Relations between different types of compactness

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!

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. $\dashv$

• 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.

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

we can place $$\sigma$$-compact and Lindelöf in the first chain of relations?

No, because it would collapse the chain, since each Tychonoff Lindelöf pseudocompact space $$X$$ is compact. Indeed, by [Eng, 3.8.11] the space $$X$$ is paracompact, that is each open cover of the space $$X$$ has a locally finite open refinement. By [Eng, 3.10.22] the space $$X$$ is feebly compact, that is if each locally finite family of non-empty open subsets of the space $$X$$ is finite. We can conclude that the space $$X$$ is compact.

PS. A stratification of different compact-like spaces is much more subtle, see, for instance, basic [Eng, Chap. 3] and general works [DRRT], [Mat], [Vau], [Ste], [Lip]. The relations between different classes of these spaces are well-studied, some of them are presented at the following diagrams: [M, D:3, p.17], [D-AS, D.1, p. 58](for Tychonoff spaces), [Ste, D. 3.6, p. 611], and [GR, p.2].

References

[BR] T. Banakh, A. Ravsky. Verbal covering properties of topological spaces, Topology Appl. ** 201** (2016), 181–205.

[D-AS]A. Dorantes-Aldama, D. Shakhmatov, Selective sequential pseudocompactness, Topology Appl., 222 (2017), 53–69.

[DRRT] E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties, Topology Appl., 39:1 (1991), 71-103.

[Eng] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

[GR] O. Gutik, A. Ravsky, On old and new classes of feebly compact spaces, Visnyk of the Lviv Univ. Series Mech. Math. 85 (2018), 48–59.

[Lip] P. Lipparini, *A very general covering property”.

[Mat] M. Matveev, A Survey on Star Covering Properties.

[Ste] R. M. Stephenson Jr., Initially $$\kappa$$-compact and related compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 603-632.

[Vau] J. E. Vaughan, Countably compact and sequentially compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 569-602.