Closed sets

Definition. $F\subseteq\mathbb{R}$ is closed iff $\mathbb{R}\setminus F$ is open.

Examples. $\emptyset$, $\mathbb{R}$, $[0,1]$, singletons $\{a\}$.

Counterexample. $]0,1]$ is neither open nor closed: it is not open (0 is a boundary point) and its complement $]-\infty,0]\cup]1,+\infty[$ is not open.

Conclusion. Some sets are neither open nor closed.