49 research outputs found
Selection principles in mathematics: A milestone of open problems
We survey some of the major open problems involving selection principles,
diagonalizations, and covering properties in topology and infinite
combinatorics. Background details, definitions and motivations are also
provided.Comment: Small update
Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures
We consider the question, which of the major classes defined by topological
diagonalizations of open or Borel covers is hereditary. Many of the classes in
the open case are not hereditary already in ZFC, and none of them is provably
hereditary. This is contrasted with the Borel case, where some of the classes
are provably hereditary. Two of the examples are counter-examples of sizes d$
and b, respectively, to the Menger and Hurewicz Conjectures, and one of them
answers a question of Steprans on perfectly meager sets
A semifilter approach to selection principles
We develop the semifilter approach to the classical Menger and Hurewicz
covering properties and show that the small cardinal g is a lower bound of the
additivity number of the family of Menger subspaces of the Baire space, and
under u< g every subset X of the real line with the property
Split(Lambda,Lambda) is Hurewicz.Comment: LaTeX 2e, 15 pages, submitted to Comment. Math. Univ. Carolina
Some new directions in infinite-combinatorial topology
We give a light introduction to selection principles in topology, a young
subfield of infinite-combinatorial topology. Emphasis is put on the modern
approach to the problems it deals with. Recent results are described, and open
problems are stated. Some results which do not appear elsewhere are also
included, with proofs.Comment: Small update
Topological diagonalizations and Hausdorff dimension
The Hausdorff dimension of a product XxY can be strictly greater than that of
Y, even when the Hausdorff dimension of X is zero. But when X is countable, the
Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers
define a natural hierarchy of properties which are weaker than ``being
countable'' and stronger than ``having Hausdorff dimension zero''. Fremlin
asked whether it is enough for X to have the strongest property in this
hierarchy (namely, being a gamma-set) in order to assure that the Hausdorff
dimensions of Y and XxY are the same.
We give a negative answer: Assuming CH, there exists a gamma-set of reals X
and a set of reals Y with Hausdorff dimension zero, such that the Hausdorff
dimension of X+Y (a Lipschitz image of XxY) is maximal, that is, 1. However, we
show that for the notion of a_strong_ gamma-set the answer is positive. Some
related problems remain open.Comment: Small update
Selective covering properties of product spaces
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 include the projection method introduced by the authors in an
earlier work, as well as several new methods. Some special consequences of our
main results are (definitions provided in the paper): \be
\item Every product of a concentrated space with a Hurewicz \sone(\Ga,\Op)
space satisfies \sone(\Ga,\Op). On the other hand, assuming \CH{}, for each
Sierpi\'nski set there is a Luzin set such that L\x S can be mapped
onto the real line by a Borel function.
\item Assuming Semifilter Trichotomy, every concentrated space is
productively Menger and productively Rothberger.
\item Every scale set is productively Hurewicz, productively Menger,
productively Scheepers, and productively Gerlits--Nagy.
\item Assuming \fd=\aleph_1, every productively Lindel\"of space is
productively Hurewicz, productively Menger, and productively Scheepers. \ee
A notorious open problem asks whether the additivity of Rothberger's property
may be strictly greater than \add(\cN), the additivity of the ideal of
Lebesgue-null sets of reals. We obtain a positive answer, modulo the
consistency of Semifilter Trichotomy with \add(\cN)<\cov(\cM).
Our results improve upon and unify a number of results, established earlier
by many authors.Comment: Submitted for publicatio