@ Appl. Gen. Topol. 16, no. 2(2015), 141-165doi:10.4995/agt.2015.3312
c© AGT, UPV, 2015

Some classes of minimally almost periodic

topological groups

W. W. Comfort a and Franklin R. Gould b,1

a
Department of Mathematics and Computer Science, Wesleyan University, Wesleyan Station,

Middletown, CT 06459, USA (wcomfort@wesleyan.edu)
b

Department of Mathematical Sciences, Central Connecticut State University, New Britain,

06050, USA (gouldfrr@ccsu.edu)

To the Memory of Ivan Prodanov on the Occasion of the
30th Anniversary of his Death

Abstract

A Hausdorff topological group G = (G, T ) has the small subgroup gen-
erating property (briefly: has the SSGP property, or is an SSGP group)
if for each neighborhood U of 1G there is a family H of subgroups of G
such that

⋃
H ⊆ U and 〈

⋃
H〉 is dense in G. The class of SSGP groups

is defined and investigated with respect to the properties usually stud-
ied by topologists (products, quotients, passage to dense subgroups,
and the like), and with respect to the familiar class of minimally al-
most periodic groups (the m.a.p. groups). Additional classes SSGP(n)
for n < ω (with SSGP(1) = SSGP) are defined and investigated, and
the class-theoretic inclusions

SSGP(n) ⊆ SSGP(n + 1) ⊆ m.a.p.

are established and shown proper.
In passing the authors also establish the presence of SSGP(1) or
SSGP(2) in many of the early examples in the literature of abelian
m.a.p. groups.

2010 MSC: Primary 54H11; Secondary 22A05.

Keywords: SSGP group, m.a.p. group; f.p.c. group.

1This paper derives from and extends selected portions of the Doctoral Dissertation [19],
written at Wesleyan University (Middletown, Connecticut, USA) by the second-listed co-
author under the guidance of the first-listed co-author.

Received 10 October 2014 – Accepted 12 July 2015

http://dx.doi.org/10.4995/agt.2015.3312


W. W. Comfort and F. R. Gould

1. Introduction

1.1. Conventions.

(a) As usual, a topological group is a pair (G, T ) with G a group and with
T a topology on G for which the maps G × G → G and G → G given
by (x, y) → xy and x → x−1 are continuous.

(b) The topological spaces we hypothesize, in particular our hypothesized
topological groups, are assumed to be completely regular and Haus-
dorff (i.e., to be Tychonoff spaces). When a topology is defined or
constructed on a set or a group, the Tychonoff property will be verified
explicitly (if it is not obvious). In this context we recall ([24](8.4)) that
in order that a topological group be a Tychonoff space, it suffices that
it satisfy the Hausdorff separation property.

(c) For X a space and x ∈ X we write

NX(x) := {U ⊆ X : U is a neighborhood of x}.

When ambiguity is unlikely we write N(x) in place of NX(x).
(d) The identity of a group G is denoted 1 or 1G; if G is known or assumed

to be abelian and additive notation is in play, the identity may be
denoted 0 or 0G.

(e) When G is a group and κ ≥ ω, we use the notations
⊕

κ
G and G(κ)

interchangeably:
⊕

κ

G = G(κ) := {x ∈ Gκ : |{η < κ : xη 6= 1G}| < ω}.

When G is a topological group,
⊕

κ
G has the topology inherited from

Gκ.

The minimally almost periodic groups (briefly: the m.a.p. groups) to which
our title refers are by definition those topological groups G for which every
continuous homomorphism φ : G → K with K a compact group satisfies
φ[G] = {1K}. It follows from the Gel

′fand-Răıkov Theorem [17] (see [24](§22)
for a detailed development and proof) that every compact group K is alge-
braically and topologically a subgroup of a group of the form Πi∈I Ui with
each Ui a (finite-dimensional) unitary group [24](22.14). Therefore, to check
that a topological group G is m.a.p. it suffices to show that each continuous
homomorphism φ : G → U(n) with U(n) the n-dimensional unitary group
satisfies φ[G] = {1Un}. Similarly, since every compact abelian group K is al-
gebraically and topologically isomorphic to a subgroup of a group of the form
TI [24](22.17), to check that an abelian topological group G is m.a.p., it suffices
to show that each continuous homomorphism φ : G → T satisfies φ[G] = {1T}.

Sometimes for convenience we denote by m.a.p. the (proper) class of m.a.p.
groups, and if G is a m.a.p. group we write G ∈ m.a.p.. Similar conventions
apply to the classes SSGP(n) (0 ≤ n < ω) defined in Definition 3.3.

Algebraic characterizations of those abelian groups which admit an m.a.p.
group topology have been achieved only recently. For a brief historical account
of the literature touching this issue, see Discussion 2.1(h),(i) below.

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Some classes of minimally almost periodic topological groups

2. m.a.p. Groups: A Brief Historical Survey

2.1. Discussion. With no pretense to completeness, we here discuss briefly
some of the literature relating to the development of the class of m.a.p. groups.

(a) In effect, the class m.a.p. was introduced in 1930 by von Neumann [27],
who then together with Wigner [28] proved that (even in its discrete
topology) the matrix group SL(2, C) is an m.a.p. group.

(b) In the period 1940–1952, several workers showed that certain real topo-
logical linear spaces are m.a.p. groups; several examples, with detailed
verification, are given by Hewitt and Ross [24](23.32).

(c) In what follows we will quote at length from the 1980 paper of Pro-
danov [33], which showed by “elementary means” that the group

⊕
ω
Z

admits an m.a.p. topology.

(d) Ajtai, Havas and Komlós [1] proved that each group G of the form
Z, Z(p∞), or

⊕
n
Z(pn) (with all pn ∈ P either identical or distinct)

admits a m.a.p. group topology.

(e) Nienhuys [29] showed that the additive group R admits an m.a.p. topol-
ogy T contained in its usual topology. Since Q remains dense in (R, T )
and the m.a.p. property is inherited by dense subgroups, this allowed
Remus [35] to infer that Q admits an m.a.p. topology, hence (since the
weak sum of m.a.p. groups is again a m.a.p. group) that every infinite
divisible abelian group admits a m.a.p. topology. In the same paper
[35], Remus showed that every free abelian group admits an m.a.p.
topology.

(f) Protasov [34] and Remus [35] asked whether every infinite abelian group
admits an m.a.p. group topology; the question was deftly settled in the
negative by Remus [36] with the straightforward observation that for
distinct p, q ∈ P, every group topology on the infinite group G :=
Z(p) × (Z(q))κ (with κ ≥ ω) has the property that the homomorphism
x → qx maps G continuously onto the compact group Z(p). (See
[3](3.J), [5](4.6) for additional discussion.)

(g) In view of the cited examples Z(p) × (Z(q))κ of Remus [36], it was nat-
ural for Comfort [3](3.J.1) to ask: (1) Does every abelian group which
is not torsion of bounded order admit an m.a.p. topology? (2) What
about the countable case?

(h) Gabriyelyan [14], [16] answered Question (g)(2) affirmatively, showing
that indeed the witnessing topology may be chosen complete in the
sense that every Cauchy net converges. Gabriyelyan [15] showed fur-
ther that an abelian torsion group of bounded order admits an m.a.p.
topology if and only if each of its leading Ulm-Kaplansky invariants
is infinite. (The reader unfamiliar with the Ulm-Kaplansky invariants
might consult [13](§77); those cardinals also play a significant role in
[4] in a setting closely related to the present paper.)

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W. W. Comfort and F. R. Gould

(i) Dikranjan and Shakhmatov [8] gave a definitive positive answer to
Question (g)(1) in its fullest generality, indeed they gave several equiva-
lent conditions on a (not necessarily abelian) group G which are neces-
sary and sufficient that G admit an m.a.p. topology. Among those con-
ditions, evidently satisfied by each abelian group not torsion of bounded
order, are these:
(1) G is connected in its Zariski topology;
(2) m ∈ Z ⇒ mG = {0} or |mG| ≥ ω;
(3) the group fin(G) is trivial, i.e., fin(G) = {0}. (The group fin(G),

whose study was initiated in [7](4.4) and continued in [4](§2), may
be defined by the relation

fin(G) = 〈
⋃

{mG : m ∈ Z, |mG| < ω}〉
)
.

Detailed subsequent analysis of the theorems and techniques of [8] have
allowed those authors to answer the following two questions in the
negative; these questions were posed in [19] and in a privately circulated
pre-publication copy of the present manuscript.
(1) Let G be a group with a normal subgroup K for which K and G/K

admit topologies U and V respectively such that (K, U) ∈ m.a.p and
(G/K, W) ∈ m.a.p. Is there then necessarily a group topology T
on G such that (K is closed in (G, T ) and) (K, U) = (K, T |K)
and (G/K, U) = (G/K, Tq) with Tq the quotient topology?

