A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {\sigma_n : n\in N} of cardinals such that w(G) = sup {\sigma_n : n \in N} and sup {2^{\sigma_n} : n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^\sigma for some cardinal \sigma, or w(G) = min {\sigma: |G| \leq 2^\sigma}; moreover, the equality |G| = 2^{w(G)} holds whenever cf (w(G)) > \omega. For a cardinal \kappa, we denote by F_\kappa the free abelian group with \kappa many generators. If F_\kappa admits a pseudocompact group topology, then \kappa \geq c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F_c is equivalent to the Lusin's Hypothesis 2^{\omega_1} = c. For \kappa > c, we prove that F_\kappa admits a (zero-dimensional) minimal pseudocompact group topology if and only if F_\kappa has both a minimal group topology and a pseudocompact group topology. If \kappa > c, then F_\kappa admits a connected minimal pseudocompact group topology of weight \sigma if and only if \kappa = 2^\sigma. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.
The following notion was introduced independently by Choquet (see Doïtchinov [14]) and Stephenson [24]. Definition 1.1. A Hausdorff group topology τ on a group G is called minimal provided that every Hausdorff group topology τ ′ on G such that τ ′ ⊆ τ satisfies τ ′ = τ . Equivalently, a Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is a topological isomorphism.
There exist abelian groups which admit no minimal group topologies at all, e.g., the group of rational numbers Q [21] or Prüfer’s group Z(p ∞ ) [11]. This suggests the general problem to determine the algebraic structure of the minimal abelian groups, or equivalently, the following Problem 1.2. [9, Problem 4.1] Describe the abelian groups that admit minimal group topologies.
Prodanov solved Problem 1.2 first for all free abelian groups of finite rank [20], and later on he improved this result extending it to all cardinals ≤ c [21]: Theorem 1.3. [20,21] For every cardinal κ ≤ c, the group F κ admits minimal group topologies.
Since |F κ | = ω • κ for each cardinal κ, uncountable free abelian groups are determined up to isomorphism by their cardinality. This suggests the problem of characterizing the cardinality of minimal abelian groups. The following set-theoretic definition is ultimately relevant to this problem. Definition 1.4.
(i) For infinite cardinals κ and σ the symbol Min(κ, σ) denotes the following statement: There exists a sequence of cardinals {σ n : n ∈ N} such that We say that the sequence {σ n : n ∈ N} as above witnesses Min(κ, σ). (ii) An infinite cardinal number κ satisfying Min(κ, σ) for some infinite cardinal σ will be called a Stoyanov cardinal. (iii) For the sake of convenience, we add to the class of Stoyanov cardinals also all finite cardinals.
The cardinals from item (ii) in the above definition were first introduced by Stoyanov in [25] under the name “permissible cardinals”. Their importance is evident from the following fundamental result of Stoyanov providing a complete characterization of the possible cardinalities of minimal abelian groups, thereby solving Problem 1.2 for all free abelian groups: Theorem 1.5. [25] (a) If G is a minimal abelian group, then |G| is a Stoyanov cardinal.
(b) For a cardinal κ, F κ admits minimal group topologies if and only if κ is a Stoyanov cardinal.
If κ is a finite cardinal satisfying (1), then κ = 2 n for some n ∈ N. On the other hand, every finite group is compact and thus minimal. Furthermore, the group F n admits minimal group topologies for every n ∈ N by Theorem 1.3. It is in order to include also the case of finite groups in Theorem 1.5(a) and finitely generated groups in Theorem 1.5(b) that we decided to add item (iii) to Definition 1. 4.
It is worth noting that the commutativity of the group in Theorem 1.5(b) is important because all restrictions on the cardinality disappear in the case of (non-abelian) free groups: Theorem 1.6. [23,22] Every free group admits a minimal group topology.
For free groups with infinitely many generators this theorem has been proved in [23]. The remaining case was covered in [22].
A subgroup H of a topological group G is essential (in G) if H ∩ N = {e} for every closed normal subgroup N of G with N = {e}, where e is the identity element of G [20,24]. This notion is a crucial ingredient of the so-called “minimality criterion”, due to Prodanov and Stephenson [20,24], describing the dense minimal subgroups of compact groups.
Theorem 1.7. ( [20,24]; see also [10,12]) A dense subgroup H of a compact group G is minimal if and only if H is essential in G.
A topological group G is pseudocompact if every continuous real-valued function defined on G is bounded [18]. In the spirit of Theorem 1.5(b) characterizing the free abelian groups admitting minimal topologies, one can also describe the free abelian groups that admit pseudocompact group topologies ( [5,13]; see Theorem 4.4). The aim of this article is to provide simultaneous minimal and pseudocompact topologization of free abelian groups. To achieve this goal, we need an alternative description of Stoyanov cardinals obtained in Proposition 3.5 as well as a more precise form of Theorem 1.5(a) given in Theorem 2.1.
We finish this section with a fundamental restriction on the size of pseudocompact groups due to van Douwen. Theorem 1.8. [26] If G is an infinite pseudocompact group, then |G| ≥ c.
Cardinality and weight of minimal abelian groups. Let κ be a cardinal. Recall that the cofinality cf(κ) of κ is defined to be the smallest cardinal κ such that there exists a transfinite sequence {τ α : α ∈ κ} of cardinals such that κ = sup{τ α : α ∈ κ} and τ α < κ for all α ∈ κ. We say that κ is exponential if κ = 2 σ for some cardinal σ, and we call κ non-exponential otherwise. Recall that κ is called a strong limit provided that 2 µ < κ for every cardinal µ < κ. When κ is infinite, we define log κ = min{σ : κ ≤ 2 σ }.
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