ArBeAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 8, No. 2, 2007

pp. 207-212

The diagonal of a first countable paratopological
group, submetrizability, and related results

A. V. Arhangel’skii and A. Bella

Abstract. We discuss some properties stronger than Gδ -diagonal.

Among other things, we prove that any first countable paratopological

group has a Gδ-diagonal of infinite rank and hence also a regular Gδ-

diagonal. This answer a question recently asked by Arhangel’skii and

Burke.

2000 AMS Classification: 54B20, 54H13

Keywords: Gδ-diagonal of rank n, semitopological and paratopological
groups, Souslin number, submetrizability.

A semitopological group is a group with a topology such that the multipli-
cation is separately continuous.

A paratopological group is a group with a topology such that the multipli-
cation is jointly continuous.

In the sequel, all spaces are assumed to be Hausdorff. For notations and
undefined notions we refer to [5].

The starting point of the present note was to answer Problem 25 in [1]. In
that paper the authors proved that a first-countable Abelian paratopological
group has a regularGδ-diagonal, raising the question of whether the Abelianity
of the group could be dropped.

A topological space X has a regular Gδ-diagonal if there exists a countable
family {Un : n < ω} of open subsets of X × X such that ∆(X) =

⋂
{Un : n <

ω}. Here ∆(X) denotes the diagonal {(x, x) : x ∈ X} of X. The star of a
collection U with respect to a set A is the set st(U, A) =

⋃
{U : U ∈ U and

U ∩ A 6= ∅}. When A = {x}, we simply write st(U, x). We put st1(U, x) =
st(U, x) and recursively define stn+1(U, x) = st(U, stn(U, x)).

We say that a space X has a Gδ-diagonal of rank n if there exists a countable
collection {Uk : k < ω} of open covers ofX such that

⋂
{stn(Uk, x) : k < ω} =

{x} for each x ∈ X. If a space has a Gδ-diagonal of any possible rank, then we
say that it has a Gδ-diagonal of infinite rank.



208 A. V. Arhangel’skii and A. Bella

Zenor has pointed out in [9] that a Gδ-diagonal of rank 3 is always regular.
For the reader’s benefit, we provide here the simple proof.

Proposition 1. A topological space X with a Gδ-diagonal of rank 3 has also
a regular Gδ-diagonal.

Proof. Let {Un : n < ω} be a sequence of open covers of X witnessing the
rank 3 of the diagonal and put Vn =

⋃
{U × U : U ∈ Un}. We will check that

∆(X) =
⋂
{Vn : n < ω}. Let (x, y) ∈ X × X \ ∆(X) and choose n in such

a way that y /∈ st3(Un, x). Then, select Ux, Uy ∈ Un such that x ∈ Ux and
y ∈ Uy. We must have Ux × Uy ∩ Vn = ∅, otherwise there would exist some
U ∈ Un such that Ux ∩ U 6= ∅ 6= Uy ∩ U and this in turn would imply that
y ∈ st3(Un, x). �

We would like to mention that the requirement for the diagonal to be an
intersection of closed neighbourhoods implies a stronger separation axiom.

Proposition 2. If the diagonal of the space X is the intersection of a collection
of closed neighbourhoods, then Xsatisfies the Urysohn separation axiom.

Proof. Let x, y ∈ X and x 6= y. Fix an open set U ⊆ X × X such that
∆(X) ⊆ U and (x, y) /∈ U . Clearly, we may find open sets V, W ⊆ X such
that (x, y) ∈ V × W ⊆ X × X \ U . Let us assume that there exists a point
z ∈ V ∩ W and fix an open set O ⊆ X such that (z, z) ∈ O × O ⊆ U . As we
have V × W ∩ O × O 6= ∅, we reach a contradiction and so V ∩ W = ∅. �

Corollary 1. Every space with a regular Gδ-diagonal is a Urysohn space.

The usual Ψ-space over the integers is a pseudocompact space with a Gδ-
diagonal of rank 2 which does not have a regular Gδ-diagonal. The latter fact
maybe derived from a result of Mc Arthur [7], stating that any pseudocompact
space with a regular Gδ-diagonal is compact.At the moment, we do not know
the answer to the following:

Problem 1. Is every regular Gδ diagonal always of rank 2?

We are interested in the above problem also because a positive answer would
allow us to deduce a recent remarkable result of Buzyakova [4], stating that
a ccc-space with a regular Gδ-diagonal has cardinality at most c, from an old
result of Bella [3], saying the same for a Gδ-diagonal of rank 2. If we assume
a diagonal of higher rank, we can weaken the hypothesis on the cellularity in
the last mentioned result.

Theorem 1. If X is a space with a Gδ-diagonal of rank 4 and cellularity at
most c, then |X| ≤ c.

