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

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 "$$\mathsf{BCP_0^{+,uni}}$$."

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.

• Comments are not for extended discussion; this conversation has been moved to chat. Dec 7 '20 at 15:19
• Just for clarification: I'm not sure, but I believe the notion of $\mathbb R$-generated space here is unrelated to the notion of $\Delta$-generated space, sometimes called "numerically-generated", where "numerical" alludes to $\mathbb R$ ($\Delta$-generated spaces are those spaces which can be build from $\mathbb R$ via disjoint unions and quotients). Apr 25 at 17:39
• @TimCampion Yes, as far as I can tell they are totally unrelated. Arguably a better name would be "second-continuous" ("continual"?), in analogy with "second-countable." Apr 25 at 17:52

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

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.

• This seems a good explanation to me! However, it's worth pointing out that you've rephrased my goal: "Noah is interested in cases where you can recover $\tau$ completely just from the restriction of $\mathcal{R}$ to a nice, small (meaning, size $\le 2^{\aleph_0}$) subclass" uses the fact that a family of continuum-many sets is bounded in the Wadge hierarchy, even though the Wadge hierarchy has length only $\Theta$; this fact is pretty trivial, since we can code continuum-many sets into a single set appropriately, but I want to point out the change in phrasing even though it's easy. Aug 12 '18 at 16:24

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 $$T_1$$.)

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).