A Dedekind cut is a partition of the rationals Q {\displaystyle \mathbb {Q} } \mathbb {Q} into two subsets A and B such that

A {\displaystyle A} A is nonempty.
A ? Q {\displaystyle A\neq \mathbb {Q} } {\displaystyle A\neq \mathbb {Q} }.
If x , y ? Q {\displaystyle x,y\in \mathbb {Q} } {\displaystyle x,y\in \mathbb {Q} }, x < y {\displaystyle x<y} {\displaystyle x<y}, and y ? A {\displaystyle y\in A} {\displaystyle y\in A}, then x ? A {\displaystyle x\in A} x\in A. ( A {\displaystyle A} A is "closed downwards".)
If x ? A {\displaystyle x\in A} x\in A, then there exists a y ? A {\displaystyle y\in A} {\displaystyle y\in A} such that y > x {\displaystyle y>x} {\displaystyle y>x}. ( A {\displaystyle A} A does not contain a greatest element.)

By relaxing the first two requirements, we formally obtain the extended real number line