Compactness and limit points

In order topology, limit points can be defined for any total order. It is natural to ask then, if certain order-topological principles can be passed on to the case of partial orders from total orders. Clearly, for example, we can define discrete and scattered partial orders based upon order topological chain conditions. To some extent, compactness generalizes lower/upper isolated points from order topology.

• A point is called lower isolated if it is an isolated point in its principal ideal.
• A point is called upper isolated if it is an isolated point in its principal filter.

Compactness is a dual-condition: an element can be either meet-compact or join-compact depending upon the semilattice operation being considered. In a total order, join-compactness coincides with lower-isolated points and meet-compactness coincides with upper-isolated points. To see this, consider that in a total order the join of all finite sets is contained within the set, so the only way to generate an element is with an infinite set. At the same time, the order topology for any total order is Frechet, which means that all limit points are generated by infinite sets.

If an element is necessarily generated by an infinite set it is not-compact by definition and it necessarily has some element in that infinite set in each of its neighbourhoods so it is also a limit point of its predecessors. Therefore, limit points and infinite joins/meets coincide. These is why the order-type of the real numbers can either be defined by its bounded-completeness as a lattice or by its topological completeness, because these two principles coincide.

Clearly limit points and non-compactness coincide in total orders, but non-compactness applies to partial orders. Its immediately clear that all join-compact elements are lower isolated in all chains ending in them and dually for meet-compact elements. If they are not, then clearly an infinite total order with no finite reduction produces them as a join/meet which means they are not compact. In the other direction, there is a process whereby we can construct a total order that has them as a join from any infinite non-compact representation which demonstrates the similarity between non-compat elements and limit points in the other direction.

Theorem. let $S$ be a countable non-compact join representation of an element $x$, then we can construct an ascending sequence that has $x$ as its join from $S$.

Proof. we will construct an infinite ascending sequence $T$ from the elements of $S$ (1) first get any element of $S$ and put it in $T$ (2) then we can always get another element $y$ of $S-T$ such that the join of that element and $T$ is greater then the current join of $T$ because if we can't then $T$ already is an upper bound for the entire set and therefore since its finite that would mean the representation $S$ is not compact which is a contradiction (3) we can applies this infinitely to get an ascending sequence of order type $\omega$ that has $x$ as its join.

Collary. we can construct descending sequences from non-compact meet representations