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On the ideal convergence of sequences of quasi-continuous functions

100%

Natkaniec T. , Szuca P.

Fundamenta Mathematicae

2016

tom 232

nr 3

269-280

For any Borel ideal ℐ we describe the ℐ-Baire system generated by the family of quasi-continuous real-valued functions. We characterize the Borel ideals ℐ for which the ideal and ordinary Baire systems coincide.

Exceptional directions for Sierpiński's nonmeasurable sets

100%

Kirchheim B. , Natkaniec T.

Fundamenta Mathematicae

1991-1992

tom 140

nr 3

237-245

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.

A big symmetric planar set with small category projections

100%

Ciesielski K. , Natkaniec T.

tom 178

nr 3

237-253

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 holds in the iterated perfect set model. We also prove that the existence of a set A as in (i) cannot be proved in ZFC alone even when we restrict our attention to homeomorphisms of ℝ. On the other hand, for the class of real-analytic functions a Bernstein set A satisfying (ii) exists in ZFC.

On Pawlak's problem concerning entropy of almost continuous functions

100%

Natkaniec T. , Szuca P.

Colloquium Mathematicum

2010

tom 121

nr 1

107-111

We prove that if f:𝕀 → 𝕀 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:𝕀 → 𝕀 with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

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

80%

Dorais F. , Filipów R. , Natkaniec T.

Open Mathematics

2013

tom 11

nr 3

487-508

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.

Universally Kuratowski–Ulam spaces

80%

Fremlin D. H. , Natkaniec T. , Recław I.

Fundamenta Mathematicae

2000

tom 165

nr 3

239-247

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: • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable π-bases); • every Baire uK-U space is ccc.

On compositions and products of almost continuous functions

70%

Natkaniec T.

Fundamenta Mathematicae

1991

tom 139

nr 1

59-74