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

Definition

Let $M$ be a $C^{r}$-manifold. The sheaf of $C^{r}$-functions' on $M$, also called the sheaf of $r$ times differentiable functions, is defined as follows:

• To each open set $U$, it associates the ring (or rather $\mathbb {R}$-algebra) of $C^{r}$-functions on $U$
• The restriction is the usual function restriction

The sheaf of $C^{r}$-functions in fact completely encodes the $C^{r}$-structure. In other words, given a topological manifold with a sheaf of functions that is supposed to be the sheaf of $C^{r}$-functions, the $C^{r}$-structure on the manifold is dictated by the sheaf.

Facts

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 $C^{r}$-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 $r$.

Further information: Hausdorffization of étale space of multiply differentiable functions