Suppose that $G$ is an open set in $\mathbb{C}$ and $(\Omega,d)$ is a complete metric space then designate by $C(G,\Omega)$ the set of all continuous functions from $G$ to $\Omega$.
The natural question is that how can we define the metric on $C(G,\Omega)$.
In Conway's book (GTM 11), he exhausts $G$ by a sequence of compact sets, i.e.$G=\cup_{n=1}^{\infty}K_n$ where each $K_n$ is compact and $K_n\subset \operatorname{int} K_{n+1}$, and define $$ \rho _n\left( f,g \right) =\sup\left\{ d\left( f\left( z \right) ,g\left( z \right) \right) :z\in K_n \right\} $$ for all functions $f$ and $g$ in $C(G,\Omega)$. Also define $$ \rho \left( f,g \right) =\sum_{n=1}^{\infty}{\left( \frac{1}{2} \right) ^n\frac{\rho _n\left( f,g \right)}{1+\rho _n\left( f,g \right)}} $$
I wonder that why not define the metric on $C(G,\Omega)$ by $$ \rho \left( f,g \right) =\sup\left\{ d\left( f\left( z \right) ,g\left( z \right) \right) :z\in G \right\} . $$ Are there any particular advantages to the metric defined by the authors?
In fact, I suspect that the metric defined by the second method is not complete. But I can't think of a good example to illustrate this point.