, \mathbb{R}^d) \; \;  & \; \;  \mathfrak{H}_W\times \mathbb{T}^{n+d}
\psset{arrows=->,linewidth=0.2pt, labelsep=1pt
,nodesep=0pt}
\ncline{1,1}{1,2}
\end{psmatrix}\end{aligned}$$ for which the image subsemigroup together with its topology are identified. Interrestingly, this semigroup remains exactly the same for model sets issued after translating, dilating, or deforming the region $W$ as long as the hyperplanes determined by the faces are unchanged. As a byproduct of the previous analysis we show that the topology of the Ellis semigroup $E(\mathbb{X},\mathbb{R}^d)$ admits a first countable topology, and thus is sequentially compact. We conclude this work by determining some algebraic features of this Ellis semigroup, as well as a picture of its underlying action on the space $\mathbb{X}$.

 {#model sets}

General definition of inter model set
-------------------------------------

In order to define what a (almost canonical) model set is in $\mathbb{R}^d$ (see [@Sch] as well as [@Mo2] for a more detailed exposition), we consider first an environment used to construct it, namely a *cut $\&$ project scheme*. It consists of a data $(\mathsf{H}, \Sigma , \mathbb{R}^d)$ with associated diagram $$\begin{aligned}
\begin{psmatrix}[colsep=1.5cm,
rowsep=0cm]
\mathsf{H} \; \; & \; \;  \mathsf{H}\times \mathbb{R}^{d} \; \; & \; \; \mathbb{R}^d \\
\cup & \cup & \cup \\
\Gamma ^* \; \;  & \; \;  \Sigma  