Perfect preimages of \omega₁ with a small diagonal
by
Oleg Pavlov
University of South Carolina

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. It has been known that under CH every compact space with a small diagonal has a G_\delta diagonal, hence metrizable. G. Gruenhage proved that existence of a nonmetrizable countably compact space with a small diagonal is consistent with and independent of ZFC. He raised the following questions:

Does CH imply that countably compact spaces having a small diagonal are metrizable?

Can there be a first-countable countably compact space with a small diagonal but no G_\delta-diagonal?

Can there be a first-countable perfect preimage of \omega₁ with a small diagonal? A 2-to-1 preimage?

(Clearly, the last question is stronger than the second one.) The last question is of interest since \omega₁ has no small diagonal. Of course, all these questions do have a consistent negative solution by a mentioned result of Gruenhage. His example is in not consistent with CH and it is not first-countable. We answer these questions in the following theorems:

Theorem 1. No finite-to-one perfect preimage of \omega₁ has a small diagonal.

Theorem 2. Assume \diamondsuit⁺, then there is a \omega-to-1 perfect preimage of \omega₁ with a small diagonal.

Date received: February 14, 2000