On order topologies and the real line
This paper deals with linearly ordered sets \(T\) with the order topology. The author wishes to find prototype spaces \(S\) with the following property: Any order space \(T\) is homeomorphic to a subspace of \(S\). Special conditions on \(T\) are required. When the spaces \(T\) are countable, then \(\mathbb Q\) serves as \(S\); if the spaces \(T\) are second countable, then \(\mathbb R\) will serve as \(S\). The main theorem applies to spaces \(T\) which are separable. In this case, \(S\) is the product space \(\mathbb R \times \{0,1\}\) with the lexicographic order. Also, the topology on \(T\) must in general be enlarged to an enhanced order topology \(T*\) before the homeomorphism can be constructed. (In the case of the real line, \(\mathbb R*\) is called the half open interval space or the Sorgenfrey line.) The construction in fact produces an order preserving homeomorphism.
scientific article; zbMATH DE number 1848799