Page 74 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 7: Continuous Functions

CONTENTS
Objectives
Introduction

7.1 Continuity
7.1.1 Continuous Map and Continuity on a Set
7.1.2 Homeomorphism

7.1.3 Open and Closed Map
7.1.4 Theorems and Solved Examples
7.2 Summary
7.3 Keywords
7.4 Review Questions

7.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the concept of continuity;
 Define Homeomorphism;
 Define open and closed map;
 Understand the theorems and problems on continuity.

Introduction

The concept of continuous functions is basic to much of mathematics. Continuous functions on
the real line appear in the first pages of any calculus look, and continuous functions in the plane
and in space follow not far behind. More general kinds of continuous functions arise as one goes
further in mathematics. In this unit, we shall formulate a definition of continuity that will
include all these as special cases and we shall study various properties of continuous functions.

7.1 Continuity

7.1.1 Continuous Map and Continuity on a Set

Definition: Let (X, T) and (Y, U) be any two topological spaces.

Let f : (X, T)  (Y, U) be a map.
The map f of said to be continuous at x  X is given any U-open set H containing f(x ), a T-open
0 0
set G containing x s.t. f(G)  H.
0
If the map in continuous at each x  X then the map is called a continuous map.

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