Prove open set is not closed The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY. 

Open set: A set $U$ is open if for every $a$ belonging to $U$, there is
  some $r = r(a) > 0$ such that the ball $B_r(a)$ is contained in $U$.
Closed set: A set $A$  is closed if it contains all of its limit
  points.

I tried using that $A$ closed and open would render $A'$ open and closed (provable using the above definitions) but that didn't lead me anywhere. 

There exists sets which are both open and closed.
For example: $\mathbb R$ with the standard topology.
http://www.youtube.com/watch?v=SyD4p8_y8Kw
