= Solution
For a point $x\in X$ and an <abelian group> $A$, the <skyscraper sheaf> $i_*A$, with $i:\{x\}\hookrightarrow X$, is
$$
(i_*A)(U)=\begin{cases}A,&x\in U,\\0,&x\notin U.\end{cases}
$$
Restrictions are identities or the map to zero, so it is a <flasque sheaf>. The global sections equal $A$ and higher cohomology vanishes:
$$
\boxed{H^0(X,i_*A)=A,\qquad H^i(X,i_*A)=0\quad(i>0).}
$$
No closed-point assumption is needed for this calculation of <skyscraper sheaf cohomology>. On an arbitrary <topological space>, its nonzero stalks are at points of $\overline{\{x\}}$ when $A\ne0$; only for a closed point is the nonzero stalk confined to $x$.
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