THE MENGER PROPERTY ON C_{p}(X, 2)

D. BERNAL-SANTOS AND Á. TAMARIZ-MASCARÚA

Abstract. A space X is said to have the Menger property (or simply X is Menger) if for every sequence ⟨ U_{n} : n ∈ ω ⟩ of open covers of X, there exists a sequence of finite sets ⟨ F_{n} : n ∈ ω ⟩ such that ⋃_{n∈ω} F_{n} is a cover of X and F_{n} ⊂ U_{n} for every n ∈ ω. We prove: (1) If X is a subspace of C_{p}(Y), where Y^{n} is Menger for every n ∈ ω, and X′ (the set of non-isolated points of X) is compact, then C_{p}(X, 2) is Menger; (2) If C_{p}(X, 2) is Menger and X is normal, then X′ is countably compact; (3) For a first countable GO-space without isolated points L, C_{p}(L, 2) is Menger if and only if C_{p}(L, 2) is Lindelöf and L is countably compact; and for a subspace L of an ordinal, C_{p}(L, 2) is Menger if and only if C_{p}(L, 2) is Lindelöf and L′ is countably compact; (4) For every F ∈ ω^{∗}, C_{p}(ω ∪ {F}, 2) is Menger if and only if F is a strong P -point;

(5) Assuming the Continuum Hypothesis, there is a maximal almost disjoint family A for which the space C_{p}(Ψ(A), 2) is Menger.

1. Introduction

M. Scheepers started the identification and classification of common prototypes for selective properties appearing in classical and modern works. In [12], [21], [23] and [24] we can find good surveys of this field of Selective Principles in Mathematics. Two of the main prototypes in the field are defined as follows [?]. Fix a topological space X, and let A and B be collections of covers of subsets of X. The following are properties which X may or may not have [?]:

S_{1}(A, B): For each sequence ⟨ U_{n} : n ∈ ω ⟩ of members of A, there exists, for
each n ∈ ω, U_{n} ∈ U_{n} such that {U_{n} : n ∈ ω} ∈ B.
S_{fin}(A, B): For each sequence ⟨ U_{n} : n ∈ ω ⟩ of members of A, there exists, for

each n ∈ ω, a finite subset F_{n} ⊂ U_{n}, such that ⋃{F_{n} : n ∈ ω} ∈ B.

When A and B coincide with the collection O of all open covers of X, then, in the case of metric spaces, S_{fin}(O, O) is the property shown by W. Hurewicz [11] to be equivalent to K. Menger’s property E [15]. And S_{1}(O, O) is F. Rothberger’s property traditionally known as C′′ [17].

So we will say that a space X has the Menger property (or simply X is Menger) if X has property S_{fin}(O, O), and a space X is Rothberger if X has property S_{1}(O, O).

2000 Mathematics Subject Classification. Primary 54C35, 54D35, 03G10, Secondary 54D45, 54C45.

Key words and phrases. Space of continuous functions with the pointwise convergence topology, Menger space, Hurewicz space, Lindelöf space, generalized ordered topological space, subspace of ordinals, strong P -point, Ψ-space.

The first author was supported by CONACyT grant No. 322416/233751. The research of the second author was supported by PAPIIT IN115312.

Every σ-compact space and every Rothberger space has the Menger property and every space with the Menger property is Lindelöf.

Naturally, the M. Scheepers’s prototypes of selective principles have been analyzed in the class of spaces of real-valued continuous functions defined on a space X with its pointwise convergence topology, C_{p}(X); see for example [9], [18], [19] and [25]. With respect to the Menger property, in [1] the following theorem is proved:

Theorem. C_{p}(X) is Menger if and only if X is finite.

For spaces of the form C_{p}(X, 2), of course, there is not an equivalent result to the previous theorem. If the space X is discrete, C_{p}(X, 2) is compact and so it is Menger. And the space C_{p}(2^{ω}, 2) is countable and thus it is Menger as well. Moreover, Á. Tamariz-Mascarúa and A. Contreras-Carreto in [7] show that if X is a EG-space and X′ is Eberlein compact, then C_{p}(X, 2) is σ-compact (hence, Menger). Therefore, the class of spaces for which C_{p}(X, 2) is Menger is not trivial.

In this article we are going to analyze when C_{p}(X, 2) is Menger for several classes of spaces X. In Section 3, we list some general remarks on Menger spaces. Section 4 is devoted to obtaining some general results about Menger property on spaces of the form C_{p}(X, 2). In Sections 5, 6, 7 and 8 we analyze sufficient and necessary conditions in order to have C_{p}(X, 2) Menger when X is a GO-space without isolated points, a subspace of ordinals, a countable space with exactly one non-isolated point and a Ψ-space, respectively.

With respect to the Rothberger property in C_{p}(X, 2), the first author of this paper made an analysis in [2].

2. Notation and terminology

All spaces under consideration are assumed to be Tychonoff, i.e., T_{3 \frac{1}{2}} . Given a space X, X′ denotes the set of non-isolated points of X. For spaces X and Y, C_{p}(X, Y) is the subspace of Y^{X} consisting of the continuous functions from X to Y (i.e., C(X, Y) with the topology of the pointwise convergence). As usual, C_{p}(X) will mean C_{p}(X, ℝ). For a space X, n ∈ ω, points x_{0}, …, x_{n} ∈ X, f ∈ C_{p}(X) and a positive real number δ, we will denote by [f; x_{0}, …, x_{n}; δ] the set {g ∈ C_{p}(X) : ∀i(0 ≤ i ≤ n → |f(x_{i}) − g(x_{i})| < δ)}. Recall that for every space X and every discrete space Y, there exists a zero-dimensional space Z such that C_{p}(X, Y) ≅ C_{p}(Z, Y). So, where reference is made to C_{p}(X, Y) where Y is discrete, we will assume that X is a zero-dimensional space. Let iw(X) be the minimal cardinal κ such that X has a weaker Tychonoff topology of weight κ; evidently, the statement iw(X) = ω is equivalent to saying that X has a weaker separable metrizable topology. A space X has countable fan tightness if for any x ∈ X and any sequence ⟨ A_{n} : n ∈ ω ⟩ of subsets of X such that x ∈ ⋂_{n∈ω} cl (A_{n}), we can choose a finite set B_{n} ⊂ A_{n} for each n ∈ ω in such a way that x ∈ cl (⋃{B_{n} : n ∈ ω}). A space X has a countable tightness (which is denoted by t(X) ≤ ω) if any x ∈ X and A ⊂ X, if x ∈ cl (A), then there is a countable set B ⊂ A such that x ∈ cl (B). Define a preorder ≤^{∗} on ω^{ω} by f ≤^{∗} g if and only if f(n) ≤ g(n) for all but finitely many n ∈ ω. The cardinal 𝔡 (resp. 𝔟) is the minimal cardinality of a cofinal (resp. unbounded) subset of (ω^{ω}, ≤^{∗}). A family P of non-empty subsets of a space X is said to be π-network at x ∈ X if every neighborhood of x contains some member of P. For any set X, [X]^{<ω} will denote the set of all finite subsets of X. The set of ordinals strictly less than an ordinal α equipped with its order topology will be denoted simply by α.

3. Properties of Menger spaces

As we have already mentioned:

Proposition 3.1. Every σ-compact space is Menger and every Menger space is a Lindelöf space.

Some other properties of Menger spaces are as follows:

Proposition 3.2. Any closed subspace of a Menger space is a Menger space and the continuous image of a Menger space is a Menger space.

For the purposes of this paper, the following equivalent formulation of Menger space will be useful.

Lemma 3.3. A space X is Menger if and only if for any sequence of open covers ⟨ U_{n} : n ∈ ω ⟩ such that, for every n ∈ ω, U_{n+1} refines U_{n}, there exists a sequence of finite sets ⟨ F_{n} : n ∈ ω ⟩ such that ⋃_{n∈ω} F_{n} is a cover of X and F_{n} ⊂ U_{n} for each n ∈ ω.

Proof. We only have to show the sufficiency. Let ⟨ U_{n} : n ∈ ω ⟩ be a sequence of open covers of X. By recursion we define a new sequence ⟨ U′_{n} : n ∈ ω ⟩ such that, for each n ∈ ω, U′_{n+1} refines U′_{n} and U_{n+1}. Let U′_{0} = U_{0}. Suppose that U′_{0}, …, U′_{n} have been defined. We define U′_{n+1} = {U ∩ V : U ∈ U_{n} ∧ V ∈ U′_{n}}. Then the sequence ⟨ U′_{n} : n ∈ ω ⟩ satisfies the required properties. □

We shall need the following results.

Proposition 3.4 ([22]). The countable union of Menger spaces is Menger

Proposition 3.5 ([22]). If X is a Menger space and Y is σ-compact, then X × Y is Menger.

