(All manifolds are smooth)
I am studying Density on a smooth manifold which is a smooth section of the Density bundle. An excercise asks
Using the fact that positive densities form an open set of Density bundle and that on each vector space positive densities form a convex set, show that any smooth manifold has a positive density.
My question is this
Why can't the solution of the above problem be used to say any smooth manifold has a positive top form? (I know it's false because there are non orientable manifolds)
Also, how does one show the existence of positive density anyway?