The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le \lw(G)$, where $\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $(\tau_\aleph)_{\aleph_0\le \aleph\le \card(G)}$, such that $(G,\tau_\aleph)\hat{\phantom{m}}\cong\hat G$. Items (2) and (3) are shown in Section 5.

Abstract. Let G be an infinite locally compact group and ℵ a cardinal satisfying ℵ 0 ≤ ℵ ≤ w(G) for the weight w(G) of G. It is shown that there is a closed subgroup N of G with w(N ) = ℵ. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup.
(2) For a locally compact group G and ℵ a cardinal satisfying ℵ 0 ≤ ℵ ≤ w 0 (G), where w 0 (G) is the local weight of G, there are either no infinite compact subgroups at all or there is a compact subgroup N of G with w(N ) = ℵ. (3) For an infinite abelian group G there exists a properly ascending family of locally quasiconvex group topologies on G, say, (τ ℵ ) ℵ 0 ≤ℵ≤card(G) , such that (G, τ ℵ ) ∼ = G. Items (2) and (3) are shown in Section 5.

Introduction
The weight w(X) of a topological space X is the smallest cardinal ℵ for which there is a basis B of the topology of X such that card(B) = ℵ. A compact group G is metric iff its weight w(G) is countable, that is, w(G) ≤ ℵ 0 . (See e.g. [8], A4.10 ff., notably, A1.16.) In particular, all compact Lie groups are metric. It is not clear a priori that a compact group of uncountable weight contains an infinite closed metric subgroup. Indeed, in Example 5.3(v) below we will show that the precompact topological group defined on Z endowed with the Bohr-topology, which it inherits from its universal almost periodic compactification, has no nonsingleton metric subgroup, while its weight is the cardinality of the continuum.
However, we shall prove the following theorem which among other things will show that every infinite compact group has an infinite metric subgroup.
Main Theorem Let G be a locally compact group of uncountable weight and let ℵ 0 ≤ ℵ < w(G). Then G has a closed subgroup N with w(N ) = ℵ.
In other words, for any infinite locally compact group G, the entire interval of cardinals [ℵ 0 , w(G)] is occupied by the weights of closed subgroups of G. If G is compact and connected, we shall see that [ℵ 0 , w(G)] is filled even with closed normal and indeed connected subgroups. It remains unsettled whether a profinite group has, in this sense, enough normal closed subgroups.
We shall deal with a proof of the Main Theorem in a piecemeal way. For reasons of presenting a stepwise proof let us call C the class of all Hausdorff topological groups G satisfying the following condition for each infinite cardinal ℵ ≤ w(G) there is a closed subgroup H of weight ℵ.
Our Main Theorem says that all locally compact groups are contained in C.
We first aim to show that all compact groups are in C and we begin with compact abelian groups.

