---
tag: SW0SN1NEJE
kind: concept
name: Lindelöf space
msc: [54D20]
sources:
  - url: https://en.wikipedia.org/wiki/Lindel%C3%B6f_space
same_as:
  wikidata: Q936155
  wikipedia: Lindelöf space
---

## Summary

A Lindelöf space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has a countable subcover.
It is a weakening of [[compactness|0TVJZYB6SH]], where finite subcovers are replaced by countable ones.
Every [[second-countable space|03PPNCM9GG]] is Lindelöf, and every regular Lindelöf space is [[paracompact|8R29G6NG9Q]].
For [[metric spaces|0VKD1TP7F3]], being Lindelöf, [[separable|0Z2P41SVDP]] and second-countable are the same.

## Explanations

### L1 human

Compactness says that finitely many members of any open cover always suffice to cover the space.
The Lindelöf property says that countably many always suffice.
The real line is not compact, since the intervals $(-n, n)$ cover it and no finitely many of them do.
It is Lindelöf, since it has a countable base of open intervals with rational end points.
An uncountable set with the discrete topology is not Lindelöf, because the cover by single points has no countable subcover.