Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

CONCERNING THE DUAL GROUP OF A DENSE SUBGROUP

W. W. COMFORT, S. U. RACZKOWSKI, AND F. JAVIER TRIGOS-ARRIETA

Abstract. Throughout this Abstract, G is a topological Abelian group and Ĝ is the space of continuous homomorphisms from G into 𝕋 in the compact-open topology. A dense subgroup D of G determines G if the (necessarily continuous) surjective isomorphism Ĝ ↠ D̂ given by h ↦ h|D is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these.

(1) There are (many) nonmetrizable, noncompact, determined groups.

(2) If the dense subgroup D_{i} determines G_{i} with G_{i} compact, then ⊕_{i} D_{i} determines Π_{i} G_{i}. In particular, if each G_{i} is compact then ⊕_{i} G_{i} determines Π_{i} G_{i}.

(3) Let G be a locally bounded group and let G^{+} denote G with its Bohr topology. Then G is determined if and only if G^{+} is determined.

(4) Let non(N) be the least cardinal κ such that some X ⊆ 𝕋 of cardinality κ has positive outer measure. No compact G with w(G) ≥ non(N) is determined; thus if non(N) = ℵ_{1} (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω.

Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is κ = non(N)? κ = ℵ_{1}?

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

Key words and phrases. Bohr compactification, Bohr topology, character, character group, Außenhofer-Chasco Theorem, compact-open topology, dense subgroup, determined group, duality, metrizable group, reflexive group, reflective group.

The first author was an invited speaker at the Ninth Prague Topological Symposium. Parts of this paper appeared in [26]. Furthermore, portions of this paper were presented

by the authors at the Ninth Prague Topological Symposium (Praha, August, 2001) and at the 2002 Annual Meeting of the American Mathematical Society (San Diego, January, 2002). A full treatment, with proofs, will appear elsewhere [9].

W. W. Comfort, S. U. Raczkowski, and F. Javier Trigos-Arrieta, Concerning the dual group of a dense subgroup, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 23–35, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204147.

0. Terminology, Notation and Preliminaries

For X a set and κ a cardinal, we write [X]^{κ} = {A ⊆ X : |A| = κ}.

For each space X = (X, T ) we write

K(X) := {K ⊆ X : K is T -compact}.

All groups considered here, whether or not equipped with a topology, are Abelian groups written additively. The identity of a group G is denoted 0 or 0_{G}, and the torsion subgroup of G is denoted tor(G). The reals, rationals, and integers are denoted ℝ, ℚ, and ℤ, respectively, and the “unit circle” group 𝕋 is the group (−\frac{1}{2}, \frac{1}{2}] with addition mod 1. Except when we specify otherwise, these groups carry their usual metrizable topology.

The symbol ℙ denotes the set of positive prime integers.

The set of homomorphisms h : G → 𝕋, a group under pointwise operation, is denoted Hom(G, 𝕋). For a subgroup A of Hom(G, 𝕋) we denote by (G, T_{A}) the group G with the topology induced by A. Evidently (G, T_{A}) is a Hausdorff topological group if and only if A separates points of G. The topology T_{A} is the coarsest topology on G for which the homomorphism e_{A} : G → 𝕋^{A} given by (e_{A}(x))_{h} = h(x) (x ∈ G, h ∈ A) is continuous. When G = (G, T ) is a topological group, the set of T -continuous functions in Hom(G, 𝕋) is a subgroup of Hom(G, 𝕋) denoted Ĝ or { \atop (G, T )}; in this case the topology TĜ is the Bohr topology associated with T , and (G, TĜ) is denoted G^{+} or

(G, T )^{+}. When { \atop (G, T )} separates points we say that G is a maximally almost periodic group and we write G = (G, T ) ∈ MAP. Whether or not (G, T ) ∈ MAP, the closure of e[G] in 𝕋Ĝ, denoted b(G) or b(G, T ), is the Bohr compactification of (G, T ).

