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   \Blatt{1}
   \Abgabe{25/04/2002}


   \Nr
   Let $\kf$ be a presheaf of sets on the topological space $X$
   satisfying condition (S1), let $U\subseteq X$ be open and
   $s,t\in\kf(U)$. Show that $s=t$ if and only if $s_x=t_x$ for all
   $x\in U$.

   \Nr
   Let $\kf$ be a presheaf of sets on $X$ and let
   $\theta:\kf\rightarrow\widehat{\kf}$  be the associated
   sheaf. Show:
   \begin{enumerate}
   \item $\widehat{\kf}$ is uniquely determined up to unique
     isomorphism.
   \item $\kf$ is a sheaf if and only if $\theta$ is an isomorphism.
   \end{enumerate}

   \Nr
   Let $\kf$ be a presheaf of abelian groups on $X$. Show:
   \begin{enumerate}
   \item If $U\subset X$ is open and $s\in\kf(U)$, then
     $\supp(s):=\{x\in U\;|\;s_x\not=0\}$ is closed in $U$.
   \item Give examples which show that $\supp(\kf):=\{x\in
     X\;|\;\kf_x\not=0\}$ may be closed or not.
   \end{enumerate}

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