(2) Let G be a group with a normal subgroup K such that both K and
G/K admit m.a.p. topologies. Must G admit a m.a.p. topology?

3. SSGP Groups: Some Generalities

Definition 3.1. Let G = (G, T ) be a topological group and let A ⊆ G. Then
A topologically generates G if 〈A〉 is dense in G.

Definition 3.2. A Hausdorff topological group G = (G, T ) has the small
subgroup generating property if for every U ∈ N(1G) there is a family H of
subgroups of G such that

⋃
H ⊆ U and

⋃
H topologically generates G.

A (Hausdorff) topological group with the small subgroup generating prop-
erty is said to have the SSGP property, or to be an SSGP group, or simply to
be SSGP.

Now for 0 ≤ n < ω the classes SSGP(n) are defined as follows.

Definition 3.3. Let G = (G, T ) be a Hausdorff topological group. Then

(a) G ∈ SSGP(0) if G is the trivial group.
(b) G ∈ SSGP(n + 1) for n ≥ 0 if for every U ∈ N(1G) there is a family H

of subgroups of G such that
(1)

⋃
H ⊆ U,

(2) H := 〈
⋃
H〉 is normal in G, and

(3) G/H ∈ SSGP(n).

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Some classes of minimally almost periodic topological groups

Remarks 3.4.

(a) For 0 ≤ n < ω the class-theoretic inclusion SSGP(n) ⊆ SSGP(n + 1)
holds, hence SSGP(n) ⊆ SSGP(m) when n < m < ω. To see this,
note that when G ∈ SSGP(n) and U ∈ N(1G) then we have, taking

H := {{1G}}, that H := 〈
⋃
H〉 = {1G} and G/H ≃ G ∈ SSGP(n), so

indeed G ∈ SSGP(n + 1).
(b) Clearly the class SSGP of Definition 3.2 coincides with the class SSGP(1)

of Definition 3.3.

A topological group G is said to be precompact if G is a (dense) topological
subgroup of a compact group. It is a theorem of Weil [40] that a topological
group G is precompact if and only if G is totally bounded in the sense that for
each U ∈ N(1G) there is finite F ⊆ G such that G = FU.

It is obvious that a precompact group G with |G| > 1 is not m.a.p. Indeed
if G is dense in the compact group G then the continuous function id : G →֒ G
does not satisfy id[G] = {1

G
}.

Theorem 3.5. The class-theoretic inclusion SSGP(n) ⊆ m.a.p. holds for each
n < ω.

Proof. The proof is by induction on n. Clearly if G ∈ SSGP(0) and φ ∈
Hom(G, U(m)) then φ[G] = {1U(m)}, so G ∈ m.a.p. Suppose now that SSGP(n) ⊆
m.a.p., let G be a topological group such that G ∈ SSGP(n+1), and let φ : G →
U(m) be a continuous homomorphism. Choose V ∈ N(1U(m)) so that V con-
tains no subgroups of U(m) other than {1U(m)}. Then U := φ

−1[V ] ∈ N(1G),
and φ maps every subgroup of U to 1U(m). Let H be a family of subgroups of G

such that
⋃
H ⊆ U and H := 〈H〉 is normal in G, with G/H ∈ SSGP(n). Since

a homomorphism maps subgroups to subgroups we have φ[H] = {1U(m)}. It

follows that φ defines a continuous homomorphism φ̃ : G/H → U(m) (given by

φ̃(xH) := φ(x)). By the induction hypothesis, φ̃ is the trivial homomorphism,
so φ is trivial as well; the relation G ∈ m.a.p. follows. �

Now in 3.6–3.18 we clarify what we do and do not know about the classes
of groups mentioned in Theorem 3.5. (The reader familiar with the literature
may recall that a group as hypothesized in Lemma 3.6 is referred to frequently
as a group with no small subgroups.)

Lemma 3.6. Let G be a nontrivial (Hausdorff) topological group for which
some U ∈ N(1G) contains no subgroup other than {1G}. Then there is no
n < ω such that G ∈ SSGP(n).

Proof. Clearly G /∈ SSGP(0). Suppose there is a minimal n > 0 such that
G ∈ SSGP(n), and let U ∈ N(1G) be as hypothesized. Then the only choice for

H (as required in Definition 3.3) is H := {{1G}}, yielding H = 〈∪H〉 = {1G}.
Thus, G/H = G ∈ SSGP(n − 1), which contradicts the assumption that n is
minimal. �

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W. W. Comfort and F. R. Gould

Evidently Lemma 3.6 furnishes a plethora of topological groups which belong
to none of the classes SSGP(n). The interested reader will readily augment the
following incomplete list.

Corollary 3.7. Let G be a nontrivial topological group which is either discrete
or a Lie group. Then there is no n < ω such that G ∈ SSGP(n).

For another statement in parallel with Corollary 3.7, see Theorem 3.16 be-
low.

Definition 3.8. Let G be a group and let 1G /∈ C ⊆ G. Then C cogenerates
G if every subgroup H of G such that |H| > 1 satisfies H ∩ C 6= ∅.

Theorem 3.9. Let G be a nontrivial finitely cogenerated topological group.
Then there is no n < ω such that G ∈ SSGP(n).

Proof. Let C be a finite set of cogenerators for G, and choose U ∈ N(1G) such
that U

⋂
C = ∅. Then U contains no subgroup other than {1G}, and the

statement follows from Lemma 3.6. �

We have noted for every n < ω the class-theoretic inclusion SSGP(n) ⊆
m.a.p. On the other hand, there are many examples of G ∈ m.a.p. such that
G ∈ SSGP(n) for no n < ω. But more is true: There are groups which admit
an m.a.p. topology which do not for any n < ω admit an SSGP(n) topology.
Indeed from Corollary 3.14 and Theorem 3.9 respectively we see that the groups
G = Z and G = Z(p∞) admit no SSGP(n) topology; while Ajtai, Havas, and
Komlós [1], and later Zelenyuk and Protasov [41], have shown the existence of
m.a.p. topologies for Z and for Z(p∞).

In Theorem 3.13 we show that in the context of abelian groups, Theorem 3.9
can be strengthened. We use the following basic facts from the theory of abelian
groups.

Lemma 3.10. A finitely cogenerated group is the direct sum of finite cyclic
p-groups and groups of the form Z(p∞), hence is torsion ([12](3.1 and 25.1)).

Lemma 3.11. A finitely generated abelian group is the direct sum of cyclic
groups and cyclic torsion groups ([12](15.5)).

Lemma 3.12. If G is a finitely generated abelian group and H is a torsionfree
subgroup then there is a decomposition G = K ⊕ T where T is the torsion
subgroup of G, K is torsionfree, and H ⊆ K ([12](Chapter III)).

Theorem 3.13. A nontrivial abelian group which is the direct sum of a finitely
generated group and a finitely cogenerated group does not admit an SSGP(n)
topology for any n < ω.

Proof. We proceed by induction on the torsionfree rank, r0(G). Suppose first
that r0(G) = 0. Then G is finitely co-generated and does not admit an SSGP(n)
topology by Theorem 3.9. Now suppose that the theorem has been proved up to
rank r−1 and we have r0(G) = r ≥ 1 and G = F ⊕T , with F finitely generated
and T finitely co-generated. Using Lemmas 3.10 and 3.11, we rewrite G in the

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 146



Some classes of minimally almost periodic topological groups

form G = F ′
⊕

T ′ where T ′ is the (finitely cogenerated) torsion subgroup and
F ′ is free. Then r0(F

′) = r0(G) = r. Let a ∈ F
′ be an element of infinite order

and choose U ∈ N(0) so that a /∈ U and so that U
⋂
C = ∅ where C is a finite

set of cogenerators of T ′ (with 0 /∈ C). If all subgroups contained in U are
torsion, then each such subgroup is a subgroup of T ′ and is therefore the zero
subgroup, since it misses C. In that case, by Lemma 3.6 there is no n < ω such
that G ∈ SSGP(n). Alternatively, if U contains a cyclic subgroup H of infinite
order, we have r0(H) > 0. Furthermore, since H ⊆ U, we have H

⋂
T ′ = {0}.

It follows from Lemma 3.12 that there is a decomposition G = F ′′
⊕

T ′ which
is isomorphic to the original decomposition and is such that H ⊆ F ′′. Since
a quotient of a finitely generated group is also finitely generated, it follows
that F ′′/H is finitely generated. Then we have G/H = (F ′′/H)

⊕
T ′. Since

r0(G) = r0(H) + r0(G/H) (see for example [12](§16, Ex. 3(d))), we also have
r0(G/H) < r. Further, the group G/H is nontrivial since H ⊆ U and a /∈ U. It
follows from the induction assumption that G/H does not admit an SSGP(n)
topology, and so by Theorem 3.15(b) (below), neither does G. �

Corollary 3.14. The group Z does not admit an SSGP(n) topology for any
n < ω.