A space X is a P -space if all G_{δ}-sets in X are open.

Proposition 3.6 ([26]). A P -space is Menger if and only if it is Lindelöf.

The typical example of a Lindelöf space which is not Menger is the space of irrationals ω^{ω}. As a consequence of this we have the following:

Proposition 3.7. For any space X, X^{ω} is Menger if and only if X is compact.

Proof. Suppose that X^{ω} is Menger and X is not compact. Since X is Lindelöf, X is not countably compact. Then X contains a closed countable discrete subspace D. In this manner D^{ω} is a closed subspace X^{ω} homeomorphic to ω^{ω}. But this is a contradiction to Proposition 3.2. □

As the Lindelöf property, the Menger property is not productive. In [14] A. Lelek gives an example (assuming the continuum hypothesis) of a Menger space X such that X^{2} is not Menger.

4. The Menger property on C_{p}(X, 2)

Recall that where reference is made to C_{p}(X, Y) where Y is discrete, we will assume that X is a zero-dimensional space.

The following is shown in [1]:

(⋆) If C_{p}(X) is a Lindelöf space, then each finite power of X has countable tightness.

A space X has countable supertightness at x ∈ X if any π-network at x consisting of finite subsets of X contains a countable π-network at x. If X has this property in each of its points we say that X has countable supertightness, and we denote this fact by st(X) ≤ ω. Clearly, countable supertightness implies countable tightness. With this new notion of tightness, we obtain a more general result than (⋆) for C_{p}(X, 2):

Proposition 4.1. If C_{p}(X, 2) is Lindelöf, then st(X^{n}) ≤ ω for any n ∈ ω.

Proof. Fix k ∈ ω, a point x = (x_{1}, …, x_{k}) ∈ X^{k} and a π-network P at x consisting of finite subsets of X^{k}. We take open neighborhoods U_{1}, …, U_{k} such that, for each i, j ∈ {1, …, k}, x_{i} ∈ U_{i}, U_{i} = U_{j} if x_{i} = x_{j}, and U_{i} ∩ U_{j} = ∅ if x_{i} ≠ x_{j}. Let U = U_{1} ×⋯×U_{k}. We can suppose that each member of P is contained in U. Since the space C_{p}(X, 2) is Lindelöf, the closed subspace

Φ = {f ∈ C_{p}(X, 2) : ∀i(1 ≤ i ≤ k → f(x_{i}) = 1)}

of C_{p}(X, 2) is Lindelöf. For each F ∈ P, we define H_{F} = ⋃{π_{i}[F] : i ∈ {1, …, k}}, where π_{i} is the projection of X^{k} over the i-th coordinate, and V_{F} = {f ∈ C_{p}(X, 2) : ∀x(x ∈ H_{F} → f(x) = 1)}. Given f ∈ Φ, for each i ∈ {1, …, k}, there is an open subset V_{i} ⊂ U_{i} such that x_{i} ∈ V_{i} and f[V_{i}] ⊂ {1}. Since P is a π-network, there is F ∈ P such that F ⊂ V_{1} × ⋯ × V_{k}. So, f[π_{i}[F]] ⊂ {1} for each i ∈ {1, …, k} and consequently f ∈ V_{F}. This shows that {V_{F} : F ∈ P} is an open cover of Φ. Therefore, there is a countable subset P′ of P such that {V_{F} : F ∈ P′} forms an open cover of Φ. Let us prove that P′ is a π-network at x.

Let W = W_{1} ×⋯×W_{k} be an open subset which contains x. We can assume that W_{i} = W_{j} if x_{i} = x_{j} and W_{i} ⊂ U_{i} for each i, j ∈ {1, …, k}. We choose f ∈ C_{p}(X, 2) such that

f[X \ ⋃_{i=1}^{k} W_{i}] ⊂ {0}

and f(x_{i}) = 1 for each i ∈ {1, …, k}. Thus f ∈ Φ, and consequently, there is F ∈ P′ such that f ∈ V_{F}. Now, if (y_{1}, …, y_{k}) ∈ F, since F ⊂ U, y_{i} ∈ U_{i} for each i ∈ {1, …, k}. Moreover, due to the fact that f ∈ V_{F}, y_{1}, …, y_{k} ∈ ⋃_{i=1}^{k} W_{i}. However, U_{i} ∩ U_{j} = ∅ if x_{i} ≠ x_{j}, then y_{i} ∈ W_{i} for each i ∈ {1, …, k}. This shows that F ⊂ W. □

M. Sakai introduces the following notion.

Definition 4.2 ([20]). A space X has countable fan tightness for finite sets if for each point x ∈ X and each sequence ⟨ P_{n} ⊂ [X]^{<ω} : n ∈ ω ⟩ of π-networks at x consisting of finite subsets of X, there is, for each n ∈ ω, a finite subfamily G_{n} ⊂ P_{n} such that ⋃{G_{n} : n ∈ ω} is a π-network at x.

The following equivalent formulation for countable fan tightness for finite sets will be useful.

Lemma 4.3. A space X has a countable fan tightness for finite sets if and only if for each point x ∈ X and any decreasing sequence ⟨ P_{n} ⊂ [X]^{<ω} : n ∈ ω ⟩ of π-networks at x, there are, for each n ∈ ω, finite subfamilies G_{n} ⊂ P_{n} such that ⋃{G_{n} : n ∈ ω} is a π-network at x.

Proof. The necessity is clear; we show the sufficiency. Let ⟨ P_{n} ⊂ [X]^{<ω} : n ∈ ω ⟩ be a sequence of π-networks at x ∈ X. For each n ∈ ω, we define P′_{n} = ⋃_{n≤k} P_{k}. Then, by hypothesis, there is a sequence of finite sets ⟨ F′_{n} : n ∈ ω ⟩ where F′_{n} ⊂ P′_{n}, such that ⋃_{n∈ω} F′_{n} is a π-network at x. Hence, if we define F_{n} = (⋃_{k≤n} F′_{k}) ∩ P_{n}, ⟨ F_{n} : n ∈ ω ⟩ is the required sequence. □

Making a modification of the proof of Proposition 4.1 we have the following.

Proposition 4.4. If the space C_{p}(X, 2) is Menger, then X^{n} has countable fan tightness for finite sets for any n ∈ ω.

Proof. We fix a k ∈ ω, a point x = (x_{1}, …, x_{k}) ∈ X^{k} and a sequence ⟨ P_{n} : n ∈ ω ⟩ of π-networks at x consisting of finite subsets of X. We take open subsets U_{1}, …, U_{k} of X such that, for each i, j ∈ {1, …, k}, x_{i} ∈ U_{i}, U_{i} = U_{j} if x_{i} = x_{j}, and U_{i}∩U_{j} = ∅ if x_{i} ≠ x_{j}. Let U = U_{1} × ⋯ × U_{k}. We can suppose that, for every n ∈ ω, each member of P_{n} is contained in U. Since the space C_{p}(X, 2) is Menger, the closed subspace

Φ = {f ∈ C_{p}(X, 2) : ∀i(1 ≤ i ≤ k → f(x_{i}) = 1)}

of C_{p}(X, 2) is Menger. For each F ∈ [X^{k}]^{<ω}, we define H_{F} = ⋃{π_{i}[F] : i ∈ {1, …, k}}, where π_{i} is the projection of X^{k} over the i-th coordinate, and we set V_{F} = {f ∈ C_{p}(X, 2) : ∀x(x ∈ H_{F} → f(x) = 1)}. For each n ∈ ω, let

U_{n} = {V_{F} : F ∈ P_{n}}.

Given f ∈ Φ, for each i ∈ {1, …, k}, there is an open subset V_{i} ⊂ U_{i} such that x_{i} ∈ V_{i} and f[V_{i}] ⊂ {1}. Since P_{n} is a π-network, there is F ∈ P_{n} such that F ⊂ V_{1} × ⋯ × V_{k}. So f[π_{i}[F]] ⊂ {1} for each i ∈ {1, …, k}. Thus f ∈ V_{F} ∈ U_{n}. This implies that U_{n} is an open cover of Φ. Therefore, since Φ is Menger, there is a sequence of finite sets ⟨ F_{n} : n ∈ ω ⟩ such that ⋃_{n∈ω} F_{n} forms a cover of Φ and F_{n} ⊂ U_{n} for every n ∈ ω. Choosing a finite subset P′_{n} ⊂ P_{n} such that F_{n} is equal to {V_{F} : F ∈ P′_{n}} and using similar argumentation to that developed in the proof of Theorem 4.1, we can prove that ⋃_{n∈ω} P′_{n} is a π-network at x. □

The converse of Propositions 4.1 and 4.4 are false. The following example can be found in [1, II.1.7].

