metrische

Metrische, in mathematics, refers to properties or objects related to a metric, a function that defines distance on a set. A metric must satisfy four axioms: non-negativity d(x,y) ≥ 0 for all x,y; identity of indiscernibles d(x,y) = 0 iff x = y; symmetry d(x,y) = d(y,x); and the triangle inequality d(x,z) ≤ d(x,y) + d(y,z). The pair (X,d) is called a metric space, and the distance function induces a topology with open balls B(x,r) = {y ∈ X : d(x,y) < r}.

Examples: on the real numbers with d(x,y) = |x - y|; Euclidean space with the usual distance; the discrete metric d(x,y) = 0 if x = y, otherwise 1; p-norm metrics on R^n with d_p(x,y) = (sum |x_i - y_i|^p)^{1/p} for p ≥ 1.

Properties and concepts: Metric spaces support notions of convergence, continuity, and compactness; completeness means every Cauchy sequence converges; compactness in metric spaces is related to Heine-Borel in Euclidean spaces; separability, total boundedness, and Lipschitz maps are common topics.

History and usage: The concept of a metric was formalized in the early 20th century, with Fréchet credited for foundational work; metric spaces underpin much of analysis and geometry. In Dutch and German mathematics, metrische ruimte or metrische is the standard phrase for metric spaces or metric properties; in English, "metric" and "metric space" are used.