# A problem of completeness of a metric space

• I
• facenian

#### 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.

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##.

• PeroK and facenian
I think ##\{1,1/2,1/3,1/4,1/5,...\}## is a counterexample for part b.

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