Great Picard's theorem and cardinality Consider a function $f:\mathbb C\to\mathbb C$ with an essential singularity at $\infty.$ Let $\alpha,\beta$ be two distinct complex "non-exceptional" numbers and consider the sets $$A=\{z\in\mathbb C\mid f(z)=\alpha\},\quad B=\{z\in\mathbb C\mid f(z)=\beta\}.$$
Can it happen that $|A|=\aleph_0,|B|=c$? Or are the level sets always either countable or uncountable?

If the set is uncountable then $f$ assumes $\alpha$ or $\beta$ on an open connected set which eventually turns $f$ into a constant function (Identity Theorem). This contradicts that $f$ has an essential sigularity at $\infty$.
