Banackag.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 5, No. 1, 2004

pp. 25- 48

The topological structure of (homogeneous)
spaces and groups with countable cs∗-character

Taras Banakh and Lubomyr Zdomsky̆ı

Abstract. In this paper we introduce and study three new cardi-
nal topological invariants called the cs∗-, cs-, and sb-characters. The
class of topological spaces with countable cs∗-character is closed under
many topological operations and contains all ℵ-spaces and all spaces
with point-countable cs∗-network. Our principal result states that
each non-metrizable sequential topological group with countable cs∗-
character has countable pseudo-character and contains an open kω-
subgroup. This result is specific for topological groups: under Martin
Axiom there exists a sequential topologically homogeneous kω-space X
with ℵ0 = cs

∗
χ(X) < ψ(X).

2000 AMS Classification: 22A05, 54A20, 54A25, 54A35, 54D55, 54E35,
54H11

Keywords: sb-network, cs-network, cs∗-network, sequential topological group,
kω-group, topologically homogeneous space, small cardinal, cardinal invariant.

Introduction

In this paper we introduce and study three new local cardinal invariants of
topological spaces called the sb-character, the cs-character and cs∗-character,
and describe the structure of sequential topological groups with countable cs∗-
character. All these characters are based on the notion of a network at a point x
of a topological space X, under which we understand a collection N of subsets
of X such that for any neighborhood U ⊂ X of x there is an element N ∈ N
with x ∈ N ⊂ U, see [Lin].

A subset B of a topological space X is called a sequential barrier at a point
x ∈ X if for any sequence (xn)n∈ω ⊂ X convergent to x, there is m ∈ ω
such that xn ∈ B for all n ≥ m, see [Lin]. It is clear that each neighborhood
of a point x ∈ X is a sequential barrier for x while the converse in true for
Fréchet-Urysohn spaces.



26 T.Banakh, L.Zdomsky̆ı

Under a sb-network at a point x of a topological space X we shall understand
a network at x consisting of sequential barriers at x. In other words, a collection
N of subsets of X is a sb-network at x if for any neighborhood U of x there is
an element N ⊂ U of N such that for any sequence (xn) ⊂ X convergent to
x the set N contains almost all elements of (xn). Changing two quantifiers in
this definition by their places we get a definition of a cs-network at x.

Namely, we define a family N of subsets of a topological space X to be a cs-
network (resp. a cs∗-network) at a point x ∈ X if for any neighborhood U ⊂ X
of x and any sequence (xn) ⊂ X convergent to x there is an element N ∈ N
such that N ⊂ U and N contains almost all (resp. infinitely many) members
of the sequence (xn). A family N of subsets of a topological space X is called a
cs-network (resp. cs∗-network) if it is a cs-network (resp. cs∗-network) at each
point x ∈ X, see [Na].

The smallest size |N| of an sb-network (resp. cs-network, cs∗-network) N
at a point x ∈ X is called the sb-character (resp. cs-character, cs∗-character)
of X at the point x and is denoted by sbχ(X,x) (resp. csχ(X,x), cs

∗
χ(X,x)).

The cardinals sbχ(X) = supx∈X sbχ(X,x), csχ(X) = supx∈X csχ(X,x) and
cs∗χ(X) = supx∈X cs

∗
χ(X,x) are called the sb-character, cs-character and cs

∗-
character of the topological space X, respectively. For the empty topological
space X = ∅ we put sbχ(X) = csχ(X) = cs

∗
χ(X) = 1. As expected the

character χ(X) of a topological space X is the smallest cardinal κ such that
each point x ∈ X has a neighborhood base of size ≤ κ.

In the sequel we shall say that a topological space X has countable sb-
character (resp. cs-, cs∗-character) if sbχ(X) ≤ ℵ0 (resp. csχ(X) ≤ ℵ0,
cs∗χ(X) ≤ ℵ0). In should be mentioned that under different names, topological
spaces with countable sb- or cs-character have already occured in topological
literature. In particular, a topological space has countable cs-character if and
only if it is csf-countable in the sense of [Lin]; a (sequential) space X has
countable sb-character if and only if it is universally csf-countable in the sense
of [Lin] (if and only if it is weakly first-countable in the sense of [Ar1] if and
only if it is 0-metrizable in the sense of Nedev [Ne]). From now on, all the
topological spaces considered in the paper are T1-spaces. At first we consider
the interplay between the characters introduced above.

Proposition 1. Let X be a topological space. Then

(1) cs∗χ(X) ≤ csχ(X) ≤ sbχ(X) ≤ χ(X);
(2) χ(X) = sbχ(X) if X is Fréchet-Urysohn;
(3) cs∗χ(X) < ℵ0 iff csχ(X) < ℵ0 iff sbχ(X) < ℵ0 iff cs

∗
χ(X) = 1 iff

csχ(X) = 1 iff sbχ(X) = 1 iff each convergent sequence in X is trivial;

(4) sbχ(X) ≤ 2
cs∗

χ
(X);

(5) csχ(X) ≤ cs
∗
χ(X) · sup{

∣

∣[κ]≤ω
∣

∣ : κ < cs∗χ(X)} ≤
(

cs∗χ(X)
)ℵ0

where

[κ]≤ω = {A ⊂ κ : |A| ≤ ℵ0}.

Here “iff” is an abbreviation for “if and only if”. The Arens’ space S2
and the sequential fan Sω give us simple examples distinguishing between



On groups with countable cs∗-character 27

some of the characters considered above. We recall that the Arens’ space
S2 is the set {(0,0),(

1
n
,0),( 1

n
, 1
nm

) : n,m ∈ N} ⊂ R2 carrying the strongest
topology inducing the original planar topology on the convergent sequences
C0 = {(0,0),(

1
n
,0) : n ∈ N} and Cn = {(

1
n
,0),( 1

n
, 1
nm

) : m ∈ N}, n ∈ N.
The quotient space Sω = S2/C0 obtained from the Arens’ space S2 by iden-
tifying the points of the sequence C0 is called the sequential fan, see [Lin].
The sequential fan Sω is the simplest example of a non-metrizable Fréchet-
Urysohn space while S2 is the simplest example of a sequential space which is
not Fréchet-Urysohn.

We recall that a topological space X is sequential if a subset A ⊂ X if closed
if and only if A is sequentially closed in the sense that A contain the limit
point of any sequence (an) ⊂ A, convergent in X. A topological space X is
Fréchet-Urysohn if for any cluster point a ∈ X of a subset A ⊂ X there is a
sequence (an) ⊂ A, convergent to a.

Observe that ℵ0 = cs
∗
χ(S2) = csχ(S2) = sbχ(S2) < χ(S2) = d while ℵ0 =

cs∗χ(Sω) = csχ(Sω) < sbχ(Sω) = χ(Sω) = d. Here d is the well-known in
Set Theory small uncountable cardinal equal to the cofinality of the partially
ordered set Nω endowed with the natural partial order: (xn) ≤ (yn) iff xn ≤ yn
for all n, see [Va]. Besides d, we will need two other small cardinals: b defined
as the smallest size of a subset of uncountable cofinality in (Nω,≤), and p equal
to the smallest size |F| of a family of infinite subsets of ω closed under finite
intersections and having no infinite pseudo-intersection in the sense that there
is no infinite subset I ⊂ ω such that the complement I \ F is finite for any
F ∈ F, see [Va], [vD]. It is known that ℵ1 ≤ p ≤ b ≤ d ≤ c where c stands
for the size of continuum. Martin Axiom implies p = b = d = c, [MS]. On the
other hand, for any uncountable regular cardinals λ ≤ κ there is a model of
ZFC with p = b = d = λ and c = κ, see [vD, 5.1].

Unlike to the cardinal invariants csχ, sbχ and χ which can be distinguished
on simple spaces, the difference between the cardinal invariants csχ and cs

∗
χ is

more subtle: they cannot be distinguished in some models of Set Theory!

Proposition 2. Let X be a topological space. Then cs∗χ(X) = csχ(X) provided
one of the following conditions is satisfied:

(1) cs∗χ(X) < p;

(2) κℵ0 ≤ cs∗χ(X) for any cardinal κ < cs
∗
χ(X);

(3) p = c and λω ≤ κ for any cardinals λ < κ ≥ c;
(4) p = c (this is so under MA) and X is countable;
(5) the Generalized Continuum Hypothesis holds.

Unlike to the usual character, the cs∗-, cs-, and sb-characters behave nicely
with respect to many countable topological operations.

Among such operation there are: the Tychonov product, the box-product,
producing a sequentially homeomorphic copy, taking image under a sequentially
open map, and forming inductive topologies.

As usual, under the box-product 2i∈IXi of topological spaces Xi, i ∈ I,
we understand the Cartesian product

∏

i∈I Xi endowed with the box-product



28 T.Banakh, L.Zdomsky̆ı

topology generated by the base consisting of products
∏

i∈I Ui where each Ui
is open in Xi. In contrast, by

∏

i∈I Xi we denote the usual Cartesian product
of the spaces Xi, endowed with the Tychonov product topology.

