# Why not use usual topology in ordered spaces ?

This is a similar question to the one about the lack of use of usual topologies in measure theory. By usual topology here is meant the Hausdorff-Kuratowski-Bourbaki concept, based on open sets, or alternatively, closed sets. In ordered spaces a variety of convergences are defined. Hoever, in their general cases they do not correspond to convergence in any topology. The question is : is there some other concept of topology so that appropriate instances of it may give precisely one or another set of convergent sequences in order spaces ?

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Can you be more explicit about what "variety of convergences" are defined in an ordered space? –  Mariano Suárez-Alvarez Jul 12 '10 at 16:23
Any better book on odered spaces given a variety of types of convergences. One of the best books is Luxemburg & Zaanen : mRiesz Spaces I. North-Holland 1971. –  Elemer E Rosinger Jul 13 '10 at 6:59