Applied General Topology c© Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85–89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract. A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) σ-compact if G is a union of countably many compact subsets, (iii) ℵ0-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F ∪ U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both ℵ0-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both ℵ0-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is σ-compact and finitely generated modulo open sets. 2000 AMS Classification: 54H11, 22A05. Keywords: Topological group, compactly generated group, dense subgroup, almost metrizable group, ℵ0-bounded group, paracompact p-space, metric space, σ-compact space, space of countable type. 1. Preliminaries All topological groups in this article are assumed to be T1 (and thus Ty- chonoff). For subsets A and B of a group G let AB = {ab : a ∈ A and b ∈ B} and A−1 = {a−1 : a ∈ A}. For a ∈ A and b ∈ B we write aB or Ab rather than {a}B or A{b}. If A is a subset of a group G, then the smallest subgroup of G that contains A is denoted by 〈A〉. Recall that a topological group G is said to be: (i) compactly generated if G = 〈K〉 for some compact subspace K of G, (ii) sigma-compact provided that there exists a sequence {Kn : n ∈ ω} of compact subsets of G such that G = ⋃ {Kn : n ∈ ω}, 86 Hiroshi Fujita and Dmitri Shakhmatov (iii) ℵ0-bounded if for every neighborhood U of the unit element there exists a countable set S ⊂ G such that US = G ([2]), (iv) totally bounded if for every neighborhood U of the unit element there exists a finite set S ⊂ G such that US = G, (v) finitely generated modulo open sets if for every non-empty open set U ⊆ G, there exists a finite set F ⊆ G such that 〈F ∪U〉 = G ([1]). Clearly, compactly generated groups are σ-compact. It is well-known that σ-compact groups, separable groups and their dense subgroups are ℵ0-bounded ([2]). 2. The results The main purpose of this note is to study the following question: When does a topological group contain a dense compactly generated subgroup? Our first result provides two necessary conditions: Theorem 2.1. If a topological group G contains a dense compactly generated subgroup, then G is both ℵ0-bounded and finitely generated modulo open sets. Proof. Let G be a topological group and K be its compact subset such that 〈K〉 is dense in G. Then G is ℵ0-bounded ([2]), so it remains only to show that G is finitely generated modulo open sets. Given a non-empty open set U, the group G is divided into pairwise disjoint left-congruence classes modulo its subgroup 〈U〉. Let X be a complete set of representatives of these congruence classes: G = ⋃ x∈X x〈U〉. Since each congruence class is an open set, finite number of those classes must cover the compact set K. Therefore there is a finite set F ⊂ X such that F〈U〉⊇ K. Since 〈K〉 is dense in G, it follows that G = U〈K〉⊆ U〈F〈U〉〉⊆ 〈F ∪U〉⊆ G. � In our future arguments we will make use of the following easy lemma: Lemma 2.2. Let X be a topological space. Let K ⊂ X be a compact set with a neighborhood base {Un}n∈ω. Suppose that we have compact sets Cn ⊂ ⋂ k≤n Uk for all n ∈ ω. Then the set C = K ∪ ⋃ n∈ω Cn is also compact. A topological group G is almost metrizable if there exist a non-empty com- pact set K ⊂ G and a sequence {Un}n∈ω of open subsets of G such that (1) K ⊂ Un for all n ∈ ω and (2) if O is an open set containing K, then there is an n ∈ ω such that K ⊂ Un ⊂ O. (Such a sequence {Un}n∈ω is called a neighborhood base of K in G.) Both metric groups and locally compact groups are almost metrizable ([3]). Our next theorem demonstrates that the necessary conditions for a topolog- ical group G to have a dense compactly generated subgroup found in Theorem 2.1 are also sufficient in case G is almost metrizable. Theorem 2.3. An almost metrizable topological group G contains a dense com- pactly generated subgroup if and only if it is ℵ0-bounded and finitely generated modulo open sets. Dense compactly generated subgroups 87 Proof. The “only if” part of our theorem follows from Theorem 2.1, so it re- mains only to prove the “if” part. Let K be a compact subset of G with a neighborhood base {Un}n∈ω. Since G is ℵ0-bounded, for each n ∈ ω there is a countable set Sn ⊂ G such that G = SnUn. The set S = ⋃ n∈ω Sn is countable, so we can fix its enumeration S = {sn}n∈ω. Let g ∈ G. Let V be any neighbor- hood of the unit element of G. Then KV −1 is an open set containing K, and so there is an n ∈ ω such that