We say that a topological space X carries the inductive topology with respect
to a cover C of X if a subset F ⊂ X is closed in X if and only if the intersection
F ∩ C is closed in C for each element C ∈ C. For a cover C of X let ord(C) =
supx∈X ord(C,x) where ord(C,x) = |{C ∈ C : x ∈ C}|. A topological space
X carrying the inductive topology with respect to a countable cover by closed
metrizable (resp. compact, compact metrizable) subspaces is called an Mω-
space (resp. a kω-space, MKω-space).

A function f : X → Y between topological spaces is called sequentially
continuous if for any convergent sequence (xn) in X the sequence (f(xn)) is
convergent in Y to f(limxn); f is called a sequential homeomorphism if f is
bijective and both f and f−1 are sequentially continuous. Topological spaces
X,Y are defined to be sequentially homeomorphic if there is a sequential homeo-
morphism h : X → Y . Observe that two spaces are sequentially homeomorphic
if and only if their sequential coreflexions are homeomorphic. Under the se-
quential coreflexion σX of a topological space X we understand X endowed
with the topology consisting of all sequentially open subsets of X (a subset U
of X is sequentially open if its complement is sequentially closed in X; equiv-
alently U is a sequential barrier at each point x ∈ U). Note that the identity
map id : σX → X is continuous while its inverse is sequentially continuous, see
[Lin].

A map f : X → Y is sequentially open if for any point x0 ∈ X and a sequence
S ⊂ Y convergent to f(x0) there is a sequence T ⊂ X convergent to x0 and
such that f(T) ⊂ S. Observe that a bijective map f is sequentially open if its
inverse f−1 is sequentially continuous.

The following technical Proposition is an easy consequence of the corre-
sponding definitions.

Proposition 3. (1) If X is a subspace of a topological space Y , then
cs∗χ(X) ≤ cs

∗
χ(Y ), csχ(X) ≤ csχ(Y ) and sbχ(X) ≤ sb(Y ).

(2) If f : X → Y is a surjective continuous sequentially open map between
topological spaces, then cs∗χ(Y ) ≤ cs

∗
χ(X) and sbχ(Y ) ≤ sbχ(X).

(3) If f : X → Y is a surjective sequentially continuous sequentially open
map between topological spaces, then min{cs∗χ(Y ),ℵ1} ≤ min{cs

∗
χ(X),ℵ1},

min{csχ(Y ),ℵ1} ≤ min{csχ(X),ℵ1}, and
min{sbχ(Y ),ℵ1} ≤ min{sbχ(X),ℵ1}.

(4) If X and Y are sequentially homeomorphic topological spaces, then
min{cs∗χ(X),ℵ1} = min{csχ(X),ℵ1} = min{csχ(Y ),ℵ1} =
= min{cs∗χ(Y ),ℵ1}, and min{sbχ(Y ),ℵ1} = min{sbχ(X),ℵ1}.

(5) For any topological space X min{sbχ(X),ℵ1} = min{sbχ(σX),ℵ1} ≤
sbχ(σX) ≥ sbχ(X) and csχ(X) ≤ csχ(σX) ≥ min{csχ(σX),ℵ1} =
min{csχ(X),ℵ1} = min{cs

∗
χ(X),ℵ1} = min{cs

∗
χ(σX),ℵ1} ≤ cs

∗
χ(σX) ≥

cs∗χ(X).



On groups with countable cs∗-character 29

(6) If X =
∏

i∈I Xi is the Tychonov product of topological spaces Xi, i ∈ I,
then cs∗χ(X) ≤

∑

i∈I cs
∗
χ(Xi), csχ(X) ≤

∑

i∈I csχ(Xi) and sbχ(X) ≤
∑

i∈I sbχ(Xi).
(7) If X = 2i∈IXi is the box-product of topological spaces Xi, i ∈ I, then

cs∗χ(X) ≤
∑

i∈I cs
∗
χ(Xi) and csχ(X) ≤

∑

i∈I csχ(Xi).
(8) If a topological space X carries the inductive topology with respect to a

cover C of X, then cs∗χ(X) ≤ ord(C) · supC∈C cs
∗
χ(C).

(9) If a topological space X carries the inductive topology with respect to a
point-countable cover C of X, then csχ(X) ≤ supC∈C csχ(C).

(10) If a topological space X carries the inductive topology with respect to a
point-finite cover C of X, then sbχ(X) ≤ supC∈C sbχ(C).

Since each first-countable space has countable cs∗-character, it is natural to
think of the class of topological spaces with countable cs∗-character as a class
of generalized metric spaces. However this class contains very non-metrizable
spaces like βN, the Stone-Čech compactification of the discrete space of positive
integers. The reason is that βN contains no non-trivial convergent sequence. To
avoid such pathologies we shall restrict ourselves by sequential spaces. Observe
that a topological space is sequential provided X carries the inductive topology
with respect to a cover by sequential subspaces. In particular, each Mω-space
is sequential and has countable cs∗-character. Our principal result states that
for topological groups the converse is also true. Under an Mω-group (resp.
MKω-group) we understand a topological group whose underlying topological
space is an Mω-space (resp. MKω-space).

Theorem 1. Each sequential topological group G with countable cs∗-character
is an Mω-group. More precisely, either G is metrizable or else G contains an
open MKω-subgroup H and is homeomorphic to the product H × D for some
discrete space D.

For Mω-groups the second part of this theorem was proven in [Ba1]. Theo-
rem 1 has many interesting corollaries.

At first we show that for sequential topological groups with countable cs∗-
character many important cardinal invariants are countable, coincide or take
some fixed values. Let us remind some definitions, see [En1]. For a topological
space X recall that

• the pseudocharacter ψ(X) is the smallest cardinal κ such that each
one-point set {x} ⊂ X can be written as the intersection {x} = ∩U of
some family U of open subsets of X with |U| ≤ κ;

• the cellularity c(X) is the smallest cardinal κ such that X contains no
family U of size |U| > κ consisting of non-empty pairwise disjoint open
subsets;

• the Lindelöf number l(X) is the smallest cardinal κ such that each open
cover of X contains a subcover of size ≤ κ;

• the density d(X) is the smallest size of a dense subset of X;



30 T.Banakh, L.Zdomsky̆ı

• the tightness t(X) is the smallest cardinal κ such that for any subset
A ⊂ X and a point a ∈ Ā from its closure there is a subset B ⊂ A of
size |B| ≤ κ with a ∈ B̄;

• the extent e(X) is the smallest cardinal κ such that X contains no
closed discrete subspace of size > κ;

• the compact covering number kc(X) is the smallest size of a cover of X
by compact subsets;

• the weight w(X) is the smallest size of a base of the topology of X;
• the network weight nw(X) is the smallest size |N| of a topological
network for X (a family N of subsets of X is a topological network if
for any open set U ⊂ X and any point x ∈ U there is N ∈ N with
x ∈ N ⊂ U);

• the k-network weight knw(X) is the smallest size |N| of a k-network
for X (a family N of subsets of X is a k-network if for any open set
U ⊂ X and any compact subset K ⊂ U there is a finite subfamily
M ⊂ N with K ⊂ ∪M ⊂ U).

For each topological space X these cardinal invariants relate as follows:

max{c(X), l(X),e(X)} ≤ nw(X) ≤ knw(X) ≤ w(X).

For metrizable spaces all of them are equal, see [En1, 4.1.15].
In the class of k-spaces there is another cardinal invariant, the k-ness intro-

duced by E. van Douwen, see [vD, §8]. We remind that a topological space
X is called a k-space if it carries the inductive topology with respect to the
cover of X by all compact subsets. It is clear that each sequential space is a
k-space. The k-ness k(X) of a k-space is the smallest size |K| of a cover K of
X by compact subsets such that X carries the inductive topology with respect
to the cover K. It is interesting to notice that k(Nω) = d while k(Q) = b, see
[vD]. Proposition 3(8) implies that cs∗χ(X) ≤ k(X) · ψ(X) ≥ kc(X) for each
k-space X. Observe also that a topological space X is a kω-space if and only
if X is a k-space with k(X) ≤ ℵ0.

Besides cardinal invariants we shall consider an ordinal invariant, called
the sequential order. Under the sequential closure A(1) of a subset A of a
topological space X we understand the set of all limit point of sequences (an) ⊂
A, convergent in X. Given an ordinal α define the α-th sequential closure A(α)

of A by transfinite induction: A(α) =
⋃

β<α(A
(β))(1). Under the sequential

order so(X) of a topological space X we understand the smallest ordinal α
such that A(α+1) = A(α) for any subset A ⊂ X. Observe that a topological
space X is Fréchet-Urysohn if and only if so(X) ≤ 1; X is sequential if and
only if clX(A) = A

(so(X)) for any subset A ⊂ X.
Besides purely topological invariants we shall also consider a cardinal invari-

ant, specific for topological groups. For a topological group G let ib(G), the
boundedness index of G be the smallest cardinal κ such that for any nonempty
open set U ⊂ G there is a subset F ⊂ G of size |F | ≤ κ such that G = F · U.
It is known that ib(G) ≤ min{c(G), l(G),e(G)} and w(G) = ib(G) · χ(G) for
each topological group, see [Tk].



