Summary: The paper aims 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 [D. Macpherson, D. Marker and C. Steinhorn, Trans. Am. Math. Soc. 352, No. 12, 5435–5483 (2000; Zbl 0982.03021)] 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 \(X \times Y\), then for a large set of tuples \(\langle \overline{a}_{1},\dots ,\overline{a}_{2^{k}}\rangle \in X^{2^{k}}\), where \(k=\dim(Y)\), the union of fibers \(S_{\overline{a}_{1}}\cup \cdots \cup S_{\overline{a}_{2^{k}}}\) is large in \(Y\). Finally, given a weakly o-minimal structure \(\mathcal M\), we find various conditions equivalent to the fact that the topological dimension in \(\mathcal M\) enjoys the addition property.