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Transfinite inductions producing coanalytic sets

Zoltán Vidnyánszky (2014)

Fundamenta Mathematicae

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that consistently there exists an uncountable coanalytic subset of the plane that intersects every C¹ curve in a countable set.

Two ideals connected with strong right upper porosity at a point

Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin (2015)

Czechoslovak Mathematical Journal

Let SP be the set of upper strongly porous at 0 subsets of ℝ + and let I ^ ( SP ) be the intersection of maximal ideals I ⊆ SP . Some characteristic properties of sets E ∈ I ^ ( SP ) are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at 0 subsets of ℝ + is a proper subideal of I ^ ( SP ) . Earlier, completely strongly porous sets and some of their properties were...

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...

Two Selection Theorems

John P. Burgess (1977)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Uniformly completely Ramsey sets

Udayan Darji (1993)

Colloquium Mathematicae

Galvin and Prikry defined completely Ramsey sets and showed that the class of completely Ramsey sets forms a σ-algebra containing open sets. However, they used two definitions of completely Ramsey. We show that they are not equivalent as they remarked. One of these definitions is a more uniform property than the other. We call it the uniformly completely Ramsey property. We show that some of the results of Ellentuck, Silver, Brown and Aniszczyk concerning completely Ramsey sets also hold for uniformly...

Universally measurable sets in generic extensions

Paul Larson, Itay Neeman, Saharon Shelah (2010)

Fundamenta Mathematicae

A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least 2 ℵ ₁ such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets of reals....

Vitali sets.

Lahiri, Benoy Kumar, Lahiri, Indrajit (2001)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

Vitali sets and Hamel bases that are Marczewski measurable

Arnold Miller, Strashimir Popvassilev (2000)

Fundamenta Mathematicae

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one...

Weak Baire measurability of the balls in a Banach space

José Rodríguez (2008)

Studia Mathematica

Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of ℓ ∞ ( ℕ ) and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised by K. Musiał....

Weak Rudin-Keisler reductions on projective ideals

Konstantinos A. Beros (2016)

Fundamenta Mathematicae

We consider a slightly modified form of the standard Rudin-Keisler order on ideals and demonstrate the existence of complete (with respect to this order) ideals in various projective classes. Using our methods, we obtain a simple proof of Hjorth’s theorem on the existence of a complete Π¹₁ equivalence relation. This proof enables us (under PD) to generalize Hjorth’s result to the classes of Π ¹ 2 n + 1 equivalence relations.

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