NiTkaIAGT.dvi
@
Applied General Topology
c© Universidad Politécnica de Valencia
Volume 6, No. 1, 2005
pp. 15-41
The character of free topological groups I
Peter Nickolas and Mikhail Tkachenko
1
Abstract. A systematic analysis is made of the character of the free
and free abelian topological groups on uniform spaces and on topologi-
cal spaces. In the case of the free abelian topological group on a uniform
space, expressions are given for the character in terms of simple cardi-
nal invariants of the family of uniformly continuous pseudometrics of
the given uniform space and of the uniformity itself. From these results,
others follow on the basis of various topological assumptions. Amongst
these: (i) if X is a compact Hausdorff space, then the character of the
free abelian topological group on X lies between w(X) and w(X)ℵ0 ,
where w(X) denotes the weight of X; (ii) if the Tychonoff space X is
not a P-space, then the character of the free abelian topological group
is bounded below by the “small cardinal” d; and (iii) if X is an infinite
compact metrizable space, then the character is precisely d.
In the non-abelian case, we show that the character of the free abelian
topological group is always less than or equal to that of the correspond-
ing free topological group, but the inequality is in general strict. It is
also shown that the characters of the free abelian and the free topolo-
gical groups are equal whenever the given uniform space is ω-narrow.
A sequel to this paper analyses more closely the cases of the free and free
abelian topological groups on compact Hausdorff spaces and metrizable
spaces.
2000 AMS Classification: Primary 22A05, 54H11, 54A25;
Secondary 54D30, 54D45.
Keywords: Free (abelian) topological group, uniform space, entourage of
the diagonal, character, cardinal invariant, compact, locally compact, pseudo-
compact, metrizable, kω-space, dominating family
1The first author wishes to thank the second author, and his department, for hospitality
extended during the course of this work. The second author was supported by Mexican
National Council of Sciences and Technology (CONACyT), Grant No. 400200-5-28411-E.
16 P. Nickolas and M. Tkachenko
1. Introduction
In 1948, Graev [5] proved that the free topological group F(X) and the free
abelian topological group A(X) on a Tychonoff space X are metrizable only
if X is discrete, in which case the groups are themselves discrete. For our
present purposes, we may rephrase Graev’s result as saying that when X is a
non-discrete Tychonoff space, the groups F(X) and A(X) have uncountable
character (= minimal size of a base at the identity of the group). It appears
that no other estimates of the characters of these groups (except for those valid
in the context of general topological groups) have been found to date.
In this paper and its sequel [14], we investigate these characters systemati-
cally and in some detail. Most of our results are in fact for free and free abelian
topological groups on uniform spaces, since this gives maximum generality and
allows the derivation at will of bounds for the characters of free and free abelian
groups over topological spaces.
In the abelian case, the free topology has a rather straightforward description
in terms of the family of uniformly continuous pseudometrics on the given
uniform space, and another in terms of the given uniformity itself, and our
initial results express the character of the corresponding group in terms of
simple cardinal invariants of the family of pseudometrics (Theorem 2.3) or of
the uniformity (Theorem 2.9). An immediate corollary of these results is that
if X is a compact Hausdorff space, then the character of A(X) lies between
w(X) and w(X)ℵ0 , where w(X) denotes the weight of X.
The well known “small cardinal” d (see [3] and our discussion below) plays
a role in several other results. For example, we show that if the Tychonoff
space X is not a P-space, then d is a lower bound for the character of A(X)
(Corollary 2.16); this applies in particular if X contains an infinite compact
subset or a proper dense Lindelöf subspace. Further, if X is an infinite compact
metrizable space, then the character of A(X) is precisely d (Corollary 2.22).
In the non-abelian case, the situation is intrinsically more complex and de-
scription of the free topology in terms of pseudometrics or entourages, for
example, is now far from straightforward (see [21] and [16]). Our main results
on the characters in the non-abelian case make use of a new description of a
neighborhood base at the identity in the free topological group on an arbitrary
uniform space (Theorem 3.6). While it is easy to see that the character of the
free abelian topological group is always less than or equal to that of the corre-
sponding free topological group (Lemma 3.1), the inequality is in general strict,
and the two characters may indeed differ arbitrarily largely (see [14]). Using
our new description of the topology of the free topological group, however, we
show that the characters of the free abelian and the free (non-abelian) topo-
logical groups are equal whenever the underlying uniform space is ω-narrow
(Theorem 3.15).
The sequel [14] to this paper analyses more closely the cases of the free and
free abelian topological groups on compact Hausdorff spaces and metrizable
spaces.
The character of free topological groups I 17
1.1. Notation and terminology. We denote by N the set of positive integers
and by R the set of real numbers.
All topological spaces are hypothesised implicitly to be completely regular,
but are not taken to be Hausdorff (and therefore Tychonoff) without explicit
indication. Similarly, our uniform spaces are not taken to be Hausdorff (or
separated) unless this is explicitly signalled.
We next establish our conventions for certain cardinal invariants of topolog-
ical spaces and uniform spaces. In some formulations, such as that of [8], for
example, cardinal invariants are taken always to be infinite, that is, to have a
minimum value of ℵ0, because this tends to simplify the statements of theo-
rems. For us, however, it is convenient not to follow this convention. Thus, if x
is a point in a space X, then χ(x, X) denotes the minimum cardinal of a local
base at x, and then χ(X), the character of X, is the supremum of the values
χ(x, X) for x ∈ X. More generally, if Y is a subspace of X, then we write
χ(Y, X) for the minimum cardinal of a base at Y in X. We introduce the ad
hoc notation χ∆(X) to denote the least cardinal of a basis at the diagonal ∆ in
X × X; that is, χ∆(X) = χ(∆, X × X). For a space X, the weight w(X) of X
is defined to be the smallest cardinal of an open base for X, and for a uniform
space (X, U), we denote by w(X, U) the least cardinal of a base of (X, U).
A space X is a kω-space if there exists a sequence of compact subsets Xn ⊆ X
for n ∈ N such that X =
⋃∞
n=1 Xn and such that X has the weak topology with
respect to the family {Xn : n ∈ N} (that is, such that U ⊆ X is open in X if
and only if U ∩ Xn is open in Xn for each n ∈ N). In this situation, we call the
collection {Xn : n ∈ N}, or the corresponding expression X =
⋃∞
n=1 Xn for X,
a kω-decomposition of X. We may take the sets Xn without loss of generality
to be non-decreasing.
If U is a uniformity on a set X, and if U ∈ U and n ≥ 1, we use nU to denote
the n-fold relational composite U ◦ U ◦ · · · ◦ U. If −U denotes the relational
inverse of U, then we call U symmetric if U = −U. For x ∈ X and U ∈ U, we
denote by B(x, U) the set {y ∈ X : (x, y) ∈ U}.
If (X, U) is a uniform space and τ ≥ ℵ0 is a cardinal, then we say that
(X, U) is τ-narrow if for each U ∈ U, there is a set {xα : α < τ} ⊆ X such that
X =
⋃
α<τ
B(xα, U). Similarly, we say that a topological group G is τ-narrow
if G can be covered by τ many translates of every neighborhood of the identity
(the groups with this property was called τ-bounded in [6]).
For a non-empty set X, we use Fa(X) and Aa(X) to denote the abstract
free group and abstract free abelian group on X. If n ∈ ω, then Fn(X) and
An(X) are the subsets of Fa(X) and Aa(X), respectively, which consist of all
elements whose length with respect to the basis X does not exceed n.
If X is a space, then F(X) and A(X) stand respectively for the free topo-
logical group and free abelian topological group on X. In this case, Fn(X) and
An(X) refer to the corresponding subspaces of F(X) and A(X), respectively.
Given a subset Y of X, F(Y, X) denotes the subgroup of F(X) generated by Y ,
and A(Y, X) has a similar meaning.
18 P. Nickolas and M. Tkachenko
If (X, U) is a uniform space, then F(X, U) and A(X, U) stand for the free
topological group and free abelian topological group on (X, U), and other re-
lated notations follow those already introduced for the free and free abelian
topological groups on a topological space.
If X and Y are sets, we write XY for the set of functions from X to Y . If
α and β are cardinals, we write cardinal exponentiation in the form βα; thus,∣∣XY
∣∣ = |Y ||X|. Also, c = 2ℵ0 is the power of the continuum.
In any topological group G, we denote by N(e) the family of all open neigh-
borhoods of the neutral element e in G.
1.2. Quasi-ordered sets. Many of the arguments below make use of the no-
tion of a quasi-ordered set, and related ideas. We establish here the relevant
terminology and notation.
We say that a pair (P, ≤) is a quasi-ordered set if ≤ is a reflexive transitive
relation on the set P . If (P, ≤) has the additional property of antisymmetry,
then it is a partially ordered set. A set D ⊆ P is called dominating or cofinal
in the quasi-ordered set (P, ≤) if for every p ∈ P there exists q ∈ D such that
p ≤ q. Similarly, a subset E of P is said to be dense in (P, ≤) if for every
p ∈ P there exists q ∈ E with q ≤ p. The minimal cardinality of a dominating
family in (P, ≤) is denoted by D(P, ≤), while we use d(P, ≤) for the minimal
cardinality of a dense set in (P, ≤). The notions of dominating and dense sets
are dual: if a set S is dense in (P, ≤), then it is dominating in (P, ≥) and
vice versa. Therefore, d(P, ≤) = D(P, ≥) and D(P, ≤) = d(P, ≥). Note in any
topological group G we have d(N(e), ⊆) = χ(G).
If (P, ≤) and (Q, ≪) are quasi-ordered sets, then a mapping f : P → Q is
called order-preserving if x ≤ y implies f(x) ≪ f(y), where x, y ∈ P . Similarly,
f is order-reversing if x ≤ y implies f(x) ≫ f(y). If the mapping f is a bijection
between P and Q and if f and f−1 both are order-preserving, then f is called
an (order-)isomorphism of (P, ≤) onto (Q, ≪).
Lemma 1.1. Let (P, ≤) and (Q, ≪) be quasi-ordered sets, and let f : P →
Q be an order-preserving mapping. If f(P) is a dominating set in Q, then
D(Q) ≤ D(P).
Proof. Let D be an arbitrary dominating set in P . For each q ∈ Q, there exists
p ∈ P such that q ≤ f(p). Also, there exists d ∈ D such that p ≤ d, from
which we have q ≤ f(p) ≤ f(d). Hence f(D) is a dominating set in Q, and
so D(Q) ≤ |f(D)| ≤ |D|. Finally, taking D to be a dominating set in P of
minimal cardinality gives D(Q) ≤ D(P), as required. �
Observe that because of the duality noted above between dominating sets
and dense sets, there are dualised versions of the lemma: a version for dense sets
rather than dominating sets, and others for order-reversing mappings rather
than order-preserving ones. We will make frequent use of the lemma and its
unstated variants, usually without explicit reference.