Compact abelian groups
For an abelian group A, let tor A denote its torsion subgroup. The first portion of our first observation is a consequence of a more precise statement due to W. R. Scott [13].
Since we prove what we shall need in a shorter way (and quite differently) we present a proof which will also establish the second part. A divisible hull of an abelian group we are using can be constructed in a very special way by using the results of E. A. Walker in [14].
Let A be an uncountable abelian group and ℵ 0 ≤ ℵ < |A|. Then A contains a subgroup B such that (A : B) = ℵ. Moreover, If (A : tor A) is at least ℵ then B may be picked so that B is pure and A/B is torsion free.
Proof. Let D be a divisible hull of A according to [8], Proposition A1.33. If D = D 1 ⊕ D 2 is any direct decomposition, and pr 1 : D → D 1 is the projection onto the first summand of D, then A/(A ∩ D 2 ) ∼ = (A + D 2 )/D 2 ∼ = pr 1 (A). Now any subgroup of D 1 is a subgroup of the divisible hull of A and therefore meets A and thus A ∩ D 1 ⊆ pr 1 (A) nontrivially (see [8], Proposition A1.33); therefore D 1 is a divisible hull of pr 1 (A). Hence either pr 1 (A) is finite or else card D 1 = card pr 1 (A) = card(A/(A ∩ D 2 )) by [8], Proposition A1.33(i).
Since we control card D 1 by choosing D 1 appropriately, we aim to set B = A ∩ D 2 and thereby prove our first assertion. We thus have to exclude the possibility that pr 1 (A) might turn out to be finite by an inappropriate choice of D 1 . We now let tor A denote the torsion subgroup of A. Then tor D is a divisible hull of tor A, and D ∼ = (tor D) × (D/ tor D) by [8], p. 657, Proposition A1.38. We now distinguish two cases: (a) Case card(tor A) = card A. Since A is uncountable, one of the p-primary components of tor A, as p ranges through the countable set of primes, say A(p), satisfies card A(p) = card(tor A) = card A. In particular card A(p) is uncountable, that is, its p-rank card A is uncountable and agrees with the p-rank of D(p) (see [8], p. 656, Corollary A1.36(iii). In view of D(p) ∼ = (Z(p ∞ ) (card A) ) by [8], p. 659, Theorem A1.42(iii), we find a direct summand D 1 of D(p) of p-rank ℵ, giving us a direct summand of tor D and thus yielding a direct sum decomposition D = D 1 ⊕ D 2 . Since the p-rank ℵ of D 1 is infinite, and D 1 is the divisible hull of pr 1 (A) we know that pr 1 (A) cannot be finite, whence ℵ = card D 1 = card(A/(A ∩ D 2 ). Our first assertion then follows with B = A ∩ D 2 .
(b) Case card(A/ tor A) = card A. Then the (torsion free) rank of D is card A (see [8], p. 656, Corollary A1.36(iii)), By the structure theorem of divisible groups (see e.g. [8], Theorem A1.42]) and elementary cardinal arithmetic, we can write D = D 1 ⊕ D 2 with a torsionfree subgroup D 1 of cardinality ℵ. Then pr 1 (A) ⊆ D 1 cannot be finite, and as in the first case, we let B = A∩D 2 and have ℵ = card D 1 = card A/B as in our first assertion.
It remains to inspect the case that card(A/ tor A) ≥ ℵ. Then we may assume tor A ⊆ D 2 and D 1 torsion free; but then tor The second part of the preceding proposition is also a consequence of Theorem 4 of [14].
Every uncountable abelian group has a proper subgroup of index ℵ 0 .
⊓ ⊔ (A different, but likewise not entirely trivial proof by Hewitt and Ross is found in [6], p. 227.) As usual, for a topological group G, the identity component of G will be denoted G 0 . Let ℵ be an infinite cardinal and let G be a compact connected group with w(G) > ℵ. Then G has a compact connected abelian subgroup T with w(T ) = ℵ.
Proof. Let T be a maximal compact connected abelian subgroup of G. Then w(T ) = w(G) by [8] While this corollary answers the question whether infinite compact groups have infinite compact metric subgroup in the affirmative, we should keep in mind, that Zelmanov's Theorem in itself is not a simple matter. It therefore appears worthwhile to pursue the question further.

Connectivity versus total disconnectivity in compact groups
Lemma 2.1. Let G be an arbitrary compact group. Then the following conclusions hold:
We recall from Corollary 1.4 that for a compact connected group G the set of all infinite cardinals ≤ w(G) is filled with the set of all infinite cardinals representing the weights of closed connected abelian subgroups. In the following we amplify this observation Let ℵ be an infinite cardinal such that ℵ < w(G) for a compact connected group G. Then G contains a closed connected and normal subgroup N such that w(N ) = ℵ.

Proof.
Following the Levi-Mal'cev Structure Theorem for Compact Connected Groups ( [8], Theorem 9.24) we have G = G ′ Z 0 (G) where the algebraic commutator subgroup G ′ is a characteristic compact connected semisimple subgroup and the identity component of the center Z 0 (G) is a characteristic compact connected abelian subgroup.
If we find a compact connected normal subgroup N of G ′ , we are done, since the normalizer of N contains both G ′ and the central subgroup Z 0 (G), hence all of G = G ′ Z 0 (G). Thus it is no loss of generality to assume that G = G ′ is a compact connected semisimple group. Case 2a. G = j∈J G j for a family of compact connected (simple) Lie groups. Then w(G j ) = ℵ 0 , and ℵ < w(G) = max{ℵ 0 , card J} (see e.g. [8], EA4.3.). Since ℵ is infinite and smaller than w(G), we have w(G) = card J. Then we find a subset I ⊆ J such that card I = ℵ, and set N = i∈I G i . Then w(N ) = card N = ℵ. Case 2b. By the Sandwich Theorem for Semisimple Compact Connected Groups ( [8], 9.20) there is a family of simply connected compact simple Lie groups S j with center Z(S j ) and there are surjective morphisms such that qf is the product j∈J p j of the quotient morphisms p j : S j → S j /Z(S j ). Now both products j∈J S j and j∈J S j /Z(S j ) have the same weight card J which agrees with the weight of the sandwiched group G. Define I as in Case 2a and set N = f ( i∈I S i ) and note that q(N ) = i∈I S i /Z(S i ). Hence N is sandwiched between two products with weight card I = ℵ and hence has weight ℵ. This proves the existence of the asserted N in the last case.

