Compact Gδ sets

In this paper we study spaces in which each compact subset is a Gδ-set and compare them to H.W. Martin's c-semi-stratifiable (CSS) spaces, i.e. spaces in which compact sets are Gδ-sets in a uniform way. We prove that a (countably) compact subset of a Hausdorff space X is metrizable and a Gδ-subset of X provided X has a δθ-base, or a point-countable, T1-point-separating open cover, or a quasi-Gδ-diagonal. We also show that any compact subset of a Hausdorff space X having a base of countable order must be a Gδ-subset of X and note that this result does not hold for countably compact subsets of BCO-spaces. We characterize CSS spaces in terms of certain functions g(n,x) and prove a “local implies global” theorem for submetacompact spaces that are locally CSS. In addition, we give examples showing that even though every compact subset of a space with a point-countable base (respectively, of a space with a base of countable order) must be a Gδ-set, there are examples of such spaces that are not CSS. In the paper's final section, we examine the role of the CSS property in the class of generalized ordered (GO) spaces. We use a stationary set argument to show that any monotonically normal CSS space is hereditarily paracompact. We show that, among GO-spaces with σ-closed-discrete dense subsets, being CSS and having a Gδ-diagonal are equivalent properties, and we use a Souslin space example due to Heath to show that (consistently) the CSS property is not equivalent to the existence of a Gδ-diagonal in the more general class of perfect GO-spaces.