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 B**air**e space is **air**.



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



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



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> So, the problem is in reversing the implication in Fact 1.
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A candidate for a counterexample is the $\mathfrak b$-scale space $X=S\cup Q$ of [Bartoszynski and Tsaban](http://Bartoszy%C5%84ski,%20Tomek;%20Tsaban,%20Boaz.%20Hereditary%20topological%20diagonalizations%20and%20the%20Menger-Hurewicz%20conjectures.%20Proc.%20Amer.%20Math.%20Soc.%20134%20(2006),%20no.%202,%20605--615.). 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 $f(i)<f(j)<\omega$ for all $i<j<n$. 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.



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> **Problem 2.** Can a $\mathfrak b$-scale space $X$ be air?
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Is there a universally meager air space?
Lyubomyr Zdomskyy (in private communication) sent me the proof of the following result giving a consistent answer to Problems 1 and 2.



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