The following theorem lists several inheritance properties for groups in the
classes SSGP(n).

Theorem 3.15.

(a) If K is a closed normal subgroup of G, with K ∈ SSGP(n) and G/K ∈
SSGP(m) then G ∈ SSGP(m + n).

(b) If G ∈ SSGP(n) and π : G ։ B is a continuous homomorphism from
G onto B, then B ∈ SSGP(n). In particular, if K is a closed normal
subgroup of G ∈ SSGP(n) then G/K ∈ SSGP(n).

(c) If K is a dense subgroup of G and K ∈ SSGP(n) then G ∈ SSGP(n).
(d) If Gi ∈ SSGP(n) for each i ∈ I then

⊕
i∈I

Gi ∈ SSGP(n) and
∏

i∈I
Gi ∈

SSGP(n).

Proof. We proceed in each case by induction on n. Each statement is trivial
when n = 0. We address (a), (b), (c) and (d) in order, assuming in each case
for 1 ≤ n < ω that the statement holds for n − 1.

(a) Let U ∈ N(1G), so that U ∩ K ∈ N(1K). Then there is a family H of
subgroups of K such that

⋃
H ⊆ U ∩ K and K/H ∈ SSGP(n − 1) where H :=

〈∪H〉
K

= 〈∪H〉
G
. Since G/K is topologically isomorphic with (G/H)/(K/H),

we have (G/H)/(K/H) ∈ SSGP(m) along with K/H ∈ SSGP(n − 1). Then
by the induction hypothesis, G/H ∈ SSGP(m + n − 1). Since

⋃
H ⊆ (U) with

U arbitrary, we have G ∈ SSGP(m + n), as required.
(b) Given G ∈ SSGP(n) and continuous π : G ։ B, let U ∈ N(1B).

Then π−1[U] ∈ N(1G) and there is a family H of subgroups of G such that⋃
H ⊆ (π−1[U]) and G/〈∪H〉 ∈ SSGP(n−1). Let H̃ be the family of subgroups

of B given by H̃ := {π[L] : L ∈ H}. Then
⋃
H̃ ⊆ (U). Set H := 〈∪H〉 and set

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W. W. Comfort and F. R. Gould

H̃ := 〈∪H̃〉. Then H̃ is normal in B since by assumption H is normal in G. By

invoking the induction hypothesis we will show that B/H̃ ∈ SSGP(n − 1) and

thus that B ∈ SSGP(n). Note that H ⊆ π−1[H̃] since 〈∪H〉 ⊆ π−1[〈∪H̃〉]; fur-

ther, π−1[H̃] is closed. We have then that G/H ∈ SSGP(n−1) so by induction,

(G/H)/
(
π−1[H̃]/H

)
∈ SSGP(n − 1) and this is topologically isomorphic with

G/π−1[H̃] by the second topological isomorphism theorem. Now, we claim that

the algebraic isomorphism π̃ : G/π−1[H̃] → B/H̃ induced by π is continuous

(though it may not be open). Clearly, π maps cosets of π−1[H̃] to cosets of H̃.

If Ṽ is an open union of cosets of H̃, then π−1[Ṽ ] is an open union of cosets

of π−1[H̃] and the claim follows. Again, by the induction hypothesis, since π̃

is continuous and surjective, we now conclude that B/H̃ ∈ SSGP(n − 1) and
thus B ∈ SSGP(n), as required.

(c) Given G and K as hypothesized, let U ∈ N(1G). Since U ∩ K ∈ N(1K),
there is a family H of subgroups of K such that

⋃
H ⊆ U ∩ K and K/H ∈

SSGP(n−1), where H := 〈
⋃
H〉

K
. Note that H = H

G
∩K. Let φ : KH/H →

K/(K ∩ H) be the natural isomorphism from the first (algebraic) isomorphism
theorem for groups. The corresponding theorem for topological groups says
that φ is an open map, i.e., φ−1 is a continuous map. Then from part (a) of
this theorem, KH/H ∈ SSGP(n − 1). Now KH/H is dense in G/H, because
the subset of G that projects onto the closure of KH/H must be closed and
must contain KH. Then G/H ∈ SSGP(n − 1) by the induction hypothesis.

Since H
G
= 〈

⋃
H〉

G
we have G ∈ SSGP(n), as required.

(d) Since
⊕

i∈I
Gi is dense in Πi∈I Gi, it suffices by part (c) to treat the

case G :=
⊕

i∈I
Gi. Let U ∈ NG(1G), say U =

⊕
i∈I

Ui where Ui ∈ N(1Gi)
and Ui = Gi for i > NU. Since each Gi ∈ SSGP(n), we have, for each i ∈ I, a
family Hi of subgroups of Gi such that

⋃
Hi ⊆ Ui and Gi/Hi ∈ SSGP(n − 1)

where Hi := 〈∪ Hi〉. Now consider the family of subgroups of G given by
H := {L ⊆ G : L =

⊕
i∈I

Li with Li ∈ Hi}. Then
⋃
H ⊆ U, 〈∪H〉 is

identical to
⊕

i∈I
〈∪Hi〉, and H := 〈∪H〉 is identical to

⊕
i∈I

Hi. We also have
that G/H is topologically isomorphic with

⊕
i∈I

Gi/Hi (cf. [24](6.9)). From
the induction hypothesis we have G/H ∈ SSGP(n − 1), so G ∈ SSGP(n), as
required. �

We give a noteworthy consequence of Theorem 3.15(b).

Theorem 3.16. Let G be a topological group which contains a proper open
normal subgroup. Then there is no n < ω such that G ∈ SSGP(n).

Proof. If G is a counterexample with proper open normal subgroup U, then
by Theorem 3.15(b) we have G/U ∈ SSGP(n) with G/U discrete, contrary to
Corollary 3.7. �

Remark 3.17. Certain other tempting statements of inheritance or permanence
type, parallel in spirit to those considered in Theorem 3.15, do not hold in
general. We give some examples.

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Some classes of minimally almost periodic topological groups

(a) We show below, using a construction of Hartman and Mycielski [22]
and of Dierolf and Warken [6], that a closed subgroup of an SSGP
group may lack the SSGP(n) property for every n < ω. Indeed, every
topological group can be embedded as a closed subgroup of an SSGP
group (Theorem 3.20).

(b) The conclusion of part (a) of Theorem 3.15 can fail when s < m + n
replaces m + n in its statement. For example, the construction used in
Lemma 4.5 shows that a topological group G /∈ SSGP(n) may have a
closed normal subgroup K ∈ SSGP(1) with also G/K ∈ SSGP(n), so
s = m + n is minimal when m = 1. We did not pursue the issue of
minimality of m + n in Theorem 3.15(a) for arbitrary m, n > 1.

(c) The converse to Theorem 3.15(c) can fail. In [19] a certain monothetic
m.a.p. group constructed by Glasner [18] is shown to have SSGP, but
we noted above in Corollary 3.14 that Z admits an SSGP(n) topology
for no n < ω.

In contrast to that phenomenon, it should be mentioned that (as has
been noted by many authors) in the context of m.a.p. groups, a dense
subgroup H of a topological group G satisfies H ∈ m.a.p. if and only
if G ∈ m.a.p. Thus in particular in the case of Glasner’s monothetic
group, necessarily the dense subgroup Z inherits an m.a.p. topology.

We now restrict our discussion to abelian groups and to the class SSGP=
SSGP(1), and examine which specific abelian groups do and do not admit an
SSGP topology. We have already noted (Theorem 3.13) that the product of a
finitely cogenerated abelian group with a finitely generated abelian group does
not admit an SSGP topology even though it may admit an m.a.p. topology.
We now give additional examples of abelian groups which admit not only an
m.a.p. topology but also an SSGP topology.

Theorem 3.18. The following abelian groups admit an SSGP topology.

(a) Q, and those subgroups of Q in which some primes are excluded from
denominators, as long as an infinite number of primes and their powers
are allowed;

(b) Q/Z and Q′/Z where Q′ is a subgroup of Q as in described in (a);
(c) direct sums of the form

⊕
i<ω

Zpi where the primes pi all coincide or
all differ;

(d) Z(ω) (the direct sum);
(e) Zω (the full product);
(f) G(α) for |G| > 1 and α ≥ ω;
(g) F α for 1 < |F | < ω and α ≥ ω;
(h) arbitrary sums and products of groups which admit an SSGP topology.

