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The Problem: Given a metric space $(X,d)$, define a new metric $d'$ on X by $$ d'(x,y)=\frac{d(x,y)}{d(x,y)+1} $$ Is the topology induced by $d'$ the same as the topology induced by d? Prove or disprove.

My Work: This problem is actually on our review sheet for the midterm. I am having trouble understanding this problem because we have not talked at all about metric induced topologies in class. Any help in understanding this concept would be greatly appreciated. Thanks

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Show the transformation between two metrics and its inverse are continuous. – Metta World Peace Feb 20 '13 at 6:30

The topology induced by a metric is given as the open sets correspond to the "open balls" relative to the metric. So given $x_0 $ and $c$ the set $U=\{ x | d(x,x_0)<c \}$ would be open.

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