Page 122 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 12: Components and Local Connectedness

CONTENTS
Objectives
Introduction

12.1 Components of a Topological Space
12.2 Local Connectedness
12.2.1 Locally Connected Spaces

12.2.2 Locally Connected Subset
12.2.3 Theorems and Solved Examples
12.3 Summary
12.4 Keywords
12.5 Review Questions

12.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the term components of a topological space;
 Solve the problems on components of a topological space;
 Define locally connectedness;
 Solve the problems on locally connectedness.

Introduction

Given an arbitrary space X, there is a natural way to break it up into piece that are connected. We
consider that process now. Given X, define an equivalence relation on X by setting x  y if there
is a connected subspace of X containing both x and y. The equivalence classes are called the
components or the “connected components” of X.
Connectedness is a useful property for a space to possess. But for some purposes, it is more
important that the space satisfy a connectedness condition locally. Roughly speaking, local
connectedness means that each point has “arbitrary small” neighbourhoods that are connected.
So, in this unit, we shall deal with two important topics components and local connectedness.

12.1 Components of a Topological Space

Definition: A subset E of a topological space X is said to be a component of X if

1. E is a connected set and
2. E is not a proper subset of any connected subspace of X i.e. if E is a maximal connected
subspace of X.

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