---
tag: H7B12XW6Y2
kind: definition
name: Compactly generated space
msc: [54D50]
defines: 4M5JXYVQ2X
uses: [0CQHFPGJ45, 0TVJZYB6SH, 0S1FJ8VEX9, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Compactly_generated_space
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *compactly generated* if it satisfies the following condition.
A subset $A \subseteq X$ is closed whenever $u^{-1}(A)$ is closed in $K$ for every [[compact|0TVJZYB6SH]] [[Hausdorff space|0S1FJ8VEX9]] $K$ and every [[continuous map|0VGFHTM75Y]] $u \colon K \to X$.

## Remarks

If $X$ is Hausdorff, the condition is equivalent to the classical one.
A subset $A \subseteq X$ is closed whenever $A \cap C$ is closed in $C$ for every compact subset $C \subseteq X$.

Example: every [[metric space|0VKD1TP7F3]] is compactly generated.
If $A$ is not closed, there is a sequence $a_n \in A$ converging to some $x \notin A$.
The set $K = \set{x} \cup \set{a_n : n \ge 1}$ is compact, and $A \cap K$ is not closed in $K$.

Non-example: an uncountable set $X$ with the cocountable topology.
Its compact subsets are finite, so every test map $u$ has finite image.
Finite subsets of $X$ are closed and discrete, so every $u^{-1}(A)$ is closed.
But not every subset of $X$ is closed.