MATEMATIQKI VESNIK

58 (2006), 57–70

UDK 515.124

originalni nauqni rad research paper

OPEN COVERS AND FUNCTION SPACES

B. A. Pansera and V. Pavlović

Abstract. We investigate some closure properties of the space C(X) of the continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology and the pointwise convergence topology.

1. Introduction

In this paper we use the standard topological notation and terminology as in [3]. All spaces are assumed to be infinite Tychonoff. Let X be a topological space. Then:

O denotes the collection of all open covers of X;

Ω denotes the collection of all open ω-covers of X. A cover U of a set Y is called an ω-cover if Y is not a member of U and every finite subset of Y is contained in a member of U [4];

K denotes the collection of all open k-covers of X. A cover U of a space X is called a k-cover if X is not a member of U and every compact subset of X is contained in a member of U [2].

A space X is called a k-Lindelöf space if for each open k-cover U of X there is a V ⊆ U such that V is countable and V ∈ K. Each k-Lindelöf space is Lindelöf, so normal, too.

For a space X and a point x ∈ X, the symbol Ω_{x} denotes the set {A ⊆ X \{x} : x ∈ \overline{A}\}. Let A and B be two arbitrary sets (usually collections of open covers of a topological space X). The symbol S_{1}(A, B) [9] denotes the selection principle:

For each sequence (A_{n} : n ∈ N) of elements of A there exists a sequence (b_{n} : n ∈ N) such that, for each n, b_{n} ∈ A_{n} and {b_{n} : n ∈ N} is an element of B.

AMS Subject Classification: 54 C 35, 54 D 20 Keywords and phrases: Reznichenko property, Pytkeev property, k-cover, countable (strong)

fan tightness, groupability, compact-open topology, pointwise convergence topology, selection principles.

The symbol S_{fin}(A, B) [9] denotes the selection principle:

For each sequence (A_{n} : n ∈ N) of elements of A there exists a sequence (B_{n} : n ∈ N) such that, for each n, B_{n} is a finite subset of A_{n} and ∪_{n∈N}B_{n} is an element of B.

Then the property S_{1}(O, O) is called the Rothberger property [5,17], and the property S_{fin}(O, O) is known as the Menger property [5,17].

For a space X by C(X) we denote the set of all continuous real-valued functions defined on X. The symbol 0 denotes the constant to zero function defined on X. Then C_{p}(X) is the set C(X) endowed with the topology of pointwise convergence (the “pc-topology” for short). Typical basic open sets of C(X) are of the form:

W (x_{1}, …, x_{k}; U_{1}, …, U_{k}) = {f ∈ C_{p}(X) : f(x_{i}) ∈ U_{i}, i = 1, …, k}

where x_{1}, …, x_{k} are points of X and U_{1}, …, U_{k} are open sets of R. For a subset S of X and a positive real number ε we let

O(S, ε) = {g ∈ C(X) : |g(x)| < ε, for all x ∈ S}.

The standard local base at the point 0 consists of the sets O(F, ε), where F is a finite subset of X and ε is a positive real number.

By C_{k}(X) we denote the set C(X) endowed with the compact-open topology (the “co-topology” for short). Typical basic open sets of C_{k}(X) are of the form:

W (K_{1}, …, K_{n}; U_{1}, …, U_{n}) = {f ∈ C(X) : f(K_{i}) ⊂ U_{i}, i = 1, …, n}

where K_{1}, …, K_{n} are compact subsets of X and U_{1}, …, U_{n} are open sets of R. The standard local base at the point 0 ∈ C_{k}(X) consists of the sets O(K, ε), where K is a compact subset of X and ε is a positive real number.

Since C_{p}(X) and C_{k}(X) are homogenous spaces we may always consider the point 0 when studying local properties of these spaces. Since we are considering two topologies of C(X) we shall use the symbol (Ω_{0})^{p} to denote Ω_{0} in the space C_{p}(X) and the symbol (Ω_{0})^{k} to denote Ω_{0} in the space C_{k}(X). Many results in the literature show that for a Tychonoff space X closure properties of function spaces C_{p}(X) and C_{k}(X) can be characterized by covering properties of X [1,9].

By standard techniques of switching between k-covers of X and subsets of C_{k}(X) one can easily see that the following holds.

Lemma 1.1.C_{k}(X) has countable tightness iff X is k-Lindelöf.

In the first section of the paper we investigate the selection principles S_{1}(K, Ω) and S_{fin}(K, Ω) and their relations with function spaces. In the remaining two sections we study a bitopological variant of the Pytkeev property and of the Reznichenko property in function spaces.

2. The selection principles S_{1}(K, Ω) and S_{fin}(K, Ω)

Note that the following relations between classes of covers defined above hold:

K ⊆ Ω ⊆ O,

hence we have:

S_{1}(K, K) ⊆ S_{1}(K, Ω);
S_{fin}(K, K) ⊆ S_{fin}(K, Ω).

The next two lemmas follow from the definition of k-covers.

Lemma 2.1. If a k-cover U of a space X is the union of finitely many subfamilies, then at least one of them is also a k-cover of X.

Lemma 2.2. If U is a k-cover of a space X, then each compact subset of X is contained in infinitely many elements of U. For any finite set S the family U \ S is also a k-cover.

In [2] the following was proved:

Lemma 2.3. If U is a k-cover of a space X^{n}, then there is a k-cover V of X such that {V^{n} : V ∈ V} refines U.

Theorem 2.1. A space X satisfies S_{1}(K, Ω) if and only if each finite power of X satisfies S_{1}(K, Ω).

