Page 35 - DMTH503_TOPOLOGY

Richa Nandra, Lovely Professional University Unit 2: Basis for Topology

Unit 2: Basis for Topology Notes

CONTENTS
Objectives
Introduction

2.1 Basis for a Topology
2.1.1 Topology Generated by Basis
2.1.2 A Characterisation of a Base for a Topology

2.2 Sub-base
2.3 Standard Topology and Lower Limit Topology
2.3.1 Standard Topology
2.3.2 Lower Limit Topology
2.4 Summary

2.5 Keywords
2.6 Review Questions
2.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the term basis for topology;
 Solve the questions related to basis for topology;
 Describe the sub-base and related theorems;

 State the standard topology.
Introduction

In mathematics, a base or basis  for a topological space X with topology T is a collection of open
sets in T such that every open set in T can be written as a union of elements of . We say that the
base generates the topology T. Bases are useful because many properties of topologies can be
reduced to statements about a base generating that topology.
In this unit, we shall study about basis, sub-base, standard topology and lower limit topology.

2.1 Basis for a Topology

Definition: Basis
A collection of subsets  of X is called a basis or a base for a topology if:
1. The union of the elements of B is X.
2. If x  B  B , B , B ,  , then there exists a B of  such that x  B  B  B .
1 2 1 2 1 2

LOVELY PROFESSIONAL UNIVERSITY 29