LinAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 1, 2006

pp. 103-107

Tightness of function spaces

Shou Lin
∗

Abstract. The purpose of this paper is to give higher cardinality
versions of countable fan tightness of function spaces obtained by A.
Arhangel’skǐı. Let vet(X), ωH(X) and H(X) denote respectively the
fan tightness, ω-Hurewicz number and Hurewicz number of a space X,
then vet(C

p
(X)) = ωH(X) = sup{H(Xn) : n ∈ N}.

2000 AMS Classification: 54C35; 54A25; 54D20; 54D99.

Keywords: Function spaces; fan tightness; Hurewicz spaces; cardinal func-
tions.

The general question in the theory of function spaces is to characterize topo-
logical properties of the space, C(X), of continuous real-valued functions on a
topological space X. A study of some convergence properties in function spaces
is an important task of general topology. It have been obtained interested
results on some higher cardinal properties of first-countability, Fréchet prop-
erties, tightness[2, 4, 6, 9]. Arhangel’skǐı-Pytkeev theorem[2] is a nice result
about tightness of function spaces: t(Cp(X)) = sup{L(X

n) : n ∈ N} for any
Tychonoff space X. The following result on countable fan tightness of function
spaces is shown by A. Arhangel’skǐı[1]: Cp(X) has countable fan tightness if
and only if X n is a Hurewicz space for each n ∈ N for an arbitrary space X. In
this paper the higher cardinality versions of countable fan tightness of Cp(X)
are obtained.

In this paper all spaces will be Tychonoff spaces. Let α be a network of
compact subsets of a space X, which is closed under finite unions and closed
subsets. Then the space Cα(X) is the set C(X) with the set-open topology as
follows[9]: The subbasic open sets of the form [A, V ] = {f ∈ C(X) : f (A) ⊂ V },
where A ∈ α and V is open in R. Then Cα(X) is a topological vector space[9].
The family of all compact subsets of X generates the compact-open topology,

∗Supported by the NNSF of China (10271026).



104 S. Lin

denoted by Ck(X). Also the family of all finite subsets of X generates the
topology of pointwise convergence, denoted by Cp(X). For each f ∈ C(X), a
basic neighborhood of f in Cp(X) can be expressed as W (f, K, ε) for each finite
subset K of X and ε > 0, here W (f, K, ε) = {g ∈ C(X) : |f (x) − g(x)| < ε for
each x ∈ K}. In this paper the alphabet λ is an infinite cardinal number, γ is
an ordinal number, and i, m, n, j, k are natural numbers.

The fan tightness of a space X is defined by vet(X) = sup{vet(X, x) : x ∈
X}, here vet(X, x) = ω + min{λ : for each family {Aγ}γ<λ of subsets of X

with x ∈
⋂

γ<λ
Aγ there is a subset Bγ ⊂ Aγ with |Bγ| < λ for each γ < λ

such that x ∈
⋃

γ<λ
Bγ}. A space X has countable fan tightness[1] if and only

if vet(X) = ω. An α-cover of a space X is a family of subsets of X such that
every member of α is contained in some member of this family. An α-cover is
called a k-cover if α is the set of all compact subsets of X. Also an α-cover
is called an ω-cover if α is the set of all finite subsets of X. The α-Hurewicz
number of X is defined by αH(X) = ω + min{λ : for each family {Uγ}γ<λ of
open α-covers of X there is a subset Bγ ⊂ Uγ with |Bγ| < λ for each γ < λ
such that

⋃
γ<λ

Bγ is an α-cover of X}. The α-Hurewicz number of X is called

the Hurewicz number of X and written H(X) if α consists of the singleton of
X. A space X is Hurewicz space[5] if and only if H(X) = ω.

Theorem 1. vet(Cα(X)) = αH(X) for any space X.

