Max-spectrum of multivariate polynomial ring over an algebraically closed field

Let $$k$$ be an algebraically closed field. The max-spectrum of $$k[x_1,x_2,\ldots,x_n]$$ is described below.

As a set
As a set, the max-spectrum is identified with $$k^n$$, where the identification is as follows:

$$(a_1,a_2,\ldots,a_n) \mapsto (x_1-a_1,\ldots, x_n-a_n)$$

The left side denotes a point in $$k^n$$ and the right side denotes the maximal ideal for that point. Equivalently, we can think of the maximal ideal as the kernel of the "evaluation map" which sends a polynomial $$p$$ to the value $$p(a_1,a_2,\ldots,a_n)$$.

The above is one of the many formulations of the weak Nullstellensatz for algebraically closed fields.

As a topological space
The topology that we give to the max-spectrum coincides with the usual Zariski topology we put on $$k^n$$.