eindintervals

Eindintervals are a concept from real analysis referring to intervals on the real line whose endpoints are real numbers. An eindinterval is determined by two real numbers a and b with a < b, and it can take one of four forms depending on whether the endpoints are included: (a, b), [a, b], (a, b], or [a, b). All such sets have finite endpoints, which is the meaning of the term eind (end) interval in this context.

The length or size of an eindinterval is given by b − a, regardless of whether the endpoints are included. An eindinterval is bounded, and its closure is the closed interval [a, b], while its interior is the open interval (a, b). As a subset of the real numbers, an eindinterval is connected and, in the case of the closed form [a, b], compact.

Eindintervals contrast with unbounded intervals, which have endpoints at infinity, such as (a, ∞), [a, ∞), (−∞, b), and (−∞, ∞). These unbounded cases are not eindintervallen because their endpoints are not finite.

Operations on eindintervals follow standard interval arithmetic. The intersection of two eindintervallen is either empty or another eindinterval. The union of two nested eindintervals is an eindinterval, while the union of disjoint eindintervals is generally not an eindinterval.

Applications of eindintervals include definite integration over [a, b], estimation and bounding problems, and probabilistic models such as the uniform distribution on [a, b]. They also serve as a fundamental building block in topology and analysis, illustrating concepts of length, compactness, closure, and interior.