Separation in Countably Paracompact Spaces

Abstract

ABSTRACT. 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

Mary Ellen Rudin's Contributions to the Theory of Nonmetrizable Manifolds

“…We do have some consistency results for Problem 1, including of course Burke's results that the product measure extension axiom (PMEA) implies that all countably paracompact Moore spaces are metrizable. Also one does not need large cardinals: Mary Ellen's argument in [36, p. 461 that V = L implies all locally compact normal Moore spaces are metrizable, works equally well for "countably paracompact" in place of "normal" if one substitutes W. S. Watson's theorem in [46] for the theorem of Fleissner used by Mary Ellen.…”

Paranormal spaces under $\diamondsuit ^{*}$

“…In fact in [W1], W. S. Watson proved that countably paracompact first countable spaces are collectionwise Hausdorff assuming ♦ SS . Theorem 1.3 (Watson, 1985). Assume ♦ SS .…”

Monotone versions of countable paracompactness

“…Moreover, it turns out that a number of set-theoretic results concerning the separation of closed discrete collections in normal spaces have direct analogues for countably paracompact spaces. For example, Burke [2] modifies Nyikos's 'provisional' solution to the normal Moore space problem by showing that countably paracompact, Moore spaces are metrizable assuming PMEA and Watson [24] shows that, assuming V=L, first countable, countably paracompact spaces are collectionwise Hausdorff. When normality is strengthened to monotone normality, pathology is reduced and the need for set-theory in such results is generally avoided.…”