The Clowder Project

Let $X$ be a set.

1. 1.

   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 $V^{\textsf{c}}\subset U^{\textsf{c}}$.
2. 2.

   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 $U,V\in \mathcal{P}\webleft (X\webright )$.

3. 3.

   Involutority. The diagram

   
   commutes, i.e. we have
   \[ \webleft (U^{\textsf{c}}\webright )^{\textsf{c}}=U \]

   for each $U\in \mathcal{P}\webleft (X\webright )$.