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Old 2013-05-21, 22:25   #5
jinydu's Avatar
Dec 2003
Hopefully Near M48

33368 Posts

Here's the partition of unity theorem:

Let X be a manifold. For any covering of X by open sets {U_alpha}, there is a sequence of smooth functions {theta_i} on X such that:
(a) 0 <= theta_i <= 1
(b) Each x\in X has a neighborhood on which all but finitely many of the theta_i are zero
(c) Each theta_i is supported on one of the U_alpha
(d) \sum_i theta_i = 1 identically

I understand the statement of the theorem. Just not seeing how to apply it...

Here's another problem that has partition of unity as a hint, by the way:

Let X be a manifold with boundary. Then there is a smooth nonnegative function f:X->R with regular value at 0 such that (boundary)X = f^-1(0)


Last fiddled with by jinydu on 2013-05-21 at 22:27
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