# 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!

## 3 Answers

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.