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1
Content available remote

Generalized projections of Borel and analytic sets

100%
Balcerzak M.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 1
47-53
EN
For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection 𝛷(A) of A given by 𝛷(A) = {x ∈ X: A_x ∉ I}, where $A_x$ ={y ∈ X: 〈x,y〉∈ A}. We study the behaviour of 𝛷 with respect to Borel and analytic sets in the case when I is a $∑_{2}^{0}$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that 𝛷 [$∑_{1}^{1}(X^2)]=∑_{1}^{1}(X)$ for a wide class of $∑_{2}^{0}$-supported σ-ideals.
2
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On c-sets and products of ideals

100%
Balcerzak M.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 1
1-6
EN
Let X, Y be uncountable Polish spaces and let μ be a complete σ-finite Borel measure on X. Denote by K and L the families of all meager subsets of X and of all subsets of Y with μ measure zero, respectively. It is shown that the product of the ideals K and L restricted to C-sets of Selivanovskiĭ is σ-saturated, which extends Gavalec's results.
3
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Some examples of true $F_{σδ}$ sets

64%
Balcerzak M., Darji U. B.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 2
203-207
EN
Let K(X) be the hyperspace of a compact metric space endowed with the Hausdorff metric. We give a general theorem showing that certain subsets of K(X) are true $F_{σδ}$ sets.
4
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On multiplication in spaces of continuous functions

64%
Balcerzak M., Maliszewski A.
Colloquium Mathematicum
|
2011
|
tom 122
|
nr 2
247-253
EN
We introduce and examine the notion of dense weak openness. In particular we show that multiplication in C(X) is densely weakly open whenever X is an interval in ℝ.
5
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Multiplying balls in the space of continuous functions on [0,1]

51%
Balcerzak M., Wachowicz A., Wilczyński W.
Studia Mathematica
|
2005
|
tom 170
|
nr 2
203-209
EN
Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ {+, min, max} then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ {+,·,min,max}, then the set $Φ^{-1}(E)$ is residual whenever E is residual in C.
6
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On Marczewski-Burstin representable algebras

51%
Balcerzak M., Bartoszewicz A., Koszmider P.
Colloquium Mathematicum
|
2004
|
tom 99
|
nr 1
55-60
EN
We construct algebras of sets which are not MB-representable. The existence of such algebras was previously known under additional set-theoretic assumptions. On the other hand, we prove that every Boolean algebra is isomorphic to an MB-representable algebra of sets.
7
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Lipschitz differences and Lipschitz functions

51%
Balcerzak M., Buczolich Z., Laczkovich M.
Colloquium Mathematicum
|
1997
|
tom 72
|
nr 2
319-324
8
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Remarks on products of σ-ideals

32%
Balcerzak M.
Colloquium Mathematicum
|
1988
|
tom 56
|
nr 2
201-209
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