In a euclidean space $\mathbb{R}^k$, is the set consisting of a single point open, closed, neither, or both?
I would say that a set $E$ consisting of a single point $p$ doesn't have any limit points, so $E$ contains all of its limit points and is therefore closed. But it might be open, too, since a ball of radius zero around $p$ is a subset of $E$. When using balls to define interior points, do balls have to have radius greater than zero?
Best Answer
One point sets are closed in $\mathbb{R}^n$. The only closed and open sets are $\emptyset,\mathbb{R}^n$.