Densely k-separable compacta are densely separable

摘要

A space has sigma-compact tightness if the closures of sigma-compact subsets determine the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between pi-weight and the density of all dense subsets, or more precisely, the cardinal invariant delta(X). (C) 2020 Elsevier B.V. All rights reserved.