*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.



When can I "draw" a topology in Baire space?
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 $dom(\delta)$.


