I was wondering if there is a canonical topology induced by a partial order on a set and how that relates to the total ordering topology (if it can be extended to a total ordering).
I thought maybe the basis would be defined as in total orderings, but this wouldn't include elements that are incomparable to everything. I noticed another question mentions the space generated by upsets and downsets; would this be the natural topology induced? Or are others used?
((I haven't been able to find much by googling, and often the pages are about all topologies on a set being partially ordered))
Thanks in advance!