@ Appl. Gen. Topol. 20, no. 1 (2019), 223-230doi:10.4995/agt.2019.10714
© AGT, UPV, 2019

A non-discrete space X with Cp(X) Menger

at infinity

Angelo Bellaa and Rodrigo Hernández-Gutiérrezb

a
Department of Mathematics and Computer Science, University of Catania, Cittá universitaria,

viale A. Doria 6, 95125 Catania, Italy (bella@dmi.unict.it)
b
Departamento de Matemáticas, Universidad Autónoma Metropolitana campus Iztapalapa, Av.

San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340, Mexico city, Mexico

(rodrigo.hdz@gmail.com)

Communicated by Á. Tamariz-Mascarúa

Abstract

In a paper by Bella, Tokgös and Zdomskyy it is asked whether there

exists a Tychonoff space X such that the remainder of Cp(X) in some
compactification is Menger but not σ-compact. In this paper we prove

that it is consistent that such space exists and in particular its existence

follows from the existence of a Menger ultrafilter.

2010 MSC: Primary 54D20; Secondary 54A35; 54C35; 54D40; 54D80;
54H11.

Keywords: Menger spaces; non-meager P-filter; pointwise convergence
topology.

1. Introduction

A space X is called Menger if for every sequence {Un ∶ n ∈ ω} of open covers
of X one may choose finite sets Vn ⊂ Un for all n ∈ ω in such a way that
⋃{Vn ∶ n ∈ ω} covers X. Given a property P, a Tychonoff space X will be
called P at infinity if βX ∖X has P.

Let X be a Tychonoff space. It is well-known that X is σ-compact at
infinity if and only if X is Čech-complete. Also, Henriksen and Isbell proved
in [7] that X is Lindelöf at infinity if and only if X is of countable type.

Received 14 September 2018 – Accepted 24 January 2019

http://dx.doi.org/10.4995/agt.2019.10714


A. Bella and R. Hernández-Gutiérrez

Moreover, the Menger property implies the Lindelöf property and is implied
by σ-compactness. So it was natural for the authors of [2] to study when X is
Menger at infinity.

Later, the authors of [4] study when a topological group is Menger, Hurewicz
and Scheepers at infinity. The Hurewicz and Scheepers properties are other
covering properties that are stronger than the Menger property and weaker
than σ-compactness (see the survey [12] by Boaz Tsaban). Essentially, [4] has
two main results.

Theorem 1.1 ([4, Theorem 1.3]). If G is a topological group and βG ∖ G is
Hurewicz, then βG∖G is σ-compact.

Theorem 1.2 ([4, Theorem 1.4]). There exists a topological group G such that
βG∖G is Scheepers and not σ-compact if and only if there exists an ultrafilter U
on ω such that, considered as a subspace of P(ω) with the Cantor set topology,
U is Scheepers.

The last section of [4] considers the specific case of the topological group
Cp(X) consisting of all continuous real-valued functions defined on X, with
the topology of pointwise convergence. It is shown that if Cp(X) is Menger
at infinity, then it is first countable and hereditarily Baire. It is a well-known
result that Cp(X) is Čech-complete (equivalently, σ-compact at infinity) if and
only if X is countable and discrete (see [1, I.3.3]). So the authors of [4] made
the natural conjecture that Cp(X) is Menger at infinity if and only if Cp(X)
is σ-compact at infinity ([4, Question 6.1]). Their conjecture is equivalent to
the statement that if Cp(X) is Menger at infinity, then X is countable and
discrete. In this paper we disprove this conjecture.

Theorem 1.3. It is consistent with ZFC that there exists a regular, countable,
non-discrete space X such that Cp(X) is Menger at infinity.

Our proof of Theorem 1.3 uses filters with a special property that is imme-
diately satisfied by Menger ultrafilters. See Theorem 3.2 for the exact property
we use.

According to [5, Theorem 3.5], Menger filters are precisely those called Can-
jar filters. Also, by [6, Proposition 2], d = c implies there is a Canjar (thus,
Menger) ultrafilter. However, a characterization of Canjar ultrafilters given in
[6] implies that a Menger ultrafilter is a P-point. Thus, Menger ultrafilters
consistently do not exist.

