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In mathematics, the order topology is a 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, and a and b are elements of X, we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. The order topology on X consists all sets that are a union of (possibly infinitely many) such open intervals. The order topology makes X into a normal Hausdorff space. The open intervals form a base for the order topology.
Several interesting variants of the order topology can be given:
- The left order topology on X is the topology whose open sets consist of intervals of the form (a, ∞).
- The right order topology on X is the topology whose open sets consist of intervals of the form (−∞, b).
The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff.
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