Questions on Completion of Metric Spaces and Isometries

azdang

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.

Dick

What does a cauchy sequence look like in a discrete metric space? Look at the definition of cauchy and think about it.

azdang

Hmm...I'm not sure if this is right, but wouldn't you pick a sequence (x_m) in the Discrete space that is Cauchy, then for every epsilon > 0 there exists N(epsilon) such that if m,n>N then d(x_m, x_n)< epsilon....but then, doesn't d(x_m, x_n)=1? But, how can we say that 1 < epsilon? I'm not sure, I really don't know what to do with this.

Dick

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?

azdang

Ahh, so in order to complete this space, x and y must always be equal. That way, d(x_m,x_n) = 0 < epsilon. Am I getting that right?

Dick

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?

azdang

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?

Dick

You don't sound confused to me. Yes, all cauchy sequences eventually repeat the same term over and over. Therefore all cauchy sequences converge to something in the space. Yes, discrete spaces are already complete. The completion is itself.