Item (h) is a special case of Theorem 3.15(d), and item (e) is demonstrated in
the second author’s paper [20]. The “coincide” case of item (c) follows from item
(f), the “differ” case is established below in Theorem 4.9. Theorem 3.22((c) and
(d)) below demonstrates the validity of item (f) for ω ≤ α ≤ c. This together

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W. W. Comfort and F. R. Gould

with (h) gives (d) and (f) in full generality. Item (g) then follows from the
relation F α ≃

⊕
2α F ([12](8.4, 8.5)). The remaining items are demonstrated

in [19].
We note that from items (a) and (h) of Theorem 3.18 and the familiar

algebraic structure theorem R =
⊕

c
Q ([12](p.1̇05), [24](A.14)) it follows that

R admits an SSGP topology.
There are many examples of nontrivial SSGP(1) groups (that is, of SSGP

groups). It has been shown by Hartman and Mycielski [22] that every topolog-
ical group G embeds as a closed subgroup into a connected, arcwise connected
group G∗; two decades later Dierolf and Warken [6], working independently
and without reference to [22], found essentially the same embedding G ⊆ G∗

and showed that G∗ ∈ m.a.p.. Indeed the arguments of [6] show with minimal
additional effort that G∗ ∈ SSGP (of course with property SSGP not yet hav-
ing been defined or named). We now describe the construction and we supply
briefly the necessary details.

Definition 3.19. Let G be a Hausdorff topological group. Then algebraically
G∗ is the group of step functions f : [0, 1) → G with finitely many steps,
each of the form [a, b) with 0 ≤ a < b ≤ 1. The group operation is pointwise
multiplication in G. The topology T on G∗ is the topology generated by (basic)
neighborhoods of the identity function 1G∗ ∈ G

∗ of the form

N(U, ǫ) := {f ∈ G∗ : λ({x ∈ [0, 1) : f(x) /∈ U}) < ǫ},

where ǫ > 0, U ∈ NG(1G), and λ denotes the usual Lebesgue measure on [0, 1).

Theorem 3.20. Let G be a topological group. Then

(a) G is closed in G∗ = (G∗, T );
(b) G∗ is arcwise connected; and
(c) G∗ ∈ SSGP.

Proof. Note first that the association of each x ∈ G with the function x∗ ∈ G∗

(the function given by x∗(r) := x for all r ∈ [0, 1)) realizes G algebraically as
a subgroup of G∗. Furthermore the map x → x∗ is a homeomorphism onto its
range, since for ǫ < 1, U ∈ N(1G) and x ∈ G one has

x ∈ U ⇔ x∗ ∈ N(U, ǫ).

(a) Let f0 ∈ G
∗ and f0 /∈ G. There are distinct (disjoint) subintervals of

[0, 1) on which f0 assumes distinct values g0, g1 ∈ G respectively. By
the Hausdorff property there is U ∈ N(1G) such that g1U ∩ g2U =
∅. Choose ǫ smaller than the measure of either of the two indicated
intervals. Then f0N(U, ǫ) is a neighborhood of f0 such that f0N(U, ǫ)∩
G = ∅. Therefore, G is closed in G∗.

(b) Let f ∈ G∗ and for each t ∈ [0, 1) define ft : [0, 1) → G by ft(x) = f(x)
for 0 ≤ x < t and ft(x) = 1G for t ≤ x < 1; and define f1 := f.
Then t 7→ ft is a continuous map from [0, 1] to G

∗ such that f0 = 1G∗

and f1 = f. To show that the map is continuous, let ftN(U, ǫ) be a

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Some classes of minimally almost periodic topological groups

basic neighborhood of ft and let s ∈ (t − ǫ/4, t + ǫ/4) ∩ [0, 1]. Then
fs ∈ ftN(U, ǫ), since

λ({x ∈ [0, 1) : fs(x) − ft(x) ∈ U}) < ǫ.

We conclude that G∗ is arcwise connected.
(c) Let N(U, ǫ) ∈ N(1G∗), and for each interval I = [t0, t1) ⊆ [0, 1) with

t1 − t0 < ǫ let

F(I) := {f ∈ G∗ : f is constant on I, f ≡ 1G on [0, 1)\I}.

Then F(I) is a subgroup of G∗ and F(I) ⊆ N(U, ǫ), and with Hǫ :=
{F(I)} we have that each f ∈ G∗ is the product of finitely many

elements from
⋃
Hǫ—i.e., f ∈ 〈

⋃
Hǫ〉 ⊆ 〈

⋃
Hǫ〉. It follows that G

∗ ∈
SSGP, as asserted.

�

For later use we identify certain subgroups G∗A of G
∗ that retain properties

(a) and (c) (but not (b)) of Theorem 3.20. When G and A are countable
the group G∗A also is countable. An equivalent definition and some related
consequences can also be found in [7] and in [19] (Definition 2.3.1). Here is the
relevant definition.

Definition 3.21. Let G be a topological group and let A ⊆ [0, 1) where A
is dense in [0, 1) and 0 ∈ A. Then G∗A = (G

∗

A, T ) is the subgroup of (G
∗, T )

obtained by restriction of step functions on [0, 1) to those steps [a, b) such that
a, b ∈ A ∪ {1}, a < b.

Theorem 3.22. Let G be a topological group. Then

(a) G is closed in G∗A = (G
∗

A, T );
(b) G∗A is dense in G

∗;
(c) G∗A ∈ SSGP; and
(d) if G is abelian, then the groups G∗A, G

(α) (with α = |A|) are isomorphic
as groups.

Proof. With the obvious required change, the proofs of (a) and (c) coincide
with the corresponding proofs in Theorem 3.20.

(b) Let f ∈ G∗ have n steps (n < ω) and let f · N(U, ǫ) ∈ NG∗(f). Then

there is f̃ ∈ f ·N(U, ǫ)∩G∗A such that f̃ has step end-points in A∪{1},
each within ǫ/n of the corresponding end-point for f.

(d) We give an explicit isomorphism. G(α) can be expressed as the set of
functions φ : A → G with finite support and pointwise addition. Each
such function is the sum of finitely many elements of the form φa,g with
a ∈ A, g ∈ G, φa,g(a) = g and φa,g(x) = 0 for x 6= a. Now we define
corresponding functions fa,g ∈ G

∗

A. Let f0,g(x) = g for all x ∈ [0, 1),
and for a > 0 let fa,g be the two-step function defined by fa,g(x) = g
for 0 ≤ x < a and fa,g(x) = 0 for a ≤ x < 1. Then the map φa,g 7→ fa,g
extends linearly to an isomorphism from G(α) onto G∗A.

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W. W. Comfort and F. R. Gould

�

Remark 3.23. Note that Theorem 3.15(c) cannot be used to prove (c) from
(b) in Theorem 3.22. Note also that the isomorphism given in the proof of
(d) provides a way of imposing an SSGP topology on G(α) for ω ≤ α ≤ c and
G an abelian group. When G is nonabelian the corresponding mapping still
exists but it need not be an isomorphism. In that case, each element of G∗A is
a product of two-step functions, but the order in which they are multiplied can
affect the product.

Some other SSGP groups arise as a consequence of the following fact.

Theorem 3.24. Let G = (G, T ) be a (possibly nonabelian) torsion group of
bounded order such that (G, T ) has no proper open subgroup. Then G ∈ SSGP.

Proof. There is an integer M which bounds the order of each x ∈ G, and then
N := M! satisfies xN = 1G for each x ∈ G.

We must show: Each U ∈ N(1G) contains a family H of subgroups such
that 〈

⋃
H〉 is dense in G. Given such U, let V ∈ N(1G) satisfy V

N ⊆ U. For
each x ∈ V we have xk ∈ U for 0 ≤ k ≤ N, hence x ∈ V ⇒ 〈x〉 ⊆ U. Thus with
H := {〈x〉 : x ∈ V } we have: H is a family of subgroups of U (that is, of subsets
of U which are subgroups of G). Then V ⊆

⋃
H, so G = 〈V 〉 ⊆ 〈

⋃
H〉—the

first equality because 〈V 〉 is an open subgroup of G. �

In Corollaries 3.25 and 3.28 we record two consequences of Theorem 3.24.

Corollary 3.25. If (G, T ) is a (possibly nonabelian) connected torsion group
of bounded order, then (G, T ) ∈ SSGP.

Proof. A connected group has no proper open subgroup, so Theorem 3.24 ap-
plies. �

Lemma 3.26. Let G ∈ m.a.p. and G abelian. Then G does not contain a
proper open subgroup.

