Suppose $\chi$ is a scheme of finite type over the field $k$; define $\overline{\chi} := \chi \otimes_{\mathrm{Spec}(k)} \mathrm{Spec}(\overline{k})$, with $\overline{k}$ an algebraic closure of $k$. Then with $G$ the absolute Galois group $\mathrm{Aut}(\overline{k}/k)$, the closed points of $\chi$ correspond to the $G$-orbits in the set of closed points of $\overline{\chi}$.
Is there a similar connection between non-closed points of $\chi$ and $\overline{\chi}$ ?
Is there a similar connection between the other closed sets of $\chi$ and $\overline{\chi}$ ?
