Showing certain subspaces of $\mathbb{R}^3$ are path connected

Question

Motivation: Let $X \subset \mathbb{R}^3$ be the union of $n$ lines through the origin. Compute $\pi_1(\mathbb{R}^3-X)$ by applying the Seifert-van Kampen theorem.

Let $U=\mathbb{R}^3-(X \cup B_{1-\epsilon}(0))$, $V=(\mathbb{R}^3-X)\cap B_{1+\epsilon}(0)$

I would like to show that $U, V, U\cap V$ are path connected and it seems obvious to me that they are. Removing a ball from $\mathbb{R}^3$ doesn't mess up path connectedness because you can find a path between any two points that goes around the ball. For the same reason, a path between two points in $\mathbb{R}^3-X$ would only have to deviate from being a straight line at most $n$ times.

I would like to compose these ideas into something that is acceptable to turn into my professor. Could anyone help me organize this using math words? Thank you.

Answer 1

You should first project your space to the 2 dimensional sphere (while this is not necessary, I think that it makes it easier to understand). You should start with the homotopy $f(x,t)=\frac{x}{(1-t)+t|x|}$. Clearly, this is continuous (because $x=0$ is not in your space), and you have that $f(x,0)=x$ and $f(x,1)=\frac{x}{|x|}$ which is on the sphere. It follows that the fundamental group of your space is the same as the fundamental group of the sphere minus 2n points, which is the same as the fundamental group of a disk minus (2n-1) point (they are homeomorphic).

Now you can use Seifert-van Kampen theorem and write you space as $U\cup V$ where $U$ is homeomorphic to a disk with one missing point, $V$ is homeomorphic to a disk with 2n-2 points and $U\cap V$ is contractible. Continue to decompose $V$ by induction on the number of missing points to get the fundamental group.