Peter Nyikos

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I have been on the faculty of the Department of Mathematics, University of South Carolina, Columbia since 1979. My main research interests are in general topology, especially the application of ZFC-independent axioms and sophisticated set-theoretic techniques to classes of topological spaces.

I have maintained a keen lifelong interest in astronomy and cosmology, and paleontology, especially mammalian paleontology.

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Serum Osmolality in Acutely Injured Patients

: The study was to determine the etiology and to evaluate the prognostic significance of serum hy... more : The study was to determine the etiology and to evaluate the prognostic significance of serum hyperosmolality in acutely injured patients. The case histories of 700 patients admitted to the Maryland Institute for Emergency Medicine were examined, and those of 72 patients with serum hyperosmolalities greater than 340 mOsm/kg were selected for this retrospective study. Of the group selected, 12.5% survived and 87.5% expired. The hyperosmolar state in 56% of the survivors resulted from their therapy (hyperalimentation or mannitol infusion). Of those that died, 79% had kidney failure and 11% had central nervous system injury. The main determinant of prognosis when hyperosmolality is present is organ system failure (particularly the kidney). If hyperosmolality is associated with renal failure, the prognosis is poor. As hyperosmolality in a patient without major organ system dysfunction is frequently a complication of therapy, it is usually reversible. Therefore, the serum osmolality of ...

The structure theory of T 5 and related locally compact , locally connected spaces under the PFA and PFA ( S ) [ S ]

A detailed structure theorem is shown for locally compact, locally connected, hereditarily normal... more A detailed structure theorem is shown for locally compact, locally connected, hereditarily normal spaces and for normal, locally compact, locally connected, hereditarily ω1-scwH spaces in models of PFA(S)[S], and for the latter kinds of spaces in models of PFA. Corollaries include a powerful refinement theorem like that for monotonically normal spaces, and the corollary that the spaces involved are [hereditarily] collectionwise normal and [hereditarily] countably paracompact. Among the problems left unsolved and discussed at the end is the ambious question of whether it is consistent that hereditarily normal, locally compact, locally connected spaces are actually monotonically normal. An affirmative solution would also solve the problem of consistency of every perfectly normal, locally compact, locally connected space being metrizable and thus also solve a 1935 problem due to Alexandroff. In [Ny3], the consistency of a strong metrization theorem for manifolds was “proven”: Statement...

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Strong Α-Favorability of the (Generalized) Compact-Open Topology

Strong α-favorability of the compact-open topology on the space of continuous functions, as well ... more

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Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271. - Andreas Blass and Saharon Shelah. There may be simple - and -points and the Rudin–Keisler ordering may be downward directed. Annals of pure and applied logic, v...

Journal of Symbolic Logic, 1992

Andreas Blass. Selective ultrafilters and homogeneity. Annals of pure and applied logic, vol. 38 (1988), pp. 215–255. - Claude Laflamme. Forcing with filters and complete combinatorics. Annals of pure and applied logic, vol. 42 (1989), pp. 125–163

Journal of Symbolic Logic, 1991

Gσ-sets in symmetrizable and related spaces

General Topology and its Applications, 1978

In this paper we answer questions of Arhangel'skii and Michael by providing an example of a regul... more In this paper we answer questions of Arhangel'skii and Michael by providing an example of a regular symmetrizable spslce which is not subparacompact and has a closed subset which is not a C&set. We also use the idea of sequential arder to obtain some positive results, and several examples are provided wfsicb show that these results are in some sense the best possible.

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Classic Problems -- 25 Years Later (Part 2)

The first article in this series [N02] (referred to below simply as “Part 1”) dealt with the four... more The first article in this series [N02] (referred to below simply as “Part 1”) dealt with the four Classic Problems that appeared in Volume 1 of this journal. This second article updates Part 1 and deals in detail with the four Classic Problems in Volume 2 and problems related to them. The most progress on any of the eight problems this past year was made by Alan Dow on Classic Problem I (“Efimov’s Problem”): Does every infinite compact space contain either a nontrivial convergent sequence or a copy of βω? [As in Volume 2, “space ” means “Hausdorff space.”] Alan Dow showed that there is a counterexample if 2 s < 2 c and the cofinality of the poset ([s] ω, ⊂) is equal to s. Roughly speaking, Dow’s construction substitutes zero-sets for points in Fedorchuk’s PH construction [F77]. The construction can be done in ZFC, and results in an infinite compact space with no convergent sequences. The purpose of the second condition is to insure that the space has cardinality 2 s, while the pu...

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METRIZABILITY AND THE FRECHLET-URYSOHN PROPERTY IN TOPOLOGICAL GROUPS&apos

ABssRAc'r. A question of Arhangel'skii, whether weakly first countable topological grou... more ABssRAc'r. A question of Arhangel'skii, whether weakly first countable topological groups are metrizable, is answered in two ways: if the Hausdorff axiom is assumed, the answer is yes, but in general a weakly first countable topological group need not be pseudometrizable. The former result is obtained as a corollary of a more general sufficient condition for a sequential group to be Fr&chet-Urysohn. A general necessary and sufficient condition for a sequential group to be Frechet-Urysohn is given, and a number of questions are raised. Examples are given to show in what respect the theorems of the paper are the "best possible". In his 1966 survey paper [1], A. V. Arhangel'skii defined the gf-axiom of countability and tossed out the question of whether every topological group which satisfies it is metrizable. Tacit in this question is the assumption of the Hausdorff (T2) separation axiom, since there are elementary examples of pseudometrizable topological grou...

