distancescontinue

Distancescontinue is a term used in mathematical discourse to denote a property of distance functions that persists under certain limit processes. It is not a universally standard term, but appears in discussions of metric geometry to describe a specific compatibility condition between sequences of metrics and a limiting metric.

Definition: Let (X_n, d_n) be a sequence of metric spaces converging to (X, d) in a chosen sense (for example, Gromov–Hausdorff convergence). Distancescontinue refers to the requirement that for each pair of points x, y in X, there exist representatives x_n, y_n in X_n converging to x, y such that lim d_n(x_n, y_n) = d(x, y). Equivalently, the distance on the limit space is the limit of a sequence of distance functions from the approximating spaces.

Meaning and use: Distancescontinue captures the idea that distances between points persist through a limiting process, ensuring stability of geometric relations. It is relevant in the study of converging sequences of spaces, metric measure spaces, and in analysis on varying domains, where preserving distances aids in transferring geometric or analytic structure. It relates to, but is not identical with, continuity of distance functions, uniform convergence of metrics, and Gromov–Hausdorff convergence.

Remarks: The term is informal and definitions vary by author. Some treatments require stronger or weaker conditions or use alternative formulations via embeddings, pullbacks, or restriction to dense subsets. Readers should consult the specific source for precise statements.

See also: distance function; metric space; convergence; Gromov–Hausdorff convergence; metric geometry.