CoRaTriAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 7, No. 1, 2006

pp. 109-124

Making group topologies with, and without,
convergent sequences∗

W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

Abstract.
(1) Every infinite, Abelian compact (Hausdorff) group K admits 2|K|-
many dense, non-Haar-measurable subgroups of cardinality |K|. When
K is nonmetrizable, these may be chosen to be pseudocompact.

(2) Every infinite Abelian group G admits a family A of 22
|G|

-many
pairwise nonhomeomorphic totally bounded group topologies such that
no nontrivial sequence in G converges in any of the topologies T ∈ A.
(For some G one may arrange w(G, T ) < 2|G| for some T ∈ A.)

(3) Every infinite Abelian group G admits a family B of 22
|G|

-many
pairwise nonhomeomorphic totally bounded group topologies, with
w(G, T ) = 2|G| for all T ∈ B, such that some fixed faithfully indexed
sequence in G converges to 0G in each T ∈ B.

2000 AMS Classification: Primary: 22A10, 22B99, 22C05, 43A40, 54H11.
Secondary: 03E35, 03E50, 54D30, 54E35.

Keywords: Haar measure, dual group, character, pseudocompact group,
totally bounded group, maximal topology, convergent sequence, torsion-free
group, torsion group, torsion-free rank, p-rank, p-adic integers.

1. Introduction

1.1. Historical background. Not long after E. Čech and M. H. Stone associ-
ated with each Tychonoff space X its maximal compactification β(X) (the so-
called Stone-Čech compactification), it was noted, denoting by ω the countably
infinite discrete space, that β(ω) contains no nontrivial convergent sequence.
This observation stimulated Efimov [10] to pose in 1969 a question which in its
full generality remains unsolved today: Does every compact Hausdorff space

∗ Portions of this paper were presented by the first-listed author at the 2004 Annual
Meeting of the American Mathematical Society (Phoenix, January, 2004).



110 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

contain either a copy of β(ω) or a nontrivial convergent sequence? (In mod-
els of ⋄ the answer is negative [12]. See [34] for several additional relevant
references.) The present paper is concerned with topological groups. In that
context, a correct and natural companion to Efimov’s question is this: Given
a class C of topological groups, does every group in C contain a nontrivial con-
vergent sequence? There is an extensive literature on questions of this form.
Here are some samples of both positive and negative results.

Positive (a) According to a result to which Šapirovskĭı, Gerlits and Efimov
have contributed (see [35] for historical details and for an “elementary” proof),
every infinite compact group K contains topologically a copy of the general-
ized Cantor space {0, 1}w(K), hence contains a convergent sequence; (b) As-
suming GCH, Malykhin and Shapiro [27] showed that every totally bounded
group G with w(G) < (w(G))ω contains a nontrivial convergent sequence;
(c) Raczkowski [30], [31] and others [1], [29], [44] have shown that for every
suitably fast-growing sequence xn ∈ Z there is a totally bounded group topol-
ogy on Z with respect to which xn → 0.

Negative. (a) Glicksberg [16] showed that when a locally compact Abelian
group (G,T ) is given its associated Bohr topology (that is, the weak topology

induced on G by (̂G,T )), no new compact sets are created; in particular, as
shown earlier by Leptin [26], the topology induced on a (discrete) Abelian group
by Hom(G, T) has no infinite compact subsets, in particular has no nontrivial
convergent sequence; (b) there are infinite pseudocompact topological groups
containing no nontrivial convergent sequence [36]; see also the more recent
papers [14] and [15] and the literature cited there for results in the same vein.

Perhaps the most celebrated unsolved question in this area of mathematics
is this: Does there exist in ZFC a countably compact topological group with no
nontrivial convergent sequence? (Many examples are known in augmented ax-
iom systems. See for example the constructions of van Douwen [9], of Hart and
van Mill [20], and of Tomita [40], [41], and see also [8] for a characterization, in a
forcing model of ZFC + CH with 2c “arbitrarily large”, of those Abelian groups
which admit a hereditarily separable pseudocompact (alternatively, countably
compact) group topology with no infinite compact subsets.)

1.2. Outline. It is well known and easily proved that every uncountable Abelian
group K admits a subgroup H of index exactly ω; if K has a compact (Haus-
dorff) group topology, such H is necessarily nonmeasurable with respect to
the associated Haar measure. It has been noticed by Stromberg [39] that R
contains a nonmeasurable subgroup of index | R|. In this spirit, we show in Sec-
tions 2 and 3 that every infinite compactly generated Abelian group K has a
family of 2|K|-many dense, nonmeasurable subgroups of index |K|. We deduce

from this that every infinite Abelian group G admits a family of 22
|G|

-many
totally bounded group topologies with no nontrivial convergent sequence (The-
orem 4.1). We show also by very different methods that such G has the same
number of totally bounded group topologies in each of which some nontrivial
sequence (fixed, and chosen in advance) does converge (Theorem 5.5). In the



Making group topologies with, and without, convergent sequences 111

obvious sense, these results are clearly optimal. An elementary cardinality ar-
gument shows that the various topological groups (G,T ) may be chosen to be
pairwise nonhomeomorphic as topological spaces.

