On well-filtered spaces and ordered sets

A topological space is well-filtered if any filtered family of compact sets with intersection in an open set must have some member of the family contained in the open set. This well-known and important property automatically satisfied in Hausdorff spaces assumes a life of its own in the T0-setting. Our main results focus on giving general sufficient conditions for a T0-space to be well-filtered, particularly the important case of directed complete partially ordered sets equipped with the Scott topology.