Proof. Suppose that H is a proper open subgroup of G. Since G/H is a nontriv-
ial abelian discrete (and therefore locally compact) group, there is a nontrivial
(continuous) homomorphism φ : G/H → T. Then the composition of φ with
the projection map from G to G/H is a nontrivial continuous homomorphism
from G to a compact group, contradicting the m.a.p. property of G. �

Remark 3.27. We are grateful to Dikran Dikranjan for the helpful reminder
that Lemma 3.26 fails when the “abelian” hypothesis is omitted. Examples
to this effect abound, samples including: (a) the infinite algebraically simple
groups whose only group topology is the discrete topology, as concocted by
Shelah [38] under [CH], and by Hesse [23] and Ol′shanskĭı [30] (and later by
several others) in [ZFC]; and (b) such matrix groups as SL(2, C), shown by von
Neumann [27] to be m.a.p. even in the discrete topology (the later treatments
[28], [24](22.22(h)) and [2](9.11) of this specific group follow closely those of
[27]). In connection with this comment, note however that Theorem 3.16 above
is an appropriate nonabelian analogue of Lemma 3.26 for SSGP(n) groups.

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Some classes of minimally almost periodic topological groups

Corollary 3.28. For an abelian torsion group G of bounded order, these con-
ditions are equivalent for each group topology T on G.

(a) (G, T ) ∈ SSGP;
(b) (G, T ) ∈ m.a.p.; and
(c) (G, T ) has no proper open subgroup.

Proof. The implications (a) ⇒ (b), (b) ⇒ (c), and (c) ⇒ (a) are given respec-
tively by Theorem 3.5, Lemma 3.26, and Theorem 3.24. �

Remark 3.29. It is worthwhile to note that connected torsion groups of bounded
order, as hypothesized in Theorem 3.25, do exist. Here we give two quite
different proofs that for 0 < n < ω there is a nontrivial connected torsion group
of exponent n. Of these (A), as remarked by the referee, uses the construction
given in Theorem 3.20; while (B), drawing freely on the expositions [37] and
[2](2.3–2.4), derives from the “free topological group” constructions first given
by Markov [25], [26] and Graev [21].

(A) Let G be a group of exponent n (for example, G = Z(n)) and define
G∗ as in Definition 3.19. Since algebraically G∗ ⊆ G[0,1), also G∗ has
exponent n; and G∗ is connected by Theorem 3.20(b).

(B) Let X be a Tychonoff space and let

G := {ΣNi=1 kixi : ki ∈ Z, N < ω, xi ∈ X}

be the free abelian topological group on the alphabet X with 0G = 0,
and for continuous f : X → H with H a topological abelian group
define f : G → H by f(ΣNi=1 kixi) = Σ

N
i=1 kif(xi) ∈ H. It is easily

checked, as in the sources cited, that (a) in the (smallest) topology

T making each such f continuous, (G, T ) is a (Hausdorff) topological
group; (b) the map x → 1 · x from X to G maps X homeomorphically
onto a closed topological subgroup of G; and (c) G is connected if (and
only if) X is connected.

Now take X compact connected and fix n such that 0 < n < ω. It suffices to
show that (1) nX is a proper closed subset of G, and (2) every proper closed
subset F ⊆ X generates a proper closed subgroup 〈F〉 of (G, T ); for then the
group G/〈nX〉 will be as desired, since a ∈ G ⇒ na ∈ 〈nX〉.

(1) nX is compact in G, hence closed. Define f0 : X → R by f0 ≡ 1;
then f0 ≡ n on nX, while for x ∈ X we have f0((n + 1)x) = n + 1, so
(n + 1)x /∈ nX.

(2) Given x ∈ X\F choose continuous f1 : X → R such that f1(x) = 1,
f1 ≡ 0 on F . Then f1 ≡ 0 on 〈F〉 and f1(x) = 1, so x = 1 · x /∈ 〈F〉;
so 〈F〉 is proper in G. If a = ΣNi=1 kixi ∈ G\〈F〉 there is i0 such that
xi0 /∈ F , and with continuous f2 : X → R such that f2(xi0) = 1,
f2(xi) = 0 for i 6= i0 and f2 ≡ 0 on F we have f2(a) = ki0 and f2 ≡ 0

on 〈F〉. Then U := f2
−1

(ki0 −1/3, ki0 +1/3) ∈ NG(a) and U ∩〈F〉 = ∅;
so 〈F〉 is closed in G.

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W. W. Comfort and F. R. Gould

4. SSGP(n) Groups: Some Specifics

The question naturally arises whether for n < ω the class-theoretic inclu-
sion SSGP(n) ⊆ SSGP(n + 1) is proper. That issue is addressed below in
Theorem 4.6. En route to that we describe one of the earliest examples of an
abelian m.a.p. group, constructed by Prodanov [33]; his proof illustrates use
of the SSGP(2) property to prove m.a.p. (Of course, the classes SSGP(n) had
not been formally defined in 1980.) We credit the referee with posing a useful
question which allowed us eventually to prove that Prodanov’s group (G, T )
satisfies not only (G, T ) ∈ SSGP(2) (Theorem 4.2) but even (G, T ) ∈ SSGP(1)
(Theorem 4.3); this corrects a misstatement given in [19] and in an early version
of this paper.

Algebraically, Prodanov’s group G is the group G := Z(ω) =
⊕

ω
Z. We

begin our verification that (G, T ) is as desired by quoting directly from Pro-
danov [33]. For this, denote by {em : 1 ≤ m < ω} the canonical basis for G
and use induction to define a sequence of finite subsets of Z(ω):

“Let A1 = {e1−e2, e2}, and suppose that the sets A1, A2, . . . , Am−1
(m = 2, 3, . . .) are already defined. By αm we denote an in-
teger so large that the s-th co-ordinates of all elements of
A1 ∪ A2 ∪ . . . ∪ Am−1 are zero for s ≥ αm. Now we define
Am to consist of all differences

(1) ei+kαm − ei+(k+1)αm (1 ≤ i ≤ m, 0 ≤ k ≤ 2
m−1 − 1)

and of the elements
(2) ei+2m−1αm (1 ≤ i ≤ m) .

Thus the sequence {Am}
∞

m=1 is defined.
Now for arbitrary n ≥ 1 we define

(3) Un := (n+1)!Z
(ω) ±An ±2An+1 ±. . .±2

lAn+l ±
. . . .”

(By the notation of (3) Prodanov means that Un consists of those elements of
Z(ω) which can be represented as a finite sum consisting of an element divisible
by (n + 1)! plus at most one element of An with arbitrary sign, plus at most
two elements of An+1 with arbitrary signs, plus at most four elements of An+2
with arbitrary signs, and so on.)

“It follows directly from that definition that the sets Un are
symmetric with respect to 0, and that Un+1 + Un+1 ⊂ Un (n =
1, 2, . . .). Therefore they form a fundamental system of neigh-
borhoods of 0 for a group topology T on Z(ω).”

Since we need it later, we give a careful proof of an additional fact outlined
only briefly by Prodanov [33].

Theorem 4.1. The group Z(ω) with the group topology T defined above is
Hausdorff.

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Some classes of minimally almost periodic topological groups

Proof. It suffices to show
⋂

n<ω
Un = {0}.

Let 0 6= g =
∑

i
aiei where the ei form the canonical basis. Then there is

a least integer r such that ai = 0 for i ≥ r + 1, and there is a least integer
s such that (s + 1)! does not divide g. Let n := max(r, s). We claim that
if n > 0 then g /∈ Un. Suppose otherwise. Since (n + 1)! does not divide
g, there is some p ≤ n such that (n + 1)! does not divide ap. Thus any
representation of g in the form (3) must include, for at least one m > n, one
or more terms of the form ±(ep − ep+αm), all with the same sign. This means
that the components ±ep+αm must be cancelled by the same components from
additional terms of the form ±(ep+αm − ep+2αm). This chain of implications
continues until k reaches its maximum value, 2m−1 − 1, with the inclusion of
terms ±(ep+(2m−1−1)αm − ep+2m−1αm). Finally, the components ±ep+2m−1αm
must be cancelled by terms of type (2) with i = p and the same value of m.
This means that we have necessarily included at least 2m−1 + 1 elements from
Am in our expansion of g, contradicting the requirement that no more than
2m−n elements of Am be included as summands for such representations of
g ∈ Un. �

Theorem 4.2. Prodanov’s group (G, T ) satisfies (G, T ) ∈ SSGP(2).

Proof. To see that (Z(ω), T ) ∈ SSGP(2), we show first that each Un generates
Z(ω). It suffices to show that each ei ∈ 〈Un〉 (1 ≤ i < ω). Choosing m such
that m ≥ i and m ≥ n, we have from (1) and (2) above that

ei = [Σ
2m−1−1
k=0 (ei+kαm − ei+(k+1)αm)] + ei+2m−1αm ∈ 〈Am〉 ⊆ 〈Um〉 ⊆ 〈Un〉.

Thus (Z(ω), T ) has no proper open subgroups, so with Hn := (n + 1)!Z
(ω) and

Gn := Z
(ω)/Hn we have that Hn ⊆ Un, Gn is of bounded order, and also

Gn has no proper open subgroups. Then Gn ∈ SSGP by Theorem 3.24, so
(Z(ω), T ) ∈ SSGP(2). �

Prodanov proves explicitly that (G, T ) ∈ m.a.p.. Theorem 4.2 exploits his
argument insofar as it applies to our more demanding context. Then of course,
the following Theorem 4.3 strengthens Theorem 4.2.

