Let $L=\left \{ (x,0)\in \mathbb{R}^{2}~:~x\geq 0 \right \}$. Prove that $\mathbb{R}^{2}\setminus L$ is path-connected.
It's easy to see why this set is path connected. I'm just not sure how to show it rigorously. Also how would I go about showing a set is not path connected?
Best Answer
For each $p\in\mathbb R^2\setminus L$, the line segment from $p$ to $(-1,0)$ is a subset of $\mathbb R^2\setminus L$. So, given a second point $q$, you have a path in $\mathbb R^2\setminus L$ from $p$ to $(-1,0)$ and another one from $(-1,0)$ to $q$.
And the simpler way to prove that a set is not path connected is to prove that it is not connected.