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From various articles in Wikipedia, I have seen some relations between several types of compactness.

In general topological spaces,

Compact $\Longrightarrow$ Sequentially compact $\Longrightarrow$ countably compact $\Longrightarrow$ pseudocompact and weakly countably compact.

I also saw that

  1. In general topological spaces,

    Compact $\Longrightarrow$ σ-compact $\Longrightarrow$ Lindelöf.

    So I was wondering if we can place σ-compact and Lindelöf in the first chain of relations? If yes, what are their positions?

  2. In metric spaces,

    Compact $\Longleftrightarrow$ Sequentially compact $\Longleftrightarrow$ countably compact $\Longleftrightarrow$ pseudocompact $\Longleftrightarrow$ Limit point compact.

    So I was wondering if we can place Limit point compact in the first chain of relations? If yes, what is its position?

Thanks and regards!

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1  
This reminded me of this old MO question I asked: mathoverflow.net/questions/37195/… –  Bruno Stonek Feb 1 '12 at 3:45
    
@BrunoStonek: Thanks! –  Tim Feb 1 '12 at 5:01

2 Answers 2

It’s not true that compactness implies sequential compactness in arbitrary spaces, even if you require them to be Hausdorff: $\beta\omega$ is compact but not sequentially compact. In fact, the sequence $\langle n:n\in\omega\rangle$ has no convergent subsequence.

Proof: Suppose that $\sigma=\langle n_k:k\in\omega\rangle$ is a subsequence of $\langle n:n\in\omega\rangle$ converging to some $p\in\beta\omega$. Clearly $p\in\beta\omega\setminus\omega$, so we can think of $p$ as a free ultrafilter on $\omega$ whose basic open nbhds are the sets $A^*=\{q\in\beta\omega:A\in q\}$ for $A\in p$. Let $A\in p$ be arbitrary, and let $A_0$ and $A_1$ be disjoint infinite subsets of $A$ whose union is $A$. Exactly one of $A_0$ and $A_1$ belongs to $p$, say $A_0$. But then $A_0^*$ is an open nbhd of $p$ that misses every term of $\sigma$ that belongs to $A_1$, so $\sigma$ is not eventually in $A_0^*$ and cannot converge to $p$ after all. $\dashv$

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Thanks, +1! What is $βω$? Does sequential compactness imply compactness? –  Tim Feb 1 '12 at 2:45
    
@Tim: $\beta\omega$ is the Čech-Stone compactification of the natural numbers. A sequentially compact space need not be compact; $\omega_1$, the first uncountable ordinal, with the order topology is a standard counterexample. –  Brian M. Scott Feb 1 '12 at 2:50

Here are a couple of counterexamples: $\mathbb R$ is $\sigma$-compact but not pseudocompact. On the other hand,$\omega_1$ with the order topology is sequentially compact but not $\sigma$-compact.

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$\mathbb{R}$ isn’t even limit point compact. –  Brian M. Scott Feb 1 '12 at 2:43

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