On groups with countable cs∗-character 31

Theorem 2. Each sequential topological group G with countable cs∗-character
has the following properties: ψ(G) ≤ ℵ0, sbχ(G) = χ(G) ∈ {1,ℵ0,d}, ib(G) =
c(G) = d(G) = l(G) = e(G) = nw(G) = knw(G), and so(G) ∈ {1,ω1}.

We shall derive from Theorems 1 and 2 an unexpected metrization theorem
for topological groups. But first we need to remind the definitions of some of
αi-spaces, i = 1, . . . ,6 introduced by A.V. Arkhangelski in [Ar2], [Ar4]. We
also define a wider class of α7-spaces.

A topological space X is called

• an α1-space if for any sequences Sn ⊂ X, n ∈ ω, convergent to a point
x ∈ X there is a sequence S ⊂ X convergent to x and such that Sn \ S
is finite for all n;

• an α4-space if for any sequences Sn ⊂ X, n ∈ ω, convergent to a
point x ∈ X there is a sequence S ⊂ X convergent to x and such that
Sn ∩ S 6= ∅ for infinitely many sequences Sn;

• an α7-space if for any sequences Sn ⊂ X, n ∈ ω, convergent to a point
x ∈ X there is a sequence S ⊂ X convergent to some point y of X and
such that Sn ∩ S 6= ∅ for infinitely many sequences Sn;

Under a sequence converging to a point x of a topological space X we un-
derstand any countable infinite subset S of X such that S \ U if finite for any
neighborhood U of x. Each α1-space is α4 and each α4-space is α7. Quite often
α7-spaces are α4, see Lemma 7. Observe also that each sequentially compact
space is α7. It can be shown that a topological space X is an α7-space if and
only if it contains no closed copy of the sequential fan Sω in its sequential
coreflexion σX. If X is an α4-space, then σX contains no topological copy of
Sω.

We remind that a topological group G is Weil complete if it is complete in its
left (equivalently, right) uniformity. According to [PZ, 4.1.6], each kω-group is
Weil complete. The following metrization theorem can be easily derived from
Theorems 1, 2 and elementary properties of MKω-groups.

Theorem 3. A sequential topological group G with countable cs∗-character is
metrizable if one of the following conditions is satisfied:

(1) so(G) < ω1;
(2) sbχ(G) < d;
(3) ib(G) < k(G);
(4) G is Fréchet-Urysohn;
(5) G is an α7-space;
(6) G contains no closed copy of Sω or S2;
(7) G is not Weil complete;
(8) G is Baire;
(9) ib(G) < |G| < 2ℵ0.

According to Theorem 1, each sequential topological group with countable
cs∗-character is an Mω-group. The first author has proved in [Ba3] that the



32 T.Banakh, L.Zdomsky̆ı

topological structure of a non-metrizable punctiform Mω-group is completely
determined by its density and the compact scatteredness rank.

Recall that a topological space X is punctiform if X contains no compact
connected subspace containing more than one point, see [En2, 1.4.3]. In par-
ticular, each zero-dimensional space is punctiform.

Next, we remind the definition of the scatteredness height. Given a topolog-
ical space X let X(1) ⊂ X denote the set of all non-isolated points of X. For
each ordinal α define the α-th derived set X(α) of X by transfinite induction:
X(α) =

⋂

β<α(X(β))(1). Under the scatteredness height sch(X) of X we under-
stand the smallest ordinal α such that X(α+1) = X(α). A topological space X is
scattered if X(α) = ∅ for some ordinal α. Under the compact scatteredness rank
of a topological space X we understand the ordinal scr(X) = sup{sch(K) : K
is a scattered compact subspace of X}.

Theorem 4. Two non-metrizable sequential punctiform topological groups G,
H with countable cs∗-character are homeomorphic if and only if d(G) = d(H)
and scr(G) = scr(H).

This theorem follows from Theorem 1 and Main Theorem of [Ba3] asserting
that two non-metrizable punctiform Mω-groups G, H are homeomorphic if and
only if d(G) = d(H) and scr(G) = scr(H). For countable kω-groups this fact
was proven by E.Zelenyuk [Ze1].

The topological classification of non-metrizable sequential locally convex
spaces with countable cs∗-character is even more simple. Any such a space
is homeomorphic either to R∞ or to R∞ × Q where Q = [0,1]ω is the Hilbert
cube and R∞ is a linear space of countable algebraic dimension, carrying the
strongest locally convex topology. It is well-known that this topology is induc-
tive with respect to the cover of R∞ by finite-dimensional linear subspaces. The
topological characterization of the spaces R∞ and R∞ ×Q was given in [Sa]. In
[Ba2] it was shown that each infinite-dimensional locally convex MKω-space is
homeomorphic to R∞ or R∞ ×Q. This result together with Theorem 1 implies
the following classification

Corollary 1. Each non-metrizable sequential locally convex space with count-
able cs∗-character is homeomorphic to R∞ or R∞ × Q.

As we saw in Theorem 2, each sequential topological group with countable
cs∗-character has countable pseudocharacter. The proof of this result is based
on the fact that compact subsets of sequential topological groups with countable
cs∗-character are first countable. This naturally leads to a conjecture that
compact spaces with countable cs∗-character are first countable. Surprisingly,
but this conjecture is false: assuming the Continuum Hypothesis N. Yakovlev
[Ya] has constructed a scattered sequential compactum which has countable
sb-character but fails to be first countable. In [Ny2] P.Nyikos pointed out that
the Yakovlev construction still can be carried under the assumption b = c.
More precisely, we have



On groups with countable cs∗-character 33

Proposition 4. Under b = c there is a regular locally compact locally countable
space Y whose one-point compactification αY is sequential and satisfies ℵ0 =
sbχ(αY ) < ψ(αY ) = c.

We shall use the above proposition to construct examples of topologically
homogeneous spaces with countable cs-character and uncountable pseudochar-
acter. This shows that Theorem 2 is specific for topological groups and cannot
be generalized to topologically homogeneous spaces. We remind that a topo-
logical space X is topologically homogeneous if for any points x,y ∈ X there is
a homeomorphism h : X → X with h(x) = y.

Theorem 5.

(1) There is a topologically homogeneous countable regular kω-space X1
with ℵ0 = sbχ(X1) = ψ(X1) < χ(X1) = d and so(X1) = ω;

(2) Under b = c there is a sequential topologically homogeneous zero-dimen-
sional kω-space X2 with ℵ0 = csχ(X2) < ψ(X2) = c;

(3) Under b = c there is a sequential topologically homogeneous totally
disconnected space X3 with ℵ0 = sbχ(X3) < ψ(X3) = c.

We remind that a space X is totally disconnected if for any distinct points
x,y ∈ X there is a continuous function f : X → {0,1} such that f(x) 6= f(y),
see [En2].

Remark 1. The space X1 from Theorem 5(1) is the well-known Arkhangelski-
Franklin example [AF] (see also [Co, 10.1]) of a countable topologically homoge-
neous kω-space, homeomorphic to no topological group (this also follows from
Theorem 2). On the other hand, according to [Ze2], each topologically ho-
mogeneous countable regular space (in particular, X1) is homeomorphic to a
quasitopological group, that is a topological space endowed with a separately
continuous group operation with continuous inversion. This shows that Theo-
rem 2 cannot be generalized onto quasitopological groups (see however [Zd] for
generalizations of Theorems 1 and 2 to some other topologo-algebraic struc-
tures).

Next, we find conditions under which a space with countable cs∗-character
is first-countable or has countable sb-character. Following [Ar3] we define a
topological space X to be c-sequential if for each closed subspace Y ⊂ X and
each non-isolated point y of Y there is a sequence (yn) ⊂ Y \ {y} convergent
to y. It is clear that each sequential space is c-sequential. A point x of a
topological space X is called regular Gδ if {x} = ∩B for some countable family
B of closed neighborhood of x in X, see [Lin].

First we characterize spaces with countable sb-character (the first three items
of this characterization were proved by Lin [Lin, 3.13] in terms of (universally)
csf-countable spaces).

Proposition 5. For a Hausdorff space X the following conditions are equiva-
lent:

(1) X has countable sb-character;



34 T.Banakh, L.Zdomsky̆ı

(2) X is an α1-space with countable cs
∗-character;

(3) X is an α4-space with countable cs
∗-character;

(4) cs∗χ(X) ≤ ℵ0 and sbχ(X) < p.
Moreover, if X is c-sequential and each point of X is regular Gδ, then the

conditions (1)–(4) are equivalent to:
(5) cs∗χ(X) ≤ ℵ0 and sbχ(X) < d.

Next, we give a characterization of first-countable spaces in the same spirit
(the equivalences (1) ⇔ (2) ⇔ (5) were proved by Lin [Lin, 2.8]).