Proof. Let {Un : n < ω} be a collection of open covers of X satisfying the
formula

⋂
{st4(Un, x) : n < ω} = {x} for each x ∈ X. Let An ⊆ X be

maximal with respect to the property thatst2(Un, x) ∩ An = {x}. As the
family {st(Un, x) : x ∈ An} consists of pairwise disjoint sets, we have|An| ≤ c.
Moreover, {st2(Un, x) : x ∈ An} is a cover of X. For any x ∈ X and any n < ω,



The diagonal of a first countable paratopological group 209

choosexn ∈ An such that x ∈ st
2(Un, xn) and let φ(x) = {st

2(Un, xn) : n < ω}.
Since we are assuming that

⋂
{st4(Un, x) : n < ω} = {x}, we have

⋂
φ(x) = {x}

and so the map x 7→ φ(x) is one-to-one. Now, an easy counting argument shows
that |X| ≤ c. �

As it is well-known [8], there are ccc-spaces withGδ -diagonal and arbitrarily
large cardinality. Nevertheless, the following question remains open.

Problem 2. Is Theorem 1 still true if we assume the diagonal to be of rank 2
or 3?

Now we move on to the case of a space with a group structure.
Let G be a semitopological group. A local π-base P at the neutral element e

is called T-linked (here T stands for translation) if P x∩xP 6= ∅ for any P ∈ P
and x ∈ G. This notion is instrumental to the following:

Lemma 1. Let G be a semitopological group and P be a T-linked local π-base
at the neutral element. Then, for any P ∈ P, the collection U(P ) = {P x∩xP :
x ∈ G} is a cover of G.

Proof. Fix P ∈ P and x ∈ G. Since P x ∩ xP 6= ∅, we may find p1, p2 ∈ P such
thatp1x = xp2. Observe that P p

−1
1 x ∋ p1p

−1
1 x = x and p

−1
1 xP ∋ p

−1
1 xp2 =

p−11 p1x = x and so x ∈ P p
−1
1 x ∩ p

−1
1 xP . �

Lemma 2. If G is a paratopological group, then for any pair of distinct points
y, z ∈ G and any integer n there exists a neighbourhood P of the neutral element
e such thatP ny ∩ zP n = ∅.

Theorem 2. If G is a paratopological group with a countable T-linked local
π-base at the neutral element, then G has a Gδ-diagonal of infinite rank.

Proof. For any P ⊆ G and any x ∈ G we putP [x] = xP ∩ P x. Let P be a
countable T-linked local π-base at the neutral element e.For any P ∈ P, let
U(P ) = {xP ∩ P x : x ∈ G} = {P [x] : x ∈ G}. By Lemma 1, each U(P ) is a
cover of G. We will show that the collection Γ = {U(P ) : P ∈ P} witnesses the
infinite rank of the diagonal. Towards this end, fix an integer n and a pair of
distinct points y, z ∈ G. We have to check that there is some P ∈ P such that

∗ z /∈ stn(U(P ), y)

By Lemma 2, we may fix P ∈ P in such a way that P ny∩zP n = ∅. The failure
of (*) for this P means that there exist x1, . . . , xn−1 ∈ G such thaty ∈ P [x1],
z ∈ P [xn−1] and P [xi]∩P [xi+1] 6= ∅ for 1 ≤ i ≤ n−1. The previous conditions
are equivalent to the existence of point s p1, . . . , pn, q0, . . . , qn−1 ∈ P satisfying
y = x1q0, z = pnxn and pixi = xi+1qi. From the equalityxi = p

−1
i xi+1qi,

we may easily arrive at the formula y = p−11 p
−1
2 · · · p

−1
n zqn−1qn−2 · · · q1q0. It

follows that ypnpn−1 · · · p2p1y = zqn−1qn−2 · · · q1q0. Hence, P
ny ∩ zP n is non-

empty, a contradiction. �



210 A. V. Arhangel’skii and A. Bella

As a local base is always T-linked, we have:

Corollary 2. Every first countable paratopological group has a Gδ-diagonal of
infinite rank.

As a local π-base in an Abelian group is clearly T-linked, we have:

Corollary 3. Every Abelian paratopological group of countable π-character has
a Gδ-diagonal of infinite rank.

Obviously, Corollaries 2 and 3 suggest the following:

Problem 3. Does a (Hausdorff, regular, Tychonoff ) paratopological group of
countable π-character have a Gδ-diagonal of rank n for each integer n?

From Corollary 2 and Proposition 1 we immediately obtain the following
recent result of C. Liu [6], answering Problem 25 in [1]:

Corollary 4. Any first countable paratopological group has a regular Gδ-diagonal.

Similarly, from Corollary 3 and Proposition 1 we obtain:

Corollary 5. Every Abelian paratopological group of countable π-character has
a regular Gδ-diagonal.

Corollary 6. Any first countable paratopological group with cellularity at most
c has cardinality at most c.

