Let $\mathbb{R}^N$ be a D5630: Set of euclidean real numbers.
Let $\mathcal{P}_{\text{interval}}(\mathbb{R})$ be the D2424: Set of real intervals.
The set of euclidean real intervals in $\mathbb{R}^N$ is the D11: Set \begin{equation} \left\{ \prod_{n = 1}^N I_n : I_1, I_2, \ldots, I_N \in \mathcal{P}_{\text{interval}}(\mathbb{R}) \right\} \end{equation}