arXiv:1507.04087v2 [] 18 Apr 2016

TWO WEAK FORMS OF COUNTABILITY AXIOMS
IN FREE TOPOLOGICAL GROUPS
FUCAI LIN, CHUAN LIU, AND JILING CAO
Abstract. Given a Tychonoff space X, let F (X) and A(X) be respectively the
free topological group and the free Abelian topological group over X in the sense of
Markov. For every n ∈ N, let Fn (X) (resp. An (X)) denote the subspace of F (X)
(resp. A(X)) that consists of words of reduced length at most n with respect to the
free basis X. In this paper, we discuss two weak forms of countability axioms in
F (X) or A(X), namely the csf -countability and snf -countability. We provide some
characterizations of the csf -countability and snf -countability of F (X) and A(X)
for various classes of spaces X. In addition, we also study the csf -countability and
snf -countability of Fn (X) or An (X), for n = 2, 3, 4. Some results of Arhangel’skiı̌
in [1] and Yamada in [22] are generalized. An affirmative answer to an open question
posed by Li et al. in [11] is provided.

1. Introduction
In 1941, Markov [16] introduced the concepts of the free topological group F (X)
and the free Abelian topological group A(X) over a Tychonoff space X, respectively.
Since then, free topological groups have been a source of various examples and also an
interesting topic of study in the theory of topological groups, see [3]. From the algebraic
point of view, the structure of F (X) or A(X) is very simple - it is the free algebraic
group over the set X. However, the topological structure of F (X) and A(X) is rather
complicated even for simple spaces X. For example, it is a well known fact that if X is
a non-discrete space, then neither F (X) nor A(X) is Fréchet-Uryshon, and hence first
countable, see [1]. This fact motivates researchers to investigate free topological groups
in two directions. The first direction of the research on free topological groups is to study
some weak forms of countability axioms in F (X) and A(X) over certain classes of spaces
X. In this line, Arhangel’skiı̌ et al. [4] considered the following questions on F (X) and
A(X) over a metrizable space X: For which spaces X, is F (X) or A(X) a k-space?
When is the tightness of F (X) or A(X) countable? They proved that F (X) is a k-space
iff X is locally compact separable or discrete; A(X) is a k-space iff X is locally compact
and X ′ is separable, where X ′ is the derived set of X. Furthermore, the tightness of
F (X) is countable iff X is separable or discrete, and the tightness of A(X) is countable
iff X ′ is separable.
2000 Mathematics Subject Classification. Primary 54H11, 22A05; Secondary 54E20; 54E35; 54D50;
54D55.
Key words and phrases. csf -countable; free Abeilain topological group; free topological group; kspace; Lašnev space; metrizable space; regular Gδ -diagonal; sequential; snf -countable.
The first author is supported by the NSFC (Nos. 11571158, 11201414, 11471153), the Natural Science
Foundation of Fujian Province (No. 2012J05013) of China and Training Programme Foundation for
Excellent Youth Researching Talents of Fujian’s Universities (JA13190) and the project of Abroad Fund
of Minnan Normal University. The third author acknowledges the support and hospitality of Minnan
Normal University, when he was appointed a Min Jiang Scholar and Visitng Professor from 2014 to
2016. This paper was partially written when the first author was visiting the School of Computer and
Mathematical Sciences at Auckland University of Technology from March to September 2015, and he
wishes to thank the hospitality of his host.
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FUCAI LIN, CHUAN LIU, AND JILING CAO

The other direction of research on free topological groups is to study (weak) countability axioms of Fn (X) or An (X), where Fn (X) (resp. An (X)) stands for the subset of F (X)
(resp. A(X)) formed by all words whose reduced length is at most n. Indeed, Yamada
[23] showed that for a metrizable space X, F3 (X) or A3 (X) is Fréchet-Uryshon iff X ′ is
compact, and F5 (X) is Fréchet-Uryshon iff X is compact or discrete. As applications,
characterizations of a metrizable space X are given such that An (X) is Fréchet-Uryshon
for each n ≥ 3, and Fn (X) is Fréchet-Uryshon for each n ≥ 3 except for n = 4. The
subspaces F4 (X) and A4 (X) are very special cases. In [22], Yamada proved that for a
metrizable space X, the following are equivalent: (i) Fn (X) is metrizable for each n ∈ N;
(ii) Fn (X) is first countable for each n ∈ N; (iii) F4 (X) is metrizable; (iv) F4 (X) is first
countable; (v) X is compact or discrete. In the same paper, Yamada also studied the first
countability of Fn (X) and An (X) for n = 2, 3. It is proved that for a metrizable space
X, the following are equivalent: (i) F3 (X) is metrizable; (ii) F3 (X) is first-countable;
(iii) F2 (X) is metrizable; (iv) F2 (X) is first-countable; (v) X ′ is compact. Furthermore,
for a metrizable space X, the following are also equivalent: (i) A2 (X) is first-countable;
(ii) A2 (X) is metrizable; (iii) An (X) is first-countable for each n ∈ N; (iv) An (X) is
metrizable for each n ∈ N; (v) X ′ is compact.
Recently, Li et al. [11] continued the study of F (X) and A(X) along the aforementioned first direction. They studied several weak forms of countability axioms of
F (X) and A(X) defined by networks over some classes of generalized metric spaces
X. More precisely, they studied the concepts of sn-networks, cs-networks, cs∗ -networks
in F (X), A(X), and their subspaces Fn (X) and An (X). Two types of countability
axioms defined by these concepts, namely snf -countability and csf -countability, were
considered. Among many other things, Li et al. established the following results: For a
metrizable and crowded space X, F (X) or A(X) is csf -countable iff X is separable; For
a stratifiable k-space X, F (X) is snf -countable iff X is discrete. However, the authors
of [11] did not consider the snf -countability and csf -countability of Fn (X) and An (X).
In the paper, we continue the study of free topological group F (X) and the free
Abelian topological group A(X) in the afore-mentioned two directions. In particular,
we investigate the csf -countability and the snf -countability of F (X), A(X), Fn (X) and
An (X) over various classes of generalized metric spaces X. In Section 2, we introduce the
necessary notation and terminologies which are used for the rest of the paper. In Section
3, we investigate the snf -countability of free (Abelian) topological groups. First, we
provide some characterizations of the snf -countability of F (X), A(X), Fn (X) and An (X)
over certain classes of topological spaces. The main theorem in this section generalizes
a result of Yamada in [22]. Section 4 is devoted to the study of the csf -countability of
F (X), A(X), Fn (X) and An (X). It is shown that for a non-discrete Lašnev space X,
F4 (X) is csf -countable iff F4 (X) is an ℵ0 -space. This result gives an affirmative answer
to an open question in [11]. It is also shown that for a sequential µ-space X, if X ′ has a
countable cs∗ -network in X, then F3 (X) and A(X) are csf -countable. Finally, we pose
several interesting open questions in the last section.
Throughout this paper, all topological spaces are assumed to be at least Tychonoff,
unless explicitly stated otherwise.
2. Notation and Terminologies
In this section, we introduce the necessary notation and terminologies. First of all,
let N, Z and R denote the sets of all positive integers, all integers and all real numbers,
respectively. For undefined terminologies, the reader may refer to [3], [6] and [8].
Let X be a topological space X and A ⊆ X be a subset of X. The closure of A in
X is denoted by A and the diagonal of X is denoted by ∆(X). The subset A is called

