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Richa Nandra, Lovely Professional University Unit 4: The Product Topology on X × Y

Unit 4: The Product Topology on X × Y Notes

CONTENTS
Objectives
Introduction

4.1 Product Topology
4.2 Projection Mappings
4.3 Summary

4.4 Keywords
4.5 Review Questions
4.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Describe the product topology;
 Solve the problems on product topology;
 Define projection mappings;

 Discuss the problems on projection mappings.
Introduction

A product space is the Cartesian product of a family of topological space equipped with a natural
topology called the product topology. This topology differs from another, perhaps more obvious,
topology called the box topology, which can also be given to a product space and which agrees
with the product topology when the product is over only finitely many spaces. However the
product topology is 'correct' in that it makes the product space a categorical product of its
factors, whereas the box topology is too fine, this is the sense in which the product topology is
natural.

4.1 Product Topology

Given two sets X and Y, their product is the set X × Y = {(x, y) : x  X and y  Y}.
For example, 2 = × , and more generally m × n = m+n .

If X and Y are topological spaces, we can define a topology on X × Y by saying that a basis
consists of the subsets U × V as U ranges over open sets in X and V ranges over open sets in Y.

The criterion for a collection of subsets to be a basis for a topology is satisfied since
(U × V ) (U × V ) = (U U ) × (V V )
1 1 2 2 1 2 1 2
This is called the product topology on X × Y.

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