On the Covering and the Additivity Number of the Real Line

We show that the real line R cannot be covered by k many nowhere dense sets iff whenever D = {D¡ : i e k} is a family of dense open sets of R there exists a countable dense set G of R such that \G\D¡\ < co for all i e k. We also show that the union of k meagre sets of the real line is a meagre set iff for every family D = {D¡ : i £ k} of dense open sets of R and for every countable dense set G of R there exists a dense set Q ç G such that \Q\Di\ <w for all i£k. 1. Notation and terminology The notation and terminology which we will use is standard and can be found in [8] or [9]. In particular, if A, B are sets and X any cardinal finite or infinite, then [A]x , [A]<Á , and [A]-À denote the sets of all subsets of A of size X, < X, and < X respectively, co03 denotes the Baire space, i.e., the set of all functions uco together with the topology, having as a base the collection of all (clopen) sets of the form [p] = {f£œco:pçf}, p£co<0J = lJ{nco:n£co}. We say that the family A C [co]03 has the countable n-base property (C7tbp), iff there exists a F g [[ca]Cí,]CÜ, call it a n-base, such that (Va G A)Cyb £ B)(3d £ B)(d Qanb). In order to avoid confusion, let us remark that the notions of orbp and a filter A of [co]01 having a countable base are not the same. B need not be included in A. In fact, B may contain disjoint sets. A ç [co]03 has the strong finite intersection property (sfip), iff | f| Q\ co for every Q £ [A]<(0. We say that a set S £ [co]w is an infinite pseudointersection of the family A ç [co]03 iff (Va G A)(\S\a\ < co). Received by the editors April 1, 1993 and, in revised form, September 10, 1993; presented to the 3rd Greek Conference on Mathematical Analysis held in Ioannina, May 28-29, 1993. 1991 Mathematics Subject Classification. Primary 03E35; Secondary 03E40.