Let (X,d) be a metric space. Show that if there exists a metric d' on X/~ such that(adsbygoogle = window.adsbygoogle || []).push({});

d(x,y) = d'([x],[y]) for all x,y in X

then ~ is the identity equivalence relation, with x~y if and only if x=y.

i have:

assume x=y

then d(x,y)=0 and [x]=[y] which implies d'([x],[y])=0 also.

now assume d(x,y) = d'([x],[y]) for all x,y in X

i now need to show that this implies x=y in order to copmlete the proof but i don't know where to go. any ideas?

thanks.

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# Homework Help: Metric space and topology help

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