Page 159 - DMTH503_TOPOLOGY

Sachin Kaushal, Lovely Professional University Unit 17: The Separation Axioms

Unit 17: The Separation Axioms Notes

CONTENTS
Objectives
Introduction

17.1 T -Axiom or Kolmogorov Spaces
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17.1.1 T -Axiom of Separation or Frechet Space
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17.2 T -Axiom of Separation or Hausdorff Space
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17.3 Summary
17.4 Keywords
17.5 Review Questions
17.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Define T -axiom and solve related problems;
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 Explain the T -axiom and related theorems;
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 Describe the T -axiom and discuss problems and theorems related to it.
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Introduction

The topological spaces we have been studying thus far have been generalizations of the real
number system. We have obtained some interesting results, yet because of the degree of
generalization many intuitive properties of the real numbers have been lost. We will now
consider topological spaces which satisfy additional axioms that are motivated by elementary
properties of the real numbers.
17.1 T -Axiom or Kolmogorov Spaces
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A topological space X is said to be a T -space if for any pair of distinct points of X, there exist at
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least one open set which contains one of them but not the other.
In other words, a topological space X is said to be a T -space if it satisfy following axiom for any
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x, y  X, x  y, there exist an open set  such that x  U but y  U.

Example 1: Let X = {a, b, c} with topology T = {, X, {a}, {b}, {a, b}} defined on X, then (X, T)
is a T -space because
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(i) for a and b, there exist an open set {a} such that a  {a} and b  {a}
(ii) for a and c, there exist an open set {b} and b  {b} and c  {b}

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