Let $\mathbf A$ be a countable model of a countable first-order language $L$, and $T$ be a first-order theory of a countable expansion $L'\supseteq L$. Let $\mathcal S$ denote the set of all expansions of $\mathbf A$ to $L'$ that are models of $T$. It is proved that $\mathcal S$ can be embedded into a metric Stone space as a $G_\delta$ subset, and therefore $k=|S|$ satisfies CH, i.e. either $k\leq\aleph_0$ or $k=2^{\aleph_0}$. Several examples that illustrates this theorem are presented, too.