Let $A$ and $B$ be separates subsets of $\mathbb R^k$ and let $a\in A$, $b\in B$

Question

$\mathbf{p}(t)=(1-t)a+tb$.

If $A_0 =\mathbf{p}^{-1} (A)$ and $B_0=\mathbf{p}^{-1}(B)$ then there is $t_0\in (0,1)$ such that $\mathbf{p}(t_0)\notin A\cup B$.

help please, I don't know how to do it.

Answer 1

Let $q:[0,1]\to\mathbb R^k$ be prescribed by $t\mapsto p(t)$.

If no such $t_0$ exists then the preimages of $A$ and $B$ under $q$ form a separation of $[0,1]$.

However, its existence contradicts that $[0,1]$ is connected.