Example 4.5. Let X be the well-known “double arrow” compact space; that is, X is the set [0, 1] × 2 endowed with the topology generated by the lexicographic order. For each a ∈ (0, 1), we define f_{a} : X → 2 as follows:

f_{a}(x) = \{ \begin{matrix}0, & if x ≤ (a, 0); \\ 1, & if x ≥ (a, 1). \end{matrix}

Then, the subspace A = {f_{a} : a ∈ (0, 1)} is closed and discrete in C_{p}(X, 2). Hence, C_{p}(X, 2) is not a Menger space. However, X^{n} has countable fan tightness for finite sets (because X^{n} satisfies the first axiom of countability) for each n ∈ ω.

Proposition 3.7 leaves out the possibility that C_{p}(X, 2)^{ω} ≅ C_{p}(X, 2^{ω}) have the Menger property when X is not a discrete space. We will only analyze the finite power of the spaces C_{p}(X, 2), and for this we have the following:

Proposition 4.6. For any space X, C_{p}(X, 2)^{n} is Menger for any n ∈ ω if and only if C_{p}(X, k) is Menger for any k ∈ ω.

Proof. This is immediate of the fact that C_{p}(X, 2)^{n} is homeomorphic to C_{p}(X, 2^{n}) for any n ∈ ω and the fact that a closed subspace of a Menger space is Menger. □

Definition 4.7. A space X is an Eberlein-Grothendieck-space, or an EG-space, if it is homeomorphic to a subspace of C_{p}(Y) for some compact space Y. We say that X is Eberlein compact if X is a compact EG-space.

M. Sakai [20] shows that C_{p}(X) has countable fan tightness for finite sets if and only if X^{n} is Menger for each n ∈ ω. Making use of this fact, it is clear that, indeed, all EG-spaces have countable fan tightness for finite sets. With everything we have already said, it is natural to conjecture that the spaces X with C_{p}(X, 2) Menger are subspaces of C_{p}(Y) where Y^{n} is Menger for each n ∈ ω. The following result shows that this is true as long as X′ is compact.

Theorem 4.8. Let X be a subspace of C_{p}(Y) where Y^{k} is Menger for every k ∈ ω. If X′ is compact, then C_{p}(X, 2)^{n} is Menger for each n ∈ ω.

Proof. We adapt, for our purposes, the respective part of the proof of Theorem 4.15 from [7]. We only show that C_{p}(X, 2) is Menger; the (n ≥ 2)-cases are shown similarly. For each n ∈ ω, we define

F_{n} = {φ ∈ 2^{X} : ∃(y_{1}, …, y_{n}) ∈ Y^{n}∀f ∈ X′(φ[[f; y_{1}, …, y_{n}; 1/n]] = {φ(f)})},

where [f; y_{1}, …, y_{n}; 1/n] = {g ∈ C_{p}(Y) : ∀i(1 ≤ i ≤ n → |f(y_{i}) − g(y_{i})| < 1/n)}. Each F_{n} is Menger. Indeed, for each n ∈ ω, F_{n} coincides with π_{2}[S_{n}], where π_{2} is the projection of Y^{n} × 2^{X} over 2^{X} and

S_{n} = {(y_{1}, …, y_{n}, φ) ∈ Y^{n}×2^{X} : ∀f ∈ X′(g ∈ [f; y_{1}, …, y_{n}; 1/n] → φ(f) = φ(g))}.

Now, Proposition 3.5 ensures that Y^{n} × 2^{X} is Menger. Then, to prove that F_{n} is Menger, by Proposition 3.2, it is sufficient to show that S_{n} is Menger. And to do this we proceed as follows: Let (y_{1}^{0}, …, y_{n}^{0}, φ_{0}) ∈ Y^{n} × 2^{X} \ S_{n}. This means that there are f_{0} ∈ X′ and g_{0} ∈ X such that g_{0} ∈ [f_{0}; y_{1}^{0}, …, y_{n}^{0}; 1/n] and φ_{0}(f_{0}) ≠ φ_{0}(g_{0}). Then, the open set

(∏_{i=1}^{n} |f_{0} − g_{0}|^{−1}[[0, 1/n)] × {φ ∈ 2^{X} : φ(f_{0}) = φ_{0}(f_{0}) ∧ φ(g_{0}) = φ_{0}(g_{0})}

of Y^{n} ×2^{X} contains the point (y_{1}^{0}, …, y_{n}^{0}, φ_{0}) and does not intersect S_{n}. Therefore, S_{n} is a closed subset of the Menger space Y^{n} × 2^{X}, and so it is Menger.

Now we will show that C_{p}(X, 2) is equal to ⋃_{n∈ω} F_{n} and, by Proposition 3.4, our theorem will be proved. First observe that C_{p}(X, 2) is contained in ⋃_{n∈ω} F_{n}. In fact, fix a function φ ∈ C_{p}(X, 2). Since φ is continuous, for each f ∈ X′ we can take a neighborhood U_{f} of f in C_{p}(Y) such that φ(g) = φ(f) if g ∈ U_{f} ∩ X. Now, for each f ∈ X′ there are n_{f} ∈ ω and y_{1}^{f}, …, y_{n_{f}}^{f} ∈ Y for which

f ∈ [f; y_{1}^{f}, …, y_{n_{f}}^{f}; 1/n_{f}] ∩ X ⊂ U_{f} ∩ X.

Since X′ is a compact space, there are points f_{0}, …, f_{k} ∈ X′ such that

X′ ⊂ [f_{0}; y_{1}^{f_{0}}, …, y_{n_{f_{0}}}^{f_{0}} ; 1/(2n_{f_{0}})] ∪ ⋯ ∪ [f_{k}; y_{1}^{f_{k}}, …, y_{n_{f_{k}}}^{f_{k}}; 1/(2n_{f_{k}})].

For each f ∈ X′ we take

V_{f} = [f; y_{1}^{f_{0}}, …, y_{n_{f_{0}}}^{f_{0}} , y_{1}^{f_{1}}, …, y_{n_{f_{1}}}^{f_{1}} , …, y_{1}^{f_{k}}, …, y_{n_{f_{k}}}^{f_{k}}; 1/l] ∩ X

with l = 2(n_{f_{0}} + ⋯ + n_{f_{k}}). It is evident that the collection V = {V_{f} : f ∈ X′} covers X′. We have that V refines {U_{f} ∩ X : f ∈ X′}. Indeed, if f ∈ X′, f must belong to

[f_{j}; y_{1}^{f_{j}}, …, y_{n_{f_{j}}}^{f_{j}}; 1/(2n_{f_{j}})]

for some j ∈ {1, …, k}. Then, if g ∈ V_{f} we have

|g(y_{i}^{f_{j}}) − f_{j}(y_{i}^{f_{j}})| ≤ |g(y_{i}^{f_{j}}) − f(y_{i}^{f_{j}})| + |f(y_{i}^{f_{j}}) − f_{j}(y_{i}^{f_{j}})| < \frac{1}{l} + \frac{1}{2n_{f_{j}}} ≤ \frac{1}{n_{f_{j}}} .

Therefore, g ∈ [f_{j}; y_{1}^{f_{j}}, …, y_{n_{f_{j}}}^{f_{j}}; 1/(n_{f_{j}})] ∩ X, and the latter set is contained in U_{f_{j}} ∩ X.

Now we prove that φ belongs to F_{l}. First note the following: if f ∈ X′, g ∈ X and they satisfy |f(x) − g(x)| < 1/l for all x ∈ {y_{1}^{f_{0}}, …, y_{n_{f_{0}}}^{f_{0}} , …, y_{1}^{f_{k}}, …, y_{n_{f_{k}}}^{f_{k}} }, then g ∈ V_{f} and, consequently, f, g ∈ U_{h} ∩ X for some h ∈ X′. Because of the choice of U_{h}, φ(g) = φ(h) = φ(f). Therefore, for each f ∈ X′, if g ∈ [f; y_{1}^{f_{0}}, …, y_{n_{f_{0}}}^{f_{0}} , …, y_{1}^{f_{k}}, …, y_{n_{f_{k}}}^{f_{k}}; 1/l], φ(f) = φ(g). This shows that φ ∈ F_{l}.

