  Descriptive set theory can be used to prove new results in classical complex
analysis. Let $\mathcal{A}(\Omega)$ be the set of complex analytic functions on
an open subset $\Omega \subseteq \mathbb{C}$ endowed with the usual topology of
uniform convergence on compact subsets. $\mathcal{A}(\Omega)$ is a Polish ring
with the operations of point-wise addition and point-wise multiplication.
Inspired by Bers' algebraic characterization of the relation of conformality,
we show that the topology on $\mathcal{A}(\Omega)$ is the only Polish topology
for which $\mathcal{A}(\Omega)$ is a Polish ring for arbitrary open $\Omega$.
In a different direction, we show that the bounded complex-analytic functions
on the unit disk admits no Polish topology for which it is a Polish ring. Along
these lines, a corollary of our general result is that the abstract field of
meromorphic functions on an open $\Omega$ cannot be made into a Polish field.
We also study the Lie ring structure on $\mathcal{A}(\Omega)$ which turns out
to be a Polish Lie ring with the usual topology. In this case, we restrict our
attention to those domains $\Omega$ that are connected. We extend a result of
Amemiya to see that the Lie ring structure is determined by the conformal
structure of $\Omega$. In a similar vein to our ring considerations, we see
that, for certain domains $\Omega$ of usual interest, the Lie ring
$\mathcal{A}(\Omega)$ has a unique Polish topology for which it is a Polish Lie
ring.
