Definition:Metric Space/Distance Function

From ProofWiki
Jump to: navigation, search

Definition

Let $\left({A, d}\right)$ be a metric space.


The mapping $d: A \times A \to \R$ is referred to as a distance function on $A$ or simply distance.


Here, $d: A \times A \to \R$ is a real-valued function satisfying the metric space axioms:

\((M1):\)     \(\displaystyle \forall x \in A:\) \(\displaystyle d \left({x, x}\right) = 0 \)             
\((M2):\)     \(\displaystyle \forall x, y, z \in A:\) \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \)             
\((M3):\)     \(\displaystyle \forall x, y \in A:\) \(\displaystyle d \left({x, y}\right) = d \left({y, x}\right) \)             
\((M4):\)     \(\displaystyle \forall x, y \in A:\) \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \)             


Also known as

The distance function $d$ is frequently referred to as a metric on $A$.

The two terms are used interchangeably on this website.


Also defined as

If $\left({A, d}\right)$ is a pseudometric space or quasimetric space, this definition still applies.

That is, a pseudometric and a quasimetric are also both found to be referred to in the literature as distance functions.


Also denoted as

Some authors use a variant of $d$ for a metric, for example $\eth$. Others use, for example, $\rho$.


Sources