Sheaf of infinitely differentiable functions

Definition

Let $M$ be a differential manifold. The sheaf of differentiable functions of $M$ 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.