I'm trying to determine some criteria for when the closure of a closed irreducible set $X$ of $\mathbb A^n$ is equal to the set itself. That is if $\overline{X}$ is the closure of $X$ in $\mathbb P^n$ and $X$ is a closed irreducible set of $\mathbb A^n$ then (by identifying $\mathbb A^n$ with $\mathbb P^n_{x_0,\dots,x_n} \setminus V_{\mathbb P^n}(x_n)$) when exactly is $X = \overline{X}$?
My thinking is this is perhaps true only when $X$ is the set of a single equivalence class of $\mathbb P^n$ with one nonzero entry, is this the case? Namely, is $X = \overline{X}$ if and only if $X = \{[a_0,\dots,a_{n-1} : 1]\}$?
We are working over a field $K$ that is algebraically closed on characteristic zero.