nLab
order topology

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

Contents

Definition

Given a linearly ordered set (S,<)(S, \lt), its order topology is the topology on SS generated from the sub-basis βP(S)\beta \subset P(S)

β{(a,),(,a)} aS. \beta \coloneqq \left\{ (a,\infty), (-\infty,a) \right\}_{a \in S} \,.

whose elements are the “open half rays”

(a,){sS|a<s}AAAA(,a){sS|s<a}. (a, \infty) \coloneqq \left\{ s \in S \,\vert\, a \lt s \right\} \phantom{AAAA} (-\infty, a) \coloneqq \left\{ s \in S \,\vert\, s \lt a \right\} \,.

Examples

References

Revised on May 3, 2017 17:44:21 by Toby Bartels (70.198.52.201)