Proposition 6. For a Hausdorff space X with countable cs∗-character the
following conditions are equivalent:

(1) X is first-countable;
(2) X is Fréchet-Urysohn and has countable sb-character;
(3) X is Fréchet-Urysohn α7-space;
(4) χ(X) < p and X has countable tightness.

Moreover, if each point of X is regular Gδ, then the conditions (1)–(4) are
equivalent to:

(5) X is a sequential space containing no closed copy of S2 or Sω;
(6) X is a sequential space with χ(X) < d.

For Fréchet-Urysohn (resp. dyadic) compacta the countability of the cs∗-
character is equivalent to the first countability (resp. the metrizability). We
remind that a compact Hausdorff space X is called dyadic if X is a continuous
image of the Cantor discontinuum {0,1}κ for some cardinal κ.

Proposition 7.

(1) A Fréchet-Urysohn countably compact space is first-countable if and
only if it has countable cs∗-character.

(2) A dyadic compactum is metrizable if and only if its has countable cs∗-
character.

In light of Proposition 7(1) one can suggest that cs∗χ(X) = χ(X) for any
compact Fréchet-Urysohn space X. However that is not true: under CH,
csχ(αD) 6= χ(αD) for the one-point compactification αD of a discrete space
D of size |D| = ℵ2. Surprisingly, but the problem of calculating the cs

∗- and
cs-characters of the spaces αD is not trivial and the definitive answer is known
only under the Generalized Continuum Hypothesis. First we note that the
cardinals cs∗χ(αD) and csχ(αD) admit an interesting interpretation which will
be used for their calculation.

Proposition 8. Let D be an infinite discrete space. Then

(1) cs∗χ(αD) = min{w(X) : X is a (regular zero-dimensional) topological
space of size |X| = |D| containing no no-trivial convergent sequence};

(2) csχ(αD) = min{w(X) : X is a (regular zero-dimensional) topological
space of size |X| = |D| containing no countable non-discrete subspace}.



On groups with countable cs∗-character 35

For a cardinal κ we put log κ = min{λ : κ ≤ 2λ} and cof([κ]≤ω) be the
smallest size of a collection C ⊂ [κ]≤ω such that each at most countable subset
S ⊂ κ lies in some element C ∈ C. Observe that cof([κ]≤ω) ≤ κω but sometimes

the inequality can be strict: 1 = cof([ℵ0]
≤ω) < ℵ0 and ℵ1 = cof([ℵ1]

≤ω) < ℵℵ01 .
In the following proposition we collect all the information on the cardinals
cs∗χ(αD) and csχ(αD) we know.

Proposition 9. Let D be an uncountable discrete space. Then

(1) ℵ1 · log |D| ≤ cs
∗
χ(αD) ≤ csχ(αD) ≤ min{|D|,2

ℵ0 · cof([log |D|]≤ω)}
while sbχ(αD) = χ(αD) = |D|;

(2) cs∗χ(αD) = csχ(αD) = ℵ1 · log |D| under GCH.

In spite of numerous efforts some annoying problems concerning cs∗- and
cs-characters still rest open.

Problem 1. Is there a (necessarily consistent) example of a space X with
cs∗χ(X) 6= csχ(X)? In particular, is cs

∗
χ(αc) 6= csχ(αc) in some model of ZFC?

In light of Proposition 8 it is natural to consider the following three cardinal
characteristics of the continuum which seem to be new:

w1 = min{w(X) : X is a topological space of size |X| = c containing no

non-trivial convergent sequence};

w2 = min{w(X) : X is a topological space of size |X| = c containing no

non-discrete countable subspace};

w3 = min{w(X) : X is a P-space of size |X| = c}.

As expected, a P-space is a T1-space whose any Gδ-subset is open. Observe
that w1 = cs

∗
χ(αc) while w2 = csχ(αc). It is clear that ℵ1 ≤ w1 ≤ w2 ≤

w3 ≤ c and hence the cardinals wi, i = 1,2,3, fall into the category of small
uncountable cardinals, see [Va].

Problem 2. Are the cardinals wi, i = 1,2,3, equal to (or can be estimated
via) some known small uncountable cardinals considered in Set Theory? Is
w1 < w2 < w3 in some model of ZFC?

Our next question concerns the assumption b = c in Theorem 5.

Problem 3. Is there a ZFC-example of a sequential space X with sbχ(X) <
ψ(X) or at least cs∗χ(X) < ψ(X)?

Propositions 1 and 5 imply that sbχ(X) ∈ {1,ℵ0}∪[d,c] for any c-sequential
topological space X with countable cs∗-character. On the other hand, for a
sequential topological group G with countable cs∗-character we have a more
precise estimate sbχ(G) ∈ {1,ℵ0,d}.

Problem 4. Is sbχ(X) ∈ {1,ℵ0,d} for any sequential space X with countable
cs∗-character?

As we saw in Proposition 7, χ(X) ≤ ℵ0 for any Fréchet-Urysohn compactum
X with csχ(X) ≤ ℵ0.



36 T.Banakh, L.Zdomsky̆ı

Problem 5. Is sbχ(X) ≤ ℵ0 for any sequential (scattered) compactum X with
csχ(X) ≤ ℵ0?

Now we pass to proofs of our results.

On sequence trees in topological groups

Our basic instrument in proofs of main results is the concept of a sequence
tree. As usual, under a tree we understand a partially ordered subset (T,≤)
such that for each t ∈ T the set ↓ t = {τ ∈ T : σ ≤ t} is well-ordered
by the order ≤. Given an element t ∈ T let ↑ t = {τ ∈ T : τ ≥ t} and
succ(t) = min(↑ t \ {t}) be the set of successors of t in T . A maximal linearly
ordered subset of a tree T is called a branch of T . By maxT we denote the set
of maximal elements of a tree T .

Definition 1. Under a sequence tree in a topological space X we understand
a tree (T,≤) such that

• T ⊂ X;
• T has no infinite branch;
• for each t /∈ maxT the set min(↑ t \ {t}) of successors of t is countable
and converges to t.

Saying that a subset S of a topological space X converges to a point t ∈ X
we mean that for each neighborhood U ⊂ X of t the set S \ U is finite.

The following lemma is well-known and can be easily proven by transfinite
induction (on the ordinal s(a,A) = min{α : a ∈ A(α)} for a subset A of a
sequential space and a point a ∈ Ā from its closure)

Lemma 1. A point a ∈ X of a sequential topological space X belongs to the
closure of a subset A ⊂ X if and only if there is a sequence tree T ⊂ X with
minT = {a} and maxT ⊂ A.

For subsets A,B of a group G let A−1 = {x−1 : x ∈ A} ⊂ G be the inversion
of A in G and AB = {xy : x ∈ A, y ∈ B} ⊂ G be the product of A,B in G.
The following two lemmas will be used in the proof of Theorem 1.

Lemma 2. A sequential subspace F ⊂ X of a topological group G is first
countable if the subspace F−1F ⊂ G has countable sb-character at the unit e
of the group G.

Proof. Our proof starts with the observation that it is sufficient to consider the
case e ∈ F and prove that F has countable character at e.

Let {Sn : n ∈ ω} be a decreasing sb-network at e in F
−1F . First we

show that for every n ∈ ω there exists m > n such that S2m ∩ (F
−1F) ⊂ Sn.

Otherwise, for every m ∈ ω there would exist xm,ym ∈ Sm with xmym ∈
(F−1F) \ Sn. Taking into account that limm→∞ xm = limm→∞ ym = e, we get
limm→∞ xmym = e. Since Sn is a sequential barrier at e, there is a number m
with xmym ∈ Sn, which contradicts to the choice of the points xm,ym.

Now let us show that for all n ∈ ω the set Sn ∩ F is a neighborhood of e in
F . Suppose, conversely, that e ∈ clF (F \ Sn0) for some n0 ∈ ω.



On groups with countable cs∗-character 37

By Lemma 1 there exists a sequence tree T ⊂ F , minT = {e} and maxT ⊂
F \Sn0. To get a contradiction we shall construct an infinite branch of T . Put
x0 = e and let m0 be the smallest integer such that S

2
m0

∩ F−1F ⊂ Sn0.
By induction, for every i ≥ 1 find a number mi > mi−1 with S

2
mi

∩F−1F ⊂
Smi−1 and a point xi ∈ succ(xi−1) ∩ (xi−1Smi). To show that such a choice
is always possible, it suffices to verify that xi−1 /∈ maxT . It follows from the
inductive construction that xi−1 ∈ F ∩ (Sm0 · · ·Smi−1) ⊂ F ∩ S

2
m0

⊂ Sn0 and
thus xi−1 /∈ maxT because maxT ⊂ F \ Sn0.

Therefore we have constructed an infinite branch {xi : i ∈ ω} of the sequence
tree T which is not possible. This contradiction finishes the proof. �

Lemma 3. A sequential α7-subspace F of a topological group G has countable
sb-character provided the subspace F−1F ⊂ G has countable cs-character at
the unit e of G.

