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A new class of functions called “ $R_{z}$ -supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_{z}$ -supercontinuous functions properly includes the class of $R_{cl}$ -supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $cl$ -supercontinuous ( $\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983), and is...

Radical ideals and coherent frames

Bernhard Banaschewski (1996)

Commentationes Mathematicae Universitatis Carolinae

It follows from Stone Duality that Hochster's results on the relation between spectral spaces and prime spectra of rings translate into analogous, formally stronger results concerning coherent frames and frames of radical ideals of rings. Here, we show that the latter can actually be obtained without Stone Duality, proving them in Zermelo-Fraenkel set theory and thereby sharpening the original results of Hochster.

Radon spaces which are not $σ$ -fragmentable

Petr Holický, Jan Pelant (1995)

Acta Universitatis Carolinae. Mathematica et Physica

Radon-Nikodým compact spaces of low weight and Banach spaces

Antonio Avilés (2005)

Studia Mathematica

We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.

Radon-Nikodym type properties for Banach spaces

Janicka, L. (1979)

Abstracta. 7th Winter School on Abstract Analysis

Raising dimension under all projections

John Cobb (1994)

Fundamenta Mathematicae

As a special case of the general question - “What information can be obtained about the dimension of a subset of $ℝ^{n}$ by looking at its orthogonal projections into hyperplanes?” - we construct a Cantor set in $ℝ^{3}$ each of whose projections into 2-planes is 1-dimensional. We also consider projections of Cantor sets in $ℝ^{n}$ whose images contain open sets, expanding on a result of Borsuk.

Ramification systems and spaces of ultrafilters

Negrepontis, S. (1972)

General Topology and its Relations to Modern Analysis and Algebra

Ramsey, Lebesgue, and Marczewski sets and the Baire property

Patrick Reardon (1996)

Fundamenta Mathematicae

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. THEOREM. In the Ellentuck topology on ${[ω]}^{ω}$ , ${(s)}_{0}$ is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is...

Ramsey Theorems and the Property of Baire

Bernd Voigt (1985)

Publications du Département de mathématiques (Lyon)

Ramsey theoretic consequences of some new results about algebra in the Stone-Čech compactification.

Carlson, Timothy J., Hindman, Neil, Strauss, Dona (2005)

Integers

Ramsey theory for structures: Nešetřil's result on finite metric spaces.

di Prisco, Carlos Augusto (2006)

Boletín de la Asociación Matemática Venezolana

Ramsey topological spaces

Nešetřil, J., Rödl, V. (1977)

General topology and its relations to modern analysis and algebra IV

Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces

Jiří Matoušek (1992)

Commentationes Mathematicae Universitatis Carolinae

Let $(X, ρ)$ , $(Y, σ)$ be metric spaces and $f : X \to Y$ an injective mapping. We put ${∥ f ∥}_{L i p} = sup {σ (f (x), f (y)) / ρ (x, y)$ ; $x, y \in X$ , $x \neq y}$ , and $dist (f) = {∥ f ∥}_{L i p} . {∥ f^{- 1} ∥}_{L i p}$ (the distortion of the mapping $f$ ). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let $X$ be a finite metric space, and let $ε > 0$ , $K$ be given numbers. Then there exists a finite metric space $Y$ , such that for every mapping $f : Y \to Z$ ( $Z$ arbitrary metric space) with $dist (f) < K$ one can find a mapping $g : X \to Y$ , such that both the mappings $g$ and ${f |}_{g (X)}$ have distortion at...

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$R_{0}$ and $R_{1}$ axioms in fuzzy topology

$R$ -continuous functions.

$r g$ -compact spaces and $r g$ -connected spaces.

$R$ -separated spaces.

$R_{z}$ -supercontinuous functions