Definition 8.1.1 (Metric Space).label Let $X$ be a set and $d: X \times X$, then $d$ is a metric if:

1. (PM1)

   For any $x \in X$, $d(x, x) = 0$.

2. (M)

   For any $x, y \in X$ with $x \ne y$, $d(x, y) > 0$.

3. (PM2)

   For any $x, y \in X$, $d(x, y) = d(y, x)$.

4. (PM3)

   For any $x, y, z \in X$, $d(x, z) \le d(x, y) + d(y, z)$.

The pair $(X, d)$ is a metric space, which comes with the metric uniformity induced by $d$, and the corresponding topology.