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@ Appl. Gen. Topol. 19, no. 2 (2018), 269-280doi:10.4995/agt.2018.9737
c© AGT, UPV, 2018

More on the cardinality of a topological space

M. Bonanzinga a, N. Carlson b, M. V. Cuzzupè a and D. Stavrova c

a
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra,

University of Messina, Italy (mbonanzinga@unime.it, mcuzzupe@unime.it)
b
Department of Mathematics, California Lutheran University, USA. (ncarlson@callutheran.edu)

c
Department of Mathematics, Sofia University, Sofia, Bulgaria. (stavrova@fmi.uni-sofia.bg)

Communicated by D. Georgiou

Abstract

In this paper we continue to investigate the impact that various sep-
aration axioms and covering properties have onto the cardinality of
topological spaces. Many authors have been working in that field. To
mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-
Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the
excellent survey by Hodel “Arhangel’skĭı’s Solution to Alexandroff’s
problem: A survey”.
Here we provide improvements and analogues of some of the results ob-
tained by the above authors in the settings of more general separation
axioms and cardinal invariants related to them. We also provide partial
answer to Arhangel’skii’s question concerning whether the continuum
is an upper bound for the cardinality of a Hausdorff Lindelöf space
having countable pseudo-character (i.e., points are Gδ). Shelah in 1978
was the first to give a consistent negative answer to Arhangel’skii’s
question; in 1993 Gorelic established an improved result; and further
results were obtained by Tall in 1995. The question of whether or
not there is a consistent bound on the cardinality of Hausdorff Lin-
delöf spaces with countable pseudo-character is still open. In this pa-
per we introduce the Hausdorff point separating weight Hpw(X), and

prove that (1) |X| ≤ Hpsw(X)aLc(X)ψ(X), for Hausdorff spaces and (2)

|X| ≤ Hpsw(X)wLc(X)ψ(X), where X is a Hausdorff space with a π-base
consisting of compact sets with non-empty interior. In 1993 Schröder
proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2c(X)χ(X) for
Hausdorff spaces, for Urysohn spaces by considering weaker invariant
- Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce
the n-Urysohn cellularity n-Uc(X) (where n ≥ 2) and prove that the
previous inequality is true in the class of n-Urysohn spaces replacing
Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2Uc(X)πχ(X)

if X is a power homogeneous Urysohn space.

2010 MSC: 54D25; 54D10; 54D15; 54D30.

Received 24 February 2018 – Accepted 19 June 2018

http://dx.doi.org/10.4995/agt.2018.9737


M. Bonanzinga, N. Carlson, M. V. Cuzzupè and D. Stavrova

Keywords: n-Hausdorff space; n-Urysohn space; homogeneous spaces; car-
dinal invariants.

1. Introduction

We will follow the terminology and notation in [16].
Firstly, we will discuss the classical Hajnal and Juhasz’s inequality |X| ≤

2c(X)χ(X) proven for Hausdorff spaces [13]. An improvement of this inequality
is obtained by Bonanzinga in [6] for the general class of n-Hausdorff spaces.
A space X is defined to be n-Hausdorff (where n ≥ 2) if H(X) = n where
H(X) is the Hausdorff number of X, i.e. the smallest cardinal τ such that
for every subset A ⊂ X, |A| ≥ τ, there exist neighborhoods Ua,a ∈ A, such
that

⋂

a∈A Ua = ∅. For every n-Hausdorff space X, the n-Hausdorff pseudo-
character of X, denoted n-Hψ(X), is defined as the smallest κ such that for each
point x there is a collection {V (α,x) : α < κ} of open neighborhoods of x such
that if x1,x2, ..,xn are distinct points from X, then there exist α1,α2, ...,αn < κ
such that

⋂n
i=1 V (αi,xi) = ∅. It was then proved that |X| ≤ 2

c(X)χ(X) holds
if replacing the character with the Hausdorff pseudo-character, and that for
every 3- Hausdorff space the inequality |X| ≤ 2c(X)3-Hψ(X) holds. In [12]
Gotchev proved that the latter inequality is true for every space X having finite
Hausdorff number. In [16] Schröder investigated the inequality of Hajnal and
Juhasz for Urysohn spaces replacing cellularity c(X) with the weaker invariant
Urysohn cellularity Uc(X) (as Uc(X) ≤ c(X)). In Section 2 we prove that
Schröder’s inequality |X| ≤ 2Uc(X)χ(X), for a Urysohn space X, can be restated
for n-Urysohn spaces provided the Urysohn cellularity is replaced by the n-
Urysohn cellularity (Theorem 2.11 below).

