• Corpus ID: 50114238

" Linearly Ordered and Generalized Ordered Spaces "

@inproceedings{RudinLO,
  title={" Linearly Ordered and Generalized Ordered Spaces "},
  author={Mary Ellen Rudin}
}

• M. Rudin
• Mathematics

For any linearly ordered set (X, <), let I(<) be the topology on X that has the collection of all open intervals of (X, <) as a base. The topology I(<) is the open interval topology of the order < and (X, <, I(<)) is a linearly ordered topological space or LOTS. For a subset Y ⊆ X, it can happen that the relative topology I(<)| Y on Y does not coincide with the open interval topology I(< | Y) induced on Y by the restricted ordering. In some cases, there might be some other ordering of Y whose open interval topology coincides with I(<)| Y , but in other cases there is not. Perhaps the best known example is the Sorgenfrey line, which is homeomorphic to the subspace Y = {(x, 1) : x ∈ R} of the lexicographically ordered LOTS X = R × {0, 1}. There is no reordering of Y whose open-interval topology coincides with the subspace topology that Y inherits from X. The study of subspaces of LOTS in their own right was pioneered by E. ˇ Cech who introduced generalized ordered spaces or GO-spaces, i.e., triples (X, <, T) where < is a linear ordering of the set X and T is a Hausdorff topol-ogy on X such that each point of X has a T-neighborhood base consisting of (possibly degenerate) intervals. (Such spaces have also been called sub-orderable spaces.) The class of GO-spaces is known to coincide with the class of topological subspaces of LOTS, and for any GO-space (X, <, T), there is a canonical linearly ordered set (X * , < *) whose ordering extends the given ordering of X, and that has the property that (X, T) embeds as a closed subspace of (X * , I(< *)). In addition, the extension space X * " inherits " many, but not all, of the topological properties of X. For example, X is metrizable if and only if (X * , I(< *)) is metrizable, but there are examples in which the GO-space X is perfect (= closed subsets of X are G δ-sets) while X * is not. Historically, LOTS and GO-spaces have been valuable sources of counterexamples in topology. The usual space [0, ω 1) of all countable ordinals, the Sor-genfrey line, and the Michael line (i.e., the usual space of real numbers with each irrational isolated) are perhaps …