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, Z, and N are the order topologies.

If Y is a subset of X, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology. As a subset of X, Y also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same.

For example, consider the subset Y = {–1} ∪ {1/n}_n∈N in the rationals. Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.

Though the subspace topology of Y = {–1} ∪ {1/n}_n∈N in the section above is shown to be not generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is an open set), and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y, simply modify the induced order on Y by defining -1 to be the greatest element of Y and otherwise keeping the same order for the other points, so that in this new order (call it say <₁) we have 1/n <₁ –1 for all n∈N. Then, in the order topology on Y generated by <₁, every point of Y is isolated in Y.

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