Observe that in any first countable paratopological group there is one count-
able family of open covers witnessing that G has a Gδ-diagonal of rank n for
each n < ω. We finish this note with a sufficient condition for the submetriz-
ability of a semitopological group, which slightly generalizes Theorem 28 in [1].
A semitopological group G is said to be ω -narrow if for every open neighbour-
hood U of the neutral element e of G there is a countable subset A ⊆ G such
that AU = G.

Theorem 3. Every separable semitopological group is ω-narrow.

Proof. Fix a countable subset A of G such that A−1 is dense in G, and let
U be any open neighbourhood of the neutral element ein G.Let us show that
AU = G. Take any b ∈ G. There is an open setW such thatb−1 ∈ W and
W b ⊆ U . Since A−1 is dense in G, there is a ∈ A such that a−1 ∈ W . Then
a−1b ∈ U and hence, b ∈ aU . Thus, AU = G. �

Lemma 3. Suppose that G is a semitopological group, and that y, z are any
two distinct elements of G. Then there is an open neighbourhood U of the
neutral element e in G such that for the family γU = {xU : x ∈ G} we have
y /∈ St(z, γU ).

Proof. Clearly, we may assume that z = e. Since G is Hausdorff, there is an
open neighbourhood U of e such that U ∩ U y = ∅. Then y /∈ U−1U .We now
show that U is the neighbourhood we are looking for. Indeed, take any x ∈ G
such that e ∈ xU . Then x ∈ U−1 and hence, xU ⊆ U−1U . It follows that y is
not in xU . Thus, y /∈ St(z, γU ). �



The diagonal of a first countable paratopological group 211

Theorem 4. Suppose that G is a Tychonoff ω-narrow semitopological group
of countable π-character. Then the space G is submetrizable.

Proof. Let P be a countable local π-base at the neutral element e. Since the
space G is Tychonoff, we may assume that every U ∈ P is a cozero-set. For
each U ∈ P, put γU = {xU : x ∈ G}. Since G is ω-narrow, there is a countable
subcover ηU ⊆ γU of G. Put B = ∪{ηU : U ∈ P}. Then B is a countable family
of cozero-sets in G. The family B is T1-separating. Indeed, fix any distinct y
and z in G. By Lemma 2, there is U ∈ P such that for γU = {xU : x ∈ G}
we have y /∈ St(z, γU ). By the choice of ηU , there is V ∈ ηU such that z ∈ V .
Then y /∈ V ∈ B. Thus, the family B is T1-separating.It remains to make a
standard step: for every V ∈ B fix a continuous real-valued function fV on
G such that V = {x ∈ G : fV (x) 6= 0}, and take the diagonal product of
functions fV where V runs over the countable set B. The resulting function
is the desired one-to-one continuous mapping of the space G onto a separable
metrizable space. �

Problem 4. Is every (Hausdorff, regular) semitopological (paratopological)
group with countable Souslin number ω-narrow?

Problem 5. Let G be a paratopological (semitopological) (Hausdorff, regular)
group of countable extent. Must G be ω-narrow?

Problem 6. Let G be a paratopological (semitopological) (Hausdorff, regular)
group of countable extent. Must G be submetrizable

In connection with the last open question, we have the following partial
result:

Theorem 5. If G is a first countable normal (weakly M -normal) paratopolog-
ical group of the countable extent, then G can be condensed onto a separable
metrizable space (hence, G is submetrizable).

Proof. Indeed, by Theorem 2, G has a rank 5-diagonal. Besides, G is star-
Lindelöf. It remains to apply a result from [2]. �

Clearly, “Countable extent” can be replaced by “star-Lindelöf” in Theorem
5.

Another open problem, already formulated in [1], is whether every first
countable regular paratopological group is submetrizable.

In connection with Problem 4, observe that if the answer is “yes”, then every
Tychonoff first countable paratopological group with countable Souslin number
is submetrizable.



212 A. V. Arhangel’skii and A. Bella

References

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[3] A. Bella, More on cellular extent and related cardinal functions, Boll. Un. Mat. Ital. 7,
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[4] R. Z. Buzyakova, Cardinalities of ccc-spaces with regular Gδ-diagonal, Topology Appl.
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[5] R. Engelking, General Topology, 1977.
[6] C. Liu, A note on paratopological groups, Comment. Math. Univ. Carolinae 47, no. 4

(2006), 633–640.
[7] V. Mc Arthur, Gδ-diagonal and metrization theorems, Pacific J. Math. 44 (1973),

213–217.
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Univ. Carolinae 25, no. 2 (1984), 257–260.
[9] P. Zenor, On spaces with regular Gδ-diagonal, Pacific J. Math. 40 (1972), 959–963.

Received March 2006

Accepted May 2006

A. V. Arhangelskii (arhangel@math.ohiou.edu)
Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio
45701, USA.

A. Bella (bella@dmi.unict.it)
Dipartimento di Matematica, Cittá Universitaria, Viale A.Doria 6, 95125, Cata-
nia, Italy.