---
tag: 4M5JXYVQ2X
kind: concept
name: Compactly generated space
msc: [54D50]
sources:
  - url: https://en.wikipedia.org/wiki/Compactly_generated_space
same_as:
  wikidata: Q1738274
  wikipedia: Compactly generated space
---

## Summary

A compactly generated space, or k-space, is a [[topological space|0CQHFPGJ45]] whose topology is determined by its compact pieces.
A set is closed as soon as its trace on every compact piece is closed.
Metric spaces and [[locally compact|0FE2SB8WVJ]] Hausdorff spaces are compactly generated.
Algebraic topology often works in the category of compactly generated spaces, because there products and function spaces behave well.

## Explanations

### L1 human

To check whether a set is closed, one may look at how it meets every compact part of the space.
In a compactly generated space, this local test is enough.
For example, in the plane a set is closed exactly when its intersection with every closed disc is closed.
So the topology of the plane is rebuilt from its compact discs.