## Proposition 5.1.3: Unions of Open Sets, Intersections of Closed Sets

- Every union of open sets is again open.
- Every intersection of closed sets is again closed.
- Every finite intersection of open sets is again open
- Every finite union of closed sets is again closed.

### Proof:

Let*{ U*be a collection of open sets, and let

_{n}}*. Take any*

**U**= U_{n}*x*in

**U**. Being in the union of all

**U**'s, it must be contained in one specific

*U*. Since that set is open, there exists a neighborhood of

_{n}*x*contained in that specific

*U*. But then that neighborhood must also be contained in the union

_{n}**U**. Hence, any

*x*in

**U**has a neighborhood that is also in

**U**, which means by definition that

**U**is open.

To prove the second statement, simply use the definition of closed sets and de Morgan's laws.

Now let
*U _{n}, n=1, 2, 3, ..., N*
be finitely many open sets. Take

*x*in the intersection of all of them. Then:

*x*is in the first set: there exists an with*( x - , x + )*contained in the first set*x*is in the second set: there is with*( x - , x + )*contained in the second set.- ....
*x*is in the*N*-th set: there is with*( x - , x + )*contained in the last set.

- let
*= min{ , , ..., }*. Then*( x - , x + )*is contained in each set*U*_{n}

*x*was arbitrary.

The last statement follows again from de Morgan's laws.