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 $f(i)<f(j)<\omega$ 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.

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

  • 2
    $\begingroup$ 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)... $\endgroup$
    – Asaf Karagila
    Oct 21, 2019 at 22:48
  • $\begingroup$ @AsafKaragila Good point. Thanks. $\endgroup$ Oct 22, 2019 at 4:21

1 Answer 1


Lyubomyr Zdomskyy (in private communication) sent me the proof of the following result giving a consistent answer to Problems 1 and 2.

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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