Editing Order topology (section)

==Left and right order topologies==
Several variants of the order topology can be given:

* The '''right order topology'''<ref>Steen & Seebach, p. 74</ref> on ''X'' is the topology having as a [[base (topology)|base]] all intervals of the form <math>(a,\infty)=\{x\in X\mid x>a\}</math>, together with the set ''X''.
* The '''left order topology''' on ''X'' is the topology having as a base all intervals of the form <math>(-\infty,a)=\{x\in X\mid x<a\}</math>, together with the set ''X''.

These topologies naturally arise when working with [[Semi-continuity|semicontinuous functions]], in that a real-valued function on a topological space is lower semicontinuous if and only if it is [[Continuous function|continuous]] when the reals are equipped with the right order.<ref name="Stromberg">Stromberg, p. 132, Exercise 4</ref>  The ([[Natural (category theory)|natural]]) [[Compact-open topology|compact open topology]] on the resulting set of continuous functions is sometimes referred to as the ''semicontinuous topology''<ref>{{Cite journal |last=Dal Maso |first=Gianni |date=1993 |title=An Introduction to Γ-Convergence |url=https://link.springer.com/book/10.1007/978-1-4612-0327-8 |journal=SpringerLink |language=en |doi=10.1007/978-1-4612-0327-8|isbn=978-1-4612-6709-6 }}</ref>''.''

Additionally, these topologies can be used to give [[counterexample]]s in general topology.  For example, the left or right order topology on a bounded set provides an example of a [[compact space]] that is not Hausdorff.

The left order topology is the standard topology used for many [[set-theoretic]] purposes on a [[Boolean algebra (structure)|Boolean algebra]].{{Clarify|Boolean algebras are not totally ordered|date=April 2021}}