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?