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

3

C ∗ -embedded in X if every bounded continuous real-valued function f defined on A has a
bounded continuous extension over X. Moreover, A is called bounded if every continuous
real-valued function f defined on A is bounded. If the closure of every bounded set in X
is compact, then X is called a µ-space. The derived set of X is denoted by X ′ . We say
that X is crowded if X = X ′ . Recall that X is said to have a Gδ -diagonal (resp. regular
Gδ -diagonal ) if ∆(X) is a Gδ -set (resp. regular Gδ -set) in X ×X. A pseudometric d on X
is said to be continuous if d is continuous as a mapping from the product space X × X to
R. The space X is called a k-space provided that a subset C ⊆ X is closed in X if C ∩ K
is closed in K for each compact subset K of X. If there exists a family of countably many
compact subsets {Kn : n ∈ N} of X such that each subset F of X is closed in X provided
that F ∩ Kn is closed in Kn for each n ∈ N, then X is called a kω -space. Note that every
kω -space is a k-space. In addition, X is called a cf -space if every compact subset of X is
finite. A subset P of X is called a sequential neighborhood [7] of x ∈ X, if each sequence
converging to x is eventually in P . A subset U of X is called sequentially open if U is a
sequential neighborhood of each of its points. The space X is called a sequential space
if each sequentially open subset of X is open. Let κ be an infinite
cardinal. For each
L
α ∈ κ, let Tα be a sequence converging to xα 6∈ Tα . Let TS:= α∈κ (Tα ∪ {xα }) be the
topological sum of {Tα ∪ {xα } : α ∈ κ}. Then Sκ := {x} ∪ α∈κ Tα is the quotient space
obtained from T by identifying all the points xα ∈ T to the point x.
Let P be a family of subsets of X. Then, P is called a cs-network [9] at a point x ∈ X
if for every sequence {xn : n ∈ N} converging to x and an arbitrary open neighborhood
U of x in X there exist an m ∈ N and an element P ∈ P such that
{x} ∪ {xn : n > m} ⊆ P ⊆ U.
The space X is called csf-countable if X has a countable cs-network at each point x ∈ X.
We call P a cs∗ -network [14] of X if for every sequence {xn : n ∈ N} converging to a
point x and an arbitrary open neighborhood U of x in X there is an element P ∈ P
and a subsequence {xni : i ∈ N} of {xn : n ∈ N} such that {x} ∪ {xni : i ∈ N} ⊆ P ⊆ U .
Furthermore, P is called a k-network [17] if for every compact subset K of X and an
′
arbitrary
S open set U containing K in X there is a finite subfamily P ⊆ P such that
K ⊆ P ′ ⊆ U . Recall that a space X is an ℵ-space (resp. ℵ0 -space) if X has S
a σ-locally
finite (resp. countable) k-network. Let P be a cover of X such that (i) P = x∈X Px ;
(ii) for each point x ∈ X, if U, V ∈ Px , then W ⊆ U ∩ V for some W ∈ Px ; and (iii) for
each point x ∈ X and each open neighborhood U of x there is some P ∈ Px such that
x ∈ P ⊆ U . Then, P is called an sn-network [13] for X if for each point x ∈ X, each
element of Px is a sequential neighborhood of x in X, and X is called snf-countable [13]
if X has an sn-network P and Px is countable for all x ∈ X. The following implications
follow directly from definitions:
first countable ⇒ snf -countable ⇒ csf -countable.
Note that none of the above implications can be reversed. It is well known that Sω is csf countable but not snf -countable. Moreover, any space without non-trivial convergent
sequences is snf -countable, see Example 3.6.
Given a group G, let eG denote the neutral element of G. If no confusion occurs,
we simply use e instead of eG to denote the neutral element of G. Let N : G → R be
a function. We call N a pre-norm on G if the following conditions are satisfied for all
x, y ∈ G: (i) N (e) = 0; (ii) N (xy −1 ) ≤ N (x) + N (y). If G is a topological space and N
is continuous, then we say that N is a continuous pre-norm on G.
Let X be a non-empty Tychonoff space. Throughout this paper, X −1 := {x−1 : x ∈
X} and −X := {−x : x ∈ X}, which are just two copies of X. For every n ∈ N, Fn (X)
denotes the subspace of F (X) that consists of all words of reduced length at most n

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FUCAI LIN, CHUAN LIU, AND JILING CAO

