A ${\bf Z}$-affine structure (of dimension $n$) on a Hausdorff topological space $Y$ is a subsheaf $Aff_{{{\bf Z}},Y}$ of the sheaf of continuous functions on $Y$, such that the pair $(Y,Aff_{{{\bf Z}},Y})$ is locally isomorphic to $({{\bf R}}^{n},Aff_{{\bf Z},{{\bf R}}^{n}})$.