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By definition, the order topology on a totally ordered set $T$ is the coarsest topology such that every subset of $T$ that can be expressed as the upward or downward closure of a singleton is closed.

Now this definition still makes sense without the condition that $T$ is totally ordered. However, there doesn't seem to be a lot of interest in this generalization. Perhaps this is evidence that we're not generalizing correctly. One thing we might try is to replace "singleton" with "antichain." So the definition becomes:

The order topology on a partially ordered set $P$ is the coarsest topology such that every subset of $P$ that can be expressed as the upward or downward closure of an antichain is closed.

This reduces to the usual definition under the assumption that $P$ is totally ordered.

What is currently known about this topology, and where can I learn more?

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  • $\begingroup$ I presume you have tried googling for "order topology" and have found, for example, heldermann.de/R&E/RAE18/ctw05.pdf and didn't like it (?). You may find results of interest by searching for "alexandrov topology". $\endgroup$ – Rob Arthan Sep 9 '13 at 12:15
  • $\begingroup$ @RobArthan: It's not clear to me how the Alexandrov topology is relevant here. $\endgroup$ – Niels J. Diepeveen Sep 9 '13 at 13:25
  • $\begingroup$ I thought it was relevant because the topology the OP has in mind is an Alexandrov topology in many interesting cases, I believe - see the book chapter that I gave a link to. $\endgroup$ – Rob Arthan Sep 9 '13 at 15:33
  • $\begingroup$ @RobArthan, thanks for the link. I think you have misread the definition given in the question. In particular, both upward and downward closures of antichains are defined to be topologically closed. Note, however, that this is a much weaker condition than simply stating that all upward closed and all downward closed sets are topologically closed. $\endgroup$ – goblin Sep 9 '13 at 15:58
  • $\begingroup$ @NielsDiepeveen, thanks for the bounty man, I owe you one. $\endgroup$ – goblin Sep 9 '13 at 15:59

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