Proof. Let X belong to the class S_{1}(K, Ω) and let (W_{n} : n ∈ N) be a sequence of k-covers of X^{m}, for a fixed natural number m. By Lemma 2.3, for each natural number n there exists a k-cover U_{n} of X such that {U^{m} : U ∈ U_{n}} refines W_{n}. Apply the fact that X ∈ S_{1}(K, Ω) to the sequence (U_{n} : n ∈ N). There is a sequence (U_{n} : n ∈ N) such that, for each n, U_{n} ∈ U_{n} and {U_{n} : n ∈ N} is a k-cover of X. For each n, let W_{n} be an element of W_{n} with U^{m} ⊂ W_{n}. Then the sequence (W_{n} : n ∈ N) shows that X^{m} is in the class S_{1}(K, Ω). Let F be a finite subset of X^{m}. The union ⋃_{i≤m} p_{i}(F) = B of the projections onto X is a finite subset of X and thus there is an n ∈ N such that B ⊂ U_{n}. Hence F ⊆ B^{m} ⊆ U_{n}^{m} ⊆ W_{n}.

In a similar way one can prove that:

Theorem 2.2. A space X satisfies S_{fin}(K, Ω) if and only if each finite power of X satisfies S_{fin}(K, Ω).

Our intent in this section is to describe how these two principles affect the bitopological space (C_{k}(X), C_{p}(X)). Let us mention here that the principles S_{1}(Ω, K) and S_{fin}(Ω, K) have already been considered in [8] where Ramsey theoretical characterizations were established.

A space X has countable fan tightness [1] if for each x ∈ X and each sequence (A_{n} : n ∈ N) of elements of Ω_{x} there exists a sequence (B_{n} : n ∈ N) of finite sets such that, for each n, B_{n} ⊆ A_{n} and x ∈ \overline{∪_{n∈N}B_{n}}, i.e. if S_{fin}(Ω_{x}, Ω_{x}) holds for each x ∈ X.

A space X has countable strong fan tightness [16] if for each x ∈ X S_{1}(Ω_{x}, Ω_{x}) holds.

The following theorem was proved in [6].

Theorem 2.3. For a space X the following are equivalent:

(1) C_{k}(X) has countable strong fan tightness;

(2) X has the property S_{1}(K, K).

Theorem 2.4. For a space X the following are equivalent:

(1) C(X) satisfies S_{1}((Ω_{0})^{k}, (Ω_{0})^{p}));

(2) X has the property S_{1}(K, Ω).

Proof. (1) ⇒ (2) : Let (U_{n} : n ∈ N) be a sequence of k-covers of X. For each pair of a compact subset K of X and an open subset U ⊇ K of X let f_{K,U} be any continuous function from X to [0, 1] such that f_{K,U}(K) ⊆ {0} and f_{K,U}(X \ U) ⊆ {1}. For each n let A_{n} = {f_{K,U} : K compact in X, K ⊆ U ∈ U_{n}}. Then for each compact subset K of X there is a f_{K,U} ∈ A_{n} so, as it can easily be verified, 0 is in the closure of each A_{n}, with respect to the compact-open topology. Since C(X) satisfies S_{1}((Ω_{0})^{k}, (Ω_{0})^{p}) there is a sequence (f_{K_{n},U_{n}} : n ∈ N) such that for each n, K_{n} is compact, U_{n} ∈ U_{n} and 0 belongs to the closure of {f_{K_{n},U_{n}} : n ∈ N} with respect to the pointwise convergence topology. We claim that {U_{n} : n ∈ N} ∈ Ω. Let F be a finite subset of X. From the fact that 0 belongs to the closure of {f_{K_{n},U_{n}} : n ∈ N} with respect to the pointwise convergence topology it follows that there is an i ∈ N such that W = O(F, 1) contains the function f_{K_{i},U_{i}}. Then F ⊆ U_{i}. Otherwise for some x ∈ F one has x ∉ U_{i} so that f_{K_{i},U_{i}}(x) = 1, contradicting f_{K_{i},U_{i}} ∈ W.

(2) ⇒ (1) : Let (A_{m} : m ∈ N) be a sequence of subsets of C(X) \ {0} the closures of which all contain 0, with respect to the compact-open topology. If X is compact then the compact-open topology coincides with the topology of uniform convergence, so C_{k}(X) is metrizable, thus first countable, which means that we can find a sequence (a_{n} : n ∈ N), a_{n} ∈ A_{n}, converging uniformly to 0 so there is nothing to be proved. Let X be a noncompact space.

For a bijection i : N^{2} → N put A_{n,m} := A_{i(n,m)}.

For each n, m ∈ N and every compact set K ⊆ X the neighborhood W = O(K, \frac{1}{n}) of 0 intersects A_{m,n}, so there exists a continuous function f_{K,m,n} ∈ A_{m,n} such that |f_{K,m,n}(x)| < \frac{1}{n} for each x ∈ K. Since f_{K,m,n} is a continuous function there is an open U_{K,m,n} such that f_{K,m,n}(U_{K,m,n}) ⊆ (−\frac{1}{n}, \frac{1}{n}). Let U_{m,n} = {U_{K,m,n} : K is a compact subset of X}.

As for any compact subset K we have K ≠ X, it can easily be achieved that none of the sets U_{K,m,n} above coincides with X. So for each m and n, U_{m,n} is a k-cover of X. To each sequence (U_{m,n} : m ∈ N) apply the fact that X is an S_{1}(K, Ω)-space to obtain sequences (U_{K_{m,n},m,n} : m ∈ N), with each K_{m,n} compact, such that {U_{K_{m,n},m,n} : m ∈ N} ∈ Ω for every n ∈ N. Let us show that 0 belongs to the closure of {f_{K_{m,n},m,n} : m, n ∈ N} with respect to the pointwise convergence topology.

Let W = O(F, ε) be a neighborhood of 0 in C_{p}(X) and let n be a positive integer such that \frac{1}{n} < ε. Since F is a finite subset of X and \{U_{K_{m,n},m,n} : m ∈ N} ∈ Ω there is a m ∈ N such that F ⊆ U_{K_{m,n},m,n}. We have

f_{K_{m,n},m,n}(F) ⊆ f_{K_{m,n},m,n}(U_{K_{m,n},m,n}) ⊆ (−\frac{1}{n}, \frac{1}{n}) ⊂ (−ε, ε)

i.e. f_{K_{m,n},m,n} ∈ W. Since f_{K_{m,n},m,n} ∈ A_{m,n} this ends the proof of the theorem. In a similar way one can show that

Theorem 2.5. For a space X the following are equivalent:

(1) C(X) satisfies S_{fin}((Ω_{0})^{k}, (Ω_{0})^{p}));

(2) X has the property S_{fin}(K, Ω).

