Order Topology

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:

    Undoubtedly we have not questions to ask which are unanswerable. We must trust the perfection of the creation so far, as to believe that whatever curiosity the order of things has awakened in our minds, the order of things can satisfy. Every man’s condition is a solution in hieroglyphic to those inquiries he would put. He acts it as life, before he apprehends it as truth.
    Ralph Waldo Emerson (1803–1882)