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2002 | 175 | 2 | 163-173
Tytuł artykułu

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

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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).
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  • Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan
  • Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Del. Iztapalapa, C.P. 09340 México D.F., México
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-2-6
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