\[ D_{X}\colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ) \]
\begin{align*} D_{X}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U,\text{Ø}\webright ]_{X}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}} \end{align*}
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Duality. We have
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Duality. The diagram
commutes, i.e. we have
\[ \underbrace{D_{X}\webleft (U\cap D_{X}\webleft (V\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [U\cap \webleft [V,\text{Ø}\webright ]_{X},\text{Ø}\webright ]_{X}}=\webleft [U,V\webright ]_{X} \]
for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images. The diagram
commutes, i.e. we have
\[ f_{*}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{!}\webleft (U\webright )\webright ) \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Inverse Images. The diagram
commutes, i.e. we have
\[ f^{-1}\webleft (D_{Y}\webleft (U\webright )\webright )=D_{X}\webleft (f^{-1}\webleft (U\webright )\webright ) \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images With Compact Support. The diagram
commutes, i.e. we have
\[ f_{!}\webleft (D_{X}\webleft (U\webright )\webright )=D_{Y}\webleft (f_{*}\webleft (U\webright )\webright ) \]
for each $U\in \mathcal{P}\webleft (X\webright )$.