@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 2, 2003

pp. 255–261

Closure properties of function spaces

Ljubǐsa D.R. Kočinac
∗

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 X endowed with the compact-open topology.

2000 AMS Classification: 54A25, 54C35, 54D20, 91A44.
Keywords: Menger property, Rothberger property, Hurewicz property, Rezni-
chenko property, k-cover, countable fan tightness, countable strong fan tight-
ness, 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 X in 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, . . . ,Kn are compact subsets of X and V1 . . . ,Vn are open in R. For
a function f ∈ Ck(X), a compact set K ⊂ X and a positive real number ε we
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 K is a compact subset of X and ε is a positive real number.

The symbol 0 denotes the constantly zero function in Ck(X). The space
Ck(X) is homogeneous so that we may consider the point 0 when studying
local properties of Ck(X).

Many results in the literature show that for a Tychonoff space X closure
properties of the function space Cp(X) of continuous real-valued functions on

∗Supported by the Serbian MSTD, grant 1233



256 Ljubǐsa D.R. Kočinac

X endowed 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 X have the Lindelöf property ([1]).

(B) [Arhangel’skii] Cp(X) has countable fan tightness if and only if all
finite powers of X have the Menger property ([1]).

(C) [Sakai] Cp(X) has countable strong fan tightness if and only if all finite
powers of X have the Rothberger property ([13]).

(D) [Kočinac–Scheepers] Cp(X) has countable fan tightness and the Rezni-
chenko property if and only if all finite powers of X have 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 U of X is called a k-cover if for each compact
set K ⊂ X there is a U ∈ U such that K ⊂ U. The symbol K denotes 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 U of X there 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 S be an infinite set and let A and 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 A there 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 A there is
a sequence (Bn : n ∈ N) of finite sets such that for each n
Bn ⊂ An, and

⋃
n∈N Bn is an element of B.

For a topological space X, let O denote 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

⋃
n∈N Bn is 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 X a space and a point x ∈ X the symbol Ωx denotes the set {A ⊂
X \{x} : x ∈ A}.

A space X has countable fan tightness [1] if for each x ∈ X and each sequence
(An : n ∈ N) of elements of Ωx there is a sequence (Bn : n ∈ N) of finite sets
such that for each n Bn ⊂ An and x ∈

⋃
n∈N Bn, i.e. if Sfin(Ωx, Ωx) holds

for each x ∈ X. A space X has countable strong fan tightness [13] if for each
x ∈ X the 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 X the following are equivalent:
(a) Ck(X) has countable strong fan tightness;
(b) X has property S1(K,K).

Proof. (a) ⇒ (b): Let (Un : n ∈ N) be a sequence of k-covers of X. For a
fixed n ∈ N and a compact subset K of X let Un,K := {U ∈Un : K ⊂ U}. For
each U ∈ Un,K let fK,U be a continuous function from X into [0,1] such that
fK,U (K) = {0} and fK,U (X \U) = {1}. Let for each n,

An = {fK,U : K compact in X,U ∈Un,K}.

Then, as it is easily verified, 0 is in the closure of An for 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 ∈ An and 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 C be a compact subset of X. From 0 ∈{fKn,Un : n ∈ N} it follows that
there is i ∈ N such that W = W(0; C; 1) contains the function fKi,Ui. Then
C ⊂ Ui. Otherwise, for some x ∈ C one has x /∈ Ui so 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 ⊂ X the neighborhood
W = W(0; K; 1/n) of 0 intersects An so 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 =

⋃
x∈KOx ⊃ K we

have fK,n(UK,n) ⊂ (−1/n, 1/n). Let Un = {UK,n : K a compact subset of X}.
For each n ∈ N, Un is a k-cover of X. Apply that X is an S1(K,K)-set: for each



258 Ljubǐsa D.R. Kočinac

m ∈ N there 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
0 in Ck(X) and let m be a natural number such that 1/m < ε. Since C is a
compact subset of X and X is an S1(K,K)-set, there is j ∈ N, j ≥ m such 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. Juhász at the IV Interna-
tional Conference “Topology and its Applications”, Dubrovnik, September 30
– October 5, 1985 (see [4]). A space X has countable T -tightness if for each
uncountable regular cardinal ρ and each increasing sequence (Aα : α < ρ) of
closed subsets of X the 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 X the following are equivalent:
(a) Ck(X) has countable T -tightness;
(b) for each regular cardinal ρ and each increasing sequence (Uα : α < ρ)

of families of open subsets of X such that
⋃
α<ρUα is a k-cover of X

there is a β < ρ so that Uβ is a k-cover of X.

