the collection $$\begin{aligned}
 \mathbb{X}:= \left\lbrace  \Lambda \text{ point pattern of } \mathbb{R}^d \; \vert \mathcal{L}_\Lambda = \mathcal{L}_{NS} \right\rbrace \end{aligned}$$ We call model set any point pattern within the hull $\mathbb{X}$ associated with some cut $\&$ project scheme and window.

The hull $\mathbb{X}$ associated with some cut $\&$ project scheme and window is also called the *local isomorphism class* (or simply $LI$-class) of any model set within this hull.

The hull as dynamical system
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There is a natural topology on the hull $\mathbb{X}$, which is metrizable and may be described by setting a basis of open neighborhoods of any point pattern $\Lambda \in \mathbb{X}$ (see for instance [@Mo2]) $$\begin{aligned}
\label{topology} \mathcal{U}_{K, \varepsilon }(\Lambda ):= \left\lbrace  \Lambda '\in \mathbb{X} \; \vert \; \exists \vert t \vert , \vert t'\vert < \varepsilon , \; (\Lambda -t)\cap K = (\Lambda '-t')\cap K \right\rbrace \end{aligned}$$

where $K$ is any compact set in $\mathbb{R}^d$ and $\varepsilon >0$. This topology roughly means that two point patterns are close if they agree on a large domain about the origin up to small shifts. Then the hull $\mathbb{X}$ endowed with this topology is a compact metrizable space, and is endowed with a natural action of $\mathbb{R}^d$ given by 