TITLE: Is a single point in euclidean space open, closed, neither or both?
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: Balls must have a radius greater than zero or else the definition would not be very useful, since that would mean any point is an interior point. Any neighborhood of radius greater than zero would include more than one point, so it cannot be a subset of E. Hence E cannot not be open.