Separation in Countably Paracompact Spaces

We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces. Two of the main directions of research in general topology in the last thirty years have been the work of Jones, Bing, Tall, Fleissner, Nyikos and others motivated by the normal Moore space problem (when are discrete families separated in normal spaces?) and the work of Rudin, Zenor and others motivated by the Dowker space problem (what is the relation between normal and countably paracompact?). This paper considers the following question: (A) Are discrete families of points separated in countably paracompact spaces? There is a related question: (B) Are discrete families of points separated in normal spaces? This question has been studied in the class of first countable spaces and in the class of separable spaces (and so that is also where we consider question (A)) and has been more or less answered in those classes: In the class of first countable spaces, question (B) is independent of the axioms of set theory: V = L and other axioms imply yes (Fleissner, Tall [1, 11]) and MA+-1CH implies no (Silver, Rothberger, Bing, Tall [11]). In the class of separable spaces, (B) is equivalent to the cardinal arithmetic 2^° < 2Hl (Jones, Heath [7, 5]). The negative results consist of the construction of counterexamples which are countably paracompact (so that these negative results also apply to (A)) while the positive results consist of proofs for which normality appears essential. Fleissner [3] and Tall have asked whether it is consistent that, in the class of first countable spaces, the answer is yes to (A). Fleissner, Przymusinski and Reed [3, 8, 9] have asked whether it is possible to show that, in the class of separable spaces, (A) is equivalent to the cardinal arithmetic 2N° < 2Kl (Fleissner was able to extend Jones' short proof to show that, in the class of separable spaces, 2N° = Ni implies yes to (A)). We show that, in the class of first countable spaces, (A) is independent of the axioms of set theory (V = L implies yes) and that, in the class of separable spaces, (A) is equivalent to a set-theoretic statement whose equivalence with 2^° < 2Nl is a special case of a well-known open problem in set theory (Steprans, Jech and Prikry [6, 10] have shown, independently, for example, that, if 2**° is a regular cardinal and there is no measurable cardinal in an inner model, then the equivalence holds). In this paper, a space is a regular topological space; a family {Aa:a < k} of subsets of a space is separated if there is a disjoint family {Oa: a < /c} of open sets Received by the editors July 9, 1982 and, in revised form, November 15, 1984. 1980 Mathematics Subject Classification. Primary 54D15; Secondary 54D18, 54A35. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page