The Bohr compactification b(G) of a topological group G is characterized by the condition that each continuous homomorphism from G into a compact Hausdorff group extends continuously to a homomorphism from b(G). From this and the uniform continuity of continuous homomorphisms it follows that if D is a dense subgroup of G then b(D) = b(G). It is conventional to suppress mention of the function eĜ and to write simply Ĝ = Ĝ^{+}. When G ∈ MAP we write G^{+} ⊆ b(G) ⊆ 𝕋Ĝ, the inclusions being both algebraic and topological.

A group G with its discrete topology is denoted G_{d}. For notational convenience, and following van Douwen [34], for every (Abelian) group G we write G^{#} = (G_{d})^{+} ⊆ b(G_{d}).

A subset S of a topological group G is said to be bounded in G if for every nonempty open V ⊆ G there is finite F ⊆ G such that S ⊆ F + V; G is locally bounded [resp., totally bounded] if some nonempty open subset of G is bounded [resp., G itself is bounded]. It is a theorem of Weil [36] that each locally bounded group G embeds as a dense topological subgroup of a locally compact group W (G), unique in the obvious sense; the group W (G) is compact if and only if G is totally bounded. We denote by LCA [resp., LBA] the class of locally compact [resp., locally bounded] Hausdorff Abelian groups. The relation LCA ⊆ MAP is a well known consequence of the Gel′fand-Raĭkov Theorem (cf. [18, 22.17]); since each subgroup S ⊆ G ∈ MAP clearly satisfies S ∈ MAP, we have in fact the relations LCA ⊆ LBA ⊆ MAP.

Lemma 0.1. Let S be a subgroup of G ∈ LBA. Then

(a) S is dual-embedded in G in the sense that each h ∈ Ŝ extends to an element of Ĝ;

(b) if h ∈ Ŝ and x ∈ G\\overline{S}^{G}, the extension k ∈ Ĝ of h may be chosen so that k(x) ≠ 0.

It follows that for each subgroup S of a group G ∈ LBA the topology of S^{+} coincides with the topology inherited by S from G^{+}. This validates the following notational convention. For S ⊆ G ∈ LBA, S not necessarily a subgroup of G, we denote by S^{+} the set S with the topology inherited from G^{+}. When G is discrete, so that G^{+} = G^{#}, we write S^{#} in place of S^{+} when S ⊆ G.

Theorem 0.2. (Glicksberg [16]). Let K ⊆ G ∈ LBA. Then K ∈ K(G) if and only if K^{+} ∈ K(G^{+}). Hence if K ∈ K(G), then K and K^{+} are homeomorphic.

Theorem 0.3. (Flor [14]. See also Reid [28]). Let G ∈ LBA and let x_{n} → p ∈ b(G) = b(W (G)) with each x_{n} ∈ G^{+} ⊆ (W (G))^{+} ⊆ b(G). Then

(a) p ∈ (W (G))^{+}, and

(b) not only x_{n} → p in (W (G))^{+} ⊆ b(G) but also x_{n} → p in W (G).

Remark 0.4. Strictly speaking, the papers cited above in connection with Theorems 0.2 and 0.3 deal with groups G ∈ LCA. Our modest generalization to the case G ∈ LBA is justified by 0.2 and 0.3 as originally given and by these facts about G ∈ LBA:

(i) G is a (dense) topological subgroup of W (G) ∈ LCA; (ii) G^{+} is a (dense) topological subgroup of (W (G))^{+}; and (iii) b(G) = b(W (G)).

In what follows, groups of the form Ĝ will be given the compact-open topology. This is defined as usual: the family

{U(K, ϵ) : K ∈ K(G), ϵ > 0}

is a base at 0 ∈ Ĝ, where for A ⊆ G one writes

U(A, ϵ) = {h ∈ Ĝ : x ∈ A ⇒ |h(x)| < ϵ}.