However, we do not know whether there are filters in ZFC that satisfy the
conditions we need. See sections 3 and 4 for a more thorough explanation and
concrete open questions.

2. Preliminaries and notation

2.1. The Menger property. The Menger property has been thoroughly stud-
ied. We state some well-known facts below:

(i) [9, Theorem 2.2] Every σ-compact set is Menger.

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A non-discrete space X with Cp(X) Menger at infinity

(ii) [9, Theorem 3.1] If X is Menger and Y ⊂ X is closed, then Y is Menger.
(iii) If X is Menger and K is σ-compact, then X ×K is also Menger.
(iv) [9, Theorem 3.1] The continuous image of a Menger space is also

Menger.
(v) The continuous and perfect pre-image of a Menger space is also Menger.
(vi) [9, p. 255] If a space is the countable union of Menger spaces, then it

is Menger as well.
(vii) ωω is not Menger.

We also mention the following observation of Aurichi and Bella.

Lemma 2.1 ([2, Corollary 1.6]). A space X is Menger at infinity if and only
if there exists a compactification of X with a Menger remainder if and only if
the remainder of every compactification of X is Menger.

2.2. Filters. A filter F on a non-empty set X is a subset F ⊂ P(X) such
that: (a) ∅∉F, (b) if x,y ∈F then x∩y ∈F, and (c) if x ∈F and x ⊂ y ⊂ X,
then y ∈ F. All filters in this paper are defined on countable sets (and most
of the times, on ω). Filters that contain the Fréchet filter of cofinite sets are
called free. Maximal filters are called ultrafilters. Let χ ∶ P(ω) → 2ω be the
function that sends each subset to its characteristic function. Using χ, a filter
on a countable set can be thought of as a subspace of the Cantor set.

For every subset Y ⊂ P(X) we may define Y∗ = {A ⊂ X ∶ X ∖ A ∈ Y}. If
F is a filter on ω, F∗ is called its dual ideal and F+ = P(X)∖F∗ is the set
of F-positive sets. Moreover, the function that takes each set in P(X) to its
complement is a homeomorphism. Thus, a filter is always homeomorphic to its
dual ideal. Also, notice that the complement P(X)∖F is then homeomorphic
to F+.

Given A ⊂ ω ×ω and n ∈ ω, define

A(n) = {i ∈ ω ∶ ⟨i,n⟩ ∈ A}, and
A(n) = {i ∈ ω ∶ ⟨n,i⟩ ∈ A}.

For Y ⊂ P(ω), let us define

Y(ω) ={A ⊂ ω ×ω ∶ ∀n ∈ ω (A(n) ∈Y))}.

There is a natural function from P(ω×ω) to P(ω)ω that takes each A ⊂ ω×ω
to {⟨n,A(n)⟩ ∶ n ∈ ω}. This function is also a homeomorphism and takes Y

(ω)

to Yω.
It is easy to see that if F is a filter on ω, then F(ω) is a filter on ω×ω. Thus,

the ω-power of a filter is always (homeomorphic to) a filter.
A filter F is a P-filter if for every {Fn ∶ n < ω}⊂F there exists F ∈F such

that F ∖ Fn is finite for all n ∈ ω. A filter is called meager if it is meager as
a topological space. It is known that ultrafilters are non-meager [3, Theorem
4.1.1]. A P-point is a P-filter that is also an ultrafilter. The existence of
P-points is independent from ZFC. For example, d = c implies there are P-
points but there are models with no P-points, see [3, section 4.4]. Non-meager
P-filters are a natural generalization of P-points; it is still an open question

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A. Bella and R. Hernández-Gutiérrez

whether they exist in ZFC but if they don’t exist, then there is an inner model
with a large cardinal, see [3, section 4.4.C].

2.3. The Hilbert cube. The Hilbert cube is the countable infinite product of
closed intervals of the reals; we will find it convenient to work with Q= [−1,1]ω.
The pseudointerior of Q is s = (−1,1)ω and the pseudoboundary is B(Q)=Q∖s.