1.3. Notation. The symbols κ and α denote infinite cardinals; ω is the least
infinite cardinal, and c := 2ω. For S a set we write [S]κ := {A ⊆ S : |A| = κ};
the symbol [S]<κ is defined analogously. Z and R denote respectively the group
of integers and the group of real numbers, often with their usual (metrizable)
topologies, and T := R/Z. General groups G are written multiplicatively, with
identity 1 = 1G, but groups G known or hypothesized to be Abelian are written
additively, with identity 0 = 0G. We identify each finite cyclic group with its
copy in T; in particular for 0 < n < ω we write Z(n) := { k

n
: 0 ≤ k < n} ⊆ T.

We denote by P the set of primes.
We work exclusively with Tychonoff spaces, i.e., with completely regular,

Hausdorff spaces. A topological group which is a Hausdorff space is necessarily
a Tychonoff space [22](8.4). A topological group (G,T ) is said to be totally
bounded if for ∅ 6= U ∈ T there is F ∈ [G]<ω such that G = FU. It is a
theorem of Weil [42] that a topological group is totally bounded if and only if
G embeds densely into a compact group; this latter group, unique in an obvious
sense, is called the Weil completion of G and is denoted G.

Given a topological group G = (G,T ), the symbol Ĝ = (̂G,T ) denotes the
set of continuous homomorphisms from G to the (compact) group T. A group

G with its discrete topology is written Gd, so Ĝd = Hom(G, T); this is a closed
subgroup of the compact group TG. It is easily seen, as in [6](1.9), that for
a subgroup H of Hom(G, T) these conditions are equivalent: (a) H separates
points of G; (b) H is dense in the compact group Hom(G, T).

For G Abelian we denote by S(G) the set of point-separating subgroups of
Hom(G, T), and by t(G) the set of (Hausdorff) totally bounded group topologies
on G. Theorem 1.2 below describes, for each infinite Abelian group G, a useful
order-preserving bijection between S(G) and t(G).

The rank, the torsion-free rank, and (for p ∈ P) the p-rank of an Abelian
group G are denoted r(G), r0(G), and rp(G), respectively. We have r(G) =
r0(G) + Σp∈P rp(G), and when |G| > ω we have |G| = r(G) (cf. [13](§16) or
[22](Appendix A)). Following those sources we write
Gp := ∪k<ω {x ∈ G : p

k · x = 0}.
We write X =h Y if X and Y are homeomorphic topological spaces, and we

write G ≃ H if G and H are isomorphic groups; it is important to note that
X =h Y conveys no information about the algebraic structure of X or Y (if
any), and G ≃ H conveys no information about the topological structure of G
or H (if any).

We assume familiarity on the reader’s part with the essentials of Haar mea-
sure λ = λK on a locally compact group K. The set of Borel sets, and the set
of λ-measurable sets, are denoted B(K) and M(K), respectively. Of course
B(K) ⊆ M(K). Our convention is that Haar measure is complete in the sense
that if S ∈ M(K) with λ(S) = 0, then each A ⊆ S satisfies A ∈ M(K) (with



112 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

λ(A) = 0). We assume for simplicity that if K is compact then λ is normalized
in the sense that λ(K) = 1.

The following three theorems are essential to our argument. Therem 1.1(a)
is due to Steinhaus [37] when G = R and to Weil [43](p. 50) in the general
case; see Stromberg [38] for a pleasing, efficient proof. Parts (b) and (c) are
the immediate consequences which we use here frequently.

Theorem 1.1 ([37, 43]). Let K be a locally compact group and let S be a
λ-measurable subset of K with λ(S) > 0. Then

(a) the difference set SS−1 contains a neighborhood of 1K;
(b) if S is a subgroup of K then S is open and closed in K; and
(c) if S is a dense subgroup of K then S = K.

Theorem 1.2 ([6]). Let G be an Abelian group.

(a) For every H ∈ S(G), the topology TH induced on G by H is a (Haus-
dorff ) totally bounded group topology such that w(G,TH ) = |H|;

(b) if (G,T ) is totally bounded then T = TH with H := (̂G,T ) ∈ S(G).

It is clear with G as in Theorem 1.2 that distinct H0,H1 ∈ S(G) induce

distinct topologies TH0,TH1 ∈ t(G)—indeed
̂(G,TH0 ) = H0 6= H1 =

̂(G,TH1 );
thus the bijection t(G) ↔ S(G) given by TH ↔ H is indeed order-preserving.

Theorem 1.3 ([7]). Let G be an infinite Abelian group with K := Hom(G, T) =

Ĝd, let (xn)n be a faithfully indexed sequence in G, and let
A := {h ∈ K : h(xn) → 0}.

Then A is a subgroup of K such that A ∈ B(K) and λ(A) = 0.

Proof. [Outline]. That A is a subgroup of K is obvious. According to the

duality theorem of Pontrjagin [28] and van Kampen [25], the map Gd ։
̂̂
Gd =

̂Hom(G, T) given by x → x̂ (with x̂(h) = h(x) for x ∈ G, h ∈ K) is a bijection.
Writing
An,m := {h ∈ K : |x̂n(h) − 1| <

1
m
},

the relation A = ∩m<ω ∪N≥m ∩n≥N An,m expresses A as a Gδσδ-subset of the
compact group K, so A ∈ B(K). If G is torsion-free, a condition equivalent to
the condition that Hom(G, T) is connected (cf. [22](24.25)), then a reference
to Theorem 1.1 completes the proof: the condition λ(A) > 0 would imply
A = Hom(G, T), so that xn → 0 in the Bohr topology of Gd, contrary to the
theorem of Leptin and Glicksberg cited above. We refer the reader to [7] for
the proof (for general Abelian G) that λ(A) = 0. �

In what follows we will frequently invoke this simple algebraic fact.

