The Clowder Project

Here are some examples of direct images with compact support.

1. The Multiplication by Two Map on the Natural Numbers. Consider the function $f\colon \mathbb {N}\to \mathbb {N}$ given by
   \[ f\webleft (n\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}2n \]

   for each $n\in \mathbb {N}$. Since $f$ is injective, we have

   \begin{align*} f_{!,\text{im}}\webleft (U\webright ) & = f_{*}\webleft (U\webright )\\ f_{!,\text{cp}}\webleft (U\webright ) & = \webleft\{ \text{odd natural numbers}\webright\} \end{align*}

   for any $U\subset \mathbb {N}$.

2. Parabolas. Consider the function $f\colon \mathbb {R}\to \mathbb {R}$ given by
   \[ f\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x^{2} \]

   for each $x\in \mathbb {R}$. We have

   \[ f_{!,\text{cp}}\webleft (U\webright )=\mathbb {R}_{<0} \]

   for any $U\subset \mathbb {R}$. Moreover, since $f^{-1}\webleft (x\webright )=\webleft\{ -\sqrt{x},\sqrt{x}\webright\} $, we have e.g.:

   \begin{gather*} f_{!,\text{im}}\webleft (\webleft [0,1\webright ]\webright ) = \webleft\{ 0\webright\} ,\\ f_{!,\text{im}}\webleft (\webleft [-1,1\webright ]\webright ) = \webleft [0,1\webright ],\\ f_{!,\text{im}}\webleft (\webleft [1,2\webright ]\webright ) = \text{Ø},\\ f_{!,\text{im}}\webleft (\webleft [-2,-1\webright ]\cup \webleft [1,2\webright ]\webright ) = \webleft [1,4\webright ]. \end{gather*}
3. Circles. Consider the function $f\colon \mathbb {R}^{2}\to \mathbb {R}$ given by
   \[ f\webleft (x,y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x^{2}+y^{2} \]

   for each $\webleft (x,y\webright )\in \mathbb {R}^{2}$. We have

   \[ f_{!,\text{cp}}\webleft (U\webright )=\mathbb {R}_{<0} \]

   for any $U\subset \mathbb {R}^{2}$, and since

   \[ f^{-1}\webleft (r\webright )= \begin{cases} \text{a circle of radius $r$ about the origin} & \text{if $r>0$,}\\ \webleft\{ \webleft (0,0\webright )\webright\} & \text{if $r=0$,}\\ \text{Ø}& \text{if $r<0$,} \end{cases} \]

   we have e.g.:

   \begin{gather*} f_{!,\text{im}}\webleft (\webleft [-1,1\webright ]\times \webleft [-1,1\webright ]\webright ) = \webleft [0,1\webright ],\\ f_{!,\text{im}}\webleft (\webleft (\webleft [-1,1\webright ]\times \webleft [-1,1\webright ]\webright )\setminus \webleft [-1,1\webright ]\times \webleft\{ 0\webright\} \webright ) = \text{Ø}. \end{gather*}