Order Topology
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High Quality Content by WIKIPEDIA articles! 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.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…mehr

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High Quality Content by WIKIPEDIA articles! 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.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, inthe order topology on Y generated by 1, every point of Y is isolated in Y.