  Ziegler introduced the idea of a good partition $\{X_{p}:p\in P\}$ of a
$T_{3}$-topological space, where $P$ is a finite partially ordered set,
satisfying $\overline{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$.
Good partitions of Stone spaces arise naturally in the study of
$\omega$-categorical structures, and a key concept for studying them is that of
a $p$-trim open set which meets precisely those $X_{q}$ for which $q\geqslant
p$. This paper develops the theory of infinite partitions of Stone spaces
indexed by a poset where the trim sets form a neighbourhood base for the
topology. We study the interplay between order properties of the poset and
topological properties of the partition, examine extensions and completions of
such partitions, and derive necessary and sufficient conditions on the poset
for the existence of the various types of partition studied. We also identify
circumstances in which a second countable Stone space with a trim partition
indexed by a given poset is unique up to homeomorphism, subject to choices on
the isolated point structure and boundedness of the partition elements. One
corollary of our results is that there is a partition $\{X_{r}:r\in[0,1]\}$ of
the Cantor set such that $\overline{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{
for all }r\in[0,1]$.
