Wyniki wyszukiwania

1

Large families of dense pseudocompact subgroups of compact groups

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Itzkowitz G. L., Shakhmatov D.

Fundamenta Mathematicae

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1995

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tom 147

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nr 3

197-212

EN

We prove that every nonmetrizable compact connected Abelian group G has a family H of size |G|, the maximal size possible, consisting of proper dense pseudocompact subgroups of G such that H ∩ H'={0} for distinct H,H' ∈ H. An easy example shows that connectedness of G is essential in the above result. In the general case we establish that every nonmetrizable compact Abelian group G has a family H of size |G| consisting of proper dense pseudocompact subgroups of G such that each intersection H H' of different members of H is nowhere dense in G. Some results in the non-Abelian case are also given.

2

Metrization criteria for compact groups in terms of their dense subgroups

100%

Dikranjan D., Shakhmatov D.

Fundamenta Mathematicae

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2013

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tom 221

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nr 2

161-187

EN

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism Ĝ → D̂ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or $G_δ$-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its $G_δ$-dense subgroups is metrizable, thereby resolving a question of Hernández, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Domínguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building $G_δ$-dense subgroups without uncountable compact subsets in compact groups of weight ω₁ (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.

3

A compact Hausdorff topology that is a T₁-complement of itself

100%

Shakhmatov D., Tkachenko M.

Fundamenta Mathematicae

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2002

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tom 175

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nr 2

163-173

EN

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = {X∖F: F ⊆ X is finite} ∪ {∅} and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X,τ_X)$ and $(Y,τ_Y)$ are called T₁-complementary provided that there exists a bijection f: X → Y such that $τ_X$ and ${f^{-1}(U): U ∈ τ_Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size $2^{𝔠}$ which is T₁-complementary to itself (𝔠 denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size 𝔠 that is T₁-complementary to itself is both consistent with and independent of ZFC. On the other hand, we construct in ZFC a countably compact Tikhonov space of size 𝔠 which is T₁-complementary to itself and a compact Hausdorff space of size 𝔠 which is T₁-complementary to a countably compact Tikhonov space. The last two examples have the smallest possible size: It is consistent with ZFC that 𝔠 is the smallest cardinality of an infinite set admitting two Hausdorff T₁-complementary topologies [8]. Our results provide complete solutions to Problems 160 and 161 (both posed by S. Watson [14]) from Open Problems in Topology (North-Holland, 1990).

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Shakhmatov D.

1 Dikranjan D.

1 Itzkowitz G. L.

1 Tkachenko M.

1 2013

1 2002

1 1995

Large families of dense pseudocompact subgroups of compact groups

Metrization criteria for compact groups in terms of their dense subgroups

A compact Hausdorff topology that is a T₁-complement of itself