I apologize in advance if this question is thought to be too soft or otherwise inappropriate for mathoverflow.net. Let M be the infinite set of all homeomorphism types of finite dimensional topological manifolds which are compact, connected and metrizable. Many years ago, the question of whether M was countable, was an open question. Eventually the answer was proved to be "Yes". Are there any examples of well known infinite sets (in this or that branch of mathematics) for which the question of whether the set is countable or uncountable is still considered an interesting open problem?



Is a particular type of question about certain infinite sets still being asked?
Here is a question of classical function theory of this sort. Let $f$ be an entire function
of one variable. Consider all factorizations $f=g\circ h$ where $g$ and $h$ are entire.
Two factorizations $f=g\_1\circ h\_1$ and $f=g\_2\circ h\_2$ are called equivalent if there is
a polynomial $L$ of degree $1$ such that $g\_2=g\_1\circ L$ and $h\_2=L^{-1}\circ h\_1$.



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> Can an entire function have uncountably many non-equivalent factorizations?
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An example which has infinitely many is $e^z$ with $g\_n(z)=z^n$ and $h(z)=e^{z/n}$.
This was unsolved when I was a student in 1970-th, and I suppose still is.


EDIT. Another question of this sort from the theory of entire functions was this
(Erdos). Consider a family $A$ of entire functions with the property that for every $z\in C$, the
set $\{ f(z):f\in A\}$ is at most countable. Does it follow that $A$ is at most countable?
But this question has been "solved": it turns out that the answer depends on the Continuum Hypothesis:-) 


