Given a smooth connected manifold $X$ of dim $\geq 2$, I need to show that $X\setminus Y$ is connected, for some $Y\subseteq X$ finite.
The claim is intuitively obvious to me, but is not finding the right argument to prove.
For smooth manifolds, I know that path-connectedness and connectedness are equivalent. To prove that $X\setminus Y$ is path-connected, I take $x,y\in X\setminus Y$. I now need to show that there exists a cont's path $\gamma:[0,1]\longrightarrow X\setminus Y$ such that $\gamma(0)=x, \gamma(1)=y$. Along the way, I need to use the finiteness of $Y$, and also the fact that $X$ is also path-connected. I don't know how?
I haven't done fundamental group yet, so hint that does not involve it would be helpful.