[Math] To prove $\mathbb{R}^n\setminus\{ 0\} $is simply connected.

Question

So far I only know that $S$ is simply connected if it is connected and every loop in $S$ can be shrunk continuously to a point.

And in order to prove simply-connectness, I only have this lemma:

Let $X=U\cup V$, with $U,V$ open and simply connected, and $U\cap V$ is path connected, then $X$ is simply connected.

Then can anyone give me a clue how to prove $\mathbb{R}^3\setminus\{ 0\}$ is simply connected?

Thanks in advance!

Answer 1

Let $L$ be a half-line. Claim : $U = R^n \backslash L$ and $V = R^n \backslash (-L)$ are simply connected, and $U \cap V = R^n \backslash (L \cup - L)$ is connected, and since $U \cup V = R^n \backslash 0$ we obtain the result.

(In the comments, I took $n=3$ for simplicity but the result holds for any $n \geq 3$.)