An analogue of the Hajnal-Juhasz inequality in the setting of homogeneous
spaces was established in [8] where Carlson and Ridderbos use the Erdös-Rado
theorem to show that if X is a power homogeneous Hausdorff space then |X| ≤
2c(X)πχ(X). In Section 2 we prove that this result can be modified in the setting
of Urysohn spaces to give the homogeneous analogue of Schröder’s result. In
particular, we prove that if X is Urysohn power homogeneous space then |X| ≤
2Uc(X)πχ(X).

In Section 3 we give a partial solution to Arhangel’skii’s problem [3] concern-
ing whether the continuum is an upper bound for the cardinality of a Hausdorff
Lindelöf space having countable pseudo-character.

In [9] Charlesworth proved that |X| ≤ psw(X)L(X)ψ(X) for every T1 space X,
where psw(X) is the mininum infinite cardinal κ such that X has an open cover
S (called separating open cover) having the property that for each distinct x
and y in X there is an S ∈ S such that x ∈ S and y /∈ S and such that each
point of X is in at most κ elements of S. Charlesworth’s result is one of the
few that provided partial answer to both of the above Arhangel’skii’s problem
and another one formulated in the same paper: “Is continuum an upper bound
for T1 Lindelöf space having countable character?”. Shelah, in an unpublished

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 270



More on the cardinality of a topological space

paper in 1978, was the first to provide a consistent negative answer to the ques-
tion of Arhangel’skii’s (whether or not 2ℵ0 is an upper bound for the cardinality
of a Hausdorff Lindelöf space of countable pseudo-characher) by constructing
a model of ZFC + CH in which there is a Lindelöf regular space of count-
able pseudo-character with cardinality c+ = ℵ2. Shelah’s paper was eventually
published in 1996 [17]. Then, Gorelic [11] proved that is consistent with CH
that 2ω1 is arbitrarily large and there is a Lindelöf, 0-dimensional Hausdorff
space X of countable pseudo-character with |X| = 2ω1, and thus improving
Shelah’s result. The question of whether or not there is a consistent bound on
the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is
still open.

We introduce an analogue of psw(X) in the class of Hausorff spaces, denoted
Hpsw(X), and prove that |X| ≤ Hpsw(X)aLc(X)ψ(X) for a Hausdorff space
X thus giving a partial answer to Arhangel’skii’s problem in ZFC by even
replacing L(X) with the weaker invariant aLc(X). This is also a partial answer
to a question in [10] if in the main result which states that for Hausdorff spaces
X, |X| ≤ 2aLc(X)χ(X), χ(X) can be replaced by ψ(X).

We also prove that |X| ≤ Hpsw(X)wLc(X)ψ(X), for a Hausdorff space X
with a π-base consisting of compact sets with non-empty interior. This result
is closely related to results in [5], [4] and [1].

2. A generalization of Schröder’s inequality

In [16], Schröder gives the following definition:

Definition 2.1 ([16]). Let X be a topological space. A collection V of open
subsets of X is called Urysohn-cellular, if O1,O2 in V and O1 6= O2 implies
O1 ∩ O2 = ∅. The Urysohn-cellularity of X, Uc(X), is defined by

Uc(X) = sup{|V| : V is Urysohn-cellular } + ℵ0.

Recall that a topological space X is said to be quasiregular provided for
every open set V , there is a nonempty open set U such that the closure of U
is contained in V . We observe the following properties.

Lemma 2.2. If X is a quasiregular space, then for every cellular family U such
that |U| = κ there exists a Urysohn cellular family U′ such that |U′| = κ.

Proof. Let X be a quasiregular space and U be a cellular family with |U| = κ.
For every U ∈ U there exists an open set VU ⊂ U such that VU ⊂ U. Clearly,
if U1 and U2 are distinct elements of U such that U1 ∩ U2 = ∅, we have
VU1 ∩ VU2 = ∅. Hence U

′ = {VU : U ∈ U} is a Urysohn cellular family for X
such that |U′| = κ. �

Property 2.3. If X is a quasiregular space, c(X) = Uc(X).