Theorem 1.4. Let K be an Abelian group, let H be a subgroup of K of index
α > ω, and let

H := {S : H ⊆ S ⊆ K,S is a proper subgroup of K, |S| = |K|}.

Then |H| = 2α.



Making group topologies with, and without, convergent sequences 113

Proof. The inequality ≤ is obvious. We have |K/H| = r(K/H) = α > ω, so
algebraically K/H ⊇ ⊕ξ<α Cξ with each Cξ cyclic. Let φ : K ։ K/H be the
canonical homomorphism, and for A ∈ [α]α\{α} set HA := φ

−1(⊕ξ∈A Cξ ×
{0ξ}ξ∈α\A). The map [α]

α\{α} → H given by A → HA is an injection, so
|H| ≥ |[α]α| = 2α, as asserted. �

Theorem 1.3 explains our interest in the existence of (many) point-separating
nonmeasurable subgroups of compact Abelian groups K (of the form K =
Hom(G, T)): each such H ∈ S(G) will induce on G a totally bounded group
topology without nontrivial convergent sequences. In order to show that such K
admit 2|K|-many such subgroups, we find it convenient to treat separately the
metrizable case (that is, w(K) = ω) and the nonmetrizable case (w(K) > ω).
We do this in Sections 2 and 3 respectively.

2. Many Nonmeasurable Subgroups: The Metrizable Case

For simplicity, and because it suffices for our applications, we take the groups
K and M in Lemma 2.1 and Theorem 2.2 to be compact; the reader may
notice that this hypothesis can be significantly relaxed. Indeed both groups are
Abelian and M is metrizable in our applications, but since those hypotheses
save no labor we omit them for now.

Lemma 2.1. Let K and M be compact groups with Haar measures λ and µ
respectively, and let φ : K ։ M be a continuous surjective homomorphism.
Then µ(E) = λ(φ−1[E]) for every E ∈ B(M).

Proof. Define m : B(M) → [0, 1] by m(E) := λ(φ−1[E]). According to
[22](15.8), and using the numbering system there, it is enough to show that (iv)
m(C) < ∞ for compact C ∈ B(M); (v) m(U) > 0 for some open U ∈ B(M);
(vi) m(a + F) = m(F) for all a ∈ M, F ∈ B(M); and

(vii) m(U) = sup{m(F) : F ⊆ U,F is compact} for open U ⊆ M, and
m(E) = inf{m(U) : E ⊆ U,U is open} for E ∈ B(M).

The verifications are routine and will not be reproduced here. In addition to
[22], the reader seeking hints might consult [17](63C and 64H, or 52G and
52H). �

Theorem 2.2. Let K and M be compact groups with Haar measures λ and µ
respectively, and let φ : K ։ M be a continuous, surjective homomorphism. If
D is a dense, non-µ-measurable subgroup of M, then H := φ−1(D) is a dense,
non-λ-measurable subgroup of K.

Proof. φ is an open map [22](5.29), so H is dense in K. Suppose now that
H ∈ M(K), so that either λ(H) > 0 or λ(H) = 0. If λ(H) > 0 then H = K
by Theorem 1.1(c) so D = φ[K] = M, a contradiction. If (H ∈ M(K) and)
λ(H) = 0 then since λ is (inner-) regular there is a sequence Kn (n < ω) of
compact subsets of K\H such that λ(∪n Kn) = λ(K\H) = 1. We write Mn :=

φ[Kn] and K̃n := φ
−1(Mn). Then Kn ⊆ K̃n ⊆ K\H and from Lemma 2.1 we

have



114 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

µ(∪n Mn) = λ(∪n K̃n) ≥ λ(∪n Kn) = 1,

so µ(∪n Mn) = 1 and hence D ∈ M(M) with µ(D) = 0, a contradiction. �

Our goal is to show that every (infinite) compact Abelian metrizable group
contains a dense, nonmeasurable subgroup of index c. We treat some special
cases first. In what follows we denote the torsion subgroup of an Abelian group
K by t(K), for s ∈ K and 0 6= n ∈ Z we write [ s

n
] = {x ∈ K : nx = s}, and for

a subgroup S of K we set

∆(S) := ∪{[
s

n
] : s ∈ S, 0 6= n ∈ Z}.

When [ s
n
] 6= ∅ we choose sn ∈ [

s
n
], and we write Λ(S) := {sn : s ∈ S, 0 6= n ∈

Z} ∪ {0G}. Then |Λ(S)| ≤ |S| · ℵ0, and ∆(S) = Λ(S) + t(K).

Lemma 2.3. Let M be an Abelian group such that |M| = κ > ω and let
S ∈ [M]<κ, E ∈ [M]κ with S a subgroup. If either

(i) |t(M)| < κ, or
(ii) there is p ∈ P such that p · M = {0},

then there is x ∈ E such that 〈x〉 ∩ S = {0}. In case (i), x may be chosen in
M\t(M).

