Etäisyysfunktio

Etäisyysfunktio, known in English as a distance function or metric, is a fundamental concept in mathematics, particularly in the fields of topology and analysis. It provides a way to quantify the "distance" between any two elements in a set. Formally, a distance function on a set X is a function d: X × X → ℝ that satisfies four properties for all x, y, and z in X.

Firstly, the non-negativity property states that the distance between any two points is always greater than or equal to zero, i.e., d(x, y) ≥ 0. Secondly, the identity of indiscernibles means that the distance is zero if and only if the two points are identical, i.e., d(x, y) = 0 if and only if x = y. Thirdly, symmetry dictates that the distance from x to y is the same as the distance from y to x, i.e., d(x, y) = d(y, x). Finally, the triangle inequality asserts that 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, i.e., d(x, z) ≤ d(x, y) + d(y, z).

A set equipped with a distance function is called a metric space. The concept of a distance function is crucial for defining notions like convergence, continuity, and completeness in various mathematical spaces. Different choices of distance functions on the same set can lead to different metric spaces with distinct topological properties. The most common example of a distance function is the Euclidean distance in Cartesian coordinate systems.