with respect to the free basis X. The subspace An (X) is defined similarly. We always
use G(X) to denote topological groups F (X) or A(X), and Gn (X) to Fn (X) or An (X)
for each n ∈ N. Therefore, any statement about G(X) applies to F (X) and A(X), and
Gn (X) applies to F (X) and A(X). Let e be the neutral element of F (X) (i.e., the
empty
L word)
Land 0 be that of A(X). For every n ∈ N and an element (x1 , x2 , · · · , xn ) of
(X
X −1 {e})n we call g = x1 x2 · · · xn a form.
L In the
L Abelian case, x1 + x2 · · · + xn
is also called a form for (x1 , x2 , · · · , xn ) ∈ (X
−X {0})n . This word g is called
reduced if it does not contains e or any pair of consecutive symbols of the form xx−1 or
x−1 x. It follows that if the word g is reduced and non-empty, then it is different from the
neutral element e of F (X). In particular, each element g ∈ F (X) distinct from the neutral
element can be uniquely written in the form g = xr11 xr22 · · · xrnn , where n ≥ 1, ri ∈ Z \ {0},
xi ∈ X, and xi 6= xi+1 for each i = 1, · · · , n − 1, and the support of g = xr11 xr22 · · · xrnn
is
S defined as supp(g) := {x1 , · · · , xn }. Given a subset K of F (X), we put supp(K) :=
the obvious changes for commutativity) are valid
g∈K supp(g). Similar assertions (with
L −1 L
for A(X). For every n ∈ N, let in : (X
X
{e})n → Fn (X) be L
the natural
L mapping
defined by in (x1 , x2 , ...xn ) = x1 x2 ...xn for each (x1 , x2 , ...xn ) ∈ (X
X −1 {e})n . We
alsoL
use theLsame symbol in the Abelian case, that is, it means the natural mapping from
(X
−X {0})n onto An (X). Clearly, each in is a continuous mapping.
3. The snf -Countability of Free Topological Groups
In this section, we discuss the snf -countability of G(X) and Gn (X) for a given topological space X. First, we provide some general characterizations for the snf -countability
of G(X). Then, we give some particular classes of spaces X for which those characterizations hold. Finally, we characterizes the snf -countability of F4 (X) for a specific class
of spaces X, namely k-spaces with a Gδ -diagonal.
The following theorem generalizes Corollary 4.14 in [1].
Theorem 3.1. For a space X, the following statements are equivalent:
(i) G(X) is snf -countable.
(ii) Each Gn (X) contains no non-trivial convergent sequences.
(iii) G(X) contains no non-trivial convergent sequences.
Proof. The implications (iii) ⇒ (ii) and (iii) ⇒ (i) are obvious.
(ii) ⇒ (iii). Assume that each Gn (X) contains no non-trivial convergent sequences.
If G(X) contains a non-trivial convergent sequence S, there is an n ∈ N such that
S ⊆ Gn (X). This implies that Gn (X) contains a non-trivial convergent sequence, which
is a contradiction.
(i) ⇒ (iii). Let G(X) be snf -countable with a countable sn-network {Un : n ∈ N} at
e, where Un+1 ⊆ Un for each n ∈ N. Assume that G(X) contains a non-trivial convergent
sequence {xi : i ∈ N} converging to e. Without loss of generality, we assume that xi 6= e
for each i ∈ N. Then there exists 2 ≤ n0 ∈ N such that {xi : i ∈ N} ⊆ Gn0 (X). We may
further assume that {xi : i ∈ N} ⊆ Gn0 (X) \ Gn0 −1 (X).
We first consider the case of F (X). For each i ∈ N, let xi (1) ∈ X ∪ X −1 be the
first letter of xi . Put A := {i ∈ N : xi (1) ∈ X} and B := {i ∈ N : xi (1) ∈ X −1 }.
Obviously, we have A ∪ B = N. Without loss of generality, we may assume that |A| = ω.
We further assume that A = N. Take an arbitrary point x ∈ X. For each m ∈ N, let
m
m
Lm := {x2 n0 xi x−2 n0 : i ∈ N}. Then Lm ∩ Ln = ∅ for any m 6= n. Indeed, take
arbitrary m, n ∈ N with m < n, and then pick any g ∈ Lm and h ∈ Ln . Since each xi
belongs to Gn0 (X) \ Gn0 −1 (X), we have
ℓ(g) > 2m+1 n0 − n0 ≥ 2m n0 + n0 ≥ 2n+1 n0 + n0 ≥ ℓ(h).

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

5

Hence g 6= h, where
and ℓ(h) denote the lengths of g and h, respectively. It is
 ℓ(g)
m
m
evident that each x2 n0 xi x−2 n0 : i ∈ N converges to e. For each n ∈ N, pick a
point yn ∈ Un ∩ Ln . Then, the sequence {yn : n ∈ N} converges to e. However, since
|{yn : n ∈ N}∩Fm (X)| < ω for each m ∈ N, by Corollary 7.4.3 in [3], the set {yn : n ∈ N}
is closed and discrete in F (X). A contradiction occurs.
Now, we consider the case of A(X). Let Lm := {mxi : i ∈ N}. It is easy to see that
Lm ∩ Ln = ∅ for any m 6= n. Obviously, each {mxi : i ∈ N} converges to e. We can
derive a contradiction by a proof similar to that for the case of F (X).

Next, we consider what conditions on a space X can guarantee G(X) to be snf countable. In the light of Theorem 3.1, one may conjecture that G(X) is snf -countable
if X contains no non-trivial convergent sequences. Unfortunately, this is not true, since
there is a countably compact and separable cf -space X such that A(X) and F (X) contain
a non-trivial convergent sequence, refer to Theorem 3.5 in [20]. Nevertheless, by applying
Theorem 3.1 to some special classes of topological spaces, we obtain the following result.
Theorem 3.2. For a space X, G(X) is snf -countable if one of the following holds:
(i) X is a cf- and µ-space;
(ii) Every countable discrete subset of X is C ∗ -embedded.
Proof. (i) It suffices to show that G(X) is a cf -space. If K ⊆ G(X) is compact, then
is L
a compact subset in X, as
K ⊆ Gn (X) for some n ∈ N. Let Z := supp(K). Then
L Z −1
X is a µ-space.
Hence
Z
is
finite.
Since
K
⊆
i
((Z
Z
{e})n ) (in Abelian case,
n
L
L
K ⊆ in ((Z
−Z {0})n )), K must be finite. Therefore, G(X) is snf -countable.
(ii) Since X is a space in which every countable discrete subset is C ∗ -embedded,
it follows from Porposition 2.4 in [20] that G(X) contains no non-trivial convergent
sequences. Therefore, by Theorem 3.1, G(X) is snf -countable.

