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𝒯 0 - and 𝒯 1 -reflections

Maria Manuel Clementino (1992)

Commentationes Mathematicae Universitatis Carolinae

In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms T 0 and T 1 . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.

-closed sets in biclosure spaces

Chawalit Boonpok (2009)

Acta Mathematica Universitatis Ostraviensis

In the present paper, we introduce and study the concept of -closed sets in biclosure spaces and investigate its behavior. We also introduce and study the concept of -continuous maps.

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 Čech function in ZFC

Fred Galvin, Petr Simon (2007)

Fundamenta Mathematicae

A nontrivial surjective Čech closure function is constructed in ZFC.

A compact ccc non-separable space from a Hausdorff gap and Martin's Axiom

Murray G. Bell (1996)

Commentationes Mathematicae Universitatis Carolinae

We answer a question of I. Juhasz by showing that MA + ¬ CH does not imply that every compact ccc space of countable π -character is separable. The space constructed has the additional property that it does not map continuously onto I ω 1 .

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...

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