  It is an interesting, maybe surprising, fact that different dense subspaces
of even "nice" topological spaces can have different densities. So, our aim
here is to investigate the set of densities of all dense subspaces of a
topological space $X$ that we call the double density spectrum of $X$ and
denote by $dd(X)$.
  We improve a result of Berner and Juhasz by showing that $dd(X)$ is always
$\omega$-closed (i.e. countably closed) if $X$ is Hausdorff.
  We manage to give complete characterizations of the double density spectra of
Hausdorff and of regular spaces as follows.
  Let $S$ be a non-empty set of infinite cardinals. Then
  (1) $S = dd(X)$ holds for a Hausdorff space $X$ iff S is $\omega$-closed and
$sup S \le 2^{2^{\min S}},$
  (2) S = dd(X) holds for a regular space X iff S is $\omega$-closed and $\sup
S \le {2^{\min S}}$.
  We also prove a number of consistency results concerning the double density
spectra of compact spaces. For instance:
  (i) If $\kappa = cf(\kappa)$ embeds in $\mathcal{P}(\omega)/fin$ and $S$ is
any set of uncountable regular cardinals $< \kappa$ with $|S| < \min S$, then
there is a compactum $C$ such that $\{\omega, \kappa\} \cup S \subset dd(C)$,
moreover $\lambda \notin d(C)$ whenever $|S| + \omega < cf(\lambda) < \kappa$
and $cf(\lambda) \notin S$.
  (ii) It is consistent to have a separable compactum $C$ such that $dd(C)$ is
not $\omega_1$-closed.
