An extreme value theorem for transitive relations

Question

The extreme value theorem says that, if a function $f:K\to \mathbb{R}$ is continuous on a compact set $K$, then it has maximum and minimum elements, i.e., there exist elements $L,U \in K$ such that, for all $x\in K$: $f(U)\geq f(x)\geq f(L)$.

I am interested in an extension of this theorem from functions to order relations. In particular, let $\succsim$ be a transitive and complete relation on a compact set $K$. What conditions on $\succsim$, analogous to continuity, guarantee that there exist elements $L,U \in K$ such that, for all $x\in K$: $U \succsim x$ and $x \succsim L$?

Answer 1

It the simplest case, $\succeq$ is a total order -- so that for all $x,y \in K$, either $x \succeq y$ or $y \succeq x$ or both -- and the right condition is continuity of $\succeq$: for all $x \in K$, the sets $\{ y : y \succeq x\}$ and $\{y: x \succeq y\}$ are both closed. If $\succeq$ is a total order that is continuous and transitive, then a continuous function exists that represents it; this is usually called Debreu's Theorem, but there is another, older result called Rader's Theorem, that guarantees the weaker conditions of upper semi-continuity, which suffices for existence of a maximizer but not a minimizer.

If you only have a pre-order, there is a result called Levin's Theorem. If the graph of $\succeq$ is closed in $K \times K$ and $\succeq$ is acyclic, then a continuous function exists that represents $\succeq$, and the Weierstrass-type conclusion follows.

References on these topics would be Nachbin's "Order and Topology", Efe Ok's "Real Analysis with Economic Applications", Trout Rader's "Theory of Microeconomics". Levin's Theorem is hard to find (it was published in a Soviet math journal and reprinted in an AMS volume), but Ok et al have a recent paper called "A Comprehensive Approach to Revealed Preference" that has a proof of it in the appendix and other references. You might also like Border's "Fixed point theorems with applications to economics" that includes lots of content about when $\succeq$ have maximizers.

Answer 2

To have a compact subset of set S with an (partial) order R, there needs to be a topology for S. Such topologies are ambiguous with no accepted definition or standard.

As every bounded subset of a Dedekind complete order has an infimum and a supremum, I suppose you could call a subsets of those orders, complete if it is convex and has maximum and minimum or is the finite union of such subsets.

As for continuity you might get ideas from dcpo or Scott orders or you might prefer to limit your question to Scott continuous function of dcpo's and the related topologies.