We will deal later with many different quasi-ordered sets, most, though
not all, of which will in fact be partially ordered sets. Generally, we will
The character of free topological groups I 19
give each ordering distinctive notation by using an appropriate subscript, since
some arguments make use of several quasi-orderings simultaneously. If ≪ is an
ordering on a set X, then we will usually denote the order defined coordinate-
wise on a set of functions YX by attaching an asterisk as a superscript, as
in ≪∗. An exception is when X = ω, when the ordering defined coordinate-
wise using ≤ on sets such as ωω and Nω will again be denoted just by ≤.
Consider ωω, the collection of all functions from ω to ω. Then following [3],
we define two quasi-orders ≤∗ and ≤ on ωω, by specifying that if f, g ∈ ωω,
then f ≤∗ g if f(n) ≤ g(n) for all except finitely many n ∈ ω, and f ≤ g if
f(n) ≤ g(n) for all n ∈ ω; the quasi-order ≤ is of course in fact a partial order.
(The use of the asterisk in ≤∗ is inconsistent with the notational convention just
established, but we will not in fact use this ordering again after this paragraph.)
The least cardinal of a dominating set in the quasi-ordered set (ωω, ≤∗) is
denoted by d, and in the partially ordered set (ωω, ≤) by d1. It is shown in
Theorem 3.6 of [3] that d = d1. It is also known that ℵ1 ≤ d ≤ c, but that the
value of d depends on extra axioms of set theory [3]. Below, we will find it most
useful to use the characterization of d as the least cardinal of a dominating set
with respect to the relation ≤: thus, d = D(ωω, ≤).
2. The character of free abelian topological groups
Graev’s proof in [5] of the fact that the free topological group on a Tychonoff
space X is Hausdorff proceeds by a construction which extends each continuous
pseudometric on X to an invariant pseudometric on the underlying abstract free
group Fa(X), and an argument which shows that the group topology induced
by all the extensions (referred to as Graev’s topology) is Hausdorff and weaker
than the free topology. It is well known that Graev’s topology is only equal
to the free topology in somewhat pathological cases [9, 19, 22]. In the abelian
case, a parallel construction was also outlined by Graev, and in this case Graev’s
topology is always the free topology (see [13, 11, 20]; cf. also 4.4 and 4.8 of [10]).
(It seems unlikely that this fact was unknown to Graev, but it is not mentioned
in his paper.)
In what follows, we deal mostly with free and free abelian topological groups
on uniform spaces, for convenience and generality, deducing applications to free
and free abelian topological groups on topological spaces when appropriate. We
therefore need the uniform space analog of the result just noted: that Graev’s
pseudometrics generate a neighborhood base at the neutral element of the free
abelian topological group on a uniform space.
For a given pseudometric d on a set X, we denote by d̂ and d̂A Graev’s
extension [5] of d to the abstract groups Fa(X) and Aa(X), respectively. Note
that algebraically the free uniform groups F(X, U) and A(X, U) are Fa(X)
and Aa(X), respectively [12, 15]. If (X, U) is a uniform space and d is a
pseudometric on X, then we write
Vd = {g ∈ A(X, U) : d̂A(g, 0) < 1},
20 P. Nickolas and M. Tkachenko
where 0 is the neutral element of A(X, U). The following theorem is the uniform
space analog of the result of [13, 20]; the arguments in the topological case apply
with minimal adjustment in the uniform space case, and we therefore omit the
proof.
Theorem 2.1. Let (X, U) be a uniform space. Then for every uniformly con-
tinuous pseudometric d on (X, U) the set Vd is open in A(X, U), and for every
neighborhood U of the neutral element in A(X, U) there exists a uniformly con-
tinuous pseudometric d on (X, U) such that Vd ⊆ U.
Instead of using uniformly continuous pseudometrics, one can directly use
the entourages U ∈ U belonging to a uniform space (X, U) to construct a
neighborhood base at the identity of A(X, U). The theorem below gives a
simple, explicit description in these terms of the topology of the free abelian
topological group on a uniform space, equivalent in a sense to Theorem 2.1.
This result has certainly been known since the early 1980s, when the first
description of the topology of the (non-abelian) free topological group on a
uniform space was published [16]. Though the paper [20] contains a result
equivalent to ours stated in the language of pseudometrics, the first occurrence
of essentially our result in the literature appears to have been in [26], though
it is formulated there in a context less general than ours.
First, we introduce the following notation. If (X, U) is a uniform space and if
〈U0, U1, . . .〉 is a sequence of elements of U, then we denote by N(〈U0, U1, . . .〉)
the set of all elements of A(X, U) of the form
x0 − y0 + x1 − y1 + · · · + xn − yn,
where n ∈ ω and where (xi, yi) ∈ Ui for i = 0, 1, . . . , n.
Theorem 2.2. Let (X, U) be a uniform space. Then the collection of all sets
of the form N(〈U0, U1, . . .〉), as 〈U0, U1, . . .〉 runs over all sequences of elements
of U, is an open basis at the neutral element in A(X, U).
Denote by P(X, U) the family of all uniformly continuous pseudometrics
on (X, U) bounded by 1. For d1, d2 ∈ P(X, U), we write d1 ≤ d2 if d1(x, y) ≤
d2(x, y) for all x, y ∈ X. We express our first result on the character of A(X, U))
in terms of the partially ordered set (P(X, U), ≤).
Theorem 2.3. If (X, U) is a uniform space, then χ(A(X, U)) = D(P(X, U), ≤).
Proof. By Theorem 2.1, there exists a natural correspondence between the
family P(X, U) and a base at the neutral element 0 of A(X, U). In fact, the
mapping d 7→ Vd from (P(X, U), ≤) to the partially ordered set (N(0), ⊇)
of open neighborhoods of 0 in A(X, U) ordered by reverse inclusion is order-
preserving and maps P(X, U) to a base at 0 in A(X, U), that is, a dominating
set in (N(0), ⊇). This immediately implies (by Lemma 1.1) that χ(A(X, U)) ≤
D(P(X, U), ≤).
Suppose that a subset Q ⊆ P(X, U) is such that {Vd : d ∈ Q} is a base at the
neutral element in A(X, U). We claim that for every ̺ ∈ P(X, U) there exists
The character of free topological groups I 21
d ∈ Q such that ̺ ≤ 2d. Indeed, given ̺ ∈ P(X, U), we can find d ∈ Q such that
Vd ⊆ V̺. Let x, y ∈ X. If d(x, y) < 1, then x−y ∈ Vd ⊆ V̺, whence ̺(x, y) < 1.
Similarly, if n ∈ N and d(x, y) < 2−n, then d̂A(0, 2
n(x − y)) = 2nd(x, y) < 1
by the linearity of the pseudometric d̂A (see [23]). Therefore, 2
n(x − y) ∈ Vd ⊆
V̺, whence ̺(x, y) < 2
−n. We have thus proved that d(x, y) < 2−n implies
̺(x, y) < 2−n, for n = 0, 1, . . .. In particular, d(x, y) = 0 implies ̺(x, y) = 0.
Therefore, the inequality ̺(x, y) ≤ 2d(x, y) holds if d(x, y) = 0. It is also
obvious that the same inequality holds if d(x, y) = 1. If 0 < d(x, y) < 1,
choose n ∈ ω such that 2−n−1 ≤ d(x, y) < 2−n. Then ̺(x, y) < 2−n, so that
̺(x, y) ≤ 2d(x, y), and the inequality again holds. Thus, ̺ ≤ 2d, proving our
claim.
For d ∈ Q, define d∗ ∈ P(X, U) by setting d∗ = min{2d, 1}, and consider
the family Q∗ = {d∗ : d ∈ Q} ⊆ P(X, U). By the claim just verified, for all
̺ ∈ P(X, U) there exists d∗ ∈ Q∗ such that ̺ ≤ d∗, and so Q∗ is a dominating
family in P(X, U). Therefore, D(P(X, U), ≤) ≤ |Q∗| ≤ |Q|.
Let B be any base at 0 in A(X, U). Then for every N ∈ B, Theorem 2.1
shows that there exists dN ∈ P(X, U) such that VdN ⊆ N. Set Q = {dN :
N ∈ B}. Then {Vd : d ∈ Q} is a base at 0 and satisfies |Q| ≤ |B|, so that
D(P(X, U), ≤) ≤ |Q| ≤ |B|. Hence D(P(X, U), ≤) ≤ χ(A(X, U)). Combining
this with the reverse inequality obtained earlier, we conclude that χ(A(X, U)) =
D(P(X, U), ≤). �
For a topological space X, denote by PX the family of all continuous pseu-
dometrics on X bounded by 1. It is clear that PX = P(X, UX), where UX is
the fine uniformity of X. Since A(X) = A(X, UX), from Theorem 2.3 we have:
Corollary 2.4. The equality χ(A(X)) = D(PX, ≤) holds for all Tychonoff
spaces X.
We record in passing a couple of consequences of these last results, not
otherwise directly related to our main concerns in this paper.
First, let (X, U) be an arbitrary uniform space, and let FG(X, U) denote
the abstract group Fa(X) equipped with the Graev topology, the topology
generated by Graev’s extensions of all the uniformly continuous pseudometrics
on (X, U). Then we have:
Theorem 2.5. If (X, U) is a uniform space, then χ(FG(X, U)) = χ(A(X, U)).
Proof. By Theorem 2.3, χ(A(X, U)) = D(P(X, U), ≤). Now if {dα : α ∈ A} is
a dominating family of pseudometrics in (P(X, U), ≤), then it is clear that the
corresponding family of Graev extensions {d̂α : α ∈ A} on the group Fa(X)
induces an open base at the identity, giving χ(FG(X, U)) ≤ χ(A(X, U)). Con-
versely, the natural continuous homomorphism from FG(X, U)) onto A(X, U)
is a quotient, from which the inequality χ(A(X, U)) ≤ χ(FG(X, U)) follows,
giving the result. �
In particular, if X is an arbitrary topological space and FG(X) denotes the
group Fa(X) equipped with the Graev topology, then we have:
22 P. Nickolas and M. Tkachenko
Corollary 2.6. χ(FG(X)) = χ(A(X)).
