Selection principles in topology: New directions

Most mathematical theories start from humble beginnings where a mathematician while examining some mathematical object makes fundamental, but ad hoc, observations. When the significance of these observations are realized some organised study begins during which these are explored in detail for some central examples. In this second part of theory building some general themes of investigation are identified. Then, it seems, one enters a third phase in which several mathematicians join in developing the theory. Typically this third phase is marked with much innovation, rapid expansion and application to other areas of mathematics. This brief discussion of the area of Selection principles in Topology will be organized according to these three phases. 1 The beginnings Selection principles appear in diagonalization arguments in mathematics. The early sources for selection principles in topology come from measure theory and basis theory in metric spaces. Basis theoretic sources In [26] K. Menger defined the Menger basis property for metric spaces: A met/-ric space (X, d) has the Menger basis property if there is for each base B of X a sequence (Bn : n ∈ N) of sets from the basis such that limn→∞ diamd(Bn) = 0, and {Bn : n ∈ N} is a cover for X. Though at first glance this concept does not seem to be related to diagonalization, evidence that it is came quickly when in [17] W. Hurewicz proved two fundamental theorems about the Menger basis property. The first of the two is that the metric space (X, d) has the Menger basis property if, and only if, for each sequence (Un : n ∈ N) of open covers of X there is a sequence (Vn : n ∈ N) 1Invited talk at the IMC “Filomat 2001”, Nǐs, August 26–30, 2001 2000 Mathematics Subject Classification: 54D20, 54B20, 54C35, 03E02, 91A44