# Extension of Baire's Theorem

Let $$X$$ be a topological space, $$\kappa$$ be a cardinal number, such that there exists a dense subset $$A\subseteq X$$ of cardinality $$\kappa$$ but there does not exist a dense subset $$A'\subseteq X$$ of cardinality less than $$\kappa$$.

Now, suppose that $$X$$ is a metric space which satisfies the above property for $$\kappa\geq \aleph_0$$. In this generality is there a Baire-type theorem stating that if $$(A_{i})_{i \in I}$$ is a collection of dense open subsets of $$X$$ each of Cardinality at most $$\kappa$$ and $$I$$ is of cardinality $$\kappa$$ then $$\cap_{i \in I} A_i \neq \emptyset?$$

• It's false for $\kappa=\aleph_0$, isn't it? Because not every separable topological space is a Baire space? Are you assuming $\kappa\gt\aleph_0$? – bof Nov 17 '19 at 12:59
• I would need $X$ to be metric and $\kappa\geq \aleph_0$. (So in particular $X$ is a Baire space, in the classical sense but it $\kappa>\aleph_0$ then it would be non-separable). – AnnieLeKatsu Nov 17 '19 at 13:07
• I vaguely recall stronger version of BCT under Martin's axiom. You can try to search for strong Baire category theorem. (I hope somebody else will be able to say a bit more about this.) – Martin Sleziak Nov 17 '19 at 13:59
• Related: Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities? I've come across several papers on this theme, but unfortunately this is not something I've ever paid attention to and thus I don't have any specific references written down anywhere. The papers I'm thinking of were mostly written independently of each other and were widely scattered in the mathematical literature. For example, I think some might have been short notes in Doklady Akademii Nauk from the 1960s and/or 1970s. – Dave L Renfro Nov 17 '19 at 17:01

Such a hypothetical Baire theorem is not true: for every cardinal $$\kappa$$ of uncountable cofinality and any cofinal subset $$C\subset \kappa$$ of cardinality $$|C|=\mathrm{cf}(\kappa)$$, the Hilbert space $$\ell_2(\kappa)$$ of density $$\kappa$$ is a complete metric space that can be written as the union $$\bigcup_{\alpha\in C}\ell_2(\alpha)$$ of $$\mathrm{cf}(\kappa)$$ many closed nowhere dense subsets $$\ell_2(\alpha)=\{x\in\ell_2(\kappa):x^{-1}(\mathbb R\setminus\{0\})\subset [0,\alpha)\}$$. Then the intersection $$\bigcap_{\alpha\in C}U_\alpha$$ of the dense open sets $$U_\alpha=\ell_2(\kappa)\setminus\ell_2(\alpha)$$ is empty.

The same concerns the closed unit ball $$B$$ of $$\ell_2(\kappa)$$. In the weak topology, $$B$$ is (uniform Eberlein) compact, which can be written as the union of $$\mathrm{cf}(\kappa)$$ many closed nowhere dense subsets. So, the Baire Theorem does not extend even to the intersection of $$\aleph_1$$ many dense open sets in nice (namely, uniform Eberlein) compact spaces.

On the other hand, there are some extensions of the Baire Theorem for topological spaces of countable cellularity, but a metrizable space has countable cellularity if and only if it is separable, so such extensions do not concern metrizable spaces. Namely, the Martin's Axiom is equivalent to the statement:

For any compact Hausdorff space $$K$$ of countable cellularity, the intersection of less than continuum many dense open sets in $$K$$ is not empty.

As a first reading of this topic, you can look at section 1 called "Topology of MA" of the survey paper "Versions of Martin's Axiom" of William Weiss in "Handbook of Set-Theoretic Topology" published by Elsevier in far 1984.

Also there exists a well-known cardinal characteristic $$\mathrm{cov}(\mathcal M)$$ of the continuum, equal to the smallest number of open dense subsets in the real line whose intersection is empty. The value of $$\mathrm{cov}(\mathcal M)$$ is between $$\omega_1$$ and $$\mathfrak c$$, but its exact position in the segment $$[\omega_1,\mathfrak c]$$ depends on additional axioms of Set Theory. The cardinal $$\mathrm{cov}(\mathcal M)$$ is one of 10 cardinal characteristics of the continuum, composing the famous Cichon's Diagram.

• This is very interesting. Would you happen to have a reference for each of the latter two points. I'd like to look into it in more detail. – AnnieLeKatsu Nov 17 '19 at 19:51
• @MrMMS I have added references to my answer. – Taras Banakh Nov 18 '19 at 6:06
• Conclusions for Baire and nonmeager usually are equivalence, since a topological space $X$ is nonmeager if and only if $X$ contains an open Baire subspace. – Taras Banakh Nov 18 '19 at 14:27
• @MrMMS It is well-known that arbitrary Tychonoff product of separable spaces is countably cellular. So for any set $X$ the space $\mathbb R^X$ of all real-valued functions on $X$ with the topology of pointwise convergence is countably cellular. Since the countable cellularity is preserved by dense subspaces, the space $C_p(X)\subset \mathbb R^X$ of continuous functions on any Tychonoff space $X$ has countable cellularity as well as many subspace of $C_p(X)$, for example smooth functions (if $X$ has a smoothness structure). – Taras Banakh Nov 19 '19 at 19:30
• @MrMMS You are welcome. Good luck then! – Taras Banakh Nov 20 '19 at 13:36