Peter Nyikos

The Theory of Nonmetrizable Manifolds

Publisher Summary This chapter discusses the theory of nonmetrizable manifolds. Manifolds have been considered too specialized, and the distinctions between them are too fine and subtle for the techniques of point-set topology to give interesting results about them. These manifolds share many basic properties with the spaces they mimic, making it possible to bring the tools of set theory into play. The structure theory of manifolds provides a fertile ground for the application of set-theoretic concepts and techniques. Manifolds also provide a convenient central category in the study of topological spaces. Manifolds can be locally compact, locally separable, locally connected, locally Lindelof, locally countable chain condition, first countable, locally second countable, locally metrizable, and completely regular. A manifold is a connected space for which there is a positive integer n such that each point has a neighborhood that is homeomorphic to Rn. A manifold is a Moore space if and only if it is submetacompact. Every submetacompact manifold is subparacompact. A manifold is metrizable if and only if it is paracompact.

A provisional solution to the normal Moore space problem

The Product Measure Extension Axiom (PMEA), whose consistency would follow from the existence of a strongly compact cardinal, implies that every normalized collection of sets in a space of character less than the continuum is well separated. Consistency of PMEA would thus solve many well-known problems of general topology, including that of whether every first countable normal space is collectionwise normal, as well as the normal Moore space problem.

On some non-Archimedean spaces of Alexandroff and Urysohn

Abstract Classical characterizations of four separable metrizable spaces are recalled, and generalized to classes of spaces which admit a uniformity with a totally ordered base. The Alexandroff-Urysohn characterization of the irrationals finds its closest analogues for strongly inaccessible cardinals, while the other three spaces, including the Cantor set, find their most natural analogues for weakly compact cardinals. In addition, A.H. Stones characterization of Baires zero-dimensional spaces is extended to give internal characterizations of all spaces γ λ × D , where D is discrete and γ λ has the initial agreement topology. The historical background for the Alexandroff-Urysohn result is briefly surveyed.

Subsets of ωω and the Fréchet-Urysohn and αi-properties

Abstract Arhangelskiǐ defined a number of related properties called α i ( i = 1, 2, 3, 4) having to do with amalgamating countably many sequences each converging to the same point. Here we use the set ω ω of functions to produce examples of Frechet spaces in the various classes and to study the relationships between the classes. We also introduce an intermediate class α 1.5 . Under various set-theoretic hypotheses we produce a countable Frechet α 1 -space that is not first countable, and several that are α 2 but not α 1 , including one which is α 1.5 and another which is not. It is now known to be consistent that none of these kinds of spaces exist, but we also construct a countable Frechet-Urysohn α 2 -space that is not first countable using only ZFC. The existence of an α 2 -space which is not α 1 in any given model of set theory is reduced to the existence of a certain kind of space whose underlying set is (ω × ω)∪∞, with neighborhoods of ∞ defined using graphs of partial functions. Alan Dow has recently shown that every α 2 -space is α 1 in the Laver model. A proof using the reduction theorem is outlined here and the result is used to obtain other information about this model. An example of a countable α 2 -topological group that is not first countable is given, and it is shown to be Frechet-Urysohn under the relatively mild assumption p = b , as is a related separable nonmetrizable topological vector space.

On spaces in which countably compact sets are closed, and hereditary properties

Abstract A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martins Axiom or 2ω0

The Cantor Tree and the Fréchet-Urysohn Property

In this paper, we will show how the Cantor tree provides an easy example of a countable FrBchet-Urysohn space that is not metrizable (equivalently, by Urysohn’s metrization theorem, not first countable). Then, we will show how it can be used to provide a countable, nonmetrizable Frichet-Urysohn topological group under certain set-theoretic hypotheses (see COROLLARY 3.5). We also will discuss two other problems for which the Cantor tree provides examples and insight. These problems were solved by A. Dow and J. Steprfms shortly after the Third New York Conference on Limits, from where they were relayed by A. Kato to Toronto.

The Scarborough-Stone problem for Hausdorff spaces

Abstract We give two methods of constructing families of sequentially compact Hausdorff spaces whose product is not countably compact.

INDICATIONS FOR EARLY HÆMODIALYSIS IN MULTIPLE TRAUMA

Abstract Renal function in 751 cases of multiple trauma was studied to define a level of function compatible with ultimate survival. Established definitions of renal failure were ignored. A daily renal index was calculated using urine volume, serum-creatinine, and blood-urea-nitrogen (B.U.N.). The data for 3600 patient-days were analysed on a computer. Probability of survival was less than 0.1 in patients with a creatinine >4 mg. per 100 ml. or a B.U.N. >80 mg. per 100 ml. or a renal index >3 on one occasion or >2 on two consecutive days. The renal index provided an earlier and more accurate prognosis in a significant number of patients when compared with the other variables. The impairment of renal function associated with death in the patient studied is considerably less than currently accepted criteria for haemodialysis. Dialysis to within the levels shown to be compatible with survival offers a method of reducing the high mortality. Clinical application of the renal index as an indication for early haemodialysis in major trauma is proposed.

Sequentially compact, Franklin-Rajagopalan spaces

A locally compact T2-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space X can be identified with a space constructed from a tower T on w (X = X(T)), and (2) for an ultrafilter u on w, a sequentially compact FR-space X(T) is not u-compact if and only if there exists an ultrafilter v on w such that v D T, and v is below u in the RudinKeisler order on w*. As one application of these results we show that in certain models of set theory there exists a family T of towers such that I T I < 2W, and nH{X(T): T E T } is a product of sequentially compact FR-spaces which is not countably compact (a new solution to the Scarborough-Stone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially m-chain compact and totally initially m-compact spaces.

Topologically orderable groups

Abstract Topological groups whose topology can be induced by a total order are characterized up to homeomorphism. In particular, a non-metrizable topological group is in this class if and only if it has a totally ordered base at the identity consisting of (closed and) open subgroups. Another characterization is a generalization of an earlier result for metrizable groups.