^{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

quickiff 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:

- If $\rho(f)\subseteq\bigcup_{g\in A}\rho(g)$, then $F(A,f)\subseteq A$ and $\rho(f)\subseteq\bigcup_{g\in F(A,f)}\rho(g)$.
- 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.

unrelatedto 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). $\endgroup$