Mazur's First Conjecture on rational points of varieties

Let $V$ be any variety over $\mathbb{Q}$, and let $V(\mathbb{Q})$ and $V(\mathbb{R})$ denote its rational and real points, respectively. Consider the topological closure of $V(\mathbb{Q})$ in $V(\mathbb{R})$. Mazur's First Conjecture. This closure has at most finitely many connected components. The conjecture is one of Mazur's proposed routes toward showing that the integers do not have a Diophantine definition over the rationals; it remains unresolved.