Equivalence of completeness statements

Question

I need to "find a metric d such that (0,1) is complete". Is it equivalent to instead find a homeomorphism f from (0,1) to a complete space, so for instance $f=\tan(-\frac{\pi }{2}+\frac{\pi x}{2})$, and then say $d(x)=|f(x)-f(y)|$

Answer 1

Let $X$ be topological space and $M$ metric space, $f\colon X\to M$ a homeomorphism. Define $d_X(x,y):=d_M(f(x),f(y))$. You can easily check that this is a metric on $X$.

Also, $f$ becomes isometry, so $(x_n)$ is a Cauchy sequence in $X$ iff $(f(x_n))$ is Cauchy sequence in $M$ and by continuity $(x_n)$ converges in $X$ iff $(f(x_n))$ converges in $M$.

Use the above to conclude that $X$ is complete iff $M$ is complete.

Answer 2

Sure. Or you can take $f(x)=\log\left(\frac x{1-x}\right)$.