# The underlying space of an affine open dense subscheme

Let $$X$$ be a Noetherian scheme, $$U\subset X$$ be an affine open dense subscheme. Is the underlying space of $$U$$ necessarily homeomorphic to the underlying space of $$X$$?

• A variant was asked here: mathoverflow.net/questions/331673/… but Francesco's answer applies to both.
– Bort
May 16 '19 at 12:16
• OK I see, I lost the assumption "affine" from the linked question.
– Bort
May 17 '19 at 12:07

Let $$k$$ be an algebraically closed field, ad take $$X=\mathbb{P}^2_k$$, $$U=\mathbb{A}^2_k$$.
Then $$X$$ and $$U$$ are not homeomorphic, since $$U$$ contains two disjoint, Zariski-closed, irreducible subsets made of more than one point (think of two parallel lines), but this is not possible in $$X$$ because of Bézout theorem.
Let $$X=\operatorname{Spec}R$$, where $$R$$ is a discrete valuation ring. It consists if two points $$x,y$$ where $$x$$ is the generic point and $$y$$ is a closed point. Then $$\{x\}$$ is an open, dense, affine subset which is not homeomorphic to $$X$$, since it has fewer points.