Pseudometric space

Definition

Let $X$ be a set. A pseudometric on $X$ is a function $d \colon X \times X \to \mathbb{R}$ such that, for all $x, y, z \in X$:

1. $d(x, x) = 0$.
2. $d(x, y) = d(y, x)$.
3. $d(x, z) \le d(x, y) + d(y, z)$.

A pseudometric space is a pair $(X, d)$, where $X$ is a set and $d$ is a pseudometric on $X$.

Remarks

Each metric is a pseudometric. A pseudometric is a metric if and only if $d(x, y) = 0$ implies $x = y$. On the set of functions $f \colon [0, 1] \to \mathbb{R}$, the rule $d(f, g) = |f(0) - g(0)|$ is a pseudometric, but not a metric. The relation $d(x, y) = 0$ is an equivalence relation, and $d$ gives a metric on the set of its classes.

Explanations

L1 human

A pseudometric measures distance, but it can fail to see the difference between two points. For example, measure the distance between two people by the difference of their ages. Then two people of the same age have distance zero.