In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms and . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
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.
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 nontrivial surjective Čech closure function is constructed in ZFC.
We answer a question of I. Juhasz by showing that MA CH does not imply that every compact ccc space of countable -character is separable. The space constructed has the additional property that it does not map continuously onto .
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 and are called T₁-complementary provided that there exists a bijection f: X → Y such that and are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...