Imposing psendocompact group topologies on Abeliau groups

Comfort, W.; Remus, I.

Comfort, W. ; Remus, I.

Fundamenta Mathematicae, Tome 142 (1993), p. 221-240 / Harvested from The Polish Digital Mathematics Library

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, $m (α) \leq 2^{α}$ . We show: Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m $(α) \leq r_{0} (G) \leq γ \leq 2^{α}$ , or α > ω and $α^{ω} \leq r_{0} (G) \leq 2^{α}$ , then G admits a pseudocompact group topology of weight α. Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies $r_{0} (G) \geq m (α)$ . Theorem 5.2(b). If G is divisible Abelian with $2^{r_{0} (G)} \leq γ$ , 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 $β \leq γ < κ \leq 2^{β}$ . Then both $\oplus_{γ} ℚ$ 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.

Publié le : 1993-01-01

Zbl 0865.54035

EUDML-ID : urn:eudml:doc:211983

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     author = {W. Comfort and I. Remus},
     title = {Imposing psendocompact group topologies on Abeliau groups},
     journal = {Fundamenta Mathematicae},
     volume = {142},
     year = {1993},
     pages = {221-240},
     zbl = {0865.54035},
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     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p221bwm}
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Comfort, W.; Remus, I. Imposing psendocompact group topologies on Abeliau groups. Fundamenta Mathematicae, Tome 142 (1993) pp. 221-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p221bwm/

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