Reference: Conway - Functions of one complex variable
Let $G$ be open in $\mathbb{C}$ and $\{K_n\}$ be a sequence of compact subsets of $G$ such that $\bigcup_n K_n = G$ and $K_{n}\subset Int(K_{n+1})$.
Let $(X,d)$ be a metric space.
Define $\rho_n(f,g)= sup_{z\in K_n} d(f(z),g(z))$ and $\rho(f,g)=\sum_{n=1}^\infty (\frac{1}{2})^n \frac{\rho_n(f,g)}{1+\rho_n(f,g)}$ for $f,g\in C(G,X)$.
Then, the compact-open topology on $C(G,X)$ is metrizable by $\rho$.
However, it is written in the text that the condition $K_n \subset int(K_{n+1})$ can be removed by applying Baire Category theorem. How?
Let $\{K_n\}$ be a sequence of compact subsets of $\mathbb{C}$ whose union is $G$.
Since $\mathbb{C}$ is a complete metric space, it is a Baire space.
Note that $G=int(G)=int(\bigcup_n K_n)$
However, this may not be equal to $\bigcup_n int(K_n)$. So I'm not sure how the above condition could be removed. How do I prove this?