Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$.
Do the meromorphic functions separate the points of $S$?
I doubt there are enough such functions, though I could also believe that this could depend on our set theory.
If not, "what is" the space $S/\!\sim$, where $p\sim q$ if $f(p)=f(q)$ for all meromorphic $f$?
Again, I could easily believe that this question has very different answers depending on one's set theory.
I don't have any especially good reason for asking this question. Feel free to retag.