Imposing psendocompact group topologies on Abeliau groups

W. Comfort; I. Remus

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 3, page 221-240
  • ISSN: 0016-2736

Abstract

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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 ⊕ 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.

How to cite

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Comfort, W., and Remus, I.. "Imposing psendocompact group topologies on Abeliau groups." Fundamenta Mathematicae 142.3 (1993): 221-240. <http://eudml.org/doc/211983>.

@article{Comfort1993,
abstract = {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.},
author = {Comfort, W., Remus, I.},
journal = {Fundamenta Mathematicae},
keywords = {pseudocompact group; $G_δ$-dense subgroup; singular cardinals hypothesis; torsion-free rank; connected topological group; 0-dimensional group; divisible hull; chain; anti-chain; -dense subgroup; pseudocompact group topology},
language = {eng},
number = {3},
pages = {221-240},
title = {Imposing psendocompact group topologies on Abeliau groups},
url = {http://eudml.org/doc/211983},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Comfort, W.
AU - Remus, I.
TI - Imposing psendocompact group topologies on Abeliau groups
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 3
SP - 221
EP - 240
AB - 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.
LA - eng
KW - pseudocompact group; $G_δ$-dense subgroup; singular cardinals hypothesis; torsion-free rank; connected topological group; 0-dimensional group; divisible hull; chain; anti-chain; -dense subgroup; pseudocompact group topology
UR - http://eudml.org/doc/211983
ER -

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