Proof. Suppose that F ⊂ G is a sequential α7-space with csχ(F
−1F,e) ≤ ℵ0.

We have to prove that sbχ(F,x) ≤ ℵ0 for any point x ∈ F . Replacing F
by Fx−1, if necessary, we can assume that x = e is the unit of the group
G. Fix a countable family A of subsets of G closed under group products
in G, finite unions and finite intersections, and such that F−1F ∈ A and
A|F−1F = {A ∩ (F−1F) : A ∈ A} is a cs-network at e in F−1F . We claim
that the collection A|F = {A ∩ F : A ∈ A} is a sb-network at e in F .

Assuming the converse, we would find an open neighborhood U ⊂ G of e
such that for any element A ∈ A with A ∩ F ⊂ U the set A ∩ F fails to be a
sequential barrier at e in F .

Let A′ = {A ∈ A : A ⊂ F ∩ U} = {An : n ∈ ω} and Bn =
⋃

k≤n Ak. Let
m−1 = 0 and U−1 ⊂ U be any closed neighborhood of e in G. By induction, for
every k ∈ ω find a number mk > mk−1, a closed neighborhood Uk ⊂ Uk−1 of e
in G, and a sequence (xk,i)i∈ω convergent to e so that the following conditions
are satisfied:

(i) {xk,i : i ∈ ω} ⊂ Uk−1 ∩ F \ Bmk−1;
(ii) the set Fk = {xn,i : n ≤ k, i ∈ ω} \ Bmk is finite;
(iii) Uk ∩ (Fk ∪ {xi,j : i,j ≤ k}) = ∅ and U

2
k ⊂ Uk−1.

The last condition implies that U0U1 · · ·Uk ⊂ U for every k ≥ 0.
Consider the subspace X = {xk,i : k,i ∈ ω} of F and observe that it is

discrete (in itself). Denote by X̄ the closure of X in F and observe that X̄ \X
is closed in F . We claim that e is an isolated point of X̄ \ X. Assuming the
converse and applying Lemma 1 we would find a sequence tree T ⊂ X̄ such
that min T = {e}, maxT ⊂ X, and succ(e) ⊂ X̄ \ X.

By induction, construct a (finite) branch (ti)i≤n+1 of the tree T and a se-
quence {Ci : i ≤ n} of elements of the family A such that t0 = e, |succ(ti) \
tiCi| < ℵ0 and Ci ⊂ Ui ∩ (F

−1F), ti+1 ∈ succ(ti) ∩ tiCi, for each i ≤ n. Note
that the infinite set σ = succ(tn) ∩ tnCn ⊂ X converges to the point tn 6= e.

On the other hand, σ ⊂ tnCn ⊂ tn−1Cn−1Cn ⊂ · · · ⊂ t0C0 · · ·Cn ⊂
U0 · · ·Un ⊂ U. It follows from our assumption on A that C0 · · ·Cn ∈ A and



38 T.Banakh, L.Zdomsky̆ı

thus (C0 · · ·Cn) ∩ F ⊂ Bmk for some k. Consequently, σ ⊂ X ∩ Bmk and
σ ⊂ {xj,i : j ≤ k, i ∈ ω} by the item (i) of the construction of X. Since e is
a unique cluster point of the set {xj,i : j ≤ k, i ∈ ω}, the sequence σ cannot
converge to tn 6= e, which is a contradiction.

Thus e is an isolated point of X̄ \ X and consequently, there is a closed
neighborhood W of e in G such that the set V = ({e} ∪X) ∩W is closed in F .

For every n ∈ ω consider the sequence Sn = W ∩ {xn,i : i ∈ ω} convergent
to e. Since F is an α7-space, there is a convergent sequence S ⊂ F such that
S ∩ Sn 6= ∅ for infinitely many sequences Sn. Taking into account that V is a
closed subspace of F with |V ∩ S| = ℵ0, we conclude that the limit point lim S
of S belongs to the set V . Moreover, we can assume that S ⊂ V . Since the
space X is discrete, limS ∈ V \ X = {e}. Thus the sequence S converges to
e. Since A′ is a cs-network at e in F , there is a number n ∈ ω such that An
contains almost all members of the sequence S. Since Sm ∩ (Sk ∪ An) = ∅ for
m > k ≥ n, the sequence S cannot meet infinitely many sequences Sm. But
this contradicts to the choice of S. �

Following [vD, §8] by L we denote the countable subspace of the plane R2:

L = {(0,0),( 1
n
, 1
nm

) : n,m ∈ N} ⊂ R2.

The space L is locally compact at each point except for (0,0). Moreover,
according to Lemma 8.3 of [vD], a first countable space X contains a closed
topological copy of the space L if and only if X is not locally compact.

The following important lemma was proven in [Ba1] for normal sequential
groups.

Lemma 4. If a sequential topological group G contains a closed copy of the
space L, then G is an α7-space.

Proof. Let h : L → G be a closed embedding and let x0 = h(0,0), xn,m =
h( 1

n
, 1
nm

) for n,m ∈ N. To show that G is an α7-space, for every n ∈ N
fix a sequence (yn,m)m∈N ⊂ G, convergent to the unit e of G. Denote by
∗ : G × G → G the group operation on G.

It is easy to verify that for every n the subspace Dn = {xn,m ∗yn,m : m ∈ N}
is closed and discrete in G. Hence there exists kn ∈ N such that x0 6= xn,m∗yn,m
for all m > kn. Consider the subset

A = {xn,m ∗ yn,m : n > 0, m > kn}

and using the continuity of the group operation, show that x0 6∈ A is a cluster
point of A in G. Consequently, the set A is not closed and by the sequentiality
of G, there is a sequence S ⊂ A convergent to a point a /∈ A. Since every
space Dn is closed and discrete in G, we may replace S by a subsequence, and
assume that |S ∩ Dn| ≤ 1 for every n ∈ N. Consequently, S can be written as
S = {xni,mi ∗ yni,mi : i ∈ ω} for some number sequences (mi) and (ni) with
ni+1 > ni for all i. It follows that the sequence (xni,mi)i∈ω converges to x0
and consequently, the sequence T = {yni,mi}i∈ω converges to x

−1
0 ∗ a. Since

T ∩ {yni,m}m∈N 6= ∅ for every i, we conclude that G is an α7-space. �



On groups with countable cs∗-character 39

Lemma 4 allows us to prove the following unexpected

Lemma 5. A non-metrizable sequential topological group G with countable cs-
character has a countable cs-network at the unit, consisting of closed countably
compact subsets of G.

Proof. Given a non-metrizable sequential group G with countable cs-character
we can apply Lemmas 2–4 to conclude that G contains no closed copy of the
space L. Fix a countable cs-network N at e, closed under finite intersections
and consisting of closed subspaces of G. We claim that the collection C ⊂ N
of all countably compact subsets N ∈ N forms a cs-network at e in G.

To show this, fix a neighborhood U ⊂ G of e and a sequence (xn) ⊂ G
convergent to e. We must find a countably compact set M ∈ N with M ⊂ U,
containing almost all points xn. Let A = {Ak : k ∈ ω} be the collection of
all elements N ⊂ U of N containing almost all points xn. Now it suffices to
find a number n ∈ ω such that the intersection M =

⋂

k≤n Ak is countably

compact. Suppose to the contrary, that for every n ∈ ω the set
⋂

k≤n Ak is
not countably compact. Then there exists a countable closed discrete subspace
K0 ⊂ A0 with K0 6∋ e. Fix a neighborhood W0 of e with W0 ∩ K0 = ∅. Since
N is a cs-network at e, there exists k1 ∈ ω such that Ak1 ⊂ W0.

It follows from our hypothesis that there is a countable closed discrete sub-
space K1 ⊂

⋂

k≤k1
Ak with K1 ∋ e. Proceeding in this fashion we construct

by induction an increasing number sequence (kn)n∈ω ⊂ ω, a sequence (Kn)n∈ω
of countable closed discrete subspaces of G, and a sequence (Wn)n∈ω of open
neighborhoods of e such that Kn ⊂

⋂

k≤kn
Ak, Wn ∩Kn = ∅, and Akn+1 ⊂ Wn

for all n ∈ ω.
It follows from the above construction that {e} ∪

⋃

n∈ω Kn is a closed copy
of the space L which is impossible. �

Proofs of Main Results

Proof of Proposition 1. The first three items can be easily derived from
the corresponding definitions. To prove the fourth item observe that for any
cs∗-network N at a point x of a topological space X, the family N ′ = {∪F :
F ⊂ N} is an sb-network at x.

The proof of fifth item is more tricky. Fix any cs∗-network N at a point
x ∈ X with |N| ≤ cs∗χ(X). Let λ = cof(|N|) be the cofinality of the cardinal
|N| and write N =

⋃

α<λ
Nα where Nα ⊂ Nβ and |Nα| < |N| for any ordinals

α ≤ β < λ. Consider the family M = {∪C : C ∈ [Nα]
≤ω, α < λ} and observe

that |M| ≤ λ · sup{|[κ]≤ω| : κ < |N|} where [κ]≤ω = {A ⊂ κ : |A| ≤ ℵ0}. It
remains to verify that M is a cs-network at x.