We have noted already that for G ∈ MAP the groups Ĝ and Ĝ^{+} are identical; that is, Ĝ = Ĝ^{+} as groups. Our principal interest in Theorem 0.2 is that for G ∈ LBA it gives a topological consequence, as follows.

Corollary 0.5. Let G ∈ LBA. Then Ĝ = Ĝ^{+} as topological groups. That is, the compact-open topology on Ĝ determined by K(G) coincides with the compact-open topology on Ĝ determined by K(G^{+}).

Acknowledgements 0.6. We acknowledge with thanks several helpful conversations relating to §6 with these mathematicians: Adam Fieldsteel, Michael Hrušák, and Stevo Todorčević.

1. The Groups Ĝ for G Metrizable

If D is a dense subgroup of an Abelian topological group G = (G, T ) then every h ∈ D̂ extends (uniquely) to an element of Ĝ; and of course, each h ∈ Ĝ satisfies h|D ∈ D̂. Accordingly, abusing notation slightly, we have D̂ = Ĝ as groups. Since groups of the form Ĝ carry the compact-open topology, it is natural to inquire whether the identity Ĝ = D̂ is topological as well as algebraic. Informally: do K(G) and K(D) induce the same topology on the set Ĝ = D̂? The question provokes this definition.

Definition 1.1. Let G be an Abelian topological group.

(a) Let D be a dense subgroup of G. Then D determines G (alternatively: G is determined by D) if Ĝ = D̂ as topological groups.

(b) G is determined if every dense subgroup of G determines G.

Remarks 1.2.

(a) It is a theorem of Kaplan [21, 2.9] (cf. Banasczyzk [4, 1.3] and Raczkowski [26, §3.1] for alternative treatments, and Außenhofer [2] and [3, 3.4] for a generalization) that for each G the family \{U(K, \frac{1}{4}) : K ∈ K(G)\} is basic at 0 ∈ Ĝ. (For notational simplicity, henceforth we write U(K) := U(K, \frac{1}{4}) ⊆ Ĝ for K ∈ K(G).) Thus the condition that a group G is determined by its dense subgroup D reduces to (i.e., is equivalent to) the condition that K(D) is cofinal in K(G) in the sense that for each K ∈ K(G) there is E ∈ K(D) such that U(E) ⊆ U(K).

(b) Let D and S be dense subgroups of a topological group G such that D ⊆ S ⊆ G. Then since K(D) ⊆ K(S) ⊆ K(G), D determines G if and only if D determines S and S determines G. In particular, a dense subgroup of a determined group is determined.

(c) The principal theorem in this corner of mathematics is the following result, obtained independently by Außenhofer [3, 4.3] and Chasco [7]. This is the point of departure of the present inquiry.

Theorem 1.3. Every metrizable, Abelian group is determined.

Discussion 1.4. Is every topological group determined? Is every MAP group determined? Are there nonmetrizable, determined groups? Is every closed (or, open) subgroup of a determined group itself determined? Is the class of determined groups closed under passage to continuous homomorphisms? Continuous isomorphisms? The formation of products? These are some of the questions we address.

2. Determined Groups: G vs. G^{+}

Lemma 2.1. Let D be a subgroup of G ∈ LBA. Then D is dense in G if and only if D^{+} is dense in G^{+}.

Corollary 2.2. Let D be a subgroup of G ∈ LBA. Then D determines G if and only if D^{+} determines G^{+}.

Theorem 2.3. Let G ∈ LBA. Then G is determined if and only if G^{+} is determined.

Theorem 2.4. Let G be an LBA group such that G^{+} determines b(G). Then

(a) G is totally bounded (and hence G = G^{+}); and

(b) if also G ∈ LCA then G is compact (and hence G = G^{+} = b(G)).

Corollary 2.5. Let G ∈ LBA. Then b(G) is determined if and only if W (G) is compact and determined; in this case W (G) = b(G).

Corollary 2.6. Let G ∈ LCA. If G is noncompact then G^{+} does not determine b(G) (and hence b(G) is not determined).

