Suppose I have a sequence of intervals $I_k = [a_k, b_k]$ for $k = 1, \ldots, n$. They may overlap each other.
I want to define a combination $\mathcal{I}$ of these intervals that is not quite a union, and not quite a direct product, but more like stitching them together like a quilt. An element of $\mathcal{I}$ would be a pair $(k, t)$ where $t \in [a_k, b_k]$.
Is this already a thing? Is there a common name or notation for it?