Sheaf of multiply differentiable functions
This article describes a sheaf on a manifold (the manifold may possess some additional structure in terms of which the sheaf is defined)
View other sheaves on manifolds
Let be a -manifold. The sheaf of -functions' on , also called the sheaf of times differentiable functions, is defined as follows:
- To each open set , it associates the ring (or rather -algebra) of -functions on
- The restriction is the usual function restriction
The sheaf of -functions in fact completely encodes the -structure. In other words, given a topological manifold with a sheaf of functions that is supposed to be the sheaf of -functions, the -structure on the manifold is dictated by the sheaf.
Étale space is non-Hausdorff
The étale space of the shead of multiply differentiable functions is not Hausdorff. This is essentially because there can be -functions which look the same in one direction but different in others.
The Hausdorffization of the étale space yields the space of polynomial functions of degree at most .
Further information: Hausdorffization of étale space of multiply differentiable functions