Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, and Lewis C. Robertson. Extremal phenomena in certain classes of totally bounded groups. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1988. <http://eudml.org/doc/268512>.

@book{W1988,
abstract = {For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly finer pseudocompact topological group topology and a proper dense pseudocompact subgroup: (1) G is O-dimensional and Abelian; (2) $G = H^α$ with α > ω, |H| > 1; (3) G is a dense subgroup of $T^\{(ω⁺)\}$.Thwarting our attempts to improve (1), (2) and (3) are examples, for every α > ω, of pseudocompact groups G₀ and G₁ of weight α such that (a) there are surjective φ ∈ Hom(G₀,K) with K compact, φ continuous and open, and a dense, pseudocompact subgroup H of K such that $φ^\{-1\}(H)$ is not pseudocompact; and (b) G₁ admits no homomorphism onto any non-trivial product.CONTENTS0. Introduction..............................................................................................51. Notation and results from the literature....................................................62. Extending a topology: Some negative results........................................103. Finding dense subgroups: Some negative results.................................124. Extensions and dense subgroups: Some positive results......................145. Extremal pseudocompact Abelian groups: The case $x^p ≡ 1$.............246. Recognizing pseudocompact groups.....................................................327. Extremal pseudocompact Abelian groups: The 0-dimensional case......37References................................................................................................41},
author = {W. W. Comfort, Lewis C. Robertson},
keywords = {Hausdorff topological group; cardinal invariants; weight; density; group topology; connected compact simple Lie group; abelian groups; pseudocompact groups; compact metrizable group; pseudocompact abelian torsion group},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Extremal phenomena in certain classes of totally bounded groups},
url = {http://eudml.org/doc/268512},
year = {1988},
}

TY - BOOK
AU - W. W. Comfort
AU - Lewis C. Robertson
TI - Extremal phenomena in certain classes of totally bounded groups
PY - 1988
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly finer pseudocompact topological group topology and a proper dense pseudocompact subgroup: (1) G is O-dimensional and Abelian; (2) $G = H^α$ with α > ω, |H| > 1; (3) G is a dense subgroup of $T^{(ω⁺)}$.Thwarting our attempts to improve (1), (2) and (3) are examples, for every α > ω, of pseudocompact groups G₀ and G₁ of weight α such that (a) there are surjective φ ∈ Hom(G₀,K) with K compact, φ continuous and open, and a dense, pseudocompact subgroup H of K such that $φ^{-1}(H)$ is not pseudocompact; and (b) G₁ admits no homomorphism onto any non-trivial product.CONTENTS0. Introduction..............................................................................................51. Notation and results from the literature....................................................62. Extending a topology: Some negative results........................................103. Finding dense subgroups: Some negative results.................................124. Extensions and dense subgroups: Some positive results......................145. Extremal pseudocompact Abelian groups: The case $x^p ≡ 1$.............246. Recognizing pseudocompact groups.....................................................327. Extremal pseudocompact Abelian groups: The 0-dimensional case......37References................................................................................................41
LA - eng
KW - Hausdorff topological group; cardinal invariants; weight; density; group topology; connected compact simple Lie group; abelian groups; pseudocompact groups; compact metrizable group; pseudocompact abelian torsion group
UR - http://eudml.org/doc/268512
ER -