I am beginning my studies on integration on manifolds and i have some theorical questions. First, in all books that I saw they says that the singular p - simplex (or p - cube) are continuous mapping but, i have to do the pull - back of formas induced by them, so, in fact they have to be differentiable, right ? What really means to say that a manifold is orientable and, a singular p - cube preserve this orientation ?
Other question is, how I know that exist a cover such that for all open in it exist a p - cube that preserves orientation ?
Finally, why we request that the set of support of functions of unity partition have to be locally finite ?
Thank you very much for the atention !