14 research outputs found
On the length of chains of proper subgroups covering a topological group
We prove that if an ultrafilter L is not coherent to a Q-point, then each
analytic non-sigma-bounded topological group G admits an increasing chain <G_a
: a of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and
For every sigma-bounded subgroup H of G there exists a such that H is a
subset of G_a. In case of the group Sym(w) of all permutations of w with the
topology inherited from w^w this improves upon earlier results of S. Thomas
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
On Milliken-Taylor ultrafilters
Abstract. We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-coloured Milliken-Taylor ultrafilters have at least k + 1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model. Milliken-Taylor ultrafilters and their projections We answer a question of López-Abad whether there can be more than two near coherence classes of ultrafilters in the core of a Milliken-Taylor ultrafilter. Then we show that in Milliken Taylor ultrafilter with k colours there are k + 1 near coherence classes in its projection to ω, generalising a result of Blass In the rest of this introductory section we review part of the relevant background. Our nomenclature follows [10] an
Lebesgue measure zero modulo ideals on the natural numbers
We propose a reformulation of the ideal of Lebesgue measure
zero sets of reals modulo an ideal on , which we denote by
. In the same way, we reformulate the ideal
generated by measure zero sets of reals modulo , which we denote
by . We show that these are -ideals and that
iff has the Baire property, which in turn is
equivalent to . Moreover, we prove that
does not contain co-meager sets and contains
non-meager sets when does not have the Baire property. We also prove a deep
connection between these ideals modulo and the notion of nearly coherence
of filters (or ideals).
We also study the cardinal characteristics associated with
and . We show their position with respect to Cicho\'n's
diagram and prove that no further inequalities can be proved in relation with
other very classical cardinal characteristics of the continuum, leaving just
very few open questions. To achieve this, we discovered a new characterization
of and . We also show
that, in Cohen model, we can obtain many different values to the cardinal
characteristics associated with our new ideals.Comment: 33 pages, 6 figure