Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers such that

(i)	$\mathcal{P}_{\text{interval}}(\mathbb{R}^N)$ is the D3037: Set of euclidean real intervals in $\mathbb{R}^N$

A D11: Set $E \subseteq \mathbb{R}^N$ is an elementary euclidean real set in $\mathbb{R}^N$ if and only if \begin{equation} \exists \, I_1, \ldots, I_N \in \mathcal{P}_{\text{interval}}(\mathbb{R}) : E = \bigcup_{n = 1}^N I_n \end{equation}