# Order Topology

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

**Example(s):****Counter-Example(s)****See:**Orderable Space, Algebraic Topology, Differential Topology, Geometric Topology, Pointless Topology, Ordinal Space, Hausdorff Space, Mathematics, Topology, Totally Ordered Set, Real Numbers, Subbase, Interval (Mathematics), Base (Topology), Union (Set Theory), Topological Space, Completely Normal Space.

## 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" :[math](a, \infty) = \{ x \mid a \lt x\}[/math]

[math](-\infty, b) = \{x \mid x \lt b\}[/math]

for all

*a,b*in*X*. This is equivalent to saying that the open intervals :[math](a,b) = \{x \mid a \lt x \lt b\}[/math]

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.

- In mathematics, an