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Exceptional directions for Sierpiński's nonmeasurable sets

B. KirchheimTomasz Natkaniec — 1992

Fundamenta Mathematicae

In [2] the question was considered in how many directions can a nonmeasurable plane set behave even "better" than the classical one constructed by Sierpiński in [6], in the sense that any line in a given direction intersects the set in at most one point. We considerably improve these results and give a much sharper estimate for the size of the sets of those "better" directions.

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz NatkaniecPiotr Szuca — 2010

Colloquium Mathematicae

We prove that if f: → is Darboux and has a point of prime period different from 2 i , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

A big symmetric planar set with small category projections

Krzysztof CiesielskiTomasz Natkaniec — 2003

Fundamenta Mathematicae

We show that under appropriate set-theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set A ⊂ ℝ such that (i) the set {c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set {c ∈ ℝ: (f+c) ∩ (A×A) = ∅} is nowhere meager for each continuous function f: ℝ → ℝ. The existence of such a set also follows from the principle CPA, which...

Universally Kuratowski–Ulam spaces

David FremlinTomasz NatkaniecIreneusz Recław — 2000

Fundamenta Mathematicae

We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following: ...

On some properties of Hamel bases and their applications to Marczewski measurable functions

François DoraisRafał FilipówTomasz Natkaniec — 2013

Open Mathematics

We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.

Cardinal inequalities implying maximal resolvability

Marek BalcerzakTomasz NatkaniecMałgorzata Terepeta — 2005

Commentationes Mathematicae Universitatis Carolinae

We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space X is maximally resolvable provided that for a dense set X 0 X and for each x X 0 the π -character of X at x is not greater than the dispersion character of X . On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.

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