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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 [itex]X[/itex] has compact closure. Show that [itex]X[/itex] is complete.

(b) Suppose that for each [itex]x\in X[/itex] there is an [itex]\epsilon>0[/itex] such as the ball [itex]B(x,\epsilon)[/itex] has compact closure. Show by means of an example that [itex]X[/itex] 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.

Let X be a metric space.

(a) Suppose that for some ϵ>0, every ϵ-Ball in [itex]X[/itex] has compact closure. Show that [itex]X[/itex] is complete.

(b) Suppose that for each [itex]x\in X[/itex] there is an [itex]\epsilon>0[/itex] such as the ball [itex]B(x,\epsilon)[/itex] has compact closure. Show by means of an example that [itex]X[/itex] 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.