3. The Pytkeev-type properties

For a space X and x ∈ X, a family F of subsets of X is called a π-network at x if every neighborhood of x contains an element of F.

A space X is called a Pytkeev space [14] if x ∈ \overline{A} \ A and A ⊆ X imply the existence of a countable π-network at x consisting of infinite subsets of A.

An open ω-cover U is said to be ω-shrinkable [14] if there is a function C such that for each U ∈ U the set C(U) is closed, C(U) ⊆ U and {C(U) : U ∈ U} is an ω-cover of X.

In [14] the following theorem was proved: Theorem 3.1. The following are equivalent:

(1) C_{p}(X) is a Pytkeev space;

(2) If U is an ω-shrinkable open nontrivial ω-cover of X, there is a sequence (U_{n} : n ∈ N) of countably infinite subfamilies of U such that {∩U_{n} : n ∈ N} is an ω-cover of X.

Nontrivial means not containing the whole space as one of its elements.

When dealing with the space C_{k}(X) we will need the “compact” version of “shrinkability”. An open cover U is said to be k-shrinkable if there is a function C such that for each U ∈ U the set C(U) is closed, C(U) ⊆ U and {C(U) : U ∈ U} is a k-cover of X; the collection of all k-shrinkable open covers of X is denoted by K_{shr}. A family U of subsets of a space X is said to be a 3-k-shrinkable cover of X if there is a function g such that for each U ∈ U g(U) = (V_{U}, Z_{U}), where V_{U} ⊆ Z_{U} ⊆ U, V_{U} is a cozero set, Z_{U} is a zero set and {V_{U} : U ∈ U} k-covers X. U is called an open 3-k-shrinkable cover provided that its elements are open subsets of X. The collection of all nontrivial open 3-k-shrinkable covers of X will be denoted by 3-K_{shr}. Obviously one has 3-K_{shr} ⊆ K_{shr} ⊆ K.

In the sequel we will need the following two lemmas which are proved exactly as Lemma 1.2 and Lemma 1.4, respectively, in [12], mostly by replacing the word finite by the word compact.

Lemma 3.1. Every open k-cover can be refined by an open 3-k-shrinkable cover.

Lemma 3.2. Let X be a k-Lindelöf space. If ε > 0 and 0 ∈ \overline{A} ⊆ C_{k}(X) then there is a B ⊆ A and a function s : B → (0, ε) such that at least one of the families {|f|^{←}[0, s(f)) : f ∈ B} or {|f|^{←}[0, s(f)] : f ∈ B} is a 3-k- shrinkable open cover of X.

What could be called “k-shr-Lindelöf” does not differ from k-Lindelöf. More exactly:

Lemma 3.3. A space X is k-Lindelöf iff every open k-shrinkable cover contains a countable k-shrinkable subcover.

Proof. If every open k-shrinkable cover contains a countable k-shrinkable subcover then Lemma 3.1 implies that X is k-Lindelöf.

Now let X be k-Lindelöf and fix an open k-shrinkable cover U and a function C confirming that. For each U ∈ U choose a f_{U} ∈ C(X) such that f_{U}[C(U)] ⊆ {0}, f_{U}[X \ U] ⊆ {1} and put T (U) := |f_{U}|^{←}[0, 1/2] ⊆ U. Clearly 0 is in the closure of A = {f_{U} : U ∈ U} with respect to the co-topology. By Lemma 1.1, C_{k}(X) has countable tightness so there is a countable B ⊆ A such that 0 ∈ \overline{B} with respect to the co-topology. By the construction of A there is a countable V ⊆ U with B = {f_{U} : U ∈ V}. Then T witnesses that V is k-shrinkable.

Indeed, let K ⊆ X be compact. There is an h ∈ B ∩ O(K, 1/2) and a U ∈ V with h = f_{U}. Then |f_{U}|[K] = |h|[K] ⊆ [0, 1/2) so K ⊆ |f_{U}|^{←}[0, 1/2] = T (U).

Using the techniques as in [12] one can show

Proposition 3.1. The following are equivalent:

(1) C_{k}(X) is a Pytkeev space;

(2) If U is a nontrivial 3-k-shrinkable cover of X, there is a sequence (U_{n} : n ∈ N) of countably infinite subfamilies of U such that {⋂ U_{n} : n ∈ N} is a k-cover of X.

For two sets A and B the formula P yt(A, B) (see [12]) abbreviates the statement:

for each sequence (A_{n} : n ∈ N) of elements of A there is a sequence (B_{n} : n ∈ N), where each B_{n} is a countably infinite subset of A_{n}, such that for each f ∈ ∏(B_{n} : n ∈ N) the set {f(n) : n ∈ N} is an element of B. For some topological properties it is possible to consider their “selective” ver-

sions (see e.g. [12,13]). In this paper we are interested in the selective bitopological versions of the Pytkeev and the Reznichenko properties.

Let τ_{1} and τ_{2} be two topologies on the same set X with τ_{2} ⊆ τ_{1}. X has the selectively (τ_{1}, τ_{2})-Pytkeev property at x ∈ X if for each sequence (A_{n} : n ∈ N) of subsets of X and x ∈ ∩_{n∈N}(\overline{A_{n}} \ A_{n}), with respect to the topology τ_{1}, there exists a sequence (B_{n} : n ∈ N) such that each B_{n} is an infinite and countable subset of A_{n} and {B_{n} : n ∈ N} is a π-network at x, with respect to the topology τ_{2}. If this holds for each point of X then we say that X has the selectively (τ_{1}, τ_{2})-Pytkeev property. This property has been already considered in the context of hyperspaces in [7]. In further text if X = C(Y) for a space Y, τ_{1} is the corresponding compactopen topology and τ_{2} is the corresponding topology of pointwise convergence, then the letters k and p will denote τ_{1} and τ_{2}, respectively, in the above notation. Obviously, P yt((Ω_{x})^{τ_{1}}, (Ω_{x})^{τ_{2}}) is another way of saying that X has the selectively (τ_{1}, τ_{2})-Pytkeev property at x ∈ X.

