Peter J. Nyikos University of South Carolina, Columbia
University of South Carolina, Columbia
Geometry and Topology, Logic and Foundations of Mathematics
Ph.D. CarnegieMellon University 1971
Publications

Article: Dowker spaces revisited
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ABSTRACT: In 1951, Dowker proved that a space X is countably paracompact and normal if and only if X ×I is normal. A normal space X is called a Dowker space if X × I is not normal. The main thrust of this article is to extend this work with regards αnormality and βnormality. Characterizations are given for when the product of a space X and (ω + 1) is αnormal or βnormal. A new definition, αcountably paracompact, illustrates what can be said if the product of X with a compact metric space is βnormal. Several examples demonstrate that the product of a Dowker space and a compact metric space may or may not be αnormal or βnormal. A collectionwise Hausdorff Moore space constructed by M. Wage is shown to be αnormal but not βnornal.  [Show abstract] [Hide abstract]
ABSTRACT: The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α4 Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α4, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α3 Fréchet spaces and its subclass of ℵ0bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α3 and Fréchet and being ℵ0bisequential.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate some combinatorial statements that are strong enough to imply that fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in settheoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindelöf spaces posed by Arhangel'skiǐ and Buzyakova (1998).  [Show abstract] [Hide abstract]
ABSTRACT: We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or fullcardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ1), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are αnormal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily βnormal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be αnormal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be βnormal; (3) hereditarily cwH spaces which fail to be αnormal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, nonnormal space. The ZFCindependence of several statements involving the hereditarily wD(ℵ1) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω1.  [Show abstract] [Hide abstract]
ABSTRACT: If X is a separable 0dimensional metrizable space in which every compact subset is countable, then C(X) with the compactopen topology is stratifiable iff X is scattered. This answers a question of Gruenhage and lends credence to a conjecture of Gartside and Reznichenko. 
Chapter: Classes of compact sequential spaces
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ABSTRACT: Without Abstract 
Article: A nonmetrizable collectionwise Hausdorff tree with no uncountable chains and no Aronszajn subtrees
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ABSTRACT: It is independent of the usual (ZFC) axioms of set theory whether every collec tionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add "or has an Aronszajn subtree," the statement remains ZFCindependent. This is done by constructing a tree as in the title, using the settheoretic hypothesis ♦∗, which holds in Godel's Constructible Universe.  [Show abstract] [Hide abstract]
ABSTRACT: We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of ω 1 with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize the iteration theorems appearing in Chapters V and VIII of [19] as well as the iteration theorem appearing in [9]. We close the paper with a ZFC example (constructed using Shelah's club–guessing sequences) that shows similar results do not hold for closed pre–images of ω 2 . 
Article: ω 1 compactness in type I manifolds
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ABSTRACT: In this paper we establish that any wellpruned ω1tree, T, admits an ω1compact type I manifold if T does not contain an uncountable antichain. If T does contain an uncountable antichain, it has been shown that whether or not T admits an ω1compact manifold is undecidable in ZFC. 
Article: Various topologies on trees
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ABSTRACT: This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in settheorist fashion as partially ordered sets in which the elements below each element are wellordered. A number of different topologies on trees are treated, some at considerable length. Two sections deal in some depth with the coarse and fine wedge topologies, and the interval topology, respectively. The coarse wedge topology gives a class of supercompact monotone normal topological spaces, and the fine wedge topology puts a monotone normal, hereditarily ultraparacompact topology on every tree. The interval topology gives a large variety of topological properties, some of which depend upon settheoretic axioms beyond ZFC. Many of the open problems in this area are given in the last section. 

