boundaryA

boundaryA is a term used in topology and related disciplines to denote the boundary, or frontier, of a subset A within a topological space X. It is often written ∂A or bd(A).

An element x belongs to boundaryA if every neighborhood of x intersects both A and the complement X\A. Equivalently, boundaryA = cl(A) ∩ cl(X\A) = cl(A) \ int(A) = cl(X\A) \ int(X\A).

boundaryA is a closed set and sits between A and its exterior in a precise sense: it is contained in the closure of A and in the closure of the complement of A, but it contains no interior points of A nor interior points of the complement. In familiar spaces, boundaryA captures the edge separating a region from its surroundings. For example, if A is the unit disk in R^2, boundaryA is the unit circle; if A is the closed interval [0,1] in R, boundaryA is the set {0,1}; if A is the half-plane { (x,y) : x > 0 } in R^2, boundaryA is the line { x = 0 }.

Applications of boundaryA appear in boundary value problems for partial differential equations, in connectivity and region classification in topology, and in computer graphics and image processing for edge detection. In spaces with regular boundaries, boundaryA typically has lower dimensional measure than the ambient space (for instance, a curve in the plane has zero area).

Variations of the concept include definitions in subspace topologies and the distinction between the topological boundary and other notions of boundary used in different contexts.