nearadjacency

Nearadjacency is a term used in mathematical topology and geometry to describe a form of closeness between two sets or objects that are not necessarily intersecting but come arbitrarily close in the sense of a defined metric or proximity structure.

In a metric space (X,d), two nonempty subsets A and B are near-adjacent, written A ≈na B, if the distance between A and B is zero: dist(A,B) = inf{d(a,b) : a ∈ A, b ∈ B} = 0. Equivalently, the closures of A and B intersect: closure(A) ∩ closure(B) ≠ ∅. This captures the idea that the two sets can be approached arbitrarily closely by points from each set, even if they do not share a point.

Examples help illustrate the concept. If A = [0,1] and B = [1,2] in the real numbers with the usual metric, dist(A,B) = 0 and closure(A) ∩ closure(B) = {1}, so A and B are near-adjacent. If instead B = [1+ε, 2] for some ε > 0, then dist(A,B) > 0 and they are not near-adjacent. This notion generalizes beyond exact intersection to a boundary-contact idea via closures.

Relation to broader theories: nearadjacency is related to, but distinct from, standard adjacency and the nearness concepts in proximity spaces. It provides a precise way to formalize when two objects can be considered immediately successive or touching in a limiting sense, which can be useful in spatial reasoning, geometric analysis, and related fields.

See also: proximity space, nearness, closure, boundary, distance, adjacency.