I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^2)$, and considering the $\omega$ power.
For example, from the title, is
$(\omega \times \omega)^{\omega}\cong (\omega \times \omega) \times (\omega \times \omega) \times... \ \cong \ (\omega \times \omega \times ...) \ \cong \ \omega^{\omega}$?
On one hand, we can analyze the function spaces and determine if they are homeomorphic. I'm wondering is there some sort of "algebra" that I can use in this type of context.