() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 2, 2012 pp. 135-150 Continuous isomorphisms onto separable groups Luis Felipe Morales López Abstract A condensation is a one-to-one continuous function onto. We give suf- ficient conditions for a Tychonoff space to admit a condensation onto a separable dense subspace of the Tychonoff cube Ic and discuss the differences that arise when we deal with topological groups, where con- densation is understood as a continuous isomorphism. We also show that every Abelian group G with |G| ≤ 2c admits a separable, precom- pact, Hausdorff group topology, where c = 2 ω . 2010 MSC: 22A05, 54H11. Keywords: Condensation, continuous isomorphism, separable groups, subtopology. 1. Introduction A condensation is a bijective continuous function. If X and Y are spaces and f : X → Y is a condensation, we can assume that X and Y have the same underlying set and the topology of X is finer than the topology of Y . In this case we say that the topology of Y is a subtopology of X or that X condenses onto Y . The problem of finding conditions under which a space X admits a subtopol- ogy with a given property Q has been extensively studied by many authors. It is known that every Hausdorff space X with nw(X) ≤ κ can be condensed onto a Hausdorff space Y with w(Y ) ≤ κ (see [7, Lemma 3.1.18]). Similar results remain valid in the classes of regular or Tychonoff spaces. In [16], the authors found several necessary and sufficient conditions for a topological space to admit a connected Hausdorff or regular subtopology. It is shown in [11] that every non-compact metrizable space has a connected Hausdorff subtopology. Druzhinina showed in [9] that every metrizable space X with w(X) ≥ 2ω and achievable extent admits a weaker connected metrizable topology. Recently, 136 L. Morales López Yengulalp [17] generalized this result by removing the achievable extent condi- tion. In topological groups (and other algebraic structures with topologies), the concept of condensation has a natural counterpart: Continuous Isomorphism, a homomorphism and a condensation at the same time. At the end of the 70’s, Arhangel’skii proved in [2] that every topological group G with nw(G) ≤ κ admits a continuous isomorphism onto a topological group H with w(H) ≤ κ. In [15] Shakhmatov gave a construction that implies similar statements for topological rings, modules, and fields. C. Hernández modified Shakhmatov’s construction and extended that result to many alge- braic structures with regular and Tychonoff topologies (see [8]). As a corollary to Katz’s theorem about isomorphic embeddings into products of metrizable groups (see [1, Corollary 3.4.24]) one can easily deduce that if G is an ω-balanced topological group and the neutral element of G is a Gδ-set, then there exists a continuous isomorphism of G onto a metrizable topological group. Pestov showed that the condition on G being ω-balanced can not be removed (see [14]). In Theorem 3.2 of this paper we present conditions that a Tychonoff space must satisfy in order to admit a condensation onto a separable dense subspace of the Tychonoff cube of weight 2ω. In Corollary 4.2 we show that those conditions are not sufficient if we want to have a continuous isomorphism from a topological group to a separable group and in Theorem 4.3 we give sufficient and necessary conditions in order for a topological isomorphism from a subgroup of the product