TITLE: Examples of Incomplete Spaces
QUESTION [6 upvotes]: A metric space is complete if every cauchy sequence is convergent.
To make space incomplete either i can change the metric or the ambient space.
For example if I change real numbers into rational number with usual metric ( absolute value ) it would be incomplete. On the other hand if have a some kind of metric on some space it would be incomplete though.
My question is: Can someone give examples of incomplete spaces such that either they have unusual metric or unusual ambient space other than rational numbers etc ?

REPLY [10 votes]: Here is an example of a metric $d$ on $\mathbf R$ such that  $(\mathbf R,d)$ is not complete.
Set $d(x,y)=\bigl\lvert\mathrm e^{-x}-\mathrm e^{-y}\bigr\rvert$. It trivially satisfies the axioms of a metric.
Now the sequence of natural numbers is a Cauchy sequence. Indeed choose $\varepsilon> 0$ and let $N$ be an integer such that  $\;\mathrm e^{-N}<\dfrac\varepsilon2$. If $p,q>N$, we have:
$$d(p,q)=\bigl\lvert\mathrm e^{-p}-\mathrm e^{-q}\bigr\rvert\le\mathrm e^{-p}+\mathrm e^{-q}<2\cdot\mathrm e^{-N}<\varepsilon. $$
This sequence does not converge. If it did converge to some $u\in \mathbf R$, we would have $\;d(n,u)=\bigl\lvert\mathrm e^{-n}-\mathrm e^{-u}\bigr\rvert<\varepsilon$ if $n>N_1$ for some $N_1$. If $n>\max(N,N_1)$, this implies
$$\mathrm e^{-u}<\mathrm e^{-n}+\varepsilon<\frac{3\varepsilon}2.$$
As $\varepsilon$ was chosen arbitrarily, this means $\mathrm e^{-u}=0$, which cannot be.