## 5.1. Open and Closed Sets

### Proposition 5.1.4: Characterizing Open Sets

**U**

**R**be an arbitrary open set. Then there are countably many pairwise disjoint open intervals

*U*such that

_{n}

**U**= U_{n}### Proof:

This proposition is rather interesting, giving a complete description of any possible open set in the real line. To prove it, we will make use of equivalence relations and classes again. First, let us define a relation on**U**:

- if
*a*and*b*are in**U**, we say that*a ~ b*if the whole line segment between*a*and*b*is also contained in**U**.

**U**equals the union of the equivalence classes, and the equivalence classes are pairwise disjoint. Denote those equivalent classes by

*U*

_{n}
Each *U _{n}* is an interval: take any two points

*a*and

*b*in

*U*. Being in the same equivalence classes,

_{n}*a*and

*b*must be related. But then the whole line segment between

*a*and

*b*is contained in

*U*as well. Since

_{n}*a*and

*b*were arbitrary,

*U*is indeed an interval.

_{n}
Each *U _{n}* is open: take any

*x U*. Then

_{n}*x*, and since

**U****U**is open, there exists an

*> 0*such that

*( x - , x + )*is contained in

**U**. But clearly each point in that interval is related to

*x*, hence this neighborhood is contained in

*U*, proving that

_{n}*U*is open.

_{n}
There are only countably many *U _{n}*:
This seems the hard part. But, each

*U*must contain at least one different rational number. Why ? Since there are only countably many rational numbers, there can only be countably many of the

_{n}*U*'s (since they are disjoint).

_{n}