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

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Shakhmatov D., Tkachenko M.

Fundamenta Mathematicae

2002

tom 175

nr 2

163-173

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

1 Fundamenta Mathematicae

1 Shakhmatov D.

1 Tkachenko M.

1 2002

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