Fix a neighborhood U ⊂ X of x and a sequence S ⊂ X convergent to x.
For every α < λ choose a countable subset Cα ⊂ Nα such that ∪Cα ⊂ U and
S ∩ (∪Cα) = S ∩ (∪{N ∈ Nα : N ⊂ U}). It follows that ∪Cα ∈ M. Let
Sα = S ∩(∪Cα) and observe that Sα ⊂ Sβ for α ≤ β < λ. To finish the proof it
suffices to show that S\Sα is finite for some α < λ. Then the element ∪Cα ⊂ U
of M will contain almost all members of the sequence S.



40 T.Banakh, L.Zdomsky̆ı

Separately, we shall consider the cases of countable and uncountable λ. If λ
is uncountable, then it has uncountable cofinality and consequently, the transfi-
nite sequence (Sα)α<λ eventually stabilizes, i.e., there is an ordinal α < λ such
that Sβ = Sα for all β ≥ α. We claim that the set S \ Sα is finite. Otherwise,
S \ Sα would be a sequence convergent to x and there would exist an element
N ∈ N with N ⊂ U and infinite intersection N ∩(S \Sα). Find now an ordinal
β ≥ α with N ∈ Nβ and observe that S ∩ N ⊂ Sβ = Sα which contradicts to
the choice of N.

If λ is countable and S \ Sα is infinite for any α < λ, then we can find
an infinite pseudo-intersection T ⊂ S of the decreasing sequence {S \ Sα}α<λ.
Note that T ∩ Sα is finite for every α < λ. Since sequence T converges to x,
there is an element N ∈ N such that N ⊂ U and N ∩T is infinite. Find α < λ
with N ∈ Nα and observe that N ∩S ⊂ Sα. Then N ∩T ⊂ N ∩T ∩Sα ⊂ T ∩Sα
is finite, which contradicts to the choice of N.

Proof of Proposition 2. Let X be a topological space and fix a point x ∈ X.
(1) Suppose that cs∗χ(X) < p and fix a cs

∗-network N at the point x such
that |N| < p. Without loss of generality, we can assume that the family N is
closed under finite unions. We claim that N is a cs-network at x. Assuming
the converse we would find a neighborhood U ⊂ X of x and a sequence S ⊂ X
convergent to x such that S \N is infinite for any element N ∈ N with N ⊂ U.
Since N is closed under finite unions, the family F = {S\N : N ∈ N , N ⊂ U}
is closed under finite intersections. Since |F| ≤ |N| < p, the family F has an
infinite pseudo-intersection T ⊂ S. Consequently, T ∩N is finite for any N ∈ N
with N ⊂ U. But this contradicts to the facts that T converges to x and N is
a cs∗-network at x.

The items (2) and (3) follow from Propositions 1(5) and 2(1). The item
(4) follows from (1,2) and the inequality χ(X) ≤ c holding for any countable
topological space X.

Finally, to derive (5) from (3) use the well-known fact that under GCH,
λℵ0 ≤ κ for any infinite cardinals λ < κ, see [HJ, 9.3.8].

Proof of Theorem 1. Suppose that G is a non-metrizable sequential group
with countable cs∗-character. By Proposition 2(1), csχ(G) = cs

∗
χ(G) ≤ ℵ0.

First we show that each countably compact subspace K of G is first-coun-
table. The space K, being countably compact in the sequential space G, is
sequentially compact and so are the sets K−1K and (K−1K)−1(K−1K) in G.
The sequential compactness of K−1K implies that it is an α7-space. Since
csχ((K

−1K)−1(K−1K)) ≤ csχ(G) ≤ ℵ0 we may apply Lemmas 3 and 2 to
conclude that the space K−1K has countable sb-character and K has countable
character.

Next, we show that G contains an open MKω-subgroup. By Lemma 5, G has
a countable cs-network K consisting of countably compact subsets. Since the
group product of two countably compact subspaces in G is countably compact,
we may assume that K is closed under finite group products in G. We can also



On groups with countable cs∗-character 41

assume that K is closed under the inversion, i.e. K−1 ∈ K for any K ∈ K.
Then H = ∪K is a subgroup of G. It follows that this subgroup is a sequential
barrier at each of its points, and thus is open-and-closed in G. We claim that
the topology on H is inductive with respect to the cover K. Indeed, consider
some U ⊂ H such that U ∩K is open in K for every K ∈ K. Assuming that U
is not open in H and using the sequentiality of H, we would find a point x ∈ U
and a sequence (xn)n∈ω ⊂ H \ U convergent to x. It follows that there are
elements K1,K2 ∈ K such that x ∈ K1 and K2 contains almost all members
of the sequence (x−1xn). Then the product K = K1K2 contains almost all xn
and the set U ∩ K, being an open neighborhood of x in K, contains almost all
members of the sequence (xn), which is a contradiction.

As it was proved before each K ∈ K is first-countable, and consequently
H has countable pseudocharacter, being the countable union of first countable
subspaces. Then H admits a continuous metric. Since any continuous metric
on a countably compact space generates its original topology, every K ∈ K is
a metrizable compactum, and consequently H is an MKω-subgroup of G.

Since H is an open subgroup of G, G is homeomorphic to H × D for some
discrete space D.

Proof of Theorem 2. Suppose G is a non-metrizable sequential topological
group with countable cs∗-character. By Theorem 1, G contains an open MKω-
subgroup H and is homeomorphic to the product H × D for some discrete
space D. This implies that G has point-countable k-network. By a result
of Shibakov [Shi], each sequential topological group with point-countable k-
network and sequential order < ω1 is metrizable. Consequently, so(G) = ω1.
It is clear that ψ(G) = ψ(H) ≤ ℵ0, χ(G) = χ(H), sbχ(G) = sbχ(H) and
ib(G) = c(G) = d(G) = l(G) = e(G) = nw(G) = knw(G) = |D| · ℵ0.

To finish the proof it rests to show that sbχ(H) = χ(H) = d. It follows
from Lemmas 2 and 3 that the group H, being non-metrizable, is not α7 and
thus contains a copy of the sequential fan Sω. Then d = χ(Sω) = sbχ(Sω) ≤
sbχ(H) ≤ χ(H). To prove that χ(H) ≤ d we shall apply a result of K. Sakai
[Sa] asserting that the space R∞ ×Q contains a closed topological copy of each
MKω-space and the well-known equality χ(R

∞ × Q) = χ(R∞) = d (following
from the fact that R∞ carries the box-product topology, see [Sch, Ch.II, Ex.12]).

Proof of Theorem 5. First we describe two general constructions producing
topologically homogeneous sequential spaces. For a locally compact space Z
let αZ = Z ∪ {∞} be the one-point extension of Z endowed with the topology
whose neighborhood base at ∞ consists of the sets αZ\K where K is a compact
subset of Z. Thus for a non-compact locally compacts space Z the space αZ
is noting else but the one-point compactification of Z. Denote by 2ω = {0,1}ω

the Cantor cube.



42 T.Banakh, L.Zdomsky̆ı

Consider the subsets

Ξ(Z) ={(c,(zi)i∈ω) ∈ 2
ω × (αZ)ω : zi = ∞ for all but finitely many indices i};

Θ(Z) ={(c,(zi)i∈ω) ∈ 2
ω × (αZ)ω : ∃n ∈ ω such that zi 6= ∞ iff i < n}.

Observe that Θ(Z) ⊂ Ξ(Z).
Endow the set Ξ(Z) (resp. Θ(Z)) with the strongest topology generating

the Tychonov product topology on each compact subset from the family KΞ
(resp. KΘ), where

KΞ = {2
ω×

∏

i∈ω Ci : Ci are compact subsets of αZ and almost all Ci = {∞}};

KΘ = {2
ω ×

∏

i∈ω Ci : ∃i0 ∈ ω such that Ci0 = αZ, Ci = {∞} for all i > i0
and Ci is a compact subsets of Z for every i < i0}.

Lemma 6. Suppose Z is a zero-dimensional locally metrizable locally compact
space. Then

(1) The spaces Ξ(Z) and Θ(Z) are topologically homogeneous;
(2) Ξ(Z) is a regular zero-dimensional kω-space while Θ(Z) is a totally

disconnected k-space;
(3) If Z is Lindelöf, then Ξ(Z) and Θ(Z) are zero-dimensional MKω-

spaces with
χ(Ξ(Z)) = χ(Θ(Z)) ≤ d;

(4) Ξ(Z) and Θ(Z) contain copies of the space αZ while Θ(Z) contains a
closed copy of Z;

(5) cs∗χ(Ξ(Z)) = cs
∗
χ(Θ(Z)) = cs

∗
χ(αZ), csχ(Ξ(Z)) = csχ(Θ(Z)) = csχ(αZ),

sbχ(Θ(Z)) = sbχ(αZ), and ψ(Ξ(Z)) = ψ(Θ(Z)) = ψ(αZ);
(6) The spaces Ξ(Z) and Θ(Z) are sequential if and only if αZ is sequential;
(7) If Z is not countably compact, then Ξ(Z) contains a closed copies of

S2 and Sω and Θ(Z) contains a closed copy of S2.

