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Unit 4: The Product Topology on X × Y

Coarser: Let T and T’ are two topologies on a given set X. If T’  T, we say that T is coarser Notes
than T’.

Hausdorff space: A topological space (X, T) is called a Hausdorff space if a given pair of distinct
points x, y  X,  G, H  T s.t. x  G, Y  H, G H = .

Interior point: Let (X, T) be a topological space and   X. A point x   is called an interior
point of  iff  an open set G such that x  G  A.

4.5 Review Questions

1. Let  ,  , ...,  be the bases for topological spaces (X , T ), (X , T ), ..., (X , T ) respectively.
1 2 n 1 1 2 2 n n
Then prove that the family {O × O × ... × O : O   , i = 1, 2, ... , n} is a basis for the product
1 2 n i i
topology on X × X × ... × X .
1 2 n
2. Prove that the product of any finite number of indiscrete spaces is an indiscrete space.
3. Let X and X be infinite sets and T and T the finite-closed topology on X and X ,
1 2 1 2 1 2
respectively. Show that the product topology, T on X × X is not the finite-closed topology.
1 2
4. Let (X , T ), (X , T ) and (X , T ) be topological spaces. Prove that
1 1 2 2 3 3
[(X , T ) × (X , T )] × (X , T )  (X , T ) × (X , T ) × (X , T )
1 1 2 2 3 3 1 1 2 2 3 3
5. (a) Let (X , T ) and (X , T ) be topological spaces. Prove that
1 1 2 2
(X , T ) × (X , T )  (X , T ) × (X , T )
1 1 2 2 2 2 1 1
(b) Generalise the above result to products of any finite number of topological spaces.

4.6 Further Readings

Books H.F. Cullen, Introduction to General Topology, Boston, MA: Heath.
K.D. Joshi, Introduction to General Topology, New Delhi, Wiley.
S. Willard, General Topology, MA: Addison-Wesley.

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