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Topic: Topology of ordered sets
Replies: 2   Last Post: Oct 22, 2011 4:30 AM

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William Elliot

Posts: 248
Registered: 10/7/08
Topology of ordered sets
Posted: Oct 19, 2011 7:00 AM
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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?

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