# Is a single point in euclidean space open, closed, neither or both?

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?

• Do you have any thoughts? In particular proving it isn't open should be quite obvious Mar 29, 2015 at 20:12
• @HBeel Proving it isn't open would not say about closed or not Jan 22, 2022 at 15:45

One point sets are closed in $\mathbb{R}^n$. The only closed and open sets are $\emptyset,\mathbb{R}^n$.
• @CharlieParker - You can clearly see its closed, as $\cap_{n \geq 1} [x-1/n, x+1/n] = \{x\}$ is a countable intersection of closed sets. As for it not also being open, note that its complement is not closed -- $x$ is a limit point. Alternatively, if you know about connectedness, it is also not hard to prove that a topological space is connected if and only if the only sets which are both closed and open are the empty set and whole set: proofwiki.org/wiki/Connected_iff_no_Proper_Clopen_Sets. Jun 11, 2018 at 1:49