4 CHAPTER L Manifolds and Vector Fields 1.4. Smooth functions. The set of smooth real valued functions on a manifold M will be denoted by C°°(M), in order to distinguish it clearly from spaces of sections which will appear later. The space C°°(M) is a real commutative algebra. The support of a smooth function / is the closure of the set where it does not vanish, supp(/) = {x G M : f(x) = £ 0}. The zero set of / is the set where / vanishes, Z(f) = {x G M : f(x) = 0}. 1.5. Theorem. Any (separable, metrizable, smooth) manifold admits smooth partitions of unity: Let {Ua)aeA be an open cover of M. Then there is a family (ipa)aeA of smooth functions on M, such that: (1) pa(x) 0 for all x G M and all a G A. (2) supp((^a) C Ua for all a G A. (3) (supp((pa))aeA ^ & locally finite family (so each x G M has an open neighborhood which meets only finitely many supp((^a)y). (4) ^2a ipa = 1 (locally this is a finite sum). Proof. Any (separable, metrizable) manifold is a 'Lindelof space\ i.e., each open cover admits a countable subcover. This can be seen as follows: Let U be an open cover of M. Since M is separable, there is a countable dense subset S in M. Choose a metric on M. For each U G U and each x G U there is a y G S and n G N such that the ball Bi/n(y) with respect to that metric with center y and radius ^ contains x and is contained in U. But there are only countably many of these balls for each of them we choose an open set U G U containing it. This is then a countable subcover ofW. Now let (Ua)aeA be the given cover. Let us fix first a and x G Ua. We choose a chart (U,u) centered at x (i.e., u(x) = 0) and e 0 such that sBn C u(UnUa), where Dn = {y G W1 : \y\ 1} is the closed unit ball. Let f e" 1 /' for t 0, hit) := [0 for t 0, a smooth function on R. Then _ Jfe(e 2 -|«(*)| 2 ) for^Gt/, / a M * J : _ \0 for zi U is a nonnegative smooth function on M with support in Ua which is positive at x. We choose such a function fa^x for each a and x G Ua. The interiors of the supports of these smooth functions form an open cover of M which refines
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