A topological space X is extremely disconnected if the closure of every open subset
is open. Since every countable discrete subset of an extremely disconnected space X is
C ∗ -embedded, by Theorem 3.2, G(X) is snf -countable. As a particular example, the
Stone-Čech compactification βD of any discrete space D is extremely disconnected, and
hence G(βD) is snf -countable.
If X is a topological group, then we can characterize the snf -countability of G(X) in
terms of the property that X contains no non-trivial convergent sequences, as it is shown
in the next result.
Theorem 3.3. For a topological group X, G(X) is snf -countable if and only if X
contains no non-trivial convergent sequences.
Proof. The necessity is obvious by Theorem 3.1. To show the sufficiency, suppose that
X contains no non-trivial convergent sequences. By Theorem 3.1, it suffices to show
that G(X) contains no non-trivial convergent sequences. We only consider the case of
F (X), since the proof for the case of A(X) is quite similar. Assume that F (X) contains
a non-trivial convergent sequence. It follows from [20, Proposition 2.1] that there are two
sequences {xn : n ∈ N} and {yn : n ∈ N} in X such that {xn : n ∈ N} is infinite, xn 6= yn
for each n ∈ N and for every continuous pseudometric d on X, and d(xn , yn ) ≥ 1 for at
most finitely many n ∈ N. Let S := {x−1
n yn : n ∈ N}. Then S is a non-trivial sequence.
We claim that S is convergent to e in X. If not, there exists an open neighborhood U of
e in X and an infinite set A ⊆ N such that x−1
n yn 6∈ U for each n ∈ A. By Theorem 3.3.9
in [3], there exists a continuous pre-norm N on X such that {g ∈ X : N (g) < 1} ⊆ U .
Define a continuous pseudometric d on X by d(x, y) = N (x−1 y), for all x, y ∈ X. It
follows that d(xn , yn ) ≥ 1 for all n ∈ A. However, this is impossible, since by the
construction of {xn : n ∈ N} and {yn : n ∈ N}, d(xn , yn ) ≥ 1 holds only for at most

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FUCAI LIN, CHUAN LIU, AND JILING CAO

finitely many n ∈ A. Thus, S is a non-trivial sequence in X converging to e. This
contradicts with the fact that X contains no non-trivial convergent sequences.

To see how Theorem 3.3 can be applied, we need to identify some classes of topological
groups that contain no non-trivial convergence sequences. By a result of [5], every hereditarily normal topological group without a Gδ -diagonal contains no non-trivial convergent
sequences, and thus G(X) is snf -countable for such a topological group X. On the other
hand, every infinite compact group X contains a non-trivial convergent sequence, as it
contains a copy of the Cantor cube {0, 1}w(X), where w(X) is the weight of X. Hence,
G(X) is not snf -countable for any infinite compact group X. Note that there are infinite
pseudocompact topological groups containing no non-trivial convergent sequences, refer
to [19] for the existence of such a topological group.
Next, we discuss the snf -countability of subspaces Fn (X) of F (X) for a space X.
Recall that a subspace Y of a space X is said to be P-embedded in X if each continuous
pseudometric on Y admits a continuous extensions over X.
Theorem 3.4. Let X be a space with a regular Gδ -diagonal. If F4 (X) is snf -countable,
then X is either a cf -space or compact.
Proof. Suppose that X contains an infinite compact subset C. We show that X must
be compact. Since X has a regular Gδ -diagonal, then C must be a metrizable subspace.
Without loss of generality, we may assume that C is a non-trivial convergent sequence
with its limit point x. Since by a result in [2] any pseudocompact space with a regular
Gδ -diagonal is metrizable and compact, we only need to show that X is pseudocompact.
Assume that X is not pseudocompact. Then X contains an infinite and discrete sequence
in X. Therefore,
of

S open subsets {Un : n ∈ N}. Note that {U n : n ∈ N} is also discreteS
U
is
closed
in
X.
Since
C
is
compact,
we
may
assume
that
C
∩
n
n∈N
n∈N U n = ∅. It
follows that {C} ∪ {Un : n ∈ N} is discrete in X. For each n ∈ N, take a point xn ∈ Un .
Then, Y := C ∪ {xn : n ∈ N} is closed, σ-compact and P -embedded in X. By a result in
[21], the subgroup F (Y, X) of F (X) generated by Y is topologically isomorphic to F (Y ).
Obviously, F (Y ) is a kω -space. We claim thatSF4 (Y ) contains a closed copy of Sω . For
each n ∈ N, put Cn := xn Cx−1 x−1
n . Let Z :=
n∈N Cn . Then Z ⊆ F (Y ) is closed. Since
F (Y ) is a kω -space, Z is also a kω -subspace. Take an arbitrary infinite subset
: m ∈ N, cm ∈ C \ {x}}.
P := {xnm cm x−1 xn−1
m
Then P is a discrete closed subset of Z, since Z is a kω -space. Therefore, the claim is
verified. It follows that Sω must be snf -countable, which is a contradiction.

The following theorem generalizes Theorem 4.9 in [22].
Theorem 3.5. For a k-space X with a regular Gδ -diagonal, the following are equivalent:
(i)
(ii)
(iii)
(iv)
(v)

F4 (X) is snf -countable.
Each Fn (X) is snf -countable.
F4 (X) is metrizable.
Each Fn (X) is metrizable.
X is discrete or compact.

Proof. Obviously, we have (ii) ⇒ (i), (iv) ⇒ (iii), (iv) ⇒ (ii) and (iii) ⇒ (i). It suffices
to show that (i) ⇒ (v) and (v) ⇒ (iv).
(i) ⇒ (v). Since X has a regular Gδ -diagonal, it follows from Theorem 3.4 that either
each compact subset of X is finite or X is compact and metrizable. Moreover, it is easy to
check that a k-space is discrete if each compact subset is finite. Therefore, X is discrete
or compact.

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

7

(v) ⇒ (iv). If X is discrete, then F (X) is discrete, hence each Fn (X) is metrizable. If
X is compact, then X is a compact metrizable space since a compact space with a Gδ diagonal is metrizable [8]. Moreover, since each in is continuous, each Fn (X) is compact.
Therefore, each Fn (X) is metrizable.

