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?