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Homework Help: Product of Two Metric Spaces

  1. Feb 27, 2010 #1
    The problem statement, all variables and given/known data[/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.
  2. jcsd
  3. Feb 27, 2010 #2
    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?
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