 Direct image sheaf

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image is defined by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$ for every open $V\subseteq Y$.

The direct image sheaf is the result of the direct image functor: $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$.