To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called [dyadic spaces][1]. There is a nice result by Haydon: every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion of the Borsuk--Dungundji theorem holds for $X$.)

>  R. Haydon, [On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim)][2],
*Studia Math.* **52** (1974), 23-31.


It is easy to see that the conclusion of the Borsuk--Dugundji theorem fails for $\beta \mathbb{N}$ (it is actually a paradigm counter-example).


  [1]: http://en.wikipedia.org/wiki/Dyadic_space
  [2]: http://matwbn.icm.edu.pl/ksiazki/sm/sm52/sm5213.pdf