Give an example of an open set in R under the usual topology that is not expressible as the union of a finite number of intervals Give an example of an open set in R under the usual topology that is not expressible as the union of a finite number of intervals.
My introduction to proofs class tutorial this week is discussing closed sets... I have feeling giving this example will lead to closed sets? 

Here is a bounded example: 
$$[0,1]\setminus\left\{\frac1n:n\in\Bbb Z^+\right\}=\bigcup_{n\in\Bbb Z^+}\left(\frac1{n+1},\frac1n\right)\;.$$
(This is what happens to Frank’s example if you subject it to the function $f(x)=\frac1x$.)
More generally, let $\{(a_n,b_n):n\in\Bbb N\}$ be any countably infinite collection of pairwise disjoint open intervals; then $\bigcup_{n\in\Bbb N}(a_n,b_n)$ is an open set that cannot be expressed as the union of finitely many open intervals.
