A combinatorial characterization of second category subsets of X^ω

Abstract

Let a finite non-empty X is equipped with discrete topology. We prove that S \subseteq X^ωis of second category if and only if for each f:ω-> \bigcup_{n \in ω} X^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i).