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Let $X$ be an affine scheme such that there exists a field $k$ and a morphism of finite type $X\rightarrow \mathrm{Spec}\,k$. Let $U\subset X$ be an open dense subscheme. Is the underlying space of $U$ necessarily homeomorphic to the underlying space of $X$? What if we assume that $k$ is algebraically closed? What if we assume that $U$ is affine?

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