Question: 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?

Answer: 
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:-)
Until very recently, it was unknown whether or not there are uncountably many topological isomorphism classes of compactly generated simple non-discrete locally compact groups. (There are, thanks to a construction of Simon Smith that appeared on the arXiv earlier this year.) I think it is still unknown whether there are uncountably many *local* isomorphism classes of them.

Also, I am not sure about this, but I think it is still unknown whether or not there are uncountably many commensurability classes of hereditarily just infinite pro-$p$ groups.