When does an "$\mathbb{R}$-generated" space have a short description?

Question

^{The following is a more focused version of the original question; see the edit history if interested. In the original version of the question, five other variants of the "simplicity" property below were discussed; I'm focusing on the strongest one I know essentially nothing about, previously called "$\mathsf{BCP_0^{+,uni}}$."}

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I'm broadly interested in ways of gauging the logical complexity of structures with operations of infinite arity. A good first step is to understand not-too-large topological spaces (or rather, their upper-complete lattices of open sets). Specifically, say that an generator for a space $\mathcal{X}=(X,\tau)$ is a map $\rho:\mathbb{R}\rightarrow\tau$ whose range is a base for $\mathcal{X}$. Fixing a space with a generator $\rho$, the entire structure of $\tau$ is determined by the "basic covering facts" about $\rho$, and analogously to group presentations it seems reasonable to ask when a relatively small number of those facts are sufficient:

Call a space quick iff there is some generator $\rho$ for the space and some map $F: \mathcal{P}(\mathbb{R})\times\mathbb{R}\rightarrow\mathcal{P}(\mathbb{R})$ such that:

1. If $\rho(f)\subseteq\bigcup_{g\in A}\rho(g)$, then $F(A,f)\subseteq A$ and $\rho(f)\subseteq\bigcup_{g\in F(A,f)}\rho(g)$.
2. There is a surjection $\mathbb{R}\rightarrow ran(F)$.

While at first glance this seems like a strong property to me, I actually know almost nothing about it. My question is whether it is in fact trivial (after making things "canonical and tame"):

Assume $\mathsf{ZF+AD+V=L(\mathbb{R})}$. Is there a space which has a generator but is not quick?

I would especially love a $T_1$ example.

Answer 1

To clarify for readers, this answer uses the notation of the previous version of this question and addresses some simplicity notions I've removed. $\quad$ - NS

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This is just a comment, but it is too long for the comment section.

Given a "generator" $\rho$, of the topology $\tau$, as Noah defined it, one could define $\varphi: \mathcal{P}(\omega^{\omega}) \rightarrow \tau$ by $\varphi(A) = \bigcup_{f \in A} \rho(f)$. $\varphi$ is a surjective $\subseteq$-homomorphism. One could then take the quotient of $\mathcal{P}(\omega^{\omega})$ by $A \sim B$ iff $\phi(A) = \phi(B)$. We can completely recover $\tau$ (considered as a pointless topology) from this quotient together with the covering relation Noah defined. We can also recover it completely from just the $\sim$-invariant binary relation $\mathcal{R}$ on $\mathcal{P}(\omega^{\omega})$ defined by $A\mathcal{R}B$ iff $\phi(A)\supseteq \phi(B)$

This allows any topology which has a "generator" to be presented as a binary covering relation $\mathcal{R} \subseteq \mathcal{P}(\omega^{\omega}) \times \mathcal{P}(\omega^{\omega})$. So we are in the realm of higher-order descriptive set theory. It wouldn't be hard to axiomatize exactly which such $\mathcal{R}$ yield a topology, but I won't do this because I would probably leave something out by accident.

A presentation of $\tau$ in this form witnesses the $BCP_0^+$ condition if: $\exists \alpha < \Theta$ $\forall g \in \omega^{\omega}$ the upward $\subseteq$ closure of $\{A : A\mathcal{R}\{g\} \text{ and } A \text{ has Wadge degree } < \alpha \}$ is $\{A : A \mathcal{R} \{g\}\}$

A presentation of $\tau$ in this form witnesses the $BCP_0^+$ condition if: $\exists \alpha < \Theta$ $\forall g \in \omega^{\omega}$ the downward $\subseteq$ closure of $\{A : \neg A\mathcal{R}\{g\} \text{ and } A \text{ has Wadge degree } < \alpha \}$ is $\{A : \neg A \mathcal{R} \{g\}\}$

To motivate Noah's question a bit for those who don't want to read between the lines: usually a relation on $\mathcal{P}(\omega^{\omega})$ is hard to get your hands around. Noah is interested in cases where you can recover $\tau$ completely just from the restriction of $\mathcal{R}$ to a nice, small (meaning, size $\leq 2^{\aleph_0}$) subclass of $\mathcal{P}(\omega^{\omega}) \times \mathcal{P}(\omega^{\omega})$. Then you can hope to represent $\mathcal{R}$ as a relation on $\omega^{\omega}$, and use all your tools from classical descriptive set theory.

Noah's first observation is that if a topological space has base of compact sets of size at most $2^{\aleph_0}$, there is a representation in which you can recover $\tau$ completely from the restriction of $\mathcal{R}$ to the finite powerset $\mathcal{P}_{Fin}(\omega^{\omega})$

His second observation is that under a certain condition, you can recover $\tau$ completely from the restriction of $\mathcal{R}$ to the $\Pi^0_1$ powerset $\mathcal{P}_{\Pi^0_1}(\omega^{\omega})$.

These conditions in $BCP_0^+$ and $BCP_0^-$ are conditions to ensure a (relatively) easy recovery process.

Answer 2

This should be a comment, but it's too long - here's a $\mathsf{ZFC}+\mathsf{CH}$ example:

Let $\mathfrak{F}$ be the set of full-measure subsets of $\mathbb{R}$ (we could also take the set of non-meager subsets, or so on), and let $\operatorname{Space}(\mathfrak{F})$ be the topological space whose points are the full-measure subsets of $\mathbb{R}$ and whose topology comes from the"canonical generator" $$\rho:r\mapsto\{X\in\mathfrak{F}: r\in X\}.$$ For this space instances of covering correspond to non-null sets: we have $\rho(g)\subseteq\bigcup_{f\in U}\rho(f)$ iff $g\in U$ or $U$ is non-null.

A standard transfinite recursion argument shows in $\mathsf{ZFC}+\mathsf{CH}$ that for every size-continuum set of non-null sets $(N_i)_{i\in\mathbb{R}}$ there is a non-null set $B$ such that $B\not\supseteq N_i$ for any $i$. This implies a weak form of non-quickness for $\operatorname{Space}(\mathfrak{F})$, namely that the canonical generator $\rho$ does not witness quickness, and it's not hard to extend this to arbitrary generators. So $\operatorname{Space}(\mathfrak{F})$ is not quick.

(Annoyingly I don't see that if a space is quick then all generators should witness that; luckily, that isn't an issue here.)

(At a glance, making each $\{X: r\in X\}$ clopen won't affect this analysis. This would result in a Hausdorff space, which would be nice; $\operatorname{Space}(\mathfrak{F})$ itself isn't even $T_1$.)

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Of course this example does not work under determinacy, since every non-null set has a non-null $F_\sigma$ subset and there are only continuum-many of those. But I suspect a more complicated ideal of sets than the null (or meager, or etc.) ideal will do the job. Note that given an $I$-indexed subbase for a $T_0$ topological space $\mathcal{X}$ there is a canonical homeomorphic copy of $\mathcal{X}$ whose points are subsets of $I$, so this isn't really a significant shift.

At the same time, the above raises a separate question:

What can we say, in $\mathsf{ZFC}$, about the minimal cardinality of a set of non-null sets $\mathfrak{A}$ such that every non-null set contains an element of $\mathfrak{A}$? (Or non-meager, or so on.)

I don't really have anything to say on this point, but I'd be interested in information about it (and may ask a separate question about it later on).