- #1

PsychonautQQ

- 784

- 10

a) If ##C## is an n-manifold and ##X## is Hausdorff, show that ##X## is an n-manifold.

b) If ##X## is an n-manifold, show that ##C##is an n-manifold

c) suppose that ##X## is a compact manifold. Show that ##C## is compact if and only if p is a finite sheeted covering.Thoughts:

a) I will have to show that ##X## is second countable and has a basis of n-euclidean balls. To show that it has a basis of n-euclidean balls, I could take the pre image of any basis element of ##X## and then use any component of the preimage which will be homeomorphic to both a euclidean n-ball and the basis element in ##X##. To show that ##X## is second countable I don't know what I would do yet.

b) The key here is using the fact that slices of the preimage of ##p## will map homeomorphically to their source neighborhood in ##X##. Perhaps if I can show that the preimage of basis elements of ##X## is surjective onto ##C## the proof will not be far away.

c) I could suppose ##C## is compact and then proceed by contradiction: Take the preimage of any neighborhood in ##X## and use this infinite preimage to construct an open covering of ##C## with no finite sub covering. I am hesitant about this strategy because I haven't seen how I can use the fact that ##X## is a compact manifold.

Any feedback is appreciated as always! Thanks PF!