rneighborhood

An r-neighborhood of a subset A in a metric space (X, d) is the set N_r(A) = { x ∈ X : d(x, A) < r }, where d(x, A) = inf{ d(x, a) : a ∈ A }. The radius r > 0 fixes how far from A points are included. The distance to a set is a continuous function, and therefore N_r(A) is an open set. The closed counterpart is N_r^cl(A) = { x ∈ X : d(x, A) ≤ r }, which is closed.

In Euclidean space, N_r({p}) is the open ball B_r(p) of radius r around p. For any subset A, N_r(A) consists of all points whose distance to A is less than r, yielding a “buffer” or tubular neighborhood around A. If A is a curve, surface, or region, N_r(A) thickens A by distance r along its boundary.

Variants and notation: In normed or metric spaces, N_r(A) can be viewed as the Minkowski sum A ⊕ B_r(0), where B_r(0) is the ball of radius r centered at the origin. Some authors use B_r(A) to denote the open r-neighborhood, while others distinguish between open and closed forms. The concept depends on the chosen metric; different metrics produce different r-neighborhoods of the same A.

Applications: r-neighborhoods are employed in analysis and topology to study proximity and convergence near a set, in geometry and measure theory for thickening or buffering regions, and in image processing and computational geometry for shape dilations, collision detection, and proximity queries in robotics and spatial databases.

Relation to general neighborhoods: While all r-neighborhoods are neighborhoods in metric spaces, the broader topological notion of a neighborhood does not require a metric and is defined differently in general spaces.