Un ⊆ KV −1. Since SnUn = G, there is an s ∈ Sn such that g ∈ sUn ⊆ sKV −1. Let g = skv−1 with k ∈ K and v ∈ V . Then gv = sk ∈ gV ∩ SK 6= ∅. Since V and g are arbitrary, it follows that SK is dense in G. Since G is finitely generated modulo open sets, there are finite sets Fn such that G = 〈Fn∪Un〉 for each n ∈ ω. Set E0 = F0∪{s0}. It follows that G = 〈E0∪U0〉. So there is a finite set E1 ⊆ U0 such that F1∪{s1}⊂ 〈E0∪E1〉. From this it follows that 〈E0∪E1∪U1〉 = G. So there is a finite set E2 ⊆ U1 such that F2 ∪{s2}⊂ 〈E0 ∪E1 ∪E2〉. In this way we obtain finite sets En+1 ⊂ Un (for n ∈ ω) such that Fn+1 ∪{sn+1} ⊆ 〈E0 ∪ E1 ∪ ·· · ∪ En+1〉. By Lemma 2.2, the set C = K ∪ ⋃ n∈ω En is compact. The subgroup 〈C〉 is dense, since it contains SK. Thus G contains a compactly generated dense subgroup. � Since every metrizable group is almost metrizable ([3]), and ℵ0-boundedness is equivalent to separability for metrizable groups, from Theorem 2.3 we obtain: Corollary 2.4. A metrizable group contains a dense compactly generated sub- group if and only if it is separable and finitely generated modulo open sets. Our next result generalizes Theorem 4 from [1]. Lemma 2.5. If a σ-compact almost metrizable group G contains a dense com- pactly generated subgroup, then G itself is compactly generated. Proof. Suppose G = ⋃ n∈ω Ln, with Ln compact. Suppose also that H = 〈L0〉 is dense in G. Let K ⊆ G be a compact set with a neighborhood base {Un}n∈ω. By regularity of the topology of G and compactness of K, we may assume without loss of generality that each Un contains the closure of Un+1. By compactness of Ln and denseness of H, there is a finite subset Fn of H such that Ln ⊂ Un+1Fn. Let Cn = LnF−1n ∩Un+1. Then Cn is compact, because it is a closed subset of the union of finitely many copies of Ln. We also have Cn ⊂ Un and Ln ⊂ CnFn ⊂ 〈Cn ∪L0〉. Therefore, setting C = L0 ∪K ∪ ⋃ n∈ω Cn, we obtain 〈C〉 = G. It follows from Lemma 2.2 that C is compact. � Combining Theorem 2.3 and Lemma 2.5, we obtain our next theorem which extends the main result of [1]: Theorem 2.6. An almost metrizable topological group is compactly generated if and only if it is σ-compact and finitely generated modulo open sets. Theorems 2.3 and 2.6 become especially simple for locally compact groups: Theorem 2.7. For a locally compact group G the following conditions are equivalent: 88 Hiroshi Fujita and Dmitri Shakhmatov (i) G has a dense compactly generated subgroup, (ii) G is compactly generated, (iii) G is finitely generated modulo open sets. Proof. Let U be an open neighbourhood of the identity element having compact closure U. (i)→(ii). Let K be a compact subset of G such that 〈K〉 is dense in G. Then U ∪K is also compact and 〈U ∪K〉 ⊇ U〈K〉 = G because 〈K〉 is dense in G and U is an open neighbourhood of the identity. (ii)→(iii) follows from Theorem 2.1. (iii)→(i). Assume that G is finitely generated modulo open sets. Then there exists a finite set F ⊆ G with 〈F ∪ U〉 = G. Now note that G = 〈F ∪ U〉 ⊆ 〈F ∪U〉⊆ G. Since 〈F ∪U〉 is compact, G is compactly generated. � Since for every non-empty open subset U of a topological group G the set 〈U〉 is an open subgroup of G, it follows that a topological group without proper open subgroups is finitely generated modulo open sets ([1]). Therefore, from Theorem 2.6 we obtain Corollary 2.8. An almost metrizable, σ-compact group without proper open subgroups is compactly generated. Corollary 2.9. A metric σ-compact group without proper open subgroups is compactly generated. Totally bounded groups are finitely generated modulo open sets, and so we get Corollary 2.10. Every σ-compact totally bounded almost metrizable group is compactly generated. References [1] H. Fujita and D. B. Shakhmatov, A characterization of compactly generated metrizable groups, Proc. Amer. Math. Soc., to appear. [2] I. Guran, Topological groups similar to Lindelöf groups (in Russian), Dokl. Akad. Nauk SSSR 256 (1981), no. 6, 1305–1307; English translation in: Soviet Math. Dokl. 23 (1981), no. 1, 173–175. [3] B.A. Pasynkov, Almost-metrizable topological groups (in Russian), Dokl. Akad. Nauk SSSR 161 (1965), 281–284. Received September 2001 Revised March 2002 Dense compactly generated subgroups 89 Hiroshi Fujita Department of Mathematical Sciences Faculty of Science Ehime University Matsuyama 790-8577 Japan E-mail address : fujita@math.sci.ehime-u.ac.jp Dmitri Shakhmatov (Corresponding Author) Department of Mathematical Sciences Faculty of Science Ehime University Matsuyama 790-8577 Japan E-mail address : dmitri@dpc.ehime-u.ac.jp