# Covering compact Hausdorff spaces with closed $G_\delta$ sets

I'm thinking about results of the form: Under assumption $$A$$, if $$X$$ is a compact Hausdorff space and $$C$$ is a cover of $$X$$ by closed $$G_\delta$$ sets, then there is a subcover of cardinality $$\leq\kappa$$.

Obviously in general we can take $$\kappa\leq|X|$$ and in some cases this is the best we can do such as in Cantor space in which every point is a closed $$G_\delta$$ set.

The strongest result I've found so far is this: If $$X$$ is a scattered compact Hausdorff space and $$C$$ is a cover of $$X$$ by closed $$G_\delta$$ sets, then there is a countable subcover.

Proof: Since $$X$$ is scattered it is totally disconnected, and we can represent each $$F\in C$$ as $$F=\bigcap_{n<\omega}Q_{F,n}$$, where $$Q_{F,n}$$ is a nested sequence of clopen sets. Partition $$X$$ into a finite collection $$X_0, \dots ,X_{n-1}$$ of clopen sets such that each $$X_i$$ has a unique point $$x_i$$ such that $$CB(x_i) = CB(X)$$ (where $$CB$$ is the Cantor-Bendixson rank). In each $$X_i$$ choose some $$F_i\in C$$ such that $$x_i \in F$$ and then consider $$X_i \setminus F_i$$, which can be written as a countable union of clopen sets. Iterate this procedure in each such clopen set. This forms a well-founded forest which is countably branching and thus countable. The subcover is the $$F_i$$'s chosen at each node of the forest.

This is certainly optimal for the general scattered case, as you can construct a cover that saturates the bound in any infinite scattered compact Hausdorff space.

A weak generalization is this: If $$X$$ is a compact Hausdorff space with perfect core $$P$$, and $$C$$ is a cover of $$X$$ by closed $$G_\delta$$ sets, then for any $$C_0 \subseteq C$$ covering $$P$$, $$C$$ has a subcover of cardinality $$\leq \omega^{|C_0| + 1}$$.

Proof: If $$P$$ is empty we can use the previous result, so assume that $$P$$ is non-empty. Let $$C_0$$ be a cover of $$P$$, so in particular $$P \subseteq \bigcup_{F\in C_0}\bigcap_{n<\omega}U_{F,n} = \bigcap_{g:C_0\rightarrow\omega}\bigcup_{F\in C_0}U_{F,g(F)}$$. By compactness, for each $$g:C_0 \rightarrow \omega$$, there is a finite set $$C_{0,g}\subseteq C_0$$ such that $$P\subseteq \bigcup_{F\in C_{0,g}}U_{F,g(F)}$$. So let $$R_g=X\setminus \bigcup_{F\in C_{0,g}}U_{F,g(F)}$$. Each $$R_g$$ is closed and scattered as a subspace of $$X$$, so we can run the previous argument to get a countable $$C_g \subseteq C$$ that covers $$R_g$$. This means that in order to cover the rest of $$X$$ we need at most $$\omega^{|C_0|}\times \omega = \omega^{|C_0|+1}$$ more sets, so since $$\omega^{|C_0|+1} \geq |C_0|$$ we need at most $$\omega^{|C_0|+1}$$ all together.

This is only clearly non-trivial in cases where $$P$$ is much smaller than $$X$$. The question is how much better can you do? In particular is there a universal bound for all compact Hausdorff spaces? Can you get a better bound in terms of $$|C_0|$$? Can you get bounds in terms of other cardinal invariants of $$P$$ or $$X$$, such as the weight or density character? Can you say anything non-trivial about perfect spaces? What is $$\kappa$$ for $$\beta \omega \setminus \omega$$ or $$2^\lambda$$ for various cardinals $$\lambda$$?

EDIT: In the negative direction I suspect you can show that if $$X$$ is a compact Hausdorff space with non-empty perfect core, then $$X$$ has a cover by closed $$G_\delta$$ sets such that no subcover has cardinality $$< 2^\omega$$.

EDIT2: Suppose that $$X$$ is a compact Hausdorff space with non-empty perfect core $$P$$. Construct a tree of closed sets $$\{F_{\sigma}\}_{\sigma \in 2^{<\omega}}$$ with the following properties: Every $$F^\circ_{\sigma}$$ has non-empty intersection with $$P$$. For every $$\sigma \in 2^{<\omega}$$, $$F_{\sigma\frown 0},F_{\sigma\frown 1} \subset F_\sigma^\circ$$ and $$F_{\sigma\frown 0}\cap F_{\sigma \frown 1} = \varnothing$$. (We can do this since $$P$$ is perfect.) By these conditions, for any $$\alpha \in 2^\omega$$, the set $$G_\alpha = \bigcap_{n<\omega} F_{\alpha \upharpoonright n}$$ is a closed $$G_\delta$$ set. If $$\alpha,\beta \in 2^\omega$$ with $$\alpha \neq \beta$$, then $$G_\alpha \cap G_\beta = \varnothing$$, and $$H = \bigcup_{\alpha \in 2^\omega} G_\alpha$$ is a closed $$G_\delta$$ set. Since $$H$$ is a closed $$G_\delta$$ set, we can find a sequence of closed $$G_\delta$$ sets $$\{D_n\}_{n<\omega}$$ such that $$\bigcup_{n<\omega} D_n = X \setminus H$$. So then the cover $$\{D_n\}_{n<\omega} \cup \{ G_\alpha \}_{\alpha \in 2^\omega}$$ is a cover by closed $$G_\delta$$ sets with no subcover of cardinality $$< 2^\omega$$.

