The direct image with compact support function associated to $f$ is the function
\[ f_{!}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]
\begin{align*} f_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, if we have}\\ & \text{$f\webleft (a\webright )=b$, then $a\in U$}\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ \text{we have $f^{-1}\webleft (b\webright )\subset U$}\webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (A\webright )$.
