Proposition 2.3.11.1.2 (0051): Properties of Complements

Style	Class	Font	Theorem Environments
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Style 4	`book`	Computer Modern	`amsthm`

Let $X$ be a set.

1. Functoriality. The assignment $U \mapsto U^{c}$ defines a functor
$(-)^{c} : P (X)^{op} \to P (X) .$

In particular, the following statements hold for each $U, V \in P (X)$ :
- If $U \subset V$ , then $V^{c} \subset U^{c}$ .
2. De Morgan’s Laws. The diagrams
commute, i.e. we have equalities of sets
$\begin{aligned} (U \cup V)^{c} & = U^{c} \cap V^{c}, \\ (U \cap V)^{c} & = U^{c} \cup V^{c} \end{aligned}$

for each $U, V \in P (X)$ .
3. Involutority. The diagram

commutes, i.e. we have

$(U^{c})^{c} = U$

for each $U \in P (X)$ .
4. Interaction With Characteristic Functions. We have
$χ_{U^{c}} \equiv 1 - χ_{U} (\mod 2)$

for each $U \in P (X)$ .
5. Interaction With Direct Images. Let $f : X \to Y$ be a function. The diagram

commutes, i.e. we have

$f_{*} (U^{c}) = f_{!} (U)^{c}$

for each $U \in P (X)$ .
6. Interaction With Inverse Images. Let $f : X \to Y$ be a function. The diagram

commutes, i.e. we have

$f^{- 1} (U^{c}) = f^{- 1} (U)^{c}$

for each $U \in P (X)$ .
7. Interaction With Direct Images With Compact Support. Let $f : X \to Y$ be a function. The diagram

commutes, i.e. we have

$f_{!} (U^{c}) = f_{*} (U)^{c}$

for each $U \in P (X)$ .