We define the $LI$ to be the long line (that is gluing $\omega_1$ copies of the space $(0,1)$ end to end), and similarly we define $LLI$ the really long line to be the gluing of $\omega_2$ copies of $(0,1)$ end to end.
Now for obvious cardinality reasons $LI$ is not homeomorphic to $LLI$. Intution would lead me to believe that they aren't homotopic either (because $LI$ isn't homotopic to the ordinary line $\mathbb{R}$), but I have no idea how to back my intuition up with a proof.
Any help would be greatly appreciated.