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Totally Bogus Axiom
by
Alan Dow
York University
Fact: if \alpha has uncountable cofinality, then wp(\alpha, \alpha+) holds.
The proof is simple (once you think of it!): Just fix a sequence {f\beta : \beta < \alpha+} of functions from \alpha into \alpha so that if \beta < \gamma, then there is a \xi < \alpha such that f\beta(\eta) < f\gamma(\eta) for all \eta > \xi.
Therefore {f\beta : \beta < \alpha+} is a subspace of the space \alpha\alpha with the product topology. To see that it is a weak P-space simply note that given any countable set A subset \alpha+, there is a coordinate \xi such that the function \beta goes to f\beta(\xi) is one-to-one on A. In fact, not only is it a weak P-space every countable set is discrete and C -embedded.
I already gave Wis a proof that it is consistent that 2\omega1 >= \omega3 (in fact any reasonable value) and such that wp(\omega1, \omega3) fails. Just add lots of Cohen reals.
I have no idea about wp(\aleph\omega, \aleph\omega+1) under any hypothesis.
Date received: May 31, 1996
Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caag-12.