Page 84 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 8: The Product Topology

CONTENTS
Objectives
Introduction

8.1 The Product Topology
8.1.1 The Product Topology: Finite Products
8.1.2 The Product Topology: Infinite Products

8.1.3 Cartesian Product
8.1.4 Box Topology
8.2 Summary
8.3 Keywords
8.4 Review Questions

8.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand the product topology;
 Define Cartesian product and box topology;

 Solve the problems on the product topology.

Introduction

There are two main techniques for making new topological spaces out of old ones. The first of
these, and the simplest, is to form subspaces of some given space. The second is to multiply
together a number of given spaces. Our purpose in this unit is to describe the way in which the
latter process is carried out.

Previously, we defined a topology on the product X × Y of two topological spaces. In present
unit, we generalize this definition to more general cartesian products. So, let us consider the
cartesian products
X ×…× X and X × X ×…,
1 n 1 2
where each X is a topological space. There are two possible ways to proceed. One way is to take
i
as basis all sets of the form × … × in the first case, and of the form × × … in the second
1 n 1 2
case, where is an open set of X for each i.
i i
8.1 The Product Topology

8.1.1 The Product Topology: Finite Products

Definition: Let (X , T ), (X , T ), …, (X , T ) be topological spaces. Then the product topology T on
1 1 2 2 n n
the set X × X × … × X is the topology having the family {O × O × … × O , O T , i = 1, …, n}
1 2 n 1 2 n i i

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