Let \(\{X_s \colon x\in S\}\) be a family of subsets of \({\mathbb R}^n\) definable in some o-minimal expansion of the real field.
Let \(\Gamma \subseteq {\mathbb R}^n\) be a lattice and \(\pi \colon {\mathbb R}^n/\Gamma \to \mathbb T\) be the quotient map.
In a series of papers (published and unpublished) together with Y. Peterzil we considered Hausdorff limits of the family \(\{ \pi(X_s)\colon s\in S\}\) and provided their description.
In this talk I describe model theoretic tools used in the description.