Exercise 1, § 3

Let X be a paracompact topological space and let $\mathscr{F}$ be a sheaf of abelian groups on X. (a) Prove that the following two properties are equivalent:

(i) The sheaf of endomorphisms of $\mathscr{F}$ is soft ;

(ii) For every pair $(A,B)$ of disjoint closed subsets of X, there exists an endomorphism $u$ of $\mathscr{F}$ which induces the identity in a neighbourhood of A and which is zero in a neighbourhood of B.

One then says that $\mathscr{F}$ is fine.

(b) Suppose that every point of X possesses a neighbourhood U such that the sheaf $\mathscr{F}|U$ induced on U is fine. Prove that $\mathscr{F}$ is fine.

(c) Let $p$ be an element of $\mathbf{N}\cup \{\infty \}$ and let M be a differentiable manifold over $\mathbf{R}$ of class $C^p$. Prove that the sheaf $\mathscr{C}^p(M;\mathbf{R}^n)$ is fine.