nLab
Sorgenfrey line

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

The Sorgenfrey line (after Robert Sorgenfrey?, a mathematical descendant of R.L. Moore?) is the real line \mathbb{R}, but topologized by taking as topological basis the collection of half-open intervals [a,b)[a, b). This is sometimes called the lower limit topology?, and the Sorgenfrey line is often denoted by l\mathbb{R}_l.

The Sorgenfrey line and spaces derived from it, notably the Sorgenfrey plane l× l\mathbb{R}_l \times \mathbb{R}_l (with the product topology), is a rich source for counterexamples in general topology. For example, the Sorgenfrey line is a (T 1T_1) normal space and even a paracompact space, but the Sorgenfrey plane fails to be normal (and thus is not paracompact either). The Sorgenfrey plane is also an example of a separable space, but admitting an uncountable discrete subspace (which is thus not separable), showing that separability is not a hereditary property.

Revised on May 22, 2017 15:20:01 by Urs Schreiber (92.218.150.85)