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Unformatted text preview: d 1 and d 2 are equivalent metrics on X then the following hold: (i) A sequence { x n } ∈ X ∞ converges to x ∈ X in ( X,d 1 ) if and only if { x n } converges to x in ( X,d 2 ) . (ii) A subset S ⊆ X is open under d 1 if and only if S is open under d 2 . 1 Exercise 3: Consider two metric spaces, ( X 1 ,d 1 ) and ( X 2 ,d 2 ) , and the product space, Z = X 1 × X 2 . De ne the following function d on Z , for any x 1 ,x 1 ∈ X 1 and x 2 ,x 2 ∈ X 2 : d [( x 1 ,x 2 ) , ( x 1 ,x 2 )] = £ d 2 1 ( x 1 ,x 1 ) + d 2 2 ( x 2 ,x 2 ) / 1 / 2 Show that d is a metric. Note: d is called the product metric. 2...
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 Fall '09
 Antoine
 Economics, Topology, Metric space, Topological space

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