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 '19 at 22:48
  • $\begingroup$ @AsafKaragila Good point. Thanks. $\endgroup$ – Taras Banakh Oct 22 '19 at 4:21

Lyubomyr Zdomskyy (in private communication) sent me the proof of the following result giving a consisting 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.