Second, as we have noted, it is well known [13, 20] that Graev’s topology is
the free topology on a free abelian topological group A(X), and Theorem 2.1
extended this to the case of a free abelian topological group A(X, U). On the
other hand, it has been noted more than once [13, 17] that the fact that a
certain family of (uniformly) continuous pseudometrics are sufficient together
to define the topology (or uniformity) of X does not imply in general that
the corresponding family of Graev extensions defines the free topology. This
is clear, for example, from the observation of Graev [5] noted at the start of
the first section: since the free abelian topological group A(X) on a Tychonoff
space X is metrizable only when X is discrete, the metrizability of X implies
the metrizability of A(X) only when X is discrete. We can adapt the proof
of Theorem 2.3 to obtain a necessary and sufficient condition on a family of
uniformly continuous pseudometrics under which their Graev extensions do
indeed define the free topology. The proof is essentially a recapitulation of the
second paragraph of the proof of Theorem 2.3, and we omit the details.
Theorem 2.7. Let (X, U) be a uniform space and let Q ⊆ P(X, U). Then the
collection of open sets
{g ∈ A(X, U) : d̂A(g, 0) < ǫ}, for d ∈ Q and ǫ > 0,
is a base at 0 for the topology of the free abelian topological group A(X, U) if
and only if for every ̺ ∈ P(X, U) there exists d ∈ Q such that ̺ ≤ 2d.
Let (X, U) be a uniform space. Given two sequences s = 〈Un : n ∈ ω〉 and
t = 〈Vn : n ∈ ω〉 in
ωU, we write s ≤ t provided that Un ⊆ Vn for each n ∈ ω.
It is easy to see that the correspondence s 7→ N(s), where N(s) is as defined
immediately before Theorem 2.2, is an order-reversing mapping of (ωU, ≤) to
(N(0), ⊇). Since by Theorem 2.2 the family {N(s) : s ∈ ωU} is a base at 0
in A(X, U), we conclude that χ(A(X, U)) ≤ d(ωU, ≤). In fact, we will show
shortly that the latter inequality is equality.
Lemma 2.8. d(ωU, ≤) ≤ D(P(X, U), ≤) for every uniform space (X, U).
Proof. For d ∈ P(X, U) and n ∈ ω, put
On(d) = {(x, y) ∈ X × X : d(x, y) ≤ 2
−n}.
It is clear that the correspondence d 7→ 〈On(d) : n ∈ ω〉 is an order-reversing
mapping of (P(X, U), ≤) to (ωU, ≤).
Consider an arbitrary sequence 〈Un : n ∈ ω〉 ∈
ωU. Clearly, there exists a
sequence 〈Vn : n ∈ ω〉 ∈
ωU such that Vn is symmetric and 3Vn+1 ⊆ Vn ⊆ Un
for each n ∈ ω. By Theorem 8.1.10 of [4], we can find d ∈ P(X, U) such
that On(d) ⊆ Vn for each n ∈ ω, so that the function d 7→ 〈On(d) : n ∈ ω〉
maps P(X, U) to a dense set in ωU. It follows, as required, that d(ωU, ≤) ≤
D(P(X, U), ≤). (We note that [4] uses the standing assumption that all uniform
spaces are Hausdorff (or separated), but it is well known and easy to see that
this assumption is unnecessary for the validity of Theorem 8.1.10.) �
The character of free topological groups I 23
Using the observation made before the lemma together with the lemma itself,
we have
χ(A(X, U)) ≤ d(ωU, ≤) ≤ D(P(X, U), ≤).
But by Theorem 2.3, we also have χ(A(X, U)) = D(P(X, U), ≤), so we have
proved both of the following results.
Theorem 2.9. If (X, U) is a uniform space, then χ(A(X, U)) = d(ωU, ≤).
Theorem 2.10. If (X, U) is a uniform space, then d(ωU, ≤) = D(P(X, U), ≤).
It is clear that every uniform space (X, U) satisfies
w(X, U) = d(U, ⊆) ≤ d(ωU, ≤) ≤ d(U, ⊆)ℵ0 = w(X, U)ℵ0 .
Hence Theorem 2.9 implies the following bounds for the character of the free
abelian topological group on (X, U):
Corollary 2.11. Let (X, U) be a uniform space. Then
w(X, U) ≤ χ(A(X, U)) ≤ w(X, U)ℵ0 .
In the case of a compact Hausdorff space X, we have w(X, U) = w(X), so
the bounds can be simplified as follows.
Corollary 2.12. If X is a compact Hausdorff space, then
w(X) ≤ χ(A(X)) ≤ w(X)ℵ0.
In Theorem 2.15 and Corollary 2.16 below we present a different lower bound
for the character of most free abelian topological groups. We shall see later
that this bound, the cardinal d, is the exact value of the character of A(X)
(and indeed of F(X)) when X is an infinite compact metrizable space.
First, we recall two useful notions. If every Gδ-set in X is open, then X
is said to be a P-space. Similarly, given a uniform space (X, U), we say that
(X, U) is a uniform P-space if the intersection of countably many elements of
U is again an element of U. Note that if (X, U) is a uniform P-space, then the
underlying topological space X is a P-space.
In some of the arguments which follow, it is natural to consider separately
the cases when (X, U) is a uniform P-space and when (X, U) is not a uniform P-
space. In fact, the character of A(X, U) in the “exceptional” case of a uniform
P-space (X, U) can be dealt with simply and conclusively in the following form.
Theorem 2.13. If (X, U) is a uniform P-space, then χ(A(X, U)) = w(X, U).
Proof. For any uniform space (X, U), the mapping U 7→ 〈U, U, . . .〉 from (U, ⊆)
to (ωU, ≤) is an order-preserving embedding, and if (X, U) is also a uni-
form P-space, then U is mapped to a dense set in ωU, since for an arbitrary
〈U0, U1, . . .〉 ∈
ωU we have 〈U, U, . . .〉 ≤ 〈U0, U1, . . .〉, where U =
⋂
n∈ω Un ∈ U.
It follows that d(U, ⊆) = d(ωU, ≤), and then the conclusion that χ(A(X, U)) =
w(X, U) follows from Theorem 2.9. �
We now turn to the “usual” case when (X, U) is not a uniform P-space.
24 P. Nickolas and M. Tkachenko
Lemma 2.14. If (X, U) is not a uniform P-space, then d ≤ D(P(X, U), ≤).
Proof. Fix a strictly decreasing sequence 〈Un : n ∈ ω〉 ∈
ωU such that U0 =
X × X and
⋂
n∈ω Un is not in U. For any s = 〈Vn : n ∈ ω〉 ∈
ωU, we define a
function fs ∈
ωω by
fs(n) = max{k ∈ ω : Vn ⊆ Uk},
for n ∈ ω. Since
⋂
n∈ω Un /∈ U, we have fs(n) < ∞ for each n ∈ ω, so our
definition of fs is valid. Moreover, it is easy to see that if 〈Vn : n ∈ ω〉 = s ≤
t = 〈Wn : n ∈ ω〉 then fs ≥ ft, so that the mapping s 7→ fs from (
ωU, ≤)
to (ωω, ≤) is order-reversing.
We claim that the set {fs : s ∈
ωU} is dominating in (ωω, ≤). Indeed, for
any f ∈ ωω, let f̂ be a strictly increasing function in ωω such that f ≤ f̂. Now
set s = 〈U
f̂(n)
: n ∈ ω〉 ∈ ωU, and note that s is a strictly decreasing sequence
of sets. Then
fs(n) = max{k ∈ ω : Uf̂(n) ⊆ Uk} = f̂(n)
for all n ∈ ω, so that fs = f̂. Therefore, f ≤ fs, proving our claim. This
immediately implies that d = D(ωω, ≤) ≤ d(ωU, ≤) = D(P(X, U), ≤), by The-
orem 2.10, as required. �
Theorem 2.3 and Lemma 2.14 imply the following lower bound, complement-
ing Theorem 2.13, for the character of A(X, U).
Theorem 2.15. If (X, U) is not a uniform P-space, then d ≤ χ(A(X, U)).
Theorem 2.15 implies several important corollaries.
Corollary 2.16. If a Tychonoff space X is not a P-space, then d ≤ χ(A(X)).
Corollary 2.17. Let (X, U) be a Hausdorff uniform space which contains an
infinite precompact subset. Then d ≤ χ(A(X, U)).
Proof. Suppose that P is an infinite precompact subset of (X, U). Let (Y, V)
be the completion of the space (X, U) and let K be the closure of P in Y
(we identify X with the corresponding dense subspace of Y ). Then K is an
infinite compact subset of Y . The group A(X, U) is topologically isomorphic to
a dense subgroup of A(Y, V) by Nummela’s theorem [15], so that χ(A(X, U)) =
χ(A(Y, V)). Since the infinite compact set K cannot be a P-space, (Y, V) fails to
be a uniform P-space. Therefore, Theorem 2.15 implies that d ≤ χ(A(Y, V)) =
χ(A(X, U)). �
Corollary 2.18. If a Tychonoff space X contains an infinite compact set (in
particular, a non-trivial convergent sequence), then d ≤ χ(A(X)).
Corollary 2.19. If a Tychonoff space X contains a proper dense Lindelöf
subspace, then d ≤ χ(A(X)).
Proof. Suppose that Y is a proper dense Lindelöf subspace of X. Then for
every point x ∈ X \ Y , there exists a Gδ-set Px in X containing x such that
Px ∩ Y = ∅. If X were a P-space, the complement X \ Y would be open in X,
The character of free topological groups I 25
thus contradicting the assumption that Y is dense in X. Hence the conclusion
follows from Corollary 2.16. �
One cannot omit the word “proper” in Corollary 2.19, since the character
of the free abelian topological group over the one-point Lindelöfication of a
discrete space of cardinality ℵ1 is exactly equal to ℵ1 (this follows from [7,
Lemma 2.9]).
Lemma 2.20. Let (X, U) be a uniform space. If d(U, ⊆) = ℵ0, then d(
ωU, ≤) =
d.
Proof. Choose a dense set {U0, U1, . . .} in (U, ⊆) such that Un strictly con-
tains Un+1 for all n ∈ ω. Then it is easy to see that the mapping (n0, n1, . . .) 7→
(Un0, Un1, . . .) from (
ωω, ≤) to (ωU, ≤) is order-reversing and maps ωω onto a
dense subset of ωU, which gives d(ωU, ≤) ≤ D(ωω, ≤) = d.
Conversely, if we map (V0, V1, . . .) ∈
ωU to the sequence (n0, n1, . . .) ∈
ωω
defined by setting nk = min{m ∈ ω : Um ⊆ Vk} for all k ∈ ω, then the mapping
is easily seen to be order-reversing and to map ωU onto a dominating subset of
ωω, giving d = D(ωω, ≤) ≤ d(ωU, ≤), and the result. �
The following result is immediate from Lemma 2.20 and Theorem 2.9.
