Today Davide gave a very interesting talk in the 2025 Spring Topology and Dynamics Conference on [Aurichi, Bonanzinga, and Giacopello 2024]. Since it reminds me of some pretty versitile techniques I’ve developed for analyzing selection principles/games in [C 2019] and [C 2020], and I’m able to answer a question the paper asks, I thought I’d write up how to apply these general techniques for this specific topic.
This note is much more technical than my typical blog posts. In particular, the reader is expected to be familiar with the basic theory of selection games and principles.
First, we recall the following.
Definition
For a set \(X\), let \(\mathbf C(X)=\{f\in(\bigcup X)^X:x\in X\Rightarrow f(x)\in x\}\) be the collection of all choice functions on \(X\).
Definition
The set \(\mathcal R\) is said to be a reflection of the set \(\mathcal A\) if
\[\mathcal A'=\{\operatorname{range} f:f\in\mathbf C(\mathcal R)\}\]is coinitial in \(\mathcal A\) with respect to \(\subseteq\); that is, \(\mathcal A'\subseteq \mathcal A\), and for all \(A\in \mathcal A\), there exists \(A'\in \mathcal A'\) such that \(A'\subseteq A\).
Definition
Two games are said to be dual if a winning strategy for Player 1 (resp. 2) in either game using a certain amount of information can be used to define a winning strategy for Player 2 (resp. 1) in the other game using corresponding information.
Note. I’ll point the reader to definitions 16-19 of [C 2020] to more carefully explain what “certain amount” and “corresponding information” mean here.
Theorem 1 (Corollary 26 of [C 2020])
If \(\mathcal R\) is a reflection of \(\mathcal A\), then \(G_1(\mathcal A,\mathcal B)\) and \(G_1(\mathcal R,\neg\mathcal B)\) are dual.
(Here, \(\neg\mathcal B\) is the complement of \(\mathcal B\), that is, Player 1 rather than Player 2 wins if the game produces an element of \(\mathcal B\).)
This provides us the machinery necessary for the following.
Definition
Let \(\mathscr N\) denote the collection of networks of a topological space.
Definition
Let
\[\mathcal P_x(U)=\{S\subseteq U:x\in S\}\]and
\[\mathcal R_N=\{\mathcal P_x(U):x\in X,U\text{ is an open neighborhood of } x\}.\]Theorem 2
\(G_1(\mathscr N,\mathcal B)\) and \(G_1(\mathcal R_N,\neg\mathcal B)\) are dual.
Proof
By Theorem 1, we need only show \(\mathcal R_N\) is a reflection of \(\mathscr N\). Let
\[\mathscr N'=\{\operatorname{range} f:f\in\mathbf C(\mathcal R_N)\}\]We must first confirm \(\mathscr N'\subseteq\mathscr N\), that is, each \(\operatorname{range} f\) is a network. To see this, let \(U\) be an open neighborhood of \(x\), and consider \(f(\mathcal P_x(U))\in\mathcal P_x(U);\) it follows \(x\in f(\mathcal P_x(U))\subseteq U.\) Since \(f(\mathcal P_x(U))\in\operatorname{range} f\), we’re done.
We now confirm that \(\mathscr N'\) is coinitial in \(\mathscr N\). So let \(\mathcal N\in\mathscr N\) be a network. Then for each \(x\in X\) and open neighborhood \(U\) of \(x\), we may choose \(f(\mathcal P_x(U))\in\mathcal N\) such that \(x\in f(\mathcal P_x(U))\subseteq U.\) Then \(f\in\mathbf C(\mathcal R_N)\) and thus \(\operatorname{range} f\in\mathscr N'.\) Finally, note \(\operatorname{range} f\subseteq\mathcal N.\)
Corollary
In the terminology of [ABG 2024], Bob \(\uparrow\) PO-Set implies Alice \(\uparrow\) a modification of R-mw-selective, where Alice is allowed to play arbitrary networks (not just countable networks).
Proof
As defined in [ABG 2024], the PO-set game is exactly \(G_1(\mathcal R_N,\neg\mathscr N):\) Player 1 chooses a point and open neighborhood, then Player 2 chooses some subset of that neighborhood containing the point; Player 1 wins provided the choices of Player 2 form a network. Since only countable collections are constructed in this game, the PO-set game is also exactly \(G_1(\mathcal R_N,\neg\mathscr N_\omega),\) where \(\mathscr N_\omega\) collects the countable networks of the space.
So by the above duality result, the PO-set game is dual to \(G_1(\mathscr N,\mathscr N_\omega),\) which is exactly the R-mw-selective game of [ABG 2024], modified to allow Alice to play arbitrary networks.
Example
[ABG 2024] notes that the R-mw-selective game with countable networks is almost dual to the PO-set game. However, it’s consistent with ZFC that Bob \(\uparrow\) PO-set while Alice \(\not\uparrow\) R-mw-selective: take a subspace of \(\mathbb R\) of cardinality \(\omega_1\) in a model where \(\omega_1<cov(\mathscr M).\) By [Ex 2.7, ABG 2024] we have Alice \(\not\uparrow\) R-mw-selective. However, Bob \(\uparrow\) PO-set (in fact, Bob \(\uparrow_{\mathrm{tact}}\) PO-set): the winning tactic given a point \(x\) and neighborhood \(U\) played by Alice each round is to simply play \(\{x\}\); it’s clear that Bob has successfully avoided constructing a network, and thus wins.
This shows Question 2.5 of [ABG 2024] cannot be answered in the affirmative in ZFC.
Question
Is there a model of ZFC where if Alice \(\uparrow\) R-mw-selective with arbitrary networks, then Alice \(\uparrow\) R-mw-selective with countable networks?