Proof. Clearly, Uc(X) ≤ c(X). Let Uc(X) = κ and suppose that c(X) > κ.
Then by Lemma 2.2 there exists a Urysohn cellular family U such that |U| > κ;
a contradiction. �

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 271



M. Bonanzinga, N. Carlson, M. V. Cuzzupè and D. Stavrova

Recall the following:

Theorem 2.4 ([12, Corollary 3.2]). Let X be a space with H(X) finite. Then
|X| ≤ 2c(X)χ(X).

The previous result together with Property 2.3 gives the following:

Corollary 2.5. If X is a quasiregular n-Hausdorff (where n ≥ 2) space, |X| ≤
2Uc(X)χ(X).

Recall that in [7] the authors define a space X to be n-Urysohn (where
n ≥ 2) if U(X) = n where U(X) is the Urysohn number of X, i.e. the smallest
cardinal τ such that for every subset A ⊂ X, |A| ≥ τ, there exist neighborhoods
Ua,a ∈ A, such that

⋂

a∈A Ua = ∅.

We introduce the following:

Definition 2.6. Let X be a topological space. A collection V of open subsets
of X is called n-Urysohn-cellular, where n ≥ 2, if O1,O2, ...,On in V and
O1 6= O2 6= ... 6= On implies O1 ∩ O2 ∩ ... ∩ On = ∅. The n-Urysohn-cellularity
of X, n-Uc(X), is defined by

n-Uc(X) = sup{|V| : V is n-Urysohn-cellular } + ℵ0.

Clearly, if V is a Urysohn cellular collection of open subsets, then V is n-
Urysohn cellular for every n ≥ 2. Also if Uc(X) ≤ κ, then n-Uc(X) ≤ κ for
every n ≥ 2.

Question 2.7. Is there a space X such that (n+1)-Uc(X) = κ and n-Uc(X) 6=
κ?

Recall that the θ-closure of a set A in the space X is the set clθ(A) = {x ∈
X : for every neighborhood U ∋ x,U ∩ A 6= ∅} [19].

Proposition 2.8. Let {Aα}α∈A be a collection of subsets of X, then
⋃

α∈A clθ(Aα) ⊆
clθ(

⋃

α∈A Aα).

Proof. If x ∈
⋃

α∈A clθ(Aα), then there exists α ∈ A such that x ∈ clθ(Aα).

Therefore for every neighborhood Ux we have Ux ∩ Aα 6= ∅. Then Ux ∩
(
⋃

α∈A Aα) 6= ∅. This implies x ∈ clθ(
⋃

α∈A Aα). �

The next lemma represents a modification of Lemma 7 in [16]:

Lemma 2.9. Let X be a topological space and µ = n-Uc(X). Let (Uα)α∈A
be a collection of open sets. Then there are B1,B2, ...,Bn−1 ⊆ A with |Bi| ≤
µ∀i = 1,2, ...,n − 1 and

⋃

α∈A Uα ⊆ clθ(
⋃

α∈B1∪B2∪...∪Bn−1
Uα).

Proof. Let V = {V ⊂ X : V is open and ∃α ∈ A such that V ⊆ Uα}. By
Zorn’s Lemma, take a maximal n-Urysohn-cellular family W ⊆ V and |W| ≤ µ.

For every W ∈ W take β = β(W) such that Uβ(W) ∈ {Uα : α ∈ A} and
W ⊆ Uβ(W). We may assume β ∈ B = B1 ⊔ B2 ⊔ ... ⊔ Bn−1,Bi ⊆ A and
|Bi| ≤ µ,∀i = 1,2, ...,n − 1.

We want to prove that
⋃

α∈A Uα ⊆ clθ((
⋃

α∈B1
Uα) ∪ ... ∪ (

⋃

α∈Bn−1
Uα)).

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 272



More on the cardinality of a topological space

Assume the contrary, then there exists x ∈
⋃

α∈A Uα and x /∈ clθ((
⋃

α∈B1
Uα)∪

... ∪ (
⋃

α∈Bn−1
Uα)).

Then we can find α0 ∈ A and a neighborhood Ux of x such that x ∈ Uα0
and Ux ∩ ((

⋃

α∈B1
Uα) ∪ ... ∪ (

⋃

α∈Bn−1
Uα)) = ∅.

