 Solution

ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2015/1/b/i/solution

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

It covers

X \times Y

, since

X \times Y \in B

. Whenever a point lies in two basis rectangles, their intersection is another basis rectangle containing it:

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

These are precisely the two basis of a topology axioms. Arbitrary unions of the rectangles therefore form a topology, the product topology. To check the finite-intersection axiom directly, distribute the intersection of two unions over all pairs of rectangles and use the displayed formula. The empty union gives the empty set.

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