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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\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$ and $f^{-1}\webleft (y\webright )\neq \text{Ø}$}\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 (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\\ & = Y\setminus \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$ and $f^{-1}\webleft (y\webright )=\text{Ø}$}\webright\} \\ & = \webleft\{ y\in Y\ \middle |\ f^{-1}\webleft (y\webright )=\text{Ø}\webright\} .\end{align*}
1Note that we have
\[ f_{!}\webleft (U\webright )=f_{!,\text{im}}\webleft (U\webright )\cup f_{!,\text{cp}}\webleft (U\webright ), \]
as
\begin{align*} f_{!}\webleft (U\webright ) & = f_{!}\webleft (U\webright )\cap Y\\ & = f_{!}\webleft (U\webright )\cap \webleft (\mathrm{Im}\webleft (f\webright )\cup \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\webright )\\ & = \webleft (f_{!}\webleft (U\webright )\cap \mathrm{Im}\webleft (f\webright )\webright )\cup \webleft (f_{!}\webleft (U\webright )\cap \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!,\text{im}}\webleft (U\webright )\cup f_{!,\text{cp}}\webleft (U\webright ). \end{align*}
2In terms of the meet computation of $f_{!}\webleft (U\webright )$ of Remark 2.6.3.1.3, namely
\[ f_{!}\webleft (\chi _{U}\webright ) =\bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=-_{1}}}\webleft (\chi _{U}\webleft (x\webright )\webright ), \]
we see that $\smash {f_{!,\text{im}}}$ corresponds to meets indexed over nonempty sets, while $\smash {f_{!,\text{cp}}}$ corresponds to meets indexed over the empty set.