Proof. Let λ = vet(C
α
(X)), and let {Uγ}γ<λ be any family of open α-covers

of X. For each γ < λ, put Aγ = {f ∈ Cα(X) : there is U ∈ Uγ such that
f (X \ U ) ⊂ {0}}. Then Aγ is dense in Cα(X). In fact, let

⋂
i≤m[Ki, Vi] be a

non-empty basic open set of Cα(X), fix f ∈
⋂

i≤m[Ki, Vi]. There is U ∈ Uγ such

that
⋃

i≤m Ki ⊂ U because Uγ is an α-cover on X. Since
⋃

i≤m Ki is compact

in Tychonoff space X, there is g ∈ Cα(X) such that g|∪i≤mKi = f|∪i≤mKi and

g(X \ U ) ⊂ {0}. Then g ∈ Aγ ∩ (
⋂

i≤m[Ki, Vi]), and Aγ = Cα(X).

Take f1 ∈ C(X) with f1(X) = {1}, then f1 ∈
⋂

γ<λ
Aγ . For each γ < λ

there is a subset Bγ ⊂ Aγ with |Bγ| < λ such that f1 ∈
⋃

γ<λ
Bγ by λ =

vet(C
α
(X)). Denote Bγ = {fκ}κ∈Φγ , here |Φγ| < λ. There is Uκ ∈ Uγ such

that fκ(X \ Uκ) ⊂ {0} for each κ ∈ Φγ . Put U
′
γ = {Uκ}κ∈Φγ . Then

⋃
γ<λ

U
′

γ is

an α-cover of X. In fact, for each A ∈ α, since f1 ∈ [A, (0, 2)], there are γ < λ
and κ ∈ Φγ such that fκ ∈ [A, (0, 2)], then A ⊂ Uκ, so

⋃
γ<λ

U′γ is an α-cover

of X. This shows that αH(X) ≤ vet(C
α
(X)).

To show the reverse inequality, let λ = αH(X). Since Cα(X) is a topological
vector space, it is homogeneous. It suffices to show that vet(Cα(X), f0) ≤ λ,
here f0 ∈ C(X) with f0(X) = {0}. Suppose that f0 ∈

⋂
γ<λ

Aγ with each

Aγ ⊂ Cα(X). For each γ < λ and n ∈ N, put Uγ,n = {f
−1(On) : f ∈ Aγ}, here

{On}n∈N is a decreasing local base of 0 in R. Then Uγ,n is an open α-cover
of X. In fact, for each A ∈ α, f0 ∈ [A, On], there is f ∈ [A, On] ∩ Aγ , thus
A ⊂ f −1(On) ∈ Uγ,n.



Tightness of function spaces 105

Case 1. λ > ω. For each n ∈ N, since {Uγ,n}γ<λ is a family of open α-
covers of X, there is a subset U′γ,n ⊂ Uγ,n with |U

′
γ,n| < λ for each γ < λ such

that
⋃

γ<λ
U′γ,n is an open α-cover of X. Denote U

′
γ,n = {Uτ }τ ∈Φγ,n . There

is fτ ∈ Aγ such that Uτ = f
−1
τ (On) for each τ ∈ Φγ,n. Let Bγ = {fτ : τ ∈

Φγ,n, n ∈ N}. Then Bγ ⊂ Aγ and |Bγ| < λ. We show that f0 ∈
⋃

γ<λ
Bγ .

For arbitrary basic neighborhood [A, V ] of f0 in Cα(X), there is n ∈ N such
that On ⊂ V . Since

⋃
γ<λ

U′γ,n is an open α-cover of X, there are γ < λ and

τ ∈ Φγ,n such that A ⊂ Uτ = f
−1
τ (On), hence fτ (A) ⊂ V , i.e., fτ ∈ [A, V ], so

f0 ∈ {fτ : τ ∈ Φγ,n, n ∈ N, γ < λ} =
⋃

γ<λ
Bγ .

Case 2. λ = ω. Put M = {n ∈ N : X ∈ Un,n}. If M is infinite, there
is m ∈ M such that Om ⊂ V for arbitrary basic neighborhood [A, V ] of f0 in
Cα(X). By the definition of Um,m, there is gm ∈ Am such that X = g

−1
m (Om),

then gm(X) ⊂ V , so gm ∈ [A, V ], thus the sequence {gm}m∈M converges to
f0. If M is finite, there is n0 ∈ N such that for each m ≥ n0 and g ∈ Am,
g−1(Om) 6= X. Since {Um,m}m≥n0 is a sequence of open α-covers of X, there
is a finite subset U′m of Um,m for each m ≥ n0 such that

