The direct image with compact support function associated to $f$ is the function
\[ f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]
\begin{align*} f_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ y\in Y\ \middle |\ \begin{aligned} & \text{for each $x\in X$, if we have}\\ & \text{$f\webleft (x\webright )=y$, then $x\in U$}\end{aligned} \webright\} \\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$}\webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (X\webright )$.
1Further Terminology: The set $f_{!}\webleft (U\webright )$ is called the direct image with compact support of $U$ by $f$.
2We also have
\begin{align*} f_{!}\webleft (U\webright ) & = f_{*}\webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}Y\setminus f_{*}\webleft (X\setminus U\webright );\end{align*}
see Item 16 of Proposition 2.6.3.1.6.
