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On some properties of squares of Sierpiński sets

100%

Nowik A.

Colloquium Mathematicum

2004

tom 99

nr 2

221-229

We investigate some geometrical properties of squares of special Sierpiński sets. In particular, we prove that (under CH) there exists a Sierpiński set S and a function p: S → S such that the images of the graph of this function under π'(⟨x,y⟩) = x - y and π''(⟨x,y⟩) = x + y are both Lusin sets.

Possibly there is no uniformly completely Ramsey null set of size $2^{ω}$

100%

Nowik A.

Colloquium Mathematicum

2002

tom 93

nr 2

251-258

We show that under the axiom $CPA_{cube}$ there is no uniformly completely Ramsey null set of size $2^{ω}$. In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

Additive properties and uniformly completely Ramsey sets

100%

Nowik A.

Colloquium Mathematicum

1999

tom 82

nr 2

191-199

We prove some properties of uniformly completely Ramsey null sets (for example, every hereditarily Menger set is uniformly completely Ramsey null).

On the structure of perfect sets in various topologies associated with tree forcings

63%

Nowik A. , Reardon P.

Open Mathematics

2013

tom 11

nr 3

509-518

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

The ideal (a) is not $G_δ$ generated

63%

Frankowska M. , Nowik A.

Colloquium Mathematicum

2011

tom 125

nr 1

7-14

We prove that the ideal (a) defined by the density topology is not $G_δ$ generated. This answers a question of Z. Grande and E. Strońska.

On nowhere weakly symmetric functions and functions with two-element range

51%

Ciesielski K. , Muthuvel K. , Nowik A.

Fundamenta Mathematicae

2001

tom 168

nr 2

119-130

A function f: ℝ → {0,1} is weakly symmetric (resp. weakly symmetrically continuous) at x ∈ ℝ provided there is a sequence hₙ → 0 such that f(x+hₙ) = f(x-hₙ) = f(x) (resp. f(x+hₙ) = f(x-hₙ)) for every n. We characterize the sets S(f) of all points at which f fails to be weakly symmetrically continuous and show that f must be weakly symmetric at some x ∈ ℝ∖S(f). In particular, there is no f: ℝ → {0,1} which is nowhere weakly symmetric. It is also shown that if at each point x we ignore some countable set from which we can choose the sequence hₙ, then there exists a function f: ℝ → {0,1} which is nowhere weakly symmetric in this weaker sense if and only if the continuum hypothesis holds.