---
tag: 09KDHB82SG
kind: definition
name: Metrizable space
msc: ["54E35"]
defines: 091GJ3FYD7
uses: [0CNQXVBHCX, 0A6YRNZB2R]
sources:
  - url: https://en.wikipedia.org/wiki/Metrizable_space
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Metrizable_space
    locator: Metrization theorems
---

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is metrizable when there is a [[metric|063W5JMY85]] $d$ on $X$ whose [[metric topology|0A6YRNZB2R]] is $\tau$.

## Remarks

The metric is not unique: equivalent metrics give the same topology.
Each metrizable space is [[Hausdorff|0MXB4CKHFW]] and [[normal|0AD0BZ49DW]].
An indiscrete space with at least two points is not metrizable.
Urysohn's metrization theorem: each [[second-countable|00RE1R4W7N]] [[regular|0JF9KW2TXB]] Hausdorff space is metrizable.

## Explanations

### L1 human

A metrizable space is a topological space where you can measure distances in a way that gives exactly the open sets you already have.
Properties that hold in all metric spaces then hold in this space.