1. The problem is to understand what it means for a set $C$ to be neither open nor closed in a topological or metric space. 2. A set is **open** if it contains none of its boundary points, meaning every point in the set has a neighborhood fully contained in the set. 3. A set is **closed** if it contains all of its boundary points, or equivalently, if its complement is open. 4. A set that is neither open nor closed does not satisfy either condition: it does not contain all its boundary points (so not closed), and it does not exclude all boundary points (so not open). 5. For example, the half-open interval $[0,1)$ in the real numbers with the usual topology is neither open nor closed. 6. This is because it contains $0$ (a boundary point) but not $1$ (another boundary point), so it is not closed. 7. Also, it does not contain an open neighborhood around $0$ or $1$, so it is not open. Final answer: A set $C$ that is neither open nor closed contains some but not all of its boundary points, like $[0,1)$ in $\mathbb{R}$.