Proof. (i) From ∆(S) = Λ(S) + t(M) follows |∆(S)| < κ, and any x ∈
E\∆(S) ⊆ E\t(M) is as required.

(ii) Since M ≃ ⊕κ Z(p) = ⊕ξ<κ Z(p)ξ, there is A ∈ [κ]
<κ such that S ⊆

⊕ξ∈A Z(p)ξ. Any x ∈ E such that 0 6= x /∈ ⊕ξ∈A Z(p)ξ is as required. �

Theorem 2.4. Let M be an infinite, compact, metrizable, Abelian group such
that either

(i) |t(M)| < c or
(ii) there is p ∈ P such that p · M = {0}.

Then M has a dense, nonmeasurable subgroup D such that |M/D| = c.
In case (i) one may arrange D ≃ ⊕c Z, in case (ii) one may arrange D ≃

⊕c Z(p).

Proof. Let {Fξ : ξ < c} be an enumeration of all uncountable, closed subsets
of M, and define Eξ := Fξ\t(M) in case (i), Eξ := Fξ\{0} in case (ii). It
is a theorem of Cantor [2](page 488) that each |Fξ| = c (see [21](VIII §9 II)
or [11](4.5.5(b)) for more modern treatments); hence each |Eξ| = c. There is
x0 ∈ E0, and by Lemma 2.3 there is y0 ∈ E0 such that 〈x0〉 ∩ 〈y0〉 = {0}.

Now let ξ < c, suppose that xη, yη have been chosen for all η < ξ, and apply
Lemma 2.3 twice to choose xξ,yξ ∈ Eξ such that

〈{xξ}〉 ∩ 〈{xη : η < ξ} ∪ {yη : η < ξ}〉 = {0}, and
〈{yξ}〉 ∩ 〈{xη : η ≤ ξ} ∪ {yη : η < ξ}〉 = {0}.

Thus xξ,yξ are defined for all ξ < c. We define D := 〈{xξ : ξ < c}〉. Clearly
D = ⊕ξ<c Z in case (i), and D = ⊕ξ<c Z(p)ξ in case (ii) since |D| = c and
p · D = {0}. For ξ < η < c we have yη + D 6= yξ + D, so c ≥ |K/D| ≥ c.



Making group topologies with, and without, convergent sequences 115

For nonempty open U ⊆ K there is by the regularity of λ a (necessarily
uncountable) compact set F = Fξ ⊆ U such that λ(Fξ) > 0. Then xξ ∈
Eξ ∩ D ⊆ Fξ ∩ D. Thus D is dense in K. If D ∈ M(K) with λ(D) > 0 then
D = K by Theorem 1.1(c), contrary to the relation |K/D| = c. If D ∈ M(K)
with λ(D) = 0 then λ(K\D) = 1 and there is Fξ ⊆ K\D such that λ(Fξ ) > 0;
then xξ ∈ Eξ ∩ D ⊆ (K\D) ∩ D = ∅, a contradiction. �

Corollary 2.5. Let M be a (compact, Abelian, metrizable) group of one of
these types.

(i) M = T;
(ii) M = ∆p (p ∈ P), the group of p-adic integers;
(iii) M = Πk<ω Z(pk), pk ∈ P, (pk)k faithfully indexed;
(iv) M = (Z(p))ω (p ∈ P).

Then M has a dense, nonmeasurable subgroup D such that |M/D| = c.

Proof. Surely |t(T)| = ω, and t(Πk<ω Z(pk)) is the countable group ⊕k<ω Z(pk).
If 0 6= h ∈ ∆p = Hom(Z(p

∞), T), then h[Z(p∞)] ≃ Z(p∞)/ ker(h) ≃ Z(p∞)
since | ker(h)| < ω, so h[Z(p∞)] is not of bounded order; thus t(∆p) = {0}. It
follows for M as in (i), (ii) and (iii) that |t(M)| = 1 < c or |t(M)| = ω < c, so
Theorem 2.4(i) applies. For M as in (iv) surely p·M = {0}, so Theorem 2.4(ii)
applies. �

Theorem 2.6. Let K be an infinite, compact, Abelian metrizable group. Then
(|K| = c and) K has a dense, nonmeasurable subgroup H such that |H| = c
and |K/H| = c.

Proof. The (discrete) dual group G = K̂ satisfies |G| = w(K) = ω. As
with any countably infinite Abelian group, G must satisfy (at least) one of
these conditions: (i) r0(G) > 0: (ii) |Gp| = ω with rp(G) < ω for some
p ∈ P; (iii) 0 < rp(G) < ω for infinitely many p ∈ P; (iv) rp(G) = ω for
some p ∈ P. According as (i), (ii), (iii) or (iv) holds we have, respectively,
G ⊇ Z, G ⊇ Z(p∞), G ⊇ ⊕k<ω Z(pk), or G ⊇ ⊕ω Z(p), so taking adjoints we

have a continuous surjection φ from K onto a group M of the form Ẑ = T,

Ẑ(p∞) = ∆p, ̂⊕k<ω Z(pk) = Πk<ωZ(pk), or ⊕̂ω Z(p) = (Z(p))
ω . According

to Corollary 2.5 the group M has a dense, nonmeasurable subgroup D such
that |M/D| = c, and then by Theorem 2.2 with H := φ−1(D) the group H
is dense and nonmeasurable in K. If a,b ∈ K with a + H = b + H then
φ(a) + D = φ(b) + D, so

c = |K| ≥ |K/H| ≥ |M/D| = c.

