Page 85 - DMTH503_TOPOLOGY

Unit 8: The Product Topology

as a basis. The set X × X × … × X with the topology T is said to be the product of the spaces Notes
1 2 n
(X , T ), (X , T ), …, (X , T ) and is denoted by (X × X × …, X , T) or (X , T ) × (X , T ) × … × (X , T ).
1 1 2 2 n n 1 2 n 1 1 2 2 n n
Proposition: Let B , B , …, B be bases for topological spaces (X , T ), (X , T ), …, (X , T ),
1 2 n 1 1 2 2 n n
respectively. Then the family {O × O × … × O : O  B , i=1, ..., n} is a basis for the product
1 2 n i i
topology on X × X × … × X .
1 2 n
Example 1: Let C , C , …, C be closed subsets of the topological spaces (X , T ), (X , T ),
1 2 n 1 1 2 2
…, (X , T ), respectively. Then C × C × … × C is a closed subset of the product space (X × X × …
n n 1 2 n 1 2
× X , T).
n
Solution: Observe that
(X × X × … × X )\(C × C × … × C )
1 2 n 1 2 n
= [ (X \C ) × X × … × X ]  [ X × (X \C ) × X × … × X ]  …  [ X × X × … × X ×
1 1 2 n 1 2 2 3 n 1 2 n–1
(X \C ) ]
n n
which is a union of open sets (as a product of open sets is open) and so is an open set in (X , T ) ×
1 1
(X , T ) × … × (X , T ). Therefore, its complement, C × C × … × C , is a closed set, as required.
2 2 n n 1 2 n

Notes

(i) We now see that the euclidean topology on R , n  2, is just the product topology on
n
n
the set R × R × … R = R .
(ii) Any product of open sets is an open set or more precisely: if O , O , …, O are open
1 2 n
subsets of topological spaces (X , T ), (X , T ) …, (X , T ), respectively, then O × O × …
1 1 2 2 n n 1 2
O is an open subset of (X , T ) × (X , T ) × … × (X , T ).
n 1 1 2 2 n n
(iii) Any product of closed sets is a closed set.
8.1.2 The Product Topology: Infinite Products

Let (X , T ), (X , T ), …, (X , T ), … be a countably infinite family of topological spaces. Then the
1 1 2 2 n n
¥
product, Õ i 1 X , of the sets X , iN consists of all the infinite sequences x , x , x , …, x , …,
i
=
1
i
n
3
2
¥
where x X for all i. (The infinite sequence x , x , …, x , … is sometimes written as Õ i 1 x ). The
i i 1 2 n = i
¥ ¥
product space, Õ i 1 (X ,T ), consists of the product Õ i 1 X with the topology T having as its basis
i
=
=
i
i
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. So a basic open set is of the form
O × O × … × O × X × X × …
1 2 n n + 1 n + 2
Note It should be obvious that a product of open sets need not be open in the product
topology T. In particular, if O , O , O , …, O , … are such that O  T and O  X for all i, then
1 2 3 n i i i i
¥
Õ i 1 O cannot be expressed as a union of members of B and so is not open in the product
=
i
¥
space (Õ i 1 X , ) T .
i
=
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