Problem: Show that $\pi_1(M\setminus${k points}$) = \pi_1(M)$, where $M$ is an n-dimensional manifold ($n\ge 3)$ and k is a positive integer.
In class, we went over the proof for the above problem for the case $k=1$. I am now trying to scale it for any positive integer $k$.
Let $D_i$ be an open ball around puncture $i$, where $i = 1,...,k$.
To use Van Kampen's theorem, I need the intersection between the balls and $M\setminus${k points} to be simply connected.
To ensure that this is indeed the case,
- Do I need to specify that the open balls around each puncture overlap?
- Since the union of open sets is open, can I just consider the open ball with all k punctures, or is this not possible for all manifolds?