Proof. (1) First we show that the space Ξ(Z) is topologically homogeneous.
Given two points (c,(zi)i∈ω),(c

′,(z′i)i∈ω) of Ξ(Z) we have to find a homeo-
morphism h of Ξ(Z) with h(c,(zi)i∈ω) = (c

′,(z′i)i∈ω). Since the Cantor cube
2ω is topologically homogeneous, we can assume that c 6= c′. Fix any disjoint
closed-and-open neighborhoods U,U′ of the points c,c′ in 2ω, respectively.

Consider the finite sets I = {i ∈ ω : zi 6= ∞} and I
′ = {i ∈ ω : z′i 6= ∞}.

Using the zero-dimensionality and the local metrizability of Z, for each i ∈ I
(resp. i ∈ I′) fix an open compact metrizable neighborhood Ui (resp. U

′
i)

of the point zi (resp. z
′
i) in Z. By the classical Brouwer Theorem [Ke, 7.4],

the products U ×
∏

i∈I Ui and U
′ ×

∏

i∈I′ U
′
i, being zero-dimensional compact

metrizable spaces without isolated points, are homeomorphic to the Cantor
cube 2ω. Now the topological homogeneity of the Cantor cube implies the
existence of a homeomorpism f : U ×

∏

i∈I Ui → U
′ ×

∏

i∈I′ U
′
i such that

f(c,(zi)i∈I) = (c
′,(z′i)i∈I′). Let

W = {(x,(xi)i∈ω) ∈ Ξ(Z) : x ∈ U, xi ∈ Ui for all i ∈ I} and
W ′ = {(x′,(x′i)i∈ω) ∈ Ξ(Z) : x

′ ∈ U′, x′i ∈ U
′
i for all i ∈ I

′}.



On groups with countable cs∗-character 43

It follows that W,W ′ are disjoint open-and-closed subsets of Ξ(Z). Let χ :
ω \ I′ → ω \ I be a unique monotone bijection.

Now consider the homeomorphism f̃ : W → W ′ assigning to a sequence
(x,(xi)i∈ω) ∈ W the sequence (x

′,(x′i)i∈ω) ∈ W
′ where

(x′,(x′i)i∈I′) = f(x,(xi)i∈I) and x
′
i = xχ(i) for i /∈ I

′.

Finally, define a homeomorphism h of Ξ(Z) letting

h(x) =











x if x /∈ W ∪ W ′;

f̃(x) if x ∈ W ;

f̃−1(x) if x ∈ W ′

and observe that h(c,(zi)i∈ω) = (c
′,(z′i)i∈ω) which proves the topological ho-

mogeneity of the space Ξ(Z).
Replacing Ξ(Z) by Θ(Z) in the above proof, we shall get a proof of the

topological homogeneity of Θ(Z).
The items (2–4) follow easily from the definitions of the spaces Ξ(Z) and

Θ(Z), the zero-dimensionality of αZ, and known properties of kω-spaces, see
[FST] (to find a closed copy of Z in Θ(Z) consider the closed embedding e :
Z → Θ(Z), e : z 7→ (z,z0,z,∞,∞, . . .), where z0 is any fixed point of Z).

To prove (5) apply Proposition 3(6,8,9,10). (To calculate the cs∗-, cs-, and
sb-characters of Θ(Z), observe that almost all members of any sequence (an) ⊂
Θ(Z) convergent to a point a = (c,(zi)) ∈ Θ(Z) lie in the compactum 2

ω ×
∏

i∈ω Ci, where Ci is a clopen neighborhood of zi if zi 6= ∞, Ci = αZ if
i = min{j ∈ ω : zj = ∞} and Ci = {∞} otherwise. By Proposition 3(6), the
cs∗-, cs-, and sb-characters of this compactum are equal to the corresponding
characters of αZ.)

(6) Since the spaces Ξ(Z) and Θ(Z) contain a copy of αZ, the sequentiality
of Ξ(Z) or Θ(Z) implies the sequentiality of αZ. Now suppose conversely that
the space αZ is sequential. Then each compactum K ∈ KΞ ∪ KΘ is sequential
since a finite product of sequential compacta is sequential, see [En1, 3.10.I(b)].
Now the spaces Ξ(Z) and Θ(Z) are sequential because they carry the inductive
topologies with respect to the covers KΞ, KΘ by sequential compacta.

(7) If Z is not countably compact, then it contains a countable closed discrete
subspace S ⊂ Z which can be thought as a sequence convergent to ∞ in αZ. It
is easy to see that Ξ(S) (resp. Θ(S)) is a closed subset of Ξ(Z) (resp. Θ(Z)).
Now it is quite easy to find closed copies of S2 and Sω in Ξ(S) and a closed
copy of S2 in Θ(S). �

With Lemma 6 at our disposal, we are able to finish the proof of Theo-
rem 5. To construct the examples satisfying the conditions of Theorem 5(2,3),
assume b = c and use Proposition 4 to find a locally compact locally count-
able space Z whose one-point compactification αZ is sequential and satisfies
ℵ0 = sbχ(αZ) < ψ(αZ) = c. Applying Lemma 6 to this space Z, we conclude
that the topologically homogeneous k-spaces X2 = Ξ(Z) and X3 = Θ(Z) give
us required examples.



44 T.Banakh, L.Zdomsky̆ı

The example of a countable topologically homogeneous kω-space X1 with
sbχ(X1) = ψ(X1) < χ(X1) can be constructed by analogy with the space
Θ(N) (with that difference that there is no necessity to involve the Cantor
cube) and is known in topology as the Ankhangelski-Franklin space, see [AF].
We briefly remind its construction. Let S0 = {0,

1
n

: n ∈ N} be a conver-
gent sequence and consider the countable space X1 = {(xi)i∈ω ∈ S

ω
0 : ∃n ∈

ω such that xi 6= 0 iff i < n} endowed with the strongest topology inducing the
product topology on each compactum

∏

i∈ω Ci for which there is n ∈ ω such
that Cn = S0, Ci = {0} if i > n, and Ci = {xi} for some xi ∈ S0 \ {0} if
i < n. By analogy with the proof of Lemma 6 it can be shown that X1 is a
topologically homogeneous kω-space with ℵ0 = sbχ(X1) = ψ(X1) < χ(X1) = d
and so(X1) = ω.

Proof of Proposition 5. The equivalences (1) ⇔ (2) ⇔ (3) were proved by
Lin [Lin, 3.13] in terms of (universally) csf-countable spaces. To prove the
other equivalences apply

Lemma 7. A Hausdorff topological space X is an α4-space provided one of the
following conditions is satisfied:

(1) X is a Fréchet-Urysohn α7-space;
(2) X is a Fréchet-Urysohn countably compact space;
(3) sbχ(X) < p;
(4) sbχ(X) < d, each point of X is regular Gδ, and X is c-sequential.

Proof. Fix any point x ∈ X and a countable family {Sn}n∈ω of sequences
convergent to x in X. We have to find a sequence S ⊂ X \ {x} convergent to
x and meeting infinitely many sequences Sn. Using the countability of the set
⋃

n∈ω Sn find a decreasing sequence (Un)n∈ω of closed neighborhoods of x in

X such that
(
⋂

n∈ω Un
)

∩
(
⋃

n∈ω Sn) = {x}. Replacing each sequence Sn by
its subsequence Sn ∩ Un, if necessary, we can assume that Sn ⊂ Un.

(1) Assume that X is a Fréchet-Urysohn α7-space. Let A = {a ∈ X : a is
the limit of a convergent sequence S ⊂ X meeting infinitely many sequences
Sn}. It follows from our assumption on (Sn) and (Un) that A ⊂

⋂

n∈ω Un.
It suffices to consider the non-trivial case when x /∈ A. In this case x is a

cluster point of A (otherwise X would be not α7). Since X is Fréchet-Urysohn,
there is a sequence (an) ⊂ A convergent to x. By the definition of A, for every
n ∈ ω there is a sequence Tn ⊂ X convergent to a and meeting infinitely many
sequences Sn. Without loss of generality, we can assume that Tn ⊂

⋃

i>n
Si

(because a ∈ A \ {x} and thus a /∈
⋃

n∈ω Sn). It is easy to see that x is
a cluster point of the set

⋃

n∈ω Tn. Since X is Fréchet-Urysohn, there is a
sequence T ⊂

⋃

n∈ω Tn convergent to x.
Now it remains to show that the set T meets infinitely many sequences Sn.

Assuming the converse we would find n ∈ ω such that T ⊂
⋃

i≤n Sn. Then

T ⊂
⋃

i≤n Tn which is not possible since
⋃

i≤n Ti is a compact set failing to
contain the point x.