�

Theorem 2.7. Let K be an infinite, compact, Abelian, metrizable group. Then
K admits a family of 2|K|-many dense, nonmeasurable subgroups, each of car-
dinality c.



116 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

Proof. Let H be as given in Theorem 2.6 and let
H := {S : H ⊆ S ⊆ K,S is a proper subgroup of K}.

Then |H| = 2c by Theorem 1.4. Theorem 1.1(c) shows for S ∈ H that S ∈
M(K) with λ(S) > 0 is impossible, and if S ∈ H with λ(S) = 0 then λ(H) = 0,
a contradiction. �

Remark 2.8. For our application in Theorem 4.1 below we do not require
that |K/S| = c, but in fact that condition does hold for 2c-many S ∈ H.

3. Many Nonmeasurable Subgroups: The Nonmetrizable Case

We turn now to the case w(K) = κ > ω. Again, our goal is to show that
such a compact Abelian group K contains 2|K|-many dense, nonmeasurable
subgroups of cardinality |K|. We find it convenient to show a bit more, namely
that K admits a family of 2|K|-many subgroups each of which is Gδ-dense in
K (and hence pseudocompact). In the transition, we will invoke the following
lemma.

Lemma 3.1. A proper, Gδ-dense subgroup H of a compact group K is non-
measurable.

Proof. As usual, using Theorem 1.1(c), H ∈ M(K) with λ(H) > 0 is impos-
sible; it suffices then to show that λ(H) = 0 is also impossible. The following
argument is from [24] and [23], as exposed by Halmos [17]. If λ(K\H) = 1 > 0
there are a compact set C and a Baire set F of K such that F ⊆ C ⊆ K\H
and λ(F) = λ(C) > 0 ([17](64H and p. 230)). As with any nonempty Baire
set, F has the form F = XB for a suitably chosen compact Baire subgroup
B of K and X ⊆ K ([17](64E)). Since every compact Baire set is a Gδ-set
([17](51D)), each x ∈ X has xB a Gδ-set. From xB ∩ H = ∅ it follows that H
is not Gδ-dense in K, a contradiction. �

Theorem 3.2. Let K be a compact, Abelian group such that w(K) = κ > ω.
Then K has a family of 2|K|-many Gδ-dense subgroups of cardinality |K|.

Proof. Let G = K̂, so that |G| = w(K) = κ, and let κ0 = r0(G) and κp = rp(G)
for p ∈ P. Since |G| = κ = κ0 + Σp∈P κp (with perhaps κi = 0 for certain
i ∈ P ∪ {0}), we have algebraically

G ⊇ ⊕κ0 Z ⊕ ⊕p∈P ⊕κp Z(p). (*)

The map ψ adjoint to the inclusion map in (*) is a continuous, surjective
homomorphism

ψ : K = Ĝ ։ M := Tκ0 × Πp∈P (Z(p))
κp = Πξ<κ Nξ (**)

(with, again, perhaps κi = 0 for certain i ∈ P ∪ {0}), each Nξ a nondegenerate
compact metric group; here w(M) = κ, |M| = 2κ. Now using κ = κ · ω+ let
{Aη : η < κ} be a partition of κ into disjoint sets of cardinality ω

+, and rewrite
(**) in the form

ψ : K ։ M = Πη<κ Mη



Making group topologies with, and without, convergent sequences 117

with Mη = Πξ∈Aη Nξ a compact group of weight ω
+. Each group Mη has

a proper Gδ-dense subgroup (for example, the Σ-product), and that in turn
extends to a proper subgroup Dη of Mη of finite or countably infinite index.

Then |Dη| = |Mη| = 2
(ω+), so D := Πη<κ Dη is Gδ-dense in M with |D| =

(2(ω
+))κ = 2κ and 2κ = |M| ≥ |M/D| = Πη<κ |Mη/Dη| ≥ 2

κ.
Now let H := ψ−1(D). Then 2κ = |K| ≥ |H| ≥ |D| = 2κ, and |K/H| = 2κ.

As noted in [3](2.2) the image under ψ of each nonempty Gδ-subset of K
contains a nonempty Gδ-subset of M, so H is Gδ-dense in K. By Theorem 1.4
there are 22

κ

-many proper subgroups of K containing H, each necessarily Gδ-
dense, as required. �

We note in passing that in general not every Gδ-dense subgroup D of a
compact, nonmetrizable group K satisfies |D| = |K|. See in this connection
Remark 6.4 below.

Corollary 3.3 is now immediate from Lemma 3.1 and Theorem 3.2; and
Theorem 3.4, which is (1) of our Abstract, is the conjunction of Theorem 2.7
and Corollary 3.3.

Corollary 3.3. Let K be a compact, Abelian group such that w(K) > ω.
Then K admits a family of 2|K|-many dense, nonmeasurable subgroups, each
of cardinality |K|.

Theorem 3.4. Every infinite, Abelian compact (Hausdorff ) group K admits
2|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K
is nonmetrizable, these may be chosen to be pseudocompact.