of compact metrizable Abelian groups onto a separable group to exist. As Arhangel’skii showed in [4], every continuous homomorphism of a count- ably compact group X onto a compact group Y of Ulam nonmeasurable car- dinality is open. In Example 4.4 we construct a condensation of a Tychonoff countably compact space with cellularity 2ω onto a separable compact space with cardinality 2c thus showing that Arhangel’skii result cannot be general- ized to arbitrary spaces. Finally, we show in Theorem 5.11 that every Abelian group of cardinality less than or equal to 2c admits a precompact separable Hausdorff group topology. 2. Notation and terminology We use I for the unit interval [0,1], T for the unit circle, N for the set of positive integers, Z for the integers, Q for the rational numbers, and R for the set of real numbers. Let X be a space. As usual, we denote by w(X), nw(X), χ(X), ψ(X), d(X) the weight, network weight, character, pseudocharacter, and density of X, respectively. We say that Z ⊂ X is a zero-set if there exists a real-valued continuous function f : X → R such that Z = f−1(0). Continuous isomorphisms onto separable groups 137 Let {fα : α ∈ A} be a family of functions, where fα : X → Yα for each α ∈ A. We denote by △{fα : α ∈ A} the diagonal product of the family {fα : α ∈ A}. Suppose that η = {Gα : α ∈ A} is a family of topological groups and Πη = ∏ α∈A Gα is the topological product of the family η. Then the Σ-product of η, denoted by ΣΠη, is the subgroup of Πη consisting of all points g ∈ Πη such that |{α ∈ A : πα(g) 6= eα}| ≤ ω and the σ-product of η, denoted by σΠη is the subgroup of Πη consisting of all points g ∈ Πη such that |{α ∈ A : πα(g) 6= eα}| < ω, where πα : Πη → Gα is the natural projection of Πη onto Gα and eα ∈ Gα is the neutral element of Gα, for every α ∈ A. It is easy to see that both ΣΠη and σΠη are dense subgroups of Πη. A description of properties of these subgroups can be found in [1, Section 1.6]. If X is a Tychonoff space and G is a topological group, we denote by βX the Čech-Stone compactification of X (see [7, Section 3.6]), and by ρG the Rǎıkov completion of G (see [1, Section 3.6]). The next definitions are standard in group theory (see [10, Section 1.1]). Let G be a group, e the neutral element of G, and g ∈ G an element of G distinct from e. We denote by 〈g〉 the cyclic subgroup of G generated by g. The order of g is o(g) = |〈g〉|. If o(g) = ∞ then 〈g〉 is isomorphic to Z. The set tor(G) of the elements g ∈ G with o(g) < ∞ is called the torsion part of G. If G is Abelian, tor(G) is a subgroup of G. We say that the group G is: • torsion-free if for every element g ∈ G \ {e}, o(g) = ∞; • a torsion group if for every element g ∈ G, o(g) < ∞; • bounded torsion if there exists n ∈ N such that gn = e for every g ∈ G; • unbounded torsion if G is torsion and for each n ∈ N there exists g ∈ G such that o(g) > n; • a divisible group if for every g ∈ G and n ∈ N, there is h ∈ G such that hn = g; • a p-group, for a prime p, if the order of any element of G is a power of p. If G is an Abelian torsion group, then G is the direct sum of p-groups Gp (see [10, Theorem 8.4]). The subgroups Gp are called the p-components of G. Let p be a prime number. The set of pnth complex roots of the unity, with n ∈ N forms the multiplicative subgroup Zp∞ of T. For every prime p, the group Zp∞ is divisible. 