Topological properties of sets definable in weakly o-minimal structures1

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [MMS] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y , then for a large set of tuples 〈a1, . . . , a2k〉 ∈ X k , where k = dim(Y ), the union of fibers Sa1 ∪ . . . ∪ Sa2k is large in Y . Finally, given a weakly o-minimal structure M, we find various conditions equivalent to the fact that the topological dimension in M enjoys the addition property.