So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions:
Theorem 1. Let $K\subset U\subseteq\mathbb{C}$, with $K$ compact and $U$ open. Then the following assertions are equivalent:
1) $K$ is $U$-convex.
2) Every holomorphic function in a neighborhood of $K$ can be uniformly approximated in $K$ by holomorphic functions in $U$.
3) Every holomorphic function in a neighborhood of $K$ can be uniformly approximated in $K$ by rational functions with poles outside of $U$.
Theorem 2. Let $G\subseteq \mathbb{C}$ be a region (open an connected). Then $G$ is simply connected if and only if every holomorphic function in $G$ can be locally uniformly approximated in $G$ by polynomials.
Where:
Definition Let $U\subset\mathbb{C}$ be an open set and $K\subset U$ compact. The $U$-convex hull (?), $\hat{K}_U$ of, $K$is the union of $K$ with every compactly contained connected component of $U\backslash K$. We say that $K$ u $U$-convex if $K = \hat{K}_U$.
Basically, $K$ is $U$-convex if it has no holes in $U$.
As we can see, both theorems relate the topology of a set ($K$ for the first Theorem and $G$ for the second) and the capacity we have to approximate functions in a given set, i.e. the density of certain family of functions in the set of holomorphic functions with a given domain.
Let $H(X)$ be the set of holomorphic functions over a set $X$. Then we can rephrase the previous theorems in the following way:
Theorem 1 Let $K\subset U\subseteq\mathbb{C}$, with $K$ compact and $U$ open. Then, the set of rational functions with poles outside of $U$ is dense in $H(K)$.
Theorem 2 Let $G\subseteq\mathbb{C}$ be a region. Then the polynomials are dense in $H(G)$ iff $G$ is simply connected.
So, my question is if there are more results of this sort which relate topological properties with the density of subsets in function spaces. For example, maybe we can say a set in the complex plane is compact iff we can approximate every holomorphic function in it with some kind of functions.
Also, I'd like to know if using these kind of ideas we can say something about more complex structures as Riemann surfaces or even complex manifolds (maybe say something about their (co)homology groups?).