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This is a problem from Munkres' Topology.

Let $X$ be a metric space.

(a) Suppose that for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete.

(b) Suppose that for each $x \in X$, there is an $\epsilon \gt 0$ such that the ball $B(x, \epsilon)$ has compact closure. Show by means of an example that $X$ need not be complete.

I'm currently stuck on (a), I'm trying to show that given a Cauchy sequence ($x_n$), some tail of the sequence belongs entirely on a ball $B(x,\epsilon)$ and thus since a compact set is sequentially compact in metric space, the sequence has a convergent subsequence and so the Caucy sequence converges to the same limit. However, I'm having trouble constructing this process. Also, for (b), I cannot think of any such example. Any solution, hint, or suggestions would be appreciated.

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  • $\begingroup$ Take a close look at the definition of a Cauchy sequence again. $\endgroup$ Commented Dec 13, 2014 at 14:19
  • $\begingroup$ Ohh... I have been mixing up (a) with (b)...Should've been so simple. $\endgroup$ Commented Dec 13, 2014 at 14:26

2 Answers 2

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Apply the definition of a Cauchy sequence to $\frac{\epsilon}{2}$. So there exists $N \in \mathbb{N}$, such that for all $n,m \ge N$, $d(x_n, x_m) < {\epsilon \over 2}$. What ball of radius $\epsilon$ does the tail lie in?

The last part follows because of the following fact:if a subsequence of a Cauchy sequence converges to $p$, the whole sequence converges to $p$.

As to an example: consider $\{\frac{1}{n}: n \in \mathbb{N}\}$.

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For part a) fix the relevant $\epsilon$ and use the definition of Cauchy. You should easily see that the some tail is contained in some epsilon ball.

For part b) consider $\{1/n:n\in \mathbb{N}\}$.

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