Order Topology - An Example of A Subspace of A Linearly Ordered Space Whose Topology Is Not An Order Topology

An Example of A Subspace of A Linearly Ordered Space Whose Topology Is Not An Order Topology

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.

We wish to define here a subset Z of a linearly ordered topological space X such that no total order on Z generates the subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.

Let in the real line. The same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.

An argument follows. Suppose by way of contradiction that there is some strict total order < = <₁ on Z such that the order topology generated by < is equal to the subspace topology on Z (note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology). In the following, interval notation should be interpreted relative to the < relation. Also, if A and B are sets, A

Let M=Z\{-1}, the unit interval. M is connected. If m,n∈M and m<-1

A space whose topology is an order topology is called a Linearly Ordered Topological Space (LOTS), and a subspace of a linearly ordered topological space is called a Generalized Ordered Space (GO-space). Thus the example Z above is an example of a GO-space that is not a linearly ordered topological space.