Theorem 2.7. Let G be a closed subgroup of a product of LBA groups. Then a dense subgroup D of G determines G if and only D^{+} determines G^{+}. Thus G is determined if and only if G^{+} is determined.

3. Determined Groups: Some Examples

Theorem 3.1. There are totally bounded, nonmetrizable, determined groups.

Proof. Let G be an arbitrary determined LBA group such that G is not totally bounded. (Appealing to Theorem 1.3, one might choose G ∈ {ℤ, ℚ, ℝ}.) That G^{+} is as required follows from three facts:

(a) G^{+} is determined (Theorem 2.3);

(b) a group with a dense metrizable subgroup is itself metrizable [6, Prop. IX §2.1.1];

(c) b(G) is not metrizable.

□

Theorem 3.2. A nondetermined group may have a dense, determined subgroup.

Theorem 3.3. The image of a nondetermined group under a continuous homomorphism may be determined.

Proof. We see in Theorem 4.8 below that compact groups of weight ≥ 𝔠 are nondetermined. Each such group maps by a continuous homomorphism onto either the group 𝕋 or a group of the form (ℤ(p))^{ω} (p ∈ ℙ) [10], and such groups are determined by Theorem 1.3. □

Discussion 3.4. Obviously an LBA group with no proper dense subgroup is vacuously determined. We mention three classes of such groups.

(i) Discrete groups.

(ii) Groups of the form G^{#} = (G_{d})^{+}. (It is well known [12, 2.1] that

every subgroup of such a group is closed.) (iii) LCA groups of the type given by Rajagopalan and Subrahmanian

[27]. Specifically, let κ ≥ ω, fix p ∈ ℙ, and topologize the group G := (ℤ(p^{∞}))^{κ} so that its subgroup H := (ℤ(p))^{κ} in its usual compact topology is open-and-closed in G.

Theorem 3.5.

(a) A determined group may contain a nondetermined open-and-closed subgroup.

(b) There are non-totally bounded, nonmetrizable, determined LBA groups.

Although compact groups of the form K^{κ} with κ ≥ 𝔠 are not determined, we see in Corollary 3.11 below that such groups do contain nontrivial determining subgroups.

Notation 3.6. Let {G_{i} : i ∈ I} be a set of groups, let S_{i} ⊆ G_{i}, and let p ∈ G := Π_{i∈I} G_{i}. Then

(i) s(p) = {i ∈ I : p_{i} ≠ 0_{i}};

(ii) ⊕_{i∈I} G_{i} = {x ∈ G : |s(x)| < ω}; and

(iii) ⊕_{i∈I} S_{i} = (Π_{i∈I} S_{i}) ∩ (⊕_{i∈I} G_{i}). In this context we often identify S_{i} with the subset S_{i} × {0_{I\{i}}} of G. In particular we write G_{i} ⊆ G and we identify Ĝ_{i} with {h|G_{i} : h ∈ Ĝ}.

We use the following property to find some determining subgroups of certain (nondetermined) products.

Definition 3.7. A topological group G has the cofinally zero property if for all K ∈ K(G) there is F ∈ K(G) such that every h ∈ U(F) satisfies h|K ≡ 0.

Remark 3.8. We record two classes of groups with the cofinally zero property.

(i) G is a determining subgroup of a compact Abelian group. (There is F ∈ K(G) such that U(F) = {0}, so each h ∈ U(F) satisfies h|K ≡ 0 for all K ∈ K(G).)

(ii) G is a torsion group of bounded order. (Given K ∈ K(G), let n > 4 satisfy nx = 0 for all x ∈ G and use Remark 1.2(a) to choose F ∈ K(G) such that U(F) ⊆ U(K, \frac{1}{n}).)

Lemma 3.9. Let {G_{i} : i ∈ I} be a set of LBA groups with the cofinally zero property and let G = Π_{i∈I} G_{i}. If D_{i} is a dense, determining subgroup of G_{i}, then D := ⊕_{i∈I} D_{i} determines G.

