 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2008/a/1/b/solution

D_{X} \subseteq X

and

D_{Y} \subseteq Y

. Their Cartesian product is countable. Every nonempty basic open set of the product topology has the form

U \times V

with

U, V

nonempty open in their respective topological spaces. Density gives

d_{X} \in D_{X} \cap U

and

d_{Y} \in D_{Y} \cap V

, hence

(d_{X}, d_{Y}) \in (D_{X} \times D_{Y}) \cap (U \times V)

Every nonempty open set in the product contains such a basic open set. Thus

D_{X} \times D_{Y}

is dense in

X \times Y

, proving topological separability of the product. If one factor is empty the product is empty and the conclusion holds directly.

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