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The Hurewicz covering property and slaloms in the Baire space

Boaz Tsaban (2004)

Fundamenta Mathematicae

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

The linear refinement number and selection theory

Michał Machura, Saharon Shelah, Boaz Tsaban (2016)

Fundamenta Mathematicae

The linear refinement number is the minimal cardinality of a centered family in [ ω ] ω such that no linearly ordered set in ( [ ω ] ω , ⊆ * ) refines this family. The linear excluded middle number is a variation of . We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that = = in all models where the continuum is at most ℵ₂, and that the cofinality of is...

Topologically invariant σ-ideals on Euclidean spaces

T. Banakh, M. Morayne, R. Rałowski, Sz. Żeberski (2015)

Fundamenta Mathematicae

We study and classify topologically invariant σ-ideals with an analytic base on Euclidean spaces, and evaluate the cardinal characteristics of such ideals.

Totally proper forcing and the Moore-Mrówka problem

Todd Eisworth (2003)

Fundamenta Mathematicae

We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.

Two point sets with additional properties

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

Czechoslovak Mathematical Journal

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 natural generalizations...

Uniformization and anti-uniformization properties of ladder systems

Todd Eisworth, Gary Gruenhage, Oleg Pavlov, Paul Szeptycki (2004)

Fundamenta Mathematicae

Natural weakenings of uniformizability of a ladder system on ω₁ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The...

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