Strongly meager sets and subsets of the plane

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Let

X \subseteq 2^{w}

. Consider the class of all Borel

F \subseteq X \times 2^{w}

with null vertical sections

F_{x}

, x ∈ X. We show that if for all such F and all null Z ⊆ X,

\cup_{x \in Z} F_{x}

is null, then for all such F,

\cup_{x \in X} F_{x} \neq 2^{w}

. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

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