Holomorphic functions on open subsets and isomorphisms of quotiens. Let $W \subset \mathbb{R}^2$ be an open subset and $U_1 \subset U_2$ two open subsets of $\mathbb{C}$ containing $W$ as a closed subset.
In the following I use the notation $\mathcal{O}$ to indicate the sheaf of holomorphic functions.
Is it true that $\mathcal{O}(U_2 \setminus W) / \mathcal{O}(U_2) $ is isomorphic to $\mathcal{O}(U_1 \setminus W) / \mathcal{O}(U_1) $?
This is a preliminary exercise that I found in some notes of algebraic topology, but I cannot deal very well with analytic stuff, so I ask you some help to make ideas clearer.
Thank you.
 A: For this problem, we use Mittag-Leffler theorem (or Cousin theorem, it depends on the authors). I mean the following theorem.

Let $U,V$ two open sets in $\mathbb{C}$. For any function $F\in \mathcal{O}(U\cap V)$ there are functions $F_1 \in \mathcal{O}(U)$ and $F_2 \in \mathcal{O}(U)$ such that $F_1(z)-F_2(z) = F(z)$ on $U\cap V.$ In a cohomological setting, this theorem says that $$H^1(U,\mathcal{O})=0$$ for any open subset $U.$

Now let us show that $$\mathcal{O}(U \setminus W) / \mathcal{O}(U)\simeq \mathcal{O}(V \setminus W) / \mathcal{O}(V)$$ with $W \subset V\subset U.$ ($W$ closed and $U,V$ open.) Since $V \subset U$ the natural restriction map gives a map $$r : \mathcal{O}(U \setminus W) / \mathcal{O}(U)\to \mathcal{O}(V \setminus W) / \mathcal{O}(V).$$ This map is clearly injective. Indeed, the restriction of $F$ on $V\backslash W$ is holomorphic on all $V$ if $F$ was already holomorphic on all $U$. To prove the surjectivity of $r$, we use Mittag-Leffler theorem. Let $F\in \mathcal{O}(V \setminus W)$ and apply the theorem to the open sets $U\backslash W$ and $V$. We get $F_1 \in \mathcal{O}(U \setminus W)$ and $F_2 \in \mathcal{O}(V)$ such that $$F(z) = F_1(z)-F_2(z)$$ on $U\backslash W \cap V = V \backslash W.$ Then the inverse image of the class $[F]$ by $r$ is exactly the class $[F_1].$ (This is because $F_2\in \mathcal{O}(V)$ so it will disappear in the quotient). Hence the conclusion.