Theorem 4.3. Prodanov’s group (G, T ) satisfies (G, T ) ∈ SSGP(1).

Proof. Since we have already shown that (G, T ) is Hausdorff, it remains only
to show that every Un ∈ N0G contains a family H of subgroups such that

G = 〈
⋃
H〉. In fact, we show that H may be chosen so that 〈

⋃
H〉 = G. With

n given, choose k such that 2k ≥ (n + 1)! − 1 and let x ∈ Am with m := n + k.
We claim that 〈x〉 ⊆ Un. Since Un contains elements formed from sums which
include up to 2k elements from An+k, each y ∈ 〈x〉 can be written in the
form y = rx + sx with r, s ∈ Z, where rx ∈ (n + 1)!Z(ω) and sx ∈ 2m−nAm.
Since ei ∈ 〈Am〉 for all i and m (by the proof of Theorem 4.2), we have that
ei ∈ 〈

⋃
H〉 for every i < ω, where H includes all subgroups of the form 〈x〉 with

x ∈ Am and 2
m−n ≥ (n + 1)! − 1. �

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W. W. Comfort and F. R. Gould

Now as promised we construct a family of topological groups demonstrating
that the class-theoretic inclusion SSGP(n) ⊆ SSGP(n + 1) is proper for each
n. For n = 0 this is clear since we already have many examples of nontrivial
groups in the class SSGP(1). The strategy is to find particular groups Gn ∈
SSGP(n) and H ∈ SSGP(1) and then construct a topology for H ⊕ Gn in
such a way that (a) H with its original topology is a closed normal subgroup,
(b) (H ⊕ Gn)/H is topologically isomorphic with Gn, (c) H ⊕ Gn /∈ SSGP(n),
and (d) H ⊕ Gn ∈ SSGP(n + 1). Such a construction in the case n = 1 was
given in the second author’s dissertation [19]; our task here is to generalize that
construction to arbitrary n.

The properties of Gn that will be required in the induction step are that
Gn = (Gn, Tn) is abelian, countable and torsionfree, with the group topology
Tn defined by a metric. For convenience, we assume that the maximum distance
is 1. These properties are satisfied in the case n = 0 by the trivial group, but
it is more illuminating to begin the induction with G1 rather than G0.

Let H be the topological group Z∗A as in Definition 3.21, where Z has the
discrete topology and

A := {x ∈ [0, 1) : x = t
2m

for m, t < ω and 0 ≤ t < 2m}.

Set G1 = H. It is clear that G1 is abelian, countable and torsionfree, and is
not the trivial group. G1 also has SSGP(1) (Theorem 3.22). The topology on
H = G1 = Z

∗

A, defined as in Definitions 3.19 and 3.21 can be seen as a metric
topology given by the norm ‖h‖ := λ(Supp(h)) for h ∈ H, where Supp(h) is
the support of h as a function on [0, 1). (Here the “norm” designation follows
historical precedent; we use it both out of respect and for convenience, but we
do not require that ‖Ng‖ = |N| · ‖g‖.)

Fix n > 1 and suppose there is a countable, torsionfree abelian group Gn−1
with a metric ρ that defines a group topology on Gn−1 such that Gn−1 ∈
SSGP(n − 1) and Gn−1 /∈ SSGP(n − 2). Now, define (algebraically) Gn :=
H ⊕Gn−1; we give Gn a metric topology Tn which is different from the product
topology, using a technique taken from M. Ajtai, I. Havas, and J. Komlós [1].
We create a metric group topology on Gn starting with a function ν : S → R

+,
where S is a specified generating set for Gn with 0 /∈ S. S will typically be
highly redundant as a generating set. We refer to ν together with the generating
set S as a “provisional norm” (in terms of which a norm on Gn will be defined).
For x ∈ Gn we write x = (h, g) with h ∈ H and g ∈ Gn−1.

We designate a double sequence of generating functions em,t ∈ H for m, t <
ω and t < 2m:

em,t(x) = 1 for x ∈
[

t
2m

, t+1
2m

)
and em,t(x) = 0 otherwise.

We note that ‖p · em,t‖ =
1
2m

for all p ∈ Z, p 6= 0. We also name a basic set
of neighborhoods of 0 ∈ Gn−1 and we label all the non-zero elements in each
neighborhood:

Um := {g ∈ Gn−1 : ‖g‖ ≤
1
2m

} for m < ω

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 156



Some classes of minimally almost periodic topological groups

Um\{0} = {gm,t : t < ω}.

In addition, let

r(m, t) : ω × ω ։ ω

be an arbitrary bijection. We set S to be the collection of elements of the
following two types:

(1) (p · em,t , 0) for p ∈ Z\{0} and m, t < ω with t < 2
m

(2) (fr , gm,t) for m < ω, 0 ≤ t < ω and r = r(m, t),

where fr =

2r−1−1∑

i=0

er,2i .

The set of functions fr are linearly independent, so the set of elements of type
(2) is also linearly independent. We use that fact along with the fact that each
fr has support of measure

1
2
. Now, we make the provisional norm assignments,

(1) ν( (p · em,t , 0) ) = ‖p · em,t‖H =
1
2m

(2) ν( (fr , gm,t) ) = ‖gm,t‖Gn−1 ≤
1
2m

with r = r(m, t).

Notice that (1) gives the same provisional norm to every nonzero element in a
subgroup of Gn, whereas the assignments hiven by (2) are for a linearly inde-
pendent set of elements of Gn.

Now we define a seminorm ‖ · ‖ on Gn in terms of the provisional norm ν.

Definition 4.4. For g ∈ Gn,

‖g‖ := inf(

{
N∑

i=1

|αi|ν(si) : g =
N∑

i=1

αisi, si ∈ S, αi ∈ Z, N < ω

}
⋃

{1}).

This defines a seminorm because S generates Gn and because the use of
the infimum in the definition guarantees that the triangle inequality will be
satisfied. Therefore, the neighborhoods of 0 defined by this seminorm will
generate a (possibly non-Hausdorff) group topology on Gn.

Now in Lemma 4.5 we use the notation and definition just introduced.

Lemma 4.5. (a) Gn is a torsionfree, countable abelian group;
(b) the seminorm on Gn is a norm (resulting in a Hausdorff metric);
(c) Gn ∈ SSGP(n); and
(d) Gn /∈ SSGP(n − 1).

Proof. (a) is clear.
(b) To show that ‖·‖ is a norm on Gn, we need to show that for 0 6= x ∈ Gn

we have ‖x‖ > 0. Let x = (h, g). If g 6= 0 then an expansion of (h, g) by ele-
ments of S must include at least one element of type (2). For those elements,
we have ν( (fr, g) ) = ‖g‖. Because the metric on Gn−1 satisfies the triangle
inequality, any expansion of (h, g) by elements of S must yield a value of ‖g‖
or greater for the expression within the curly brackets in Definition 4.4. We
conclude that ‖(h, g)‖Gn ≥ ‖g‖Gn−1. On the other hand, if g = 0 then there
is an expression for (h, 0) in terms of elements of S of type (p · em,t, 0) such

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W. W. Comfort and F. R. Gould

that
∑N

i=1 |αi|ν(si) = ‖h‖ = λ(supp(h)) and this value is minimal. If, instead,
the expansion includes elements of type s = (fr , gm,t) then there is a minimal
ν-value such a term can have. This is because there is a minimal size, 1

2M
, for

an interval on which h is constant. An expansion of (h, 0) by elements of S that
includes an element s = (fr , gm,t) such that r(m, t) > M would also have to
include 2r−1 elements of S of the form (er,i, 0), each with coefficient −1. The

contribution of these terms to the sum
∑N

i=1 |αi|ν(si) is greater than or equal
to 1

2
. Such an expansion cannot affect the infimum. So if ‖(h, 0)‖ 6= ‖h‖H

then ‖(h, 0)‖ ≥ min({‖gm,t‖Gn : r(m, t) < M}). We conclude that ‖(h, g)‖ is
bounded away from 0 except when (h, g) = (0, 0).

(c) We show next that Gn ∈ SSGP(n). We claim first that the subgroup
H×{0} of Gn is an SSGP group in the topology inherited from Gn. This is clear
because from the provisional norm assignment, ν( (p · em,t , 0) ) = ‖p · em,t‖H,
it follows that ‖(h, 0)‖ ≤ ‖h‖H for each h ∈ H, so any ǫ-neighborhood of (0, 0)
contains a family H of subgroups such that 〈

⋃
H〉 = H×{0} = {(h, 0) : h ∈ H}.

