The Stacks project

Example 6.11.4. Suppose $X = \mathbf{R}^ n$ with the Euclidean topology. Consider the presheaf of $\mathcal{C}^\infty $ functions on $X$, denoted $\mathcal{C}^\infty _{\mathbf{R}^ n}$. In other words, $\mathcal{C}^\infty _{\mathbf{R}^ n}(U)$ is the set of $\mathcal{C}^\infty $-functions $f : U \to \mathbf{R}$. As in Example 6.7.3 it is easy to show that this is a sheaf. In fact it is a sheaf of $\mathbf{R}$-vector spaces.

Next, let $x \in X = \mathbf{R}^ n$ be a point. How do we think of an element in the stalk $\mathcal{C}^\infty _{\mathbf{R}^ n, x}$? Such an element is given by a $\mathcal{C}^\infty $-function $f$ whose domain contains $x$. And a pair of such functions $f$, $g$ determine the same element of the stalk if they agree in a neighbourhood of $x$. In other words, an element if $\mathcal{C}^\infty _{\mathbf{R}^ n, x}$ is the same thing as what is sometimes called a germ of a $\mathcal{C}^\infty $-function at $x$.

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