nonoverlapping

Nonoverlapping describes a relationship among objects or regions that do not share any interior points. It is a common notion in mathematics, computer science, and design, used to indicate that elements are distinct in space, time, or content. In many contexts nonoverlapping is synonymous with disjointness, meaning that the intersection of the involved sets is empty.

Formally, a collection of sets {A_i} is nonoverlapping when A_i ∩ A_j = ∅ for all i ≠ j. Note that the treatment of boundaries varies by convention: some definitions require the intersection to be empty, which forbids even a shared boundary point; others allow boundaries to touch while keeping interiors disjoint.

Examples include time intervals: (1,2) and (2,3) are nonoverlapping since they do not contain a common point; by using half-open intervals such as [1,2) and [2,3) one can also ensure nonoverlap. Geometric tiling also uses nonoverlapping regions: tiles meet along edges or vertices but do not overlap in their interiors.

Applications include scheduling resources, memory management, and spatial indexing. In each case nonoverlap simplifies analysis by preventing conflicts or ambiguity. Related concepts include disjoint sets, partitions, tiling, and packing, all of which rely on the idea that distinct components occupy exclusive portions of a larger space.