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

• Autorzy: Dmitri Shakhmatov , Michael Tkachenko
• Języki publikacji: EN

Abstrakty

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).

Kategorie tematyczne

• 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
• 54D30: Compactness
• 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
• 54D20: Noncompact covering properties (paracompact, Lindel\"of, etc.)
• 06B99: None of the above, but in this section
• 54G20: Counterexamples
• 54A35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2002
• Tom: 175
• Numer: 2
• Strony: 163-173

Daty

wydano

2002

Twórcy

autor

Dmitri Shakhmatov

• Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan

autor

Michael Tkachenko

• Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Del. Iztapalapa, C.P. 09340 México D.F., México

Identyfikatory

DOI

10.4064/fm175-2-6