 Solution

The restriction maps of this point sheaf satisfy the presheaf composition identities: every possible restriction is either the identity on $\mathbb{C}$ or the unique map to the zero group. We check the sheaf identity and gluing axioms for any open cover $U={\bigcup }_i{U}_i$.

If $p\notin U$, every section group in the cover is zero, so a compatible family has the unique zero gluing. If $p\in U$, at least one $U_i$ contains $p$. For any two members containing $p$, their intersection also contains $p$, and compatibility forces their complex values to be equal, since the restriction maps there are identities. Call the common value $c$. Members not containing $p$ have only the zero section. The section $c\in {\mathbb{C}}_p(U)$ restricts to every member of the family, proving existence of a gluing; its restriction to any member containing $p$ determines $c$, proving uniqueness.

This also proves local identity for sections, and ${\mathbb{C}}_p(\varnothing )=0$ handles the empty open set. HenceIt is the point-pushforward sheaf, usually called a skyscraper sheaf. Its restrictions are all surjective, so it is also a flasque sheaf. No separation assumption on $X$ is needed for the preceding sheaf-axiom argument.