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Unformatted text preview: Lecture 34 6.6 Orientation of Manifolds Let X be an ndimensional 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 dont 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 k ( X ). Then supp = { p X : p = 0 } . (6.71) We use the notation k supp 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 { i C ( X ) : i = 1 , 2 , 3 , . . . } is a partition of unity if 1. 0 i , 2. For every compact set A X , there exists N > 0 such that supp i A = for all i > N , 3. i = 1. Suppose the collection of sets U = { U : I } is a covering of X by open subsets U of X . Definition 6.23. The partition of unity i , i = 1 , 2 , 3 , . . . , is subordinate to U if for every i , there exists I such that supp i U . Claim. Given a collection of sets U = { U : I } , there exists a partition of unity subordinate to U . 1 Proof. For each I , let U be an open set in R N such that U = U X . We define the collection of sets U = { Let U : I } . U = U . (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 X : X U be the inclusion map. i...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
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