The next example shows that the condition that X is “a k-space” in Theorem 3.5
cannot be dropped.
Example 3.6. There is an infinite, non-discrete, cf -space X with a regular Gδ -diagonal
such that F4 (X) is snf -countable. Let βN be the Stone-Čech compactification of N
(equipped with the discrete topology). Take an arbitrary point p ∈ βN \ N, and consider
the subspace X := N ∪ {p} of βN. It is well known that X is not compact. Indeed, X is
a non-discrete cf -space with a regular Gδ -diagonal. However, as shown in [11, Example
3.12], F (X) is snf -countable. Thus F4 (X) is also snf -countable.
By Theorem 4.12 in [22], it is easy to see that “F4 (X)” in Theorem 3.5 cannot be
replaced by “F3 (X)”. However, we have the following result.
Theorem 3.7. Let X be a stratifiable k-space. If G2 (X) is snf -countable, then X ′ is
compact.
Proof. We only consider the case of A2 (X), as the proof of the F2 (X) case is quite similar.
Suppose that X ′ is not compact. Then, X ′ is not countably compact since X is
stratifiable. Therefore, there is a closed, countable, infinite and discrete subset {xn : n ∈
N} in X ′ . Since X is paracompact, we can choose a discrete family {Un : n ∈ N} of open
subsets in X such that xn ∈ Un for each n ∈ N. For each n ∈ N, since X is sequential
and {xn : n ∈ N} ⊆ X ′ , we can choose a non-trivial sequence {xn (m) : m ∈ N} such that
{xn (m) : m ∈ N} converges to xn and {xn (m) : mS
∈ N} ⊆ Un . Let Yn := {xn (m) : m ∈
N} ∪ {xn }, Cn := {xn (m) − xn : m ∈ N} and C := {Cn : n ∈ N} ∪ {0}. Obviously, each
sequence {xn (m) − xn : m ∈ N} converges to 0 in A2 (X) and {Yn : n ∈ N} is a discrete
family in X.
We claim that the subspace C is a copy of Sω . Indeed, the subspace S, defined by
S := {(xn (m), −xn ) : m, n ∈ N} ∪ {(xn , −xn ) : n ∈ N},
is closed in (X ∪−X)×(X ∪−X). Since X is paracompact, it follows from Proposition 4.8
in [22] that i2 is a closed map. Then i2 ↾ S is a quotient mapping. Therefore, C ⊆ A2 (X)
is homeomorphic to Sω , and this verifies the claim. Since A2 (X) is snf -countable, then
C is snf -countable. This contradicts with the fact that Sω is not snf -countable.

Note that in general, the converse of Theorem 3.7 does not hold. To see this, consider
the space Sω . It is easy to check that Sω is a stratifiable k-space whose set of non-isolated
points is compact. However, neither Sω nor G2 (Sω ) is snf -countable.
4. The csf -Countability of Free Topological Groups
In this section, we discuss the csf -countability of G(X) and Gn (X) for a given space
X. First of all, we have the following simple observation.
Proposition 4.1. For a space X, G(X) is csf -countable iff Gn (X) is csf -countable for
all n ∈ N.
Proof. It is clear that if G(X) is csf -countable, then each Gn (X) is csf -countable.
Conversely, if each Gn (X) is csf -countable
with a cs-network Pn = {Pn,i : i ∈ N} at
S
e, then it is easy to check that P = n∈N Pn is a cs-network for G(X) at e. Hence, if
Gn (X) is csf -countable for all n ∈ N, then G(X) is csf -countable.


8

FUCAI LIN, CHUAN LIU, AND JILING CAO

In the light of Proposition 4.1, one of our purposes in this section is to identify those
classes of spaces X for which the csf -countability of Gn (X) at certain level n ∈ N will
be adequate to guarantee the csf -countability of G(X).
Theorem 4.2. For a paracompact, crowded, k- and ℵ-space X, the following are equivalent:
(i) G(X) is csf -countable.
(ii) G2 (X) is csf -countable.
(iii) X is separable.
Proof. Since (i) ⇒ (ii) is trivial, we only need to show (ii) ⇒ (iii) and (iii) ⇒ (i). Further,
we only consider A(X), as the proof of the case of F (X) is quite similar.
(ii) ⇒ (iii). Suppose X is not separable. Since X is an ℵ-space, there is a closed,
uncountable and discrete subset {xα : α < ω1 } in X. Since X is paracompact, we can
choose a discrete family of open subsets {Uα : α < ω1 } in X such that xα ∈ Uα for each
α < ω1 . For each α < ω1 , since X is sequential, we can choose a non-trivial sequence
{xn (α) : n ∈ N} ⊆ Uα , convergent to xα . Put
Yα := {xn (α) : n ∈ N} ∪ {xα }, Cα := {xn (α) − xα : n ∈ N}
S
and let C := {Cα : α < ω1 } ∪ {0}. Obviously, {xn (α) − xα : n ∈ N} is convergent to 0
in A2 (X) and {Yα : α < ω1 } is a discrete family in X. We claimed that C is a copy of
Sω1 . Indeed, let
S := {(xn (α), −xα ) : n ∈ N, α < ω1 } ∪ {(xα , −xα ) : α < ω1 }.
Then S is closed in (X ∪ −X) × (X ∪ −X). Since X is paracompact, by Proposition
4.8 in [22], i2 is a closed map. It follows that i2 ↾ S is a quotient mapping, and thus
C ⊆ A2 (X) is homeomorphic to Sω1 . Since A2 (X) is csf -countable, then C is csf countable. However, Sω1 is not csf -countable, which is a contradiction.
(iii) ⇒ (i). Let X be separable. Then X is an ℵ0 -space. By Theorem 4.1 in [4], A(X)
is an ℵ0 -space. Thus, A(X) is csf -countable.

The following theorem provides an affirmative answer to Question 3.9 in [11]. Recall
that a topological space X is said to be Lašnev if it is the closed image of some metric
space.
Theorem 4.3. Let X be a non-discrete Lašnev space. Then F4 (X) is csf -countable if
and only if F (X) is an ℵ0 -space.
Proof. The sufficiency is obvious. To show the necessity, let F4 (X) be csf -countable.
Then X contains no copy of Sω1 , and hence X is an ℵ-space. By Theorem 4.1 in [4],
it suffices to show that X is an ℵ0 -space. Since X is an ℵ-space, the proof will be
completed if we can show that each closed and discrete subset of X is at most countable.
Assume that X contains an uncountable, closed and discrete subset D = {dα : α < ω1 }.
Since X is a non-discrete Lašnev space, there exists aLnon-trivial convergent sequence
{xn : n ∈ N} with the limit point x inLX such that D S is a closed copy of X,Lwhere
S := {xn : n ∈ N} ∪ {x}. Since F4 (D S) is a closed subspace of F4 (X), F4 (D S) is
csf -countable.
L
Next, we shall show that F4 (D S) contains a copy of Sω . To this end, for each
α < ω1S
, define Cα := {dα xn x−1 dα −1 : nL∈ N}. Then Cα converges to e. Let C :=
S), which implies that C is csf -countable.
{e} ∪
α<ω1 Cα . Obviously, C ⊆ F4 (D
Then C has a countable cs-network {Pn : n ∈ N} at e. Put
N1 := {n ∈ N : |{α < ω1 : Pn ∩ Cα 6= ∅}| ≤ ℵ0 }

