Topological statements independent of CH
by
Peter J. Nyikos
University of South Carolina

Recent work of Todd Eisworth in set theory has resulted in significantly new tools for obtaining results independent not only of ZFC but of CH as well. A joint result of Eisworth and Nyikos is that CH is compatible with the statement that every first countable, countably compact Hausdorff space is either compact or contains a copy of \omega₁. Using strengthenings of an axiom of Abraham and Todorcevic due to Eisworth, a number of strong results are obtained about trees and manifolds that are compatible with CH; for example, the following are compatible with and independent of CH: Statement 1:Every perfectly normal manifold is either metrizable or contains a separable nonmetrizable subspace. Statement 2:Every collectionwise Hausdorff tree of height \omega₁ either contains an uncountable branch or else is metrizable. Statement 3:If X is a locally compact space in which every countable subset has Lindelöf closure, then either (1) every uncountable subset of X has a condensation point or (2) X contains a perfect preimage of \omega₁ or (3) X contains an uncountable closed discrete subspace.

Date received: February 27, 1998