# Is there a universally meager air space?

Let $$\mathcal P$$ be a family of nonempty subsets of a topological space $$X$$. A subset $$D\subset X$$ is called $$\mathcal P$$-generic if for any $$P\in\mathcal P$$ the intersection $$P\cap D$$ is not empty.

A topological space $$X$$ is called air if $$X$$ has a countable family $$\mathcal P$$ of infinite subsets of $$X$$ such that $$X$$ contains no disjoint $$\mathcal P$$-generic $$G_\delta$$-sets $$A,B\subset X$$. In the opposite case the space $$X$$ is called airless.

It is easy to see that each second countable crowded Baire space is air.

Problem 1. Can an air subspace of the real line be universally meager?

A subset $$A\subset \mathbb R$$ is called universally meager if for any continuous injective map $$f:P\to X$$ from a perfect Polish space $$P$$ the preimage $$f^{-1}(A)$$ is meager in $$P$$. A Polish space is perfect if it contains no isolated points.

Fact 1. Each airless space $$X\subset\mathbb R$$ is universally meager.

Proof. If $$X\subset \mathbb R$$ is not universally meager, then for some continuous inejctive map $$f:P\to X$$ from a perfect Polish space the preimage $$f^{-1}(A)$$ is not meager in $$P$$. Then for some non-empty open set $$U\subset P$$ the intersection $$U\cap f^{-1}(A)$$ is a dense Baire subspace of $$U$$. Replacing $$P$$ by $$U$$, we can assume that $$B:=f^{-1}(A)$$ is a dense Baire subspace of $$P$$.

Let $$\{U_n\}_{n\in\omega}$$ be a countable base of the topology of the Baire space $$B$$. Since $$B$$ is perfect and $$f$$ is injective, the family $$\{f(U_n):U\in\mathcal B\}$$ consists of infinite sets. Then the family $$\mathcal P$$ witnesses that $$X$$ is viscous. Indeed, for any $$\mathcal P$$-generic $$G_\delta$$-sets $$G_1,G_2\subset X$$ the preimages $$f^{-1}(G_1)$$ and $$B\cap f^{-1}(G_2)$$ are dense $$G_\delta$$-sets in the Baire space $$B$$. So $$f^{-1}(G_1)\cap f^{-1}(G_2)\ne\emptyset$$ and finally $$G_1\cap G_2\supset f(f^{-1}(G_1)\cap f^{-1}(G_2))$$ is not empty.

So, the problem is in reversing the implication in Fact 1.

A candidate for a counterexample is the $$\mathfrak b$$-scale space $$X=S\cup Q$$ of Bartoszynski and Tsaban. Let us recall that $$X=S\cup Q$$ is a subspace of the countable power $$(\omega+1)^\omega$$ of the convergent sequence. The set $$Q$$ consists of functions $$f:\omega\to\omega+1$$ such that $$f(n)=\omega$$ for some $$n\in\omega$$ and $$f(i) for all $$i. The set $$S$$ is a $$\mathfrak b$$-scale $$S=\{f_\alpha\}_{\alpha\in\mathfrak b}$$, i.e., an unbounded well-ordered transfinite sequence of length $$\mathfrak b$$ in the preordered space $$(\omega^\omega,\le^*)$$. It is known that each $$\mathfrak b$$-scale space $$X=S\cup Q$$ is universally meager.

Problem 2. Can a $$\mathfrak b$$-scale space $$X$$ be air?

• Just a note on terminology, $\cal P$-dense should be $\cal P$-generic. At least if you wish to be somehow compatible with the standard forcing notation (and its topological origins)... Oct 21 '19 at 22:48
• @AsafKaragila Good point. Thanks. Oct 22 '19 at 4:21

Theorem (Zdomskyy). Under $$\mathfrak b=\mathfrak c$$ there exists a $$\mathfrak b$$-scale space $$X$$ which is air and universally meager.