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A category Ψ-density topology

Władysław Wilczyński, Wojciech Wojdowski (2011)

Open Mathematics

Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open...

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

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

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...

A hit-and-miss topology for 2 X , Cₙ(X) and Fₙ(X)

Benjamín Espinoza, Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano (2009)

Colloquium Mathematicae

A hit-and-miss topology ( τ H M ) is defined for the hyperspaces 2 X , Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between τ H M and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

A poset of topologies on the set of real numbers

Vitalij A. Chatyrko, Yasunao Hattori (2013)

Commentationes Mathematicae Universitatis Carolinae

On the set of real numbers we consider a poset 𝒫 τ ( ) (by inclusion) of topologies τ ( A ) , where A , such that A 1 A 2 iff τ ( A 1 ) τ ( A 2 ) . The poset has the minimal element τ ( ) , the Euclidean topology, and the maximal element τ ( ) , the Sorgenfrey topology. We are interested when two topologies τ 1 and τ 2 (especially, for τ 2 = τ ( ) ) from the poset define homeomorphic spaces ( , τ 1 ) and ( , τ 2 ) . In particular, we prove that for a closed subset A of the space ( , τ ( A ) ) is homeomorphic to the Sorgenfrey line ( , τ ( ) ) iff A is countable. We study also common properties...

A reverse viewpoint on the upper semicontinuity of multivalued maps

Marcio Colombo Fenille (2013)

Mathematica Bohemica

For a multivalued map ϕ : Y ( X , τ ) between topological spaces, the upper semifinite topology 𝒜 ( τ ) on the power set 𝒜 ( X ) = { A X : A } is such that ϕ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map ϕ : Y ( 𝒜 ( X ) , 𝒜 ( τ ) ) . In this paper, we seek a result like this from a reverse viewpoint, namely, given a set X and a topology Γ on 𝒜 ( X ) , we consider a natural topology ( Γ ) on X , constructed from Γ satisfying ( Γ ) = τ if Γ = 𝒜 ( τ ) , and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map ϕ : Y ( X , ( Γ ) ) ...

A sufficient condition for maximal resolvability of topological spaces

Jerzy Bienias, Małgorzata Terepeta (2004)

Commentationes Mathematicae Universitatis Carolinae

We show a new theorem which is a sufficient condition for maximal resolvability of a topological space. We also discuss some relationships between various theorems about maximal resolvability.

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