Approximal

Approximal is a term used in the study of metric spaces to describe a notion of approximate connectivity at variable scales. It treats connectivity not as a single dichotomy (connected or not) but as a scale-sensitive relation based on short steps.

Definition and concept. In a metric space (X, d), two points x and y are called approximal if for every positive radius ε > 0 there exists a finite sequence x = x0, x1, ..., xn = y such that the distance between successive points is always less than ε (d(xi, xi+1) < ε for all i). Equivalently, for every ε > 0 there is an ε-chain from x to y. The relation is reflexive, symmetric, and transitive, so it partitions X into equivalence classes called approximal components or approximate components.

Relation to other notions. Approximal components refine the traditional idea of connected components and relate to chain connectivity. If a space is path-connected, then any two points can be connected by a path, and subdividing that path yields ε-chains for every ε > 0, so path-connected spaces are approxinal. However, the converse need not hold in general. In common spaces such as the real line, the entire space is one approximal component; in a discrete metric space with d(x, y) = 1 for x ≠ y, only identical points are approxinal to each other, so every point forms its own approximal component.

Applications and status. The concept is used in coarse geometry, topological data analysis, and multi-scale studies of connectivity. It provides a framework for examining connectivity properties that persist under changes of scale and under uniformly continuous mappings.

See also. Proximal, chain connectivity, ε-chain, uniform continuity.