On order topologies and the real line
type: Item
instance of: scholarly article
title: On order topologies and the real line
zbMATH Open document ID: 1014.54020
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: 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.
reviewed by: Mark Mandelkern
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 06A05
zbMATH DE Number: 1848799
zbMATH Keywords: linearly ordered sets
zbMATH Keywords: order topology
zbMATH Keywords: enhanced order topology
zbMATH Keywords: order preserving homeomorphism
author: Frank S. Cater
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1848799