A problem of completeness of a metric space

[facenian]

TL;DR Summary

This seems to be a contradictory topology problem

Hi, I found this problem in Munkres' topology book, and it seems to be contradictory:
Let X be a metric space.
(a) Suppose that for some ϵ>0, every ϵ-Ball in $X$ has compact closure. Show that $X$ is complete.
(b) Suppose that for each $x\in X$ there is an $\epsilon>0$ such as the ball $B(x,\epsilon)$ has compact closure. Show by means of an example that $X$ need not be complete.

I believe that (a) can be proved. But then, (b) must be impossible to prove. Am I crazy? or it is a typo. Any help will be much appreciated.

 

[George Jones]

Summary:: This seems to be a contradictory topology problem

(a) Suppose that for some ϵ>0, every ϵ-Ball in $X$ has compact closure.
(b) Suppose that for each $x\in X$ there is an $\epsilon>0$ such as the ball $B(x,\epsilon)$ has compact closure.

a) $\exists \epsilon$ such that $\forall x$ ,,,

b) $\forall x$, $\exists \epsilon$

Note that in b), each $\epsilon$ could depend on $x$.

 

[Infrared]

I think $\{1,1/2,1/3,1/4,1/5,...\}$ is a counterexample for part b.

 

[facenian]

I think you are both, George and Infrared, right. Thank you very much guys.