The generating degree
Now we have to reach beyond connectivity, all the while still staying within the class of compact groups. We refer to a cardinal invariant for compact groups G which is one of several alternatives to the weight w(G), namely, the so called generating degree s(G) (see [8], Definition 12.15). The definition relies on the Suitable Set Theorem, loc. cit. Theorem 12.11, which in turn invokes the so called Countable Layer Theorem (see [7] or [8], Theorem 9.91). Indeed recall that in a compact group G a subset S is called suitable iff it does not contain 1, is closed and discrete in G \ {1}, and satisfies G = S . The Suitable Set Theorem asserts, that every compact group G has a suitable set. A suitable set is called special iff its cardinality is minimal among all suitable subsets of G. The cardinality s(G) of one, hence every special suitable set is called the generating degree of G.
The relevance of the generating degree in our context is the following Let G be a profinite group with uncountable weight. Then w(G) = s(G).
Proof. By Proposition 12.28 of [8], for an infinite profinite, that is, compact totally disconnected group we have This implies the assertion immediately in the case of w(G) > ℵ 0 .
⊓ ⊔ The next step, namely, proving that every profinite group is in C will be facilitated by a lemma on suitable sets for which all ingredients are contained in [8].  From this and (a), w(G) = card S follows. Now assume that S is infinite. Then either S is uncountable or card S = ℵ 0 . In the first case, w(G) is uncountable by (a) and w(G) = card S holds. If, on the other hand, card S = ℵ 0 and w(X) = ℵ 0 , then card S = w(X) as well.
⊓ ⊔ The significance of this lemma is that for a profinite group, infinite suitable subsets all have the same cardinality, namely, the weight of the group.
It is instructive to take note of the following remarks which are pertinent to this context:

Remark b.
If H is a precompact group whose Weil completion G is profinite and if H has an infinite relatively compact suitable subset, then card H ≥ w(H).

Proof.
Assume that S is an infinite relatively compact suitable subset of H. Then by [8], p. 616, Lemma 12.4, S is a suitable subset of G and by Lemma 3.2 it follows that card H ≥ card S = w(G) = w(H).
(ii) Let G be a compact group of weight ℵ 1 . Then G contains countable dense subgroups. If G is profinite, then none of these contains an infinite relatively compact suitable set.
Proof. (i) This follows from the equation d(G) = log w(G) valid for any compact group G, see [1]. (ii) Let G be a compact group of weight ℵ 1 . Then by (i) it has a countable dense subgroup H. Suppose that G is profinite and H has an infinite relatively compact suitable subset S. Then by Remark b we would have ℵ 0 = card H ≥ w(H) = w(G) = ℵ 1 , a contradiction.
⊓ ⊔ Now we show that every profinite group of uncountable weight is in C.

Lemma 3.3. Let G be a profinite group of uncountable weight and let ℵ < w(G)
be an infinite cardinal. Then there is a closed subgroup H such that w(H) = ℵ.

Proof.
Let T be a suitable subset of G with card T = w(G) according to Proposition 3.1. Then T contains a subset S of cardinality ℵ. We set H = S . Proof. Let G be an infinite metric group. If w(G) = ℵ 0 then G itself is metric. If G has uncountable weight, then we apply Lemma 3.3 with ℵ = ℵ 0 .
⊓ ⊔ Now we are ready to prove that every compact group is in C which is the main portion of the following result: Let ℵ be an infinite cardinal and G a compact group such that ℵ < w(G). Then there is a closed subgroup H such that w(H) = ℵ, and if G is connected, H may be chosen normal and connected.
Proof. Let G 0 denote the identity component of G. The case that w(G) = w(G 0 ) is handled in Proposition 2.2. So we assume w(G 0 ) < w(G). By Lemma 2.1 we may assume that G is totally disconnected, that is, profinite. Then Lemma 3.3 proves the assertion of the theorem.

⊓ ⊔
In particular, we have the following conclusion: Corollary 3.6. Every infinite compact group contains an infinite closed metric subgroup.

⊓ ⊔
Recall that by Corollary 1.7 we know that we find even a closed abelian metric subgroup. For a compact connected group G, Corollary 1.4 shows that for any infinite cardinal ℵ ≤ w(G) there is in fact a closed connected abelian subgroup of weight ℵ.
Problem. If G is a compact group and ℵ is an infinite cardinal ≤ w(G). Is there a normal closed subgroup H such that w(H) = ℵ?
The answer is affirmative if it is affirmative for profinite groups. A profinite group has weight ℵ iff its set of open-closed normal subgroups has cardinality ℵ. This observation points into the direction of an affirmative answer to the problem.

The weights of closed subgroups of a locally compact group
Now we finish the proof of the Main Theorem by showing that every locally compact group is in C. A first step is the following:
In order to prove the reverse inequality, let D be a dense subset of H of cardinality d(H). For each finite tuple F = (Hg 1 , Hg 2 , . . . , Hg n ) ∈ X n , n ∈ N, the set P F def = Hg 1 Hg 2 · · · Hg n has a dense subset ∆ F def = Dg 1 Dg 2 · · · Dg n of cardinality d(H) with the density of H. Then K = F P F , with the union extended over the set of finite n-tuples F , has a dense subset ∆ = F ∆ F whose cardinality is ≤ card X ·d(H) ≤ max{card(X ), w(H)} (see [8], p. 620, Proposition 6.20 and its proof). It follows that ⊓ ⊔

An application of the Main Theorem
Let (G, τ ) be an arbitrary topological abelian group. We shall continue to write abelian groups additively, unless specified otherwise such as in the case of the multiplicative circle group S 1 = {z ∈ C : |z| = 1}. A character on (G, τ ) is a continuous morphism χ: G → T = R/Z. The pointwise sum of two characters is again a character, and the set G of all characters is a group with pointwise multiplication as the composition law. If G is equipped with the compact open topology τ , it becomes a topological group ( G, τ ) which is called the dual group of (G, τ ). Typically, if G is a locally compact abelian group and G its Pontryagin dual, that is, its character group, then G and G are in duality, but there are group topologies τ on G which are in general coarser than the given locally compact topology on G but which are nevertheless compatible with this duality. One of the best known is the so-called Bohr topology τ + on G which we discuss in the next example for the sake of completeness and because it plays a role in our subsequent discussion which, at least in the case that G is discrete, provides a substantial cardinality of topologies on G which are compatible with the Pontryagin duality of G.

Example 5.3.
Let G = (A, τ ) be a locally compact abelian group and let η: G → G α be the Bohr compactification morphism and let τ + be the pull-back topology on A (that is, the topology which makes η an embedding ǫ: G + → G α ). Equivalently, τ + is the topology of pointwise convergence when A is considered as the space of characters of G via Pontryagin duality. We write G + for the topological group (A, τ + ). Then (i) If (a n ) n∈N is a sequence of A, and a ∈ A, then a = τ -lim n a n iff a = τ + -lim n a n . If τ is the discrete topology, then (a n ) n∈N converges w.r.t. τ + iff it is eventually constant.
(ii) τ + is compatible with the duality between A and G, that is, G + = G. Also, G + = (G α ) .
(iii) If σ is any group topology on A such that τ + ⊆ σ ⊆ τ , then σ is compatible with the duality between A and G.
(v) If G has no nonsingleton compact subgroups, such as G = Z or G = R, then G + has no nonsingleton metric subgroups.