Corollary 3.10. Let {G_{i} : i ∈ I} be a set of determined LBA groups with the cofinally zero property and let G = Π_{i∈I} G_{i}. If D_{i} is a dense subgroup of G_{i}, then ⊕_{i∈I} D_{i} determines G.

Corollary 3.11. Let {G_{i} : i ∈ I} be a set of compact Hausdorff groups and let G = Π_{i∈I} G_{i}. Then ⊕_{i∈I} G_{i} determines G.

Theorem 3.12. The image under a continuous homomorphism of a compact determined group is determined.

Remark 3.13. It is easily checked that if a locally compact space X is σcompact then it is hemicompact, i.e., some countable subfamily {K_{n} : n < ω} of K(X) is cofinal in K(X) in the sense that for each K ∈ K(X) there is n < ω such that K ⊆ K_{n}. It follows that if an LCA group G is σ-compact (equivalently: Lindelöf) then w(Ĝ) ≤ ω, so Ĝ in this case is determined by Theorem 1.3.

4. Nondetermined groups: Some Examples

The principal result of this section is that compact Abelian groups of weight ≥ 𝔠 are nondetermined.

Lemma 4.1. Let G be an LBA group with a proper dense subgroup D such that each K ∈ K(D) satisfies either

(i) K is finite or (ii) ⟨K⟩ is closed in D.

Then D does not determine G.

We noted in Theorem 2.4(b) that if G^{+} determines b(G) with G ∈ LCA, then G is compact (in fact G = G^{+} = b(G)). Lemma 4.1 allows a more direct proof in the case that G is discrete.

Corollary 4.2. Let G be an infinite Abelian group. Then G^{#} does not determine b(G_{d}).

Lemma 4.3. ([11]). Let G be an Abelian group.

(a) If A is a point-separating subgroup of Hom(G, 𝕋), then (G, T_{A}) is a totally bounded, Hausdorff topological group with { \atop (G, T_{A})} = A;

(b) for every totally bounded Hausdorff topological group topology T on G the subgroup A := { \atop (G, T )} of Hom(G, T ) is point-separating and satisfies T = T_{A}.

Discussion 4.4. It is easily checked that for each Abelian group G the set Hom(G, 𝕋) is closed in the compact space 𝕋^{G}. Thus Hom(G, 𝕋), like every Hausdorff (locally) compact group, carries a Haar measure. Our convention here is that Haar measure is complete, so in particular every subset of a measurable set of measure 0 is itself measurable (and of measure 0).

Concerning Haar measure λ on a LCA group G we appeal frequently to the Steinhaus-Weil Theorem: If S ⊆ G is λ-measurable and λ(S) > 0, then the difference set S − S := {x − y : x, y ∈ S} contains a nonempty open subset of G; thus S, if a subgroup of G, is open in G.

Lemma 4.5. ([13, 3.10]). Let G be an Abelian group, let {x_{n} : n < ω} be a faithfully index sequence in G, and let

S := {h ∈ Hom(G, 𝕋) : h(x_{n}) → 0 ∈ 𝕋}.

Let λ be the Haar measure of Hom(G, 𝕋). Then S is a λ-measurable subgroup of Hom(G, 𝕋), with λ(S) = 0.

Theorem 4.6. Let X be a compact Hausdorff space such that |X| < 2^{ℵ_{1}}. Then

(a) ([17]) X contains a closed, countably infinite subspace; and

(b) X contains a nontrivial convergent sequence.

Theorem 4.7. Let G be an Abelian group such that |G| < 2^{ℵ_{1}} and let A be a dense subgroup of Hom(G, 𝕋) such that either

(i) A is non-Haar measurable, or (ii) A is Haar measurable, with λ(A) > 0.

Then (G, T_{A}) does not determine W (G, T_{A}).

Theorem 4.8. Let G be a compact, Abelian group such that w(G) ≥ 𝔠. Then G is not determined.

