Page 52 - DMTH503_TOPOLOGY

Topology

Notes  Given a pair of distinct elements (x , x ), (y , y )  X there are disjoint open subsets X × G ,
1 2 1 2 1 2
X × H of X s.t. (x , x )  X × G , (y , y )  X × H .
1 2 1 2 1 2 1 2 1 2
The leads to the conclusion that (X, T) is a Hausdorff space.

Example 3: Let T = {, {1}, X } be a topology on X = {1, 2, 3} and T = { , X , {a}, {b} {a, b},
1 1 1 2 2
{c, d}, {a, c, d}, {b, c, d}} be a topology for X = {a, b, c, d}.
2
Find a base for the product topology T.
Solution: Let B be a base for T and B be a base for T . Then B = {B × B : B  B , B  B } is a base
1 1 2 2 1 2 1 1 2 2
for the product topology T.
We can take B = {{1}, X }
1 1
B = {{a}, {b}, {c, d}}.
2
The elements of B are
{1} ×{a}, {1} ×{b}, {1} ×{c, d}, {1, 2, 3} ×{a}, {1, 2, 3} ×{b}, {1, 2, 3} ×{c, d}.

That is to say
ì {(1,a)}, {(1,b)}, {(1,c),(1,d)} ü
ï ï
B = í {(1,a),(2,a),(3,a)}, {(1,b),(2,d),(3,b)} ý
ï ï
î {(1,c),(2,c),(3,c),(1,d),(2,d),(3,d)} þ
is a base for T.

Self Assessment

1. Let X and X’ denote a single set in the topologies T and T’ respectively let Y and Y’ denote
a single set in the topologies U and U’ respectively. Assume these sets are non-empty.

(a) Show that if T’  T and U’  U, then the product topology on X’ × Y’ is finer than the
product topology on X × Y.
(b) Does the converse of (a) hold? Justify your answer.

4.2 Projection Mappings

Definition:
The mappings,

 : X × Y  X s.t.  (x, y) = x  (x, y)  X × Y
x x
 : X × Y  Y s.t.  (x, y) = y  (x, y)  X × Y
x y
are called projection maps of X × Y onto X and Y spaces respectively.
Theorem 6: If (X, T) is the product space of topological spaces (X , T ) and (X , T ), then the
1 1 2 2
projection maps  and  are continuous and open.
1 2
Proof: Let (X, T) be a product topological space of topological spaces (X , T ) and (X , T ). Then
1 1 2 2
X = X × X .
1 2
Define maps
 : X  X s.t.  (x , x ) = x  (x , x )  X
1 1 1 1 2 1 1 2
 : X  X s.t.  (x , x ) = x  (x , x )  X.
2 2 2 1 2 2 1 2

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