Theorem 2.21. Let (X, U) be a uniform space satisfying w(X, U) = ℵ0. Then
χ(A(X, U)) = d.
Since a uniform space is pseudometrizable if and only if it has a countable
base, we have in particular:
Corollary 2.22. If X is an infinite compact metrizable space, then χ(A(X)) =
d.
Theorem 2.23. Let X be a Tychonoff space satisfying χ∆(X) ≤ ℵ0. Then
either X and A(X) are discrete or χ(A(X)) = d.
Proof. By Theorem 14 in [18], from χ∆(X) ≤ ℵ0 it follows that the set X
′ of
all non-isolated points in X is compact and χ(X′, X) ≤ ℵ0. If X
′ = ∅, then
both X and A(X) are discrete. Suppose, therefore, that X′ 6= ∅. Clearly,
X admits a perfect mapping onto a space Y with a single non-isolated point
(map X′ to a point). Therefore, both Y and X are paracompact, so that
every neighborhood of the diagonal in X2 belongs to the fine uniformity U
of X. In particular, w(X, U) = χ∆(X) ≤ ℵ0. The result now follows from
Theorem 2.21. �
3. The character of free topological groups
The next lemma establishes a simple relation between the characters of the
groups A(X, U) and F(X, U).
Lemma 3.1. If (X, U) is a uniform space, then χ(A(X, U)) ≤ χ(F(X, U)).
26 P. Nickolas and M. Tkachenko
Proof. Since A(X, U) is a quotient group of F(X, U) and continuous open ho-
momorphisms do not raise the character, we have χ(A(X, U)) ≤ χ(F(X, U)).
�
Similarly, of course, for a topological space X, we have χ(A(X)) ≤ χ(F(X)).
Thus, each of the lower bounds we have derived above for χ(A(X, U)) or
χ(A(X)) yields automatically a corresponding lower bound for χ(F(X, U))
or χ(F(X)), respectively. Corollary 2.19, for example, gives us the bound
d ≤ χ(A(X)) ≤ χ(F(X)) for a space X containing a proper dense Lindelöf
subspace.
In fact, however, the conclusion of Corollary 2.19 can be strengthened to
d ≤ χ(A(X)) = χ(F(X)) (see Corollary 3.18 below), but this is not at all
straightforward. Our aim now is to establish the equality χ(A(X)) = χ(F(X))
for the wide class of ℵ0-narrow spaces, which are also known as pseudo-ω1-
compact spaces. By definition, a space X is pseudo-ω1-compact if every discrete
family of open sets in X is countable. Our choice of the new name for this class
of spaces is motivated (apart from the aesthetic reason) by the fact that X is
pseudo-ω1-compact if and only if the uniform space (X, UX) is ℵ0-narrow, where
UX is the fine uniformity of X (see [24, Assertion 1.2]).
Our arguments require some unpleasant work describing a neighborhood
base in a free topological group. Since the cases of the free topological group
on a topological space and a uniform space are very similar, we prefer to present
the description in the most general form, for free topological groups on uniform
spaces.
First we recall some notions related to trees. A partially ordered set (P, ≤)
is a tree if the set Px = {y ∈ P : y < x} is well ordered by < for each x ∈ P
(where we write y < x if and only if y ≤ x and y 6= x). The height of an
element x ∈ P , denoted by h(x), is the order type of the set (Px, <). For an
ordinal α, we call the set P(α) = {x ∈ P : h(x) = α} the αth level of (P, ≤).
Finally, the height of (P, ≤) is defined to be the smallest ordinal α such that
P(α) = ∅.
In what follows we shall work only with trees of height ω. Clearly, if the
height of P is ω, then every x ∈ P has only finitely many predecessors with
respect to ≤.
Definition 3.2. Let (X, U) be a uniform space. We say that a tree (P, ≤) of
height ω is U-covering if it satisfies the following conditions:
(i) each element x ∈ P has the form x = (U0, . . . , Un), where U0, . . . , Un
are non-empty open sets in X and n ∈ ω;
(ii) if x = (U0, . . . , Un) and y = (V0, . . . , Vm) are in P, then x ≤ y if and
only if n ≤ m and Ui = Vi for each i = 0, . . . , n;
(iii) the family γP = {V : (V ) ∈ P(0)} is a U-uniform cover of X;
(iv) if x = (U0, . . . , Un) ∈ P, then the family γP (x) = {V : (U0, . . . , Un, V ) ∈
P} is a U-uniform cover of X.
The character of free topological groups I 27
Denote by T (X, U) the family of all U-covering trees. For (P, ≤) ∈ T (X, U),
we define a subset WP of Fa(X) as follows:
(3.1) WP =
⋃
(U0,...,Un)∈P,
ε0,...,εn=±1,
n∈ω
U−ε00 · · · U
−εn
n U
εn
n · · · U
ε0
0
The next lemma is the first step towards the promised description of a neigh-
borhood base at the identity of free topological groups.
Lemma 3.3. For every neighborhood O of the identity e in F(X, U), there
exists (P, ≤) ∈ T (X, U) such that WP ⊆ O. In addition, if the space (X, U) is
τ-narrow for some τ ≥ ℵ0, then one can choose (P, ≤) satisfying |P | ≤ τ.
Proof. Denote by O(e) the family of all open symmetric neighborhoods of e
in the group F(X, U). For a given O ∈ O(e), choose W ∈ O(e) such that
W 3 ⊆ O. Suppose that the uniform space (X, U) is τ-narrow for some τ ≥ ℵ0.
Then the group F(X, U) is τ-narrow by [6] or [2, Lemma 3.2]. For every x ∈ X,
put Ux = x · W ∩ X ∩ W · x. Since the two-sided uniformity of F(X, U) induces
on X its original uniformity U [15], we can find a set Y ⊆ X with |Y | ≤ τ
such that X =
⋃
x∈Y Ux. Put P(0) = {(Ux) : x ∈ Y }. Then |P(0)| ≤ τ and
γP = {Ux : x ∈ Y } is a U-uniform cover of X. This defines the initial level of
the required tree P .
Let us describe the second step of the construction. For every x ∈ Y , choose
Vx, Wx ∈ O(e) such that x
−ε · Vx · x
ε ⊆ W for ε = ±1 and W 3x ⊆ Vx, and put
Ux,y = y · Wx ∩ X ∩ Wx · y for each y ∈ X. Since the space (X, U) is τ-narrow,
we can choose, given any x ∈ Y , a set Y (x) ⊆ X with |Y (x)| ≤ τ satisfying
X =
⋃
y∈Y (x) Ux,y. Put γP (x) = {Ux,y : y ∈ Y (x)}. Then γP (x) is a U-uniform
cover of X for each x ∈ Y , and we define the level P(1) of the tree P by
P(1) = {(Ux, Ux,y) : x ∈ Y, y ∈ Y (x)}.
Clearly, |P(1)| ≤ τ.
At the third step, for each x ∈ Y and for each y ∈ Y (x), choose Vx,y, Wx,y ∈
O(e) such that y−ε · Vx,y · y
ε ⊆ Wx for ε = ±1 and W
3
x,y ⊆ Vx,y. For z ∈ X,
put Ux,y,z = z · Wx,y ∩ X ∩ Wx,y · z and choose a subset Y (x, y) of X such that
|Y (x, y)| ≤ τ and X =
⋃
z∈Y (x,y) Ux,y,z. Put γP (x, y) = {Ux,y,z : z ∈ Y (x, y)}.
Then γP (x, y) is a U-uniform cover of X for each x ∈ Y and each y ∈ Y (x),
and we define the level P(2) of the tree P by
P(2) = {(Ux, Ux,y, Ux,y,z) : x ∈ Y, y ∈ Y (x), z ∈ Y (x, y)}.
Clearly, |P(2)| ≤ |P(1)| · τ ≤ τ. Continuing this process, we finally obtain
the set P =
⋃
n∈ω P(n), partially ordered according to (ii) of Definition 3.2,
and such that |P | ≤ τ. One easily verifies that (P, ≤) satisfies (i)–(iv), so that
(P, ≤) ∈ T (X, U).
It remains to show that WP ⊆ O. Let (U0, U1, . . . , Un) be an element of P .
We have to verify that U−ε00 · · · U
−εn
n U
εn
n · · · U
ε0
0 ⊆ O for arbitrary ε0, . . . , εn =
28 P. Nickolas and M. Tkachenko
±1. There exist points x0 ∈ Y, x1 ∈ Y (x0), . . . , xn ∈ Y (x0, . . . , xn−1) such
that U0 = Ux0, U1 = Ux0,x1, . . . , Un = Ux0,x1,...,xn. By definition, we have
U0 ⊆ (x0 · W) ∩ (W · x0),
U1 ⊆ (x1 · Wx0) ∩ (Wx0 · x1),
. . . . . .
Un ⊆ (xn · Wx0,...,xn−1) ∩ (Wx0,...,xn−1 · xn).
We claim that
(3.2) U
−εk+1
k+1 · · · U
−εn
n U
εn
n · · · U
εk+1
k+1 ⊆ Vx0,...,xk
for each k = 0, 1, . . . , n − 1. Indeed, if k = n − 1, then
U−1n Un ⊆ (xnWx0,...,xn−1)
−1 · xnWx0,...,xn−1 = W
2
x0,...,xn−1
⊆ Vx0,...,xn−1,
and, similarly,
Un · U
−1
n ⊆ Wx0,...,xn−1xn · (Wx0,...,xn−1xn)
−1 = W 2x0,...,xn−1 ⊆ Vx0,...,xn−1,
giving (3.2) for k = n − 1. Suppose that (3.2) holds for some k > 0. If εk = 1,
we have Uk ⊆ xk · Wx0,...,xk−1, whence
U−1
k
U
−εk+1
k+1 · · · U
−εn
n U
εn
n · · · U
εk+1
k+1 Uk
⊆ U−1
k
Vx0,...,xkUk ⊆ W
−1
x0,...,xk−1
· x−1
k
· Vx0,...,xk · xk · Wx0,...,xk−1
⊆ W 3x0,...,xk−1
⊆ Vx0,...,xk−1.
Similarly, if εk = −1, then we use the inclusion Uk ⊆ Wx0,...,xk−1 · xk to deduce
that
UkU
−εk+1
k+1 · · · U
−εn
n U
εn
n · · · U
εk+1
k+1 U
−1
k
⊆ Vx0,...,xk−1.
