---
tag: 0A6YRNZB2R
kind: definition
name: Metric topology
msc: ["54E35"]
defines: 0YVKNB7V5F
uses: [063W5JMY85, 0DC3MPR7C9, 0CNQXVBHCX]
sources:
  - url: https://en.wikipedia.org/wiki/Metric_space
    locator: The topology of a metric space
---

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]].
The metric topology of $(X, d)$ is the set $\tau_d$ of all $U \subseteq X$ such that for each $x \in U$ there is a real $\varepsilon > 0$ with $B(x, \varepsilon) \subseteq U$.
Here $B(x, \varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\varepsilon$.

## Remarks

The set $\tau_d$ is a [[topology|0CNQXVBHCX]] on $X$: it contains $\varnothing$ and $X$, and it is closed under [[unions|0HEF46RPEK]] and [[finite intersections|09NDE9V7X1]].
Each open ball is in $\tau_d$, by [[open balls are open|0B2305GNAZ]]. The open balls form a [[basis|06EX0FHBTF]] of $\tau_d$.
For $\mathbb{R}$ with the usual metric, $\tau_d$ is the usual topology, generated by the open intervals.
Different metrics can give the same topology. Such metrics are [[equivalent|0TWQ5PRKWS]].

## Explanations

### L1 human

The metric topology is how a metric space becomes a topological space.
A set is open when each of its points has a small ball about it inside the set.