Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / E / 1 / b / i

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- Solution i

Solution

The following open rectangles form a basis of a topology for the product topology:

{U \times V : U \in T_{X}, V \in T_{Y}} .

These open rectangles in a finite product cover

X \times Y

, and intersections remain rectangles since

(U_{1} \times V_{1}) \cap (U_{2} \times V_{2}) = (U_{1} \cap U_{2}) \times (V_{1} \cap V_{2})

. One may equivalently take products of elements from a chosen basis of a topology in each factor.

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