The inclusion (3.2) now follows. Finally, from U0 ⊆ x0 · W , and using (3.2)
with k = 0, it follows that
U−10 U
−ε1
1 · · · U
−εn
n U
εn
n · · · U
ε1
1 U0 ⊆ W
−1 · x−10 · Vx0 · x0 · W
⊆ W 3
⊆ O,
and similarly, from U0 ⊆ W · x0 it follows that
U0U
−ε1
1 · · · U
−εn
n U
εn
n · · · U
ε1
1 U
−1
0 ⊆ O.
Since WP is the union of the sets of the form U
−ε0
0 · · · U
−εn
n U
εn
n · · · U
ε0
0 , we have
proved the inclusion WP ⊆ O. �
Remark 3.4. Given the statement of the lemma just proved, it is worth re-
marking that the family {WP : P ∈ T (X, U)} does not in general constitute
a base at the identity in the group F(X, U), or indeed in any group topol-
ogy on the group Fa(X). In fact, for certain P ∈ T (X, U), one cannot find
Q ∈ T (X, U) with WQ · WQ ⊆ WP .
We will shortly show how a rather more elaborate family of sets constructed
using the sets WP do form an open base in F(X, U).
The character of free topological groups I 29
Let us first establish some other properties of the sets WP .
Lemma 3.5. Suppose that P, Q ∈ T (X, U) and g ∈ Fa(X). Then one can find
R ∈ T (X, U) such that WR ⊆ WP ∩ WQ and g
−1 · WR · g ⊆ WP .
Proof. The existence of R ∈ T (X, U) satisfying WR ⊆ WP ∩ WQ is immediate.
Hence it suffices to construct a U-covering tree R such that g−1 · WR · g ⊆ WP .
The existence of the required tree R is clear if g is the identity e of Fa(X).
If g 6= e, then it suffices to consider the case when g ∈ X ∪ X−1 and then
apply induction on the length of g in the general case. So we assume that
g ∈ X ∪ X−1. Since γP = {U : (U) ∈ P(0)} is a cover of X, there exist
U0 ∈ γP and ε0 = ±1 such that g ∈ U
ε0
0 . Put
R = {(U1, . . . , Un) : (U0, U1, . . . , Un) ∈ P, n ≥ 1}.
It is easy to see that R ∈ T (X, U), and we claim that g−1 · WR · g ⊆ WP .
Indeed, if (U1, . . . , Un) ∈ R and ε1, . . . , εn = ±1, then
g−1 · U−ε11 · · · U
−εn
n U
εn
n · · · U
ε1
1 · g ⊆ U
−ε0
0 U
−ε1
1 · · · U
−εn
n U
εn
n · · · U
ε1
1 U
ε0
0 ⊆ WP .
This proves the inclusion g−1 · WR · g ⊆ WP . �
Now we present our description of a neighborhood base at the identity of
F(X, U) in terms of U-covering trees. Again, we need some definitions. Let
s = 〈Pn : n ∈ N〉 ∈
NT (X, U) be a sequence of U-covering trees. Then we define
a set Os ⊆ Fa(X) as follows:
(3.3) Os =
⋃
n∈N
⋃
π∈Sn
WPπ(1) · · · WPπ(n),
where Sn is the group of permutations of the set {1, . . . , n}.
Theorem 3.6. The family Σ = {Os : s ∈
NT (X, U)} forms a base at the
identity e of the group F(X, U).
Proof. Our argument is close to that of [21, Th. 1.1]. It suffices to verify that
the following assertions are valid:
(a) Σ is a base for a group topology T ∗ on Fa(X);
(b) T ∗ is finer than the topology T of the group F(X, U);
(c) the two-sided uniformity V of the group G = (Fa(X), T
∗) induces on X
a uniformity coarser than U.
Since F(X, U) carries the finest group topology whose two-sided uniformity
induces on X the uniformity U, from (a)–(c) it follows that T ∗ = T . Let us
start with (a).
(a) To verify that Σ is a base at e for a group topology on Fa(X), it suffices
to show that Σ has the following four properties:
(1) for every U, V ∈ Σ there exists W ∈ Σ with W ⊆ U ∩ V ;
(2) for every U ∈ Σ there exists V ∈ Σ with V −1 · V ⊆ U;
(3) for every U ∈ Σ and g ∈ U there exists V ∈ Σ with V · g ⊆ U;
(4) for every U ∈ Σ and g ∈ Fa(X) there is V ∈ Σ such that g
−1 ·V ·g ⊆ U.
30 P. Nickolas and M. Tkachenko
We only check (2), (3) and (4), since (1) is immediate from Lemma 3.5. Note
that by definition, the set WP is symmetric for each P ∈ T (X, U), and so is
Os for each s ∈
NT (X, U). Let U ∈ Σ be arbitrary. Then U = Os for some
s ∈ NT (X, U), say s = 〈Pn : n ∈ N〉.
Let us check (2). By Lemma 3.5, we can find a sequence t = 〈Qn : n ∈ N〉 ∈
NT (X, U) such that WQn ⊆ WP2n−1 ∩ WP2n for each n ∈ N. We claim that
O−1t ·Ot ⊆ Os. Thus, we take m, n ∈ N and π ∈ Sm and ̺ ∈ Sn, and show that
WQπ(1) · · · WQπ(m) WQ̺(1) · · · WQ̺(n) ⊆ Os.
In fact, however, it suffices to assume that m = n here, since each WQp con-
tains e. Thus, let n ∈ N and π, ̺ ∈ Sn. Define σ ∈ S2n by σ(i) = 2π(i) if
1 ≤ i ≤ n and σ(i) = 2̺(i − n) − 1 if n < i ≤ 2n. Then from our definition of
σ and t it follows that
WQπ(1) · · · WQπ(n)WQ̺(1) · · · WQ̺(n)
⊆ WPσ(1) · · · WPσ(n) WPσ(n+1) · · · WPσ(2n) ⊆ Os.
This along with (3.3) implies that O−1t ·Ot = Ot ·Ot ⊆ Os, as claimed, and the
set V = Ot ∈ Σ is as required.
To verify (3), take an arbitrary g ∈ U = Os. Then g ∈ WPπ(1) · · · WPπ(k)
for some k ∈ N and π ∈ Sk. Put Qn = Pn+k for each n ∈ N and consider
t = 〈Qn : n ∈ N〉. Then t ∈
NT (X, U) and the set V = Ot satisfies V · g ⊆ U.
Indeed, for n ∈ N and σ ∈ Sn, define ̺ ∈ Sn+k by ̺(i) = σ(i) + k if i ≤ n and
̺(i) = π(i − n) if n < i ≤ n + k. Then
WQσ(1) · · · WQσ(n) · g ⊆ WQσ(1) · · · WQσ(n) WPπ(1) · · · WPπ(k)
= WP̺(1) · · · WP̺(n+k)
⊆ Os,
so that Ot · g ⊆ Os or, equivalently, V · g ⊆ U.
The verification of (4) is similar. Let g be an arbitrary element of Fa(X). By
Lemma 3.5, for every n ∈ N there exists Qn ∈ T (X, U) such that g
−1 ·WQn ·g ⊆
WPn. Put t = 〈Qn : n ∈ N〉. Then t ∈
NT (X, U) and g−1 · Ot · g ⊆ Os. Indeed,
if n ∈ N and π ∈ Sn, then we have
g−1 · WQπ(1) · · · WQπ(n) · g = (g
−1 · WQπ(1) · g) · · · (g
−1 · WQπ(n) · g)
⊆ WPπ(1) · · · WPπ(n)
⊆ Os.
This implies that g−1 · Ot · g ⊆ Os, and so the set V = Ot ∈ Σ is as required.
We conclude, therefore, that Σ is a base for a group topology T ∗ on Fa(X).
This proves (a).
(b) Let O be an arbitrary neighborhood of e in F(X, U). Choose a sequence
〈Vn : n ∈ ω〉 of open symmetric neighborhoods of e in F(X, U) such that
V0 ⊆ O and V
3
n+1 ⊆ Vn for each n ∈ ω. By Lemma 3.3, for every n ∈ N there
The character of free topological groups I 31
exists Pn ∈ T (X, U) such that WPn ⊆ Vn. Note that if n ∈ N and π ∈ Sn, then
Vπ(1) · · · Vπ(n) ⊆ V0 by Lemma 1.3 of [21]. This immediately implies that
WPπ(1) · · · WPπ(n) ⊆ Vπ(1) · · · Vπ(n) ⊆ V0,
whence it follows that Os ⊆ V0 ⊆ O, where s = 〈Pn : n ∈ N〉. This proves that
the topology T ∗ generated by the family Σ is finer than T .
(c) Let V = Os, where s = 〈Pn : n ∈ N〉 ∈
NT (X, U) is arbitrary. Put
W =
⋃
(U)∈P1(0)
U × U.
Then W ∈ U, and from the definition of WP1 it follows that
W ⊆ {(x, y) ∈ X × X : x−1y ∈ WP1, xy
−1 ∈ WP1 }.
Since WP1 ⊆ Os, we conclude that
W ⊆ {(x, y) ∈ X × X : x−1y ∈ Os, xy
−1 ∈ Os}.
Therefore, the restriction of the two-sided uniformity of the group (Fa(X), T
∗)
to the set X ⊆ Fa(X) is coarser than U. The proof is complete. �
Suppose that (X, U) is an ℵ0-narrow uniform space. Roughly speaking,
Theorem 3.6 and Lemma 3.3 show that one has to use only countably many
elements of U to produce a basic neighborhood of the identity in F(X, U). We
use this fact as well as the next two lemmas in the proof of Theorem 3.10.
For any uniform space (X, U), we denote by C(U) the family of all U-uniform
covers of X, and for γ, λ ∈ C(U), we write γ ≺ λ provided that γ refines λ.
Lemma 3.7. If (X, U) is a uniform space, then each of the partially ordered
sets (U, ⊆) and (C(U), ≺) admits an order-preserving mapping onto a dense
subset of the other, and hence d(U, ⊆) = d(C(U), ≺).
Proof. For each U ∈ U, we set γU = {B(x, U) : x ∈ X}, where we recall that
B(x, U) denotes the set {y ∈ X : (x, y) ∈ U} for each x ∈ X. Clearly, γU is a
U-uniform cover of X. It is easy to see that the mapping U 7→ γU of (U, ⊆) to
(C(U), ≺) is order-preserving, and the set {γU : U ∈ U} is obviously dense in
(C(U), ≺).