We now characterize this bitopological property considered on the set C(X).

Theorem 3.2. For a k-Lindelöf space X the following are equivalent:

(1) C(X) satisfies the selectively (k, p)-Pytkeev property;

(2) P yt(K_{shr}, Ω) holds.

Proof. (1) ⇒ (2). Let us first remark that X is normal.

Let U be a nontrivial k-shrinkable open cover of X and C a function such that for each U ∈ U, C(U) ⊆ U, C(U) is closed and such that {C(V) : V ∈ U} ∈ K. List injectively the finite subsets of X as (F_{α} : α < |X|). Choose a U_{0} ∈ U with F_{0} ⊆ C(U_{0}) and f_{0} ∈ C(X) with f_{0}[C(U_{0})] ⊆ {0}, f_{0}[X \ U_{0}] ⊆ {1}. If (U_{β}, f_{β}) have been defined for all β < α, so that f_{β} ∈ C(X), C(U_{β}) ⊆ f\overleftarrow{β}\{0\}, X \ U_{β} ⊆ f\overleftarrow{β}\{1\} and for all β_{1} < β_{2} < α f_{β1} ≠ f_{β_{2}}, U_{β_{1}} ≠ U_{β_{2}}, proceed the recursive definition as follows: if \{f\overleftarrow{β}\{0\} : β < α\} k-covers X then end defining; if \{f\overleftarrow{β}\{0\} : β < α\} does not k-cover X take a finite T_{α} ⊆ X with T_{α} ⊆ f\overleftarrow{β}\{0\} for no β < α, a U_{α} ∈ U with T_{α} ∪F_{α} ⊆ C(U_{α}) and f_{α} ∈ C(X) such that C(U_{α}) ⊆ f\overleftarrow{α}\{0\}, X \ U_{α} ⊆ f\overleftarrow{α}\{1\}; it is clear that f_{α} ≠ f_{β} for all β < α and also, for each β < α we must have that U_{α} ≠ U_{β} because otherwise T_{α} ⊆ C(U_{α}) = C(U_{β}) ⊆ f\overleftarrow{β}\{0\} for a β < α, which is impossible. Having finished this recursive definition there is a β_{0} < α such that \{f\overleftarrow{β}\{0\} : β < β_{0}\} k-covers X and with U_{β_{1}} ≠ U_{β_{2}} and f_{β1} ≠ f_{β_{2}} for each β_{1} < β_{2} < β_{0}. Therefore, the function g_{U} such that for every β < β_{0} (g_{U}(U_{β}) = f_{β}) and dom(g_{U}) ⊆ U is correctly defined. For g_{U} the following hold: dom(g_{U}) ⊆ U, ran(g_{U}) ⊆ C(X) , {g_{U}(U)^{←}{0} : U ∈ dom(g_{U})} k-covers X and X \ U ⊆ g_{U}(U)^{←}\{1\} for all U ∈ dom(g_{U}). Clearly 0 ∈ \overline{ran(g_{U})} with respect to the co-topology and as X ∉ U, ran(g_{U}) does not contain the function 0.

Now let C(X) satisfy the selectively (k, p)-Pytkeev property and let (U_{n} : n ∈ N) be a sequence of elements of K_{shr}. For each n associate to U_{n} a function g_{U_{n}} as described above. 0 ∈ \overline{ran(g_{U_{n}})} \ ran(g_{U_{n}}) with respect to the co-topology for each n, so by the selectively (k, p)-Pytkeev property, there is a sequence (B_{n} : n ∈ N) with each B_{n} an infinite subset of ran(g_{U_{n}}) such that {B_{n} : n ∈ N} is a π-network at 0 with respect to the pointwise-convergence topology. For each n there is an infinite V_{n} ⊆ dom(g_{U_{n}}) ⊆ U_{n} with B_{n} = {g_{U_{n}}(U) : U ∈ V_{n}}. We show that {∩V_{n} : n ∈ N} ∈ Ω.

Let F be a finite subset of X. As {B_{n} : n ∈ N} is a π-network at 0 with respect to the pointwise-convergence topology there is n_{0} ∈ N such that B_{n_{0}} ⊆ O(F, 1). This actually means that g_{U_{n_{0}}} (U)[F] ⊆ (−1, 1) for all U ∈ V_{n_{0}}. But g_{U_{n_{0}}} (U)[X \ U] ⊆ {1} for each U ∈ V_{n_{0}}. Therefore F ⊆ U for every U ∈ V_{n_{0}}, i.e. F ⊆ ∩V_{n_{0}}.

(2) ⇒ (1). Let (A_{n} : n ∈ N) be a sequence of subsets of C(X) with 0 ∈ \overline{A_{n}}\A_{n}, with respect to the co-topology, for each n.

If A ⊆ C_{k}(X) and 0 ∈ \overline{A} \ A, call A small if for every ε > 0 there is an f ∈ A such that |f|[X] ⊆ [0, ε). For a small A, given any δ > 0 one can find an injective sequence (f_{n}(A, δ) : n ∈ N) of elements of A such that |f_{n}(A, δ)|[X] ⊆ [0, δ) for all n ∈ N.

If A ⊆ C_{k}(X), 0 ∈ \overline{A} \ A and A is not small, there is a positive real number δ(A) > 0 such that for each f ∈ A we have |f|^{←}[0, δ(A)) ≠ X.

Put S := {n ∈ N : A_{n} is small}.

Case 1. S is infinite. For each n ∈ S let B_{n} := {f_{m}(A_{n}, 1/n) : m ∈ N}. If n ∉ S choose arbitrarily an infinite B_{n} ⊆ A_{n}. Obviously for each n ∈ N B_{n} is an infinite subset of A_{n}. We show that {B_{n} : n ∈ N} is a π-network at 0 with respect to the pc-topology.

