < ε. Thus |fm (x) − f0 (x)| < ε, x ∈ C, so there exists m n0 such that C ⊂ Um = fm f0 ∈ {fm : m n0 }. 2 The T-tightness T (X) of a space X [2,3,6] is defined by T (X) = ω + min τ : {Fα : α < κ} is an increasing family of closed subsets of X and cf (κ) > τ, then {Fα : α < κ} is closed in X . Definition 3.6. A space X satisfies property Tc (τ ) if whenever κ is a regular cardinal greater than τ and U = α<κ Uα an open c-cover of X such that Uα ⊂ Uβ (α < β), there exists α < κ such that Uα is an open c-cover of X. Theorem 3.7. For a space X, the followings are equivalent: (1) T (Cc (X)) τ ; (2) X satisfies the property Tc (τ ). Proof. (1) ⇒ (2) Let T (Cc (X)) τ , κ is a regular cardinal and τ < κ, U = X such that Uα ⊂ Uβ (α < β). For α < κ, let
α<κ
Uα is an open c-cover of
Fα = f ∈ Cc (X): there exists U ∈ Uα , such that f (X − U ) ≡ 1 .
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Then α<κ Fα = α<κ Fα . Indeed, α<κ Fα ⊂ α<κ Fα since Fα ⊂ α<κ Fα . On the other hand, Uα ⊂ Uβ (α < β), so {Fα : α < κ} is an increasing family of closed subsets. Because Cc (X) is a T-tightness space and {Fα : α < κ} is closed, so α<κ Fα ⊂ α<κ Fα = α<κ Fα . We want to prove that f0 ≡ 0 ∈ α<κ Fα . For each basic neighborhood V (f0 , C, ε) of f0 , there exist α < κ and U ∈ Uα such that C ⊂ U since U is an open c-cover of X. Take f ∈ Cc (X) with f (C) = {0}, f (X − U ) ⊂ 1, then f ∈ V (f0 , C, ε) ∩ Fα . So f0 ∈ α<κ Fα = α<κ Fα . We choose α < κ such that f0 ∈ Fα . Let C ∈ C(X), then V (f0 , C , 1) is a neighborhood f0 , so there exist α < κ, fα ∈ Fα , U ∈ Uα such that fα ∈ V (f0 , C , 1) ∩ Fα and fα | (X − U ) ≡ 1, fα (C ) ⊂ (−1, 1). Thus C ⊂ U , so Uα is an open c-cover of X. (2) ⇒ (1) Assume the condition (2). κ is a regular cardinal and τ < cf (κ). {Fα : α < κ} is an increasing family of closed subsets of Cc (X). Since Cc (X) is homogeneous, we suppose f0 ∈ α<κ Fα . For n ∈ N, α < κ, let Uα(n) =
f −1
1 1 − , : f ∈ Fα , n n
Un =
Uα(n) .
α<κ
Then Un is an open c-cover of X. Indeed, let C ∈ C(X), then V (f0 , C, n1 ) is a neighborhood of f0 . By (n) f0 ∈ α<κ Fα , there exists α < κ such that fα ∈ V (f0 , C, n1 ) ∩ Fα . So C ⊂ fα−1 ((− n1 , n1 )) ∈ Uα . For (n) n ∈ N, since X satisfies the property Tc (τ ), there exists αn < κ such that Uαn is an open c-cover of X. Let (n) β = sup{αn : n ∈ N}, then β < κ since κ is a regular cardinal. For n ∈ N, Uβ is an open c-cover. We claim f0 ∈ Fβ . Let V (f0 , C , ε) is a neighborhood of f0 , take n ∈ N such that n1 < ε. Since Uβ is an open c-cover of X, there exists f ∈ Fβ such that C ⊂ f −1 ((− n1 , n1 )). So f ∈ Fβ ∩ V (f0 , C , ε), hence f0 ∈ Fβ = Fβ . Thus f0 ∈ α<κ Fα . So {Fα : α < κ} is closed in Cc (X). 2 (n)
The set-tightness ts (X) of a space X [3,6,10] is defined by ts (X) = ω + min τ : if A ⊂ X, x ∈ A − A, there is {Aα : Aα ⊂ A, α < τ } such that for any α < τ, then x ∈ / Aα and x ∈
{Aα : α < τ } .
