Is the Maximum Metric Valid for Product Topology on Combined Metric Spaces?

[ideas]

Let (X, dX ) and (Y , dY ) be metric spaces. The product of X and Y (written X × Y ) is the set of pairs {(x, y) : x ∈ X, y ∈ Y } with the metric:
d((x1 , y1 ), (x2 , y2 )) = max {dX (x1 , x2 ), dY (y1 , y2 )}
1)How to prove that d is a metric on X × Y?
2)Prove that d induces the product topology on X × Y.

 

[Studiot]

(1) Check the triangle inequality.

(2) Check the bases - one from each set - are satisfied

Is this coursework?