Sheaf of infinitely differentiable functions

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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.