Page 112 - DMTH503_TOPOLOGY

Topology

Notes Unit 11: Connected Spaces, Connected
Subspaces of Real Line

CONTENTS
Objectives

Introduction
11.1 Connected Spaces
11.2 Connected Subspaces of Real Line
11.3 Summary

11.4 Keywords
11.5 Review Questions
11.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Define connected spaces;
 Solve the questions on connected spaces;
 Understand the theorems and problems on connected subspaces of the real line.

Introduction

The definition of connectedness for a topological space is a quite natural one. One says that a
space can be “separated” if it can be broken up into two “globs” – disjoint open sets. Otherwise,
one says that it is connected. Connectedness is obviously a topological property, since it is
formulated entirely in terms of the collection of open sets of X. Said differently, if X is connected,
so is any space homeomorphic to X.

Now how to construct new connected spaces out of given ones. But where can we find some
connected spaces to start with? The best place to begin is the real line. We shall prove that R is
connected, and so are the intervals.

11.1 Connected Spaces

Definition: A topological space X is said to be disconnected iff there exists two non-empty
separated sets A and B such that E = A B.
In this case, we say that A and B form a partition or separation of E and we write, E = A|B.

A topological space X is said to be connected if it cannot be written as the union of two disjoint
non-empty open sets.
A subspace Y of a topological space X is said to be connected if it is connected as a topological
space it its own right.

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