# Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort; Lewis C. Robertson

- Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1988

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topW. 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 -

## Citations in EuDML Documents

top- Gerald Itzkowitz, Dmitri Shakhmatov, Large families of dense pseudocompact subgroups of compact groups
- Constancio Hernández, Mihail G. Tkachenko, Subgroups and products of $\mathbb{R}$-factorizable $P$-groups
- William Wistar Comfort, Jan van Mill, Extremal pseudocompact Abelian groups: A unified treatment

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