Page 42 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 3: The Order Topology

CONTENTS
Objectives
Introduction

3.1 The Order Topology
3.1.1 Intervals
3.1.2 Order Topology

3.1.3 Rays
3.1.4 Order Topology on the Linearly Ordered Set
3.1.5 Lemma (Basis for the Order Topology)
3.2 Summary
3.3 Keywords

3.4 Review Questions
3.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand the order topology;

 Solve the problems on order topology;
 Describe the open intervals, closed intervals and half-open intervals.

Introduction

If X is a simply ordered set, there is a standard topology for X, defined using the order relation.
It is called the order topology; in this unit, we consider it and study some of its properties.

3.1 The Order Topology

3.1.1 Intervals

Suppose that X is a set having a simple order relation <. Given elements a and b of X such that
a < b, there are four subsets of X that are called the intervals determined by a and b. They are the
following:
(a, b) = {x|a < x < b}

(a, b] = {x|a < x  b}
[a, b) = {x|a  x < b}
[a, b] = {x|a  x  b}

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