A NOTE ON NORMALITY AND COLLECTIONWISE NORMALITY

In this note we make several unrelated observations concerning circumstances under which normality implies col lectionwise normality. All spaces are assumed Hausdorff. I. Screenable Spaces In [Tll I proved that screenable normal spaces are collectionwise Hausdorff. I can now improve this. Theorem 1. Screenable normal spaces are collection ~ise normal ~ith respect to countably metacompact closed sets. Proof. Let mbe a discrete collection of countably metacompact closed sets. Let 5 = u < 5 be a a-disjoint n w n refinement of the canonical cover. The closed subspace Um is normal and countably metacompact. Since point-finite open covers of normal spaces can be shrunk, there are open {Tn}n<w ' Tn C u5 n um. Um is closed so the closure signn is unambiguous. {M n T: M E m} for each n is a discrete n collection separated by the open sets Sn,M = U{S E 5n: s n M ~ ~}. By normality the S MiS may be shrunk to a n, discrete separation {S ' M: M Em}. Then U{S ' : M E m} is n,l: n,M a-discrete and so yields a separation by standard arguments. Theorem 1 cannot be improved in ZFC to get collection wise normality; assuming 0++, M. E. Rudin [Rl] constructe