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Heine-Borel Theorem; If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded.

I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'. (There exists $r\in \mathbb{R}$ such that for every $x\in E$, $|x|<r$)

The proof in Rudin PMA p.40 uses 'countable axiom of choice'

I have googled it and found some proofs, but they all used some weaker form of AC.

Please help me how to show that $compact⇒closed$ in ZF..

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2 Answers 2

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We will prove that compact implies closed by contraposition.

Suppose that $E$ is not closed. Then there is some $x\notin E$ such that every neighbourhood of $x$ has a nonempty intersection with $E$. In particular, the collection $E_n:=E\setminus \overline {B(x,1/n)}$ (where $\overline {B(x,1/n)}$ is the closed ball centered at $x$ with radius $1/n$) is an infinite, nondecreasing open cover of $E$ (because for any $p\in E$ and $n>1/d(x,p)$ we have $p\in E_n$).

It is enough to show that $E_n$ does not stabilize. But if it did, then we would have for some $N<\omega$ that $E_N=E$, so $B(x,1/N)$ would be disjoint from $E$, so $x$ would not belong to the closure of $E$, so we're done.

This argument should work in an arbitrary Hausdorff space, though without countable character the cover will not be a sequence, but a directed set.

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Can you give me a hint why $E_n$ is nonempty and infinite? –  Katlus Jul 29 '12 at 23:07
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@Katlus The family $\{E_n:n\in\mathbb{N}\}$ is nonempty and infinite. –  Michael Greinecker Jul 29 '12 at 23:44
    
Oh.. I thought he meant each $E_n$ is infinite. I understand the last contradiction leads that $\{E_n\}$ doesn't cover $E$, but i can't understand the part $\{E_n\}$ is an open cover of $E$. Plus, does 'open' here mean that 'open relative to $E$'?? –  Katlus Jul 29 '12 at 23:48
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It is open in the big space. That is why he uses the closed Ball in the construction. If $p\in E$, there is $n$ such that $d(x,p)<1/n$. Then $p\notin \overline{B(x,1/n)}$ and hence $p\in E\backslash\overline{B(x,1/n)}$. Since $p$ wa arbitrary, the family is a cover. –  Michael Greinecker Jul 29 '12 at 23:53
    
@Katlus It's open relative to $E$. If you want them open in $\mathbf R^k$, you can replace $E_n$ with $\mathbf R^k\setminus \ldots$, but it does not really matter for compactness. The rest is as Michael Greinecker said, except for the typo where it's supposed to be $d(x.p)>1/n$, naturally. :) –  tomasz Jul 29 '12 at 23:55

Any compact subset of a Hausdorff space is closed.

Proof: Let $A \subset X$ be compact. We show that $X -A$ is open. Let $x \in X - A$. Since $X$ is Hausdorff, for every $a \in A$, there are disjoint open sets $U_a$ and $V_a$ such that $x \in U_a$ and $a \in V_a.$ So $\{V_a\}_{a \in A}$ is an open cover which has a finite subcover $\{V_{a_1},...,V_{a_n}\}$. Let $V =\bigcup\limits_{i=1}^n V_{a_i}$ and $U = \bigcap\limits_{i=1}^n U_{a_i}$. Clearly, $U \cap V = \emptyset$, so $x \in U \subset X -A$. Hence, $X - A$ is open.

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How do you choose the $U_a$ and $V_a$ without the axiom of choice? –  Michael Greinecker Jul 29 '12 at 23:45
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In metric spaces, you can choose $U_\alpha$ and $V_\alpha$ without axiom of choice (by choosing suitable radius for a ball), but in general this looks like your run-of-the-mill application of AC. –  tomasz Jul 30 '12 at 0:00

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