Marion Scheepers

Combinatorics of open covers I: Ramsey theory

Abstract We study several schemas for generating from one sort of open cover of a topological space a second sort of open cover. Some of these schemas come from classical literature, others are borrowed from the theory of ultrafilters on the set of positive integers. We show that the fact that such a schema actually succeeds in producing a cover imposes strong combinatorial structure on the family of open covers of a certain sort. In particular, we show that certain analogues of Ramseys theorem characterize some of these circumstances.

The Combinatorics of Open Covers

The combinatorics of open covers is a study of Cantor’s diagonal argument in various contexts. The field has its roots in a few basic selection principles that arose from the study of problems in analysis, dimension theory, topology and set theory. The reader will also find that some familiar works are appearing in new clothes in our survey. This is particularly the case in connection with such problems as determining the structure of compact scattered spaces and a number of classical problems in topology. We hope that the new perspective in which some of these classical enterprises are presented will lead to further progress. In this article we also attempt to give the reader an overview of the problems and techniques that are currently fueling much of the rapidly increasing current activity in the combinatorics of open covers.

Combinatorics of Open Covers VI: Selectors for Sequences of Dense Sets

Abstract We consider the following two selection principles for topological spaces: Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space; Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space. We show that for separable metric space X one of these principles holds for the space Cp (X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was st...

The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets

We prove the following theorems:(1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property S0.(2) If X has Hurewiczs covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set.(3) If X has strong measure zero and Hurewiczs covering property then its algebraic sum with any set in is a set in . ( is included in the class of sets always of first category, and includes the class of strong first category sets.)These results extend: Fremlin and Millers theorem that strong measure zero sets having Hurewiczs property have Rothbergers property, Galvin and Millers theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszyfnski and Judahs characterization of -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts.

Function spaces and a property of Reznichenko

Abstract In this paper we show that for a set X of real numbers the function space C p ( X ) has both a property introduced by Sakai in [Proc. Amer. Math. Soc. 104 (1988) 917–919] and a property introduced by Reznichenko (see [Topology Appl. 104 (2000) 181–190]) if and only if all finite powers of X have a property that was introduced by Gerlits and Nagy in [Topology Appl. 14 (1982) 151–161]. It follows that the minimal cardinality of a set of real numbers for which the function space does not have the properties of Sakai and Reznichenko is equal to the additivity of the ideal of first category sets of real numbers.

Combinatorics of open covers (V): Pixley–Roy spaces of sets of reals, and ω-covers

Daniels (1988) started an investigation of the duality between selection hypotheses for XR and selection hypotheses for the Pixley‐Roy space ofX. Daniels, Kunen and Zhou (1994) introduced the “open‐open game”. We extend some results of Daniels (1988) by connecting the relevant selection hypotheses with game theory (Theorems 2, 3, 14 and 15) and Ramsey theory (Theorem 10, Corollary 11, Theorem 23 and Corollary 24). Our results give answers to some of the questions asked by Daniels et al. (1994).

Finite Powers of Strong Measure Zero Sets

In a previous paper-[17]-we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove: 1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3). 2. A subspace X of a a-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9). 3. A subspace X of a a-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12). Then 2 and 3 yield characterizations of strong measure zeroness for a-totally bounded metric spaces in terms of Ramseyan theorems. ?

Cp(X) and Arhangel'skiǐ's αi-spaces

Abstract Nogura showed that whereas Arhangelskiǐs properties α1, α2 and α3 are preserved by finite products, the property α4 is not. It is shown here that for each space X the properties α2, α3 and α4 are the same for the function space C p (X) . As a consequence, α4 is closed under finite products of such function spaces.

The length of some diagonalization games

Abstract. For X a separable metric space and

OPEN COVERS AND PARTITION RELATIONS

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