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.

A tree T on ω is said to be cofinal if for every $\alpha \in {\omega }^{\omega }$ there is some branch β of T such that α ≤ β, and quasi-bounded otherwise. We prove that the set of quasi-bounded trees is a complete Σ¹₁-inductive set. In particular, it is neither analytic nor co-analytic.

We characterize those classes 𝓒 of separable Banach spaces for which there exists a separable Banach space Y not containing ℓ₁ and such that every space in the class 𝓒 is a quotient of Y.

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.  THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.  THEOREM. In the Ellentuck topology on ${\left[\omega \right]}^{\omega }$, ${\left(s\right)}_0$ is a proper subset of the hereditary ideal associated with (s).  We construct an example in the Ellentuck topology of a set which is...

Various ordinal ranks for Baire-1 real-valued functions, which have been used in the literature, are adapted to provide ranks for Baire-1 multifunctions. A new rank is also introduced which, roughly speaking, gives an estimate of how far a Baire-1 multifunction is from being upper semicontinuous.

Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible to a countable...

An example of a non-zero non-atomic translation-invariant Borel measure ${\nu }_p$ on the Banach space ${\ell }_p\left(1\le p\le \infty \right)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "${\nu }_p$-almost every element of ${\ell }_p$ has a property P" implies that “almost every” element of ${\ell }_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.