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Appl. Gen. Topol. 21, no. 1 (2020), 81-85

doi:10.4995/agt.2020.12065

c© AGT, UPV, 2020

A note on rank 2 diagonals

Angelo Bella and Santi Spadaro

Dipartimento di Matematica e Informatica, University of Catania, Città Universitaria, Viale A.

Doria 6, 95125 Catania, Italy (bella@dmi.unict.it, santidspadaro@gmail.com)

Communicated by S. Garćıa-Ferreira

Abstract

We solve two questions regarding spaces with a (Gδ)-diagonal of rank 2.
One is a question of Basile, Bella and Ridderbos about weakly Lindelöf
spaces with a Gδ-diagonal of rank 2 and the other is a question of
Arhangel’skii and Bella asking whether every space with a diagonal of
rank 2 and cellularity continuum has cardinality at most continuum.

2010 MSC: 54D10; 54A25.

Keywords: cardinality bounds; weakly Lindelöf; Gδ-diagonal; neighbour-
hood assignment; dual properties.

1. Introduction

A space is said to have a Gδ-diagonal if its diagonal can be written as
the intersection of a countable family of open subsets in the square. This
notion is of central importance in metrization theory, ever since Sneider’s 1945
theorem [14] stating that every compact Hausdorff space with a Gδ-diagonal
is metrizable. Sneider’s result was later improved by Chaber [8] who proved
that every countably compact space with a Gδ-diagonal is compact and hence
metrizable.

Around the same time, Ginsburg and Woods [10] showed the influence of Gδ-
diagonals in the theory of cardinal invariants for topological spaces by proving
that every space with a Gδ-diagonal without uncountable closed discrete sets
has cardinality at most continuum. Their result led them to conjecture that
every ccc space with a Gδ-diagonal must have cardinality at most continuum.

Received 09 July 2019 – Accepted 22 October 2019

http://dx.doi.org/10.4995/agt.2020.12065


A. Bella and S. Spadaro

Shakhmatov [13] and Uspenskii [15] gave a pretty strong disproof to this con-
jecture by constructing Tychonoff ccc spaces with a Gδ-diagonal of arbitrarily
large cardinality. However, in the meanwhile, several strengthenings of the no-
tion of a Gδ-diagonal had been introduced, leading several researchers to test
Ginsburg and Woods’s conjecture against these stronger diagonal properties.
That culminated in Buzyakova’s surprising result [7] that a ccc space with a
regular Gδ-diagonal has cardinality at most continuum. A space has a regular
Gδ-diagonal if there is a countable family of neighbourhoods of the diagonal in
the square such that the diagonal is the intersection of their closures.

Another way of strengthening the property of having a Gδ-diagonal is by
considering the notion of rank. Recall that given a family U of subsets of
a topological space and a point x ∈ X, St(x,U) :=

⋃
{U ∈ U : x ∈ U}.

The set Stn(x,U) is defined by induction as follows: St1(x,U) = St(x,U) and
Stn(x,U) =

⋃
{U ∈U : U ∩Stn−1(x,U) 6= ∅} for every n > 1. A space is said

to have a diagonal of rank n if there is a sequence {Uk : k < ω} of open covers
of X such that

⋂
{Stn(x,Uk) : k < ω} = {x}, for every x ∈ X. By a well-

known characterization, having a diagonal of rank 1 is equivalent to having a
Gδ-diagonal. Note also that a space with a Gδ-diagonal of rank 2 is necessarily
T2.

Zenor [17] observed that every space with a diagonal of rank 3 also has a
regular Gδ-diagonal so by Buzyakova’s result, every ccc space with a diagonal
of rank 3 has cardinality at most continuum. In [3], the first author proved the
stronger result that every ccc space with a Gδ-diagonal of rank 2 has cardinality
at most 2ω. The following question is still open though:

Question 1.1 (Arhangel’skii and Bella [1]). Is every regular Gδ-diagonal al-
ways of rank 2?

A positive answer would lead to a far-reaching generalization of Buzyakova’s
cardinal bound.

Arhangel’skii and the first-named author proved in [1] that every space with
a diagonal of rank 4 and cellularity ≤ c has cardinality at most continuum, and
leave open whether this is also true for spaces with a diagonal of rank 2 or 3.

Question 1.2. Let X be a space with a diagonal of rank 2 or 3 and cellularity
at most c. Is it true that |X| ≤ c.

From Proposition 4.7 of [4] it follows that |X| ≤ c(X)ω for every space X
with a diagonal of rank 3, which in turn that the answer to Arhangel’skii and
Bella’s question is yes for spaces with a diagonal of rank 3. We show that
the answer to their question is no for spaces with a diagonal of rank 2, by
constructing a space with a diagonal of rank 2, cellularity ≤ c and cardinality
larger than the continuum. That leads to a complete solution to Arhangel’skii
and Bella’s question.

Recall that space X is weakly Lindelöf provided that every open cover has
a countable subfamily whose union is dense in X. This notion is a common
generalisation of the Lindelöf property and the countable chain condition (ccc).

