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...

Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for $G_{\delta }$ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

We show that if $X$ is a ${\Sigma }_1^1$ separable metrizable space which is not $\sigma$-compact then $C_p^{*}\left(X\right)$, the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_1^1$-complete. Assuming projective determinacy we show that if $X$ is projective not $\sigma$-compact and $n$ is least such that $X$ is ${\Sigma }_n^1$ then $C_p\left(X\right)$, the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_n^1$-complete. We also prove a simultaneous improvement of theorems of Christensen...

We show that under the axiom $CP{A}_{cube}$ there is no uniformly completely Ramsey null set of size $2^{\omega }$. In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.

Let X be a Borel subset of the Cantor set C of additive or multiplicative class α, and f: X → Y be a continuous function onto Y ⊂ C with compact preimages of points. If the image f(U) of every clopen set U is the intersection of an open and a closed set, then Y is a Borel set of the same class α. This result generalizes similar results for open and closed functions.

We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^{\omega }=\aleph ₂$ together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [“Proper and Improper Forcing”, Ch. V]) and < ω₁-proper ${}^{\omega }\omega$-bounding forcings adding reals. We show...

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.