Conversely, for every γ ∈ C(U), put Wγ =
⋃
{V × V : V ∈ γ}. It is clear
that the mapping γ 7→ Wγ of (C(U), ≺) to (U, ⊆) is order-preserving. To show
that {Wγ : γ ∈ C(U)} is dense in (U, ⊆), let U ∈ U and take V ∈ U which is
symmetric and satisfies 2V ⊆ U. Then
WγV =
⋃
x∈X
B(x, V ) × B(x, V ),
and it is easy to check that WγV ⊆ U, as required.
The equality d(U, ⊆) = d(C(U), ≺) now follows from (a version of) Lemma
1.1. �
32 P. Nickolas and M. Tkachenko
Suppose that s = 〈γn : n ∈ ω〉 and t = 〈λn : n ∈ ω〉 are two sequences of
U-uniform covers of X, that is, that s, t ∈ ωC(U). We write s ≺ t if γn ≺ λn
for each n ∈ ω. This defines the partially ordered set (ωC(U), ≺), and the next
result follows directly from Lemma 3.7 and Theorem 2.10.
Corollary 3.8. d(ωC(U), ≺) = d(ωU, ≤) = D(P(X, U), ≤) for every uniform
space (X, U).
Our next step is to show that the difference between the characters of the
groups A(X, U) and F(X, U) cannot be too big for any ℵ0-narrow uniform
space (X, U) (see Theorem 3.10). First, we deal with the special case when the
uniform space has a countable base. Technically, this is the most difficult part
of the work.
Lemma 3.9. Let an ℵ0-narrow uniform space (X, U) satisfy w(X, U) ≤ ℵ0.
Then χ(F(X, U)) ≤ d.
Proof. (I) Let {Un : n ∈ ω} be a countable base for the uniformity U. Since
(X, U) is ℵ0-narrow, we can choose a sequence Γ = 〈γn : n ∈ ω〉 of countable
U-uniform covers of X such that γn+1 ≺ γn and
⋃
{V × V : V ∈ γn} ⊆ Un
for each n ∈ ω. Note that the space (X, U) is pseudometrizable (and hence
paracompact, using the term without the assumption of separation which most
authors include in the definition of paracompactness), and so each cover γn
can be additionally chosen to be locally finite. We can further assume that if
U ∈ γm and V ∈ γn where m > n, then U does not properly contain V , that
is, that U ⊇ V implies U = V .
(II) We define an order ≤t on the collection T (X, U) of U-covering trees
by specifying that for P, Q ∈ T (X, U), we have P ≤t Q if WP ⊇ WQ (see
equation (3.1)). Clearly, ≤t is a quasi-order on T (X, U), though not a partial
order. Denote by T (Γ) ⊆ T (X, U) the set of all U-covering trees P with the
property that γP ∈ Γ and γP (x) ∈ Γ for each x ∈ P (see (iii) and (iv) of
Definition 3.2). Also, set γ∗ =
⋃
n∈ω γn. Then our definition of the sequence
Γ = 〈γn : n ∈ ω〉 implies that every V ∈ γ
∗ is contained only in finitely many
distinct elements of γ∗.
Claim 1. T (Γ) is a dominating subset of (T (X, U), ≤t).
For subsets U1, . . . , Un−1, Un of X, we write
π̂(U1, . . . , Un−1, Un) = (U1, . . . , Un−1)
if n ≥ 2, and we write
π(U1, . . . , Un−1, Un) = Un
if n ≥ 1.
To prove the claim, let P ∈ T (X, U). We construct a tree Q as follows.
Since γP is a U-uniform cover of X, there is δ ∈ Γ such that δ ≺ γP . We
set Q(0) = {x ∈ (γ∗)1 : π(x) ∈ δ}. Since δ ≺ γP , we can choose a function
p0 : Q(0) → P(0) such that π(x) ⊆ π(p0(x)) for all x ∈ Q(0). Now for each
x ∈ Q(0), we pick δx ∈ Γ such δx ≺ γP (p0(x)). Then we put Q(1) = {y ∈
The character of free topological groups I 33
(γ∗)2 : x = π̂(y) ∈ Q(0), π(y) ∈ δx}. Since δx ≺ γP (p0(x)) for each x ∈ Q(0),
we can choose a function p1 : Q(1) → P(1) such that π̂(p1(y)) = p0(π̂(y)) and
π(y) ⊆ π(p1(y)) for all y ∈ Q(1). Now for each y ∈ Q(1), we pick δy ∈ Γ such
δy ≺ γP (p1(y)), and then put Q(2) = {z ∈ (γ
∗)3 : y = π̂(z) ∈ Q(1), π(z) ∈ δy}.
Continuing in this way, we finally obtain the tree Q =
⋃
n∈ω Q(n), ordered
according to (ii) of Definition 3.2. We clearly have Q ∈ T (Γ). Also, if u =
(U0, . . . , Un) ∈ Q(n), then, by construction, pn(u) ∈ P(n), and if we write
pn(u) = (V0, . . . , Vn), say, then we have U0 ⊆ V0, . . . , Un ⊆ Vn, so that each
product in the union of the form (3.1) defining WQ is contained in some product
in the corresponding union defining WP , giving WQ ⊆ WP , and hence P ≤t Q,
as required to prove the claim.
(III) If we define
E = {(V0, . . . , Vn) : V0, . . . , Vn ∈ γ
∗, n ∈ ω},
then it is clear that |E| ≤ ℵ0. Let F =
Eω be the family of all mappings from
E to ω. Define a partial order ≤e on E as follows: for p = (U0, . . . , Uk), q =
(V0, . . . , Vl) ∈ E, we define p ≤e q if and only if k ≤ l and Ui ⊇ Vi for
i = 0, . . . , k. Now we define a “strong” partial order ≤s on F =
Eω, as follows
(the name and notation distinguish the relation from another that we will define
later on the same set). For f, g ∈ F, we write f ≤s g if f = g or if p ≤e q in E
always implies f(p) ≤ g(q) in ω. It is straightforward to verify that ≤s is indeed
a partial order. We also define a partial order ≪s on ω × F coordinate-wise,
using the usual order ≤ on ω and the order ≤s on F.
We define a function Π: ω × F → T (X, U). For a pair (m, f) ∈ ω × F, the
U-covering tree P = Π(m, f) is defined as follows. Define the initial level of P
by P(0) = {(U) : U ∈ γm}. Suppose that we have defined P(n) ⊆ (γ
∗)n+1 for
some n ∈ ω. Given an element p = (U0, . . . , Un) ∈ P(n), put
p+ = {(U0, . . . , Un, V ) : V ∈ γf(p)}
and then set
P(n + 1) =
⋃
{p+ : p ∈ P(n)}.
To finish the construction, we put P =
⋃
n∈ω P(n) and define the partial order
≤ on P as in Definition 3.2. Note that the tree P = Π(m, f) is in fact an
element of T (Γ) ⊆ T (X, U).
Claim 2. The mapping Π is order-preserving as a mapping from (ω×F, ≪s)
to (T (X, U), ≤t).
Thus, we suppose that m, n ∈ ω and m ≤ n and that f, g ∈ F and f ≤s g,
and we show that WΠ(n,g) ⊆ WΠ(m,f). Indeed, by (3.1), WΠ(n,g) is the union
over all k of all the sets of the form
V −ε00 · V
−ε1
1 · · · V
−εk
k
· V εk
k
· · · V ε11 · V
ε0
0 ,
where
V0 ∈ γn, V1 ∈ γg(q1), V2 ∈ γg(q2), . . . , Vk ∈ γg(qk),
34 P. Nickolas and M. Tkachenko
where
q1 = (V0), q2 = (V0, V1), . . . , qk = (V0, V1, . . . , Vk−1),
and where ε0, ε1, . . . , εk = ±1. Fix one such set. Since γn refines γm, there
exists U0 ∈ γm such that V0 ⊆ U0. Put p1 = (U0). Then from f ≤s g it
follows that f(p1) ≤ g(q1). Hence, γg(q1) refines γf(p1), so there exists U1 ∈
γf(p1) such that V1 ⊆ U1. From f ≤s g it follows that f(p2) ≤ g(q2), where
p2 = (U0, U1). Again, γg(q2) refines γf(p2), so there exists U2 ∈ γf(p2) such
that V2 ⊆ U2. Continuing this way, we finally obtain Uk ∈ γf(pk) such that
Vk ⊆ Uk, where pk = (U0, U1, . . . , Uk−1). Clearly, pk ∈ Π(m, f), so that the set
U−ε00 · U
−ε1
1 · · · U
−εk
k
· Uεk
k
· · · Uε11 · U
ε0
0 is a summand in the union of the form
(3.1) corresponding to WΠ(m,f). By construction, we have Vi ⊆ Ui for each
i = 0, . . . , k, whence it follows that
V −ε00 · · · V
−εk
k
· V εk
k
· · · V ε00 ⊆ U
−ε0
0 · · · U
−εk
k
· Uεk
k
· · · Uε00 .
We conclude, therefore, that WΠ(n,g) ⊆ WΠ(m,f), proving our claim.
We claim next that Π(ω × F) = T (Γ). That Π(ω × F) ⊆ T (Γ) is clear. For
the reverse inclusion, let P ∈ T (Γ). Now γP ∈ Γ, so that γP = γn ∈ Γ for some
n ∈ N. Also, for all x ∈ P , we have γP (x) ∈ Γ, so that γP (x) = γnx ∈ Γ for
some nx ∈ N. Define f : E → ω, that is, f ∈ F, by setting f(x) = nx for all
x ∈ P and f(x) = 0 for all x /∈ P . Then it is easy to see that Π(n, f) = P ,
proving that Π(ω × F) ⊇ T (Γ), and hence our claim.
We now consider the quasi-ordered sets
(N(ω × F), ≪∗s) and (
NT (X, U), ≤∗t ),
where the orders ≪∗s and ≤
∗
t are defined coordinate-wise in terms of ≪s and
≤t, respectively. We likewise consider the mapping
Π∗ : (N(ω × F), ≪∗s) → (
NT (X, U), ≤∗t )
defined coordinate-wise in terms of Π. It is immediate from what we have
shown above that the mapping Π∗ is order-preserving, and maps N(ω × F) to
a dominating subset of (NT (X, U), ≤∗t ). Thus, we conclude that
(3.4) D(NT (X, U), ≤∗t ) ≤ D(
N(ω × F), ≪∗s).
(IV) We wish to obtain a different expression for the right-hand side of (3.4).
