# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 103: Dedekind Cuts

The Dedekind cut is a method of constructing $\mathbb{R}$ from $\mathbb{Q}$. A cut is first defined to be a partition of $\mathbb{Q}$ into 2 non-empty subsets $A, B$ such that

1. if $p \in A$ and $q < p$, then $q \in A$
2. if $p \in A$, then there is a $q \in A$ such that $q > p$ (meaning $A$ has no maximum element)

A cut can be equivalently determined solely by $A$ alone, rather than the pair $(A, B)$.

Cuts can then be used to construct $\mathbb{R}$ by defining any $r \in \mathbb{R}$ to be the cut where $A$ is the set of all members $q$ of $\mathbb{Q}$ such that $q < r$. That is, $r$ is simply defined as the subset of rationals smaller than itself, and $\mathbb{R}$ is the set of all such subsets.