On groups with countable cs∗-character 45

(2) If X is Fréchet-Urysohn and countably compact, then it is sequentially
compact and hence α7, which allows us to apply the previous item.

(3) Assume that sbχ(X) < p and let N be a sb-network at x of size |N| < p.
Without loss of generality, we can assume that the family N is closed under
finite intersections. Let S =

⋃

n∈ω Sn and FN,n = N ∩ (
⋃

i≥n Si) for N ∈ N
and n ∈ ω. It is easy to see that the family F = {FN,n : N ∈ N , n ∈ ω}
consists of infinite subsets of S, has size |F| < p, and is closed under finite
intersection. Now the definition of the small cardinal p implies that this family
F has an infinite pseudo-intersection T ⊂ S. Then T is a sequence convergent
to x and intersecting infinitely many sequences Sn. This shows that X is an
α4-space.

(4) Assume that the space X is c-sequential, each point of X is regular
Gδ, and sbχ(X) < d. In this case we can choose the sequence (Un) to satisfy
⋂

n∈ω Un = {x}. Fix an sb-network N at x with |N| < d. For every n ∈ ω
write Sn = {xn,i : i ∈ N}. For each sequential barrier N ∈ N find a function
fN : ω → N such that xn,i ∈ N for every n ∈ ω and i ≥ fN(n). The family
of functions {fN : N ∈ N} has size < d and hence is not cofinal in N

ω.
Consequently, there is a function f : ω → N such that f 6≤ fN for each N ∈ N .
Now consider the sequence S = {xn,f(n) : n ∈ ω}. We claim that x is a cluster
point of S. Indeed, given any neighborhood U of x, find a sequential barrier
N ∈ N with N ⊂ U. Since f 6≤ fN, there is n ∈ ω with f(n) > fN(n). It
follows from the choice of the function fN that xn,f(n) ∈ N ⊂ U.

Since S \ Un is finite for every n, {x} =
⋂

n∈ω Un is a unique cluster point
of S and thus {x} ∪ S is a closed subset of X. Now the c-sequentiality of X
implies the existence of a sequence T ⊂ S convergent to x. Since T meets
infinitely many sequences Sn, the space X is α4. �

Proof of Proposition 6. Suppose a space X has countable cs∗-character.
The implications (1) ⇒ (2,3,4,5) are trivial. The equivalence (1) ⇔ (2) fol-
lows from Proposition 1(2). To show that (3) ⇒ (2), apply Lemma 7 and
Proposition 5(3 ⇒ 1).

To prove that (4) ⇒ (2) it suffices to apply Proposition 5(4 ⇒ 1) and observe
that X is Fréchet-Urysohn provided χ(X) < p and X has countable tightness.
This can be seen as follows.

Given a subset A ⊂ X and a point a ∈ Ā from its closure, use the count-
able tightness of X to find a countable subset N ⊂ A with a ∈ N̄. Fix any
neighborhood base B at x of size |B| < p. We can assume that B is closed
under finite intersections. By the definition of the small cardinal p, the family
{B ∩N : B ∈ B} has infinite pseudo-intersection S ⊂ N. It is clear that S ⊂ A
is a sequence convergent to x, which proves that X is Fréchet-Urysohn.

(5) ⇒ (2). Assume that X is a sequential space containing no closed copies
of Sω and S2 and such that each point of X is regular Gδ. Since X is sequential
and contains no closed copy of S2, we may apply Lemma 2.5 [Lin] to conclude
that X is Fréchet-Urysohn. Next, Theorem 3.6 of [Lin] implies that X is an



46 T.Banakh, L.Zdomsky̆ı

α4-space. Finally apply Proposition 5 to conclude that X has countable sb-
character and, being Fréchet-Urysohn, is first countable.

The final implication (6) ⇒ (2) follows from (5) ⇒ (2) and the well-known
equality χ(Sω) = χ(S2) = d.

Proof of Proposition 7.

The first item of this proposition follows from Proposition 6(3 ⇒ 1) and
the observation that each Fréchet-Urysohn countable compact space, being
sequentially compact, is α7.

Now suppose that X is a dyadic compact with cs∗χ(X) ≤ ℵ0. If X is not
metrizable, then it contains a copy of the one-point compactification αD of an
uncountable discrete space D, see [En1, 3.12.12(i)]. Then cs

∗
χ(αD) ≤ cs

∗
χ(X) ≤

ℵ0 and by the previous item, the space αD, being Fréchet-Urysohn and com-
pact, is first-countable, which is a contradiction.

Proof of Proposition 8. Let D be a discrete space.
(1) Let κ = cs∗χ(αD) and λ1 (λ2) is the smallest weight of a (regular zero-

dimensional) space X of size |X| = |D|, containing no non-trivial convergent
sequence. To prove the first item of proposition 8 it suffices to verify that
λ2 ≤ κ ≤ λ1. To show that λ2 ≤ κ, fix any cs

∗-network N at the unique
non-isolated point ∞ of αD of size |N| ≤ κ. The algebra A of subsets of D
generated by the family {D \ N : N ∈ N} is a base of some zero-dimensional
topology τ on D with w(D,τ) ≤ κ. We claim that the space D endowed with
this topology contains no infinite convergent sequences. To get a contradiction,
suppose that S ⊂ D is an infinite sequence convergent to a point a ∈ D \ S.
Then S converges to ∞ in αD and hence, there is an element N ∈ N such that
N ⊂ αD\{a} and N ∩S is infinite. Consequently, U = D\N is a neighborhood
of a in the topology τ such that S \ U is infinite which contradicts to the fact
that S converges to a. Now consider the equivalence relation ∼ on D: x ∼ y
provided for every U ∈ τ (x ∈ U) ⇔ (y ∈ U). Since the space (D,τ) has no
infinite convergent sequences, each equivalence class [x]∼ ⊂ D is finite (because
it carries the anti-discrete topology). Consequently, we can find a subset X ⊂ D
of size |X| = |D| such that x 6∼ y for any distinct points x,y ∈ X. Clearly
that τ induces a zero-dimensional topology on X. It rests to verify that this
topology is T1. Given any two distinct point x,y ∈ X use x 6∼ y to find an
open set U ∈ A such that either x ∈ U and y /∈ U or x /∈ U and y ∈ U. Since
D \ U ∈ A, in both cases we find an open set W ∈ A such that x ∈ W but
y /∈ W . It follows that X is a T1-space containing no non-trivial convergent
sequence and thus λ2 ≤ w(X) ≤ |A| ≤ |N| ≤ κ.

To show that κ ≤ λ1, fix any topology τ on D such that w(D,τ) ≤ λ1 and
the space (D,τ) contains no non-trivial convergent sequences. Let B be a base
of the topology τ with |B| ≤ λ1, closed under finite unions. We claim that
the collection N = {αD \ B : B ∈ B} is a cs∗-network for αD at ∞. Fix
any neighborhood U ⊂ αD of ∞ and any sequence S ⊂ D convergent to ∞.
Write {x1, . . . ,xn} = αD \ U and by finite induction, for every i ≤ n find a



On groups with countable cs∗-character 47

neighborhood Bi ∈ B of xi such that S \
⋃

j≤i Bj is infinite. Since B is closed

under finite unions, the set N = αD \ (B1 ∪ · · · ∪ Bn) belongs to the family N
and has the properties: N ⊂ U and N ∩ S is infinite, i.e., N is a cs∗-network
at ∞ in αD. Thus κ ≤ |N| ≤ |B| ≤ λ1. This finishes the proof of (1).

An obvious modification of the above argument gives also a proof of the item
(2).

Proof of Proposition 9. Let D be an uncountable discrete space.
(1) The inequalities ℵ1 · log |D| ≤ cs

∗
χ(αD) ≤ csχ(αD) follows from Propo-

sitions 7(1) and 1(2,4) yielding |D| = χ(αD) = sbχ(αD) ≤ 2
cs∗

χ
(αD). The

inequality csχ(αD) ≤ c · cof([log |D|]
≤ω) follows from proposition 8(2) and the

observation that the product {0,1}log |D| endowed with the ℵ0-box product
topology has weight ≤ c · cof([log |D|]≤ω). Under the ℵ0-box product topology
on {0,1}κ we understand the topology generated by the base consisting of the
sets {f ∈ {0,1}κ : f|C = g|C} where g ∈ {0,1}ω and C is a countable subset
of κ.

The item (2) follows from (1) and the equality ℵ1·log κ = 2
ℵ0·min{κ,(log κ)ω}

holding under GCH for any infinite cardinal κ, see [HJ, 9.3.8]

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Received October 2002

Accepted April 2003

Taras Banakh and Lubomyr Zdomsky̆ı (tbanakh@franko.lviv.ua, lubomyr@opari.ltg.lviv.ua)

Instytut Matematyki, Akademia Świȩtorzyska, Kielce, and Department of Math-
ematics, Ivan Franko Lviv National University, Universytetska 1, Lviv, 79000,
Ukraine