- #1
azdang
- 84
- 0
I actually have two questions I am having trouble with.
What is the completion of a discrete metric space X?
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
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.
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 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.