Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 9

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

Property C'', strong measure zero sets and subsets of the plane

100%
Pawlikowski J.
Fundamenta Mathematicae
|
1997
|
tom 153
|
nr 3
277-293
EN
Let X be a set of reals. We show that  • X has property C" of Rothberger iff for all closed F ⊆ ℝ × ℝ with vertical sections $F_x$ (x ∈ X) null, $∪_{x ∈ X}F_x$ is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections $F_x$ (x ∈ ℝ) null, $∪_{x ∈ X}F_x$ is null.
2
Content available remote

Strongly meager sets and subsets of the plane

100%
Pawlikowski J.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 3
279-287
EN
Let $X ⊆ 2^w$. Consider the class of all Borel $F ⊆ X×2^w$ with null vertical sections $F_x$, x ∈ X. We show that if for all such F and all null Z ⊆ X, $∪_{x ∈ Z}F_x$ is null, then for all such F, $∪_{x ∈ X}F_x≠2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].
3
Content available remote

Selivanovski hard sets are hard

100%
Pawlikowski J.
Fundamenta Mathematicae
|
2015
|
tom 228
|
nr 1
17-25
EN
Let $H ⊆ Z ⊆ 2^{ω}$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{ω}$ to Z give as preimages of H all Σₙ¹ subsets of $2^{ω}$, then so do continuous injections.
4
Content available remote

Undetermined sets of point-open games

100%
Pawlikowski J.
Fundamenta Mathematicae
|
1994
|
tom 144
|
nr 3
279-285
EN
We show that a set of reals is undetermined in Galvin's point-open game iff it is uncountable and has property C", which answers a question of Gruenhage.
5
Content available remote

On a problem of saturation of certain reduced direct products

64%
Laradji A., Pawlikowski J.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 2
189-191
6
Content available remote

Parametrized Cichoń's diagram and small sets

64%
Pawlikowski J., Recław I.
Fundamenta Mathematicae
|
1995
|
tom 147
|
nr 2
135-155
EN
We parametrize Cichoń's diagram and show how cardinals from Cichoń's diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w×2^w$ and continuous functions $e, f:w^w → w^w$ such that  • N is $G_δ$ and ${N_x:x ∈ w^w}$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_σ$ and ${M_x:x ∈ w^w}$ is a basis for the ideal of meager subsets of $2^w$;  •$∀x,y N_{e(x)} ⊆ N_y ⇒ M_x ⊆ M_{f(y)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_δ$) sets $B ⊆ X×2^w$ with all vertical sections null, $∪_{x ∈ X}B_x$ is null, then for all Borel (resp. $F_σ$) sets $B ⊆ X×2^w$ with all vertical sections meager, $∪_{x ∈ X}B_x$ is meager;  •if there exists a Borel (resp. a "nice" $G_δ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for measure zero sets, then there exists a Borel (resp. $F_σ$) set $B ⊆ X×2^w$ such that ${B_x:x ∈ X}$ is a basis for meager sets
7
Content available remote

Covering Property Axiom $CPA_{cube}$ and its consequences

64%
Ciesielski K., Pawlikowski J.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 1
63-75
EN
We formulate a Covering Property Axiom $CPA_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than 𝔠. (c) cof(𝓝) = ω₁ < 𝔠, i.e., the cofinality of the measure ideal 𝓝 is ω₁. (d) For every uniformly bounded sequence $⟨fₙ ∈ ℝ^{ℝ}⟩_{n<ω}$ of Borel functions there are sequences: $⟨P_ξ ⊂ ℝ: ξ < ω₁⟩$ of compact sets and $⟨W_ξ ∈ [ω]^ω: ξ < ω₁⟩$ such that $ℝ = ⋃_{ξ<ω₁}P_ξ$ and for every ξ < ω₁, $⟨fₙ ↾ P_ξ⟩_{n∈W_ξ}$ is a monotone uniformly convergent sequence of uniformly continuous functions. (e) Total failure of Martin's Axiom: 𝔠 > ω₁ and for every non-trivial ccc forcing ℙ there exist ω₁ dense sets in ℙ such that no filter intersects all of them
8
Content available remote

Uncountable γ-sets under axiom $CPA_{cube}^{game}$

51%
Ciesielski K., Millán A., Pawlikowski J.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 2
143-155
EN
We formulate a Covering Property Axiom $CPA_{cube}^{game}$, which holds in the iterated perfect set model, and show that it implies the existence of uncountable strong γ-sets in ℝ (which are strongly meager) as well as uncountable γ-sets in ℝ which are not strongly meager. These sets must be of cardinality ω₁ < 𝔠, since every γ-set is universally null, while $CPA_{cube}^{game}$ implies that every universally null has cardinality less than 𝔠 = ω₂. We also show that $CPA_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets.
9
Content available remote

The Hahn-Banach theorem implies the Banach-Tarski paradox

44%
Pawlikowski J.
Fundamenta Mathematicae
|
1991
|
tom 138
|
nr 1
21-22
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.