Page 60 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 5: The Subspace Topology

CONTENTS
Objectives
Introduction

5.1 Subspace of a Topological Space
5.1.1 Solved Examples on Subspace Topology
5.1.2 Basis for the Subspace Topology

5.1.3 Subspace of Product Topology
5.2 Summary
5.3 Keywords
5.4 Review Questions
5.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Describe the concept of subspace of topological space;
 Explain the problems related to subspace topology;
 Derive the theorems on subspace topology.

Introduction

We shall describe a method of constructing new topologies from the given ones. If (X, T) is a
topological space and Y  X is any subset, there is a natural way in which Y can “inherit” a
topology from parent set X. It is easy to verify that the set  Y, as runs through T, is a
topology on Y. This prompts the definition of subspace or relative topology.

5.1 Subspace of a Topological Space

Definition: Let (X, T) be a topological space, V be a non empty subset of X and T be the class of
Y
all intersections of Y with open subsets of X i.e.
T = {Y  :  T}
Y
Then T is a topology on Y is called the subspace topology (or the relative topology induced on
Y
Y by T. The topological space (Y, T ) is said to be a subspace of (X, T).
Y

Note Let A  Y  X
(1) It A is open in Y, Y is open in X, then A is open in X.
(2) It A is closed in Y, Y is closed in X, then A is closed in X.

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