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1
Content available remote

Some additive properties of special sets of reals

100%
Recław I.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 2
221-226
2
Content available remote

Every Lusin set is undetermined in the point-open game

100%
Recław I.
Fundamenta Mathematicae
|
1994
|
tom 144
|
nr 1
43-54
EN
We show that some classes of small sets are topological versions of some combinatorial properties. We also give a characterization of spaces for which White has a winning strategy in the point-open game. We show that every Lusin set is undetermined, which solves a problem of Galvin.
3
Content available remote

On spaces with the ideal convergence property

64%
Jasinski J., Recław I.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 1
43-50
EN
Let I ⊆ P(ω) be an ideal. We continue our investigation of the class of spaces with the I-ideal convergence property, denoted 𝓘𝓒(I). We show that if I is an analytic, non-countably generated P-ideal then 𝓘𝓒(I) ⊆ s₀. If in addition I is non-pathological and not isomorphic to $I_{b}$, then 𝓘𝓒(I) spaces have measure zero. We also present a characterization of the 𝓘𝓒(I) spaces using clopen covers.
4
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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
5
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Ideal limits of sequences of continuous functions

64%
Laczkovich M., Recław I.
Fundamenta Mathematicae
|
2009
|
tom 203
|
nr 1
39-46
EN
We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{σδ}$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.
6
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Strong Fubini properties of ideals

64%
Recław I., Zakrzewski P.
Fundamenta Mathematicae
|
1999
|
tom 159
|
nr 2
135-152
EN
 Let I and J be σ-ideals on Polish spaces X and Y, respectively. We say that the pair ⟨I,J⟩ has the Strong Fubini Property (SFP) if for every set D ⊆ X× Y with measurable sections, if all its sections $D_x = {y: ⟨x,y⟩ ∈ D}$ are in J, then the sections $D^y = {x: ⟨x,y⟩ ∈ D}$ are in I for every y outside a set from J (``measurable" means being a member of the σ-algebra of Borel sets modulo sets from the respective σ-ideal). We study the question of which pairs of σ-ideals have the Strong Fubini Property. Since CH excludes this phenomenon completely, sufficient conditions for SFP are always independent of ZFC.  We show, in particular, that:  • if there exists a Lusin set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a non-meager set, then ⟨MGR(X), J⟩ has SFP for every J generated by a hereditary $п^1_1$ (in the Effros Borel structure) family of closed subsets of Y (MGR(X) is the σ-ideal of all meager subsets of X),  • if there exists a Sierpiński set of cardinality the continuum and every set of reals of cardinality the continuum contains a one-to-one Borel image of a set of positive outer Lebesgue measure, then $⟨NULL_μ, J⟩$ has SFP if either $J= NULL_ν$ or J is generated by any of the following families of closed subsets of Y ($NULL_μ$ is the σ-ideal of all subsets of X having outer measure zero with respect to a Borel σ-finite continuous measure μ on X):  (i) all compact sets,  (ii) all closed sets in $NULL_ν$ for a Borel σ-finite continuous measure ν on Y,  (iii) all closed subsets of a $п^1_1$ set A ⊆ Y.
7
Content available remote

Universally Kuratowski–Ulam spaces

51%
Fremlin D. H., Natkaniec T., Recław I.
Fundamenta Mathematicae
|
2000
|
tom 165
|
nr 3
239-247
EN
We introduce the notions of Kuratowski-Ulam pairs of topological spaces and universally Kuratowski-Ulam space. A pair (X,Y) of topological spaces is called a Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X× Y. A space Y is called a universally Kuratowski-Ulam (uK-U) space if (X,Y) is a Kuratowski-Ulam pair for every space X. Obviously, every meager in itself space is uK-U. Moreover, it is known that every space with a countable π-basis is uK-U. We prove the following:  • every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uK-U (so there are uK-U Baire spaces which do not have countable π-bases);  • every Baire uK-U space is ccc.
8
Content available remote

Very small sets

51%
Judah H., Lior A., Recław I.
Colloquium Mathematicum
|
1997
|
tom 72
|
nr 2
207-213
9
Content available remote

On small sets in the sense of measure and category

38%
Recław I.
Fundamenta Mathematicae
|
1989
|
tom 133
|
nr 3
255-260
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