I don't get why the boundary of a clopen set is empty. If you take $A = [0,1)$ in $\mathbb R$, then isn't the closure of this the smallest closed super set that contains $A$ which is $[0,1]$. Isn't the interior, $(0,1)$? So the boundary would be the closure minus the interior, aka, ${0}$ and ${1}$?
I'm confused. Thanks for all of your great and quick replies!! No wonder I was so confused, I was completely misunderstanding the definition of clopen. I've learned a lot from all of you!
Best Answer
The half-open interval $[0,1)$ is not clopen. A set is clopen if it is both closed and open, and $[0,1)$ is neither closed nor open.
If a set is clopen, then it is open so it is equal to its own interior; and it is closed so it is equal to its closure. The difference between a set and itself is empty.