The double density spectrum of a topological space

Abstract

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.