Cardinality of Schemes

Question

I was thinking about set theoretic considerations of scheme theory and a question came to me.

I was wondering if there is a way to bound the cardinality of a scheme $S$ (the underlying set of the underlying topological space) if we know a bound on the cardinality of its rings of regular functions ($\sup_{U\subset S\;\text{open}}\lvert\Gamma(U,\mathcal{O}_S)\rvert\leq\alpha$) and let’s say a bound on the cardinality of its topology ($\rvert\{U\text{open of}\;S\}\rvert\leq\alpha$).

Any suggestion is much appreciated,

Thanks in advance,

Max

Answer 1

In fact the problem is quite trivial as $\Gamma(U,\mathcal{O}_S)=A$ for all affine of S written $U=Spec(A)$.