Editing Preorder (section)

==Uses==
Preorders play a pivotal role in several situations:
* Every preorder can be given a topology, the [[Alexandrov topology]]; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
* Preorders may be used to define [[interior algebra]]s.
* Preorders provide the [[Kripke semantics]] for certain types of [[modal logic]].
* Preorders are used in [[Forcing (mathematics)|forcing]] in [[set theory]] to prove [[consistency]] and [[independence (mathematical logic)|independence]] results.<ref>{{citation
 | last = Kunen | first = Kenneth
 | title = Set Theory, An Introduction to Independence Proofs
 | publisher = Elsevier
 | publication-place = Amsterdam, the Netherlands
 | series = Studies in logic and the foundation of mathematics
 | volume = 102
 | year = 1980
}}.</ref>