# The underlying space of open dense subscheme

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?