To this end, we define a “weak” partial order ≤w on F =
Eω, as follows. For
f, g ∈ F, we write f ≤w g if f(p) ≤ g(p) for each p ∈ E. It is clear that ≤w is
a partial order. It is also clear that f ≤s g implies f ≤w g. Now we define ≪w
on ω × F and then ≪∗w on
N(ω × F) in exact analogy to the earlier definitions
of ≪s and ≪
∗
s.
Claim 3. D(N(ω × F), ≪∗s) = D(
N(ω × F), ≪∗w).
Indeed, it is clear that D(N(ω × F), ≪∗w) ≤ D(
N(ω × F), ≪∗s), since every
dominating set in (N(ω×F), ≪∗s) remains dominating in (
N(ω×F), ≪∗w). Next,
we have to verify that D(N(ω × F), ≪∗s) ≤ D(
N(ω × F), ≪∗w). For q ∈ E, put
E(q) = {p ∈ E : p ≤e q}.
The character of free topological groups I 35
From our assumptions about the covers γn, it follows that E(q) is finite for
every q ∈ E. In fact, suppose that there exists q = (V0, . . . , Vl) ∈ E such that
E(q) is infinite. Then for some k ≤ l, the set
Ek(q) = {p ∈ E : p = (U0, . . . , Uk), p ≤e q}
is infinite. Then for some i ≤ k, we can find a sequence of elements
pn = (U
(n)
0 , . . . , U
(n)
i , . . . , U
(n)
k
) ∈ Ek(q)
for n ∈ N such that the sets U
(n)
i are distinct for all n and such that U
(n)
i ⊇ Vi
for all n. If n0 ∈ N is such that Vi ∈ γn0, then U
(n)
i must properly contain Vi
for infinitely many n, and it follows that there is n1 ≤ n0 such that infinitely
many of the U
(n)
i are in γn1, contradicting the local finiteness of γn1. This
contradiction shows that E(q) is finite, as claimed. It is clear furthermore that
each E(q) is also non-empty.
Now for every f ∈ F and q ∈ E, put
f̃(q) = max{f(p) : p ∈ E(q)},
noting that by the argument above the value f̃(q) is defined validly. Therefore,
we obtain a function f̃ : E → ω, that is, f̃ ∈ F. It is easy to see that the
mapping f 7→ f̃ from (F, ≤w) to (F, ≤s) is order-preserving, and from our
definition of the order ≤s and the function f̃, it is also easily checked that
f ≤s f̃ for each f ∈ F, so that the image of the mapping is a dominating
subset of (F, ≤s).
For ϕ ∈ N(ω × F), we have ϕ(n) ∈ ω × F for each n ∈ N, and we have
ϕ(n) = (π1(ϕ(n)), π2(ϕ(n))), where π1 and π2 are the projections from ω × F
into ω and F, respectively. We then define ϕ̃ ∈ N(ω × F) by setting ϕ̃(n) =
(π1(ϕ(n)), ˜π2(ϕ(n))) for each n ∈ N. Then by the argument above, the mapping
ϕ 7→ ϕ̃ from (N(ω × F), ≪∗w) to (
N(ω × F), ≪∗s) is order-preserving and has as
its image a dominating set, proving the inequality D(N(ω×F), ≪∗s) ≤ D(
N(ω×
F), ≪∗w), and hence our claim.
(V) We can now complete the proof of the lemma. Equation (3.3) (immedi-
ately preceding Theorem 3.6) defines for us a mapping s 7→ Os from
NT (X, U) to
N(e), the family of open neighborhoods of the identity e in F(X, U). Further, it
is immediate from the relevant definitions that this mapping is order-reversing
from (NT (X, U), ≤∗t ) to (N(e), ⊆), where ≤
∗
t is defined coordinate-wise in terms
of ≤t. Moreover, rephrased in this terminology, Theorem 3.6 states that the
mapping has a dense subset of (N(e), ⊆) as its image. Hence we have
(3.5) d(N(e), ⊆) ≤ D(NT (X, U), ≤∗t ).
36 P. Nickolas and M. Tkachenko
In addition, we have obvious order-isomorphisms as follows:
(N(ω × F), ≪∗w)
∼= (Nω, ≤) × (NF, ≤∗w)
∼= (Nω, ≤) × (N×Eω, ≤)
∼= (
ωω, ≤) × (ωω, ≤)
∼= (ωω, ≤)(3.6)
(where ≤∗w is the coordinate-wise extension of ≤w from F to
NF). Therefore,
from (3.5), (3.4), Claim 3 and (3.6), we have
χ(F(X, U)) = d(N(e), ⊆)
≤ D(NT (X, U), ≤∗t )
≤ D(N(ω × F), ≪∗s)
= D(N(ω × F), ≪∗w)
= D(ωω, ≤)
= d.
This finishes the proof. �
The conclusion of the next theorem will be strengthened in Theorem 3.15.
Theorem 3.10. If (X, U) is an ℵ0-narrow uniform space, then χ(F(X, U)) ≤
d · χ(A(X, U)).
Proof. According to Theorem 2.3 and Theorem 2.10, we have
(3.7) χ(A(X, U)) = D(P(X, U), ≤) = d(ωU, ≤),
where P(X, U) is the family of uniformly continuous pseudometrics on (X, U)
bounded by 1. Therefore, all we have to verify is that
(3.8) χ(F(X, U)) ≤ d · d(ωU, ≤).
Denote by S the family of all sequences V = 〈Un : n ∈ ω〉 ∈
ωU such that
3Un+1 ⊆ Un for each n ∈ ω. It is clear that S is dense in (
ωU, ≤), whence
d(S, ≤) = d(ωU, ≤). Choose a dense subset D of (S, ≤) of the minimal car-
dinality. Then D is also dense in (ωU, ≤). Note that the set of terms of the
sequence V is a base for a (non-Hausdorff) uniformity Ṽ on X for each V ∈ S,
and hence Lemma 3.9 implies that χ(F(X, Ṽ)) ≤ d. So, for every V ∈ D, we
can find a base B(V) at the identity in F(X, Ṽ) such that |B(V)| ≤ d. It is clear
that the family B =
⋃
V∈D B(V) satisfies |B| ≤ d · d(
ωU, ≤), and we claim that
B is a base at the identity in the group F(X, U).
By assumption, the space (X, U) is ℵ0-narrow, so the group F(X, U) is ℵ0-
narrow according to [2, Lemma 3.2] or [6]. Hence the topology of F(X, U)
is generated by continuous homomorphisms to second countable topological
groups (see [6] or [25, Lemma 3.7]). In other words, given a neighborhood U of
the identity in F(X, U), one can find a continuous homomorphism f : F(X, U) →
G to a second countable topological group G and an open neighborhood V of
the identity in G such that f−1(V ) ⊆ U. Choose a countable base {Vn : n ∈ ω}
The character of free topological groups I 37
at the identity of G such that V0 = V and V
3
n+1 ⊆ Vn for each n ∈ ω. For
n = 0, 1, . . ., put
(3.9) Un = {(x, y) ∈ X × X : f(x)
−1 · f(y) ∈ Vn, f(x) · f(y)
−1 ∈ Vn}.
Evidently 3Un+1 ⊆ Un for each n ∈ ω, so that V = 〈Un : n ∈ ω〉 ∈ S. Since
D is dense in (D, ≤), we can assume that V ∈ D. Our choice of V (see (3.9))
guarantees that the restriction of f to X is a uniformly continuous mapping of
(X, Ṽ) to (G, ∗V∗), where ∗V∗ is the two-sided uniformity of the group G. Hence
the homomorphism f : F(X, Ṽ) → G remains continuous. Take an element
W ∈ B(V) such that f(W) ⊆ V . Then W ⊆ f−1(V ) ⊆ U, and hence B is a
base at the identity in F(X, U). This proves (3.8) and the theorem. �
Combining Corollary 2.17 and Theorem 3.10, we obtain the following result,
which will be given its final form in Theorem 3.15.
Corollary 3.11. If an ℵ0-narrow Hausdorff uniform space (X, U) contains an
infinite precompact set, then χ(A(X, U)) = χ(F(X, U)).
Now we proceed to show, in Theorem 3.15, that the existence of an infinite
precompact set in (X, U) can be omitted in the assumptions of Corollary 3.11.
The main additional information we need is given in the following result.
Lemma 3.12. If (X, U) is an ℵ0-narrow uniform P-space, then the group
F(X, U) has a base at the identity consisting of open normal subgroups. In
particular, the topology of F(X, U) is generated by Graev’s extensions of the
uniformly continuous pseudometrics on (X, U).
Proof. The first assertion follows from the uniform analog of [22, Th. 4]. For
the second assertion, suppose that (X, U) is a uniform P-space, and let U
be an open neighborhood of the identity e in F(X, U). Then there exists an
open normal subgroup V of F(X, U) with V ⊆ U. Consider the open cover
γ = {X ∩ xV : x ∈ X} of the space X. It is clear that γ is a U-uniform
cover. Since V is a subgroup of F(X, U), the family γ is a partition of X,
i.e., every two elements of γ are disjoint or coincide. Define a pseudometric
̺ on X by setting ̺(x, y) = 0 if x, y ∈ O for some O ∈ γ, and ̺(x, y) = 1
otherwise. Clearly, ̺ is uniformly continuous. Let ̺̂ be Graev’s extension of
̺ to the maximal invariant pseudometric on Fa(X) (see [5, Section 3]). Then
the set
W̺ = {g ∈ Fa(X) : ̺̂(g, e) < 1}
is an open neighborhood of e in F(X, U) by the continuity of ̺̂ on F(X, U). It
remains to show that W̺ ⊆ V .
From the fact that the pseudometric ̺ is {0, 1}-valued, it is immediate from
Graev’s construction that the extension ̺̂ is integer-valued. We therefore have
W̺ = {g ∈ Fa(X) : ̺̂(g, e) = 0}.
38 P. Nickolas and M. Tkachenko
(Indeed, it follows that W̺ is an open normal subgroup of F(X, U).) Further,
Graev’s construction shows straightforwardly that for g ∈ W̺, there exist (non-
reduced) representations
g = xε11 · · · x
ε2n
2n and e = y
ε1
1 · · · y
ε2n
2n
of g and e, for some n ∈ N, some x1, . . . x2n, y1, . . . y2n ∈ X, and ε1, . . . ε2n =
±1, such that ̺(xi, yi) = 0 for i = 1, . . . , 2n. Now we clearly have y
εi
i y
εi+1
i+1 = e
for some i, from which it follows that xεii x
εi+1
i+1 ∈ V . Set g1 = x
ε1
1 · · · x
εi−1
i−1
and g2 = x
εi+2
i+2 · · · x
ε2n
2n , and put ĝ = g1g2. Note that ĝ ∈ W̺. If we assume
inductively that ĝ ∈ V , we also have g2g1 ∈ V by the normality of V , and
then we have g−11 gg1 = x
εi
i x
εi+1
i+1 g2g1 ∈ V , from which we have g ∈ V , again by
normality. It follows by induction that W̺ ⊆ V , as required. �
This allows us to extend Theorem 2.13 to the non-abelian case, assuming
additionally that our uniform space is ℵ0-narrow.
