---
tag: 00RE1R4W7N
kind: definition
name: Second-countable space
msc: [54D70]
defines: 03PPNCM9GG
uses: [0CNQXVBHCX, 06EX0FHBTF, 0ZG61TFBES]
sources:
  - url: https://en.wikipedia.org/wiki/Second-countable_space
    locator: lead section
---

## Definition

A [[topological space|0CNQXVBHCX]] $(X, \tau)$ is second-countable when $\tau$ has a [[basis|06EX0FHBTF]] $\mathcal{B}$ that is [[countable|0ZG61TFBES]].

## Remarks

The space $\mathbb{R}^n$ is second-countable: the open balls with rational centres and rational radii form a countable basis.
Each second-countable space is [[separable|0GKJ78948V]] and [[first-countable|0HG11FW8MJ]].
A metric space is second-countable if and only if it is separable.
An uncountable set with the [[discrete topology|0S2TEHX4E1]] is not second-countable.

## Explanations

### L1 human

A second-countable space has a countable list of open sets from which you can build every open set by unions.
For $\mathbb{R}$, the open intervals with rational endpoints form such a list.