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Descriptive set-theoretical properties of an abstract density operator

100%
Gła̧b S.
Open Mathematics
|
2009
|
tom 7
|
nr 4
732-740
EN
Let $$ \mathcal{K} $$(ℝ) stand for the hyperspace of all nonempty compact sets on the real line and let d ±(x;E) denote the (right- or left-hand) Lebesgue density of a measurable set E ⊂ ℝ at a point x∈ ℝ. In [3] it was proved that $$ \{ K \in \mathcal{K}(\mathbb{R}):\forall _x \in K(d^ + (x,K) = 1ord^ - (x,K) = 1)\} $$ is ⊓11-complete. In this paper we define an abstract density operator ⅅ± and we generalize the above result. Some applications are included.
2
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Descriptive properties of density preserving autohomeomorphisms of the unit interval

76%
Głąb S., Strobin F.
Open Mathematics
|
2010
|
tom 8
|
nr 5
928-936
EN
We prove that density preserving homeomorphisms form a Π11-complete subset in the Polish space ℍ of all increasing autohomeomorphisms of unit interval.
3
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On the structure of perfect sets in various topologies associated with tree forcings

76%
Nowik A., Reardon P.
Open Mathematics
|
2013
|
tom 11
|
nr 3
509-518
EN
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.
4
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F-limit points in dynamical systems defined on the interval

52%
Szuca P.
Open Mathematics
|
2013
|
tom 11
|
nr 1
170-176
EN
Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider a question about continuity of the multivalued map x → ω fF(x). We point out some connections between the Baire class of f p and tame dynamical systems, and give some open problems.
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