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Suppose $(X,d)$ is an incomplete metric space. Prove that there exist a uniformly continuous function $$f\ \colon\ \ X\rightarrow (0,\infty)$$ such that $$\inf_{x\in X} f(x)=0.$$ This is what I did. Now since completion of $X$ does exist so taking any cauchy sequence $\{x_n\}$ we can find the limit, say, $x_0$ in the completion . Define the function $$f(x)=x-x_0.$$ Then $$\lim_{n\rightarrow \infty} f(x_n)=0$$

But what assures that it will not take values less than $0$? And what about uniformity of the function?

So I think what I came up with is useless.

How to prove it then? Thanks for any help.

Glinka
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1 Answers1

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Let $(X’, d’)$ be a completion of the space $(X,d)$. Since the space $(X,d)$ is incomplete, there exists a point $x_0\in X'\setminus X$. Put $f(x)=d’(x,x_0)$ for all $x\in X$. Then $f(X)\subset (0,\infty)$ and the triangle inequality for the metric $d’$ and condition $d’|X\times X=d$ imply uniform continuity of the function $f$. The density of the set $X$ in the space $X'$ imply that $\inf_{x\in X}f(x)=0$.

Alex Ravsky
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