Let \(X\) be a metric space.
If \(U\) is a subset of \(X\), show that there is a unique open set \(\neg U\) disjoint from \(U\) and containing all open sets disjoint from \(U\).
Give an example of an open set \(U\) with \(U\ne\neg\neg U\).
Prove that for every open set \(U\), \(\neg U=\neg\neg\neg U\).
Solution
Take \(\neg U\) to be the union of all open sets disjoint from U , which is open as the union of any collection of open sets is open.
Take X to be the real line and U to be the nonzero reals. Then \(\neg U\) is empty so \(\neg \neg U\) is the real line.
We have \(A \subseteq \neg \neg A\) and applying this to \(A = \neg U\) we get \(\neg U \subseteq \neg \neg \neg U\) . On the other hand, if \(A \subseteq B\) then \(\neg B \subseteq \neg A\) , and applying this to \(A = U , B = \neg \neg U\) we get \(\neg \neg \neg U \subseteq \neg U\)