3. Problems

Problem S-1. State and prove a simple correspondence between closure systems and closure operators on the same set.

Problem S-2. Prove this fact due to Tarski:

Theorem. Let $L$ be a complete lattice and let $f:L\rightarrow L$ be isotone (i.e., order-preserving). Then $f$ has a fixed point. In other words, there is an element $p \in L$ with $f(p) = p$. (Suggestion: Let $A = \{x \in L: f(x) \geq x\}$ and let $p = \sup A$.)

Problem S-3. The completion by cuts CC$(P)$ of a partially ordered set $P$ is the lattice of closed subsets of $P$ under the polarity $(P,P, \leq )$. It can be shown that $P$ is isomorphically embedded in CC$(P)$ by $p \mapsto ( p ] = \{q \in P: q \leq p\}$, with all existing meets and joins preserved. Draw diagrams of the completion by cuts of the two partially ordered sets shown in Figure . Notice that the two middle elements in the second diagram are not comparable. (Suggestion: Start by identifying the closures of singletons. Their meets become set intersections. Joins can be found dually.)

Figure: Two partially ordered sets

text/edir/cc.eps

Problem S-4. Find a simple description (up to isomorphism) of the completion by cuts of the chain of rational numbers.

Problem S-5. For these contexts $(X,Y,\rho)$, describe the closed subsets in these examples, stating whether your description is both a necessary and sufficient condition or just necessary, and giving proof where asked:

(a) $X =$   R (the reals), $Y$ is the set of all continuous functions $f:$R$\rightarrow$   R, and $x \rho f$ means $f(x) = 0$. Prove your description of the $X$-closed subsets.

(b) $X = Y =$R$^ 3$ and $x \rho y$ means $x \cdot y \geq 0$.

Problem S-6. In the context $(X,Y,\rho)$, the statement that a particular subset $A$ of $X$ is closed actually has the force of an existence statement and so can be highly nontrivial. Specifically, if $A$ is closed, then for each $x \not \in A$ there exists $y \in Y$ such that $a \rho y$ for all $a \in A$ but not $x \rho y$. Write such existence statements in these more specific contexts from §, using your knowledge of what the closed subsets of $X$ are:

(a) Example 2(a).

(b) Example 4(a).

(c) Example 7(a).