@
Applied General Topology

c© Universidad Politécnica de Valencia
Volume 13, no. 1, 2012

pp. 33-38

On density and π-weight of Lp(βN, R, μ)

Giuseppe Iurato

Abstract

The new topological concept of selective separability is able to give some
estimates for density and π-weight of the Lebesgue space Lp(βN, R, μ)
with 1 ≤ p < +∞. In particular, we deduce a purely topological proof
of the non-separability of such a space.

2010 MSC: 28A33, 28B20, 54C65, 54D65

Keywords: separability, selective separability, Lebesgue space

In Integration Theory, it is important to establish the separability or not
of Lebesgue spaces of the type Lp, with 1 ≤ p < +∞. In general, the usual
proof of this type of results for certain Lebesgue spaces, is conducted through
methods of Real Analysis.

In this work, we use some concepts and methods of pure General Topology
in proving the non-separability of a particular Lebesgue space. Further, we
provide some estimates for density and π-weight of such a space.

1. Introduction

In the context of infinite-combinatorial topology, M. Scheepers (see [14]) has
introduced and studied a particular selection principle Sfin(D,D) connected
with some problems of topological diagonalization of covers of a given com-
pletely regular topological space X, whose dense set family is D. Subsequently,
A. Bella and coworkers (see [3]) have extended and generalized some formal
properties introduced by M. Scheepers in [14], into a pure topological context,
proposing the notion of selective separability, defined as follows.
A completely regular topological space (or a T3 1

2
-space), say (X,τ), is said to

be selectively separable (or M-separable) if, for any sequence {Dn;n ∈ N} of
dense sets of X, there exists a sequence {Fn;n ∈ N} of finite subsets of X such
that Fn ⊆ Dn for each n ∈ N and

⋃
n∈N Fn is dense in X.



34 G. Iurato

A selectively separable space is a separable space too. In fact, if we put Dn = X
for any n ∈ N, then there exists a finite subset Fn ⊆ X for each n ∈ N such
that

⋃
n∈N Fn = X, with

⋃
n∈N Fn countable.

There exist however separable spaces that are not selectively separable, as, for
example, the Tychonov cube Ic of weight c = 2ℵ0, when I = [0,1] is equipped
with the usual Euclidean topology.
One of the purposes of this note is that of discussing some elementary properties
of selectively separable spaces in view of their applications.

2. Preliminary concepts

Given an arbitrary topological space (X,τ), we put τ∗ = τ \ {∅}.
A base of X is a family (∅ �=)B ⊆ τ∗ such that every element of τ∗ is the union
of elements of B. A π-base of X is a family (∅ �=)B ⊆ τ∗ such that, for every
A ∈ τ∗, there is a B ∈ B such that B ⊆ A. Any base of X is also a π-base, but
the inverse, in general, is not true. The minimal cardinality of a base [π-base]
of X, is called the weight [π-weight] of X, say w(X) [πw(X)]. The property
πw(X) ≤ w(X) holds true; moreover, if βN is the Čech-Stone compactification
of the discrete space N, then it is possible to prove that πw(βN) = ℵ0 < c =
w(βN).
If d(X) is the density of X, it is immediate to prove that d(X) ≤ πw(X). In
fact, if B is a π-base of X such that card B = πw(X), and A is a dense subset
of X, then we have A ∩ B �= ∅ for any B ∈ B. In addition, if we choose a
xB ∈ A ∩ B for every B ∈ B, then the set {xB;B ∈ B} is dense in X, so that
d(X) ≤ card {xB;B ∈ B} ≤ card B = πw(X).
Therefore, we have the following relation between cardinal functions

(1) d(X) ≤ πw(X) ≤ w(X).

If (X,τ) is a Hausdorff topological space, then it is known that it has an infinite

disjoint base, say B. It follows that ℵ0 ≤ π(X), because, if B̃ is any π-base of X,
then, by the definition of π-base, there exists a AB ∈ B̃, for each B ∈ B, such
that AB ⊆ B, for which card B̃ ≥ ℵ0 being B an infinite disjoint family. Since
B̃ is an arbitrary π-base of X, it follows that πw(X) ≥ ℵ0. In particular, if
(X,τ) is a metrizable space, then it follows that d(X) = πw(X) = w(X) ≥ ℵ0
(see [11], Section 6, Remark 6.2, and [7]).
The space (X,τ) has countable fan tightness if, for any x ∈ X and any sequence
{An;n ∈ N} of subsets of X such that x ∈

⋂
n∈N An, there exists a finite set

Bn ⊆ An, for each n ∈ N, such that x ∈
⋃

n∈N Bn. In this case, we write
concisely vet (X) ≤ ℵ0.
The space (X,τ) has countable tightness if, for any x ∈ X and A ⊆ X with
x ∈ A, there exists a countable set B ⊆ A such that x ∈ B. In this case, we
write concisely t(X) ≤ ℵ0. We have (vet(X) ≤ ℵ0) ⇒ (t(X) ≤ ℵ0), but the
inverse is, in general, not true.