Locally compact, $ω_1$-compact spaces

This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo... more This paper is centered on an extremely general problem: Problem. Is it consistent (perhaps modulo large cardinals) that a locally compact space X must be the union of countably many ω-bounded subspaces if every closed discrete subspace of X is countable [in other words, if X is ω1-compact]? A space is ω-bounded if every countable subset has compact closure. This is a strengthening of countable compactness. Assuming the consistency of a supercompact cardinal, it is shown that the answer is affirmative for the specialized class of monotonically normal spaces and the more general class of hereditarily ω1-scwH spaces, where “scwH” stands for “strongly collectionwise Hausdorff”. Even for monotonically normal spaces, this is only a consistency result: Theorem. If ♣, then there exists a locally compact, ω1-compact monotonically normal space that is not the union of countably many ω-bounded subspaces. These two results together are unusual in that most independence results on monotonically ...

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Trees , and an Answer to a Problem of Buzyakova

A ZFC example is given of a Tychonoff space in which the extent of every subspace equals its Lind... more A ZFC example is given of a Tychonoff space in which the extent of every subspace equals its Lindelöf degree, yet the space is not a D-space, answering a question of Raushan Z. Buzyakova. The example is the tree of compact subsets of a stationary, co-stationary subset of ω1. A simple method is given of embedding any tree as a dense open subspace of a tree which is a D-space. Some classes of trees that are D-spaces are discussed, along with some open problems about D-spaces.

Topology Proceedings 3 (1978) pp. 461-471: A SURVEY OF TWO PROBLEMS

Compactifications and remainders of monotonically normal spaces

Topology and its Applications, 2016

Monotonically normal spaces have many strong properties, but poor preservation properties. For ex... more Monotonically normal spaces have many strong properties, but poor preservation properties. For example, there are locally compact, monotonically normal spaces whose one-point compactifications are not monotonically normal, and hence have no monotonically normal compactifications. We give two classes of such spaces, and give a pair of necessary conditions for spaces of pointwise countable type to have, respectively, compactifications or remainders that are monotonically normal. We show that a monotonically normal, locally compact space has a monotonically normal compactification if it is either locally connected or countably compact, and show that this latter condition cannot be weakened to "σ-countably compact."

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A provisional solution to the normal Moore space problem

Proceedings of the American Mathematical Society, 1980

The Product Measure Extension Axiom (PMEA), whose consistency would follow from the existence of ... more 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.

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Research Announcement: THE TOPOLOGICAL STRUCTURE OF THE TANGENT AND COTANGENT BUNDLES ON THE LONG LINE

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Monotone Normality and Paracompactness in Scattered Spaces

Annals of the New York Academy of Sciences, 1989

... for a locally compact scattered space of countable derived length that hereditary paracompact... more

Normality in X 2 for Compact X

Transactions of the American Mathematical Society, 1993

In 1977, the second author announced the following consistent negative answer to a question of Ka... more In 1977, the second author announced the following consistent negative answer to a question of Katêtov: Assuming MA+-GH , there is a compact nonmetric space X such that X1 is hereditarily normal. We give the details of this example, and construct another example assuming CH . We show that both examples can be constructed so that X2\A is perfectly normal. We also construct in ZFC a compact nonperfectly normal X such that X2\A is normal.

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Spaces with bases of countable rank

General Topology and its Applications, 1978

Countable tightness and proper forcing

Bulletin of the American Mathematical Society, 1988

One of the most basic and natural generalizations of first countability is countable tightness: t... more One of the most basic and natural generalizations of first countability is countable tightness: the condition that, whenever^ is in the closure of A, there is a countable subset B of A such that x 6 B. Countably tight spaces include sequential spaces, i.e., those in which closure is obtainable by iteration of the operation of taking limits of convergent sequences. The two classes are distinct, since there are easy examples of countable, nondiscrete spaces with only trivial convergent sequences. On the other hand, it was long a major unsolved problem whether every compact Hausdorff space of countable tightness is sequential. First posed in [1] and motivated by the main results of [2], it gained importance from subsequent discoveries on the strong structural properties enjoyed by compact sequential spaces (see [3] and its references). A negative answer was shown to be consistent by Ostaszewski [4], who used Gödel's Axiom of Constructibility (V = L) to construct a countably compact space X whose one-point compactification is countably tight; of course, no sequence from X can converge to the extra point. Now (Theorem 2) we have shown that a positive answer follows from the Proper Forcing Axiom (PFA), introduced in [5]. Our research has uncovered many other striking consequences of PFA, numbered below. None was known to be consistent until now, nor was the following consequence of Corollary 1 and Theorem 3: the ordinal space oj\ embeds in every first countable, countably compact, noncompact Ti space (in particular, in every countably compact nonmetrizable T<i manifold), assuming PFA. This remains true if "first countable" is weakened to "character < ui," meaning every point has a local base of cardinality < u)i. This leads to a remarkable structure theorem for regular, countably compact spaces of character < ui (e.g. the product space [0, l] Wl): under PFA, the closure of every set A can be taken by first adjoining all limits of convergent sequences, and then adjoining to the resulting set A" all points x for which there is a copy W of w\ in A~ such that W U {x} is homeomorphic to ui + 1. Although large cardinals are needed to prove the consistency of PFA, all our PFA results are consistent if ZF is consistent. This is established by using u^-p-i-C-[6, Chapter VIII] posets and a ground model with a o W2-sequence to capture approximations to possible counterexamples in a countable support

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On first countable, countably compact spaces III: The problem of obtaining separable noncompact examples Peter Nyikos1

Classic Problems III

Topology and its Applications

Peter Nyikos

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