Remark 3.5. (a) In an earlier version of this manuscript privately circulated
to colleagues we expanded the scope of arguments from [5] to associate with
each uniform ultrafilter q over κ a homomorphism hq : K ։ F with F = T
or F = Z(p) in such a way that Mq := ker(hq) is Gδ-dense in K. (The cases
cf(κ) > ω, cf(κ) = ω required separate treatment.) The present alternative
proof of Theorem 3.2, contributed anonymously, seems briefer and conceptually
simpler.

(b) It is known [39] that R contains a dense non-measurable subgroup of
both cardinality and index c. Arguing as in Theorem 2.7, we see then that
R has 2c-many dense non-measurable subgroups of both cardinality and index
c. We say as usual that a compactly generated group is a topological group
generated, in the algebraic sense, by a compact subset. By [22] (9.8) for every
(Hausdorff) locally compact Abelian compactly generated group G there are
non-negative integers m and n and a compact Abelian group K such that G is
of the form Rm × K × Zn. If G is not discrete, then either m > 0 or K is not
discrete, so w(G) = ω+w(K). It follows that a non-discrete (Hausdorff ) locally
compact Abelian compactly generated group G has 2|G|-many dense non-Haar-
measurable subgroups of both cardinality and index |G|. The result cannot be
generalized to arbitrary locally compact Abelian groups since such groups may
fail to have a proper dense subgroup [32].



118 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

Note added in proof. We are grateful to Professor James D. Reid for calling
to our attention this theorem of W. R. Scott [Proc. Amer. Math. Soc. 5 (1)
(1954), 19–22], which is closely related to our arguments in Sections 2 and 3:
If G is an abelian group with |G| = α > ω, and if κ is a cardinal satisfying
α ≥ κ ≥ ω, then G has 2α-many subgroups of index κ. It is clear (using
Theorem 1.1) that if in addition G has a compact group topology then those
subgroups of index ω are necessarily non-measurable; but it is not obvious to us
whether simple additional algebraic arguments show that they must be, or may
be chosen to be, dense in G (or Gδ-dense, in the case that G is non-metrizable).

4. Group Topologies Without Convergent Sequences

Here we pull together the threads of Sections 2 and 3.

Theorem 4.1. Every infinite Abelian group G admits a family A of totally

bounded group topologies, with |A| = 22
|G|

, such that no nontrivial sequence in
G converges in any of the topologies in A. One may arrange in addition that

(i) w(G,T ) = 2|G| for each T ∈ A, and
(ii) for distinct T0, T1 ∈ A the spaces (G,T0) and (G,T1) are not homeo-

morphic.

Proof. By Theorem 2.7 when |G| = ω, and by Corollary 3.3 when |G| > ω,

the compact group K := Hom(G, T) = Ĝd admits a family H of dense, non-

measurable subgroups such that |H| = 22
|G|

and |H| = |K| for each H ∈ H.
According to Theorems 1.2 and 1.3 the family A := {TH : H ∈ H} satisfies
all requirements except (perhaps) (ii). A homeomorphism between two of the
spaces (G,T0), (G,T1) with Ti ∈ A is realized by a permutation of G, and
there are just 2|G|-many such functions, so for each T ∈ A there are at most
2|G|-many T ′ ∈ A such that (G,T ) =h (G,T

′). Statement (ii) then follows
(with A replaced if necessary by a suitably chosen subfamily of cardinality

2|K| = 22
|G|

). �

Remark 4.2. The case G = Z of Theorem 4.1 is not new. See in this connec-
tion [30] and [31], which were the motivation for much of the present paper.

5. Topologies With Convergent Sequences

We turn now to the complementary or opposing problem, that of finding on

an arbitrary infinite Abelian group G the maximal number (that is, 22
|G|

) of
totally bounded group topologies in which some nontrivial sequence converges.
It was shown in [29], [44] concerning the group Z that if 0 < n ∈ Z with
xn+1/xn → ∞ then there is a topology in t(Z) with respect to which xn → 0;
independently Raczkowski [30], [31] proved that if xn+1/xn ≥ n + 1 then there
are 2c-many such topologies in t(Z); later the authors of [1] obtained the same
conclusion assuming only that xn+1/xn → ∞.

To handle the case of general Abelian G, our strategy is to show first that
certain “basic” countable groups accept 2c many such topologies. We begin



Making group topologies with, and without, convergent sequences 119

with technical results concerning groups of the form ⊕k<ω Z(p
rk
k ) and of the

form Z(p∞).
We remark for emphasis that in Theorem 5.1 the given sequence (pk)k in P

is not necessarily faithfully indexed. Indeed the case pk = p ∈ P (a constant
sequence) is not excluded. For x ∈ A = ⊕k<ω Z(p

rk
k ) we write x = (x(k))k<ω .

Theorem 5.1. Let pk ∈ P and A = ⊕k<ω Z(p
rk
k ) with 0 < rk < ω, and let

(xn)n<ω be a faithfully indexed sequence in A such that
(i) there is S ∈ [ω]ω such that xn(k) = 0 for all n < ω, k ∈ S, and
(ii) |{n < ω : xn(k) 6= 0}| < ω for all k < ω.

Let {Aξ : ξ < c} enumerate P(S) ∪ [ω]
<ω, and for ξ < c define hξ ∈ Hom(A, T)

by

hξ(x) = Σk∈Aξ x(k).

