The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable number of half open intervals glued together.
My question is about not-so-long rays, where you take a countable ordinal $\alpha$, and make the space $\alpha \times [0,1)$. This is claimed to be homeomorphic to $[0,1)$. How does one define this homeomorphism? I see how this can be defined for $\omega$, $\omega^2$, and others, but I don't know how to do it in general.
This property is used to prove, for example, that every interval on the long line is homeomorphic to an interval on the real line.