Dense-Kernel and Closed-Core Reductions in the D-space Problem for Charming Spaces

Abstract

Let X be a charming space with a Lindelof Sigma kernel Y, and let B be the closure of Y in X. We show that the question whether every charming space is a D-space can be reduced first to the dense-kernel case and then to a closed core. We define Obs_cc(Y,B) as the set of boundary points x in B minus Y such that, for every open neighborhood U of x in X, the intersection of U and Y is not countably compact, and let H be the closure of Obs_cc(Y,B) in B. We prove that the intersection of H with B minus Y is exactly Obs_cc(Y,B), and our main reduction theorem shows that X is a D-space if and only if H is. We also prove the following sufficient condition. If there is a closed set S contained in B minus Y with compact covering number less than the dominating number d, such that every point of B minus Y minus S has an open neighborhood U in X for which the intersection of U and Y is countably compact, then X is a D-space.