Stone space partitions indexed by a poset

Stone space partitions $$\{X_{p}\mid p\in P\}$$ satisfying conditions like $$\overline{X_{p}}=\bigcup _{q\leqslant p}X_{q}$$ for all $$p\in P$$ , where P is a poset or PO system (poset with distinguished subset), arise naturally in the study both of primitive Boolean algebras and $$\omega $$ -categorical structures. A key concept studying such that p-trim open set which meets precisely those $$X_{q}$$ $$q\geqslant p$$ ; spaces, this topological equivalent pseudo-indecomposable set. This paper develops theory infinite spaces indexed by trim sets form neighbourhood base topology. We interplay between order properties poset/PO partition, examine extensions completions partitions, derive necessary sufficient on existence various types partition studied. also identify circumstances second countable given unique up to homeomorphism, subject choices isolated point structure boundedness elements. One corollary our results there $$\{X_{r}\mid r\in [0,1]\}$$ Cantor $$\overline{X_{r}}=\bigcup _{s\leqslant r}X_{s}\text { }r\in [0,1]$$ .