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1
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Exceptional directions for Sierpiński's nonmeasurable sets

100%
Kirchheim B., Natkaniec T.
Fundamenta Mathematicae
|
1991-1992
|
tom 140
|
nr 3
237-245
XX
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.
2
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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
EN
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.
3
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On Pawlak's problem concerning entropy of almost continuous functions

100%
Natkaniec T., Szuca P.
Colloquium Mathematicum
|
2010
|
tom 121
|
nr 1
107-111
EN
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.
4
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A big symmetric planar set with small category projections

100%
Ciesielski K., Natkaniec T.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 3
237-253
EN
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.
5
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Universally Kuratowski–Ulam spaces

81%
Fremlin D. H., Natkaniec T., Recław I.
Fundamenta Mathematicae
|
2000
|
tom 165
|
nr 3
239-247
EN
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.
6
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On some properties of Hamel bases and their applications to Marczewski measurable functions

81%
Dorais F., Filipów R., Natkaniec T.
Open Mathematics
|
2013
|
tom 11
|
nr 3
487-508
EN
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.
7
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On compositions and products of almost continuous functions

70%
Natkaniec T.
Fundamenta Mathematicae
|
1991
|
tom 139
|
nr 1
59-74
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