Corollary 4 (Zariski)

Let $k$ be a field and $A$ a $k$-algebra essentially of finite type or a complete noetherian local ring. The set of prime ideals $\mathfrak{p}$ of $A$ such that the local ring $A_{\mathfrak{p}}$ is regular is open in $\mathrm{Spec}(A)$.

It is enough to apply Remark 1, taking for $k_0$ the prime subfield of $k$.