0 N ov 1 99 9 A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category

Let M denote the ideal of first category subsets of R. We prove that min{card X : X ⊆ R,X 6∈ M} is the smallest cardinality of a family S ⊆ {0, 1} with the property that for each f : ω −→ ⋃ n∈ω{0, 1} n there exists a sequence {an}n∈ω belonging to S such that for infinitely many i ∈ ω the infinite sequence {ai+n}n∈ω extends the finite sequence f(i). We inform that S ⊆ {0, 1} is not of first category if and only if for each f : ω −→ ⋃ n∈ω{0, 1} n there exists a sequence {an}n∈ω belonging to S such that for infinitely many i ∈ ω the infinite sequence {ai+n}n∈ω extends the finite sequence f(i). LetM denote the ideal of first category subsets ofR. LetM({0, 1}) denote the ideal of first category subsets of the Cantor space {0, 1}. Obviously: (∗) non(M) : = min{card X : X ⊆ R, X 6∈ M} = min{card X : X ⊆ {0, 1}, X 6∈ M({0, 1})} Let ∀ abbreviate ”for all except finitely many”. It is known (see [1], [2] and also [3]) that: non(M) = min{card F : F ⊆ ω and ¬ ∃g ∈ ω ∀f ∈ F ∀k g(k) 6= f(k)} Mathematics Subject Classification 2000. Primary: 03E05, 54A25; Secondary: 26A03.