Skyscraper sheaf

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

For any point inclusion $i:\{x\}\hookrightarrow X$, the skyscraper sheaf $i_{\ast }A$ has restriction maps that are identities or maps to zero. It is flasque, so its higher sheaf cohomology vanishes and its global sections are $A$. This requires no assumption that $x$ is closed.