When can I "draw" a topology in Baire space? 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.
 A: I have two things to offer, the first of which could help with getting better characterizations, the latter should give ample of examples.
Since I am not aware of standard terminology, call $(X,\tau')$ a topological weakening of $(X,\tau)$ if $\tau' \subseteq \tau$.
Theorem: A space is pictorial iff it is homeomorphic to a topological weakening of a subspace of $\mathcal{N}$.
For the forward implication, note that for every $b \in B$ the set $\{a \mid aRb\}$ is open in the subspace topology that $A$ inherits from $\mathcal{N}$, hence the entire topology we induce on $A$ is a subset of its subspace topology.
For the converse implication, take the standard coding of open subsets of $\mathcal{N}$ over $\mathcal{N}$, and note that p is contained in the open set coded by q is an open relation on $\mathcal{N}$. So to see that any topological weakening $(A,\tau)$ of a subspace $\mathbf{A}$ is pictorial, just take the underlying set $A$, and for $B$ the set of all codes for extensions of opens in $\tau$.
We get plenty of preservation results as corollaries, namely everything that commutes with weakening of the topology and preserves subspaces of $\mathcal{N}$.
Theorem: Every sequential $\mathrm{QCB}_0$ space[1] is pictorial.
The sequential $\mathrm{QCB}_0$-spaces are precisely those topological spaces that arise as represented spaces in computable analysis. A represented space is just a pair $(X,\rho)$, where $\rho : \subseteq \mathcal{N} \to X$ is a partial surjection onto the set $X$.
Whenver $\mathbf{X}$ is a represented space, then so is the space $\mathcal{O}(\mathbf{X})$ of its open subsets[2]. The relationship $x \in U$ is open on $\mathbf{X} \times \mathcal{O}(\mathbf{X})$, which can be pulled back to the level of $\mathcal{N}$. Now just choose a single representative from each $\rho_\mathbf{X}^{-1}(\{x\})$ for $x \in \mathbf{X}$ to construct $A$. For $B$, we can just take the domain of the representation of $\mathcal{O}(\mathbf{X})$.
The second observation accounts for all sequential separable spaces "I have come across naturally" as being pictorial.
[1] $\mathrm{QCB}_0$ spaces are $T_0$ quotients of countably-based spaces. See https://www.sciencedirect.com/science/article/pii/S0304397501001098 for more.
[2] A subset $U \subseteq \mathbf{X}$ of a represented space $(X,\delta)$ is open, if $\delta^{-1}(U)$ is open in $dom(\delta)$.
