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Let $S$ be a subset of $\mathbb R^n $. Then the following statements are equivalent
a) $S$ is compact
b) $S$ is closed and bounded
c) Every infinite subset $S$ has an accumulation point in $S$

I think I terribly misunderstood the compactness and open covering. It's clear that $b) \implies a)$, I am having hard time to understand why $a) \implies b)$, isn't the open interval $(1,3)$ covered by finite open sub-covering of the form $(n,n +1)$.

The proof of my book can be summarized as follows.

Let $y$ be accumulation point in $S| y \notin S , \forall x \in S,$ choose $r_x = \frac12||x-y||$, then $ \{ B(x, r_x): x \in S\} $ is open sub-covering of $S$.i.e $S \subseteq \cup_{k=1}^p B(x_k, r_x )$, choosing $r = \frac 12 \min\{r_k\}$, we can show that $\cup_{k=1}^nB(x_k, r) \cap B(y, r) = \emptyset$.

But what's the point? don't we have freedom to choose our $r$? If we define the covering this way, no matter how many open sub-covers we make, we will never cover this the region between $r$ and $y$??

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$(1,3)$ is indeed covered by a finite subcover of $(n,n+1)$. But this doesn't make it compact, as you need every open cover to have a finite subcover –  Nameless Dec 30 '12 at 9:07
    
@Nameless suppose, we have a closed set instead $[1, 3]$, still there would be infinite subcovers. But we would be able to extract a fine subcovers that also covers it? Is that the point? Could you explain a bit more? –  Santosh Linkha Dec 30 '12 at 9:20
    
$[1,3]$ is compact. Thus, for every (infinite) open cover there is a finite subcover. If you find an infinite cover, then you can find a finite subcover as well –  Nameless Dec 30 '12 at 9:22
    
@Nameless I'm sorry again :(( –  Santosh Linkha Dec 30 '12 at 9:33
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3 Answers

up vote 4 down vote accepted

You wrote:

The proof of my book can be summarized as follows.

Let $y$ be accumulation point in $S\mid y \notin S , \forall x \in S,$ choose $r_x = \frac12\|x-y\|$, then $ \{ B(x, r_x): x \in S\} $ is open sub-covering of $S$.i.e $S \subseteq \cup_{k=1}^p B(x_k, r_x )$, choosing $r = \frac 12 \min\{r_k\}$, we can show that $\cup_{k=1}^nB(x_k, r) \cap B(y, r) = \emptyset$.

Part of the problem is that the proof in the book cannot be summarized that way (unless the book is so badly written that you’d be better off throwing it away!).

  • $\{B(x,r_x):x\in S\}$ is not an ‘open sub-covering of $S$’; it’s the open cover of $S$ with which you’re starting.

  • You never say what $x_1,\dots,x_p$ are.

  • The notation $B(x_k,r_k)$ is undefined: either you must write $B(x_k,r_{x_k})$, or for $k=1,\dots,p$ you must define $r_k=r_{x_k}$ and then write $B(x_k,r_k)$.

  • There is no need for the factor of $\frac12$ in the definition of $r$ (though I suppose that the author might have added this unnecessary touch).

Here’s a correct version, with a little extra explanatory detail:

Let $y$ be an accumulation point of $S$ that is not in $S$. For each $x\in S$ let $r_x=\frac12\|x-y\|$; since $y\notin S$, $\|x-y\|>0$ for all $x\in S$, and therefore $r_x>0$ for each $x\in S$. Thus, $\{B(x,r_x):x\in S\}$ is an open cover of $S$. Since $S$ is compact, this cover has a finite subcover; that is, there is a finite set $\{x_1,\dots,x_p\}\subseteq S$ such that $\{B(x_k,r_{x_k}):k=1,\dots,p\}$ covers $S$, i.e., $S\subseteq\bigcup_{k=0}^pB(x_k,r_{x_k})$. Now $\{r_1,\dots,r_p\}$ is a finite set of positive real numbers, so $\min\{r_1,\dots,r_p\}>0$. Let $r=\min\{r_1,\dots,r_p\}$; then $B(y,r)\cap S\subseteq B(y,r)\cap\bigcup_{k=1}^pB(x_k,r_{x_k})=\varnothing$, because $B(y,r)\cap B(x_k,r_{x_k})=\varnothing$ for $k=1,\dots,p$. To see this, suppose that $z\in B(y,r)\cap B(x_k,r_{x_k})$; then $$\|x_k-y\|\le\|y-z\|+\|z-x_k\|<r+r_{x_k}\le 2r_{x_k}=\|x_k-y\|\;,$$ which is absurd. (The step $r+r_{x_k}\le 2r_{x_k}$ is justified by the fact that $r\le r_k$.)

But then $B(y,r)$ is an open neighborhood of $y$ disjoint from $S$, contradicting the hypothesis that $y$ is an accumulation point of $S$. This contradiction shows that every accumulation point of $S$ must in fact belong to $S$ and hence that $S$ is closed. $\dashv$

Note that this is essentially the same argument that copper.hat gives in his answer; it’s just arranged to stay a bit closer to the one that you summarized incorrectly.

