Page 48 - DMTH503_TOPOLOGY

Topology

Notes
Example 1: A basis for the product topology on × consists of the open rectangles
(a , b ) × (a , b ). This is also a basis for the usual topology on 2 , so the product topology
1 1 2 2
coincides with the usual topology.

Example 2: Take the topology T = {, {a, b}, {a}} on X = {a, b}.
Then the product topology on X × X is
{, X × X, {(a, a)}, {(a, a), (a, b)}, {(a, a), (b, a)}, {(a, a), (a, b), (b, a)}} where the last open set in the list
is not in the basis.
Theorem 1: If (X , T ) and (X , T ) are any two topological spaces, then the collection
1 1 2 2
 = {G × G : G  T , G  T }
1 2 1 1 2 2
is a base for some topology on X = X × X .
1 2
Proof: Suppose, (X , T ) and (X , T ) be any two topological spaces.
1 1 2 2
Write X = X × X ,
1 2
 = {U × U : U  T , U  T }.
1 2 1 1 2 2
To show:  is a base for some topology on X.
(i) To prove: U {B : B  } = X.
X  T , X  T  X × X  
1 1 2 2 1 2
X

 X = U {B : B  }
(ii) Let U × U , V × V   and let
1 2 1 2
(x , x )  (U × U ) (V × V )
1 2 1 2 1 2
To prove:  W × W   s.t.
1 2
(x , x )  W × W  (U × U ) (V × V )
1 2 1 2 1 2 1 2
(x , x )  (U × U ) (V × V )
1 2 1 2 1 2
 (x , x )  U × U and (x , x )  V × V
1 2 1 2 1 2 1 2
 x  U , x  U ; x  V , x  V
1 1 2 2 1 1 2 2
 x  U V ; x  U V
1 1 1 2 2 2
 x  W ; x  W
1 1 2 2
On taking W = U V ,
1 1 1
W = U V
2 2 2
 (x , x )  W × W
1 2 1 2
U × U  , V × V  
1 2 1 2
 U  T , U  T ; V  T , V  T
1 1 2 2 1 1 2 2
 U V  T , U V  T
1 1 1 2 2 2
 W  T , W  T
1 1 2 2
 W × W  
1 2

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