Proof. (Outline) Step 1. 𝕋^{𝔠} is not determined. [Proof. There is a nonmeasurable subgroup A of 𝕋 algebraically of the form ⊕_{𝔠} ℤ. Apply Theorem 4.7(i) to the (dense) subgroup e_{A}(ℤ) of 𝕋^{A} = 𝕋^{𝔠}.]

Step 2. F^{𝔠} is not determined (F a finite Abelian group, |F | > 1). [Proof. We have b((⊕_{ω} F)_{d}) = F^{𝔠}, so Corollary 4.2 applies.]

Step 3. There is a continuous isomorphism ϕ : G ↠ K^{𝔠} with either K = 𝕋 or K = F as in Step 2, so Theorem 3.12 applies. □

Corollary 4.9 (CH). Let G be a compact Abelian group. Then G is determined if and only if G is metrizable.

Corollary 4.10 (CH). Let {G_{i} : i ∈ I} be a set of compact Abelian groups with each |G_{i}| > 1, and let G = Π_{i∈I} G_{i}. Then G is determined if and only if |I| ≤ ω and each G_{i} is determined.

We close this section with an example indicating that the intersection of dense, determining subgroups may be dense and nondetermining.

Theorem 4.11. There are dense, determining subgroups D_{i} (i = 0, 1) of 𝕋^{𝔠} such that D_{0} ∩ D_{1} is dense in 𝕋^{𝔠} and does not determine 𝕋^{𝔠}.

Proof. Let Z be a dense, cyclic, nondetermining subgroup of 𝕋^{𝔠}, let A_{i} (i = 0, 1) be dense, torsion subgroups of 𝕋 such that A_{0} ∩ A_{1} = {0_{𝕋}}, and set D_{i} := Z + ⊕_{𝔠} A_{i} ⊆ 𝕋^{𝔠} (i = 0, 1). Then A_{i} determines 𝕋 by Theorem 1.3 so ⊕_{𝔠} A_{i} determines 𝕋^{𝔠} by Lemma 3.9, so D_{i} determines 𝕋^{𝔠} (i = 0, 1); but the dense subgroup Z = D_{0} ∩ D_{1} of 𝕋^{𝔠} does not determine 𝕋^{𝔠}. □

5. Concerning Topological Linear Spaces

Remark 5.1. Let κ be a cardinal number and denote by l_{κ}^{1} the space of real κ-sequences x = {x_{ξ} : ξ < κ} such that ||x||_{1} := ∑_{ξ<κ} |x_{ξ}| < ∞. The additive topological group l_{κ}^{1} respects compactness (cf. Remus and Trigos-Arrieta [29]).

The group { \atop (l_{κ}^{1})^{+}} is not discrete, so the Weil completion W ((l_{κ}^{1})^{+}) is another example of a compact nondetermined group.

Definition. A topological group G is (group) reflective if the evaluation

mapping Ω_{G} : G → ˆ Ĝ defined by Ω_{G}(x)(h) := h(x) for x ∈ G, h ∈ Ĝ is a

topological isomorphism of G onto ˆ Ĝ.

Theorem 5.2. Let G be a noncomplete, reflective group and let R(G) be its Raĭkov completion. Then R(G) ∈ MAP and G does not determine R(G).

Example 5.4 infra illustrates Theorem 5.2.

Definition 5.3. A reflexive locally convex vector space (LCS) in which every closed bounded subset is compact is called a Montel space.

Reflexivity and boundedness (Schaefer[30] §I.5, §IV.5) are meant here in the sense of topological vector spaces. By a Montel group we mean the underlying (additive) topological group of a Montel space. Since by definition these are reflexive LCS, Montel groups are reflective as proven by Smith [31].

