Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.All topological spaces are assumed to be Hausdorff.First, a definition and three standard lemmata are needed.

Abstract. It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

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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

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