3. The example

According to [4, Proposition 6.2], if Cp(X) is Menger at infinity, then it
is first countable and hereditarily Baire. From [1, I.1.1], it follows that X is
countable. In [10], Witold Marciszewski studied countable spaces X such that
Cp(X) is hereditarily Baire. We will consider one specific case: when X has a
unique non-isolated point.

Given a filter F ⊂ P(ω), consider the space ξ(F) = ω ∪{F}, where every
point of ω is isolated and every neighborhood of F is of the form {F}∪A with
A ∈F. All our filters will be free, that is, they contain the Fréchet filter. In this
case, F is not isolated. When F is the Fréchet filter, ξ(F) is homeomorphic
to a convergent sequence. It is easy to see that a space X is homeomorphic to
a space of the form ξ(F) if and only if X is a countable space with a unique
non-isolated point.

Theorem 3.1 ([10]). For a free filter F on ω, the following are equivalent.

(a) F is a non-meager P-filter,
(b) F is hereditarily Baire, and
(c) Cp(ξ(F)) is hereditarily Baire.

Thus, it is natural to ask when Cp(ξ(F)) is Menger at infinity. By Lemma
2.1 it is sufficient to look at any compactification of Cp(ξ(F)) and try to decide
whether the remainder is Menger.

Consider the set of functions in the Hilbert cube that F-converge to 0:

KF ={f ∈Q ∶∀m ∈ ω {n ∈ ω ∶ ∣f(n)∣< 2
−m} ∈F},

and those that only take values in the pseudointerior

CF = KF ∩ s.

By [11, Lemma 2.1], it easily follows that CF is homeomorphic to Cp(ξ(F)).
Also, CF is dense in Q. So our problem becomes equivalent to finding a filter F
such that Q∖CF is Menger. In fact, we will prove the following characterization
of those filters.

Theorem 3.2. Let F be a free filter on ω. Then Cp(ξ(F)) is Menger at
infinity if and only if F+ is Menger.

Recall that since F+ is homeomorphic to P(ω)∖F, the property in The-
orem 3.2 is equivalent to saying that F is Menger at infinity. So the problem
is reduced to the existence of such filters. As discussed in the introduction,
Menger ultrafilters have the desired properties. Indeed, an ultrafilter coincides
with its set of positive sets. Thus, we conclude the following.

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A non-discrete space X with Cp(X) Menger at infinity

Corollary 3.3. If F is a Menger ultrafilter on ω, then Cp(ξ(F)) is Menger
at infinity.

Now we proceed to the proof of Theorem 3.2. The argument is based on two
classical theorems that relate the Cantor set and the unit interval: [0,1] has a
subspace homeomorphic to 2ω and is a continuous image of 2ω. We just have
to take the filter into account and the proof will follow naturally.

A family of closed, non-empty subsets {Js ∶ s ∈ 2<ω} of a space X will be
called a Cantor scheme on X if

(i) for every s ∈ 2<ω, Js ⊃ Js⌢0 ∪Js⌢1, and
(ii) for every f ∈ 2ω, Jf =⋂{Jf↾n ∶ n < ω} is exactly one point.

Let 1 = ω ×{1} and σ ={1↾n∶ n ∈ ω}.

Lemma 3.4. For every free filter F on ω there is a closed embedding of F(ω)

into Cp(ξ(F)).

Proof. As explained earlier we shall work on CF ⊂ Q instead of the function
space. Recursively, construct a Cantor scheme {Js ∶ s ∈ 2<ω} on the interval
(−1,1) such that:

(i) J∅ = [−12,0],
(ii) for every s ∈ 2<ω, Js is a non-degenerate closed interval of length < 2

−∣s∣,
(iii) for every s ∈ 2<ω, Js⌢0 ∩Js⌢1 =∅, and
(iv) for every s ∈ σ, 0 ∈ Js.

Now define the function ϕ ∶ P(ω ×ω)→Q such that for all A ⊂ ω ×ω and
n ∈ ω, ϕ(A)(n) is the unique point in Jχ(A(n)). So informally speaking, the
n-th row of A is used to define the value of the function ϕ(A) ∶ ω → 2 at n.

From standard arguments, it is easy to see that ϕ is an embedding. Now we
shall prove that A ∈F(ω) if and only if ϕ(A) ∈ KF.

