An interval is an under category, over category, or under-over category in a poset.
Specifically, given a poset and an element of , the upwards unbounded interval (also , , etc) is the subset
the downwards unbounded interval (also , , etc) is the subset
and given an element of , the bounded interval (also ) is the subset
Thinking of as a category and subsets of as subcategories, is the coslice category , is the slice category , and is the bislice category .
Besides the closed intervals above, we also have the open intervals
as well as the half-open intervals
These are important in analysis, and more generally whenever the quasiorder is at least as important as the partial order .
The entire poset is also considered an unbounded interval in itself.
Intervals of real numbers are important in analysis and topology. The bounded closed intervals in the real line are the original compact spaces.
The unit interval is primary in homotopy theory; a homotopy from to (themselves continuous maps from to ) is a continuous map such that and always. This generalises to the notion of interval object in an arbitrary category.
The usual integral in ordinary calculus is done over an interval in the real line, a compact interval for a ‘proper’ integral, or any interval for an ‘improper’ integral. The theory of Lebesgue measure? removes this restriction and allows integrals over any measurable subset of the real line. Still, the Lebesgue measure on intervals (even compact intervals) generates all of the rest.
To integrate a -form on the real line requires orienting an interval; the standard orientation is from to in . If , then (which by the definition above would be empty) may also be interpreted as with the reverse orientation. This also matches the traditional notation for the integral.