Then (Uα0 ∩ Ux) ⊆ Ux and (Uα0 ∩ Ux) ∪ W is a n-Urysohn cellular family
containing W; this contradicts the maximality of W. �

Corollary 2.10 ([16]). Let X be a topological space and µ = Uc(X). Let
(Uα)α∈A be a collection of open sets. Then there is B ⊆ A with |B| ≤ µ and
⋃

α∈A Uα ⊆ clθ
⋃

α∈B Uα.

Theorem 2.11. Let X be a n-Urysohn space. Then |X| ≤ 2n−Uc(X)χ(X).

Proof. Set µ = n-Uc(X)χ(X). For every x ∈ X let B(x) denote an open
neighbourhood base of x with |B(x)| ≤ µ. Construct an increasing sequence
{Cα : α < µ

+} of subsets of X and a sequence {Vα : α < µ
+} of open collections

of open subsets of X such that:

(1) |Cα| ≤ 2
µ for all α < µ+.

(2) Vα =
⋃

{B(c) : c ∈
⋃

τ<α
Cτ},α < µ

+.
3 If {Gγ1,γ2,...,γn−1 : (γ1,γ2, ...,γn−1) ⊆ µ} is a collection of subsets of X
and each Gγ1,γ2,...,γn−1 is the union of closures of ≤ µ many elements
of Vα and

⋃

{γ1,γ2,...,γn−1}⊆µ

clθGγ1,γ2,...,γn−1 6= X

then

Cα \
⋃

{γ1,γ2,...,γn−1}⊆µ

clθGγ1,γ2,...,γn−1 6= ∅.

The construction is by transfinite induction. Let x0 be a point of X and
put C0 = {x0}. Let 0 < α < µ

+ and assume that Cβ has been constructed for
each β < α. Note that Vα is defined by (2) and Vα ≤ 2

µ. For each collection
{Gγ1,γ2,...,γn−1 : (γ1,γ2, ...,γn−1) ⊆ µ} of subsets of X where each Gγ1,γ2,...,γn−1
is the union of closures of ≤ µ many elements of Vα and

⋃

{γ1,γ2,...,γn−1}⊆µ

clθGγ1,γ2,...,γn−1 6= X,

choose a point of X \
⋃

{γ1,γ2,...,γn−1}⊆µ
clθGγ1,γ2,...,γn−1. Let Hα be the set of

points chosen in this way, (clearly, |Hα| ≤ 2
µ) and let Cα = Hα ∪ (

⋃

β<α
Cβ).

It is clear that the family {Cα : 0 < α < µ
+} constructed in this way satisfies

condition (1),(2) and (3).
Let C =

⋃

α<µ+
Cα. We shall show that C = X. Assume there is y ∈ X \C.

For every Bγ1,Bγ2, ...,Bγn−1 ∈ B(y), with |Bγi| > 1∀i = 1,2, ...,n − 1 and
γ1,γ2, ...,γn−1 ⊆ µ define

Fγ1,γ2,...,γn−1 = {Vc : c ∈ C,Vc ∈ B(c),Vc ∩ Bγ1 ∩ Bγ2 ∩ ... ∩ Bγn−1 = ∅}.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 273



M. Bonanzinga, N. Carlson, M. V. Cuzzupè and D. Stavrova

Since X is n-Urysohn, we have

C ⊆
⋃

{γ1,γ2,...,γn−1}⊆µ

⋃

Fγ1,γ2,...,γn−1.

By Lemma 2.9 we find for every {γ1,γ2, ...γn−1} ⊆ µ subcollections

Gγ1,Gγ2, ...,Gγn−1 ⊆ F{γ1,γ2,...,γn−1}, |Gγi| ≤ µ∀i = 1,2, ...,n − 1

such that
⋃

F{γ1,γ2,...,γn−1} ⊆ clθ

n−1
⋃

i=1

(
⋃

Gγi).

Note y /∈ clθ
⋃n−1
i=1 (

⋃

Gγi). Indeed, since

(

n−1
⋃

i=1

(
⋃

Gγi)) ∩ Bγ1 ∩ Bγ2 ∩ ... ∩ Bγn−1 = ∅

and then

(

n−1
⋃

i=1

(
⋃

Gγi)) ∩ (Bγ1 ∩ Bγ2 ∩ ... ∩ Bγn−1) = ∅.