First, assume that A ∈ F(ω), that is, A(i) ∈ F for all i ∈ ω. Let m ∈ ω. By
the definition of ϕ(A), for every n ∈⋂{A(i) ∶ i ≤ m} we have ϕ(A)(n) ∈ J1↾m.
Since J1↾m has diameter less than 2

−m and contains 0, we obtain that

⋂{A(i) ∶ i ≤ m}⊂{n ∈ ω ∶ ∣ϕ(A)(n)∣< 2
−m}.

Thus the set {n ∈ ω ∶ ∣ϕ(A)(n)∣< 2−m} is an element of F. Since this holds for
every m < ω, we obtain that ϕ(A) is F-convergent to 0.

Now assume that ϕ(A) ∈ KF and fix m ∈ ω. Let k ∈ ω be such that the length
of J1↾m is greater than 2

−k. By our hypothesis the set {n ∈ ω ∶ ∣ϕ(A)(n)∣< 2−k}
is an element of F and

{n ∈ ω ∶ ∣ϕ(A)(n)∣< 2−k}⊂{n ∈ ω ∶ ϕ(A)(n) ∈ J1↾m}

so {n ∈ ω ∶ ϕ(A)(n) ∈ J1↾m} ∈F. Finally, notice that by the definition of ϕ

{n ∈ ω ∶ ϕ(A)(n) ∈ J1↾m}⊂{n ∈ ω ∶ ⟨m,n⟩ ∈ A}= A(m),

so we obtain that A(m) ∈ F. Since this is true for all m ∈ ω, we conclude
that A ∈F(ω).

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A. Bella and R. Hernández-Gutiérrez

This concludes the proof that A ∈ F(ω) if and only if ϕ(A) ∈ KF. Also,
notice that the image of P(ω × ω) under ϕ is a subset of s. Thus, we can
even say that A ∈ F(ω) if and only if ϕ(A) ∈ CF. Thus, ϕ↾F(ω) is the closed
embedding we wanted. �

Lemma 3.5. Let F be a free filter on ω. Then there exists a continuous
surjective function ϕ ∶ P(ω ×ω)→Q such that ϕ←[KF]=F(ω).

Proof. As before, we work on CF ⊂ Q instead of the function space. Recur-
sively, construct a Cantor scheme {Js ∶ s ∈ 2<ω} on the interval [−1,1] according
to the following conditions:

(1) J∅ = [−1,1],
(2) for every s ∈ 2<ω, Js is a non-degenerate closed interval of length ≤ 2

−∣s∣,
(3) if s ∈ σ, there are 0 < x < y with Js = [−y,y], Js⌢1 = [−x,x] and

As⌢0 = [−y,−x]∪ [x,y],
(4) if s ∈ σ, there are 0 < x < y with Js⌢⟨0,0⟩ = [−y,−x] and Js⌢⟨0,1⟩ = [x,y],
(5) if s ∈ 2<ω ∖ σ and there are a < b with Js = [a,b], then there exists

x ∈ (a,b) such that Js⌢0 = [a,x] and Js⌢1 = [x,b].
Again define the function ϕ ∶ P(ω × ω) → Q such that for all A ⊂ ω × ω and
n ∈ ω, ϕ(A)(n) is the unique point in Jχ(A(n)).

Another standard argument implies that ϕ is a continuous, surjective func-
tion. Also, the equality ϕ←[KF]=F(ω) can be proved in a manner completely
analogous to the corresponding equality from Lemma 3.4. Thus, we will leave
this argument to the reader. �

Finally, the following allows us to simplify the characterization we will ob-
tain.

Lemma 3.6. Let F be a free filter. Then (F(ω))
+
is Menger if and only if F+

is Menger.

Proof. First, assume that F+ is Menger. For each n ∈ ω, consider Mn = {A ⊂
ω×ω ∶ A(n) ∈F

+}, which is homeomorphic to the product F+×P(ω)ω. Since
the product of a Menger space and a compact space is Menger, it follows that
Mn is Menger for every n ∈ ω. Then notice that (F(ω))+ =⋃{Mn ∶ n ∈ ω} is a
countable union of Menger spaces so it is Menger.

