*The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free to email me.*

I'm playing around with various notions of describing topological spaces, and I've found the following fun:

Let Baire space $\mathcal{N}$ be $\mathbb{N}^\mathbb{N}$ with the usual topology. Say that a space $\mathcal{X}=(X,\tau)$ is *pictorial* if there are $A,B\subseteq\mathcal{N}$ arbitrary and $R\subseteq\mathcal{N}^2$ open such that $$\{\{a\in A: aRb\}: b\in B\}$$ generates a topology $\sigma$ on $A$ with $\mathcal{X}\cong(A,\sigma)$.

*Of course, we could replace $\mathcal{N}$ with an arbitrary space, or even (distinguishing between "point part" and "set part") an arbitrary pair of spaces, but at present I don't see what that added generality gives us.*

My question is simply:

What spaces are pictorial?

Or even better:

Are some good sources on this topic/type of problem?

I haven't been able to find any, but I'm not familiar with the relevant literature.

There are a couple easy observations:

All pictorial spaces are separable.

*(And truly trivially, all pictorial spaces have at most continuum many points and a base of size at most continuum.)*Subspaces, countable products, and $T_0$-ifications of pictorial spaces are pictorial.

There are Hausdorff non-first-countable pictorial spaces, and there are $T_1$ pictorial spaces such that no point is characterized uniquely by countably many open sets; on the other hand, if $(X,\tau)$ is Hausdorff and pictorial, then for every point $p\in X$ there is a countable family $(U_i)_{i\in\mathbb{N}}$ of open sets characterizing $p$ completely.

*(A point $p$ is characterized uniquely by countably many open sets if there is a sequence $(U_i)_{i\in\mathbb{N}}$ of open sets such that for all points $q$ we have $\{i: q\in U_i\}=\{i: p\in U_i\}\iff p=q$).*

However, I have no idea how to prove any *nontrivial* results about pictorial spaces, the main issue being the difficulty of proving any negative results. For example, I don't know whether the continuous image of a pictorial space need be pictorial: removing points from the $B$-part is tempting at first but the pullback of a subbase of the target may not look anything like the subbase witnessing the pictoriality of the starting space.

minus finitely many points. $\endgroup$ – Noah Schweber Aug 7 '18 at 20:38