John Kulesza

An example in the dimension theory of metrizable spaces

Abstract A dense G δ (completely) metrizable subspace Z of ω ω 1 is described which satisfies Ind( Z ) = 1 while (trivially) ind( Z ) = 0. Necessarily, Z has weight equal to ℵ 1 . Furthermore, it is shown that if C is a separable closed subspace of Z , and K is any other closed subspace of Z disjoint from C , then there is a clopen set O with C ⊂ O ⊂ Z ⧹ K .

Metrizable spaces where the inductive dimensions disagree

A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roys technique, these spaces can often be shown to have positive large inductive dimension. Examples of N-compact, complete metrizable spaces with ind = 0 and Ind = 1 are provided, answering questions of Mrowka and Roy. An example with weight c and positive Ind such that subspaces with smaller weight have Ind = 0 is produced in ZFC. Assuming an additional axiom, for each cardinal A a space of positive Ind with all subspaces with weight less than A strongly zero-dimensional is constructed.

On weak compactness and countable weak compactness in fixed point theory

We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk. Let (X, d) be a metric space. Let F be the family of subsets of X consisting of X and sets which are complements of closed balls of X . The weak topology (also called ball topology) on X is the topology whose open sets are generated by F , i.e. F forms a subbase for the open sets of X . If X is a subset of a Banach space, the weak topology defined in this sense is generally weaker than the usual weak topology on X . For details on this topology in a Banach space setting, see [1] and [2]. We say X is weakly compact if it is compact in the weak topology. A subset of X is called a ball-intersection if it is an intersection of closed balls. X is said to have normal structure if every ball-intersection D of X containing more than one point contains a non-diametral point, i.e. a point r ∈ D such that sup{d(r, x) : x ∈ D} < diam(D). It is clear that X is weakly compact if and only if every nonempty family of ballintersections with the finite intersection property has a nonempty intersection. And by the Alexander subbase theorem, one can replace ‘ball-intersections’ with ‘closed balls’. X is countably weakly compact if every decreasing sequence of nonempty ball-intersections in X has a nonempty intersection. Obviously every weakly compact metric space is countably weakly compact. In this paper we show that the converse is false. However, if the metric space has normal structure, the converse is true. This shows that the assumption of countable weak compactness in some theorems in the literature is not really weaker than that of weak compactness. It also shows that every complete metric space with uniform normal structure is weakly compact, thereby a fixed point theorem of Khamsi [3] actually follows from that of Kirk [4]. Throughout this paper, Br(p) will denote the closed ball centered at p with radius r. Received by the editors January 3, 1995 and, in revised form, March 27, 1995. 1991 Mathematics Subject Classification. Primary 47H10, 47H09; Secondary 54E50, 54D30.

Connectifications of metrizable spaces

Abstract We answer a question of Alas, Tkacenko, Tkachuk, and Wilson by constructing a metrizable space with no compact open subsets which cannot be densely embedded in a connected metrizable (or even perfectly normal) space. We also obtain a result that implies that every nowhere locally compact metrizable space can be densely embedded in a connected metrizable space.

Separation in Ψ-spaces

Abstract A Ψ -space is the topological space usually associated with a maximal almost disjoint family of subsets of the integers. In this paper, we study properties that an infinite set of nonisolated points in a Ψ -space might possess. These properties are 2-embeddedness, C ∗ -embeddedness, and another weaker property, that of being solidly normalized. It is known from [2,3] that the existence of a Ψ -space with an infinite set of nonisolated points with any of these properties is independent of ZFC. Here we show that these three properties are distinct by providing examples, assuming less than Martins axiom, of Ψ -spaces with infinite sets of nonisolated points which in one case are C ∗ - but not 2-embedded, and in the other case solidly normalized but not C ∗ -embedded. Additionally, Martins axiom implies the existence of a Ψ -space with a set S of nonisolated points of cardinality c such that every subset of S with cardinality less than c is 2-embedded.

More examples of metric spaces where the inductive dimensions disagree

We give three examples of metric spaces where the inductive dimensions disagree. The two main examples are both N-compact. The first has closed sets which are not clopen Borel (Borel in the σ-algebra generated by the clopen sets). The second has weight ω1 and, assuming all sets of cardinality ω1 in the interval are Q-sets, contrasts the first by having all closed sets clopen Borel. The third example provides, for each α with ω1⩽α⩽c, a metric space of weight α with noncoinciding dimensions for which all subsets of weight less than α are strongly zero-dimensional. Each example answers a question posed by Mrowka.

Some problems from George Mason University

Publisher Summary This chapter discusses some of Kuleszas problems, Levys problems, and Matveevs problems raised at George Mason university. The behavior of dimension for nonseparable metric spaces is not well understood, despite having been studied for well over half a century; there are no analogs for several important theorems regarding dimension in separable spaces and generally the results and examples are quite complicated. Almost any new theorem or example relating to the covering dimension dim would be interesting; there are several problems in the chapter that are of interest. The focus is on two fundamental problems that remain largely unsolved. The relatively recent remarkable example νμ0 of Mrowka gives a consistent solution to one of the great problems in dimension theory. Its finite powers give examples of metric spaces for which dim-ind, the discrepancy between covering and the small inductive dimension, can be any positive integer. The chapter elaborates problems in dimension theory of nonseparable metric spaces. A question about weak P-points is also discussed in the chapter.

A counterexample to the extension of a product theorem in dimension theory to the noncompact case

Abstract We produce, for each n > 0, a subspace X n of R 2 n + 1 which does not embed in R 2 n and whose square has dimension n . This shows that a recent theorem about products of compact spaces cannot be extended to noncompact spaces.

Remainders of normal spaces

Abstract Fleissner, W., J. Kulesza and R. Levy, Remainders of normal spaces, Topology and its Applications 49 (1993) 167-174. If X is totally compact, then X is the Stone-tech remainder of a normal space. A partial converse: if X is first countable and the Stone-tech remainder of a normal space, then X is locally compact. Every metric space, but not every first countable space, is the remainder of a normal space. For countable spaces, or even countable spaces which are locally compact except at one point, there are examples, but few theorems. We show that a construction of Porter and Woods applies to certain examples only if b = b. We investigate the property that all normal images are compact, and show that large products minus small subsets have this property. Keywords: Remainder, normal, locally compact, countable type, Stone-tech, Z-product, count- able spaces.

The dimensional of a subset of lp as p varies

Abstract We investigate how the dimension of a set X contained in lp can change as p is varied. Assume 1 ⩽ s dimr X, then dimp X = dimr X + 1, dimq X = dimr X, and X is not a subset of ls. Another example shows that it is possible that dimr X > dimp X. It is shown, for example, that the dimension of the rational points in l2 is zero when this set is viewed as a subset of lp where p > 2. There is a positive dimensional closed subset of lp whose projection onto each coordinate axis is a two point set; this subset admits a natural topological group structure.