Page 88 - DMTH503_TOPOLOGY

Topology

Notes product topology on X is a set of the form  - 1 ( ) , where  is some index and is open X .
    
Now
-
1
f - 1 ( -  1 (  ) ) = f (  ) ,

because f =  of. Since f is continuous, this set is open in A, as desired.
  
8.2 Summary

 Let (X , T ), (X , T ), …, (X , T ) be topological spaces. Then the product topology T on the set
1 1 2 2 n n
X × X × … × X is the topology having the family {O × O × … × O , O  T , i = 1, …, n} as
1 2 n 1 2 n i i
a basis. The set X × X × … × X with the topology T is said to be the product of the space
1 2 n
(X , T ), (X , T ), …, (X , T ) and is denoted by (X × X × …, X , T).
1 1 2 2 n n 1 2 n
 The product space, Õ ¥ i 1 (X , T ), consists of the product Õ ¥ i 1 X with the topology T having
=
i
=
i
i
as its basis the family
{ ¥ }
B = Õ O : O  T i and O  X for all but a finite number of i.
i
i
i
i
i 1
=
The topology T is called the product topology.
 The cartesian product of this index family, denoted by Õ A , is defined to be the set of all

 J
J-tuples (x ) of elements of X such that x  A for each   J.
 J  
 Let {X } be an indexed family of topological spaces. Let us take as a basis for a topology
   J
on the product space Õ X the collection of all sets of the for Õ , where is open in X ,
   
 J  J
for each   J. The topology generated by this basis is called the box topology.
8.3 Keywords
Discrete Space: Let X be any non empty set and T be the collection of all subsets of X. Then T is
called the discrete topology on the set X. The topological space (X, T) is called a discrete space.
Indiscrete Space: Let X be any non empty set and T = {X, }. Then T is called the indiscrete
topology and (X, T) is said to be an indiscrete space.
Open & Closed Set: Any set A  T is called an open subset of X or simply a open set and X – A is
a closed subset of X.
Topological Space: Let X be a non empty set. A collection T of subsets of X is said to be a topology
on X if
(i) XT, T

(ii) AT, BTABT
(iii) A T A T where  is an arbitrary set.
 
8.4 Review Questions

1. If (X , T ), (X , T ), …, (X , T ) are discrete spaces, prove that the product space (X , T ) × (X , T )
1 1 2 2 n n 1 1 2 2
× … × (X , T ) is also a discrete space.
n n
2. Let X and X be infinite sets and T and T the finite-closed topology on X and X ,
1 2 1 2 1 2
respectively. Show that the product topology, T, on X × X is not the finite-closed topology.
1 2

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