EDIT3: There's a much better bound in the second case that I missed: If $$X$$ is a compact Hausdorff space with perfect core $$P$$, and $$C$$ is a cover of $$X$$ by closed $$G_\delta$$ sets, then for any $$C_0 \subseteq C$$ covering $$P$$, $$C$$ has a subcover of cardinality $$\leq |C_0|+\omega$$.

Proof: Same proof as before, just notice that while there are $$\omega^{|C_0|}$$ many functions $$g:C_0 \rightarrow \omega$$, there are only $$(|C_0|\times \omega)^{<\omega} = |C_0|+\omega$$ many finite subsets of $$\{U_{F,n}\}_{F\in C,n<\omega}$$, so there are only at most $$|C_0|+\omega$$ many $$R_g$$'s to consider, and we get that $$C$$ has a subcover of size at most $$|C_0|+(|C_0|+\omega)\times\omega = |C_0|+\omega$$.

By the same argument as in the scattered case, this is basically optimal relative to the size of $$|C_0|$$, so now the questions really is about perfect spaces.

EDIT4: Combining the content of Taras and Anonymous's comments, you get a more refined bound in terms of generalized Cantor-Bendixson ranks (these probably have a name somewhere), specifically for any cardinal $$\kappa$$ we can can define

$$X^\prime_{\kappa}=X \setminus \bigcup \{U\subseteq X : U\text{ open, has density character }\leq\kappa\}$$

$$X^{(\alpha)}_{\kappa} = \bigcap_{\beta < \alpha} (X^{(\beta)}_{\kappa})^\prime _{\kappa}$$

where $$\bigcap\varnothing=X$$, for any $$\alpha \in \text{Ord}\cup\{\infty\}$$. And $$CB_{\kappa}(X)$$ is the smallest ordinal $$\alpha$$ for which $$X^{(\alpha + 1)}_{\kappa}=\varnothing$$, if it exists, and $$\infty$$ otherwise. Then if we let $$S(X)$$ be the smallest $$\kappa$$ such that $$CB_{\kappa}(X)<\infty$$, we should get a better bound in terms of $$S(X)$$.

If the density character of $$X$$ is $$\kappa$$, then every closed $$G_\delta$$ cover has a subcover of cardinality $$\leq 2^\kappa$$, since there are only at most that many closed $$G_\delta$$ sets in $$X$$.

Assume that for some cardinal $$\kappa$$ and some ordinal $$\alpha$$, we've shown that if $$CB_{\kappa}(X)<\alpha$$, then every closed $$G_\delta$$ cover has a subcover of cardinality $$\leq 2^\lambda$$ for some $$\lambda < \kappa$$. Let $$X$$ be a compact Hausdorff space such that $$CB_{\kappa}(X) = \alpha$$ and let $$C$$ be a cover of $$X$$ by closed $$G_\delta$$ sets. This implies that the density character of $$P = X^{(CB_{\kappa}(X))}_{\kappa}$$ is $$\leq\kappa$$, so there is some set $$C_0 \subseteq C$$ of cardinality $$\leq 2^\kappa$$ such that $$C_0$$ covers $$P$$. By the same argument as before, $$X\setminus \bigcup C_0$$ can be written as the union of at most $$2^\kappa$$ many closed subspaces, each of which has strictly smaller $$CB_{\kappa}$$. So by the induction hypothesis each of these has a subset of $$C$$ of cardinality $$\leq 2^\kappa$$ that covers it, so overall $$X$$ is covered by a subset of $$C$$ of cardinality $$\leq 2^\kappa$$.

So by induction, if $$CB_{\kappa}(X)<\infty$$, then every closed $$G_\delta$$ cover of $$X$$ has a subcover of cardinality $$\leq 2^\kappa$$. So in general any closed $$G_\delta$$ cover of a compact Hausdorff space has a subcover of cardinality $$\leq 2^{S(X)}$$.

You can probably get slightly tighter bounds at singular cardinals using a CB derivative for density characters $$<\kappa$$ rather than just $$\leq \kappa$$.

