A T 1 -Complement for the Reals

Steiner, A. K.; Steiner, Eberhard

doi:10.2307/2036164

Cited by 15 publications

(15 citation statements)

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“…The study of complementarity in L 1 (X) was initiated by A. Steiner and E. Steiner in [6,8,7]. Later on, S. Watson used an elaborated combinatorics in [10] to prove that a set X of cardinality c + , where c = 2 ω , admits a Tychonoff self T 1 -complementary topology τ .…”

Section: Introductionmentioning

confidence: 99%

Compact self T1-complementary spaces without isolated points

Tkachenko

2009

Appl.Gen.Topol.

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Section: Introductionmentioning

confidence: 99%

Compact self T1-complementary spaces without isolated points

Tkachenko

2009

Appl.Gen.Topol.

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“…Steiner and Steiner [26] presented more classes of T 1 topologies which have T 1 -complements. In [27], they showed that the usual topology on a countable dense subset of the reals has a T 1 -complement. This entails, along with previous results in [26], that the usual topology on the real numbers has a T 1 -complement.…”

Section: Introductionmentioning

confidence: 99%

On transversal group topologies

Dikranjan¹,

Tkachenko²,

Yaschenko³

2005

Topology and its Applications

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Two non-discrete Hausdorff group topologies τ 1 , τ 2 on a group G are called transversal if the least upper bound τ 1 ∨ τ 2 of τ 1 and τ 2 is the discrete topology. We show that an infinite totally bounded topological group never admits a transversal group topology and we obtain a new criterion for precompactness in lattice-theoretical terms (the existence of transversal group topologies) for a large class of Abelian groups containing all divisible and all finitely generated groups. A full description is given of the class M of all abstract Abelian groups G where this criterion is valid (i.e., when the non-precompact group topologies on G are precisely the transversable ones). It turns out that M is the class of groups G for which the submaximal topology is precompact.We also give a complete description of the structure of the transversable locally compact Abelian groups. (D. Dikranjan), mich@xanum.uam.mx (M. Tkachenko), ivan@mccme.ru (I. Yaschenko).

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“…The topological notion of T 1 -complementarity of topologies on X corresponds to the algebraic notion of complementarity in the lattice (L(X), ≤). We refer the reader to [10], [9], [1], [5] and [14,Sec. 12] for details and relevant discussions.…”

Section: Introductionmentioning

confidence: 99%

“…The classical notion of T 1 -complementarity has been studied for a long time (see [10], [9], [1], [5] and [11]). …”

Section: Introductionmentioning

confidence: 99%

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

Shakhmatov

Tkachenko

2002

Fund. Math.

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Abstract. Topologies τ 1 and τ 2 on a set X are called T 1 -complementary if τ 1 ∩ τ 2 = {X \ F : F ⊆ X is finite} ∪ {∅} and τ 1 ∪ τ 2 is a subbase for the discrete topology on X.We provide an example of a compact Hausdorff space of size 2 c which is T 1 -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 T 1 -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 T 1 -complementary to itself and a compact Hausdorff space of size c which is T 1 -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 T 1 -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). Introduction.Recall that a topology τ on a set X is called a T 1 topology, and the pair (X, τ ) is called a T 1 -space, provided that all singletons {x} of X are τ -closed. The cofinite topology {X \ F : F ⊆ X is finite} ∪ {∅} on X is the smallest (with respect to set inclusion) T 1 topology on X.Two topologies τ 1 and τ 2 on an infinite set X are called:

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A T 1 -Complement for the Reals

Cited by 15 publications

References 1 publication

Compact self T1-complementary spaces without isolated points

Compact self T1-complementary spaces without isolated points

On transversal group topologies

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

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A T 1 -Complement for the Reals

Cited by 15 publications

References 1 publication

Compact self T1-complementary spaces without isolated points

Compact self T1-complementary spaces without isolated points

On transversal group topologies

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

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