does the supremum of a complement exist

does the supremum of a complement exist

We have that A has a supremum $\sup (A)$. Do $\sup \overline{A}$ and $\inf
\overline{A}$ exist?
If both $\sup A$ and $\inf A$ exist, do $\sup \overline{A}$ and $\inf
\overline{A}$ still exist?