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 82



A note on rank 2 diagonals

In view of the results by Ginsburg-Woods and Bella mentioned above it is
natural to consider the following question:

Question 1.3 ([4]). Let X be a weakly Lindelöf space with a Gδ-diagonal of
rank 2. Is it true that |X| ≤ 2ω?

The above question was explicitly formulated in [4] and two partial positive
answers were obtained there under the assumptions that the space is either
Baire or has a rank 3 diagonal. Here we will prove that Question 1.3 has a
positive answer assuming that the space is normal.

All undefined notions can be found in [12].

2. Spaces with a diagonal of rank 2

Recall that a neighbourhood assignment for a space X is a function φ from
X to its topology such that x ∈ φ(x) for every x ∈ X. A set Y ⊆ X is a
kernel for φ if X =

⋃
{φ(y) : y ∈ Y}. Following [11], we say that a space X

is dually P if every neighbourhood assignment in X has a kernel Y satisfying
the property P. Of course, P implies dually P. A dually ccc space may fail to
be even weakly Lindelöf.

Here we need the countable version of a well-known result of Erdös and
Rado:

Lemma 2.1. Let X be a set with |X| > 2ω. If [X]2 =
⋃
{Pn : n < ω},

then there exist an uncountable set S ⊆ X and an integer n0 ∈ ω such that
[S]2 ⊆ Pn0.

Theorem 2.2. If X is a dually weakly Lindelöf normal space with a Gδ-
diagonal of rank 2, then |X| ≤ 2ω.

Proof. Let {Un : n < ω} be a sequence of open covers of X such that {x} =⋂
{St2(x,Un) : n < ω} for each x ∈ X. Assume by contradiction that |X| > 2ω

and for any n < ω put Pn = {{x,y} ∈ [X]2 : St(x,Un) ∩ St(y,Un) = ∅}.
The assumption that the sequence {Un : n < ω} has rank 2 implies that
[X]2 =

⋃
{Pn : n < ω}. By Lemma 2.1 there exists an uncountable set S ⊆ X

and an integer n0 such that [S]
2 ⊆ Pn0 . The collection {St(x,Un0 ) : x ∈ S}

consists of pairwise disjoint open sets. From that it follows that, for any z ∈ X,
the set St(z,Un0 ) cannot meet S in two distinct points, which implies that the
set S is closed and discrete.

We define a neighbourhood assignment φ for X as follows: if x ∈ S let
φ(x) = St(x,Un0 ) and if x ∈ X \S let φ(x) = X \S. Since X is dually weakly
Lindelöf, there exists a weakly Lindelöf subspace Y such that X =

⋃
{φ(y) :

y ∈ Y}. By the way φ is defined, it follows that S ⊆
⋃
{φ(y) : y ∈ Y ∩ S}

and hence S ⊆ Y . As X is normal, we may pick an open set V such that
S ⊆ V and V ⊆

⋃
{St(x,Un0 ) : x ∈ S}. The trace on Y of the open cover

{St(x,Un0 ) : x ∈ X} ∪ {X \ V} witnesses the failure of the weak Lindelöf
property on Y . This is a contradiction and we are done. �

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 83



A. Bella and S. Spadaro

Related results for the classes of dually ccc spaces and for that of cellular-
Lindelöf spaces were proved in [16] and [6].

Finally we will construct a space with a diagonal of rank 2, cellularity at most
continuum and cardinality larger than the continuum, thus solving Problem 2
from [1]. Recall that a κ-Suslin Line L is a continuous linear order (endowed
with the order topology) such that c(L) ≤ κ < d(L). The existence of a κ-
Suslin Line for every κ ≥ ω is consistent with ZFC (Jensen proved that it
follows from V = L).

Theorem 2.3. (V = L) There is a space X with a diagonal of rank 2 such
that c(X) ≤ c and |X| ≥ c+.

Proof. Let T be an ω1-Suslin Line. Let S be the set of all points of L which
have countable cofinality. Since T is a continuous linear order the set S is
dense in T and hence d(S) > ℵ1. In particular, |S| > ℵ1. Let τ be the
topology on S generated by intervals of the form (x,y], where x < y ∈ S. Note
that c((S,τ)) = ℵ1 and that the space (S,τ) is first-countable and regular.
So applying Mike Reed’s Moore Machine (see, for example [9]) to (S,τ) we
obtain a Moore space M(S) such that |M(S)| = |S| > ℵ1 and c(M(S)) ≤ℵ1.
Recalling that Moore spaces have a diagonal of rank 2 (see Proposition 1.1
of [2]) and that c = ℵ1 under V = L, we see that X = M(S) satisfies the
statement of the theorem. �

The above theorem also shows that the assumption that the space is Baire
is essential in Proposition 4.5 from [4], thus solving a question asked by the
authors of [4] (see the paragraph after the proof of Lemma 4.6).

3. Acknowledgements

The authors acknowledge support from INdAM-GNSAGA.

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c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 84



A note on rank 2 diagonals

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c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 85