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Sachin Kaushal, Lovely Professional University Unit 16: The Countability Axioms

Unit 16: The Countability Axioms Notes

CONTENTS
Objectives
Introduction
16.1 Countability Axioms
16.1.1 First Axiom of Countability
16.1.2 Second Axiom of Countability
16.1.3 Hereditary Property
16.1.4 Theorems and Solved Examples on Countability Axioms
16.1.5 Theorems Related to Metric Spaces
16.2 Summary

16.3 Keywords
16.4 Review Questions
16.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define countability axioms;

 Understand and describe the theorems on countability axioms;
 Discuss the theorems on countability axioms related to the metric spaces.

Introduction

The concept we are going to introduce now, unlike compactness and connectedness, do not arise
naturally from the study of calculus and analysis. They arise instead from a deeper study of
topology itself. Such problems as imbedding a given space in a metric space or in a compact
Hausdorff space are basically problems of topology rather than analysis. These particular
problems have solutions that involve the countability and separation axioms. In this unit, we
shall introduce countability axioms and explore some of their consequences.
16.1 Countability Axioms

16.1.1 First Axiom of Countability

Let (X, T) be a topological space. The space X is said to satisfy the first axiom of countability if X
has a countable local base at each x X. The space X, in this case, is called first countable or first
axiom space.

Example 1: Consider x R

 1 1 
A =  x  ,x    x  N
n  n n 

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