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

9

and
B := {α < ω1 : Cα ∩ Pn 6= ∅ for some n ∈ N1 }.
Clearly, N \ N1 is a countable infinite set and B is a countable set. It is easy to see that
{Pn : n ∈ N \ N1 } is a countable cs-network at e in
D1 := {e} ∪ {dα xn x−1 d−1
α : α ∈ ω1 \ B, n ∈ N}.
Moreover, for each n ∈ N \ N1 , the set Pn intersects uncountably many Cα . Inductively,
we can find a countable infinite subset R := {αn ∈ ω1 \ B : n ∈ N \ N1 } ofSω1 such that
αn 6= αm if n 6= m and L
Pn ∩ Cαn 6= ∅ for each n ∈ N \ N1 . Let Y := {e} ∪ α∈R Cα . It
is clear that Y ⊆ F4 (D S). We claim that Y is homeomorphic to Sω . Indeed, letL
Z=
supp(Y ). Then, Z is a countable, infinite, locally compact and closed subspace in D S.
Hence, F
F (Z) is a subspace
L(Z) is a sequential
L space by Theorem 7.6.36 in [3]. Moreover,
L
of F (D S), since D S is metrizable and Z is closed in D S. Assume that Y is not
a copy of Sω . Then there is a non-trivial convergent sequence {dαnk xmk x−1 d−1
αnk : k ∈ N}
:
k
∈
N} in Z is
in Y . Since Z is paracompact, the closure of supp{dαnk xmk x−1 d−1
αnk
compact. However, {dαnk : k ∈ N} is an infinite, closed and discrete subspace in Z,
which is a contradiction. Therefore, the claim is verified.
Finally, pick an arbitrary point an ∈ Pn ∩ Cαn for each n ∈ N \ N1 . Since the family
{Pn ∩ Y : n ∈ N \ N1 } is a countable cs-network at e in Y , e is a cluster point of the set
{an : n ∈ N \ N1 }, which is a contradiction. Therefore, each closed and discrete subset
of X is at most countable.

Corollary 4.4. For a non-discrete metrizable space X, the following are equivalent:
(i)
(ii)
(iii)
(iv)

X is separable.
F (X) is an ℵ0 -space.
F (X) is csf -countable.
F4 (X) is csf -countable.

Proof. By Theorem 4.1 in [4], we have (i) ⇔ (ii). Moreover, (ii) ⇒ (iii) and (iii) ⇒ (iv)
are obvious. Finally, (iv) ⇒ (ii) follows from Theorem 4.3.

Remark 4.5. (i) Theorem 4.3 does’t hold for the Abelian case, refer to Theorem 4.5 in
[22].
L
(ii) In Corollary 4.4, F4 (X) cannot be replaced by F3 (X). Let X := D S, where D
is an uncountable discrete space and S is a non-trivial convergent sequence with its limit.
Then, F3 (X) is metrizable by Theorem 4.12 in [22], hence it is csf -countable. However,
X is not separable.
(iii) The conclusion of Corollary 4.4 does not hold for the snf -countability. Consider
R with the usual Euclidean topology. By Corollary 4.4, F (R) is an ℵ0 -space. However,
it follows from Theorem 3.5 that F4 (R) is not snf -countable.
Next, we shall consider the question when G3 (X) is csf -countable for a given space
X. First, we recall an important lemma in [18].
Lemma 4.6 ([18]). If P is a countable cs∗ -network at x ∈ X, then the family
n[
o
F : F ⊆ P, F is finite
is a countable cs-network at x.
Theorem 4.7. Let X be a sequential and µ-space. If X ′ has a countable cs∗ -network in
X, then F3 (X) is csf -countable.

10

FUCAI LIN, CHUAN LIU, AND JILING CAO

∗
Proof. Let P be a countable
for X ′ in X. By Lemma 4.6, X is csf L −1 Lcs -network
n
countable. Hence, (X
X
{e}) is csf -countable for each n ∈ N. L
Moreover,
L since
F3 (X) \ F1 (X) is open in F3 (X) and homeomorphic to a subspace of (X
X −1 {e})3 ,
F3 (X) is csf -countable at each point of F3 (X) \ F1 (X). In what follows, we shall show
that F3 (X) is also csf -countable at each point of F1 (X). To this end, take an arbitrary
point g ∈ F1 (X). By Lemma 4.6, we are done if we can prove that F3 (X) has a countable
cs∗ -network at g. We divide the proof in three cases.
Case 1. The point g is isolated in X ∪ X −1 .
In this case, let

B(g) := {gP1ε1 P2ε2 , P1ε1 P2ε2 g : P1 , P2 ∈ P ∪ {{g}}, ε1, ε2 ∈ {1, −1}, ε1 6= ε2 }.
Obviously, |B(g)| ≤ ω. We verify that B(g) is a cs∗ -network for F3 (X) at g. Take an
arbitrary sequence {xn : n ∈ N} converging to g in F3 (X) and an open neighborhood U of
g in F3 (X). Without loss of generality, we may assume that {xn : n ∈ N} is a non-trivial
convergent sequence such that that xn 6= xm whenever n 6= m. By Theorem 1.5 in [4],
supp({xn : n ∈ N}∪{g}) is bounded in X. Since X is a µ-space, supp({xn : n ∈ N} ∪ {g})
is compact in X. Let

