Definition – Metric Space. – Math-reference.net

Contents
1. Definition
2. Fourth Condition

Definition

Let $X$ be a set, and let $d: X\times X \rightarrow \mathbb{R}$ be a function. We say that the pair $(X, d)$ is a metric space, if, $\forall x_1, x_2, x_3 \in{X}$ ,

$d(x_1, x_2) = 0 \Leftrightarrow x_1 = x_2$
$d(x_1, x_2) = d(x_2, x_1)$
$d(x_1, x_3) \leq d(x_1, x_2) + d(x_2, x_3)$

The 3rd condition, $d(x_1, x_3) \leq d(x_1, x_2) + d(x_2, x_3)$ , is referred to as the triangle inequality.

The function $d$ , if $(X, d)$ forms a metric space, is referred to as a metric (or a distance) on $X$ .

Fourth Condition

Many textbooks will include a fourth ‘condition’, which actually is a consequence of the first and third conditions (when taken from this webpage; as textbooks may not have them in the same order as they appear here). If we take $x_1 = x_3$ in condition 3, we have

$\begin{aligned} d(x_1, x_1) &\leq d(x_1, x_2) + d(x_2, x_1) \\ \Leftrightarrow ~ 0 &\leq 2d(x_1, x_2) \\ \Leftrightarrow ~ 0 &\leq d(x_1, x_2) \end{aligned}$

So the fourth condition is that, $d(x_1, x_2) \geq 0$ for all $x_1, x_2 \in X$