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

Dmitri Shakhmatov ; Michael Tkachenko

Fundamenta Mathematicae, Tome 173 (2002), p. 163-173 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $2^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff space of size 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 which is T₁-complementary to itself and a compact Hausdorff space of size 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 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).

Détail

Publié le : 2002-01-01

Zbl 1020.54002

EUDML-ID : urn:eudml:doc:283217