186 research outputs found
O-minimal cohomology: finiteness and invariance results
We prove that the cohomology groups of a definably compact set over an
o-minimal expansion of a group are finitely generated and invariant under
elementary extensions and expansions of the language. We also study the
cohomology of the intersection of a definable decreas-ing family of definably
compact sets, under the additional assumption that the o-minimal structure
expands a field.Comment: 28 pages, 7 figures and diagrams Added the hypothesis that singletons
are construcible to section 3. Corrected misprint
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Narrow coverings of omega-product spaces
Results of Sierpinski and others have shown that certain finite-dimensional
product sets can be written as unions of subsets, each of which is "narrow" in
a corresponding direction; that is, each line in that direction intersects the
subset in a small set. For example, if the set (omega \times omega) is
partitioned into two pieces along the diagonal, then one piece meets every
horizontal line in a finite set, and the other piece meets each vertical line
in a finite set. Such partitions or coverings can exist only when the sets
forming the product are of limited size.
This paper considers such coverings for products of infinitely many sets
(usually a product of omega copies of the same cardinal kappa). In this case, a
covering of the product by narrow sets, one for each coordinate direction, will
exist no matter how large the factor sets are. But if one restricts the sets
used in the covering (for instance, requiring them to be Borel in a product
topology), then the existence of narrow coverings is related to a number of
large cardinal properties: partition cardinals, the free subset problem,
nonregular ultrafilters, and so on.
One result given here is a relative consistency proof for a hypothesis used
by S. Mrowka to construct a counterexample in the dimension theory of metric
spaces
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