I A problem of completeness of a metric space

facenian
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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.
 
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facenian said:
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##.
 
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Likes PeroK and facenian
I think ##\{1,1/2,1/3,1/4,1/5,...\}## is a counterexample for part b.
 
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I think you are both, George and Infrared, right. Thank you very much guys.
 
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