Unbounded Filters on ω

Abstract

Publisher Summary This chapter reviews unbounded filters on w = {0, 1, 2 . . .}. Filters will be non-principal and the Frechet filter F = {a ⊆ w: w–a is finite} will be a subset of filters. The study of these objects has given answers to many interesting problems in descriptive set theory and in the theory of forcing. The chapter describes the connection between filters on w and sets of real numbers, which are not Lebesgue messurable, do not have the Baire property, are not Ramsey and Kτ-regular. These properties involve a different characterization of the real line. In the case of the Lebesgue measure and the Baire property, the set of reals will be the set 2 w , the set of all w-sequences of 0's and 1's. The Lebesgue measure for 2w is obtained by taking the product measure over the equidistributed probability for {0, l} = 2. The topology for 2w is achieved by taking the product topology for the discrete space 2.