A combinatorial characterization of second category subsets of the

We prove that S ⊆ {0, 1} is of second 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). Let M denote the ideal of first category subsets of R. Let M({0, 1}) denote the ideal of first category subsets of the Cantor discontinuum {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)}. Theorem 1 yields information about sets S ⊆ {0, 1} with the following property (22): Mathematics Subject Classification 2000. Primary: 03E05, 54E52.