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"
for all a,b in X. This is equivalent to saying that the open intervals
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.
The order topology makes X into a completely normal Hausdorff space.
The standard topologies on R, Q, and N are the order topologies.
Read more about Order Topology: Induced Order Topology, An Example of A Subspace of A Linearly Ordered Space Whose Topology Is Not An Order Topology, Left and Right Order Topologies, Ordinal Space
Famous quotes containing the word order:
“Where mass opinion dominates the government, there is a morbid derangement of the true functions of power. The derangement brings about the enfeeblement, verging on paralysis, of the capacity to govern. This breakdown in the constitutional order is the cause of the precipitate and catastrophic decline of Western society. It may, if it cannot be arrested and reversed, bring about the fall of the West.”
—Walter Lippmann (1889–1974)