The double density spectrum of a topological space

It is an interesting, maybe surprising, fact that different dense subspaces of even “nice” topological spaces can have densities. So, our aim here to investigate the set densities all a space X we call double density spectrum and denote by dd(X). We improve result from [1] showing dd(X) always ω-closed (i.e., countably closed) if Hausdorff. manage give complete characterizations spectra Hausdorff regular as follows. Let S be non-empty infinite cardinals. Then also prove number consistency results concerning compact spaces. For instance: