Abstract
In this chapter, we will introduce the interval domain \(\mathbb {I\hspace {1.1pt} R}\), which is a topological space that represents the line of time in our work to come. The points of this space can be thought of as compact intervals [a, b] in \(\mathbb {R}\). The specialization order on points gives \(\mathbb {R}\) a non-trivial poset structure—in fact it is a domain—and as such it is far from Hausdorff.
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Notes
- 1.
(0, 1)-sheaves are also sometimes called ideals, but this terminology clashes with two other notions we use in this book that are also called ideals—namely, ideals in a poset and rounded ideals in a predomain (see Definitions 2.5 and A.6)—thus we call them (0, 1)-sheaves to avoid confusion.
- 2.
- 3.
Throughout this book, we often use d or δ to stand for “down” and u or υ (upsilon) to stand for “up.”
- 4.
We will neither need this nor prove this, but
is the largest coverage for which I is a sheaf.
is the largest coverage for which I is a sheaf.References
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Schultz, P., Spivak, D.I. (2019). The Interval Domain. In: Temporal Type Theory. Progress in Computer Science and Applied Logic, vol 29. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-00704-1_2
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