Theorem 3.13. If (X, U) is an ℵ0-narrow uniform P-space, then χ(F(X, U)) =
w(X, U).
Proof. By Lemma 3.12, the topology of F(X, U) is generated by Graev’s exten-
sions of the uniformly continuous pseudometrics on (X, U). It follows, there-
fore, that χ(F(X, U)) ≤ D(P(X, U), ≤) = w(X, U). However, Theorem 2.3 and
Lemma 3.1 together imply that D(P(X, U), ≤) = χ(A(X, U)) ≤ χ(F(X, U)).
Combining these inequalities, we obtain the required conclusion. �
In contrast to Theorem 2.13, the assumption of ℵ0-narrowness cannot be
removed in Theorem 3.13, as is shown by the following example.
Example 3.14. For every cardinal τ > ℵ1, there exists a Hausdorff uniform
P-space (X, U) such that w(X, U) = ℵ1 < τ < χ(F(X, U)).
Indeed, let X = L⊕D be the topological sum of the one-point Lindelöfication
L of a discrete space Y with |Y | = ℵ1 and a discrete space D of cardinality
τ > ℵ1. Denote by x0 the unique non-isolated point of L (and of X). Then a
base of open neighborhoods of x0 in L (and in X) consists of the sets L \ C,
where C is an arbitrary countable subset of Y . Since |Y | = ℵ1, it is easy to see
that χ(x0, L) = χ(x0, X) = ℵ1. Let U be the fine uniformity of the space X.
Then a basic entourage of the diagonal ∆ in X × X has the form
UC = {(x, y) ∈ L × L : x, y ∈ L \ C} ∪ ∆,
where C ⊆ Y is countable. Therefore, w(X, U) = ℵ1. Let us show that
χ(F(X, U)) ≥ τ.
For every a ∈ D, put La = a
−1x−10 La, and consider the subspace Z =⋃
a∈D La of F(X, U)
∼= F(X). Apply an argument similar to that in [1,
Prop. 3.2] to show that Z is homeomorphic to the fan V (τ) obtained from
the topological sum T of τ copies of L by identifying to a point the set T ′
of all non-isolated points of T . Since each of the τ distinct spines of the fan
V (τ) is homeomorphic to L, a straightforward diagonal argument implies that
τ < χ(e, Z) ≤ χ(F(X)).
The character of free topological groups I 39
Finally, we have the main result of this section.
Theorem 3.15. The equality χ(A(X, U)) = χ(F(X, U)) holds for every ℵ0-
narrow uniform space (X, U).
Proof. If (X, U) is a uniform P-space, then the required equality is given by
Theorems 2.13 and 3.13, while if (X, U) is not a uniform P-space, then the
equality follows from Theorems 2.15 and 3.10 and Lemma 3.1. �
The above theorem has several applications; the following four are immedi-
ate.
Corollary 3.16. χ(A(X)) = χ(F(X)) for every ℵ0-narrow space X.
Using Corollary 2.22, we have in particular:
Corollary 3.17. If X is an infinite compact metrizable space, then χ(F(X)) =
d.
Corollary 3.18. Suppose that a space X contains a dense Lindelöf subspace.
Then χ(A(X))=χ(F(X)).
Proof. It is easy to see that the space X is ℵ0-narrow. Indeed, let Y be a
dense Lindelöf subspace of X. If γ is a discrete family of non-empty open sets
in X, then the family µ = {U ∩ Y : U ∈ γ} is a discrete family of non-empty
open subsets of Y . However, every such family of subsets of Y is countable, so
|γ| = |µ| ≤ ℵ0. Now the necessary conclusion follows from Corollary 3.16. �
Clearly, every space of countable cellularity is ℵ0-narrow. Therefore, we have
the following.
Corollary 3.19. If a space X has countable cellularity, then χ(A(X)) =
χ(F(X)).
We believe that Corollary 3.21 below compared with Theorem 2.3 or Corol-
lary 2.4 gives a more comprehensive expression for the character of the groups
F(X) and A(X) on a Lindelöf space X. It is based on a simple relation between
dense subsets of (ωUX, ≤) and the character of the diagonal in (X ×ω)
2, where
UX is the fine uniformity of the space X and the set ω carries the discrete
topology.
We recall that if s = {Un : n ∈ ω} and t = {Vn : n ∈ ω} are elements of
ωU,
where U is a uniformity on some set, then s ≤ t means that Un ⊆ Vn for each
n ∈ ω.
Lemma 3.20. If X is a paracompact topological space and UX is the fine
uniformity on X, then d(ωUX, ≤) = χ∆(X × ω).
Proof. We can assume that X is not discrete—otherwise, the equality becomes
trivial. Denote by UY the fine uniformity on Y ≡ X ×ω. The paracompactness
of X implies that every neighborhood of the diagonal ∆X in X
2 belongs to the
fine uniformity UX, and similarly for the diagonal ∆Y in Y
2. The family of all
neighborhoods of ∆Y in Y
2 contains a base which can be naturally identified
40 P. Nickolas and M. Tkachenko
with the family of all sequences {Un : n ∈ ω}, where Un ∈ UX for each n ∈ ω.
It is now immediate that d(ωUX, ≤) = χ∆(X × ω). �
Finally, Theorem 3.15, Corollary 2.4, Theorem 2.10 (with U = UX) and
Lemma 3.1 imply the following result.
Corollary 3.21. Let X be a Lindelöf space. Then χ(F(X)) = χ(A(X)) =
χ∆(X × ω).
It may be worth remarking that the conclusion of the corollary holds in
particular if X is a kω-space or a compact space. In the sequel [14] to this
paper, we will investigate the compact case in more depth.
References
[1] A.V. Arhangel’skii, O.G. Okunev, and V.G. Pestov, Free topological groups over
metrizable spaces, Topology Appl. 33 (1989), 63–76.
[2] D. Dikranjan and M.G. Tkachenko, Weakly complete free topological groups,
Topology Appl. 112 (2001), no. 3, 259–287.
[3] E.K. Douwen, van, The Integers and Topology, In: Handbook of Set-Theoretic
Topology, K. Kunen and J.E. Vaughan, Eds., Elsevier Science Publ. B. V., 1984,
pp. 111–167.
[4] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
[5] M. I. Graev, Free topological groups, In: Topology and Topological Algebra,
Translations Series 1, vol. 8 (1962), pp. 305–364, American Mathematical Society.
Russian original in: Izvestiya Akad. Nauk SSSR Ser. Mat. 12 (1948), 279–323.
[6] I. Guran, On topological groups close to being Lindelöf, Soviet Math. Dokl. 23
(1981), 173–175.
[7] C. Hernández, D. Robbie, and M. Tkachenko, Some properties of o-bounded and
strictly o-bounded groups, Applied General Topology 1 (2000), 29–43.
[8] R. Hodel, Cardinal functions I, In: Handbook of Set-Theoretic Topology,
K. Kunen and J.E. Vaughan, Eds., Elsevier Science Publ. B. V., 1984, pp. 1–61.
[9] M. S. Khan, S. A. Morris, and P. Nickolas, Local invariance of free topological
groups, Proc. Edinburgh Math. Soc. 29 (1986), 1–5.
[10] D. Marxen, Neighborhoods of the identity of the free abelian topological groups,
Math. Slovaca 26 (1976), 247–256.
[11] S. A. Morris and P. Nickolas, Locally invariant topologies on free groups, Pacific
J. Math. 103 (1982), 523–537.
[12] T. Nakayama, Note on free topological groups, Proc. Imp. Acad. Sci. 19 (1943),
471–475.
[13] P. Nickolas, Free topological groups and free products of topological groups, PhD.
thesis, University of New South Wales, Australia, 1976.
[14] P. Nickolas and M. G. Tkachenko, The character of free topological groups II,
this volume.
[15] E. C. Nummela, Uniform free topological groups and Samuel compactifications,
Topology Appl. 13 (1982), 77–83.
[16] V. G. Pestov, Neighborhoods of unity in free topological groups, Mosc. Univ.
Math. Bull. 40 (1985), 8–12.
The character of free topological groups I 41
[17] V. G. Pestov, Topological groups: where to from here?, In: Proceedings, 14th
Summer Conf. on General Topology and its Appl. (C.W. Post Campus of Long
Island University, August 1999), Topology Proceedings 24 (1999), 421–502.
[18] V. I. Ponomarev and V. V. Tkachuk, Countable character of X in βX in com-
parison with countable character of the diagonal of X in X × X, Vestnik Mosk.
Univ. no. 5 (1987), 16–19 (in Russian).
[19] O. V. Sipacheva and M. G. Tkachenko, Thin and bounded subsets of free topo-
logical groups, Topology Appl. 36 (1990), 143–156.
[20] M. G. Tkachenko, On the completeness of free Abelian topological groups, Soviet
Math. Dokl. 27 (1983), 341–345. Russian original in: Dokl. AN SSSR 269 (1983),
299–303.
[21] M. G. Tkachenko, On topologies of free groups, Czech. Math. J. 33 (1984), 57–69.
[22] M. G. Tkachenko, Some properties of free topological groups, Math. Notes 37
(1985), 62–66. Russian original in: Mat. Zametki 37 (1985), 110–118.
[23] M. G. Tkachenko, On the linearity of the Graev extension of pseudometrics, In:
Algebraic and logical constructions, Tver. Gos. Univ., Tver 1994, pp. 84–90 (in
Russian).
[24] M. G. Tkachenko, On group uniformities on square of a space and extending
pseudometrics II, Bull. Austral. Math. Soc. 52 (1995), 41–61.
[25] M. G. Tkachenko, Introduction to topological groups, Topology Appl. 86 (1998),
179–231.
[26] K. Yamada, Characterizations of a metrizable space X such that every An(X) is
a k-space, Topology Appl. 49 (1993), 75–94.
Received November 2002
Accepted June 2004
P. Nickolas (peter−nickolas@uow.edu.au)
Department of Mathematics and Applied Statistics, University of Wollongong,
NSW 2522, Australia
M. Tkachenko (mich@xanum.uam.mx)
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av.
San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, México,
D.F.