Now assume that (F(ω))+ is Menger. The diagonal in a product is always a
closed subspace and the diagonal of (F(ω))+ is equal to the set

{A ⊂ ω ×ω ∶ ∃F ∈F+ ∀n ∈ ω (A(n) = F)},

which is homeomorphic to F+. Then F+ is Menger because it is a closed
subspace of a Menger space. �

Proof of Theorem 3.2. By Lemma 3.6, it is sufficient to prove that CF is Menger
at infinity if and only if F(ω) is Menger at infinity.

First, assume that CF is Menger at infinity. This means that Q∖ CF is
Menger. By Lemma 3.4, F(ω) can be embedded as a closed set F in CF. Let

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A non-discrete space X with Cp(X) Menger at infinity

F denote the closure of F in Q. Then F ∖F is a closed subset of Q∖CF. Then
F is a compactification of F(ω) with Menger remainder.

Now, assume that F(ω) is Menger at infinity. Let ϕ ∶ P(ω × ω) → Q be
the continuous surjection from Lemma 3.5. Then it follows that ϕ[P(ω×ω)∖
F(ω)] = Q∖ KF. Since the Menger property is preserved under continuous
functions, Q∖KF is Menger. Notice that

Q∖CF = (Q∖KF)∪B(Q).

Since the boundary B(Q) is σ-compact and the union of countably many
Menger spaces is Menger, Q∖ CF is Menger. This concludes the proof of
the Theorem. �

4. Questions

Let F be a free filter on ω such that F+ is Menger. We have just proved that
Cp(ξ(F)) is Menger at infinity. By the observations of Aurichi and Bella from
[2], Cp(ξ(F)) is a hereditarily Baire space. Then, by Marciszewski’s result
from [10] it follows that F is a non-meager P-filter. Thus, inadvertently we
proved the following, which supersedes [4, Observation 3.4] (for filters only).

Corollary 4.1. Let F be a free filter on ω. If F+ is Menger, then F is a
non-meager P-filter.

Here is another more direct proof: Assume that F is a free filter that is not
a non-meager P-filter. By Lemmas 2.1 and 2.2 from [10], the space F has a
closed subset Q homeomorphic to the rationals. The closure Q of Q in P(ω)
must be homeomorphic to the Cantor set. Also, Q∖Q is homeomorphic to ωω,
contained in P(ω)∖F and closed in P(ω)∖F. Since ωω is not Menger and
the Menger property is hereditary to closed sets, P(ω)∖F is not Menger.

By [8, 2.7] every filter of character < d is a Menger filter. However, it not
hard to conclude from [3, 4.1.2] that any filter of character < d is meager. So
indeed none of these filters can have its positive set Menger.

As discussed earlier, the existence of a non-meager P-filter in ZFC is still
an open question. But so far the only example of a filter F with F+ Menger is
a Menger ultrafilter, which we know that consistently does not exist. So it is
natural to ask about the consistency of filters that are Menger at infinity.

Question 4.2. Does ZFC imply that there exists a free filter F on ω such that
F+ is Menger?

Question 4.3. Is the existence of a free Menger ultrafilter on ω equivalent to
the existence of a free filter F on ω such that F+ is Menger?

Finally, regarding the original question from [4], we could ask whether there
exist examples with other properties. In [10, Proposition 3.3], Marciszewski
gave general conditions for a countable space X to guarantee that Cp(X) is
hereditarily Baire. So we may ask what happens in general.

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A. Bella and R. Hernández-Gutiérrez

Question 4.4. Does there exist a countable, regular, crowded space X such
that Cp(X) is Menger at infinity?

Question 4.5. Does there exist a countable, regular, maximal space X such
that Cp(X) is Menger at infinity?

Question 4.6. Characterize all countable regular spaces X such that Cp(X)
is Menger at infinity.

Acknowledgements. The research that led to the present paper was partially
supported by a grant of the group GNSAGA of INdAM. The second-named
author was also supported by the 2017 PRODEP grant UAM-PTC-636 awarded
by the Mexican Secretariat of Public Education (SEP).

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