# 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 $$X$$?

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• A variant was asked here: mathoverflow.net/questions/331673/… but Francesco's answer applies to both. – Bort May 16 at 12:16
• OK I see, I lost the assumption "affine" from the linked question. – Bort 2 days ago

## 2 Answers

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.