Are there only 2 clopen sets on real plane?

Question

How can I prove that the only open and closed sets on the real plane are empty set and real plane itself? Preferably by using order theory.

Thanks.

Answer 1

You can try something like: If there exists non-trivial clopen set $U$, then $U^c$ is also a nontrivial clopen set. Make a path from $x\in U$, to $y\in U^c$.

$l(t) = ty+(1-t)x$ for $t\in [0,1]$. Since both $U,U^c$ are closed you can show that $x_0 = l(\alpha)$ lies in both $U,U^c$ where $\alpha = \sup\{t:l(t)\in U\}$