- #1

azdang

- 84

- 0

## Homework Statement

What is the completion of a discrete metric space X?

## Homework 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

## Homework Statement

If X

_{1}and X

_{2}are isometric and X

_{1}is complete, show that X

_{2}is complete.

I know that since they are isometric, there is a mapping T such that d

_{2}(Tx,Ty) = d

_{1}(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.