Thread: Both open and closed set proof

1. Both open and closed set proof

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.

2. What do you know about connected sets? The space $\displaystyle \mathbb{R}$ is connected.
If $\displaystyle F\subseteq \mathbb{R}$, is non-trivial, and is both open and closed then $\displaystyle F\cup F^c=\mathbb{R}$.

3. If $\displaystyle F$ is open then $\displaystyle F^c$ is closed. So if a set is both open and closed, what does that make the complement of that set? Neither open nor closed?

4. No. These are known as clopen sets (both open and closed).
So if $\displaystyle F$ is clopen then $\displaystyle F^c$ is clopen.
There is a well known result about connected sets and clopen sets.
Otherwise, I have not seen this topic discussed.

5. 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 $\displaystyle \phi$?

6. It is completely obvious that those two sets are both open and closed.
They are the only two sets in $\displaystyle \mathbb{R}$ that are.

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