Then
(a) the set {hξ : ξ < c} is faithfully indexed;
(b) the set {hξ : ξ < c} separates points of A; and
(c) hξ(xn) → 0 for each ξ < c.

Proof. (a) If ξ,ξ′ < c with ξ 6= ξ′, say k ∈ Aξ\Aξ′ , then any x ∈ A such that
0 6= x(k) ∈ Z(p

rk
k ) and x(m) = 0 for k 6= m < ω satisfies

hξ(x) = x(k) 6= 0 = hξ′ (x).
(b) Let x,x′ ∈ A with, say, x(k) 6= x′(k). There is ξ < c such that {k} = Aξ,

and then
hξ(x) = x(k) 6= x

′(k) = hξ(x
′).

(c) If Aξ ∈ P(S) then hξ(xn) = 0 for all n by (i), so hξ(xn) → 0. If
Aξ ∈ [ω]

<ω then by (ii) there is N < ω such that hξ(xn) = 0 for all n > N, so
again hξ(xn) → 0. �

Next, following [22] and [13], we identify the elements of the compact group
∆p = Hom(Z(p

∞), T) with those sequences h = (h(k))k<ω of integers such that
0 ≤ h(k) ≤ p − 1 for all k < ω. For a

pn
∈ Z(p∞) (with 0 ≤ a ≤ pn − 1) we have

h
(

a
pn

)
= a · h

(
1

pn

)
= a · Σn−1k=0

h(k)
pn−k

= a
pn

· Σn−1k=0 h(k) · p
k (mod 1).

In what follows we write Fac := {n! : n < ω}.

Theorem 5.2. Let (an)n<ω be a sequence of integers such that 0 ≤ an ≤ p− 1
for all n < ω, and let xn =

an
pn!

∈ Z(p∞). Let {Aξ : ξ < c} enumerate

P(Fac) ∪ [ω]<ω, and for ξ < c define hξ = (hξ(k))k<ω ∈ ∆p by

hξ(k) =

{
1 if k ∈ Aξ,
0 otherwise.

Then

(a) the set {hξ : ξ < c} is faithfully indexed;
(b) the set {hξ : ξ < c} separates points of A;
(c) if an > 0 for all but finitely many n, then the sequence (xn)n<ω is

faithfully indexed; and
(d) hξ(xn) → 0 for each ξ < c.



120 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

Proof. The proofs of (a) and (b) closely parallel their analogues in Theorem 5.1,
and (c) is obvious. We prove (d). If Aξ ∈ [ω]

<ω there is N < ω such that

hξ(xn) =
an
pn!

· Σn!−1k=0 hξ(k) · p
k ≤ p−1

pn!
· ΣNk=0 p

k for all n > N,

so hξ(xn) → 0. If Aξ ∈ P(Fac) then

hξ(xn) =
an
pn!

· Σn!−1k=0 hξ(k) · p
k =

an
pn!

· Σk∈Fac,0≤k<n!−1 hξ(k) · p
k ≤

≤
p − 1

pn!
· Σn−1m=0 p

m! ≤
p − 1

pn!
· n · p(n−1)! <

(p − 1) · n

pn
,

and again hξ(xn) → 0. �

Theorem 5.3. Let A = Z or A = ⊕k<ω Z(pk) (pk ∈ P, repetitions allowed),
or A = Z(p∞) (p ∈ P). Then there are a faithfully indexed sequence (xn)n<ω
in A and a point-separating subgroup H ⊆ Hom(A, T) such that |H| = c and
xn → 0 in the space (A,TH ).

Proof. We refer the reader for a detailed proof in the case A = Z to [30], [31].
When A = ⊕k<ω Z(pk) or A = Z(p

∞) there are, according to Theorem 5.1
or 5.2 respectively, a sequence (xn)n<ω in A and a faithfully indexed family
{hξ : ξ < c} ⊆ Hom(A, T) such that hξ(xn) → 0 for each ξ < c. It is then clear,
as in [7], that with H := 〈{hξ : ξ < c}〉 ⊆ Hom(A, T) we have h(xn) → 0 for
each h ∈ H. �

Corollary 5.4. Let G be an infinite Abelian group. There are a faithfully
indexed sequence (xn)n<ω in G and a topology TH∗ ∈ t(G) such that |H

∗| = 2|G|

and xn → 0 in (G,TH∗ ).

Proof. As indicated earlier, G contains algebraically a group A such that A ≃ Z
or A ≃ ⊕k<ω Z(pk) or A ≃ Z(p

∞). Let H ⊆ Hom(A, T) and (xn)n<ω in A be as
in Theorem 5.3, and set H∗ := {k ∈ Hom(G, T) : k|A ∈ H}. Clearly k(xn) → 0
for each k ∈ H∗, so xn → 0 in (G,TH∗ ). If |G| = ω then |H

∗| = c = 2ω since
c = |H| ≤ |H∗| ≤ |TG| = c. If |G| > ω we write

A(Ĝ,A) := {k ∈ Hom(G, T) : k ≡ 0 on A} ⊆ H∗;

then |A(Ĝ,A)| = 2|G/A| = 2|G| since algebraically A(Ĝ,A) = Hom(G/A, T)
and |G| = |G/A|, so |H∗| = 2|G| in this case also. �

We have arrived at the final result of this Section, which we view as a
companion or “echo” to Theorem 4.1.

