Density Spectra of Topological Groups

Abstract

We study density spectra associated with dense and closed subgroups of topological groups. For a topological group $G$, let $\dd^*(G)$ denote the set of densities of its dense subgroups. We prove that every compact group satisfies $\dd^*(G)=[d(G),w(G)]$. We also investigate the density spectrum $\cd(G)$ of infinite closed subgroups. In $\mathbf{ZFC}$, we construct a separable countably compact Boolean group containing a closed non-separable subgroup, answering a question of Leiderman, Morris, and Tkachenko. We further show that a free pro-$p$ group of uncountable rank has no non-trivial metrizable closed normal subgroup, giving a negative answer to a problem of Hern\'andez, Hofmann, and Morris. Finally, we study when $\cd(G)$ is an interval. Among other results, we show that under Shelah's Strong Hypothesis ($\mathbf{SSH}$), the closed density spectrum of every infinite $\omega$-bounded group of countable tightness is an interval of cardinals.