Intervallsegmente

Intervallsegmente are the subintervals of a given interval on the real number line. If I = [α, β] is an interval, then an Intervallsegment is any nonempty set of the form [a,b], (a,b), [a,b) or (a,b], with α ≤ a ≤ b ≤ β. Degenerate cases where a = b yield a single point. These sets are the convex, connected pieces that lie inside I.

Properties and structure: The intersection of two Intervallsegmente is either another Intervallsegment or empty. The union of two Intervallsegmente is an Intervallsegment if and only if the segments overlap or touch; otherwise the union is not an interval. The collection of all Intervallsegmente of I is ordered by inclusion and forms a lattice, with meet given by intersection and join by the smallest interval containing both segments.

Types and topology: Intervallsegmente can be closed, open, or half-open, depending on whether their endpoints are included. Closed intervals are compact, open intervals are not compact, and half-open intervals possess properties intermediate between the two. In many contexts, the type of endpoints chosen affects convergence, measure, and continuity considerations.

Relation to partitions and analysis: A partition of I is a finite set of nonempty pairwise disjoint Intervallsegmente whose union is I; such partitions are central to the construction of Riemann sums and piecewise definitions of functions. Intervallsegmente also play a role in the study of the order topology on the real line, where they form a natural basis for open sets.

Generalization: The concept extends to any totally ordered set with an order topology, where the subintervals are the connected, convex subsets of the order.