⋃
m≥n0

U′m is an

open α-cover of X. Denote U′m = {Um,j}j≤i(m). There is fm,j ∈ Am such

that Um,j = f
−1
m,j

(Om) for each m ≥ n0, j ≤ i(m). Next, we shall show that

f0 ∈ {fm,j : m ≥ n0, j ≤ i(m)}. For arbitrary basic neighborhood [A, V ] of f0
in Cα(X), let F = {(m, j) ∈ N

2 : m ≥ n0, j ≤ i(m) and A ⊂ Um,j}. Obviously,
F 6= ∅. If F is finite, take xm,j ∈ X \ Um,j for each (m, j) ∈ F because
Um,j 6= X. There is K ∈ α with A ∪ {xm,j : (m, j) ∈ F } ⊂ K. Then K is not
contained by any element of

⋃
m≥n0

U′m, so
⋃

m≥n0
U′m is not an α-cover of X,

a contradiction. Hence F is infinite, and there are m ≥ n0 and j ≤ i(m) such
that A ⊂ Um,j = f

−1
m,j(Om) and Om ⊂ V , so fm,j (A) ⊂ V , i.e., fm,j ∈ [A, V ].

Thus f0 ∈ {fm,j : m ≥ n0, j ≤ i(m)}.
This shows that vet(Cα(X)) ≤ αH(X). �

By Theorem 1, Cp(X) has countable fan tightness if and only if for each
sequence {Un} of open ω-covers of X there is a finite subset U

′
n ⊂ Un for each

n ∈ N such that
⋃

n∈N U
′
n is an ω-cover of X.

Theorem 2. vet(Cp(X)) = sup{H(X
n) : n ∈ N} for any space X.

Proof. Let λ = vet(C
p
(X)) and n ∈ N. Suppose that {Uγ}γ<λ is a family of

open covers of the space X n. For each γ < λ, a family V of subsets of X is
called having a property Pn,γ if for each {Vi}i≤n ⊂ V there is U ∈ Uγ such that∏

i≤n Vi ⊂ U . Denote by Γn,γ the family of the all finite sets, which has the

property Pn,γ , of open sets in X. For each V ∈ Γn,γ , let FV = {f ∈ Cp(X) :
f (X \

⋃
V) ⊂ {0}}. We show that the set Aγ =

⋃
{FV : V ∈ Γn,γ} is dense in

Cp(X).
Let W (f, K, ε) be any basic neighborhood of f in Cp(X). Since K is

finite, there is a finite family W of open subsets in X such that for any
(x1, x2, ..., xn) ∈ K

n there are U ∈ Uγ and a finite subset {Wi}i≤n ⊂ W
such that (x1, x2, ..., xn) ∈

∏
i≤n Wi ⊂ U . Then K ⊂

⋃
W. For each x ∈ K,



106 S. Lin

put Vx =
⋂
{W ∈ W : x ∈ W }, and V = {Vx : x ∈ K}. Then K ⊂

⋃
V and the

family V has the property Pn,γ . In fact, take an arbitrary (x1, x2, ..., xn) ∈ K
n,

there are {Wi}i≤n ⊂ W and U ∈ U such that (x1, x2, ..., xn) ∈
∏

i≤n Wi ⊂ U .

Since each Vxi ⊂ Wi,
∏

i≤n Vxi ⊂ U . Now, take g ∈ Cp(X) such that f|K = g|K
and g(X \

⋃
V) = {0}, then g ∈ FV ⊂ Aγ , so W (f, K, ε) ∩ Aγ 6= ∅. Thus

Aγ = Cp(X).

Let f1 ∈ C(X) with f1(X) = {1}. Then f1 ∈
⋂

γ<λ
Aγ . There is a subset

Bγ ⊂ Aγ with |Bγ| < λ for each γ < λ such that f1 ∈
⋃

γ<λ
Bγ . Then there

is a subset ∆n,γ ⊂ Γn,γ with |∆n,γ| < λ such that Bγ ⊂
⋃
{FV : V ∈ ∆n,γ}.

Let V ∈ ∆n,γ . For each ξ = (V1, V2, ..., Vn) ∈ V
n, take Gξ ∈ Uγ such that∏

i≤n Vi ⊂ Gξ. Put Gγ = {Gξ : ξ ∈ V
n, V ∈ ∆n,γ}. Clearly, |Gγ| < λ and

Gγ ⊂ Uγ . We show that
⋃

γ<λ
Gγ covers X.

