Show that if $X$ is a connected Riemann surface and $S$ is a set of finite points in $X$, then $X \setminus S$ is connected.
Here is my work so far: since $S$ is a finite set, by induction it suffices to assume that $ = \{ x \}$ is a singleton. Suppose $X \setminus \{x\} = A \cup B$ where $A, B$ are disjoint relatively open sets of $X \setminus \{x\}$. Then there exist open subsets $\tilde{A}$ and $\tilde{B}$ of $X$ such that $A = X \setminus \{x\} \cap \tilde{A}$ and $B = X \setminus \{x\} \cap \tilde{B}$. Because $X$ is a Hausdorff space, $\{x\}$ is closed and thus $X \setminus \{x\}$ is open, so that both $A$ and $B$ are open subsets of $X$. We also have $X = A \cup (B \cup \{x\})$. $B \cup \{x\}$ is a closed subset of $X$. I am not sure how to show that $A$ is a closed subset of $X$ also. Any other ways to solve this problem and any generalization would be greatly appreciated.