Proof.
For easy reference we provide proofs. (i) The fact that G and G + have the same converging sequences is based on two implications of which "a = τ -lim n a n implies a = τ + -lim n a n " is immediate from the continuity of η and the definition of τ + . The other implication has fascinated several authors (see e.g. [3], [4], [9], [11]); the following argument was credited by Reid to originate from Varopoulos in the 1960s but may have been around before: Let a = τ + -lim n a n . We consider G as the character group of G. For any f ∈ L 1 ( G) we calculate the value of the Fourier transform f ∈ C 0 (G) at a n as f (a n ) = G f (χ) exp(−i χ, a n ) dχ By the definition of τ + as topology of pointwise convergence, the sequence of functions f (•) exp(−i •, a n ) is dominated by |f | since the absolute value of the exponential is 1, and it converges pointwise to f (•) exp(−i •, a . Hence by the Lebesgue Dominated Convergence Theorem we have lim n f (a n ) = f (a).
(v) Let H + be a first countable nonsingleton subgroup of G + . Then H + is precompact since G + is precompact, and its topology is determined by its convergent sequences since it is first countable. On the other hand, by (i) above, it has the same convergent sequences as the subgroup H which has the same underlying group as H + but whose topology is the one induced by the topology of G. Hence H + = H as topological groups. Since H + is precompact, the locally compact group H is precompact and thus is compact. Now assume that G is a locally compact abelian group without compact nondegenerate subgroups, say, G = Z or G = R. In those cases G + cannot have any nonsingleton metric subgroup.
⊓ ⊔ By its very definition, the Bohr topology τ + is a precompact topology. Indeed, a topological group G is said to be precompact or totally bounded if for any neighborhood U of the neutral element in G, there is a subset F ⊆ G with card(F ) < ℵ 0 such that F U = G. Next we need to generalize the concept of total boundedness: Definition 5.4 Let ℵ be a cardinal number. A topological group G is said to be ℵ-bounded when for every neighborhood U of the neutral element in G, there is a subset S ⊆ G with card(S) < ℵ such that SU = G.

⊓ ⊔
According to this definition, a group G is totally bounded iff it is ℵ 0 -bounded. It is uniformly Lindelöf if it is ℵ 1 -bounded. For a topological abelian group Γ and a cardinal ℵ we let K ℵ (Γ) denote the set of all compact subsets K ⊆ Γ with w(K) < ℵ.
Lemma 5.5. Let G be the character group of an abelian topological group Γ. Let ℵ be a cardinal ≤ w(Γ), and let G have the topology of uniform convergence on compact subsets K ∈ K ℵ (Γ). Then G is ℵ-bounded.
Proof. From Theorem 3.4 in [2] we have the following information due to Ferrer and Hernández who deal with the following set-up: Let X be a set, let M be a metrizable space, and let Y be a subset of M X that is equipped with some bornology B consisting of pointwise relatively compact sets. Denote by µ B the uniformity on X defined as sup{µ F : F ∈ B}. The concept of an ℵ-bounded topological group generalizes rather immediately to that of an ℵ-bounded uniform space. Now we have The ℵ-Boundedness Theorem. If ℵ is a cardinal such that w(M ) < ℵ, then the following statements are equivalent: (i) (∀F ∈ B) w(F ) < ℵ. Let G be a topological group, the local weight (or character) w 0 (G) is the smallest among the cardinals of neighbourhood bases at the neutral element. We say that a collection {K i : i ∈ I} is a compact cover of G when ∪ i∈I K i = G and K i is compact for all i ∈ I. The compact covering number κ(G) of G is defined as the smallest of the cardinals of the members of the set of compact covers of G.
If G is any locally compact group, let H be an almost connected open subgroup and let C be a maximal compact subgroup of H. Then G is homeomorphic to R n ×C ×G/H and w 0 (G) = max{ℵ 0 , w(C)} and κ(G) = card G/H. If G is abelian and Γ = G, then w 0 (Γ) = κ(G) and κ(Γ) = w 0 (G). In this sense, w 0 and κ are "dual" cardinals. In view of our Main Theorem we may summarize: The Local Weight Lemma for Locally Compact Groups. For a locally compact nondiscrete group G select any almost connected open subgroup H and any maximal compact compact subgroup C of H. Then (ii) for ℵ 0 ≤ ℵ < ℵ ′ ≤ κ(G) one has τ ℵ ⊂ = τ ℵ ′ ⊆ τ . ⊓ ⊔ Thus, if [ℵ 0 , κ(G)] denotes the full interval of infinite cardinals up to the cardinality of G, then Theorem 4.7 provides card[ℵ 0 , κ(G)]-many locally quasiconvex group topologies on the abelian group G all of which are coarser than the original topology of G and yield the (locally compact abelian ) group G as character group. This means that all topologies τ ℵ have the same compact subsets as τ .