Are These Metric Spaces Topologically Equivalent but Not Both Complete?

[zilla]

Homework Statement

Give an example of two metric spaces (X₁, d₁) and (X₂, d₂) which are topologically equivalent and for which (X₁, d₁) is complete and (X₂, d₂) is not.


2. The attempt at a solution
The open unit disc and R². They are homeomorphic, but there are Cauchy sequences in the disc which will converge to limits outside of the disc, so it's not complete.

 

[Dick]

I can't argue with your reasoning.

 

[zilla]

Thanks - I was pretty sure, but wanted to check.