Show the following set is connected For any $x \in \Bbb R^n$ how do I show that the set
$B_x := \{{kx\mid k \in \Bbb R}$} is connected.
It should also be concluded that $\Bbb R^n$ is connected.
I was thinking of starting by assuming that the set is not connected.Then there exist $U,V$ relatively open such that $\varnothing =U \cap V$ and $E=U \cup V$??

Prove that it is path-connected by proving that every point in $B_x$ is path-connected with $\vec 0$.
