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A scheme $X$ is separated if the daiganoal morphism $\Delta:X \rightarrow X \times_{\Bbb Z} X$ is a closed immersion. I know how to show that all affine schemes are separated. So

Are open subschemes of affine schemes separated? If not, what is an example?

For example, I would be interested to know if the open subscheme $$ D= Spec\, k[x.y] \setminus \{0\} \rightarrow Spec \,k[x.y] $$ is separated.

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They are. In fact, any open subscheme of a separated scheme is separated.

Let $U$ be an open subscheme of X. Base change $X\to X\times_\mathbb{Z}X$ by the map from $U\times_\mathbb{Z}U$ induced by the inclusion $U\to X$. The fiber product is just $U$, and the induced morphism is the diagonal of $U$. Since closed immersions are stable under base change, $U$ is also separated.

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