TOPOLOGICAL PROPERTIES OF SETS DEFINABLE IN WEAKLY O-MINIMAL STRUCTURES

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 [2] 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 x Y. then for a large set of tuples <(a) over bar (1),...,(a) over bar (2k)> is an element of X(2k). where k = dim(Y), the union of fibers S((a) over bar1) boolean OR...boolean OR S((a) over bar 2k) is large in Y. Finally, given a weakly o-minimal structure M. we find various conditions equivalent to the fact that the topological dissension in M enjoys the addition property