The intuitive idea is really fairly simple. Since $y\notin S$, we can cover $S$ with open balls so small that $y$ isn’t even in the closure of any of them; that’s what we’re doing when we choose the radii $r_x$ to be less than the distance from $x$ to $y$. (We actually took $r_x$ to be half the distance from $x$ to $y$, but that was just a convenience; anything less than $\|x-y\|$ would work.) Then we use the compactness of $S$ to say that we don’t actually need all of these open balls $B(x,r_x)$: finitely many of them are already enough to cover $S$. Let these finitely many open balls be $B(x_k,r_{x_k})$ for $k=1,\dots,p$. For $k=1,\dots,p$ let $\overline{B}(x_k,r_{x_k})=\{z\in\Bbb R^n:\|z-x_k\|\le r_{x_k}\}$, the closed ball of radius $r_{x_k}$ centred at $x_k$. We chose each $r_x$ to be less than the distance from $x$ to $y$, so $y$ is not in any of these closed balls, and therefore it’s not in their union:

$$y\notin\bigcup_{k=1}^p\overline{B}(x_k,r_{x_k})\;.\tag{1}$$

The righthand side of $(1)$ is the union of finitely many closed sets, so it is itself a closed set, and it contains $S$. Every closed set that contains $S$ also contains the closure of $S$, so $y$ is not in the closure of $S$, contradicting the hypothesis that $y$ is an accumulation point of $S$. (Or, if you arrange the argument as copper.hat did, showing directly that the complement of $S$ is open and hence that $S$ is closed.)


Finally, as copper.hat says, ‘$S$ is compact’ does not mean that $S$ has a finite open cover. It means that no matter what open cover of $S$ you start with, that open cover has a finite subcover. If you can find just one open cover of $S$ that has no finite subcover, then $S$ is not compact. Finding one open cover of $S$ that does have a finite subcover, on the other hand, tells you absolutely nothing about whether $S$ is compact.

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is it possible to have an example of an example of finite sub-cover of open covering closed set? Like $\{(1/n, 2/n): n \ge 2\}$ forms open infinite covering, similarly can we have for $[0,1]$ but finite one? –  Santosh Linkha Dec 31 '12 at 7:36
    
@experimentX: I’m not sure what you’re asking: $[0,1]$ is compact, so every open cover of $[0,1]$ has a finite subcover. –  Brian M. Scott Dec 31 '12 at 15:22
    
I mean example of open cover of [0,1] having finite subcovers –  Santosh Linkha Dec 31 '12 at 16:18
    
@experimentX: Pick any open cover of $[0,1]$ that you like: it has a finite subcover. For instance, the open cover $$\left\{\left(-1,\frac1{100}\right)\right\}\cup\left\{\left(\frac1n,2\right):n \ge 2 \right\}$$ has $$\left\{\left(-1,\frac1{100}\right),\left(\frac1{101},2\right)\right\}$$ as a finite subcover. –  Brian M. Scott Dec 31 '12 at 16:25
    
but this also covers (0,1). What is the difference? Sorry for asking unnecessary questions. –  Santosh Linkha Dec 31 '12 at 16:32
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$S$ is covered by $\bigcup_{x\in S}B(x,r_x)$ and as $S$ is compact, $S\subseteq \bigcup_{k=1}^p B(x_k,r_{x_k})$. If we choose $r=\min\left\{r_{x_k}\right\}>0$ then it is true that $B(x_k,r)\cap B(y,r)=\emptyset$ and so $$S\cap B(y,r)\subseteq \bigcup_{k=1}^p B(x_k,r_{x_k})\cap B(y,r)=\emptyset$$ which is a contradiction as $y$ is a limit point of $S$.

I sincerely do not understand your questions. Would you please elaborate?

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sorry .. fixed again. –  Santosh Linkha Dec 30 '12 at 9:30
    
@experimentX Now that there are no more typos, lets concentrate on the problem. What is it you don't understand in this proof? –  Nameless Dec 30 '12 at 9:34
    
Could you explain your first comment? If i understand that, i might understand the most of it. –  Santosh Linkha Dec 30 '12 at 9:36
    
$(1,3) \subset \cup_0^3 (n, n+1)$ but why is it not compact? –  Santosh Linkha Dec 30 '12 at 9:38
    
Compact means every open subcover has a finite subcover. You have picked just one open cover. Another would be $\{\mathbb{R}^n\}$, which 'works' for all subsets of $\mathbb{R}^n$. –  copper.hat Dec 30 '12 at 9:39
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I think it is easier and more instructive to show a) $\Rightarrow$ b) directly.

The key is that every open cover has a finite subcover. $(1,3)$ has the open cover $\{(1+\frac{1}{n},3)\}_n$, but it has no finite subcover, hence is not compact.

If $S$ is compact, then every open cover has a finite subcover. So take $\{B(0,n)\}_n$. This is an open cover of $S$, hence has a finite subcover, hence $S \subset B(0,n')$ for some $n'$. Hence $S$ is bounded.

Now suppose $y \notin S$. For each $x \in C$, there exists some $\epsilon_x>0$ such that $|y-x| \geq \epsilon_x$, or in other words, $y \notin B(x,\epsilon_x)$. Now consider the open cover $\{B(x,\frac{1}{2} \epsilon_x)\}_{x \in S}$. Since $S$ is compact, there is a finite subcover $\{B(x_i, \frac{1}{2} \epsilon_{x_i})\}_i$, and so $C \subset \cup_i B(x_i, \frac{1}{2} \epsilon_{x_i})$. By construction, if $x \in B(x_i, \frac{1}{2} \epsilon_{x_i})$, then $|y-x| \geq \frac{1}{2} \epsilon_{x_i}$, hence if we take $\delta = \frac{1}{2}\min_i \epsilon_{x_i}$, we have $B(y,\delta) \cap S = \emptyset$, and hence we have that $S^C$ is open. Hence $S$ is closed.

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Brian, thanks for the corrections. I was thinking under the influence. –  copper.hat Dec 30 '12 at 18:04
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