Imposing psendocompact group topologies on Abeliau groups

W. W. Comfort ¹   ; Dieter Remus ²

¹ Department of Mathematics Wesleyan University Middletown, Connecticut 06459 U.S.A.
² Institut für Mathematik Universität Hannover Welfengarten 1 d-3000 Hannover, Germany

Fundamenta Mathematicae, Tome 142 (1993) no. 3, pp. 221-240

Résumé

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, $m(α) ≤ 2^α$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m$(α)≤ r_0 (G) ≤ γ ≤ 2^α$, or α > ω and $α^ω ≤ r_0(G) ≤ 2^α$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_0(G) ≥ m(α)$.  Theorem 5.2(b). If G is divisible Abelian with $2^{r_{0}(G)} ≤ γ$, then G admits at most $2^γ$-many pseudocompact group topologies.  Theorem 6.2. Let $β = α^ω$ or $β = 2^α$ with β ≥ α, and let $β ≤ γ κ ≤ 2^β$. Then both $⊕_γℚ$ and the free Abelian group on γ-many generators admit exactly $2^κ$-many pseudocompact group topologies of weight κ. Of these, some $κ^+$-many form a chain and some $2^κ$-many form an anti-chain.

Détail

DOI : 10.4064/fm-142-3-221-240

Keywords: pseudocompact group, $G_δ$-dense subgroup, singular cardinals hypothesis, torsion-free rank, connected topological group, 0-dimensional group, divisible hull, chain, anti-chain