Topological aspects of poset spaces (Q977107)

For any poset \(P\), let \(\mathrm{UF}(P)\) denote the set of unbounded filters on \(P\) topologized with the basis \(\{ N_p\mid p\in P\}\), where \(N_p=\{F\in \mathrm{UF}(P)\mid p\in F\}\), and let \(\mathrm{MF}(P)\) denote the set of maximal filters on \(P\) topologized with the subspace topology inherited from \(\mathrm{UF}(P)\). These spaces are referred to as \textit{poset spaces} and include (up to homeomorphism) all complete metric spaces and all locally compact Hausdorff spaces. The motivation for them arose in recent work in mathematical logic. This paper studies the general topology of poset spaces in some detail.