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I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove the following

Theorem: A topological space X is compact iff every net has a convergent subnet.

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Suppose that $x_\alpha$ is a net in $X$ with no convergent subnet. Then $(x_\alpha)$ do not have accumulation point in $X$. For each $x\in X$, let $V_x$ be an open neighbourhood of $x$ that excludes all the part of the net from some term onward. Let $V=\{V_x:\ x\in X\}$ and note that $V$ is an open cover of $X$. Can you prove that it is impossible to find a finite subcover of $X$ in $V$?

On the other hand, let $V$ be a open cover of $X$ such that every finite subcover of $V$ do not cover $X$. Consider the open cover $U$ of $X$ consisting of finite unions of elements of $V$. If $A,B\in U$, we say that $A\leq B$ when $A\subseteq B$. With this relation, $V$ is an directed set. For $A\in V$, let $x_A\in X\setminus A$. Can you show that the net $x_A$ does not have any convergent subnet?

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I suppose you mean for me to prove those claims by contradiction. Can you give me a hint about where the contradictions stem from. – Student Mar 13 '13 at 18:08
Where do you got stucked? – Tomás Mar 13 '13 at 18:43
I'm about as far as nowhere. – Student Mar 13 '13 at 19:41

See Theorem $15.3$ in this excellent PDF, Translating Between Nets and Filters, by Saitulaa Naranong; it’s well worth reading the whole thing.

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An interesting paper, but I was a bit confused, because they switched $\Psi$ and $\Phi$ in the definition of subnet on page 11. – Stefan Hamcke Mar 9 '13 at 18:29
@Stefan: I’m not sure what you mean: on that page he consistently uses $\Psi$ for the subnet and $\Phi$ for the original net. – Brian M. Scott Mar 9 '13 at 18:32
I mean $\Psi$ is subnet of $\Phi$ iff $\Psi$ is eventually in all the sets in which $\Phi$ is eventually. So in the second line of Definition 10.2 the arrow should point in the other direction. – Stefan Hamcke Mar 9 '13 at 18:39
@Stefan: No, the definition goes the other way: $\Phi$ is eventually in all of the sets in which $\Psi$ is eventually. The subnet may be in more sets eventually. – Brian M. Scott Mar 9 '13 at 18:41
@Stefan: Aaargghh! You’re right: I’m so used to the ideas that I kept reading what it meant instead of what it said. The one-sentence paragraph under Definition $11.1$ is right, and $\Phi$ and $\Psi$ are indeed reversed in the displayed line of Definition $10.2$. – Brian M. Scott Mar 9 '13 at 19:06

If you are studying Kelley, and you want to prove this subnet characterization of compactness, then a hint is Problem 2 J.

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