Consider the following statement:
Let $C \subset \mathbb{R}^{n}$ be a closed convex set. For $x \in \mathbb{R}^{n}$ the projection of $x$ on $C$, denoted by $\text{Proj}_{C}(x)$, is defined as follows
$$ \text{Proj}_{C}(x) = \text{argmin}_{y \in C} ||x-y||_{2}^{2}. $$ Then it can be shown that $\text{Proj}_{C}(x)$ is unique for all $x \in \mathbb{R}^{n}$. Why the set $C$ must be closed or convex for the uniqueness?
If it's not closed, the projection might not exist. Consider e.g. $C = (0,1) \subseteq \mathbb{R}$ and $x = 2$.
If it's not convex the projection might not be unique. For this case consider e.g. $C = \{0\} \cup \{2\} \subseteq \mathbb{R}$ and $x = 1$. All elements in $C$ minimize the distance to $x$.