Let $\Lambda$ be a lattice in $(\mathbb{R}^n,+)$, $T=\mathbb{R}^n/\Lambda$ a torus, and $\pi\colon \mathbb{R}^n\to T$ the quotient map. For a family $(X_s,s\in S)$ of subsets of $\mathbb{R}^n$ definable in an o-minimal an structure on the reals, we consider Hausdorff limits of the family $(\pi(X_s),s\in S)$ and describe when a limit is the whole $T$. The main goal of this talk is to demonstrate model-theoretic tools involved.