(i) State the Knaster-Tarski fixed point theorem. Use it to prove the Cantor-Bernstein Theorem; that is, if there exist injections $A \rightarrow B$ and $B \rightarrow A$ for two sets $A$ and $B$ then there exists a bijection $A \rightarrow B$.

(ii) Let $A$ be an arbitrary set and suppose given a subset $R$ of $P A \times A$. We define a subset $B \subseteq A$ to be $R$-closed just if whenever $(S, a) \in R$ and $S \subseteq B$ then $a \in B$. Show that the set of all $R$-closed subsets of $A$ is a complete poset in the inclusion ordering.

Now assume that $A$ is itself equipped with a partial ordering $\leqslant$.

(a) Suppose $R$ satisfies the condition that if $b \geqslant a \in A$ then $(\{b\}, a) \in R$.

Show that if $B$ is $R$-closed then $c \leqslant b \in B$ implies $c \in B$.

(b) Suppose that $R$ satisfies the following condition. Whenever $(S, a) \in R$ and $b \leqslant a$ then there exists $T \subseteq A$ such that $(T, b) \in R$, and for every $t \in T$ we have (i) $(\{b\}, t) \in R$, and (ii) $t \leqslant s$ for some $s \in S$. Let $B$ and $C$ be $R$-closed subsets of $A$. Show that the set

$$[B \rightarrow C]=\{a \in A \mid \forall b \leqslant a(b \in B \Rightarrow b \in C)\}$$

is $R$-closed.