Definition:Grothendieck Universe

Definition
A Grothendieck universe is a set (not a class) which has the properties expected of the universe $\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties:


 * $(1): \quad \mathbb U$ is a transitive set: If $u \in \mathbb U$ and $x \in u$ then $x \in \mathbb U$


 * $(2): \quad$ If $ u, v \in \mathbb U$ then $\set {u, v} \in \mathbb U$


 * $(3): \quad$ If $u \in \mathbb U$ then the power set $\powerset u \in \mathbb U$


 * $(4): \quad$ If $A \in \mathbb U$, and $\set {u_\alpha: \alpha \in A}$ is a family of elements $u_\alpha \mathop \in \mathbb U$ indexed by $A$, then $\displaystyle \bigcup_{\alpha \mathop \in A} u_\alpha \in \mathbb U$

Motivation
A Grothendieck universe allows us to work with something "like" the collection of all sets without having to consider classes, which helped in his studies of algebraic geometry.

One can check that if $u, v \in \mathbb U$ and $f: u \to v$ is a mapping, then $f \in \mathbb U$, and similarly the Cartesian product $u \times v \in \mathbb U$, etc. In other words, it is closed under the algebra of sets.

A Grothendieck universe $\mathbb U$ is closed under many set-theoretical operations, some of them listed below.

Also defined as
Some authors require additionally, that $\mathbb U$ is not empty.