Suppose $C=\bigcup_{1}^{n} F_i\cap O_i$ is a constructible subset of $X$.
$\mathcal{F}$ is a sheaf of abelian groups on $C$, how can we extend it by zero to a sheaf $\mathcal{F'}$ of $X$? That is, how to write out $\mathcal{F'}(U)$ with $\mathcal{F}(V)$?
Also is there an example of a constructible set which cannot be written as a $\mathbb disjoint$ union of locally closed set?
I think if it can be expressed as disjoint union, then we can extend step by step on locally closed subsets and sum up.
I am not sure if it is meaningful to consider the extension of an arbitrary subset.Can this be done?