Let F be a finite subset of X and ε > 0. Take n_{0} ∈ N with 1/n_{0} < ε and n ∈ S with n ≥ n_{0}. Then B_{n} ⊆ O(F, ε): if h ∈ B_{n} then h = f_{m}(A_{n}, 1/n) for a m ∈ N, so h[X] ⊆ (−1/n, 1/n) ⊆ (−ε, ε), i.e. h ∈ O(F, ε).

Case 2. S is finite. Let m_{0} := max S. Fix n > m_{0}. The set A_{n} is not small thus X ≠ |f|^{←}[0, δ(A_{n})) for each f ∈ A_{n}. Let δ_{n} := min{δ(A_{n}), 1/n}. Then δ_{n} ≤ 1/n and if f ∈ A_{n} we have that X ≠ |f|^{←}[0, δ_{n}). By Lemma 3.2 there is a function U_{n} with dom(U_{n}) ⊆ A_{n} such that U_{n} := {U_{n}(f) : f ∈ dom(U_{n})} is an open 3-k-shrinkable cover of X and such that U_{n}(f) ⊆ |f|^{←}[0, δ_{n}) for each f ∈ dom(U_{n}). X ∉ {|f|^{←}[0, δ_{n}) : f ∈ A_{n}} so U_{n} is nontrivial.

Apply the principle P yt(K_{shr}, Ω) to (U_{n} : n > m_{0}) to get a sequence (V_{n} : n > m_{0}) with V_{n} ⊆ U_{n}, |V_{n}| = ω for each n > m_{0} and such that {∩V_{n} : n > m_{0}} ∈ Ω. Then for each n > m_{0} there is an infinite B_{n} ⊆ A_{n} with {U_{n}(f) : f ∈ B_{n}} = V_{n}. For n ≤ m_{0} choose any infinite B_{n} ⊆ A_{n}. We show that {B_{n} : n ∈ N} is a π-network at 0 with respect to the pc-topology.

Let F be a finite subset of X and ε > 0. Take n_{0} ∈ N with 1/n_{0} < ε and a finite F_{0} ⊆ X such that there is no n ∈ N, m_{0} < n < n_{0}, with F_{0} ⊆ ∩V_{n}. As {∩V_{n} : n > m_{0}} ∈ Ω there is a k > m_{0} with F ∪ F_{0} ⊆ ∩V_{k}. By the construction of F_{0} we have k ≥ n_{0}. Also, F ⊆ ∩{U_{k}(f) : f ∈ B_{k}}, so for each f ∈ B_{k} we have F ⊆ U_{k}(f) ⊆ |f|^{←}[0, δ_{k}) ⊆ |f|^{←}[0, 1/k) ⊆ |f|^{←}[0, 1/n_{0}) ⊆ |f|^{←}[0, ε), i.e. f ∈ O(F, ε). In other words B_{k} ⊆ O(F, ε).

A space X is called a (τ_{1}, τ_{2})-Pytkeev space, if whenever x ∈ \overline{A}\A with respect to the τ_{1} topology, there is a countable π-network at x with respect to the τ_{2} topology consisting of infinite subsets of A.

Theorem 3.3. For a k-Lindelöf space X the following are equivalent:

(1) C(X) is a (k, p)-Pytkeev space;

(2) If U is a k-shrinkable open nontrivial cover of X, there is a sequence (U_{n} : n ∈ N) of subfamilies of U such that |U_{n}| = ω for each n and {∩U_{n}}_{n∈N} is an ω-cover of X.

Proof. Practically repeat the proof of the previous theorem.

4. The Reznichenko-type properties

In 1996 Reznichenko (at a seminar at the Moscow State University) introduced the following property of a space X:

For each x ∈ X and A ⊆ X with x ∈ \overline{A}\A, there is a countably infinite pairwise disjoint family F of finite subsets of A such that for every neighborhood V of x the family {F ∈ F : F ∩ V = ∅} is finite. This property is referred to as the weakly Fréchet-Urysohn property [14,15],

or the Reznichenko property [7,10]. Let us remark that every Pytkeev space is a Reznichenko space (see [11]).

In [14] it was shown

Theorem 4.1. For a space X the following are equivalent:

(1) C_{p}(X) is a Reznichenko space;

(2) If U is a nontrivial ω-shrinkable open cover of X, then there is a sequence (U_{n} : n ∈ N) of pairwise disjoint finite subsets of U such that for each finite F the set {n ∈ N : F ⊆ U for some U ∈ U_{n}} is cofinite in N.

In a similar way one can prove

Theorem 4.2. For a space X the following are equivalent:

(1) C_{k}(X) is a Reznichenko space;

(2) If U is a nontrivial k-shrinkable open cover of X, there is a sequence (U_{n} : n ∈ N) of pairwise disjoint finite subsets of U such that for each compact set K of X the set {n ∈ N : K ⊆ U for some U ∈ U_{n}} is cofinite in N.

Let τ_{1} and τ_{2} be two topologies on the same set X with τ_{2} ⊆ τ_{1}. A space X satisfies the selectively (τ_{1}, τ_{2})-Reznichenko property if for each x ∈ X and each sequence (A_{n} : n ∈ N) of subsets of X with x ∈ ∩_{n∈N}(\overline{A_{n}} \ A_{n}) with respect to the topology τ_{1}, there exists a sequence (B_{n} : n ∈ N) such that B_{n} is a finite subset of A_{n} for each n, B_{n} and B_{m} are disjoint for distinct m and n and for every neighborhood V of x, with respect to the topology τ_{2}, the set {n ∈ N : B_{n} ∩V = ∅} is finite. In further text if X = C(Y) for a space Y, τ_{1} is the corresponding compactopen topology and τ_{2} is the corresponding topology of pointwise convergence, then the letters k and p will stand for τ_{1} and τ_{2}, respectively, in the above notation.

In [7] this property has been considered in the context of hyperspaces.

