0 A combinatorial characterization of second category subsets of X ω

Let 2 ≤ cardX < ω and X is equipped with discrete topology. We prove that S ⊆ X is of second category if and only if for each f : ω → ⋃ n∈ω X 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). Theorem 1 yields information about sets S ⊆ X with the following property (2): (2) for each infinite J ⊆ ω and each f : J → ⋃ n∈ω X n there exists a sequence {an}n∈ω belonging to S such that for infinitely many i ∈ J the infinite sequence {ai+n}n∈ω extends the finite sequence f(i). Theorem 1. Assume that 2 ≤ cardX < ω and X is equipped with discrete topology. We claim that if S ⊆ X is of second category then S has the property (2). Proof. Let us fix f : J → ⋃ n∈ω X . Let Sk(f) (k ∈ ω) denote the set of all sequences {an}n∈ω belonging to X ω with the property that there exists i ∈ J such that i > k and the infinite sequence {ai+n}n∈ω extends Mathematics Subject Classification 2000. Primary: 03E05, 54E52.