Let $U$ be an arbitrary set, and $\mathcal{P}(U)$ the power set of $U$. For $X$ a subset of $\mathcal{P}(U)$, the dual $X^{\vee}$ of $X$ is the set $\{y \subseteq U:(\forall x \in X)(y \cap x \neq \emptyset)\}$.

(i) Show that $X \subseteq Y \rightarrow Y^{\vee} \subseteq X^{\vee}$.

Show that for $\left\{X_{i}: i \in I\right\}$ a family of subsets of $\mathcal{P}(U)$

$$\left(\bigcup\left\{X_{i}: i \in I\right\}\right)^{\vee}=\bigcap\left\{X_{i}^{\vee}: i \in I\right\}$$

(ii) Consider $S=\left\{X \subseteq \mathcal{P}(U): X \subseteq X^{\vee}\right\}$. Show that $S$, $\subseteq$ is a chain-complete poset.

State Zorn's lemma and use it to deduce that there exists $X$ with $X=X^{\vee}$.

Show that if $X=X^{\vee}$ then the following hold:

$X$ is closed under superset; for all $U^{\prime} \subseteq U, X$ contains either $U^{\prime}$ or $U \backslash U^{\prime}$.