How do I show that the following set is not simply connected?
$M= \mathbb{R}^3\setminus\\{(x,y,z)\in \mathbb{R}^3|y=z=1\\} $
I am aware that if a set has a hole, then it isn't simply connected. But how can I show this? In the example I gave, I know that the set isn't simply connected because there is a "hole" which is the line y=z=1, basically there exists a loop on $\mathbb{R}^3$ which "contains" this hole. But I don't know how to formalize this.
An example of the loop would be something like.
$\left(
\begin{array}{c}
x\\\\1+\cos(\phi)\\\\1+\sin(\phi)
\end{array}
\right),\phi\in[0,2\pi[$
kreijstal
·
2021-05-23T20:44:27Z (over 3 years ago)