Find α < µ+ such that
⋃

{γ1,γ2,...,γn−1}⊆µ
(Gγ1 ∪ Gγ2 ∪ ... ∪ Gγn−1) ⊆ Vα. Then

y /∈
⋃

{γ1,γ2,...,γn−1}⊆µ

clθ

n−1
⋃

i=1

(
⋃

Gγi)

but

Cα ⊆ C ⊆
⋃

{γ1,γ2,...,γn−1}⊆µ

⋃

Fγ1,γ2,...,γn−1 ⊆
⋃

{γ1,γ2,...,γn−1}⊆µ

clθ

n−1
⋃

i=1

(
⋃

Gγi).

Put Gγ1,γ2,...,γn−1 =
⋃n−1
i=1 (

⋃

Gγi). This contradicts 3. �

Corollary 2.12 ([16]). Let X be a Urysohn space. Then |X| ≤ 2Uc(X)χ(X).

We end this section with a new cardinality bound for power homogeneous
Urysohn spaces involving the Urysohn cellularity Uc(X). Recall that a space X
is homogeneous if for every x,y ∈ X there exists a homeomorphism h : X → X
such that h(x) = y. X is power homogeneous if there exists a cardinal κ
for which Xκ is homogeneous. It is well established that cardinality bounds
on a topological space can be improved if the space possesses homogeneous-
like properties. For example, while |X| ≤ 2c(X)χ(X) holds for any Hausdorff
space X, Carlson and Ridderbos [8] have shown that if X is additionally power
homogeneous then |X| ≤ 2c(X)πχ(X), where πχ(X) denote the π-character of
the space X. By modifying this result, we show below that an analogous result
holds for Urysohn power homogeneous spaces when Uc(X) is used in place of
c(X).

It is first helpful to establish this result in the homogeneous setting. To
prove the following result we will use the well-known Erdös-Rado Theorem

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 274



More on the cardinality of a topological space

which states that if f : [X]2 → κ is a function and |X| > 2κ, then there is some
subset Y of X with |Y | ≥ κ+ such that f(y) = f(z) for all y,z ∈ [Y ]2, where
[Y ]2 denotes the family of all subsets of Y of cardinality = 2.

Theorem 2.13. If X is a homogeneous Hausdorff space that is Urysohn or
quasiregular then |X| ≤ 2Uc(X)πχ(X).

Proof. If X is quasiregular, then, by Property 2.3 (following Lemma 2 of this
section), Uc(X) = c(X) and the result follows from Proposition 2.1 in [8]. So
we assume X is Urysohn.

Let κ = Uc(X)πχ(X), fix p ∈ X, and let B be a local π-base at p such that
|B| ≤ κ. As X is homogeneous, for all x ∈ X there exists a homeomorphism
hx : X → X such that hx(p) = x.

As X is Urysohn, for all x 6= y ∈ X there exist open sets U and V such that
x ∈ U, y ∈ V and U ∩V = ∅. Then p ∈ h−1x [U]∩h

−1
y [V ], an open set. As B is a

local π-base at p, there exists B(x,y) ∈ B such that B(x,y) ⊆ h−1x [U]∩h
−1
y [V ].

Thus, hx[B(x,y)] ⊆ U, hy[B(x,y)] ⊆ V , and

hx[B(x,y)] ∩ hy[B(x,y)] = ∅.

The existence of B(x,y) for each x 6= y ∈ X defines a function B : [X]2 → B.
Suppose by way of contradiction that |X| > 2κ. As |B| ≤ κ, we can apply

the Erdös-Rado Theorem to the function B. Thus, there exists a subset Y of
X with |Y | = κ+ and A ∈ B such that B(x,y) = A for all distinct x,y ∈ Y .

Observe that for every x 6= y ∈ Y , we have

hx[A] ∩ hy[A] = hx[B(x,y)] ∩ hy[B(x,y)] = ∅.

