F U N D a M E N T a Mathematicae the Arkhangel'ski˘ I–tall Problem: a Consistent Counterexample

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skĭı and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in [ω] , and a version of an (ω, 1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration. 0. Introduction. In 1971, A. V. Arkhangel’skĭı [A] proved that every perfectly normal, locally compact, metacompact space is paracompact. This suggests the question, stated in print by Arkhangel’skĭı (see [AP], Chapter 5, p. 309) and Tall [T] three years later, and oft-repeated since then, whether “perfectly normal” can be reduced to “normal”: Problem. Is every normal , locally compact , metacompact space paracompact? Recall that a space is metacompact if every open cover has a point-finite open refinement. Standard topological arguments show that if there is a counterexample to the problem, then there is one which is not collectionwise Hausdorff (CWH). Bing’s famous Example G [Bi] is a ZFC example of a normal space which is not CWH, and Michael’s metacompact subspace of this example (see [Mi]), which is not locally compact, shows that the assumption of local compactness is essential for this problem. 1991 Mathematics Subject Classification: 54A35, 03E35, 54D15, 54D20. Research of the first author partially supported by NSF grants DMS-9102725 and DMS-9401529. The second author was an NSERC of Canada postdoctoral fellow at York University and the Hebrew University of Jerusalem while the research leading to this paper was