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.

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- Oct 18th 2010, 10:29 AMZalrenBoth 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. - Oct 18th 2010, 10:49 AMPlato
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 $\displaystyle F\cup F^c=\mathbb{R}$.

Is that a contradiction? WHY? - Oct 18th 2010, 11:10 AMZalren
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?

- Oct 18th 2010, 11:17 AMPlato
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. - Oct 18th 2010, 11:29 AMZalren
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 $\displaystyle \phi$?

- Oct 18th 2010, 11:57 AMPlato
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.