We work in an algebraically closed field $k$. Let $B$ a proper subset of $\Bbb{A}^1$. Apparently we get the result by comparing the coordinate rings of the two sets, and showing they’re not isomorphic as $k$-algebras. But here’s my questions:
I only know the definition of coordinate ring in the case if an affine algebraic set. Apparently we can define the coordinate ring of $B$ as the ring $k[X]/I(B)$, even if $B$ is not an affine algebraic set?
Supposedly, the two coordinate rings are not isomorphic, but we have $I(\Bbb{A}^1)={0}$, and if we take $B=\Bbb{A}^1-{0}$, we have $I(B)={0}$, and the two rings are isomorphic.
It is easier to solve this problem without coordinate rings. If $B$ is a proper subset of $\mathbb A^1$, then it follows from the Fundamental theorem of algebra that there are no nonconstant regular (i.e. polynomial) functions from $\mathbb A^1$ to $B$. And certainly, there are no isomorphism.