Page 86 - DMTH503_TOPOLOGY

Topology

Notes
Example 2: Let (X , T ), …, (X , T ), … be a countably infinite family of topological spaces.
1 1 n n
Then the box topology T on the product Õ ¥ i 1 X is that topology having as its basis the family
i
=
B = { Õ O : O Î T i }
¥
i
i
=
i 1
It is readily seen that if each (X , T ) is a discrete space, then the box product (Õ ¥ i 1 X , T ) is a
i i = i
discrete space. So if each (X , T) is a finite set with the discrete topology, then (Õ ¥ i 1 X , T ) is an
i = i
infinite discrete space, which is certainly not compact. So, we have a box product of the compact
spaces (X , T ) being a non-compact space.
i i
Example 3: Let (X , T ), …, (Y , T), iÎN, be countably infinite families of topological
i i i i
spaces having product spaces (Õ ¥ i 1 X , ) T and (Õ ¥ i 1 Y , T ) respectively. If the mapping h : (X , T )
=
=
i
i
i
i
i
 (Y , T ) is continuous for each i Î N, then so is the mapping h: (Õ ¥ i 1 X , ) T  (Õ ¥ i 1 Y , T ) given
i i = i = i
by h : (Õ ¥ i 1 x i ) = Õ ¥ i 1 h (x ); that is, h (x , x , …, x , …) = h (x ), h (x ), …, h (x ), ….
=
i
i
=
2
2
n
n
1
1
1
2
n
–1
Solution: It suffices to show that if O is a basic open set in (Õ ¥ i 1 Y , T ) , then h (O) is open in
=
i
¥
(Õ i 1 X , ) T . Consider the basic open set × × … × Y × Y × … where Î T, for i = 1,
i
=
n+1
n
i
n+2
1
2
…, n. Then
h ( × … × × Y × Y × …)
–1
1 n n+1 n+2
–1
–1
–1
= h ( ) × … × h ( )× h (Y ) × h (X ) × …
–1
1 1 n n n+1 n+2
–1
and the set on the right hand side is in T, since the continuity of each h implies h ( ) Î T , for
i 1 i i
i = 1, …, n. So h is continuous.
8.1.3 Cartesian Product
Definition: Let {A } be an indexed family of sets; let X = A . The cartesian product of this
 ÎJ ÎJ 
index family, denoted by Õ A , is defined to be the set of all J-tuples (x ) of elements of X such

 ÎJ
Î J
that x Î A for each  Î J. That is, it is the set of all functions
 
x : J  A 
Î J
such that x() Î A for each  Î J.

8.1.4 Box Topology
Let {X } be an indexed family of topological spaces. Let us take as a basis for a topology on the
  Î J
product space Õ X the collection of all sets of the for Õ  , where is open in X , for each  Î J.

Î J Î J  
The topology generated by this basis is called the box topology.
Example 4: Consider euclidean n-space  . A basis for  consists of all open intervals in
n
n
; hence a basis for the topology of  consists of all products of the form
(a , b ) × (a , b ) × … × (a , b )
1 1 2 2 n n
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