Finally we prove that, for each n ∈ ω, F_{n} ⊂ C_{p}(X, 2); that is, each element of F_{n} is a continuous function. Let φ ∈ F_{n} and f ∈ X. If f is an isolated point of X then φ is continuous at f. Suppose f ∈ X′. By definition of F_{n}, there is (y_{1}, …, y_{n}) ∈ Y^{n} such that φ[[f; y_{1}, …, y_{n}; 1/n] ∩ X] = {φ(f)}. Since [f; y_{1}, …, y_{n}; 1/n] ∩ X is an open subset of X containing f, φ is continuous at f. □

Given a space X, C_{p}^{∗}(X, ω) denotes the subspace of C_{p}(X) consisting of all bounded functions with values in ω. 4

Corollary 4.9. Let X be a space and suppose that X′ is compact. Then the following statements are equivalent:

(a) C_{p}(X, 2)^{n} is Menger for each n ∈ ω;

(b) X ⊂ C_{p}(Y) for some space Y such that Y^{n} is Menger for each n ∈ ω;

(c) C_{p}^{∗}(X, ω) is Menger.

Proof. The equivalence of (a) and (c) is immediate from the fact that C_{p}^{∗}(X, ω) = ⋃_{n∈ω} C_{p}(X, n) (see Propositions 3.2 and 3.4). (a) implies (b) follows from Theorem 4.8 and the fact that C_{p}(X, 2)^{n} is homeomorphic to C_{p}(X, 2^{n}) for each n ∈ ω. And the proof of (b) implies (a) is as follows: For each x ∈ X, we define x̃ : C_{p}(X, 2) → 2 as x̃(f) = f(x). It is not difficult to show that the function x ↦ x̃ is an embedding of X into C_{p}(C_{p}(X, 2)). Then Y = C_{p}(X, 2) is the required space. □

A subspace Y of a space X is bounded in X if for every continuous function f : X → ℝ, f ↾ Y is a bounded function, or equivalently, if every sequence of open sets in X, which meets Y, has an accumulation point in X.

Since ℚ and ω^{ω} are second countable, C_{p}(ℚ, 2) and C_{p}(ω^{ω}, 2) are Lindelöf. The following result rules out the possibility that C_{p}(ℚ, 2) and C_{p}(ω^{ω}, 2) satisfy the Menger property.

Theorem 4.10. If C_{p}(X, 2) is Menger, then X′ is bounded in X.

Proof. We proceed by contradiction. Suppose that there exists a sequence ⟨ O_{n} : n ∈ ω ⟩ of open subsets of X without accumulation points in X such that O_{n} ∩X′ ≠ ∅. We can suppose without loss of generality that each element of the sequence is open and closed, and that any two different elements of this sequence are disjoint. Let Y = X\⋃_{n∈ω} O_{n}. Since the sequence ⟨ O_{n} : n ∈ ω ⟩ does not have accumulation points, Y is open and closed. Moreover, the family {O_{n} : n ∈ ω} ∪ {Y } forms a partition of X in clopen subsets of X. Then C_{p}(X, 2) is homeomorphic to

( ∏_{n∈ω} C_{p}(O_{n}, 2)) × C_{p}(Y, 2).

For each n ∈ ω, since C_{p}(X, 2) is Menger, C_{p}(O_{n}, 2) is Menger, and hence, Lindelöf. On the other hand, since each O_{n} contains a non-isolated point of X, C_{p}(O_{n}, 2) is a proper dense subspace of 2^{O_{n}}. So, since C_{p}(O_{n}, 2) is Lindelöf, then C_{p}(O_{n}, 2) is not countably compact; in particular, it contains a countable discrete closed subspace D_{n}. In this manner, ∏_{n∈ω} D_{n} is a closed subspace of C_{p}(X, 2), and given that C_{p}(X, 2) is Menger, ∏_{n∈ω} D_{n} is Menger, which is impossible since it is homeomorphic to ω^{ω}. □

Corollary 4.11. If X is a normal space and C_{p}(X, 2) is Menger, then X′ is countably compact.

Proof. By Theorem 4.10, X′ is bounded in X. Since X is a normal space and X′ is a closed subset of X, X′ is pseudocompact. Again, by the normality of X, X′ is countably compact. □

Corollary 4.12. Let X be a Lindelöf space. Then C_{p}(X, 2)^{n} is Menger for any n ∈ ω if and only if X′ is compact and X ⊂ C_{p}(Y) for some space Y such that Y^{n} is Menger for each n ∈ ω.

Proof. If C_{p}(X, 2)^{n} is Menger for any n ∈ ω, then C_{p}(X, n) is Menger for each n ∈ ω and, by Corollary 4.9, X ⊂ C_{p}(Y) for some space Y such that Y^{n} is Menger for each n ∈ ω. Furthermore, applying Corollary 4.11, X′ is countably compact and hence compact since X is a Lindelöf space. The proof of the converse is a consequence of Theorem 4.8. □

A space is σ-pseudocompact if it is the countable union of pseudocompact subspaces. The following theorem appears in [1, III.4.23].

Theorem 4.13 ([1]). If X contains a dense σ-pseudocompact subspace, then every countably compact subspace of C_{p}(X) is compact.

Corollary 4.14. Let X be a space with iw(X) = ω. Then the following statements are equivalent.

(a) X′ is compact and X ⊂ C_{p}(Y) for some space Y such that Y^{n} is Menger for any n ∈ ω;

(b) C_{p}(X, 2)^{n} is Menger for any n ∈ ω and X is a normal space.

Proof. By Theorem 4.8, (a) implies that C_{p}(X, 2)^{n} is Menger for any n ∈ ω. Moreover, since X is regular and X′ is compact, then X is normal. Now suppose (b), since X ⊂ C_{p}(C_{p}(X, 2)), to prove (a) it is sufficient to show that X′ is compact. The normality of X and Corollary 4.11 imply that X′ is a countably compact space. Given that iw(X) = ω, C_{p}(X, 2) is separable. By Theorem 4.13, X′ is compact. □

On metric spaces we have the following.

Theorem 4.15. Let X be a metrizable space. Then the following statements are equivalent.

(a) C_{p}(X, 2) is Menger;

(b) C_{p}(X, 2)^{n} is Menger for each n ∈ ω;

(c) C_{p}(X, 2) is σ-compact;

(d) X′ is compact.

Proof. Since every metrizable space is an EG-space (see [1, IV.1.25]), X is an EGspace. Then, if X′ is countably compact, it is compact being X metrizable; so, X′ is Eberlein compact, and by Corollary 4.12 in [7], C_{p}(X, 2) is σ-compact. This proves that (d) implies (c). Clearly (c) implies (b) and (b) implies (a). Finally, by the Corollary 4.11, (a) implies (d). □

5. The Menger property on C_{p}(L, 2) when L is a GO-space

A space L is a GO-space (Generalized Ordered space) if it is a subspace of a linearly ordered topological space. Observe that if L is a countable GO-space, then L is zero-dimensional, separable and metrizable (and C_{p}(L, 2) is Lindelöf). Then, by the Theorem 4.15 we obtain:

Proposition 5.1. Let L be a countable GO-space. Then the following statements are equivalents.

(a) C_{p}(L, 2) is Menger;

(b) C_{p}(L, 2) is σ-compact;

(d) L′ is compact.

Now we are going to characterize the Menger property on C_{p}(L, 2) when L is an uncountable GO-space. We will follow some notations, terminology and constructions due to R.Z. Buzyakova in [6]. First we will review a construction of the Dedekind completion of a given GO-space L.

Definition 5.2. An ordered pair ⟨ A, B ⟩ of disjoint closed subsets of a GO-space L is called a Dedekind section if A ∪ B = L, max A or min B do not exist, and A is to the left of B; that is, for every a ∈ A and b ∈ B, a < b holds. A pair ⟨ L, ∅ ⟩ (⟨ ∅, L ⟩) is also a Dedekind section if max L (min L) does not exist.

Definition 5.3. The Dedekind completion of L, denoted by cL, is constructed as follows. The set cL is the union of L and the set of all Dedekind sections of L. The order on cL is natural: the order on cL among elements of L coincides with the order on L of these elements. If x ∈ L and y = ⟨ A, B ⟩ ∈ cL \ L then x is less (greater) than y if x ∈ A (x ∈ B). If x = ⟨ A_{1}, B_{1} ⟩ and y = ⟨ A_{2}, B_{2} ⟩ are elements of cL \ L, then x is less than y if A_{1} is a proper subset of A_{2}. Consider now cL with the order topology generated by the order just defined. We will denote by ∞ and −∞ the supremum and infimum, respectively, of cL.

Observe that for every GO-space L, cL is a compact linearly ordered space. For a given GO-space L we consider the space T (L):

Definition 5.4. An element x ∈ cL is in T (L) if and only if x ∈ cL \ L, or x ∈ L and either x is the smallest or the greatest element in L or x has an immediate succesor in L. Points of T (L) that are in L are declared isolated. The other points inherit base neighborhoods from the Dedekind completion cL.

