How Does the Maximum Metric Define Topology on the Product of Two Metric Spaces?

[ideas]

Homework Statement [/b]
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.

 

[VeeEight]

To prove it is a metric, just go through the axioms of being a metric, taking into account that dX and dY are metrics on X and Y respectively (and satisfy all the axioms).

Notice that X x Y is a finite product. What is a basis for the metric topology that d induces? What is a basis for the product topology on X x Y?