pointsets

In mathematics, a point set is a set whose elements are points of a given ambient space, typically Euclidean space R^n with the standard topology. A point set can be finite or infinite, and it may be discrete, dense, closed, or open as a subspace. The term is used broadly for any collection of points, without implying additional structure.

In topology, a point set is a subset of a topological space. Fundamental notions include interior, closure, boundary, and accumulation points. The interior of a finite point set in R^n is empty; its closure is the set itself, while more complex sets may acquire limit points under closure.

Measures and dimension: Finite point sets have Lebesgue measure zero in R^n. The affine hull of a point set is the smallest affine subspace containing it, and its dimension equals the size of the largest affinely independent subset. The convex hull of a point set is the smallest convex set containing it and is a polytope when the set is finite.

Constructions and operations: Point sets underpin many constructions in geometry and analysis. Common operations include unions, intersections, translations, and dilations, as well as computing the convex hull, affine span, closures, and boundaries.

Applications: In computational geometry, problems on point sets include nearest neighbor search, clustering, triangulations, Voronoi diagrams, and Delaunay graphs. In analysis and topology, point sets model spaces, study measure-zero phenomena, and serve as subspaces used to explore topological and geometric properties.