Hereditary Weak Perfectness in Generalized Ordered Spaces

A space is weakly perfect if every closed subset contains a dense Gδ subset of the ambient space. Bennett,Hosobuchi and Lutzer asked whether weak perfectness is hereditary in generalized ordered (GO) spaces, observingthat it would suffice to prove preservation by dense subspaces. We prove in ZFC that every weakly perfect GO-spaceis hereditarily weakly perfect. The principal ingredient is a synchronized trace lemma for first-countable GO-spaces.A decreasing sequence of open sets with empty total intersection but dense trace on a closed set generates, throughconvex components of successive nowhere-dense complements, a point-finite family of open supports whose selectedpoints form a dense Gδ set. An ambient weak-perfect core supplies exactly such a decreasing sequence after thepart already captured by its trace on a dense subspace is removed. The isolated-point step is handled using thehereditary collectionwise Hausdorff property of GO-spaces. The dense-subspace theorem then yields the hereditaryconclusion.