---
tag: E8XKSQY76M
kind: concept
name: Limit point compact space
msc: [54D30]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_point_compact
same_as:
  wikidata: Q6549451
  wikipedia: Limit point compact
---

## Summary

A limit point compact space is a topological space in which every infinite subset has a [[limit point|0E3X7TXS67]].
It is the form of compactness that appears in the Bolzano–Weierstrass theorem.
Every [[countably compact|S3N3CHZV89]] space is limit point compact, and the two notions agree for T1 spaces.
For metric spaces, it is equivalent to compactness.

## Explanations

### L1 human

If you put infinitely many points in a limit point compact space, they must pile up somewhere.
There is a point whose every neighbourhood contains points of the set other than itself.
For example, in $[0, 1]$ any infinite set of points accumulates somewhere, by the Bolzano–Weierstrass theorem.
In $\R$ this fails: the integers are infinite but do not accumulate anywhere.