On Countably Compact, Perfectly Normal Spaces

Abstract

The main aim of this paper is to give an example of a non-compact, hereditarily separable, locally compact, perfectly normal, countably compact space; the construction is based on Jensen's Combinatorial Principle <>. This principle is known to follow from Godel's Axiom of Constructibility and so our result is in fact a consistency theorem. It is not known whether such an example can be constructed on the assumption of the continuum hypothesis CH alone (note that 0 implies CH). Juhasz, Kunen and Rudin [14] obtained a very beautiful modification of the technique presented here which allows the construction to be derived from CH alone but with loss of countable compactness. This is most perplexing since the situation in our construction is quite the reverse: countable compactness is derived from CH and the remaining properties from a hypothesis which is apparently weaker than <>• It has since been shown, however, by W. Weiss [30] that if CH be negated and Martin's Axiom assumed, every perfectly normal, countably compact space is compact.A more complicated example of a non-compact, perfectly normal, countably compact space was independently and a little earlier given by Fedorcuk in [4] (see also [5, 6]) to illuminate certain problems in dimension theory. Further applications of the 0 technique and the CH-counterpart were studied by Rudin and Zenor [22], Fedorcuk [7, 8], J. Vaughan [28], E. Pol [19], M. Wage [29] and also by the present author in [17] and [18]. Other related matter may be found in papers of Rudin [21], van Douwen [26] and van Douwen and Wicke [27]. We wish also to call to attention R. Pol's most elegant construction in [20], which comes very close to the results of this paper without assuming any additional set-theoretic axioms. The question whether perfectly normal, countably compact spaces are compact was raised by a number of authors in various contexts e.g.: Malyhin and Shapirovskii in a draft of [16], Stephenson [24], Hager, Reynolds and Rice [10], who weakened the hypotheses of the question.In §2 we prove some theorems about countably compact spaces whose open sets are !F a and deduce a strengthening of Stephenson's main theorem in [24]. The theorems allow us to predict many properties of our example.It is a great pleasure to thank Professor M. E. Rudin for bringing the problem treated here to my attention and to thank Professor Roy O. Davies, for many, very interesting and stimulating discussions. The author gratefully acknowledges the financial support of the U.K. Science Research Council and the kind hospitality of Leicester University during the tenure of an S.R.C. Fellowship.Notation. We follow set-theoretic terminology as in, e.g., [1]; in particular, every ordinal is identified with the set of preceding ordinals. Topological terminology is as in [3] or [13]; in particular, x is a point of condensation of a set S if every neighbourhood of x meets S in a set of cardinality \S\. Other definitions are recalled in §2.

Countable tightness and proper forcing

“…With the exception of Corollary 2(i) [one of us has a model of MA where it fails to hold], this can be demonstrated by a number of examples of the 1970s using o, a consequence of V -L\ (a) Ostaszewski's space [4] which is locally compact, noncompact, hereditarily separable (hence admits no continuous map onto w\, yet has countably tight compactification), countably compact, perfectly normal, and of cardinality wi; (b) A principal 5 1 -bundle over the long ray (6.17 of [10]) with a perfect preimage (but no copy) of w\ and (c) Fedorchuk's [11] compact hereditarily separable space of cardinality 2 C with no nontrivial convergent sequences.…”

On the problem of compact totally disconnected reflection of nonmetrizability

“…To take care of continuous images of the subspaces, one needs to modify however, the construction so that, for example, the closed cometrizable subspaces look like the entire space. Let us sketch such a simplified construction of a connected version of an Ostaszewski space from ♦ ( [35]) which works for our purpose. It can be described in the language similar to our unordered split interval: define an inverse limit system K α ⊆ [0, 1] α with α ≤ ω 1 containing the point 0 α as a nonisolated point.…”

Normality versus countable paracompactness in perfect spaces

“…Now, define a base 8 for the desired topology on X* as follows 9 let {x} G B; and (2) if U is an open set in X and a < X such that U C (IP U D), let THEOREM 1. If X is a normal, noncollectwise normal T 2 -space then X* is a regular, nonnormal T 2 -space and The construction of the example in Theorem 2 depends heavily on Jensen's <^> and the technique used by Ostaszewski [9].…”