Let $k$ be an algebraically closed field (of characteristic zero, if it helps).


Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the cardinality of $k$.


For every $i$ in $I$, let $S\_i$ be a proper closed subset of $X$.



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I see this fact used quite often when $k$ is the field of complex numbers and $I$ is a countable set. But how does one prove this? I would be happy with a reference.



Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$
If $dim X=1$ then it follows from cardinalities, and if $dimX>1$ then apply induction on dimension to $X \cap H, S\_i \cap H$ where $H$ is a hyperplane such that dimensions of $X \cap H, S\_i \cap H$ all drop by 1 (such $H$ exists by cardinality). (Embed an affine open of $X$ in projective space in general.)


