317 research outputs found
A notion of selective ultrafilter corresponding to topological Ramsey spaces
We introduce the relation of "almost-reduction" in an arbitrary topological
Ramsey space R, as a generalization of the relation of "almost-inclusion" on
the space of infinite sets of natural numbers (the Ellentuck space). This leads
us to a type of ultrafilter U on the set of first approximations of the
elements of R which corresponds to the well-known notion of "selective
ultrafilter" on N, the set of natural numbers. The relationship turns out to be
rather exact in the sense that it permits us to lift several well-known facts
about selective ultrafilters on N and the Ellentuck space to the ultrafilter U
and the Ramsey space R. For example, we prove that the Open Coloring Axiom
holds in M[U], where M is a Solovay model. In this way we extend a result due
to Di Prisco and Todorcevic which gives the same conclusion for the Ellentuck
space.Comment: 24 pages; submitted to Mathematical Logic Quarterl
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