Definition

1. Noun:
   • A mathematical function that defines distance: A "metric function" is a function that takes any two points from a set (often a topological space) and returns a non-negative real number representing the distance between them. This function must satisfy specific mathematical properties to be considered a valid metric.

Usage and Examples

• Noun:
  • In Euclidean geometry, the standard metric function is the straight-line distance between two points.
  • The mathematician defined a new metric function for the abstract space.
  • To analyze the shape, we first need to specify the metric function on the surface.

Advanced Usage

• "Induce a topology": A metric function can be used to define a topology on a set, leading to the concept of a "metric space."

  • The usual metric function on the real numbers induces the standard topology.

• "Satisfy the metric axioms": A valid metric function must satisfy four conditions: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

  • We must verify that our proposed formula satisfies all the metric axioms to be a proper metric function.

Variants and Related Words

• Metric (noun): Often used synonymously with "metric function."

  • The taxicab metric is a different way to measure distance in a city grid.

• Metric Space (noun): A set equipped with a metric function.

  • A complete metric space is one where every Cauchy sequence converges.

• Pseudometric (noun): A function similar to a metric function but where the distance between two distinct points can be zero.

  • A pseudometric function relaxes the condition of identity of indiscernibles.

Synonyms

• Distance function: A direct synonym for metric function.
• Metric: A common shorthand term in mathematics.

Related Phrases and Concepts

• Triangle inequality: A critical property of a metric function. It states that for any three points, the distance from the first to the third is less than or equal to the sum of the distances from the first to the second and the second to the third.

  • Any proposed metric function must obey the triangle inequality.

• Continuity of a metric function: In a metric space, the metric function itself is a continuous function with respect to the topology it generates.

  • The continuity of the metric function is essential for many analytical proofs.

Noun

1. a function of a topological space that gives, for any two points in the space, a value equal to the distance between them