On the minimal cardinality of a subset of R which is not of first category

Abstract

Let M be the ideal of first category subsets of R and non(M)=min{card X: X \subseteq R, X \not\in M}. We consider families \Phi of sequences converging to \infty, with the property that for every open set U \subseteq R that is unbounded above there exists a sequence belonging to \Phi, which has an infinite number of terms belonging to U. We present assumptions about \Phi which imply that the minimal cardinality of \Phi equals non(M).