
Topology of ordered sets
Posted:
Oct 19, 2011 7:00 AM


Let (S,<=) be a (partial) order. Assume S has a topological base of order convex sets. Let R be the relation { (x,y) in SxS  x <= y }.
A set K, is order convex when for all a,b in K, if a <= x <= b, then x in K.
If R is closed within SxS, then S is Hausdorff.
If S is Hausdorff, is R closed?
If S is a linear order then yes, R is closed. What's the situation when S isn't linear?

