Why are minimal irreducible closed sets in $A^n$ single points?

Question

In Hartshorne's Algebraic Geometry example 1.4.4, he says

A maximal ideal $m$ of $A = k[x_1,\cdots,x_n]$ corresponds to a minimal irreducible closed subset of $A^n$, which must be a point ...

I don't understand why must it be a point, namely why are points closed in $A^n$? Suppose $p = (a_1,\cdots,a_n)\in A^n$, what's the ideal that $p$ is the common zero of?

Answer 1

The given point is cut out by the maximal ideal $(x_1 - a_1, \dots, x_n - a_n)$. One of the versions of the Nullstellensatz is that all maximal ideals of $k[x_1, \dots, x_n]$ are of this form. Anyway, the Zariski topology is not Hausdorff but it is, at least, $T_1$.