Observe that T (L) is a GO-space. Indeed, T (L) can be obtained from cL as follows. For each x ∈ L that has an immediate succesor x^{+} in L, insert a new point p_{x} between x and x^{+}. If x ∈ L is the smallest element of L, we add a point p_{−∞} to the left of x; and if x ∈ L is the greatest element of L, we add a point p_{∞} to the right of x. The resulting space is a compact linearly ordered topological space containing cL as a closed subspace. The subspace of this structure that consists of all p_{x}’s and cL \ L is a copy of T (L). Thus we can think of T (L) as a GO-space with the order inherited from cL. R.Z. Buzyakova presents in [6] some examples of T (L) for some particular GO-spaces.

If x_{1}, …, x_{n} ∈ cL and −∞ ≤ x_{1} ≤ ⋯ ≤ x_{n} ≤ ∞ then by f = f_{x_{1},...,x_{n}}^{0} we denote the function from L to 2 defined by

f[(x_{i}, x_{i+1}] ∩ L] = {i mod 2},

for each i ∈ {1, …, n}. The rightmost formula is simply {1} if n is odd and {0} otherwise. The functions f_{x_{1},...,x_{n}}^{1} are defined similarly by changing {0} with {1} in the above formulas.

Definition 5.5. A function f from X to 2 belongs to S_{p}(L, 2) if and only if there exists −∞ ≤ x_{1} ≤ ⋯ ≤ x_{n} ≤ ∞ in T (L) such that f = f_{x_{1},...,x_{n}}^{0} or f = f_{x_{1},...,x_{n}}^{1}

Observe that [−∞, x_{1}] ∩ L, (x_{1}, x_{2}] ∩ L, …, (x_{n}, ∞] ∩ L are clopen because x_{1}, …, x_{n} ∈ T (L). Therefor f_{x_{1},...,x_{n}}^{0} is continuous and S_{p}(L, 2) ⊂ C_{p}(L, 2). The topology of S_{p}(L, 2) is the topology of subspace of C_{p}(L, 2). For each n ≥ 1, we define

S_{n} = {f ∈ S_{p}(L, 2) : ∃x_{1}, …, x_{n} ∈ T (L)(f = f_{x_{1},...,x_{n}}^{0} ∨ f = f_{x_{1},...,x_{n}}^{1})}.

Observe that S_{p}(L, 2) = ⋃_{1≤n} S_{n}, and S_{p}(L, 2) = C_{p}(L, 2) if L is countably compact.

We are going to denote by S^{∗} the subspace {f_{x}^{1} : x ∈ T (L)} of S_{1}.

Lemma 5.6 ([6]). Let L be a GO-space. Then for any f ∈ S_{p}(L, 2), there exist f_{1}, …, f_{k} ∈ S^{∗} such that f = f_{1} + ⋯ + f_{k}.

More properties on S^{∗} are given in [6]. One of these is the following:

Theorem 5.7 ([6]). The subspace S^{∗} of S_{p}(L, 2) is homeomorphic to T (L).

For a countably compact GO-space L, R.Z. Buzyakova proves in [6] that C_{p}(L, 2) is Lindelöf if and only if T (L) is Lindelöf. We show that in fact this is a sufficient condition in order to have C_{p}(L, 2) Menger.

Theorem 5.8. Let L be a first countable GO-space without isolated points. The following statements are equivalent.

(a) C_{p}(L, 2) is Lindelöf and L is countably compact,

(b) T (L) is Lindelöf and L is countably compact,

(c) T (L)^{n} is Menger for each n ∈ ω and L is countably compact,

(d) C_{p}(L, 2)^{n} is Menger for each n ∈ ω,

(e) C_{p}^{∗}(L, ω) is Menger,

(f) C_{p}(L, 2) is Menger.

Proof. The equivalence (a) ⇔ (b) is Theorem 4.1 in [6]. We suppose (b) and we are going to prove (c). Let us show that T (L) is a P -space. For each n ∈ ω, let U_{n} be an open subset of T (L). We are going to prove that F = ⋂_{n∈ω} U_{n} is open. Take any x in this intersection. If x ∈ L then x is isolated in T (L). If x ∉ L then, due to the countably compactness of L, x is unreachable by nontrivial countable sequence in cL, and therefore, in T (L). In both cases, we conclude that x is in the interior of F. This shows that T (L) is a P -space. Then T (L)^{n} is a P -space for n ∈ ω. Applying the Noble’s theorem [16], a countable power of a Lindelöf P -space is Lindelöf, T (L)^{ω} is Lindelöf, and hence, T (L)^{n} is Lindelöf for any n ∈ ω. But Lindelöf property agrees with Menger property in P -spaces (see Proposition 3.6). Then T (L)^{n} is Menger for any n ∈ ω.

Now suppose (c). Given that T (L) is homeomorphic to S^{∗} (see Theorem 5.7) and the countable union of Menger spaces is Menger, the topological sum ⨁_{k∈ω}(S^{∗})^{k} is Menger. Moreover, every finite power of this space is Menger. Besides, if we define the continuous function F : ⨁_{k∈ω}(S^{∗})^{k} → S_{p}(L, 2) as F(F) = f_{1} + ⋯ + f_{k} where F = (f_{1}, …, f_{k}) ∈ (S^{∗})^{k}. Then by Lemma 5.6, F is surjective. Then, for each n ∈ ω, the function F^{n} : (⨁_{k∈ω}(S^{∗})^{k})^{n} → S_{p}(L, 2)^{n} defined by F^{n}(F_{1}, …, F_{n}) = (F(F_{1}), …, F(F_{n})) is a surjective continuous function. Thus, S_{p}(L, 2)^{n} = C_{p}(L, 2)^{n} is Menger. This shows that (c) implies (d).

(d) implies (e) is trivial. Since C_{p}(L, 2) is a close subspace of C_{p}^{∗}(L, ω), by the Proposition 3.2, (e) implies (f). Finally, if we suppose (f), then, by Corollary 4.11, L = L′ is countably compact and clearly C_{p}(L, 2) is Lindelöf. This proves that (f) implies (a). □

Corollary 5.9. Let L be a first countable countably compact GO-space without isolated points. Then, C_{p}(L, 2) is Lindelöf if and only if C_{p}(L, 2) is Menger.

Problem 5.10. Determine when C_{p}(L, 2) is Menger when L is a first countable GO-space (without any restriction about the isolated points in L).

6. The Menger property on C_{p}(X, 2) when X is a subspace of ordinals

By a subspace of ordinals we are refering to a subspace of an ordinal α. As we have already said, the set of ordinals lower than an ordinal α endowed with its order topology is denoted by α. As a corollary of Proposition 5.1 we have the following.

Corollary 6.1. Let α ∈ ω_{1}. Then the following statements are equivalent. (a) C_{p}(α, 2) is Menger,

(b) C_{p}(α, 2) is σ-compact,

(c) α is a succesor ordinal.

If X is normal and C_{p}(X, 2) is Menger, then X′ is countably compact and X has countable fan tightness (see Corollary 4.11 and Proposition 4.1). Then, when X is a subspace of ordinals and C_{p}(X, 2) is Menger, X′ must be countably compact and X is first countable. In the following statements we see that these properties are enough.

The proof of the following theorem was suggested to the referee by Professor Piotr Szewczak. His proof is simpler than one we gave in a previous version of this paper.

Theorem 6.2. Let X be a subspace of ordinals and n ∈ ω \ {0}. Then C_{p}(X, 2)^{n} is Menger if and only if C_{p}(X, 2)^{n} is Lindelöf and X′ is countably compact.

Proof. Remember that C_{p}(X, 2)^{n} is homeomorphic to C_{p}(X, 2^{n}). We are going to show our theorem when n = 1 (the proof for 2^{n} instead of 2 is similar). It is obvious that if C_{p}(X, 2) is Menger, then C_{p}(X, 2) is Lindelöf. Moreover, by Corollary 4.11, X′ is countably compact. Reciprocally, we suppose that C_{p}(X, 2) is Lindelöf and X′ is countably compact. We will show that C_{p}(X, 2) is Menger. We will prove this fact by induction over α = sup X. Let us assume that the statement is true for every β < α. Let δ = sup X′.

Case I. If δ < α, then X′ ⊂ Z = X ∩(δ+1), Z is clopen in X and X \Z is clopen and discrete in X. Therefore X = Z ⊕(X \Z) and C_{p}(X, 2) ≅ C_{p}(Z, 2)×2^{X\Z}. It is easy to see that C_{p}(Z, 2) is Lindelöf as a closed subspace of C_{p}(X, 2) and Z′ = X′ so it is countably compact. From the assumption we have that C_{p}(Z, 2) is Menger and by compactness of 2^{X\Z} the space C_{p}(X, 2) is also Menger.

