  Let $\Gamma=\prod_{i\in\mathbb{N}}\Gamma_i$ be an infinite product of
nontrivial finite groups, or let $\Gamma=(\mathbb{R}/\mathbb{Z})^n$ be a
finite-dimensional torus. For a countably infinite set $S\subseteq \Gamma$, an
$\aleph_0$-domatic partition is a partial $\aleph_0$-coloring on $\Gamma$ such
that the $S$-neighborhood $S\cdot x$ of every vertex $x\in \Gamma$ contains at
least one instance of each color. We show that an open $\aleph_0$-domatic
partition exists, iff a Baire measurable $\aleph_0$-domatic partition exists,
iff the topological closure of $S$ is uncountable. We also investigate domatic
partitions in the general descriptive combinatorics setting.