We will show that the quotient topology for Gn/(H × {0}) coincides with
the original topology for Gn−1 (which also implies that H × {0} is closed in
Gn). For each g ∈ Gn−1 there is an h ∈ H such that ‖(h, g)‖ = ‖g‖Gn−1,
namely h = fr where g = gm,t and r = r(m, t). (There are, in general,
many such pairs m, t and corresponding fr.) On the other hand, as we showed
above, ‖(h, g)‖ ≥ ‖g‖Gn−1 for each h ∈ H. We conclude that g is in the ε-
neighborhood of 0 ∈ Gn−1 if and only if there is h ∈ H such that (h, g) is in
the ε-neighborhood of (0, 0) ∈ Gn. In other words, the neighborhoods of 0 in
Gn−1 coincide with the projections onto Gn/(H, ×{0}) of the neighborhoods of
(0, 0) in Gn. Thus the topologies of Gn/(H × {0}) and Gn−1 coincide. (Note,
however, that the subgroup topology on Gn−1 does not coincide with its origi-
nal topology.) Since by assumption Gn−1 ∈ SSGP(n − 1), we have indeed that
Gn ∈ SSGP(n).

(d) It remains to show Gn /∈ SSGP(n − 1). Suppose the contrary. Then
every ǫ-neighborhood Uǫ of (0, 0) ∈ Gn (say with ǫ <

1
4
) contains a family Kǫ

of subgroups such that Gn/ 〈
⋃
Kǫ〉 ∈ SSGP(n − 2). Let G ∈ Kǫ and (h, g) ∈ G

with g 6= 0. For |N| < ω we must have ‖(Nh, Ng)‖ < ǫ, so each (Nh, Ng) has
an expansion

(Nh, Ng) =
∑MN

i=1 αN,i(hi, 0) +
∑LN

j=1 βN,i(h
′

i, gi) such that

∑MN
i=1 η(αN,i) ν((hi, 0)) +

∑LN
j=1 |βN,i| ν((h

′

i, gi)) < ǫ

where each (hi, 0), (h
′

i, gi) ∈ S and where η : Z → {0, 1} is defined by
η(k) := 1 when k 6= 0 and η(k) := 0 when k = 0.

We consider two cases.
Case 1. In the expansion above, each coefficient βN,i has the form βN,i =

Nβ1,i. Then clearly for sufficiently large |N| we have ‖(Nh, Ng)‖ >
1
4
> ǫ.

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Some classes of minimally almost periodic topological groups

Case 2. There is N < ω for which, for some k, the expansion for (Nh, Ng)
is such that βN,k 6= Nβ1,k. For the H-component of the given expansion of
(Nh, Ng), we can write

Nh =
∑M

i=1 αN,ihi +
∑L

i=1 βN,ih
′

i

where M = max(M1, MN) and L = max(L1, LN). Multiplying the specified
expansion of (h, g) by the number N, we also have

Nh =
∑M

i=1 Nα1,ihi +
∑L

i=1 Nβ1,ih
′

i.

Equating the two expansions and rearranging, we can write
∑L

j=1(βN,j − Nβ1,j)h
′

j =
∑M

i=1(Nα1,i − αN,i)hi

where, for the specified index k, we have (βN,k − Nβ1,k)h
′

j 6= 0. Since the
h′j are linearly independent, each h

′

j that has a nonzero coefficient in the ex-
pression above must be balanced by terms on the right. This implies that∑M

i=1 η(Nα1,i−αN,i) ≥
1
2
, which in turn means that either

∑M
i=1 η(α1,i) ≥

1
4
or

∑M
i=1 η(αN,i) ≥

1
4
. We conclude that ‖(h, g)‖ ≥ 1

4
> ǫ or ‖(Nh, Ng)‖ ≥ 1

4
> ǫ,

contradicting G ∈ Kǫ.
We conclude that for ǫ < 1

4
we must have

⋃
Kǫ ⊆ H, so that 〈∪Kǫ〉 is

a closed subgroup of H. In such cases we have Gn/〈∪Kǫ〉 /∈ SSGP(n − 2)

because, by Theorem 3.15(b), Gn/〈∪Kǫ〉 ∈ SSGP(n − 2) would imply that

(Gn/〈∪Kǫ〉)/(H/〈∪Kǫ〉) ∈ SSGP(n− 2) or, equivalently, that Gn/H ≃ Gn−1 ∈
SSGP(n − 2), contrary to the induction assumption for Gn−1. �

We emphasize the essential content of Lemma 4.5.

Theorem 4.6. For 1 ≤ n < ω there is an abelian topological group G such
that G ∈ SSGP(n) and G /∈ SSGP(n − 1).

In the paragraph following Theorem 3.9 we noted the existence of abelian
topological groups G ∈ m.a.p. such that G ∈ SSGP(n) for no n < ω. (Z and
Z(p∞), appropriately topologized, are examples.) In the following corollary we
note the availability of other examples to the same effect.

Corollary 4.7. There is a group G of the form G = Πk<ω Gk such that
(a) for each k < ω there is nk < ω such that Gk ∈ SSGP(nk);
(b) there is no n such that G ∈ SSGP(n).

Proof. Using Theorem 4.6, for k < ω choose Gk ∈ SSGP(k + 1)\SSGP(k), and
set G := Πk<ω Gk. Then (a) holds with nk = k + 1. If there is n < ω such
that G ∈ SSGP(n) then since the projection πn : G ։ Gn is continuous we
would from Theorem 3.15(b) have the contradiction Gn ∈ SSGP(n); thus (b)
holds. �

Remark 4.8. Any group G satisfying the conditions of Corollary 4.7 is neces-
sarily an m.a.p. group. This is the case since each Gk ∈ SSGP(nk) ⊆ m.a.p.
and since the m.a.p. property is preserved by products.

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W. W. Comfort and F. R. Gould

While Theorem 3.13 furnishes an ample supply of well-behaved abelian
groups which admit no SSGP(n) topology, we have found that an SSGP topol-
ogy can be constructed for many of the standard building blocks of infinite
abelian groups. We now verify Theorem 3.18(c), that is, we give a construc-
tion of an SSGP topology for groups of the form G := ⊕pi Zpi (with (pi) a
sequence of distinct primes); this illustrates the method used throughout the
second-listed co-author’s thesis [19].

Using additive notation, write 0 = 0G and let {ei : i = 1, 2, ...} be the
canonical basis for G, so that p1e1 = p2e2 = . . . = piei . . . = 0. We define a
provisional norm ν, just as in the description preceding Lemma 4.5. This will
generate a norm || · || via Definition 4.4 in such a way that in the generated
topology every neighborhood of 0 contains sufficiently many subgroups to gen-
erate a dense subgroup of G. Suppose we can show that G is Hausdorff and that
each U ∈ N(0) contains a family of subgroups H such that G/H is torsion of
bounded order, where H := 〈∪H〉. Then if also G/H has no proper open sub-
group, we have from Theorem 3.24 that G/H ∈ SSGP, so that G ∈ SSGP(2).
Our plan is to choose a norm so that G/H, and thus G/H, is finite. Then if G
contains no proper open subgroup, it is necessarily the case that H = G. We
will, then, define a norm ‖ · ‖ so that

(1) Every neighborhood of 0 contains a set of subgroups of G whose union
generates a subgroup H such that G/H is finite;

(2) G has no proper open subgroups, or equivalently, every neighborhood
of 0 generates G; and

(3) G is Hausdorff.

To that end, define ν(men) =
1
n
for every m < ω such that m 6≡ 0 mod pn.

We then obtain (1) because the neighborhood of 0 defined by ||g|| < 1
n
contains

subgroups which generate H := p1p2...pn−1G. Thus, G/H is finite, as desired.
To satisfy (2) define ên :=

∑n
i=1 ei for n < ω, and define ν(ên) :=

1
n
for each

n < ω. What then remains (the most difficult piece) is to show that G with
this topology is Hausdorff. We will then have the following result.

Theorem 4.9. Let G =
⊕

i<ω
Zpi where p1 < p2 < p3 < ... are primes. Let

S = {men : n < ω, 0 < m < pn}
⋃
{ên : n ∈ N}, with en, ên defined as above.

Let ν(men) =
1
n
for 0 < m < pn, and let ν(ên) =

1
n
. Then the norm defined

by

‖g‖ = inf

{
n∑

i=1

|αi|ν(si) : g = α1s1 + ... + αnsn, si ∈ S, αi ∈ Z, n < ω

}

generates an SSGP topology on G.

Proof. As noted above, our construction for the norm ‖ · ‖ guarantees that
every ǫ-neighborhood U of 0 generates G and also contains subgroups whose
union generates an H such that G/H is finite. Then, as also noted, if G is
Hausdorff we are done.

