derived smooth geometry
One reason the category SmthMfd of smooth manifolds and smooth functions is important is that a good supply of bump functions exists; this fact accounts for the great flexibility of smooth objects, in stark contrast to analytic geometry or algebraic varieties (since non-zero analytic or algebraic functions with compact support exist only on compact spaces). For instance, bump functions can be used to construct partitions of unity, which can in turn be used to smoothly patch together local structures into a global structure without obstruction, as for example Riemannian metrics.
For , define , and define
so that the family is a smooth approximation to the identity? (of convolution, the Dirac functional), compactly supported on the closed ball of radius and having an -norm equal to . Then, it is standard that for any pair with compact and open in a cartesian space , we can choose an open containing and with compact closure contained in , and then taking the convolution product
of with the characteristic function , for sufficiently small, we obtain a smooth function equal to on and equal to outside .
By performing the above construction in charts, we obtain, in any smooth manifold, a smooth function equal to on any compact subspace and equal to outside any neighbourhood of . (This is a smooth regularity property.)