Q:

For which $m,n$, are the affine spaces $\mathbb A^n(k)$ and $\mathbb A^m(k)$ , with Zariski topology, are homeomorphic?

Let $k$ be an algebraically closed field. For each $n \in \mathbb N$, we can equip the affine space $\mathbb A^n(k)$ with Zariski topology. For which $m,n$, are the affine spaces $\mathbb A^n(k)$ and $\mathbb A^m(k)$ , with Zariski topology, are homeomorphic ?

A:

The maximal length of a chain of closed irreducible subsets is a topological invariant. Therefore, if and only if $n=m$.

