Prove that the only sets that are both open and closed are $\displaystyle \mathbb{R}$ and the empty set $\displaystyle \phi$.
How do you go about proving that every other set is either open, closed, or neither? There are infinitely many other sets.
Prove that the only sets that are both open and closed are $\displaystyle \mathbb{R}$ and the empty set $\displaystyle \phi$.
How do you go about proving that every other set is either open, closed, or neither? There are infinitely many other sets.
So if I can prove that the empty set, $\displaystyle \phi$, is both open and closed (or clopen) then I can say that the set of all reals, $\displaystyle \mathbb{R}$, is clopen because $\displaystyle \mathbb{R}$ is the complement to $\displaystyle \phi$?