This shows {hx[A] : x ∈ Y } is a Urysohn cellular family. However,

|{hx[A] : x ∈ Y }| = |Y | = κ
+ > Uc(X),

which is a contradiction. Thus, |X| ≤ 2κ = 2Uc(X)πχ(X). �

To establish the more general theorem in the power homogeneous case, we
adapt the proof of Theorem 2.3 in [8]. Importantly, we adopt the following
notation: If X is a power homogeneous space, let µ be a cardinal such that Xµ

is homogeneous. Fix a projection π : Xµ → X and a point p in the diagonal
∆(X,µ). Let κ be a cardinal such that πχ(X) ≤ κ and fix a local π-base U at
π(p) in X such that |U| ≤ κ. We may assume without loss of generality that
κ ≤ µ. For any B ⊆ A ⊆ µ, let πA→B be the projection of X

A to XB, and for
A ⊆ µ, define U(A) by

U(A) =

{

π−1
A→B

[

∏

b∈B

Ub

]

: B ∈ [A]<ω, and Ub ∈ U for all b ∈ B

}

.

Observe that the family A is a local π-base at pA in X
A.

The following is Lemma 2.2 in [8]. This lemma establishes that if Xµ is
homogeneous that not only are there homeomorphisms hx : X

µ → Xµ such
that hx(p) = x for all x ∈ X, but that we can guarantee these homeomorphisms
have special properties.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 275



M. Bonanzinga, N. Carlson, M. V. Cuzzupè and D. Stavrova

Lemma 2.14. For every x ∈ ∆(X,µ) there is a homeomorphism hx : X
µ →

Xµ such that hx(p) = x and the following conditions are satisfied;

(1) For all z ∈ Xµ, if zκ = pκ then π(hx(z)) = π(x),
(2) For all U ∈ U(κ), there is a point q(U) ∈ π−1κ [U] and a basic open

neighbourhood Ux of hx(q(U))κ in X
κ such that;

(a) q(U)α = pα for all α ∈ µ \ κ,
(b) π−1κ [Ux] ⊆ hx[π

−1
κ [U]].

We now prove a generalization of Theorem 2.13.

Theorem 2.15. If X is a power homogeneous Hausdorff space that is Urysohn
or quasiregular then |X| ≤ 2Uc(X)πχ(X).

Proof. If X is quasiregular then again Uc(X) = c(X) and the proof follows
directly from Theorem 2.3 in [8]. So as in the last proof we assume X is
Urysohn.

Let κ = Uc(X)πχ(X). For every x ∈ ∆(X,µ), fix a homeomorphism hx :
Xµ → Xµ as in Lemma 2.14 above. Now, for x ∈ ∆(X,µ) and U ∈ U(κ), the
set Ux obtained from Lemma 2.14 is a basic open set in X

κ. We may thus fix
a collection {Ux,α : α < κ} of open sets in X such that

Ux =
⋂

α<κ

π−1α [Ux,α].

For every α ∈ κ we can fix a local π-base {V (x,U,α,β) : β < κ} of the point
hx(q(U))α in X.

In Claim 1 in the proof of Theorem 2.3 in [8], it is shown that whenever
x 6= y ∈ ∆(X,µ) there exists U ∈ U(κ) and α,β < κ such that V (x,U,α,β) ⊆
Ux,α\Uy,α. We make a related Claim in our proof. We omit the proof as it is
very similar.

Claim. Whenever x 6= y ∈ ∆(X,µ), there is U ∈ U(κ) and α,β < κ such
that

V (x,U,α,β) ⊆ Ux,α and V (x,U,α,β) ∩ Uy,α = ∅.

Continuing with our main proof, by way of contradiction, assume that |X| >
2κ. Define a a map G : [X]2 → U(κ) × κ × κ as follows. For {x,y} ∈ [X]2

and x 6= y, we apply the Claim and let G({x,y}) = 〈U,α,β〉 be such that the
conclusion of the Claim is satisfied. Here we have identified ∆(X,µ) with X. As
|U(κ)×κ×κ| = κ and |X| > 2κ, we can apply the Erdös-Rado Theorem to find
Y ⊂ X and 〈U,α,β〉 ∈ U(κ)×κ×κ such that |Y | = κ+ and for all {x,y} ∈ [Y ]2,

G({x,y}) = 〈U,α,β〉. Thus for all y ∈ Y we have V (y,U,α,β) ⊆ Uy,α.
Let C = {V (x,U,α,β) : x ∈ Y } and note that C is a collection of open

subsets of X. If x 6= y ∈ Y then

V (x,U,α,β) ∩ Uy,α = ∅ and V (y,U,α,β) ⊆ Uy,α,

and therefore V (x,U,α,β) and V (y,U,α,β) are disjoint. This means C is a
Urysohn cellular family. However, |C|= |Y | = κ+ > Uc(X), which is a contra-
diction. Thus |X| ≤ 2κ.

