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Extremal phenomena in certain classes of totally bounded groups

100%
Comfort W. W., Robertson L. C.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 0. Introduction..............................................................................................5 1. Notation and results from the literature....................................................6 2. Extending a topology: Some negative results........................................10 3. Finding dense subgroups: Some negative results.................................12 4. Extensions and dense subgroups: Some positive results......................14 5. Extremal pseudocompact Abelian groups: The case $x^p ≡ 1$.............24 6. Recognizing pseudocompact groups.....................................................32 7. Extremal pseudocompact Abelian groups: The 0-dimensional case......37 References................................................................................................41
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Imposing psendocompact group topologies on Abeliau groups

93%
Comfort W. W., Remus I.
Fundamenta Mathematicae
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1993
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tom 142
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nr 3
221-240
EN
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.
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Correction to the paper “The Bohr compactification, modulo a metrizable subgroup” (Fund. Math. 143 (1993), 119–136)

75%
Comfort W. W., Javier Trigos-Arrieta F., Wu T.-S.
Fundamenta Mathematicae
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1997
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tom 152
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nr 1
97-98
4
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The Bohr compactification, modulo a metrizable subgroup

75%
Comfort W. W., Trigos-Arrieta F. J., Wu S. L.
Fundamenta Mathematicae
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1993
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tom 143
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nr 2
119-136
EN
The authors prove the following result, which generalizes a well-known theorem of I. Glicksberg [G]: If G is a locally compact Abelian group with Bohr compactification bG, and if N is a closed metrizable subgroup of bG, then every A ⊆ G satisfies: A·(N ∩ G) is compact in G if and only if {aN:a ∈ A} is compact in bG/N. Examples are given to show: (a) the asserted equivalence can fail in the absence of the metrizability hypothesis, even when N ∩ G = {1}; (b) the asserted equivalence can hold for suitable G and N with N closed in bG but not metrizable; (c) an Abelian group may admit two topological group topologies U and T, with U totally bounded, T locally compact,U ⊆ T, with U and T sharing the same compact sets, and such that nevertheless U is not the topology inherited from the Bohr compactification of ⟨ G, T⟩. There are applications to topological groups of the form kG for G a totally bounded Abelian group.
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Extending continuous functions on X x Y to subsets of βX x βY

74%
Comfort W. W., Negrepontis S.
Fundamenta Mathematicae
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1966
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tom 59
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nr 1
1-12
6
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Some topological properties associated with measurable cardinals

74%
Comfort W. W., Negrepontis S.
Fundamenta Mathematicae
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1970
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tom 69
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nr 3
191-205
7
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Compact-covering numbers

56%
Baloglou G., Comfort W. W.
Fundamenta Mathematicae
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1988
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tom 131
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nr 1
69-82
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On families of large oscillation

56%
Comfort W. W., Negrepontis S.
Fundamenta Mathematicae
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1972
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tom 75
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nr 3
275-290
9
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Topologies induced by groups of characters

47%
Comfort W. W., Ross K. A.
Fundamenta Mathematicae
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1964
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tom 55
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nr 3
283-291
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