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M he the inclusion map.

Defined by o i s Coo. Similarly, there is a Coo function k : elsewhere. :::: 0, increasing on (0,8), and � g: �n � g(x) j(XI ) ... j(xn ) (4) The function = defined by je is Coo; it is positive on (-e, e) /e x . " x 1 x for -E > , 2: 8. e (-e, e) and 0 elsewhere. On a Coo manifold M we can now produce many non-constant Coo func­ tions. The closure is called the support of f, and denoted simply by support f (or sometimes supp f)· {x : /(x) oft O} 2. LEMMA. Let C c U e M with C compact and U open.

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