https://en.wikipedia.org/wiki/Order_topology Order topology (抜粋) 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,∞)={x ｜ a<x}} (-∞,b)={x ｜ 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}} 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.
Contents 1 Induced order topology 2 An example of a subspace of a linearly ordered space whose topology is not an order topology 3 Left and right order topologies 4 Ordinal space 5 Topology and ordinals 5.1 Ordinals as topological spaces
