Page 56 - DMTH503_TOPOLOGY

Topology

Notes Thus we have shown that
–1
any y  f (G)   an open set V  Y s.t. y  V  f (G).
–1
–1
–1
 y is an interior point of f (G) and hence every point of f (G) is an interior point, showing
thereby f (G) is open in Y.
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Theorem 8: The product topology is the coarser (weak) topology for which projections are
continuous.
Proof: Let (X × Y, T) be product topological space of (X, T ) and (Y, T ).
1 2
Let  be a base for T. Then
 = {G × G : G  T , G  T }
1 2 1 1 2 2
The mappings,  : X × Y  X s.t.  (x, y) = x
x x
and  : X × Y  Y s.t.  (x, y) = y
y y
are called projection maps.
These maps are continuous. [Refer theorem (4)]
Let T* be any topology on X × Y for which  and  are continuous.
x y
To prove: T is the coarest (weakest) topology for which projections are continuous, we have to
show that T  T*.
For this, we have to show that

any G  T  G  T*
Let G  T, by definition of base,
G  T  B  B s.t. G = {B : B   }
1 1
 G = {G × G : G × G   }
1 2 1 2 1
G  X  G X = G
1 1 1
G  X  G X = G
2 2 2
Then G = {(G X) × (G Y) : G × G   }
1 2 1 2 1
= {(G × G ) (X × Y) : G × G   }
1 2 1 2 1
[For (a × b) (c × d) = (a c) × (b d)]
or G = {  x 1 (G )   y 1 (G ) : G  G   1 } … (1)
2
1
2
1
 : X × Y  X, G  T,  is continuous
x 1 x
   1 (G )  T*
x 1
Similarly,   1 (G )  T*
y 2
This implies   1 (G )   1 (G ) T*, be definition of topology.

x 1 y 2
In this event (1) declares that G is an arbitrary union of T* open sets and hence G is T* open set
and so G  T*.
any G  T  G  T*

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