3. Condensations and subtopologies Not every space has a separable subtopology. For example, a compact Haus- dorff space X has a separable Hausdorff subtopology only if X is separable. Let us extend this fact to a wider class of spaces. We recall that a Hausdorff space X is ω-bounded if the closure of any count- able subset of X is compact. 138 L. Morales López Proposition 3.1. Let X be an ω-bounded non-separable space. Then X does not admit a condensation onto a separable Hausdorff space. Proof. By our assumptions, for every countable subset S of X we have that X \ S̄ 6= ∅. Let f : X → Y be a condensation onto a Hausdorff space Y and D a countable subset of Y . Then S = f−1(D) is a countable subset of X, and S̄ is compact. Take an element x ∈ X \ S̄. Observe that f(S̄) is compact, D ⊂ f(S̄) and f(x) 6∈ f(S̄), so D cannot be dense in Y . � The next theorem gives sufficient conditions on a Tychonoff space to admit a condensation onto a separable dense subspace of Ic, where c = 2ω. Theorem 3.2. Let X be a Tychonoff space with nw(X) ≤ 2ω. Suppose that X contains an infinite, closed, discrete, and C∗-embedded subset A. Then X can be condensed onto a separable dense subspace of Ic. Proof. We can assume that |A| = ω. By the Hewitt-Marczewski-Pondiczery theorem, we know that d(Ic) = ℵ0. Let D = {dn : n ∈ ω} be a countable dense subset of Ic, N a network for X, |N| ≤ 2ω, and A = {xn : n ∈ ω} an enumeration of A. Let g : A → D be a bijection, where g(xn) = dn for each n ∈ ω. For every α < c, let fα = pα ◦ g, where pα : I c → I(α) denotes the natural projection of Ic to the α-th factor. Our goal is to construct a family of continuous functions {gα : X → I}α ω, there is no continuous isomorphism of G onto a separable topological group. Theorem 4.1 can be generalized if we replace Tκ by the product of any family of compact metrizable Abelian groups: Theorem 4.3. Let η = {Gα : α ∈ κ} be a family of compact metrizable groups with κ ≤ 2ω, and G be a subgroup of Σ = ΣΠη. Then there exists a continuous isomorphism ϕ : G → H of G onto a separable topological group H if and only if ψ(G) ≤ ω. The proof of this fact is almost the same as in the Theorem 4.1, and we omitted. Arhangel’skii showed in [4, Corollary 12] that every continuous homomor- phism of a countably compact topological group onto a compact group of Ulam nonmeasurable cardinality is open. In particular, if there exists a continuous isomorphism of a countably compact topological group G onto a compact group of Ulam nonmeasurable cardinality, then G is compact. The next example shows that one cannot extent this result to topological spaces. Example 4.4. There exists a condensation of a countably compact non-separable Tychonoff space onto a separable compact space of Ulam nonmeasurable car- dinality, 2c. Let Y = βN be the Čech-Stone compactification of the natural numbers and Z = Y \ N. By [7, Example 3.6.18], Z contains a family A of cardinality c consisting of pairwise disjoint non-empty open sets. Let π1 : Y × Z → Y and π2 : Y × Z → Z be the natural projections to the first and the second factor respectively. Since Y is compact, π2 is a closed mapping. By [7, Theorem 3.5.8], Z is a compact space because it is the remainder of a locally compact space and so, π1 is a closed mapping too. By [7, Theorem 3.6.14], every infinite closed subset S of both Y and Z has cardinality equal to 2c. Let M be an infinite subset of Y × Z. It is clear that 142 L. Morales López at least one of the set, π1(M) or π2(M), is