II.1.14

This problem has Hartshorne Height 1.

HAPPY

 * Show the complement of $$\mbox{Supp} s$$ is open. If $$ s \in F(U) $$ then since the stalk is a direct limit, for s to map to 0 in the stalk means it maps to 0 in some small enough open neighborhood of the point.  So the any point in the complement is an interior point.
 * One example for $$ \mbox{Supp} F$$ not being closed. Take $$ X = \mathbb{R}$$ with the usual topology.  The for any open set $$U$$ consider the set of real valued functions which are 0 if $$ 0 \in U$$.  Then $$ \mbox{Supp} F$$ is $$ \mathbb{R} \backslash 0$$