Normal subspaces of the density topology

The density topology on the real line is a strengthening of the usual Euclidean topology which is intimately connected with the measure-theoretic structure.The space itself is not normal; we are interested in characterizing its normal subspaces.This leads us to the consideration of various set-theoretic axioms, and yields a consistent example of a homogeneous normal non-collectionwise Hausdorff space and indeed a general method for producing normal non-collectionwise Hausdorff spaces.(A space is collectionwise Hausdorff if for each closed discrete subset Y there exist pairwise disjoint open sets, one about each element of Y.)