Displaying similar documents to “A poset of topologies on the set of real numbers”

Addition theorems for dense subspaces

Aleksander V. Arhangel'skii (2015)

Commentationes Mathematicae Universitatis Carolinae

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We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space X which is the union of two dense metrizable subspaces need not be a p -space. However, if a normal space X is the union of a finite family μ of dense subspaces each of which is metrizable by a complete metric, then X is also metrizable...

Topologies on groups determined by right cancellable ultrafilters

Igor V. Protasov (2009)

Commentationes Mathematicae Universitatis Carolinae

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For every discrete group G , the Stone-Čech compactification β G of G has a natural structure of a compact right topological semigroup. An ultrafilter p ∈ G * , where G * = β G ∖ G , is called right cancellable if, given any q , r ∈ G * , q p = r p implies q = r . For every right cancellable ultrafilter p ∈ G * , we denote by G ( p ) the group G endowed with the strongest left invariant topology in which p converges to the identity of G . For any countable group G and any right cancellable ultrafilters p , q ∈ G * , we show that G ( p ) is homeomorphic to G ( q ) if...

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

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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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...

Compacta are maximally G δ -resolvable

István Juhász, Zoltán Szentmiklóssy (2013)

Commentationes Mathematicae Universitatis Carolinae

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It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum X contains Δ ( X ) many pairwise disjoint dense subsets, where Δ ( X ) denotes the minimum size of a non-empty open set in X . The aim of this note is to prove the following analogous result: Every compactum X contains Δ δ ( X ) many pairwise disjoint G δ -dense subsets, where Δ δ ( X ) denotes the minimum size of a non-empty G δ set in X .

ω H-sets and cardinal invariants

Alessandro Fedeli (1998)

Commentationes Mathematicae Universitatis Carolinae

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A subset A of a Hausdorff space X is called an ω H-set in X if for every open family 𝒰 in X such that A ⊂ ⋃ 𝒰 there exists a countable subfamily 𝒱 of 𝒰 such that A ⊂ ⋃ { V ¯ : V ∈ 𝒱 } . In this paper we introduce a new cardinal function t s θ and show that | A | ≤ 2 t s θ ( X ) ψ c ( X ) for every ω H-set A of a Hausdorff space X .

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

More reflections on compactness

Lúcia R. Junqueira, Franklin D. Tall (2003)

Fundamenta Mathematicae

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We consider the question of when X M = X , where X M is the elementary submodel topology on X ∩ M, especially in the case when X M is compact.