 
−1
Z := supp{xn : n ∈ N} ∪ {g} ∪ supp{xn : n ∈ N} ∪ {g}
.
Then Z is sequentially compact. For each n ∈ N, pick a point yn ∈ Z 3 ∩ i−1
3 (xn ). Then
it follows from the sequential compactness of X that {yn : n ∈ N} has a subsequence
{(zi (1), zi (2), zi (3)) : i ∈ N} converging to z = (z1 , z2 , z3 ) for some z ∈ Z 3 . Since xn 6= xm
for any n 6= m, the sequence {yn : n ∈ N} is a non-trivial sequence. Hence, we have that
′
′
−1
z ∈ i−1
for each i ∈ {1, 2, 3}, |{i : i ≤ 3, zi 6= g}| is an
3 (g), zi ∈ (X ∪ {g}) ∪ (X ∪ {g})
even number, z1 = g or z3 = g, and z2 ∈ X ′ ∪(X ′ )−1 . Then z ∈ {(g, z2 , z2−1 ), (z2−1 , z2 , g)}.
Without loss of generality, we assume that z1 = g, z2 ∈ X ′ and z = (g, z2 , z2−1 ). Pick an
open neighborhood V of z2 in X such that {g} × V × V −1 ⊆ i−1
3 (U ). Since g is isolated in
X ∪X −1 and the sequence {zi (1) : i ∈ N} converges to g in X ∪X −1 , zi (1) = g for all but
finitely many i. Then, since P is a cs∗ -network for X ′ , it is easy to see that there exist
P1 , P2 ∈ P such that z2 ∈ P1 ⊆ V , z2−1 ∈ P2−1 ⊆ V −1 and z ∈ {g} × P1 × P2−1 contains
a subsequence of {(zi (1), zi (2), zi (3)) : i ∈ N}. Hence, g ∈ gP1 P2−1 ⊆ gV V −1 ⊆ U and
gP1 P2−1 contains a subsequence of {xn : n ∈ N}.
Case 2. The point g is non-isolated in X ∪ X −1 .
Without loss of generality, we assume that g ∈ X. In this case, let
B(g) := {P1ε1 P2ε2 P3ε3 : P1 , P2 , P3 ∈ P ∪ {{e}}, ε1, ε2 , ε3 ∈ {1, −1}, ε1 + ε2 + ε3 = 1} .
Obviously, |B(g)| ≤ ω. We verify that B(g) is a cs∗ -network for F3 (X) at g. Take an
arbitrary sequence {xn : n ∈ N} converging to g in F3 (X) and an open neighborhood U
of g in F3 (X). Next we shall show that there exists a B ∈ B(g) such that g ∈ P ⊆ U
and P contains a subsequence of {xn : n ∈ N}. Without loss of generality, we assume
that {xn : n ∈ N} is a non-trivial convergent sequence such that that xn 6= xm whenever
n 6= m.
Subcase 2.1. The sequence {xn : n ∈ N} contains a subsequence {xni : i ∈ N} which
is contained in X.
Since P is a cs∗ -network for X ′ in X, there exists a P ∈ P such that P contains a
subsequence of {xni : i ∈ N} and g ∈ P ⊆ U . Hence, P = P {e}{e} ∈ B(g) contains a
subsequence of {xni : i ∈ N} and g ∈ P ⊆ U .
Subcase 2.2. The sequence {xn : n ∈ N} does not contain any subsequence which is
contained in X.

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

11

Without loss of generality, we may assume that {xn : n ∈ N} ⊆ F3 (X) \ F2 (X), since
(F2 (X) \ F1 (X)) ∪ {e} is clopen in F3 (X). Similar to the proof of Case 1, let
−1

 
.
Z := supp{xn : n ∈ N} ∪ {g} ∪ supp{xn : n ∈ N} ∪ {g}
Then Z 3 is sequentially compact. For each n ∈ N, pick a point yn ∈ Z 3 ∩ i−1
3 (xn ). Then
{yn : n ∈ N} has a subsequence {(zi (1), zi (2), zi (3))}i∈N converging to z = (z1 , z2 , z3 ) for
some z ∈ Z 3 . Since g is assumed to be non-isolated in X ∪ X −1 , we have
(X ′ ∪ {g}) ∪ (X ′ ∪ {g})−1 = X ′ ∪ (X ′ )−1 .
Moreover, since xn 6= xm for any n 6= m, the sequence {yn : n ∈ N} is a non-trivial
′
′ −1
for each i ∈ {1, 2, 3}, z1 = g
sequence. Hence, we have z ∈ i−1
3 (g), zi ∈ X ∪ (X )
′
′ −1
or z3 = g, and z2 ∈ X ∪ (X ) . Then z ∈ {(g, z2 , z2−1 ), (z2−1 , z2 , g)}. Without loss of
generality, we assume that z1 = g, z2 ∈ X ′ and z = (g, z2 , z2−1 ). Pick open neighborhoods
V1 , V2 of g and z2 in X, respectively, such that V1 × V2 × V2−1 ⊆ i−1
3 (U ). Since P is
a cs∗ -network for X ′ in X, it is easy to see that there exist P1 , P2 , P3 ∈ P such that
g ∈ P1 ⊆ V1 , z2 ∈ P2 ⊆ V2 , z2−1 ∈ P3−1 ⊆ V2−1 and z ∈ P1 × P2 × P3−1 contains a
subsequence of {(zi (1), zi (2), zi (3)) : i ∈ N}. Hence, g ∈ P1 P2 P3−1 ⊆ V1 V2 V2−1 ⊆ U and
P1 P2 P3−1 contains a subsequence of {xn : n ∈ N}.
Case 3: g = e.
In this case, let
B(e) := {P1ε1 P2ε2 : P1 , P2 ∈ P ∪ {{e}}, ε1, ε2 , ∈ {1, −1}, ε1 + ε2 = 0} .
Obviously, |B(g)| ≤ ω. We verify that B(g) is a cs∗ -network for F3 (X) at e. Take
an arbitrary sequence {xn : n ∈ N} converging to e and an open neighborhood U of
e in F3 (X). Without loss of generality, we assume that {xn : n ∈ N} is a non-trivial
convergent sequence such that xn 6= xm whenever n 6= m. Since (F2 (X) \ F1 (X)) ∪ {e}
is clopen in F3 (X), we assume that
{xn : n ∈ N} ⊆ F2 (X) \ F1 (X)).
By an argument similar to that in Case 2, we can show that there exist P1 , P2 ∈ B(e)
such that e ∈ P1ε1 P2ε2 ⊆ U and P1ε1 P2ε2 contains a subsequence of {xn : n ∈ N}.

