John Lee,Smooth mfds exercise 2.16

[seydunas]

Hi,

i have tried to solve exercise2.16 in Lee' s book, smooth manifolds: Let X be a top. mfd with the propert that for every open cover O there exist partition of unity subordinate to O. Show X is paracompact.

I think there is a trick to construct locally refinement open cover by using the conditions of partition of unity. But i can not find it. Can you help me?

 

[mathwonk]

I don't even remember the definitions, but isn't this trivial? re-read the definition of partition of unity and see if it doesn't involve a locally finite condition. remember the set where a continuous function does not vanish is open. i believe this is a tautology, i.e. like proving A+B implies B. see if you agree.

 

[seydunas]

If we can take the interior of support coming from partition of unity, we are done.