7,716 research outputs found
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
Combinatorial images of sets of reals and semifilter trichotomy
Using a dictionary translating a variety of classical and modern covering
properties into combinatorial properties of continuous images, we get a simple
way to understand the interrelations between these properties in ZFC and in the
realm of the trichotomy axiom for upward closed families of sets of natural
numbers. While it is now known that the answer to the Hurewicz 1927 problem is
positive, it is shown here that semifilter trichotomy implies a negative answer
to a slightly weaker form of this problem.Comment: Small update
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