Finite powers of strong measure zero sets

Abstract

AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspaceXof a σ-compact metric spaceYhas strong measure zero if, and only if, a certain Ramseyan partition relation holds forY(Theorem 9).3. A subspaceXof a σ-compact metric spaceYhas strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds forY(Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.