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The excercise 1.4.8(a) of Hartshorne's Algebraic Geometry says

Show that any variety of positive dimension over $k$ has the same cardinality as $k$.

Using Hartshorne's notation, we define a quasi-affine variety as an open subset of an affine variety. I was able to prove the hint given in the excercise based on this claim

Claim. Let $X$ be an affine variety and $W\subset X$ a quasi-affine variety, then $|X| = |W|$

Over $\mathbb A^1$ its obvious, but I can't prove the general case. Any help would be appreciated.

ADR
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    Since a positive-dimensional variety $X$ is contained in a projective space, and it is easy to see that the latter have the cardinality of the field, it is enough to show that $X$ has a least the cardinality of the field. If you show that there is a an open set $U\subseteq X$ and a surjective regular function $f:U\to k$, then you are done. – Mariano Suárez-Álvarez Jul 20 '13 at 03:43

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