We now borrow some terminology from [12]. Let A and F be two sets (here one may look at F as a “list of certain properties”). HL_{0}(A, F) denotes the following statement:

for each sequence (U_{n} : n ∈ N) of elements of A there is a sequence (V_{n} : n ∈ N) such that each V_{n} is a finite subset of U_{n}, if n ≠ m then V_{n} ∩ V_{m} = ∅, and for every F ∈ F there exists n_{0} such that for every n ≥ n_{0} there is a U ∈ V_{n} such that U ∈ F. If we do not require that the V_{n}-s must be pairwise disjoint we obtain the

principle denoted by HL(A, F). The corresponding games GameHL_{0}(A, F) and GameHL(A, F) are defined as it is customary with selection principles.

The next two general results will considerably simplify further study in this section.

Proposition 4.1. [12] If for every X ∈ A there exists Y ⊆ X such that Y ∈ A and card(Y) = ω and if HL(A, F) holds, then ONE has no winning strategy in the game GameHL(A, F).

Proposition 4.2. [12] Let for each X ∈ A and each finite set Y, X \ Y ∈ A hold. Then: ONE has no winning strategy in the game GameHL(A, F) iff he has no winning strategy in the game GameHL_{0}(A, F).

Note 4.1. Obviously, if ONE has no winning strategy in the game GameHL(A, F) (GameHL_{0}(A, F)), then HL(A, F) (HL_{0}(A, F)) holds. Also, HL_{0}(A, F) implies HL(A, F). Thus, if A satisfies both the condition of Proposition 4.1 and Proposition 4.2, then HL(A, F) is equivalent to HL_{0}(A, F). As a consequence of this we have that the following holds:

Let τ_{1} and τ_{2} be two topologies on the same set X with τ_{2} ⊆ τ_{1} such that (X, τ_{1}) has countable tightness. If x ∈ X, for A = (Ω_{x})^{τ_{1}} and a suitable F, we obtain that X has the selectively (τ_{1}, τ_{2})-Reznichenko property at x iff for each sequence (A_{n} : n ∈ N) of subsets of X and x ∈ ∩_{n∈N}(\overline{A_{n}} \ A_{n}), with respect to the topology τ_{1}, there exists a sequence (B_{n} : n ∈ N) such that B_{n} is a finite subset of A_{n} for each n and for every neighborhood V of x, with respect to the topology τ_{2}, the family {n ∈ N : B_{n} ∩ V = ∅} is finite, i.e. such that the sequence (B_{n} : n ∈ N) converges to x with respect to the τ_{2} topology. Note that the B_{n}-s do not have to be pairwise disjoint.

We state our next bitopological result.

Theorem 4.3. Let X be k-Lindelöf. Then the following are equivalent:

(1) C(X) satisfies the selectively (k, p)-Reznichenko property;

(2) If (U_{n} : n ∈ N) is a sequence of nontrivial k-shrinkable open covers of X, there is a sequence (V_{n} : n ∈ N) such that V_{n} is a finite subset of U_{n} for each n, V_{n} and V_{m} are disjoint for distinct m and n and for each finite F ⊆ X the set {n ∈ N : K ⊆ U for some U ∈ V_{n}} is cofinite in N.

Proof. (1) ⇒ (2): As X is k-Lindelöf, by Note 4.1 and Lemma 3.3 we only need to show that for each sequence (U_{n} : n ∈ N) of nontrivial k-shrinkable open covers of X there is a sequence (R_{n} : n ∈ N), with each R_{n} a finite subset of U_{n}, such that for each finite F ⊆ X for all but finitely many n ∈ N the set {U ∈ R_{n} : F ⊆ U} is not empty.

Let (U_{n} : n ∈ N) be a sequence of nontrivial k-shrinkable open covers of X. For each n ∈ N associate to U_{n} in the way we did in Theorem 3.2 a function g_{n} such that dom(g_{n}) ⊆ U_{n}, ran(g_{n}) ⊆ C(X) , {g_{n}(U)^{←}{0} : U ∈ dom(g_{n})} k-covers X and X \ U ⊆ g_{n}(U)^{←}\{1\} for all U ∈ dom(g_{n}). Clearly 0 ∈ \overline{ran(g_{n})} with respect to the co-topology and as X ∉ U_{n}, ran(g_{n}) does not contain the function 0. Apply the selectively (k, p)-Reznichenko property of C(X) to the sequence (ran(g_{n}) : n ∈ N) to obtain a sequence (R_{n} : n ∈ N), with each R_{n} a finite subset of ran(g_{n}), converging to 0 with respect to the pc-topology. For each n there is a finite subset R_{n} of dom(g_{n}) ⊆ U_{n} with {g_{n}(U) : U ∈ R_{n}} = R_{n}. C_{k}(X) has countable tightness, so by Note 4.1 it suffices to show that for each finite F ⊆ X for all but finitely many n ∈ N the set {U ∈ R_{n} : F ⊆ U} is not empty, so fix such an F. As (R_{n} : n ∈ N) converges to 0 with respect to the pc-topology there is n_{0} ∈ N such that for all n > n_{0} the set O(F, 1) ∩ R_{n} is not empty. Fix n > n_{0}. There is an f ∈ O(F, 1) ∩ R_{n} and a U ∈ R_{n} with g_{n}(U) = f. Since g_{n}(U)[F] = f[F] ⊆ (−1, 1) and X \ U ⊆ g_{n}(U)^{←}{1}, it follows that F ⊆ U. Thus {U ∈ R_{n} : F ⊆ U} is not empty.

(2) ⇒ (1): As X is k-Lindelöf, by Note 4.1 and Lemma 1.1 we only need to show that for each sequence (A_{n} : n ∈ N) of subsets of C(X) with x ∈ ∩_{n∈N}(\overline{A_{n}} \ A_{n}) with respect to the co-topology, there exists a sequence (B_{n} : n ∈ N) converging to 0 with respect to the pc-topology, such that for each n, B_{n} is a finite subset of A_{n}.