�

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More on the cardinality of a topological space

This above result shows that Schröder’s cardinality bound 2Uc(X)χ(X) for
Urysohn spaces can be improved in the power homogeneous setting.

Question 2.16. If X is power homogeneous and n-Urysohn, is

|X| ≤ 2n−Uc(X)πχ(X)?

3. The Hausdorff point separating weight

Recall the following properties which represent weaker forms of the Lindelöf
degree L(X):

• the almost Lindelöf degree of X with respect to closed sets, denoted
aLc(X), is the smallest infinite cardinal κ such that for every closed
subset H of X and every collection V of open sets in X that covers H,
there is a subcollection V′ of V such that |V′| ≤ κ and {V : V ∈ V′}
covers H;

• the weak Lindelöf degree of X with respect to closed sets, denoted
wLc(X), is the smallest infinite cardinal κ such that for every closed
subset H of X and every collection V of open sets in X that covers H,
there is a subcollection V′ of V such that |V′| ≤ κ and H ⊆

⋃

V′.

The following holds:

wLc(X) ≤ aLc(X) ≤ L(X)

Recall the following definition:

Definition 3.1 ([9]). A point separating open cover S for a space X is an
open cover of X having the property that for each distinct points x and y in
X there is S in S such that x is in S but y is not in S. The point separating
weight of a space X is the cardinal

psw(X) = min{τ : X has a point separating cover S

such that each point of X is contained in at most τ elements of S} + ℵ0

Definition 3.2. A Hausdorff point separating open cover S for a space X is
an open cover of X having the property that for each distinct points x and y
in X there is S in S such that x is in S but y is not in S. The Hausdorff point
separating weight of a Hausdorff space X is the cardinal

Hpsw(X) = min{τ : X has a Hausdorff point separating cover S

such that each point of X is contained in at most τ elements of S} + ℵ0

Recall the following:

Theorem 3.3 ([9, Theorem 2.1]). If X is T1, then nw(X) ≤ psw(X)
L(X).

Following the proof of Theorem 2.1 in [9], we prove the following:

Theorem 3.4. If X is a Hausdorff space, then nw(X) ≤ Hpsw(X)aLc(X).

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 277



M. Bonanzinga, N. Carlson, M. V. Cuzzupè and D. Stavrova

Proof. Let aLc(X) = κ and let S be a Hausdorff point separating open cover
for X such that for each x ∈ X we have |Sx| ≤ λ, where Sx denotes the
collection of members of S containing x. We first show that d(X) ≤ λκ. For
each α < κ+ construct a subset Dα of X such that:

(1) |Dα| ≤ λ
κ.

(2) If U is a subcollection of
⋃

{Sx : x ∈
⋃

β<α
Dβ} such that |U| ≤ κ and

X \
⋃

U 6= ∅, then Dα \
⋃

U 6= ∅.

Such a Dα can be constructed since the number of possible U’s at the αth stage
of construction is ≤ (λκ · κ · λ)κ = λκ. Let D =

⋃

α<κ+
Dα. Clearly |D| ≤ λ

κ.

Furthermore D is a dense subset of X. Indeed, if there is a point p ∈ X \ D,
since Hpsw(X) ≤ λ, for every x ∈ D there exists an open set Vx ∈ Sx such
that x ∈ Vx and p /∈ Vx. Since x ∈ D, we have Vx ∩ D 6= ∅. Fix y ∈ Vx ∩ D.
Then Vx ∈

⋃

{Sy : y ∈ D}. Put W = {Vx : x ∈ D} ⊆
⋃

{Sy : y ∈ D}. Clearly,
W is an open cover of D. Since aLc(X) ≤ κ, we can select a subcollection
W′ ⊆ W with |W′| ≤ κ such that D ⊆

⋃

{V : V ∈ W′} and p /∈
⋃

{V :
V ∈ W′}; this contradicts 2. Since d(X) ≤ λκ we have that |S| ≤ λκ. Let
N = {X \ S : S is the union of at most κ members of S}. Then |N| ≤ λκ and
N is a network for X.

