Let $E$ a vector normed space with $\dim (E)\ge 2$. Prove that $E\setminus\{0\}$ is connected.
I think I have a counterexample. Let the vector space $\Bbb Q^2$ with any $p$-norm. Then observe that the sets $\Bbb Q\times ((-\infty,\sqrt 2)\cap\Bbb Q)$ and $\Bbb Q\times ((\sqrt 2,\infty)\cap\Bbb Q)$ are open and disjoint (is easy to check that any point of these subsets is an interior point and it union is $\Bbb Q^2$).
Now, the set $\Big(\Bbb Q\times ((-\infty,\sqrt 2)\cap\Bbb Q)\Big)\setminus\{(0,0)\}$ is still open because it is the intersection of open sets. Then the set $\Bbb Q^2\setminus\{(0,0)\}$ is not connected.
This is correct?