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Perfect preimages of \omega1 with a small diagonal.
by
Oleg Pavlov
Ohio University
A space X has a small diagonal, if for every uncountable Y subset X2 \\Delta(X), there is a neighborhood U of \Delta(X) such that Y\U is uncountable. G. Gruenhage proved that every countably compact space with a small diagonal is metrizable under certain set-theoretic assumption. In particular, this is true for any perfect preimage of \omega1. He asked if ZFC is enough for every perfect preimage of \omega1, at least for every 2-to-1 such preimage. We answer this question in the following theorems:
Theorem 1. No finite-to-one perfect preimage of \omega1 has a small diagonal.
Theorem 2. Assume diamond, then there is a perfect preimage of \omega1 with a small diagonal.
Theorem 2 is also of interest because every compact space with a small diagonal is metrizable under CH.
Date received: June 3, 1999
Copyright © 1999 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 # cacl-42.