# Sheaf of infinitely differentiable functions

From Diffgeom

## Definition

Let be a differential manifold. The **sheaf of differentiable functions** of is defined as follows:

- To every open set, we associate the ring of all differentiable functions from that open set to the real numbers (the ring structure arises from pointwise operations)
- The restriction map is simply function restriction

In fact, a differential manifold is completely characterized by its sheaf of differentiable functions. In other words, given a topological manifold and the sheaf of differentiable functions arising from some choice of differential structure on it, the differential manifold structure can be recovered from the sheaf.