Let $X$ be a set.
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Functoriality. The assignment $U\mapsto U^{\textsf{c}}$ defines a functor
\[ \webleft (-\webright )^{\textsf{c}}\colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\to \mathcal{P}\webleft (X\webright ). \]
In particular, the following statements hold for each $U,V\in \mathcal{P}\webleft (X\webright )$:
- If $U\subset V$, then $V^{\textsf{c}}\subset U^{\textsf{c}}$.
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De Morgan’s Laws. The diagrams commute, i.e. we have equalities of sets
\begin{align*} \webleft (U\cup V\webright )^{\textsf{c}} & = U^{\textsf{c}}\cap V^{\textsf{c}},\\ \webleft (U\cap V\webright )^{\textsf{c}} & = U^{\textsf{c}}\cup V^{\textsf{c}} \end{align*}for each $U,V\in \mathcal{P}\webleft (X\webright )$.
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Involutority. The diagram
commutes, i.e. we have
\[ \webleft (U^{\textsf{c}}\webright )^{\textsf{c}}=U \]for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Characteristic Functions. We have
\[ \chi _{U^{\textsf{c}}}\equiv 1-\chi _{U}\ \ (\mathrm{mod}\ 2) \]
for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have
\[ f_{*}\webleft (U^{\textsf{c}}\webright )=f_{!}\webleft (U\webright )^{\textsf{c}} \]for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have
\[ f^{-1}\webleft (U^{\textsf{c}}\webright )=f^{-1}\webleft (U\webright )^{\textsf{c}} \]for each $U\in \mathcal{P}\webleft (X\webright )$.
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Interaction With Direct Images With Compact Support. Let $f\colon X\to Y$ be a function. The diagram
commutes, i.e. we have
\[ f_{!}\webleft (U^{\textsf{c}}\webright )=f_{*}\webleft (U\webright )^{\textsf{c}} \]for each $U\in \mathcal{P}\webleft (X\webright )$.
