The Constructible Universe

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's axiom with the negation of CH implies the existence of a separable nonmetrizable normal Moore space. In this paper we consider the situation under Gödel's axiom of constructibility {V = L). Bing [l] shows that a normal Moore space is metrizable iff it is collectionwise normal. Moore spaces have character XQ (i.e. are first countable). The following is then a partial result towards proving the normal Moore space conjecture in L. Theorem (V = L). If X is a normal Hausdorff space of character < X., then X is collectionwise Hausdorff. Definition. A space is collectionwise Hausdorff {CWH) iff every closed discrete set of points can be simultaneously separated by disjoint open sets. Let CWH(k) be CWH restricted to sets of cardinality < k. Remarks. 1. The Theorem shows consistent Bing's conjecture [2] that normal Moore spaces be CWH. It shows in fact that in a normal character X, space, a discrete collection of countable closed sets can be simultaneously separated by the device of shrinking each closed set to a point, which then is of character X,. Received by the editors August 2, 1973 and, in revised form, September 30, 1973. AMS iMOS) subject classifications (1970). Primary 02K05, 04A20, 54D15, 54E30.