Page 31 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Let G =  {G :   } Notes

Then G =  {G :   }, (By De Morgan’s law)

By (iii), e (G)  G = G or e (G)  G …(3)
G  G  G  G  e (G)  e (G) by (1)
a a a
 G  e (G) by (2)

  G  e (G)

 G  e (G) …(4)
From (3) & (4),

e (G) = G so that G  T
So, G  T   {G :   }  T
 
This shows that T is a topology on X.
It remains to prove that

e (A) = T-exterior of A.
By (iv), e (A) = e [(e (A))]
 e (A)  T [By (iii)],
e (A)  A

Thus, e (A) is an open set contained in A.
Also, e (A) is the largest open set contained in A.
 T-interior of A = e (A)
or T-exterior of A = e (A)

Self Assessment

14. Let X = {a, b, c} and let T = {, X, {b}, {a, c}}, find the interior of the set {a, b}.

15. If T = {, {a}, {a, b}, {a, c, d}, {a, b, e}, {a, b, c, d}, X} be a topology on X = {a, b, c, d, e} then find
the interior points of the subset A = {a, b, c} on X.

1.8 Summary

 Topology deals with the study of those properties of certain objects that remain invariant
by stretching or bending.
 Let X be any non-empty set and T be the collection of all subsets of X. Then T is called
discrete topology.
 Let X be any non-empty set and T = {X, }, then T is called indiscrete topology.
 Let T be a collection of subset of X where complements are finite along with , forms a
topology on X is called cofinite topology.
 Let (X, T) be a topological space. Any set A  T is called an open set and X – A is called
closed set.
 Closure of a set is the intersection of all closed sets containing A where A is subset of X.

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