Sheaf of infinitely 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)
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Let be a differential manifold. The sheaf of infinitely differentiable functions of is defined as follows:
- To every open set, we associate the ring of all infinitely differentiable functions (-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 infinitely 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.
- Sheaf of multiply differentiable functions
- Sheaf of real-analytic functions
- Sheaf of continuous functions
The sheaf defined here is a soft sheaf: it has the property that any section on a closed subset that extends to a section on an open subset containing it, can in fact extend to a section on the whole space
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The sheaf of infinitely differentiable functions on any differential manifold is a soft sheaf; in other words, given a differentiable function in an open neighbourhood of a closed subset, there is a differentiable function on the whole manifold whose germ on the closed subset is the same as that of the function on the open neighbourhood.
The etale space associated to the sheaf of infinitely differentiable function on a manifold is not a Hausdorff space. This is a consequence of the fact that for any point , there exists an infinitely differentiable function on a neighbourhood of such that:
- does not have the same germ as the zero function, at the point
- For any open set containing , there exists a nonempty open set contained in such that is identically zero on
The Hausdorffization of the sheaf looks locally like the sheaf of real-analytic functions (though there may be no global interpretation of this if the manifold does not have a real-analytic structure).