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lecture34 - Lecture 34 6.6 Orientation of Manifolds Let X...

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Lecture 34 6.6 Orientation of Manifolds Let X be an n -dimensional manifold in R N . Assume that X is a closed subset of R N . Let f : X R be a C map. Definition 6.21. We remind you that the support of f is defined to be supp f = { x X : f ( x ) = 0 } . (6.69) Since X is closed, we don’t have to worry about whether we are taking the closure in X or in R n . Note that f C 0 ( X ) ⇐⇒ supp f is compact. (6.70) Let ω Ω k ( X ). Then supp ω = { p X : ω p = 0 } . (6.71) We use the notation ω Ω k supp ω is compact. (6.72) c ( X ) ⇐⇒ We will be using partitions of unity, so we remind you of the definition: Definition 6.22. A collection of functions { ρ i C 0 ( X ) : i = 1 , 2 , 3 , . . . } is a partition of unity if 1. 0 ρ i , 2. For every compact set A X , there exists N > 0 such that supp ρ i A = φ for all i > N , 3. ρ i = 1. Suppose the collection of sets U = { U α : α I } is a covering of X by open subsets U α of X . Definition 6.23. The partition of unity ρ i , i = 1 , 2 , 3 , . . . , is subordinate to U if for every i , there exists α I such that supp ρ i U α . Claim. Given a collection of sets U = { U α : α I } , there exists a partition of unity subordinate to U . 1
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˜ ˜ Proof. For each α I , let U α be an open set in R N such that U α = U α X . We ˜ define the collection of sets U = { ˜ Let U α : α I } . ± ˜ ˜ U = U α . (6.73) From our study of Euclidean space, we know that there exists a partition of unity ˜ 0 ( ˜ U ) , i = 1 , 2 , 3 , . . . , subordinate to U ˜ . Let ι X : X U ˜ be the inclusion map. ρ i C Then ρ i = ρ ˜ i ι X = ι ρ ˜ i , (6.74) X which you should check.
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