---
date: 2022-04-05 23:42
modification date: Friday 27th May 2022 22:00:13
title: "universally closed"
aliases: [universally closed, closed]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #AG/basics 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# universally closed

- A morphism $f: X\to Y$ is **universally closed** iff the following is a closed map for all $Z$:
$$
X\times Z\mapsvia{(f, \id_Z)} Y\times Z
$$
	- Equivalently when $Y$ is Hausdorff: for any map $Z\to Y$, the following pullback is a closed map:
$$
X\fiberprod{Y}Z\to Z
$$
- For $X$ Hausdorff and $Y$ locally compact, $f:X\to Y$ is **universally closed** iff $f$ is [proper](Unsorted/proper%20morphism.md). 

![](attachments/Pasted%20image%2020220914155931.png)