As shown by the following result, for the Abelian case, the conclusion of Theorem 4.7
can be strengthened significantly.
Theorem 4.8. Let X is be a sequential and µ-space. If X ′ has a countable cs∗ -network
in X, then A(X) is csf-countable.
Proof. Since A(X) is a topological group, we only need to prove that A(X) has a countable cs-network at 0. Let P be a countable cs∗ -network for X ′ and


2k
2k

X
X
εj = 0 .
εi Pi : Pi ∈ P ∪ {{0}}, εi ∈ {1, −1}, i ≤ 2k,
B(0) :=


j=1

i=1

Obviously, |B(0)| ≤ ω. Next, we verify that B(0) is a cs -network for A(X) at 0.
Take an arbitrary sequence {xn : n ∈ N} converging to 0 in A(X) and an arbitrary
open neighborhood U of 0 in A(X). Without loss of generality, we may assume that
{xn : n ∈ N} is a non-trivial convergent sequence. Further, we may assume that xn 6= xm
for any n 6= m. Obviously, we have {xn : n ∈ N} ∪ {0} ⊆ Al (X) for some l ∈ N. Then
there exists some m ≤ l such that Am \ Am−1 contains a subsequence of {xn : n ∈ N}.
Therefore, we may assume that {xn : n ∈ N} is contained in Am \ Am−1 . Let
 


Z := supp{xn : n ∈ N} ∪ {0} ∪ − supp{xn : n ∈ N} ∪ {0} .
∗

12

FUCAI LIN, CHUAN LIU, AND JILING CAO

By the proof of Theorem 4.7, Z m is sequentially compact. For each n ∈ N, pick a point
yn ∈ Z m ∩ i−1
m (xn ). Then {yn : n ∈ N} has a subsequence {(zi (1), zi (2), ...zi (m)) :
i ∈ N} converging to z = (z1 , z2 , ...zm ) for some z ∈ Z m . Obviously,
zL∈ i−1
m (0) and
L
′
′
zj ∈ X ∪ (−X) for each j ≤ m. Pick an open subset Vj in X
−X {0} for each
j ≤ m such that
V1 × V2 × · · · × Vm ⊆ i−1
m (U ).
For each zj ∈ X ′ ∪ (−X)′ , choose a Pj ∈ P inductively such that zj ∈ εj Pj ⊆ Vj , εj Pj
contains a subsequence {zik (j) : k ∈ N} of {zi (j) : i ∈ N} and {(zik (1), zik (2), ...zik (m)) :
k ∈ N} is a subsequence of {(zi (1), zi (2), ...zi (m)) : i ∈ N}, where each εj = 1 or −1. Let
B := ε1 P1 + ε2 P2 + · · · + ε2k P2k .
Then 0 ∈ B ⊆ U , which verifies that B(0) is a cs∗ -network at 0. By Lemma 4.6, A(X)
is csf -countable.

Corollary 4.9. For a Lašnev space X, if G2 (X) is csf -countable, then so is G3 (X).
Proof. Since X is a Lašnev space, it is sequential. By a proof analogous to that of the
implication (ii) ⇒ (iii) in Theorem 4.2, one can show that X ′ is a separable subspace
of X. Hence it follows from [10] that X is a paracompact ℵ-space, since G2 (X) is
csf -countable and X is a Lašnev space. Since X ′ is closed and separable in X, X ′ is
Lindelöf, which shows that X ′ is an ℵ0 -subspace in X. Hence X ′ has a countable cs∗ network in X. Therefore, F3 (X) and A3 (X) are csf -countable by Theorems 4.7 and 4.8,
respectively.

L
Remark 4.10. (i) Let X := D K, where D is an uncountable discrete space and
K is a compact metric space. Then A(X) is csf -countable. However, F3 (X) and each
An (X) are first-countable by Theorem 4.5 and Theorem 4.12 in [22], and F4 (X) is not
csf -countable by Corollary 4.4.
(ii) In general, the converse of Theorem 4.7 or Theorem 4.8 does not hold. Indeed,
let X be an uncountable pseudocompact topological group containing no nontrivial convergent sequences, as given in [19]. By Theorem 3.3, G(X) is snf -countable, and hence
G(X) is csf -countable. Since X is an uncountable pseudocompact topological group, X ′
is uncountable. Then X ′ must not have a countable cs∗ -network in X. Indeed, assume
on the contrary that X ′ has a countable cs∗ -network in X. Then X ′ = X is an ℵ0 -space,
hence it is submetrizable. Since a pseudocompact submetrizable space is metrizable, it
follows that X is metrizable, which is a contradiction with the assumption.
5. Open Questions
We conclude this paper by posing some open questions. Our first open question
concerns about the Abelian case of Theorem 3.5. Theorem 3.5 establishes relationships
among the snf -countability and the metrizability of F4 (X) and each Fn (X), as well as
some properties of X for a fairly large class of topological spaces. It is nice to know
whether a similar result on A4 (X) and An (X) holds for the same class of spaces. Thus,
the following question is of interests.
Question 5.1. Let X be a k-space with a regular Gδ -diagonal.
countable, must every An (X) be snf -countable?

If A4 (X) is snf -

In the light of Theorem 4.5 and Theorem 4.12 in [22], Theorem 3.5 and Theorem 4.3,
it is natural to pose the following two questions.
Question 5.2. Let X be a k-space with a regular Gδ -diagonal.
countable, must G3 (X) be snf -countable?

If G2 (X) is snf -

TWO WEAK FORMS OF COUNTABILITY AXIOMS IN FREE TOPOLOGICAL GROUPS

13

Question 5.3. Let X be a non-discrete Lašnev space. If A4 (X) is csf -countable, must
each An (X) be csf -countable?
Note that the answers to Question 5.2 and Question 5.3 are not known even when X
is a metrizable space. Our last open question concerns about how to characterize the
csf -countability of F (X) and A(X) in term of properties of X.
Question 5.4. Let X be a space. Is there a topological property A of X which characterizes the csf -countability of F (X) or A(X)?
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(Fucai Lin): School of mathematics and statistics, Minnan Normal University, Zhangzhou
363000, P. R. China
E-mail address: linfucai2008@aliyun.com
(Chuan Liu): Department of Mathematics, Ohio University Zanesville Campus, Zanesville,
OH 43701, USA
E-mail address: liuc1@ohio.edu
(Jiling Cao): School of Engineering, Computer and Mathematical Sciences, Auckland
University of Technology, Private Bag 92006, Auckland 1142, New Zealand
E-mail address: jiling.cao@aut.ac.nz

