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If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite.

This is of course true for $X$ is a scheme. I wish it is also true for the case when $X$ is an algebraic space.

I guess this is because any rational point of $X$ is "scheme-like" i.e. there is an open neighborhood $U\subset X$ of the rational point such that $U$ is a scheme. But I don't know why rational points are scheme -like. What do you think?

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Take the complement of a non-empty open subscheme, and use noetherian induction.

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  • $\begingroup$ This is really a nice way to see the problem, thank you very much! $\endgroup$ – Lei Jan 2 '11 at 20:28

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