John Lee,Smooth mfds exercise 2.16

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In summary, The conversation is discussing how to prove that a topological manifold X is paracompact, given the condition that for every open cover O there exists a partition of unity subordinate to O. The speaker suggests using the locally finite condition of partition of unity to construct a locally refinement open cover. They also mention that the set where a continuous function does not vanish is open and that this may be a tautology. The conversation concludes by stating that if the interior of support from the partition of unity can be taken, then the proof is complete.
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seydunas
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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?
 
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  • #2
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.
 
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  • #3
If we can take the interior of support coming from partition of unity, we are done.
 

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