486 research outputs found
Orderability of all noncompact images
AbstractWe characterize the noncompact spaces whose every noncompact image is orderable as the noncompact continuous images of ω1. We find other useful characterizations as well. We also characterize the continuous images of ω1 + 1
P-remote points of X
AbstractA new inductive technique is given for the construction of certain kinds of special points in the ÄŒech-Stone compactification of X. This technique significantly simplifies the proofs of some known results on remote points and is also used to show that noncompact first-countable spaces without isolated points have far points
P-remote points of X
AbstractA new inductive technique is given for the construction of certain kinds of special points in the ÄŒech-Stone compactification of X. This technique significantly simplifies the proofs of some known results on remote points and is also used to show that noncompact first-countable spaces without isolated points have far points
Better closed ultrafilters on Q
AbstractCMA (Martin's Axiom for countable posets) implies that for each nϵN there is a free maximal closed filter on the space Q such that the filter it generates on the set Q is the intersection of n ultrafilters
The combinatorics of splittability
Marion Scheepers, in his studies of the combinatorics of open covers,
introduced the property Split(U,V) asserting that a cover of type U can be
split into two covers of type V. In the first part of this paper we give an
almost complete classification of all properties of this form where U and V are
significant families of covers which appear in the literature (namely, large
covers, omega-covers, tau-covers, and gamma-covers), using combinatorial
characterizations of these properties in terms related to ultrafilters on N.
In the second part of the paper we consider the questions whether, given U
and V, the property Split(U,V) is preserved under taking finite unions,
arbitrary subsets, powers or products. Several interesting problems remain
open.Comment: Small update
A decomposition theorem for compact groups with application to supercompactness
We show that every compact connected group is the limit of a continuous
inverse sequence, in the category of compact groups, where each successor
bonding map is either an epimorphism with finite kernel or the projection from
a product by a simple compact Lie group. As an application, we present a proof
of an unpublished result of Charles Mills from 1978: every compact group is
supercompact.Comment: 12 page
On disjoint Borel uniformizations
Larman showed that any closed subset of the plane with uncountable vertical
cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that
Larman's result is best possible: there exist closed sets with uncountable
cross-sections which do not have more than aleph_1 disjoint Borel
uniformizations, even if the continuum is much larger than aleph_1. This
negatively answers some questions of Mauldin. The proof is based on a result of
Stern, stating that certain Borel sets cannot be written as a small union of
low-level Borel sets. The proof of the latter result uses Steel's method of
forcing with tagged trees; a full presentation of this method, written in terms
of Baire category rather than forcing, is given here
Amalgams, connectifications, and homogeneous compacta
We construct a path-connected homogenous compactum with cellularity 2^omega
that is not homeomorphic to any product of dyadic compacta and first countable
compacta. We also prove some closure properties for classes of spaces defined
by various connectifiability conditions. One application is that every infinite
product of infinite topological sums of T_i spaces has a T_i pathwise
connectification, where i is 1, 2, 3, or 3.5.Comment: 10 pages; corrected typo
Reduced Coproducts of Compact Hausdorff Spaces
By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the reduced coproduct , which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the ultracoproduct can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces
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