Article: Applications of some strong settheoretic axioms to locally compact T 5 and hereditarily scwH spaces
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ABSTRACT: Under some very strong settheoretic hypotheses, hereditarily normal spaces (also referred to as T5 spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω1compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ א1. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact subspaces in certain continuous preimages of ω1. It also exposes (Theorem 2) the fine structure of perfect preimages of ω1 which are T5 and hereditarily collectionwise Hausdorff. In these theorems, "T5 and hereditarily collectionwise Hausdorff" is weakened to "hereditarily strongly collectionwise Hausdorff." Corollaries include the consistency, modulo large cardinals, of every hereditarily strongly collectionwise Hausdorff manifold of dimension > 1 being metrizable. The concept of an alignment plays an important role in formulating several of the structure theorems.  [Show abstract] [Hide abstract]
ABSTRACT: Strong αfavorability of the compactopen topology on the space of continuous functions, as well as of the generalized compactopen topology on continuous partial functions with closed domains is studied.  [Show abstract] [Hide abstract]
ABSTRACT: A manifold is a connected Hausdorff space in which every point has a neighborhood homeomorphic to Euclidean nspace (n is unique). A space is collectionwise Hausdorff (cwH) if every closed discrete subspace D can be expanded to a disjoint collection of open sets each of which meets D in one point. There are exactly two examples of 1dimensional nonmetrizable hereditarily normal, hereditarily cwH manifolds: the long line and the long ray. The main new result is that if it is consistent that there is a supercompact cardinal, it is consistent that every hereditarily normal, hereditarily cwH manifold of dimension greater than 1 is metrizable.  [Show abstract] [Hide abstract]
ABSTRACT: This chapter focuses on a problem related to first countable, countably compact, noncompact spaces. This problem asks"Does ZFC imply the existence of a separable, first countable, countably compact, noncompact Hausdorff(T2) space?" The usual topology on ω1 satisfies everything except separability. The NovakTeresaka space described in Vaughan's article satisfies everything except first countability. If the cofinite topology on ω1 is refined by making initial segments open, then the resulting space satisfies everything except T2 and is T1. The remaining two properties are obviously necessary also to have an open problem. The problem mentioned is one of a small but growing number of topological problems for which a negative answer is known to entail (2ω =)c ≥א3, yet c =א3 has not been ruled out. The chapter discusses that a space X is ωbounded if every countable subset has compact closure, and strongly ωbounded if every σcompact subset has compact closure. The chapter describes basic concepts related to good spaces. It also provides details about other consistent constructions for the stated problem.  [Show abstract] [Hide abstract]
ABSTRACT: Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFCindependent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.  [Show abstract] [Hide abstract]
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 AlexandroffUrysohn 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. Stone's characterization of Baire's zerodimensional spaces is extended to give internal characterizations of all spaces γλ × D, where D is discrete and γλ has the initial agreement topology. The historical background for the AlexandroffUrysohn result is briefly surveyed. 
Article: Gently Killing Sspaces
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ABSTRACT: We produce a model of ZFC in which there are no locally compact first countable Sspaces, and in which 2^{aleph_0}<2^{aleph_1}. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size aleph_1, answering a question of the second author.  [Show abstract] [Hide abstract]
ABSTRACT: In 1975 E. K. van Douwen showed that if (Xn)n∈ω is a family of Hausdorff spaces such that all finite subproducts ∏n<m Xn are paracompact, then for each element x of the box product □n∈ωXn the σproduct σ(x) = {y ∈ □n∈ωXn : {n ∈ ω : x(n) ≠ y(n)} is finite} is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Theorem. Let κ be an infinite cardinal number, and let (Xα)α∈κ be a family of compact Hausdorff spaces. Let x ∈ □ = □α∈κXα be a fixed point. Given a family □ of open subsets of □ which covers σ(x), there exists an open locally finite in □ refinement S of □ which covers σ(x). We also prove a slightly weaker version of this theorem for Hausdorff spaces with "all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question. 
Article: Hereditary normality of γ N spaces
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ABSTRACT: A γNspace is a locally compact Hausdorff space with a countable dense set of isolated points, and the rest of the space homeomorphic to ω1. We show that under the Open Coloring Axiom (OCA) no γNspace is hereditarily normal. This is the key to showing that some sweeping statements are consistent with (and independent of) the usual axioms of set theory, including: 1.(1) Every countably compact, hereditarily normal space is sequentially compact.2.(2) Every separable, hereditarily normal, countably compact space is compact and FréchetUrysohn.3.(3) The arbitrary product of countably compact, hereditarily normal spaces is countably compact. Not all of these conclusions follow just from MA + ¬ CH: a forcing construction is given of a model of MA + c = κ where κ is any cardinal ⩾ ℵ2 satisfying κ = 2
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