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?