Let (A_{n} : n ∈ N) be a sequence of subsets of C(X) with 0 ∈ \overline{A_{n}} \ A_{n}, with respect to the co-topology, for each n. By Lemma 3.2 for each n ∈ N there is a function U_{n} with dom(U_{n}) ⊆ A_{n} such that U_{n} := {U_{n}(f) : f ∈ dom(U_{n})} is an open 3-k-shrinkable cover of X and such that U_{n}(f) ⊆ |f|^{←}[0, 1/n) for each f ∈ dom(U_{n}). Set S := {n ∈ N : X ∉ U_{n}}.

Case 1. S is finite. Then there would be a sequence (f_{n} : n > max S), with f_{n} ∈ A_{n} for each n > max S, uniformly converging to 0 so this would end the proof.

Case 2. S is infinite. For each n ∉ S pick an f_{n} ∈ A_{n} with f_{n}[X] ⊆ (−1/n, 1/n). As for each n ∈ S the cover U_{n} is nontrivial, we can apply the condition (2) of this theorem to the sequence (U_{n} : n ∈ S) to get a sequence (V_{n} : n ∈ S), where for each n ∈ S, V_{n} is a finite subset of U_{n}, such that for each finite F ⊆ X for all but finitely many n ∈ S the set {U ∈ V_{n} : F ⊆ U} is not empty. For each n ∈ S there is a finite C_{n} ⊆ dom(U_{n}) ⊆ A_{n} with V_{n} = {U_{n}(f) : f ∈ C_{n}}. Put B_{n} := C_{n} if n ∈ S and B_{n} := {f_{n}} if n ∉ S. We show that (B_{n} : n ∈ N) is as required.

Let F be a finite subset of X and ε > 0. By the construction of (V_{n} : n ∈ S) there is n_{0} ∈ S such that for each n ∈ S with n > n_{0} the set {U ∈ V_{n} : F ⊆ U} is not empty. Without loss of generality we may suppose that 1/n_{0} < ε. Fix n > n_{0}. If n ∉ S then f_{n}[F] ⊆ f_{n}[X] ⊆ (−1/n, 1/n) ⊆ (−ε, ε), so O(F, ε)∩B_{n} is not empty. If n ∈ S then there is a U ∈ V_{n} with F ⊆ U and an f ∈ C_{n} with U = U_{n}(f). Since F ⊆ U = U_{n}(f) ⊆ |f|^{←}[0, 1/n) ⊆ f^{←}(−ε, ε), thus again O(F, ε)∩B_{n} = O(F, ε)∩C_{n} is not empty.

A space X has the (τ_{1}, τ_{2})-Reznichenko property if A ⊆ X and x ∈ \overline{A} \ A with respect to τ_{1} topology imply the existence of a countably infinite disjoint family F of subsets of A such that for every neighborhood V of x with respect to the τ_{2} topology, the family {F ∈ F : F ∩ V = ∅} is finite.

Theorem 4.4. For a k-Lindelöf space X the following are equivalent:

(1) C(X) has the (k, p)-Reznichenko property;

(2) If U is a nontrivial k-shrinkable open k-cover of X, there is a sequence (U_{n} : n ∈ N) of pairwise disjoint finite subsets of U such that for each finite set F the set {n ∈ N : F ⊆ U for some U ∈ U_{n}} is cofinite in N.

Proof. (1) ⇒ (2): If U ∈ K_{shr} then as in Theorem 3.2 construct a function g with dom(g_{U}) ⊆ U, ran(g) ⊆ C(X), X \ U ⊆ g(U)^{←}{1} for all U ∈ dom(g) and \{g(U)^{←}\{0\} : U ∈ dom(g)\} k-covers X. We have that 0 ∈ \overline{ran(g_{U})} \ ran(g) with respect to the co-topology, so we can apply the condition (1) to ran(g) to get a sequence (R_{n} : n ∈ N) of finite pairwise disjoint subsets of ran(g) converging to 0 with respect to the pc-topology. Pick an h : ran(g) → dom(g) with h(g(U)) = U for each U ∈ dom(g). Put U_{n} := {h(a) : a ∈ R_{n}}. Then (U_{n} : n ∈ N) is a sequence of finite pairwise disjoint subsets of U. It is not difficult to check using the methods of previous theorems that this sequence is as required.

(2) ⇒ (1): Let the condition (2) be satisfied.

Claim. Let 0 ∈ \overline{A} \ A with respect to the co-topology and δ > 0. Then there is a sequence (A_{n} : n ∈ N) of pairwise disjoint finite subsets of A and a B ⊆ A, with 0 ∈ \overline{B} \ B with respect to the co-topology, such that (⋃_{n∈N} A_{n}) ∩ B = ∅ and the sequence (A_{n} : n ∈ N) is δ-converging to 0, i.e. for each finite subset F of X for all but finitely many n the set A_{n} ∩ O(F, δ) is not empty.

Proof of the claim. If for each ε > 0 there is an f ∈ A with f[X] ⊆ (−ε, ε), there exists an injective sequence (f_{n} : n ∈ N) of elements of A uniformly converging to 0 so, in this case there is nothing to prove. Thus we may suppose that there is a δ_{1} > 0 with f[X] ⊆ (−δ_{1}, δ_{1}) for no f ∈ A. Put δ_{0} := min{δ, δ_{1}}.

By Lemma 4.1, let U be a function with dom(U) ⊆ A, U(f) ⊆ |f|^{←}[0, δ_{0}) for every f ∈ dom(U), such that ran(U) is a 3-k-shrinkable open cover of X. By the assumption made above ran(U) is nontrivial so by the condition (2) there is a sequence (U_{n} : n ∈ N) of pairwise disjoint finite subsets of ran(U) such that for each finite subset F of X for all but finitely many n the set {V ∈ U_{n} : F ⊆ V } is not empty. For each n there is a finite subset A_{n} of dom(U) ⊆ A with U_{n} = {U(f) : f ∈ A_{n}}. If n ≠ m and g ∈ A_{n} ∩ A_{m} then U(g) ∈ U_{n} ∩ U_{m}, which is impossible. Thus A_{n} ∩ A_{m} = ∅ for distinct n and m.