infinite. Suppose that π1(M) is infinite. Since the projection π1 is closed, π1(M) is a closed subset of Y , so π1(M) and M have cardinality equal to 2 c. Our goal is to construct a countably compact non-separable subspace X ⊂ Y ×Z such that π1(X) = Y , π1|X is a one-to-one mapping, and π2(X)∩A 6= ∅ for every A ∈ A. Recall that [Y ]ω is the family of subsets of Y with cardinality ω. Let A = {Aα : α < c} be a faithful enumeration of A and choose zα ∈ Aα for each α < c. Let also Y = {yβ : β < 2 c} and [Y ]ω = {Fγ : c ≤ γ < 2 c} be faithful enumerations of Y and [Y ]ω respectively such that Fc ⊂ {yβ : β < c}. We shall define a transfinite sequence {Xγ : c ≤ γ < 2 c} of subsets of Y × Z satisfying the following conditions for each γ with c ≤ γ < 2c: (iγ): Xβ ⊂ Xγ if c ≤ β < γ; (iiγ): the restriction of π1 to Xγ is a one-to-one mapping; (iiiγ): Fγ ⊂ π1(Xγ); (ivγ): π −1 1 (Fγ) ∩ Xγ has an accumulation point in Xγ; (vγ): |Xγ| ≤ |γ|. For every α < c put xα = (yα,zα) and let X ′ c = {xα : α < c}. By our enumeration of [Y ]ω, Fc ⊂ π1(X ′ c ). Put Bc = π −1 1 (Fc) ∩ X ′ c . Since π1 is closed and Fc ⊂ π1(Bc), the cardinality of π1(Bc) is equal to 2 c, so we can choose xc ∈ Bc such that π1(xc) 6∈ π1(X ′ c ). Put Xc = X ′ c ∪ {xc}. Conditions (iic), (iiic), (ivc), and (vc) are clearly satisfied, condition (ic) is vacuous. Suppose that for some γ with c ≤ γ < 2c, Xξ are defined for all ξ, c ≤ ξ < γ. Let X̃γ = ⋃ c≤ξ<γ Xξ. We have two possibilities. If Fγ ⊂ π1(X̃γ), then put X′γ = X̃γ. If Fγ \ π1(X̃γ) 6= ∅, then choose an arbitrary point xy ∈ π −1 1 (y) for each y ∈ Fγ \ π1(X̃γ) and put X ′ γ = X̃γ ∪ {xy : y ∈ Fγ \ π1(X̃γ)}. In both cases, Fγ ⊂ π1(X ′ γ). Since conditions (iξ) and (vξ) are satisfied for all c ≤ ξ < γ, |X ′ γ| ≤ |γ| < 2 c. Let Bγ = π −1 1 (Fγ) ∩ X ′ γ. Since π1 is a closed mapping, |π1(Bγ)| = 2 c, so there exists xγ ∈ Bγ such that π1(xγ) 6∈ π1(X ′ γ). Let Xγ = X ′ γ ∪ {xγ}. Clearly condition (iγ) is satisfied. Since conditions (iξ) and (iiξ) are satisfied for every ξ with c ≤ ξ < γ, π1|X̃γ is a one-to-one mapping. By our definition of X ′ γ, π1|X′γ is a one-to-one mapping too. Finally, by our choose of xγ, π1(xγ) 6∈ π1(X ′ γ), so π1|Xγ is a one-to-one mapping by (iiγ). As Fγ ⊂ π1(X ′ γ) and xγ ∈ Xγ, (iiiγ) and (ivγ) are satisfied. Since (vξ) and (iξ) are satisfied for every ξ with c ≤ ξ < γ, |X̃γ| ≤ |γ|. As |Xγ \ X̃γ| ≤ ω, we conclude that |Xγ| ≤ |γ| Put X = ⋃ c≤γ<2c Xγ and let f : X → Y be the restriction of π1 to X. Since conditions (iγ) and (iiγ) are satisfied for all γ with c ≤ γ < 2 c, f is a continuous one-to-one function. Let y ∈ Y be an arbitrary element of Y and F ∈ [Y ]ω be a subset of Y with y ∈ F . Then there exists γ, c ≤ γ < 2c such Continuous isomorphisms onto separable groups 143 that F = Fγ. By (iiiγ), y ∈ F = Fγ ⊂ π1(Xγ) ⊂ π1(X) = f(X), so f(X) = Y . Therefore f is a condensation of X onto Y . Let B be an arbitrary infinite countable subset of X. Then F = f(B) is an infinite countable subset of Y and there exists γ < 2c such that F = Fγ. By (ivγ), B = f −1(F) = π−11 (Fγ) ∩ Xγ has an accumulation point in Xγ and in X. This means that X is countably compact. Since A∩π2(X) ⊃ A∩π2(Xc) 6= ∅ for every A ∈ A, X cannot be separable. 