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Dense Subgroups of Compact Groups Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved Dense subgroups of compact groups W. W. Comfort 1. Introduction The symbol G here denotes the class of all infinite groups, and TG denotes the class of all infinite topological groups which satisfy the T0 separation prop- erty. Each element of TG is, then, a Tychonoff space (i.e., a completely regular, Hausdorff space) [50, 8.4]. We say that G = (G, T ) ∈ TG is totally bounded (some authors prefer the expression pre-compact) if for every U ∈ T \ {∅} there is a finite set F ⊆ G such that G = FU. Our point of departure is the following portion of Weil’s Theorem [77]: Every totally bounded G ∈ TG embeds as a dense topological subgroup of a compact group G; this is unique in the sense that for every compact group Ge containing G densely there is a homeomorphism-and- isomorphism ψ : G Ge fixing G pointwise. As usual, a space is ω-bounded if each of its countable subsets has compact closure. For G = (G, T ) ∈ TG we write G ∈ C [resp., G ∈ Ω; G ∈ CC; G ∈ P; G ∈ TB] if (G, T ) is compact [resp., ω-bounded; countably compact; pseudocompact; totally bounded]. And for X ∈ {C, Ω, CC, P, TB} and G ∈ G we write G ∈ X0 if G admits a group topology T such that (G, T ) ∈ X. The class-theoretic inclusions C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG and C0 ⊆ Ω0 ⊆ CC0 ⊆ P0 ⊆ TB0 ⊆ TG0 = G are easily established. (For P ⊆ TB, see [31]. For TG0 = G, impose on an arbitrary G ∈ G the discrete topology.) We deal here principally with (dense) subgroups of groups G ∈ C, that is, with G ∈ TB. Given G ∈ G we write tb(G) := {T :(G, T ) ∈ TB}. It is good to remember that tb(G) = ∅ and |tb(G)| = 1 are possible (for different G); see 2.3(a) and 5.8(a)(2) below. The symbol A is used as a prefix to indicate an Abelian hypothesis. Thus, for emphasis and clarity: The expression G ∈ AG may be read “G is an infinite Abelian group”, and G ∈ ACC0 may be read “G is an infinite Abelian group which admits a countably compact Hausdorff group topology.” For G, H ∈ G, we write G =alg H to indicate that G and H are algebraically isomorphic. For (Tychonoff) spaces X and Y , we write X =top Y to indicate that the spaces X and Y are homeomorphic. The relation G =alg H promises nothing whatever about the underlying topologies (if any) on G and H; similarly, the rela- tion X =top Y is blind to ambient algebraic considerations (if any). We say that G, H ∈ TG are topologically isomorphic, and we write G =∼ H, if some bijection between G and H establishes simultaneously both G =alg H and G =top H. 377 378 §40. Comfort, Dense subgroups of compact groups We distinguish between Problems and Questions. As used here, a Problem is open-ended in flavor, painted with a broad brush; different worthwhile contri- butions (“solutions”) might lead in different directions. In constrast, a Question here is relatively limited in scope, stated in narrow terms; the language suggests that a “Yes” or “No” answer is desired—although, as we know from experience, that response may vary upon passage from one axiom system to another. With thanks and appreciation I acknowledge helpful comments received on preliminary versions of this paper from: Dikran Dikranjan, Frank Gould, Kenneth Kunen, G´abor Luk´acs, Jan van Mill, Dieter Remus, and Javier Trigos-Arrieta. 2. Groups with topologies of pre-assigned type 848–852? Problem 2.1. Let X ∈ {C, Ω, CC, P, TB}. Characterize algebraically the groups in X0. 853–857? Problem 2.2. Let X ∈ {AC, AΩ, ACC, AP, ATB}. Characterize algebraically the groups in X0. Discussion 2.3. (a) That ATB0 = AG is easily seen (as in Theorem 3.1(a) below, for example, using the fact that Hom(G, T) separates points of G whenever G ∈ AG). That the inclusion TB0 ⊆ G is proper restates the familiar fact that there are groups G ∈ G whose points are not distinguished by homomorphisms into compact (Hausdorff) groups; in our notation, these are G such that tb(G) = ∅. For example: according to von Neumann and Wigner [59], [50, 22.22(h)], every homomorphism h from the (discrete) special linear group G := SL(2, C) to a compact group satisfies |h[G]| = 1. (b) It is a consequence of the Cech–Pospisilˇ Theorem [9] (see also [43, Prob- lem 3.12.11], [51, 28.58]) that every G ∈ C satisfies |G| = 2w(G). Thus in order that G ∈ G satisfy G ∈ C0 it is necessary that |G| have the form |G| = 2κ. (c) The algebraic classification of the groups in AC0 is complete. The full story is given in [50, §25]. (d) It is well known that every G ∈ P0 satisfies |G| ≥ c. See [28] or [13, 6.13] for an explicit proof, and see [8], [42, 1.3] for earlier, more general results. (e) The fact that every pseudocompact space satisfies the conclusion of the Baire Category Theorem has two consequences relating to Problems 2.1 and 2.2. (1) If G ∈ P0, the cardinal number |G| = κ cannot be a strong limit cardinal with cf(κ) = ω [42]. (2) every torsion group in AP0 is of bounded order [29, 7.4]. (f) No complete characterization of the groups in P0 (nor even in AP0) yet exists, but the case of the torsion groups in AP0 is well understood ([26, 40, 41]): A torsion group G ∈ AG of bounded order is in P0 iff for each of its p-primary constituents G(p) each infinite cardinal number of the form κ := |pk ·G(p)| satisfies L 0 κ Z(p) ∈ P . (Thus for example, as noted in [26], if p is prime and κ is a strong L 2 L 0 limit cardinal of countable cofinality, then 2κ Z(p ) ⊕ κ Z(p) ∈ AP while L L 2 0 2κ Z(p) ⊕ κ Z(p ) ∈/ P .) Topologies induced by groups of characters 379 (g) An infinite closed subgroup H of G ∈ X ∈ {C, Ω, CC, TB} satisfies H ∈ X, but the comparable assertion for X = P is false [32]. Indeed every H ∈ TB embeds as a closed topological subgroup of a group G ∈ P ([20, 72, 75]). (h) Examples are easily found in ZFC showing that the inclusions AC ⊆ AΩ ⊆ ACC ⊆ AP ⊆ ATB are proper. (See also in this connection 3.3(h) below.) As is indicated in [15, 3.10], the inclusions AC0 ⊆ AΩ0 and ACC0 ⊆ AP0 ⊆ ATB0 are proper in ZFC, but the examples cited there from the literature to show AΩ0 6= ACC0 rest on either CH [70, 71] or MA [73]. 0 0 Question 2.4. Is there in ZFC a group G ∈ ACC \ AΩ ? 858? 3. Topologies induced by groups of characters For G ∈ G, we use notation as follows. •H(G) := Hom(G, T), the set of homomorphisms from G to the circle group T. •S(G) is the set of point-separating subgroups of H(G). • When A ∈ S(G), TA is the smallest topology on G with respect to which each element of A is continuous. • When (G, T ) ∈ TG, (\G, T ) is the set of T -continuous funtions in H(G). These symbols are well-defined for arbitrary groups G, but (in view of the privileged status of the group T) typically they are useful only when G is Abelian. The fact that the groups A ∈ S(G) are required to separate points ensures that the topology TA satisfies the T0 separation property required throughout this article; A indeed, the evaluation map eA : G → T (given by eA(x)h = h(x) for x ∈ G, h ∈ A) is an injective homomorphism, and TA is the topology inherited by G (identified A in this context with eA[G]) from T . When H(G) is given the (compact) topology inherited from TG, a subgroup A ⊆ H(G) satisfies A ∈ S(G) iff A is dense in H(G) (cf. [30, 1.9]). The point of departure for our next problem is this theorem. Theorem 3.1 ([30]). Let G ∈ AG. Then (a) A ∈ S(G) ⇒ (G, TA) ∈ ATB; (b) (G, T ) ∈ ATB ⇒ ∃ A ∈ S(G) such that T = TA; (c) A ∈ S(G) ⇒ (G,\TA) = A; and (d) A ∈ S(G) ⇒ w(G, TA) = |A|. It follows from Theorem 3.1(c) that the map A 7→ TA from S(G) to tb(G) is an order-preserving bijection between posets, so tb(G) is large. That theme is noted and developed at length in [4, 5, 25, 27, 30, 34, 63], where the following results (among many others) are given for such G: (a) From |H(G)| = 2|G| ([53], |G| |G| [44, 47.5], [50, 24.47]) and |S(G)| = 22 ([54], [5, 4.3]) follows |tb(G)| = 22 ; 2|G| 2|G| (b) from any set of 2 -many elements TA ∈ tb(G), some 2 -many of the spaces (G, TA) are pairwise nonhomeomorphic; (c) each of the two posets (tb(G), ⊆) and (P(P(|G|)), ⊆) embeds into the other, so any question relating to the existence of a chain or anti-chain or well-ordered set in tb(G) of prescribed cardinality is 380 §40. Comfort, Dense subgroups of compact groups independent of the algebraic structure of G and is equivalent to the corresponding strictly set-theoretic question in the poset (P(P(|G|)), ⊆).
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