Induced Topologies on the Poset of Finitely Generated Saturated Sets

Abstract

Abstract In [R. Heckmann, K. Keimel, Quasicontinuous Domains and the Smyth Powerdomain, Electronic Notes in Theoretical Computer Science 298 (2013), 215–232], Heckmann and Keimel proved that a dcpo P is quasicontinuous iff the poset Fin P of nonempty finitely generated upper sets ordered by reverse inclusion is continuous. We generalize this result to general topological spaces in this paper. More precisely, for any T 0 space ( X , τ ) and U ∈ τ , we construct a topology τ F generated by the basic open subsets U F = { ↑ F ∈ Fin X : F ⊆ U } . It is shown that a T 0 space ( X , τ ) is a hypercontinuous lattice iff τ F is a completely distributive lattice. In particular, we prove that if a poset P satisfies property DINTop, then P is quasi-hypercontinuous iff Fin P is hypercontinuous.