After reading:
Why is the fundamental group a sheaf in the etale topology?
I followed read the linked survey paper by Minhyong Kim: http://people.maths.ox.ac.uk/kimm/papers/leeds.pdf
where at the top of page 8, he says that if $X$ is a variety over $\mathbb{Q}$ with a $\mathbb{Q}$-rational point $b\in X(\mathbb{Q})$, then the universal cover $\tilde{X}$ (defined as the inverse limit of all finite etale covers of $X$ by geometrically connected $\mathbb{Q}$-varieties) must also have a rational point $\tilde{b}\in\tilde{X}$.
This seems patently false. For example, let $X = \mathbb{P}^1_{\mathbb{Q}} - \{0,1,\infty\}$, then by Belyi's theorem every curve over $\mathbb{Q}$ admits an unramified map to $X$, including curves without any rational points (for example, conic sections like $x^2 + y^2 = -1$), which would imply that the universal cover of $X$ doesn't have any rational points.
Perhaps I'm misinterpreting his statement? Could he be assuming that $X$ is proper? Even so, I don't see why every cover of $X$ must have a rational point.
