Displaying similar documents to “Compact covering mappings and cofinal families of compact subsets of a Borel set”

Some topological properties of ω -covering sets

Andrzej Nowik (2000)

Czechoslovak Mathematical Journal

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We prove the following theorems: There exists an ω -covering with the property s 0 . Under c o v ( 𝒩 ) = there exists X such that B o r [ B X is not an ω -covering or X B is not an ω -covering]. Also we characterize the property of being an ω -covering.

Perfect set theorems

Otmar Spinas (2008)

Fundamenta Mathematicae

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We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for G δ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

Finite-to-one continuous s-covering mappings

Alexey Ostrovsky (2007)

Fundamenta Mathematicae

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The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction f | f - 1 ( S ) is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets Y i such that every restriction f | f - 1 ( Y i ) is an inductively perfect map.

Quandle coverings and their Galois correspondence

Michael Eisermann (2014)

Fundamenta Mathematicae

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This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π ( Q , q ) : = A u t ( p ) = g A d j ( Q ) ' | q g = q , where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble...

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

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Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space κ κ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in κ κ for uncountable regular κ is however consistent (with...

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ( A ) n ω of X such that f | A is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

F σ -mappings and the invariance of absolute Borel classes

Petr Holický, Jiří Spurný (2004)

Fundamenta Mathematicae

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It is proved that F σ -mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any F σ -mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable F σ -set X X satisfying f ( X ) = Y .

Perfect mappings in topological groups, cross-complementary subsets and quotients

Aleksander V. Arhangel'skii (2003)

Commentationes Mathematicae Universitatis Carolinae

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The following general question is considered. Suppose that G is a topological group, and F , M are subspaces of G such that G = M F . Under these general assumptions, how are the properties of F and M related to the properties of G ? For example, it is observed that if M is closed metrizable and F is compact, then G is a paracompact p -space. Furthermore, if M is closed and first countable, F is a first countable compactum, and F M = G , then G is also metrizable. Several other results of this kind are...

Borel sets with σ-compact sections for nonseparable spaces

Petr Holický (2008)

Fundamenta Mathematicae

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We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections E x = y Y : ( x , y ) E , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.

Lifting of homeomorphisms to branched coverings of a disk

Bronisław Wajnryb, Agnieszka Wiśniowska-Wajnryb (2012)

Fundamenta Mathematicae

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We consider a simple, possibly disconnected, d-sheeted branched covering π of a closed 2-dimensional disk D by a surface X. The isotopy classes of homeomorphisms of D which are pointwise fixed on the boundary of D and permute the branch values, form the braid group Bₙ, where n is the number of branch values. Some of these homeomorphisms can be lifted to homeomorphisms of X which fix pointwise the fiber over the base point. They form a subgroup L π of finite index in Bₙ. For each equivalence...

Two point sets with additional properties

Marek Bienias, Szymon Głąb, Robert Rałowski, Szymon Żeberski (2013)

Czechoslovak Mathematical Journal

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A subset of the plane is called a two point set if it intersects any line in exactly two points. We give constructions of two point sets possessing some additional properties. Among these properties we consider: being a Hamel base, belonging to some σ -ideal, being (completely) nonmeasurable with respect to different σ -ideals, being a κ -covering. We also give examples of properties that are not satisfied by any two point set: being Luzin, Sierpiński and Bernstein set. We also consider...

Uncountable γ-sets under axiom C P A c u b e g a m e

Krzysztof Ciesielski, Andrés Millán, Janusz Pawlikowski (2003)

Fundamenta Mathematicae

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We formulate a Covering Property Axiom C P A c u b e g a m e , which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < , since every γ-set is universally null, while C P A c u b e g a m e implies that every universally null has cardinality less than = ω₂. We also show that C P A c u b e g a m e implies the existence of a partition of ℝ into ω₁ null...

A new lower bound for the football pool problem for 7 matches

Laurent Habsieger (1996)

Journal de théorie des nombres de Bordeaux

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Let K 3 ( 7 , 1 ) denote the minimum cardinality of a ternary code of length 7 and covering radius one. In a previous paper, we improved on the lower bound K 3 ( 7 , 1 ) 147 by showing that K 3 ( 7 , 1 ) 150 . In this note, we prove that K 3 ( 7 , 1 ) 153 .

Factors of a perfect square

Tsz Ho Chan (2014)

Acta Arithmetica

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We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between n - n ( l o g n ) 1 / 7 and n + n ( l o g n ) 1 / 7 .

On Ponomarev-Systems

Ying Ge, Lin Shou (2007)

Bollettino dell'Unione Matematica Italiana

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In this paper the relations of mappings and families of subsets are investigated in Ponomarev-systems, and the following results are obtained. (1) f is a sequence-covering (resp. 1-sequence-covering) mapping iff 𝒫 is a csf -network (resp. snf -network) of X for a Ponomarev-system ( f , M , X , 𝒫 ) ; (2) f is a sequence-covering (resp. 1-sequence-covering) mapping iff every 𝒫 n is a cs-cover (resp. wsn-cover) of X for a Ponomarev-system ( f , M , X , { 𝒫 n } ) . As applications of these results, some relations between sequence-covering...