Definition:Metric Space/Distance Function

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

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

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

Also known as
The distance function $d$ is frequently referred to as the 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, $\rho$.