Definition 3.8. Let U be a collection of cozero subsets of a space X, and Z(U) = {Z(U ): U ∈ U} be a collection of zero-sets of a space of X indexed by U. (U, Z(U)) is called a tc -pair if (1) for every U ∈ U, Z(U ) ⊂ U = X; (2) Z(U) is a c-cover of X. Definition 3.9. A space X satisfies property tc (τ ) if for every tc -pair (U, Z(U)), there exists Uα ⊂ U(α < τ ) such that Z(Uα ) is not a c-cover of X, but α<τ Uα is a c-cover of X. Theorem 3.10. For a space X, the followings are equivalent: (1) ts (Cc (X)) τ ; (2) X satisfies the property tc (τ ). Proof. (1) ⇒ (2) Let ts (Cc (X)) τ , (U, Z(U)) is a tc -pair of X. For each U ∈ U, since Z(U ) is a zero-set, there exists f1 ∈ Cc (X), such that Z(U ) = f1−1 (0). Since U is a cozero set, there exists f2 ∈ Cc (X) such f1 (x) , then fU ∈ C(X) and fU (Z(U )) = 0, fU (X − U ) = 1. Set that X − U = f2−1 (0). Let fU (x) = f1 (x)+f 2 (x) F = {fU : U ∈ U}, then f0 ∈ F − F . By ts (Cc (X)) τ , there exists {Uα : Uα ⊂ U, α < τ } such that / {fU : U ∈ Uα }, and f0 ∈ {fU : U ∈ α<τ Uα }. If Z(Uα ) is a c-cover of X, for any α < τ , we have f0 ∈
Z. Li / Topology and its Applications 161 (2014) 321–329
329
then f0 ∈ {fU : U ∈ Uα }. This is a contradiction. So each Z(Uα ) is not a c-cover of X. Let C ∈ C(X), then V (f , C, 1) is a neighborhood of f0 , there exists U ∈ α<τ Uα such that fU ∈ V (f0 , C, 1). So C ⊂ U , thus 0 α<τ Uα is a c-cover of X. (2) ⇒ (1) Let X be of the property tc (τ ), F ⊂ Cc (X), g ∈ F − F . Since Ck (X) is homogeneous, we can take g = f0 ≡ 0. For each f ∈ F , n ∈ N, let 1 1 − , ; =f n n (n) Un = Uf : f ∈ F ;
(n) Uf
−1
1 1 − , =f n+1 n+1 (n) Zn = Zf : f ∈ F . (n) Zf
−1
;
Let M = {n ∈ N: X ∈ Un }. If M is infinite, then {fn }n∈M ⊂ F uniformly converges f0 . So {fn }n∈M converges f0 in Cc (X). Let Fn = {fn }, then f0 ∈ / Fn , and f0 ∈ n∈M Fn . Thus ts (Cc (X)) ω τ . Without a loss of generality, suppose M = ∅. Then for any n ∈ N, we have X ∈ / Un . For each C ⊂ C(X), a basic 1 1 1 1 , n+1 ) ⊂ open neighborhood V (f0 , C, n+1 ) of f0 , there exists f ∈ V (f0 , C, n+1 ) ∩ F . So f (C) ⊂ (− n+1 1 1 1 1 [− n+1 , n+1 ], thus C ⊂ f −1 ([− n+1 , n+1 ]) = Zf . Obviously, Zf (n)
a tc -pair of X. By (2), there exists (n)
(n) {Fα
(n)
(n)
(n)
⊂ Uf , and Uf
= X. Hence (Un , Zn ) is
⊂ F : α < τ } such that
(n)
(a) {Zf : f ∈ Fα } is not a c-cover of X. (n) (n) (b) {Uf : f ∈ α<τ Fα } is a c-cover of X. (n)
(n)
The condition (a) means f0 ∈ / Fα . Indeed, there exists C ∈ C(X), such that for any n ∈ N, f ∈ Fα , (n) (n) (n) 1 ) ∩ Fα = ∅. The condition (b) means f0 ∈ {Fα : α < τ, n ∈ N}. we have C ⊂ Zf . Then V (f0 , C, n+1 (n) (n) Indeed, for any C ∈ C(X), ε > 0, there exists n ∈ N such that n1 < ε. Since {Uf : f ∈ α<τ Fα } is a (n) (n) (n) c-cover of X, there exists f ∈ α<τ Fα such that C ⊂ Uf . So f ∈ V (f0 , C, n1 ) ∩ ( α<τ Fα ). Hence ts (Cc (X)) τ . 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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