  For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its
\emph{set of multiples} $\mathcal M_{\mathcal B}:=\bigcup_{b\in\mathcal
B}b\mathbb Z$ and the set of \emph{$\mathcal B$-free numbers} $\mathcal
F_{\mathcal B}:=\mathbb Z\setminus\mathcal M_{\mathcal B}$. Tautness of the set
$\mathcal B$ is a basic property related to questions around the asymptotic
density of $\mathcal M_{\mathcal B}\subseteq\mathbb Z$. From a dynamical
systems point of view (originated by Sarnak) one studies $\eta$, the indicator
function of $\mathcal F_{\mathcal B}\subseteq\mathbb Z$, its shift-orbit
closure $X_\eta\subseteq\{0,1\}^{\mathbb Z}$ and the stationary probability
measure $\nu_\eta$ defined on $X_\eta$ by the frequencies of finite blocks in
$\eta$. In this paper we prove that tautness implies the following two
properties of $\eta$: (1) The measure $\nu_\eta$ has full topological support
in $X_\eta$. (2) If $X_\eta$ is proximal, i.e. if the one-point set
$\{\dots000\dots\}$ is contained in $X_\eta$ and is the unique minimal subset
of $X_\eta$, then $X_\eta$ is hereditary, i.e. if $x\in X_\eta$ and if $w$ is
an arbitrary element of $\{0,1\}^{\mathbb Z}$, then also the coordinate-wise
product $w\cdot x$ belongs to $X_\eta$. This strengthens two results from
[Bartnicka et al. 2015] which need the stronger assumption that $\mathcal B$
has light tails for the same conclusions.
