Related papers: Additivity of the Gerlits--Nagy property and conce…
The_additivity_number_ of a topological property (relative to a given space) is the minimal number of subspaces with this property whose union does not have the property. The most well-known case is where this number is greater than…
We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods…
Menger's basis property is a generalization of $\sigma$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $\sigma$-compact sets of reals with Menger's property.…
We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than…
We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are…
We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every…
The theorem we prove is a slight strengthening of some results by Just, Miller, Scheepers and Szeptycki [JMSS]. We use the Michael technique instead of the combinatorial approach in the literature. Comments by the submitter: This short…
We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped…
We provide simplified solutions of Menger's and Hurewicz's problems and conjectures, concerning generalizations of sigma-compactness. The reader who is new to this field will find a self-contained treatment in Sections 1, 2, and 5. Sections…
According to a result of Kocinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover…
We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy, which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a \delta-set that is not a \gamma-set.…
We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in…
We continue to investigate various diagonalization properties for sequences of open covers of separable metrizable spaces introduced in Part I. These properties generalize classical ones of Rothberger, Menger, Hurewicz, and Gerlits-Nagy. In…
A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…
We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number…
We prove that assuming $\mathfrak{b}=\mathfrak{d}$, in the class of hereditarily Lindel\"of spaces, each productively Scheepers space is productively Hurewicz. The above statement remains true in the class of all general topological spaces…
For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is…
Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods…
The Hurewicz property is a classical generalization of $\sigma$-compactness and Sierpi\'nski sets (whose existence follows from CH) are standard examples of non-$\sigma$-compact Hurewicz spaces. We show, solving a problem stated by Szewczak…
We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily sigma-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods,…