Advanced Search

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, ${\cup }_{x\in X}{F}_x$ is null;  • X has strong measure zero iff for all closed F ⊆ ℝ × ℝ with all vertical sections $F_x$ (x ∈ ℝ) null, ${\cup }_{x\in X}{F}_x$ is null.

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.

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, ${\cup }_{x\in X}{F}_x\ne 2^w$. The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

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 $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets $B\subseteq X\times 2^w$ with all...

We formulate a Covering Property Axiom $CP{A}_{cube}$, 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) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨fₙ\in {ℝ}^{ℝ}{⟩}_{n<\omega }$ of Borel functions there are sequences:...

We formulate a Covering Property Axiom $CP{A}_{cube}^{game}$, 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 < , since every γ-set is universally null, while $CP{A}_{cube}^{game}$ implies that every universally null has cardinality less than = ω₂. We also show that $CP{A}_{cube}^{game}$ implies the existence of a partition of ℝ into ω₁ null compact sets....