5. Separable Group Topologies for Abelian Groups In this section we prove that every Abelian group G with |G| ≤ 2c admits a separable precompact Hausdorff group topology. To do this, we divide the job in three parts: Case 1.: There is x ∈ G with o(x) = ∞. Case 2.: G is a bounded torsion group. Case 3.: G is an unbounded torsion group. We say that a topological group is monothetic if it has a dense cyclic sub- group. The next result is proved in [12, Corollary 25.15]: Lemma 5.1. The group Tκ is monothetic if and only if κ ≤ c. Let us begin with the case when G is a non-torsion group (Case 1). Theorem 5.2. Let G be an Abelian group. Suppose that |G| ≤ 2c and there is an element x ∈ G of infinite order. Then there exists a separable precompact Hausdorff group topology on G. Proof. The main idea of the proof is to define a monomorphism ϕ : G → Tc such that ϕ(G) will be separable. First we do this in the case when G is divisible. Let H be a minimal divisible subgroup of G with x ∈ H. Since o(x) is infinite, H is isomorphic to Q. By Lemma 5.1, there exists a ∈ Tc such that 〈a〉 = Tc. Let ϕ : H → Tc be a monomorphism such that ϕ(x) = a. For every β < c, put ϕβ = pβ ◦ ϕ, where pβ = T c → T(β) is the projection of T c to the β’s factor. Let κ = |G| > ω. Since G is divisible, it is isomorphic to the direct sum H ⊕ ⊕ α∈A Gα, where each Gα is a subgroup of G isomorphic either to Q or Zp∞ for some prime number p, and A is an index set of cardinality κ (see [10, Theorem 23.1]). For each α ∈ A, let ̺α : Gα → F be the isomorphism of Gα onto F , where F is either Q or Zp∞ for some prime number p. Consider A as a subspace of the space 2c with the product topology. Let B be the canonical base of 2c, we know that |B| = c. For each g ∈ G, let hg ∈ H and k ∈ ⊕ α∈A Gα be such that g = hg + k. If g ∈ G \ H, then k 6= e and there exists a non-empty finite subset c(g) ⊂ A such that k ∈ ⊕ α∈c(g) Gα. For every α ∈ c(g), take kα ∈ Gα such that k = 144 L. Morales López ∑ α∈c(g) kα. Choose an arbitrary α(g) ∈ c(g) such that kα(g) is not the identity of the group Gα(g). Let Ug ∈ B be an open set satisfying Ug ∩ c(g) = {α(g)}. Thus for each g ∈ G \ H we have defined a pair (hg,Ug) ∈ H × B. The cardinality of the set P = {(hg,Ug) : g ∈ G \ H} is less than or equal to |H × B| = ω · c = c. Let P = {Pβ : β < c} be an enumeration of P , where Pβ is a pair (hβ,Uβ) with hβ ∈ H and Uβ ∈ B. For each β < c, we define a homomorphism ψβ : ⊕ α∈A Gα → T as follows: If ϕβ(hβ) = 1, we can define ψβ such that ψβ|Gα = ̺α if α ∈ Uβ, and ψβ|Gα = 1, otherwise. If ϕβ(hβ) 6= 1, we define ψβ ≡ 1. Let ϕβ be the homomorphism defined by ϕβ = ϕβ ⊕ ψβ. It is clear that, for each β < c, ϕβ is an extension of ϕβ, therefore ϕ = △β ω. Suppose first that for every g ∈ G, the order of g is a power of a fixed prime number p. Since G is a bounded torsion group, we can find k ∈ N with o(g) ≤ pk for each g ∈ G. Hence there exists a set {gα : α ∈ A} ⊂ G such that G = ⊕ α∈A〈gα〉 (see [10, Theorem 17.2]). For each α ∈ A, let ̺α : 〈gα〉 → T be the monomorphism defined by ̺α(gα) = e 2πi/nα, where nα = o(gα). For every n ≤ k, put An = {α ∈ A : o(gα) = p n} and m = max{n : |An| ≥ ω}. Let J0 = {αj : j ∈ ω} be an infinite countable subset of Am, G0 = ⊕ α∈J0 〈gα〉, J = ( ⋃ n≤m An) \ J0 and F = ⋃ n>m An. Observe that G′ = ⊕ α∈F 〈gα〉 is finite and |G0| = ω. So G = G ′ ⊕ G0 ⊕ ⊕ α∈J〈gα〉. Let H = Pc, where P is the subgroup of T consisting of all pm-th complex roots of unity. By the Hewitt-Marczewski-Pondiczery theorem, H is separable. Let D be a countable dense subgroup of H. Since D is a bounded torsion group, it is direct sum of cyclic groups, i.e., D = ⊕ j∈ω〈dj〉. By Lemma 5.4 we can assume that