On density and π-weight of Lp(βN, R, μ) 35

3. Selective separability: basic properties

We here recall some properties of selectively separable spaces.

Theorem 3.1. Every dense subspace of a selectively separable space is also
selectively separable.

Proof. If S is a dense subspace of a selectively separable space (X,τ) and
{Dn;n ∈ N} is a sequence of dense subsets of (S,τS), it follows that each Dn is
also dense in X (as a consequence of the density of S in X), so that, for every

n ∈ N, there exists a finite set Fn ⊆ Dn such that
⋃

n∈N Fn
X
= X. Hence, we

have ⋃
n∈N

Fn
S

=
⋃
n∈N

Fn
X

∩ S = X ∩ S = S.

�

Theorem 3.2. A topological space (X,τ) has the property that

(1) if πw(X) = ℵ0, then X is selectively separable,
(2) if X is separable and vet (X) ≤ ℵ0, then X is selectively separable.

Proof. If πw(X) = ℵ0, let B = {Bn;n ∈ N} be a π-base of X. If {Dn;n ∈ N} is
a sequence of dense subsets of X, then we have Dn ∩ Bm �= ∅ for all n,m ∈ N,
so that we can choose a point xn ∈ Dn ∩Bn for each n ∈ N. Hence, {xn;n ∈ N}
is dense in X, and thus X is selectively separable. This proves (1).

If vet (X) ≤ ℵ0, and X is separable, let {an;n ∈ N} be a countable dense
subset of X, and let {Dn;n ∈ N} be a sequence of dense subsets of X. Since
Dk = X = {an;n ∈ N0} for each k ∈ N, we have, for each n ∈ N, an ∈ Dk for
infinitely many k ∈ N, so that an ∈ Dk ∀k ∈ Ln, with Ln infinite subset of
N. Therefore, we may always choose a disjoint family {Ln;n ∈ N} of infinite
subsets of N in such a way that1 N =

⋃
n∈N Ln. Hence an ∈

⋂
k∈Ln Dk for every

n ∈ N, and since X has countable fan tightness, it follows that there exists a
finite set Fk ⊆ Dk, for each k ∈ Ln, such that an ∈

⋃
k∈Ln Fk, whence we have

{an;n ∈ N} ⊆
⋃
n∈N

( ⋃
k∈Ln

Fk
)
⊆
⋃
n∈N

⋃
k∈Ln

Fk,

so that(
X = {an;n ∈ N} ⊆

⋃
n∈N

⋃
k∈Ln

Fk ⊆ X
)

⇒
(
X =

⋃
n∈N

⋃
k∈Ln

Fk

)
.

The countability of
⋃

n∈N
⋃

k∈Ln Fk proves the selective separability of X. This
proves (2). �

Remark 3.3. In general, the proposition (1) of Theorem 3.2, is not valid if
we suppose πw(X) < ℵ0. Moreover, since w(X) = ℵ0 for every separable

1For instance, if {pm; m ∈ N} is the infinite sequence of the prime numbers of N, then
Ln = {pnm; m ∈ N} ∀n ∈ N, verifies as required.



36 G. Iurato

metrizable spaces X, from what has been said in Section 2, it follows that
w(X) = πw(X) = d(X) = ℵ0 for such spaces2.

If X is a completely regular topological space, and Cp(X) is the space of
continuous functions f : X → R, equipped with the pointwise topology (see
[6], Section 2.6.), then it is possible to prove the following

Theorem 3.4. Cp(X) is selectively separable if and only if X is separable and
vet (X) = ℵ0.

The proof is given in [3].

Nevertheless, the study of C(X) from this viewpoint has been only done with
respect to the pointwise topology (see [2]).

4. The Lebesgue space Lp(βN,R,μ)

Following [15], Counterexamples 110 and 111, if we endow N with the dis-
crete topology, said βN the Čech-Stone compactification of the discrete space
N, then βN is a compact Hausdorff space. Let us recall that ℵ0 = πw(βN) <
w(βN) = c.

The definition of Lp(βN,R,μ) can be given in three ways3 as follows.