For an arbitrary (x1, x2, ..., xn) ∈ X
n, let F = {f ∈ Cp(X) : f (xi) > 0

for each i ≤ n}. Then F is an open neighborhood of f1 in Cp(X). Since

f1 ∈
⋃

γ<λ
Bγ , there is γ < λ such that F ∩ Bγ 6= ∅. Then F ∩ FV 6= ∅ for

some V ∈ ∆n,γ . Let g ∈ F ∩ FV . Then g(X \
⋃
V) = 0 and g(xi) > 0 for each

i ≤ n. Take Vi ∈ V such that xi ∈ Vi for each i ≤ n, then there is Gξ ∈ Gγ such
that (x1, x2, ..., xn) ∈

∏
i≤n Vi ⊂ Gξ. So (x1, x2, ..., xn) ∈

⋃
(
⋃

γ<λ
Gγ ). Hence

H(X n) ≤ vet(C
p
(X)).

Conversely, suppose λ = sup{H(X n) : n ∈ N}. Fix f ∈ Cp(X) and a family

{Aγ}γ<λ of subsets in Cp(X) such that f ∈
⋂

γ<λ
Aγ . For each n ∈ N, γ < λ

and x = (x1, x2, ..., xn) ∈ X
n, there is gx,γ ∈ W (f, {x1, x2, ..., xn}, 1/n)

⋂
Aγ .

For each i ≤ n, since |gx,γ (xi) − f (xi)| < 1/n, by the continuity of f and gx,γ,
there is an open neighborhood Oi of xi in X such that |gx,γ (yi) − f (yi)| < 1/n
if yi ∈ Oi. The set Ux,γ =

∏
i≤n Oi is a neighborhood of x in X

n. Thus

Un,γ = {Ux,γ : x ∈ X
n} covers X n, and |gx,γ (yi) − f (yi)| < 1/n for each

(y1, y2, ..., yn) ∈ Ux,γ .
Case 1. λ > ω. Since H(X n) ≤ λ, there is a family {Sn,γ}γ<λ of subsets in

X n with |Sn,γ| < λ for each γ < λ such that
⋃

γ<λ
Sn,γ covers X

n, here each

Sn,γ = {Ux,γ : x ∈ Sn,γ}. For each γ < λ, let Bn,γ = {gx,γ : x ∈ Sn,γ}, and

Bγ =
⋃

n∈N Bn,γ . Then Bγ ⊂ Aγ , |Bγ| < λ, and f ∈
⋃

γ<λ
Bγ .

In fact, let W (f, {y1, y2, ..., yn}, ε) be a basic neighborhood of f in Cp(X)
with 1/n < ε. There is γ < λ such that (y1, y2, ..., yn) ∈

⋃
Sn,γ , thus there is

x ∈ Sn,γ such that (y1, y2, .., yn) ∈ Ux,γ, so gx,γ ∈ Bn,γ and |gx,γ (yi) − f (yi)| <
1/n < ε for each i ≤ n, hence gx,γ ∈ W (f, {y1, y2, ..., yn}, ε) ∩ Bγ . This shows

that f ∈
⋃

γ<λ
Bγ .

Case 2. λ = ω. Replace γ < λ by k ≥ n, and let Bk =
⋃

n≤k Bn,k in the

proof of Case 1, then Bk is finite subset of Ak and f ∈
⋃

k∈N Bk.
In a word, vet(C

p
(X)) ≤ sup{H(X n) : n ∈ N}. �

The following result obtained by A. Arhangel’skǐı[1] is generalized: Cp(X)
has countable fan tightness if and only if X n is a Hurewicz space for each n ∈ N.



Tightness of function spaces 107

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[5] W. Hurewicz, Über folgen stetiger fukktionen, Fund. Math. 9(1927), 193-204.
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Received August 2004

Accepted May 2005

Shou Lin (linshou@public.ndptt.fj.cn)
Department of Mathematics, Zhangzhou Teachers’ College, Fujian 363000, P.
R. China
Department of Mathematics, Ningde Teachers’ College, Fujian 352100,P. R.
China