147 research outputs found
Irredundant Sets in Atomic Boolean Algebras
Assuming GCH, we construct an atomic boolean algebra whose pi-weight is
strictly less than the least size of a maximal irredundant family.Comment: This version corrects some errors in the original arXiv versio
Dissipated Compacta
The dissipated spaces form a class of compacta which contains both the
scattered compacta and the compact LOTSes (linearly ordered topological
spaces), and a number of theorems true for these latter two classes are true
more generally for the dissipated spaces. For example, every regular Borel
measure on a dissipated space is separable.
A product of two compact LOTSes is usually not dissipated, but it may satisfy
a weakening of that property. In fact, the degree of dissipation of a space can
be used to distinguish topologically a product of n LOTSes from a product of m
LOTSes.Comment: 34 page
The Complex Stone-Weierstrass Property
C(X) denotes the space of continuous complex-valued functions on the compact
Hausdorff space X. X has the CSWP if every subalgebra of C(X) which separates
points and contains the constant functions is dense in C(X). W. Rudin showed
that all scattered X have the CSWP. We describe a class of non-scattered X with
the CSWP; by another result of Rudin, such X cannot be metrizable.Comment: 15 pages This version extends the main result of the previous version
from separable compact ordered spaces to general compact ordered space
Characterizing Subgroups of Compact Abelian Groups
We prove that every countable subgroup of a compact metrizable abelian group
has a characterizing set. As an application, we answer several questions on
maximally almost periodic (MAP) groups and give a characterization of the class
of (necessarily MAP) abelian topological groups whose Bohr topology has
countable pseudocharacter.Comment: 12 page
Properties of the Class of Measure Separable Compact Spaces
We investigate properties of the class of compact spaces on which every
regular Borel measure is separable. This class will be referred to as MS.
We discuss some closure properties of MS, and show that some simply defined
compact spaces, such as compact ordered spaces or compact scattered spaces, are
in MS. Most of the basic theory for regular measures is true just in ZFC. On
the other hand, the existence of a compact ordered scattered space which
carries a non-separable (non-regular) Borel measure is equivalent to the
existence of a real-valued measurable cardinal less or equal to c.
We show that not being in MS is preserved by all forcing extensions which do
not collapse omega_1, while being in MS can be destroyed even by a ccc forcing
Locally Constant Functions
Let X be a compact Hausdorff space and M a metric space. E_0(X,M) is the set
of f in C(X,M) such that there is a dense set of points x in X with f constant
on some neighborhood of x. We describe some general classes of X for which
E_0(X,M) is all of C(X,M). These include beta N - N, any nowhere separable
LOTS, and any X such that forcing with the open subsets of X does not add
reals. In the case that M is a Banach space, we discuss the properties of
E_0(X,M) as a normed linear space. We also build three first countable Eberlein
compact spaces, F,G,H, with various E_0 properties: For all metric M: E_0(F,M)
contains only the constant functions, and E_0(G,M) = C(G,M). If M is the
Hilbert cube or any infinite dimensional Banach space, E_0(H,M) is not all of
C(H,M), but E_0(H,M) = C(H,M) whenever M is a subset of RR^n for some finite n
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