o(dj) = p m for every n ∈ ω. Let ϕ be a monomorphism ϕ : G0 → H such that ϕ(gαj ) = dj for every n ∈ ω. We will extend this monomorphism to Ḡ = G0 ⊕ ⊕ α∈J〈gα〉. For every β < c, let ϕβ = pβ ◦ ϕ, where pβ : H → P(β) is the natural projection of H onto the β-th factor. Consider J as a subspace of the space 2c endowed with the product topology. Let B be the base of canonical open sets in 2c, |B| = c. For every ḡ ∈ Ḡ, there exists ḡ0 ∈ G0 and a finite set c(ḡ) ⊂ J such that ḡ = ḡ0+lḡ, where lḡ ∈ ⊕ α∈c(ḡ)〈gα〉. If ḡ ∈ Ḡ\G0, then c(ḡ) 6= ∅. Let αḡ ∈ c(ḡ) be an arbitrary element of c(ḡ) and choose Uḡ ∈ B such that Uḡ ∩ c(ḡ) = {αḡ}. The set S = {(ḡ0,Uḡ) : ḡ ∈ Ḡ \ G0} has cardinality less than or equal to |G0 × B| = ω · c = c. Let S = {Sβ : β < c} be an enumeration of S, where Sβ is a pair (aβ,Uβ) with aβ ∈ G0 and Uβ ∈ B. If ϕβ(aβ) = 1, then let ψβ : ⊕ α∈J〈gα〉 → P be a homomorphism such that ψβ|〈gα〉 = ̺α for each α ∈ Uβ and ψβ(gα) = 1 if α ∈ J \ Uβ. If ϕβ(aβ) 6= 1, put 146 L. Morales López ψβ ≡ 1. Let ϕβ = ϕβ ⊕ ψβ. It is clear that, for each β < c, ϕβ is an extension of ϕβ. Therefore ϕ = △β 4π/m, there are two distinct m-th roots of z1 in V . Let y1, y2 be two elements of V such that my1 = my2 = z1 and the distance between y1 and y2 is 2π/m. Note that y1 and y2 can not be both n-th roots of z2, otherwise the distance between them would be greater than or equal to 2π/n, and it would follow that m ≤ n contradicting the assumptions of the lemma. � Continuous isomorphisms onto separable groups 147 Lemma 5.7. Let K be a countable subgroup of Tc and f′ ∈ K, m ≥ 2. Suppose that {Vα : α < c} is a family of open arcs of T such that 4π/m < l(Vα), for every α < c. Then there exists f ∈ ∏ {Vα : α < c} such that mf = f ′ and nf 6∈ K for each n with 1 ≤ n < m. Proof. Let K × {1, ...,m − 1} = {(hk,nk) : k ∈ ω} be an enumeration of K ×{1, ...,m−1}. For each k < ω we will define αk < c and xαk ∈ T satisfying the following conditions: (ik): αk 6= αj if j < k; (iik): xαk ∈ Vαk ; (iiik): mxαk = f ′(αk); (ivk): nkxαk 6= hk(αk). Let α0 < c be an arbitrary ordinal. By Lemma 5.6 (with V = Vα0, z1 = f′(α0), z2 = h0(α0), n = n0) we can choose an element xα0 ∈ Vα0 that satisfies (ii0), (iii0) and (iv0). Condition (i0) is vacuous. Suppose that for every j < k we have chosen αj and xαj such that conditions (ij) - (ivj) are satisfied. We can pick αk < c that satisfies (ik). By Lemma 5.6 with V = Vαk , z1 = f ′(αk), z2 = hk(αk), and n = nk, we can choose xαk that satisfies (iik) - (ivk). Finally, for each α ∈ c \ {αk : k ∈ ω} we use Lemma 5.6 again with V = Vα, z1 = f ′(α), z2 = 1, n = 1 to select xα ∈ Vα such that mxα = f ′(α). We define f ∈ Tc by f(α) = xα for each α < c. Then: • f(α) ∈ Vα for each α < c, therefore f ∈ ∏ {Vα : α < c}. • mf(α) = mxα = f ′(α) for each α < c, so mf = f′. • Given n ∈ {1, ...,m − 1} and h ∈ K, there exists k ∈ ω such that (h,n) = (hk,nk). By conditions (iiik) and (ivk), we have that nf(αk) = nkxαk 6= hk(αk) = h(αk). Since h ∈ K is arbitrary, nf 6∈ K for every n < m. � The proof of the following lemma can be found in [1, Lemma 1.1.5]: Lemma 5.8. Let G and G∗ be Abelian topological groups, K and K∗ subgroups of G and G∗, respectively. Suppose that there exist x ∈ G, x∗ ∈ G∗, m ∈ N, m ≥ 2 and an isomorphism ψ : K → K∗ that satisfy the following conditions: • mx ∈ K and mx∗ ∈ K∗; • nx 6∈ K and nx∗ 6∈ K∗ for every n ∈ N, 1 ≤ n < m; • ψ(mx) = mx∗. Then there exists a unique isomorphism ϕ : K + 〈x〉 → K∗ + 〈x∗〉 extending ψ such that ϕ(x) = x∗. Now we are going to give some definitions from group theory. A system {a1, ...,ak} of a group G is called independent if n1a1 + ... + nkak = 0 (ni ∈ Z) implies 148 L. Morales López n1a1 = ... = nkak = 0. We say that an infinite system L of the group G is independent if any finite subset of L is independent. By the rank r(G) of an Abelian group G is meant the cardinal number of a maximal independent system in G. The torsion-free rank r0(G) is the cardinal of the maximal independent system which contains only elements of infinite order. For each prime number p, the p-rank rp(G) of G is the cardinal of a maximal independent system which contains only elements whose orders are powers of p. The next lemma can be found in [6, Lemma 3.17]. Lemma 5.9. Let G and G∗ be Abelian groups such that |G| ≤ r(G∗) and |G| ≤ rp(G ∗) for every prime number p. Suppose that H is a subgroup of G satisfying r(H) < r(G∗) and rp(H) < rp(G ∗) for every prime p. If G∗ is a divisible group, then every monomorphism ϕ : H → G∗ can be extended to a monomorphism ψ : G → G∗. Now we are in position to prove the following theorem. Theorem 5.10. Let G be an unbounded torsion Abelian group with |G| ≤ 2c. Then G admits a separable, precompact, Hausdorff group topology. Proof. Let V be a countable base for the topology of T consisting of open arcs such that T ∈ V. Since G is an unbounded torsion group, we can choose a subset S ⊂ G \ {e} such that |nS| = ω for every n ∈ N, where e is the unity of G. Consider c as the topological space 2ω and let B be the canonical base for 2ω consisting of non-empty clopen subsets of 2ω. Then |B| = ω. Let U be the set of all finite open covers of 2ω formed by pairwise disjoint sets. For U ∈ U and α < c let Uα,U denote the unique U ∈ U such that α ∈ U. Put E = {(U,υ) : U ∈ U and υ : U → V is a function}. For (U,υ) ∈ E, let F(U,υ) = ∏ {υ(Uα,U) : α < c}. Clearly E is countable. Let E = {(Uk,υk) : k ∈ ω} be an enumeration of E such that U0 = {2 ω} and υ0(2 ω) = T. For each k < ω, choose nk ∈ N such that 4π/nk < min {l(υ(U)) : U ∈ Uk}. By recursion on k ∈ ω we will choose an element xk ∈ S and define a map ϕk : Hk = 〈{xj : j ≤ k}〉 → T c satisfying the following conditions: (ik): ϕk(xk) ∈ F(Uk,υk); (iik): ϕk is a monomorphism; (iiik): ϕk|Hj = ϕj for all j < k. Pick any element x0 in S and let ϕ0 : 〈x0〉 → T c be an arbitrary monomor- phism. Then conditions (i0) and (ii0) are satisfied, while condition (iii0) is vacuous. Now let k ∈ N, and suppose that xj ∈ S and a map ϕj satisfying (ij), (iij) and (iiij) have already been constructed for every j < k. Put H′k = ⋃ j nk and 4π/m < 4π/nk < l(Vα) for every α < c. Note that m ≥ 2. By Lemma 5.7 we can find f ∈ F(U,υ) = ∏ {υ(Uα,U) : α < c} such that mf = f ′ and nf 6∈ K for n < m. Put Hk = 〈{xj : j ≤ k}〉. By Lemma 5.8 we can extend ϕ ′ k to a monomorphism ϕk : Hk → T c with ϕk(xk) = f. We are going to verify that ϕk satisfies (ik), (iik) and (iiik). As ϕk(xk) = f ∈ F(U,υ), the condition (ik) is satisfied. By Lemma 5.8, ϕk is a monomorphism that extends ϕ′k, so (iik) and (iiik) are satisfied. Let H = ⋃ k∈ω Hk and ϕ = ⋃ k∈ω ϕk. Since (iik) and (iiik) are fulfilled for every k ∈ ω, we have that ϕ : H → Tc is a monomorphism. We claim that ϕ(H ∩S) is a dense subset of Tc. Let W be a non-empty open set of Tc. Then there exist a finite subset I = {α1, ...,αn} of c and non-empty open arcs Vα1, ...,Vαn ∈ V such that ∏ α