\[ -_{1}\boxtimes _{X\times Y}-_{2}\colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\times Y\webright ) \]
\begin{align*} U\boxtimes _{X\times Y}V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{pr}^{-1}_{1}\webleft (U\webright )\cap \text{pr}^{-1}_{2}\webleft (V\webright )\\ & = \webleft\{ \webleft (u,v\webright )\in X\times Y\ \middle |\ \text{$u\in U$ and $v\in V$}\webright\} . \end{align*}
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Interaction With Direct Images. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be functions. The diagram
commutes, i.e. we have
\[ \webleft [f_{*}\times g_{*}\webright ]\webleft (U\boxtimes _{X\times Y}V\webright )=f_{*}\webleft (U\webright )\boxtimes _{X'\times Y'}g_{*}\webleft (V\webright ) \]
for each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright )$.
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Interaction With Inverse Images. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be functions. The diagram
commutes, i.e. we have
\[ \webleft [f^{-1}\times g^{-1}\webright ]\webleft (U\boxtimes _{X'\times Y'}V\webright )=f^{-1}\webleft (U\webright )\boxtimes _{X\times Y}g^{-1}\webleft (V\webright ) \]
for each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X'\webright )\times \mathcal{P}\webleft (Y'\webright )$.
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Interaction With Direct Images With Compact Support. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be functions. The diagram
commutes, i.e. we have
\[ \webleft [f_{!}\times g_{!}\webright ]\webleft (U\boxtimes _{X\times Y}V\webright )=f_{!}\webleft (U\webright )\boxtimes _{X'\times Y'}g_{!}\webleft (V\webright ) \]
for each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (Y\webright )$.
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Interaction With Diagonals. The diagram
i.e. we have
\[ U\cap V=\Delta ^{-1}_{X}\webleft (U\boxtimes _{X\times X}V\webright ) \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.