Let $\mathcal{P}_{\text{open}} = \mathcal{P}_{\text{open}}(\mathbb{R})$ be the D2615: Set of open real intervals.
Let $\mathcal{P}_{\text{closed}} = \mathcal{P}_{\text{closed}}(\mathbb{R})$ be the D2425: Set of closed real intervals.
Let $\mathcal{P}_{\text{right-closed}} = \mathcal{P}_{\text{right-closed}}(\mathbb{R})$ be the D3004: Set of right-closed real intervals.
Let $\mathcal{P}_{\text{left-closed}} = \mathcal{P}_{\text{left-closed}}(\mathbb{R})$ be the D3005: Set of left-closed real intervals.
The set of real intervals is the D11: Set \begin{equation} \mathcal{P}_{\text{open}} \cup \mathcal{P}_{\text{closed}} \cup \mathcal{P}_{\text{right-closed}} \cup \mathcal{P}_{\text{left-closed}} \end{equation}