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

Dmitri Shakhmatov; Michael Tkachenko

Fundamenta Mathematicae (2002)

- Volume: 175, Issue: 2, page 163-173
- ISSN: 0016-2736

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topDmitri 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 -

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