126 research outputs found
Covering the Baire space by families which are not finitely dominating
It is consistent (relative to ZFC) that the union of max{b,g} many families
in the Baire space which are not finitely dominating is not dominating. In
particular, it is consistent that for each nonprincipal ultrafilter U, the
cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model
is constructed by oracle chain condition forcing, to which we give a
self-contained introduction.Comment: Small update
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
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
A semifilter approach to selection principles II: tau*-covers
In this paper we settle all questions whether (it is consistent that) the
properties P and Q [do not] coincide, where P and Q run over selection
principles of the type U_fin(O,A).Comment: 9 pages; Latex2e; 1 table; Submitted to CMU
Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations
A classical theorem of Hurewicz characterizes spaces with the Hurewicz
covering property as those having bounded continuous images in the Baire space.
We give a similar characterization for spaces X which have the Hurewicz
property hereditarily. We proceed to consider the class of Arhangel'skii
alpha_1 spaces, for which every sheaf at a point can be amalgamated in a
natural way. Let C_p(X) denote the space of continuous real-valued functions on
X with the topology of pointwise convergence. Our main result is that C_p(X) is
an alpha_1 space if, and only if, each Borel image of X in the Baire space is
bounded. Using this characterization, we solve a variety of problems posed in
the literature concerning spaces of continuous functions.Comment: To appear in Jouranl of the European Mathematical Societ
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
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
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