# Order Topology

An Order Topology is a Point-Set Topology that can be defined on any totally ordered set.

## References

### 2019

• (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Order_topology Retrieved:2019-4-25.
• In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays" :

$(a, \infty) = \{ x \mid a \lt x\}$

$(-\infty, b) = \{x \mid x \lt b\}$

for all a,b in X. This is equivalent to saying that the open intervals :

$(a,b) = \{x \mid a \lt x \lt b\}$

together with the above rays form a base for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays.

A topological space X is called orderable 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 Hausdorff space.

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