I'm trying to prove that if you remove a finite number of points from $\Bbb R^2$, you get a connected set. Let $X\subset R^2$ be a finite set. Every open set in the induced topology of $\Bbb R^2 - X$ is of the form $O-X$, where $O$ is open in $\Bbb R^2$. We must show that if $O-X$ is also closed in $\Bbb R^2 - X$, then it's either the whole space, or the empty set. So assume it's closed (in the induced topology).
I was hoping to show that therefore $O$ would have to be closed in $\Bbb R^2$, and since $\Bbb R^2$ is connected, therefore $O$ is either the plane or the empty set. So if $x\not\in O$, then if $x\not\in X$, we can wrap $x$ in a neighborhood that doesn't touch $O$, since $O-X$ is closed in the induced topology. However, if $x\in X$, we can't use that argument, and this is where I get stuck.