The Clowder Project

Let $U$ be a subset of $X$.¹²

1. 1. The image part of the direct image with compact support $f_{!}\left(U\right)$ of $U$ is the set $f_{!,\text{im}}\left(U\right)$ defined by
   $$\begin{array}{rl}{f}_{!,\text{im}}\left(U\right)& \stackrel{\text{def}}{=}{f}_{!}\left(U\right)\cap \mathrm{Im}\left(f\right)\\ & =\{y\in Y|\text{we have }{f}^{-1}\left(y\right)\subset U\text{ and }{f}^{-1}\left(y\right)\ne \text{Ø}\}.\end{array}$$
2. 2. The complement part of the direct image with compact support $f_{!}\left(U\right)$ of $U$ is the set $f_{!,\text{cp}}\left(U\right)$ defined by
   $$\begin{array}{rl}{f}_{!,\text{cp}}\left(U\right)& \stackrel{\text{def}}{=}{f}_{!}\left(U\right)\cap (Y\setminus \mathrm{Im}\left(f\right))\\ & =Y\setminus \mathrm{Im}\left(f\right)\\ & =\{y\in Y|\text{we have }{f}^{-1}\left(y\right)\subset U\text{ and }{f}^{-1}\left(y\right)=\text{Ø}\}\\ & =\{y\in Y|f^{-1}\left(y\right)=\text{Ø}\}.\end{array}$$