intervallene

Intervallene, or intervals, are subsets of the real line that consist of all points between two endpoints. An interval is usually described by endpoints a and b with a ≤ b, and can be written as {x ∈ R | a ≤ x ≤ b} or, more commonly in notation, as (a, b), [a, b], (a, b], or [a, b). There are several common types: open intervals (a, b) exclude the endpoints; closed intervals [a, b] include both endpoints; and half-open intervals (a, b] or [a, b) include exactly one endpoint. Intervals may be bounded, with finite endpoints, or unbounded, for example (a, ∞), (-∞, b], and (-∞, ∞), which is the entire real line. Degenerate cases include [a, a], a single point, and (a, a), the empty set.

On the real line, intervals are connected and convex; between any two points in an interval lies the entire subinterval. The intersection of any family of intervals is either an interval or empty; finite intersections are intervals as well. The union of overlapping intervals is an interval as long as they share a common overlap; the union of disjoint intervals is not an interval.

Notation is central: brackets indicate whether endpoints are included. In higher dimensions, intervals generalize to axis-aligned boxes, formed by Cartesian products of one-dimensional intervals. Intervallene are foundational in analysis and applied contexts: they define domains of functions, describe convergence and continuity properties, and underlie numerical methods such as interval arithmetic, which keeps bounds on quantities to account for uncertainty.