Proof. (a) ⇒ (b): Let ρ be a regular uncountable cardinal and let (Uα : α < ρ)
be an increasing sequence of families of open subsets of X such that

⋃
α<ρUα

is a k-cover for X. For each α < ρ and each compact set K ⊂ X let Uα,K :=
{U ∈Uα : K ⊂ U}. For each U ∈Uα,K let fK,U be a continuous function from
X into [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 =

⋃
α<ρ Aα is 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 0 intersects some
of the sets Aα, α < ρ, which means that 0 belongs to the closure of the set⋃
α<ρ Aα. Since this set is actually the set A it follows there exists a β < ρ

with 0 ∈ Aβ. We claim that the corresponding family Uβ is a k-cover of X.
Let C be 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 ρ a regular uncountable cardinal. We shall prove that the set
A :=

⋃
α<ρ Aα is closed. Let f ∈ A. For each n ∈ N and each compact set

K ⊂ X the neighborhood W(f; K; 1/n) of f intersects 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, Un is a k-cover of X. Let K be a compact
subset of X. The neighborhood W := W(f; K; 1/n) of f intersects A, i.e.
there is g ∈ A such 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 ⊂Un which is a k-cover of X. Put β0 = sup{βn : n ∈
N}. Since ρ is a regular cardinal, β0 < ρ. It is easy to verify that for each n the
set Un,β0 is a k-cover of X. Let us show that f ∈ Aβ0 . Take a neighborhood
W(f; C; ε) of f and let m be a positive integer such that 1/m < ε. Since Um,β0
is a k-cover of X one can find g ∈ Aβ0 such that C ⊂ g←(−1/m, 1/m). Then
g ∈ W(f; C; 1/m) ∩ Aβ0 ⊂ W(f; C; ε) ∩ Aβ0 , i.e. f ∈ Aβ0 = Aβ0 and thus
f ∈ A. So, A is closed. �

4. The Reznichenko property of Ck(X).

In this section we shall need the notion of groupability (see [7]).

1. A k-cover U of a space X is groupable if there is a partition (Un : n ∈ N)
of U into pairwise disjoint finite sets such that: For each compact subset
K of X, for all but finitely many n, there is a U ∈Un such that K ⊂ U.

2. An element A of Ωx is groupable if there is a partition (An : n ∈ N)
of A into 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;
• Ωgpx – 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 Ωx is a member of Ωgpx .
This property was further studied in [6] and [7] (see Introduction). In [6] it was
called the Reznichenko property at x. When X has the Reznichenko property
at each of its points, then X is said to have the Reznichenko property.

We study now the Reznichenko property in spaces Ck(X).

Theorem 4.1. Let X be a Tychonoff space. If ONE has no winning strategy
in the game G1(K,Kgp), then Ck(X) has property S1(Ω0, Ω

gp
0 ) (i.e. Ck(X) has

countable strong fan tightness and the Reznichenko property).



260 Ljubǐsa D.R. Kočinac

Proof. Evidently, from the fact that ONE has no winning strategy in the game
G1(K,Kgp), it follows that X satisfies S1(K,Kgr) and consequently X is 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 Ω0 is groupable (i.e. that Ck(X) has the Reznichenko property).

Let A be a countable subset of Ck(X) such that 0 ∈ A. We define the
following strategy σ for ONE in G1(K,Kgp). For a compact set K ⊂ X the
neighborhood W = W(0; K; 1) of 0 intersects A. Let fK ∈ A ∩ W . As fK
is continuous, for every x ∈ K there is a neighborhood Ox of x such that
fK(Ox) ⊂ (−1, 1). From the open cover {Ox : x ∈ K} of K choose a finite
subcover {Ox1, . . . ,Oxm} and let UK = Ox1∪·· ·∪Oxm. Then fK(UK) ⊂ (−1, 1)
and the set U1 = {UK : K a 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 K of X ONE chooses a function fK ∈ A ∩
W(0; K; 1/2) and a neighborhood UK of K such that fK(UK) ⊂ (−1/2, 1/2).
The set U2 = {UK : K a compact subset of X}\{UK1} is a k-cover of X. ONE
plays σ(UK1 ) = U2. Suppose that UK2 ∈ U2 is TWO’s response. ONE first
considers the function fK2 ∈ A1 with 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 X and for each n Un =
σ(U1, . . . ,Un−1);

(ii) For each n, Un ∈Un and Un /∈{U1, . . . ,Un−1)};
(iii) For each n, fn is a member of A\{f1, . . . ,fn−1);
(iv) For each n, fn(Un) ⊂ (−1/n, 1/n).

Since σ is 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 ⊂ X there is k0 such that for each k > k0 there is H ∈Hk
with K ⊂ H. Define also Mk := {fi : nk ≤ i < nk+1}. Then the sets Mk are
finite pairwise disjoint subsets of A. One can also suppose that A =

⋃
k∈N Mk;

otherwise we distribute countably many elements of A\
⋃
k∈N Mk among Mk’s

so that after distribution new sets are still finite and pairwise disjoint. We
claim that the sequence (Mk : k ∈ N) witnesses that A is groupable (i.e. that
Ck(X) has the Reznichenko property).

Let W(0; K; ε) be a neighborhood of 0 and let m be the smallest natural
number such that 1/m < ε. There is n0 such that for each n > n0 one can
choose an element Hn ∈Hn with K ⊂ Hn; choose also a corresponding function
fn ∈ Mn satisfying 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)∩Mn 6= ∅.
The theorem is shown. �

In a similar way one can prove

Theorem 4.2. For a space X the 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, Ω
gp
0 )).

Problem 4.3. Is the converse in Theorem 4.1 and in Theorem 4.2 true?

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Received November 2001

Revised November 2002

Ljubǐsa D.R. Kočinac

Faculty of Sciences, University of Nǐs, 18000 Nǐs, Serbia

E-mail address : lkocinac@ptt.yu