• Your question is about the Lindelof number of the $G_\delta$-modification of a topological space. You can consider scatteredness with respect to the property $P$ you are interested in: a topological space is P-scattered if each closed non-empty subasace contains a non-empty relatively open subspace with property $P$. In your case the property P is the existence of small subcover of any cover by $G_\delta$-sets. – Taras Banakh Feb 4 at 6:55
• For $\beta \omega \setminus \omega$ the cardinal $\kappa$ is $2^\omega$. The reason is that for compact Hausdorff spaces, closed $G_\delta$ sets are the same as zero sets and a compact subset of a separable space has only $2^\omega$ continuous functions and, therefore, only $2^\omega$ zero sets. – Anonymous Feb 4 at 14:48

Your question is related to a pair of old questions of A.V.Arhangel'skii. First of all note that in a regular space, for every point $$x$$ and every $$G_\delta$$ set G containing $$x$$ there is a closed $$G_\delta$$ $$H$$ contained in $$G$$ such that $$x \in H$$. So the topology generated by the closed $$G_\delta$$ sets of a compact Hausdorff space $$X$$ coincides with the "$$G_\delta$$-topology" $$X_\delta$$ (that is, the topology generated by the $$G_\delta$$ subsets of $$X$$).

The Lindelof degree of $$X$$ ($$L(X)$$) is defined as the minimum cardinal $$\kappa$$ such that every open cover of $$X$$ has a $$\leq \kappa$$-sized subcover and the weak Lindelof degree of $$X$$ ($$wL(X)$$) is defined as the minimum cardinal $$\kappa$$ such that every open cover of $$X$$ has a $$\leq \kappa$$ subcollection whose union is dense in $$X$$.

Question (A.V. Arhangel'skii, 1970): Let $$X$$ be a compact Hausdorff space.

1. Is it true that $$L(X_\delta) \leq 2^{\aleph_0}$$?
2. Is it true that $$wL(X_\delta) \leq 2^{\aleph_0}$$?

Various partial positive answers to these questions can be found in the literature. For example (see the references below):

• In 1972 Juhász proved that $$wL(X_\delta) \leq 2^{\aleph_0}$$ for every compact ccc space $$X$$.
• In 1974 Fleischmann and Williams proved that $$L(X_\delta) \leq 2^{\aleph_0}$$ for every compact linearly ordered space $$X$$.
• In 1985 Pytkeev proved that $$L(X_\delta) \leq 2^{\aleph_0}$$ for every compact space of countable tightness $$X$$.
• In 2016 I proved that $$wL(X_\delta) \leq 2^{\aleph_0}$$ for every compact space $$X$$ where player II has a winning strategy in the "weak Lindelof game of length $$\omega_1$$" (which among other things implies Juhász's 1972 result).

However both questions have a negative answer. A construction of a compact space $$X$$ such that $$wL(X_\delta) > 2^{\aleph_0}$$ can be found in my paper with Szeptycki (item 5 from the reference list). To find a compact space $$X$$ such that $$L(X_\delta) > 2^{\aleph_0}$$ it's enough to take the Cantor cube $$X=2^{\mathfrak{c}^+}$$ (just note that $$X$$ is homeomorphic to $$(2^\omega)^{\mathfrak{c}^+}$$, so $$X_\delta$$ is homeomorphic to the $$G_\delta$$ topology on $$D^{\mathfrak{c}^+}$$, where $$D$$ is a discrete space of size continuum and a $$G_\delta$$ cover without subcovers of size continuum for the latter space is given by $$\{[\sigma]: \sigma \in Fn(\mathfrak{c}^+, D, \omega_1) \wedge \sigma$$ is not one-to one on its domain $$\}$$ where $$[\sigma]=\{f \in D^{\mathfrak{c}^+}: f \supset \sigma \}$$).

Finally, to address your question regarding the existence of a bound, Toshimichi Usuba very recently proved that an $$\omega_1$$-strongly compact cardinal is a precise upper bound on both the Lindelof degree and the weak Lindelof degree of the $$G_\delta$$-topology on a compact space. So it's consistent that there is no bound to $$wL(X_\delta)$$ for $$X$$ compact!

References:

1. Juhász, István, On two problems of A.V.Arkhangel’skij, General Topology Appl. 2, 151-156 (1972). ZBL0237.54002.

2. Williams, Scott; Fleischman, William, The $$G_\delta$$-topology on compact spaces, Fundam. Math. 83 (1974), pp. 143-149. ZBL0278.54021.

3. Pytkeev, E. G., About the $$G_\lambda$$-topology and the power of some families of subsets on compacta, Topology theory and applications, 5th Colloq., Eger/Hung. 1983, Colloq. Math. Soc. János Bolyai 41, 517-522 (1985). ZBL0604.54007.
4. Spadaro, Santi, Infinite games and chain conditions, Fundam. Math. 234 (2016), pp. 229-239. ZBL1360.54010.
5. Spadaro, S.; Szeptycki, P., $$G_{\delta}$$ covers of compact spaces, Acta Math. Hung., 154 (2018) pp. 252-263. ZBL06850215.
6. T. Usuba, "$$G_\delta$$-topology and compact cardinals", preprint, arXiv:1709.07991.

P.S.: A related question is: "how big can a partition of a compact space by closed $$G_\delta$$s be?". In this case the continuum is a bound (see my answer to the following Mathoverflow question: A generalization of the Arhangelskii Theorem).