avståndsmetrikfunktion

An avståndsmetrikfunktion, often translated as a distance metric function, is a fundamental concept in mathematics, particularly in the fields of topology, geometry, and analysis. It is a function that defines a distance between any two points in a given set. For a set X, an avståndsmetrikfunktion d is a function d: X × X → ℝ that satisfies the following properties for all points x, y, and z in X:

1. Non-negativity: d(x, y) ≥ 0. The distance between two points is always greater than or equal to zero.

2. Identity of indiscernibles: d(x, y) = 0 if and only if x = y. The distance is zero if and only if the two points are identical.

3. Symmetry: d(x, y) = d(y, x). The distance from point x to point y is the same as the distance from point y to point x.

4. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z). The direct distance between two points is always less than or equal to the sum of the distances via an intermediate point.

A set equipped with an avståndsmetrikfunktion is called a metrisk rum, or metric space. This concept allows for a rigorous definition of concepts like convergence of sequences, continuity of functions, and the notions of open and closed sets in a generalized way, applicable to a wide variety of mathematical structures beyond Euclidean space. Common examples of avståndsmetrikfunktioner include the Euclidean distance, the Manhattan distance (or taxicab distance), and the Hamming distance, each defining distance in a different manner depending on the context and the nature of the set.