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The restriction of the natural map from \beta(\omega×\omega) to \beta\omega×\beta\omega is very badly not an isomorphism between the smallest ideals.
by
Gugu Moche
Howard University
The ``natural map'' from \beta(\omega×\omega) to \beta\omega×\beta\omega is the continuous extension of the identity function. It has been known for some time that some points in \beta\omega×\beta\omega have infinite preimages (and therefore have 2c elements in their preimage). However, even if p and q are in the smallest ideal K(\beta\omega) of \beta\omega, the known proofs do not produce more than one point in the preimage of (p, q) lying in K(\beta(\omega×\omega)). This left open the fascinating possibility that the natural map might be an isomorphism from K(\beta(\omega×\omega)) onto K(\beta\omega) ×K(\beta\omega). We eliminate this possibility by showing that the preimage of each point of K(\beta\omega) ×K(\beta\omega) has infinite intersection with K(\beta(\omega×\omega)).
Date received: May 15, 2001
Copyright © 2001 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 # cagw-43.