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:
“Russian forests crash down under the axe, billions of trees are dying, the habitations of animals and birds are layed waste, rivers grow shallow and dry up, marvelous landscapes are disappearing forever.... Man is endowed with creativity in order to multiply that which has been given him; he has not created, but destroyed. There are fewer and fewer forests, rivers are drying up, wildlife has become extinct, the climate is ruined, and the earth is becoming ever poorer and uglier.”
—Anton Pavlovich Chekhov (1860–1904)