You can use conectedness of $\mathbb R^n \setminus 0$ for $n\geq 2$ to show that doesn't exist a an $\mathbb R$-division algebra of any odd dimension $n\geq 3$.

Take any odd $n\geq 1$ and a $\mathbb R$-algebra $A$ of dimension $n$. For $a\in A$ denote by $f(a)$ the determinant of the linear map $A\to A$ given by $x\mapsto ax$. This is a continuous function on $A$ and we have $f(1)=1$ and $f(-1)=-1$ because $n$ is odd. If $A$ is a division algebra, then $f(a)$ is nonzero for all $a\neq 0$, what forces $A\setminus 0$ to be disconnected. Hence $n=1$.