Theorem 5.5. Every infinite Abelian group G admits a family B of totally

bounded group topologies, with |B| = 22
|G|

, such that some (fixed) nontrivial
sequence in G converges in each of the topologies in B. One may arrange in
addition that

(i) w(G,T ) = 2|G| for each T ∈ B, and
(ii) for distinct T0, T1 ∈ B the spaces (G,T0) and (G,T1) are not homeo-

morphic.



Making group topologies with, and without, convergent sequences 121

Proof. Let H∗ be a subgroup of Hom(G, T) such that |H∗| = 2|G| and some
nontrivial sequence (xn)n<ω in G satisfies xn → 0 in (G,TH∗ ). There is a
subgroup H of H∗ such that H separates points of G and |H| = |G|, and
since |H∗/H| = 2|G| there is by Theorem 1.4 a faithfully indexed family {Hξ :

ξ < 22
|G|

} of (point-separating) groups such that H ⊆ Hξ ⊆ H
∗ and |Hξ| =

|H∗| = 2|G| for each ξ < 22
|G|

. Then from Theorem 1.2(a) we have w(G,THξ ) =

|Hξ| = 2
|G| for each ξ < 22

|G|

, and the family B := {THξ : ξ < 2
2|G|} satisfies

(i); condition (ii) is then achieved as in the final sentences of the proof of
Theorem 4.1. �

Remark 5.6. (a) If A and B are as in Theorems 4.1 and 5.5, and if T0,T1 ∈
A∪B with T0 6= T1, then the spaces (G,T0) and (G,T1) are not homeomorphic.
For if both Ti ∈ A or both Ti ∈ B this is already proved, while if (say) T0 ∈ A
and T1 ∈ B then (G,T1) has a nontrivial convergent sequence and (G,T0) does
not.

(b) We emphasize that for an infinite Abelian group G, the algebraic struc-
ture of a point-separating subgroup H ⊆ Hom(G, T) by no means determines
the topology TH on G. It is noted explicitly in [30], [31] that when G = Z
then every one of the topologies in the families A and B (as in Theorems 4.1
and 5.5) can be chosen of the form TH with H ⊆ T = Hom(G, T) and with
H ≃ ⊕ξ<c Zξ.

6. Concluding Remarks

Remark 6.1. In the direction converse to Theorem 1.3 above, it is natural to

ask whether, given an infinite Abelian group G with K = Ĝd, there exist a dense
subgroup H of K with λ(H) = 0 such that no nontrivial sequence converges in
(G,TH ). An affirmative answer was given in [1] for the case G = Z assuming
MA, in [18] for G = Z in ZFC, and in [19](1.9) in ZFC for every such G.

Our convergent sequences in Theorems 5.1 and 5.2 were quite “thin”. View-
ing those results from another perspective, a natural question arises.

Question 6.2. Let G be an infinite Abelian group. For which faithfully in-
dexed sequences (xn)n<ω in G is there a topology TH ∈ t(G) with w(G,TH ) =
|H| = 2|G| such that xn → 0 in (G,TH )?

Many questions arise in the non-Abelian context. Perhaps this one of Saeki
and Stromberg [33] bears repeating.

Question 6.3. [33]. Does every infinite (not necessarily Abelian) compact
group have a dense, nonmeasurable subgroup?

Remark 6.4. Malykhin and Shapiro [27] showed by a direct argument that
for every faithfully indexed sequence (xn)n<ω in an Abelian group G there is
hx ∈ Hom(G, T) such that hx(xn) 6→ 0. Thus every Abelian G with |G| = α
admits a topology TH ∈ t(G) such that |H| = w(G,TH ) = α

ω and no nontrivial
sequence converges in (G,TH ). This statement can be improved slightly using



122 W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta

Theorem 1.3 above and this result from [4](2.2): Every compact group K with
w(K) = α contains a dense, countably compact, subgroup H such that |H| =
(log α)ω . Since such H is pseudocompact and hence nonmeasurable, we have
(beginning with Abelian G such that |G| = α and taking K := Hom(G, T) =

Ĝd) that w(G,TH ) = (log α)
ω and (G,TH ) has no convergent sequences. (We

note in this connection that (log α)ω < 2α for every strong limit cardinal α
such that cf(α) > ω.) This discussion suggests a question.

Question 6.5. Given an infinite Abelian group G, what is the minimal weight
of a topology in T ∈ t(G) such that no nontrivial sequence converges in (G,T )?
For which G is this 2|G|?

Acknowledgements. The second listed author acknowledges partial sup-
port from the University Research Council at CSU Bakersfield. She also wishes
to thank Mrs. Mary Connie Comfort for her encouragement, without which this
paper would never see the daylight. Thank you.

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Received September 2004

Accepted February 2005

W. W. Comfort (wcomfort@wesleyan.edu)
Department of Mathematics, Wesleyan University, Middletown, CT 06459.

S. U. Raczkowski (racz@csub.edu)
Department of Mathematics, California State University, Bakersfield, Bakers-
field, CA, 93311-1099.

F. J. Trigos-Arrieta (jtrigos@csub.edu)
Department of Mathematics, California State University, Bakersfield, Bakers-
field, CA, 93311-1099.