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:
“Put shortly, these are the two views, then. One, that man is intrinsically good, spoilt by circumstance; and the other that he is intrinsically limited, but disciplined by order and tradition to something fairly decent. To the one party man’s nature is like a well, to the other like a bucket. The view which regards him like a well, a reservoir full of possibilities, I call the romantic; the one which regards him as a very finite and fixed creature, I call the classical.”
—Thomas Ernest Hulme (1883–1917)