Example 5.4. Kōmura [23] and Amemiya and Kōmura [1] construct by induction three different noncomplete Montel spaces, the completion of each being a “big product” of copies of ℝ, and one of them being exactly ℝ^{𝔠}. These groups indicate that Theorem 5.2 is not vacuous. One of the groups constructed in [1] is separable. Thus in particular, again by Theorem 5.2, we see that ℝ^{𝔠} has a countable dense subgroup which does not determine ℝ^{𝔠}.

The remarks above show again that the property of being determined is not 𝔠-productive.

6. Cardinals κ Such That ω < κ ≤ 𝔠

It is well known (cf. for example [24, 2.18] or [8, 8.2.4]) that under Martin’s Axiom [MA] every cardinal κ with ω ≤ κ < 𝔠 satisfies 2^{κ} = 𝔠. In particular under MA +¬CH it follows from Theorem 4.6(b) that every compact Hausdorff space X such that |X| < 2^{ℵ_{1}} = 𝔠 contains a nontrivial convergent sequence. Malykhin and Šapirovskiĭ [25] have achieved a nontrivial extension of this result: Under MA, every compact Hausdorff space X with |X| ≤ 𝔠 contains a nontrivial convergent sequence.

Theorem 6.1 (MA). Let G be a group with |G| ≤ 2^{ω}, and let A be a dense nonmeasurable subgroup of Ĝ_{d}. Then every compact subset of (G, T_{A}) is finite, so its completion W (G, T_{A}) is not determined.

If we denote by λ_{G} the (completed) Haar measure on a LCA group G, let λ_{G}^{∗} stand for the associated outer measure. The existence of a nonmeasurable subset X of 𝕋 (with |X| = 𝔠) is well known, so the case κ = 𝔠 of the following theorem generalizes the statement in Step 1 of the proof of Theorem 4.8.

Theorem 6.2. Let ω < κ ≤ 𝔠. If there is X ∈ [𝕋]^{κ} such that λ_{𝕋}^{∗}(X) > 0, then there is a nonmeasurable, free Abelian subgroup A of 𝕋 algebraically of the form A = ⊕_{κ} ℤ.

Responding to a question on a closely related matter, Stevo Todorčević [33] proposed and proved the above result for κ = ℵ_{1}. His proof additionally yields that X\tor(𝕋) can be broken into ω-many pairwise disjoint independent sets, each of cardinality ℵ_{1}.

For torsion groups of prime order, we obtain the following.

Theorem 6.3. Let F be a finite group of prime order p, and let κ_{1}, κ_{2} be infinite cardinals such that κ_{1} ≤ 2^{κ_{2}}. If there is X ∈ [F^{κ_{2}}]^{κ_{1}} such that λ_{F^{κ_{2}}}^{∗} (X) > 0, then there is a nonmeasurable subgroup A of F^{κ_{2}} algebraically of the form A = ⊕_{κ_{1}}F.

Discussion 6.4. For an ideal I of subsets of a set S we write as usual

non(I) = min{|Y | : Y ⊆ S, Y ∉ I}.

Let F be a finite group (|F | > 1), let N (𝕋) and N (F^{ω}) denote the σ-algebra of λ_{𝕋}- and λ_{F^{ω}}-measurable sets of measure zero. As with any two compact metric spaces of equal cardinality equipped with atomless (“continuous”) probability measures, the spaces 𝕋 and F^{ω} are Borel-isomorphic in the sense that there is a bijection ϕ : 𝕋 ↠ F^{ω} such that the associated bijection \overline{ϕ} : P(𝕋) ↠ P(F^{ω}) carries the Borel algebra B(𝕋) onto the Borel algebra B(F^{ω}) in such a way that λ_{F^{ω}}(\overline{ϕ}(B)) = λ_{𝕋}(B) for each B ∈ B(𝕋). (See [22, 17.41] or [32, 3.4.23] for a proof of this “Borel isomorphism Theorem for measures”.)

Lemma 6.5. The cardinals non(N (𝕋)) and non(N (F^{ω})) are equal.

We write

non(N) := non(N (𝕋)) = non(N (F^{ω})),

a definition justified by Lemma 6.5.

