TITLE: Topology on the set of analytic functions
QUESTION [33 upvotes]: Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$.
Everyone who worked with this set knows that there is only one reasonable topology
on it: the uniform convergence on compact subsets of $D$.

Is this statement correct?

If yes, can one state and prove a theorem of this sort: 

If a topology on $H(D)$ has such
  and such natural properties, then it must be the topology of uniform convergence on
  compact subsets.

One natural property which immediately comes in mind is that the point
evaluatons must be continuous. What else?
EDIT: 1. On my first question, I want to add that other topologies were also studied.
For example, pointwise convergence (Montel, Keldysh and others). Still we probably all feel
that the standard topology is the most natural one.

If the topology is assumed to come from some metric, a natural assumption would be
that the space is complete. But I would not like to assume that this is a metric space
a priori.
As other desirable properties, the candidates are continuity of addition and multiplication. However it is better not to take these as axioms, because the topology
of uniform convergence on compacts is also the most natural one for the space of
meromorphic functions (of one variable), I mean uniform with respect to the spherical metric.

REPLY [2 votes]: Given any topological space $(T,\tau)$ and a mapping 
$\varphi :T\to T$, it is natural to ask 

What is the coarsest topology, finer than $\tau$, such 
  that $\varphi$ is continuous. 

This topology is the supremum of all $\tau_n$, which is constructed by adding the inverse images, through $\varphi$, of all open subsets of $\tau_{n-1}$. Note it $\hat{\tau}$.
If one takes the space $T_0$ of infinitely differentiable functions on $\mathbb{R}$, $\tau_0$, the topology of local uniform convergence (which is reasonable to preserve continuity) and $\varphi=\frac{d}{dx}$, then $\hat{\tau_0}$ is the topology of compact uniform convergence of functions and all their derivatives (one can check that the sequence $\tau_{n}$ is strictly increasing). The process is stationary iff $\varphi$ is already continuous (obvious). What is remarkable is that, due to Cauchy, $\varphi$ is continuous for $\tau_0$. The process can be applied to a set $(\varphi_i)_{i\in I}$ of functions, $\tau_n$ being  constructed by adding the inverse images, through $\varphi_{i}$ of all open subsets of $\tau_{n-1}$ and taking their unions and finite intersections. If, on $C^{\infty}(D;\mathbb{C})$, one takes $\varphi_1=\frac{\partial}{\partial x}$ and $\varphi_2=\frac{\partial}{\partial y}$, one get the topology of compact uniform convergence of functions and all their (partial) derivatives. The process described above is, again, strictly increasing, but the restriction of it to $H(D)$ is stationary which is IMHO remarkable.