�

Theorem 3.5. If X is Hausdorff space, then |X| ≤ Hpsw(X)aLc(X)ψ(X).

Proof. It is known that if X is a T1 space, |X| ≤ nw(X)
ψ(X). Then by Theorem

3.4, we have |X| ≤ Hpsw(X)aLc(X)ψ(X). �

Corollary 3.6. If X is a Hausdorff space with L(X) = ω,ψ(X) = ω and
Hpsw(X) ≤ c, then |X| ≤ c.

The previous corollary gives a partial solution to Arhangel’skii’s problem
[2, Problem 5.2] concerning whether the continuum is an upper bound for the
cardinality of a Hausdorff Lindelöf space having countable pseudo-character.

Remark 3.7. Using Remark 2.5 in [9] we note that countable pseudo-character
is essential in Corollary 3.6: if X is the product of 2ω copies of the two point
discrete space, then X is Hausdorff, Lindelöf and ψ(X) > ω but |X| > 2ω.

The following theorem, under additional hypothesis, gives a result similar
to Theorem 3.4 in which the weakly Lindelöf degree with respect to closed sets
takes the place of the almost Lindelöf degree with respect to closed sets.

Theorem 3.8. If X is Hausorff space with a π-base consisting of compact sets
with non-empty interior, then nw(X) ≤ Hpsw(X)wLc(X).

Proof. Let wLc(X) = κ and let S be a Hausdorff point separating open cover
for X such that for each x ∈ X we have |Sx| ≤ λ, where Sx denotes the
collection of members of S containing x. Without loss of generality, we can
suppose that the family Sx is closed under finite intersection. We first show
that d(X) ≤ λκ. For each α < κ+ construct a subset Dα of X such that:

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 278



More on the cardinality of a topological space

(1) Dα ≤ λ
κ.

(2) If U is a subcollection of
⋃

{Sx : x ∈
⋃

β<α
Dβ} such that |U| ≤ κ and

X \
⋃

U 6= ∅, then Dα \
⋃

U 6= ∅.

Such a Dα can be constructed since the number of possible U’s at the αth
stage of construction is (≤ λκ · κ · λ)κ = λκ. Let D =

⋃

α<κ+
Dα. Clearly

|D| ≤ λκ. Furthermore D is a dense subset of X. Indeed if D 6= X, X \ D is
a non-empty open set. Since X has a π-base consisting of compact sets with
non-empty interior, we can find a non empty open subset W ⊆ X such that W
is compact and W ⊂ X \ D, hence W ∩ D = ∅. Fix x ∈ D. For every p ∈ W
there exists an open subset Vp ∈ Sx such that p /∈ Vp. Then, we can find a
family {Vp : p ∈ W} of open subsets of X such that

⋂

{Vp : p ∈ W} ∩ W = ∅.
So, for the compactness of W the family {Vp ∩ W : p ∈ W} can not have the
finite intersection property. So put Fx = Vp1 ∩ ... ∩ Vpk , where p1, ...,pκ ∈ W
are such that Fx ∩ W = ∅. Put Gx = Vp1 ∩ ... ∩ Vpk . Since Sx is closed
under finite intersection, Gx ∈ Sx and Gx ∩ W = ∅. Since Gx ∈ Sx then
Gx ∈ Sy for some y ∈ D. Clearly, V = {Gx : x ∈ D} is an open cover
of D. Using wLc(X) ≤ κ we can select a subcollection V

′ ⊆ V, |V′| ≤ κ

such that D ⊆
⋃

{V : V ∈ V′}. For every U ∈
⋃

V′, U ∩ W = ∅, hence
⋃

V′ ∩ W = ∅. Since W is a nonempty open set,
⋃

V′ ∩ W = ∅ and then

X \
⋃

V′ 6= ∅. This contradicts 2. Since d(X) ≤ λκ we have that |S| ≤ λκ.
Let N = {X \ S|S is the union of at most κ members of S}. Then |N| ≤ λκ

and N is a network for X. �

Then we have the following result:

Theorem 3.9. If X is a Hausdorff space with a π-base consisting of compact
sets with non-empty interior, then |X| ≤ Hpsw(X)wLc(X)ψ(X).

Acknowledgements. The authors are strongly indebted to the referee for

the very careful reading of the paper.

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