For each finite F ⊆ X we have that for all but finitely many n the set A_{n} ∩ O(F, δ) is not empty: there is an n_{0} with {V ∈ U_{n} : F ⊆ V } ≠ ∅ for each n > n_{0}. Fix an n > n_{0} and a V ∈ U_{n} with F ⊆ V. Then V = U(f) for some f ∈ A_{n}. Hence F ⊆ V = U(f) ⊆ |f|^{←}[0, δ_{0}) ⊆ |f|^{←}[0, δ), i.e. f ∈ O(F, δ) ∩ A_{n}.

If 0 belongs to the closure of the set (A \ ⋃_{n∈N} A_{n}) ∪ (⋃_{n∈N} A_{2n}) with respect to the co-topology then let B := (A\⋃_{n∈N} A_{n})∪(⋃_{n∈N} A_{2n}) and let C_{n} := A_{2n−1}. If 0 belongs to the closure of the set (A\⋃_{n∈N} A_{n})∪(⋃_{n∈N} A_{2n−1}) with respect to the co-topology then let B := (A \ ⋃_{n∈N} A_{n}) ∪ (⋃_{n∈N} A_{2n−1}) and let C_{n} := A_{2n}. Then it easy to see that the sequence (C_{n} :∈ N) and the set B are as required.

Now we prove the theorem. Let 0 ∈ \overline{A} \ A with respect to the co-topology. By the above Claim let (H_{n}^{1} : n ∈ N) be a sequence of pairwise disjoint finite subsets of A which 1−converges to 0 and B_{1} ⊆ A with 0 ∈ \overline{B_{1}} \ B_{1}, (⋃_{n∈N} H_{n}^{1}) ∩ B_{1} = ∅. If the sequences (H_{n}^{i} : n ∈ N) and sets B_{i} ⊆ A have been defined for 1 ≤ i ≤ k so that:

(i) ⋃{H_{n}^{i} : 1 ≤ i ≤ k, n ∈ N} ∩ B_{k} = ∅; (ii) 0 ∈ \overline{B_{k}} \ B_{k}; (iii) (H_{n}^{i} : n ∈ N) (1/i)−converges to 0 for each 1 ≤ i ≤ k, then let by the above Claim (H_{n}^{k+1} : n ∈ N) be a sequence of pairwise disjoint finite subsets of B_{k} which 1/(k+1)−converges to 0 and B_{k+1} ⊆ B_{k} with 0 ∈ \overline{B_{k+1}}\B_{k+1}, (⋃_{n∈N} H_{n}^{k+1}) ∩ B_{k+1} = ∅.

Having finished the construction we set A_{n} := ⋃{H_{n}^{i} : 1 ≤ i ≤ n}. Obviously for distinct n and m the sets A_{n} and A_{m} are disjoint finite subsets of A. We show that (A_{n} : n ∈ N) converges to 0 with respect to the pc-topology.

Let F be a finite subset of X and ε > 0. Fix an m_{0} > 1/ε and an n_{0} > m_{0} such that H_{n}^{m_{0}} ∩ O(F, 1/m_{0}) ≠ ∅ for every n ≥ n_{0}. If n ≥ n_{0} then H_{n}^{m_{0}} ⊆ ⋃{H_{n}^{i} : 1 ≤ i ≤ n} = A_{n}, so ∅ ≠ A_{n} ∩ O(F, 1/m_{0}) ⊆ A_{n} ∩ O(F, ε).

REFERENCES

[1] A. V. Arhangel’skiĭ, Topological Function Spaces, Kluwer Academic Publishers, 1992.

[2] G. Di Maio, Lj.D.R. Kočinac and E. Meccariello, Applications of k-covers, Acta Mathematica Sinica, English Series, to appear.

[3] R. Engelking, General Topology, PWN, Warszawa, 1977.

[4] J. Gerlits and Zs. Nagy, Some properties of C(X),I, Topology Appl. 14 (1982), 151–161.

[5] W.Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of open covers (II), Topology Appl., 69 (1996), 31–62.

[6] Lj.D.R. Kočinac, Closure properties in function spaces, Applied General Topology, 4, 2 (2003), 255–261.

[7] Lj.D.R. Kočinac, The Reznichenko property and the Pytkeev property in hyperspaces, Acta Math. Hungar., 107, 3 (2005), 231–239.

[8] Lj.D.R. Kočinac, Two covering properties and partition relations, Proceedings of the 3rd Seminar on Geometry and Topology, July 15–17, 2004, Tabriz, Iran (Sh. Rezapour, ed.), 105–109.

[9] Lj.D.R. Kočinac and M. Scheepers, Combinatoric of open covers(VII): Groupability, Fund. Math., 179, 2 (2003), 131-155.

[10] Lj.D.R. Kočinac and M. Scheepers, Function spaces and a property of Reznichenko, Topology Appl., 123 (2002), 135–143.

[11] V.I. Malykhin and G. Tironi, Weakly Frèchet-Urysohn and Pytkeev spaces, Topology Appl., 104 (2000), 181–190.

[12] V. Pavlović, A selective version of the property of Reznichenko in function spaces, submitted.

[13] V. Pavlović, Selectively A-function spaces, East-West J. Math., (to appear).

[14] M. Sakai, The Pytkeev property and the Reznichenko property in function spaces, Note di Matematica, 22, 2 (2003), 43–52.

[15] M. Sakai, Weakly Fréchet-Urysohn property in function spaces, preprint.

[16] M. Sakai, Property C′′ and function spaces, Proc. Amer. Math. Soc., 104 (1988), 917–919.

[17] M. Scheepers, Combinatorics of open covers (I): Ramsey theory, Topology Appl., 73 (1996), 241–266.

[18] M. Scheepers, Combinatorics of open covers (III): C_{p}(X) and games, Fund. Math., 152 (1997), 231–254.

(received 07.04.2005, in revised form 07.09.2005)

Bruno A. Pansera, Dipartimento di Matematica, Universita di Messina, Messina, Italy

Vladimir Pavlović, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia