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Generalized projections of Borel and analytic sets

100%

Balcerzak M.

Colloquium Mathematicum

1996

tom 71

nr 1

47-53

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.

On c-sets and products of ideals

100%

Balcerzak M.

Colloquium Mathematicum

1991

tom 62

nr 1

1-6

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.

Some examples of true $F_{σδ}$ sets

63%

Balcerzak M. , Darji U. B.

Colloquium Mathematicum

2000

tom 86

nr 2

203-207

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.

On multiplication in spaces of continuous functions

63%

Balcerzak M. , Maliszewski A.

Colloquium Mathematicum

2011

tom 122

nr 2

247-253

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.

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

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.

On Marczewski-Burstin representable algebras

51%

Balcerzak M. , Bartoszewicz A. , Koszmider P.

Colloquium Mathematicum

2004

tom 99

nr 1

55-60

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.

Lipschitz differences and Lipschitz functions

51%

Balcerzak M. , Buczolich Z. , Laczkovich M.

Colloquium Mathematicum

1997

tom 72

nr 2

319-324

Remarks on products of σ-ideals

32%

Balcerzak M.

Colloquium Mathematicum

1988

tom 56

nr 2

201-209