A problem of completeness of a metric space

In summary, the conversation discusses a topology problem that appears to be contradictory. Part (a) suggests that a metric space X is complete if every ϵ-ball has compact closure, while part (b) provides a counterexample where X is not complete even though every point x has an ϵ-ball with compact closure. The conversation ends with the suggestion that there may be a typo in part (b) or that the problem may be impossible to prove.
  • #1
facenian
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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 [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.
 
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  • #2
facenian said:
Summary:: This seems to be a contradictory topology problem

(a) Suppose that for some ϵ>0, every ϵ-Ball in [itex]X[/itex] has compact closure.
(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.
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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  • #3
I think ##\{1,1/2,1/3,1/4,1/5,...\}## is a counterexample for part b.
 
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  • #4
I think you are both, George and Infrared, right. Thank you very much guys.
 

Related to A problem of completeness of a metric space

What is a metric space?

A metric space is a mathematical concept that consists of a set of objects and a function called a metric, which measures the distance between any two objects in the set. This function satisfies certain properties, such as non-negativity, symmetry, and the triangle inequality.

What is the problem of completeness in a metric space?

The problem of completeness in a metric space is a fundamental issue in mathematics that deals with whether or not all Cauchy sequences in a metric space converge to a limit within that space. In other words, it concerns the existence of a "missing point" in the space that would make it complete.

What is the importance of completeness in a metric space?

Completeness is an essential property of a metric space because it guarantees the existence of limits for all Cauchy sequences. This is crucial in various areas of mathematics, such as analysis, topology, and differential equations, as it allows for the development of rigorous proofs and solutions.

How is completeness related to compactness in a metric space?

Completeness and compactness are closely related properties in a metric space. A metric space is compact if and only if it is complete and totally bounded. This means that every Cauchy sequence in a compact metric space converges to a limit within that space, and the space is "bounded" in the sense that it can be covered by a finite number of open balls.

Can a metric space be complete and not compact?

Yes, a metric space can be complete but not compact. An example of such a space is the real numbers with the standard metric. It is complete because all Cauchy sequences converge to a limit within the space, but it is not compact because it is unbounded. In general, a metric space can be complete, compact, both, or neither.

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