Skyscraper sheaf cohomology ($H^0(X,i_{\ast }A)=A,H^{i>0}(X,i_{\ast }A)=0$)

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.