## Questions on Completion of Metric Spaces and Isometries

I actually have two questions I am having trouble with.

1. The problem statement, all variables and given/known data
What is the completion of a discrete metric space X?

2. Relevant equations
d(x,x) = 0
d(x,y) = 1 if x does not = y

I don't really understand how to complete a metric space that is incomplete. I just know that every Cauchy sequence in X would have to converge to something in X itself, but I'm not sure how to manipulate it to ensure that this happens.

AND

1. The problem statement, all variables and given/known data
If X1 and X2 are isometric and X1 is complete, show that X2 is complete.

I know that since they are isometric, there is a mapping T such that d2(Tx,Ty) = d1(x,y). Other than that, I'm not sure how to prove it. It just kind of seems intuitive.

Any suggestions would be GREATLY appreciated. Thank you SO much.
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 Recognitions: Homework Help Science Advisor What does a cauchy sequence look like in a discrete metric space? Look at the definition of cauchy and think about it.
 Hmm...I'm not sure if this is right, but wouldn't you pick a sequence (xm) in the Discrete space that is Cauchy, then for every epsilon > 0 there exists N(epsilon) such that if m,n>N then d(xm, xn)< epsilon....but then, doesn't d(xm, xn)=1? But, how can we say that 1 < epsilon? I'm not sure, I really don't know what to do with this.

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## Questions on Completion of Metric Spaces and Isometries

Pick epsilon=1/2. If d(xm,xn)<1/2, what does that tell you about xm and xn if the metric is discrete? The elements in a sequence don't HAVE to be different. But being in a discrete metric might force them to be the same. Hmm?
 Ahh, so in order to complete this space, x and y must always be equal. That way, d(xm,xn) = 0 < epsilon. Am I getting that right?
 Recognitions: Homework Help Science Advisor You are getting closer. In order for a sequence to be cauchy, there must be an N such that xn=xm for all n,m>N. Agree? Does the sequence have a limit? What is it?
 Okay, now I'm confused. Since x_m is NOT cauchy unless x_m = x_n, doesn't that say that every Cauchy sequence in the discrete space converges? So, isn't the discrete space already complete, and therefore, why would it need a completion?

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