Theorem 6.6. Let G be a compact Abelian group such that w(G) ≥ non(N). Then G is nondetermined.

7. Questions

Question 7.1. Is there a compact group G with a countable dense subgroup D such that w(G) > ω and D determines G?

Question 7.2. If {G_{i} : i ∈ I} is a set of topological Abelian groups and D_{i} is a dense determining subgroup of G_{i}, must ⊕_{i∈I} D_{i} determine Π_{i∈I} G_{i}? In particular, does ⊕_{i∈I} G_{i} determine Π_{i∈I} G_{i}? In particular, does ⊕_{𝔠} ℝ determine ℝ^{𝔠}?

Discussion 7.3. Consider the following cardinals:

(a) 𝔪_{𝕋} := the least cardinal κ such that 𝕋^{κ} is nondetermined;

(b) 𝔪_{f∃} [resp., 𝔪_{f∀}] := the least cardinal κ such that some [resp., each] finite group F has F^{κ} nondetermined;

(c) 𝔪_{c∃} [resp., 𝔪_{c∀}] := the least cardinal κ such that some [resp., each] compact abelian group of weight κ is nondetermined;

(d) 𝔪_{p∃} [resp., 𝔪_{p∀}] := the least cardinal κ such that some [resp., each] product of κ-many compact determined groups is nondetermined.

It follows from Theorems 1.3 and 6.6 that each 𝔪_{x}, with the possible exception of 𝔪_{p∃}, satisfies ℵ_{1} ≤ 𝔪_{x} ≤ non(N). Further if non(N) = ℵ_{1}, then all seven cardinals 𝔪_{x} are equal to ℵ_{1}. The condition non(N) = ℵ_{1} is clearly consistent with CH, and it has been shown to be consistent as well with ¬CH (see for example [5], [15] and [20, Example 1, page 568]), so in particular there are models of ZFC + ¬CH in which every compact (Abelian) group G satisfies: G is determined if and only if G is metrizable. (Without appealing to the cardinal non(N), Michael Hrušák [19] in informal conversation suggested the existence of models of ZFC + ¬CH in which {0, 1}^{ℵ_{1}} is nondetermined.)

Question 7.4. Are the various cardinal numbers 𝔪_{x} equal in ZFC? Are they equal to one of the familiar “small cardinals” conventionally noted in the Cichoń diagram (cf. [5], [35])? Is each 𝔪_{x} = non(N)? Is each 𝔪_{x} = ℵ_{1}? Is each cf(𝔪_{x}) > ω?

We know of no models of ZFC in which 𝕋^{ℵ_{1}}, or some group of the form F^{ℵ_{1}} (F finite, |F | > 1), is determined, so we are forced to consider the possibility that the following questions have an affirmative answer.

Question 7.5. Are the following (equivalent) statements theorems of ZFC?

(a) The group 𝕋^{ℵ_{1}} and groups of the form F^{ℵ_{1}} (F finite, |F | > 1) are nondetermined.

(b) A compact abelian group G is determined if and only if G is metrizable.

The following question is suggested by those above.

Question 7.6. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ?

Question 7.7. Is it consistent with ZFC that 𝔪_{p∃} = 2? Is it consistent with ZFC that 𝔪_{p∃} = ω?

Question 7.7 has analogues in the context of groups which are not assumed to be compact, as follows.

Question 7.8. In ZFC alone or in augmented axiom systems: Is the product of finitely many determined groups necessarily determined? If G is determined, is G × G necessarily determined?

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W. W. Comfort, Department of Mathematics, Wesleyan University, Middletown, CT 06459

   E-mail address: wcomfort@wesleyan.edu

S. U. Raczkowski and F. Javier Trigos-Arrieta, Department of Mathematics, California State University, Bakersfield, 9011 Stockdale Highway, Bakersfield, CA, 93311-1099

   E-mail address: racz@cs.csubak.edu
   E-mail address: jtrigos@cs.csubak.edu