Case II. If δ = α and there is in X an increasing countable sequence ⟨ α_{n} : n ∈ ω ⟩ which converges to α. Then by δ = α and countably compactness of X′ we infer that α ∈ X′. Let us observe that family {X ∩ (α_{n}, α] : n ∈ ω} forms a base at α. Since every f ∈ C_{p}(X, 2) is continuos at α and α ∈ X′ so there is k ∈ ω such that f[(α_{n}, α]] = {f(α)}. Let A_{n}^{j} = {f ∈ C_{p}(X, 2) : ∀x(x ∈ (α_{n}, α] → f(x) = j)} for every n ∈ ω and j ∈ 2. We have that

C_{p}(X, 2) = ⋃{A_{n}^{j} : n ∈ ω ∧ j ∈ 2}

Every A_{n}^{j} is homeomorphic to C_{p}(Z_{n}, 2) where Z_{n} = X ∩(α+1). Since Z_{n} is clopen in X we can easily verify that C_{p}(Z_{n}, 2) is Lindelöf and Z′_{n} is countably compact. Now it follows from inductive assumption that A_{n}^{j} ≅ C_{p}(Z_{n}, 2) is Menger. Because C_{p}(X, 2) = ⋃{A_{n}^{j} : n ∈ ω ∧ j ∈ 2} we conclude that C_{p}(X, 2) is Menger.

Case III. If δ = α and the cofinality of α in X is not countable. Let ⟨ U_{n} : n ∈ ω ⟩ be a sequence of countable open covers of C_{p}(X, 2) consisting of open baisis sets. Let us observe that each element U ∈ U_{n} is in the form U = ∏_{x∈X} U(x) ∩ C_{p}(X, 2) where U(x) ≠ 2 only if x ∈ X_{U} ⊂ X, where X_{U} is finite. Let us observe that there is some β ∈ X such that ⋃{X_{U} : ∃n (n ∈ ω ∧ U ∈ U_{n})} ⊂ X ∩(β +1) = Z. Clearly β = sup Z. Then

(∗) ∀U ∀x (∃n (n ∈ ω ∧ U ∈ U_{n}) ∧ x ∈ X \ Z → U(x) = 2).

It is easy to see that Z is a clopen subset of X and Z′ = X ∩ (β + 1). Hence Z′ is countably compact as a closed subset of countably compact space X′. Since C_{p}(X, 2) is homeomorphic to C_{p}(Z, 2) × C_{p}(X \ Z, 2) we have that C_{p}(Z, 2) as a closed subset of C_{p}(X, 2) is Lindelöf. By inductive assumption C_{p}(Z, 2) is Menger. Now let U′_{n} = {U ∩ C_{p}(Z, 2) : U ∈ U_{n}} for every n ∈ ω. Then ⟨ U′_{n} : n ∈ ω ⟩ is a sequence of open cover of C_{p}(Z, 2). Therefore, there are V′_{n} ∈ [U′_{n}]^{<ω} such that ⋃{V′_{n} : n ∈ ω} covers C_{p}(Z, 2). For every n ∈ ω pick V_{n} ∈ [U_{n}]^{<ω} such that V_{n} = {U ∩ C_{p}(Z, 2) : U ∈ V′_{n}}. By (∗) we have that ⋃{V_{n} : n ∈ ω} covers C_{p}(X, 2). □

It is shown in [4] that for every countably compact first countable subspace X of ordinals, C_{p}(X, 2)^{n} is Lindelöf for each n ∈ ω. Therefore:

Corollary 6.3. For any countably compact first countable subspace X of ordinals, C_{p}(X, 2)^{n} is Menger for each n ∈ ω.

It is shown in [5] that every first countable subspace X of ordinals with countable extent has C_{p}(X, 2)^{n} is Lindelöf for each n ∈ ω. So, we obtain:

Corollary 6.4. Let X be a first countable subspace of ordinals with countable extent. Then C_{p}(X, 2)^{n} is Menger for each n ∈ ω if and only if X′ is countably compact.

Corollary 4.11 shows that in the class of normal spaces X, if C_{p}(X, 2) is Menger, then X′ is countably compact. With the same hypotheses we cannot imply the compactness of X′. Indeed, by Corollary 6.3, C_{p}(ω_{1}, 2) is Menger.

Observe, on the one hand, that the ordinal number X = ω · ω is countable, metrizable ordinal subspace such that C_{p}(X, 2) is Lindelöf but C_{p}(X, 2) is not Menger (see Theorem 6.2). So, it is not possible to add the statement “C_{p}(X, 2) is Lindelöf” in the list of equivalent claims neither in Theorem 4.15 nor in Proposition 5.1, nor in Corollary 6.1 (compare with Corollary 5.9). On the other hand, the converse of Corollary 6.3 is not true. Indeed, C_{p}(ω, 2) is Menger and ω is not countably compact. A non discrete example of the same fact is the countable metrizable ordinal subspace Y = (ω · ω + 1) \ {ω}.

Moreover, it is natural to conjecture that the class of subspaces of ordinals X for which C_{p}(X, 2) is Menger is equal to the class of ordinal subspaces which are the topological sum of two subspaces, one of them a discrete subspace and the other a first countable countably compact ordinal subspace. This is not true. In fact, consider X = {ω_{1} · n : n ≤ ω} ∪ ⋃_{n∈ω}{(ω_{1} · n) + m : m ∈ ω}. We have that C_{p}(X, 2) is Menger but X cannot be expressed as the sum of a discrete subspace plus a countably compact first countable ordinal subspace. Also, it is natural to conjecture that Corollary 6.3 is valid for any GO-space (or LOT S) not only for subspaces of ordinals, but Example 4.5 shows that this is false.

In [5] R. Z. Buzyakova asks if C_{p}(X, 2) is Lindelöf when X is a first countable subspace of ordinals and X′ is countably compact. We ask the same question in the following form (see Theorem 6.2):

Problem 6.5. Is there a first countable ordinal subspace X with X′ countably compact such that C_{p}(X, 2) is not Menger?

7. The Menger property on C_{p}(X, 2) when X is a countable simple space

A space X is called simple if X has exactly one non-isolated point. For any filter F on ω, we define the space ω ∪ {F} as follows: any n ∈ ω is declared isolated and the sets A ∪ {F}, where A ∈ F, form a base of neighborhoods of F. Any countable simple space is homeomorphic to ω ∪ {F} for a filter F on ω.

It is proved in [1, III.3.3] that if A_{τ} denotes the one-point compactification of the discrete space of cardinality τ, then A_{τ} is Eberlein compact. And therefore C_{p}(A_{τ} , 2) is Menger. It is also shown in [1, III.1.7] that if F ∈ ω^{∗}, ω ∪ {F} is not an EG-space; that is, ω ∪ {F} cannot be embedded in a space C_{p}(Y) where Y is compact. The following results shows that, under certain conditions, ω ∪ {F} can be embedded in a space C_{p}(Y) for some space Y for which Y^{n} is Menger for every n ∈ ω. These conditions are set in the following definitions.

Definition 7.1 ([13]). An ultrafilter F ∈ ω^{∗} is a strong P -point if for any sequence ⟨ C_{n} : n ∈ ω ⟩ of compact subspaces of F (considering F as a subset of 2^{ω} with the product topology) there is an interval partition ⟨ I_{n} : n ∈ ω ⟩ of 2^{ω} such that for each choice of X_{n} ∈ C_{n} we have

⋃_{n∈ω}(I_{n} ∩ X_{n}) ∈ F.

Given a filter F on ω we define F^{<ω} to be the filter on [ω]^{<ω} \ {∅} generated by {[F]^{<ω} \ {∅} : F ∈ F}. Note that the filter F^{<ω} on [ω]^{<ω} \ {∅} is not an ultrafilter even if F is.

Definition 7.2 ([3]). A filter F on a countable set S is a P^{+}-filter if for any ⊂- descending sequence ⟨ X_{n} : n ∈ ω ⟩ ⊂ F^{+}, there is an X ∈ F^{+} such that X ⊂^{∗} X_{n} for all n ∈ ω, where F^{+} = {X ⊂ S : S \ X ∉ F}.

The elements of F^{+} are called positive sets (with respect to F). Then, a filter F is a P^{+}-filter if every decreasing sequence of positive sets has a positive pseudointersection. The definition of a strong P -point that we will use is the following.

Theorem 7.3 ([3]). An ultrafilter F ∈ ω^{∗} is a strong P -point if and only if F^{<ω} is a P^{+}-filter.

The following result was conjectured by M. Huršák and is the key to characterize the Menger property on C_{p}(ω ∪ {F}, 2).

Proposition 7.4. Let F be a filter on ω. The space X = ω ∪ {F} has countable fan tightness for finite sets if and only if F^{<ω} is a P^{+}-filter.

