Nearextension

Nearextension is a term used in some areas of mathematics to describe a relaxed form of extending a function or structure from a subset to a larger domain. The concept appears chiefly in functional analysis, topology, and approximation theory, where exact extensions may be impossible or undesirable, and one instead seeks extensions that are close to the original data in a controlled way.

In the metric-space setting, a common informal definition is as follows. Let X and Y be metric spaces, A a subset of X, and f: A → Y a given map. A near extension of f with tolerance ε ≥ 0 is a map F: X → Y such that for every a ∈ A, the distance d_Y(F(a), f(a)) ≤ ε. If one can obtain a family of extensions F_ε parameterized by ε > 0, with the property that the bound holds for each ε and ε can be taken arbitrarily small, then an exact extension (ε = 0) may be recovered in favorable cases. Near extensions thus generalize the extension problem by allowing a controlled, quantitative deviation on the original domain.

Near extensions are studied in approximation theory and related fields because they provide a framework for stability and feasibility when exact extension is obstructed by topological or geometric constraints. They relate to, but are distinct from, exact extension results such as the Tietze or McShane–Whitney theorems, which guarantee precise extensions under certain conditions. In other contexts, the idea appears in proximity spaces and coarse geometry, where preserving “nearness” or closeness relations is more important than preserving exact values.

See also: extension problem, Lipschitz extension, proximity spaces, approximation theory.