---
tag: 0H3W614GEC
kind: definition
name: Pseudometric space
msc: ["54E35"]
defines: 0A3HJX3KTF
uses: [063W5JMY85]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudometric_space
    locator: Definition
---

## 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|063W5JMY85]] 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.