1. Following [10], Chap. 7, and4 [4], Chapter IX, Section 5, let μ be a pre-
scribed Radon measure on βN, so that we can consider the Lebesgue
space Lp(βN,R,μ) of the Lp-sommable functions f : βN → R, with
1 ≤ p < +∞.
It is known that the space of functions with compact support C0(βN,R)
is dense in Lp(βN,R,μ), and since βN is compact, we have that C0(βN,R) =
C(βN,R) is dense too in Lp(βN,R,μ).

2. Taking into account that βN is a Hausdorff compact space, let μ be a
positive Borel measure on βN (see [13], Chap. 3, and [9], Chap. 10),
defined over the Borel σ-algebra generated by the topology of βN, and
having a countable base (see [9], Section 37.1, and [8], Chap. III, Sec-
tion 3, Problems 418 - 420).
Hence, reasoning as in 1., it follows that C(βN,R) is dense in Lp(βN,R,μ).

3. Let X be a compact space with a countable base (w(X) = ℵ0), and let
μ be a Baire measure on X. Then Lp(X,R,μ) is a separable space for
each 1 ≤ p < +∞. Indeed (see [12], Chap. IV, Section 4, and Problem
43 (a)), if {Bn;n ∈ N} is a countable base of X, for all n,m ∈ N, n �= m,

2From here, it follows that the concepts of separability and selective separability are the
same, in the case of metric spaces.

3There exists a fourth way, based on integration theory over separable measure spaces
(see [5], and [16], Chap. 7, Section 5), leading to the same results.

4Section 5 of N. Bourbaki’s Chapter IX, deals with measures on a completely regular
space. These results can be applied to βN since this is a Hausdorff compact space, and hence
a completely regular space (see [6], Section 3.3.).



On density and π-weight of Lp(βN, R, μ) 37

with Bn ∩ Bm = ∅, it is possible to define fn,m ∈ C(X,R) in such a
way that fn,m = 0 on Bn and fn,m = 1 on Bm. Hence, the fn,m can be
used to define a countable dense set in C(X,R), whence a countable
dense set in Lp(X,R,μ), being C(X,R) dense in Lp(X,R,μ).
Nevertheless, this proof is no longer valid when πw(X) = ℵ0, hence for
X = βN since ℵ0 = πw(βN) < w(βN) = c, so that we cannot say that
such a Lebesgue space is separable, but only that it has a dense subset
(namely C(X,R)).

On the other hand, it is well-known too as the Banach space (respect to the
supremum norm) of all real bounded sequences, namely l∞(N,R), is not sepa-
rable as metric space. Since each f ∈ l∞(N,R) is continuous and bounded, by
the universal properties of βN (see [6], Section 3.6.), it is possible, in a unique
manner5, to extend it to a continuous function ψ(f) : βN → R. It is possible
to prove (see [1], Sections 2.17 and 2.18) that the correspondence f → ψ(f)
is an isometric isomorphism from l∞(N,R) to C(βN,R), hence a homeomor-
phism. It follows that C(βN,R) is not separable, but dense in Lp(βN,R,μ),
with 1 ≤ p < +∞.

Finally, it also follows that the space Lp(βN,R,μ) cannot be separable:
indeed, as metric space, taking into account what has been said in Section 2
and in Remark 3.3, we have the following density and π-weight estimates for
such a space

(2) ℵ0 < d(Lp(βN,R,μ)) = πw(Lp(βN,R,μ)) = w(Lp(βN,R,μ)).

Hence, if Lp(βN,R,μ) were separable, then we would have w(Lp(βN,R,μ)) =
ℵ0 (as separable metric space), hence πw(Lp(βN,R,μ)) = ℵ0, so that, by (1)
of Theorem 3.2, Lp(βN,R,μ) would be selectively separable, whence it would
follow that every dense subspace of it would be selectively separable too (by
Theorem 3.1), and this is impossible since C(βN,R) is a no separable dense
subspace of Lp(βN,R,μ), with 1 ≤ p < +∞.

Following the lines of this paper, it would be of a certain interest to analyze
the possible role played by the various notions of tightness, fan tightness (see
[2] for the topological function spaces case) and others properties of selectively
separable spaces, in the case of a generic Lebesgue space Lp(X,R,μ), with
1 ≤ p < +∞.

5Recalling, on the other hand, that N is dense in βN.



38 G. Iurato

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(Received July 2010 – Accepted August 2011)

Giuseppe Iurato (giuseppe.iurato@unipa.it)
Department of Physics and Related Technologies, University of Palermo, Ave-
nue of Science, Building 18, I-90128, Palermo, Italy


	On density and -weight of Lp(N,R,). By G. Iurato