Page 79 - DMTH401_REAL ANALYSIS

Richa Nandra, Lovely Professional University Unit 5: Connectedness

Unit 5: Connectedness Notes

CONTENTS
Objectives
Introduction
5.1 Connected, Separated
5.2 Connectedness in Metric Spaces

5.3 Connected Spaces from Others
5.4 Connected Components
5.5 Path Connectedness
5.6 Open Sets in  n
5.7 Summary
5.8 Keywords
5.9 Review Questions

5.10 Further Readings
Objectives

After studying this unit, you will be able to:
 Define Connectedness
 Discuss the Connectedness in metric spaces
 Explain connected spaces from others
 Describe connected components and Path connected

Introduction

In last unit you have studied about the compactness of the set. As you all come to know about the
compactness and continuity. After understanding the concept of compactness let us go through
the explanation of connectedness.

5.1 Connected, Separated

Definition: Topological T connected if for every decomposition T = AB into disjoint open A, B
either A or B is empty.
Definition: T  S separated by sets U, V  S if T  UV, U  V  T = , U  T , V  T .

Proposition: T  C S disconnected if T is separated by some U, V  S.
Proof: () If disconnected  A, B  T, A, B  s.t. T = A  B and A  B = . T  S so U, V open in
S s.t. A = U T, B = V T. Then U, V separate T.

() If U, V separate T let A = U T, B = V T then T not connected.
Proposition: TFAE:
1. T disconnected

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