The combinatorial study of subsets of the set N of natural numbers and of functions from N to N leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many infinite subsets X of N must I take so that every subset Y of N or its complement includes one of my X’s? Or how many functions f from N to N must I take so that every function from N to N is majorized by one of my f’s? The main results about these cardinal characteristics of the continuum are of two sorts: inequalities involving two (or sometimes three) characteristics, and independence results saying that other such inequalities cannot be proved in ZFC. Other results concern, for example, the cofinalities of these cardinals or connections with other areas of mathematics. This survey concentrates on the combinatorial set-theoretic aspects of the theory.

Combinatorial Cardinal Characteristics of the Continuum

The Combinatorics of Open Covers

Topological Homogeneity.- Some Recent Progress Concerning Topology of Fractals.- A biased view of topology as a tool in functional analysis.- Large scale versus small scale.- Descriptive aspects of Rosenthal compacta.- Minimality conditions in topological groups.- Set-Theoretic update on Topology.- Topics in Dimension Theory.- Representations of dynamical systems on Banach spaces.- Generalized metrizable spaces.- Permanence in Coarse Geometry.- Selections and Hyperspaces.- Continuum Theory.- Almost disjoint families and topology.- Some Topics in Geometric Topology II.- Topological aspects of dynamics of pairs, tuples and sets.- Continuous selections of multivalued mappings.- The combinatorics of open covers.- Covering properties.- Paratopological and semitopological groups vs topological groups.

Recent progress in general topology.

On subgroups of minimal topological groups

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit interval I = [0,1] into itself. More recently, the notion of Baire class-1 function turned out to also be important in some areas of functional analysis (see [R3]). For example, Odell and Rosenthal [OR] showed that the double dual of a separable Banach space E with the weak* topology consists only of Baire class-1 functions defined on the unit ball of E* if and only if the space E contains no subspace isomorphic to ?1. This resulted in a renewed interest in this class of spaces. For example, building on the work of Rosenthal [R2], Bourgain, FYemlin and Talagrand [BFT] proved analogues of the two theorems of Helly for the whole first Baire class. Using their results Godefroy [Go] showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space K is representable as a compact set of Baire class-1 functions, then so is P(K), the space of all Radon probability measures on K with the weak* topology. Some further permanence properties of this class of spaces were obtained by Marciszewski ([Ml], [M2]) and an excellent survey of the early results is given by R. Pol [Po2]. Our paper is an attempt towards a fine structure theory of compact subsets of first Baire class. The first result that we give is a positive answer to a natural question one usually asks in such a context.

/pdf/compact-subsets-of-the-first-baire-class-3g8q0g20mk.pdf

Compact subsets of the first Baire class

This is a cumulative status report on the 1100 problems listed in the volume Open Problems in Topology (North-Holland, 1990), edited by J. van Mill and G.M. Reed.

G.M. Reed

Papers

Open Problems in Topology