Intervallset

Intervallset is a mathematical concept representing a collection of intervals. An interval is a set of real numbers that includes all numbers between two given endpoints. For example, the closed interval [a, b] includes all real numbers x such that a <= x <= b. The open interval (a, b) includes all real numbers x such that a < x < b. Half-open intervals [a, b) and (a, b] are also common.

An intervallset can be a single interval, or it can be a union or intersection of multiple intervals. For instance, the set { [1, 3], [5, 7] } is an intervallset consisting of two disjoint closed intervals. The set { (0, 5), (2, 7) } is an intervallset whose union is the interval (0, 7).

The properties of an intervallset depend on the intervals it comprises. For example, an intervallset can be bounded or unbounded, connected or disconnected. Its measure, which is a generalization of length, can be calculated by summing the lengths of its constituent intervals, taking care to avoid double-counting if the intervals overlap.

Intervallsets are used in various areas of mathematics, including real analysis, measure theory, and numerical analysis. They provide a convenient way to describe sets of numbers with specific structural properties, which can simplify proofs and calculations. For instance, they are fundamental in defining integrals and understanding the behavior of functions over certain domains.