Let $U$ be a subset of $A$.[1][2]
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The image part of the direct image with compact support $f_{!}\webleft (U\webright )$ of $U$ is the set $f_{!,\text{im}}\webleft (U\webright )$ defined by
\begin{align*} f_{!,\text{im}}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright )\cap \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ b\in B\ \middle |\ \text{we have $f^{-1}\webleft (b\webright )\subset U$ and $f^{-1}\webleft (b\webright )\neq \emptyset $}\webright\} .\end{align*}
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The complement part of the direct image with compact support $f_{!}\webleft (U\webright )$ of $U$ is the set $f_{!,\text{cp}}\webleft (U\webright )$ defined by
\begin{align*} f_{!,\text{cp}}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright )\cap \webleft (B\setminus \mathrm{Im}\webleft (f\webright )\webright )\\ & = B\setminus \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ b\in B\ \middle |\ \text{we have $f^{-1}\webleft (b\webright )\subset U$ and $f^{-1}\webleft (b\webright )=\emptyset $}\webright\} \\ & = \webleft\{ b\in B\ \middle |\ f^{-1}\webleft (b\webright )=\emptyset \webright\} .\end{align*}
