  This is the first in a sequence of three papers in which we prove the
following generalization of Thomassen's 5-choosability theorem: Let $G$ be a
finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored,
where $L$ is a list-assignment for $G$ in which every vertex has a 5-list
except for a collection of pairwise far-apart components, each precolored with
an ordinary 2-coloring, as long as the face-width of $G$ is $2^{\Omega(g)}$ and
the precolored components are of distance $2^{\Omega(g)}$ apart. This provides
an affirmative answer to a generalized version of a conjecture of Thomassen and
also generalizes a result from 2017 of Dvo\v{r}\'ak, Lidick\'y, Mohar, and
Postle about distant precolored vertices.