Suppose 0 6= g ∈ G and n is the largest nonzero coordinate index for g. We
show that ‖g‖ ≥ 1

n
. For convenience we extend the domain of ν to all formal

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Some classes of minimally almost periodic topological groups

finite sums of elements from S with coefficients from Z:

For ϕ =
∑N

i=M (aiei + biêi), let ν(ϕ) =
∑N

i=M (ηi + |bi|)
1
i

where each ηi is either 0 or 1, according as to whether or not ai ≡ 0 mod pi .
In addition, we will assume

(1) g = val(ϕ), so the formal sum ϕ evaluates to g ∈ G;
(2) each ei and each êi appears at most once in any formal sum; and
(3) 0 ≤ ai < pi for each i,

item (2) being justified by the fact that we are ultimately interested in the
norm, which minimizes ν(ϕ).

Let F(M, N) be the set of such formal sums where M is the smallest coordi-
nate index for a nonzero coefficient aM or bM and where N is the largest such
index. (Here for bi, “nonzero” indicates that bi is not a multiple of p1p2...pi.)

We want to show that ν(ϕ) ≥ 1
n
where g = val(ϕ), where ϕ =

∑N
i=M (aiei +

biêi) and where either aM or bM is nonzero and either aN or bN is nonzero. In
other words, ϕ ∈ F(M, N). Clearly ν(ϕ) ≥ 1

n
when M ≤ n.

Suppose that N = M = n + 1. Then, since the n + 1 component of g is 0
we have that an+1 + bn+1 ≡ 0 mod pn+1. Both coefficients are 0 only if g = 0,
so either both are nonzero or an+1 = 0 and bn+1 = mpn+1 for some m 6= 0.
In the first case we have ν(ϕ) ≥ 2

n+1
> 1

n
and in the second case we have

ν(ϕ) ≥
pn+1
n+1

> 1 ≥ 1
n
.

Suppose instead that M = N > n+1. In this case, we know that the (N −1)
component of g is 0. Then, since g 6= 0 can be written as ϕ = aNeN +bN êN, we
have bN = mpN−1 for some m 6= 0. But then we have ν(ϕ) ≥

pN−1
N

≥ 1 ≥ 1
n
.

Finally, we fix M and use induction on N. Assume that we have already
shown that ν(ϕ) ≥ 1

n
when ϕ ∈ F(M, Q) for n < M ≤ Q ≤ N −1, and suppose

that ϕ ∈ F(M, N). We treat three cases separately.

(a) Case 1. |bN−1 + bN| ≥ pN−1. Then

ν(ϕ) ≥
|bN−1|

N − 1
+

|bN|

N
≥

pN−1
N

≥ 1 ≥
1

n
.

(b) Case 2. bN−1 + bN = 0. Recalling that all coordinates of g vanish after
the nth, we note that (aN + bN)eN = 0, and so

bN−1êN−1 + bN êN + aNeN = 0.

This means that we can delete these terms from ϕ without affecting its
value, and with that done, our induction assumption can be applied.

(c) Case 3. |bN−1 + bN| < pN−1 and bN−1 + bN 6= 0. Again, from (aN +
bN)eN = 0 we obtain the equality

bN−1êN−1 + bN êN + aNeN = (bN−1 + bN)êN−1.

Let ϕ′ be the formal sum obtained from ϕ by replacing the three terms
on the left with the one on the right, and compare the provisional

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W. W. Comfort and F. R. Gould

norms:

ν(ϕ′) − ν(ϕ) =
|bN−1 + bN|

N − 1
−

(
|bN−1|

N − 1
+

|bN|

N
+

1

N

)
≤

|bN|

N(N − 1)
−

1

N
.

We see that this difference is negative or zero as long as |bN| ≤ N − 1.
Then, since ϕ′ ∈ F(M, N − 1), our induction assumption applies. If,
on the contrary, |bN| ≥ N, we already have ν(ϕ) ≥ 1 ≥

1
n
.

We conclude in each case that ν(ϕ) ≥ 1
n
, so G is Hausdorff, as desired. �

5. Concluding Comments

Here we discuss briefly some other classes of groups which are closely related
to the classes SSGP(n) and the class m.a.p.

Remark 5.1. In the dissertation [19], the second-listed co-author found it conve-
nient to introduce the class of weak SSGP groups (briefly, the WSSGP groups),
that is, those topological groups G = (G, T ) which contain no proper open sub-
group and have the property that for every U ∈ N(1G) there is a family H of

subgroups of G such that
⋃
H ⊆ (U), H = 〈∪H〉 is normal in G, and G/H is

torsion of bounded order. Subsequent analysis (as in Theorem 3.24 above) along
with the definitions of the classes SSGP(n) has revealed the class-theoretic in-
clusions SSGP(1) ⊆ WSSGP ⊆ SSGP(2). A consequence of Theorem 3.24
is that the Markov-Graev-Remus examples (as in Remark 3.29) are not just
WSSGP but are, in fact, SSGP. The same is true of Prodonov’s group G:
G ∈ SSGP = SSGP(1) (Theorem 4.3). From these facts we conclude that the
class of WSSGP groups contributes little additional useful information to the
present inquiry, and we have chosen to suppress its systematic discussion in
this paper.

In Theorems 3.9 and 3.13 we have identified several classes of groups which
do not admit an SSGP topology. That suggests the following natural question.

Question 5.2. What are the (abelian) groups which admit an SSGP topology?

Our work also leaves open this intriguing question:

Question 5.3. Does every abelian group which for some n > 1 admits an
SSGP(n) topology also admit an SSGP topology?

There is another important and much-studied class of groups related to the
class of m.a.p. groups, namely the class of groups whose every continuous action
on a compact space has a fixed point, the so-called fixed point on compacta
groups (hereafter, the f.p.c. groups); for basic facts, some recent developments
and a bit of history, see for example [18], [32] and [11]. (The reader will recall
that a continuous action of a topological group G on a space X is a continuous
map φ : G × X ։ X such that (a) φ(g, ·) : X ։ X is a bijection for each
g ∈ G with φ(eG, ·) = idX and (b) φ(g, φ(h, x)) = φ(gh, x) for all g, h ∈ G and
x ∈ X. A fixed point for the configuration (G, X, φ) is a point x ∈ X such that
φ(g, x) = x for all g ∈ G.) It is easy to see that every f.p.c. group is a m.a.p.

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Some classes of minimally almost periodic topological groups

group (we write simply f.p.c. ⊆ m.a.p.): Given a nontrivial homomorphism h
from a (non-m.a.p.) group G into a compact group K, the continuous map
φ : G × K ։ K given by φ(g, x) := h(g) · x ∈ K is a G-action on K with no
fixed point. The class-theoretic inclusion f.p.c. ⊆ m.a.p. is proper since, as we
remarked in Discussion 2.1(a), the group SL(2, R) is an m.a.p. group even in its
discrete topology, while Veech [39](2.2.1) has shown that every locally compact
group, m.a.p. or not, has a continuous action on a compact space such that
each non-identity element of the group moves every element of the compact
space. (This is called a “free action”.) Whether or not every abelian f.p.c.
group is a m.a.p. group, however, is a difficult long-standing open question in
abelian topological group theory raised in 1998 by Glasner [18]:

Question 5.4. Do the f.p.c. abelian groups constitute a proper subclass of
the m.a.p. abelian groups?

Even the characterization of abelian m.a.p. groups and abelian f.p.c. groups
by different “big set” conditions (see [31] and [10]) did not settle Question 5.4.
Unfortunately, and contrary to our hopes, our own work with the SSGP prop-
erty also has so far not shed light on this question. It is known [18], how-
ever, that there are f.p.c. topologies for Z, so the class-theoretic inclusion
f.p.c. ⊆ SSGP fails. We have not successfully addressed the issue of the re-
versed inclusion, so we list it as another question to be resolved:

Question 5.5. Do the SSGP groups constitute a subclass of the f.p.c groups?
What about the abelian case?

Note added in proof

Shortly after this paper had been completed in final form and accepted
for publication, we received a preprint of [9] from its authors. They build
substantially on our results, reformulating and possibly generalizing SSGP(n)
with the use of some pleasing algebraic characterizations, and extending the
concept to families SSGP(α) for ordinals α. In the process they have provided a
positive solution to our Question 5.3, and they have made considerable progress
in answering Question 5.2 for abelian groups.

Acknowledgements. We gratefully acknowledge helpful comments received
from Dieter Remus, Dikran Dikranjan, and Saak Gabriyelyan. Each of them
improved the exposition in pre-publication versions of this manuscript, and
enhanced our historical commentary with additional bibliographic references.

We are grateful also for an unusually thoughtful and detailed referee’s report,
which helped us (a) to correct an error and several minor expository ambiguities
in our early draft and (b) to reorganize and rewrite some of the proofs, most
notably the proof of Theorem 4.6.

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W. W. Comfort and F. R. Gould

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