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Bad Wadge-like reducibilities on the Baire space

Luca Motto Ros (2014)

Fundamenta Mathematicae

We consider various collections of functions from the Baire space ω ω into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on ω ω (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of...

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if σ ⊆ D ∈ M is a support, M being an inner model of ZFC, and 𝒫 ( D ∖ σ ) ∩ M = r ` ` σ with r ∈ M , then r determines a preorder " ⪯ " of D such that σ becomes a filter on ( D , ⪯ ) generic over M . We show that if the relation r is replaced by a function 𝒫 ( D ∖ σ ) ∩ M = f - 1 ( σ ) , then there exists an equivalence relation " ∼ " on D and a partial order on D / ∼ such that D / ∼ is a complete Boolean algebra, σ / ∼ is a generic filter and [ f ( u ) ] ∼ = - ∑ ( u / ∼ ) for any u ⊆ D , u ∈ M .

Banach spaces without minimal subspaces – Examples

Valentin Ferenczi, Christian Rosendal (2012)

Annales de l’institut Fourier

We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.

Base-base paracompactness and subsets of the Sorgenfrey line

Strashimir G. Popvassilev (2012)

Mathematica Bohemica

A topological space X is called base-base paracompact (John E. Porter) if it has an open base ℬ such that every base ℬ ' ⊆ ℬ has a locally finite subcover 𝒞 ⊆ ℬ ' . It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.

Bell-type inequalities for parametric families of triangular norms

Saskia Janssens, Bernard De Baets, Hans De Meyer (2004)

Kybernetika

In recent work we have shown that the reformulation of the classical Bell inequalities into the context of fuzzy probability calculus leads to related inequalities on the commutative conjunctor used for modelling pointwise fuzzy set intersection. Also, an important role has been attributed to commutative quasi-copulas. In this paper, we consider these new Bell-type inequalities for continuous t-norms. Our contribution is twofold: first, we prove that ordinal sums preserve these Bell-type inequalities;...

Beyond Lebesgue and Baire II: Bitopology and measure-category duality

N. H. Bingham, A. J. Ostaszewski (2010)

Colloquium Mathematicae

We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions...

Biequivalence vector spaces in the alternative set theory

Miroslav Šmíd, Pavol Zlatoš (1991)

Commentationes Mathematicae Universitatis Carolinae

As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field Q of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total...

Binary and ternary relations

Vítězslav Novák, Miroslav Novotný (1992)

Mathematica Bohemica

Two operators are constructed which make it possible to transform ternary relations into binary relations defined on binary relations and vice versa. A possible graphical representation of ternary relations is described.

Binary Relations-based Rough Sets – an Automated Approach

Adam Grabowski (2016)

Formalized Mathematics

Rough sets, developed by Zdzisław Pawlak [12], are an important tool to describe the state of incomplete or partially unknown information. In this article, which is essentially the continuation of [8], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library [11]). Here we drop the classical equivalence- and tolerance-based models of rough...

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