Are the affine and projective spaces $k^n$ and $P^n(k)$ , homeomorphic w.r.t. to the Zariski topology on them , where $k$ is an algebraically closed field ?
1 Answer
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Yes if $n=1$.
No if $n \ge 2$. One reason: inside $k^n$ there are two closed subsets of dimension $n-1$ with void intersection. This is not true for $P^n(k)$. ( note that the dimension of a topological space is a topological invariant).
answered Mar 26, 2018 at 5:59