Order topology

In mathematics, an order topology is a specific 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" { x ∣ a < x } {\displaystyle \{x\mid a<x\}} { x ∣ x < b } {\displaystyle \{x\mid x<b\}} for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals ( a , b ) = { x ∣ a < x < b } {\displaystyle (a,b)=\{x\mid a<x<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 or linearly 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.