We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

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:...

For any three noncollinear points c₀,c₁,c₂ ∈ ℝ², there are sprays S₀,S₁,S₂ centered at c₀,c₁,c₂ that cover ℝ². This improves the result of de la Vega in which c₀,c₁,c₂ were required to be the vertices of an equilateral triangle.

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

We develop the theory of topological Hurewicz test pairs: a concept which allows us to distinguish the classes of the Borel hierarchy by Baire category in a suitable topology. As an application we show that for every ${\Pi }_{\xi }^0$ and not ${\Sigma }_{\xi }^0$ subset $P$ of a Polish space $X$ there is a $\sigma$-ideal $ℐ\subseteq 2^X$ such that $P\notin ℐ$ but for every ${\Sigma }_{\xi }^0$ set $B\subseteq P$ there is a ${\Pi }_{\xi }^0$ set $B^{\text{'}}\subseteq P$ satisfying $B\subseteq B^{\text{'}}\in ℐ$. We also discuss several other results and problems related to ideal generation and Hurewicz test pairs.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.

The resemblance relation is used to reflect some real life situations for which a fuzzy equivalence is not suitable. We study the properties of cuts for such relations. In the case of a resemblance on a real line $E$ we show that it determines a special family of crisp functions closely connected to its cut relations. Conversely, we present conditions which should be satisfied by a collection of real functions in $E$ in order that this collection determines a resemblance relation.