7.1 Metrics

Definition 7.1.1 (Metric Space). Let $X$ be a set and $d: X \times X$, then $d$ is a metric if:

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

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

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

4. 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.