Editing Preorder (section)

===Other===
Further examples:
* Every [[finite topological space]] gives rise to a preorder on its points by defining <math>x \lesssim y</math> if and only if ''x'' belongs to every [[Neighborhood (mathematics)|neighborhood]] of ''y''. Every finite preorder can be formed as the [[Specialization (pre)order|specialization preorder]] of a topological space in this way. That is, there is a [[one-to-one correspondence]] between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.

* A [[net (mathematics)|net]] is a [[directed set|directed]] preorder, that is, each pair of elements has an [[upper bound]].  The definition of convergence via nets is important in [[topology]], where preorders cannot be replaced by [[partially ordered set]]s without losing important features.

* The relation defined by <math>x \lesssim y</math> if <math>f(x) \lesssim f(y),</math> where ''f'' is a function into some preorder.
* The relation defined by <math>x \lesssim y</math> if there exists some [[Injective function|injection]] from ''x'' to ''y''. Injection may be replaced by [[surjection]], or any type of structure-preserving function, such as [[ring homomorphism]], or [[permutation]].
* The [[embedding]] relation for countable [[total order]]ings.

Example of a [[strict weak ordering#Total preorders|total preorder]]:
* [[Preference]], according to common models.<ref>{{Citation |last1=Hansson |first1=Sven Ove |title=Preferences |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/preferences/ |access-date=2025-03-16 |edition=Winter 2024 |publisher=Metaphysics Research Lab, Stanford University |last2=Grüne-Yanoff |first2=Till |editor2-last=Nodelman |editor2-first=Uri}}</ref>