Editing Order topology (section)

{{Short description|Certain topology in mathematics}}
{{Distinguish|Order topology (functional analysis)}}

In [[mathematics]], an '''order topology''' is a specific [[topological space|topology]] that can be defined on any [[totally ordered set]].  It is a natural generalization of the topology of the [[real number]]s to arbitrary totally ordered sets. <!-- why this restriction to totally ordered sets? Partial order relations generate nice non-Hausdorff counter-examples //-->

If ''X'' is a totally ordered set, the '''order topology''' on ''X'' is generated by the [[subbase]] of "open rays"
:<math>\{ x \mid a < x\}</math>
:<math>\{x \mid x < b\}</math>
for all ''a,&nbsp;b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open [[interval (mathematics)|interval]]s
:<math>(a,b) = \{x \mid a < x < b\}</math>
together with the above rays form a [[base (topology)|base]] for the order topology. The [[open set]]s in ''X'' are the sets that are a [[union (set theory)|union]] of (possibly infinitely many) such open intervals and rays.

A [[topological space]] ''X'' is called '''orderable''' or '''linearly orderable'''<ref>{{cite journal |last1=Lynn |first1=I. L. |title=Linearly orderable spaces |journal=[[Proceedings of the American Mathematical Society]] |date=1962 |volume=13 |issue=3 |pages=454–456 |doi=10.1090/S0002-9939-1962-0138089-6 | doi-access=free}}</ref> if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a [[completely normal space|completely normal]] [[Hausdorff space]].

The standard topologies on '''R''', '''Q''', '''Z''', and '''N''' are the order topologies.