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Theorem 2.34 Compact subsets of a metric space are closed.

Proof.

Suppose $K\subseteq X$, $K$ compact. Let $p\in K^c$, $q\in K$. Let $V_q,W_q$ be neighborhoods of $p$ and $q$ with radius less than $\frac 1 2 d(p,q)$.

Since $K$ is compact, we have $K\subseteq W_{q_1}\cup \cdots\cup W_{q_n}=W$ for some $q_1,...,q_n\in K$.

If V=$V_{q_1}\cap \cdots V_{q_n}$, $V$ is a neighborhood of p which does not intersect $W$, then $V\subseteq K^c$ so $p$ is an interior point of $K^c$. QED.

My understading of this proof is that $V$ is actually $$V= \text {'$\min$'}\left\{V_{q_i}\right\}$$

Where the minimum is to be taken in terms of inclusion.

Also, we have that $V$ doesn't intersect $W$ because $V_{q_i}$ doesn't intersect $W_{q_i}$ for every $i$, right (How do I prove that)?

I'm uploading a picture of my understanding of the situation, am I understanding this correctly?

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  • $\begingroup$ What is your $W$? $\endgroup$
    – xpaul
    Commented Jan 6, 2016 at 21:42
  • $\begingroup$ The Union of all $W_{q_i}$s, forgot to add that in, thanks. $\endgroup$ Commented Jan 6, 2016 at 21:45

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Since each $V_{q}$ is a neighborhood of $p$, any finite intersection of them is again a neighborhood of $p$; yes, $V=V_{q_1}\cap\dots\cap V_{q_n}$ is a ball with the radius which is the minimum, but it's mostly irrelevant: the important fact is that it is a neighborhood of $p$.

Now, since $V\subseteq V_{q_i}$, we have by construction that $$ V\cap W_{q_i}\subseteq V_{q_i}\cap W_{q_i}=\emptyset $$ and therefore $$ V\cap W=V\cap(W_{q_1}\cup\dots\cup W_{q_n})= (V\cap W_{q_1})\cup\dots\cup(V\cap W_{q_n})=\emptyset $$ Hence $V\subseteq W^c\subseteq K^c$.

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  • $\begingroup$ The fact that $V$ is the ball with minimum radius is so irrelevant that this result and its proof holds mutatis mutandis to a compact subspace of a Hausdorff space. $\endgroup$
    – Aloizio Macedo
    Commented Jan 6, 2016 at 22:31
  • $\begingroup$ @AloizioMacedo Yes, that's the idea. I don't think Rudin should put so much emphasis on balls, in these proofs. $\endgroup$
    – egreg
    Commented Jan 6, 2016 at 22:32
  • $\begingroup$ Sorry for the late comment. But isn't the fact that $V \subseteq V_{q_i}$ intimately connected with the fact that $V$ is the neighborhood of smallest radius? $\endgroup$ Commented Mar 4, 2023 at 0:03
  • $\begingroup$ @SillyGoose If the $V_i$ are balls, yes, and I write so in my answer. However, insisting on balls is, in my opinion, misleading, because the argument works in any Hausdorff space with no mention of metric and balls. $\endgroup$
    – egreg
    Commented Mar 4, 2023 at 9:21
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I believe here you mean $W=W_{q_1}\cup\ldots\cup W_{q_n}$. So, let $x\in V$, for each $1\le k\le n$, $$\begin{align} x\in V & \implies x\in V_{q_k}\\ & \implies x\notin W_{q_k}\\ & \implies x\notin W. \end{align}$$

On the other hand, if $x\in W$, $$\begin{align} x\in W & \implies x\in W_{q_k},\quad\text{for some }1\le k\ne n\\ & \implies x\notin V_{q_k}\\ & \implies x\notin V. \end{align}$$

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$V$ doesn't intersect $W$ because $V$ doesn't intersect $W_{q_i}$ for every $i$. I think that is the same as what you said:

"Also, we have that $V$ doesn't intersect $W$ because $V_{q_i}$ doesn't intersect $W_{q_i}$ for every $i$, right (How do I prove that)?"

Perhaps this follows simply from the fact that $V$ has radius less than $\frac{1}{2}d(p,q_i), \forall i$, and $W$ is at most less than $\frac{1}{2}d(p,q_i)$ away from any $q_i$.

If that does not lead to a simple, direct proof, then I think perhaps you can prove this BWOC (if they did intersect, then some $W_{q_i}$ would need to intersect every $V_{q_j}$, but it can't intersect the $V_{q_i}$ by definition).

Also, I believe you are correct about $V$, the intersection of the neighborhoods, simply being the smallest neighborhood (since they are all neighborhoods of the same point, they different only in radius).

Lastly, your picture seems correct to me.

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    $\begingroup$ Despite the subscript, $V_{q_i}$ is a neighborhood of $p$, of radius $d(p,q_i)/2$. $\endgroup$
    – BrianO
    Commented Jan 6, 2016 at 21:50
  • $\begingroup$ Yes, my mistake. I will edit my answer and improve it. $\endgroup$
    – majmun
    Commented Jan 6, 2016 at 21:52
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Yeah, $ V $ is the smallest set from $V_{q_i}$ in the sense that $ x\in V $ if $ d(x,p) < min\{d(p,q_i)\}$ as $ q_i $ are finite the minimum exist, call it $ d(p,q_m) $. The point here is to show that every $ x\in V $ is not in $ W_{q_i} $ for every $ i={1,...,n} $, that is, if $ x\in V $ then $ d(x,q_i)\geq\frac{1}{2}d(p,q_i) $.

  1. $ d(p,q_i)\leq d(p,x)+d(x,q_i) $, by metric space definition.
  2. $ d(x,p)<\frac{1}{2}d(p,q_m)\leq \frac{1}{2}d(p,q_i) $.

From (2) $ -\frac{1}{2}d(p,q_i)<-d(x,p) $, then adding this to (1) we get

$ d(p,q_i)-\frac{1}{2}d(p,q_i)< d(p,x)-d(p,x)+d(x,q_i) \rightarrow \frac{1}{2}d(p,q_i)<d(x,q_i)$ for any $i=\{1,...,n\}$.

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Not exactly relevant to your post, but possibly helpful to some searching for guidance on Theorem 2.34. One subtlety that had me scratching my head the first time I read it is the reason why we can apply the definition of compactness to conclude that $K \subset W_{q_1} \cup \dots \cup W_{q_n}$ for some finite set of points.

By construction $K \subset \cup_\alpha W_\alpha$ where each $W_\alpha$ is an open subset (namely, the neighborhood $N_r(q)$ where $r = \frac{1}{2} d(p,q))$. Hence $W_\alpha$ is an open cover of $K$. Thus, because $K$ is compact, $K \subset W_{q_1} \cup \dots \cup W_{q_n}$ for some finite set of points.

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Using this construction, you can show that if $V$ intersects $W$, then you will have violated the triangle inequality (because if $x$ is in $X$ and $V$, then $x$ is in $V_x$ which is impossible due to the constraint of the triangle inequality). The key part of the proof is that compactness allows you to construct $V$ by taking the intersection of a finite collection of neighborhoods of $p$, this allows us to conclude that $V$ is open (this would not follow necessarily given an infinite collection of open balls) and disjoint from $K$. Then, since $p$ was an arbitrary element of the complement, this implies that every point in the complement is an interior point. Therefore $K$ is closed because its complement is open.

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