Exercise 5, § 3

Let $A$ and $B$ be topological spaces and let $u: A\rightarrow B$ be a continuous mapping. For every sheaf $\mathscr{F}$ on B and every element $s\in \mathscr{F}(B)$, denote by $s_A$ the element of $(u^*\mathscr{F})(A)$ deduced from $s$.

a) Let $\mathscr{F}$ be a sheaf on B, let $s$ and $s'$ be sections of $\mathscr{F}$ on B. Suppose that, for every pair $(U,U')$ of open sets of B such that $\overset{-1}{u}(U) =\overset{-1}{u}(U')$, one has $U = U'$. Prove that $s=s'$ if $s_A=s'_A$.

b) Prove that $s=s'$ if $u$ is surjective and $s_A=s'_A$.

Suppose henceforth that, for every sheaf $\mathscr{F}$ on B and for every pair $(s, s')$ of sections of $\mathscr{F}$ on B such that $s_A=s'_A$, one has $s=s'$.

c) Suppose that B is an accessible topological space (i.e., the subsets of B reduced to one point are closed, cf. TG, I, p. 100, §8, exerc. 1). Prove that $u$ is surjective.

d) Prove that, for every pair $(U,U')$ of open subsets of B such that $\overset{-1}{u}(U) =\overset{-1}{u}(U')$, one has $U = U'$.

e) Take for B the topological space $\{1,2\}$ endowed with the topology for which the open sets are $\emptyset ,\{1\}$ and $\{1,2\}$. Prove that if $s$ and $s'$ are sections of a sheaf $\mathscr{F}$ on B such that $s_2=s'_2$, then $s=s'$.