Proof. First note that P ∈ (F^{<ω})^{+} if and only if P is a π-network at F in X. Suppose that F^{<ω} is a P^{+}-filter. Let ⟨ P_{n} ⊂ [X]^{<ω} : n ∈ ω ⟩ be a decreasing sequence of π-networks at F (see Lemma 4.3). Given that F^{<ω} is a P^{+}-filter, and ⟨ P_{n} ⊂ [X]^{<ω} : n ∈ ω ⟩ is a decreasing sequence of positive sets with respect to F^{<ω}, ⟨ P_{n} : n ∈ ω ⟩ has a positive pseudointersection P ∈ (F^{<ω})^{+}. Since P \ P_{0} is finite, P ∩P_{0} is a π-network at F. Then, if we suppose that ⋂_{n∈ω} P_{n} = {p_{n} : n ∈ ω}, and define K_{n} = (P ∩P_{n}\P_{n+1})∪{p_{n}} for each n ∈ ω, ⋃_{n∈ω} K_{n} = P ∩P_{0} is a π-network at F. Observe that K_{n} is finite because (P ∩ P_{n} \ P_{n+1}) ∪ {p_{n}} ⊆ (P \ P_{n+1}) ∪ {p_{n}} and P is a pseudointersection of the family {P_{n} : n < ω}.

Reciprocally, suppose that ω ∪ {F} has countable fan tightness for finite sets. Let ⟨ P_{n} : n ∈ ω ⟩ be a decreasing sequence of positive sets. Since ⟨ P_{n} : n ∈ ω ⟩ is a sequence of π-networks at F, there is a sequence of finite sets ⟨ F_{n} : n ∈ ω ⟩ such that P = ⋃_{n∈ω} F_{n} is a π-network at F and F_{n} ⊂ P_{n} for each n ∈ ω. Then P is a positive set and, since P \ P_{n+1} ⊂ F_{0} ∪ ⋯ ∪ F_{n}, P is a pseudointersection of ⟨ P_{n} : n ∈ ω ⟩. □

Corollary 7.5. Let F ∈ ω^{∗}. The subspace ω ∪ {F} of βω has countable fan tightness for finite sets if and only if F is a strong P -point.

Theorem 7.6. Let F be a filter on ω and X = ω ∪ {F}. Then, the following statements are equivalent:

(a) C_{p}(X, 2) is Menger;

(b) C_{p}(X, 2)^{n} is Menger for any n ∈ ω;
(c) C_{p}^{∗}(X, ω) is Menger;
(d) F^{<ω} is a P^{+}-filter.

Proof. If C_{p}(X, 2) is Menger, then, by Propositions 4.4 and 7.4, F^{<ω} is a P^{+}-filter. Now assume that F^{<ω} is a P^{+}-filter in ω. We are going to show that C_{p}(X, 2) is Menger (the proof for n instead of 2 is similar). For each k ∈ 2 and F ⊂ ω, we define A_{F}^{k} = {f ∈ 2^{ω} : ∀x(x ∈ F → f(x) = k)}. Note that C_{p}(X, 2) is homeomorphic to the subspace ⋃{A_{F}^{k} : F ∈ F ∧ k ∈ 2} of 2^{ω}. Then, to see that C_{p}(X, 2) is Menger, by Proposition 3.4, it is enough to show that ⋃{A_{F}^{k} : F ∈ F} is Menger for each k ∈ 2. However, ⋃{A_{F}^{k} : F ∈ F} is homeomorphic to ⋃{A_{F}^{m} : F ∈ F} for k, m ∈ 2. So, it is enough to prove that A = ⋃{A_{F}^{0} : F ∈ F} is Menger. To simplify the notations, we write A_{F} to mean A_{F}^{0} and, if F is a single point x, we write A_{x} instead of A_{F}.

Let ⟨ U_{n} : n ∈ ω ⟩ be a sequence of countable covers of A such that U_{n+1} refines U_{n} for each n ∈ ω (see Lemma 3.3). We can suppose that each U_{n} is closed under finite unions. For each open subset of A, let Y_{U} = {H ∈ [ω]^{<ω} : A_{H} ⊂ U}. And we define Z_{n} = ⋃_{U∈U_{n}} Y_{U} for each n ∈ ω. In view of the fact that U_{n+1} refines U_{n}, Z_{n+1} ⊂ Z_{n}. Moreover, Z_{n} is a positive set. Indeed, if F ∈ F, since A_{F} is compact, there is an element U ∈ U_{n} containing A_{F}. Given that A_{F} = ⋂_{x∈F} A_{x} and A_{x} is compact, there is H ∈ [F]^{<ω} such that A_{H} = ⋂_{x∈H} A_{x} ⊂ U. Then [F]^{<ω} ∩Z_{n} ≠ ∅. Now, since F^{<ω} is a P^{+}-filter, the sequence of positive sets ⟨ Z_{n} : n ∈ ω ⟩ has a positive pseudointersection Z̃ ∈ (F^{<ω})^{+}. Suppose that ⋂_{n∈ω} Z_{n} = {b_{n} : n ∈ ω}, then we define, for each n ∈ ω, P_{n} = (Z̃ ∩ Z_{n} \ Z_{n+1}) ∪ {b_{n}}. In the same way as in Proposition 7.4 we infer that P_{n} is finite for every n ∈ ω. In this manner Z = ⋃_{n∈ω} P_{n} = Z̃∩Z_{0} ∈ (F^{<ω})^{+}. For each n ∈ ω and H ∈ P_{n}, we choose U_{H} ∈ U_{n} such that A_{H} ⊂ U_{H}. We define F_{n} = {U_{H} : h ∈ P_{n}} for each n ∈ ω. Given f ∈ A, there is F ∈ F such that f ∈ A_{F}. Since Z is a positive set, [F]^{<ω} \ {∅} intersects Z and, hence, intersects some P_{n}. Consequently, if H ∈ ([F]^{<ω} \ {∅}) ∩ P_{n}, then A_{F} ⊂ A_{H} ⊂ U_{H}. This proves that f ∈ U_{H}. That is, ⋃_{n∈ω} F_{n} is a cover of A.

The implication (b) ⇒ (c) is a consequence of Proposition 3.4 and the equality C_{p}^{∗}(X, ω) = ⋃_{n∈ω} C_{p}(X, n). The implication (c) ⇒ (b) is a consequence of Proposition 3.2 and the fact that each C_{p}(X, n) is a closed subset of C_{p}^{∗}(X, ω). □

As a consequence of Theorems 7.3 and 7.6 we conclude:

Corollary 7.7. Let F ∈ ω^{∗} and X the subspace ω ∪{F} of βω. Then the following statements are equivalent

(a) C_{p}(X, 2) is Menger;

(b) C_{p}(X, 2)^{n} is Menger for any n ∈ ω;

(c) C_{p}^{∗}(X, ω) is Menger;

(d) F is a strong P -point.

As previously mentioned, Example III.1.7 in [1] shows that ω ∪ {F} is not an EG-space and, by the Theorem 4.16 in [7], C_{p}(ω ∪ {F}, 2) is not σ-compact. Then, by Corollary 7.7, if F is a strong P -point, C_{p}(ω ∪ {F}, 2) is a Menger space which is not σ-compact.

8. The Menger property on C_{p}(Ψ(A), 2)

An almost disjoint family of subsets of ω is a collection A of subsets of ω such that each element in A is infinite, and if A, B ∈ A, |A ∩ B| < ℵ_{0}. An almost disjoint family A is maximal if it is not proper subfamily of an another almost disjoint family. For an infinite maximal almost disjoint family (mad) A on ω, a Ψ-space is a space Ψ(A) whose underlying set is ω ∪ A and the topology is given by: All points of ω are isolated, and the neighborhood base at A ∈ A consists of all sets {A} ∪ A \ F where F is a finite subset of ω. A. Dow shows in [8] that if 𝔟 > ω_{1}, for each mad family A, C_{p}(Ψ(A), 2) is not Lindelöf and, hence, in this case, C_{p}(Ψ(A), 2) is not Menger. M. Hrušák, P.J. Szeptycki and Á. Tamariz-Mascarúa show in [10], assuming CH, the existence of a Mrówka mad family A (that is, the one-point compactification of Ψ(A) coincides with its Stone-Čech compactification) such that C_{p}(Ψ(A), 2) is Lindelöf.

