Definition

Noun: A mathematical structure consisting of a set of points, along with a function (called a metric) that defines a distance between any two points. This distance must be a non-negative real number, be symmetric, satisfy the triangle inequality, and be zero only when the two points are identical.

Usage

A metric space is the fundamental setting for studying concepts of convergence, continuity, and limits in analysis and topology. It provides a rigorous way to talk about "closeness" and "distance" in abstract sets.

Examples

• In a sentence:
  • The set of real numbers with the usual absolute value distance forms a metric space.
  • To define continuity for functions between abstract sets, one must first equip each set with the structure of a metric space.
• In a definition:
  • A metric space is an ordered pair (M, d) where M is a set and d is a metric on M.

Advanced Usage

• Complete metric space: A metric space in which every Cauchy sequence converges to a point within the space.
  • The real numbers are a complete metric space, which is a crucial property for calculus.
• Metric space topology: The collection of all open sets induced by the metric, which defines the topological structure of the space.
  • Many topological properties, like compactness, can be defined within the framework of a metric space.

Variants and Related Words

• Metric (n): The distance function itself, which must satisfy specific axioms (non-negativity, identity of indiscernibles, symmetry, triangle inequality).
  • The Euclidean metric is the most familiar example.
• Pseudometric space (n): A generalization where the distance between two distinct points can be zero.
• Normed vector space (n): A vector space with a norm, which automatically induces a metric space structure.

Synonyms

• Distance space (less common but descriptive).

Related Concepts (Not Phrasal Verbs or Idioms)

• Triangle inequality: A key axiom for a metric, stating that for any three points x, y, z, the distance from x to z is less than or equal to the sum of the distances from x to y and from y to z.
  • The triangle inequality is what makes the concept of distance in a metric space coherent.
• Open ball: Given a point p and a radius r > 0, the set of all points whose distance from p is less than r. These balls form a basis for the metric topology.
  • Convergence in a metric space can be defined using open balls.

Noun

1. a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality