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

Dmitri Shakhmatov, and Michael Tkachenko. "A compact Hausdorff topology that is a T₁-complement of itself." Fundamenta Mathematicae 175.2 (2002): 163-173. <http://eudml.org/doc/283217>.

@article{DmitriShakhmatov2002,
abstract = {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).},
author = {Dmitri Shakhmatov, Michael Tkachenko},
journal = {Fundamenta Mathematicae},
keywords = {-complementary; transversal; -independent; compact; countably compact; self-complementary; sequential; subsequential; convergent sequence},
language = {eng},
number = {2},
pages = {163-173},
title = {A compact Hausdorff topology that is a T₁-complement of itself},
url = {http://eudml.org/doc/283217},
volume = {175},
year = {2002},
}

TY - JOUR
AU - Dmitri Shakhmatov
AU - Michael Tkachenko
TI - A compact Hausdorff topology that is a T₁-complement of itself
JO - Fundamenta Mathematicae
PY - 2002
VL - 175
IS - 2
SP - 163
EP - 173
AB - 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).
LA - eng
KW - -complementary; transversal; -independent; compact; countably compact; self-complementary; sequential; subsequential; convergent sequence
UR - http://eudml.org/doc/283217
ER -