For a mad family A and j ∈ 2, we define the closed subspace σ_{n}^{j}(A) = {f ∈ C_{p}(Ψ(A), 2) : |f^{−1}(j) ∩ A| ≤ n} of C_{p}(Ψ(A), 2). If A is a Mrówka family, then C_{p}(Ψ(A), 2) = ⋃_{n∈ω,j∈2} σ_{n}^{j}(A). For every n ∈ ω, σ_{n}^{0}(A) is homeomorphic to σ_{n}^{1}(A). We are going to write σ_{n}(A) instead of σ_{n}^{1}(A). Thus, by Proposition 3.4:

Lemma 8.1. If A is a Mrówka mad family then C_{p}(Ψ(A), 2) is Menger if and only if σ_{n}(A) is Menger for each n ∈ ω.

To characterize when σ_{n}(A) is Menger, we need certain terminology and notation. For a, b ∈ P(ω), a△b will denote their symmetric difference; that is a△b = (a ∪ b) \ (a ∩ b). Given a mad family A and Y ⊂ P(ω), we will say that A^{n} is concentrated on Y [10], if for each open U of the Cantor set 2^{ω} containing χ_{Y} = {χ_{y} : y ∈ Y}, there is a countable B ⊂ A such that χ_{∪x} ∈ U for all x ∈ [A \ B]^{n}. And we will say that A^{n} + [ω]^{<ω} is concentrated on Y if for each open subset U of 2^{ω} containing χ_{Y}, there is a countable subset B ⊂ A such that χ_{(∪x)△b} ∈ U for all x ∈ [A \ B]^{n} and for all b ∈ [ω]^{<ω}.

Lemma 8.2. Let A be a mad family. If A^{n+1} + [ω]^{<ω} is concentrated on [ω]^{<ω} and σ_{n}(A) is Menger, then σ_{n+1}(A) is Menger.

Proof. The proof depends on two claims.

Claim 1. If V is an open subset of σ_{n+1}(A) containing σ_{n}(A), then there is a countable subset B ⊂ A such that f^{−1}(1) ∩ B ≠ ∅ for any f ∈ σ_{n+1}(A) \ V.

Indeed, since σ_{0}(A) is a countable subset of σ_{n}(A), we can choose a sequence of finite functions s_{k} ⊂ Ψ(A)×2 such that σ_{0}(A)∩[s_{k}] ≠ ∅ and σ_{0}(A) ⊂ ⋃_{k∈ω}[s_{k}] ⊂ V, where [s_{k}] = {f ∈ σ_{n+1}(A) : s_{k} ⊂ f} for each k ∈ ω. Note that s_{k}^{−1}(1) ⊂ ω and s_{k} ↾ A is the constant zero for each k ∈ ω. We define the open subset U of 2^{ω} to be ⋃_{k∈ω}{f ∈ 2^{ω} : s_{k} ↾ ω ⊂ f} and note that χ_{[ω]^{<ω}} ⊂ U. Then, by hypothesis, there is a countable subset B′ ⊂ A such that χ_{⋃ x△b} ∈ U for all x ∈ [A \ B′]^{n+1} and for all b ∈ [ω]^{<ω}. Let B = B′ ∪ ⋃_{k∈ω}(s_{k}^{−1}(0) ∩ A) and show that B is the required set by Claim 1. Let f ∈ σ_{n+1}(A) \ V and x = f^{−1}(1) ∩ A. Since V contains σ_{n}(A), |x| = n + 1. Proceed by contradiction, suppose that x ∩ B = ∅. We choose b ∈ [ω]^{<ω} such that f^{−1}(1) ∩ ω = ⋃ x△b. By the choice of B, χ_{⋃ x△b} ∈ U and consequently, there is k ∈ ω such that s_{k}^{−1}(1) ⊂ ⋃ x△b = ω∩f^{−1}(1) and s_{k}^{−1}(0) ∩ ω ⊂ ω \ (ω ∩ f^{−1}(1)) = f^{−1}(0) ∩ ω. Given that x ∩ s_{k}^{−1}(0) = ∅, s_{k}^{−1}(0) ⊂ f^{−1}(0). Then f ∈ [s_{k}] which is a contradiction, and Claim 1 is proved.

Claim 2. If V is an open subset of σ_{n+1}(A) containing σ_{n}(A), then there is a countable subset Y of σ_{1}(A) such that σ_{n+1}(A) \ V ⊂ ⋃_{h∈Y} (h + σ_{n}(A)), where h + σ_{n}(A) = {h + g : g ∈ σ_{n}(A)}.

Let B be the countable subset of A given by Claim 1, and define Y = {f ∈ σ_{1}(A) : f^{−1}(1) ∩ A ⊂ B}. Then Y is countable. Let f ∈ σ_{n+1}(A) \ V. Again, by the choice of B, there is an element a ∈ f^{−1}(1) ∩ B. We define a continuous function g : Ψ(A) → 2 as follows

g(x) = \{ \begin{matrix}1, & if x ∈ a ∪ \{a\}; \\ 0, & otherwise. \end{matrix}

Then g ∈ Y and f + g ∈ σ_{n}(A) and consequently f = g + (f + g) ∈ ⋃_{h∈Y} (h + σ_{n}(A)). This concludes the proof of Claim 2.

Now, we are going to finish the proof of our lemma. Let ⟨ U_{k} : k ∈ ω ⟩ be a sequence of covers of σ_{n+1}(A). Since σ_{n}(A) is Menger, there is a finite subset F′_{k} ⊂ U_{k} for each k ∈ ω such that σ_{k}(A) ⊂ ⋃ ⋃_{k∈ω} F′_{k}. Then, by Claim 2, there is a countable subset Y ⊂ σ_{1}(A) such that σ_{k+1}(A)\⋃ ⋃_{k∈ω} F′_{k} ⊂ ⋃_{h∈Y} (h+σ_{n}(A)). Since σ_{n}(A) is homeomorphic to h + σ_{n}(A) for each h ∈ Y and Y is countable, ⋃_{h∈Y} (h + σ_{n}(A)) is Menger. Then, there is a finite subset F′′_{k} ⊂ U_{k} for each k ∈ ω such that ⋃_{k∈ω} F′′_{k} is a cover of σ_{n+1}(A) \ ⋃ ⋃_{k∈ω} F′_{k}. Therefore, the sequence ⟨ F′_{k} ∪ F′′_{k} : k ∈ ω ⟩ is the required choice. □

As we have already mentioned in the previous paragraphs, σ_{0}(A) is countable. Then, by Lemma 8.2, if A^{k} + [ω]^{<ω} is concentrated on [ω]^{<ω} for each k ≤ n, then σ_{n}(A) is Menger. However, in [10, Corollary 4.3] M. Hrušák, M., P.J. Szeptycki and Á. Tamariz-Mascarúa proves the following two results.

Proposition 8.3 ([10]). Let A be a mad family and n ∈ ω. Then, A^{n} + [ω]^{<ω} is concentrated on [ω]^{<ω} if and only if A^{n} is concentrated on [ω]^{<ω}.

Corollary 8.4 ([10]). Suppose that A is a mad family and n ∈ ω. Then, σ_{n}(A) is Lindelöf if and only if A^{k} is concentrated on [ω]^{<ω} for all k ≤ n.

These last two results with Lemma 8.2 imply the following result.

Proposition 8.5. Let A be a mad family and n ∈ ω. Then the following statements are equivalent.

(a) σ_{n}(A) is Lindelöf;

(b) σ_{n}(A) is Menger;
(c) A^{k} is concentrated on [ω]^{<ω} for every k ≤ n.

Proof. It is clear that (b) implies (a) and by Corollary 8.4, (a) implies (c). Finally, Lemma 8.2 and Proposition 8.3 prove that (c) implies (b). □

A corollary of the previous result is:

Theorem 8.6. Let A be a Mrówka mad family. Then the following are equivalent. (a) C_{p}(Ψ(A), 2) is Lindelöf;

(b) C_{p}(Ψ(A), 2) is Menger;

(c) A^{n} is concentrated on [ω]^{<ω} for every n ∈ ω.

Theorem 4.5 in [10] shows, assuming CH, the existence of a Mrówka mad family A for which A^{n} is concentrated on [ω]^{<ω} for all n ∈ ω. Then we have the following:

Theorem 8.7 (CH). There is a mad family A such that C_{p}(Ψ(A), 2) is Menger.

Problem 8.8. Let A be the Mrówka mad family whose existence is guaranteed by Theorem 4.5 in [10]. Is C_{p}(Ψ(A), 2)^{n} Menger for every n ≥ 2?

Problem 8.9. Are there a topological space X and a natural number n > 2 such that C_{p}(X, 2) is Menger and C_{p}(X, 2)^{n} is not Menger?

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Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito exterior s/n, Ciudad Universitaria, CP 04510, México D. F., México

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Departamento de Matemáticas, Facultad de Ciencias, Circuito exterior s/n, Ciudad Universitaria, CP 04510, México D. F., México

E-mail address: atamariz@unam.mx